Mathematicians' Reflections on Teaching: A Symbiosis with Mathematics Education Theories (Advances in Mathematics Education) 3031342941, 9783031342943

Table of contents :
Introduction
References
Acknowledgments
Contents
Chapter 1: Reflecting on Teaching Mathematics at University: Collaboration Among Mathematics Educators and Mathematicians
1.1 Introduction
1.2 Our Collaborative Narrative Research Studies (2009-2022)
1.2.1 The Language and Visualization in Linear Algebra (2009-2016)
1.2.2 Accommodating in the Formal World: Abstract Algebra (2012-2014)
1.2.3 Balancing Embodied, Symbolic, and Formal World Thinking in Teaching First-Year Calculus (2013-2014)
1.2.4 Physics Bridging the Embodied and Symbolic Worlds of Mathematical Thinking (2014-2015)
1.2.5 From Intuition to the Formal World of Mathematical Thinking: A Geometric Topologist´s Thought Processes (2014-2016)
1.2.6 Moving Between the Worlds of Mathematical Thinking in Linear Algebra (2016-Present)
1.2.7 Language and Spoken Words in Linear Algebra (2021-Present)
1.2.8 Other Examples of Collaborative Activities with Mathematicians (2016-Present)
1.2.8.1 Algebra Issue in Calculus Classrooms
1.2.8.2 Challenges and Strategies in Teaching Linear Algebra
1.2.8.3 Linear Algebra Curriculum Study Group (LACSG 2.0)
1.2.8.4 Resources for Teaching Linear Algebra
1.2.8.5 Teaching Linear Algebra Proof
1.2.9 Summary
1.3 The Design of Our Collaborative Studies
1.4 A Vision for a Fruitful and Sustainable Collaboration Model Among Mathematics Educators and Mathematicians
References
Chapter 2: Promoting Effective Collaboration in the Mathematics Community
2.1 Introduction
2.2 Collaboration
2.3 Collaboration Within the Mathematics Community
2.4 Concluding Remarks
References
Chapter 3: Synergy Between Mathematicians and Mathematics Educators: Stories of Many, and Potent, Facets
3.1 Introduction
3.2 Part I: The Crucial, Yet Fragile, Relationship Between Mathematicians and Mathematics Educators
3.3 Part II: Propelling and Deepening the Relationship Between Mathematicians and Mathematics Educators
3.3.1 Teaching
3.3.2 Professional Development
3.4 Part III: Mathematicians and Mathematics Educators: Has the ``Beautiful Friendship´´/Joint and Multi-faced Enterprise Alre...
References
Chapter 4: Mind the Gap: Reflections on Collaboration in Research and Teaching
4.1 Background
4.2 The Research Projects
4.2.1 The First Project
4.2.2 A Second Project
4.2.3 Illustrative Lecture Vignettes
4.2.4 An Illustrative Boundary Encounter
4.2.5 General Outcomes
4.2.6 The Value to Individuals of the Boundary Activities
4.2.7 Final Comments
References
Chapter 5: Students Enjoying Transformed and Improved Learning Experiences of Mathematics in Higher Education
5.1 Introduction
5.2 Part 1: The Context for Transformation and Improvement
5.2.1 The Case for Change
5.2.2 Higher Education Mathematics Teachers: A Community of Practice
5.2.3 Mathematics Education Researchers: Another Community of Practice
5.2.4 Influencing the HEMT-CoP from Outside
5.3 Part 2: Effective Intervention
5.3.1 An Emphasis on Learning: Students Need the Opportunity to Learn
5.3.2 Prior Knowledge: The Fallacy of Accelerated Remediation
5.3.3 The Challenge of Learning Mathematics
5.3.4 Attitudes Towards Mathematics: Motivation, Attributions, and Approaches to Learning
5.3.5 Learning to Learn Mathematics
5.3.6 Transforming Teaching 1: Feedback
5.3.7 Transforming Teaching 2: Active Learning
5.4 Conclusion
5.5 Postscript
References
Chapter 6: Identifying Minimally Invasive Active Classroom Activities to Be Developed in Partnership with Mathematicians
6.1 Introduction
6.2 Relevant Literature
6.2.1 What Do We Know About Lecturing in Advanced Mathematics?
6.2.2 Why Do Mathematicians Choose to Lecture?
6.3 An Alternative Model for Mathematics Education Innovation
6.3.1 A Partial Example of the Model from Previous Work
6.3.2 A Proof of Concept: Clicker Questions
6.3.3 Another Proof of Concept: Partial Notes
6.4 Summary
References
Chapter 7: Didactics of Mathematics as a Field of Mathematical Research: The Anthropological Approach
7.1 A Personal Introduction
7.2 Theoretical Framework
7.3 Historical Milestones Revisited
7.4 University Schizophrenia
7.5 Conclusion and Perspective
References
Chapter 8: Collaborative Evaluation of Teaching and Assessment Interventions: Ideas From Realistic Evaluation
8.1 Introduction
8.2 Teaching Excellence and Student Outcome Framework and Evaluation of Teaching Interventions
8.2.1 A Brief Note on Evaluation
8.3 Two Case Studies of Collaborative Evaluation of Teaching Interventions
8.3.1 Case Study 1: Summative Assessment of Mathematics
8.3.2 Case Study 2: Learning Proof with Lean
8.4 Evaluating Teaching Interventions: The Importance of Mechanism
8.5 Concluding Remarks
References
Chapter 9: The Development of Inquiry-Based Mathematics Teaching and Learning
9.1 Background and Introduction
9.2 Vignette: Investigating the Teaching of Linear Algebra
9.3 Investigating Mathematics Teaching in Schools and Classrooms
9.4 A Constructivist Perspective on Teaching
9.5 Teaching Development Emerging from Research into Teaching-Challenges to Radical Constructivism
9.6 Research Collaboration Between Teachers and Researchers
9.7 Theoretical Development
9.8 Inquiry in Learning, Teaching and Development
9.8.1 Developmental Research
9.8.2 The Role of Inquiry
9.8.3 A Sociocultural Perspective
9.9 Developing Mathematics Teaching and Learning at University Level
9.10 Concluding Thoughts
9.11 Post-script
References
Chapter 10: The Intimate Interplay Between Developing Teaching and Exploring Mathematics Through Reflecting in, on and for Tea...
10.1 Introduction
10.1.1 Foundations
10.2 Chordal Mid-Points
10.2.1 Initial problem
10.2.1.1 Pedagogical Observation
10.2.2 First Use With Learners
10.2.2.1 Pedagogic Realisations
10.2.2.2 Pedagogic Remarks on Learners as Narrators
10.3 Diversion: Tangent Power of a Point
10.3.1 Pedagogic Remarks on Themes
10.3.2 Pedagogic Remarks on Multiplicity
10.3.3 Pedagogic Remarks on Invariance
10.3.4 Pedagogic Remarks on Concept Images
10.4 Returning to Chordal Mid-Points
10.4.1 Pedagogic Comment: Different Ways of `seeing´ and (Re)presenting
10.4.2 Pedagogical Comment: Personal Narratives
10.4.3 Further Investigation
10.4.3.1 Pedagogical Remark: Stimulating a Disposition to Enquire Mathematically
10.4.4 Quartics
10.4.4.1 Pedagogic Comment: Example Construction
10.5 Pseudo Cubics: Glued Quadratics
10.5.1 Pedagogical Comment
10.5.2 Pedagogic Comment
10.5.3 Mathematical Comment
10.5.4 Pedagogical Comment
10.5.5 Pseudo Quartics: Glued Cubics
10.5.6 Reflection
10.6 Taylor-Powers
10.6.1 Mathematical Comment
10.6.2 Pedagogical Comment
10.7 Overview
References
Chapter 11: Signatures of Teaching Mathematics
11.1 Introductory Inventory
11.1.1 The Dilemma of Literature Research
11.1.2 Living with Ambiguities
11.1.3 Methodological Constraints
11.2 Mathematics, Philosophy and World Views: First Observations
11.2.1 The Impact of the Reference Science: Mathematics
11.2.2 Teaching Mathematics: The Classroom Inducing Further Variables
11.2.3 Epistemological Obstacles
11.2.4 Philosophy of Mathematics
11.2.5 World Views of Mathematics
11.3 Signatures of Mathematics Teaching and Learning
11.3.1 Dimension: The Content-Mathematics
11.3.2 Dimension: Teaching Mathematics
11.3.3 Dimension: Classroom
11.3.4 Dimension: Immanent Philosophy
11.4 Conclusion
References
Chapter 12: Long-Term Principles for Meaningful Teaching and Learning of Mathematics
12.1 Introduction
12.2 Enlightenment, Transgression and Multi-contextual Overview
12.2.1 Questioning Personal Beliefs
12.2.2 Long-Term Historical Evolution of Mathematical Thinking
12.2.3 Long-Term Development of Mathematical Thinking in the Individual Over a Lifetime
12.3 Long-Term Supportive Principles and Resolution of Problematic Transitions
12.3.1 Conservation of Counting Number and General Principles of Arithmetic
12.3.2 How Humans Make Sense Reading and Speaking Text
12.3.3 Giving Meaning to Expressions Through Spoken Articulation
12.4 Interpreting the Duality and Flexibility of Expression as Operation or Object
12.4.1 Making Sense of Equations and the Equals Sign
12.4.2 Practical and Theoretical Limits in the Calculus
12.5 Making Sense of Constants and Variables
12.5.1 Variables as Infinitesimals
12.5.2 Infinitesimals as Fixed Points on a Number Line
12.6 Making Sense of the Calculus Using New Technological Tools
12.6.1 Embodying Integration and the Fundamental Theorem of Calculus
12.7 Is the Calculus About Quantities With Dimensions, or Is It About Numbers?
12.7.1 Practical Continuum, Theoretical Closeness and Formal Completeness
12.8 Making Sense Reading a Mathematical Proof
12.9 Discussion
References
References

Citation preview

Advances in Mathematics Education

Sepideh Stewart   Editor

Mathematicians’ Reflections on Teaching A Symbiosis with Mathematics Education Theories

Advances in Mathematics Education Series Editors Gabriele Kaiser, University of Hamburg, Hamburg, Germany Bharath Sriraman, University of Montana, Missoula, MT, USA Editorial Board Members Marcelo C. Borba, São Paulo State University (UNESP), São Paulo, Brazil Jinfa Cai, Newark, NJ, USA Christine Knipping, Bremen, Germany Oh Nam Kwon, Seoul, Korea (Republic of) Alan Schoenfeld, University of California, Berkeley, CA, USA

Advances in Mathematics Education is a forward looking monograph series originating from the journal ZDM – Mathematics Education. The book series is the first to synthesize important research advances in mathematics education and welcomes proposals from the community on topics of interest to the field. Each book contains original work complemented by commentary papers. Researchers interested in guest editing a monograph should contact the series editors Gabriele Kaiser ([email protected]) and Bharath Sriraman ([email protected]) for details on how to submit a proposal.

Sepideh Stewart Editor

Mathematicians’ Reflections on Teaching A Symbiosis with Mathematics Education Theories

Editor Sepideh Stewart David and Judi Proctor Department of Mathematics University of Oklahoma Norman, OK, USA

ISSN 1869-4918 ISSN 1869-4926 (electronic) Advances in Mathematics Education ISBN 978-3-031-34294-3 ISBN 978-3-031-34295-0 (eBook) https://doi.org/10.1007/978-3-031-34295-0 © Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.

Introduction

It is a privilege to be invited to write the introduction for this groundbreaking book, bringing together the reflections of university mathematicians and mathematics educators on the teaching and learning of their students. The last half century has seen phenomenal change. In 1957, Russia launched Sputnik and the teaching of mathematics and science was revolutionised in the USA and other Western countries to seek to remain competitive and gain an advantage by introducing ‘new math’. When I first began to contemplate the mathematical thinking of my own undergraduates as a young lecturer in mathematics some 50 or so years ago, there was little theoretical basis available to address the issues. Educational research in mathematics included studies of young children’s arithmetic and more general aspects of school mathematics, but there was no mathematical education research at university level. The first significant consideration of education in the professional mathematics community took place at the 1900 International Congress of Mathematicians (ICM) in Paris in a section entitled ‘Teaching and History of Mathematics’. This included the famous lecture in which David Hilbert listed the 23 ‘Mathematical Problems’ that shaped much of twentieth-century research mathematics. Major countries in Europe and North America were seeking to introduce significant reforms in school mathematics. In the fourth ICM in Rome 1908, it was decided to establish wideranging cooperation between countries through an ‘International Commission on the Teaching of Mathematics’ under its first president Felix Klein. The commission held four international meetings before the next ICM in 1912, on topics such as ‘What mathematics should be taught to students studying sciences?’, ‘What is the place of rigour in mathematics teaching?’ and ‘How can the teaching of the different branches of mathematics best be integrated?’ It continued with a vast survey of teaching practices in over 300 reports from 18 member countries. Then the First World War intervened, Klein died in 1923, and it was only in 1928 that the commission was re-established under its modern name, the ‘International Commission of Mathematical Instruction’ (ICMI).

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Its first task was to collect data on teacher training methods to be presented at the next ICM in 1932 in Zurich. This was in the great depression between the wars and there was little interest in educational innovation. The Second World War caused the ICM to be cancelled and ICMI ceased activity until it was re-constituted in 1952, now with a permanent secretariat under the auspices of the International Mathematics Union, but still reporting every 4 years to the ICM. Apart from the two world wars, ICM meetings continued every 4 years with a subsection devoted to mathematics education. As the range of interests proliferated, this became inadequate and Hans Freudenthal, as President of ICMI from 1967 to 1970, proposed that a full conference be held every 4 years between the mathematical ICMs and organised the first International Congress of Mathematics Education (ICME) at Utrecht in August 1967. Since then, the last half century has seen phenomenal change. When I first began to contemplate the mathematical thinking of my own undergraduates as a young lecturer in mathematics some 50 or so years ago, educational theory of mathematics teaching and learning focused mainly at the school level. As a lecturer in mathematics at Sussex University, I saw it as my duty to present mathematical ideas to undergraduates in ways that made sense to them. I was also integrated into the undergraduate community through musical activities in which I participated as a choral and orchestral conductor. Motivated by my increasing pleasure in working with students in both mathematics and music, I saw the link between the two in terms of the joy that arises as an individual working in a group for a common purpose. This contrasted with a growing sense of alienation in mathematical research in K-theory which I felt arid and meaningless. This conflict was addressed when I made the transition from a ‘Lecturer in Mathematics’ at Sussex University to a ‘Lecturer in Mathematics with Special Interests in Education’ at Warwick University in 1969. My first experience of an international conference in mathematics education was at the second ICME conference at Southampton in 1972. Here I encountered the parlous state of mathematics education at the time. I attended two working groups, one on ‘the teaching of calculus and analysis’, the other on ‘history and mathematics teaching’. The first included mathematics professors debating as to whether it was proper to calculate the derivative of sine x using a visual diagram in a unit circle or whether it required the formal definition of the limit of the function as a power series. The second working group had no theoretical content relating to teaching and learning. One professor suggested that many students found mathematics options in the final year too difficult and suggested that history of mathematics could provide them with an alternative where they could gain sufficient credit to be awarded a degree. Another showed his collection of mathematicians on postage stamps, and a third showed his photographs of Euler’s birthplace. I left the conference early and returned home to my family. I remained a mathematician giving mathematical support to the local teacher training college as part of my job specification. I attended ICME in Karlsruhe, 1976, still in my role as a mathematician, but did not choose to go to the new working group on ‘Psychology of Mathematics Education’ which featured at that conference. Travelling back with Richard

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Skemp, I learnt that the group had proposed an annual conference of PME as a working group of the ICME, with the first arranged by Freudenthal in Utrecht in 1978, which I was fortunate to attend. Most of the topics related to school mathematics with a handful of participants interested in the transition from high school to university and on to college and undergraduate levels. Gontran Ervynck, from Belgium, organised a working group to study this transition into undergraduate mathematics, which produced the first multi-author book on undergraduate mathematics teaching and learning, which I was fortunate to edit (Tall, 1991). In 1990, President H. W. Bush declared 1990–1999 to be ‘the decade of the brain’ providing huge resources to enhance public awareness of the benefits to arise from brain research, leading to greater insight into the structure and operation of the brain. Around this time, the mathematical community began to reflect on the issues. In a 1990 article on ‘Mathematics Education’, Fields Medalist William Thurston observed: Mathematics is amazingly compressible: you may struggle a long time, step by step, to work through some process or idea from several approaches. But once you really understand it and have the mental perspective to see it as a whole, there is often a tremendous mental compression. You can file it away, recall it quickly and completely when you need it, and use it as just one step in some other mental process. The insight that goes with this compression is one of the real joys of mathematics. After mastering mathematical concepts, even after great effort, it becomes very hard to put oneself back in the frame of mind of someone to whom they are mysterious. (Thurston, 1990, p. 1)

In a second article ‘On Proof and Progress in Mathematics’ in 1994, he reflected on how a sub-community of mathematicians (say, analysts) may easily share ideas within their specialism which are opaque to another (such as topologists) and vice versa. He offered a detailed analysis of how communities of mathematicians operate and related this to the difficulties encountered by students: The transfer of understanding from one person to another is not automatic. It is hard and tricky. Therefore, to analyze human understanding of mathematics, it is important to consider who understands what, and when. Mathematicians have developed habits of communication that are often dysfunctional. Organizers of colloquium talks everywhere exhort speakers to explain things in elementary terms. Nonetheless, most of the audience at an average colloquium talk gets little of value from it. Perhaps they are lost within the first 5 minutes, yet sit silently through the remaining 55 minutes. Or perhaps they quickly lose interest because the speaker plunges into technical details without presenting any reason to investigate them. At the end of the talk, the few mathematicians who are close to the field of the speaker ask a question or two to avoid embarrassment. This pattern is similar to what often holds in classrooms, where we go through the motions of saying for the record what we think the student “ought” to learn, while the students are trying to grapple with the more fundamental issues of learning our language and guessing at our mental models. Books compensate by giving samples of how to solve every type of homework problem. Professors compensate by giving homework and tests that are much easier than the material “covered” in the course, and then grading the homework and tests on a scale that requires little understanding. We assume that the problem is with the students rather than with the communication: that the students either just don’t have what it takes, or else just don’t care. Outsiders are amazed at this phenomenon, but within the mathematical community we dismiss it with shrugs. (Thurston, 1994, pp. 5, 6)

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In North America in 1992, Ed Dubinsky encouraged the Mathematical Association of America (MAA) to form a joint Committee on Research in Undergraduate Mathematics Education (CRUME) with the American Mathematical Society (Selden, 2012). One of the first projects of CRUME was the CBMS series of ‘occasional volumes of papers’ on RUME, called Research in Collegiate Mathematics Education (RCME), producing seven volumes in the next decade. In England in 1992, the London Mathematical Society produced a report (Neumann, 1992) to respond to the increasing number of students studying a university degree with a far wider range of achievement and the explosion of differing mathematical needs in society. The plan was to reform the current 3-year degree with a 3-year bachelor’s degree covering a wider range of material than the first half of the current degree and a 4-year master’s degree going beyond the current curriculum. Both degrees were to be ‘taught in such a way that students achieve a markedly fuller understanding than they do at present’. However, the term ‘understanding’ was interpreted in very different ways by mathematicians and mathematics educators. At an LMS conference to celebrate the life of my colleague the late Rolph Schwarzenberger, who embraced both mathematics and mathematics education, the presentations covered both aspects and I was invited to give a lecture on ‘Mathematicians thinking about students thinking about mathematics’ (summarised in Tall, 1993). At the same time, members of the LMS were invited to update their areas of research interest and I replied ‘Advanced Mathematical Thinking’. The committee reluctantly refused to accept it because it was not an accepted heading in the American Mathematical Society’s mathematical subject classification. A formal request to the AMS from CRUME was also rejected. I gave an invited presentation in the Mathematics Education section of ICM in Strasbourg (Tall, 1994a) in which I said: I cannot believe that mathematicians can continue to ignore the study of mathematical thinking as part of the totality of the profession, for if it is not done by mathematicians, others surely lack the mathematical knowledge to research it in depth. I suggest that the study of mathematical thinking be given a place in the canons of mathematical activity comparable with other areas of mathematics. Just as a topologist will defend a number-theorist’s right to do research within the umbrella of mathematics I hope that specialists in mathematical research will similarly defend the right of mathematicians to do research into mathematical thinking. Respect will have to be earned by mathematics educators. But if opportunities to earn respect are not honoured then mathematics itself can only be the poorer. (Tall, 1994b, p. l6)

The issue was finally resolved after a meeting of RUME in 1996 when Hyman Bass took up the matter and ‘mathematics education’ was added to the AMS Mathematics Subject Classification as Topic 97. Now it became possible for an individual applying for a post in a college or university to specify their mathematical area of interest as ‘mathematics education’. Even so, it still remained necessary for mathematics educators to gain respect within the mathematical community. The editor of this book, Sepideh Stewart, was fortunate to study for a PhD in mathematics education in a university in New Zealand where the mathematics

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department took mathematics education seriously and integrated their work within a single community. Even so, when a mathematician and a mathematics educator were given equal support in recommendation for promotion, the university promotion committee chose the mathematician over the educator. Having obtained and held a position in an American university for 10 years, Sepideh opens the first chapter of this book reporting the development of her research working with mathematics professors willing to reflect on their teaching and sharing their experience with other members of their department. In the second chapter, she collaborates with Bharath Sriraman to review theories and models for collaboration between mathematicians and mathematics educators at college and university level. Around the world, there has been a widespread activity in seeking to link activities of university and college mathematicians, mathematics educators and teachers. These include the 24-year-old annual conference of Research in Undergraduate Mathematics Education, Special Interest Group of the Mathematical Association of America; the biennial Delta conferences since 1997 nurturing exchanges between mathematicians, educators and researchers in the southern hemisphere; the Thematic Working Group on University Mathematics Education in the Congress of European Researchers in Mathematics Education (CERME) evolving into ERME Topic Conferences (Montpellier, 2016; Kristiansand, 2018; Bizerte, 2020); and a range of national activities on teaching university mathematics. Of particular interest is the development of bilingual conferences in English and the language of the hosting country in the International Network for Didactics Research in University Mathematics (INDRUM), first held in France in 2016. These have the advantage of directly linking international research to the local community which has the potential to advance the link between theory and practice nationally and internationally. The chapters which follow in this book report individual researchers’ developments in undergraduate teaching and learning. In Chap. 3, Elena Nardi traces the relationship between mathematicians and mathematics educators in research on the teaching and learning of mathematics, recounting examples of initiatives that developed over time in research, teaching, professional development and public engagement. She re-imagines this not just as a story of paths crossing, but as paths meeting at a vanishing point in the future where boundaries between mathematics and mathematics education may fade into insignificance to become a joint, multi-faceted enterprise. In Chap. 4, Michael Thomas reports his experience in developing collaborative work between mathematicians and mathematics educators. His title ‘mind the gap’ arose from the 1960s London underground rail warning of the inherent danger in the gap stepping between the train and the platform. Thomas uses Schoenfeld’s Framework of ‘Resources, Orientations and Goals’ (ROG) as part of an extensive collaboration between mathematicians and mathematics educators as equal partners in his university mathematics department to show how interchange can be of mutual benefit. Techniques include the lecturer writing up his experiences after giving a lecture to form the basis for discussion and lecturers choosing a short selection from one of their lectures to illustrate aspects of interest.

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In Chap. 5, Simon Goodchild writes of his experience as the founding director (2013-2020) of the Norwegian National Centre for Research, Innovation and Coordination of University Mathematics Teaching (MatRIC). The Centre was formed to promote the vision of ‘students enjoying transformed and improved learning experiences of mathematics in higher education’. He categorises Higher Mathematics Education Teachers (HEMT) to include mathematicians and specialists in service subjects. These adopt a variety of instructional approaches which are not resolved by the current complexity of Mathematical Education Research (MER). He reports initiatives in his own institution to address these problems directed at teaching staff and students. Essentially the staff seek practical strategies that they can use to improve the effectiveness of their teaching in their own terms without the need to translate from technical terms in educational theory. University and college lecturers often find that they are constrained in how they can teach. They are part of a system which is subject to a range of differing demands, from preparing students of varying abilities for future employment to encouraging highly able students to become research mathematicians of the future. In Chap. 6, Paul Christian Dawkins and Keith Weber observe that, although some mathematics educators have developed radically new approaches for students to take an active part through some form of ‘inquiry-based learning’, most mathematicians still see the lecture as a central form of teaching. They are unlikely to take on new approaches that are unfamiliar and do not guarantee success in their own teaching. Nevertheless, based on a synthesis of the research literature, they suggest that mathematicians and mathematics educators would agree that Students need opportunities to reflect on central ideas and understandings in their advanced mathematics courses.

They propose that lectures can be enhanced by including activities that encourage students to reflect on their understanding. As an example, clickers can be used to allow students to choose from contrasting multi-choice options that can then be displayed to form a basis for discussion of their meaning. Or lecture notes can be printed with gaps so that the lecturer can speak about ideas and the students can fill in the details. This involves a general principle to develop ‘minimally invasive classroom activities’ in partnership between mathematicians and educators. In Chap. 7, Carl Winsløw begins by outlining his personal development. He was taught the ‘New Mathematics’ in school, based on the structural approach of the French Bourbaki group. After completing his undergraduate degree, he became a mathematics researcher in an algebraic area of functional analysis, and then an associate professor in mathematics teaching undergraduates, using this experience to develop an integrated perspective encompassing mathematics education research (which he terms the Didactics of Mathematics) and the mathematical sciences. He bases his approach on the Anthropological Theory of the Didactic (Chevallard, 1992), which encompasses two major components in historical and personal development of knowledge in mathematical communities: praxis (practical knowledge) and logos (theoretical knowledge). Praxis involves recognising a problem and knowing the technique to solve it. Logos uses words, pictures and diagrams

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to build logical relationships in increasingly sophisticated forms of deduction and proof. A praxeology is a theory of the relationship between the two. This leads to an analysis of the evolution of mathematical knowledge in history and in mathematical communities where individuals play different roles. There is a particular focus on Felix Klein’s distinction between a curriculum that separates different topics into self-contained units and his preference for mathematical science as a great connected whole (Klein, 1908/2016, vol. I, pp. 82–83). Winsløw proposes the need for ‘mathematics teacher educators who can effectively pursue Klein’s vision, we need to prepare mathematicians-didacticians who are both acquainted with contemporary mathematics, with creative mathematical work, and with modern methods and results from the Didactics of Mathematics’. In Chap. 8, Paola Iannone, a mathematician who developed a deep interest in teaching and learning, describes her shift to mathematics education and subsequent collaboration with mathematicians who wish to design and evaluate new approaches to their own teaching and assessment. She notes growing evidence in the literature that written examinations, as they are currently structured, fail to assess types of reasoning valued by the mathematics community such as conceptual understanding and problem solving. This is investigated in a summative question and answer session with the student writing on the board (or using pen and paper) answering questions which can be theoretical (stating known definitions, theorems or proofs) or applied (working out examples, tackling unseen problems or proofs, or using algorithms appropriately). A second study considers students using theorem-proving software to investigate how this changes students’ understanding of proof. In both cases, the mathematicians and educators involved learnt a great deal about the subtleties of their own perceptions of mathematics and the understandings of the students taking part but questioned how these specific experiences could be generalised for wider dissemination. This was related to the chapter by Dawkins and Weber (Chap. 6) that questioned why teaching and curriculum innovations proposed by mathematics educators have had little impact on how university mathematics is taught and echoed the need for mathematicians to be aware of tools to study the impact of the transfer of interventions into their own context. Chapter 9 by Barbara Jaworski reviews the development of inquiry-based mathematics and learning, starting from her own experiences which have influenced the development of several other authors in this collection. After a first career as a schoolteacher in a comprehensive school for students aged 13–18, she took an active interest in the Open University where she was introduced to the ideas of mathematical investigations as a means of developing mathematical thinking and learning. In subsequent university posts, she developed local communities of practice in which researchers and teachers shared their expertise to their mutual benefit. Several authors in this book have taken part in these developments, including Elena Nardi, Simon Goodchild and Carl Winsløw. A common thread is the development of Inquiry Based Learning, which also features in Chap. 6 written by Paul Christian Dawkins and Keith Weber. Jaworski formulates three layers of inquiry represented diagrammatically as an inner layer where students engage in inquiry with a teacher in the classroom, a

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middle layer of teachers engaging in professional inquiry and an outer layer of didacticians inquiring with teachers to research processes, practices and issues in developing mathematics teachers and learning. The chapter includes the story of development of international organisations in which she has taken an active part. In Chap. 10, John Mason focuses on how individuals can keep themselves mathematically alive by being attentive to their own thinking, to sensitise themselves to the struggles faced by learners and to improve their own pedagogy. Based on this attitude, he recommends readers to think through mathematical questions he poses before reading his account which follows so that they have personal experience as a foundation to consider his observations. Chapter 11 sees Günter Törner incorporating Shulman’s notion of a ‘signature of teaching mathematics’ that characterises each teacher’s approach to teaching and learning. He offers a range of examples in terms of four overarching aspects: the mathematical content as such and its structure, the underlying understanding of teaching and learning, the characteristics of the partly socially acting classroom, and the immanent philosophies of mathematics. My own chapter on ‘long-term principles for meaningful teaching and learning of mathematics’ is appropriately placed last as it is my own personal attempt to formulate how mathematical thinking evolves in sophistication over time in history and in the individual, taking account of different approaches appropriate for differing specialisms, experts, teachers and learners. In particular, it formulates how different communities of practice may have approaches that are appropriate for some yet be problematic for others and proposes a ‘multi-contextual overview’ where each community is aware of the values shared between the two, to build confidence based on their communalities while respecting and addressing their differences. Its main purpose is not to conflate a highly complicated theory. Its objective is to find fundamental ideas that can be observed meaningfully by most readers. This includes how we speak and hear mathematical expressions that can be interpreted both as operations and as mental objects, how we see moving objects as variables – which allows the imagination of an infinitesimal as a quantity that grows arbitrarily small – and how we read mathematical proofs to make sense of them. This offers new possibilities for readers to reflect on their own experiences and beliefs, taking into account other chapters in the book. University of Warwick, Coventry England, UK

David Tall

References Chevallard, Y. (1992). Fundamental concepts in didactics: Perspectives provided by an anthropological approach. In R. Douady & A. Mercier (Eds.), Research in didactique of mathematics, selected papers (pp. 131–167). La Pensée Sauvage.

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Klein, F. (1908/2016). Fundamental mathematics from a higher standpoint, I-III (G. Schubring, Trans.). Springer. Neumann, P. (1992). The future for honours degree courses in mathematics. Journal of the Royal Statistical Society. Series A (Statistics in Society), 155(2), 185–189. https://doi.org/10.2307/ 2982954. Selden, A. (2012). A home for RUME: The story of the formation of the mathematical association of America’s special interest group on research in mathematics education. https://www.tntech. edu/cas/pdf/math/techreports/TR-2012-6.pdf Tall, D. O. (Ed.) (1991). Advanced mathematical thinking. Kluwer. Tall, D. O. (1993). Mathematicians thinking about students thinking about mathematics. Newsletter of the London Mathematical Society, 202, 12–13. Tall, D. O. (1994a). Understanding the processes of advanced mathematical thinking. In Abstracts of invited talks, International Congress of Mathematicians, Zurich, August 1994, pp. 182–183. Tall, D. O. (1994b). Understanding the processes of advanced mathematical thinking. In International Congress of Mathematicians, Zurich, August 1994. Full Lecture at: http://homepages. warwick.ac.uk/staff/David.Tall/pdfs/dot1996i-amt-icm.pdf Thurston, W. P. (1990). Mathematical education. Notices of the American Mathematical Society, 37 (7), 844–850. https://arxiv.org/pdf/math/0503081.pdf (Note: page numbers in this text refer to the available pdf, not the original pagination.) Thurston, W. P. (1994). On proof and progress in mathematics. Bulletin of the American Mathematical Society, 30(2), 161–177. https://arxiv.org/pdf/math/9404236.pdf (page numbers in the text refer to the latter pdf.)

Acknowledgments

I would like to thank each author for their valuable collaboration, embracing this book’s theme, and producing such superior work. I would also like to thank the mathematicians who collaborated in all the studies in this book. Our sincere thanks for sharing your thought processes, teaching resources, and valuable time. These studies would not have been possible without your input and collaboration. Many thanks also to all the students who participated in our studies.

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Reflecting on Teaching Mathematics at University: Collaboration Among Mathematics Educators and Mathematicians . . . . . . . . . . . Sepideh Stewart

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Promoting Effective Collaboration in the Mathematics Community . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sepideh Stewart and Bharath Sriraman

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Synergy Between Mathematicians and Mathematics Educators: Stories of Many, and Potent, Facets . . . . . . . . . . . . . . . . . . . . . . . . Elena Nardi

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Mind the Gap: Reflections on Collaboration in Research and Teaching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Michael O. J. Thomas

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Students Enjoying Transformed and Improved Learning Experiences of Mathematics in Higher Education . . . . . . . . . . . . . . Simon Goodchild

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Identifying Minimally Invasive Active Classroom Activities to Be Developed in Partnership with Mathematicians . . . . . . . . . . . 103 Paul Christian Dawkins and Keith Weber

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Didactics of Mathematics as a Field of Mathematical Research: The Anthropological Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Carl Winsløw

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Collaborative Evaluation of Teaching and Assessment Interventions: Ideas From Realistic Evaluation . . . . . . . . . . . . . . . . 139 Paola Iannone

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The Development of Inquiry-Based Mathematics Teaching and Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 Barbara Jaworski

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The Intimate Interplay Between Developing Teaching and Exploring Mathematics Through Reflecting in, on and for Teaching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 John Mason

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Signatures of Teaching Mathematics . . . . . . . . . . . . . . . . . . . . . . . . 199 Günter Törner

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Long-Term Principles for Meaningful Teaching and Learning of Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 David Tall

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

Chapter 1

Reflecting on Teaching Mathematics at University: Collaboration Among Mathematics Educators and Mathematicians Sepideh Stewart

Abstract Studies concerning collaboration among research mathematics educators and mathematicians at the university level have had limited attention in mathematics education literature. This chapter reopens the case to promote productive collaborations among the two communities and discusses my collaborative research studies from 2009 to 2022. Research teams in these narrative studies consisted of mathematics educators, mathematicians, and several undergraduate research assistants. The data for these studies came from mathematicians’ reflections and written journals after each class as well as students’ reflections, surveys, interviews, and regular discussions at research meetings. Mathematicians reflected on pedagogy, mathematics, their students, and themselves. Their dual role as mathematics teachers and education researchers placed them in a unique position of involvement in every aspect of the research, including analyzing the results, writing manuscripts, and sometimes presenting at conferences. The research team employed relevant mathematics education theories to make sense of the data. The results revealed mathematicians’ reflections on teaching various mathematics subjects (linear algebra, abstract algebra, calculus, and algebraic topology). The outcomes of these studies, mathematically and pedagogically intense, were often challenging to establish by observing the mathematicians’ classes and interviewing them as participants. The results showed that while mathematics theories and their communicating and explanatory capabilities strengthened the research studies, more work on evolving and creating new frameworks is needed to capture their complexities. Keywords Reflection · Collaboration · Mathematics · Teaching · Research · University · Mathematics education · Mathematician

S. Stewart (✉) The University of Oklahoma, Norman, OK, USA e-mail: [email protected] © Springer Nature Switzerland AG 2023 S. Stewart (ed.), Mathematicians’ Reflections on Teaching, Advances in Mathematics Education, https://doi.org/10.1007/978-3-031-34295-0_1

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1.1

Introduction

As a vital topic, the collaboration among mathematicians and researchers in mathematics education at the university level has had limited attention in the literature. Over three decades ago, Dreyfus (1991) emphasized that “one place to look for ideas on how to find ways to improve students’ understandings is the mind of the working mathematician. Not much has been written on how mathematicians actually work” (p. 29). The initial series of efforts to understand the mind of the working mathematicians was interviewing them. For example, Sfard (1994) interviewed three renowned mathematicians (a logician (ML), a set theorist (ST), and a specialist on ergodic theory (ET)). Sfard found that: In the answer to the question about what happens in their minds when they feel that they have arrived at a deep understanding of a mathematical idea, they unanimously claimed that the basis of this unique feeling is not a manipulative power but an ability to “identify a structure that [one is] able to grasp somehow” (ST), or “to see an image” (ET), or “to play with some unclear images of things” (ML). (p. 48)

In a book titled, Mathematicians as Enquirers: Learning about Learning Mathematics, Burton (2004) interviewed 70 mathematicians with the goal of building a model that approaches “the learning and teaching of mathematics as a meaningmaking, rather than meaning-transferral, enterprise” (p. 27). Burton hoped to identify and bridge the gap between “how mathematicians themselves came to know and how they promoted learning in others” (p. 27). In another book titled, Amongst Mathematicians: Teaching and Learning Mathematics at University Level, Nardi (2008) examined dialogues between mathematicians and mathematics educators in the form of interviews, having conversations about these fragile yet crucial relationships (see also Nardi, Chap. 3. this volume). Although interviewing mathematicians has continued and remains a datagathering method in some studies, others started collaborating with mathematicians. For example, Bass (2005) encouraged collaboration and stated that “as practitioners of the discipline, research mathematicians can bring valuable mathematical knowledge, perspectives, and resources to the work of mathematics education. This is a tradition worthy of continued development and support” (p. 430). More than two decades ago, Speer et al. (2010) declared that “very little research has focused directly on teaching practice and what teachers do and think daily, in class and out, as they perform their teaching work” (p. 111). Soon after, there was a slight change in the trend. For example, a study by Paterson et al. (2011) described a supportive and positive association of two groups of mathematicians and mathematics educators in the same department, which allowed the “cross-fertilization of ideas” (ibid, p. 359). The group met regularly and discussed teaching strategies while watching small clips of each other’s videos during a teaching episode. Hodgson (2012), in his plenary lecture at ICME-12, raised the point about the need for a community and forum where mathematicians and mathematics educators can work as closely as possible on teaching and learning mathematics. It was encouraging to attend this plenary, but how one starts on such a journey? The

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following book (Fried & Dreyfus, 2014), resulting from a symposium dedicated to Ted Eisenberg, sought to delve deeper into this matter. In this edited book titled, Mathematics and Mathematics Education: Searching for Common Ground, Michelle Artigue, Gunter Torner, Ehud de Shalit, and Pat Thompson wrote about their individual collaborative experiences (Thompson, 2014). For example, Michele Artigue, who has experience in many collaborative projects, wrote: Collaboration between mathematicians and didacticians is necessary, and I am personally convinced that no substantial and sustainable improvement of mathematics education can be obtained without building on the complementarity of their respective expertise, without their common engagement and coordinated efforts. However, I am perfectly aware that productive collaboration is not easy to create and that maintaining it, once established, requires continued effort. (Thompson, p. 314)

In the same book, wrote a chapter titled, “Mutual Expectations Between Mathematicians and Mathematics Educators,” and examined four authors’ perspectives on the issue. One of those authors, Mamona-Downs, expressed her views as follows: Particularly over this last decade, there have been quite a few educational studies where the researcher interviews lecturers about their ways of thinking whilst doing mathematics, their teaching principles and practice, and other facets that impinge on their professional life. The output; usually a published paper in an educational journal, aimed mostly for a readership of educators. Hence the interaction between educators and mathematicians largely runs one way. (Dreyfus, 2014, p. 64)

Mamona-Downs expressed her dissatisfaction and concerns, a fair point and reminder that more work is needed to unite the two communities. In another edited book titled Developing Research in Mathematics Education: Twenty Years of Communication, Cooperation, and Collaboration in Europe, many authors reflected on the three Cs from a variety of perspectives in mathematics education. In particular, (Winsløw et al., 2018) delved into the progress in research on University Mathematics Education (UME) and pointed out that: A special characteristic of UME is that many of the teachers are research mathematicians, and research presented at CERME conferences has drawn on different forms of engagement with mathematicians, often in the form of intimate interviews that explore their epistemological and pedagogical perspectives. (p. 62)

Despite this, in their view, “much less research exists when it comes to university teachers’ pedagogical knowledge and its development through formalized education” (p. 69), and they concluded that “organized, deliberate development of UME teachers, based on RUME, is still rare” (p. 70). On the theme of collaboration and reflection, Hospesova et al. (2018) commented that “collaboration has been a central concept, being one element of the CERME spirit the three Cs of communication, cooperation, and collaboration” (p. 186). In their view, “collaborative environments, in which mathematics researchers, teachers, and prospective teachers work together, reflecting on and designing classroom situations, emerged as an appropriated context to build a bridge between theory and practice” (pp. 185–186).

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However, the authors pointed out that “although collaboration is a familiar concept between ERME members, it has to be noticed that the meaning of collaboration is, in general, not defined or clarified” (p. 186). A couple of definitions they found were: “a positive inter-relationship amongst the people involved, not necessarily connected to a specific modality of acting in groups” (Persci, 2010, p. 1987, cited in Hospesova et al., 2018, p. 186) or “teachers and researchers work together as co-learners” (Berg, 2011, p. 2588, cited in Hospesova et al., 2018, p. 186). The authors also noted that some authors envisaged collaboration under Wenger’s (1998) notion of a community of practice, and others also used Jaworski’s (2008) inquiry community. On the topic of reflection, the authors noted that “most papers dealt with pre- or in-service teachers in grades from 1 to 12, preschool and university teachers being almost absent” (p. 183). Reports on collaboration show some limited progress in other parts of the world. For example, in a recent paper by Bardini et al. (2021), the authors reported on three cases of more positive experiences of collaboration in Australia, the USA, and Mexico, highlighting the existence of different forms of collaborative projects. The authors concluded with some open questions. For example, while some collaborative studies emerge to solve a particular problem, “to what extent this collaboration will be only temporary or will take a more regular shape remains an open question” (p. 55). While the literature primarily focuses on school teachers’ reflections on teaching, systematic studies to hear and examine mathematicians’ voices and reflections on teaching university-level mathematics remain sparse. What are the motivations for such collaborative works, and how to start a fruitful conversation and help develop this research area further? In this chapter, I will present our collaborative research studies from 2009 to 2022 with mathematicians who reflected on mathematics, pedagogy, students, and themselves while teaching a course. I will only refer to the results from the currently published studies. Depending on each study’s goal and nature, mathematics education frameworks that effectively captured them were employed to guide the investigations. The overall objectives of this chapter are to: 1. Present our collaborative research studies over a decade. 2. Introduce a vision for a sustainable model to support fruitful collaborations among mathematics educators and mathematicians.

1.2

Our Collaborative Narrative Research Studies (2009–2022)

This section will briefly discuss our collaborative studies (see Table 1.1). The mathematicians in these studies were teaching the following topics: linear algebra, calculus, abstract algebra, and algebraic topology. I also worked with a physicist in

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Table 1.1 The collaborative studies (2009–2022) Topic/year Linear algebra 2009–2016

Context Visualization, the language of linear algebra

Paper Hannah et al. (2011, 2013a, b, 2014, 2015, 2016)

Abstract algebra 2012–2014

Accommodating in the formal world

Calculus 2013–2014

Balancing embodied, symbolic, and formal thinking

Cook et al. (2013, 2014), Stewart et al. (2015b, c) and Stewart and Schmidt (2017) Stewart et al. (2015a)

Physics 2014–2015

Physics bridging embodied and symbolic worlds

Thompson et al. (2016)

Algebraic topology 2014–2016

From intuition to the formal world of mathematical thinking: A geometric topologist’s thought processes

Stewart et al. (2017, 2018b)

Linear algebra 2016–2019

Moving between the three worlds

Stewart et al. (2018a)

Linear algebra 2018–2022 Linear algebra 2018– present

The linear algebra curriculum study group (LACSG 2.0) recommendations Reaching the formal world, linking the worlds

Stewart et al. (2022a)

Linear algebra 2019– present

Linear algebra proofs Linear algebra II tutors’ tactics

Stewart et al. (2022b), Stewart and Tran (2022) and Cronin and Stewart (2022)

Stewart et al. (2019a, b) and Stewart and Epstein (2020), Epstein et al. (under review)

Theoretical lens Three worlds of mathematical thinking (Tall, 2008, 2013) Resources, orientations, and goals (Schoenfeld, 2011) Three worlds of mathematical thinking (Tall, 2008, 2013) Three worlds of mathematical thinking (Tall, 2008, 2013) Three worlds of mathematical thinking (Tall, 2008, 2013) Three worlds of mathematical thinking (Tall, 2008, 2013) Teaching tactics (Mason, 2002a, b) Three worlds of mathematical thinking (Tall, 2008, 2013) Resources, orientations, and goals (Schoenfeld, 2011)

Three worlds of mathematical thinking (Tall, 2008, 2013) Concept images and concept definitions (Tall & Vinner, 1981) Tall’s three worlds (2008, 2013) Harel’s ways of thinking and ways of understanding (2008) Teaching tactics (Mason, 2002a, b) (continued)

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Table 1.1 (continued) Topic/year Linear algebra 2021– present

Context Language including the mathematical symbols in linear algebra; visualization with GeoGebra

Paper Madden et al. (2023)

Linear algebra 2022– present

Linear algebra Proofs; curriculum

McDonald et al. (under review)

Theoretical lens Tall’s three worlds (2008, 2013) Harel’s ways of thinking and ways of understanding (2008) Ways of thinking (Harel, 2008) Needs and epistemological justification (Harel, 2018)

one study. These studies were not initiated based on a pressing problem or crisis. Since linear algebra is my area of research, naturally, I collaborated more with mathematicians who teach this topic. The objectives of these studies were to: (a) Systematically study the nature of teaching mathematics at the university level through the mathematicians’ reflections, detailed journals, handouts, recorded lectures, and other teaching resources and tools, regular research meeting discussions, and a few students’ reflections. (b) Analyze the data using mathematics education theories and literature. (c) Expand the knowledge of the field on collaboration and build a robust model to study and research teaching mathematics at the university level. Depending on our initial goals, several mathematics education frameworks were employed (see Table 1.1). The frameworks provided a structure for designing each study and an appropriate language to communicate and explain our thoughts to each other and ultimately helped to analyze the data.

1.2.1

The Language and Visualization in Linear Algebra (2009–2016)

My collaborative work with mathematicians started as a PhD student at The University of Auckland in New Zealand. My advisor, Mike Thomas, and I were examining students’ conceptual understanding of linear algebra and were interested in using technology in learning and teaching the topic. We started collaborating with John Hannah, a mathematician from the University of Canterbury, New Zealand, using MATLAB to teach linear algebra. My thesis was built on two mathematics education theoretical frameworks, three worlds of mathematical thinking developed by David Tall (2004, 2008, 2013), and Ed Dubinsky’s Action, Process, Object, and Schema (APOS) theory (Dubinsky & McDonald, 2001).

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The goal was to examine linear algebra students’ action, process, and object thinking in embodied, symbolic, and formal mathematical worlds. This was exciting and challenging. Tall’s (2004) three-world model was just published, and Mike Thomas and I found his framework an ideal theoretical framework for analyzing linear algebra students’ thought processes. David Tall (2004, 2008) is the founder of a framework to capture the long-term development of mathematical thinking and look at the human journey from a toddler to a research mathematician. He also applied the theory to make sense of learners’ struggles in learning various mathematics topics (e.g., calculus, analysis, and study of vectors). We were interested in applying his framework as a lens to linear algebra students embodied, symbolic, and formal thought processes at a specific sector of their mathematical journey. In 2008, soon after my PhD defense, we decided to work on another collaborative project with John Hannah that lasted till 2016. The focus of this research was initiated based on my PhD thesis titled “Understanding Linear Algebra Concepts Through the Embodied, Symbolic and Formal Worlds of Mathematical Thinking.” As we started working with John Hannah, we needed a theoretical framework to capture the professor’s beliefs and actions. Hence, we employed Schoenfeld’s (2011) Resources, Orientations, and Goals (ROGs) theory to analyze John’s teaching in conjunction with Tall’s (2004, 2008) three worlds to analyze his students’ reactions (see Mike Thomas’s Chap. 4, this volume). Schoenfeld’s theory was designed for teachers’ ROGs, and we found it applicable to teaching at the university level. Collaborating with Mike Thomas and John Hannah was a rich experience and enlightenment. I saw the enormous value of including a mathematician in every aspect of the research process, from designing the study to publications and presenting together at conferences. Around the same time, I joined a research team involving a mathematician and several mathematics educators at The University of Auckland. The mathematician in this study was teaching a first-year mathematics course, a service course that included both calculus and linear algebra. The study examined teachers’ beliefs and identity, and the team acted as the community of practice to provide support and research expertise (Kensington-Miller et al., 2013, 2014). Other similar research studies were also taking place at the University of Auckland, led by the Development and Analysis of the Teaching of Undergraduate Mathematics (DATUM) research team. This newly formed community provided an excellent environment for growth, building, and sharing new ideas and knowledge (see Mike Thomas’s Chap. 4, this volume). In 2012, I moved to the USA and continued this research with mathematicians at the University of Oklahoma.

1.2.2

Accommodating in the Formal World: Abstract Algebra (2012–2014)

The overarching aim of this narrative study was to investigate how mathematicians live and dwell in the formal world of mathematical thinking and, at the same time, communicate their knowledge to their students. We employed Tall’s (2004, 2008,

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2010, 2013) three-world model to guide this research and to help us understand more about mathematicians as formal thinkers. Although, in theory, we have some understanding of Tall’s formal world, we wanted to understand more about what happens in this world. Our working research questions were: What are some features of the formal world of mathematical thinking? Given that the mathematician is a formal thinker, how does he invite students to his world, and to what extent is he willing to help students to reach a higher level of mathematical thinking? What pedagogical challenges does a mathematician face in communicating formal mathematics to students? The research team consisted of two mathematicians and two mathematics educators. One of the mathematics educators was a former student, and the other mathematician was his former postdoctoral, so their research areas were close. The data for this study came from the instructor’s (Ralf Schmidt) daily reflections on his teaching of an abstract algebra course, which were made available to the research team after each class; the team members’ observations of the classes and their comments; weekly discussion meetings of the whole team after reading each of these reflections; the audio recordings of each session which were later transcribed. In addition, a graduate student took daily journals. The main themes emerging from the data were (a) pedagogical challenges of communicating the “greatness” of a concept (e.g., Galois Theory) to a beginner, (b) difficulties of teaching very abstract concepts (e.g., tensor products) that are hard to explain or break down, (c) having a dynamical class while still being traditional, and (d) mediating the disconnect between the desire for mathematical elegance and the struggles of a student learning complex material. The narrative of the 4-day event discussed in this paper and many conversations with the instructor throughout the semester illustrated that Ralf made the path through the three worlds and made it possible for students to walk through it. In one of the weekly research meetings, Ralf said: “Mathematics drives the class. I don’t even think about pedagogy in some sense.” During this 4-day event, it seemed the student did not have sufficient experience and expertise to follow the path leading up to the main proof. Our qualitative data analyses indicated the mathematician’s and student’s disparate thought processes. It was as if they described completely different classes and, at times, operated in separate worlds of mathematical thought (see an example in Fig. 1.1). The results of investigating mathematician’s courses over two semesters and spending another two semesters analyzing and making sense of data and writing conference papers provided us with some insight into the formal world (Cook et al., 2013, 2014; Stewart et al., 2015b, c; Stewart & Schmidt, 2017). In the mathematician’s view: “The air in the formal world is much thinner, but also much clearer.” The two-semester course in abstract algebra discussed in this study was designed to provide beginning graduate students with a solid foundation in group, ring, and field theories. I visited his classes a number of times. During these visits, on the surface, the class was mainly lecture-style, and everything was done the way I expected and experienced as a graduate student myself. However, reading the mathematician and his student’s journals revealed so much more. For example, although I saw no emotions observing the classes, both the instructor and the student

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Fig. 1.1 Ralf and Kim’s journals. (Stewart & Schmidt, 2017, p. 46)

expressed their emotions in the journals. During the time of this collaboration, the mathematician initiated at least 7–8 activities which he called “Teaching Experiments.” I was present in class during one of those sessions. He turned his lecturestyle class into a problem-solving session and asked a student to go to the board and, with the help of her classmates, solve several problems. In his journals (November 2, 2012), he wrote: I contributed to the discussion, but more like a regular participant, not as a teacher. I had also changed places with the moderating student, meaning I was sitting in the middle of the classroom. Of course, the old conundrum reared its head. While this was a very “cool” and instructive class, we made zero progress on the material we are supposed to cover in this course.

One of the research team members with the same research interest as Ralf, Number Theory, visited the class on one occasion. He said, “mathematics was very interesting, so I did not pay much attention to the pedagogy.” It was interesting to see the interactions between the two mathematicians after the visit compared to the description of the accounts written in the student’s journals. At a conference after my talk, several young mathematicians requested that we make the mathematician’s journals available to everyone. Also, on another occasion, one conference attendee asked, “if you wanted to see the mathematician’s type of thinking, why did you look at his teaching and not his research?” Although I attended one of Ralf’s research seminars and saw the beauty of the mathematics created live, his mathematics was out of my reach. On another occasion, I asked Ralf the next time he went through discovering mathematics to write down the process of how it all happened and make

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it available to the research team, to which he generously spent the time and produced a very detailed description. Sometime after the end of the semester and publishing a couple of papers together, Ralf gave a seminar about this particular research at the mathematics department. Many of his graduate students attended the talk, wanted to know more, and especially liked reading his journals. We shared our publications with the graduate students, and one of those students took a reading course with me the following semester, in which we read and discussed David Tall’s (2013) book weekly for the entire semester. This study set the scene for all the studies that followed. Although the subsequent studies had commonalities in other ways, they also differed.

1.2.3

Balancing Embodied, Symbolic, and Formal World Thinking in Teaching First-Year Calculus (2013–2014)

In another study with a mathematician (Keri Kornelson), we examined her thought processes while teaching calculus (Stewart et al., 2015a). Through an in-depth study of her thought processes, we found that the instructor and her students placed differing values on the three worlds of mathematical thinking, creating a pedagogical challenge. Based on her reflections, she decided to depart from her initial emphasis on providing extensive theoretical background before offering students the opportunity to explore the concept through an example. Although it is not always clear in which order (embodied, symbolic, formal) the concepts should be introduced in different mathematics courses (Hannah et al., 2014), our study aligned with Tall’s (2010) belief that the instructor must decide which order to present the material (given the particular course/audience). However, we believe more research in this area is needed.

1.2.4

Physics Bridging the Embodied and Symbolic Worlds of Mathematical Thinking (2014–2015)

In a study by Thompson et al. (2016), we conjectured that Physics must “bridge” the embodied and symbolic worlds (Fig. 1.2). Our hypothesis was that novice students struggle to embody the symbols and symbolically express the embodiments. We believed that the Physics instructor created a bridge for his students to move between the embodied and symbolic worlds. He put several connected support pillars in place, including classroom demonstrations of physical phenomena, a student response system that allowed real-time communication with the instructor, and peer instruction. The experienced instructor acted as a guide for his novice students as they crossed uncharted territory. He often broke more complex problems down

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Fig. 1.2 Physics: a bridge between the embodied and symbolic worlds. (Thompson et al., 2016, p. 1341)

into smaller, more manageable pieces. He noted the importance of students creating visualizations regularly. He believed: “Sometimes students’ main obstacle to crossing the embodied–symbolic bridge is simply a lack of mathematical knowledge. I wish I could guarantee that my students had vector calculus when we were talking about some of this” (p. 1345). Students’ self-generated drawings on the final exam revealed gaps in their embodied understanding even though their overall exam grades showed that they had a firm grasp on symbolically solving related problems.

1.2.5

From Intuition to the Formal World of Mathematical Thinking: A Geometric Topologist’s Thought Processes (2014–2016)

In this study, we examined a geometer’s thought processes while teaching Algebraic Topology over a semester (Stewart et al., 2017). We spent the following semester analyzing his teaching journals, which he wrote after each class and examined them in the weekly research meetings by asking the geometer further questions. We noticed that he could confidently move between the worlds of mathematical thinking (see Fig. 1.3). During the research meetings, the mathematician would describe some of the key points of his past week’s lecture and often wrote them on the board or drew pictures on paper. The research team members often asked questions about what made him move, for example, from embodied to symbolic. It was unclear at what point the mathematician decided to move, for example, from the embodied to the symbolic. I believe that he never left the embodied world. I conjectured that he moved to symbolic because, generally, the symbolic enables us

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Fig. 1.3 The three-lens view of homology theory. (Stewart et al., 2017, p. 2264)

to operate on objects and manipulate them. I also believe he gained more understanding, so the switch was worthwhile. He claimed he could also work and manipulate his pictures and embodied objects. When I think of the mathematical world of algebra, I have examples in my mind, many of which are very embodied, and many of which are symbolic; I also know the axiomatic definitions of concepts in this world like “group,” “ring,” “field,” etc. So, when I think of the world of algebra, all three lenses (embodied, symbolic, formal) kick into gear. Likewise, for the mathematical world of topology (Stewart et al., 2017, p. 2264). The mathematician “refused to give students proofs that were pre-packaged. More specifically, he wanted to provide students with intuitions and pictures that would help them understand the conceptual nature of the proof and ultimately lead them to it” (p. 2262). In addition, the instructor reported that students experienced the most difficulty in moving from the embodied world into the formal world. He also gave handouts to his students at the start of each class. The instructor often said, if you really want to know how I think, you need to look at my handouts. One student in his class also took daily journals. On one occasion, the student wrote: Dr. Brady’s way of proving results that come from concepts we’re already supposed to have come across before his class is nice, I think. He gives a detailed outline verbally, which is helped along visually by his pictures and hand gestures. For the most part, I’ll watch without writing almost anything, but I definitely get a lot out of reviewing concepts in this way. I’m a little worried, however, that when we get to brand new material, Dr. Brady’s way of proving results might remain in the same verbal/hand-waving/picture-drawing style and that this won’t be enough for me to follow the proof right there and then. He tends to speak and write very quickly, which is fine when we’re reviewing. But since I can either copy furiously what

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he writes on the board or listen to him, but not both, this could become a problem (Stewart et al., 2017, p. 2262). The student later mentioned that during the oral part of the General Exam (a requirement for PhD), although he couldn’t remember all the definitions and concepts discussed in class, he remembered the pictures that the mathematician often drew (by hand) in the margins of the handouts (Stewart et al., 2018b).

1.2.6

Moving Between the Worlds of Mathematical Thinking in Linear Algebra (2016–Present)

Linear algebra consists of many languages and representations, and instructors often move between these languages and modes fluently. Employing Tall’s (2008, 2013) three-world model, I created a set of initial linear algebra tasks that are designed to encourage students to move between the embodied, symbolic, and formal worlds of mathematical thinking (Stewart, 2018). My hypothesis was that by creating opportunities to move between the worlds, we would encourage students to think in multiple modes of thinking, resulting in a richer conceptual understanding. Theoretically, we believe that movements between different worlds (embodied, symbolic, and formal) of mathematical thinking are beneficial. Having these ideas in mind, we conducted two studies. In the first project, the research team examined a mathematics educator’s (David Plaxco) teaching journals as he taught linear algebra (Stewart et al., 2018a; Stewart et al., 2019c). We created a model (see Fig. 1.4) of instruction to investigate his movements between Tall’s (2013) three worlds of mathematical thinking at the micro-level. The study revealed that despite the instructor’s intentions to move the class between the worlds, his students were reluctant to move.

Fig. 1.4 A model of instructional decision-making (Stewart et al., 2019c, p. 1260).

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In the second project, the research team examined Jonathan Epstein’s teaching journals as he taught linear algebra (Stewart et al., 2019a, b; Stewart & Epstein, 2020; Epstein et al., under review). We introduced Jonathan Epstein, a geometer, to Tall’s (2013) three worlds model. As usual, he taught his class as planned, and the research team did not comment on how and what he should teach. We believe that knowing about Tall’s worlds of mathematical thinking helped him to articulate his thoughts and gave him a language to express his teaching journals as he reflected. The key difference in this study was that Jonathan Epstein, who was teaching the course, introduced his students to Tall’s worlds. This allowed the students to communicate their thoughts in surveys and interviews using the same lens. The result was that the teacher and mathematician in this study, who had gone through the mathematical journey himself, knew the path well. He saw the formal world as the final destination and wished to get his students there. However, this was not so straightforward in the class scenario and took many efforts. In light of our goal and the way this study unfolded, some of our research questions were: (a) When and why did the teacher decide to move between the three worlds in teaching the eigentheory? (b) Were the students comfortable moving with him, and what were some of their challenges? (c) What were some challenges for the mathematician to reach the formal world? The diagram (see Fig. 1.5) shows the progression of his moves between the worlds (Stewart et al., 2019c). Later the team studied and discussed the concept of blending. Fauconnier and Turner (2002) define blending as a process where “two input mental spaces [are] projected to a new space, the blend” (p. 47). I had pondered the idea of moving between the worlds since my PhD defense when John Hannah asked me, “How do the students get to the symbolic world if they do not have the embodied?” These thoughts and conversations have led us to the following modified research questions: In what ways, if any, did the students work in multiple worlds? In what ways did the mathematician work in multiple worlds? (Epstein et al., under review).

Fig. 1.5 An overview of the mathematician’s three worlds emphasis in class

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1.2.7

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Language and Spoken Words in Linear Algebra (2021– Present)

The study aimed to examine the complexities of a mathematician’s ways of thinking while moving between the three worlds of mathematical thinking while teaching linear combination, subspaces, and span (Madden et al., 2023). The research team analyzed a segment of the instructor’s lecture, which was video-recorded and was easy to follow. However, when we transcribed the audio and came to analyze the instructor’s spoken words, the complexities of the languages used were striking. The instructor’s choice of language moved from informal to geometric to formal and back again. Weaving formal language with informal and familiar language was intended to make the topics more accessible and was detected in his lecture and spontaneous responses to questions. The team is currently working on the spoken and written mathematical language in two of the instructor’s video-recorded lectures.

1.2.8

Other Examples of Collaborative Activities with Mathematicians (2016–Present)

In addition to the specific research studies mentioned above, we have had other collaborative activities that all led to publications and joint presentations at conferences and workshops.

1.2.8.1

Algebra Issue in Calculus Classrooms

The edited book titled, And the Rest is Just Algebra (Stewart, 2017), is a collaborative project with mathematics education researchers primarily working in mathematics departments, several research mathematicians, a mathematics education researcher and elementary school teacher, a mathematics education researcher and former high school teacher from the college of education, and a group of cognitive psychologists. The chapters in this book reveal the students’ challenges with algebra at school that continue to college and the unsatisfactory outcomes from the remedial courses, such as college algebra, that can work as barriers to students’ success. The book initiated a number of research studies that examined the consequences of not dealing with school algebra (e.g., Stewart et al., 2018d; Reeder et al. 2019).

1.2.8.2

Challenges and Strategies in Teaching Linear Algebra

In 2016, I co-chaired a discussion group (DG) on Teaching Linear Algebra at the 13th International Congress in Mathematics Education (ICME 13) in Hamburg,

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Germany. The ICME 13 series volume, Challenges and Strategies in Teaching Linear Algebra (Stewart et al., 2018c), resulted from this DG. It featured papers from 18 authors (12 mathematics educators and 6 mathematicians) from 9 countries. Some of the key questions discussed in the DG included the role of applications of linear algebra, some advantages of proving results, how to use challenging problems in teaching, the role of visualization, and the order (pictures, symbols, definitions, and theorems) in which we should teach concepts.

1.2.8.3

Linear Algebra Curriculum Study Group (LACSG 2.0)

In the early 1990s, the Linear Algebra Curriculum Study Group (LACSG) was formed and later published a set of recommendations, including a core syllabus for the first course in linear algebra (Carlson et al., 1993). In 2018, almost three decades later, with support from NSF, I invited a group of mathematicians and mathematics educators to a 2-day workshop at the University of Oklahoma to revise those recommendations (IMAGE, 2018). Soon after the workshop, we formed a committee (LACSG 2.0) to rethink many issues surrounding linear algebra education and preparing students for the future job market. The committee also met with a panel from the industry. We offered a vision for the future of linear algebra through this collaboration and created a new set of recommendations and a core syllabus for both first and second courses in linear algebra. The new recommendations which were published in The Notices of the American Mathematical Society (Stewart et al., 2022a) suggest the following: Teaching linear algebra sooner in the curriculum, removing calculus as a prerequisite, considering the needs of industry, being aware of the latest research in linear algebra education, taking advantage of technology in teaching, motivating concepts with applications, and developing second courses in linear algebra. (p. 814)

In 2021, the LACSG 2.0 members presented the recommendations at two conferences (Mathematical Association of America (MAA) Virtual Panel; Society for Industrial and Applied Mathematics (SIAM)).

1.2.8.4

Resources for Teaching Linear Algebra

The International Linear Algebra Society (ILAS) education committee comprises mathematicians interested in teaching linear algebra. I am currently serving as a chairperson of this committee. We are embarking on a new project inviting the ILAS community to submit lesson plans for linear algebra at any level. The lesson plans will be peer-reviewed and stored on the ILAS website. A description of a lesson plan and an example can be found in IMAGE 69, the Bulletin of ILAS December 2022 issue. In addition, the ILAS education committee members are the guest editors of a special issue of Problems, Resources, and Issues in Mathematics Undergraduate Studies (PRIMUS) titled Linear Algebra Education: An International Perspective.

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Since ILAS is an international community, collaborative linear algebra education projects will benefit from this group of mathematics researchers’ content, curriculum, and instructional expertise.

1.2.8.5

Teaching Linear Algebra Proof

In a research project with Anthony Cronin, we examined the tutors’ responses to students’ queries in linear algebra (Cronin & Stewart, 2022). Employing Mason’s (2002b) tactics, we built a model and analyzed the data gathered over 6 years. We also worked on an exploratory research project investigating linear algebra students’ perspectives on proof as we were both teaching a second linear algebra course at the same time (Stewart et al., 2022b; Stewart & Tran, 2022). Investigating linear algebra students’ (Ireland and the USA) perspectives on proof and our collaborative work have been fruitful in the sense that we are fine-tuning our research questions and, practically speaking, mindful of how our students deal with the complexities of linear algebra proofs.

1.2.9

Summary

Throughout many of these research studies, one of our main goals was a deeper understanding of the nature of each world of mathematical thinking in real time with the help of the experts who had journeyed through these worlds. We were interested to see how mathematicians were dwelling inside each world, as well as to see how they were moving between the worlds and how they were enabling the learner to make suitable pathways between the worlds. According to Duval (2006), moving between registers is essential in understanding mathematics. One of our central research questions has been moving between the three worlds of mathematical thinking. If you ask a mathematician why you move between the worlds, the response would be, I don’t know, I just do. In the study with David Plaxco, we tried to capture the exact incident and moments that the instructor moved between the worlds and tried to analyze those on a micro-level. We found that although David Plaxco tried to move students between the worlds, the students did not necessarily want to move (Stewart et al., 2018a, p. 1263): We saw how a formal goal was set, then when symbolic reasoning was unsuccessful, a shift to the embodied world enabled the instructor to create meaning and use it to connect back to the symbolic and finally to the formal world. Also, this time, he anticipated that students will not be able to make connections and made provision for it.

In the study on Abstract Algebra, the instructor mainly worked in the formal world (Stewart & Schmidt, 2017), and it was evident that the student could not think within that world. We also found that the most challenging move for students in Algebraic Topology was embodied to formal (Stewart et al., 2017). In another Linear

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Algebra study, we saw that the teacher was determined to reach the formal world (Stewart et al., 2019a, b). In some studies (Stewart & Schmidt, 2017; Stewart et al., 2017; Thompson et al., 2016), we looked at bridging the gap between teachers and students. In their own way, each person who reflected tried to bridge the gap. For example, in one study, the teacher used his handouts to reach out to students (Stewart et al. 2018b). The physicist tried to communicate with his students in a variety of ways. He put several connected support pillars in place, including classroom demonstrations of physical phenomena, a student response system that allowed real-time communication with the instructor, and peer instruction. In another study, the teacher gave his students the definitions for Tall’s three worlds of mathematical thinking. Data from surveys and interviews showed that students used those theoretical notions and language (embodied, symbolic, formal) to communicate and express their thoughts (Stewart et al., 2019a, b).

1.3

The Design of Our Collaborative Studies

This section will explain the design and structure of our collaborative research studies, which were developed in 2012 and carried over to the consequent studies (with minor changes). The collaborative research studies were considered qualitative narrative studies. According to Creswell (2013), narrative researchers collect stories from individuals about their lived and told experiences, which shed light on their identities as individuals and how they see themselves. Narrative research is best for capturing the detailed stories or life experiences of a single individual or a small number of individuals. As Creswell states, in analyzing the qualitative data (e.g., interviews, observations), the researchers may take an active role and “restory” the stories into a framework that makes sense. Creswell (2013, p. 74) defines restorying as, “the process of reorganizing the stories into some general type of framework. This framework may consist of gathering stories, analyzing them for key elements of the story (e.g., time, place, plot, and scene), and then rewriting the stories to place them within a chronological sequence” (Ollerenshaw & Creswell, 2002). Narrative studies also often collaborate with participants and actively involve them in the research (Clandinin & Connelly, 2000). Nardi (2016) noted that “mathematicians have their own ‘stories’, their own ways of articulating how they make sense of their students’ learning and their own pedagogical practices” (p. 366). The nature of the collaboration was such a way that the mathematicians were not the subjects. They were co-researchers and equipped with the language of mathematics education to communicate their thought processes, experiences, and beliefs about teaching and learning. There are many advantages to doing so, as Heron (1985, p. 128) placed it:

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Co-operative inquiry (Reason & Rowan, 1981) is primarily a way of doing research with people rather than on people... It is only when the subjects of research start to participate in the thinking that generates, manages and draws conclusions from the research – that is, when they become co-researchers – then they manifest within the research situation as fully selfdetermining people.

At the start of each study, we set up a shared folder to store all our data, mathematics education papers, draft papers, and other resources and made it available to everyone on the research team. We read and discussed many papers and book chapters and shared our views during the meetings. This process was ongoing but significantly increased while analyzing the data and later writing and publishing the papers. Each research team consisted of a mixture of mathematicians and mathematics educators, a cognitive psychologist (three studies), and a physics educator (one study). Occasionally, I invited other mathematicians to participate in the meetings to encourage conversations. In each study, a mathematician had dual roles as a teacher and a researcher who took daily journals about each class and shared them with the team before the research meetings. This dual role was indispensable in every aspect of the research. The mathematician was free to write anything that they felt noteworthy. Each mathematician reflected on mathematics, students, and self while teaching. Reflection was indispensable in these studies and made it possible for us to look into the mind of the working mathematician. In some sense, it made everyone on the research team think about our classes and teaching. The literature maintains that reflection is essential to teaching mathematics (e.g., Davis, 2006; Moore-Russo & Wilsey, 2014). According to Dewey (1933), reflection is “active, persistent, and careful consideration of any belief or form of knowledge in the light of the grounds that support it and the further conclusions to which it tends” (p. 9). Fund (2010) adds that “teachers need to develop particular skills, such as observation and reasoning, in order to reflect effectively and should have qualities such as open-mindedness and responsibility” (p. 680). According to Moon (1999), “reflection is the tool for service rather than being part of the service itself” (p. 19). Some writers contented that, as adults, we cannot learn from experience without reflection (Pearson & Smith, 1985; Burnard, 1991, Cited in Moon, 1999, p. 21). Our research meetings always started with an uninterrupted time for the mathematician to talk about the classes taught in the past week. Mathematicians’ written journals were projected on the screen during the meeting, and the team was encouraged to read them before coming to the meetings. The meeting was open for everyone to ask questions. Sometimes, the team members would ask clarifying questions during those initial times allocated for the mathematicians. When mathematicians talked about mathematics, our attention often shifted solely to the beauty and complexity of mathematics. The nature of these conversations was also very delicate and intense. The mathematicians could easily get sidetracked from the teaching, and our education research objectives and possibly missed to convey vital information. On one occasion, one mathematician said, “that’s all I could remember,” so we concluded the meeting. On another occasion, after some general conversation about his course, the mathematician suddenly remembered more about the events in class and asked to turn on the recorder to

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express his thoughts. The key was always to stay focused and allow for times of possible chaos. In Heron’s view (1985, p. 135): It really is important for the group to be able to tolerate intermittent confusion, ambiguity, uncertainty, chaotic profusion of issues and possibilities and apparent difficulties. Otherwise, there may be a tendency in group reflection phases for members to press for premature intellectual closure as a defense against the anxiety of the whole process. Clearly this is not in the interests of valid reflection. The group therefore needs to abide with the chaotic profusion for a while, and wait for a genuinely creative and illuminating order to emerge in its own time.

Reading the teaching journals alone could not replace our rich pedagogical and stimulating discussions during the weekly meetings. In my opinion, we were engaging in deep conversations, and no single mathematics education theory could perhaps capture its complexity. Even though we could not theoretically explain everything about the teaching process, we benefited from talking about teaching a particular mathematics subject and gained ideas that could be easily applied to other mathematical topics. One feature of our meetings was to be open to new ideas, pause and reflect on more profound thoughts, and not treat the meeting like a rigid business meeting with a set agenda. The mathematicians often wrote their ideas on the board and talked about them to the group. In two of the studies, some research team members made classroom visits. These were, at times, helpful and produced insights and other conversations during the research meetings. Hospesova et al. (2018, p. 187) noted that: Although teachers and researchers seem, in general, to have identical objectives, that is not always really true: “Researchers look for answers to theoretical questions, while teachers deal with practical problems” (Hospesova et al., 2007, p. 1914). In contrast, mathematicians in our studies often welcomed the theories and gravitated toward them. Throughout the studies, mathematicians’ dual roles as instructors and researchers were crucial and indispensable to the studies. A summary of mathematicians’ roles as a teacher versus their role as a researcher is shown in Tables 1.2 and 1.3. All meeting conversations were audio-recorded and later transcribed. Some mathematicians also shared their notes, handouts, slides, exams, and recorded lectures. Table 1.2 Mathematician’s teaching roles

Reflection Mathematics Teaching Students Self

Teaching journals Taking daily journals and sharing them with the research team

Weekly team meetings Describing teaching thoughts and actions in the past week Explaining the mathematics Answering questions

Designing teaching materials Teaching experiments Technologybased apps and tools

Mathematics education literature and theories Getting to know the mathematics education theories and their role in education research Using theoretical language in communicating ideas

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Table 1.3 Mathematician’s research roles

Reflection Mathematics Teaching Students Self

Designing research tools Surveys, tests, interview questions

Analyzing Teaching journals, tests, interview questions Themes and coding

Writing and presenting at conferences Help with research questions A first draft of the results Help with outlining a discussion section and writing a portion of it Mathematical and technical Parts of the paper Help address the reviewers’ comments Help prepare slides and present at conferences

Mathematics education literature and theories Studying the mathematics education theories Discussing them in weekly meetings Relating the theory to the study Helping to choose the appropriate theory (or theories)

At the end of each semester, the research team analyzed the journals and strived to make sense of the data. This activity involved reading the journals, establishing common themes, and open coding (Strauss & Corbin, 1998). Although timeconsuming, the process is highly beneficial since we see the data as big pictures and feel it in fine grains. The mathematicians were involved in the entire process of coding and shed light on situations that were not easy to interpret from the data since they taught the course and took journals. The collaborative process of writing and publishing is vital. The team members were influenced by their disciplines and had different strategies. The reviewers often had questions directly for the mathematician and asked for more details. We discussed those carefully and often addressed them during our team meetings. Inspired by Dreyfus’s (1991) statement, in the beginning, our working research question was about what goes on in the mind of the working mathematician. This was an important starting place, but ideally, we needed to be more specific to make progress. Throughout the studies, we revisited and refined our research questions and sought to articulate our thoughts better, proving that we need time for our ideas to settle and new ideas to emerge. Feedback from the reviewers and eventually publishing papers helped to articulate and focus our research questions as the years went by. Access to online platforms also helped our collaboration as we continued collaborating with colleagues from different institutions and time zones. Notably, the presence of students and their role in the studies were very significant. Their daily journals and contributions as research assistants were indispensable to the research. All undergraduate research assistants were our former linear algebra students, and the research teams highly valued their input. Two graduate students took journals about their classes and their perspectives on learning mathematics. We also collected data from students’ surveys and interviews and shared them with the mathematicians after the end of the semester. One of the graduate

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students who took daily journals also took a couple of mathematics education courses with me in the following semesters.

1.4

A Vision for a Fruitful and Sustainable Collaboration Model Among Mathematics Educators and Mathematicians

Involving mathematicians in every aspect of the research significantly impacted our research studies. Unlike rigid business meetings, allowing time for reflection, sharing, and generating new ideas was indispensable to our work. The weekly meetings and reflections were significantly more productive than just observing a classroom. Simply interviewing mathematicians and not involving them in every aspect of the research will result only in a small fraction of what goes on in the mind of the working mathematician. Reflection was the most important component in our research studies and played a crucial role, and reading mathematicians’ reflections was the most rewarding part of these collaborative studies. Although many mathematics teachers at the university level often reflect, the systematic reflections supported by a research team who were excited to hear and learn more were instrumental in our research studies. In some sense, the mathematicians in these studies never stopped reflecting. Moreover, their reflections were recorded in their journals, allowing the research team to refer back to them if needed. Another important component was that, throughout the studies, we collaboratively strived to make sense of the mathematics education theories we were employing. As we applied them to the data, we often struggled to communicate and articulate our emerging ideas to each other, our audience, and the reviewers of our papers. Initially, as the mathematics educator, I thought my responsibility was introducing the papers to the team and reading and interpreting them for everyone. As time passed, these initial responsibilities changed significantly. We spent months reading papers together, reflecting, and revising our research questions while preparing papers for publication, learning and relying on each other’s take on the theories. Reading and discussing mathematics education theories together and dealing with their complexities and subtleties were highly beneficial and critical to the success of each study. Mathematicians are immensely interested in theories and delve deeply into a careful discussion about them. Hence evolving and generating new theories has enormous potential. Due to their discipline and training, mathematicians pay careful attention to details and bring their strength and expertise into education research studies. Theories are remarkable as they travel over boundaries and, in some ways, glue us together and strengthen our claims. Our studies indicated that productive interactions between the two communities could generate deep conversations and, consequently, new research questions that

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could significantly change how we teach mathematics at the university level. The mathematics departments must promote, value, and reward such collaborations. Furthermore, the collaboration among mathematicians and mathematics educators is vital in advancing mathematics education at the university level. Hence, designing systematic research studies to understand the nature of teaching at the university level should be prioritized in mathematics education research. While some work is available (e.g., the authors in this volume), further ongoing research is desperately needed. Collaboration should come naturally. Our studies were not initiated based on a pressing problem or crisis. The structure is needed so that progress will be made. The commitment of one semester of writing reflections, 6 months or more analyzing the data, and 1 year of preparing manuscripts and publishing is required. The most challenging part of collaborative studies has been analyzing the data and determining meaningful theoretical and practical results. Writing stories about what takes place in class and trying to communicate unspoken thought processes is valuable in our discipline. However, regrettably, since we are not systematically reading and writing reflections, I believe that our analyzing capabilities need to be improved, and we require better mechanisms to analyze mathematically and pedagogically complex and dense reflections. Still, many concerns related to collaboration among mathematicians and mathematics educators remain. Some have been raised and discussed in this book. An even more significant concern is that the more we do not collaborate, the further behind we will get from knowing how to collaborate, examine our work, and continuously develop our collaborative endeavors. As Artigue puts it, “There is no alternative because, as already stressed above, each community alone will not produce sustainable and large-scale improvement of mathematics education” (Thompson, 2014, p. 318). Overall, the unique design of our collaborative work, reflection on teaching, and collaboratively studying mathematics education theories are the main pillars of our studies. Many of our activities with mathematicians are ongoing, as research papers are in the writing stage, and new research collaborations are just starting. Without truly collaborating, our research products would not have happened, and if they did would have been a completely different product. By reflecting on these studies and the incredible insights from all the authors in this volume, we must strive to explore new directions in the future. Acknowledgments I would like to sincerely thank my collaborators John Hannah, Ralf Schmidt, Keri Kornelson, Noel Brady, Bruce Mason, David Plaxco, Jonathan Epstein, and Jeff Meyer. While teaching a mathematics class, they took journals, shared their thought processes, teaching resources, and, most importantly, their valuable time for many years. Many thanks to my collaborators, John Paul Cook, Ameya Pitale, Mike Thomas, Clarissa Thompson, and Jonathan Troup, for their valuable work, help, and support over the past decade. I would like to thank my undergraduate research assistants for their work on some of the research studies and all students who wrote daily journals and participated in our surveys and interviews.

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References Bardini, C., Bosch, M., Rasmussen, C., & Trigueros, M. (2021). Current interactions between mathematicians and researchers in university mathematics education. In V. Durand-Guerrier, R. Hochmuth, E. Nardi, & C. Winslow (Eds.), Research and Development in university mathematics education: Overview produced by the international network for didactic research in university mathematics. Routledge. Bass, H. (2005). Mathematics, mathematicians, and mathematics education. American Mathematical Society, 42, 417–430. https://doi.org/10.1090/S0273-0979-05-01072-4 Burton, L. (2004). Mathematicians as enquirers: Learning about learning mathematics. Kluwer. Carlson, D., Johnson, C. R., Lay, D. C., & Porter, A. (1993). The linear algebra curriculum study group recommendations for the first course in linear algebra. The College Mathematics Journal, 24(1), 41–46. Clandinin, D. J., & Connelly, F. M. (2000). Narrative inquiry: Experience and story in qualitative research. Jossey-Bass. Cook, J. P., Pitale, A., Schmidt, R., & Stewart, S. (2013). Talking mathematics: An abstract algebra professor’s teaching diaries. In S. Brown, G. Karakok, K. H. RoH, & M. Oehrtman (Eds.), Proceedings of the 16th annual conference on research in undergraduate mathematics education (pp. 633–536). Denver. Cook, J. P., Pitale, A., Schmidt, R., & Stewart, S. (2014). Living it up in the formal world: An abstract algebraist’s teaching journey. In T. Fukawa-Connolly, G. Karakok, K. Keene, & M. Zandieh (Eds.), Proceedings of the 17th annual conference on research in undergraduate mathematics education (pp. 511–516). Denver. Creswell, J. W. (2013). Qualitative inquiry and research design: Choosing among five approaches (3rd ed.). Sage. Cronin, A., & Stewart, S. (2022). Analysis of tutors’ responses to students’ queries in a second linear algebra course at a mathematics support center. The Journal of Mathematical Behavior, 67, 100987. https://doi.org/10.1016/j.jmathb.2022.100987 Davis, E. (2006). Characterizing productive reflection among preservice elementary teachers: Seeing what matters. Teaching and Teacher Education, 22, 281–301. Dewey, J. (1933). How we think: A restatement of the relation of reflective thinking to the educative process. D. C. Heath & Company. Dreyfus, T. (1991). Advanced mathematical thinking processes. In D. O. Tall (Ed.), Advanced mathematical thinking (pp. 25–41). Kluwer. Dreyfus, T. (2014). Mutual expectations between mathematicians and mathematics educators. In M. N. Fried & T. Dreyfus (Eds.), Mathematics & Mathematics Education: Searching for common ground (Advances in mathematics education) (pp. 57–70). Springer. Dubinsky, E., & McDonald, M. (2001). APOS: A constructivist theory of learning. In D. Holton et al. (Eds.), The teaching and learning of mathematics at university level: An ICMI study (pp. 273–280). Kluwer. Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61, 103–131. Epstein, J., Powers, A., Stewart, S., & Troup J. (under review). Working in multiple worlds of embodied, symbolic, and formal thinking in linear algebra: The experiences of a mathematician and his linear algebra students. Fauconnier, G., & Turner, M. (2002). The way we think: Conceptual blending and the mind’s hidden complexities. Perseus Books Group. Fund, Z. (2010). Effects of communities of reflecting peers on student-teacher development, including in-depth case studies. Teachers and Teaching: Theory and Practice, 16, 679–701. Hannah, J., Stewart, S., & Thomas, M. O. J. (2011). Analysing lecturer practice: The role of orientations and goals. International Journal of Mathematical Education in Science and Technology, 42(7), 975–984.

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Hannah, J., Stewart, S., & Thomas, M. O. J. (2013a). Conflicting goals and decision making: The deliberations of a new lecturer. In A. M. Lindmeier & A. Heinze (Eds.), Proceedings of the 37th conference of the international group for the psychology of mathematics education (Vol. 2, pp. 425–432). PME. Hannah, J., Stewart, S., & Thomas, M. O. J. (2013b). Emphasizing language and visualization in teaching linear algebra. International Journal of Mathematics Education in Science and Technology, 44(4), 475–489. Hannah, J., Stewart, S., & Thomas, M. O. J. (2014). Teaching linear algebra in the embodied, symbolic, and formal worlds of mathematical thinking: Is there a preferred order? In S. Oesterle, P. Liljedahl, C. Nicol, & D. Allan (Eds.), Proceedings of the joint meeting of PME 38 and PME-NA 36 (Vol. 3, pp. 241–248). PME. Hannah, J., Stewart, S., & Thomas, M. (2015). Linear algebra in the three worlds of mathematical thinking: The effect of permuting worlds on students’ performance. In Proceedings of the 18th annual conference on research in undergraduate mathematics education (pp. 581–586). Pittsburgh. Hannah, J., Stewart, S., & Thomas, M. O. J. (2016). Developing conceptual understanding and definitional clarity in linear algebra through the three worlds of mathematical thinking, teaching. Mathematics and its Applications: An International Journal of the IMA, 35(4), 216–235. https:// doi.org/10.1093/teamat/hrw001 Harel, G. (2008). What is mathematics? A pedagogical answer to a philosophical question. In B. Gold & R. Simons (Eds.), Proof and other dilemmas: Mathematics and philosophy (pp. 265–290). Mathematical Association of America. Harel, G. (2018). The learning and teaching of linear algebra through the lenses of intellectual need and epistemological justification and their constituents. In S. Stewart, C. Andrews-Larson, A. Berman, & M. Zandieh (Eds.), Challenges and strategies in teaching linear algebra (pp. 3–27). Springer. Heron, J. (1985). The role of reflection in a co-operative inquiry. In D. Boud, R. Keogh, & D. Walker (Eds.), Reflection: Turning experience into learning. Kogan Page. Hodgson, B. R. (2012). Whither the mathematics/didactics interconnection? Evolution and challenges of a kaleidoscopic relationship as seen from an ICME perspective. In ICME conference, Plenary Presentation, Seoul, South Korea. Hospesova, A., Carrillo, J., & Santos, L. (2018). Mathematics teacher education and professional development. In T. Dreyfus, M. Artigue, D. Potari, S. Prediger, & K. Ruthven (Eds.), Developing research in mathematics education – Twenty years of communication, cooperation and collaboration in Europe (New perspectives on research in mathematics education series) (Vol. 1, pp. 181–195). Routledge. Jaworski, B. (2008). Building and sustaining inquiry communities in mathematics teaching development. In K. Krainer & T. Wood (Eds.), Participants in MTE (pp. 30–330). Sense Publishers. Kensington-Miller, B., Yoon, C., Sneddon, J., & Stewart, S. (2013). Changing beliefs about teaching in large undergraduate mathematics classes. Mathematics Teacher Education and Development., 15(2), 52–69. Kensington-Miller, B., Sneddon, J., & Stewart, S. (2014). Crossing new uncharted territory: Shifts in academic identity as a result of modifying teaching practice in under-graduate mathematics. International Journal of Mathematics Education in Science and Technology, 45(6), 827–838. McDonald, J., Stewart, S., & Harel, G. (under review). A student-Centered lesson on eigenvalues and eigenvectors. Madden, A., Stewart, S., & Meyer, J. (2023). A linear algebra instructor’s ways of thinking of moving between the three worlds of mathematical thinking within concepts of linear combination, span, and subspaces. In Proceedings of the 25th annual conference on research in undergraduate mathematics education. Mason, J. (2002a). Reflection in and on practice. In P. Kahn & J. Kyle (Eds.), Effective learning & teaching in mathematics & its applications (pp. 117–128). ILT & Kogan Page.

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Mason, J. (2002b). Mathematics teaching practice: A guide for university and college lecturers. Horwood Publishing. Moon, J. (1999). Reflection in learning and professional development: Theory and practice. Kogan Page. Moore-Russo, D., & Wilsey, J. (2014). Delving into the meaning of productive reflection: A study of future teachers’ reflections on representations of teaching. Teaching and Teacher Education, 37, 76–90. Nardi, E. (2008). Amongst mathematicians: Teaching and learning mathematics at university level. Springer. Nardi, E. (2016). Where form and substance meet: using the narrative approach of re-storying to generate research findings and community rapprochement in (university) mathematics education. Educational Studies in Mathematics, 92(3), 361–377. Paterson, J., Thomas, M. O. J., & Taylor, S. (2011). Decisions, decisions, decisions: What determines the path taken in lectures? International Journal of Mathematical Education in Science and Technology, 42(7), 985–995. Reason, P., & Rowan, J. (Eds.). (1981). Human inquiry: A sourcebook of new paradigm research. Wiley. Reeder, S., Stewart, S., Raymond, K., Troup, J., & Melton, H. (2019). Analyzing the nature of university students’ difficulties with algebra in calculus: Students’ voices during problem solving. In A. Weinberg, D. Moore-Russo, M. Wawro, & H. Soto (Eds.), Proceedings of the 22nd annual conference on research in undergraduate mathematics education (pp. 501–508). Oklahoma. Schoenfeld, A. H. (2011). How we think: A theory of goal-oriented decision making and its educational applications. Routledge. Sfard, A. (1994). Reification as the birth of metaphor. For the Learning of Mathematics, 14(1), 44–55. Speer, N. M., Smith, J. P., & Horvath, A. (2010). Collegiate mathematics teaching: An unexamined practice. Journal of Mathematical Behavior, 29, 99–114. Stewart, S. (Ed.). (2017). And the rest is just algebra. Springer. Stewart, S. (2018). Moving between the embodied, symbolic and formal worlds of mathematical thinking in linear algebra. In S. Stewart, C. Andrews-Larson, A. Berman, & M. Zandieh (Eds.), Challenges and strategies in teaching linear algebra (pp. 51–67). Springer. Stewart, S., & Epstein, J. (2020). Linear algebra thinking in the embodied, symbolic and formal worlds: Students’ reasoning behind preferring certain worlds. In S. S. Karunakaran, Z. Reed, & A. Higgins (Eds.), Proceedings of the 23rd annual conference on research in undergraduate mathematics education (pp. 546–553). Boston. Stewart, S., & Schmidt, R. (2017). Accommodation in the formal world of mathematical thinking. International Journal of Mathematics Education in Science and Technology, 48(1), 40–49. https://doi.org/10.1080/0020739X.2017.1360527 Stewart, S., & Tran, T. (2022). Linear algebra proofs: Ways of understanding and ways of thinking in the formal world. In C. Fernández, S. Llinares, A. Gutiérrez, & N. Planas (Eds.), Proceedings of the 45th conference of the International Group for the Psychology of mathematics education (Vol. 4, pp. 43–50). Alicante. Stewart, S., Thompson, C. A., Kornelson, K., Lifschitz, L., & Brady, N. (2015a). Balancing formal, symbolic, and embodied world thinking in first year calculus lectures. In T. Fukawa-Connolly, N. Engelke Infante, K. Keene, & M. Zandieh (Eds.), Proceedings of the 18th annual conference on research in undergraduate mathematics education (pp. 970–976). Pittsburgh. Stewart, S., Schmidt, R., Cook, J. P., & Pitale, A. (2015b). Living in the formal world of mathematical thinking. In K. Beswick, T. Muir, & J. Fielding-Wells (Eds.), Proceedings of 39th psychology of mathematics education conference (Vol. 1, p. 201). PME. Stewart, S., Schmidt, R., Cook, J. P., & Pitale, A. (2015c). Pedagogical challenges of communicating mathematics with students: Living in the formal world of mathematical thinking. In Proceedings of the 18th annual conference on research in undergraduate mathematics education (pp. 964–969). Pittsburgh.

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Stewart, S., Thompson, C., & Brady, N. (2017). Navigating through the mathematical world: Uncovering a geometer’s thought processes through his handouts and teaching journals. In T. Dooley & G. Gueudet (Eds.), Proceedings of the Tenth Congress of the European Society for Research in Mathematics Education (CERME10, February 1–5) (pp. 2258–2265). DCU Institute of Education and ERME. Stewart, S., Troup, J., & Plaxco, D. (2018a). Teaching linear algebra: Modeling one instructor’s decisions to move between the worlds of mathematical thinking. In A. Weinberg, C. Rasmussen, J. Rabin, M. Wawro, & S. Brown (Eds.), Proceedings of the 21st annual conference on research in undergraduate mathematics education (pp. 1014–1022). San Diego. Stewart, S., Thompson, C., & Brady, N. (2018b). Examining a mathematician’s goals and beliefs about course handouts. In A. Weinberg, C. Rasmussen, J. Rabin, M. Wawro, & S. Brown (Eds.), Proceedings of the 21st annual conference on research in undergraduate mathematics education (pp. 1069–1075). San Diego. Stewart, S., Andrews-Larson, C., Berman, A., & Zandieh, M. (Eds.). (2018c). Challenges and strategies in teaching linear algebra. Springer. Stewart, S., Reeder, S., Raymond, K., & Troup, J. (2018d). Could algebra be the root of problems in calculus courses? In A. Weinberg, C. Rasmussen, J. Rabin, M. Wawro, & S. Brown (Eds.), Proceedings of the 21st annual conference on research in undergraduate mathematics education (pp. 1023–1030). San Diego. Stewart, S., Epstein, J., Troup, J., & McKnight, D. (2019a). An analysis of a Mathematician’s reflections on teaching eigenvalues and eigenvectors: Moving between embodied, symbolic and formal worlds of mathematical thinking. In A. Weinberg, D. Moore-Russo, M. Wawro, & H. Soto (Eds.), Proceedings of the 22nd annual conference on research in undergraduate mathematics education (pp. 586–593) Oklahoma. Stewart, S., Epstein, J., Troup, J., & McKnight, D. (2019b). A mathematician’s deliberation in reaching the formal world and students’ world views of the eigentheory. In U. T. Jankvist, M. Van den Heuvel-Panhuizen, & M. Veldhuis (Eds.), Proceedings of the eleventh congress of the European Society for Research in mathematics education (CERME11, February 6–10, 2019) (pp. 4835–4842). Freudenthal group \& Freudenthal institute, Utrecht University and ERME. Stewart, S., Troup, J., & Plaxco, D. (2019c). Reflection on teaching linear algebra: Examining one instructor’s movements between the three worlds of mathematical thinking. ZDM Mathematics Education, 51(7), 1253–1266. Springer. https://doi.org/10.1007/s11858-019-01086-0. Stewart, S., Axler, S., Beezer, R., Boman, E., Catral, M., Harel, G., McDonald, J., Strong, D., & Wawro, M. (2022a). The linear algebra curriculum study group (LACSG 2.0) recommendations. The Notices of American Mathematical Society, 69(5), 813–819. https://doi.org/10.1090/ noti2479 Stewart, S., Cronin, A., Tran, T., & Powers, A. (2022b). Ways of thinking and ways of understanding in the formal world: Students’ perspectives on nature of proofs in a second course in linear algebra. In S. Karunakaran & A. Higgins (Eds.), Proceedings of the 24th Annual Conference on Research in Undergraduate Mathematics Education (pp. 1152–1157). Boston. Strauss, A. L., & Corbin, J. (1998). Basics of qualitative research: Grounded theory procedures and techniques (2nd ed.). Sage. Tall, D. O. (2004). Building theories: The three worlds of mathematics. For the Learning of Mathematics, 24(1), 29–32. Tall, D. O. (2008). The transition to formal thinking in mathematics. Mathematics Education Research Journal, 20(2), 5–24. Tall, D. O. (2010). Perceptions operations and proof in undergraduate mathematics, community for undergraduate learning in the mathematical sciences (CULMS). Newsletter, 2, 21–28. Tall, D. O. (2013). How humans earn to think mathematically: Exploring the three worlds of mathematics. Cambridge University Press.

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Tall, D. O., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12(2), 151–169. Thompson, P. W. (2014). Reflections on collaboration between mathematics and mathematics education. In M. N. Fried & T. Dreyfus (Eds.), Mathematics and mathematics education: Searching for common ground, advances in mathematics education (pp. 313–333). Springer. Thompson, C. A., Stewart, S., & Mason, B. (2016). Physics: Bridging the embodied and symbolic worlds of mathematical thinking. In T. Fukawa-Connolly, N. E. Infante, M. Wawro, & S. Brown (Eds.), Proceedings of the 19th annual conference on research in undergraduate mathematics education (pp. 1340–1347). Pittsburgh. Wenger, E. (1998). Communities of practice: Learning, meaning, and identity. Cambridge University Press. Winsløw, C., Gueudet, G., Hochmut, R., & Nardi, E. (2018). Research on university mathematics education. In T. Dreyfus, M. Artigue, D. Potari, S. Prediger, & K. Ruthven (Eds.), Developing research in mathematics education – Twenty years of communication, cooperation and collaboration in Europe (New perspectives on research in mathematics education series) (Vol. 1, pp. 60–74). Routledge.

Chapter 2

Promoting Effective Collaboration in the Mathematics Community Sepideh Stewart and Bharath Sriraman

Abstract Collaboration is vital for the growth and advancement of organizations and communities, in particular when confronted with challenging tasks. To promote positive and productive collaboration within the mathematics community, we must fully understand its capabilities and complexities. In this chapter, we will define the concept of collaboration and distinguish it from other notions, such as cooperation and coordination, that are sometimes used interchangeably. Furthermore, we will take a close look at the history of collaboration in the mathematics and mathematics education communities. Keywords Reflection · Collaboration · Mathematics education · Mathematician

2.1

Introduction

Collaboration between mathematics educators and mathematicians today is perceived as an ever-widening gulf to be bridged, whereas a historical view suggests that mathematicians were ubiquitous with mathematics educators. The birth of ICMI was forged by mathematicians like Felix Klein at the 4th International Congress of Mathematicians in Rome in 1908. The goal of ICMI was to recognize problems relevant to the teaching and learning of school mathematics, particularly in the way mathematics could be presented in textbooks to make it accessible to both teachers and students. The Stoffdidaktik tradition in Germany follows in this vein with numerous mathematicians (e.g., Arnold Kirsch) with the goal of writing textbooks that follow this style of teaching mathematics (Kirsch, 1977a, b, 1978). In other words, textbooks were written with the goal of making mathematical topics

S. Stewart (✉) The University of Oklahoma, Norman, OK, USA e-mail: [email protected] B. Sriraman University of Montana, Missoula, MT, USA e-mail: [email protected] © Springer Nature Switzerland AG 2023 S. Stewart (ed.), Mathematicians’ Reflections on Teaching, Advances in Mathematics Education, https://doi.org/10.1007/978-3-031-34295-0_2

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accessible by using multiple representations, activating student’s previous knowledge, and with attention to contextual knowledge. An entire genre of textbooks in Germany and in different AMS series can be traced back to this style of teaching mathematics, which is compatible with mathematics education. Sriraman and Törner (2008) also point to this type of collaboration in other parts of Europe. “Furinghetti (2006) reflected on this phase of Italian mathematics education history and mused that although Peano’s goals in the Formulario Mathematico project were utopian, “his enthusiasm and good willingness attracted secondary teachers who collaborated with him and. . .[c]onstitutes an early example of a mixed group of university professors and school teachers working on didactic problems” (p. 102). It is important to note that even reform-based curricular efforts in the USA, like the School Mathematics Study Group (SMSG) during the cold war, focused on producing mathematics textbooks that could be used by teachers to delve deeper into the architecture of mathematics. In this historical view, mathematics is placed at the center of mathematics education, with mathematicians and school teachers collaborating to make mathematics accessible and relevant to school children. This seems to be a far cry from the ongoing math wars that pit teachers and teacher’s organizations against mathematicians and the popular rhetoric that the goals of mathematics educators are divorced from that of mathematicians. If this is true, how is collaboration possible?

2.2

Collaboration

What are the characteristics of a sound model of collaboration, and why is it essential that we in the mathematics community collaborate? According to Bergstrom et al. (1995, p. 1): The goal of community collaboration is to bring individuals and members of communities, agencies and organizations together in an atmosphere of support to systematically solve existing and emerging problems that could not be solved by one group alone. While this is easily “said,” experience shows that it is not easily “done.” It has been likened to “teaching dinosaurs to do ballet.” (Schlechty in DeBevoise, 1986, p. 12)

Hogue (1993) suggested a model based on the Community Linkages-Choices and Decisions (see Fig. 2.1) that defines five levels of relationships and the purpose, structures, and processes of each level. Note that the purpose of the collaboration is “to accomplish shared vision,” and the process involves high leadership and trust, resulting in high productivity. To assist communities in collaborating and achieving shared goals, the National Network for Collaboration designed a collaboration framework “as a comprehensive guide to form new collaboration, enhance existing efforts and/or evaluate the progress of developing collaboration” (Bergstrom et al., 1995, p. 2). The common elements of the framework are grounding, core foundation, outcome and process, and contextual factors:

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Fig. 2.1 Community linkages, choices, and decisions (Bergstrom et al., 1995, p. 3). (Source: Community Based Collaborations – Wellness Multiplied 1994, Teresa Hogue, Oregon Center for Community Leadership)

Grounding, the bedrock of collaboration, is the diversity with which the people, groups, organizations and communities share a desire to collaborate; Core Foundation, the shared purpose and destiny of the collaborative efforts; Outcome, that which is achieved by implementing a collaboration; and the Process and Contextual Factors, those which affect the everyday activities of the collaboration.

Bergstrom et al. (1995, p. 5) describe the core foundation as: The core represents the common ground of understanding. It focuses on creating a sense of common purpose that binds people together and inspires them to fulfill their deepest aspirations. Building the core takes time, care and strategy. The discipline of building a core is centered around a never-ending process, whereby people in the collaboration articulate their common interests-around vision, mission, values and principles. Vision, an image of the desired future - A vision is a picture of the future, described in the present tense, as if it were happening now. Mission, defines the purpose of the collaboration. The Mission represents the fundamental reason for the collaboration’s existence. Values and Principles, the beliefs the individual and groups hold, values and the principles are the guides for creating working relationships and describe how the group intends to operate on a day-byday basis.

In their book titled, Collaboration: What Makes It Work, Mattessich et al. (2001) defined collaboration as: . . .a mutually beneficial and well-defined relationship entered into by two or more organizations to achieve common goals. The relationship includes a commitment to mutual

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Mattessich et al. (2001, p. 60) distinguished collaboration as high risk, coordination as low risk, and cooperation as no risk and defined each as: Cooperation is characterized by informal relationships that exist without any commonly defined mission, structure, or planning effort. Information is shared as needed, and authority is retained by each organization so there is virtually no risk. Resources are separate as are rewards. Coordination is characterized by more formal relationships and an understanding of compatible missions. Some planning and division of roles are required, and communication channels are established. Authority still rests with the individual organizations, but there is some increased risk to all participants. Resources are available to participants and rewards are mutually acknowledged. Collaboration connotes a more durable and pervasive relationship. Collaborations bring previously separated organizations into a new structure with full commitment to a common mission. Such relationships require comprehensive planning and well-defined communication channels operating on many levels. Authority is determined by the collaborative structure. Risk is much greater because each member of the collaboration contributes its own resources and reputation. Resources are pooled or jointly secured, and the products are shared.

Mattessich et al. (2001) identified “twenty factors that influence the success of collaborations formed by nonprofit organizations, government agencies, and other organizations” (p. 7). These factors emerged from 133 research-based health, social science, education, and public affairs studies. An overview of the twenty factors, which are grouped into six categories, Environment, Membership Characteristics, Process and Structure, Communication, Purpose, and Resources (p. 7), is shown in Table 2.1. The authors also posed other questions, such as: “What is the proper “mix” of factors? Are some more important and some less important? Can a project succeed if it has most, but not all, of the factors?” (p. 55). In reinforcing their ideas on collaboration, Mattessich et al. (2001, p. 5) used the analogy of a garden: Let’s say this book focused on gardening rather than collaboration. In that case we would inform you, as a reader and prospective gardener, about the basics of growing a healthy, productive garden. For example, we’d talk about soil conditions, the length of the growing season, and how much sunlight and water are needed to grow various plants. We would not, however, offer detailed instructions on how to plant and tend your own garden.

The authors noted that “sunlight is a factor necessary for a garden. If sunlight is totally absent, the garden will not grow at all. However, if sunlight is present to some degree, the garden will still produce results” (p. 55). Mattessich et al. (2001, p. 33) presented the following questions for future research: What is the relative importance of each factor? Can we assign a “weight” to each, indicating levels of attention that each deserves? What are the costs and benefits of collaboration? Do the benefits outweigh the costs, or vice versa, in certain types of situations, or for certain types of groups? Is it possible to access the net grains of collaboration?

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Table 2.1 Factors influencing the success of collaboration Environment History of collaboration or cooperation in the community Collaborative group seen as a legitimate leader in the community Favorable political and social climate

Membership characteristics Mutual respect, understanding, and trust Appropriate cross section of members Members see collaboration as in their selfinterest Ability to compromise

Process and structure Members share a stake in both process and outcome Multiple layers of participation Flexibility Development of clear roles and policy guidelines Adaptability Appropriate pace of development

Communication Open and frequent communication Established informal relationships and communication links

Purpose Concrete, attainable goals and objectives Shared vision Unique purpose

Resources Sufficient funds, staff, materials, and time Skilled leadership

Mattessich et al. (2001)

The authors also noted that “trust” is a very important factor in collaboration (p. 55). In addition, “partners should continuously monitor whether new groups or individuals should be bought into the ongoing process. A formal integration and education plan for new members should be developed” (p. 16). When to collaborate? According to Hansen (2009), we should not collaborate for the sake of it. Mattessich et al. (2001, p. 34) asserted that: “Those who feel that joint efforts offer promise for getting important work done always need to assess the pros and cons of alternative arrangements to determine which will best suit their needs at a particular time.”

2.3

Collaboration Within the Mathematics Community

The history of mathematics and mathematics education has its genesis in the turn of the late nineteenth–early twentieth century when mathematicians like Felix Klein and Hans Freudenthal became interested in the teaching and learning of mathematics. In both the Romantic and Germanic languages of Europe, mathematics education translates directly to didactics of mathematics. The key to this translation is the preposition “of” and what the preposition denotes, namely, mathematics. Therefore mathematics was the central focus, out of which teaching and learning the stuff (Stoffdidaktik in German) was realized through the writing of textbooks that would “open” up small arenas of mathematics that pupils could pursue at university. The goal of these textbooks like Klein’s classic Elementary Mathematics from a Higher Standpoint: Arithmetic, Algebra, and Analysis was to expose prospective

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mathematics teachers to topics that allowed a glimpse at advanced mathematics. It is no fluke that the didactic tradition of writing mathematics textbooks resulted in high school mathematics teachers being thought of as capable of pursuing their own small research strands in mathematics, known in Germany as Klein’s proverbial garden. A garden is where a teacher of mathematics or mathematician cultivates their skill and competence with a specific area of mathematics and contributes to it by creating illustrative examples. This tradition has been followed in Germany by Arnold Kirsch and colleagues well into the late twentieth century. A new English translated edition of Klein’s classic released a century after the original by the Mathematical Association of America garnered high praise in numerous reviews. One of the reviews said: The first, on arithmetic, algebra, and analysis combines an interest in the logical development of the subject with extensive discussion of pedagogy. Each section asks first “what is the state of our knowledge?” and then “what are the implications for teaching?”. The geometry volume focuses rather on giving an overall “take” on geometry as a whole (Klein even uses the word “encyclopedic”), which of course reflects Klein’s famous idea that the subject should be organized in terms of the groups of isometries attached to various geometries. (Gouvêa, 2004)

Klein himself, in his numerous talks on the “state of mathematics,” invariably linked it to the “teaching of mathematics,” a sentiment echoed by Burton’s (2004) study on mathematicians as enquirers and most recently by Sriraman (2022) on the role of uncertainty in research mathematics, and how this obstacle is removed by catalysts such as accessible textbooks or a teacher/professor who leads them in a fruitful direction. The tension or dialectic between mathematicians as researchers and mathematicians as teachers is not new. Sriraman (2022) presents a model that uses Celucci and Hersh’s philosophy of mathematics to point to the underlying dialectic between intuition and logic (rigor) that also plays out in the classroom when mathematicians are in the role of a teacher. Klein (1887) drew attention to this dialectic between intuition and rigor or rather, the contradictions inherent in teaching mathematics differently from how one approaches it in research mathematics. In his own words: I must add a few words on mathematics from the point of view of pedagogy. We observe in Germany at the present day a very remarkable condition of affairs in this respect; two opposing currents run side by side without affecting one another appreciably. Among the teachers in our Gymnasia the need of mathematical instruction based on intuitive methods has now been so strongly and universally emphasized that one is compelled to enter a protest, and vigorously insist on the necessity for strict logical treatment. This is the central thought of a small pamphlet on elementary geometrical problems which I published last summer. Among the university professors of our subject exactly the reverse is the case; intuition is frequently not only undervalued, but as much as possible ignored. This is doubtless a consequence of the intrinsic importance of the arithmetizing tendency in modern mathematics. But the result reaches far beyond the mark. It is high time to assert openly once for all that this implies, not only a false pedagogy, but also a distorted view of the science. . . . Through this one sided adherence to logical form we have lost among these classes of men much of the prestige properly belonging to mathematics, and it is a pressing and urgent duty to regain this prestige by judicious treatment.

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Concluding Remarks

The relationship of mathematics education with mathematics has been the subject of very few investigations. While regrettably, the communication between the two disciplines has been limited, our research and experiences show that: • It is fruitful to collaborate with mathematicians and share our expertise to seek ways of solving the mathematics education problems both communities are facing. • As professors working in mathematics departments, we want to convey that issues between the two disciplines are not as contentious as they are made to be. There are some false notions about mathematicians that need to be demystified. • We encourage young mathematics educators and mathematicians to work together as early as possible. • New questions will arise that set a new path for the community to consider in the future. Collaboration in the mathematics community should not be like teaching the dinosaurs how to do ballet. In this book, many authors will provide examples of collaboration between the two communities and encourage the readers to get involved in future collaborations with many possibilities yet to be realized and discovered.

References Bergstrom, A., Clark, R., Hogue, T., Iyechad, T., Miller, J., Mullen, S., et al. (1995). Collaboration framework: Addressing community capacity. The National Network for Collaboration. Retrieved from https://www.uvm.edu/sites/default/files/media/Collaboration_Framework_ pub.pdf Burton, L. (2004). Mathematicians as enquirers: Learning about learning mathematics. Kluwer. Gouvêa, F. Q. (2004, October 1). Review of the book Elementary mathematics from an advanced standpoint: Arithmetic, algebra, and analysis (F. Kelin). MAA Reviews. https://www.maa.org/ press/maa-reviews/elementarymathematics-from-an-advanced-standpoint-arithmetic-algebraand-analysis Hansen, M. T. (2009). Collaboration: How leaders avoid the traps, create unity, and reap big results. Harvard Business Press. Hogue, T. (1993). Community-based collaboration: Community wellness multiplied. Oregon Center for Community Leadership, Oregon State University. Kirsch, A. (1977a). Didaktik Math, 5(2), 87–101. Kirsch, A. (1977b). Westermanns Paedagog (Beitraege 29/1977/4, 151–157). ABDE Unterrichtliche Vereinfachung. Kirsch, A. (1978). Aspects of simplification in mathematics teaching. Aspekte des Vereinfachens im Mathematikunterricht (German). Kassel Univ. (Gesamthochschule) (West Germany). Klein, F. (1887). The arithmetizing of mathematics. In B.W. Ewald (Ed.) 1996, From Kant to Hilbert: A source book in the foundations of mathematics (pp. 965–971). Oxford University Press.

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Klein, F. (1908). Elementary mathematics from an advanced standpoint. Arithmetic, Algebra, Analysis (E.R. Hedrick & C.A. Noble, Trans.) (3rd edn., German edn.). Dover Publications. Mattessich, P. W., Murray-Close, M., & Monsey, B. R. (2001). Collaboration: What makes it work (2nd ed.) Sriraman, B. (2022). Uncertainty as a catalyst and condition for creativity: The case of mathematics. ZDM – Mathematics Education, 54(1), 19–33. Sriraman, B., & Törner, G. (2008). Political union/ mathematical education disunion: Building bridges in European didactic traditions. In L. English (Ed.), The handbook of international research in mathematics education (2nd ed., pp. 660–694). Routledge, Taylor & Francis.

Chapter 3

Synergy Between Mathematicians and Mathematics Educators: Stories of Many, and Potent, Facets Elena Nardi

Abstract In this chapter, I draw on my experiences as mathematics education researcher collaborating with research mathematicians in order to tell a story of paths crossing at four points: in research, teaching, professional development and public engagement. I discuss these four tiers of examples to propose a re-imagining of this story, not merely as a story of paths crossing – but as a story of paths “meeting” at a vanishing point, a point where the boundaries between the two communities fade into insignificance, recede and may even be replaced by a strong sense of joint and multi-faceted enterprise. I conclude with indicating how this joint enterprise may look like in the near future. Keywords University mathematics education · Mathematicians’ teaching practices · Collaboration between mathematics education mathematics researchers

3.1

Introduction

I entered the field of mathematics education research as a doctoral student in 1992. I position myself as a (non-research) mathematician who chose to become a researcher in mathematics education. For more than 20 years, I have been collaborating with research mathematicians, with our paths intersecting at various points – at least four: research, teaching, professional development and public engagement. This chapter aims to recount the story of these intersections in order to illustrate that the mathematics and mathematics education research communities have much in common; that these commonalities are evidenced in a multiplicity of contexts; and that experiences of recent years are starting to indicate what the minimal and optimal conditions of productive synergy might be. The mathematicians referred to in this chapter are mostly academic mathematicians, employed by universities to carry out research in mathematics as well as teach

E. Nardi (✉) University of East Anglia, Norwich, England, UK e-mail: [email protected] © Springer Nature Switzerland AG 2023 S. Stewart (ed.), Mathematicians’ Reflections on Teaching, Advances in Mathematics Education, https://doi.org/10.1007/978-3-031-34295-0_3

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mathematics at university level. The community of mathematicians is of course much more diverse: as evidenced, for example, in the membership of the Institute for Mathematics and Its Applications (IMA) in the UK, mathematicians are employed in a range of positions and across many sectors. The story has three parts. In the first, and relatively short, part, I trace the relationship between mathematicians and mathematics educators in research on the teaching and learning of mathematics, and I offer a potted account of issues that have been helping this relationship grow, grow slowly or occasionally stall. In the second, and longer, part, I recount examples of initiatives that have been propelling the deepening of this relationship, from research, teaching, professional development and public engagement. In the third, and final, part, I discuss the four tiers of examples presented in the second part in order to put forward the proposition that these stories of mathematicians and mathematics educators intersecting at various points in research, teaching, professional development and public engagement activities can be re-imagined, not merely as a story of paths crossing – but as a story of paths “meeting” at a vanishing point, a point where the boundaries between the two communities fade into insignificance, recede and may even be replaced by a strong sense of joint and multi-faceted enterprise. I note that, to make my point, I centre on examples of activity across the four fields (research, teaching, professional development and public engagement) initiated by myself and my close collaborators over the years, and that the selection is also heavily UK-based. However, I do so through embedding these examples in the rather impressive bulk of recent activity around the world in this area. Examples of this impressive, recent activity include the annual conference of the RUME SIGMAA (Research in Undergraduate Mathematics Education, Special Interest Group of the Mathematical Association of America); the biennial Delta conferences, the southern hemisphere symposia which have been nurturing exchanges between mathematicians, educators and researchers committed to improving undergraduate mathematics and statistics education since 1997; the rapid growth of the University Mathematics Education Thematic Working Group at CERME, the congress of European Researchers in Mathematics Education and its evolution into the first ERME Topic Conference (Montpellier, 2016; Kristiansand, 2018; Bizerte (online), 2020; Hannover, 2022) of the newly constituted INDRUM (International Network for Didactics Research in University Mathematics); the presence of several university mathematics education Topic Study Groups at recent and imminent International Congresses of Mathematics Education (ICME) including Mathematics education at tertiary level, The Teaching and Learning of Calculus and Mathematics for non-specialist/mathematics as a service subject at tertiary level; the launch of IJRUME, the International Journal of Research in Undergraduate Mathematics Education, by Springer in 2015; and the growing number of Special Issues in international, peer-reviewed journals (such as the 2014 Research in Mathematics Education, RME, issue mentioned later; the 2014 and 2019 ZDM issues focused on

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Calculus and Linear Algebra, respectively; the 2022 and 2023 IJRUME Special Issues focused on Calculus; and so on); special events with a focus on university mathematics education research organised in well-established mathematics fora such as the Mathematisches Forschungsinstitut Oberwolfach (MFO) workshop in December 2014, and its follow-up in Hannover in December 2015. I see these as developments which evidence that university mathematics education research is thriving. They also lend some scope and validity to the point I make: that initiatives which propel and deepen the relationship between mathematicians and mathematics educators can and should encompass teaching and collaborative research on teaching, as well as bring the two communities together on other, forward-facing (university mathematics teachers’ professional development) and outward-facing (public engagement) activities. I note that, while these developments indicate that a burgeoning of fora where the communities might intersect – there is explicit interest in this in the programmes of most conferences and events listed earlier – the existence of such fora per se does not imply a burgeoning relationship between the two communities. As I elaborate in Part I, this crucial relationship remains fragile.

3.2

Part I: The Crucial, Yet Fragile, Relationship Between Mathematicians and Mathematics Educators

The relationship between mathematicians and mathematics educators has been the focus of debate in the writings of several mathematics education authors at least since the 1990s. Anna Sfard’s (1998) discussion with Shimshon A. Amitsur – poignantly presented in the form of a dialogue – is one of the first. Writings by authors from a variety of geographical and institutional contexts such as Michèle Artigue (1998), Anthony Ralston (2004) and Gerry Goldin (2003) have portrayed this relationship as at best fragile. Artigue (ibid.), in her account of mathematics education “through the eyes of mathematicians” (p. 477), summarises the issues to be tackled as follows. Mathematics educators need to maintain a close and comfortable relationship with mathematics on issues to be tackled. Mathematicians are often bemused by the explosion of theories in mathematics education research that they may not be methodologically and epistemologically equipped to understand. Mathematicians may see mathematics educators as inferior mathematicians and may seek from them help in the shape of pedagogical quick fixes that educators simply cannot offer. Brousseau’s (1997) criteria for mathematics education output such as relevance, immediacy, and freedom from jargon are not always met (and one may understandably wonder whether they can or should). Mathematics education researchers need then to engage more systematically with disseminating outputs to less specialist, but interested audiences. Amongst Mathematicians: Teaching and Learning Mathematics at University Level (Nardi, 2008) resonates and extends these earlier accounts, setting out from acknowledging the fragility of this relationship and, particularly in Chapter 8,

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focusing on a systematic account of how the two communities perceive the benefits, obstacles and desires regarding this relationship. The chapter, and the book overall, presents the debate in the form of a dialogue between a mathematician and a mathematics educator composed out of lengthy interviews with 21 mathematicians based in the UK. In this account, the interviewed mathematicians cite the potential benefits for their pedagogical practice ensuing from using the findings and recommendations of mathematics education research – as well as from engaging themselves with collaborative educational research. However they also reflect on, and often critique, what they see as unhelpful practices of mathematics education research: findings take long to emerge and are often not generalisable or immediately applicable; educational theory is often indulgent in its diversity, yet built on less than solid grounds; the epistemological and methodological understandings of what constitutes evidence and findings are far from shared; findings are written in lengthy accounts and in often inaccessible, jargon-ridden language; research outputs are disseminated in outlets (journals, conferences, books) that mathematicians would not necessarily reach out to. Finally, in the account in Amongst Mathematicians, there is acknowledgement of the stereotypical narratives about mathematics, mathematicians and educational research that tantalise the relationship of the two communities – and of the need for a more considerate, thoughtful reflection on these stereotypes. The provisos from this group of mathematicians for a more fruitful relationship are clear – if not always feasible or desirable on both sides: mathematics educators need to stay as close to mathematics as possible; mathematics education research findings must be presented in more accessible and succinct formats; and, mathematics education researchers must conduct their research with a much more overt intention to support and impact pedagogical practice. At the same time, the account in Amongst Mathematicians acknowledges the precarious position in which mathematics education researchers often find themselves. Their research may expose weaknesses of teaching or complicity in the malfunctioning and ineffectiveness of an educational system. Their findings can therefore be unpleasant, even disturbing, and thus be ignored or disregarded by longestablished, institutional parts of the mathematics community. In the search for immediate answers to often perennial pedagogical questions, heavily and otherwise preoccupied mathematicians may also disregard the complexity of learning issues. Sometimes, convoluted recommendations from mathematics education researchers may even annoy mathematicians. While accessibility and clarity in academic writing is unequivocally a virtue, elaborate writing – and some specialist terminology – is often necessary in order to address the complexity of learning and teaching issues. A little tolerance towards the genre of writing in mathematics education – an academic community that is closely related to, but also distinct from, the community of mathematics – can go a long way in bringing pedagogical insight and recommendations for practice to the fore. At the end of the day, as Hyman Bass (2005) has stressed, mathematics education research as a field is both basic and applied, and its outputs are better understood when approached with this in mind.

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As the increasing number of mathematicians with interest in, and appreciation of, mathematics education research participating, as attendees and/or co-authors, in the RUME, Delta, ICME, CERME and INDRUM conferences suggests, the growing volume of work produced in the field – see the notable syntheses in (Artigue et al., 2007) and (Thomas et al., 2014) – is starting to reach hitherto harder-to-reach communities. Artigue (2016) noted so in her INDRUM2016 plenary lecture. In this, she also notes the often breath-taking theoretical and methodological range that now characterises the field of university mathematics education research: the field now embraces research on teaching practices and teachers’ perspectives (in contrast to earlier, narrow foci on often deficit accounts of students’ learning). I agree with Artigue that these are highly promising signs of maturity and growth. Artigue also, however, alerts us to weaknesses and challenges this rapprochement between the two communities faces. “How can we maintain some connection between the living field of mathematics, so dynamic and diverse, and undergraduate mathematics education, both in terms of content and practice?” (p. 22), she wonders. If nothing else, she observes pertinently, the practices of mathematics and the practices of mathematics education keep evolving all the time. What types of synergy can keep both communities on their toes as this evolution paces on?

3.3

Part II: Propelling and Deepening the Relationship Between Mathematicians and Mathematics Educators

Having painted a landscape of challenges and potential obstacles in the relationship between mathematicians and mathematics educators in the previous section, I now wish to turn towards a less deficit-oriented, and much more proactive and upbeat, discourse. To do so, I offer examples of initiatives that have been propelling and deepening the relationship between mathematicians and mathematics educators. The examples are from research, teaching and professional development. Research With regard to research, I would like to draw on examples of studies dating from the 1990s, to trace the evolution of collaborative research between mathematics educators and mathematicians from studies of university mathematics students’ learning of particular mathematical topics (Example 3.1) to a progressively shifting focus on university mathematics teachers’ perspectives on mathematics and mathematics teaching (Example 3.2) and, then, more recent, and more specialised, studies of a smaller grain size (Example 3.3). I note that I see this specialisation as a tendency that reflects the growth of the field, and a sign of paradigmatic maturity. I exemplify briefly from each, also noting briefly the shifting characteristics of the collaborative work between the two communities. Example 3.1 A study of university mathematics students’ learning of particular mathematical topics, my doctoral research (Nardi, 1996). With the sterling support

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of mathematics tutors from several Oxford colleges, I gained access to 20 first-year mathematics undergraduates whom I observed in their weekly tutorials in four colleges during their first two terms (200 h) on Foundational Analysis, Calculus, Topology, Linear Algebra and Group Theory. I interviewed the students twice at the end of each term, and the analysis of typical learning episodes led to an account of their first encounters with mathematical abstraction described as a personal meaningconstruction process as well as an enculturation process. The thesis described the new culture as “advanced mathematics” introduced by an expert, the tutor. The students’ concept image construction was described in two ways: students tend to construct meaningful metaphors, and they also tend to engage with the new concepts by exploring the raison d’ être of the new concepts and the new reasoning. The analyses – see, for example, Nardi (2000) for samples of these analyses in the context of Abstract Algebra – highlighted also tensions between informal and formal ways of doing mathematics across all the mathematical topics from which data was collected. Allowing, in fact trusting, a novice researcher with access to the intimate teaching setting of the Oxford tutorial over a lengthy period of time – and when recording technologies were a far cry from today’s discreet, almost invisible devices – was a first step towards a collaboration between the departments of Education and Mathematics at Oxford that deepened (in research) and expanded (in teaching and professional development) in the years that followed. Example 3.2 provides a flavour of how the student-focussed findings of the doctoral study became the basis for a suite of collaborative, teaching-focussed studies. Example 3.2 The study that followed the doctorate was also conducted in collaboration with colleagues at Oxford’s Mathematical Institute and marked a shift of focus from the students’ experiences to those of their teachers. In the University Mathematics Teaching Project (UMTP), led by Barbara Jaworski, weekly observations of six Oxford mathematics tutors were followed by in-depth interviews which discussed critical incidents from the observed tutorials. Data analysis led to the emergence of the Spectrum of Pedagogical Awareness (Nardi et al., 2005), a fourlevel descriptor of pedagogical awareness (naive and dismissive; intuitive and questioning; reflective and analytic; confident and articulate, p. 293) that has the capacity to propel subtle discussions of university mathematics pedagogy: • “Naive and dismissive” captures incidents seen largely an ignorance of pedagogy involving recognition of student difficulties with little reasoned attention to their origin or to teaching approaches that might enable students to overcome difficulty. • “Intuitive and questioning” captures evidence of implicit and hard to articulate but identifiable pedagogic thinking as well as of recognition of students’ difficulties with intuition into their resolution, and of questioning of what approaches might help students. • “Reflective and analytic” captures evidence of awareness in starting to articulate pedagogical approaches and of reflection that enables making strategies explicit as well as clearer recognition (than in previous levels) of teaching issues related to students’ difficulties and analysis of possibilities in addressing them.

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• “Confident and articulate” captures evidence of considered and developed pedagogical approaches designed to address issues, recognition and articulation of students’ difficulties with certain well-worked-out teaching strategies for addressing them, and critiquing of practice. Aforementioned Amongst Mathematicians (Nardi, 2008) is based on studies that followed the work at Oxford, and expanded that work from a single to multiple institutional contexts (spanning six mathematics departments) and a team now including a research mathematician-cum-mathematics education researcher, Paola Iannone. I would like to propose this project as a foundation for the type of research that has the potential to bring the two communities together in productive ways. There are five key characteristics of this type of research: • Collaborative: it engages mathematicians and mathematics educators in as many phases of the research as possible. • Mathematically focussed: it maintains a sharp focus on the mathematics in the university syllabus. • Context-specific: it sets out from discussions that pertain to the learning and teaching in the institutions within which the research takes place. • Non-deficit: it stays clear of apportioning blame, e.g. to student lack of mathematical ability or lecturer lack of pedagogical sensitivity or skill. • Non-prescriptive: its primary aim is not to offer quick-fix solutions to longstanding pedagogical problems. As noted earlier, data analyses in the book are presented in the form of a dialogue between two fictional, yet data-grounded characters, a mathematician and a mathematics educator, composed out of lengthy group interview data with 21 mathematicians based in the UK. The composition follows the methodological principles and processes of the narrative approach of re-storying (Ollerenshaw & Creswell, 2002; as detailed in Nardi, 2008, 2016). I claim that this particular form of generating insights into university mathematics pedagogy (collaborative, mathematicallyfocused, context-specific, non-deficit, non-prescriptive) addresses some of the longstanding tensions between the two communities and offers an alternative way in which the communication between them can take place. A key feature, for example, in the dialogues in Amongst Mathematicians is that they are jargon-free, even though their construction is fundamentally driven by the mathematics education research findings cited in the footnotes that are present on almost every page. In this sense, these dialogues are intended as an effective communicative tool: their constitutive elements are the mathematicians’ insights into university mathematics pedagogy contributed over a lengthy period of elaborate discussions with mathematics educators, woven together with the mathematics educators’ insights emerging out of their knowledge of the research literature in this field. In other words, I propose re-storying as a vehicle for community rapprochement achieved through generating and sharing research findings – the substance of research – in forms that reflect the five key characteristics that underpin this research. The accessibility of storytelling in this format has facilitated the use of the book in under/postgraduate

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teaching as well as professional development courses for university mathematics teachers. The format is also being used in other, more recent collaborative studies. Example 3.3 A recent study of a smaller grain size, is a doctorate (Thoma, 2019) that explores the transition from school to university through commognitive (Sfard, 2008) analyses of mathematics undergraduates’ examination scripts, examination tasks and assessment practices that lecturers enact as they design and deploy these examination tasks. The analyses (reported, e.g. in Thoma & Nardi, 2018) trace manifestations of unresolved commognitive conflict in the students’ scripts – particularly in relation to lecturer expectations – and draw pedagogical implications for a smoother transition from school to university mathematics. The study’s findings are part of an ongoing dialogue with the mathematics department in which it was conducted on how to address the challenges that incoming students face, especially at the start of their studies. As the grain becomes smaller though – and I am making this observation on research in this field also with my hat on as CERME University Mathematics Education Thematic Working Group between 2011 and 2015 inaugural leader (Nardi 2017), INDRUM2016 co-chair (Nardi et al., 2016) and IJRUME editor since 2019 – there are significant epistemological (theoretical, substantive and methodological) evolutionary steps in the field. I see the most notable of these as being the gradual emancipation from a relatively limited initial focus on cognitive aspects of the student learning experience in university mathematics to the grander vista of issues – also inclusive of pedagogical, institutional, affective and social issues – that studies nowadays address. What this broadening and deepening of vista entails, in fact is predicated upon more systematic engagement with theory. The RME Special Issue Institutional, sociocultural and discursive approaches to research in university mathematics education Research in Mathematics Education (Nardi et al., 2014a, b, c) aimed to contribute exactly to this through collating reviews of recent university mathematics education research studies that deploy a selection of approaches: Anthropological Theory of the Didactic, Theory of Didactic Situations, Instrumental and Documentational Approaches, Communities of Practice and Inquiry and the Theory of Commognition. I note that elaborating the focus of research beyond sometimes simplistically used cognitive approaches – which also tend to steer attention towards individual student traits such as ability or work ethic, and away from systemic deficits in terms of resources, teacher preparation, equity, etc. – may seem at first as complicating the accessibility of mathematics education research findings to practitioners of mathematics teaching: language may be more theory-laden, the focus may be more on specific contexts and pedagogical recommendations may be more elaborate. However, much like mathematics research, mathematics education research is a complex enterprise, and, as such, its toolbox – of theories and methodologies – needs to be sophisticated. With the caveat that the mathematics education research community needs to invest more effort in establishing accessible ways to communicate research outcomes, I cannot see the tangible maturity in the field as anything else other than a strengthening of its position to negotiate and effect change.

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Teaching

With regard to teaching within mathematics departments, I would like to offer examples that trace two developments within university mathematics curricula: towards a more inclusive approach to mathematics education modules within university mathematics programmes; and towards engaging mathematics education researchers with university mathematics teaching (as is the case in my own institution as well as elsewhere in the UK, for example, Loughborough University’s Department of Mathematics Education). In my institution, two mathematics education modules (one led by Irene Biza, and formerly by Paola Iannone, and one led by me) are available to mathematics undergraduates. Both introduce students to the field of mathematics education research: one with a stronger slant on the mathematics that the students are currently engaged in their university studies; and one with a slant towards mathematics as a field in need of more robust modes of engagement with the public (see also later in the chapter a little more on the latter).

3.3.2

Professional Development

With regard to professional development, I offer examples that trace the evolution from generally unpopular, non-discipline-specific training of new lecturers in mathematics to more appealing, mathematics-specific training formats. In these formats, that we have begun to see in several institutions, university mathematics education researchers are involved in increasingly mathematics-specific training of new mathematics lecturers, offering them opportunities to familiarise themselves with mathematics education research findings and to reflect on their teaching practice. One early, and rather rudimentary, example of the attempt to communicate mathematics education research results to new mathematics lecturers is How To Prove It: A Brief Guide for Teaching Proof to Mathematics Undergraduates (Nardi & Iannone, 2006). Another example of the involvement of mathematics educators in the professional development of new mathematics lecturers in my institution, is in the shape of a module run as part of an MA in Higher Education Practice that new lecturers are expected to complete parts of. The module involves the evaluation of the new lecturer’s teaching portfolio through in-person and in-writing conversations. Here is an example of feedback to a newly appointed lecturer: [. . .] Particularly impressive is the way in which you have chosen to introduce the concept of group to [second year] students in [course name]. Contrary to the typical definition-lemmatheorem approach to this introduction, you have chosen to build students’ encounter with this novel concept carefully – and with remarkable sensitivity to its often overwhelming nature – from mathematical objects they are already familiar with (such as natural numbers, integers, fractions, matrices and polynomials). Chapter 4 of your portfolio, particularly pages 23–26, offer an account of your formidable efforts in this matter: constructing the notion of group, its properties and a good array of examples of it, through gradually enriching

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Public engagement (mathematicians and mathematics educators working together towards a more accurate and appealing image of mathematics) Beyond the collaborative activities that I have exemplified so far (research, teaching, professional development), I now draw on examples of initiatives where mathematicians and mathematics educators have been working together towards strengthening another, very crucial, and often very fragile, relationship: that of the public with mathematics. I exemplify from teaching and engagement activities that aim to draw non-mathematical audiences into the world of mathematics. With regard to teaching (this time outside mathematics departments), while mathematics has always been taught to students of other disciplines – STEM subjects, Economics, etc. – here I draw on an example from a less common area: teaching within the social sciences and the humanities in order to illustrate the increasing interest in engagement with the world of mathematics of students hitherto resisting, or at best being sceptical about, such engagement. The example is from a module I teach to Year 3 students on the BA in Education in my institution. The module is called Children, teachers and mathematics: Changing public discourses about mathematics and explores young children’s learning of what is acknowledged in the module’s outline as “one of the most important, yet notoriously feared and misunderstood, subject: mathematics!”. The module aims to share some of the excitement experienced by those who love mathematics – such as enthusiastic teachers and mathematicians across the professions – but also examines some of the key challenges that children face when they engage with mathematical learning. The focus is mostly on primary school, given the course’s core demographic of future applicants to primary teacher education programmes. The module investigates where the social and psychological stigma of mathematics comes from – the fear that prevents many adults and children from building a good relationship with mathematics. The module also juxtaposes this “stigma” with results from neuroscience that show that mathematical thinking is quite natural and that mathematical ability is innate to all human beings. A further juxtaposition that the module offers is between these research findings and examples from popular culture (TV, films, pop music), media and the arts that seem to perpetuate largely “math-o-phobic” images – such as the portrayal of mathematicians and scientists as mad, unattractive and socially inept, mostly male and white, etc. The module closes with considering how education, particularly in the crucial years of primary school, can work against the tide of such images and introduce children to the creativity and excitement of mathematics and science. Since its inception in 2012 the module is well-received by some cohorts and with some scepticism by others: those who choose to take it go as far as declaring it a game-changer in their attitudes towards mathematics; of those who do not, most

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say – most emphatically – that they would never associate voluntarily with anything to do with mathematics. The challenge to convince those students otherwise is steep, but I am encouraged by some longitudinal evidence: some course graduates return to train as primary teachers, even opting for a mathematics specialism or joining, mathematics education initiatives years later. With regard to engagement, here, I put forward one example, that was in place in my institution between 2006 and 2015 and has now become an example of a whole stream of analogous activities: MAUD, the Maths At Uni Days, illustrates some of the ways in which departments of education and mathematics have been coming together to organise events that showcase the importance and appeal of mathematics as well as its capacity to open windows to a wide range of professions. Launched in 2006 – through a small internal research, teaching and development grant (Bills et al., 2006) – MAUD offered a day of lectures that bring to the fore the vast array of ways in which mathematics is used across disciplines; mathematics problem-solving workshops; and a discussion with a panel of mathematics experts from a range of professions. Word of mouth and the zest of the teams across the Schools of Mathematics and Education sustained and grew the reputation of this event which became part of the mainstreaming of such events in my institution, including a now strong presence of mathematics activities, for members of the public of all ages at the annual Norwich Science Festival.

3.4

Part III: Mathematicians and Mathematics Educators: Has the “Beautiful Friendship”/Joint and Multi-faced Enterprise Already Begun?

In the light of the four tiers of examples presented in Parts I and II, sketching a story of mathematicians and mathematics educators intersecting at various points in research, teaching, professional development and public engagement activities, can we possibly re-imagine this as a story of paths “meeting” at a vanishing point, a point where the boundaries between the two communities may fade away and morph into a sense of joint and multi-faceted enterprise? I see forward-facing (university mathematics teachers’ professional development) and outward-facing (public engagement) activities as two growing facets of this enterprise. The panel discussion at INDRUM2018 (Winsløw et al., 2018) evidenced the former rather well. Back in 2004, in one of the first reports (Nardi & Iannone, 2004) from the interviews with 21 UK mathematicians that eventually became the underpinning data for Amongst Mathematicians, the fragility of the relationship between mathematicians and mathematics educators had been duly noted. The obstacles highlighted there amounted to issues of “trust, access, priority, communicability, applicability and subtlety” (p. 407). But this critique was also accompanied by acknowledgement of the potential benefits from engaging with the conduct and the results of mathematics education research – and a genuine intellectual curiosity for its ways of

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working. In a nutshell, I had certainly sensed in these lengthy, zesty conversations about the teaching and learning of mathematics the potent “beginnings of a beautiful friendship” (Epstein et al., 1942). I daresay that the examples of the many and varied collective activities sampled in this chapter indicate that the “friendship” may indeed have begun: in research, as the various initiatives in the introduction indicate, the meetings of the two communities are now far more regular and substantial; teaching within mathematics departments, both of mathematics and mathematics education, is becoming more inclusive and respectful of each discipline’s distinctive characteristics and potential offerings; professional development, particularly with regard to the training of new lecturers in mathematics, is becoming more content-specific and attuned to the findings of mathematics education research; and engaging the public is fast becoming an area of intense, interdisciplinary and collective work. In fact, science communication is emerging as a new discipline, at the crossroads of science and social science. A browse through the 11 teams from across the UK involved in RCUK’s 2013–2018 Schools and Universities Partnership Initiative shows up the fast emerging cross-community work in this area. At UEA, for example, the project brought together UEA’s first Chair of Science Communication (from the School of Biological Sciences), faculty from the School of History and the Norwich Medical School, and myself. A vanishing point is a trompe l’oeil; it gives the illusory impression that two paths that start out as parallel, meet somewhere very long down the way. Illusory part aside, I use this metaphor here as a way of saying that, over the years, the two communities have been building quite a bit of collaborative work. This work now goes beyond the occasional, short-lived crossing of two paths that meet briefly and then diverge from each other again. I have been thinking about this work as starting to resemble something more than a series of isolated encounters – and much more as an emergent, and vastly promising, interdisciplinary common enterprise. The two communities of mathematics and research in mathematics education – which intersect in at least one juncture, in the joint enterprise (Biza et al., 2014) of mathematics teaching at university level – need to meet, confer and generate negotiated, mutually acceptable perspectives more often (Artigue, 1998). I return to Amongst Mathematicians (Nardi, 2008) to make my points more tangible. Through a demonstration of the rich pedagogical canvas that is evident in the utterances of the character of mathematician in the dialogues, this emphatically evidence-based approach is intended not only as a contribution to the rapprochement of the two communities, but also as a riposte to stereotypical views that see university mathematics teaching practitioners as non-reflective actors who rush through content-coverage in ways often insensitive to their students’ needs, and who have no pedagogical ambition other than that related to success in examinations and audits. Simultaneously, it challenges presentations of mathematics education researchers as having a suspiciously loose commitment to the cause of mathematics, and whose irrelevant theorising renders them incapable of relatability to practitioners of mathematics. The dialogues that came into being through the research design of

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the study behind Amongst Mathematicians – of which the re-storying approach is a key component – are intended as an embodiment1 of these ripostes. One of the objectives of the re-storying approach proposed is what Pais (2013) describes as superseding the traditional macro/micro divide: overcoming this dichotomy to realise how the universal (macro) manifests itself in concrete situations and to acknowledge how the universal operates within the particular which, in return, colours its very universality and accounts for its efficiency. In tune with Pais, the re-storying approach attempts to capture what the universal (the claim, e.g. that mathematics education research can provide quick-fix, water-tight pedagogical prescriptions) secretly excludes; and, to observe how epistemological belief and institutional practice/policy is enacted through the situation-specific, context-bounded utterances of individuals, all involved with mathematical pedagogy but who may come from different, but often crossing, disciplinary and institutional paths. In the dialogues constructed out of the focus group interview data, knowledge (mostly about mathematical pedagogy) is relocated distinctly away from typical mathematical epistemologies but, even more crucially, as far away as possible from decontextualised pedagogical prescription. The proposition made here is that this new form of knowledge about mathematical pedagogy, co-constructed by members of two often separated communities (mathematicians and mathematics educators), is relocated to a novel third space which welcomes the collaborative, mathematically focused, context-specific, non-deficit and non-prescriptive discourses that govern the production and communication of this knowledge. This chapter was initiated as an invitation to respond to the editor’s work so far in developing partnerships between the communities of mathematicians and mathematics educators. The vision which underpins the study that I deployed as a centrepiece example in this chapter resonates with this work very strongly: the study provided a safe, open space – away from quotidian institutional pressures as well as steering clear of judgemental, deficit narratives about dominant pedagogical practices in university mathematics – where colleagues across the communities of Mathematics and Mathematics Education research interacted productively and reflected critically on the what, when, why and how we can further improve, even reform, said practices. Acknowledgements The chapter builds on recent syntheses of the reported work presented and or published as follows: two plenary talks (one at Mathematisches Forschungsinstitut Oberwolfach (MFO) Workshop entitled Mathematics in undergraduate study programs: challenges for research and for the dialogue between mathematics and didactics of mathematics (2014); CERME10 (Nardi, 2017)) and two public lectures (one for the Norfolk and Norwich association of the British Federation of Women Graduates (2018) and my UEA professorial inaugural lecture (2015) entitled Mathematics and Mathematics Education: A story of paths just crossing or of meeting at a vanishing point? A version of this lecture was published in (Nardi, 2015).

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One of so many possible embodiments, as Chris Rasmussen and I discovered when we worked towards a synthesis of such efforts for (Lerman, 2020).

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References Artigue, M. (1998). Research in mathematics education through the eyes of mathematicians. In A. Sierpinska & J. Kilpatrick (Eds.), Mathematics Education as a research domain: A search for identity (pp. 477–490). Kluwer Academic Publishers. Artigue, M. (2016). Mathematics education research at university level: Achievements and challenges. In E. Nardi, C. Winsløw, & T. Hausberger (Eds.), Proceedings of the 1st INDRUM (International Network for Didactic Research in University Mathematics) Conference: an ERME Topic Conference (pp. 11–27). Montpellier. Artigue, M., Batanero, C., & Kent, P. (2007). Mathematics thinking and learning at post-secondary level. In F. K. Lester (Ed.), The second handbook of research on mathematics teaching and learning (pp. 1011–1049). IAP. Bass, H. (2005). Mathematics, mathematicians and mathematics education. Bulletin of the American Mathematical Society, 42(4), 417–430. Bills, L., Cooker, M., Huggins, R., Iannone, P., & Nardi, E. (2006). Promoting mathematics as a field of study: Events and activities for the sixth-form pupils visiting UEA’s Further Mathematics Centre (A UEA Teaching Fellowship Report). Available from Elena Nardi. Biza, I., Jaworski, B., & Hemmi, K. (2014). Communities in university mathematics. Research in Mathematics Education, 16(2), 161–176. Brousseau, G. (1997). Theory of didactical situations in mathematics. Kluwer. Epstein, J. J., Epstein, P. G., Koch, H., Burnett, M., Alison, J., Robinson, C., & (Writers). (1942). Casablanca [Motion picture]. Warner Bros. Pictures. Goldin, G. A. (2003). Developing complex understandings: On the relation of mathematics education research to mathematics. Educational Studies in Mathematics, 54(2/3), 171–202. Lerman, S. (Ed.). (2020). Encyclopedia of Mathematics Education. Springer. Living Edition | Editors: Steve Lerman Nardi, E. (1996). The novice mathematician’s encounter with mathematical abstraction: Tensions in concept-image construction and formalisation Unpublished doctoral thesis. University of Oxford. Available at http://www.uea.ac.uk/~m011 Nardi, E. (2000). Mathematics undergraduates’ responses to semantic abbreviations, ‘geometric’ images and multi-level abstractions in Group Theory. Educational Studies in Mathematics, 43(2), 169–189. Nardi, E. (2008). Amongst mathematicians: Teaching and learning mathematics at university level. Springer. Also available as eBook. Nardi, E. (2015, August). The many and varied crossing paths of mathematics and mathematics education. In Mathematics today (Special issue: Windows on advanced mathematics) pp. 212–215. Nardi, E. (2016). Where form and substance meet: Using the narrative approach of re-storying to generate research findings and community rapprochement in (university) mathematics education. Educational Studies in Mathematics, 92(3), 361–377. Nardi, E. (2017). From advanced mathematical thinking to university mathematics education: A story of emancipation and enrichment. In T. Dooley & G. Gueudet (Eds.), Proceedings of the 10th conference of European research in mathematics education (CERME) (pp. 9–31). Ireland. Nardi, E., & Iannone, P. (2004). On the fragile, yet crucial relationship between mathematicians and researchers in mathematics education. In Proceedings of the 28th annual conference of the international group for psychology in mathematics education (Vol. 3, pp. 401–408). ERIC Clearinghouse. Nardi, E., & Iannone, P. (2006). How to prove it: A brief guide for teaching proof to mathematics undergraduates (No. 978-0-9539983-8-8). Commissioned by the Higher Education Academy (Mathematics, Statistics and Operational Research branch). ISBN 978-0-9539983-8-8. Available at http://mathstore.ac.uk/publications/Proof%20Guide.pdf Nardi, E., Jaworski, B., & Hegedus, S. (2005). A spectrum of pedagogical awareness for undergraduate mathematics: From ‘tricks’ to ‘techniques’. Journal for Research in Mathematics Education, 36(4), 284–316.

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Nardi, E., Biza, I., González-Martin, A., Gueudet, G., & Winsløw, C. (2014a). Remarks on institutional, sociocultural and discursive approaches to research in (university) mathematics education: (Dis)connectivities, challenges and potentialities. In P. Barmby (Ed.), Proceedings of the British society for research into learning mathematics 34(1) (pp. 89–94). BSRLM. Nardi, E., Biza, I., González-Martin, A., Gueudet, G., & Winsløw, C. (2014b). Institutional, sociocultural and discursive approaches to research in university mathematics education. Research in Mathematics Education, 16(2), 91–94. Nardi, E., Ryve, A., Stadler, E., & Viirman, O. (2014c). Commognitive analyses of the learning and teaching of mathematics at university level: the case of discursive shifts in the study of Calculus. Research in Mathematics Education, 16(2), 182–198. Nardi, E., Winsløw, C., & Hausberger, T. (2016). Editorial. In Proceedings of the 1st INDRUM (International Network for Didactic Research in University Mathematics) conference: An ERME topic conference (pp. 6–9). Montpellier. Ollerenshaw, J. A., & Creswell, J. W. (2002). Narrative research: A comparison of two restorying data analysis approaches. Qualitative Inquiry, 8(3), 329–347. Pais, A. (2013). An ideology critique of the use-value of mathematics. Educational Studies in Mathematics, 84, 15–34. Ralston, A. (2004). Research mathematicians and mathematics education: A critique. Notices of the American Mathematical Society, 51, 403–411. Sfard, A. (1998). A mathematician’s view of research in mathematics education: An interview with Shimshon A. Amitsur. In A. Sierpinska & J. Kilpatrick (Eds.), Mathematics Education as a research domain: A search for identity (pp. 445–458). Kluwer Academic Publishers. Sfard, A. (2008). Thinking as communicating. Human development, the growth of discourse, and mathematizing. Cambridge University Press. Thoma, A. (2019). Transition to university mathematical discourses: A commognitive analysis of first year examination tasks, lecturers’ perspectives on assessment and students’ examination scripts. Doctoral thesis: University of East Anglia, UK. Thoma, A., & Nardi, E. (2018). Transition from school to university mathematics: Manifestations of unresolved commognitive conflict in first year students’ examination scripts. International Journal for Research in Undergraduate Mathematics Education, 4(1), 161–180. Thomas, M., de Freitas Druck, O., Huillet, D., Ju, M. K., Nardi, E., Rasmussen, C., & Xie, J. (2014). Key mathematical concepts in the transition from secondary school to university. In S. J. Cho (Ed.), Proceedings of the 12th International Congress on Mathematical Education (pp. 265–284). Springer. Winsløw, C., Biehler, R., Jaworski, B., Rønning, F., & Wawro, M. (2018). Education and professional development of university mathematics teachers. In V. Durand-Guerrier, R. Hochmuth, S. Goodchild, & N. M. Hogstad (Eds.), Proceedings of the 2nd INDRUM (International Network for Didactic Research in University Mathematics) Conference: an ERME Topic Conference (p. 12). Kristiansand.

Chapter 4

Mind the Gap: Reflections on Collaboration in Research and Teaching Michael O. J. Thomas

Abstract In this chapter, I will consider my personal experience of many years of collaborative work in research and teaching with mathematicians. The phrase “mind the gap”, originally arising in 1968 from the gap between the platform and the train on the London underground, has become synonymous with the idea of avoiding the danger inherent in such a gap. However, I would like to propose a different perspective related to minding of the gap that can (often does?) exist between mathematics educators and mathematicians. I believe we all have a responsibility to mind, in the sense of to take care of or to think carefully about and work on, the gap that emerges in terms of differing goals and perspectives on mathematics and how it should be taught and learned. I will describe some research projects I have been involved in with collaborators both outside and within my own institution, the University of Auckland, in which mathematics educators and mathematicians engaged in professional development (PD), reflecting together on university teaching of mathematics. I will also present an analysis of the reasons for the successful outcomes from these projects. This will include a brief presentation of Schoenfeld’s (Schoenfeld, How we think. A theory of goal-oriented decision making and its educational applications. Routledge, New York, 2010) theoretical Resources, Orientations and Goals (ROG) framework of teacher practice. Overall, I believe there are important lessons to be learned from minding the gap that can inform PD strategies in tertiary institutions. Keywords Mathematics · Tertiary · University · Teaching · Professional development · Collaboration · Boundary

M. O. J. Thomas (✉) University of Auckland, Auckland, New Zealand e-mail: [email protected] © Springer Nature Switzerland AG 2023 S. Stewart (ed.), Mathematicians’ Reflections on Teaching, Advances in Mathematics Education, https://doi.org/10.1007/978-3-031-34295-0_4

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4.1

Background

Mathematics teaching at all levels is usually seen as of utmost importance to the economic and social development of a country. Hence, it is of little surprise that several communities of practice, in the sense of Wenger (1998), have taken an interest in the structure and processes of mathematics education. Foremost among these communities have been mathematics educators, with their training in pedagogical ideas, and mathematicians with their training in mathematics. Unfortunately, for many years, especially in the USA, and to some extent in the UK and other places, this has led to strong disagreements between the two groups (Klein, 2007; Schoenfeld, 2004). The arguments mounted for and against each position are not of interest to us here. I simply want to acknowledge that there have been entrenched positions leading to difficulties and what we may describe as a gap between the two communities on this issue (and maybe others), so that we have two related, but distinct, communities of professional practice (Goos, 2015) each of whom has often failed to understand the perspective of the other. However, rather than viewing this gap as in any sense dangerous and something to be warned about, as is true of the gap between the platform and the trains on the London underground, I strongly believe that it is something we should be sufficiently concerned about to recognise that we bear a responsibility to work at understanding and lessening it. The gap has been described in terms of the theoretical construct of a boundary between two social groups. Presenting the term boundary from the literature, Akkerman and Bakker (2011, p. 133) speak of how “A boundary can be seen as a sociocultural difference leading to discontinuity in action or interaction. Boundaries simultaneously suggest a sameness and continuity in the sense that within discontinuity two or more sites are relevant to one another in a particular way”. Interactions at such a boundary, called boundary encounters (Wenger, 1998), give community members an understanding of how meaning is negotiated within another community and a two-way connection involves several participants from each community in an encounter (Goos, 2015). Akkerman and Bakker (2011) further analyse what is involved when some individuals from either side of the boundary take part in what is termed a boundary crossing (Suchman, 1994). This idea has been used to denote how professionals may reach out to those in another, nearby, community. However, in doing so they can expect to experience, and need to deal with, some difficulties. In those cases where boundary crossings occur, we discover that crossing boundaries involves encountering difference, entering onto territory in which we are unfamiliar and, to some significant extent therefore, unqualified. For those of us who have spent a lifetime building up our competence within a domain of specialized professional practice, placing ourselves on unknown ground is a difficult thing to do, particularly insofar as it may lead to painful reflections on our own lives and positions. (Suchman, 1994, p. 25)

Akkerman and Bakker (2011, Table 1 on p. 151) have analysed the learning mechanisms and processes that characterise such boundary crossings. Their list of mechanisms is identification, coordination, reflection, and transformation. In turn the characteristic processes of these they list as:

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Identification: Othering; and Legitimising coexistence. Coordination: Communicative connection; Efforts of translation; Increasing boundary permeability; and Routinization. Reflection: Perspective making; and Perspective taking. Transformation: Confrontation; Recognising shared problem space; Hybridization; Crystallization; Maintaining uniqueness of intersecting practices; and Continuous joint work at the boundary.

They further explain (Akkerman & Bakker, 2011, pp. 144, 145) what the reflection mechanism involves “In addition to identification and coordination, we find studies, often proposing or evaluating an intervention, that focus on the potential of the boundary in terms of reflection. These studies emphasize the role of boundary crossing in coming to realize and explicate differences between practices and thus to learn something new about their own and others’ practices”. These characteristics of boundary crossings, and especially the concept of reflection on practices, correspond well with attributes of the research programmes outlined below that I have been involved in conducting, along with my students and colleagues in the Mathematics Department at Auckland University, and elsewhere, over the last 15 years or so. Our studies have focused on the boundary between mathematics education and mathematics with the goal of assisting participants to benefit by learning about our own practices and those based in the other community and hence to lessen the gap between the communities.

4.2

The Research Projects

One of my first boundary encounters was in the early 2000s when I began to work with a pure mathematician from a New Zealand university, looking at the use of computer-based algebra systems in university mathematics courses (Stewart et al., 2005). At that stage the role of digital technology in learning mathematics had been an area of interest of mine for 17 years and was the area of my doctoral research. This initial study progressed into a major consideration of the teaching and learning of linear algebra (e.g., Hannah et al., 2011, 2012, 2013a, b; Stewart et al., 2005). A primary theoretical construct used in this, and most of my research at the boundary, has been Schoenfeld’s (2008, 2010) goal-oriented decision-making framework. This links in-the-moment pedagogical decisions with teacher Resources, Orientations and Goals (ROG). The framework explains how an individual’s orientations, which include dispositions, beliefs, values, tastes and preferences, etc. (Schoenfeld, 2010), shape the short- and long-term goals set, but also the prioritisation of those goals and the resources used to try to attain them. These resources comprise primarily teachers’ procedural and conceptual knowledge, but also include available physical entities, such as pens and whiteboards, textbooks, models, digital technology, etc., as well as their time and energy. Decision-making is crucial since the quality of decisions affects success in attaining goals. While Schoenfeld’s framework had previously been used to investigate how teacher

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orientations can influence practice, including goal setting (Aguirre & Speer, 2000), and to describe how subject matter goals and beliefs can dominate pedagogical content goals and beliefs (Törner et al., 2010), it had not previously been used at the university level. A second theoretical strand was related to the three dimensions of communities of practice Wenger (1998) identifies, namely, mutual engagement, a joint enterprise that involves members’ complementary expertise, and development of joint resources for making meaning (Goos, 2015). The intention in the projects below was to build a boundary co-learning community of practice (Jaworski, 2001) in which both the mathematics educators and the mathematicians would be learners, engaged both in active practice and reflection on that practice. In this way a community of inquiry (Jaworski, 2006) would be developed where members could reflect critically on the process of teaching practice (Jaworski, 2003; Wells 1999) enabling a positive, supportive critique of each other’s practice.

4.2.1

The First Project

Much of the work in the first project centred around the lecturing of a pure mathematician, L1. There was no intent to evaluate in any way what he was doing but rather to use a small community of inquiry (Jaworski, 2006) to share both mathematical and pedagogical ideas in both directions and critically examine them. The mathematics educators passed on some mathematics education theory, such as Schoenfeld’s (2010) ROG ideas and Tall’s Three Worlds framework (Tall, 2004, 2008) along with their knowledge of linear algebra and received in return expert ideas on the mathematics content being taught and how it might best be approached mathematically, along with ideas on potential teaching perspectives. In one phase of the research the lecturer spent up to an hour after each lecture writing a very detailed diary reflecting on events that happened in each of 24 lectures and emailed them to the mathematics educators on the same day (Hannah et al., 2011). There were also two interviews with him and fortnightly Skype sessions after the lectures had finished as part of our community discussion. All these were audiorecorded and transcribed for analysis. The mathematician’s ROG recorded his primary goal “My ultimate goal was I want them to think like a mathematician”. He not only had a very good overview of the mathematics, but in his linear algebra lecturing he wanted students to share what he called the “big picture” of mathematics, the connections between ideas and concepts with his students: So what I’m trying to do is get them away from that and see that actually there’s patterns there. That this sort of process that you’ve used here is another one over here that looks a bit like it and there’s a reason why they look like one another. I’m trying to get them into thinking big picture’. . .One of the other things I’ve been telling students is that while I’m telling them this ‘big picture’, the story [of linear algebra], I’m going to use that story to

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introduce new words so they become a way of describing features of a story. [These quotes are from two separate occasions]

This big picture idea is related to what Ball et al. (2005) in their Mathematical Knowledge for Teaching framework have called horizon content knowledge; the ability to relate current mathematics knowledge to that which will be studied later. His other important goals were to “expose students to the structure of mathematical ideas” and to “help students make connections between mathematical ideas”. A fuller analysis of his ROG is found in Hannah et al. (2011). He was also clear (Hannah et al., 2013a, b) that he wanted students to be able to think about mathematics “I’d like to show them other ways of thinking, but I’d really prefer that they went out into the world thinking for themselves, and if I could give them some tools that will do that, that will be really nice”. To achieve this, in line with Tall’s (2008) worlds of thinking, he emphasised what he saw as the crucial role of language and visualisation in learning. As part of this he asked questions where the students were required to write a paragraph of text in response. One illustrative example of the use of his ROG for an in the moment decision occurred in a lecture when he had just reviewed the three spaces, null(A), row(A) and col.(A). In Hannah et al. (2011, p. 982) he describes how he came to a decision point: On the spur of the moment, my memory jogged by the mention of pictures, I decided to remind them of our earlier picture [Figure 4.1] of the action of a 2 × 2 matrix A. We see now that what was called the ‘range of the transformation given by A’ is actually col(A) and what was called the ‘solution to Ax=0’ is actually null(A). The other line in that diagram is not a subspace as it does not go through the origin (or zero vector). But our third subspace, row(A) can be pictured in this diagram too.

Commenting on this decision he wrote that: Here it seems I intended all along to discuss pictures that represent the three fundamental subspaces, but that during my discussion the mention of pictures has jogged my memory about another picture I had shown them earlier. Features in that earlier picture have since acquired the names we are now discussing, so I’ve decided it would be worthwhile to update the picture by including these new names I guess I want them to make connections, a rich network of them, and this was a spur of the moment idea for connecting an earlier lecture with this one.

One of the advantages the mathematician saw in engaging in this project was the opportunity to reflect on his practice in the manner we see above. He said that he found sharing in the community of inquiry an enriching experience. Having discussions with people who had taught similar courses gave him the opportunity to reflect on his teaching practice. Further reflection came through writing the lecture notes after each lecture, where he said: “It’s been interesting. . .It takes about an hour to write it down. . .I can see you’re only getting my view of what happened. I’m trying to be dispassionate and stand back from it and say this is what happened and so I can tell you things that didn’t go quite the way I wanted”. Overall, he said he viewed the research as a form of professional development, and in (Hannah et al., 2013a, b, p. 487) we read how he felt about it “On reflection, the lecturer feels quite pleased with the result of his pedagogical experiment, and it is our hope that documenting his

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Fig. 4.1 The mathematician’s visual representations of linear transformations

experience in the above manner may contribute in some way to the continuing dialogue on models of professional development for academic mathematicians”.

4.2.2

A Second Project

The confluence of a number of factors led to the establishment of a second key boundary crossing project in 2010. As described in Barton et al. (2014), these included the acceptance of mathematics educators within a Mathematics Department, the external pressure to address teaching issues in the university, a visit from Alan Schoenfeld from UC Berkeley and a research grant opportunity. This development was DATUM: a project for the Development and Analysis of Teaching in Undergraduate Mathematics, which involved a group of eight staff (four mathematicians and four mathematics educators) from Auckland University’s mathematics department. What was particularly novel about this research was the decision to employ videos of the participants’ lecturing and to use short sections of these during in-group discussions. A decision was made (Barton et al., 2014), once again, that we would work to build a supportive community of practice (Lave & Wenger, 1991)

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through which knowledge could develop and be critically evaluated (Jaworski, 2003). This initial research project lasted 2 years, during which 6 participants were videotaped, with a total of 19 full lectures captured (Barton et al., 2014). An observer also took notes during each of these lectures. A key part of the process was to assist the lecturers to be more aware of their ROG. Hence they were encouraged to record, before and after the lectures (signalling intent or reflecting on what happened), either a list of resources, orientations and goals for that lecture, or notes based on such a ROG. A complete example of a mathematician’s ROG can be seen in Barton et al. (2014). There were 15 full meetings of 1 h each, three per semester, held to discuss the lecturing. For these meetings one lecturer selected for discussion a 2–4-minute excerpt from his/her videotape, to be watched by the group and the ROG was circulated. While we expected that the lecturers might have chosen for discussion parts of the lecture that they thought had gone well, instead they chose sections where they felt less comfortable. For the mathematics educators this often involved their mathematical knowledge, and the mathematics researchers it related to moments of pedagogical concern. This contributed in no small measure to boundary crossings that assisted with minding the gap. Analysis of the videotapes, voice recordings and transcripts of the meetings, was put alongside that of interviews with the mathematicians. Some of the outcomes of the project for individuals have been recorded in papers. For example, the experience of one of the mathematics educators in this research has been recorded in Barton (2011), and three mathematicians have been co-authors of papers outlining part of their experience (Paterson et al., 2011a, b, c; Paterson & Evans, 2013). I will describe here a few vignettes emerging from the lecture excerpts and some of the lessons arising from reflection on them.

4.2.3

Illustrative Lecture Vignettes

The first example illustrates the importance of helping to create awareness of the effects of decisions made “in the moment”. The lecturer, L2, an applied mathematician, was teaching a first year undergraduate service course, and the lecture was on trigonometric ratios of angles of any magnitude. As part of the ROGs he provided, he said that “I see my role as adding value to the [slides]” [these were provided since the course was taught by a team]. Thus he believed that the slides could be improved on and his goal was to add to the students’ experience provided by additional insight. However, at the start he was unsure how this would be achieved, since “I have never tried to teach this stuff before so I am not exactly sure how I plan to explain each point and will work on this at the time”. So his intent was to make decisions in-themoment as the need arose. This demonstrated sufficient confidence in the mathematical content that he could be free to add value pedagogically at certain points. During the lecture, after about 19 minutes of the 50-minute period, the slide contained a version the diagram below:

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Fig. 4.2 (a, b) The lecturer reflects on aspects of the unit circle used for trig ratios of angles of any magnitude

He used the overhead projector and a pen to draw this unit circle himself several times, saying in the group discussion “If they see me do it then I can say this is point P they can see it rather than just talking about it, showing them it”. It was clearly a goal of his to show the students how the diagram was constructed. This was based on his beliefs about learning, as he commented “Maybe it reflects how I learn and I want to draw it and see the point and okay this is what is there, to re-copy. I’m creating the figure”. During the lecture, he commented on this diagram (see Fig. 4.2a) that: Ok so there’s my unit circle. It goes.. The radius of the circle is always equal to one. Not the diameter but the radius and the centre’s at zero. And then we have some point on this curve determined by the angle. So we have some magic point out here, which is P of x y [P(x, y)], which depends on the angle.

Hence, on the spur of the moment, he made a decision to call the general point P “magic”. During the group discussion he reflected on this decision “I suppose I just chose a point at random and said now it’s become a magic point because I’ve chosen it. I guess that’s what I was meaning. This point had suddenly become special because I’d chosen it and then”. So, on reflection he saw a potential mathematical conflict between whether the point “chosen” in the diagram is special, “magic” or general. This concern had been reinforced when a student had asked him after the lecture about his choice of a point: Because the students, it’s funny how the students have, like I wish I could remember what it was, but a student came and asked me something a bit along the lines like: Why have you

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chosen and does the point always have to have x equal to y? They didn’t see that it was arbitrary, they’d missed that. . .It would have been better to have made it clearer what I was meaning by that term, why I was saying it. The interesting point is to think about that.

It appears that the discussion on this lecture excerpt made him aware of how he might improve his practice. He recognised the potential confusion about a general point he may have provoked simply by calling P a magic point. “You’re trying to say that it could be any point and really make that clear because that’s something that the students could probably miss.” Most importantly he was aware of what could lie behind students’ confusion: That’s really confused the students because they’ve lost track of where the. . . I wish I could remember what the student asked me but it was very much along those lines, they were muddled about that thing where it had shifted from being something that was arbitrary to something that was specific because I’d already given it some value, which unconsciously..

Reflecting further on the lecture during the discussion he was unsure if he had achieved his goal “. . .hopefully [I] gave them a feeling that they could draw this picture and from this gain understanding”, but he had considered how he might have improved his teaching: “I also know that some simple pictures, etc would have helped to break up the lecture and give students a chance to refocus”. This brief account illustrates an important step in changing our pedagogical practice. Any shift in practice requires a conscious level of awareness of the practice and a desire to achieve more than the practice seems to make possible (Jaworski, 2006). The lecturer’s awareness of what he did and why he did it raised an inner tension or conflict between his orientations and his goals. This is an important part of reflecting on our role as university teachers. The hypothesis is that when awareness is coupled with a desire to make changes, it may promote incremental growth in pedagogy, which has been described an effective approach to improving teaching (Speer, 2008). The second example concerns an applied mathematician (L3) presenting an applied mathematics lecture to first-year students and illustrates what can happen when you are in the middle of a lecture and you find yourself doing something you hadn’t thought you were going to do (see Paterson et al., 2011a, b). The primary purpose of the lecture was to consider solutions, for various values of the parameter q, of a difference equation that reduced to the form xm = qxm - 1(1 - xm - 1). At a particular point in the lecture he made the decision to move to a computer projected graph and show the students the periodicity of a function. It was during the process of counting the function’s local minima that a crucial decision point arose. The following lecture transcription is taken from Paterson et al. (2011c, p. 358): 1. What’s happening here it looks even more complicated, 3, 6. . .[3 to 4 secs] yeh so you can see that if you look at it closely. . .[walks to screen] 2. Suppose you start by looking at this value here [pointing at the graph on the projection] then there’s going to be 1, 2, 3, 4, 5, 6, 7, you can count to 8 I think maybe.. do I ever get back to where I started, maybe not [he realises that there is a problem] 3. 9, 10, 11, 12,13, 14 um.. how many values? So it looks like there’s a period of um.. let’s see 1, 2, 3, 4, 5, 6, 7, 8.. [starts to count again] it looks like there’s a period of 14. Whether that’s the case or not I’m not sure.

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M. O. J. Thomas 4. We might not have got to the limiting value yet. But it looks like we’ve settled down to a period of 14. By a period of 14 I mean that it takes 14 um.. we need n to change by 14 to get back to where you started from. It looks like that.. something like that is happening anyway. So it seems to be settling down to some complicated periodic um.. solution. 5. All this.. all these things happen just by changing q. In fact if you do analyse this a bit further you can you can look at.. There are critical values of q where you do get a change. And this allows you to draw what is called a bifurcation diagram. Bifurcation diagrams are diagrams that show how the solution splits up into different solutions as you change the parameter.

Although he counted the period to be 14, the lecturer knew from theory that the actual value was 16, as he later explained “. . .because the period doubles each time, so it goes from 2 to 4 to 8 to 16, so.. and so on, so there’s a theory that actually says the period has to double”. This mathematical discrepancy was not expected, and he commented “I guess the thing that I was probably concerned about was um.. observing something that I didn’t expect and not being about to explain it immediately”. Hence, in-the-moment, he had to decide whether to address it or not. He decided to press on with the rest of the lecture. His reasons for doing so were given in this way: I couldn’t figure out at that particular time what the problem was, uh.. I think I know now but even then it’s not obvious. . . I think I actually probably could have shown them that it was periodic with period 16. . .Part of me wanted to address this at the time but I had already gone over time with this part of the lecture and had achieved the goals I desired Actually I would have liked to have pursued it a bit but we had already spent more than the allotted amount of time on this demonstration and I had shown them periodicity for shorter periods already so I think they had grasped the concept quite well so the fact that I didn’t actually get a period of 16 bugged me a bit but not enough to ruin the rest of the lecture.

He was able to decide that, in this situation, with these students, his pedagogical goals as a teacher should win out over his desire as a mathematician to have things “right”. His primary goal to demonstrate that the solution of the difference equation is a periodic function had been met, and so he decided he didn’t need to address the mathematical anomaly (Paterson et al., 2011a). The final vignette presented here involved a pure mathematician, L4. He was giving a lecture to a graduate class where the primary aim was to introduce the students to continued fractions. His stated goals in his ROG were “To help students to understand the theory and do proofs”, “To provide good general preparation for post-graduate study in number theory”, “To increase the students’ mathematical maturity” (c.f. Artigue et al., 2007) and “To give exposure to different proof techniques”. He believed that the proof he was presenting was interesting and important, what he called “a cool proof”, since it proved a more general result. However, during the proof he suddenly realised that he was going to encounter a “notational conundrum” while proving the correctness of the recurrence formulae for computing the convergents. The lecture transcription, as given in Paterson et al. (2011a, p. 362) was as follows:

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1 plus . . . [indicates a succession of terms 1. This is the sum of a0 plus, this thing is ai þ aiþ1 1 on board] plus ai þ aiþ1 . So when you consider that separate extra term and we bundle that whole thing into one piece [circles it on the board] They are the same object. [Stands back from the board and looks at class and then back at the board] 2. So, by the inductive hypothesis, [starts to write] I know what this [gestures in swirl over previous line] It is some hi over ki. [looks at board] [looks as if he is thinking] 3. I’m going to call it . . . Did I give it a name? [Looks at paper] I didn’t give it a name. It’s just some hi over ki. But whatever that hi over ki is it apparently satisfies the recurrence formula [points to paper he is holding and looks at class] [Stands back and looks at the board] [Pauses a moment] [Moves] [Appears to come to decision] 4. Yeah I mean this is an hi over ki but it’s not the hi over ki that I am really thinking of [gestures back to previous expression] This is a very subtle point [Said very quietly – thinking aloud?] 5. [Comes back to the board and writes above previous 2 expressions] Let’s define hi over . . . Let’s define hj over kj to be these things up to aj where I have worked these out already all the way up to i [writes down and puts in rectangular box above previous 2 expressions] Right so the symbols hi over ki will from now on mean precisely one of these things for the specific numbers I am interested in. This thing I have written down here [gestures to bottom expression] is not the hi over ki in that notation because this end term is wrong.

What caused him to make this in-the-moment decision to address the notational inconsistency? During the group discussion he said it was really the mathematician within going “Oh this is not really right”, adding “at that point the whole world disappeared and it’s just me and the mathematics”. Later, when interviewed about it, he added “This is exactly the point where I suddenly realise that it is sort of not quite fitting how I was using those symbols previously. I was thinking ahead to where the proof was going and suddenly it becomes clear to me that there is a problem ahead”. While I could have glossed over the point and “swept it under the carpet”, telling the students “It’s in the notes . . .it would be a good exercise for you to do carefully”, instead he chose to sort it out then and there. In this case after reflection he could see that his personal need as a mathematician to make sure it was right won the day.

4.2.4

An Illustrative Boundary Encounter

The following selective extract taken from a transcript of one of the DATUM meetings is provided to illustrate some differences in perspectives and goals of the participants, as well as the potential two-way benefits of boundary encounters. This extract comprises part of a 50-minute discussion between three mathematicians and three mathematics educators that arose from a lecture by M1 on recurrence relations. Initially, after 20 minutes and 5 seconds of the meeting, M1 explains the role and importance in applied mathematics of the recurrence relation, how it differs from a formula and the potential difficulties involved in solving one: M1 20:05 Because there’s quite a big difference between recurrence relation and a formula and often you’re presented with a physical law or something and if it’s not a recurrence

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M. O. J. Thomas relation you want to translate it into one. There’s sort of three steps: you have a system and then you have the recurrence relation and you have the solution kind of solving it. . .Well you’re given a problem like I want to know the number of birds that are going to be, the number of penguins in the Antarctic next year and you model it as a recurrence relation and then you write down a mathematical equation, but you don’t have the solution to it and then you solve it by some means.

Immediately, the second mathematician added to the explanation, stressing the difficulty of finding solutions: M2 There’s difference between the recurrence relation and the formula, so your recurrence relation says an + 1= the function of an and then your formula says an = the function. M1 21:19 There’s quite a big difference, it’s much more difficult.

Two of the mathematics educators became interested in the pedagogical implications of this for school students, and a realisation began to dawn that these mathematicians saw the recurrence relation as more important in their research than the function: ME1 22:24 But they would have at school have met that one where you’re adding four as an example, they’ve quite often would have plotted it as a series of points. As a linear relationship between the two things. They won’t have. . . M1 22:39 They’ve probably never done the recurrence relation. ME2 22:44 You’re saying the recurrence relation is primary are you? M1 22:50 Yeah definitely.

This new understanding raises pedagogical questions in the minds of the mathematics educators, who begin to see a potential failing in the manner of teaching of general formulas in school, while the mathematicians at this stage seem to remain focused on the process and the relative difficulty of the mathematics: ME2 25:03 But effectively for the students coming through our school system they actually experience the general formula before the recurrence. M2 25:28 That’s because general formulas are harder. ME3 25:30 But they’re not taught to do that and the whole idea is that teachers are trying to steer them away from the recurrence relation because they don’t think it’s a good thing. They want to try and go straight to the general formula. M1 25:39 Yeah because that’s sort of the answer, that’s also called answer. You can get one from the other. M2 25:43 Yeah but it’s hard. If you’ve got a linear recurrence relation. M1 25:48 It’s quite hard to see how to go from one to the other, that’s the problem. There’s no simple way to connect them. . . M2 28:54 If you’ve got the answer it’s nice but in terms of understanding how you go there I feel like you need this step and the point is that most recurrence calculations you can’t write down a general formula, you could write it down for a linear one. M1 29:16 No you can’t absolutely you can’t. You can very easily write one with no, well I mean the notion that there’s, I mean it’s just like anything, unless it’s a sine or a cosine there’s basically no other function or a polynomial we don’t have a name for it right. So imagine.. so in a sense the recurrence relation becomes the..

Towards the end of this discussion, a third mathematician relates recurrence relations to a specific type called difference equations and again stresses how important they are in research, since they are a prime means of solving differential equations, through converting from continuous to discrete variables. The first mathematician

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agrees and clearly values the beauty of the method of using differential equations to solve a practical, banking interest problem: M3 34:29 That’s the type of thing that we deal with in applied maths though. Differential equations are solved by exactly converting them to difference equations [a form of recurrence relation] so what starts off as continuous ends up being discrete, albeit a very small time involved but to get accuracy it’s discrete. So difference equations are all over the place. M1 37:21 It’s really interesting how the banks use a discrete calculation . . . if you go onto the mortgage calculator at the [bank name] and you say I’ll take a mortgage for $100,000 at 7% what’s the weekly repayment and they’ll calculate it and the computer churns away and does it discretely but if you model as a differential equation you get almost exactly that answer with a simple formula. It’s quite beautiful. You can never write that formula down.

Finally in this particular section of the discussion, a mathematics educator tries to summarise what has been something of a pedagogical revelation to the three, namely, the global importance of the recurrence relation in mathematics coupled with the fact that this is often ignored, or seen as of secondary importance, in the school curriculum: ME2 38:34 Well I’m just thinking on what is it that we’re actually saying today and what we’re talking about . . . is the depth of the ideas that are behind the whole issue of recurrence formula and solution and the idea is not represented at all in the school curriculum. . .It’s just not there. It’s not even previewed, it’s not.. If anything it’s done back to front in school with the analytical formula as your introduction to algebra. . . You’re saying that probably by now we ought to be introducing them to the idea that the formula is the analytical solution of the recurrence relation.

In addition to this revelation for the mathematics educators arising from the boundary encounter, the mathematicians became aware that their first year students were unlikely to have had recurrence relations emphasised in any way in school and this was very relevant to their presentation of this material.

4.2.5

General Outcomes

There have been a number of positive outcomes from the boundary research projects briefly described above. One crucial outcome was that the DATUM project was considered so successful by the participants that it continued beyond the initial research project, and grew to encompass two groups of mathematicians, teaching fellows and mathematics educators, each meeting six to eight times per year. It continues up to the time of writing, and over 26 teaching practitioners have participated, with the two groups evolving in membership over the years. A second aspect of professional development emerged from this success. In 2013, the mathematics department decided, in response to an imperative from the university, to establish a voluntary, mathematics-based teaching peer review process to produce formal documentation for performance evaluations of lecturers that they could use in applications for establishing tenure and for promotion applications. The mathematics

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education team was approached to begin this initiative, called PROMT: Peer Review and Observation of Mathematics Teaching, which had the goal of engaging lecturers in reflection on their teaching. In an email to all academic staff, the head of department strongly recommended that staff provide evidence to substantiate the claim that they are a good teacher, and supported the PROMT initiative by saying he particularly welcomed evidence provided by this group. This PROMT boundary crossing activity has also proven to be highly valued in the department. It involves a five-step process. The lecturer identifies goals for the review, standard observation categories are agreed with the lecturer and then observation of a lecture (chosen by the lecturer) takes place. A report is produced after the observation, including both positive observations and critical feedback. This report is shared only with the lecturer, who is invited to respond to the observer’s comments, correct mistakes and offer explanations. This leads to joint reflection on the observed teaching, suggestions for professional development and the opportunity for the lecturer to request a subsequent observation and report. In just a few years more than 34 reports were produced for over 20 teachers, ranging from postdoctoral fellows in their first semester of teaching to full professors with prior teaching awards. In addition to PROMT, an annual daylong event called Conversations between Mathematicians, Talking about Teaching was instituted to bring mathematicians and mathematics educators together, with members of both groups giving short 15-minute talks about their teaching. The attendance at this 1 day event each year has been more than 75% of the department, and over the first 3 years there were 22 different speakers. Again, this activity at the boundary has enabled valuable joint reflection on teaching practice. Another outcome from paying attention to the gap has been on a theoretical level. One thing the decision-making video situations and discussions taught us was that for many lecturers there seems to be an inner dialogue that takes place between the role of teacher and mathematician (Paterson et al., 2011b). Sometimes the teacher would win out and at other times the mathematician would. The lecturers commented on this dilemma, saying “So should you just ignore that corner [of important mathematics] and just hope that it’s not noticed? But then is that bad because you’ve somehow told them something incorrect?” and “I do this kind of thing all the time, I think it’s really distracting because you’ve gone out and tried to make your big point and then you get all flustered over some detail [of the mathematics] and. . .you have to get it right and the students go ‘. . .now I’m completely confused’”. This conflict was enlarged on in a later paper (Schoenfeld et al., 2016), where, based on some of the data above, we suggested that while all teachers draw upon mathematical and pedagogical considerations to varying degrees as they teach university mathematicians’ in-the-moment decision-making is significantly shaped by their mathematical knowledge and considerations. In fact, for many “instruction will not proceed unless the instructor is convinced that the mathematics being discussed is absolutely correct” (ibid, p. 3). In addition, we claimed that a core pedagogical orientation of mathematicians is the understanding that claims must be justified and justifiable, so that if we make a claim in an instructional context, we must be able to back it up (this is depicted in Fig. 4.3). However, Kember and Kwan

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Fig. 4.3 A frame for characterising university mathematician’s decisionmaking. (From Schoenfeld et al., 2016)

(2000) argue that while university lecturers may adopt either a focus on content or on student learning approaches, fundamental changes to teaching and learning are only likely to result from changes to conceptions of teaching. Thus when making in-themoment decisions, there is a tension for the mathematician teaching in a university. Thus the mathematician’s decisions are governed by “the richness (depth, breadth and connectivity) of their mathematical content knowledge (MCK), the extent of their pedagogical knowledge (PK); and the quality of the synergistic relationship between the two” (Schoenfeld et al., 2016, p. 13).

4.2.6

The Value to Individuals of the Boundary Activities

One of the key indicators of the success of any programme is the response from the participants. We saw above that the mathematician L1 said he saw it as a form of professional development and found it an enriching experience that gave him the opportunity to reflect on his teaching practice. The mathematicians and mathematics educators who took part in the DATUM project also expressed positive thoughts on it. One of the mathematician participants in the second iteration of DATUM was affected sufficiently by the experience to want to record the benefits as she saw them. She valued the format of the boundary encounters in DATUM, where she had the opportunity to observe videos of others teaching as well as to examine via video and discuss her own practice within a supportive community. This process proved to be a valuable opportunity for productive professional development for her (Oates & Evans, 2017). She explains in Paterson and Evans (2013, p. 134) the shock she experienced when she first saw her video in contrast with another lecturer’s: “When I watched my own lecture it was a complete shock to me. I saw that I wasn’t as cheerful as I thought and certainly I was not coming across as an enthusiastic person

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who smiles easily. I only smiled a few times during the entire lecture. This was a revelation to me the image that I had in my head was not matching the reality”. Placed in a state of dissonance by this event during the boundary encounters, she quickly came to the conclusion that she needed to change her teaching style completely in order to be able to smile more often, and to show the students her energetic engaging self. She explains that she put this into practice, transforming her teaching through injecting more of herself into lectures, employing chocolate rewards for students to encourage engagement and using physical props, such as an umbrella to represent a given 3-D surface. Oh yes, and she smiled a lot more! She was much happier with her teaching after these changes and so were the students— and it showed in her positive teaching evaluations. Some of the other mathematician participants also expressed their personal feelings on the benefits to them and the department of the initiatives described here. In particular they appreciated the pedagogical insights that they were able to benefit from during the boundary crossings they were involved in. Some of their specific comments related to both the process and the outcomes are presented here: It was very useful for me to be able to talk to other lecturers in an informal setting and have the opportunity for others and myself to reflect on exactly what is going on in the classroom. I am sure that both my teaching, and my ability to think about my teaching, improved because of it. The analysis of the clip is encouraging in that I gained a mathematics education perspective of the clip, which clarified in my own mind what I do when I teach. . .It’s good to get that feedback from other people and in some cases people identify things that I do that I wasn’t even aware of . . .you come in with your mathematics education theory from time to time explaining some of the things that we all do and that’s very useful as well. It’s good to have some of the theory behind it. The experience has been overwhelmingly positive for me, and has definitely changed the way I think about teaching. One crucial aspect of the project is that it creates a space where a small group can have a frank, deep and ongoing discussion about teaching. I always felt very comfortable sharing recordings of my lectures with the group. The main impacts on my practice have been a much greater awareness of the different modes of teaching: sometimes one is giving motivation, other times explaining a crucial idea or showing how to solve problems of a certain type, other times modelling how a mathematician behaves. I have much greater awareness when teaching of the transition between these modes, and I make greater effort to communicate to students the transition from one mode to another. I think it was very useful. Having had no training, or very little training in teaching. . .having these discussions allowed me to actually think about what I am doing and in some cases to actually realise what I’m doing too because some things I do subconsciously. And as well as that it was also good to get feedback about teaching. . .to find about the things I’m doing well.

These comments show that the mathematicians in this project recognised their lack of training in teaching and appreciated the structure of the process of boundary encounters. In particular they liked the informal but frank nature of the discussions, the time given to reflection on teaching and the positive feedback from those with educational perspectives, including the variety of modes of teaching. In addition they were able to benefit from the input of educational theory without it being couched in inaccessible language and from seeing the teaching skills of others:

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An important feature of these activities is that they are fully informed and guided by scholarship in Mathematics Education. But, and this is crucial for their success, one can participate fully in the activities and benefit from them without having to “speak the language" of mathematics education research. Hence, these activities are perfectly designed to be engaging and beneficial. I have benefited from [DATUM] to learn excellent teaching skills from the group’s members. This simply opened my mind to potential teaching improvements.

4.2.7

Final Comments

My personal experience of minding the gap by engaging in boundary crossing activities with mathematics researchers is that it has been highly valuable and productive. My involvement in mathematical and pedagogical discussions has enabled me to reflect on my own practice of lecturing mathematics at university. For a number of years I was entrusted sufficiently by the department to be able to teach linear algebra and calculus on a first year course designed for mathematics majors, rather than only on a service course. This was considered a flagship course for the department and engaging in professional development activity with the mathematicians certainly focussed my energy on what was really important in the course. I hope it helped me do a better job of sharing the important ideas with students. Having seen the benefits firsthand, for me and for others, I feel I can strongly recommend the professional development model described above. In doing so I think it would be beneficial to recapitulate some of the key lessons that we have learned from our boundary crossings. The studies described here had a strong professional development theme, and this needs to take into account both the mathematical and pedagogical orientations of the practitioner participants. That is, there needs to be a focus on key pedagogical understandings—but taking into account the mathematical context (Schoenfeld et al., 2016). Our experience was that the ROG framework (Schoenfeld, 2010) worked really well. It was effective in focusing reflective attention of group members on their practice and enhanced the discussions. The development of a community of inquiry (Jaworski, 2006) works very well, but groups should be quite small (say around six members), should be based on mutual trust and it is better to induct new members into a group 1, or at most 2, at a time (Barton et al., 2014). We have also suggested that the groups’ members should all be focused on an interest in mathematics but be heterogeneous across variables such as research field (mathematicians and mathematics educators); mathematics background (pure and applied, different perspectives were in evidence on some issues); years of experience; and seniority (Barton et al., 2014; Schoenfeld et al., 2016). The key suggestions for implementing the process of professional development in the communities of inquiry are to use a short (3–4 min) slice of the lecture video to focus group attention and provide a sufficient springboard for discussion; to allow the lecturer full control over the selection of the lecture to be videoed, the slice

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chosen, and whether or not it is used after they have reviewed it; and to encourage participants to develop awareness of their own teaching style and to use this as a catalyst for pedagogical change (Barton et al., 2014). Although it is easy to record this process, it needs to be noted that it is not trivial to implement. It involves a significant commitment of time and energy from all the participants in order to be successful. The kind of PD described in this chapter is still reasonably rare in universities and other tertiary establishments around the world. I don’t believe that the situation in Auckland was special in any way. The mathematics educators there are appointed as full members of the department and are respected as such. The respect of the research mathematicians for the ideas that we have brought to the department was gained over a number of years, and the support of senior members of the department was essential in bringing the PD described here to fruition. One of my hopes in documenting this PD journey, and providing references to fuller accounts elsewhere, is that others will not only see the potential value of the boundary encounters and crossings but be motivated to give it a go themselves. Yes, we can all work to mind the gap! Acknowledgements I would like to thank my fellow researchers for all their many contributions to the projects described here and my mathematics department colleagues who willingly and generously gave of their time and energy to participate in the research.

References Aguirre, J., & Speer, N. M. (2000). Examining the relationship between beliefs and goals in teacher practice. Journal of Mathematical Behavior, 18(3), 327–356. Akkerman, S., & Bakker, A. (2011). Boundary crossing and boundary objects. Review of Educational Research, 81, 132–169. Artigue, M., Batanero, C., & Kent, P. (2007). Mathematics thinking and learning at post-secondary level. In F. K. Lester (Ed.), The second handbook of research on mathematics teaching and learning (pp. 1011–1049). Information Age. Ball, D. L., Hill, H. C., & Bass, H. (2005). Knowing mathematics for teaching: Who knows mathematics well enough to teach third grade, and how can we decide? American Educator, 29(1), 4–17, 20–22, 43–46. Barton, B. (2011). Growing understanding of undergraduate mathematics: A good frame produces better tomatoes. International Journal of Mathematical Education in Science and Technology, 42(7), 963–974. Barton, B., Oates, G., Paterson, J., & Thomas, M. O. J. (2014). A marriage of continuance: Professional development for mathematics lecturers. Mathematics Education Research Journal, 27(2), 147–164. Goos, M. (2015). Learning at the boundaries. In M. Marshman, V. Geiger, & A. Bennison (Eds.), Mathematics education in the margins (Proceedings of the 38th annual conference of the mathematics education research group of Australasia) (pp. 269–276). MERGA. Hannah, J., Stewart, S., & Thomas, M. O. J. (2011). Analysing lecturer practice: The role of orientations and goals. International Journal of Mathematical Education in Science and Technology, 42(7), 975–984.

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Hannah, J., Stewart, S., & Thomas, M. O. J. (2012). Student reactions to an approach to linear algebra emphasising embodiment and language. In Proceedings of the 12th international congress on mathematical education (ICME-12) topic study group, Seoul, Korea (Vol. 2, pp. 1386–1393). Hannah, J., Stewart, S., & Thomas, M. O. J. (2013a). Emphasizing language and visualization in teaching linear algebra. International Journal of Mathematical Education in Science and Technology, 44(4), 475–489. https://doi.org/10.1080/0020739X.2012.756545 Hannah, J., Stewart, S., & Thomas, M. O. J. (2013b). Conflicting goals and decision making: The influences on a new lecturer. In A. M. Lindmeier & A. Heinze (Eds.), Proceedings of the 37th conference of the International Group for the Psychology of mathematics education (Vol. 2, pp. 425–432). PME. Jaworski, B. (2001). Developing mathematics teaching: Teachers, teacher-educators and researchers as co-learners. In F.-L. Lin & T. J. Cooney (Eds.), Making sense of mathematics teacher education. Kluwer. Jaworski, B. (2003). Research practice into/influencing mathematics teaching and learning development: Towards a theoretical framework based on co-learning partnerships. Educational Studies in Mathematics, 54, 249–282. Jaworski, B. (2006). Theory and practice in mathematics teaching development: Critical inquiry as a mode of learning in teaching. Journal of Mathematics Teacher Education, 9, 187–211. Kember, D., & Kwan, K.-P. (2000). Lecturers’ approaches to teaching and their relationship to conceptions of good teaching. Instructional Science, 28, 469–490. Klein, D. (2007). A quarter century of US ‘math wars’ and political partisanship. BSHM Bulletin, 22(1), 22–33. https://doi.org/10.1080/17498430601148762 Lave, J., & Wenger, E. (1991). Situated learning: Legitimate peripheral participation. Cambridge University Press. Oates, G., & Evans, T. (2017). Research mathematicians and mathematics educators: Collaborating for professional development. In K. Patterson (Ed.), Focus on mathematics education research (pp. 1–30). Nova Science Publishers, Inc. Paterson, J., & Evans, T. (2013). Audience insights: Feed forward in professional development. In D. King, B. Loch, & L. Rylands (Eds.), Proceedings of Lighthouse Delta, the 9th Delta conference of teaching and learning of undergraduate mathematics and statistics through the fog (pp. 132–140). Delta. Paterson, J., Thomas, M. O. J., Postlethwaite, C., & Taylor, S. (2011a). The internal disciplinarian: Who is in control? In S. Brown, S. Larsen, K. Marrongelle, & M. Oehrtman (Eds.), Proceedings of the 14th annual conference on research in undergraduate mathematics education (Vol. 2, pp. 354–368). Oregon. Paterson, J., Thomas, M. O. J., & Taylor, S. (2011b). Decisions, decisions, decisions: What determines the path taken in lectures? International Journal of Mathematical Education in Science and Technology, 42(7), 985–996. Paterson, J., Thomas, M. O. J., & Taylor, S. (2011c). Reaching decisions via internal dialogue: Its role in a lecturer professional development model. In B. Ubuz (Ed.), Proceedings of the 35th conference of the international group for the psychology of mathematics education (Vol. 3, pp. 353–360). Middle East Technical University. Schoenfeld, A. H. (2004). The math wars. Educational Policy, 18, 253–286. Schoenfeld, A. H. (2008). On modeling teachers’ in-the-moment decision-making. In A. H. Schoenfeld (Ed.), A study of teaching: Multiple lenses, multiple views (Journal for research in mathematics education monograph 14, pp. 45–96). NCTM. Schoenfeld, A. H. (2010). How we think. A theory of goal-oriented decision making and its educational applications. New York. Schoenfeld, A., Thomas, M. O. J., & Barton, B. (2016). On understanding and improving the teaching of university mathematics. International Journal of STEM Education, 3(4), 17. Available from https://stemeducationjournal.springeropen.com/articles/10.1186/s40594-0160038-z

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Speer, N. M. (2008). Connecting beliefs and practices: A fine-grained analysis of a college mathematics teacher’s collections of beliefs and their relationship to his instructional practices. Cognition and Instruction, 26(2), 218–267. Stewart, S., Thomas, M. O. J., & Hannah, J. (2005). Towards student instrumentation of computerbased algebra systems in university courses. International Journal of Mathematical Education in Science and Technology, 36(7), 741–750. Suchman, L. (1994). Working relations of technology production and use. Computer Supported Cooperative Work, 2, 21–39. Tall, D. O. (2004). Building theories: The three worlds of mathematics. For the Learning of Mathematics, 24, 29–32. Tall, D. O. (2008). The transition to formal thinking in mathematics. Mathematics Education Research Journal, 20, 5–24. Törner, G., Rolke, K., Rösken, B., & Sririman, B. (2010). Understanding a teacher’s actions in the classroom by applying Schoenfeld’s theory teaching-in-context: Reflecting on goals and beliefs. In B. Sriraman & L. English (Eds.), Theories of mathematics education, advances in mathematics education (pp. 401–420). Springer-Verlag. Wells, G. (1999). Dialogic inquiry: Towards a sociocultural practice and theory of education. Cambridge University Press. Wenger, E. (1998). Communities of practice, learning, meaning and identity. Cambridge University Press.

Chapter 5

Students Enjoying Transformed and Improved Learning Experiences of Mathematics in Higher Education Simon Goodchild

Abstract In this chapter, the author reflects on his professional experiences that span more than five decades working in mathematics education as mathematics teacher and mathematics education researcher. He draws on experience from working in schools, teacher education, and a university department of mathematical sciences, latterly leading a national centre for excellence in higher education. Reflecting on this experience and engagement in mathematics teaching developmental research and supporting evidence from published sources, it is asserted that many students in higher education would enjoy improved learning experiences of mathematics if there were a transformation of teaching approaches. Framing the argument within Community of Practice Theory, the author sets out obstacles that need to be overcome if there is to be an effective collaboration and knowledge exchange between mathematics education researchers/teaching developers and mathematicians/mathematics teachers in higher education. In the second part of the chapter, consideration is given to the types of intervention in teaching approaches that might make a positive difference to students’ experiences. This begins with a brief summary of a small part of what is known about learning mathematics. Attention to two key issues in teaching is recommended, providing constructive feedback to students, and approaching teaching in a way that promotes students’ active engagement with the mathematics to be learned. Keywords Mathematics teaching development · Active learning · Community of Practice Theory · Interventions in teaching · Constructive feedback

This chapter title is the vision statement of MatRIC, Centre for Research, Innovation and Coordination of Mathematics Teaching, which is the background for this reflective essay. S. Goodchild (✉) University of Agder, Kristiansand, Norway e-mail: [email protected] © Springer Nature Switzerland AG 2023 S. Stewart (ed.), Mathematicians’ Reflections on Teaching, Advances in Mathematics Education, https://doi.org/10.1007/978-3-031-34295-0_5

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5.1

S. Goodchild

Introduction

The thesis I argue in this reflective essay is: if mathematics teaching and learning in higher education is to change, it is up to mathematics teachers to be the change agents. I will argue based on my own experience, informed by others’ published work, that mathematics education researchers (MERs) and Higher Education Mathematics Teachers (HEMTs)1 represent disjoint communities of practice (CoP). In general I assert, members of one CoP will not be accepted or effective as change agents in the other. However, some scholars participate in both CoPs, and these can be effective as brokers between the two communities. Someone who participates as an established member in just the Mathematics Education Research Community of Practice (MER-CoP) can only act as a facilitator of brokering and disseminating evidence of transformative practice within the Higher Education Mathematics Teaching Community of Practice (HEMT-CoP). Throughout this essay, I will refer to MERs and HEMTs, CoP, MER-CoP, and HEMT-CoP. I will aim to avoid the use of other acronyms.

5.2

Part 1: The Context for Transformation and Improvement

Mathematics Education Researchers (MERs) can act as catalysts for change. MERs can listen, question, discuss, and even propose changes in practice. Irrespective of how deeply rooted are MERs’ propositions in research evidence or substantive theory, Higher Education Mathematics Teachers (HEMTs) are unlikely to welcome MERs’ telling, instructing, criticising, advising, and even exemplifying. It may seem a statement of the obvious, but the assertion central to my argument is that change in higher education mathematics teaching will only occur when HEMTs themselves initiate changes in their practice. If HEMTs are to transform their practice, at least three conditions need to be met. First, HEMTs must be convinced of the need for change, and they need to be motivated to make changes in their educational practices. Second, HEMTs must trust the agents of change (the brokers); the agents must be experienced HEMTs and convincing. Third, HEMTs must believe in proposed changes to practice. They should observe (maybe in a virtual sense) and experience change in teaching and learning in situations they perceive as authentic and like their own. The starting point is to make the case that change needs to occur

1

I refer to higher education mathematics teachers rather than the simpler, mathematician or mathematics researcher because many academics teaching mathematics in higher education have backgrounds in other disciplines with substantial mathematical content such as physics, engineering, or economics. Further, I refer to “teaching and teachers” as embracing practices and practitioners who may adopt a variety of instructional approaches including lecturing, mentoring, supervising, etc.

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in higher education mathematics teaching and learning practices. It could be a HEMT is already alert to the need for change. S/he might be concerned with students’ participation in class, motivation, attendance, performance, evaluations of teaching, or many other issues. However, being alert to a problem does not necessarily mean that a solution is available; it is possible for a HEMT to perceive all these issues as lying outside the domain of their influence or scope of transformation. In such cases, it must be demonstrated that the change needed lies within the HEMT’s domain of practice and that the HEMT has the agency to make effective changes.

5.2.1

The Case for Change

Most students who take courses in mathematics in higher education do so as part of another programme of study, for example, engineering, science, and economics. In these programmes, mathematics is studied as a “service subject”. Thus, unlike most other disciplines and fields of inquiry studied at higher education, many students study mathematics because it is a requirement of their programme of study rather than because they want to. Further, acceptance onto their chosen programme of study may not be heavily dependent upon their mathematics performance in high school. These two factors, their motivation and the security of their prior knowledge, can be decisive in terms of students’ future performance (Arnold & Straten, 2012; Ausubel, 1968). In many universities, service-mathematics courses include students from several loosely related programmes of study; the composition of mathematics courses can exert a strong influence on the educational context. First, cohorts in mathematics courses can be very large. An experienced HEMT may take responsibility for the main lectures held in large auditoriums. However, if the course includes group work, this part is likely to be supervised or tutored by teaching/learning assistants who have limited subject knowledge and little pedagogical or didactical education or experience. Given the large number of students, the provision of timely and informative feedback at an individual level is scarce. Further, because of the diversity of students’ study programmes, it is difficult for the HEMT to demonstrate convincingly how the mathematics taught is relevant and might be applied in each student’s field of interest. I will explain in the second part of this essay how the combination of students’ personal characteristics and educational context puts students at risk and leads to poor performance. The risks are transformed into realised threats in educational contexts that are experienced by too many students taking service mathematics courses. The conditions of teaching and learning mathematics on service courses, outlined above, have emerged over the last 50 years, especially due to increasing participation in higher education. However, mathematics teaching has remained largely unchanged for a much longer period. New didactical strategies and educational technologies that have emerged from mathematics education research could be

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implemented to transform and improve the learning experiences of students; however, these remain somewhat underused.

5.2.2

Higher Education Mathematics Teachers: A Community of Practice

Wenger (1998) describes a community of practice in terms of three intersecting components: engagement, enterprise, and repertoire. Members of a CoP are mutually engaged in a joint enterprise and share a repertoire of discourse, history, and artefacts. If the enterprise is mathematical education in which the HEMTs share the common goal of students’ learning mathematics, the HEMTs’ shared repertoire will include lectures and tutorials/problem-solving classes, assessments, texts, and histories of shared experiences of students’ performance. Wenger also describes three distinct modes of belonging to a CoP: First, through mutual engagement in mathematical education, possibly for engineering or some other profession. Second, through imagination; HEMTs will share a sense of similar experiences with other HEMTs as similar institutional structures influence the nature of their practice, and because they will likely share similar experiences of teaching and learning in the mathematical education they received. Through this sense of shared experience, they can imagine other mathematics teachers’ practice operating within similar affordances and constraints as their own. The third mode of belonging described by Wenger is alignment, which Wenger explains as “doing what it takes” in this case to be recognised as a competent teacher of mathematics in the arena of higher education. Entry to the HEMT-CoP most often happens as a process of “apprenticeship of observation” (Lortie, 1975). Before becoming HEMTs, members of this CoP will have spent many years studying mathematics and consequently observing mathematics teachers at work. HEMTs will know what to do because they have watched others engaged in the practice for hundreds, possibly thousands of hours. Full participation in the HEMT-CoP is achieved after a period of “legitimate peripheral participation” (Lave & Wenger, 1991). The apprenticeship begins as students commence their university studies, and their participation is limited to observing. As students advance through their studies, so they have opportunities to support their peers in group-work, maybe take on the role of learning/teaching assistants and eventually as junior lecturers. Specialised mathematics teacher education courses for practice in higher education are uncommon. In Norway, the Centre for Research, Innovation and Coordination of Mathematics Teaching (MatRIC) has developed such a course that is intended to complement the general university-pedagogy course offered by most universities in the country. Until recently, the general pedagogy course entailed about 150 h of commitment over a year (this is increasing in response to changes in national requirements to 200 h minimum). The MatRIC course entails 100 h of commitment. In the context of the hundreds of hours of apprenticeship of

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observation, the formal provision is minimal, whether it is only the general pedagogy course or in combination with the MatRIC course; the effect of these general and specialised educational provisions for HEMTs is likely to be no more than marginal (Postareff et al., 2008).2 I assert that apprenticeship by observation and legitimate peripheral participation comprise the widely shared preparation for and trajectory into the HEMT-CoP. I also point to strong evidence that there is little variation across institutions and countries in the educational practices that will have been observed by HEMTs. This evidence comes, for example, from research by Artemeva and Fox (2011) whose study included 50 mathematics lecturers working in 10 universities in 7 countries. The sample of HEMTs included in their study additionally shared between them 16 first languages. It is thus reasonable to assume that the lecturers represented a wider range of national and cultural contexts than planned for in the research design, and consequently, the findings more globally representative. Artemeva and Fox studied the practices of these lecturers; they observed that there were commonalities of instructional practices shared by the lecturers across institutions and countries, suggesting a global practice of teaching mathematics in higher education. This practice is characterised by the lecturer at the front of an auditorium writing mathematics text (statements, theorems, proofs, figures, etc.) on a chalkboard while simultaneously saying what is being written. Concurrently with writing, the lecturer will also utter occasional meta-comments, which are not written, to add some explanation or information to that which is written. Artemeva and Fox refer to this instructional practice as “chalk-talk”. Artemeva and Fox observed some variations in practices that might be attributed to cultural differences, for example, the use of and concurrent reference to teaching notes. In some milieux it was common practice to read and copy from notes to the chalkboard; at the other extreme, it was expected that the lecturer would present the mathematics without any visible evidence of preparation or notes. Chalk-talk is not a mindless reproduction of observed or received practice. Mathematics teachers rationalise the approach in terms of providing students with a sound model of mathematical argument and development. Mathematicians’ board work makes mathematics explicit, as argued by the London Mathematical Society (LMS, 2010). In the teaching/learning process, it must always be remembered that: Mathematics [involves] strict logical deduction with conclusions that follow with certainty and confidence from clear starting points.

and one needs to see someone else, the lecturer, working through and creating the results. (LMS, 2010, p. 3)

2

Supporting evidence for this assertion is also emerging from the analysis of a Norwegian survey of active learning approaches used in higher education mathematics classes. The findings are yet to be published; a preliminary report from the survey is available; see Bjørkestøl et al. (2021).

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Thus at least one influential community of mathematicians rationalises chalk-talk both epistemologically and didactically. The readers is also referred to the chapter in this volume by Dawkins and Weber, which offers a much more developed account of mathematician’s rationale for lecturing. The same statement from the London Mathematical Society compares studying passively mathematics presented in a textbook with the active exposition of mathematics in a lecture, introducing a further pedagogical argument. However, active instruction or exposition is not (necessarily) the same as active learning. There is growing evidence of superior performance by students taught using approaches that encourage active learning (e.g. Freeman et al., 2014). Confronted by such evidence, some mathematics lecturers will respond that successful students are actively engaged in thinking and reflection as mathematics is presented in a lecture. Such a claim is not unreasonable or without foundation given the generations of students that have succeeded in learning mathematics from lectures and chalk-talk approaches. Poor performance and failure are explained by other (student) factors, as outlined above. The efficacy of practice is demonstrated because there is minimal evidence of Type 1 errors. Type 1 errors would be evident when individuals achieve full participation within the CoP but lack the necessary competence in mathematics. However, there is little evidence available about Type 2 errors; these are people who have the potential to develop the necessary competencies in mathematics but are prevented from doing so because lecturers do not employ approaches that motivate or facilitate active learning. I return to the theme of active learning later.

5.2.3

Mathematics Education Researchers: Another Community of Practice

As a community of practice, mathematics education researchers engage in a common enterprise of inquiry into teaching and learning mathematics, the educational contexts in which mathematics is taught, learned, and applied, the reasons for success and failure, and patterns of performance in mathematics, and the search for improvement. It is a diverse and eclectic enterprise and rightly described as a field of research rather than, like mathematics, a discipline. MERs similarly share a repertoire that is composed of philosophies, paradigms, theories, and arguments, of methodologies and critiques of validity, reliability, trustworthiness, and authenticity. There is no single approach to developing new knowledge in mathematics education; methodologies include positivistic and naturalistic approaches. MERs do not make claims for truth like mathematicians. The MER’s repertoire also includes histories of disputes with other MERs, either as witnesses or participants. MERs identify themselves with the MER-CoP but may characterise the field in a variety of ways, in a spectrum extending from social-science to applied mathematics. In many respects, the MER-CoP is quite different from the HEMT-CoP.

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Most mathematicians consider themselves, at least to some degree, to be both educators and researchers. Yet few embrace or even respect the subject at the intersection of these fields: research in mathematics education. (Steen, 1999, p. 235)

If a person were competent in mathematics, she/he would aspire to be a research mathematician! I am not sure how many people would agree with such a statement, but it could have some validity from the perspective of the person who is both passionate about and highly competent in mathematics, such as those who aspire to mathematics research positions in universities. From such a position, the person who has devoted themselves to mathematics education research or school teaching (the professional background of many mathematics education researchers) might be viewed as someone who has failed to make the grade. Anecdotally, some years ago, I met a mathematics education researcher who recalled advice he was given by the supervisor of his PhD in pure mathematics. He was strongly advised against pursuing a career as a research mathematician with a remark that went along the lines, although you have been successful in your PhD, do not look for a career in mathematics research because you are not up to it! Further evidence of the disdain in which mathematicians may hold mathematics education researchers is reported by Nardi following her study of mathematicians: [I]t is still the case that the image of a mathematics department that pays a lot of attention and contributes to research in mathematics education would be poor from other mathematicians’ point of view: the mathematics community does not in its bulk look to this type of research as a source of knowledge or ideas about mathematics teaching. It just doesn’t . . . whether it should or not is a different matter of course. (Mathematician’s voice in Nardi, 2008, p. 268)

Mathematics education researchers (MERs) enter their community of practice via several converging trajectories. As in the anecdote above, for some, the point of entry is a doctorate in mathematics. Others may have bachelor or masters’ qualification in mathematics as the foundation for school teaching and, from that point, enter the MER-CoP. Others may enter from a variety of disciplines, such as psychology, sociology, linguistics, philosophy, or physical sciences, or possibly commerce or industry. The diversity of trajectories into the MER-CoP is evidence of the field’s eclectic nature and that a competent MER requires the intellectual qualities of a polymath. It is sometimes interesting to observe an experienced mathematics researcher’s struggle to enter the field of MER. On the other hand, MERs might be embarrassed by the response of the mathematician who is bewildered by the abundance of theoretical positions occupied by MER, for example, reflecting on the work of a group of MERs, set up by the International Commission on Mathematics Instruction, one mathematician reflected: So what do these international experts tell us about their own field? Pretty much the same thing as the skeptical mathematicians. Indeed, the most striking conclusion of this ICMI study is that in spite of much thoughtful work by individual researchers, there is no agreement among leaders in the field about goals of research, important questions, objects of study, methods of investigation, criteria for evaluation, significant results, major theories, or usefulness of results. The papers in these two volumes - five working group reports, 33 expert papers, and the editors’ summary (significantly titled “Continuing the Search”)— document a field in disarray, a field whose high hopes for a science of education have been

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Although this statement is now 20 years old, MER has become more complex, and the diversity of theories adopted has grown. However, perhaps there have also been some positive influences, such as in the form of the production of texts and courses rooted in MER (e.g. Rasmussen et al., 2018). MER is a complex field of scholarship, and participation in the MER-CoP requires a deep and broad understanding of epistemology and didactics that is informed from theoretical engagement and empirical evidence. Nevertheless, I am acutely aware of the fact that this opinion is not widely shared amongst the CoP of mathematics teachers that I want to influence.3 My concern here is with two largely disjoint communities of practice. First, there is the CoP in which the participants are mathematicians, mathematics researchers, and HEMTs. The trajectory of participation as a member of this CoP takes place through a form of apprenticeship, which can be characterised as legitimate peripheral participation. Second, there is the CoP in which the participants are mathematics education researchers. There is no single, well-defined trajectory into full participation in this community. The opportunities for an apprenticeship of observation, such as through being a junior member of a research group, are few. Where opportunities exist, for example, in post-doctoral positions, they are not extended over many years as experienced by participants in the HEMT-CoP. Perhaps the common element that initially marks individuals as legitimate peripheral participants of the MER-CoP is their application of education and professional competencies to systematic inquiry within the broad and varied field of mathematics education research.

5.2.4

Influencing the HEMT-CoP from Outside

External conditions can make an impact on the enterprise in which the HEMT-CoP is engaged. Such an impact has been evident recently when universities were required to “lockdown” to reduce the spread of the COVID-19 virus. Within a few days, HEMTs were required to switch to online provision. It must be admitted that some mathematics teachers were either so uncomfortable, unprepared, or ill-equipped to meet the demands that the switch was to no-teaching rather than online-teaching. Despite this evidence of external forces’ potential to impact upon the HEMT-CoP, I assert that meaningful, sustainable change must be prompted from within the CoP, not by a participant in another CoP. The question then arises, how can a participant in the MER-CoP exert any influence within the HEMT-CoP? This was the question I had to address at the end of 2013 when informed that my

3

I admit that I am making an unsupportable generalisation here, and I therefore acknowledge that many HEMTs are interested to learn about, inquire into, and make sense of theories espoused by MERs.

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institution’s proposal for establishing a national centre for excellence in (higher) education for mathematics was successful, and I became the first Director of this Centre, MatRIC. MatRIC, Centre for Research, Innovation and Coordination of Mathematics Teaching, is a Norwegian Centre for Excellence in Education. The Norwegian Centre for Excellence programme sets three fundamental goals for Centres awarded the Centre for Excellence status. They must: provide excellent R&D-based education; develop innovative ways of working with R&D-based education; and contribute to the development and dissemination of knowledge about educational methods that are conducive to learning (NOKUT, 2013). I had led the preparation for the proposal for MatRIC, and it was based upon my prior experience of working within mathematics teaching developmental research with schoolteachers. Like the earlier school-teaching based projects, MatRIC was framed within Wenger’s CoP theory. However, I was aware of the differences between working with schoolteachers and MatRIC’s proposed engagement with HEMTs. There were also contrasts between the goals and purposes of the earlier work and MatRIC. The earlier projects were focused primarily on research and financed by the Research Council of Norway. In contrast, MatRIC was being financed initially as an educational initiative administered by the Norwegian Agency for Quality Assurance in Education. In the former, the creation of research-based knowledge was the core goal; in the latter, the core goal is educational development. Although a strategy of developmental research is appropriate for meeting both goals, there are differences in emphasis and the measures of impact and outcomes. The former were local projects composed of university didacticians4 and teachers in schools (8 schools including students from grades 1 to 13) and four kindergartens. MatRIC is a national enterprise in which all participants are employed in the same type of institution but within disjoint albeit similar CoPs. In the former, the project leaders were mathematics education researchers with a background in school teaching, teacher education, and schoolbased research. There was an abundance of common experiences shared with the school mathematics teachers with whom we were working. The goals included the development of a collaborative community of inquiry composed of school and university participants. In MatRIC, I, as the leader, was firmly established as a member of the MER-CoP, and MatRIC’s goals were focused within the HEMTCoP. My earlier higher education experience teaching mathematics was within a teacher education programme in another country, and it was insufficient to ensure that I would be considered a participant in the Norwegian HEMT-CoP, and indeed unlikely to be accepted. My trajectory into the MER-CoP had begun with undergraduate studies in mathematics; in other words, it coincided with the legitimate peripheral participation within the HEMT-CoP. However, following a bachelor’s degree, my trajectory

4

In the school-focused mathematics teaching development research, we referred to the universitybased participants as “didacticians” because we wanted to emphasise that all participants, whether based in school or university, were researchers.

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diverted, first into school teaching and later mathematics teacher education. I knew that in leading MatRIC, the most influential role I could play within the HEMT-CoP was that of a facilitator. Wenger et al. (2002) acknowledge a developmental role for outsiders of the CoP. They describe them as “shepherding their [CoP’s] evolution” and that “design elements should be catalysts for a community’s natural evolution” (p. 51). I do not use the metaphor “shepherding”, because I would not want to imply any similarity between the purposeful and independent participants within the HEMT-CoP, and the rather timid, follow the flock instinct of sheep! However, I would possibly characterise the elements of MatRIC that are designed to support the HEMT-CoP as catalysts. MatRIC’s design elements are principally events to bring HEMTs together to focus on teaching and learning mathematics. The events are made possible because of MatRIC’s funding as a Centre for Excellence. In this respect, like catalysts, the HEMT-CoP should evolve without long-term dependence on MatRIC’s events. Nevertheless, I remain hopeful that some events will have proved to be so valuable to the HEMT-CoP that they will continue beyond the life and funding opportunities of MatRIC. In “Cultivating communities of practice”, Wenger and colleagues also identify the role of “coordinator”: Alive communities, whether planned or spontaneous, have a “coordinator” who organizes events and connects community members. (Wenger et al., 2002, p. 55)

MatRIC was structured with individuals in such roles and referred to as “coordinators”. These individuals are members of the HEMT-CoP, and they have an essential role in establishing MatRIC as a national centre. The role of “coordinator” in MatRIC has had an impact either on the individuals’ career trajectories or MatRIC’s networking – or both. However, it is not evident after nearly 7 years that the HEMTCoP cultivated by MatRIC is sustainable solely on the contributions of the “coordinators”. In MatRIC the “coordinators” were recognised as part of the process of proposing the Centre. The coordinators stood out as colleagues that had demonstrated a commitment over many years to develop their own mathematics teaching. In one case, through the development of a vast collection of digital visualisations that he used to illustrate his teaching (mathematics, statistics, and physics5), he became the coordinator of a “simulation and visualisation” network. In another case, the coordinator had engaged deeply in using digital assessment for mathematics, and he had visited many other HEIs to inform other HEMTs about this approach to assessment. This same HEMT had also spent time developing short videos to support his teaching, and thus he found himself coordinating two networks focusing on areas of teaching innovation: digital assessment and mathematics video. A third colleague was deeply involved in mathematical modelling and keen to influence teaching and learning mathematics through modelling. The coordinators were “insider” members of the HEMT-CoP who were committed to developing innovative approaches to teaching and learning mathematics. As coordinators within MatRIC their role was to be expanded to be the core of a network through which

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See https://grimstad.uia.no/perhh/phh/MatRIC/SimReal/Menu/Science_eng.htm

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information and expertise about the innovation might be shared and developed. They provided an essential core of knowledge and expertise for the growth of the networks. The coordinators were identified because of their highly creative and innovative expertise as inspiring teachers. However, they were not naturally network builders or the event organisers that were needed to ensure networks were nourished. Therefore, it was also necessary to have someone in the role of facilitator. As the leader of MatRIC, I assumed the role of facilitator, and in this role, I set out to create networks of practitioners with the coordinators as nexus and strengthen the HEMT-CoP as an outcome. I further looked out for innovative and creative mathematics education researchers (MERs) who were also engaged as HEMTs; in other words, they were accepted participants in both communities of practice. This strategy, too, is consistent with the suggestions made by Wenger and colleagues: “Good community design brings information from outside the community into the dialogue about what the community could achieve” (Wenger et al., 2002, p. 54). It should be noted that I was not (and am not) a member of the HEMT-CoP; membership of the HEMT-CoP is, in my experience, neither necessary nor sufficient to take on the role of facilitator. Wenger (1998) refers to outsiders who can bridge across disjoint CoPs, as “brokers”. In MatRIC, brokers served an important function in crossing the boundary between the MER and HEMT CoPs. The broker’s task is to take the scientific product of mathematics education research and presenting it in a manner that will convince HEMTs. From my perspective, an ideal broker is an active participant in both HEMT and MER CoPs; they will teach undergraduate mathematics and be active researchers of teaching and learning the subject. The brokers should be able to talk convincingly from their own evidence-based, innovative practice that results in students’ experiencing transformed and improved learning of mathematics. The people I seek as brokers need to be good communicators, and they need to share MatRIC’s vision. The combined set of criteria means the pool of potential brokers is relatively small. Fortunately, a reasonable budget has made it possible to bring in brokers from other countries. Over the 6 years that I led MatRIC, brokers were introduced to the Norwegian mathematics teaching CoP from Europe and North and Central America. One of the enduring impacts of MatRIC is that many of the brokers have taken on a form of honorary membership of the Norwegian HEMT-CoP and are called upon to contribute to the community outside MatRIC’s direct activity. The identification of suitable brokers is challenging; it is possible to ensure the meeting of many of the criteria for extending invitations to authors of publications in the journals (so-called boundary objects) that seek to bridge the CoP boundaries. However, a printed text does not necessarily mean the author will be an effective communicator at a personal level. Therefore, part of my work as a facilitator required that I built a network of contacts so that I could seek and obtain personal recommendations. Another dimension of my role as facilitator entailed the creation of events to which brokers could be invited to present their innovative practice to the Norwegian HEMT-CoP. MatRIC thus led me to adopt the role of “events’ organiser” and promoter of conferences, workshops, and seminars throughout Norway. These

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events serve the dual purpose of bringing HEMTs together (i.e. networking) and stimulating transformation and change in the HEMT-CoP through the presentations of the external, international speakers (i.e. brokering). MatRIC’s home page and a Newsletter help to publicise these events, but the most effective means of ensuring participation is by personal invitation. Although I was not a member of the HEMTCoP, I was accepted as a facilitator, and the personal invitations I made were effective in eliciting a response. It seems that the role of a facilitator is quite different from that of a broker, and it is a role that is not considered in Wenger’s development of communities of practice or learning communities. It is, however, an essential role if one wishes to provoke or incite change in a CoP of which one is not engaged or accepted as a full participant. It may be worthwhile to briefly compare the three roles in MatRIC: facilitator, coordinator, and a broker. The facilitator acts as a catalyst, enables things to happen by organising events, creates meeting opportunities, and establishes networks. Most of the work of the facilitator can be done in the background. The facilitator needs to have a range of contacts that can advise, for example, about possible brokers or issues upon which to focus within the events, but the facilitator does not need to be highly visible. Coordinators need to be more or less permanent members of their networks, and HEMT-CoP, they bring the flame of innovative practice to the network. Brokers need to be active in both HEMT- and MER-CoPs and inspire HEMTs to act and reflect and theorise their practice. It is possible that all three roles could be undertaken by the same person, but given the differing profiles necessary for each, it is possible to pick out individuals ideally suited for each role. MatRIC provided a context in which the three roles could be developed, but the functions of each exist outside a specific developmental framework, and perhaps any department of mathematical sciences needs to have facilitators, coordinators, and brokers and a budget to import coordinators and brokers as necessary.

5.3

Part 2: Effective Intervention

If I were invited into the HEMT-CoP and challenged to make suggestions about how to improve student performance, my starting point would be to consider students’ learning rather than HEMT’s teaching. HEMTs will have considerable experience of teaching, both observing teaching as they were learners and later as teachers. They are likely to consider themselves to possess some degree of expertise as defined within the HEMT-CoP. I do not set out to challenge their sense of expertise. Challenging HEMTs to reflect on their practice begins, I believe, by accepting that they are experts in their current practices. Further, I do not call on theories of learning that have evolved within mathematics education research. These theories set out to make sense of the observations that MERs make in classrooms, and they provide a foundation for a practical pedagogy of mathematics. However, the principled explanations that these theories offer are complex and deeply rooted in paradigms; that is, the contested area of philosophical reasonings about ontology and epistemology.

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They can also confuse and bewilder the mathematician; consider the quote from Steen (1999) included in Part 1. I believe HEMTs seek practical solutions to the challenges of teaching and learning they experience. This is not to say that a piecemeal, inadequately theorised approach based on practical tips, could result in sustainable self-generating transformation and improvement. They might enjoy philosophical debate, but not necessarily as the foundation for transforming their teaching practice. I believe the HEMT wants substantive and practical suggestions to address the issues they experience with students’ learning. They want empirical evidence for the effectiveness of interventions, not a theoretical argument. I am not sure that it is helpful to point to the evidence that active learning approaches or inquiry-based mathematics education result in superior student performance without addressing what these approaches might entail practically. My starting point, therefore, is empirical evidence arising from cognitive psychology rather than the theories in which I frame my research into mathematics learning, teaching, and teaching development.6

5.3.1

An Emphasis on Learning: Students Need the Opportunity to Learn

In any evaluation of teaching, I believe a good starting point is to look for the opportunities that students are given to learn, especially those which emphasise students’ meaning-making. Are students encouraged to adopt effective learning strategies that will support the development of understanding and fluency in concepts and procedures? Are students likely to develop habits of mind, self-awareness, and attitudes that will support learning? The emphasis is on learning because a person can only learn for her/himself. A teacher cannot learn for her/his students. I admit this is a philosophical issue, and it will be evident that I adopt a broad constructivist view of meaningful learning. I emphatically reject theories that entail transmission or behaviouristic notions of operant conditioning if the desired outcome is conceptual understanding. To be honest, in the case of developing mathematical understanding, I am not convinced by metaphors such as “internalisation” and “appropriation” as used in socio-cultural theories. These metaphors do not seem to embrace the notion of metacognition,7 including critical reflection (on what is to be learned and one’s present understanding) and meaning-making that I believe to be essential in learning mathematics. I want to allow one of the reviewers of this chapter to have a voice at this point. “I agree that mathematicians may not like some of the theories (I guess we all have our own preferred ones), but in my experience, they also enjoy employing them and using them as a lens. Sharing suitable theories with them enables the community to communicate and share the same language. We hope that in the future, through productive collaboration, more theories (that are tailored to our needs) would emerge from these CoPs.” 7 More about “metacognition” follows below. 6

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Suppose one considers the learning processes and remarkable growth in knowledge and competencies of young children. In that case, learning appears to be a natural, unconscious process of appropriating behaviour from the socio-cultural environment (e.g. a young child learning language and other social behaviours). I believe young children are capable of abstract thought. For example, evidence that emerged from the Cognitively Guided Instruction research (Fennema et al., 1996) reveals young children’s thinking at deeper levels than suspected possible by their teachers. However, developmental psychologists describe changes in cognitive processes that occur around the onset of adolescence. In his socio-cultural account of cognition, Vygotsky describes a transformation in thinking that occurs around adolescence (Vygotsky, 1978): For the young child, to think means to recall; but for the adolescent, to recall means to think. Her memory is so “logicalized” that remembering is reduced to establishing and finding logical relations . . . (Vygotsky, 1978, p. 51, emphasis in original)

In complementary work, Piaget’s biological maturation-constructivist account of cognition (Richardson, 1998) also describes a transition around the same stage of development; this is the onset of the cognitive stage of “formal operations”, which is marked by the emergence of abstract, logical, theoretical thinking characteristic of adults. The theories of cognitive development separately proposed by Vygotsky and Piaget are argued to be inconsistent, incompatible, and incommensurable8; nevertheless, they both mark this development in thinking. The point I want to make is that meaningful learning in mathematics by students in higher education requires the engagement of logical, conceptual, and connected thinking. I believe it is incorrect to think that mathematics is learned in much the same way that a young child learns language. For most people learning mathematics requires effort and can be enhanced if the learner exercises some agency to control their learning behaviour. The entailments of adult learning of mathematics are the theme I principally focus on below. A further point I want to make is that, because of their maturity, university students and adults are much better equipped to take control of their thinking and learning actions. It is essential that students in higher education take control of their learning in mathematics. Mathematical knowledge, although a cultural artefact, is more than a collection of cultural behaviours that can be appropriated and internalised. For example, Duval (2006) argues that to understand a mathematical concept fully, one needs to experience the concept through more than one representation and coordinate several representations of the concept. Coordination of representations entails critical engagement in which the learner examines the representations to determine what they have in common and what differs between

8

This is not the place to reiterate extensive and deep philosophical arguments behind these assertions. I will simplistically add that they are based upon whether mind and cognition are purely biological products of an individual organism, or social-cultural artefacts that emerge temporarily as amalgamations of the individual and social in a particular socio-cultural context. For more on this, I refer the reader to Cobb (1994) and Lerman (1996, 2010).

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them.9 Differences must then be examined to determine whether they illuminate the concept or point to the boundaries and what is not entailed in the concept. Duval draws attention to the fact that the mathematical meaning does not reside at the surface level of the representation but at a deeper level, and meaning is not explicit in a single form of representation. Skemp (1982) also draws attention to this in his writing about syntactic (surface) and semantic (deep) level cognitive structures. Usually, learning higher mathematics is not the outcome of natural, unconscious processes that can characterise young children’s learning. Learning higher mathematics requires effort; it entails grappling with cognitive challenges and sustained mental engagement. HEMTs cannot do this for their students; students must be enabled to take responsibility for their own learning. However, HEMTs can motivate and facilitate deep level engagement through posing meaningful and interesting problems that do not have routine algorithmic solutions. Further, the teacher can create work contexts (such as in small project groups) in which students can engage in a collaborative enterprise to find solutions. Many students enter university believing that good teachers make mathematics easy. This belief is not supported by research (Stein et al., 1996). Reducing the cognitive demand of tasks does not support the development of understanding. The reduction in demand effectively removes the element that the student needs to engage with to learn; the student might be able to memorise a procedure. However, the opportunity to make sense for her/himself is reduced, and there is a reduced possibility to develop a deep conceptual understanding. On the other hand, the effort students need to make is much more than going through “the motions” such as attending lectures, taking notes, doing mathematical tasks with the hope that something will “stick”. This approach is not an efficient or effective way to learn higher mathematics or arrive at a deep understanding of mathematical meanings. Effort must be applied in taking measures that will make learning more effective and efficient, and a lot is known about what can make learning higher mathematics more effective. If the purpose is for students to learn, then they must be confronted with cognitively challenging mathematical tasks. If a student is not challenged, i.e. the task can be dealt with using routines familiar to the student, then there is nothing new to learn; the student is only rehearsing existing knowledge. Rehearsal is not a bad thing, especially if one is seeking fluency, but here the concern is learning new mathematics.10 Further, if any learning is likely to take place, it will be where the challenge lies. I construct the following rather absurd example to illustrate. Consider the learning purpose of a task is to understand how the Newton-Raphson iterative solution of equations is rooted in the way successive linear approximations to a function y = f(x) 9

A crucial issue for teachers is how students might be motivated towards and facilitated in such critical engagement. 10 There is a growing literature describing efforts to develop students’ motivation and meaningful engagement in undergraduate mathematics, especially based on the setting of non-routine projects. One good entry point to this literature is the collection of papers “Transitions in undergraduate mathematics education” (Croft et al., 2015).

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intersect with the x-axis. Suppose the task given to students is to program a spreadsheet with the iteration. In that case, the student may be challenged by the programming but may not be led to reflect deeply on the underlying explanation of why the iterations approach the desired root, or in some cases approach a different root or possibly do not converge to any root. A lot depends upon how the task is presented and the student’s competence in programming spreadsheets. Perhaps the most striking thing is that none of the knowledge about effective learning is particularly new; the basic ideas have roots in research done in the 1970s and 1980s, that is, nearly half a century ago. Nevertheless, there seems little evidence of this knowledge being applied widely or in any systematic way to transform and improve students’ learning experiences of mathematics in higher education.

5.3.2

Prior Knowledge: The Fallacy of Accelerated Remediation

It is not unusual to hear HEMTs remarking that students do not have the prerequisite knowledge they need to succeed in a course. The significance of a students’ prior knowledge for future success is summarised in the pithy aphorism: The most important single factor influencing learning is what the learner already knows. Ascertain this and teach him [/her] accordingly. (Ausubel, 1968, p. vi)

Many mathematics courses that immediately follow a transition from one phase of education to another (secondary to high school; high school to university) will begin with a crash course to ensure students have the prerequisite knowledge they need to progress. This type of remedial action seems to be based on two fallacies, first that students who may have struggled with learning mathematics over several years can make up any deficit in their knowledge within the space of a few hours or days. The second fallacy is the insidious and erroneous implication that the post-transition teacher is somehow more competent than the students’ earlier teachers. It is possible to address weaknesses in students’ prior knowledge, but it requires substantial and sustained effort. I will illustrate with an example from MatRIC. The intervention I describe here is not the type of “quick fix” that I have just criticised. I will describe a specially designed pre-course that is taught over a whole semester, the course is carefully structured, and students are given information about where they might benefit from participation. The provision also combines learning support as well as direct instruction. In Norway, university economics programmes will include at least one course in mathematics, but the entry requirement to the programmes often does not specify the prerequisite mathematics needed. There are several different routes through high school mathematics courses in Norway. Some include a substantial element of calculus, some are better described as pre-calculus, and at a minimum, some high school courses do not take the student much further than grade 10 algebra. A team of researchers at one of Norway’s largest and high-ranking universities has

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demonstrated that the course taken at high school can have a decisive impact on the student’s performance in the university mathematics course (Opstad et al., 2017). At the University of Agder, MatRIC recruited a doctoral fellow to research this issue (Landgärds, 2019). Landgärds’ developed an introductory mathematics course that was implemented concurrently with transformations within the Economics Programme. The mathematics course in the Economics Programme was moved from the first to the second semester. This enables students to complete Landgärds’ non-compulsory pre-mathematics course before starting the regular mathematics course. An online diagnostic test based upon the necessary starting knowledge for the regular Mathematics for Economics course was also created and made a compulsory part of new students’ introduction to the economics study programme. The students receive immediate feedback from the diagnostic test about where they might benefit from further instruction and learning in advance of the regular mathematics course. Langärds designed and structured the pre-mathematics course to enable students to join at any time, with different points corresponding to information received from the diagnostic test. The course is composed of lectures, problem-solving sessions, and online work. A team of student learning assistants supports the delivery of the course. Although attendance at the pre-course is voluntary and carries no academic credit, the uptake is very high, at a level similar to courses with compulsory attendance. The overall impact on student performance has been marked; the pre-course is now in its third year. In the second year, the pre-course was taught by a substitute teacher without any noticeable difference in performance than when Langärds taught the course herself, thus indicating some degree of robustness of the measures taken.

5.3.3

The Challenge of Learning Mathematics

Mathematics is a difficult subject to learn. There are several reasons for the difficulty, for example: • Mathematics is dependent on symbols, representations, and abstraction. • Mathematics is a conceptual subject, which is composed of a complex network of ideas. • Mathematical understanding and activity entail engagement with representations of concepts, which are not entirely manifest in a single representation. • Mathematical communication, especially between teacher and learner, takes place at the surface level of symbols, graphical representations, text, and language. At the same time, the conceptual meanings exist at a deep level, as ideas in the minds of individuals. • Competence in mathematics is not measured only in terms of mastery of the surface level nor with added attention to the mastery of individual ideas, but rather in the competence to use both surface and deep levels fluently in communication, proof, problem-solving, and modelling.

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• Development of mathematical competencies entails substantial engagement in mathematical activity – communication, proof, problem-solving, and modelling.

5.3.4

Attitudes Towards Mathematics: Motivation, Attributions, and Approaches to Learning

Learning mathematics requires sustained effort. Learners need to be motivated to learn and believe in themselves that they can learn. There exists research evidence that points to the importance and consequences of students’ motivation, attributions, and approaches to learning. Students need to be motivated to engage in sustained mathematical activity. Researchers have approached motivation from contrasting epistemologies. On the one hand, Mellin-Olsen (1987) considers students’ learning from the social-cultural perspective of Cultural Historical Activity Theory. Mellin-Olsen sets out a case that students need to possess effective rationales for learning. He describes an S-Rationale, leading the student to perceive personal (social) meaning in the material to be learned, such as an engineering student recognising the importance of mathematics within his/her field of engineering. Mellin-Olsen also describes an I-rationale in which the student realises that learning may pay off in terms of examination grades or some other reward extrinsic to the subject matter. From the perspective of cognitive science, Deci and Ryan (2000) developed Self Determination Theory, and within this, they have explored the effect of students’ motivation to learn. They describe the intrinsic motivation and several kinds of extrinsic motivation. Intrinsic motivation appears to be similar to Mellin-Olsen’s S-rationale and extrinsic to the I-rationale. Deci and Ryan note that intrinsically motivated students who engage with learning because they have a personal interest in the material to be learned are likely to study for longer and engage with the material with deeper thinking and reflection. Although it appears that Mellin-Olsen and Deci and Ryan agree about the superiority of S-rationale/intrinsic motivation, they also agree that both forms of motivation are valuable. Simply, examinations have a place to motivate students’ engagement in addition to their purpose of assessing the learning that has taken place. In addition to being motivated to learn, students must be convinced that their effort will pay-off; in other words, that they are capable of achieving mastery of the material to be learned. Again, it is possible to point to complementary strands of research. Dweck (1999) engaged in several decades of research that points to the impact of students’ attributions of success. When confronted with a problem, a student who sees it as an opportunity to learn is more likely to engage with the problem than a student who fears failure, and that failure would make the student appear less intelligent. Further, a student who believes intelligence is a characteristic that can be developed through mental exercise is more likely to engage with

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challenging problems than one who believes that intelligence is a fixed attribute that cannot be developed. Another perspective of students’ willingness to engage with challenging tasks has been investigated by Bandura (1982), who refers to students’ sense of “self-efficacy”. A student is likely to be more ready to engage with a task if she/he perceives a high degree of self-efficacy to succeed in the task and is likely to persevere longer and deeper when confronted with unforeseen challenges while engaging with the task. It follows that it will benefit students if their teachers can support students’ development of attributions of success and perceptions of self-efficacy, in addition to presenting subject content. Students need to be given problems in which they are likely to experience success and a sense of personal achievement and growth when the problem is solved. At the same time, problems must be sufficiently challenging to require students to think deeply about the mathematics involved and their engagement supported in ways that facilitate progress without reducing the cognitive challenge. Selecting and posing problems and supporting students places a heavy demand on teachers who need deep knowledge of subject matter and learning processes in addition to the extensive experience of students. The development of these didactic and pedagogical competencies needs more attention in terms of developmental opportunities for HEMTs, rewards for teaching expertise, and a culture that promotes these competencies within the HEMT-CoP. So far, I have considered students’ readiness for learning in terms of their prior knowledge, their motivation to engage with new challenges, and their willingness to take on fresh learning challenges. Assuming these conditions are met, the approach taken by learners as they engage with challenging tasks is also significant. Students’ approaches to learning have been characterised in several ways. I suggest that the fundamental distinction rests in whether the student aims for being able to recall facts and procedures from memory or seeks meanings that emerge and evolve through the development of rich interconnected conceptual schema. Skemp (1976), for example, writes about instrumental and relational understanding. The former, instrumental understanding, is achieved quickly and is dependent upon memory; it diminishes quite rapidly over time and contributes little to further learning. In contrast, relational understanding takes some time to achieve; the learner needs to be intent on meaningful engagement and making connections to existing ideas. Relational understanding supports further learning and endures over time. Marton and Säljö (1976) describe students’ approaches to learning. Adopting a surface approach, students engage with new material intending to memorise, perhaps for a test or examination. In contrast, students can take a deep approach that seeks meaning and understanding. Although the deep approach is likely to entail more effort, the additional rewards of increased interest, better understanding, and more resilient learning outweigh the cost of the additional effort involved. Another characterisation of learning or the goal of learning suggests that learners need to aim at both memorising that supports fluency and understanding that supports meaning. Hiebert and Lefevre (1986) distinguish between procedural and conceptual learning. In mathematics, it is necessary to develop procedural fluency as well as competencies founded on conceptual learning, meaning, and strategic thinking.

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A PhD fellow at the University of Agder has pursued a quantitative study that applies structural equation models to test causal hypotheses that relate engineering students’ prior mathematical knowledge, self-efficacy, and approaches to learning to their performance in their end of first-semester mathematics examination (Zakariya, 2021a; b). The models are complex, and the provisional results reflect the complexity. As expected, there is evidence that a student’s prior knowledge is a crucial factor associated with later examination performance. In Zakariya’s data, there is evidence of the positive effect of students’ self-efficacy in mediating prior knowledge with later performance. However, whereas surface approaches to learning are seen to have a statistically significant negative impact on performance, there appears to be only a small, statistically non-significant positive influence of deep approaches to learning. Zakariya’s research is only a small study in the context of quantitative research and includes only one cohort of students; nevertheless, the cohort size meets the statistical requirements for the psychometric analyses undertaken. The tests of prior knowledge and end of semester performance were adopted from naturally occurring situations. Although Zakariya evaluated the tests for their measurement properties and reliability, they cannot be assumed to meet criteria of construct validity. Nevertheless, despite the reservations, the results of the study offer confirmation of the foregoing assertions and lay the foundations for further work that aims to explore the relationships between these factors.

5.3.5

Learning to Learn Mathematics

I believe we fail many students by assuming that because they have managed to secure a university place, maybe one which requires very high grades from high school, that therefore they must be competent learners. Many students succeed at school as a result of their competence in preparing for and performing well in examinations. They may have succeeded on a foundation of instrumental understanding, achieved through surface learning and the development of procedural fluency. Proficiency in taking examinations will not be sufficient for the student to develop the critical competencies in mathematics on any study programme they are following at the undergraduate level. Possibly, students do not expect mathematics to be fundamentally meaningful. Many years ago, I came across the following assertion, which astonished me because I had no idea what was meant by “metaconcept” and when I came across it, I thought I had been successfully teaching mathematics for many years: The (. . .) development of a proper metaconcept, then, relieves the teacher of the necessity of doing ‘interesting’ things every lesson: it makes teaching mathematics (. . .) possible. (Howson & Mellin-Olsen, 1986, 30)

Following much thought, I offer the following as a working definition of metaconcept of mathematics and learning mathematics:

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(i) A person’s metaconcept of mathematics is a belief about the nature, structure, and coherence of mathematics, it provides (the learner with) a concept-context within which the interpretation of new (mathematical) experiences might be made. (ii) A person’s metaconcept of learning mathematics is about the nature and purpose of learning experiences and what constitutes appropriate responses to experiences with ‘learning potential’. (Goodchild, 2001, p. 207) I believe that many students arrive at university and study mathematics as a service subject without having a fully developed metaconcept of mathematics. In other words, they neither expect mathematics to be fundamentally meaningful nor do they engage with mathematical activity with the disposition to seek meaning. All mathematics teachers need to take time to develop students’ metaconcept of mathematics and their learning approaches. I have already emphasised the value of a problem-based approach, and my recipe for success remains the same, except that at this point, I will add that students need to have the “big” ideas of mathematics emphasised. These are the ideas that are fundamental to mathematical meaning, such as identity, equality, invariance, variability, relation, function, etc. Students need not only to learn the techniques of analysis, calculus, and linear algebra, say, but they need to be deeply conscious of the unifying fundamental principles on which the techniques rest. To this point, I have considered characteristics relating to students’ competences, motivation, attributions, approaches, and their personal relationship with mathematics. These characteristics suggest that first semester courses could be strengthened by the inclusion of efforts to develop students’ study skills as well as their prerequisite mathematical knowledge. A focus on study skills should also include guidance about optimal behaviour in following lectures and note-taking, effective approaches to reading mathematics text, and watching tutorial videos. Also, collaborative group work, modelling, and problem-solving can introduce an essential element of usefulness and sense of application that could support the development of intrinsic motivation. On the one hand, much is taken for granted about students possessing competency in learning mathematics. On the other hand, a lot is known about effective study skills that are often not shared with incoming students. For example, there is a body of research evidence relating to metacognition (Schoenfeld, 1992); metacognition means being conscious of and thereby monitoring and controlling one’s own thinking processes. Metacognition includes reflection, creating, following, reviewing, and revising a plan for learning and problem-solving. It would be a strange engineering project that started with a finished goal in mind but did not initially plan how to reach it. Further, once the plan begins to be executed, it will be continuously under review. If problems or issues arise that interfere with the original plan, the plan will be modified and adapted to align with the new information and conditions as they become known. Why should it be assumed by learners that their work towards achieving a learning goal in mathematics should be any different?

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There are, also, some key teaching actions that can also have a decisive and positive impact on students’ performance. I draw attention to just two here; both are well-supported by evidence.

5.3.6

Transforming Teaching 1: Feedback

First, learners require well-targeted constructive feedback about their work (Black & Wiliam, 1998). Learners need to be encouraged to sustain their effort when challenged and brought to reflect on their work when it is heading in an unproductive direction. Such feedback is not the same as reward and punishment in a behaviouristic sense of operant conditioning. In the context of meaningful learning, feedback is intended to connect with a student’s understanding and will lead the student to metacognitive action. Feedback should avoid telling answers; instead, useful feedback provides sufficient information to the student that will enable the student to make an informed judgement about their progress and direction within a task in addition to facilitating meaningful critical reflection on the task and their thinking. I see it this way: if I try to explain an idea or tell a student what to do, they have to try to make sense of what is going on in my mind. If I ask a student to explain what they are doing or trying to do, they have to make sense of what is going on in their own mind, and if it does not make sense to me, then I can explore by asking more questions. This can be very frustrating for the student because their questions are only ever met by my questions, but my goal is to enable the student to be more aware of their own thinking. Such question-posing must not be confused with the type of questioning that gradually breaks a problem down into a sequence of small steps that lead the student to the correct answer, but the orchestration of the solution remains with the teacher. As Bruner wrote many years ago, “it is a truism that there are very few single or simple adult acts that cannot be performed by a young child. In short, any more highly skilled activity can be decomposed into simpler components, each of which can be carried out by a less skilled operator. What higher skills require is that the component operations be combined” (Bruner, 1964, 2). Competence to provide constructive feedback and the educational context that facilitates constructive feedback should not be assumed. In many situations, constructive feedback is conditional upon identifying the thinking that underlies a student’s response to a task. The pedagogic competence to do this can take a teacher many years to develop out of regular engagement with students, and the expertise of experienced teachers needs to be recognised, rewarded, nurtured, and used. In service-mathematics courses with large student-cohorts, the provision of support and individual feedback to students is often devolved to relatively inexperienced teaching/learning assistants. Although assistants are usually enthusiastic and caring, they are probably selected because of their competence in mathematics, and they will not have built up any form of a repertoire of responses to students’ difficulties.

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Possibly the best feedback comes from the task on which a student is working and concurrently with her/his engagement rather than hours, days, or even weeks later. The feedback that comes from engagement with a task has a neutral emotional valence in that it comes free from any possibility of a personal judgement of competence by another person. Further, the feedback is not loaded with any assumptions about the students’ underlying reasoning. However, it is necessary to be cautious about leaving a student entirely on their own in making sense of mathematics without interacting with a competent teacher in ways, such as questionposing, that I have suggested above. The fallacy of completely independent learning was laid bare in the early 1970s when Erlwanger (1973) reported his encounter with Benny, who had been left to his own devices to learn about fractions using individualised programmed instruction material. Benny had not found any underlying rationale connecting calculations with fractions, and each new calculation entailed learning a new rule that he invented or deduced. One might suggest that Benny had not developed a proper metaconcept of mathematics that prepared him to look for meanings and connections. Erlwanger’s paper has been recognised as making a seminal contribution in the transition from mindless behaviourist notions of learning in education. It is necessary to be very careful that the advent of powerful applications for digital assessment in mathematics does not lead to a return to a form of mindless learning.

5.3.7

Transforming Teaching 2: Active Learning

The importance of active learning has been recognised for some decades (Bonwell & Eison, 1991). Nevertheless, it remains necessary to make a case for transforming teaching to emphasise students’ active role in their learning. Advocates of active learning approaches widely cite the meta-study by Freeman et al. (2014). The metastudy included 255 research papers that considered the impact of active learning approaches in STEM subjects. The meta-analysis revealed that students’ performance was significantly improved when their instructors implemented teaching approaches that motivated students’ active learning. Active learning means cognitively active, which theoretically could be accompanied by physical inactivity. Thus, mathematics teachers who argue for the “chalktalk” approach to mathematics instruction will argue that it is essential that students actively engage cognitively during lectures. They will assert that such active participation is far more critical than the frenetic activity of note-taking that occupies many students in lectures. Some students may have learned successful techniques for cognitive activity in lectures; however, many have not, and for these, teachers must present mathematics in ways that will promote students’ learning activity. There are several teaching approaches described in the literature. I will here only mention them by name to facilitate a search to find out more: inquiry-based mathematics education (see Jaworski’s chapter, this volume), research-based mathematics education, problem-based mathematics (see Mason’s chapter, this volume),

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blended learning approaches, flipped classroom approaches, use of audience response software (i.e. clickers as described in the chapter by Dawkins and Weber in this volume) during lectures, mathematical modelling, projects, and group work, peer tutoring, and peer-based comparative judgement. These approaches and innovations very often diverge more in the way they are theorised than the actions undertaken by students. The common ground is that they are designed to motivate students’ active engagement with the mathematics to be learned, and they facilitate students to exercise control over their learning.

5.4

Conclusion

The title of this reflective essay is taken from MatRIC’s vision statement: Students enjoying transformed and improved learning experiences of mathematics in higher education. The vision is established on a realisation that many students of mathematics need (and deserve) transformed and improved learning experiences. The vision is also based on a belief that it is possible to transform and improve students’ learning experiences of mathematics. The realisation of need and belief in the possibility of remedial action are established on a moral imperative. It is unacceptable to recruit students into programmes in which there is an awareness of structural and cultural conditions that facilitate failure, and that leads students to believe that the reason for poor performance always lies with them. I do not mean to deny that some students must share the burden of responsibility for poor performance in learning mathematics. There is no magic pill that can be swallowed to remove the (pain of11) need for sustained effort and engagement. Success in learning mathematics is the outcome of a partnership between institutions, instructors, and learners. Unfortunately, it seems that often the message communicated to students through the institution and prevailing educational culture of higher education is that success is so far removed from them that it is unattainable. The student is left asking why he/she should make any effort because it is unlikely to pay off. I have focused on what HEMTs might do to improve and transform students’ learning. However, instructors are constrained by the institutional structures and national professional frameworks within which they work. I have shown how, in the case of the Economics Programme at my university, a modest change to the programme has enabled the provision of an effective pre-course in mathematics that ensures students have the prerequisite knowledge to begin the mathematics for the economics course. However, I believe that there is a need for much smaller teaching groups and more mathematically and professionally highly qualified and

11 “Pain of” avoids the mixing of the metaphor of “magic pill”; however, fundamentally I believe that if the engagement is motivated by intrinsic interest, a belief in one’s own self-efficacy, and a deep approach to learning, sustained effort in learning mathematics and grappling with the challenges is not a matter of pain, it is a cause for joy!

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experienced teachers engaging directly with students. The proposal for smaller teaching groups is loaded with financial implications that suggest solutions lie not just at the programme or institutional levels but with national policies that determine institutions’ finances. National policies rightly, in my view, open higher education to be available to a large proportion of the population and not just the elite. Those same policies should accept the cost implications and the moral imperative to use education as a route, not to personal failure as it is for so many now, but to success for all.

5.5

Postscript

I have been challenged to address two questions by readers of early versions of this chapter. The questions might be summarised thus: 1. What have I learned from my experience with MatRIC that would be of value to a local community of higher education mathematics teachers (i.e. a group within a single faculty or institution) who want to transform and improve their students’ experiences of learning mathematics, but without a centre like MatRIC? 2. What advice can I offer new participants to the HEMT- and MER-CoPs to encourage them to build bridges between their communities early in their careers? I will try to address these questions together. I remain convinced that the HEMT-CoP has the intellectual resources, competencies, imagination, and creativity to be able to develop their teaching practices without the involvement of MERs. Consequently, I very strongly encourage HEMTs to develop their “learning communities” and focus on educational transformation. Newcomer-participants in the HEMT-CoP could then be assimilated within this learning community. A local learning community could meet the three conditions I set out in the first paragraph of Part 1 by interpreting “broker” as a mathematics developmental-research focused conference such as RUME in the USA, INDRUM in Europe, or an appropriate working group at the CERME conference.12 Notwithstanding the above claim that the HEMT-CoP has the necessary and sufficient resources to transform practice, I believe there is a role for the MER to act as a kind of “agent provocateur”. There needs to be some form of provocation, an irritant such as the grain of sand that irritates the oyster to create a pearl. MatRIC is a provocateur. In addition, the recurrent poor performance of students, high rates of failure, and drop-out resulting, especially those which result in funding cuts and poor

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RUME: Research in Undergraduate Mathematics Education, a Special Interest Group of the Mathematical Association of America, holds an annual conference, http://sigmaa.maa.org/rume/ Site/News.html INDRUM: International Network for Didactics Research in University Mathematics, https:// indrum2020.sciencesconf.org/ CERME: Congress of the European Society for Research in Mathematics Education (ERME) https://www.mathematik.uni-dortmund.de/~erme/ (INDRUM is a “Topic Conference” or ERME).

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recruitment, can provoke transformation and change. The recent lockdowns caused by the COVID-19 pandemic have been a provocation motivating HEMTs to develop and implement online lectures and learning support and reflect deeply about alternative forms of assessment. It could be that without such repeated provocation, there will be no glowing embers to sustain a quest for transformation of practice. I will illustrate using one of the less successful actions I tried to motivate within MatRIC. The department of mathematical sciences in which I have the privilege to work has evolved over the last three decades. In the early 1990s, the department was composed from groups of mathematicians and mathematics teacher educators that were brought together by the amalgamation of several smaller institutions to form the university’s foundation. National and local circumstances led to the strong development of mathematics education as a research domain. In 2002, the department’s mathematics education research group developed Norway’s first PhD programme in mathematics education. An imbalance of influence and composition has developed in which the mathematics education research group and the mathematics teacher educators significantly outnumber the mathematicians. In an attempt to strengthen the HEMT-CoP independently of the undue influence and voice of the MER-CoP, as a MatRIC action, I initiated “Mathematicians Lunches”. These lunch meetings were set up with two conditions: First, they were intended to be exclusively for the mathematicians, although they could agree to invite a MER for meetings if they wanted. Second, the meetings could occur as often as the mathematicians wanted, but at least 10 min of their meeting time was to focus on teaching mathematics. There would be no monitoring of the discussion. One mathematician was asked to coordinate the meetings, reserve a meeting room, and inform an administrator of the numbers who would attend for placing an order for sandwiches and drinks, for which MatRIC would cover the cost. The organisation was intended to create little demand. The meetings were additionally intended to address an issue raised with me by one of the mathematicians. It was claimed they had no opportunity or common space to meet and talk informally about mathematics and professional matters. The mathematicians gathered for these lunches about once a month for a year or so, and discussions about teaching would often last more than 30 min. Then the meetings ceased. When I inquired about why they had stopped, I was told they had run out of things (about teaching) to talk about. In retrospect, I believe it would have been useful to include a limited MER presence. A similar exercise in a different part of the faculty on another campus has been more successful and enduring because one person in the HEMT community accepted responsibility for arranging lunch meetings regularly, although attendance varies a lot. It should be evident from the foregoing that I consider the involvement of a MER in the HEMT-CoP can be useful, but it is not essential. Not all HEMT-CoPs have MERs that they can invite into their learning community, especially MERs that are concerned with teaching and learning mathematics at the undergraduate level. An HEMT-CoP determined to be a learning community focusing on the transformation of teaching and learning can make use of the conferences named above and other national meetings; also they can refer to journals such as the International Journal of

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Research in Undergraduate Mathematics Education (IJRUME),13 and the journal Teaching Mathematics and its Applications (TMA).14 These conferences and journals provide a valuable bridge between the HEMT- and MER-CoPs. The only ingredient that possibly might need adding is a deep and sustained desire to transform and improve students’ experiences of learning mathematics in higher education.

References Arnold, I. J. M., & Straten, J. T. (2012). Motivation and math skills as determinants of first-year performance in economics. The Journal of Economic Education, 43(1), 33–47. Artemeva, N., & Fox, J. (2011). The writing’s on the board: The global and the local in teaching undergraduate mathematics through chalk talk. Written Communication, 28(4), 345–379. Ausubel, D. P. (1968). Educational psychology: A cognitive view. Holt, Rinehart & Winston. Bandura, A. (1982). Self-efficacy mechanism in human agency. American Psychologist, 37(2), 122–147. Bjørkestøl, K., Borge, I. C., Goodchild, S., Nilsen, H. K., & Tonheim, O. H. M. (2021). Educating to inspire active learning approaches in mathematics in Norwegian universities. Nordic Journal of STEM Education, 5(1) Online https://www.ntnu.no/ojs/index.php/njse/article/view/3972/3 620. https://doi.org/10.5324/njsteme.v5i1.3972 Black, P., & Wiliam, D. (1998). Assessment and classroom learning. Assessment in Education: Principles, Policy & Practice, 5(1), 7–74. Bonwell, C. C., & Eison, J. A. (1991). Active Learning: Creating excitement in the classroom. In Association for the study of higher education; ERIC clearinghouse on higher education. George Washington University, School of Education and Human Development. On-line https://files. eric.ed.gov/fulltext/ED336049.pdf Bruner, J. S. (1964). The course of cognitive growth. American Psychologist, 19(1), 1–15. Cobb, P. (1994). Where is the mind? Constructivist and sociocultural perspectives on mathematical development. Educational Researcher, 23(7), 13–20. Croft, A. C., Grove, M. J., Kyle, J. & Lawson, D. A. (Eds.) (2015) Transitions in undergraduate mathematics education. Online Lulu.com. Deci, E. L., & Ryan, R. M. (2000). Intrinsic and extrinsic motivations: Classic definitions and new directions. Contemporary Educational Psychology, 25, 54–67. Retrieved from http:// selfdeterminationtheory.org/SDT/documents/2000_RyanDeci_IntExtDefs.pdf November 23, 2020 Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61, 103–131. Dweck, C. (1999). Self-theories: Their role in motivation, personality, and development. Essays in social psychology. Psychology Press. Erlwanger, S. H. (1973). Benny’s conceptions of rules and answers in IPI mathematics. Journal of Children’s Mathematical Behavior, 1(2), 7–26. Fennema, E., Carpenter, T., Franke, M., Levi, L., Jacobs, V., & Empson, S. (1996). A longitudinal study of learning to use children’s thinking in mathematics instruction. Journal for Research in Mathematics Education, 27(4), 403–434.

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IJRUME: https://www.springer.com/journal/40753 TMA: https://academic.oup.com/teamat

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Freeman, S., Eddy, S. L. McDonough, M., Smith, M. K., Okoroafor, N., Jordt, H., & Wenderoth, M. P. (2014). Active learning increases student performance in science, engineering, and mathematics. Proceedings of the National Academy of Sciences of the United States of America 111(23) June 10, 2014, 8410–8415. On-line. Retrieved May 14, 2018 from http://www.pnas. org/content/pnas/111/23/8410.full.pdf Goodchild, S. (2001). Students’ goals. Caspar Forlag. Hiebert, J., & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 1–27). Lawrence Erlbaum Associates, Inc. Howson, A. G., & Mellin-Olsen, S. (1986). Social norms and external evaluation. In B. Christiansen, A. G. Howson, & M. Otte (Eds.), Perspectives on mathematics education (pp. 1–48). Reidel. Landgärds, I. M. (2019). Providing economics students opportunities to learn basic mathematics. Nordic Journal of STEM Education, 3(1), 185–189. Lave, J., & Wenger, E. (1991). Situated learning: Legitimate peripheral participation. Cambridge University Press. Lerman, S. (1996). Intersubjectivity in mathematics learning: A challenge to the radical constructivist paradigm? Journal for Research in Mathematics Education, 27(2), 133–150. Lerman, S. (2010). Theories of mathematics education: Is plurality a problem? In B. Sriraman & L. English (Eds.), Theories of mathematics education (pp. 99–109). Springer. LMS. (2010). Mathematics degrees, their teaching and assessment. The London Mathematics Society. Retrieved October 14, 2020 from https://www.lms.ac.uk/sites/lms.ac.uk/files/ Mathematics/Policy_repors/2010%20teaching_position_statement.pdf Lortie, D. (1975). Schoolteacher: A sociological study. University of Chicago Press. Marton, F., & Säljö, R. (1976). On qualitative differences in learning I – Outome and process; II – Outcome as a function of the learner’s conception of the task. British Journal of Educational Psychology, 46(4–11), 115–127. Mellin-Olsen, S. (1987). The politics of mathematics education. Reidel. Nardi, E. (2008). Amongst mathematicians: Teaching and learning mathematics at university level. Springer. NOKUT. (2013). Standards and guidelines for centres and criteria for the assessment of applications. Norwegian Agency for Quality Assurance in Education. Opstad, L., Bonesrønning, H., & Fallan, L. (2017). Tar vi opp de rette studentene ved økonomiskadministrative studier?: En analyse av matematikkbakgrunn og resultater ved NTNU Handelshøyskolen. [Do we enrol the right students on economics-administration studies?: An analysis of mathematics background and results at NTNU Business School]. Samfunnsøkonomen, 1, 21–29. Postareff, L., Lindblom-Ylänne, S., & Nevgi, A. (2008). A follow-up study of the effect of pedagogical training on teaching in higher education. Higher Education, 56, 29–43. Rasmussen, C., Keene, K. A., Dunmyre, J., & Fortune, N. (2018). Inquiry oriented differential equations: Course materials. Available at https://iode.wordpress.ncsu.edu. This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. Richardson, K. (1998). Models of cognitive development. Psychology Press. Schoenfeld, A. (1992). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics. In D. A. Grouws (Ed.), Handbook of research in mathematics teaching and learning (pp. 334–370). MacMillan/National Council of Teachers of Mathematics. Skemp, R. R. (1976). Relational understanding and instrumental understanding. Mathematics Teaching, 77, 20–26. Skemp, R. R. (1982). Communicating mathematics: Surface structures and deep structures. Visible Language, 16(3), 281–288.

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Steen, L. A. (1999). Review: Theories that Gyre and Gimble in the Wabe reviewed work(s): Mathematics education as a research domain: A search for identity by Anna Sierpinska; Jeremy Kilpatrick. Journal for Research in Mathematics Education, 30(2), 235–241. Stein, M. K., Grover, B. W., & Henningsen, M. (1996). Building student capacity for mathematical thinking and reasoning: An analysis of mathematical tasks used in reform classrooms. American Educational Research Journal, 33(2), 455–488. Vygotsky, L. S. (1978). Mind in society. Harvard University Press. Wenger, E. (1998). Communities of practice. University of Cambridge Press. Wenger, E., McDermott, R., & Snyder, W. M. (2002). Cultivating communities of practice: A guide to managing knowledge. Harvard Business School Press. Zakariya, Y. F. (2021a). Undergraduate students’ performance in mathematics: Individual and combined effects of approaches to learning, self-efficacy, and prior mathematics knowledge. Doctoral Dissertation, University of Agder. Zakariya, Y. F. (2021b). Self-efficacy between previous and current mathematics performance of undergraduate students: An instrumental variable approach to exposing a causal relationship. Frontiers in Psychology, Section Educational Psychology, 11, 556607. https://doi.org/10.3389/ fpsyg.2020.556607

Chapter 6

Identifying Minimally Invasive Active Classroom Activities to Be Developed in Partnership with Mathematicians Paul Christian Dawkins and Keith Weber

Abstract To improve tertiary mathematics education, researchers in undergraduate mathematics education often use the following approach. The researchers first design student-centered instruction that privileges active learning and demonstrate that this instruction leads to greater learning outcomes. They then persuade mathematicians to use this innovative instruction in their classroom. In this chapter, we question this dominant approach to instructional innovation by considering both the low uptake of existing, high-quality products and the rational and reasonable commitments that keep many mathematicians lecturing during most of their class time. This chapter proposes alternative models of mathematics education research and intervention that might support more incremental improvements to instruction that could be taken up on a wider scale. The literature already provides precedents and examples of such minimally invasive modifications of lecture. This model of mathematics education research and innovation entails a more respectful view of mathematician instructors, whose commitments to student learning lead them to be committed to lecture instruction. Keywords Teaching innovations · Interactive lecture · Teacher goals

6.1

Introduction

Consider the following five premises about the teaching of undergraduate mathematics.

P. C. Dawkins (✉) Department of Mathematics, Texas State University, San Marcos, TX, USA e-mail: [email protected] K. Weber Graduate School of Education, Rutgers University, New Brunswick, NJ, USA e-mail: [email protected] © Springer Nature Switzerland AG 2023 S. Stewart (ed.), Mathematicians’ Reflections on Teaching, Advances in Mathematics Education, https://doi.org/10.1007/978-3-031-34295-0_6

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(i) Most advanced mathematics courses are taught via lecture. (ii) The typical student learns less than they are capable of in their advanced mathematics courses. (iii) These undesirable learning outcomes are due, at least in part, to the way that advanced mathematics courses are taught. (We cannot simply ascribe students’ undesirable learning outcomes to inadequate mathematical preparation or a poor work ethic. The teachers themselves bear some of the blame.) (iv) If we hope to improve students’ learning in advanced mathematics, we need to substantially change the way advanced mathematics is taught. If we continue to teach by lecture, we will continue to see poor learning outcomes. (v) If mathematics educators can develop effective instructional pedagogies that differ substantially from the lecture format and disseminate these pedagogies to the mathematicians who teach advanced mathematics, this might lead to better student learning outcomes at scale. We believe that many researchers in undergraduate mathematics education would agree with these premises. We further maintain that these premises, especially premises (iv) and (v), have had a significant impact on how researchers in undergraduate mathematics education conduct research. Indeed, large amounts of effort and time are given to premise (v) as it constitutes a primary hope for how we can improve students’ learning outcomes at scale. For instance, we are aware of many scholars who have advocated for inquirybased instructional approaches (e.g., Laursen & Rasmussen, 2019) and who have developed curricula in which lecturing plays a limited role (e.g., Larsen, 2013; Leron & Dubinsky, 1995; Swinyard & Larsen, 2012). However, we are aware of few scholars who focus on how lecturing in advanced mathematics can be improved (see Gabel & Dreyfus, 2017, for a notable exception). Like most mathematics educators, we fully agree with premises (i), (ii), and (iii).1 However, in this chapter, we will challenge premises (iv) and (v). To motivate the ideas that will follow, let us consider one of the most well-known research programs in undergraduate mathematics education: Leron and Dubinsky’s (1995) Learning Abstract Algebra with ISETL, an abstract algebra curriculum, which is based on collaborative learning and computer programming. In an article in the American Mathematical Monthly, a widely respected expository journal for mathematicians, Leron and Dubinsky baldly asserted that “the teaching of abstract algebra is a disaster” (p. 227). Leron and Dubinsky declared that students’ poor learning outcomes are due to mathematicians holding “a too narrowly conceived view of instruction” (p. 227) and their consequent reliance on lecturing. Mathematicians’ reliance on lecturing is misguided because “telling students about mathematical process, objects, and relations is not sufficient to induce meaningful learning

1 Indeed, rejecting premise (ii) or rejecting premise (iii) would imply that we cannot hope to improve students’ learning outcomes by changing instruction, which in turn would suggest that undergraduate mathematics education has, at best, a limited role to play in the teaching of advanced mathematics.

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(hence the sorry state of affairs even with the best of lecturers” (p. 241, italics are the authors’ emphasis). But there is hope: “replacing the lecture method with constructive, interactive methods involving computer activities and cooperative learning, can change radically the amount of meaningful learning achieved by average student” (p. 227). We see in these quotations strong agreement with the five premises above. In particular, lectures cannot lead to good learning outcomes, but a very different set of constructive interactive methods can. Ed Dubinsky and his colleagues did substantial work in designing their curriculum. They developed a course textbook (Dubinsky & Leron, 1994), conducted studies demonstrating that students who were taught in a traditional manner developed impoverished understandings of basic group theory concepts, but students who were taught with Dubinsky’s instruction developed more sophisticated understandings, published these studies in highly respected mathematics education outlets (e.g., Asiala et al., 1997; Brown et al., 1997; Dubinsky et al., 1994), and advertised their curriculum to mathematicians in the American Mathematical Monthly. Dubinsky and his colleagues’ work was invaluable to mathematics educators, providing foundational research about the nature of concepts in group theory and how students can come to understand them. Nonetheless, this impressive work has had only a modest impact in how abstract algebra is taught. In a recent survey with 131 algebraists, Johnson, Keller, and Fukawa-Connelly (2018) found only a single algebraist was using the curriculum. A similar story can be told about Sean Larsen’s innovative Teaching Abstract Algebra for Understanding curriculum (sometimes called Inquiry-Oriented Abstract Algebra). Larsen and his colleagues carefully documented how students using their curriculum developed impressive understandings of difficult concepts and reported these results in respected outlets (e.g., Larsen et al., 2013), presented their findings at national mathematics conferences, and even developed a sophisticated system of instructional supports for mathematicians who might want to adopt the curriculum (Lockwood et al., 2013). Nonetheless, only one of the algebraists in Johnson et al.’s (2018) survey was using Larsen’s curriculum. To avoid misinterpretation, the intent of the previous paragraph is decidedly not to criticize the work of Dubinsky’s team and Larsen’s team, which we think is excellent. Rather, we note the lack of uptake of their curricula to call attention to a model that is implicit in undergraduate mathematics education research. The standard model for improving instruction in advanced mathematics is roughly this: (a) Mathematics educators note a problem. (b) Mathematics educators design non-lecture-based instruction to fix the problem. (c) Mathematics educators persuade mathematicians to use their instruction (in research reports, expository articles, workshops, etc.). (d) Mathematicians use the instruction, with mathematics educators providing scaffolding as needed. (e) The problem is ameliorated. We offer two reasons why we think this approach is problematic. The first is the role of the mathematicians. We agree with Alcock (2010), who argued that (i) undergraduate mathematics educators need to be aware of mathematicians’

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most pressing concerns and they need to take these concerns seriously; and (ii) that many mathematicians, based on their decades of teaching experience, have pedagogical knowledge that mathematics educators should draw upon and capitalize. Where in the model above, do we see attention to mathematicians’ most pressing concerns or seek their input into how we should develop instruction? The mathematicians’ input into the process is minimal, primarily relegated to providing input as to what scaffolding they may need. The second problem is with the viability of step (c). If mathematicians aren’t willing to use the documented high-quality curricula that Dubinsky and Larsen developed, perhaps undergraduate mathematics educators should reconsider whether their model for improving instruction is viable. In this chapter, we propose another model with more mathematician involvement to improve pedagogy in advanced mathematics at scale. We also highlight some alternative formats for classroom innovations that could be adapted more easily and independently by mathematicians.

6.2 6.2.1

Relevant Literature What Do We Know About Lecturing in Advanced Mathematics?

In Johnson et al.’s (2018) survey, the authors found that 85% of their algebraists claimed to teach by lecture. These survey results, as well as many observations of mathematicians’ teaching (e.g., Artameva & Fox, 2011; Fukawa-Connelly et al., 2017), strongly suggest that premise (i) above is correct: Most advanced mathematics courses are indeed taught by lecture. However, lecturing is more sophisticated than some mathematics educators think. A comprehensive review of this literature is presented in Melhuish et al. (2022), but for the sake of brevity, we constrain ourselves to several relevant findings from the research literature that we present below. First, mathematicians are thoughtful about how they lecture in advanced mathematics. Mathematicians are intentional in their pedagogical actions, and they do what they do for what they believe are good reasons (e.g., Gabel & Dreyfus, 2017; Lew et al., 2016; Weber, 2004). Many mathematicians care deeply about their teaching. Second, there is considerable intersection between mathematicians’ pedagogical goals and mathematics educators’ pedagogical goals (e.g., Alcock, 2010; Nardi, 2007; Weber, 2012). Third, mathematicians make an active effort in their lectures to highlight the types of understandings and activities that mathematics educators think are important (e.g., Fukawa-Connelly et al., 2017). For instance, mathematicians provide informal ways of thinking about mathematical concepts, exemplify their concepts, and explicitly model productive mathematical dispositions.

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Importantly, mathematicians who identify as lecturers do not only lecture. Johnson et al. (2018) reported that self-identified lectures collectively reported spending about 75% of their time lecturing. Many respondents also reported doing some types of student-centered activities, such as asking students questions, having students explain their reasoning, having students collaboratively to solve problems, and holding whole class discussion, at least some of the time. Our final point about lecturing in advanced mathematics concerns lecturers’ use of questioning. Mathematicians believe questioning as a crucial method to increase student engagement (Woods & Weber, 2020). Paoletti et al. (2018) found that questioning in lectures in advanced mathematics was common. However, Paoletti et al. reported that mathematicians’ questions typically asked students to recall factual information (rather than asking student about the meaning of concepts or why statements were true), and the lecturers provided little wait time for students to formulate an answer. While the questioning practices described in Paoletti et al. (2018) are not sufficient to increase engagement, perhaps different questioning practices could be. None of the points that we raise vindicates mathematicians’ decisions on how they teach. Numerous studies demonstrate that students’ learning outcomes in mathematics are indeed poor; most students complete their advanced mathematics courses with a limited understanding of the core concepts covered in the course (e.g., Asiala et al., 1997; Rasmussen & Wawro, 2017) and lacking the ability to read and write proofs effectively (e.g., Mejia-Ramos & Weber, 2019; Stylianides, Stylianides, & Weber, 2017). These results strongly suggest that premise (ii)—that students learn less advanced mathematics than we would like—is correct. Further, researchers have shown that students miss the main points of lectures, even when these lectures are of high quality (Gabel, 2019; Krupnik et al., 2018; Lew et al., 2016). Researchers have also identified causal mechanisms for why students struggle to grasp the points that lecturers are trying to convey, such as inefficient note-taking practices, not paying attention to the most important parts of the lecture (e.g., Fukawa et al., 2017; Lew et al., 2016), and a lack of opportunity for meaningful reflection (e.g., Paoletti et al., 2018). These results suggest that premise (iii), that students’ poor learning outcomes in advanced mathematics are due at least in part to the instruction that they receive, is correct. However, on a positive note, mathematicians are amenable to incorporating activities in their lecture that increase student engagement. Indeed, mathematicians may even value these activities.

6.2.2

Why Do Mathematicians Choose to Lecture?

In a recent study, Woods and Weber (2020) asked eight mathematicians for their reflections on common teaching practices in advanced mathematics. Based on this analysis, Woods and Weber identified several goals that the mathematicians in their sample possessed that we hypothesize are common in the mathematical community. Woods and Weber argued that one reason that mathematicians choose to lecture is

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because lecturing is consistent with their goals. We extend Woods and Weber’s analysis and argue below that many forms of instruction are inconsistent with some of the goals that Woods and Weber reported. We consider five goals below: Engagement Mathematicians want students to be engaged in their advanced mathematics classes and they acknowledge that lecturing is not an engaging form of instruction. They address this limitation by supplementing lecturing with other activities, such as asking questions, but they still choose to lecture because lecturing satisfies other goals beyond engagement, which we discuss below. Coverage of Content A common goal of mathematicians is to cover the content in the course. Indeed, mathematicians frequently cite lecture as the only viable way for them to cover all the content in an advanced mathematical course. Ordinarily reformoriented curricula in advanced mathematics substantially reduce the number of topics that are covered. For instance, Dubinsky and his colleagues’ description of their instruction ends with the concepts of normality and quotient groups. Their treatment ignores topics in abstract algebra that many algebraists will feel are essential for an abstract algebra course (e.g., homomorphisms and the first isomorphism theorem). If a reform-oriented curriculum omits concepts that mathematicians believe are central to a theory, mathematicians are unlikely to use that curriculum. Preparing Future Mathematicians Many mathematicians view a central goal of their advanced mathematics course as preparing future mathematicians. In particular, mathematicians believe a key requirement of their teaching is to prepare their strongest students to gain entry to, and succeed in, graduate programs in mathematics. Woods and Weber found that although some mathematicians observed that the lecturing that they used might not work for all students, these mathematicians felt that lecturing was sufficient to prepare their strongest students to be future mathematicians. Mathematics educators frequently advocate for abandoning lecture and adopting reform-oriented instruction on the grounds that reform-oriented instruction is better for the typical student or even the struggling student.2 Often, little to no attention is paid to the strong student who might go on to study mathematics.3 Mathematics educators have very good reasons to focus on the average student and the struggling student,4 but their focus ignores a population that is of central interest

Consider Leron and Dubinsky’s remark about lecturing: “Experience, theory and research all point to the fact that verbal explanations that do not relate to the student’s prior experience are quite ineffective (except for a few individuals with special talent in mathematics.)” (p. 231). To some mathematicians, the special students dismissed in the parenthetical remark are the most important population in the class. 3 An important exception is some types of IBL instruction like the Modified Moore Method, whose proponents claim, are particularly effective at preparing future mathematicians. It is interesting to note that these curricula have been adopted by a sizeable collection of mathematicians. 4 Mathematics educators are more likely than mathematicians to value equity and teacher preparation. Valuing equity often leads to an emphasis on the opportunities to learn for all students, and it may be implicitly assumed that opportunities to learn for students who struggle constitute opportunities to learn for all. The needs of preservice teachers in advanced mathematics courses are 2

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to many mathematicians. If mathematicians are not convinced that reform-oriented curricula will not help their strongest students become future mathematicians, many will be reluctant to adopt those curricula. Precision and Language Some mathematicians value mathematical precision. Indeed, some mathematicians insist on writing definitions, propositions, and proofs on the board, fearing that if they merely say these things aloud, students will write them down incorrectly and leave the class with misconceptions. Indeed, in our (non-research-based) experience, we found that some mathematicians are deeply concerned that inaccuracies in language and notation can lead students to develop pernicious misconceptions. Other mathematicians view it as their professional pedagogical obligation to avoid mathematical inaccuracies and to correct mathematical inaccuracies when they arrive. A virtue of lecturing is that it allows the mathematician to carefully control the language and the notation in the course. This is exactly the opposite perspective of many of the reform-oriented curricula that mathematics educators develop. Reform-oriented curricula, particularly those developed with a Realistic Mathematics Education framework (Van den HuevelPanhuizen & Drijvers, 2014) such as Larsen’s Teaching Abstract Algebra For Understanding curriculum, explicitly invite students to invent their own notation and have a high tolerance for inaccuracy, at least as students are first learning the material. To be clear, we personally support Larsen’s (2013) contention that a premature focus on logical precision can be a barrier to learning and that his curriculum works well because students learn to use standard notations by the end of his course (Kuster et al., 2018). Nonetheless, we can see how such curricula might not be a palatable option for mathematicians who insist on precision at all times. Efficiency When mathematicians have to choose how to teach, they tend to satisfice (in the sense of Simon, 1972). That is, rather than exerting considerable time and cognitive resources in trying to find the best form of pedagogy, many mathematicians will select the first acceptable form of instruction that they encounter (“acceptable” meaning meeting a minimal threshold for achieving their goals). Using reform-oriented curricula often requires mathematicians to make substantial effort to develop a skill set that they do not possess where the promised learning gains are not guaranteed. Many mathematicians prefer the tried-and-true method of lecturing on the grounds that it has been successful enough for generations. To avoid misinterpretation, in describing mathematicians’ goals and values, we are not endorsing these goals and values. We acknowledge that mathematics educators have good reasons to object to these goals and values or to place others above them. Nonetheless, these goals and values are not prima facia wrong. We would not accuse a mathematician who espoused these goals and values to be irrational or arguing/teaching in bad faith. As we see it, if we want mathematicians to use reform

different since they are unlikely to proceed to a graduate program in pure mathematics and instead may benefit by learning the connections between advanced content and high school content (e.g., Wasserman et al., 2017).

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curricula, we have two options. Either we can attempt to persuade mathematicians to replace the goals and values that they currently hold with ones that mathematics educators endorse, or we can attempt to design instruction that is compatible with the goals and values that mathematicians currently hold. Johnson et al. (2013) documented at least one mathematician who used Larsen’s Abstract Algebra curriculum because of his deep dissatisfaction with lecture instruction, showing that there are some mathematicians who share goals and orientations more consistently with those of mathematics educators. However, mathematics educators should not restrict our focus to that subset of mathematicians. Changing the broader mathematical community’s goals and beliefs may be possible, but it is a daunting task. In the next section, we describe a model that approaches the problem from the other direction: designing instructional reforms that are more compatible with the way that mathematicians currently lecture (and their reasons for doing so).

6.3

An Alternative Model for Mathematics Education Innovation

Based on our synthesis of the research literature, we believe that mathematicians and mathematics educators would agree on the following instructional principle: Students need opportunities to reflect on central ideas and understandings in their advanced mathematics courses.

Mathematicians evince this principle by discussing important understandings in their classrooms and asking students questions about central ideas. However, too often this fails to lead to meaningful learning because students ignore the mathematicians’ discussion in this regard, and mathematicians’ questioning practices are not suited to lead to deep engagement. Mathematics educators evince this principle by having students engage in inquiry-oriented activity in which students construct and/or grapple with central ideas and understandings. But such instruction is not viable to many mathematicians because it is at variance with other instructional goals that they have. In this section, we outline a different model for how mathematics educators can partner with mathematicians to develop more accessible instructional innovations that may be consistent with mathematicians’ practices and goals. We further illustrate with some examples and discuss the benefits of this alternative model. I. Clear identification of learning goals in cooperation/negotiation with mathematicians. The research literature suggests that there is a considerable overlap between mathematicians’ and mathematics educators’ goals in advanced mathematics. For instance, both groups of scholars want students to be actively engaged in their mathematics classes, have a conceptual understanding of the material that they are studying, and hold an appreciation for, understanding about, and an ability to

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produce mathematical proofs. Nonetheless, a key challenge to true partnership between mathematicians and mathematics educators is the fact that mathematics educators have often adopted or emphasized a different set of learning goals (e.g., students’ classroom experiences predominantly consist of engaging them in authentic mathematical activity) that lead them away from teacher-led presentations. As noted above, many other mathematicians have viable goals that mathematics educators can and should respect. Instead, our instructional innovations may yield greater use if they focus on learning goals that are identified in cooperation with mathematicians. II. Identify methods of instruction that are lecture-friendly (or “minimally invasive”) and promote these learning goals. There are many such methods that have been identified and tested in the educational psychology literature outside of mathematics that are appropriate for this purpose: clicker questions, partial notes, and electronic applets for homework. As we will argue below, findings about common student ways of thinking identified by mathematics education research can be effectively leveraged for use in these types of interchanges. Here we address one potential rebuttal that a mathematics educator might offer: the types of “minimally invasive” instructional interventions that educational psychologists offer are not sufficiently nuanced to lead to meaningful, deep, or conceptual learning. Here we offer Hodds et al. (2014) study on proof comprehension as a counterexample to this rebuttal. In their study, Hodds et al. (2014) used Chi et al.’s (1994) self-explanation training to improve the ways that mathematics undergraduates read proofs. The authors assessment measure was based on Mejia-Ramos et al.’s (2012) model of proof comprehension and included questions that asked students to justify why steps in a proof were true, evaluate summaries of the proof, transfer the methods of the proof to a new domain, and explain how the proof worked in the context of specific examples. In short, the simple method of self-explanation led to improvement on a conceptually based, nuanced assessment. Whether this would work for other “minimally invasive” interventions is an open empirical question, but we do not think it is one that mathematics educators should dismiss out of hand. III. Mathematics educators develop instruments and tools with feedback and guidance from mathematicians. We imagine that both mathematics educators and mathematicians play an important role in designing “minimally invasive” activities that lead students to focus on central understandings. Mathematics educators have two resources for designing tasks about central understandings. First, they have engaged in theoretical analyses about what it means to really understand key concepts (see, for instance, Dubinsky et al., 1994). Second, as a discipline, we have a large corpus of literature in which students are interviewed about their understanding, so we have a robust collection of productive ways of thinking about key concepts as well as unproductive understandings that many students hold and epistemological obstacles that students need help to overcome. Finally, and not trivially, mathematics educators have the time to generate

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such tasks, as item generation can serve as part of their research work. Such work does not contribute directly to mathematicians’ research. We believe mathematicians have an important role to play here as well. While mathematics educators are perfectly capable of identifying core ideas in the context of a specific course such as introductory abstract algebra, mathematicians are better equipped to identify core ideas in the broader discipline being discussed, such as what constitutes “algebraic thinking.”5 Further, mathematicians can identify mathematical ambiguities and inaccuracies in the tasks that mathematics educators develop. Finally, many mathematicians are quite clever and creative in generating tasks in their own right. IV. Mathematics educators test and refine the developed interventions and measures. One example of mathematician and mathematics educator collaborations that the first author participated in was a workshop hosted by the American Institute of Mathematics in December of 2015. This workshop brought together practitioners of IBL instruction with mathematics educators interested to codevelop agendas of research on IBL (https://aimath.org/pastworkshops/iblanalysisrep.pdf). This workshop did not follow the model we describe here because members of the mathematics community developed the instructional innovation over the past few decades. Nevertheless, two patterns that became clear over the course of the workshop were that (1) mathematics education research gave name to many ideas and practices that the mathematicians used regularly and (2) the mathematicians needed much guidance in formulating research questions that could be adequately studied. They were often convinced of the efficacy of their method and needed to be cautioned to apply scientific skepticism to any research thereupon. This anecdote emphasizes that mathematics educators bring a number of different kinds of expertise to such partnerships beyond just methodology and knowledge of literature. V. Mathematicians and mathematics educators negotiate how the innovations can be used and disseminated. We are advocating two key changes in the way that mathematics educators disseminate their instructional innovations to mathematicians. The first involves flexibility. Mathematics educators often provide fixed and highly structured curricula to mathematicians. One alternative approach is for mathematics education research to develop heuristics for opening the classroom rather than prepackaged curricula. This approach opens the possibility that such products could become generative for instructors who use those tools (i.e., they could adapt them in different ways in different classes), and such adaptation could be the object of future study.

5

In the mathematics teacher education parlance, mathematicians can capitalize on their greater “horizon knowledge” (Ball et al., 2008) to document “profound understanding of fundamental (advanced) mathematics” (Ma, 1999).

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Tools such as clicker questions, partial notes, and self-explanation training can all be adapted in various ways by end users. The second is how mathematicians’ unsuccessful use of a curriculum or their refusal to use a curriculum can be interpreted. In the past, some mathematics educators have not viewed this as a shortcoming of the curriculum. Rather, they viewed this as an impetus for developing more persuasive arguments for why their curriculum should be used or designing better instructional supports to facilitate the effective implementation of their curriculum. Occasionally, this has generated animosity against mathematicians for being too irrational, obstinate, or selfish to acknowledge the limitations of lecture and adopt the innovative curriculum. What we suggest is that in a true collaboration, mathematicians’ inefficient use of the curriculum or their refusal to use a curriculum could also be interpreted as a fault of the curriculum and an impetus to change the curriculum, so it is more palatable to the mathematicians who are to use it.

6.3.1

A Partial Example of the Model from Previous Work

In a previous work, the second author and his colleague designed multiple-choice proof comprehension tests. We summarize this work to illustrate the aforementioned Stages I and II of our model. This example shows both how collaboration with mathematicians can proceed and the benefits of engaging mathematicians in our mathematics education work. Our motivation for developing these tests is that proof comprehension is a topic that mathematicians have identified as important (Cowen, 1991; Conradie & Firth, 2000; Weber, 2012), assessments are a common tool to promote and assess learning (Resnick & Resnick, 1992), and the use of multiple-choice tests is compatible with the lecture format (e.g., they can be used during exams or outside of class as homework). We designed the tests so that by completing the tests, students are led to reflect upon key aspects of proof that mathematics educators and mathematicians have identified as important. A summary of the research methodology to develop these tests is described in Mejia-Ramos et al. (2017). Stage I We identified the dimensions of our proof comprehension model based on interviews with mathematicians about what they thought it meant to understand a proof (Weber & Mejia-Ramos, 2011) and what they wanted students to understand when they presented a proof (Weber, 2012). If a particular aspect of understanding was highlighted as important both by the mathematicians in our interviews and by mathematics educators in the research literature, we included it in our proof comprehension model (Mejia-Ramos et al., 2012). By choosing to focus on the intersection of mathematicians’ and mathematics educators’ learning goals, we ignored some aspects of proof comprehension that were important to each group of scholars. Nonetheless, we identified a collection of understandings that were sufficiently rich and robust to be accepted by both communities. (This capitalizes on the fact that

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although mathematicians and mathematics educators may have or prioritize different learning goals, there is enough intersection to have a true collaboration). Stage II In generating questions and multiple-choice items, we used the proof comprehension assessment model that we developed to identify broad categories of types of questions. To create actual questions from each category, we made use of extensive empirical work with students, both from interviews that we conducted with students and from common unproductive student understandings reported in the research literature. However, we also received feedback from mathematicians in three ways. First, we conducted workshops with mathematicians at conferences in which they helped us to generate items.6 These workshops were invaluable. In addition to specific comprehension questions that mathematicians developed for particular proofs, we identified schemes for generating questions that could be applied to any proof. Second, as part of several National Science Foundation grants for developing these multiple-choice tasks, we included a mathematician, Hyman Bass, on our Advisory Board. Bass provided several types of feedback, including identifying subtle inaccuracies and ambiguities in the items that we developed and highlighting valuable insights that could be gleaned from the proof that we did not assess. Third, prior to testing our tests with students, we interviewed mathematicians while they completed the tests, removing any items that they felt were problematic, did not assess useful understandings, or answered incorrectly.7 In summary, we see that including mathematicians in the generation of our questions provided us with useful heuristics for generating questions, removed inaccuracies, increased precision, and increased the construct validity of our questions. We also believe the process made the questions more appealing to other mathematicians. Stage III and IV Stage III, testing and refining the items, is not germane to this chapter, but is described in Mejia-Ramos et al. (2017). We have not undertaken Stage IV yet, but as we see it, it would not be sufficient for us to simply provide the tests to mathematicians and tell them how they should be used. Rather, we would need to work with mathematicians and refine our tests in a manner that mathematicians would find them to be viable with their instruction.

6

For instance, we conducted a special RUME workshop in the 2012 Mathfest meeting in Madison, WI. 7 If a professional mathematician did not answer an item incorrectly, this is a warning that the item lacks construct validity.

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6.3.2

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A Proof of Concept: Clicker Questions

In our previous research, we observe that even in highly interactive lecture settings in which the professor guides the discussion, students may still experience the learning processes and benefits commonly associated with inquiry-driven instruction (e.g., Dawkins 2012, 2014). We mentioned above three accessible examples of lessinvasive interventions that could be developed to open lecture-format classrooms: clicker questions, partial notes, and self-explanation training. Let us first consider how the use of clickers might augment instruction in advanced mathematics. While different scholars use clickers in different ways, they are commonly used in the following manner. At various points in a lecture, the lecturer presents students with a multiple-choice conceptual question. Students are given one minute to think about, and then answer, the question, which they register using a clicker. They are then given a minute to discuss the question with their peer, after which they again register their (possibly revised) answer with a clicker. The lecturer can then see what percentage of students answered the question correctly, both individually and after peer interaction. This type of intervention has many desirable features. Students are provided time to engage with the conceptual content that the instructor believes is important. Students get safe and immediate feedback on their comprehension. They have the chance to discuss ideas with their peers. Both teachers and students gain waypoints where they can check students’ current understandings. If implemented well, they also provide students with the time they need to think about difficult class concepts with a focus on ideas that the instructor views as most important. Studies outside of mathematics education have demonstrated the general efficacy of this type of instruction (e.g., Crouch & Mazur, 2001). We also believe that such a “minimally invasive” intervention will be seen as viable by mathematicians. They can continue to lecture, and this is consistent with their practice of using questioning to increase student engagement (Paoletti et al., 2018; Woods & Weber, 2020). From cognitive research and mathematician input, mathematics educators could develop banks of clicker questions for use. Much previous work in the literature (e.g., Shipman 2012, 2013) lends itself to this kind of use quite naturally. As a plausible illustration of how we can utilize the existing mathematics education research, consider Roh’s work on students’ understanding of limit. Roh (2010) described a sequence of thoughtful activities in which students use a physical manipulative (an ε-strip) and graphical computer software to explore whether various sequences converge. The purpose of these activities is for students to build robust understandings of limits. For the reasons cited in this chapter, we do not believe that most mathematicians would be willing to adopt this method of instruction, as this requires extensive time for student investigations (thereby limiting content coverage) and deviates from lecture-based instruction in important ways, such as requiring computers and manipulatives (thereby adding substantial work for the lecturers). However, we think that Roh’s excellent work in implementing her instructional technique, including her analyses of students’ response to her ε-strip

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activities, can be valuable in generating conceptual questions that can be used for clicker questions. For example, consider the following question:

Which explanation best summarizes what is displayed in the picture above? (A) 1 is one of the limits of the sequence because it is eventually contained in the ε-strip centered at 1. (B) 1 is the limit with this epsilon strip because after 5, all of the terms are inside the ε-strip. (C) 1 is not one of the limits because the first two terms of the subsequence fall outside of the ε-strip. (D) 1 is not a limit because infinitely many terms are outside of the ε-strip. The question itself was one of Roh’s tasks. The multiple-choice foils (a, b, and c) were generated based on Roh’s (2008) analyses of common stable unproductive ways of thinking that students use to understand the limit concept. Of course, it would be against the themes of the chapter to use this specific clicker question without consultation with mathematicians. In our own class, we have streamlined Roh’s task to the following pair of true-false prompts: A sequence converges to L whenever every epsilon of L contains infinitely many terms of the sequence. A sequence converges to L whenever for every epsilon neighborhood of L, there are only finitely many terms of the sequence outside the neighborhood.

What we hope this illustrates is how we can capitalize on mathematics education research to augment lectures in “minimally invasive” ways that have a real chance to improve student comprehension. Another efficient way to create true-false statements that help students reflect on key concepts is to present upcoming theorems alongside closely related variants. For instance: Given any sequence, if it converges, then it is bounded. Given any sequence, if it is bounded, then it converges. Given any monotone sequence, if it is unbounded, then it diverges. Given any monotone sequence, if it is bounded, then it converges.

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These sequences help students explore the definitions and example spaces while demonstrating how theorems depend on each particular hypothesis. We often use such tasks before we prove the statement that is a theorem. True-false prompts can also be easily produced from theorem statements by switching quantifiers (8∃ to ∃8) or the entities quantified (8x 2 D ∃ y 2 f(D) to (8y 2 f(D) ∃ x 2 D).

6.3.3

Another Proof of Concept: Partial Notes

Research reveals that mathematicians often convey important information in advanced mathematics lectures orally, including heuristics for solving problems and informal ways to represent mathematical concepts. Research also shows that students do not write this content in their notes, because they are focusing on the formal mathematics written on the blackboard and because they do not recognize the formal content as important (e.g., Fukawa-Connelly et al., 2017; Lew et al., 2016). A technique for addressing this issue from the educational psychology literature is the use of partial notes. Partial notes are based on the idea that the main ideas of the lecture are present in notes handed out to students, but there are gaps in the notes where students must fill in particularly important information. Again, outside of mathematics education, there are studies demonstrating the efficacy of this strategy (see Putnam et al., 2016), and an important exploratory study in advanced mathematics suggests that this method could improve comprehension there as well (Iannone & Miller, 2019). The strategy for developing these notes would benefit greatly from collaboration with mathematicians to learn what precisely they hope students will learn from their lectures. This strategy has the benefit that faculty still control the pace of the course, and they still maintain mathematical accuracy and precision in everything they present to students. However, since the students have the formal mathematics given to them in advance, they will be directed to focus on the mathematics that mathematicians identify as most important while they complete their partial notes.

6.4

Summary

We recognize that what we are proposing is in some ways a radical departure from the traditions of mathematics education intervention in instruction. Our proposed changes are informed by the recent research on mathematicians’ instructional practices and motivations. One can interpret our proposals in analogy with the constructivist revolution in mathematics education that occurred decades ago. Research into student thinking shifted mathematics educators from viewing student ideas as mistakes toward seeing that students are often rational and motivated in their (mathematically incorrect) thinking. This led to a revolution in how we engaged students in the classroom and gave credence to their ways of thinking. More recent research in

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collaboration with mathematicians is similarly working to help us see them as rational and motivated in their instructional thinking and practice. We anticipate that this can similarly lead to a revolution in how we engage with them as thoughtful colleagues and practitioners. As such, we propose the above models to give them rightful space within the practices of mathematics education research as well as to give rightful respect to their goals and motivations for their classrooms. Mathematicians largely lecture for rational and valuable reasons, so in addition to our more radical innovations, we can work to help make lectures as productive as possible. We also observe that some of the example interventions we propose may be less attractive to mathematics educators because they are not rooted in mathematical thinking, but rather are imported from other fields (ourselves included, as demonstrated in Dawkins & Karunakaran, 2016). In a recent review of mathematics education research over the last decades, Inglis and Foster (2018) demonstrate how experimental methods have fallen out of practice in mathematics education. Our proposals, in addition to being ripe for partnerships with mathematicians, have been tested using experimental methods that are (1) less theory-driven than much other work in our field, (2) more defensible to colleagues who do not share our presuppositions, and (3) have been unduly ignored for too long in our community. We believe that our model has other benefits as well. First, more consistently inviting the partnership and input of mathematicians will not only make our products more palatable and useful to them but also help us produce better products in the first place. Second, it will naturally sensitize us to their goals and orientations, which until now have largely been ignored in mathematics educators’ innovative curricula. A recurring conversation that we have with our mathematician colleagues relates to why mathematics education research in many cases does not answer the questions they want to ask. Our research will always be oriented to the questions we choose to ask, but it is worth our effort to have those questions informed by the practitioners we hope to aide. Third, producing more generative practices that mathematicians can adapt on their own has a higher chance of creating larger impact than whole curricula that mathematicians must either adopt or avoid. We expect that all three of our example interventions could become just such generative tools that mathematicians adapt in various settings. We recognize that there are disadvantages to the model we propose. First, we expect that these types of interventions within the lecture framework will only produce incremental differences and will not get to what many mathematics educators believe is the root of the problem with conventional instruction (e.g., rooting students’ mathematical understanding in their own activity). The natural counterpoint is that incremental changes in more classrooms may yield more net benefit than radical changes that are used less often. Second, we have not provided evidence of the efficacy of this model and the widespread adoption we discuss. While we cannot deny this, we have several decades of undergraduate instructional innovations within the old model that we know have struggled to be adopted at scale. We do not advocate that we abandon old models for developing mathematics education innovations, because we think our field benefits from a diversity of models. Teaching experiments are highly productive in learning about student

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thinking and providing existence proofs for how ideas develop for novices. Developing inquiry-oriented curricula yields a number of benefits to our field as we learn about many facets of the teaching and learning ecology. What we propose in this chapter is to give mathematicians a more natural place in our work and to adopt more focus on improving the lectures that continue to be delivered across our universities each semester.

References Alcock, L. (2010). Mathematicians’ perspectives on the teaching and learning of proof. Research in Collegiate Mathematics Education, 7, 63–92. Artemeva, N., & Fox, J. (2011). The writing’s on the board: The global and the local in teaching undergraduate mathematics through chalk talk. Written Communication, 28(4), 345–379. Asiala, M., Dubinsky, E., Mathews, D. M., Morics, S., & Oktac, A. (1997). Development of students’ understanding of cosets, normality, and quotient groups. The Journal of Mathematical Behavior, 16(3), 241–309. Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching. Journal of Teacher Education, 59(5), 389–407. Brown, A., DeVries, D. J., Dubinsky, E., & Thomas, K. (1997). Learning binary operations, groups, and subgroups. The Journal of Mathematical Behavior, 16(3), 187–239. Chi, M. T., De Leeuw, N., Chiu, M. H., & LaVancher, C. (1994). Eliciting self-explanations improves understanding. Cognitive Science, 18(3), 439–477. Conradie, J., & Frith, J. (2000). Comprehension tests in mathematics. Educational Studies in Mathematics, 42(3), 225–235. Cowen, C. (1991). Teaching and testing mathematics reading. American Mathematical Monthly, 98(1), 50–53. Crouch, C. H., & Mazur, E. (2001). Peer instruction: Ten years of experience and results. American Journal of Physics, 69(9), 970–977. Dawkins, P. C. (2012). Metaphor as a possible pathway to more formal understanding of the definition of sequence convergence. Journal of Mathematical Behavior, 31(3), 331–343. Dawkins, P. C. (2014). How students interpret and enact inquiry-oriented defining practices in undergraduate real analysis. The Journal of Mathematical Behavior, 33, 88–105. Dawkins, P. C., & Karunakaran, S. S. (2016). Why research on proof-oriented mathematical behavior should attend to the role of particular mathematical content. The Journal of Mathematical Behavior, 44, 65–75. Dubinsky, E., Dautermann, J., Leron, U., & Zazkis, R. (1994). On learning fundamental concepts of group theory. Educational Studies in Mathematics, 27(3), 267–305. Dubinsky, E., & Leron, U. (1994). Learning abstract algebra with ISETL. Springer. Fukawa-Connelly, T., Weber, K., & Mejía-Ramos, J. P. (2017). Informal content and student notetaking in advanced mathematics classes. Journal for Research in Mathematics Education, 48(5), 567–579. Gabel, M. (2019). The flow of proof—Rhetorical aspects of proof presentation. Doctoral dissertation. Tel Aviv University. Gabel, M., & Dreyfus, T. (2017). Affecting the flow of a proof by creating presence—A case study in number theory. Educational Studies in Mathematics, 96(2), 187–205. Hodds, M., Alcock, L., & Inglis, M. (2014). Self-explanation training improves proof comprehension. Journal for Research in Mathematics Education, 45(1), 62–101. Iannone, P., & Miller, D. (2019). Guided notes for university mathematics and their impact on students’ note-taking behaviour. Educational Studies in Mathematics, 1–18.

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Inglis, M., & Foster, C. (2018). Five decades of mathematics education research. Journal for Research in Mathematics Education, 49(4), 462–500. Johnson, E., Caughman, J., Fredericks, J., & Gibson, L. (2013). Implementing inquiry-oriented curriculum: From the mathematicians’ perspective. The Journal of Mathematical Behavior, 32(4), 743–760. Johnson, E., Keller, R., & Fukawa-Connelly, T. (2018). Results from a survey of abstract algebra instructors across the United States: Understanding the choice to (not) lecture. International Journal of Research in Undergraduate Mathematics Education, 4(2), 254–285. Krupnik, V., Fukawa-Connelly, T., & Weber, K. (2018). Students’ epistemological frames and their interpretation of lectures in advanced mathematics. The Journal of Mathematical Behavior, 49, 174–183. Kuster, G., Johnson, E., Keene, K., & Andrews-Larson, C. (2018). Inquiry-oriented instruction: A conceptualization of the instructional principles. Primus, 28(1), 13–30. Larsen, S. P. (2013). A local instructional theory for the guided reinvention of the group and isomorphism concepts. The Journal of Mathematical Behavior, 32(4), 712–725. Larsen, S. P., Johnson, E., & Weber, K. (2013). The teaching abstract algebra for understanding project: Designing and scaling up a curriculum innovation. Special Issue Journal of Mathematical Behavior, 32(4). Laursen, S. L., & Rasmussen, C. (2019). I on the prize: Inquiry approaches in undergraduate mathematics. International Journal of Research in Undergraduate Mathematics Education, 5(1), 129–146. Leron, U., & Dubinsky, E. (1995). An abstract algebra story. The American Mathematical Monthly, 102(3), 227–242. Lew, K., Fukawa-Connelly, T. P., Mejía-Ramos, J. P., & Weber, K. (2016). Lectures in advanced mathematics: Why students might not understand what the mathematics professor is trying to convey. Journal for Research in Mathematics Education, 47(2), 162–198. Lockwood, E., Johnson, E., & Larsen, S. (2013). Developing instructor support materials for an inquiry-oriented curriculum. The Journal of Mathematical Behavior, 32(4), 776–790. Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States. Routledge. Mejía-Ramos, J. P., Lew, K., de la Torre, J., & Weber, K. (2017). Developing and validating proof comprehension tests in undergraduate mathematics. Research in Mathematics Education, 19(2), 130–146. Mejía-Ramos, J. P., Fuller, E., Weber, K., Rhoads, K., & Samkoff, A. (2012). An assessment model for proof comprehension in undergraduate mathematics. Educational Studies in Mathematics, 79(1), 3–18. Mejía-Ramos, J. P., & Weber, K. (2019). Mathematics Majors’ diagram usage when writing proofs in calculus. Journal for Research in Mathematics Education, 50(5), 478–488. Melhuish, K., Fukawa-Connelly, T., Dawkins, P. C., Woods, C., & Weber, K. (2022). Collegiate mathematics teaching in proof-based courses: What we now know and what we have yet to learn. The Journal of Mathematical Behavior, 67, 100986. Nardi, E. (2007). Amongst mathematicians: Teaching and learning mathematics at university level. Springer Science & Business Media. Paoletti, T., Krupnik, V., Papadopoulos, D., Olsen, J., Fukawa-Connelly, T., & Weber, K. (2018). Teacher questioning and invitations to participate in advanced mathematics lectures. Educational Studies in Mathematics, 98(1), 1–17. Putnam, A. L., Sungkhasettee, V. W., & Roediger, H. L., III. (2016). Optimizing learning in college: Tips from cognitive psychology. Perspectives on Psychological Science, 11(5), 652–660. Rasmussen, C., & Wawro, M. (2017). Post-calculus research in undergraduate mathematics education. In J. Cai (Ed.), The compendium for research in mathematics education. National Council of Teachers of Mathematics.

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Resnick, L. B., & Resnick, D. P. (1992). Assessing the thinking curriculum: New tools for educational reform. In B. R. Gifford & M. C. O’Connor (Eds.), Changing assessments: Alternative views of aptitude, achievement, and instruction (pp. 37–75). Kluwer. Roh, K. H. (2008). Students’ images and their understanding of definitions of the limit of a sequence. Educational Studies in Mathematics, 69(3), 217–233. Roh, K. H. (2010). How to help students conceptualize the rigorous definition of the limit of a sequence. Primus, 20(6), 473–487. Shipman, B. A. (2012). Determining definitions for comparing cardinalities. Primus, 22(3), 239–254. Shipman, B. A. (2013). Convergence and the Cauchy property of sequences in the setting of actual infinity. Primus, 23(5), 441–458. Simon, H. A. (1972). Theories of bounded rationality. Decision and organization, 1(1), 161–176. Stylianides, G. J., Stylianides, A. J., & Weber, K. (2017). Research on the teaching and learning of proof: Taking stock and moving forward. In J. Cai (Ed.), The compendium for research in mathematics education. National Council of Teachers of Mathematics. Swinyard, C., & Larsen, S. (2012). Coming to understand the formal definition of limit: Insights gained from engaging students in reinvention. Journal for Research in Mathematics Education, 43(4), 465–493. Van den Heuvel-Panhuizen, M., & Drijvers, P. (2014). Realistic mathematics education. In S. Lerman (Ed.), Encyclopedia of mathematics education (pp. 521–525). Springer. Wasserman, N. H., Fukawa-Connelly, T., Villanueva, M., Mejia-Ramos, J. P., & Weber, K. (2017). Making real analysis relevant to secondary teachers: Building up from and stepping down to practice. Primus, 27(6), 559–578. Weber, K. (2004). Traditional instruction in advanced mathematics courses: A case study of one professor’s lectures and proofs in an introductory real analysis course. The Journal of Mathematical Behavior, 23(2), 115–133. Weber, K. (2012). Mathematicians’ perspectives on their pedagogical practice with respect to proof. International Journal of Mathematical Education in Science and Technology, 43(4), 463–482. Weber, K., & Mejia-Ramos, J. P. (2011). Why and how mathematicians read proofs: An exploratory study. Educational Studies in Mathematics, 76, 329–344. Woods, C., & Weber, K. (2020). The relationship between mathematicians’ pedagogical goals, orientations, and common teaching practices in advanced mathematics. The Journal of Mathematical Behavior, 59, 100792.

Chapter 7

Didactics of Mathematics as a Field of Mathematical Research: The Anthropological Approach Carl Winsløw

Abstract We develop the thesis that mathematics is an essentially didactical field of human activity, and that, therefore, Didactics of Mathematics is (or should be) a central part of mathematics itself. The basic ideas of the anthropological theory of the didactic are invested in this argument, as are some personal experiences from 25 years of research in von Neumann algebra theory, as well as in the didactics of analysis at university level. Keywords Anthropological Theory of the Didactic · Didactics of Mathematics

7.1

A Personal Introduction

The relationship between, on the one hand, the scholarly field of activity that is known, in the Anglo-Saxon world, as mathematics education research, and, on the other hand, the mathematical sciences, is a topic which may evoke a number of personal sentiments in people involved in either of these endeavors, depending on the person’s experience and position in relation to both. It is the main goal of this chapter to propose a theoretical framework that will allow for a more depersonalized analysis of both the two “objects”—mathematics education research (which I will, for reasons explained later, refer to as the Didactics of Mathematics), and the mathematical sciences—and their mutual relation. Appreciation of the relevance of this goal requires certain sensitivity to theoretical precision, which I trust to share with most readers, whether they identify mostly with one or the other of the two fields mentioned above. Of course, any perspective on these matters will appear more or less attractive and reasonable depending on personal experience, knowledge, and interests. I will begin by briefly explaining the trajectory that led me to adopt the perspective developed in

C. Winsløw (✉) University of Copenhagen, Copenhagen, Denmark e-mail: [email protected] © Springer Nature Switzerland AG 2023 S. Stewart (ed.), Mathematicians’ Reflections on Teaching, Advances in Mathematics Education, https://doi.org/10.1007/978-3-031-34295-0_7

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this chapter—not so much to explain it but to prepare the final section, which deals with how to further it in practice. My research career so far has been roughly tripartite: 1. Becoming a researcher in mathematics, more precisely von Neumann algebra theory (a field of research situated within Functional Analysis), roughly in the period 1986–1998; 2. Becoming a researcher in Didactics of Mathematics, roughly in the period 1994–2003; 3. Developing the integrative perspective on the two fields that I hold today (from 2001). When I attended Danish high school (1983–1986), mathematics was still taught more or less according to the scheme of “New Mathematics.” It began, from the very first weeks, with informal set theory and logic and proceeded to the rigorous study of elements of Algebra (including abstract operations, groups, etc.), Geometry (based on matrices and mappings), and Analysis (including rigorous definitions and proofs, going up to the Fundamental Theorem relating the Riemann integral with antiderivatives). The last domain impressed me in a very special way: not only can area and volume of spooky sets be given a rigorous definition, but also these quantities can be computed by the simple (albeit sometimes a bit tricky) algebraic determination of antiderivatives, for a very general class of subsets of the plane and of space. Certainly, the theory is difficult and abstract for a teenager, but we were lucky to have a great teacher, Mrs Grønlund, who was not only able to open up the many difficult quantified expressions but also to lead some of us to share her enthusiasm about the theory. I was entirely sold! And I rushed to study mathematics at the university in Odense immediately after high school. In a sense, my way toward research mathematics and Analysis thus began even before 1986. However, during the first study years, it was still my intention to become a high school teacher; there was, and still is, no difference in the educational trajectories toward these professions, at least at the bachelor level. Also at university, Analysis continued to attract me the most; and again I had the fortune of meeting several excellent teachers. Among these, Uffe Haagerup became the most influential—in a biographical article about him, written some years ago (Moslehian et al., 2017), I had the opportunity to describe this in more detail. It will suffice here to say that Uffe encouraged me to pursue a PhD in the field he excelled in, namely von Neumann algebra theory. Upon his recommendation, I did so under the direction of Yasuyuki Kawahigashi at the University of Tokyo. The following years were filled with hard work and reasonably successful results, including a classification of strongly free actions of discrete amenable groups on strongly amenable subfactors of certain types (Winsløw, 1993). I completed my PhD in 1994 and got a job as assistant professor at the University of Copenhagen. Already in Odense, I had been a teaching assistant; but now I was given full responsibility for larger courses, such as “Mathematics for biologists” (about 250 students every year), and “Complex Analysis” (about 150 students). Around the same time, a formal training on pedagogy had become mandatory for young faculty in Danish

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universities; I attended and was surprised by the practical irrelevance of the general material we were given to read. Luckily, there was also a “practical” part of the course, with supervision by peers and more experienced colleagues. I was lucky to be supervised by yet another excellent teacher: Niels Grønbæk, who later became my coauthor on a long series of papers on the Didactics of Analysis at university level. However, at this stage, I was mainly on my own seeking for reading material that could really support my efforts to teach the courses I had in an efficient way, while pursuing my research on von Neumann algebras. Slowly I discovered that there is a scholarly literature on the teaching of mathematics, much of what seemed to me almost as distant from university mathematics as the texts from the pedagogy course. One name, however, began to attract my attention: Guy Brousseau, the founder of the theory of didactical situations in mathematics (Brousseau, 1997). For Brousseau, Didactique des Mathématiques is clearly a mathematical endeavor—just as Didaktik der Mathematik had been for Felix Klein a century before. We shall return to this point later in the chapter. Various other circumstances—in particular the opportunity to lead a “Didactical seminar” for mathematics students who were aiming at a career in high school— gave me opportunities to pursue this interest further. In 1998, I moved to an associate professorship at what was then the Royal Danish School of Education. The main task there was still to teach mathematics courses (mainly calculus, linear and abstract algebra)—but the research environment favored my emerging efforts in Didactics of Mathematics. And then, in 2001 and 2003, I came to attend the Summer Schools in this field, organized every second year by the ARDM (French association for research in Didactics of Mathematics). There, I met several great scholars, including Artigue, Brousseau and Chevalllard—the “ABC” of French Didactics. What I learned from them in these years became fundamental for my subsequent research in the field. Finally, we move on from the second to the third moment I listed above. This move becomes gradually visible in the publications I was able to produce after 2003, so I will close this already lengthy account by yet another change of position. In 2003, I moved back to the University of Copenhagen, where a new Centre for Science Education had been established at the Faculty of Science. There I got to occupy the first professorship in the field ever established at this relatively traditional institution. Again, good colleagues and opportunities favored my development; in particular, the Centre (which became an ordinary department in 2007) had, and has, a special focus on the Didactics of Science, including Mathematics, at the university level. Engaging in development and research projects on university education with colleagues from the faculty, and increasingly working with high school teacher education, necessitated for me a new kind of expertise in which content knowledge is closely associated with frameworks and methods to carry out experimental and otherwise empirical work on teaching the contents. Naturally, my background made me want to focus mostly on the mathematical field—and fortunately, I gradually got colleagues with specialties in other sciences who made it possible to recover this focus.

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I will now turn to a brief outline of the theoretical framework, which underlies the main content of this chapter: a historical and philosophical argument that “the didactic” is everywhere dense in mathematics and that Didactics of Mathematics is therefore essentially a mathematical field. The next section will require careful reading if you are new to the theory.

7.2

Theoretical Framework

We now introduce some basic terms and ideas from the Anthropological Theory of the Didactic (ATD), which were formulated by Yves Chevallard (1992) almost 30 years ago and which are fundamental to a rapidly expanding research program in mathematics education research. In the summer of 2019, a two-month “special research intensive period” on ATD research gathered more than 100 scholars from around 20 countries and was hosted by the Centre de Recerca Matemàtica (an international center for mathematical research) in Barcelona, Spain. The most original and important asset of ATD is to furnish a theoretical tool— called praxeological analysis, and explained below—for modeling mathematical knowledge. This tool is usually easy to adopt for students and practitioners of mathematics, and it focuses our attention on elements and relations within mathematics that are of vital importance to the teaching of mathematics. To introduce the tool, let us begin with an example of a mathematical task: What is the length of diagonal in a rectangle with side lengths 20 and 40? Or, on the practical side: what is the distance between towns A and B, given that A is 20 km west and 40 km north of B? This is of course an example of a wider type of tasks, which might be described more abstractly as “find the length of the diagonal in a rectangle with side lengths a and b.” Let us call this type of tasks T. It could be described differently, for instance, in terms of rectangular triangles. What really characterizes tasks of type T is that all tasks in T can be solved by the same a2 þ b2 . If we call the technique τ, we have technique, essentially computing what is called a praxis block Π = (T, τ). This is a minimal element of practical knowledge: a type of task and a corresponding technique. Even for practical knowledge outside of mathematics, it is often useful to think of the essence as knowing to recognize the type of task (for instance, “do the dishes”) and knowing to apply a corresponding technique. Notice that, typically, it is useful to consider, as one unit, several pairs (T, τ)—for instance, several cases of a problem such as “find the derivative of a function in closed form.” Such a unit will still be called a praxis block. The mathematical praxis block mentioned above has in some sense been known since Antiquity; but the details of the technique have certainly evolved, as have the ways the technique is described and justified. Even the algebraic symbolism a2 þ b2 was unknown to the Euclidean mathematicians who thought of what we call “squared numbers” in terms of areas and conversely, of “square roots” in terms

Didactics of Mathematics as a Field of Mathematical Research. . .

Fig. 7.1 A common way to illustrate the structure of a praxeology, as a unit of praxis and logos

THEORY TECHNOLOGY TECHNIQUE TYPE OF TASK

LOGOS BLOCK: “know that”, “know why” - explicit knowledge PRAXIS: “Know that” - practical knowledge

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of diagonals. Even before that, elaborate techniques to compute diagonals were known to Babylonian mathematicians more than 1000 years before Euclid. It goes without saying that these were formulated in ways only experts of mathematical history can read today. What has developed, then, is not only the techniques but also the ways to describe and justify them—that is, what ATD calls technology (technologos: “words about techniques”). The development of mathematics is not simply a refinement of techniques to solve tasks and ways to describe those techniques. A hallmark of both Euclidean and contemporary mathematics is a strong focus on theory. Unlike the term technology, ATD defines the term “theory” roughly in the commonplace sense: as discourse superseding technology, which serves to unify and clarify it. Theory can be both ad hoc and developed almost for its own sake. In scholarly mathematics, it involves concise and abstract definitions, statements, and proofs, which go beyond (but also unify and justify) the mere explanations of techniques to do tasks. In taught mathematics, the theory is sometimes less concise and complete, especially at elementary levels—but it is important to note that, just as techniques and types of tasks cannot exist separately, there is always some form of theory that accompanies a technology—for instance, some kind of “rough definition” or explicit meaning given to the terms we use to describe techniques. In relation to the example of diagonals, school mathematics in many countries includes some version of Pythagoras’ theorem, but not an elaborate theory in which it can be proved from first principles, such as the Euclidean theory. Together, a technology θ and a theory Θ are called a logos block Λ= (θ, Θ). Here, the Greek word logos (literally, word) refers human discourse based on words, diagrams, symbols, and so on. Finally, a praxeology consists of a praxis block together with a logos block: ℘= (Π, Λ). The name explains itself—it is the integrated whole of a praxis with the corresponding logos (cf. Fig. 7.1). In fact, just as the elements of the blocks described above are inseparable, so are praxis and logos. Mathematical praxis that depends crucially on logos (while we may imagine a monkey somehow “doing the dishes”): without theory and technology of some sort, we could not even think, let alone speak about the diagonal task and the corresponding techniques. On the other hand, the technology is about the praxis, and the theory is about the technology; thus, the theory block is meaningless without the praxis blocks it describes, unifies, justifies, and so on.

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There is evidently a strongly dialectic relation between praxis and logos: one the one hand, solving problems (which evolve into types of tasks as techniques are developed) both requires and supports the development of theoretical discourse and reasoning; on the other hand, the development of theory often leads to new techniques for solving old problems. As has transpired from the above explanation of praxeologies, one may also use this notion to model other kinds of human practice and knowledge than the ones involving strictly mathematical tasks. While not many actual studies have applied this idea, it has been proposed to distinguish didactical praxeologies as those in which the tasks relate to allow others to acquire mathematical praxeologies—most notably, through formal teaching. An example of such a task is to construct a set of examples or exercises that can help students to become familiar with the computation of diagonals. It appears, however, that the techniques and the logos blocks related to such tasks are considerably less straightforward to observe and study than those directly related to mathematical praxeologies. Mathematical technology and theory have been published in written forms since Antiquity, and more elementary mathematical praxeologies are relatively stable over time and widely shared. The techniques and logos related to mathematics teaching are typically much less stable and shared among practitioners, in spite of text books and mathematics education literature. The Didactics of Mathematics investigates and develops mathematical and didactical praxeologies and especially how they relate to each other. To do so, explicit models of the mathematical praxeologies to be taught are needed—notice that didactic tasks are always related to (in a way, derived from) the mathematical praxeologies to be taught. The latter may be more or less precisely described by official documents. But as a result of didactic praxis, a slightly different mathematical praxeology may be developed by the pupils or students, reflecting what is actually learnt. Research in Didactics of Mathematics studies both the mathematical praxeologies to be taught, and those learnt—and how they both relate to the didactical praxeology. Because an intention of teaching is involved, all three praxeologies belong to what ATD calls “the didactic”—phenomena related to sharing praxeologies. From this comes the slightly strange name of ATD itself and also the widespread preference to use the name “Didactics of X” for the study of didactic phenomena related to a discipline X (this corresponds to the terminology in almost all European languages; a more detailed argument for its use in English was provided by Chevallard, 1999). Praxeologies are not individual but are shared and conveyed in larger groups of people. In particular, special institutions are set up to support the construction and dissemination of mathematical (and other) praxeologies; in particular, schools and universities also support didactic praxeologies. What is an institution? In ATD, it is not a building or brand, but a specific configuration of positions one may occupy in relation to a definite set of praxeologies. For instance, in a university, “professor of mathematics” is such a position. The mathematical praxeologies are developed, conveyed, taught, and learned in ways that are to a large extent determined by the positions people occupy in the institution. The institution functions, metaphorically,

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as a habitat for certain praxeologies. Of course, institutions change over time, as do the praxeologies whose construction and dissemination they favor. It is important to note here that although institutions and praxeologies can of course only be studied through the action and discourse of concrete persons, the study objects of ATD are not those people but their praxeologies and the institutions in which they, temporarily, occupy certain positions.

7.3

Historical Milestones Revisited

Some of the oldest evidence of mathematical activity dates from about 2000 BC— the so-called papyri from Egypt, which contain a number of questions, which we might today understand as exercises. The topics range from simple what we might interpret as arithmetic of fractions to geometry and combinatorics. While the functions of these and similar texts are not quite clear, they seem to support at least three hypotheses: (1) humans have had mathematical needs and aspirations wherever civil societies have developed, and they were not always strictly limited to practical ends; (2) along with the development of writing systems to record texts in natural language, also mathematical productions appear, very often with specialized sign system, to organize the work with numbers, figures, and so on; (3) while “written” mathematics may not have been equally important in humankind’s first mathematical activities, we of course cannot know much about what has not somehow been retained in written accounts—and whenever we have such, “tasks” in a broad sense play a major role, and whether or not they are extended to “worked examples” (i.e., a technique is also demonstrated), their function has most likely been didactical—for use in sharing or developing mathematical praxeologies with people who did not know them. Moving onto Euclid’s Elements (300 BC), the contents is certainly much more abstract and structured, but also, from the first Proposition, often takes the form of a type of tasks: “To construct an equilateral triangle on a given line segment” (i.e., with the segment as base). The proof demonstrates a technique that most school children of a certain age will not fail to remember even today. But the Elements represent also a remarkable leap into explicit mathematical theory, more or less as we know it today: it contains axioms, explicit assumptions, definitions, and proofs. The techniques are not just simple artifacts to solve independent (types of) tasks, but they contribute to progression of justified, general results. Through the theoretical structure, the mathematical results known at the time are presented in a unified, coherent, and economical fashion. Both the economy of exposition and the knowledge we have of Greek didactic institutions lead us to assume a clearly didactical purpose of the text, which it certainly had for centuries and, to some extent, retains. According to Wilson (2006, p. 278), “Euclid’s Elements subsequently became the basis of all mathematical education, not only in the Roman and Byzantine periods, but right down to the mid-20th century, and it could be argued that it is the most successful textbook ever written.” Whether the originally intended public of the text

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was an elite of mathematically learned people, or it was also meant for use in instructing more general classes of citizens, we cannot know; but already a contemporary text, the Socratic dialogue Menon (380 BC), provides an attempt to demonstrate that even a slave could have (or, against the author’s opinion, be taught) mathematical knowledge. And certainly Plato’s Academy, where mathematics was taught, was open to a general public of citizens (Wilson, 2006, p. 2). This and other evidence support that the first major development of mathematical logos at a level of precision that we can accept even today went hand in hand with institutions for its dissemination, hosting substantial didactical praxis. Indeed, the Socratic dialogues, along with other philosophical texts of the period, contain both didactical demonstrations and explicit didactical logos. Euclid’s Elements is an example of a synthesis of mathematical knowledge, focusing essentially on mathematical logos. Can one say that putting together such a synthesis is only a didactical endeavor, which merely collects preexisting knowledge? I believe most mathematicians would agree that writing a monograph in a novel area is a valuable research effort, even if the main results were already published in journal papers. This is not only because of the value of the connected exposition but also because they know that in editing such a text, it is almost inevitable that the exposition even of the individual results becomes more efficient and accessible than when it is scattered throughout the literature. Very often the editing process will also lead to more efficient and elegant proof routes, more general results, and so on. So it seems obvious that at least syntheses of recent mathematical achievements are genuine scholarly works. In fact, according to the philosopher and mathematician Bouligand (1957), the mathematical sciences move forward through a dialectic interaction between problems and syntheses. Bosch and Winsløw (2016), drawing on ATD and Bouligand, demonstrate how important historical positions and patterns in mathematics teaching can be classified according to the role they give to problems and syntheses in the work assigned to students. For instance, one may privilege the acquisition of mathematical logos, by having students memorize a synthesis like one or more of Euclid’s books, while students do little or no work with problems (even tasks for which the techniques are already provided by the synthesis). Or one may stipulate that all student work should be based on problems, without ever requiring a synthesis, let alone providing students with one. Of course, all sorts of intermediate positions exist and have in fact been proposed and practiced in historical didactical institutions. What is important to notice here is the parallel between the ways in which mathematical praxeologies develop in the scholarly (research) and didactical (teaching) setting, somehow combining solving problems while drawing on syntheses of results, and subsequently producing new syntheses, leading to new problems. One can think of mathematical research as the action of walking on two legs: solving problems (based on syntheses) and producing syntheses (based on problems solved). Teaching does not frequently reproduce this interaction in a similar, forwardoriented manner; as the synthesis of new problems already exists, it is frequently presented first, which makes problem-solving a mere “application” of the synthesis,

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without any progress added. This is what Freudenthal (1983, p. 305) called antididactical inversion: “the final result of the developmental process is chosen as the starting point for the logical structure in order to finish deductively at the start of the development.” This, of course, is not possible in mathematical research. While it may often appear as the most economical solution in teaching, it has the disadvantage to present the problems that are or could be made intelligible to learners as mere applications of more general answers, which in themselves appear to come out of the blue. In terms of mathematical praxeologies, the task comes, in principle, before the techniques, while the technology develops along with them; in the same way, theory supports new technology but is also, progressively, extended in order to unify and generalize the discourse focused on solving particular tasks, or—as far as techniques are developed—types of tasks. The dialectics between problems and syntheses is therefore, also, a kind of progressive development of mathematical praxeologies, in a dialogue between praxis and logos. New theory gives rise to new tasks, or problems—and hence, to new praxis, new problem-solving. Subsequently, new praxis is theorized and abstracted, through the process of synthesis. Freudenthal, in his day and age, opposed the emerging “new mathematics” for teaching that became attached to another monumental work of synthesis, the Éléments de mathématiques (a work developing since 1939). This work was initially undertaken by a collective of French researchers who adopted the pseudonym of the (mythical) figure of Nicolas Bourbaki. Of course, this project was, at least initially, to produce a purely scholarly synthesis—to unite, in a self-contained series of volumes, the principal elements of modern mathematics, beginning with (and based on) the theory of sets. As the famous “instructions for use,” included in every volume, stipulate, the goal was “to take mathematics from its beginnings” (Archives Bourbaki, n.d.). The history of its becoming, in various “transpositions” (Chevallard, 1991), also a model of mathematical instruction (“New Math”), cannot be developed in detail here; it is often presented as a decisive failure, while more recent accounts have produced a less simplistic picture (Philips, 2015). It should be noted that Bourbaki’s ambitious project of syntheses of mathematical achievements has indeed proven a most productive one in the scholarly world and is not limited to present closed, complete state of the discipline. The volumes also signal a range of open problems, in addition relatively challenging “exercises,” which encourage the reader to perform known deductions that are not included in the text. Felix Klein (1908/2016), in his masterpiece on “Elementary mathematics from a higher standpoint”, defines two strategies for developing mathematical knowledge— whether in teaching or in research—which he calls “Plan A” and “Plan B”: Plan A is based upon a more particularistic conception of science which divides the total field into a series of mutually separated parts and attempts to develop each part for itself, with a minimum of resources and with all possible avoidance of borrowing from neighboring fields. Its ideal is to crystallize out each of the partial fields into a logically closed system. On the contrary, the supporter of Plan B lays the chief stress upon the organic combination of the partial fields, and upon the stimulation which these exert one upon another. He prefers, therefore, the methods which open for him an understanding of several fields under a uniform point of view. His ideal is the comprehension of the entire mathematical science as a great connected whole. (Klein, 1908/2016, vol. I, pp. 82–83)

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Both the Elements of Euclid and those of Bourbaki are evident examples of Plan A, although Klein of course did not know of the latter. Klein himself cites the historical developments related to the most common special functions, in particular exponential, logarithmic, and trigonometric functions. In the period just around 1600, Copernicus developed the first trigonometric tables, while Napier’s logarithms were initially just seen as a clever technical aid to trigonometric calculations; this field, then, was developed according to Plan A. By contrast, the following century clearly witnessed dramatic developments according to Plan B, in which the aforementioned special functions appear as, to begin with, functions, and also power series, derivatives, integrals, and so on. Previously unrelated elements of geometry, algebra, and trigonometry were now freely combined, certainly at the cost of the relative precision and explicit foundations, which the three fields had enjoyed in separation. The gradual emergence of rigorous analysis in the nineteenth and early twentieth century, eventually leading to an axiomatic foundation, of course marks a new instance of Plan A—extended and culminating in the Bourbaki project, as we have just seen. What we should above all retain here is how Klein—one of the greatest mathematicians in recent times and, at the same time, a pioneer in the field of mathematics education—often insisted on the similarities and necessary connections between the development of mathematics as a science and the ways in which mathematics is or can be taught in school. Mathematics teaching, in both schools and universities, sometimes degenerates into training isolated little bits of praxis, each with their independent logos, or at least fails to teach students some of the deeper links among larger domains, such as trigonometry and calculus (see e.g., Kondratieva and Winsløw, 2018). Klein points out the same need, at his time, to pursue curriculum reforms that would lead to more connectivity: Teaching in the secondary schools, however, as I have already indicated, has long been under the one-sided domination of the Plan A. Any movement toward reform of mathematical teaching must, therefore, press for more emphasis upon direction B. (Klein, 1908/2016, vol. I, p. 88)

Klein’s vision of stronger connections between the different fields of mathematics taught in school and university, and in fact, between school mathematics and mathematics as a science, is clearly entirely independent from the later New Math movement. Klein’s view of mathematics as a science was also considerably broader than the Bourbakist one and included what we could roughly call “applied mathematics”; he certainly considered this to be of great relevance to school mathematics as well (Klein, 1908/2016, vol. III). Klein also recognized the importance of historical and cultural variations of mathematics teaching, and as founding president of the International Commission on Mathematical Instruction, he actively undertook and initiated early comparative studies in the field. By today’s standards, these studies—such as his own inter-European survey on the teaching of elementary geometry (Klein, 1908/2016, vol. II, pp. 236–276)—are somewhat anecdotal. We must bear in mind that while Klein was certainly a master of one of the oldest and most well-established scholarly endeavors—mathematics itself—he was also among

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the very early frontrunners and pioneers of the Didactics of Mathematics. Even today, one cannot fail to notice the relatively primitive state of the latter field, in comparison with the former.

7.4

University Schizophrenia

At an intellectual level, the mathematical sciences have developed standards of explicit reasoning that enable a very high level of consensus among experts, when it comes to validating concrete results. The difference states of research on mathematics and on didactical phenomena related to mathematics come in part from the different objectives of research: on the one hand, to produce “new” mathematical praxeology, and on the other hand, studying the relations between mathematical and didactical praxeologies as well as the institutional conditions and constraints under which they develop. The latter involves, necessarily, empirical evidence and thus forms of justification that are not common in the mathematical sciences, and which are often much more partial and controversial than formal proof. Even theoretical analyses in Didactics of Mathematics can only to some extent draw explicitness and solidity from scholarly mathematics itself. Most of the time, the mathematical praxeologies we study are quite far from those pursued and developed in presentday research institutions, and simply using the latter as a model of the former is rarely relevant or possible—even when it comes to university teaching of, say, Calculus. These intellectual distances and differences are most likely at the root of the alienation that many researchers in mathematics have expressed, over the years, in relation to research on mathematical education. It is no doubt reinforced by the wealth of publications and projects on mathematical education that are undertaken with little or no theoretical framework (Kilpatrick, 1981, 25)—in particular, with little explicit and analytic attention to mathematical praxeologies. The absence of explicit models of the subject of research evidently leaves the door wide open to implicit assumptions and normative choices and to replacing scientific debate by confrontations of beliefs and opinions. But I claim that these intellectual and academic discontinuities—which in part can be explained historically and philosophically—are perhaps not the main reasons why we seem to be so far, today, from Klein’s ideal of aligning the Mathematical Sciences and School Mathematics better, with an interface provided by the new scholarly field, which already at his time was referred to as Didactics of Mathematics (Simon, 1895, quoted in Klein 1908/2016, vol. I, p. 4). The particular situation of German Didactics of Mathematics, after Klein’s death in 1925, was of course profoundly affected by the historical developments surrounding the two world wars. The international development has been even more lastingly affected by a general development of university institutions, which Cuban (1999) describes through the case study of Stanford University in the United States, in a book with the expressive title How Scholars Trumped Teachers. In short, this development has

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seen universities pass from von Humboldt’s nineteenth century ideal of Einheit von Lehre und Forschung (“unity of teaching and research”), in which elites for certain state employments studied under the direction of eminent scholars, to institutions with two quite separate aims: mass training of still wider populations (cf. also Verret, 1975), and research in still more specialized areas. At so-called “research-intensive” universities such as Stanford, teaching clearly holds a secondary place in the priorities and career development of professors. These priorities can be traced, at least in part, to institutional and societal factors, such as expanding schemes of funding for research excellence in certain politically prioritized areas that clearly outperform the often-shrinking budgets for undergraduate education. We find a moving expression of this situation, at the individual level, in the “automathography” of the American mathematician Paul Halmos (1985, p. 321f): Despite my great emotional involvement in work, I just hate to start doing it; it’s a battle and a wrench every time. Isn’t there something I can (must?) do first? Shouldn’t I sharpen my pencils, perhaps? (. . .) Yes, yes. I may not have proved any new theorems today, but at least I explained the law of sines pretty well, and I have earned my keep.

It is clear from the context, as well as from his series of seminal textbooks on Analysis, that Halmos was an outstanding university mathematics teacher, who enjoyed the teaching part of his profession thoroughly. At the same time, in this context, by “work,” he means research—proving new theorems. In relation to this more prestigious endeavor, teaching appears as a distraction, “sharpening pencils.” Falling into the trap of postponing research for the easier and more immediately satisfying tasks related to elementary teaching fills the scholar with a sense of guilt. Studies of the links that mathematicians perceive between their double duty as teachers and researchers (e.g., Madsen and Winsløw, 2009) have indeed revealed similar experiences of conflict, in some cases bordering frustration if not schizophrenia, but also (in some) a strong adherence to the ideal that (university) teaching and research draw, or should draw, from each other. Neither Halmos nor Klein could reasonably be accused for having neglected research at the expense of teaching. But Klein (1908/2016, vol. I, p. 1) probably described the future as much as the past as he described the priorities of “university men” in relation to their newly acquired interest in mathematics teacher education: For a long time prior to its appearance, university men were practicing exclusively research of optimal quality, without giving a thought to the needs of the schools, without even caring to establish a connection with school mathematics.

Because of their direct or at least indirect implication in teacher education, mathematicians’ priorities in and of teaching general courses to future teachers could indeed have lasting profound effects on the degree to which the mathematics as practiced in schools and mathematics in universities get separated, or on the contrary nurture each other (the Kleinian ideal). In many countries, mathematicians in the classical sense are no longer involved in teacher education at all, even when it comes to preparing teachers for the secondary level. This development, paradoxically at first sight, runs parallel to the increasing academization of teacher education in many countries, with specialized,

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non-research teacher colleges often merging into (or becoming) university institutions. These institutional developments have specific and complex reasons and variations across the world. A common trend seems to be that the “research basis” for teacher education is increasingly taken from the new education sciences. The latter are often understood as comprising the Didactics of Mathematics, which can then mean that the mathematical element is quite weak, to the point where “mathematics educators” do not even have an undergraduate degree in mathematics. We do not aim to trace such variations here, but there is no doubt that battles over the contents of teacher education are often related both to these and to the institutional interests of scholarly communities who, naturally, emphasize the importance of their contribution to future teachers. One relatively well-documented case will suffice. In the United States, where universities (unlike many universities in continental Europe) are often equipped with faculties of education, which also host scholars in mathematics education, this tendency is perhaps particularly evident, despite the many variations among the different states—and not unrelated to the wider “math wars” over the methods and contents of school mathematics itself (Schoenfeld, 2008). While such conflicts are of course always driven by individuals, they can in this case also become conflicts between institutions (or rather cross segments of university institutions)—in this case, between “mathematicians” and “mathematics educators.” The most fierce combatants have at times not hesitated to question the legitimacy and intellectual honesty of not only individuals but also their disciplines, at least when it comes to their contributions to school mathematics. Similar, less vocal conflicts undoubtedly exist in many other countries. Not only can one perceive an increasing discontinuity between the ever-expanding field of scholarly mathematical knowledge and the form and contents of school mathematics, but also a tendency to divorce of the first field from the scholarly field founded by great mathematicians such as Klein and Freudenthal. At the same time, the tendency that scholarly mathematics loses terrain when it comes to defining the school mathematical curriculum and ensuring the mathematical education of teachers has undoubtedly also—as at the time of Klein, but for new reasons—contributed to a renewed interest and engagement in this field among research mathematicians and their institutions. As for the American case, we read in a recent report from the US National Research Council (2013, pp. 122–123): It is critical that the mathematical sciences community actively engage with STEM discussions going on outside the mathematical sciences community and not be marginalized in efforts to improve STEM education, especially since those plans would greatly affect the responsibilities of mathematics and statistics faculty members. This committee knows of no evidence that teaching lower-division college mathematics and statistics or providing a mathematical background for K-12 mathematics teachers can be done better by faculty from other subjects but it is clear that the mathematicsintensive disciplines are full of creative people who constitute a valuable resource for innovative teaching ideas.

Moreover, several American reports and position papers have appeared over the past decades (e.g., CBMS, 2012), which take on a somewhat more refined approach

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to the issues. They are typically coauthored by scholars who identify as mathematicians and mathematics educators or both. These and many other contributions to the literature certainly modify the picture of conflicting institutions and individuals, painted above, and suggest that the scholarly debate is far from limited to individuals who accept or even promote the divorce between mathematics at school and mathematical research in universities or between the scholarly communities engaged in mathematics and its didactics at universities.

7.5

Conclusion and Perspective

In the larger historical perspective laid out in the preceding sections, it is clear that mathematics as a whole is a profoundly didactical science and derives a vital part of its development from ideas and results that are themselves as old as the pyramids, through the dialectics of problems and syntheses. Think of the Pythagorean formula shows up in linear algebra, or of how a handful of relatively simple number theoretical problems continue to generate fascinating investigations in a number of other fields, without themselves being solved. Computer technology, itself rendered possible by mathematical inventions, gradually makes new mathematical subjects more relevant and approachable in schools and in principle could also facilitate a more logos-oriented approach to old ones, such as the Calculus, which before was often reduced to routine praxis. To prevent such ambitious developments of school mathematics from repeating the mistakes associated with anti-didactical inversion, we will need still more the creative people with a mathematics-intensive background to design problems and situations in which students can develop more complete mathematical praxeologies, in the natural order—from task to theory. Also the teachers, in order to direct and sometimes invent such situations, need a solid mathematical background, with a real experience of mathematics as a creative rather than dead discipline. Klein already formulated this vision of reforming school mathematics so as to give pupils an experience of mathematics as a living subject rather than as an inventory of routine praxises, a subject in which notions are connected to meaningful problems and indeed connected between them. In his inaugural lecture, Klein (Klein, 1873, published and translated in Rowe, 1985, p. 139) also noted that this was far from the situation then, and we might add from the situation in many schools today: Instead of developing a proper feeling for mathematical operations, or promoting a lively, intuitive grasp of geometry, the class time is spent learning mindless formalities or practicing trivial tricks that exhibit no underlying principle. One learns to reduce with virtuosity long expressions that are devoid of meaning, or to apply one’s diligence to the solution of artificially constructed equations that are contrived in such a fashion that one cannot even begin to make progress unless one knows some special trick in advance. When, however, the student with this sort of training is required to develop an independent idea or answer a question that is unfamiliar to him, he lacks all trace of individual initiative. It is here that we, as university teachers of mathematics have a wide and hopefully rewarding field for our activity. At stake is the task, precisely in the sense just mentioned, of raising the standards of the mathematical education for later teaching candidates.

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To engage in the Kleinian program of preparing teachers with a creative, rather than museological relation to school mathematics, may require university teachers with a profile that is different from both the typical PhD in mathematics and the average PhD in Didactics of Mathematics (or mathematics education). Both lack some the other has. In my personal experience, it is extremely rare for a doctoral student in pure mathematics to take an advanced course on Didactics, let alone engage in even a small empirical research project on school mathematics; and it is probably no less rare for PhD students in Didactics of Mathematics to take advanced mathematics courses or to prove even a lemma. Especially if the same department or faculty harbors the two fields, one could even develop doctoral programs in both fields that require such mixed experiences. To prepare mathematics teacher educators who can effectively pursue Klein’s vision, we need to prepare mathematicians-didacticians who are acquainted with contemporary mathematics, with creative mathematical work, and with modern methods and results from the Didactics of Mathematics. Such Kleinians could also help the scholarly fields rediscover that “the didactic” is everywhere dense in mathematics and that Didactics is really a form of experimental or applied mathematics. In particular, design research and curriculum development could draw directly on contemporary mathematical ideas and methods, and the teaching and dissemination of advanced mathematics would become more efficient—and, indeed, Kleinian—while drawing on new ideas and methods from Didactics.

References Archives Bourbaki. (n.d.). Mode d’emploi de ce traité. Manuscript, located April 2, 2019, on: http:// sites.mathdoc.fr/archives-bourbaki/PDF/031ter_delr_007bis.pdf Bosch, M., & Winsløw, C. (2016). Linking problem solving and mathematical contents: the challenge of sustainable study and research processes. Recherches en Didactique des Mathématiques 35(3), 357–401. Bouligand, G. (1957). L’activité mathématique et son dualisme. Dialectica, 11, 121–139. Brousseau, G. (1997). Theory of didactical situations in mathematics. Kluwer. CBMS. (2012). The Mathematical Education of Teachers II. American Mathematical Society and Mathematical Association of America. Chevallard, Y. (1991). La transposition didactique – du savoir savant au savoir enseigné (first edition, 1985). La Pensée Sauvage. Chevallard, Y. (1992). Fundamental concepts in didactics: Perspectives provided by an anthropological approach. In R. Douady & A. Mercier (Eds.), Research in didactique of mathematics, selected papers (pp. 131–167). La Pensée Sauvage. Chevallard, Y. (1999). Didactique? Is it a plaisanterie? You must be joking! A critical comment on terminology. Instructional Science, 27(1), 5–7. Cuban, L. (1999). How scholars trumped teachers - Change without reform in university curriculum, teaching, and research, 1890-1990. Teachers College Press. Freudenthal, H. (1983). Didactical phenomenology of mathematical structures. Kluwer. Halmos, P. (1985). I want to be a mathematician. An automathography. Springer. Kilpatrick, J. (1981). The reasonable ineffectiveness of research in mathematics education. For the Learning of Mathematics, 2(2), 22–28.

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Klein, F. (1908/2016). Fundamental mathematics from a higher standpoint, I–III (G. Schubring, Trans.). Springer. Kondratieva, M., & Winsløw, C. (2018). Klein’s Plan B in the early teaching of analysis: Two theoretical cases of exploring mathematical links. International Journal of Research in Undergraduate Mathematics Education, 4(1), 119–138. Madsen, L., & Winsløw, C. (2009). Relations between teaching and research in physical geography and mathematics at research intensive universities. International Journal of Science and Mathematics Education, 7, 741–763. Moslehian, M. E., Størmer, M., Thorbjørnsen, S., & Winsløw, C. (2017). Uffe Haagerup - His life and mathematics. Advances in Operator Theory, 3(1), 295–325. National Research Council. (2013). The mathematical sciences in 2025. The National Academies Press. Philips, C. (2015). The new math – A political history. University of Chicago Press. Schoenfeld, A. (2008). The math wars. Educational Policy, 18(1), 253–286. Verret, M. (1975). Le temps des études. Paris. Wilson, N. (2006). Encyclopedia of ancient Greece. Routledge. Winsløw. (1993). Strongly free actions on subfactors. International Journal of Mathematics, 4, 675–688.

Chapter 8

Collaborative Evaluation of Teaching and Assessment Interventions: Ideas From Realistic Evaluation Paola Iannone

Abstract Evaluation of small teaching interventions in university mathematics is an exciting area for collaboration between mathematicians and mathematics educators and one that is often overlooked in mathematics education. In this chapter, I will discuss how I became involved in such collaborations. I will do so by first introducing my background and my research focus and explain how this influenced the trajectory that my collaborative research with mathematicians has taken. I will then discuss how one policy change concerning UK Higher Education, the introduction of the Teaching Excellence and Student Outcome Framework (TEF), has caused UK universities, especially those with big STEM courses, to invest resources into supporting and evaluating innovations in teaching and assessment of mathematics. I will then draw from two collaborative research projects I have been involved in to reflect on what shape could the evaluations of (small) teaching interventions take and how these can become a fruitful collaborative enterprise for mathematicians and mathematics educators. Keywords University mathematics · Summative assessment · Assessment interventions · Realistic evaluation · Case study

8.1

Introduction

I have written elsewhere (Iannone, 2014) how a chance encounter started my career in mathematics education and moved me away from pure mathematics research. What I have not described in that writing is the subtle disappointment of my mathematician friends (colleagues were too polite to let this show) when it was obvious that I had moved away from pure mathematics to another discipline and a fluffy one at that! This transition was happening around the year 2000. Twenty years on, I welcome the opportunity given to me by the Editor of this book to reflect on some aspects of my research, which, inevitably, intersected with that of the many P. Iannone (✉) School of Mathematics, The University of Edinburgh, Edinburgh, UK e-mail: [email protected] © Springer Nature Switzerland AG 2023 S. Stewart (ed.), Mathematicians’ Reflections on Teaching, Advances in Mathematics Education, https://doi.org/10.1007/978-3-031-34295-0_8

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mathematicians I have collaborated with. My collaborative work with mathematicians has taken different forms and has had a variety of objectives. I have worked with mathematicians to: • Analyze and reflect on students’ work, involving mathematicians as co-researchers (e.g., Iannone & Nardi, 2005; see also Nardi’s chapter in this book); • Build a community of enquiry with mathematicians and mathematics educators to reason on and learn about teaching and learning mathematics (e.g., the PLATINUM Project, https://platinum.uia.no—see Jaworski’s chapter in this book); • Design and evaluate small-scale teaching and assessment interventions. In this chapter, I will focus on the latter: the design and evaluation of teaching and assessment interventions. I believe this to be a very interesting aspect of collaborative work with mathematicians, which is often not given due attention in the research literature. By design and evaluation of teaching interventions, I do not mean the evaluation of large programs of intervention, but rather the evaluation of smaller interventions such as the ones a practitioner may implement in their own practice. If a mathematician introduces a new assessment method in their module, how will they find out what impact this change had on their students? It will be apparent as I write this chapter why I consider those interventions and their evaluation important for mathematicians, mathematics educators and for the practice of teaching mathematics. I start by observing a change—at least in the United Kingdom where I work—in the perceptions that the community of mathematicians holds of mathematics education. Interest is growing in the community of mathematicians not only for the teaching of mathematics at university level but also for the research and findings that the community of mathematics educators has to offer. There are some tangible indicators of this change. Professional associations such as the London Mathematical Society (LMS) and the Institute for Mathematics and its Applications (IMA) in the United Kingdom, but also the American Mathematical Society (AMS) and the European Mathematical Society (EMS), are dedicating space in their newsletters to education issues, mostly regarding learning and teaching mathematics at university level, but also regarding mathematics in school. Examples such as the article by Alcock (2018) recently appeared in the Newsletter of the LMS, and the survey released by the EMS (Koichu & Pinto, 2019) aimed at understanding the transition between school and university mathematics (see also Gregorio, Di Martino & Iannone, 2019), are manifestations of the interest of these learned societies in mathematics education matters. This shift of perceptions has also meant that mathematicians are starting (albeit very slowly) to experiment with changes in their teaching practice and, more crucially, are starting to ask what impact these changes have on their students. In what follows, I will briefly discuss what are the contextual factors that, in the United Kingdom, have led to interest in innovation of teaching and evaluation at university level and what I mean by evaluation of small teaching interventions. I will then present two case studies of evaluation of teaching interventions I have been

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involved in recently, with particular emphasis on how the collaboration was shaped. I will conclude the chapter with some reflection on this evaluation work and why it is important, as mathematics educators, to be part of it.

8.2

Teaching Excellence and Student Outcome Framework and Evaluation of Teaching Interventions

It can be argued that the increased interest of mathematicians in the United Kingdom to experiment with new teaching and assessment methods and to understand the impact of teaching innovations on their students has in a first instance been motivated by a teaching accountability exercise recently introduced by the UK government (the Teaching Excellence and Student Outcome Framework—TEF). TEF’s outcomes have the potential to impact on many aspects of the way in which universities are funded, and therefore, it is important, for universities, to be able to show evidence in the TEF portfolio submission that they have engaged in teaching innovations and have in some way explored their outcomes (i.e., they are interested in pedagogy). This is especially true for STEM subjects, and in particular mathematics, as the teaching of these subjects is still perceived as old fashioned and teacher-centered. These small-scale innovations of mathematics teaching mostly originate from mathematicians themselves and come either from their reading of the mathematics education literature (especially journals aimed at practitioners such as the MSOR Connections journal in the United Kingdom) or from their own pedagogical experience. However, through taking part in a workshop organized by the IMA on effective assessment in university mathematics in May 2019, I realized that very motivated teachers of mathematics are often unsure of what to do when it comes to investigate the impact of the interventions they implement in their teaching on their students. In short, how can mathematicians become informed evaluators of their pedagogical interventions? What could such evaluations look like?

8.2.1

A Brief Note on Evaluation

In a very naïve way, Evaluation is that process by which, when introducing changes in a setting, we collect and analyze data to understand what the impact of the changes introduced was in that setting and if the changes achieved the desired outcomes. For example, if a community program for supporting teenagers to quit smoking is introduced in a community, evaluators would want to find out whether the program has achieved its aims. Evaluators will ask questions such as: following the introduction of this program, is the number of teenagers quitting smoking increasing (in that community)? If the same program was introduced in another community, would it still be effective? In the case of (very expensive) social interventions such as the ones related to public health, results of evaluations often guide the related policy (is the

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government going to increase the funding for community centeres supporting teenagers to stop smoking?). It is not difficult to see that evaluations are important in education too. The What Works agenda in the United States and the work of the Educations Endowment Fund in the United Kingdom (https:// educationendowmentfoundation.org.uk/news/the-teaching-and-learning-toolkit-acomplex-summary/) are examples of how evaluations (Experimental Evaluations, mostly using randomized control trials in the field and summarizing outcomes and effectiveness of interventions through meta-analysis) are informing policymakers and practitioners alike and attract most education research funding, at least in the United Kingdom. Evaluations of these type are large scale and require significant investment of time and resources and have attracted criticism both for the methodology used (e.g., Simpson, 2018) and for their theoretical underpinnings. In this chapter, I am not concerned with evaluations of large-scale interventions in university mathematics. I am instead concerned with the desire that those who teach mathematics at university have to find out whether small-scale interventions introduced in their own teaching have any impact on their students’ learning and what this impact may be. Both case studies described below are outcomes of collaborations with mathematicians and started from the wish to investigate the impact of a teaching intervention. It is important to note that the nature of such interventions was informed by the mathematicians’ teaching experience (as students and as mathematicians with teaching duties), by their reading of some of the educational research and by their own research in mathematics.

8.3 8.3.1

Two Case Studies of Collaborative Evaluation of Teaching Interventions Case Study 1: Summative Assessment of Mathematics

Summative assessment is a hot topic for university mathematics departments in the United Kingdom both because of its uniformity (Iannone & Simpson, 2022) with the written exam dominating assessment and because there is a growing literature (e.g., Bergqvist, 2007; Darlington, 2014) suggesting that written exams, as they are currently structured, fail to assess the types of reasoning valued by the mathematics community such as conceptual understanding and problem-solving. This realization, and the pressures of TEF, has motivated some mathematics departments to make funds available to mathematicians to develop innovative summative assessment methods and evaluate their impact. One type of summative assessment that has recently gained some attention (and that has been the focus of some of my recent research, e.g., Iannone & Simpson, 2015) is oral performance assessment. Oral performance assessment is a dialogic form of summative assessment: a question and answer session with the student writing at the board (or using pen and paper) while answering questions that can be of a theoretical nature (state seen definitions, proofs, or theorems) or of a more applied one (working out examples, tackling

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unseen problems or proofs, or using algorithms appropriately). Because this method can be labor-intensive for the assessors, and for historical reasons linked to the development of universities in the United Kingdom (Stray, 2001), it is very rarely used in this country, at any educational level. In the winter of 2017, I was contacted by colleagues from the mathematics department of a large and successful university in the south east of England asking me to collaborate in designing an oral performance assessment for two financial mathematics modules1 offered in the final year of a degree course in financial mathematics and evaluate its impact on the students’ experience of and engagement with the modules. My colleagues had the possibility to apply for an internal grant made available by their university in view of the TEF assessment and had come across some of my previous research on the affordances and drawbacks of this type of assessment for university mathematics. Crucial to their decision to trial this assessment method was the mathematicians’ experiences with the oral performance assessment, both as undergraduate students and as assessors when working outside the United Kingdom. These past experiences motivated my colleagues to trial oral performance assessment as an assessment that they believed would give them information about their students’ conceptual understanding of mathematics, information they felt was difficult to obtain from the current way of writing written exam questions. The study that followed, due to the small cohort sizes, was necessarily predominantly a qualitative one. We constructed a studentcentered instrumental case study (Stake, 2000) that allowed us to trace the impact of the new high-stakes assessment on students’ engagement and experiences. This case study was theory-driven: the overwhelming evidence from the work on deep and surface approaches to learning (Marton & Säljö, 1997) predicts a very strong link between the students’ perceptions of what the assessment requires in order to be successful and their approaches to learning. Therefore, we hypothesized that a change in high-stakes assessment would impact on students’ approaches to learning. The case study design was necessary in order to account for the context in which the intervention was carried out: would the same intervention in a different university give the same outcomes? The collaboration consisted of co-designing each stage of the intervention and the materials used. In order to do so, we met several times during the teaching period at the mathematicians’ university, where I observed lectures and some of the preparation sessions for the oral exams. In these meetings, we also discussed what was needed for the study in terms of material design. Such materials were both documents such as assessment grids for the students to help them make sense of the new assessment, and instruments for the research, such as the adaptation of an assessment preferences questionnaire and the design of the data collection. Each of us contributed our expertise and our professional knowledge. I contributed templates for the

1 University degrees in the United Kingdom are typically modularized. To progress, each student needs to accrue a certain number of credits by taking appropriate modules. The intervention of this case study was carried out in two such modules, which were optional for students in the third year of study.

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assessment documents, which had proved to be useful in my previous research (e.g., the assessment grid) and research tools (e.g., the questionnaire) related to the research design. My colleagues decided, for example, on the structure of the mock exams and the questions within those. Every resource, once drafted, was up for discussion. Particularly interesting were the exchange of ideas that happened in one of our meetings around the general assessment grid as included in the students’ handbook and how this could be reflected in the assessment grid for the oral performance assessment. Such discussions helped me to focus on what my colleagues valued when they assessed their students and how they felt they could pinpoint conceptual understanding of the mathematics they had taught. The one difficulty in this process was that I could not share the students’ data with my colleagues before the end of the academic year when the students left the university as the data collected could potentially have influenced the assessment process. I worried that this would create friction between myself and my colleagues but as it happened, it did not, and we all agreed that this was the best course of action. Interestingly, the two mock oral performance assessments, one for each of the two modules, were designed differently and followed the preference of each of my colleagues. We all agreed early on in the study that the students needed to experience the new assessment method in a formative way during the teaching period as none of them were likely to have experienced oral performance assessment before. We therefore decided to offer a mock assessment for each module. However, there was no agreement on how to structure this mock assessment: while one of my colleagues prioritized similarity to the real oral performance assessment experience over exposure to the type of questioning, the other opted to prioritize exposure over realism. This meant that the two mock exams were very different. The one for the first module was held beyond closed doors to replicate the final oral performance assessment conditions; therefore, each student only experienced their own mock assessment and could not witness the questioning associated to the mocks of their peers. The second one was carried out during each seminar, one for each student, which meant that each student could see all the mock assessments, but they were not as formal as the final oral performance assessment would be and probably not as intimidating for the students. This was one of the instances where the difference in focus between myself and my colleagues revealed itself: while I worried about the solidity of the research design, my colleagues were bold, trying to allow the students to experience several conditions for comparison. During this process, we learned a lot about the ways in which we each work as teachers of mathematics interested in pedagogy: my colleagues saw that organizing a case study is more difficult than asking a few students some questions and have appreciated the need to resort to a diverse data collection. I witnessed how some of my best research-informed intention may just be impractical in a classroom and will need to be adjusted to the context in which they are to be implemented. One example of the latter was the inability to collect an end of the module questionnaire: students were just not interested in this exercise, and this part of the data collection had to be abandoned. Following the case study research tradition, we tried to understand the role of the context in the evaluation and the mechanism behind the findings. Would

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students in a different university with much lower employability rates and much lower emphasis on employability react differently to this assessment innovation? As a result of this study, we found that although we could see some desirable effects in terms of student study habits and engagement, for some students the novelty of the assessment prevented them to employ successful revision strategies. Some students found thinking about a different way of being assessed that late in their studies so hard that they could not see how to change their preparation practices in order to be successful. Of note here is that the findings contribute to the original theory in that they help explain what are the barriers that prevent students engaging in deep learning even when they perceive the assessment to require such learning in order to be successful. The resulting publication (Iannone et al., 2020) was coauthored with my colleagues in mathematics. This also was a collaboration that taught me much about different social norms related to writing in the two communities. The paper we wrote is published in a mathematics education journal, and my mathematician colleagues found that style of writing somewhat verbose!

8.3.2

Case Study 2: Learning Proof with Lean

Again motivated by the demands of TEF, one of the largest London universities, which has top research ratings and top research income for STEM subjects in the United Kingdom, but that often scores very low in league tables for student experience, has recently invested considerable resources in the implementation and evaluation of teaching innovations. One such evaluation started when a mathematician there, whom I had met at a meeting of the LMS, contacted me regarding one intervention he was implementing in a first-year transition to proof module. This module had been taught very traditionally for a number of years, and my colleague had recently started to introduce the students, on a voluntary basis, to the software Lean (https://leanprover.github.io) in connection to the module material. Lean is an open-source interactive theorem prover in which it is possible to compute mathematical objects with precise formal semantics, and it is built on a verified mathematics library. The development of Lean belongs to the part of mathematics concerned with proof automation, aiming at producing and programming a language that eventually may be used to verify existing proofs. Lean had been used for teaching undergraduate mathematics before (see Avigad, 2019, for the use of Lean in a basic logic module), but so far there had been no systematic investigation of the impact of teaching with Lean on students’ proof production. The year we carried out the evaluation, my colleague had designed all the module materials (exercise sheets and lecture notes) so that they could be used both in a traditional way (solving exercises with pen and paper) and by using Lean. He also offered voluntary workshops on programming Lean that students on the module could attend. My colleague had run the module this way once already, and he was motivated to do so by his own research: the workshops with the undergraduate students were helping in

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constructing a richer library for Lean so that the capabilities of the software could be enhanced and eventually it could be used to check proofs. Therefore, the original motive for introducing this software was a research motive—the advancement of a tool that was used in mathematics research—rather than a pedagogical one. At the time this mathematician contacted me he had, however, started to ask himself what impact being familiar with Lean programming had on his students’ understanding of the demands of formal proof and on their engagement with the module. By his own admission in one of our early conversations, he told me that it was only recently that he had started to think about the way in which he taught and what effect his pedagogical choices had on his students’ learning. This collaboration was different in nature from the one described in the previous case study. In designing alternative assessment, the mathematicians involved in the previous case study reflected on their own experience as students and teachers and intended to prioritize the assessment of aspects of mathematics they found valuable (e.g., conceptual understanding). In this case study, a tool originating from mathematics research was used to support teaching, and our study proposed to find out whether its use would have any pedagogical value. For the Lean study, my contribution to the design of the teaching resources was minimal, and my lack of knowledge of the software did not allow me to keep up with the lecturer and was mostly limited to the design of the data collection. The mathematician teaching the module was invaluable in helping with the design of the proof tasks included in the material we gave the students and the proof tasks included in the interviews. With his help, myself and my colleague Athina Thoma, who worked with me on this project, were able to gauge the difficulty and nature of the tasks appropriately to the students. Also this case study can be seen to be guided by theory: the frameworks of Selden and Selden (2009) and MejiaRamos et al., (2012) allowed us to think plausible that engagement with a software such Lean would help students improving the quality of their proof writing and proof structure. Much of our collaboration revolved around the exchange of ideas concerned with the use and potential of the software as a learning tool. While it is plausible to think that learning to program a software designed for testing proofs would help students overcome some of the difficulties that are well documented in the literature, the software as it was in 2018 when we collected the data appeared to be too complex for many students to engage with. Those students who had decided to attend the voluntary workshops gave up much of their time to this activity, and such a large time investment would not be an option for many students. The supporting materials accompanying the software (e.g., manuals and library) were also in the early stages of development at that time, and this was a big factor in the low students’ uptake, as we found out during the data collection when we enquired why most students did not engage with the voluntary sessions. I debated these issues with my mathematician colleague at great length during our meetings, but it was really hard for me to explain how, if we wanted to find some impact on students’ performance, we would have wanted to have a large number of students engaged with the software across the full achievement spectrum. It could be possible otherwise that using this software would benefit proof writing of high-achieving students but could be detrimental for

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low-achieving students. At the time of the study, only 18 out of 300 students in the cohort attended the voluntary programming workshops with regularity, which meant that the original idea of investigating the impact of software use on achievement in proof tasks was not viable anymore. Moreover, the students who attended the workshops were obviously highly motivated and enthusiastic, and any impact on their achievement could have been justified by more study time spent attending the workshops. As a researcher in mathematics education, I perceived this as problematic for the study we wanted to develop, while my colleague did not perceive this as a problem as, he reasoned, the students were not selected according to achievement (but they self-selected). Once we gathered the uptake data I mentioned previously, our collaboration changed focus and became an explorative study on accessibility and use of the software as well as students’ proof production and proof writing. The mathematician brought the enthusiasm of the big development in sight for mathematics to the collaboration, we brought the concern for the need of the students, all students, not only the ones who were able to give up time and effort to engage in non-compulsory activities. One of the outcomes of these exchanges of ideas was the design of the Natural Numbers Game. In our discussions, we agreed that it is hard to teach new and challenging mathematics while at the same time teaching how to program a very complex language, which is what had happened while teaching the transition to proof course. For such programming activities to be successful and to engage large numbers of students, the mathematics involved had to be easy so that the students could focus on the programming aspect. Therefore, our colleague and some of his undergraduate students designed the Natural Number Game (http:// wwwf.imperial.ac.uk/~buzzard/xena/natural_number_game/): an online game students can engage with and where they can learn how to program in Lean. In this game, the mathematics is very familiar to the students, and the focus of the learning is programming of the software. The study that was eventually designed had a different aim from what we had thought initially. It investigated common characteristics of proof produced by students who had engaged in the voluntary workshops by comparing those proofs to those produced by students who did not attend the workshops but produced proof at the same achievement level. We also investigated what were the most common barriers to the use of the software, as reported by the students. We are still in the process of analyzing the large data set2 we collected, but early findings suggest that students who had engaged with the software tended to use better mathematical formalism when writing proofs than students who had not engaged at all, but on occasion, this formalism could get in the way of semantic thinking about the mathematical objects involved in a proof. We also realized that students found the interface of the software Lean very hard to use, and this was a big component of the low engagement with the intervention we observed. As in the previous case study, we cannot help ask whether we would have found the same enthusiasm and engagement from the part of some of the students if this

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Much of the research that was in progress at the time of writing has now been published.

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had not been a research-intensive university, the teacher had not been a very charismatic mathematician, or if the students had not been among the best qualified in the country. In short, what could a mathematician in a different university take from our research findings?

8.4

Evaluating Teaching Interventions: The Importance of Mechanism

In the previous part of this chapter, I have outlined the contextual reasons that have encouraged some mathematicians in the United Kingdom to implement small-scale changes to their teaching and their institution to support these changes. In two examples of small-scale collaborative evaluations, I explained how I became involved in this type of research and what this research may entail. These smallscale studies have the potential to contribute to basic knowledge, through, for example, the detailed analysis of the proof data in the second intervention, and to curriculum and assessment design, through suggesting which aspects of the intervention have been effective and why. I will now discuss some aspects of such collaborative evaluation of teaching/assessment interventions, and I will do so by referring to some of the principles of realistic evaluation (Pawson & Tilley, 1997). While experimental evaluation is, in its simplest form, designed on a cycle of Observation, Experiment, Observation (Pawson & Tilley, 1997) and relies predominantly on large-scale randomized control trials in the field, realistic evaluation differs from experimental evaluation in that focuses on context and mechanisms. Realistic evaluators ask what is the mechanism that allowed or prevented one intervention to be effective in the specific context in which it was implemented. As Tilley puts it in his address to the Danish Evaluation Society in 2000: Whereas the question which was asked in traditional experimentation was, “Does this work?” or “What works?”, the question asked by us in realistic evaluation is “What works for whom in what circumstances?” (Tilley, 2000, p. 4)

The design of realistic evaluations starts from theory, which shapes the hypothesis that guides the design of the intervention (what might work for whom in what circumstances). The design of the intervention is then tested via observations and data collection of various kinds, quantitative (including randomized control trials if relevant) and qualitative. These observations inform generalization where the emphasis is on what works for whom in what circumstances (Pawson & Tilley, 1997). These observations of the implementation of the intervention are bound to involve multiple methods of data collection, as in the two case studies above. Pawson and Tilley (1997) call mechanism the changes to the context where the intervention is implemented that cause changes in the set of outcomes that is observed and that the intervention wishes to change. Therefore, the focus of realistic evaluation is not just to ascertain whether an intervention has achieved the desired results, but above all to understand the mechanism that has originated such change and that has led to the results observed in the context where the intervention was

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implemented. As it is clear from the work of Pawson and Tilley, carried out mostly in the evaluation of large interventions concerning health policy, realistic evaluation requires time and resources. Its progress is slow and usually requires the work of many researchers and the collaboration of many practitioners. None of these was true in the two case studies I have described earlier in this chapter, yet the construction of that studies was inspired by the principles of realistic evaluation in the way I will describe below. • The case studies had a consistent focus on context, and because these were evaluation of small interventions, the data collected were mostly (but not only) qualitative. Efforts were also made to design the research tools (questionnaires, interview schedules, mathematics tasks) coherently and above all collaboratively with the mathematicians. Each tool was designed to answer a specific (research) question, and some of the tools were designed sequentially (i.e., the design followed the data analysis of the previous round of data collection). • The mathematicians who had implemented the interventions were not considered subjects in the studies but rather collaborators who brought their practitioner expertise to the study. They had a big part in every step of the research, especially in the design of the tasks and the resources for the students. This point highlights the importance of the professional knowledge that the practitioners bring to the evaluation. • Both small evaluations asked relatively bold questions: could a change of assessment change the approach and engagement with the subject in a student cohort where non-attendance and disengagement are often the norm (as Moore, Armstrong & Pearson, 2008, found)? Could the introduction of an automated prover such as Lean help students overcome some of the difficulties with writing proofs, which have been highlighted in the literature (e.g., Moore, 1994)? This idea is in line with the realistic evaluation suggestions and with Pawson and Tilley (2001) who argue that even small interventions, if guided by theory, can contribute to the refinement of that theory. Both case studies were indeed guided by theory: Marton and Säljö’s (1997) approaches to learning for the first case study on oral performance assessment, and the proof writing (Selden & Selden, 2007) and proof comprehension (Mejia-Ramos et al., 2012) frameworks we used for guiding our analysis in the second. • Both case studies aimed not only to find the impact of the interventions but also to understand the mechanism that originated what was observed so that practitioners who would want to consider the same interventions in their context could infer what their context meant in terms of intervention outcomes. Description of the mechanism that produced the outcomes we observed will be integral part of any dissemination of the two case studies. It comes therefore as a natural next step, having highlighted some of the points above, that some collaborative discussion with mathematicians could be open regarding issues of evaluation. In the United Kingdom, there are already forums where this discussion could happen: as reported earlier, the mathematics learned societies are gaining progressively interest in education-related issues. The IMA, for example, funds workshops for those who teach mathematics at university level to

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share practice, and it is investigating, at the time of writing, the possibility to support workshops dedicated to the evaluation of teaching interventions. These workshops could offer a natural forum to discuss collaboratively between mathematicians and mathematics educators such issues. Mathematics educators (like myself) could bring the theoretical tools to conceptualize such studies and on which to base the design of the interventions, mathematicians could bring the professional and discipline knowledge to make such studies work in context. The interest of this learned society in evaluation issues has originated from its membership signaling the importance for UK universities of a good TEF performance, but above all a real change in the mathematics’ perception of university teaching and its pedagogy.

8.5

Concluding Remarks

In this chapter, I have drawn on my own research experience to focus on the design and evaluation of teaching innovations in university mathematics and related issues of adaptability and transferability of such interventions. I have also outlined some principles inspired by the work of realistic evaluators, which could be useful to mathematics practitioners and mathematics education researchers alike who may wish to understand the impact of innovations in their own teaching and the transferability to their own context of the innovations of others. I believe this to be an important issue: in the chapter by Dawkins and Weber in this book, the authors discuss the reasons why so little of the teaching and curriculum innovations that have been proposed by mathematics educators have had an impact on the way in which university mathematics is taught. Dawkins and Weber offer a very detailed and accurate analysis of these reasons and make very insightful suggestions to overcome this problem. I would add to their suggestions that it is also necessary to give mathematicians the tools to study the impact of the transfer of interventions into their own context; otherwise such interventions may not be as successful as predicted by the research that originated them, and teaching and assessment may eventually revert to its traditional form largely ignoring mathematics education findings. Acknowledgments I would like to thank all the mathematicians who have collaborated in the studies I have conducted for my research, and in particular Christoph Czichowsky and Johannes Ruf, with whom I collaborated for the oral performance assessment case study, and Kevin Buzzard and Athina Thoma, without whom the second case study would not have happened. I would also like to thank all the students who have been participants in my research.

References Alcock, L. (2018). Tilting the classroom. London Mathematical Society Newsletter, 474, 22–27. Avigad, J. (2019). Learning logic and proof with an interactive theorem prover. In Proof technology in mathematics research and teaching (pp. 277–290). Springer.

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Bergqvist, E. (2007). Types of reasoning required in university exams in mathematics. The Journal of Mathematical Behavior, 26(4), 348–370. Darlington, E. (2014). Contrasts in mathematical challenges in A-level Mathematics and Further Mathematics, and undergraduate mathematics examinations. Teaching Mathematics and its Applications: An International Journal of the IMA, 33(4), 213–229. Gregorio, F., Di Martino, P., & Iannone, P. (2019). The Secondary-Tertiary Transition in mathematics. Successful students in crisis. EMS Newsletter, 9(113), 45–47. Koichu, B., & Pinto, A. (2019). The secondary-tertiary transition in mathematics. What are our current challenges and what can we do about them? EMS Newsletter, 6(112), 34–35. Iannone, P., & Nardi, E. (2005). On the pedagogical insight of mathematicians: Interaction and transition from the concrete to the abstract. Journal of Mathematical Behavior, 24, 191–215. Iannone, P., & Simpson, A. (2022). How we assess mathematics degrees: The summative assessment diet a decade on. Teaching Mathematics and its Applications: An International Journal of the IMA, 41(1), 22–31. Iannone, P. (2014) Mathematics & mathematics education: Searching for common ground, edited by M. Fried and T. Dreyfus, New York, Springer, 2014, 402 pp., 90, ISBN978-94-007-7472-8. Research in Mathematics Education, 16(3), 330–334. Iannone, P., & Simpson, A. (2015). Students’ views of oral performance assessment in mathematics: Straddling ‘assessment of’ and ‘assessment for’ learning divide. Assessment and Evaluation in Higher Education, 40, 971–987. Iannone, P., Czichowsky, C., & Ruf, J. (2020). The impact of high stakes oral performance assessment on students’ approaches to learning: A case study. Educational Studies in Mathematics, 103(3), 313–337. Marton, F., & Säljö, R. (1997). Approaches to learning. In F. Marton, D. Hounsell, & N. Entwistle (Eds.), The experience of learning. Implications for teaching and studying in higher education (pp. 36–55). Scottish Academic Press. Mejia-Ramos, J. P., Fuller, E., Weber, K., Rhoads, K., & Samkoff, A. (2012). An assessment model for proof comprehension in undergraduate mathematics. Educational Studies in Mathematics, 79(1), 3–18. Moore, R. C. (1994). Making the transition to formal proof. Educational Studies in Mathematics, 27(3), 249–266. Moore, S., Armstrong, C., & Pearson, J. (2008). Lecture absenteeism among students in higher education: A valuable route to understanding student motivation. Journal of Higher Education Policy and Management, 30(1), 15–24. Pawson, R., & Tilley, N. (1997). Realistic evaluation. Sage. Pawson, R., & Tilley, N. (2001). Realistic evaluation bloodlines. American Journal of Evaluation, 22(3), 317–324. Selden, A., & Selden, J. (2007). Overcoming students’ difficulties in learning to understand and construct proofs (Tech. Rep. No. 2007-1). Tennessee Technological University, Department of Mathematics. Simpson, A. (2018). Princesses are bigger than elephants: Effect size as a category error in evidence based education. British Educational Research Journal, 44(5), 897–913. Stake, R. E. (2000). Qualitative case studies. In N. K. Denzin & Y. S. Lincoln (Eds.), Handbook of qualitative research (pp. 443–466). Sage publications. Stray, C. (2001). The shift from oral to written examination: Cambridge and Oxford 1700–1900. Assessment in Education: Principles, Policy & Practice, 8(1), 33–50. Tilley, N. (2000). Realistic evaluation: An overview. In Founding conference of the Danish evaluation society (8). Retrieved on the 04.06.2019 from http://healthimpactassessment. pbworks.com/f/Realistic+evaluation+an+overview+-+UoNT+England+-+2000.pdf

Chapter 9

The Development of Inquiry-Based Mathematics Teaching and Learning Barbara Jaworski

Abstract This chapter focuses on a long-term development of insights into the teaching and learning of mathematics from personal, professional and research perspectives. Its objectives are to communicate from situations and events that led to these insights and to share theoretical perspectives related to developments in mathematics teaching and learning at different times. The idea of inquiry as a way of being permeates the developmental process, with relationships between participants (students, teachers and researchers) within communities of inquiry and associated critical alignment emerging as central constructs. The chapter uses vignettes from professional and research practice to provide examples that can illuminate development. Keywords Mathematics education development · Inquiry as a way of being · Communities of inquiry · Critical alignment · Professional and research practice

9.1

Background and Introduction

I am a mathematics educator and a researcher in mathematics education. I have taught mathematics at secondary and tertiary levels. I have been a teacher-educator, working with prospective teachers and practising teachers. I have helped to create a new doctoral programme and worked with doctoral students in mathematics education, teaching and supervising. I am interested fundamentally in researching mathematics teaching development: this includes the ways in which teaching develops when teachers become involved in researching their teaching practice. I have engaged with this research for many years with secondary-school teachers, where I have been the researcher (from the outside), and the teacher has been teacher-researcher (from the inside). More recently, I have

B. Jaworski (✉) Loughborough University, Loughborough, UK e-mail: [email protected] © Springer Nature Switzerland AG 2023 S. Stewart (ed.), Mathematicians’ Reflections on Teaching, Advances in Mathematics Education, https://doi.org/10.1007/978-3-031-34295-0_9

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been involved in research into teaching development at university level, in some cases where I myself have been the teacher-researcher. Being a mathematics educator, for me, includes all of the above. The one central part of it all, the reason why it all is important, is the sincere desire to enable learners of mathematics to engage deeply with mathematical ideas and to know what it means to understand mathematical concepts. They need to know, to be aware, that it is worth struggling to understand. In addition, I follow Paola Valero (UTube: 1.8.19), who says ‘Mathematics is always considered as a subject which is abstract and has nothing to do with people. [However, it] has all to do with how we think of ourselves and how we think of ourselves in relation to society’. My research reflects this position. In her introduction, editor Sepideh Stewart quotes Tommy Dreyfuss (1991) in suggesting that ‘one place to look for ideas on how to find ways to improve students’ understandings is the mind of the working mathematician’. This suggests a constructivist perspective on mathematics learning focusing on the individual mind of the learner. Although my own PhD thesis (Jaworski, 1991) was conducted from a (radical) constructivist perspective, I have moved since then into researching within a sociocultural frame. So, my current research is from a sociocultural perspective and reflects the position quoted from Paola Valero above and the ways in which this fits with seeking mathematical understandings for all students. A sociocultural perspective looks beyond the mind of the learner to consider the whole sociocultural setting in which the learner engages. Where mathematics is concerned, this includes the history and culture in which mathematics has been experienced and in which a learner’s mathematical progress is configured. I include the teacher also as a learner: not only as a learner of mathematics but also as a learner of mathematics teaching, creating the ways in which mathematics is framed for the student (mathematics didactics) and the ways in which the environment facilitates the student’s mathematics engagement (pedagogy). However, I am interested not only in studying teaching development but also in influencing that development through the research process. In this respect, I take an inquiry perspective to the development. This includes the use of inquiry-based tasks for students through which they develop mathematical understandings; inquiry processes for teachers through which they explore and develop teaching approaches; and research inquiry, into inquiry-based learning and teaching, which not only charts the development as it takes place but also simultaneously supports and contributes to that development. Thus, within the sociocultural perspective there is an inquiry perspective that defines inquiry, inquiry community and other associated concepts. I expand on these starting points in the sections below, but first a vignette to capture the essence of what I shall write about below.

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Vignette: Investigating the Teaching of Linear Algebra

Shortly after starting to work in the Mathematics Education Centre at Loughborough University, I persuaded a colleague in mathematics, a research mathematician, Thomas Bartsch, to allow me to observe his teaching of linear algebra and talk with him about his perspective on teaching linear algebra to first-year mathematics students. I wanted to start to understand how mathematicians regarded the process of teaching mathematics at university level. A new PhD student, Stephanie Thomas, joined me in this task. Stephanie and I talked with Thomas throughout the first semester of his teaching, and we observed most of his lectures, asking him questions related to what we observed. Thomas was very articulate about his teaching; one thing we observed over and again was that what he told us in conversation, we subsequently heard him say to students at some later stage. There was certainly a consistency between what he said he would do and what we actually saw (or heard) him do in lectures. Our conversations occurred in his office or around the campus as we walked with him to a lecture, listened to his reflections after a lecture or sat in the cafeteria eating lunch together and questioned him on his mathematical examples or his perspectives on how students responded to his tasks. This was a rich experience for Stephanie and myself, and Stephanie ultimately wrote her thesis with an Activity Theory perspective based on her analyses of Thomas’s teaching (Thomas, 2012). One of the most obvious characteristics, emerging from our inquiry into his practice, was the way in which Thomas talked to us and to his students; this was a distinction between talking mathematics, talking about mathematics and advising students as to how they should work with mathematics. For example, (Jaworski et al., 2009): (Talking about mathematics) Thursday is about defining the characteristic polynomial, understanding that its zeroes are the eigenvalues, and I’ll show an example of an eigenvalue that has algebraic and geometric multiplicity 2. (Talking mathematics) . . . algebraic multiplicity, meaning this is the power with which the factor lamda minus eigenvalue appears in the characteristic polynomial, and geometric multiplicity is the number of linearly independent eigenvectors. (Talking about mathematics) And these are the important concepts for determining if a matrix is diagonalisable because, for that, we need sufficiently many linearly independent eigenvectors. (Advising students on how they should work with mathematics) But it’s important that you be able to understand the language that we’re using and to use it properly. So please, pay attention to the new terms and the new ideas that we’re going to introduce over this chapter.

I present the above ideas to illustrate what we learned about Thomas’s approaches to teaching expressed through our analysis of the three different modes. However, as well as presenting an analysis of his style of teaching, we also paid attention to Thomas’s reflections on the effects his talk to students had on their understanding of the concepts involved. He said on one occasion:

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I do think, however, that didn’t go very well because . . . many students aren’t sufficiently familiar with the idea of a linear transformation. We have discussed that many times, that a linear transformation is a function that is defined by a matrix. But my impression is that very many students haven’t absorbed that idea of reading a matrix as a function. And whenever I talk about the transformation that is defined by a matrix in a small group tutorial, or when going around in class, quite often I get a blank stare. Now that being as it is there seems not much point in trying to express that function in a different basis, so that is . . . probably most students haven’t really absorbed that section.

Here we see the teacher, confident in his own exposition of concepts in linear algebra, and what students themselves need to be aware of and do in their work on LA, expressing uncertainty in a situation where his way ahead with students is not clear. Inevitably, since teaching does not wait for endless reflection and deliberation, he would resolve the dilemma in how he approached the topic of change of basis with students. However, he acknowledged on several occasions that talking with us, presenting an opportunity to reflect aloud and discuss a tricky situation, was helpful in seeing his way through and resolving his dilemma. There are many issues here of importance. First, a mathematician and mathematics educators inquiring together into the teaching of LA resulted in all of us gaining insights and learning from the experience. Second, the research here started to characterise one version of the teaching of LA. When this is published, it becomes available in the public domain for other teachers of mathematics to compare what they do and their goals for doing it with what is presented in the characterisation. Third, the inquiry process here, an example of developmental research, enables a form of professional development for all participants, in a practice that to date offers few such opportunities. In what follows, I chart a lengthy process through which I came to the experiences reflected on in this vignette.

9.3

Investigating Mathematics Teaching in Schools and Classrooms

I studied for my PhD at the Open University in the United Kingdom where I worked for 5 years in the Centre for Mathematics Education (CME) alongside John Mason, David Pimm and other mathematics educators. Prior to this, I had been a full-time teacher of secondary school mathematics, with a part-time job as tutor to students in the first-year Open University mathematics course (M101). In my school teaching, I had worked in a comprehensive ‘upper’ school (students aged 13–18, pre-university) teaching mathematics and as head of the Mathematics Department (HoD). In my Open University teaching, I tutored students in introductory mathematics, both at a distance and in summer schools. At summer school, I was introduced to the ideas of mathematical investigations as a means of developing mathematical thinking and learning.

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In the comprehensive school, our students came from a wide cross-section of society: some seemed very able in mathematics; others struggled. As teachers, we struggled to know how best to work with students who struggled; there were very different perspectives on this among department members. While it was a delight to teach those students who aimed for university studies in mathematics, it was a constant challenge as to what approaches were best for the other students. In those days, most of our teaching fitted a description of ‘exposition with consolidation and practice’ with some (directed) ‘problem-solving’ where appropriate. We were not experienced with other modes of pedagogy (or associated didactics), such as investigation, practical activities, classroom discussion and open problem-solving. The Cockcroft Report (1982) of the Committee of Inquiry into the Teaching of Mathematics in Schools had been recently published, and we were starting to be aware of these ‘extra’ modes of teaching. My experience with using investigational tasks (i.e. tasks in which the student is challenged to engage, explore, ask questions, make conjectures, justify and prove) in Open University summer schools provided insights to the value of such activity in mathematics. As HoD in my school, I invited educators from the Open University Centre for Mathematics Education (CME) to talk with teachers in our department about these ways of working with students. It became clear that while they, the OU educators, were more theoretically knowledgeable, they also were not experienced in the practical day-to-day classroom activity which was needed to fulfil the recommendations of the Cockcroft Report (HMSO, 1982, pp. 66 ff.) Shortly after these discussions, I was invited to take up a lecturing position in CME. My first task was to collect video material from regular classrooms in which students engaged with the range of activities recommended by Cockcroft, especially investigation, discussion and practical work. This was a highly privileged position in which I was able to visit many schools, observe teachers with their students and talk with the teachers, often with a whole department, about their understanding of the Cockcroft recommendations and their ways of implementing these with students in their classrooms. I was on a steep learning curve from these new experiences which had not been possible when I was a teacher myself. The excitement this generated led me to start my own PhD in which I focused on what it means to teach mathematics through investigative approaches. The environment was ideal since I had relationships with dedicated teachers who were interested themselves in exploring these ideas with their students. The data I collected consisted largely of classroom observations (audio and some video-recorded) and audio recordings of conversations with teachers. With Open University colleagues, I spent many hours viewing and analysing video recordings of secondary classrooms as well as developing theoretical insights into the practical nature of investigative approaches to teaching mathematics. After 5 years of work, I submitted a thesis entitled Interpretations of a Constructivist Philosophy in Mathematics Teaching’ and gained my PhD (Jaworski, 1991).

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A Constructivist Perspective on Teaching

In those days, I was very committed to constructivism. I had met it for the first time at PME 1986 in London where I joined a group focusing on in-service education for secondary teachers and heard an impassioned account on constructivist teaching from Robert Underhill from Virginia Tech in the United States. This introduced me to the works of Ernst von Glasersfeld, Les Steffe, Paul Cobb, Jere Confrey and others, all committed constructivists, and I read everything I could get my hands on. Glasersfeld’s work was particularly compelling. I was very ready to believe in the activity of the (individual) human mind ‘constructing’ knowledge through experience, rather than having it transmitted directly from other scholars. The big question that emerged from my reading and thinking, and which I discussed with many scholars over my PhD years, was this: If individuals construct their own mathematical knowledge based on their own experience, how can it be that they all come to the same (established) meanings of mathematical concepts?

Or, in the words of Alan Bishop (1988): Given that each individual constructs his own mathematical meaning how can we share each other’s meanings?

I was attracted by Les Steffe’s notion of construction of ‘second-order models’ in which a teacher, seeking to understand a student’s mathematical conceptions, has to construct a model from what she discerns from the student’s activity. Thus, listening to students, asking them about their mathematical perceptions, encouraging discussion between them, and getting them to write their own mathematical explanations became central to a so-called constructivist pedagogy. The idea of investigational tasks which involved students in exploring mathematical ideas for themselves in discussion with their colleagues and with the teacher seemed to offer a mode of activity that would engage students with mathematics, get them really thinking mathematically and give a teacher access to this thinking. Thus, the teacher could learn (construct a model) about a student’s conceptions from which her own questions and explanations could be informed. As this activity proceeded, the theory was that student and teacher could become increasingly close in their mathematical conceptions with the student approaching the established version of a concept. Of course, in this vein, established concepts are no more than a convergence of meaning through millennia of mathematical discussion between mathematicians. My PhD examiners encouraged me to convert my thesis into a book in which I would link theory and practice through case studies from the teachers with whom I worked over long periods. This was published under the title ‘Investigating Mathematics Teaching – a constructivist enquiry’ (Jaworski, 1994).

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Teaching Development Emerging from Research into Teaching—Challenges to Radical Constructivism

One of the most significant learning experiences from my PhD work was my awareness of what certain teachers called ‘hard questions’. These were questions that I asked as a researcher which a teacher found difficult to answer, largely because s/he had never before been challenged to articulate such ideas (Jaworski, 1998). From me, they were genuine questions. I sincerely wanted to know why a teacher had acted in a certain way. As we talked about teaching actions, and teachers tried to explain the ‘why’ of acting with students, my own awareness of classroom practice developed along with that of the teacher. Teachers acknowledged that their thinking about teaching became more explicit, that such reflection revealed their own knowledge previously unspoken. This knowledge might be thought of as finger-tip knowledge of teaching, that is, practice knowledge on which a science of teaching could develop. Such a science of teaching emerged for me too and was theorised within a radical constructivist philosophy (Jaworski, 1994). The essence of Radical Constructivism, as presented by Ernst von Glasersfeld, was two-fold: first, that knowledge is not passively received but actively built up by the learner; second, that the learning process is adaptive and enables learners to make sense of their experiences rather than to discover external truths about the world (Glaserfeld, 1987). Reading Glasersfeld and other constructivists took me back to Piaget, and the concepts of assimilation, accommodation and reflective abstraction in trying to make sense myself of what I was seeing and hearing and what I was learning from it. In those days (mid-1980s), the CSMS research (e.g. Hart, 1981), rooted in Piagetian psychology, provided many examples of children’s strategies and errors (or misconceptions as they were sometimes called) in responding to mathematical questions. This challenged the readers, who might think about these errors in constructivist terms, to consider the basis by which these learner conceptions were misconceptions. By what basic truths could a conception be termed a misconception? Such ‘basic truths’ or external truths were precisely what (radical) constructivism abhorred. Being construed as a constructivist denied one the possibility of external truths to explain the errors. This led to another serious issue: in communication, could we ever claim any conception to be ‘true’? Could we therefore justify something to be an error? And as teachers working with students in mathematics, how might we justify our own knowledge of mathematics as being the right knowledge to which our students should aspire? These thoughts led to a questioning of the idea of ‘intersubjectivity’—the belief that two or more people could believe they had common understandings, such as the mathematical understandings of number or spatial relationships. Such questioning came from scholars who saw intersubjectivity in sociocultural terms referring to the work of Vygotsky in his ‘general genetic law of cultural development’ in which he postulated learning as occurring on two planes, the first socially, between people (inter-mental), and the second, developing as part of the first, in the individual plane (intra-mental) (Wertsch, 1991, p. 26). Lerman (1996), writing about

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intersubjectivity, challenged the writings of Glasersfeld, Steffe and others regarding the nature of intersubjectivity. In return, Steffe challenged these ideas robustly from his radical constructivist perspective (e.g. Steffe & Thompson, 2000). There seemed to be an impasse between the two theoretical factions. I was fortunate enough at this time to have long conversations with Lerman and Steffe (separately). An upshot of this was my own movement towards more sociocultural perspectives. I was able to look back on my own writings (e.g. Jaworski, 1994), in which I had been strongly aware of interactions between teacher and students in classrooms I observed, and start to see these analysed from a sociocultural perspective taking more account of the wider sociocultural setting than a constructivist analysis encouraged and presenting a more believable account of learning and development.

9.6

Research Collaboration Between Teachers and Researchers

Over the next decade, three projects brought me more overtly to a dimension of my research that has permeated my work since then. These were (a) The Mathematics Teacher Enquiry Project (Oxford, 1994–6, Jaworski, 1998); (b) The Teaching Triad Project (Oxford 1997–2000. Potari & Jaworski, 2002) and (c) Learning Communities in Mathematics (Kristiansand, Norway, 2003–2007, e.g. Jaworski, 2006, 2008). In these projects, my study of teaching in classrooms extended to a study with teachers of their teaching and its development. These projects still focused on ideas of investigational activity involving both investigations in mathematics with students and investigation of teaching approaches and processes. The idea of investigation continued to be a focus of interest, and it became part of a theoretical perspective on inquiry (or enquiry).1 I had cited various authors on inquiry (Jaworski, 1994, pp. 10–11), Inquiry teaching forces students in articulating theories and principles that are critical to deep understanding of a domain (Collins, 1988); [Inquiry is] fallible, socially constructed, contextualised, and culture dependent. . . driven by the human desire to reduce uncertainty but without the expectation of ever totally eliminating it (Borasi, 1992); The term inquiry suggests that the teacher is exclusively oriented towards ‘enabling independent reasoning, and therefore implies the teacher has unstructured aims in mind (Elliott & Adelman, 1975).

and from here started what would become later a theoretical perspective on inquiry. The Mathematics Teacher Enquiry (MTE) Project involved a collaboration between six teachers, myself and one of my PhD students at that time. With a

1

Enquiry is an alternative spelling of inquiry; inquiry is the more generally used spelling in educational literature.

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small research grant from the university, we invited the school-based mentors in our initial teacher education programme to join us for the research—six of them accepted. The funding paid for them to come out of school for an afternoon once a month for a year. We invited them to focus on one aspect of their teaching that they would like to develop and, with our support, to conduct a small developmental enquiry. We visited them in their schools and provided support as they wished. The regular meetings (in the university) were exceptionally fruitful, raising issues related to teaching and learning which the teachers discussed avidly. This resulted in teachers supporting each other in their individual enquiries. Where it seemed relevant to us, we introduced theoretical ideas, but, although these were discussed, they did not generally figure in the teachers’ ongoing enquiry. The teachers were all very busy in their schools and were keen to share the practical details and issues, as well as their modes of enquiry, but less keen to consider theoretical perspectives (Jaworski, 1998). Two of the teachers were motivated to write an article for a professional journal about their enquiry and what they learned from it. Nevertheless, one area of theory that we discussed with the teachers was a theoretical construct, the Teaching Triad, with elements of Management of Learning, Sensitivity to Students and Mathematical Challenge, which had emerged from my earlier research on investigations, studying the teachers’ teaching practice. Two of the teachers were interested to use the TT as a tool to help develop their teaching—one focusing on sensitivity to students and the other on mathematical challenge. We were joined by Despina Potari, then on sabbatical leave in Oxford, in what we called The Teaching Triad Project. Despina and I observed the two teachers in their classrooms, and we all met frequently at my house. We formed what I would later refer to as a community of inquiry—each teacher inquiring into their own teaching practice using the TT as a developmental tool and we educators using the TT as an analytical tool to analyse data from observations of teaching and conversations with the teachers (Potari & Jaworski, 2002; Jaworski & Potari, 2009). I have referred briefly to these two projects because they were forerunners of future projects in which inquiry was central to project design, implementation and analysis. The projects above had included inquiry (or investigation) in mainly two ways—inquiry in mathematics, with mathematical tasks designed and used in inquiry mode to promote students’ mathematical understanding, and inquiry in mathematics teaching through which teaching developed as teachers became more aware of engaging students through inquiry in mathematics. A significant feature in these projects was the collaborative relationship between the teachers and the researcher(s) studying their teaching development. It was clear throughout that teachers valued having a researcher interested in their teaching; they enjoyed opportunities to talk about how and why they worked with students in the ways we observed; they acknowledged that the encouragement to talk and reflect led to new ways of acting in the classroom to support their students. This collaborative, inquiry-based activity between teachers and researchers became more overtly the basis for a new project Learning Communities in Mathematics with colleagues and teachers in Norway. A group of teacher-educatorresearchers—didacticians—designed a project to promote inquiry-based teaching

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development in schools ranging from lower primary to upper secondary near to the university. A chief aim was to improve the mathematical understandings of students at all levels through inquiry activity and to study the processes and outcomes. We were successful in attracting substantial funding from the Norwegian Research Council. This project, which is well documented (e.g. Jaworski et al., 2007), provided the environment and culture for theoretical development around the focus on inquiry in teaching and learning.

9.7

Theoretical Development

We worked with a number of theoretical constructs at that time. The first was the idea of inquiry activity in mathematics. It was a central concept for most of the didacticians in the project, all of whom had some experience with a mathematics curriculum embodying open-ended problems for mathematical exploration. The second was the notion of collaboration in the form of inquiry communities. We didacticians saw ourselves as an inquiry community, inquiring into both mathematics and the learning and teaching of mathematics in our schools. We saw the teachers in each school as also forming an inquiry community, the nature of which varied from school to school. The whole project was conducted to develop a large inquiry community in which we were all inquirers, albeit with differing focuses. In the beginning, teachers asked the didacticians, ‘what exactly is inquiry’—there was no one word for it in Norwegian. Our early take on a meaning for inquiry was that it was intended to engage students with mathematics in ways that would draw them deeply into mathematical concepts and their own mathematical thinking; it would involve them in asking questions, seeking solutions, solving problems, exploring and investigating, looking critically at what they were learning. We drew on the work of Gordon Wells who had written: Inquiry does not refer to a method . . . still less to a generic set of procedures for carrying out activities. Rather it indicates a stance towards experiences and ideas – a willingness to wonder, to ask questions, and to seek to understand by collaborating with others in the attempt to make answers for them. (Wells, 1999)

The third was critical alignment. This was derived from a sequence of ideas reflecting around community of inquiry (CoI) and its relations with community of practice (CoP) as introduced by Lave and Wenger (1991). Many of the ideas surrounding CoP fitted well with our concept of CoI; for example, Wenger’s (1998) constructs of mutual engagement, joint enterprise and shared repertoire (pp. 73 ff) to describe community. It seemed to us that these also fitted well with CoI. Wenger also claimed that belonging to a CoP involved members in three modes of belonging: those of engagement, imagination and alignment (pp. 173 ff). We engage together in the practice, use imagination in developing individual trajectories and align with the norms and expectations of the practice. These too seemed to fit

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with CoI, except in one respect. Alignment, aligning with norms and expectations, might result in the perpetuation of elements of the practice which were not what, ideally, we would like to see. For example, some students were encouraged to work with mathematics in procedural ways, learning by memorisation and reproducing rules or formulae according to standard mathematical questions. This could be seen as contrary to an inquiry view of teaching and learning and therefore not to be perpetuated at the expense of more investigative approaches. However, we were aware that teachers seeking to develop their teaching could not change to new directions overnight. Thus, it seemed appropriate to think of a stance of looking critically at our practice while engaging in and with it; this could lead to the visulisation of change. Hence, we started to talk of critical alignment. We saw critical alignment to develop from a reflective view on teaching in which teachers reflected on their teaching and thought about what worked well and why and what perhaps needed to be changed. Reflective teaching has a long history: I have personally drawn on Dewey, Kemmis, Polanyi and Schon (Jaworski, 1994, pp. 186 ff). Particularly, Dewey writes: . . . reflective thinking, in distinction to other operations to which we apply the name of thought, involves (1) a state of doubt, hesitation, perplexity, mental difficulty, in which thinking originates, and (2) an act of searching, hunting, inquiring, to find material that will resolve the doubt, settle and dispose of the perplexity (1933, p. 12).

In the projects described above, the researcher, working with a teacher or teachers, encouraged teachers’ reflection while reflecting herself on her own activity and experiences. When such reflection became mutual, in a collaborative setting, the result in many cases might have been described by the concept of critical alignment, although the term was not used at that stage. The LCM project brought us to an overt awareness of three layers of inquiry interacting in our practice. Two of these, I have articulated already: inquiry in doing and learning mathematics in the classroom; inquiry in reflecting on and developing our teaching of mathematics to enable students’ conceptual learning. The third involves overt, systematic, reflective inquiry into the processes and outcomes of inquiry in the other two layers. This third layer embodies what we have called developmental research, our fourth theoretical construct. I draw on Laurence Stenhouse’s (1984) definition of research as ‘Systematic inquiry made public’. This succinct definition seems to capture the important relationship between research and inquiry. In our three-layer model, inquiry in the first two layers needs not be ‘systematic’, and it often was not. For example, in MTE, the teachers’ research was described as ‘evolutionary’ rather than systematic. This was because it seemed to ‘evolve’ reflectively over time rather than emerge as a result of systematic actions by the teachers. Because LCM was overtly a research project, as set out in the proposal we made to the research funding organisation, it involved, from the start, collecting data from all activity in a variety of forms and then analysing these data rigorously according to clear research questions. We were a large team in the university (12–15 at different times), so different researchers took responsibility for different research questions,

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Didacticians inquiring with teachers in systematic ways to research Teachers engaging in professional inquiry … Students engaging in inquiry in mathematics in the classroom with a teacher to learn more about creating mathematical opportunities for students processes, practices and issues in developing mathematics teaching and learning Fig. 9.1 Representation of the three-layer model of inquiry

according to their interests, and involved a huge data base to which we all had access. In a few cases, teachers joined us in the more formal research inquiry, although many did not wish to engage at this level. A diagrammatic representation of the three layers follows: inquiry in the middle layer looks into and seeks to promote inquiry in the central layer. Inquiry in the outer layer explores (conducts research into) inquiry in the central and middle layers (Fig. 9.1).

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Inquiry in Learning, Teaching and Development Developmental Research

Developmental research is research that not only studies the developmental process but also contributes to development. The process is usually cyclic and iterative as in action or design research; in fact, action and design research can be seen as forms of developmental research. Such research can be undertaken by a lone individual who researches her own practice while seeking to develop that practice. However, this is very difficult; since teaching is a very demanding activity: at any level of education, it is hard for a teacher to engage alone in research. Thus, developmental research is most effective when it is conducted in an inquiry community in which the members support each other in inquiring into practices. For example, in the LCM project,

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teachers and didacticians had very different roles, but their activity together allowed teachers to inquire into their own practice and their students’ learning, while didacticians conducted more formal research into the processes and practices within the project as a whole. In fact, just to complicate things, the didacticians also investigated their own practice as practitioners working with teachers (Goodchild, 2008; Jaworski, 2008).

9.8.2

The Role of Inquiry

According to Stenhouse, research is (systematic) inquiry. Thus, developmental research is inquiry that seeks development of some form of activity or practice. It follows that inquiry is central to developmental activity. Inquiry can take many different forms. Clearly inquiry in mathematics differs from inquiry in mathematics teaching. Differences can be seen in who is involved, what they do and what goals they have for their actions. A teacher engaging in development of teaching is inquiring into the teaching process and possibly looking for ways to improve students’ mathematical understandings. One way of doing this might be to design, and try out in practice, mathematical tasks that are inquiry-based. Teachers working together on development of teaching might share examples from their teaching (possibly in video form) and discuss teaching actions and goals with their fellows (e.g. Barton et al., 2014). Here inquiry is invested in choosing extracts from teaching, sharing these with colleagues and discussing the associated goals and outcomes. This developmental approach is not necessarily involved with the design of inquiry-based mathematical tasks.

9.8.3

A Sociocultural Perspective

Research that is collaborative, inquiry-based and developmental requires a frame, paradigm or world view that makes sense with the cultures involved—perhaps seeing inquiry as a way of being in practice with colleagues. A sociocultural perspective takes as its basic premise Vygotsky’s general genetic law of cultural development, that any function of cultural development appears on two planes: first on the inter-mental plane (between people) and then within the individual in the intra-mental plane (Wertsch, 1991). So, according to Wertsch (1991, p. 27), rather than ‘the idea that mental functioning in the individual derives from participation in social life’, ‘the specific structures and processes of intramental processing can be traced to their genetic precursors on the intermental plane’:

While it would be possible to argue development for each individual from a constructive perspective, sociocultural theory’s arguments, as indicated above, emphasise the social nature of learning and the structures whereby human

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development takes place. Thus, for the fundamentally social engagement involved in collaborative inquiry activity, it makes sense to analyse interactions from a Vygotskian perspective. In fact, in much recent work, I and colleagues have used Activity Theory, developed by A. N. Leont’ev from Vygotskian perspectives.

9.9

Developing Mathematics Teaching and Learning at University Level

Developmental research into the teaching and learning of mathematics at university level, for me, built on all the experiences detailed above. The vignette presented at the beginning of the chapter was my first experience of working with a mathematician to characterise teaching and to address corresponding issues and tensions for the teacher. Another of my PhD students, Angeliki Mali, worked with mathematicians teaching in small group tutorials in our first year of the mathematics degree. Persuading 26 mathematicians or mathematics educators to allow her to observe a tutorial, Angeliki was able to synthesise characteristics of tutorial teaching, from which she selected three teachers who allowed her to observe them over a semester. Her detailed study of their thinking and actions takes us further in understanding perspectives on teaching at university level (Mali, 2016). It has been important for me to try to understand how teachers at this level think about teaching. Most do not have the educational background of mathematics educators who do research into learning and teaching, who have expertise in educational research and are steeped in the educational research literature. So the scholarship of teaching at university level has very different roots from that at school levels. Elena Nardi’s book (2008), Amongst Mathematicians, presents composed dialogues and arguments, based on her data, between mathematician and mathematics educator to reach for the different ways in which these professionals regard teaching, the needs of their students and ways of seeing students’ mathematical development. Some forms of teaching have different labels to distinguish their special forms. My earlier experiences of teaching with the Open University were referred to as tutoring since the Open University teaches through its distance learning materials. The mathematics tutorials observed by Angeliki were led by tutors; teaching was done in lectures by lecturers. I met another term at Loughborough, that of mathematics support, a form of teaching pioneered by Tony Croft and Duncan Lawson (Solomon et al., 2010). Drop-in, mathematics support centres provide a welcoming environment where students from across the university can bring their questions, problems, difficulties and concerns with mathematics and get one-to-one responsive help. This is referred to as support, rather than teaching. In my experience as a school teacher, all of these modes of working with students would be referred to as teaching and would have elements of didactics and pedagogy related to the particular

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situations and their needs. Fundamentally, for me, a teacher is someone who works with students in ways that accord with the situation and context of the teachinglearning relationship, seeking to discern the student needs along with the expectations of the situation. More recently, I have been involved in two projects which have brought students into the teaching arena employing them to design resources for second-year mathematics modules (Duah, 2017; Duah et al., 2014) or designing tasks for Foundation students (Jaworski et al., 2018). As well as demonstrating the value for students in contributing to these projects, the projects have provided teachers/lecturers with insights into students’ perspectives on learning and learning needs. We have seen the value of working with students in these ways that have suggested that university teachers and students, both, can benefit from collaborating in teaching design. This is yet another mode of teaching and learning for both groups. However, these ‘ways of working’ are not well defined. Preparing to teach goes beyond writing a mathematical script for a lecture. In some way, every teaching event can be seen as an exercise in inquiry from which, through the critical reflective process, a teacher develops insights and understanding for teaching. When this is made overt and shared with others, through some form of learning community, teaching knowledge develops and enables more knowledgeable practice. Bill Barton and colleagues (Barton et al., 2014) write about their DATUM project in which mathematics teachers in a New Zealand university meet regularly to discuss their teaching according to theory of resources, orientations and goals (Schoenfeld, 2010) and from which teaching develops. This is a community of inquiry in university mathematics teaching and provides a form of professional development for the teachers involved. I have written elsewhere about research projects in university teaching-learning which have contributed to professional development of the teachers involved (Jaworski, 2019a, b). In Europe, through CERME, the bi-annual congress of ERME, the society for European Research in Mathematics Education, a group focusing on University Mathematics Education has been working for a decade sharing and developing ideas for teaching and learning at university level. More recently, a specialised Topic Conference has emerged—INDRUM—the International Network in Didactic Research in University Mathematics. Led by international scholars, Viviane Durand-Guerrier, Elena Nardi, Carl Winslow, Gislaine Gueudet and Chris Rasmussen, this group is devoted to research that explores and develops the teaching and learning of mathematics at university level (Winslow et al., 2018). Such activity and publication are establishing inquiry in university mathematics teaching and learning. Scholars can share in conference communication or read the resulting literature to start to think about engaging in educational development. In the spirit of these areas of development, a 3-year European project in the ERASMUS+ programme was established in 2018 under the title of Partnership in Learning and Teaching IN University Mathematics—PLATINUM. Eight universities in seven countries are involved, led from Norway and including universities in Czech Republic, Germany, Netherlands, Spain, United Kingdom and Ukraine. The project takes as its central theoretical perspective the idea of inquiry in mathematics

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learning and teaching in three layers as discussed above. A central theme is the idea of Inquiry Community. Each university team in the project has formed an inquiry community of mathematics teachers (mathematicians and mathematics educators) interested in developing teaching and learning. Each inquiry community is specific to its own culture and context; we collect data from the activity of each and aim to write case studies about their development including the thinking of the participants and the ways in which they have explored teaching and learning. We hope to capture the particular issues which have arisen and ways in which these have been addressed (See PLATINUM.uia.no). A second area of focus is the design of inquiry-based tasks in many areas of mathematics: that is tasks which do not just test students’ recall of standard functions, formulae, rules and so on, but also engage students in exploration, investigation and collaborative inquiry into mathematical questions and situations; in Dewey’s terms, in acts of searching, hunting, inquiring, to find material that will resolve doubt, settle and dispose of perplexity. Our aims here in PLATINUM are to explore the nature of inquiry-based tasks, share the tasks that are developed in different universities and produce a bank of such tasks for sharing more widely. An expectation is that the design of such tasks will raise questions about inquirybased practice including the ways in which tasks will be included in established courses, bringing an inquiry element to the course. We will track such questions and the experiences of the teachers who use the tasks. A third area of inquiry concerns professional development for university mathematics teachers, both those coming new to teaching and others who have been teaching for many years. Three universities designed and trialled a course, in inquiry mode, relevant to their context; experiences from one fed into the next, and we collected data overall to address practices, goals, experiences, issues and tensions. These feed into a summer school in Spain where a course is offered building on what we have learned from the three inquiries. Our learning from these inquiries will feed into a package for designing such a course which can be shared more widely. It can perhaps be seen that inquiry-based practice in mathematics learning and teaching at university level is developing widely in Europe and beyond. In parallel with this, the International Journal of Research in University Mathematics, IJRUME, is publishing papers from scholars around the world researching such activity. I am very happy to feel a part of this development.

9.10

Concluding Thoughts

In the sections above, I have traced developmental practice and associated theoretical perspectives in mathematics education over several decades. An aim throughout, at any educational level, has been to develop teaching in order to promote conceptual learning of students in mathematics. Fundamentally, the principle notion underpinning developmental practice has been that of inquiry: inquiry in learning mathematics, in developing the teaching of mathematics and in the research

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process through which development can be charted, characterised and evidenced. Central to the activity in all the examples above has been collaboration. From sociocultural perspectives, rooted in Vygotskian theory, collaboration in inquiry is seen to be a highly fruitful means to learn and develop. Although associated practices are particular to the levels of education, cultures and contexts in which collaborative inquiry takes place, the underpinning principles are the same. The formation of communities of inquiry enables teachers to share activity and developments in teaching and the associated challenges and issues. We learn with and from each other within the practices in which we engage. The idea of critical alignment allows us to continue to align with the processes and practices in established teaching and learning while looking critically at what we do as we do it and exploring new elements. This inquiry, in collaboration with peers, enables us to address issues and tensions and to find support among our colleagues. Thus, we become more knowledgeable about forms of practice and more secure in our inquiry being. Bringing inquiry into the classroom, lecture or workshop engages students in inquiry in mathematics, collaborating with each other to address mathematical questions and develop conceptual understandings. For both students and teachers, the aim is to develop inquiry as a way of being, so that it becomes natural to inquire and to share our learning through inquiry. In these ways, teaching and learning develop in deep and meaningful ways over time. There is no given way of proceeding or succeeding, although reflection on and analysis of inquiry activity enable us to document and share more widely, for others to appreciate and critique, what we have learned. The following three points can help new practitioners to make a start with inquiry-based activity in collaboration with peers. 1. Consider forming a community of inquiry, with one or more colleagues, for sharing ideas of activity with students, issues and tensions that arise and developing knowledge of practice 2. Look at the mathematical tasks you give to students and discuss them with colleagues. Consider how they might be modified to make them more inquirybased to encourage greater student involvement and conceptual learning 3. Form a reading group to read and discuss some of the sources referenced above and to think about ways in which this can encourage new activity in mathematics teaching and learning.

9.11

Post-script

In his review of this chapter, Simon Goodchild, posed the following issue regarding this chapter. ‘A notion that recurs throughout the paper is that of ‘understanding’. I think it would be helpful to explain this from the sociocultural perspective that is adopted later, especially in the context of concepts such as activity, practice, appropriation,

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knowing and intersubjectivity’. I thank him for this suggestion since it has helped me to be clearer as to what understanding can mean (for me) socioculturally. I begin from Vygotsky’s two planes and the quotation from Wertsch (above): . . . rather than ‘the idea that mental functioning in the individual derives from participation in social life’, ‘the specific structures and processes of intramental processing can be traced to their genetic precursors on the intermental plane’.

Students (and indeed their teachers) participate in the social (intermental) activity in teaching-learning situations in classrooms, lectures, tutorials and so on. As they participate, they engage with what they see and hear (and sense in other ways) and make sense dialectically with their peers, to evolve their own perceptions intramentally. As a researcher, one of the important data-gathering functions is to track this process to gain insights into the ways in which individuals participate in the social and how they recognise their own growth of understanding of what is in focus. As a teacher, addressing overtly how students experience their social involvement is important. As such a focus is developed, students may become aware of their learning through participation and engage through questioning and reflection. Through such overt intersubjectivity personal understanding develops concomitantly.

References Barton, B., Oates, G., Paterson, J., & Thomas, M. (2014). A marriage of continuance: Professional development for mathematics lecturers. Mathematics Education Research Journal, 27(2), 147–164. Bishop, A. J. (1988). Mathematics education in its cultural context. Educational Studies in Mathematics, 19, 179–191. Borasi, R. (1992). Learning mathematics through inquiry. Heinemann. Collins, A. (1988). Different goals of inquiry teaching. Questioning Exchange, 2(1), 39–45. Dewey, J. (1933). How we think. D. C. Heath & Co. Duah, F. (2017). Students as partners and students as change agents in the context of university mathematics. Unpublished PhD thesis, Loughborough University, UK. Duah, F., Croft, T., & Inglis, M. (2014). Can peer assisted learning be effective in undergraduate mathematics? International Journal of Mathematical Education in Science and Technology, 45(4), 552–565. Dreyfus, T. (1991). Advanced mathematical thinking processes. In D. O. Tall (Ed.), Advanced mathematical thinking (pp. 25–41). Kluwer. Elliott, J., & Adelman, C. (1975). The language and logic of informal teaching, in The Ford Teaching Project, Unit 1, Patterns of teaching. University of East Anglia, Centre for applied research in education. Glasersfeld, von E. (1987). Learning as a constructive activity. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics. Lawrence Erlbaum. Goodchild, S. (2008). A quest for ‘good’ research: The mathematics teacher educator as practitioner researcher in a community of inquiry. In B. Jaworski & T. Wood (Eds.), The mathematics teacher educator as a developing professional. Sense Publishers. Hart, K. (Ed.). (1981). Children’s learning of mathematics 11–16. John Murray. HMSO. (1982). Mathematics counts: The Cockcroft report. HMSO. Jaworski, B. (1991). Interpretations of a constructivist philosophy in mathematics teaching. Unpublished PhD thesis, Open University, UK.

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Jaworski, B. (1994). Investigating mathematics teaching: A constructivist enquiry. Falmer Press. Jaworski, B. (1998). Mathematics teacher research: Process practice and the development of teaching. Journal of Mathematics Teacher Education, 1(1), 3–31. Jaworski, B. (2006). Theory and practice in mathematics teaching development: Critical inquiry as a mode of learning in teaching. Journal of Mathematics Teacher Education, 9(2), 187–211. Jaworski, B. (2008). Building and sustaining inquiry communities in mathematics teaching development. Teachers and didacticians in collaboration. In K. Krainer (Volume Ed.) & T. Wood (Series Ed.) International handbook of mathematics teacher education: Vol. 3. Participants in Mathematics Teacher Education: Individuals, teams, communities and networks (pp. 309–330). Sense Publishers. Jaworski, B. (2019a). Inquiry-based practice in university mathematics teaching development. In D. Potari & O. Chapman (Eds.), International handbook of mathematics teacher education: Volume 1: Knowledge, beliefs, and identity in mathematics teaching and teaching development (2nd ed., pp. 275–302). Koninklijke Brill NV. Jaworski, B. (2019b). Preparation and professional development of university mathematics teachers. In S. Lerman (Ed.), Encyclopedia of mathematics education. Springer. https://doi. org/10.1007/978-3-319-77487-9_100027-1 Jaworski, B., & Potari, D. (2009). Bridging the macro-micro divide: Using an activity theory model to capture complexity in mathematics teaching and its development. Educational Studies in Mathematics, 72, 219–236. Jaworski, B., Fuglestad, A.-B., Bjuland, R., Breiteig, T., Grevholm, B., & Goodchild, S. (2007). Learning communities in mathematics. Caspar. Jaworski, B., Treffert-Thomas, S., & Bartsch, T. (2009). Characterising the teaching of university mathematics: A case of linear algebra. In M. Tzekaki, M. Kaldrimidou, & C. Sakonidis (Eds.), Proceedings of the 33rd Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 249–256). PME. Jaworski, B., Treffert-Thomas, S., Hewitt, D., Feeney, M., Shrish-Thapa, D., Conniffe, D., Dar, A., Vlaseros, N., & Anastasakis, M. (2018). Student Partners in Task Design in a computer medium to promote foundation students’ learning of mathematics. In Proceedings of INDRUM: Second conference of the International Network for Didactic Research in University Mathematics (pp. 316–325). University of Agder and INDRUM. Lave, J., & Wenger, E. (1991). Situated learning: Legitimate peripheral participation. Cambridge University Press. Lerman, S. (1996). Intersubjectivity in mathematics learning: A challenge to the radical constructivist paradigm. Journal for Research in Mathematics Education, 27(2), 133–150. Mali, A. (2016). Unpublished PhD thesis, Loughborough University, UK. Nardi, E. (2008). Amongst mathematicians: Teaching and learning mathematics at the university level. Springer. Potari, D., & Jaworski, B. (2002). Tackling Complexity in Mathematics Teacher Development: Using the teaching triad as a tool for reflection and enquiry. Journal of Mathematics Teacher Education, 5(4), 351–380. Schoenfeld, A. H. (2010). How we think. A theory of goal-oriented decision making and its educational applications. Routledge. Solomon, Y., Croft, T., & Lawson, D. (2010). Safety in numbers: Mathematics support centres and their derivatives as social learning spaces. Studies in Higher Education, 35(4), 421–431. ISSN: 0307-5079. Steffe, L. P., & Thompson, P. (2000). Interaction or intersubjectivity? A reply to Lerman. Journal for Research in Mathematics Education, 31(2), 191–209. Stenhouse, L. (1984). Evaluating curriculum evaluation. In C. Adelman (Ed.), The politics and ethics of evaluation. Croom Helm. Thomas, S. (2012). An activity theory analysis of linear algebra teaching within university mathematics. Unpublished doctoral dissertation. Loughborough University, UK.

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Wells, G. (1999). Dialogic inquiry: Toward a sociocultural practice and theory of education. Cambridge University Press. Wenger, E. (1998). Communities of practice: Learning, meaning and identity. Cambridge University Press. Wertsch, J. V. (1991). Voices of the mind: A sociocultural approach to mediated action. Harvard University Press. Winsløw, C., Gueudet, G., Hochmut, R., & Nardi, E. (2018). Research on university mathematics education. In T. Dreyfus, M. Artigue, D. Potari, S. Prediger, & K. Ruthven (Eds.), Developing research in mathematics education – Twenty years of communication, cooperation and collaboration in Europe (New perspectives on research in mathematics education series) (Vol. 1, pp. 60–74). Routledge.

Chapter 10

The Intimate Interplay Between Developing Teaching and Exploring Mathematics Through Reflecting in, on and for Teaching John Mason

Abstract Reflecting on some 60 years of mathematical thinking and teaching leads me to suggest that, in order to be freshly aware of opportunities to support and stimulate the mathematical thinking of others, it has proved greatly valuable to me to continue to explore mathematics for myself. Conversely, maintaining contact with pedagogical issues in my teaching has served to refresh, stimulate and inform my personal awareness of the conduct of mathematical thinking in my mathematical explorations, as well as suggest questions worthy of further exploration. This chapter offers a few examples of mathematical explorations arising from teaching of particular topics and how they bring pedagogical issues to the surface. Similar ideas from other authors have emerged in recent years as interest in both mathematically based pedagogical actions and topic-specific didactical actions has grown. Keywords Mathematical themes · Mathematical powers · Awareness · Pedagogical actions · Exploration · Taylor-Powers · Chordal mid-points

10.1

Introduction

My aim here is to offer instances in which, through mathematical exploration of my own, I have been reminded of or come into contact with pedagogical issues and actions, and in which, through considering pedagogical issues, I have been led to mathematical exploration. My claim is that an interest in my own lived experience of thinking mathematically has both informed and been informed by concern for the lived experience of learners. This relationship is not obvious to everyone (Madsen & Winsløw, 2009). It is common experience for lecturers that learners often ignore necessary conditions in theorems or in procedures and that learners have a tendency not to think of special cases (such as ruling out dividing by zero). An antidote to this is to look out

J. Mason (✉) University of Oxford, Oxford, UK e-mail: [email protected] © Springer Nature Switzerland AG 2023 S. Stewart (ed.), Mathematicians’ Reflections on Teaching, Advances in Mathematics Education, https://doi.org/10.1007/978-3-031-34295-0_10

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for tasks in which there are opportunities to bring this essential form of variation and specialisation to their attention and to do it in a pedagogically effective manner (see scaffolding and fading, later). What I am offering here are specific examples of exploiting ubiquitous mathematical themes and use of my natural human powers in my mathematical explorations. I claim that recognising how I have exploited themes and used powers has sharpened and enriched my awareness to be able to be more explicit and effective in my teaching. In particular, recognising experiences of getting stuck and unstuck has sensitised me to similar learners’ experiences, enabling me to adjust my pedagogy and my didactics (for differences between these, see later). However, I have neither the room nor the data to justify this claim. Rather it is a conjecture for others to test in their own experience, which calls upon a method of enquiry elaborated in Mason (2002a). Conversely, paying close attention to pedagogical and didactical issues arising from teaching serves to sharpen my awareness, which then has impact on more insightful exploration of mathematical questions for myself. For more detailed methodological remarks, see Mason (2002a). Talking with respected and respectful ‘others’ can greatly enrich reflection and so inform future practice (Nardi 2008). Because what I value most highly is lived experience upon which to reflect and from which to learn, I recommend to readers to undertake the mathematical questions posed here before reading my accounts of what I noticed. That way you have personal experience as a foundation against which to consider my observations.

10.1.1

Foundations

When as a graduate student, I first saw George Pólya’s film ‘Let Us Teach Guessing’ (Pólya 1965), it both released in me aspects of the way I had been taught in high school and aligned with my own experiences of exploring mathematics while at school and as an undergraduate. When, in 1970, I took up my first post at the Open University, I was assigned the task of devising and organising summer schools lasting 6 days, for some 7000 adult learners on three different sites and spread over 11 weeks. I based the summer-school activities on what I had experienced myself as a student and as a teaching assistant, including Pólya’s film (1965), and on what I learned from primary teachers about how they structured tasks and engaged learners. This aligned with the thinking of colleagues whom I met through the Association of Teachers of Mathematics, whose main thrust over three decades in the mid-twentieth century was promoting mathematical exploration. These were called investigations, after John Wallis’ reference to ‘my method of investigation’ which involved seeking patterns in multiple examples, in response to tasks sent to him from Pierre Fermat. For a partial history of ‘inquiry based learning’ and its relation to more recent frameworks of distinctions, see Artigue & Blömhoj (2013).

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My thinking was also informed by reading Pólya (1945, 1962), and eventually, my experiences led to Thinking Mathematically (Mason, Burton & Stacey 1982; see also Mason 2008b). The language we used was the discourse of the times, and so the whole was expressed in terms of ‘processes’ involved in mathematical thinking. When we came to revise it (Mason et al., 2010), the language of processes was no longer in favour, but I found that treating the same experiences as the exercise of natural human powers struck a chord both with teachers and with learners, as well as with established philosophers. My approach takes inspiration from Leonardo Da Vinci, who said: Avoid the teachings of speculators whose judgements are not confirmed by experience (Zammattio et al., 1980, p. 133) ‘Experience’, for [Leonardo] meant for him a form of knowledge based on direct observation of natural phenomena” (Barone, 2019, p. 12) Similarly, Roger Bacon (1267: see Bridges, 1897, p. 167) based his novel approach to enquiry and research which forms the basis for scientific enquiry, on the adage that Without experience nothing can be known sufficiently. My interest and concern have always been phenomenological, focused on the lived experience of thinking mathematically, detecting it in myself and offering opportunities to others to recognise it in themselves (Mason et al., 1982, 2010; Mason 2011). In this I found myself in alignment with the aims and practices of the Association of Teachers of Mathematics when I encountered them in the early 1970s. I use the term phenomenological, not as a reference to philosophical discourse, but in its essential meaning, which later turned out to have enormous resonances with more theoretical philosophical writings (see, for example, Westbury et al., 2000). I take, therefore, as my starting point here, awareness of natural powers evidenced by very young children and essential for mathematical thinking at all ages, which include among others, Imagining and Expressing Specialising and Generalising

Conjecturing and Convincing Organising and Characterising

Human powers are exploited in pervasive mathematical themes (see also Gardiner, 1987), amongst which are Doing and Undoing Invariance in the Midst of Change Freedom and Constraint

Extending and Restricting Exchanging and Substituting

Rather than try to recreate conditions here which imitate how these powers and themes came to my explicit attention, I choose to consider pedagogical phenomena arising from one particular line of exploration. Thus, not all of these themes and

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powers will play a central role. Each of the following explorations was itself triggered by pedagogical questions arising either in particular topics informed by the gradual assimilation of ideas encountered in the literature and not always recognised as such, or by pedagogical issues which emerged from exploration. An important awareness for me came from Bill Brookes (1976): ‘a problem’ is not an object but rather a state which someone experiences (see also Mason, 2019). So my first thought when designing tasks for others is to create some intrigue, often by presenting a mathematical or material world phenomenon. It may be worth noting that I use pedagogical to refer to actions that teachers can initiate in almost every mathematical situation and didactic to refer to actions which seem to be specific to a particular mathematical concept, topic or domain. My interest here is pedagogical in that sense.

10.2

Chordal Mid-Points

Most introductions to the calculus involve chords (sometimes called secants) drawn between two points on the graph of a function. One day it occurred to me to ask the following:

10.2.1

Initial problem

What is the locus of mid-points of chords of a polynomial? I chose polynomials because I could generate these in dynamic geometry software through offering the user choice of polynomial degree and then the opportunity to adjust the positions of degree + 1 points thorough which the Lagrange polynomial would pass. Many years ago, I did this synthetically, using Euclidean constructions; with fast computers, it is easier to do it algebraically. Constructing the Lagrange polynomial through a collection of points in a dynamic geometry programme is a good exercise in the exploitation of the additive and multiplicative properties of zero. This emerged for me when I prepared an applet to illustrate how the Lagrange construction works. Varying the position of a single point and watching how the graph changes added richness both to my appreciation of polynomials and to learners’ appreciation, as well as expanding my example space (Watson & Mason, 2005). For chordal mid-points, I did some thinking for myself by specialising in quadratics and then moved immediately to cubics (a step towards generalising). My conjecture for cubics surprised me for a few seconds, until I had thought about it more carefully. This experience suggested to me that the task had both pedagogical and mathematical potential. I therefore offered the task to some Secondary School mathematics teachers. For them, I did the specialising by posing the problem for

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quadratics, which I expected to be a quick introduction, anticipating an immediate move on to cubics where I knew some interesting pedagogic and mathematical ideas were likely to emerge.

10.2.1.1

Pedagogical Observation

It is so tempting to do the mathematically natural thing and specialise for learners, taking them through a sequence of tasks of increasing complexity to a generality. However, this can actually inhibit learners. Such a pedagogic action is an instance of the transposition didactiquec (Chevellard, 1985), in which expert awareness is transposed into instruction in behaviour, allowing, even encouraging, learners to carry out a sequence of instructions without really engaging their natural powers to think mathematically. Such a task sequence can even train learners in dependency so that they come to expect the teacher (or the text) to do the specialising for them. The potential and value of specialising then fade into the background rather than becoming second nature to them as thinkers. The result can be that learners do not even become aware of it as a natural power which they possess and which they can use in order to make progress on difficult problems that they encounter. I was once invited by a mathematics department to give a seminar on mathematical pedagogy and chose to work on the importance and role of example construction as an aspect of specialising. Some days later, I was told that a group of mathematicians who happened to be meeting together to work on some difficult problems in their area around the same time and decided to come to my seminar were inspired to try constructing some specific examples related to their problems, and that this had assisted their explorations. The point is that even experienced mathematicians may not always have specialising become available as an action, perhaps because they have confidence at or in a higher level of abstraction. The terms scaffolding and fading (Brown et al., 1989) and directed-promptedspontaneous (Love & Mason, 1992) have both been used to describe a process of introducing a mathematical action such as specialising and then supporting learners in internalising it by using increasingly indirect prompts to action, so that it becomes available for them to initiate for themselves when they get stuck on a problem. For more details of these and many other strategies and tactics, see Mason (2002b).

10.2.2

First Use With Learners

The teachers in the workshop were clearly stumped by the chordal mid-points question. They seemed not to know what to do. The fact of having two things that could vary (the ends of the chords) seemed to act as a smoke screen obscuring the possible mathematical action of specialising by fixing one end and considering all the chords through that point.

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This resonated with an experience I had had when posing primary teachers the task of predicting the number of matches needed to make a configuration of r rows and c columns of squares made from unit-length segments as in the figure below1.

Three rows of 8 columns of unit-stick squares

Many of them seemed unable to cope with two things varying at the same time. Furthermore, they did not think of fixing one variable, say the number of rows, and developing experience and facility in generalising that count before, then varying the number of rows as well. Note that to ponder a specific diagram but to think in general requires an act of imagining and expressing what is imagined, if only to oneself. For more details, see Mason (2002b). Thus, my past experience alerted me, in the face of no progress with the chords, to suggest specialising by fixing one end of the chords. This is an example of having been sensitised both to a state (being stuck) and to an action (specialising) which could inform my pedagogical choices, whether in the moment, as in this case, or in task design. Unfortunately, the teachers were still stumped, apparently because they were not thinking geometrically. Some had a ‘sense’ that it was probably the interior of the quadratic, but could not see how to justify this in any way. At this point, I intervened again, inviting them to imagine a point P running around a circle, together with a fixed point F, and the mid-point M between F and P. Putting attention on the mid-point M, what path does it follow? Eventually, they realised that the mid-point was really a scaling of the original and applied this to the quadratic. Even so they found it difficult to justify the property that every point on or ‘inside’ the quadratic could be a mid-point of some chord. Here I recognised a state (being stuck needing some way to justify a conjecture) and felt further pedagogical action was required. One possible action is to attend to what is actually being claimed and to choose a representative instance: in this case, a point P but in general position. This is another version of specialising but with a view to generality, an action manifested by Hilbert (Courant, 1981). I cannot now recall in what way precisely I intervened, but I drew attention to a single point P (specialising again) and drew a chord through P. The fact that P may 1

An interesting follow-up is to ask in how many different ways a given number N could be the number of sticks required to make such a rectangular grid. It leads to factoring of bilinear forms, among other things.

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not be the mid-point bothered them, whereas I was content to give the chord more freedom and then to try to see if I could guarantee the existence of some chord for which P was the mid-point. This for me is an instance of exploiting the theme of freedom and constraint. Rotating the chord through 180° about P provides a family of chords, and by the intermediate value theorem, for some chord, P will be indeed be the mid-point. However, it had taken so long to get to this point of the exploration that I merely indicated the possibility of going on to cubics and shifted to a different task in order to bring different aspects of mathematical thinking to the surface.

10.2.2.1

Pedagogic Realisations

What can be done when one or more learners are stuck? Fortunately, I had plenty of experience of invoking mental imagery and promoting the use of this power as a way of stimulating activity. I see the power of mental imagery both as a vital contribution to learning mathematics and as an important component of mathematical thinking, not to say preparing to teach a topic (Mason & Johnston-Wilder, 2004). Not surprisingly, therefore, it came to mind2 when the teachers remained unsure as to how to think about the chordal mid-points of quadratics. Mental imagery is a vital power to use when modelling, whether with algebraic word problems or with material world situations. Mental imagery (by which I mean all internalised senses, including an inchoate ‘sense-of’) supports and sustains Bruner’s notion of iconic presentation (Bruner, 1966) which mediates between enaction (doing, specialising, manipulating as if physically) and symbolising. These for me form three inter-related worlds of action, similar to but not the same as those of Tall (2004). Through tutoring and teaching over many years, I have become sensitised to the action, when seeking to justify a conjecture, to concentrate on a representative object and to list all that I know about that object (see Mason et al., 1982, 2010). The mathematical theme of freedom and constraint can be helpful, as here, where removing some constraints (that P be a mid-point) and exploring the new-found freedom open up the possibility of justifying a conjecture which initially looked difficult to prove. But how often, when this theme is used, is attention drawn to it? For me, the power of the theme was enhanced by me becoming explicitly aware of its use in what to most mathematicians seems a trivial manner. This explicitness influenced and informed my future teaching. It seemed to me that the task had potential because of the need to shift from thinking of mid-points of all possible chords to specialising to a single family of chords with a fixed end-point and from there to shift again to see the locus of mid-points as a scaling of the original quadratic, combined with fixing a single

It was not simply a passing thought. Rather, strictly speaking, it ‘came to action’, that is, became available as an action, which is what I think Gattegno (1970) meant by educating awareness. See also Mason & Davis, 1987; Mason, 1994. 2

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point and using the intermediate value theorem to show that all ‘interior’ points could be mid-points. There is potential for learners to experience a shift from a conjectured ‘sense-of’ to an articulation which justifies the conjecture. Experiences like these have alerted me to be on the lookout for tasks which can be approached in two or more ways. I can then draw attention to different learners’ approaches or introduce a second approach so as to call upon flexibility and connection making, often seen as signs of creativity (Levenson et al., 2018). A further duality is available by developing an algebraic approach (see later) involving considering all chords whose mid-point has a constant x-coordinate. This opens up the action of using calculus to find extreme values. Notice that it too is an instance of the themes of freedom and constraint, restricting and extending and invariance in the midst of change: here restricting attention by constraining the chords to a sub-class characterised by keeping some aspect invariant, which can be more readily analysed algebraically.

10.2.2.2

Pedagogic Remarks on Learners as Narrators

Relatively recently, I have been becoming aware of the importance of prompting learners to try to formulate and articulate their own personal narrative of their work on an exploration or indeed of a suite of exercises. Of course, this has been known for thousands of years and prompted by many authors such as Bruner (1990). But as with any insight, it only has ‘bite’ when it is reconstructed personally, linked to personal experience, rather than simply passed on as part of folklore or encountered in academic articles. In mathematics, the specific proof is not usually so important as the method of proof, the approach and the reasoning. Most mathematicians are interested in a new proof for how it works rather than simply that it works, so that it can be used in other situations. So too, reflecting on moving from being stuck to finding some action to undertake may be the most important thing to learn from experience. As I have often said (Mason 1998), One thing we do not seem to learn from experience, is that we don’t often learn from experience alone: something else is required.

That ‘something else’ is personal narrative construction, re-flection on recent past experience coupled with pro-flection forward, imagining ourselves in a similar situation in the future and making use of the same action. With effort, re-flection and pro-flection inform flection, that is, experiencing a choice in-the-moment when it is needed, perhaps when preparing for a session or in the midst of one. This is part of what I called the Discipline of Noticing (Mason 2002a, 2009) which offers techniques for turning what Schön (1983) called reflection-on-action (after the event) into reflection-for-action (Killion & Todnem, 1991) to enable an action to become available and reflection-in-action (in the moment). In Mason (2002a), I called this reflecting-through-action or flection (suddenly changing from being immersed in the details of the moment to also simultaneously being aware of choices you might

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have). Reflecting-through-action is initiated by the process described above of mentally placing yourself in some future event and imagining yourself using some tactic. This cyclic process of learning, famously described by Kolb (1984), has, as is the way of descriptions, been turned into a theoretical frame for use in professional development. In essence, it advocates entering into situations as openly as possible, then standing back to reflect upon them, developing a framework or theory to account for them and using this to inform and take action resulting in yet further experiences. However, once such a sequence of actions becomes mechanical, it is subject to criticism as being at best partial (see, for example, the critique by Letiche, 1988 pp. 22–25). The critical features of all learning on which most theorists are agreed involve exposure to stimulus which creates some sort of disturbance, which draws attention to some feature, aspect or issue not previously salient and which, supported and informed by analysis, by logic, by recourse to analogy and so on, gives rise to conjectures to be tested in experience. These apply equally well to ourselves as teachers, to ourselves as learners and to students as learners as well. General descriptions are of little value unless they are related to specific experiences, as pointed out by Leonardo Da Vinci (quoted earlier) and by Hiller (2000 p218–219), who reports that over a period of some 20 years, he has noticed little change in the form of lesson plans constructed at the end of a teacher education course, though the ‘apologies’, the justifications for proposed actions have changed in line with the fashionable discourse of the time. He also reports that pedagogic and didactic abstractions are quickly abandoned in the face of the practical need to act in real time. Neubrand (2000 p251–266) articulates the value of reflection on mathematics and on processes undertaken which align with Pólya (1962, 1965) and with Mason et al. (1982).

10.3

Diversion: Tangent Power of a Point

At around this time, I became aware of a lack of clarity among some learners, teachers and colleagues as to what actually happens to points on a curve such as a quadratic when the x-coordinate becomes large in absolute value. Specifically, what happens to tangents to a quadratic through different points on the x-axis as those points tend to ± 1? My experience with mid-points and rotating chords made me wonder what could be said not about chords, but about lines through a given point and tangent to a given polynomial. This is an example of a pedagogic issue turning into an exploration with pedagogic consequences. I defined the tangent-power of a point P with respect to a curve to be the number of tangents to the curve through the point P. The word ‘power’ was based on a partial analogy with the power of a point with respect to a circle, which is actually the length of the tangent rather than the number of tangents!

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For a quadratic, the tangent-power of a point is 0, 1 (for points on the curve) or 2. It was the realisation that for large |x|, the tangent power of [x, 0] with respect to a quadratic is 2 that is missing for some people. Of course, thinking projectively, a quadratic looks like a closed curve with a point ‘at infinity’, and so the tangentpower of a point must be 0 (if interior), 1 (if on the curve) or 2 (if exterior). Perhaps then this task would bring the issue out into the open and provide a context in which to clarify people’s concept image of tangents to curves through points on the x-axis. I was also aware that there are differences in how tangents to a curve are defined and differences in learners’ concept images of tangents (Tall, 1987, Vincent et al., 2015). This is a serious matter when the calculus is based on thinking about the slopes of tangents at a point on a curve! From my own thinking on examples, I soon realised that again there are two approaches: rotating a line through the point while counting the number of times it becomes tangent to the curve, and running a tangent along the curve while counting the number of times it passes through the point. This brought to mind again the notion of two things varying in the chordal mid-points and the matchstick problems (Fig. 10.1). Notice how it is possible to see these two images as single frames from an animation, using the power to imagine and to express to yourself what you are seeing. It is not clear that all learners use this power when shown a static picture, and sometimes it is not even clear what might be animated and what held fixed. Once there are two or more ways of looking at things, it can be helpful to become flexible in their use. From the tangent running along the curve, you get a sense from the way the tangent swings about, that there are regions of the plane in which the tangent-power is constant. This is enhanced by the thought of fixing a point and swinging a line through that point, where the line becomes tangent only every so often.

Fig. 10.1 Driving a tangent along the curve and rotating a line through the point

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The regions of invariant tangent-power are bounded by places where the tangent reverses its direction of rotation or, in other words, the inflection-tangents. This in turn provides an enactive contribution to learners’ sense of the second derivative, part of what has been called their concept image (Tall & Vinner, 1981). A third consequence is that as the point moves to larger and larger absolute values of its xcoordinate, the tangent-power doesn’t tend to change (as long as what is visible on the screen has all the turning points). For many people this serves to ‘educate their awareness’. It also makes use of the ubiquitous theme of modern mathematics to look for and to exploit invariance in the midst of change. Fried (2011) considered convex curves and then showed how to combine these with parts of the curve extended by inflection tangents to provide one general way of calculating the tangent-powers in different regions. Some years later, Mark & Schramm (2016) studied the region(s) with tangent-power 0.

10.3.1

Pedagogic Remarks on Themes

Again we have an instance of restricting attention (restricting and extending) to potential tangents through a fixed point (keeping something invariant and considering what can change) and considering what freedom is available when a constraint is added or removed.

10.3.2

Pedagogic Remarks on Multiplicity

As fractions, both 34 and 68 denote the same ‘number’, because as operators acting on the unit interval or on any ‘shape’, they end up with the same result. So numbers can have multiple names. This sometimes comes as a shock to learners, even though they are well used to having multiple ‘names’ themselves and for their friends. However, it is well known that learners stumble over the fact that terminating decimals also have a non-terminating name, so some decimals have two names while others have only 1. More significantly, in arithmetic, the symbol 34 can be read in many different ways: as a fraction which acts upon things, including numbers; as the operation of dividing 3 by 4; as the answer to dividing 3 by 4; as the rational number consisting of all fractions of the form 3k/4k for k ≠ 0 (k may be required to be positive); as the position on a number line ¾ of the way from 0 to 1; as a ratio of ¾ : 1 or 1 : 4/3 or 3 : 4; as the value of the ratio of 3 : 4.

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Similarly, in algebra, the expression 2n+1 can also be read in many different ways: as an instruction to multiply something by 2 and add 1; as the result of multiplying some as-yet unknown number by 2 and adding 1; as a specific but unspecified odd number; as a general odd number; as referring to all possible odd numbers. Being aware of these multiplicities, recognising situations in which multiplicity is possible and drawing student attention to that multiplicity can enrich their sense of mathematics and of how mathematical thinking develops, as well as contributing to the growth of flexibility in their thinking.

10.3.3

Pedagogic Remarks on Invariance

Almost any theorem (other than pure existence theorems) can be cast as the statement of an invariant together with the conditions under which it remains invariant. Indeed in his book Proofs and Refutations, Lakatos (1976) can be interpreted as claiming that a considerable amount of mathematical thinking involves extending the boundaries of the constraints within which some quality or property remains invariant. Sometimes an invariant can be perceived through an animation in which some feature remains fixed. However, there are many mathematical invariants which are difficult to perceive as such. Consequently, student experience of mathematics can be enriched by drawing attention to invariants and their associated ranges of permitted change (Watson & Mason, 2005), thereby helping to link apparently disparate mathematical ideas as instances of one ubiquitous theme. Where a lecturer fails to be explicit about this despite being aware, perhaps only subconsciously in their own thinking, learners may not recognise the theme and so not develop possible richness in their own thinking. Furthermore, the theme provides a guide for mathematical exploration: seek out invariants and probe the ranges of permissible change in relevant aspects.

10.3.4

Pedagogic Remarks on Concept Images

Formal definitions are necessary for precision, but informal intuitions and ‘sensesof’ play a significant role in thinking. Most learners find that actions become available to them arising from associated feelings, senses and images, as well as internalised incantations and narratives. For example, the swinging tangents provide a kinaesthetic sense (whether actual or virtual) of the meaning of inflection points of a curve and as such provide an introduction to the second derivative and its meaning.

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Returning to Chordal Mid-Points

Encouraged by the pedagogic potential of the chordal mid-point problem, I then sought to use dynamic geometry to display the locus. Generating the locus by fixing one end of a chord and driving that fixed point along the curve seemed a sensible way to proceed. So what was displayed was a family of scaled copies of the original (see Fig. 10.2). It was only when I was trying to formulate this algebraically for quadratics and then cubics that I realised there was another way to generate the locus, namely by specialising in the sub-family of chords all of whose mid-points have the same x– coordinate, say t. This meant that calculus ideas (which are well within secondary school curriculum) could be used to find the extrema of the family of mid-points all of whose x-coordinates were fixed at t. The result was a completely different image and way of thinking and an algebraic approach that could clearly be generalised further (see Fig. 10.3).

10.4.1

Pedagogic Comment: Different Ways of ‘seeing’ and (Re)presenting

Finding a different way of ‘seeing’ or ‘sensing’ or (re)presenting3 a mathematical object (here a sub-family of chords) is a fundamental strategy in mathematics.

Fig. 10.2 Quadratic with single fixed-end locus of chordal mid-points and a family of these

3

I prefer presenting, so I place the prefix in parentheses in order to draw attention to the fact that each presentation highlights certain features and downplays others. Each is a presentation, but none capture the totality by themselves.

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Fig. 10.3 Quadratic with single chord with specified mid-point x-coordinate and locus, and family of these

Fig. 10.4 Cubic with single chord; single scaled cubic; family of loci

Recognising myself using it makes it more likely that I can exploit it with learners, first in how I construct tasks, and second, in how and when I make explicit reference to this as a strategy. With a calculus-based method of locating components of the locus available, it was a simple matter to sort out cubics. Figure 10.4 shows the scaled-copies version of the chordal mid-points. It was evident that the vertical line through the inflection point formed a boundary, which, as mentioned earlier, was a bit of a surprise until I realised that the rotational symmetry of a cubic forces this result. The locus of mid-points is a region which is open along x = 0 except at [0, 0] and closed along the original curve. Exploiting the alternative calculus-affording approach produces the diagrams in Fig. 10.5. There are details to consider, such as cubics which are monotonic, but these bring to light no further surprises.

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Fig. 10.5 Cubic with single chord and chordal mid-point locus, and the family of such loci

10.4.2

Pedagogical Comment: Personal Narratives

Articulating these phases of exploration is of value to me, much more so than to anyone reading this. It crystallises what I have learned and opens up possible mathematical extensions as well as potential pedagogic actions for use in situations with learners which are in some way parallel to my experiences. It is the formulating of a personal narrative that is so important in learning mathematics effectively. Temptation to try to internalise someone else’s narrative intrudes on, even stymies the growth of your own story.

10.4.3

Further Investigation

I was now aware of a choice: it might be interesting to see what happens when the mid-point is changed to dividing the chord in some fixed ratio (and requiring one end to remain to the left of the other end so as not to confound the image with two versions!). Another choice was to go on to quartics, and yet a third was to mimic cubics by gluing together two quadratics, so as to probe more deeply the role of cubic symmetry. I chose to look at quartics.

10.4.3.1

Pedagogical Remark: Stimulating a Disposition to Enquire Mathematically

My hope is to enculturate learners into asking such questions for themselves. To do this, they need to be in the presence of someone who does this and given the time, space and enciuragement to do it for themselves.

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Quartics

Quartics do not have a symmetry like the quadratic and the cubic, so I pursued this direction to see what would happen (see Fig. 10.6). The algebra proved rather complicated, despite using a computer algebra system. Resorting to special cases using dynamic geomtry software (specialising yet again), I began to see a possible relationship between the boundary of the mid-point locus outside of the function and the function itself. I found myself detecting a possible relationship. However, it is of little or no use to ask learners what they notice, because they do not have the immediate recent experience of enquiry, choosing to vary some feature. Inspired, I returned to the algebra but now with a conjecture, which soon turned out to be valid. What emerged is that for any quartic with two inflection points, scaling the curve by ½, rotating through 180° and then translating allow the transformed graph to fit perfectly between the two inflection points of the quartic, where the tangents of the original and of the transformed version match. I call this transformed function the inflection-tangent function. The boundary of the locus is, for quartics with positive leading coefficient, the function itself outside of the inflection-points interval and otherwise, the inflectiontangent function. This fact is a kind of pseudo-symmetry for quartics (see Fig. 10.7).

Fig. 10.6 Quartic with mid-point locus generated in two ways Fig. 10.7 Quartic with scaled and rotated copy nestling at the inflection points

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Of course, there are special cases to consider, for example, where there is only one or even no inflection points4. I have found that few learners appreciate the range of ‘shapes’ that a quartic can take, which has alerted me to taking time over displaying that range before lauunching in to work with quartics. This is an example of becoming pedagogically aware through realising my own impoverished awareness and of that awareness having an influence on my subsequent teaching. Curious about this unexpected pseudo-symmetry property of quartics, I was further encouraged to consider quartic-like graphs arising from gluing polynomials together to see what it is about a quartic that leads to this pseudo-symmetry.

10.4.4.1

Pedagogic Comment: Example Construction

It was by trying multiple examples, indeed being able to adjust examples apparently continuously, that revealed a possible relationship between the original function and the mid-chord boundary. It is possible that gazing at one single example might have eventually revealed it, but the point is that constructing your own examples is itself beneficial. Even more, using examples in order to try to get a sense of what might be going on is what specialising is really about, as John Wallis seems to have found. It is fodder for conjectured generalisations which might then be justified. Using dynamic geometry to enable multiple examples to be displayed has the advantage that it can bring to the surface calculations that need to be done, even ways of thinking (such as chords with a fixed x-coordinate of the mid-point). It can also divert attention to mathematical or programming difficulties and take attention away from the original question! There is, however, a stark contrast between examples provided by someone else and examples constructed for oneself. Attention is not only differently focused, but differently structured. Many learners seem unsure as to what to do with an example, so it behoves lecturers to ask themselves what they expect learners to do with examples, then model that behaviour themselves. To be clear to oneself about the role and use of examples, it helps enormously to catch oneself using an example, whether working through one provided by others or constructing ones’ own.

10.5

Pseudo Cubics: Glued Quadratics

Many learners struggle at first to accept that a function specified piecewise, that is, formed by gluing two formulae together at a point (and more generally), actually constitutes a valid function. They are in good company with early mathematicians who likewise wanted a single formula. I have long been intrigued by the subtle difference between

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Possible exploration for learners: characterise quartics that have a single inflection point.

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f ðxÞ =

x2 for x ≥ 0 and f ðxÞ ¼ x3 : - x2 for x < 0

Knowing that a curve that looks somewhat like a cubic can be formed by gluing two quadratics together, using opposite signs in the coefficients of highest degree and agreeing at their common point in value and slope made me curious about the role of cubic symmetry for the chordal mid-points. For a long time, I forbore working on this, feeling that the algebra and calculus would be too complicated. I felt I needed to be able to generate diagrams in dynamic geometry, which required extensive modifications to my applets. While preparing this chapter, I decided to make the applet modifications, which took me even longer than I had imagined. First, I worked on gluing Lagrange polynomials through specified points while agreeing at the glue point with a single Taylor polynomial (also specified by points). This construction was then imported into mid-point calculations. Generating a generalised Lagrange polynomial specified piecewise, with agreement at glue points with specified Taylor polynomials, is an excellent challenge for learners to support them in appreciating how gluing works when there are conditions to fulfil at the glue point. I started with a quadratic and its rotation through 180°. However, in constructing the applet, I had the option of requiring the left quadratic to be specified by two points and the glue point or by one point and both the glue point and the slope at the glue point (see Fig. 10.8). The diagrams suggest a possible task for learners to show that gluing a quadratic and its rotation through 180° at a point forces their slopes to agree at the glue point, or as a variation, asking what is the same and what different about the family of functions formed by the two constructions.

Fig. 10.8 Rotation-symmetric quadratics constructed so as to agree to degree 0 at the glue point and rotation-symmetric quadratics constructed so as to agree to degree 1 at the glue point

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Pedagogical Comment

This is an example of a possible task emerging as a side line from an exploration, where a phenomenon appears in examples before it had been thought through theoretically. The task then invites that theoretical thinking on the part of the learners. Although comparatively trivial, the task requires considerable care in specifying algebraically what it means for the quadratics to be rotationally symmetric. Many learners balk at the notion of functions being specified by being glued at a point, so the task can serve to deepen appreciation and comprehension of glued functions. If the construction is being done via Lagrange polynomials, as in the case of my applet, it is likely to lead to appreciation of how Lagrange polynomials are constructed in order to take into account more than passing through specified points. Having constructed the applet to permit gluing Lagrange polynomials together, agreeing with a polynomial of specified degree acting as a Taylor polynomial at the glue point, it was easy to form curves with less symmetry than cubics, in order to explore the effect of the symmetry (see Fig. 10.9). Attracted by the role of the inflection points for quartics, I used my applet to display inflection points as well as chordal mid-points, only to detect an unexpected phenomenon which explains part of the quartic property.

10.5.2

Pedagogic Comment

My aim is to foster and support mathematical thinking amongst learners. One way to do this is to indicate possible avenues for exploration, without succombing to the temptation to do the exploring for them. Consequently, I am choosing not to be

Fig. 10.9 Examples of glued quadratics with the Taylor polynomial of agreement at the glue point, together with their chordal mid-point regions

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explicit about the role of inflection points with chordal mid-points, nor about what might be explored when mid-points are generalised to ‘μ-points’.

10.5.3

Mathematical Comment

One of the features of the original and the first version of the glued applets was that the boundaries were simply calculated as the extremes of the calculated individual vertical rays corresponding to a fixed x-coordinate. When I tried to calculate these theoretically, I ran into considerable difficulty in handling the constraints. Fix the x coordinate at t. Let the glue point occur at x = t0. Let the chord have width w. Then the range of the y-coordinate of the chordal mid-points involves the union of the range when w > |t - t0|, where the chord spans the glue point, and the bounded range when w ≤ |t - t0|, where the chord joins points both of which are on one side or the other of the glue point. I began to appreciate difficulties learners might have when finding extrema on restricted domains, where end points have to be considered. I may even now be more sensitised to learner struggles with restricted domains.

10.5.4

Pedagogical Comment

One of the pedagogical opportunities arising from personal mathematical exploration arises from paying attention to my own experience. Where some useful action becomes available, trying to become aware of what triggered that action can alert me to actions that might be worth bringing to student attention and invoking specifically. Where I struggle, however briefly, might point to analogous struggles experienced by learners and give me greater sensitivity to what learners go through. This could suggest looking out for parallel situations where some pedagogical action, including simply allowing more time, could be of significant benefit for learners, rather than rushing through expecting them to catch up somehow. Despite the tendency to want to keep failed conjectures and computational mistakes to oneself, if learners never see someone modify their conjecture or check and correct their calculations, they are likely to form the impression that correct mathematics flows out of a mathematician’s pen, so that when they start to struggle, they may give up as being inadequate. Resilience is developed through experience of modifying and correcting, not from trying to be perfect all the time (Mason 2008). This aligns with the notion popularised by Vygotsky (1978) that higher psychological processes are first encountered in the behaviour of others. Note: I am not advocating mid-points of glued quadratics as a task for undergraduates. I am, however, saying that investigating it for myself made me appreciate more fully and deeply what it might be like for learners struggling with similar or parallel tasks, such as finding local extrema on restricted domains.

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Pseudo Quartics: Glued Cubics

When finally my applet for gluing Lagrange polynomials together was functioning, I turned to gluing cubics with various degrees of agreement at the glue point so that the glued function is differentiable, shown in Fig. 10.10. The role of the inflection points is intriguing!

10.5.6

Reflection

The idea of considering chords with a fixed mid-chord x-coordinate crossed over to the idea of fixed-width chords for approximating derivatives. It is traditional to discuss the derivative of a function at a single point so that functions which do not have derivatives everywhere can still be considered. This fixed-point perspective has some drawbacks for learners however, because they get little or no support for assembling the slope of the tangent at a point over a range of points in order to experience the derivative as a whole (where it exists). Thinking instead of fixedwidth chords offers an image of a curve (the w-width chord-slope function) which everywhere (where it makes sense) approximates the derivative. As the chord width w goes to 0, the chord-slope function approaches the derivative everywhere that the function is differentiable. Put another way, as the chord width goes to 0, the chord height also goes to 0. How fast does it go to zero? It turns out that dividing by the chord width goes to a limit. This thinking gave rise to a collection of tasks which can be used to underpin learners’ use of calculus and calculus-inspired questions in order to internalise some basic ways of using the calculus. For example, what can be said about the gap between the mid-point of a chord and the function value there as the chord width goes to 0? It clearly goes to 0, but how fast? Dividing by the chord width still goes to 0. But dividing by the square of the chord width goes to half the second derivative at the point. This really surprised me, even though I could, after some thought, connect it with other things I knew about the second derivative and Taylor approximations.

Fig. 10.10 Two cubics agreeing to degrees 1 and 2, respectively, at the glue point

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My explorations gave rise not simply to a task about the chordal gap, as an application or instance of second derivatives, but to an enriched concept image associated with second derivatives. It also fed an interest in exploring ‘how quickly’ various constructs associated with a chord go to 0 as the chord width goes to 0. For example, the ex-radii, inradius and circumradius of the chordal triangle and the area between the chord and the function, among others.

10.6

Taylor-Powers

At some point, I realised that the notion of tangent-power was a special case of what could be called the Taylor-powers of a point with respect to a curve. For each point X on a curve, and for each t, construct the Taylor approximation of degree t at X. Now ask for how many points X the t-Taylor at X passes through some given point P. Thus, for each P, there is a tuple of ‘powers’, one for each degree of Taylor approximation. I found that predicting these led me to a deeper appreciation of Taylor approximations (Fig. 10.11). One consequence of the two ideas, Taylor-powers of a point and fixed-width chords was to construct an applet in which the user can select the Taylor approximation and run it along the curve in order to count the number that pass through a given point. The images encountered go a considerable way to augment the purely algebraic notion of Taylor approximations, adding to and enriching learners’ concept image of Taylor series. The applet also does the count, for all the Taylor approximations, so patterns can be sought amongst them.

Fig. 10.11 3 Instances of Taylor polynomials of degree 2

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The Intimate Interplay Between Developing Teaching and Exploring. . .

10.6.1

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Mathematical Comment

I found the 0-th Taylor power somewhat confusing at first, partly because I had never really stopped to think about it, partly because it is of little interest algebraically and partly because of the sheer unexpectedness, until I realised that it was simply counting the number of times the curve crosses the horizontal line through the point.

10.6.2

Pedagogical Comment

It is so easy to rush through the mechanics of important ideas without giving learners sufficient exposure to enrich their concept image and to extend their accessible example space (Watson & Mason, 2002, 2005; Mason 2008a). Inviting them to explore side lines is one way to foster such enrichment. Of course, it takes time, but if learners are to shift from passive recipients of strings of theorems into mathematical thinkers who take initiative to explore, to construct examples and to build their own narratives around their concept images, then time must be taken. It is a good investment, because their learning, and consequently your teaching, becomes more efficient. And connecting your experiences with their struggles contributes to that greater efficiency.

10.7

Overview

Through describing and in some cases recounting personal explorations, usually triggered by some pedagogical issue but sometimes by curiosity, I hope to have shown how being attentive to my own thinking has sensitised me to parallel struggles learners may be undergoing and so informed my subsequent pedagogy. Personal explorations keep me mathematically alive and active and so in a position to work with others on developing their mathematical thinking. A common excuse for not engaging learners meaningfully is the shortage of time. But are learners studying efficiently and effectively? What do they actually do with examples they are given? How are they stimulated to think mathematically rather than to gain competence in procedures sufficiently in order to get through an examination? Are lecturers preparing their sessions effectively? Surely getting learners to think mathematically, to appreciate and comprehend what lies behind the major theorems and procedures is more efficient than throwing definitions, lemmas and theorems onto boards from which learners copy them down. I suggest that it is time management, not lack of time itself, which holds people back, in parallel with sensitivity to the use of human powers and ubiquitous mathematical themes. Being sensitised to learner difficulties through becoming aware of your own struggles can make

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teaching more efficient, because you can get to the heart of the struggle more quickly and effectively. What could be more thrilling as a lecturer than to see learners developing a rich and extensive sense of a topic, rather than simply trying to complete homework exercises on time? What could be more thrilling for learners than to feel confidence growing through their use of mathematical powers to make sense of mathematical topics, theorems and procedures?

References Artigue, M., & Blomhøj, M. (2013). Conceptualizing inquiry-based education in mathematics. ZDM, 45(6), 797–810. Barone, J. (2019). Leonardo Da Vinci: A mind in motion. British Library. Bridges, J. (Ed.) (1897). The ‘Opus Majus’ of Roger Bacon with introduction and analytical tables. Clarendon Press. Brookes, B. (1976). Philosophy and action in education: When is a problem? ATM Supplement, 19, 11–13. Brown, S., Collins, A., & Duguid, P. (1989). Situated cognition and the culture of learning. Educational Researcher, 18(1), 32–41. Bruner, J. (1966). Towards a theory of instruction. Harvard University Press. Bruner, J. (1990). Acts of meaning. Harvard University Press. Chevallard, Y. (1985). La Transposition Didactique. La Pensée Sauvage. Courant, R. (1981). Reminiscences from Hilbert’s Gottingen. Mathematical Intelligencer., 3(4), 154–164. Fried, M. (2011). Tangent powers. Private communication. Gardiner, A. (1987). Discovering mathematics: The art of investigation. Oxford Science Publications, Clarendon Press. Gattegno, C. (1970). What we owe children: The subordination of teaching to learning. Routledge & Kegan Paul. Hiller, G. (2000). Levels of classroom preparation. In I. Westbury, S. Hopmann & K. Riquarts (Eds.), Teaching as a reflective practice: The German didaktik tradition (pp. 207–221). Psychology Press. Killion, J., & Todnem, G. (1991). A process of personal theory building. Educational Leadership, 48(6), 14–17. Kolb, D. (1984). Experiential learning: Experience as the source of learning and development. Prentice Hall. Lakatos, I. (1976). In J. Worral & E. Zahar (Eds.), Proofs and refutations: The logic of mathematical discovery. Cambridge University Press. Letiche, H. (1988). Interactive experiential learning in enquiry courses. In J. Nias & S. Groundwater-Smith (Eds.), The enquiring teacher: Supporting and sustaining teacher research (pp. 15–39). Falmer Press. Levenson, E., Swisa, R., & Tabach, M. (2018). Evaluating the potential of tasks to occasion mathematical creativity: Definitions and measurements. RME, 20(30), 273–294. Love, E., & Mason, J. (1992). Teaching mathematics: Action and awareness. Open University. Madsen, L., & Winslow, C. (2009). Relations between teaching and research in physical geography and mathematics at research-intensive universities. International Journal of Science and Mathematics Education., 7(4), 741–763.

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Mark, M., & Schramm, M. (2016). The College Mathematics Journal., 47(5), 334–339. Mason, J. (1994). Only awareness is educable. In A. Bloomfield & T. Harries (Eds.), Teaching, learning and mathematics (pp. 28–29). Association of Teachers of Mathematics. Mason, J. (1998). Enabling teachers to be real teachers: necessary levels of awareness and structure of attention. Journal of Mathematics Teacher Education., 1(3), 243–267. Mason, J. (2002a). Researching your own practice: The discipline of noticing. RoutledgeFalmer. Mason, J. (2002b). Mathematics teaching practice: A guidebook for university and college lecturers. Horwood Publishing. Mason, J. (2008a). From concept images to pedagogic structure for a mathematical topic. In C. Rasmussen & M. Carlson (Eds.), Making the connection: Research into practice in undergraduate mathematics education. MAA Notes (pp. 253–272). Mathematical Association of America. Mason, J. (2008b). Being mathematical with & in front of learners: Attention, awareness, and attitude as sources of differences between teacher educators, teachers & learners. In T. Wood (Series Ed.) & B. Jaworski (Vol. Ed.), International handbook of mathematics teacher education: vol.4. the mathematics teacher educator as a developing professional (pp. 31–56). Sense Publishers. Mason, J. (2009). Teaching as disciplined enquiry. Teachers and Teaching: Theory and Practice, 15(2-3), 205–223. Mason, J. (2011). Phenomenology of example construction. ZDM Mathematics Education, 2011(43), 195–204. Mason, J. (2019). Pre-parative and post-parative play as key components of mathematical problem solving. In P. Felmer, P. Lilljedahl, & B. Koichu (Eds.), Problem solving in Mathematics instruction and teacher professional development. Springer. Mason, J., & Davis, J. (1987). Only awareness is educable. Mathematics Teaching, 120, 30–31. Mason, J., & Johnston-Wilder, S. (2004). Fundamental constructs in mathematics education. RoutledgeFalmer. Mason, J., Burton, L., & Stacey, K. (1982). Thinking mathematically. Addison Wesley. Mason, J., Burton L. & Stacey K. (2010). Thinking mathematically (2nd edn). Addison Wesley. Neubrand, M. (2000). Reflecting as a Didaktik Construction: speaking about mathematics in the mathematics classroom. In I. Westbury, S. Hopmann & K. Riquarts (Eds.), Teaching as a reflective practice: The German didaktik tradition (pp. 251–266). Psychology Press. Nardi, E. (2008). Amongst Mathematicians: Teaching and learning mathematics at university level. Springer. Pólya, G. (1945). How to solve it: A new aspect of mathematical method. Princeton University Press. Pólya, G. (1962). Mathematical discovery: On understanding, learning, and teaching problem solving (combined edition). Wiley. Pólya, G. (1965). Let us teach guessing (film). Mathematical Association of America. Schön, D. (1983). The reflective practitioner: How professionals think in action. Temple Smith. Tall, D. (1987). Constructing the concept image of a tangent. In Proceedings of the 11th conference of the international group for the psychology of mathematics education (Vol. 3, pp. 69–75). Tall, D. (2004). Thinking through three worlds of mathematics. In M. Johnsen Joines & A. Fuglestad (Eds.), Proceedings of the 28th conference of the international group for the psychology of mathematics education (Vol. 4, pp. 281–288). Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12(2), 151–169. Vincent, B., LaRue, R., Sealey, V., & Engelke, N. (2015). Calculus students’ early concept images of tangent lines. International Journal of Mathematical Education in Science and Technology., 46(5), 641–657. https://doi.org/10.1080/0020739X.2015.1005700 Vygotsky, L. (1978). Mind in Society: The development of the higher psychological processes. Harvard University Press.

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Watson, A., & Mason, J. (2002). Extending example spaces as a learning/teaching strategy. In A. Cockburn & E. Nardi (Eds.), Mathematics. Proceedings of PME 26 (Vol. 4, pp. 377–385). University of East Anglia. Watson, A., & Mason, J. (2005). Mathematics as a constructive activity: Learners generating examples. Erlbaum. Westbury, I., Hopmann, S., & Riquarts., K. (Eds.). (2000). Teaching as a reflective practice: The German Didactics tradition. Erlbaum. Zammattio, C., Marinoni, A., & Brizio, A. (1980). Leonardo the scientist. McGraw-Hill.

Chapter 11

Signatures of Teaching Mathematics Günter Törner

Abstract The title of this book ‘Mathematicians’ Reflections on Teaching: A Symbiosis with Mathematics Education Theories raises the question of How can mathematicians’ teaching be understood by mathematics education theories? for all the authors of articles within the book. Since the terms mathematics, mathematician and teaching are by no means well-defined notions, they are open and coloured by subjective philosophies. Thus, philosophical embeddings play a decisive role in the understanding of the behavior of mathematical lecturers. Further influencing factors are addressed. It is the aim of this chapter to show hitherto barely discussed hidden variables of teaching mathematics. In this sense, following some ideas of Lee Shulman, we introduce the notion of a ‘signature of teaching mathematics’ and justify this concept formation through numerous examples. At least four dimensions are relevant to us: the mathematical content as such and its structure, the underlying understanding of teaching and learning, the characteristics of the partly socially acting classroom and the immanent philosophies of mathematics. Keywords Signatures of teaching · Classroom realities · Philosophy of mathematics · Philosophy of mathematics teaching · Epistemological obstacles · Didactical contracts

11.1

Introductory Inventory

Before the author reveals his views, he wants to highlight his subjectivity. First, the author is a research mathematician and a mathematics educator, also engaged in international research within mathematics education, and thus—as Lee Shulman once stated—he is a commuter between two different worlds. Evidently, he was lifelong occupied through teaching mathematics. His teaching in mathematics education differs completely from his teaching of mathematical research topics. In

G. Törner (✉) Faculty of Mathematics, University of Duisburg–Essen, Essen, Germany e-mail: [email protected] © Springer Nature Switzerland AG 2023 S. Stewart (ed.), Mathematicians’ Reflections on Teaching, Advances in Mathematics Education, https://doi.org/10.1007/978-3-031-34295-0_11

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retrospect, when he reflects on his teaching in these two distant areas, he realizes that his endeavors to improve himself in these areas have given him different answers. This makes something actually trivial clear, teaching styles, e.g. in mathematics versus mathematics education, are inseparable from the context. He begins to write this article as he began to write every mathematical article in the past: he deliberately starts a literature search first. When starting a study of mathematical literature, the author sensed that actually everything is known and already said by outstanding scientists, as the bibliography shows and how it should be for a mathematician to admit openly. What does remain to be explained about mathematics teaching at all?

11.1.1

The Dilemma of Literature Research

However, since he was convinced by the basic concern of the editor’s issues, he agreed with a contribution and went on a deeper search in the literature at the interface between mathematics and mathematics education. Here, he encounters another problem that everyone knows who is researching at the interface between mathematics and mathematics didactics. Some important mathematical articles are listed in the two well-known mathematical databases, MathSciNet of the American Mathematical Society and zbMath of the European Mathematical Information Service (EMIS), but the didactic articles could be found incompletely in the small database MathEduc, which was unfortunately closed on December 31, 2019, as the author observed when revising this article. While researching the topic requested, the author became increasingly aware that the old masterminds in their still interesting fundamental works (see e.g. Courant & Robbins, 2001) on mathematics over several editions did not say anything about the teaching of mathematics. By contrast, Freudenthal (e.g. Freudenthal, 1973), the prominent mathematician, is gratifyingly self-reflective in his basic statements, but his publications are already 40 years back. A few publications with the keyword Advanced Mathematical Thinking could be added to these articles, e.g. the not very new articles by Dreyfus (1990) or Tall’s contributions (1994, 2019) or a few basic articles by Mason (1992). What bothers a little is the fact that they rarely address university teachers and teacher educators. In addition, their illuminating examples come largely from elementary mathematics. Finally, articles from the International Journal of Research in Undergraduate Mathematics Education and other teacher education journals sometimes contain small paragraphs, but rarely fundamental papers. Such publications are hardly noticed by the mathematical community. In fairness, admittedly these articles are not listed in the mathematical databases. It is rarely possible to find important reflective insights from ‘hard mathematics’ everyday university life beyond the third semester. It should not be overlooked that the word ‘mathematics’ comes from the Greek and is freely translated as the art of learning. Teaching and learning are paired

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categories. The complementary act of teaching, the other side of the same coin, is often treated by research mathematicians as neglected, perhaps because on the teaching-side of university one sees the a priori more knowledgeable mathematicians, the professors, the graduates and the teachers. While considerable progress in researching individual mathematics learning is achieved, progress in understanding teaching in university classrooms has been less prominent. Probably, it is much easier to research learning processes with students than goings-on regarding teaching related to the teachers. The author became more and more aware that the teachings of mathematics are by no means uniform, all-in-one process, further teachings take place on many stages, different actors act, the content varies, the contexts are very different and the audience is very complex. It quickly becomes clear that lessons in a historical perspective are subject to a continuous process of change. The longer you reflect on teaching, the more you realize that you should understand the term ‘teaching’ more openly and further. To use a metaphor, classic teaching has several ‘siblings’. Publishing is, strictly speaking, also a further teaching, one could speak of a frozen, virtual teaching. Instructing in a specific context, thus the ‘teaching’ of mathematics, e.g. via hand books and instructions, is also essentially a teaching form for a specific audience.

11.1.2

Living with Ambiguities

One quickly realizes that mathematics has specific teaching styles that are not known in this form in other subjects. The teaching of mathematics involves further deeper lying, partly unknown and explored dilemmas and obstacles, because mathematics is characterized by many ambiguities: Ambiguity, which implies the existence of multiple, conflicting frames of references, is the environment that gives rise to new mathematical ideas. [. . .] Now one might think that mathematics is characterized by the clarity and precision of its ideas to understand a given mathematical situation or concept. On the contrary, I maintain that what characterizes important ideas is precisely that they can be understood in multiple ways; this is the way to measure the richness of the idea (Byers, 2007, p. 23).

The book by Hersh (1997) published for the first time in 1997 goes as far as the ‘lowlands’ and with it other authors influenced by it, such as Byers (2007), see some examples from calculus already in Byers (1984), and the essential statements are also found in Artigue (1999) and Bass (2015), which tackle the questions from another point of view. The longer one deals with the subject, the clearer it becomes that teaching is not primarily a didactically perfected and optimized acting of a mediation process, which is a well-balanced application of appropriate mathematical-didactical theories, but teaching of mathematics as devised by mathematics must be probably stronger than subjects in humanities disciplines. I do not agree with the opinion of some mathematicians that an established mathematician while teaching mathematics is automatically a competent teacher.

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Explaining mathematics is not the same as teaching mathematics, although it is not uncommon for mathematics lessons to be reduced to explanations formulated by the teacher.

11.1.3

Methodological Constraints

This chapter will not deliver an empirically based theory. It is based on the author’s (albeit longer term) observations. Maybe, it seems almost impossible to quantify and ensure my statements. The author does not believe the conclusions of questionnaire surveys, which often respond to the expectations of the questioner: adult respondents want to cause the asker no difficulties. Many variables are linguistic. The answers of mathematicians to a Likert-scale questionnaire are often of the type ‘yes and no’. And a further argument should be considered. In my experience (and my country), it is not customary to sit in the lecture of a colleague as an institute member and observe lecturing. The lecturer on the board would be very irritated. If you explained after the lecture you were interested in his/her style of teaching, it would increase his/her irritation. Exceptions may confirm the rule here. What is self-evident to the author, however, is the reasonably continuous participation in faculty colloquia, and the observations are based on this form of mediation. The author’s statements are based on many observations and references in the literature.

11.2 11.2.1

Mathematics, Philosophy and World Views: First Observations The Impact of the Reference Science: Mathematics

Mathematics is growing, and highly successful, expanding its body of knowledge. Mathematics is increasingly penetrating other domains of knowledge, such as life sciences, geothermal energy and materials science, just to name three fields. Mathematics is internationally positioned and structured like no other field of knowledge, and the subdisciplines of the International Mathematical Union in the Mathematics Subject Classification1 refined into over clarify it to your colleagues what interests you in mathematics; one’s research domain is more or less locatable on a large virtual map of mathematics. Finally, there is no school subject than mathematics with the same international curriculum canon that has the same international curriculum canon by which 1

The current version, MSC2010, consists of 63 areas classified with two digits.

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repeatedly initiated performance comparisons can be run. In this respect, mathematics is officially well established internationally and cultural practices do not play a dominant role, and however they should be ignored. Thus, in our field of knowledge, we are dealing with a proud science that, in some areas, produces results that, in the first approximation, provide eternally valid insights unlike other sciences in which theories sometimes exhibit ‘expiration dates’. To cite Atiyah (1978), § 8 since the aim of mathematics is to explain as much as possible in simple basic terms, mathematicians strive for simplicity. This is one example for the ‘unity of mathematics’, a prominent overarching goal of math, in which many mathematicians believe. This implies that progress in mathematics is continuous, marked by more than 100.000 research articles per year, even increasing and this should excite any lecturer. However, is this known to all lectures of mathematics? To be honest, partly a philosophical question.

11.2.2

Teaching Mathematics: The Classroom Inducing Further Variables

Wherever mathematics is used, there is an urgent need for detailed instructions, mathematics must be adopted and mathematics has to be taught on many levels. This teaching of mathematics is subject to discipline-specific norms: calculus in a lecture for mathematicians looks different from a lecture of the same name among engineering students. Many colleagues live personal traditions. The university’s teaching of research mathematics on the one hand and teaching mathematics in school on the other are just only two of many (partly opposite) samples. The tutor’s tutoring is a teaching and learning process, the presentation of new research insights in the faculty colloquium is another, the informal acquaintance with mathematical processes and contexts another. Obviously—and it is our first insight—the teaching process is always paired with different intensive learning processes and their requirements, goals and expectations. Primarily, the teacher is responsible for the optimal tuning, so we are dealing with two sides of a coin, but occasionally some stakeholders are only interested in one side. This teaching process may be based on a divide between two levels of people involved (the teacher in relation to his/her students, the pupils in school and their teacher), but it may also be superficially on an equal footing if colleagues in the Faculty Council speak. There are differences when looking at the proven expert in the circle of his curious colleagues. By simply watching and imitating, one acquires no competence in mathematics and the drudgery essential for the understanding of mathematics. Mathematics may be communicated in small or larger circles, the ‘Socratic Dialogue’ (see Rényi, 2006) with Menon is an example of intrusion into mathematical relationships in the

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smallest group, neatly in an exchange of two people. In Western Europe, in time, with the growth of the universities, who have mostly conducted, their knowledge transfer has been performed via lectures. Today, this form of teaching is becoming increasingly critical. The digitalization of our world allows for other contacts and the medially completed lecture and smartphones in the homes of the listeners massively change the forms of communication. So much for the general organizational conditions of modern teaching and learning. The results so far have clarified that the teaching of mathematics is very complex and takes place in different frameworks, general organizational conditions of modern teaching and learning. What the didactical research has deliberately made us aware by international comparative investigations (see Clarke et al., 2006), which we cannot refer exclusively to the old tripartite interactive ‘teacher-student-content’ teaching model, but that we must not overlook the social context, in short the ‘classroom’ in a changing society must be in the focus. We use the term ‘classroom’ in the more general sense. The auditorium of a lecture is for us a ‘classroom’ and the audience of a colloquium lecture very different types of ‘classrooms’. The lessons at school take place not only in a real classroom but also in a virtual social one. Classrooms have different possibilities of participation, in school this is selfevident; in many lectures at a university, the audience is largely silent. The lecturer writes quickly at critical points to keep students from embarrassing questions. The classroom must be considered as a variable in describing the teaching of mathematics. Here is another didactic theory, that of the didactical contracts, which goes back to Brousseau (1984), Brousseau and Warfield (2014) which offers to be applied. Silent arrangements are formed between a teacher and the people in a generalized classroom, which rules should be followed by mutual interactions. If a teacher repeatedly ‘irones out’ queries or irones questions, the willingness of the listeners to ask intermediate questions decreases, since the procedure is not considered effective. Conversely, a teacher who deals openly with his own mistakes implicitly creates a positive encouraging culture of errors. The didactical contracts created in this way are not random, limited to the individual event, but depending on the acting players (teachers, students, classrooms) and traditional accepted norms beyond the individual case, the mutual interaction establishes as a characteristic.

11.2.3

Epistemological Obstacles

Where educational processes are perceived by the recipient to be less than optimal, failure will be attributed to a teacher who seeks blame. While understanding mathematics rarely is considered easy, there are not a few situations where you

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have to look for ‘guilt’ in the listener, namely when you have to overcome so-called epistemological barriers. Hardly any mathematician knows the term, which didacticians carefully analyzed in some publications over the years. The classical mathematician is content and states that mathematics is inherently difficult. It is arguably Bachelard who, in 1934, first exposes the phenomenon of epistemological obstacles; Sierpinska (1987) and others have worked intensively on it (see Schneider, 2014; Job & Schneider, 2014; Brousseau, 1984) and clarified that they can be found in all areas of mathematics, in Calculus, in Algebra (e.g. Subroto & Suryadi, 2019) or in Geometry. Schneider points out that the problem is explained by the effect of mental presetting which has to be proved as helpful. Subjective images of learning are based on these knowledge constructs and experience they succeed until contexts emerge, where these representations are explained as wrong or at least inadequate, and thus they have to be overcome and changed. Changes in hitherto successful mental attitudes might be often tedious. Schneider (2014) states: Brousseau distinguishes indeed the ontogenic obstacles, related to the genetic development of intelligence, the didactical obstacles, that seem to only depend on the choice of a didactic system, and the epistemological obstacles from which there is no escape due to the fact that they play a constitutive role in the construction of knowledge.

If, according to our explanations, the actual causes lie in the changing context, i.e. with the listener, it is essential that the teacher also knows about these obstacles and counsels assistance.

11.2.4

Philosophy of Mathematics

It is the central statement of the following analysis that we cannot leave out the philosophical side of mathematical science. The author is convinced that in this abstract science, nevertheless personal philosophies play a dominant role, since each world view of mathematics is coloured by personal philosophies, unless this thesis is not all everywhere reflected or accepted. As part of a mathematical colloquium at the author’s university 15 years ago, the guest, a Finnish colleague, had given his approximately 30 listeners three categorizations of mathematics (mathematics is a system of rules, mathematics is a formal system, mathematics is problem solving, see Dionne, 1984) and they were asked to allocate 30 points to these different aspects. The audience was then very surprised by how strongly the subjective assessments of all those present deviated from each other. Here we should make a digression on different philosophies of mathematics that delineate Platonist versus formalist philosophy and expound Hersh’s proposal for a humanist philosophy, at least at a high level of math, but not in younger learners: What’s the connection between philosophy of mathematics and teaching of mathematics? Each influences the other. The teaching of mathematics should affect the philosophy of mathematics, in the sense that philosophy of mathematics must be compatible with the fact that mathematics can be taught. (Hersh, 1997, p. 237)

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In 1973, over 46 years ago, the young Fields medalist René Thom (1923–2002) pointed out: In fact, whether one wishes it or not, all mathematical pedagogy, even if scarcely coherent, rests on a philosophy of mathematics. (Thom, 1973, p. 204)

It was the time when it was expected that ‘modern mathematics’ could reform lessons. Today we know that not a few who advocate modern mathematics have had a limited understanding of mathematics (see in particular Sect. 11.2.5). Ultimately, this reform failed. It was the highly competent mathematician Thom who loudly commented loud and clear the time of ‘modern mathematics’ (Thom, 1973). If we accept his statement, then we maintain a close link between mathematics and philosophy. The author is deeply convinced of this: our often hidden philosophy of mathematics has an essential effect on our self-image of the teaching of mathematics. Hersh states, as the author says, aptly: Yet, as we all know, the dogmatic style of textbooks and teaching is pervasive and deeply rooted. Mathematical subjects appear as isolated and inhuman piles of axioms and algorithms. The student gets a letter, brutal message: Here it is, swallow it down! (Hersh, 1990, p. 105)

Philosophy of mathematics is a universal key to understand the widely varying appearance of teaching of mathematics—from school to university and from the research field to communication in industry and economy. The author is convinced that he believes that the teaching of mathematics is not primarily exclusively anchored not only in educational theories but also in the philosophy of mathematics that the person concerned has.

11.2.5

World Views of Mathematics

If we address philosophies of mathematics here and refer the reader to the Hersh book (Hersh, 1990) for brevity, then these philosophies are not add-ons that you have to look up next to the lecture manuscript and deliberately incorporate them into places. These are mostly unreflective attitudes and subjective impressions alias theories that constantly influence our justification and argumentation. They are ‘colouring’ your teaching. In the didactics of mathematics, such ideas are shortened as beliefs (see Leder et al., 2002). There is an enormous uncountable number of publications on beliefs, every major mathematical conference has its beliefs section. It should be noted critically that many publications do not make their definition of beliefs explicit, as noted already 37 years ago, see Pajares (1992). What is known is that almost everything serves as a ‘beliefs object’, namely as the object of subjective ideas around that mathematical object. These beliefs cannot be ignored, we have ‘to live by beliefs’, modifying a famous title of a booklet of Lakoff and Johnson (1980).

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Individual beliefs usually fit into larger complex beliefs systems (Törner & Pehkonen, 1996) (sometimes called worldviews (Schoenfeld, 1985) and influence our thinking massively, as Schoenfeld (2010) has worked out in 2010, using the unspent term orientation. These worldviews decide on the success or failure, when for example implementing curricular changes, as the observations in the 80s in establishing a problem-solving curriculum in American schooling (Frank, 1990), and here are again interfaces to philosophy. The teaching of a mathematician, for whom mathematics is a huge playground of problem-solving, completely differs from those who see mathematics constituted by formalism or, alternatively, as a system. By no means is there a mutually exclusive either–or spoken. To be more precise, Loos and Ziegler (2016), Ziegler and Loos (2017) point out that sometimes at least three different mathematics are to be taught in school. Citing the world views (for teachers) from Ziegler and Loos and slight extending these characteristics, we should mention: • Mathematics can be regarded as a large toolbox for the everyday world of a working mathematician. • A large field of research central in pure as well as applied mathematics • A discipline with a long history and a prominent part of culture and simultaneously a key for modern technologies (Hoffmann et al., 1998) Often while you are lecturing, you have to switch between these positions.

11.3

Signatures of Mathematics Teaching and Learning

It is Lee Shulman (born 1938) who, with only a few contributions fourteen years ago, deals with the question of how, in individual scientific disciplines, the education of young students is permanently shaped by the discipline-specific environment. His argument refers to the imprinting of children by nurseries. We cite Shulman: . . . if you wish to understand why professions develop as they do, study their nurseries, in this case, their forms of professional preparation. When you do, you will generally detect the characteristic forms of teaching and learning that I have come to call signature pedagogies. These are types of teaching that organize the fundamental ways in which future practitioners are educated for their new professions. (Shulman, 2005a, p. 52)

In a lecture in which the author participated, he demonstrated typical training situations with photos: engineering students sit around a table, communicate intensively and reciprocally work in coordination with materials lying on the table. In another picture, a head physician in the foreground was standing at a hospital bed, surrounded by the hierarchical classification of a few senior physicians, other assistant physicians and finally nurses; there was hardly anything visible from the patient. Finally, a horseshoe-shaped auditorium was shown, in which law students were involved in violent verbal exchanges.

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Images from a mathematical school were not shown. But Shulman mentioned that teaching in departments of mathematics is often practiced as a dorsal teaching. The teacher turned to the large board and is busy writing. The back of this person pointed to the auditorium, so that one could speak of a ‘dorsal teaching’. Based on these and further observations, he founded the concept of so-called Signature Pedagogies (Shulman, 2005a,b, 2007; Falk, 2006 etc.). At first glance, his concern and his intentions are understandable, further advance formulation in the scientific literature of this internationally respected senior among educational psychologists is still outstanding. This approach by Lee Shulman suggests that, especially when restricted to teaching mathematics, it is possible and interesting to characterize multidimensional characteristics of teaching mathematics. The above comments have been the direction of an approach. We want to call this conceptual construct the ‘signature of teaching mathematics’. Below we specify possible dimensions of the ‘signatures of teaching’ beyond the introductory remarks above. What are the essential dimensions of such a signature of teaching, to confront teaching of mathematics with?

11.3.1

Dimension: The Content—Mathematics

The variable ‘content’ contains many facets and has different effects on the teaching units, which we will not elaborate here. If we want to better understand the teaching of mathematics or more precisely ‘learning of mathematics’, then we must also go to the material level. Content implies material-organizational problems that cannot necessarily be decided canonically. Known are the usual procedures: from the ‘general to the specific’ or in the opposite direction, from axioms to theorems, although in the historical development the analysis was made in the opposite direction (see below). Sometimes mathematicians believe that axiomatics answers the problems of sequencing. But only a few areas reduce axiomatic scaffolding. It appears that calculus, as a school-related area, is well suited to highlighting the same problems. Calculus is not an axiomatically representable field of knowledge. Normally, one opens up the essential contents, especially the central propositions, that one conveys the real number with its completeness properties; then, continuity and differentiability are conveyed, around the ‘central sentences’ (Rolle’s Theorem, Mean Value Theorem, Theorem about the existence maximum and minimum of a continuous function etc.) as this procedure gives the impression of a logical progression, of a progressive ascendancy in knowledge. Thus, lowering layers is a process of accrediting content. In the history of linear algebra, rounding up and expanding are known as ways of precising the content. In the work Toeplitz (1909) by Toeplitz from 1909, you will almost casually find a remark at the end of his work, which shows that calculations

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are correct, when the linear systems of equations have real coefficients, but that the argumentation works over every number field. Scheduling theory has another cut at the content level. Since almost every example has its own theory, it is important to find relevant differentiating parameters again and again to specify the problems or areas favorably. Historically however, in calculus, it was sometimes the other way round: for a long time, there had been an uncritical, implicit understanding of continuous functions, and it was only with such (continuous) functions that one wanted to protect oneself. Intermediate value properties, minima, and maxima for such restricted functions were regarded as self-evident for more than a century in history. It took some time to realize that the underlying number concept was to be presupposed and that we are today familiar with the sequencing of the analysis content is gradually disappearing. Students usually take a few semesters to recognize why the number field of the reals is indispensable. We need (exactly) the properties of the real numbers to have these classical statements. An analysis of the rational numbers does not allow us to make the same statements. This is what Freudenthal (1973), p. 101, has pointed out and what he calls ‘antididactic inversion’: The mathematics is presented as a deductively ordered product and we conceal the thoughts that led us to the result.

He points out that the only thing didactically relevant, the analysis of the subject, is suppressed, and the author is given the result of the analysis, and he may watch as the teacher knows, where it goes, put it together.

11.3.2

Dimension: Teaching Mathematics

Since we have described above publishing as a ‘sister’ of teaching, then it becomes clear what an anti-didactic inversion at the teaching level means; this access, however, can appear tolerable to necessary for another form of communication. To be more precise, we start this section with an often overlooked dependency and refer to Byers (1984), p. 35: Of course, every teacher is faced with the problem of teaching for understanding as opposed to teaching facts or techniques. Also we are all aware that there are different levels of understanding. However, while I was preparing to teach this course I was made aware of a dilemma which I had never before seen so clearly and which is shared, I believe, by many teachers. This dilemma involved certain conflicting goals which existed for me in this particular course and which I suspect may be present in many teaching situations. . . . In the past, I had dealt with this problem by presenting my own understanding to the students in a very patient and detailed manner, hoping in this way to provide them with a path through the material which was relatively free of conceptual ambiguity. . . . In the past, I had dealt with this problem by presenting my own understanding to the students in a very patient and detailed manner, hoping in this way to provide them with a path through the material which was relatively free of conceptual ambiguity.

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Byers (2007) is continuing by presenting examples from number theory. However, teaching could not be understood as a uniquely determined process, contrary the input variables are dependent on the output of the teaching activity. The last remark clarified that teaching does not only take place when a visible teacher is facing a student or a class. A book or journal can take over the role of the teacher and the laws around the teaching are different. Common to all styles or ‘siblings of teaching’ is that a sustainable learning process is initiated. It is trivial to note this, but it should be reissued. Anyone who reflects on teachings needs to understand a lot about learning. It is the often cited coin with two sides. When talking to mathematics professors about the design principles of their teaching—occasionally such inquiry raises amazement—the following insights quickly emerge in my experience—as professors believe they can perform this task without any further instruction help. (1) Make the lesson units small. (2) Repeat what has been said as often as necessary. (3) Try to make the process slow. The underlying metaphor in the end is that learning in a lecture has a lot to do with ‘swallowing’, so one might think that swallowing is indispensable for the learner, the chunks must be small so that they do not get stuck in the learner’s mind. Much has to be swallowed up in teaching. Byers (2007), p. 363, affirms this assessment [. . .] The professor goes through some intricate proof of a mathematical theorem. The class is silent. If the professor were to tum around and look at the students, their blank faces would demonstrate that, in the vast majority of cases, they do not understand what he has been saying. [. . .] The students say nothing, probably because they are afraid to reveal their lack of understanding but possibly because they have learned that it can be dangerous to ask questions. Finally some brave soul picks on some particular aspect of the argument and asks the professor to explain it. The professor is amazed that anyone could not understand something that is so obvious to him. Being a good sport he proceeds to ‘explain’ by going over the section in question. But what does he do? He merely repeats the same words, the same argument, maybe talking more slowly and possibly even filling in a detail that he had omitted. ‘There,’ he says, ‘you can see that it’s trivial!’ The student remains in the same state of ignorance she was in before she asked the question and will think twice about asking another such question in the future.

From this text it can be seen that ‘didactical contracts’ are real (see Sect. 11.2.2), but they may not automatically lead to a sustainable learning. What good teachers emphasize in mathematics is not to use the word ‘trivially’ while teaching. But there are arguments for a different approach. If the author tries to understand a scientific journal article, then for him the interplay between global and local is meaningful. First you want to recognize the big lines, but then understand the local details. It is a continual shift from ‘local’ to ‘global’ and vice versa, just as one arranges for problem solving. When classical teaching is done linearly in lectures, moving in circles, asking questions, deep understanding, and reinterpreting relationships are more productive.

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Modern media make such a procedure possible, and the classic linear organization of blackboards—the writing of text—can only adequately reflect such a process.

11.3.3

Dimension: Classroom

There are very different classrooms in which teaching of mathematics takes place, e.g. the panel of experts in a company where a mathematical project has to be discussed, the standard classroom in the school, which is structured differently depending on the culture tradition in individual countries, the large lecture hall at the university, the exercise in the smaller practice room, in which the lecture material is to be deepened, the prominent hall in the faculty in which the colloquia take place. We have to leave it for reasons of space with a few remarks. What I learned when I joined the university: mathematicians love many and big boards because they— rightly think that they need them. In the lecture hall assignment, I have often experienced that some rooms were rejected because the black boards were too small. The overhead projector is not considered a suitable medium because the projected page is too small. You are delighted to find many black boards—sometimes even six boards you could move over elevators, further a big (wet) wiper and good white chalk. The old lecture hall buildings in the universities fulfilled these requirements. Some teachers at my university bring such personal tools to their lectures as table wipers and sponges since they do not know whether they may be present in the lecture room. Students who had to give a lecture in such a room have to be taught how to skillfully deal with such a huge writing room at the board, e.g. large wide panels should be divided into two segments. Finally, the boards should be organized so that the last text remains visible from the previous board; some are small scheduling problems. Let us talk about the classroom in which the auditorium comprises faculty members and you want to listen to a colloquium lecture. There are local traditions that the colloquium guest should not ignore. In previous colloquia, most faculties would have been frowned upon to reduce a lecture to a PowerPoint presentation, which would have been considered unfair to the co-authors. Gradually, however, young (mostly applied) mathematicians, with a great deal of experience in project recruitment and applied this condensed style of communication from their company visits automatically to university events, appeared on the mathematical stage. Regardless of the presentation layout, it was almost a success in the colloquium lecture if you could follow the speaker for about 10 minutes—it depends—unless you were an insider in the same field of research. We note that there are teaching opportunities where even the mathematically qualified listener cannot but understand everything.

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Dimension: Immanent Philosophy

If we believe in Thom’s statement (Thom, 1973), we should not ignore the connection to the philosophy of mathematics: our secret philosophy of mathematics has an essential effect on our understanding of the teaching of mathematics. Again and again, there are colleagues who emphasize the priority significance of proving in mathematics, which we fundamentally do not want to question: proving is central in mathematics, undoubtedfully, but its role should not be singled out in isolation. It quickly becomes a position you often encounter. • Teaching is deducing and proving. This position is ‘executed’ in not a few lectures, but, not in school math lessons. Our ‘ancestors’ were deeply convinced: everything in a lecture has to be proven, whatever the cost. The author still remembers a conversation with a colleague fifteen years ago. He just came out of his lecture and reported proudly: Today I proved the theorem of . . . I took two hours, but I did it. Whether this is worthwhile would have to be discussed in more detail. In an advanced lecture about Algebra you will mention the fundamental theorem of algebra, but are you able to prove this statement? The author, having run many introductory and advance algebra courses in the past had never done it. It would be timely and expensive. . . It should not be overlooked that the classical proving rarely contains traces of how these statements were originally obtained and justified. The layouts in the textbooks and in lecturing have often been optimized, primarily for elegant demonstrations, less for deeper understanding. I like to remember a statement by an experienced teacher who once said houses need not to be built exclusively on solid walls, sometimes even pillars anchored in the underground are enough, truly a philosophical statement for teaching mathematics which convinces me. For me, a prominent textbook author is Gilbert Strang. Many successful editions of his textbook on Linear Algebra were published starting 30 years ago—and it also convinced the author to do the same: . . . The emphasis is on understanding - I try to explain rather than deduce. (Strang, 2006, p. viii)

To put it in a sentence: • Teaching is last not least explaining. The term ‘explaining’ involves more facets than the word ‘proving’. Wherever we explain, we comment on the historical development and do not break down the meaning. Next, humans are in front of you and are listening to you, do not forget to motivate them! Much more is addressed. A position that Hersh would classify as humanistic is the statement

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• Teaching is problem solving. It is questionable whether this teaching method is suitable for the big events. Such a teaching style is time-consuming and can be realized only to a limited extent in front of a large audience. But it should not remain unknown to students. Ultimately, it would be fair, for example, for an in-service-teacher-training, if both the problem and the problem solver would be on the same level of information, so if problematic had no time advantage. However, such a procedure is difficult to implement. Problem solving creates the implicit impression there are serious and easy problems, sometimes a problem cannot be solved in the short term or even is insoluble. That is a true description for the world of mathematics.

11.4

Conclusion

I summarize that my research has shown me that far from everything that should be understood is not illuminated. For teaching, subject didactics mostly only focuses on their clientele in their environment: teachers, fresh men at the universities, students. But what happens beyond the third semester, in seminars, in colloquia, at specialist conferences, and at international congresses, is rarely attracted attention, not least because there is little communication about this, a matter for many mathematicians. A second field has also not yet been discussed: the expanding digitalization of our communication culture and the diversity of new media. The importance of new media—also for teaching mathematics—will increase beyond its present scope, user-friendly systems will be further developed. That is just the side of ‘hardware’. The software side, however, is far more complex and these considerations would be discussed in more detail in a separate paper, so I can be brief here. My observations, however, are more restrained. It is probably a question of generations as in school, to what extent one can discard old action scripts and is willing to make use of newer routines. Progress is slow for evolving traditional systems. The older classical university teachers were rather hesitant in the past and difficult to convince of changes compared to the traditional ‘Chalk-Black-Board’ method. Application-oriented mathematicians knew well that they had to adapt their performance when acquiring projects in the industry to the local conditions and the audiences. They know that they have to be flexible and adaptable. The reluctance of the elders not only explains that they see skeptical new techniques but also explains the predicted. If one follows Thom’s (Thom, 1973) basic position, one can only change teachings if one influences philosophy. However, personal attitudes and built worldviews are very resistant to change. (Hersh, 1990)

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The author assumes that hybrid forms of teaching are developed especially at universities. Unfortunately, at American and Canadian universities, for example, the setting up of e-platforms in the sense of a Moodle learning is often introduced under rationalization aspects. For cost reasons, you save expensive mediation (by professors) and replace them with cheaper wizards. Such basic requirements complicate honestly meant further developments. But mathematics learning (and thus the teaching of mathematics) rarely comes without mediators who are expelled in the terrain itself. From experience, the author knows: just what is ‘unsaid’ in the lecture is crucial. How did a student testify? That which is not in the script, that which you have not written on the board, is often more meaningful than what you can read. In this respect, not much will change in the constitutive features of the human learning process, but the outward forms of the designs may adapt. Here every mode of teaching has to reflect itself. How will the teaching of mathematics develop? To do this, we need to discuss the four dimensions mentioned above, which should not least be understood as reflecting lines: • The level of mathematics: Mathematics will develop enormously and open up new areas of application. Hence, new applicants will enter our teaching stage and they have to be addressed. It would not be surprising for me that the new impulses may come from applied mathematics. • The teaching factor: The author believes that teaching will obtain more significant siblings. Publishing is teaching poured into a file. By making mathematics accessible, further variants will be opened up. • New forms of classrooms using the Internet will also open up. • In this context, new philosophical principles are also opened up. More jobs in mathematics will be taken over by machines and software. We wish the next generation wisdom and courage, but old obstacles and epistemological barriers will not fall by themselves. Last but not least, mathematics remains a difficult discipline. However the media will evolve, and many mountains must be laboriously climbed. For example, epistemological difficulties cannot be easily optimized by using other means. Therefore, using good mathematicians cannot be rationalized.

References Artigue, M. (1999). The teaching and learning of mathematics at the university level. Crucial questions for contemporary research in education. Notices of the American Mathematical Society, 46(11), 1377–1385. Atiyah, M. F. (1978). The unity of mathematics. Bulletin of the London Mathematical Society, 10, 69–79.

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Bachelard, G. (1938). La formation de ‘lesprit scientifique. Paris: J Vrin. Bass, H. (2015). Mathematics and teaching. Notices of the American Society, 62(6), 630–636. Brousseau, G. (1984). The crucial role of the didactical contract in the analysis and construdion of situations in teaching and learning mathematics. In H. G. Steiner (Ed.), u.a.: Theory of mathematics education. Occasional Paper Nr. 54, 110–119. Bielefeld: Institut für Didaktik der Mathematik. Brousseau, G., & Warfield, V. (2014). Didactical contract and the teaching and learning of science. In Encyclopedia of science education. Dordrecht: Springer Science+Business Media. https:// doi.org/10.1007/978-94-007-6165-0-93-2 Byers, B. (1984). Dilemmas in teaching and learning mathematics. For the Learning of Mathematics, 4(1), 35–39. Byers, W. (2007). How mathematicians think: Using ambiguity, contradiction and paradox to create mathematics. Princeton: Princeton University Press. Clarke, D., Emanuelsson, J., Jablonka, E., & Mok, I. A. C. (Eds.). (2006). Making connections: Comparing mathematics classrooms around the world. Rotterdam: Sense Publishers. Courant, R., & Robbins, H. (1941). What is mathematics? New York: Oxford University Press. Courant, R., & Robbins, H. (2001). Was ist mathematik? (5th ed.). Berlin: Springer. Davis, P. J., & Hersh, R. (1986). The ideal mathematician. In T. Tymoczko (Ed.), New directions in the philosophy of mathematics (pp. 177–184). Boston: Birkhäuser. Dionne, J. J. (1984). The perception of mathematics among elementary school teachers. In J. M. Moser (Ed.), Proceedings of the 6th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (PME). (pp. 223–228). Madison, WI, USA: University of Wisconsin. Falk, B. (2006). A conversation with Lee Shulman - signature pedagogies for teacher education: Defining our practices and rethiniking our preparation. The New Educator, 2, 73–82. Frank, M. L. (1990). What myths about mathematics are held and convoyed by teachers? The Arithmetic Teacher, 37(5), 10–12. Freudenthal, H. (1973). Mathematik als pädagogische Aufgabe. 2 Bd. Stuttgart: Klett. Halmos, P. R. (1994). What is teaching? The American Mathematical Monthly, 101(9), 848–854. Hersh, R. (1979). Some proposals for reviving the philosophy of mathematics. Advances in Mathematics, 31, 31–50. Hersh, R. (1990). Let’s teach philosophy of mathematics. The College Mathematics Journal, 21(2), 105–111. Hersh, R. (1997). What is mathematics, really? New York: Oxford University Press. Job, P., & Schneider, M. (2014). Empirical positivism, an epistemological obstacle in the learning of calculus. ZDM - The International Journal on Mathematics Education, 46(4), 635–646. Kirsch, A. (2000). Aspects of simplification in mathematics teaching. In I. Westbury (Ed.), Teaching as a reflective practice: The German Didaktik tradition (pp. 267–284). Mahwah, NJ, USA: Erlbaum. Lakoff, G., & Johnson, M. (1980). Metaphors we live by. Chicago: Chicago University Press. Leder, G. C., Pehkonen, E., & Törner, G. (2002). Beliefs: A hidden variable in mathematics education? Dordrecht: Kluwer Academic Publishers. Loos, A., & Ziegler, G. M. (2016). Was ist Mathematik lernen und lehren. Mathematische Semesterberichte, 63(2016), 155–169. Mason, J. (1992). Developing Teaching through thinking mathematically and developing through reflecting on teaching. Draft SV, not yet published. Pajares, M. F. (1992). Teachers’ beliefs and educational research: Cleaning up a messy construct. Review of Educational Research, 62(3), 307–332. Rényi, A. (2006). A Socratic dialog on mathematics (Chapter 1). In R. Hersh (Ed.), 18 unconventional essays on the nature of mathematics (pp. 1–16). New York: Springer. Schneider, M. (2014). Epistemological obstacles in mathematics education. In S. Lerman (Ed.), Encyclopedia of mathematics education (pp. 214–217). Dordrecht: Springer. Schoenfeld, A. H. (1985). Mathematical problem solving. Orlando, FL, USA: Academic.

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G. Törner

Schoenfeld, A. H. (2010). How we think: A theory of goal-oriented decision making and its educational applications. New York: Routledge. Shulman, L. S. (2005a). Signature pedagogies in the profession. Daedalus Summer, 134(3), 52–59. Shulman, L. S. (2005b). The signature pedagogies of the profession of law, medicine, engineering, and the clergy: potential lessons for the education of teachers. In Manuscript Delivered at the Math Science Partnerships (MSP) Workshop ‘Teacher Education for Effective Teaching and Learning’ Hosted by the National Research Council’s Center for Education, February 6–8, 2005, Irvine, California. Shulman, L. (2007). Signature Pedagogies in the professions. In D. Lemmermöhle (Hrsg.), Professionell lehren – erfolgreich lernen (pp. 13–22). Münster: Waxmann. Sierpinska, A. (1987). Humanities students and epistomological obstacles related to limits. Educational Studies in Mathematics, 18, 371–397. Strang, G. (2006). Linear algebra and its applications (4th ed.). Belmont, CA, USA: Thomson Higher Education. Subroto, T., & Suryadi, D. (2019). Epistemological obstacles in mathematical abstraction on abstract algebra. In 3rd International Conference on Mathematical Sciences and Statistics. IOP Conference Series: Journal of Physics: Conference Series (Vol. 1132). https://doi. org/10.1088/1742-6596/1132/1/012032 Tall, D. (Ed.). (1994). Advanced mathematical thinking. Dordrecht: Kluwer Academic Publishers. Tall, D. (2019). Long-term principles form meaningful teaching and learning of mathematics. To be published. Thom, R. (1973). Modern mathematics. Does it exist? In A.G. Howson (Ed.), Developments on Mathematical Education. Proceedings of the Second International Congress on Mathematical Education (pp. 194–209). Cambridge: Cambridge University Press. Toeplitz, O. (1909). Über die Auflösung unendlich vieler linearer Gleichungen mit unendlich vielen Unbekannten. Rendiconti del Circolo Matematico di Palermo, 28(1909), 88–96. Törner, G. & Pehkonen (1996). On the structure of mathematical belief systems. International Reviews on Mathematical Education, 28(4), 109–112. Ziegler, G. M., & Loos, A. (2017). What is mathematics? and why we should ask, where one should experience and learn that, and how to teach it. In Proceedings of the 13th International Congress on Mathematical Education (pp. 63–77). Berlin: SpringerLink.

Chapter 12

Long-Term Principles for Meaningful Teaching and Learning of Mathematics David Tall

Abstract As mathematicians reflect on their teaching of students, they have their own personal experience of mathematics that they seek to teach. This chapter offers an overall framework to consider how different mathematical specialisms may require different approaches depending on the nature of the specialism and the needs of the students. It involves not only problematic transitions for learners but also fundamental theoretical differences between specialisms. At university and college level, mathematicians may have sophisticated knowledge that they believe will offer enlightenment to their students, but these insights may not be shared either by learners in their own community, nor by experts in other communities. The proposed framework reveals simple insights related to mathematical thinking that are visible to both teachers and learners, offering new insights into long-term meaningful growth from practical to theoretical mathematics and on to formal definition and proof. It invites readers to challenge their own beliefs to make an informed choice of strategies that respects the needs and choices of different communities that are essential parts of a complex society. Keywords Reflection on teaching · Insights related to mathematical thinking · Long-term meaningful growth · Meaning of operational symbolism · Interpretation of visual information · Making sense of mathematical proof · Meaningful long-term framework · Transitions

12.1

Introduction

Every one of us has our own personal history and different forms of expertise. To formulate a long-term overview of the ways in which different individuals make sense of mathematics in different ways requires us to be prepared to question our own beliefs, at least to the extent of gaining insight into the reasoning of others.

D. Tall (✉) University of Warwick, Coventry, UK e-mail: [email protected] © Springer Nature Switzerland AG 2023 S. Stewart (ed.), Mathematicians’ Reflections on Teaching, Advances in Mathematics Education, https://doi.org/10.1007/978-3-031-34295-0_12

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The mathematics required for a pure mathematician, an engineer, an economist or a participant in any other profession depends on the needs and purposes of each specialism. The last half century or so has seen the development of technological tools that enable us to operate in ways that were unthinkable a century ago. While the tools have changed, the essential structure of the human brain has had little time to develop in a Darwinian evolutionary sense. It is reasonable to assume that human brain structure has remained essentially the same over the last five or ten thousand years which covers the main development of modern mathematics. To make sense of the differing approaches, it is therefore of value to focus on fundamental aspects that underpin human thinking. It is even more valuable to identify features that are easily observed by teachers and learners that may lead to a deeper understanding of longterm development of sophistication in mathematics. Aspects considered here will include: • • • • •

How we speak mathematical expressions to make precise sense of them. How we hear someone else speak mathematically. How we interpret expressions flexibly as operations in time or as mental objects. How we see static and moving objects as constants and variables in the calculus. How we read mathematical proofs to make sense of them.

In How Humans Learn to Think Mathematically (Tall, 2013), three strands of development termed conceptual embodiment, operational symbolism and axiomatic formalism were formulated, which develop in sophistication in the long-term. The first focuses on (physical and mental) objects and their properties, the second on operations on objects and the third on formally defined properties. These develop long-term from practical mathematics interacting with the world we live in, to theoretical mathematics used in society to model and predict outcomes and axiomatic formal mathematics using axiomatic definitions and formal proof. Our species uses tools to enhance our abilities and means of expression—tools to paint pictures, marks on clay tablets to represent numbers, ruler and compass in Euclidean geometry, quill pens to write books, printing, telescopes, microscopes, logarithm tables, slide rules and now the development of digital technology, with arithmetic calculation, dynamic visual representations, symbol manipulators and smart phones with retina displays that we can control by moving our fingers or using our voice. This chapter will show how features that can be observed by teachers and learners offer new insight into the meaning of operational symbolism and interpretation of visual information. How we speak and hear mathematical expressions leads to new ways of giving precise meaning to operational symbolism and its communication. This, together with how we interpret expressions as operations or objects, offers a comprehensive theory of meaning of mathematical expressions throughout the whole of mathematics. How we see static and moving objects allows us to imagine points on a number line where static points are constants and moving objects are variables. At a more advanced level of axiomatic formalism, we can prove theorems that provide more sophisticated levels of embodiment and symbolism.

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Mathematical thinking has been enhanced by digital technology, performing algorithms in arithmetic and algebra and in providing dynamic graphical displays. The recent development of retina displays and new knowledge of how our eyes and brain interpret visual information offer an insightful resolution of foundational difficulties that have confounded our understanding of the calculus for three and a half centuries. Mathematical thinking changes in different contexts, and what may work well in one context may become problematic in another, causing conflict that can impede progress. The usual approach to the long-term curriculum is to break it down into sub-stages which may be designed by different communities with specified criteria to move from one stage to the next. Summative assessment of successive stages can be helpful. (In my own case, I welcomed the need to review what I had studied, to link the whole together in a more coherent framework.) However, it can also cause the teacher or learner to focus on rote-learnt procedures, to pass the test in a shortterm manner that may cause increasingly problematic aspects in new contexts over the longer term. The framework formulated here takes a long-term view, seeking principles that are supportive over several successive changes in context to build a sense of confidence in them. The plan is to use these principles as a firm foundation to rethink the problematic aspects that arise to make explicit sense in new contexts. The mathematical knowledge we have available today is the product of development of previous generations in different cultural contexts over the centuries, yet we expect our children to grasp sufficient ideas for their own purposes in society in their own lifetime. We live in a complex society that requires different individuals to use different kinds of mathematics in productive ways. Some may require practical mathematics for everyday situations, some may require specific kinds of technical mathematics in different professions, some may require more theoretical mathematics that enables them to model real-life situations to predict possible outcomes, some may go on to more formal aspects of pure mathematics and logic involving set-theoretic axioms, definitions and formal proof. As professional mathematicians reflect on their own teaching practice, it is important to take into account not only the current learning of their students but also the knowledge structures their students have developed from previous experience. This chapter will reveal details of student difficulties at college that arise from learning at all levels in school and will offer new ways of approach that can be of value to improve long-term learning at all levels. Different communities may have differing beliefs and needs, to such an extent that their views may be considered to be incompatible, leading not only to so-called ‘math wars’ but also to disrespect of one community for another. In a complex society, it is essential to have an interconnected web of differing talents. To address the social aspect of differing communities, it is necessary to compare different communities of practice, which Lave and Wenger (1991) describe as ‘group[s] of people who share a craft or a profession’. Individuals in a community

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may have differing personal viewpoints, but overall, they agree (or believe that they agree) to certain shared principles. Each community has ‘experts’, well-versed in the practices and ‘novices’ who are being introduced to them as learners developing over time. Some learners may focus on copying routine practices of the experts, others may develop sophisticated personal knowledge structures that go beyond the practices of the community. It is also important for experts themselves to reflect on their experiences and act as learners seeking new insights.

12.2

Enlightenment, Transgression and Multi-contextual Overview

To make sense of the developments that occur in a community of practice, it is essential to consider the views of both individual experts and individual learners. Experts are likely to see their role as offering enlightenment into the practices of the community. While some learners may make personal sense of this enlightenment, others may find the situation problematic as if they are faced by a conceptual boundary that they are unable to cross. The same phenomenon may also occur between communities of practice with radically differing belief structures, each of which makes sense in its own context. For example, a pure mathematician may regard the real line as a complete ordered field which cannot include infinitesimal elements while an applied mathematician may work with ‘arbitrarily small’ variable quantities which they imagine in their own version of ‘infinitesimal calculus’. The beginning of the twentieth century saw a splintering of mathematical beliefs into different communities of practice, such as intuitionism, formalism, logicism and various later versions of standard, non-standard and constructivist views of mathematics. To formulate a framework that incorporates these phenomena within the development of individuals in a single community and also between different communities, it is useful to consider what happens when one or more individuals from a given community are faced with the possibility of transition to a second community with radically different views. Initially there is a boundary which acts as an impediment to those fixed in the first community, unable to comprehend the views of the other. However, if one or more individuals shift between communities, those in the community that the individuals leave may see the move as a transgression, while those in the community that is moved to may see it as an enlightenment. This may involve strong differences of opinion between the communities. However, these differences may be considered in a more cooperative light in a multi-contextual overview where each community is aware of the values shared between the two, to build confidence based on their communalities while respecting and addressing their differences (Tall, 2019). They may still continue to hold strong opinions about their differences but may now be in a position to listen to each other. This framework applies not only to the differences between communities, it also applies to a learner in a given community faced with a

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Fig. 12.1 Transition over a boundary

problematic change in meaning from one context to another (Fig. 12.1). Some fortunate learners may benefit from the enlightening insight provided for them, while others are faced with an impenetrable boundary. In the case of differences between one community and another, the possibilities may be characterised as:

Impediment: inability to leave the current community to cross over a boundary. Transgression: crossing out of the current community over a boundary. Enlightenment: crossing into a new community over a boundary. Overview: encouraging communication between communities. Examples include differences between communities of pure and applied mathematicians, between mathematicians and educators, between politicians who prescribe the curriculum, curriculum designers, teachers and assessors, between different levels of teaching in early learning, primary, secondary, university and different forms of expertise in mathematics. In the case of an individual seeking to make a change in context within a single community, the possibilities are: Impediment: inability to change context. Transgression: unwillingness to change context. Enlightenment: ability to change context. Overview: ability to switch between contexts. Examples include generalising number systems from counting numbers to fractions, to signed numbers, to rational numbers, reals, complex numbers, from arithmetic to algebra, from practical drawing to Euclidean proof, through changes in meaning in geometry (van Hiele, 1986), from school mathematics to university and so on.

12.2.1 Questioning Personal Beliefs It is natural for academic mathematicians to focus on aspects relevant to their own research and teaching, but this is not sufficient to deal with the broader long-term issues that arise. Mathematics at all levels builds on what was experienced before

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and affects what happens after. Making sense of university mathematics depends on students’ previous experience in school, and mathematicians affect what happens in school through their participation in mathematics education. To grasp an overview of the wider picture requires communication between the various participants. To be able to participate in such a discourse, it is important for all participants, including the reader, to identify our own sources of personal knowledge and beliefs. After reading this section, it would be useful for the reader to pause and question his or her personal beliefs. It is helpful if this includes the mathematical topics where we have expertise, the level at which we have familiarity of learning, teaching and research and to contrast these with aspects where we have had little engagement to seek to identify our own personal experiences and prejudices. In my own case, I grew up in a working-class family, encouraged to be competitive in a boys-only Grammar School selected by an 11-plus examination, obtaining a scholarship to Oxford University, being awarded a university prize in the finals as an undergraduate, then obtaining a Doctorate with Fields medallist, Sir Michael Atiyah. This gave me a bias towards high-level pure mathematics which colours my views. It includes not only the pleasure of achievement but also the tensions arising from struggling to make sense of boundaries, some of which I still consider to be problematic. My first position as a university lecturer made me realise how much I enjoyed thinking about mathematics in a way that could make sense to students, and I moved to a position as ‘Lecturer in Mathematics with Special Interests in Education’ at Warwick University in 1969, where I continued as a mathematician and began to research undergraduate mathematical thinking, leading to a second PhD in Psychology of Mathematics Education with Richard Skemp in 1986. I spent a year as a school teacher with 8–12-year-olds, and then, as I shared ideas with other colleagues with varied professional experience and took on PhD students from around the world, I gained a sense of differing practices in a wide range of countries. Over the years, I worked on building a long-term framework covering the development of the individual from birth to adulthood (Tall, 2013). Instead of performing a detailed comparison of many available theoretical frameworks, I took these theories into account by seeking underlying fundamentals that contribute to longer-term development of sophistication in mathematical thinking. This proved to have relevance both in the growth of the individual child and also over the generations in history. However, the analysis implicitly accepted the broad sequence of curriculum development designed to build ideas over the years with summative assessment that set standards to be attained at key points in transition from one stage of schooling to the next. Could it be that the chosen sequence has an adverse effect on the long-term outcome? More seriously, could the approach chosen by a particular community of practice actually impede long-term growth of sophistication? To investigate this possibility, the next step is to study mathematical growth in a manner that is not bound by today’s curriculum decisions. This begins by considering longterm historical evolution before turning to the lifetime development of individuals.

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Long-Term Historical Evolution of Mathematical Thinking

The historical evolution of our species has developed at an accelerating pace over the last two million years or so, with many aspects that are shared with other species. These relate to how we use our perception to input information, our action for output and internal processes in our brain to make immediate, often unconscious, decisions. I refer to this as conceptual embodiment. It is initially independent of language, though language is essential to formulate increasingly subtle levels of sophistication. The development of language itself is difficult to date as it leaves no physical evidence. However, some forms of proto-language using vocalised sounds evolved over time, possibly in species of Homo two million years or so ago, continuing to function in various ways in many other species today. Mathematical symbolism evolved in Homo Sapiens in the last fifty thousand years or so (which coincides approximately with the appearance of constructed artefacts), with the earliest known tally marks for counting dating back around twenty-five to thirty-five thousand years.1 As archaeological evidence continues to be discovered, these dates may be modified to some extent. Arithmetic symbols proliferated in various communities in Egypt, Babylon, India and China around five thousand years ago. The first flowering of mathematical proof arose in Greek geometry two and a half thousand years ago. Meanwhile, arithmetic problems for unsigned numbers were described verbally in various cultures leading to al-Khwarizmi’s book on Al-jabr in 720 AD which solved problems related to areas of rectangles and squares expressed as linguistic equations. On each side of the equation was a verbal expression to add, subtract and square unsigned numbers, and two methods of operation were introduced to give a solution: one to move subtracted numbers to the other side to become an addition, the other to perform the same operation on both sides of the equation. When Descartes linked geometry to algebra to deal with curves in the plane in algebraic form in 1635, his quantities were still unsigned lengths. Algebraic methods that led to negative and complex solutions of equations were rejected. They continued to be regarded with suspicion even when Argand (1806) and Gauss interpreted them as points in the plane at the turn of the nineteenth century as a visual representation of solutions of polynomial equations. Eventually, they became so useful in the complex analysis of Cauchy (1821) that they could no longer be resisted. Then, at the turn of the twentieth century, new approaches using quantified set-theoretic axioms and definitions introduced a more sophisticated axiomatic formal approach that applies not just to ‘naturally occurring’ real-world problems, but also to any, as yet undiscovered, contexts that satisfy the specified set-theoretic axioms and definitions. In this sense, ‘axiomatic formal mathematics’ is ‘future-

1

For more detail, look up ‘Tally marks’ and ‘History of mathematical notation’ in Wikipedia.

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proofed’ and can be used to build up new mathematical theories, even though there may be subtle implicit meanings that later require more careful consideration. The evolution of sophistication in mathematical thinking continues today. The need for a formal proof to be given in a finite number of steps led to limitations in working with infinite systems including the counting numbers: Gödel’s incompleteness theorem shows that there are quantified statements that may be true for all whole numbers yet may not have a finite proof. New forms of logical systems have been proposed such as the hyperreal numbers and the theory of non-standard analysis. Again, these new structures involve problematic boundaries to be crossed, and they evoke a sense of transgression in some pure mathematicians, who prefer to stay in the context of standard analysis where they feel comfortable because it still works for them. Meanwhile, applied mathematicians, who find it useful to think in terms of ‘arbitrarily small quantities’, continue to use ideas that they refer to as ‘infinitesimal calculus’. In my own personal transition from the Mathematics Department at Warwick University to the Mathematics Education Research Centre in the Science Education Department, I experienced strong differences between the two cultures. I was contracted to continue to teach one mathematics course and one mathematics education course in the Mathematics Department each year. The mathematics education involved two courses which alternated in successive years so that secondand third-year students could study both options. One was a ‘problem solving’ course following the approach of Mason et al. (1982), and the other was a course on ‘development of mathematical concepts’ in which I compared long-term cognitive, historical and logical sequences of development. The undergraduates attending the mathematical concepts course, including a young David Pimm, persuaded me to run a course on Keisler’s approach to infinitesimal calculus based on his book for instructors (Keisler, 1976). I translated the logical formulation into set theory and offered a new mathematics option called ‘Infinitesimal Calculus’. This ran for 1 year but was rejected for the next by a vote of the Mathematics Department (whose expertise at the time focused on standard analysis, algebra and topology) claiming that the course was not pure mathematics. Nevertheless, it was accepted as an ‘education option’ and was followed by large numbers of mathematics students who had previously studied standard mathematical analysis and found the ideas supportive (Tall, 1980). The framework of set-theoretic definition and formal proof also provided a context to introduce infinitesimals in any proper ordered field extension of the real numbers by proving properties that have embodied and symbolic interpretations that can be pictured visually and manipulated algebraically (Tall, 2013; Stewart & Tall, 2014, 2018). The general approach taken is widely applicable in other axiomatic formal contexts. It involves proving a structure theorem that an axiomatic system has properties that can be interpreted in embodied and symbolic ways, thus evolving more sophisticated levels of embodied, symbolic and formal modes of thinking. For example, in mathematical analysis, the real numbers, as a complete ordered field, can be proved to have a visual representation as the number line and a symbolic

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representation as infinite decimal expansions. In the theory of vector spaces, the elements of a finite dimensional vector space over a field F can be represented using coordinates (x1, . . ., xn) in Fn, which could be operated upon symbolically or imagined visually (for instance, when F is the field of real or rational numbers and n = 2 or 3). The theory of finite groups can be interpreted as operations on a visual figure with n vertices where the operations are permutations of a set of n objects. Stewart and Tall (2014) show how these different structure theorems lead to different kinds of formal theory, including one with a single structure unique up to isomorphism (the real numbers in mathematical analysis), another involving specific generic cases (a finite dimensional vector space, Fn) and others seeking the classification of different cases (finite groups). This gives a spiralling evolution of embodiment, symbolism and formalism where embodiment and symbolism inspire axiomatic formal theories which in turn give sophisticated forms of embodiment and symbolism, leading to more sophisticated formal structures.

12.2.3

Long-Term Development of Mathematical Thinking in the Individual Over a Lifetime

How Humans Learn to Think Mathematically (Tall, 2013) focused on the long-term cognitive and affective development of the individual. This considered the development of embodiment, symbolism and formalism in topic areas including arithmetic, algebra, geometry, trigonometry, calculus, in school and on to undergraduate and research mathematics at university. The three long-term threads of development (that I termed ‘three worlds of mathematics’) reveal distinct roles played by objects (initially physical, later as mental representations), operations on objects, (initially involving actions on physical objects, then on mental objects) and properties (of objects and of operations). Each thread incorporates all three aspects in different ways. Conceptual embodiment focuses on objects and their properties, with operations on (physical and mental) embodied objects performed to reveal more sophisticated properties of the objects. As more sophisticated contexts are encountered, the language used at one level may change in meaning at the next. Operational symbolism focuses more on operations and the symbols specifying operations can be used as objects to manipulate at a higher level. Formalism is seen to arise at two distinct levels, one involving theoretical mathematics, to apply to objects that can be imagined in a thought experiment to deduce that one property implies another, as occurs in Euclidean geometry. The other is a higher axiomatic formal level where specific properties are selected as a basis for a theory using quantified set-theoretic definitions, and all other properties are deduced by manipulating the quantified statements using formal proof. As noted in the historical development, formal proof is not the final summit of mathematical thinking. When axiomatic formal proof is attained, the individual may

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prove a wider range of embodiment, symbolism and formalism using structure theorems. The original study (Tall, 2013) looked in detail at the growth of mathematical thinking. In attempting to assess progress of individuals, not only do different individuals diverge in their ability to succeed in successive stages of the curriculum, the same individual can operate at different levels in different aspects at a given time. To gain insight into the broad development of mathematical thinking in individuals and in communities over time, Fig. 12.2 distils the broad outline of development of mathematical thinking through practical, theoretical and axiomatic formal levels of development in the three worlds of mathematics. Our purpose now is to reflect on this overall picture, to compare the coherence of practical mathematics with the consequence of definition and deduction in theoretical mathematics and with set-theoretic definition and deduction in axiomatic formal mathematics. Different readers may see this framework in differing ways depending on their insight into various aspects. Since this book is entitled Mathematicians’ Reflections on Teaching: A Symbiosis with Mathematics Education Theories, the plan is to focus on more advanced mathematics in college and university. Even so, students (and mathematicians) base their mathematical thinking on knowledge, beliefs and attitudes built up over years of previous experience: fundamental aspects of human thinking over a lifetime are relevant at all levels. It is essential to realise that the framework does not represent a strict sequential development. Although embodiment begins before operational symbolism, practical mathematics begins before theoretical mathematics, which precedes axiomatic

Fig. 12.2 Long-term development of mathematical thinking. (Based on Tall, 2013, p. 403)

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formal mathematics, as the individual’s knowledge structure becomes more sophisticated, all aspects in the framework may be relevant in any order. In the embodied world, practical mathematics involves not only perception and operation with physical objects but also imaginative thinking related to coherent recognition and description of properties of objects, such as those arising in ruler and compass constructions in Euclidean geometry. By contrast, theoretical mathematics involves definition and proof using carefully chosen definitions of naturally occurring objects and properties that follow as a consequence of a deductive argument. In arithmetic and algebra, practical mathematics involves observed recognition of operations in arithmetic such as the fact that adding a list of numbers gives the same total regardless of the order of addition. Theoretical mathematics selects specific rules such as the commutative, associative and distributive laws of addition and subtraction in their simplest forms and deduces general properties. This involves minimal definitions but requires more sophisticated proofs. For example, general properties familiar from practical mathematics may require theoretical proof by induction in a potentially infinite form, with a starting statement and a succession of deductions that if the property holds at one stage, then it holds at the next, so the statement can be proved for any specific stage after a finite number of steps. In axiomatic formal mathematics, induction uses the Peano postulates and takes the form of a finite proof: prove the first stage, then prove the general deduction that if it is true at one stage, it is true at the next, then quote the induction axiom that asserts the truth of all stages. Dealing with the infinite reveals a crucial distinction between practical, theoretical and axiomatic formal levels of thinking. There is boundary between practical and theoretical mathematics in the transition from geometry and algebra to the calculus. In Euclidean geometry, the tangent to a circle is a practical construction that can be described as ‘a straight line that touches the curve at precisely one point’. It can be constructed by drawing the radius from the centre of the circle to a point on the circle, and the tangent is the line at right angles to the radius. But in the calculus, the tangent to a more general curve is no longer given by this definition. This is a van-Hiele-type change in context where the language in geometry no longer applies in the context of calculus. Meanwhile, the practical algebraic definition of the tangent at y = f(x) involves calculating the limit of the expression ( f(x + h) - f(x))/h as h tends to zero. This is a potentially infinite process where h gets as small as desired but cannot be taken equal to zero. This causes a conflict because the quotient only exists if h is non-zero, but it is calculated by putting h equal to zero. A visual resolution of this difficulty can be seen by plotting points x,

f ð x þ hÞ - f ð x Þ , h

for fixed h and variable x (Tall, 1985). These build up points on a graph that can be called the practical slope function. For small values of h, the graph stabilises on the graph that can be called the

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theoretical slope function, which is the derivative of f. The infinite limit process arises by varying h in the practical slope function, which, visually, stabilises on the derivative as a graphical object. The difference between practical and formal mathematics arises in the capacity of the learner to think of the expression lim

h→0

f ð x þ hÞ - f ð x Þ h

flexibly as an infinite process or a limit concept. Some individuals find this a powerful insight, often implicit in their use of symbolism, others find it to be an impediment causing conflict. A further transition occurs between calculus and analysis using the quantified set-theoretic definition of limit. The definition is still a potentially infinite process (‘given an epsilon, however small, there exists a delta such that . . .’), but the derivative is now defined to be the limit object. The devil lies in the detail. Axiomatic formal mathematics involves multi-quantified definitions that are highly problematic for many learners. The same framework of practical, theoretical and formal development in embodiment, symbolism and formalism applies in other areas of mathematics. For example, the arithmetic of whole numbers begins with practical mathematics including embodied aspects that allow visualisation of properties of triangular, square, rectangular numbers and non-rectangular prime numbers that have no proper factors. Theoretical mathematics includes the uniqueness of prime factorisation and the potential infinity of primes. Formal mathematics develops set-theoretic definitions, distinguishing prime and irreducible numbers in algebraic extensions of the whole number system and the vast machinery required to address Fermat’s last theorem. (Stewart & Tall, 2015). We now underpin this framework to provide evidence that can be observed by teachers and learners and trust that experts will be willing to consider evidence that comes from a different area of expertise.

12.3 Long-Term Supportive Principles and Resolution of Problematic Transitions In this section, we focus on simple observations that are visible to the teacher and learner relating to how we speak, hear, see, read, interpret and communicate mathematical ideas. The plan is to seek supportive principles that extend over several changes in context to provide a stable foundation to give learners confidence to deal explicitly with problematic aspects. While the focus of our study will be on more advanced mathematics studied in college and university, many problematic aspects at this level arise from experiences in school.

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12.3.1

229

Conservation of Counting Number and General Principles of Arithmetic

The foundational idea of counting lies in Piaget’s Principle of Conservation which gives: The principle of conservation of number: The number of elements in a collection of objects is independent of the way it is counted. Gray and Tall (1991, 1994) studied the development of children aged 5–12 and revealed how various methods of counting (count-all, count-on, known facts, derived facts) compress different procedures for counting into the concepts of number and arithmetic, revealing the way in which a symbol such as 3 + 2 operates dually as a process in time (addition) or as a mental concept (the sum). The dual use of such a symbol flexibly as process or concept was named a ‘procept’ in which many different operations give the same mental object. The notion of procept is a foundational idea throughout operational symbolism. For example, the principle of conservation of number underpins the idea that whatever operation is used to add a collection of numbers, the result is always the same. This can be formulated as: The principle of conservation of addition: The sum of any collection of numbers is independent of the order of addition. This applies not only to whole numbers but also to fractions, signed numbers, rational numbers, real numbers and complex numbers. For example, 3 þ 1=4 þ ð–2Þ þ √2 is the same as √2 þ ð–2Þ þ 1=4 þ 3: The principle therefore operates over several changes of context to provide the learner with a stable foundation for long-term learning. However, the situation will look very different to learners, who may be impeded by their current experience with number systems, compared to experts, who already have wider experience of the more sophisticated ideas. In practice, learners will spend long periods working with whole numbers and encounter other operations such as subtraction and multiplication which present them with problematic aspects. For example, learners may read the expression 2 þ 2×2 from left to right in the normal reading order to get 2 þ 2 is 4 and 4 × 2 is 8

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yet they are told they must respect a convention that says, ‘multiplication takes precedence over addition’, so they must first calculate 2 × 2, is 4, then add 2 þ 4 to get the‘ correct’ answer 6: This violates their fundamental experience of reading from left to right and can often lead to rote-learning of the rule of precedence which is not understood and is likely to lead to more and more errors over the long term. The literature is full of student errors and ‘misconceptions’ with sophisticated theories about what is going wrong. This chapter sets out not only to offer reasons why these errors may occur but also to provide principles to improve learning at all levels.

12.3.2

How Humans Make Sense Reading and Speaking Text

To give meaning to mathematical expressions, it is useful to understand how humans make sense when reading and speaking. This may be done by becoming aware of what happens as we read a line of text. The reader should select any paragraph on the page and notice what is happening as you read. Do this now... You will find that your eye does not move smoothly over the text as you read, instead it moves in a sequence of jumps (called ‘saccades’). Read any paragraph again to make sure you are aware of this. This happens because the central area in the retina of the eye (the fovea) that focuses on detail is only around 1.5 mm in diameter and only takes in a few characters of text at a time. A larger area (the macula) around 5.5 mm in diameter surrounds the fovea and gives surrounding detail that is not as clearly in focus while the blind spot which links to the optic nerve gives none (Fig. 12.3).2 The brain interprets each chunk of information in turn and fits them together to give sequential meaning to the text. Each chunk is interpreted as a whole, so short stretches such as the number 123 in German may be processed as ‘ein hundert, drei und zwanzig’, reading the digits in the order 1, 3, 2, while the chunks as a whole are read successively in a direction appropriate for the language (left to right in Western languages, right to left in Hebrew or Arabic, top to bottom in columns written from right to left in Chinese). If the text is spoken out loud, then it will be heard by another person in the same spoken sequence in time. However, when text is spoken, there are other aspects that communicate additional meaning, such as the tone of voice or the way in which the language is articulated. This enables us to give more precise meaning to mathematical expressions.

2

See: https://en.wikipedia.org/wiki/Macula_of_retina.

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Fig. 12.3 The central part of the retina seen through the pupil of the eye

12.3.3

Giving Meaning to Expressions Through Spoken Articulation

The manner in which we speak mathematics can radically clarify its meaning. We can give different meanings to an expression depending on how we say it. In particular, by leaving short gaps in speech, we can distinguish between two different meanings of 2 + 2 × 2 as: ‘Two plus [gap] two times two’ gives 2 + 4, which is 6 ‘Two plus two [gap] times two’ gives 4 × 2, which is 8. I joked about this with my 11-year-old grandson, who contacted me 2 days later using FaceTime on his iPad (Tall et al., 2017). He asked, ‘What is the square root of nine times nine?’ I knew he was aware of the properties of negative numbers and thought he was trying to test me, so I replied, ‘It can be plus or minus nine.’ He smiled and said, ‘No, its twenty-seven!’ I had no idea what he was talking about until he explained: It is the square root of nine [gap] times nine.

Over the weeks that followed, I steadily realised that this offered a simpler interpretation of the difficulties with symbolism expressed in the wider literature and also in my own publications. For example, it now gives a meaning to the need to introduce brackets. The reader is invited to give different meanings to the following by speaking of them in different ways: five minus four plus one, two plus three times four, minus two squared.

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Does ‘five minus four plus one’ mean ‘five minus four [gap] plus one’, which is 5–4 plus 1, giving 1 + 1, which is 2, or is it ‘five [gap] minus four plus one’, which is 5 minus 4 + 1, giving 5–5, which is 0. The reader is invited to reflect on the meaning of the other expressions given above. This led me to propose: The Articulation Principle: The meaning of a sequence of operations can be expressed by the manner in which the sequence is articulated. (Tall, 2019) This is not a definition in a mathematical sense. It is a principle to encourage us to reflect on the meaning of mathematical expressions as they develop in sophistication over the long term. More importantly, it leads to a more natural way of expressing meaning through the use of brackets (or ‘parentheses’ in American English). For instance, 2 + 3 [gap] × 4 may be written as (2 + 3) × 4 2 + [gap] 3 × 4 may be written as 2 + (3 × 4). The use of brackets leads us to consider the distributive law involving the product of a number times a sum in brackets, such as the expression 3 × (2 + 4), where the sum 2 + 4 must be performed first. This may be related to the expression 3 × 2 + 3 × 4 using embodied pictures, one calculating with whole numbers, the other calculating areas where lengths need not be whole numbers (Fig. 12.4). In the absence of brackets, it is natural to perform operations in the sequence spoken in time or read in the standard order from left to right. For instance, the expression 5 - 4 + 1 gives 2. When an expression just adds numbers together, the order of addition does not matter. If the operations are a mixture of addition and subtraction, there is a possible problematic aspect. If the terms in 5 - 4 + 1 are reordered as 1 - 4 + 5, then this works if negative numbers are allowed, but not with whole numbers, because you can’t take four objects away if you only start with one.

2

4 3

2

4 3

4+2 3×(4 + 2) = 3×4 + 3×2 as whole numbers

4+2 3×(4 + 2) = 3×4 + 3×2 as areas

Fig. 12.4 Embodied examples of the distributive law: counting and measuring

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Over the longer term, when contexts arise that include negative numbers (such as bank balances or temperatures above or below zero), then an extended general principle arises that applies not only to whole numbers but also to any quantities in arithmetic or algebra: The General Principle of Addition and Subtraction: A finite sequence of additions and subtractions of quantities is independent of the order of calculation. For an expert with sophisticated experience, this may appear as an enlightenment. For the learner encountering successive number systems including signed numbers, real numbers and complex numbers, it may (and often does) involve problematic transitions. A similar principle holds for multiplication and division, though, in this case, division has different properties for whole numbers (in terms of quotients and remainders) than for real and complex numbers, where division by a non-zero number is always possible. In the latter case, for fractions, rational, real and complex numbers, we have: The General Principle of Multiplication and Division: A finite sequence of multiplications and divisions of quantities is independent of the order of calculation. These two principles need to be put together using the distributive law. In practical mathematics, the expression in brackets may be a finite sequence of additions and subtractions, and there may be occasions when several brackets are multiplied together. Learning arithmetic involves both specific knowledge of number bonds and general principles of relationships. Both aspects need to be made explicit so that the learner not only has technical facility but also is able to apply general principles meaningfully in a variety of situations. As the expressions become more complicated, more complex issues arise, such as the order of precedence of operations, involving Parentheses (Brackets), Exponents (Indices), Multiplication, Division, Addition, Subtraction. This is often given by a rote-learned mnemonic, such as PEMDAS (Please Excuse My Dear Aunt Sally) in the USA or BIDMAS (Brackets, Indices, Division, Multiplication, Addition and Subtraction) in the United Kingdom, with the added complication that the order of precedence is P > E > M = D > A = S, or equivalently, B > I > D = M > A = S. Frankly, few people make sense of these conventions. To make sense of mathematical ideas requires more than rote-learning rules, as explained by my doctoral supervisor: I always want to try to understand why things work. I’m not interested in getting a formula without knowing what it means. I always try to dig behind the scenes, so if I have a formula,

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I understand why it’s there [. . .] to understand why it works, you have to have a kind of gut reaction to the thing. You’ve got to feel it. (Michael Atiyah, quoted in Roberts, 2016, with italics added.)

To feel the different strengths of bonds in mathematical expressions relates to being able to realise that sub-expressions can be seen flexibly as operations or as objects and also that an operation of higher order is bound more strongly than one of lower order. For instance, the expression 3x2 has the power 2 bound more strongly to the variable x than 3 is to x. It is therefore seen as ‘3 times the object x2’ rather than ‘the square of the object 3x’.

12.4

Interpreting the Duality and Flexibility of Expression as Operation or Object

To aid a learner to see an expression built up in various ways with some parts as operations and other parts as objects, Tall (2019) proposed drawing (or imagining) an object in an expression being placed in a box. For example, the expression 2 + 3 can be expressed as: 2 þ 3 as the operation of addition of two mental objects (numbers) or 2 þ 3 as the single mental object (the sum of the two numbers). In the case of the expression 3x2 (also written as 3 × x ^ 2), the stronger bond for the power allows it to be written as 3 × x2 or as 3 x2 . In practice, it is not necessary to physically draw boxes. In an expression such as 3x2 þ 2x þ 1 it may be sensed that the expression x2 is bound together more strongly than the other terms, and the whole expression is a sum of three objects, 3x2, 2x and 1, where 3x2 and 2x can be seen flexibly as objects or operations. More general expressions that are written spatially such as 3x2 þ 2x þ 1 x2 þ 2 can also be interpreted in the same way: as an object

3x2 þ2xþ1 x2 þ2

or as an operation

3x2 þ2xþ1 x2 þ2

,

where the sub-expressions as objects can recursively be seen as processes.

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235

Making Sense of Equations and the Equals Sign

The study of equations features widely in mathematics education research. What may be less well understood is that the meaning of equations depends very much on whether the expressions are interpreted as operations (processes) or objects (concepts) or dually as procepts which can be flexibly interpreted as either. Initially, an arithmetic equation such as 4 + 3 = 7 is seen as an addition giving the answer as a number, so the left-hand side is an operation and the right-hand side is an object. In this interpretation, the eq. 7 = 4 + 3 may be problematic because 7 does not ‘make’ 4 + 3. Research has shown that an equation of the form ‘expression = number’ such as 4x - 1 = 7 is more easily solved than an equation with expressions on both sides such as 3x + 2 = 4x - 1 (Filloy & Rojano, 1989). The first equation can be interpreted as an operation giving a number which can be ‘undone’ by reversing the steps of the operation 4x - 1 by adding 1 to the result 7 to get 4x is 8, so, dividing by 4 gives x = 2. The second equation involves difficulties of a higher order, catalogued widely in the literature that often involves rote-learnt procedures applied in inappropriate ways. In terms of the framework offered here, if the two sides are seen to give the same numerical object, then applying the same operation to both sides will maintain the equality, and a suitably chosen sequence of operations can lead to a solution. There are two distinct possibilities: either the equation is true for all values of x, which gives an identity, or it is not. The latter case leads to the study of techniques to solve equations, while an identity is the notion of ‘procept’ where different operations represent a single object. Visually, different algebraic expressions that represent the same function are pictured as the same graph, offering an embodied ‘gut reaction’ that they feel like a single mental object. The equal sign can also represent the definition of a quantity. For instance, if a variable is written as being equal to an expression, such as y = x2, then it expresses the variable y as a dependent variable given in terms of the independent variable x, or, in set-theoretic terms, it expresses the notion of a function. In more advanced mathematics, a power series such as sinðxÞ = x -

x2n - 1 x3 þ ... þ ... þ 3! ð2n - 1Þ!

offers a practical definition of a function as a potential process of calculation to any desired accuracy. Drawn over a finite interval, the approximations stabilise on the limit graph. Again, there is a sense that the infinite process stabilises on the limit object. In all these cases, it is possible to either focus on the potentially infinite process that gives a practical limit, as close as is desired, or switch attention to the theoretical limit, which is the limit object itself. Axiomatic formal mathematics takes the theory to a more sophisticated level by formulating the definition of the limit as an object in

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terms of a quantified set-theoretic definition which can be used to prove relationships using formal proof.

12.4.2

Practical and Theoretical Limits in the Calculus

The arrival of graphical displays in the late seventies and early eighties offered new possibilities to distinguish between practical and theoretical limits by drawing practical approximations and considering what happens as they go through the process of approaching the limit object. The Graphic Calculus software that I programmed in the mid-1980s offered a number of facilities. For differentiation the student is encouraged to magnify a graph to see that, under sufficiently high magnification, a differentiable function will look ‘locally straight’, giving (within the limitations of the primitive graphics) a straight line. Then, looking along the graph of the function y = f(x), it becomes possible to imagine the changing slope and to draw the practical slope function f ðx þ cÞ - f ðxÞ c for a small, fixed values of c. The limiting process can be studied by considering what happens to the practical slope function as c gets small (Fig. 12.5). In the picture, the practical slope function of f(x) = x2 is 2x + c, and for small values of c, this stabilizes on the theoretical slope function f′(x) = 2x. This technique can be used to investigate all the derivatives of standard functions, using visualisation. The software also allows the user to stretch the graph horizontally while maintaining the vertical scale. If this is done for a continuous function, then the graph ‘pulls flat’ (Fig. 12.6).

Fig. 12.5 Magnifying a differentiable function and its practical slope

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Fig. 12.6 A continuous graph pulled flat

This can be translated into the definition of continuity. If the pixel height represents a scaled value of ±ε for a real number ε > 0, then the graph of y = f(x) ‘pulls flat’ around a point x0, means a real number δ > 0 can be found such that for points within δ of x0, the points (x, f(x)) lie in the horizontal line of pixels. If this is true for any positive ε > 0, then this is the theoretical ε – δ definition of continuity. As a pure mathematician intent on giving meaning to learners, I used the embodied ideas of local straightness and pulling flat to build a practical theory of differentiability and continuity that includes an embodied version of the fundamental theorem. This was designed to be shared by beginners yet has the capacity to form the initial introduction to a wide variety of approaches to calculus and analysis appropriate for different communities. However, in reality, technology was evolving so fast that curriculum change could not keep up with the technological advances. As we will see in §6, the latest developments in enactive retina displays offer new tools to make sense of sophisticated embodied ideas. The direction taken in the future will depend on whether various communities get to know of these ideas and whether they consider them to be an enlightenment or an impediment.

12.5

Making Sense of Constants and Variables

In our discussion so far, we have often spoken about quantities that are ‘variable’. This builds on how our eyes and brain interpret the dynamic movement of objects. Again, this arises as our eyes jump in saccades to focus on specific detail, as can be seen by holding a finger at a distance from the eye and moving it to the left and right. In this case, there is an initial saccade to fix on the finger, and then the eye follows the

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x moving point (variable)

c fixed point (constant)

Fig. 12.7 Embodied constant and variable

finger, keeping it in focus. The notion of ‘variable’ is therefore a natural feature of our human perception and has been with us for centuries. It underpins our fundamental mathematical thinking of variable quantities. On a number line, we can imagine two different kinds of point. One, which remains in a fixed place, can be considered a constant and another, which moves around, we can call a variable. This supports the interpretation of graphs and calculus in terms of constants and variables. Such an idea can be imagined in our mind’s eye, but with a dynamic picture on a retinal display, we can now offer a visual representation of a variable approaching a fixed point (Fig. 12.7). Although the modern approach to analysis speaks in terms of sets and relations in which points are represented as being fixed elements on a number line, thinking of quantities as constants or variables is deeply embedded in our human thought processes and is a more ‘natural’ way of thinking about the variation of quantities in the calculus.

12.5.1

Variables as Infinitesimals

A variable that tends to zero can be visualised dynamically as an infinitesimal. In Fig. 12.8, points are marked on the vertical line x = t, where it is crossed by the constant line y = k (for k > 0), the straight line, y - x, and the parabola, y = x2. The points are marked on the line x = t as k, t, t2, respectively. The order of the points on the vertical line is determined by their relative height as t moves down to zero. In this case, once t is less than k, the height of the points is 0 < t2 < t < k. This gives a linear order on the field ℝ(t) of rational expressions in an indeterminate t in which 0 < t2 < t < k for any positive real number k and so t and t2 are infinitesimal. Furthermore, t2 tends to zero faster than t, so it is, in this sense, an infinitesimal of higher order. Given two non-zero elements u, v, we say that u is ‘a higher order infinitesimal than v’ if u/v is infinitesimal and ‘of the same order’ if u/v is finite. In this sense, t2 is of higher order than t. Their inverses 1/t and 1/t2 are larger than any real number and so are infinite, with t2 a higher order of infinity than t. This new way of conceptualising infinitesimals may be problematic for some students and even for some mathematicians. However, the idea of an infinitesimal as a variable tending to zero arose at various times in history, including the approach to real and complex analysis introduced by Cauchy (1821). (See Tall & Katz, 2014.)

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y = x2 y=x constant k

t

infinitesimals t2 as variables

x=t Fig. 12.8 Variables as infinitesimals

12.5.2

Infinitesimals as Fixed Points on a Number Line

Conceptualising infinitesimals may be problematic for some. However, for an individual familiar with formal axiomatic mathematics, it is possible to use the completeness of ℝ to prove a structure theorem that shows that for any ordered extension field K of ℝ, every finite element x in K (meaning a < x < b for some real numbers a, b) is uniquely of the form x = c + ε where c 2 ℝ and ε is infinitesimal or zero. The proof is elementary. Let L be the set of real numbers less than x, then L is a non-empty subset of real numbers (because a 2 K ) and is bounded above by b, so it has a unique least upper bound c 2 ℝ, and it is straightforward to show that ε = x - c is zero or infinitesimal. If K is any ordered extension field of the real numbers, we may call the elements of K ‘quantities’, and elements of ℝ may be called ‘constants’. The structure theorem now says: Every finite quantity is either a constant or a constant plus an infinitesimal. For a finite quantity x, the unique constant c is called the standard part of x and written as c = st(x). The map m : K → K given by m(x) = (x - c)/ε for any c, ε 2 K, (c ≠ 0) is called the ε-lens pointed at c. This can be defined for any c, ε 2 K for ε ≠ 0, but it is of particular interest when ε is a positive infinitesimal: in this case, m is said to be the ε-microscope pointed at c.

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c –2

–1

0

1

2

P

K

field of view, V

P(c) P (c  ) P(c  2 ) P (c   3 2 ) Fig. 12.9 Optical microscope mapping infinitesimal detail onto a real line

The subset V given by V = fx 2 Kjðx - cÞ=ε is finiteg is called the field of view of m and μ : V → ℝ given by μðxÞ = st

x-c ε

maps the field of view to the real numbers. In general, it is called the optical ε-lens pointed at c. If ε is a positive infinitesimal, then μ is said to be the optical ε-microscope pointed at c. This allows us to see infinitesimals as fixed points on an extended number line (Fig. 12.9). For any λ 2 ℝ, the image μ(c + λε) = λ, so μ maps onto the whole of ℝ. If two points in V differ by a value that is infinitesimal and of the same order as ε, then they will be distinguished in the real image, but if they differ by a higher order of infinitesimal, then they will be mapped to the same point. The same representations generalise to two or more dimensions. All that is necessary is to operate with an optical lens on each axis. It is possible to use the same scale on the axes to see local straightness or different scales to see local flatness. With these formal possibilities in mind, we now return to reflect on how we humans interpret visual information using enactive retinal quality graphics. We will find that it enables us to bridge the transition between practical mathematics and theoretical mathematics in calculus and, if required, to transition further to axiomatic formal analysis.

12.6

Making Sense of the Calculus Using New Technological Tools

The early software available for drawing pictures on a computer or on a graphic calculator had relatively large pixels, so that the picture of a straight line looks like a line of visible blocks, as shown earlier in Fig. 12.5. The increasing pixel density on a

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modern retina display allows the human eye to imagine zooming in to ‘see’ local straightness (Fig. 12.10). This is because the magnified graph is redrawn to the same thickness, maintaining the illusion that the magnification is zooming in on the graph itself, rather than magnifying a picture of a static graph on a page in a book where magnification would thicken the graph by the same magnification factor. Figure 12.11 shows a graph drawn on a smart phone using the software Desmos (2011): beside it is part of this printed picture reprinted at a higher resolution. This reveals that the graph is drawn with pixels at the edges coloured in lighter shades so that it appears smooth to the human eye when viewed from an appropriate distance. The illusion of smoothness is related to the structure of the eye as illustrated in Fig. 12.3. The part of the eye that detects the highest resolution detail lies in the fovea, which is a circular disk containing approximately 200,000 cones (Kolb, 2007). From this, the diameter of the fovea can be calculated as approximately

Fig. 12.10 Magnifying a graph on a retina screen

Fig. 12.11 Graph drawn in Desmos on an iPhone, with part of the picture enlarged

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250 cones across. This places a severe restriction on the amount of detail that the eye can perceive. Holding a smart phone at a comfortable distance sees a graph as in Fig. 12.8, but presented on a large high-resolution screen, close-up, it may be possible to see individual pixels (as in Fig. 12.10) or even tiny arrays of red, green and blue dots that at a distance are seen as a combined colour. The result is that, when seen highly magnified, a differentiable function may be seen as being locally straight, and the observer may look along the graph to see its changing slope to give a visual meaning to the derivative.

12.6.1

Embodying Integration and the Fundamental Theorem of Calculus

Visually, the area A between a graph y = f(x) and the horizontal x-axis from x = a to x = b can be seen (Fig. 12.12). On the left, the area A is being approximated by the sum of rectangular strips, and on the right, it is being extended by an extra strip width dx. In calculus courses, the Riemann integral is usually introduced by using a partition a = x1 < x2 < . . . < xr < . . . xnþ1 = b and calculating the approximate area as n r = 1 yr dxr ,

where dxr = xr + 1 - xr.

The important information here is not the number of strips, but the endpoints a, b and the maximum width m of the strips (called the mesh of the partition). Denoting the

y f(x)

y f(x)

y1 y2

yr

yn

dx1 dx2

dxr

dxn

a= x

1

x2

xr

A b=x

Approximating area A as a finite sum of strips

n+1

x

Fig. 12.12 The area A between a graph and the x-axis

a

y

dA dx b

Increasing area A by a thin strip width dx

x

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width of a strip by the symbol dx and interpreting y, dx as variable quantities defined by the partition, we can denote the Riemann approximation by the notation b a

y dx:

For the expert in analysis, Fig. 12.11 requires two formal proofs for a continuous function: 1. The limit A of Σba y dx exists as the mesh tends to zero. 2. dA/dx = y. For the learner encountering the calculus at a practical or theoretical level, it is not necessary to prove the existence of the area A as it can be seen as a physical object: it is only necessary to calculate it as accurately as required using the Riemann sum and then to prove (2). In a modern presentation, the value of A is often written as b

ydx a

and spoken as ‘the integral of y with respect to x’. It is also useful to use other letters to allow other quantities to vary, for instance, the expression AðxÞ =

x a ydt

where y = f(t).

allows the area A(x) from a to x to vary for a ≤ x ≤ b. The area A(x) for a continuous function y = f(t) can be calculated by showing that the finite sum xa ydt can be made as close as desired to A(x) using a partition with a suitably small mesh m. This is usually introduced in a first course on calculus as an informal version of the formal definition. It involves an implicit potentially infinite process as the mesh becomes arbitrarily small but not zero. x The meaning of a y dt has changed from the original idea of Leibniz, who used the ‘elongated s’ to signify an infinite sum of infinitesimal quantities. The idea of visualising an arbitrarily thin strip on a retinal display can be performed by stretching the graph of the function horizontally while maintaining the same vertical scale. Figure 12.13 shows a possible picture of this idea where the graph of y = f(x) in the left box with a thin strip width dx is stretched horizontally to fill the right box. The numerical value of the area calculated by multiplying height y by width dx has a practical value dA = ydx, and its error is contained in the horizontal line of pixels, which can be made as small as required. This allows the learner to visualise a practical interpretation that can be extended theoretically to give the formal fundamental theorem of calculus: y = dA/dx which may also be written as A′(x) = f(x).

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Fig. 12.13 Visual representation of the fundamental theorem

12.7

Is the Calculus About Quantities With Dimensions, or Is It About Numbers?

The calculus is presented in at least two distinct forms which may operate in different ways with a problematic transition between them. One, which I will call ‘real calculus’, involves quantities that are real numbers, such as y = ax2, another, which occurs in applications, I will call ‘dimensional calculus’ involving quantities with dimensions, such as expressing a distance s as a function of time t, say s = ut. In the first case, x and y are real variables, and a is a real constant where the dependent variable y depends on the independent variable x. In the second case, t represents time, say in seconds, s is the dependent variable, representing distance, say in metres, u is a velocity in metres per second. These two forms of calculus have distinct properties. In real calculus, successive derivatives are again real functions. In dimensional calculus, if distance is a function of time, its first derivative is a velocity, its second derivative is an acceleration, but what is the third derivative? Some say the sudden change of an acceleration is a ‘jerk’. For s = sin(t), the third derivative is –cos(t), and this certainly does not feel like a jerk. And what is the meaning of the fourth derivative? A solution of this impasse is to deal mainly with real calculus and to use dimensional calculus in specific applications with a clear contextual meaning, such as the relationship between time, distance, velocity, acceleration and force in Newtonian dynamics. Using real calculus is particularly valuable in dealing with successive differentiation and integration where the derivatives and integrals of real functions are again real functions. Even so, there is a fundamental impediment in elementary calculus that affects almost all of us. When real calculus is represented visually as graphs, the variables are drawn as lengths. This is not much of a problem in differentiation as the

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derivative is a length divided by a length, which is a dimensionless number, and this has a simple interpretation in real calculus using real numbers. But it is a subtle x impediment in integration, where the integral AðxÞ = a f ðt Þ dt is represented as an area. If the scales on the axes are changed, then the visual area changes while the numerical area remains the same. This is particularly problematic when different scales are used as in the transformation of the picture to stretch the horizontal scale while maintaining the vertical scale, as in the case of ‘pulling the graph of a continuous function flat’. It is therefore little wonder that the notion of ‘pulling flat’ has not been integrated into contemporary courses. ‘Local straightness’ on the other hand has been incorporated in some courses, often in the symbolic form of ‘local linearity’, as the numerically ‘best linear approximation’ to a curve, as in Harvard Calculus (Hughes-Hallett & Gleason, 1994). Local linearity involves sophisticated theoretical symbolism. Local straightness offers a more fundamental embodied meaning to differentiation, not only as an intuitive beginning but also in more sophisticated situations throughout the longterm development of practical, theoretical and formal mathematics. For instance, it is possible to give a simple example of a function that is everywhere continuous but nowhere differentiable, and so its integral is differentiable once everywhere (with a continuous derivative) but not differentiable twice anywhere. Integrating a function n times gives a function that is continuously differentiable n times but not n + 1 times. This offers an embodied meaning relating continuity and differentiability at a more sophisticated level than is usually encountered in formal analysis.

12.7.1

Practical Continuum, Theoretical Closeness and Formal Completeness

There are changes in meaning between practical, theoretical and formal versions of mathematics relating to the difference between what we perceive with our practical human senses, what we imagine theoretically in our mind and how we move on to a set-theoretic formal approach. A practical drawing arises from the dynamic movement of a pen or pencil, and the final static curve is a continuum that, as our eye moves along it, we see it as dynamically continuous. Yet we know from our experience with factorization of whole numbers that √2 is distinct from any rational number. In theoretical mathematics, real numbers can be conceived as separate infinite decimals. Later, axiomatic formal mathematics uses the completeness axiom to formulate a set-theoretic interpretation of the real line. Practical, theoretical and axiomatic formal mathematics have their own characteristics. Practical arithmetic involves making numerical approximations that are ‘good-enough’ for a particular practical purpose. For example, a good-enough approximation for π might be 3.142 to four significant figures or 3.14159 to 5 decimal places. Solving the simple harmonic motion of a pendulum through an angle θ radians might be performed for small values of θ by replacing sin(θ) by θ.

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In calculus, the practical slope function of y = f(x) may be calculated for variable x and a small constant value c by calculating ( f(x + c) - f(x))/c and (for differentiable functions) the limit object is the theoretical slope function f′(x) = dy/dx. The practical integral function for a variable x on an interval [a,b] for y = f(x) may be calculated as the finite sum Σxa f ðt Þ dt for a partition of [a,b], and as the mesh size gets x suitably small, the limit object is the theoretical integral function I ðxÞ = 0 f ðt Þ dt, and the fundamental theorem says I′(x) = f(x). The visual link between the practical approximation and the limit object in calculus can be represented using a retinal interface as in Figs. 12.8 and 12.12, which builds on the natural working of the human eye and brain. This allows the integral to be imagined as a sum of arbitrarily thin strips, neglecting higher-order details in a manner favoured by many engineers modelling physical problems.

12.8

Making Sense Reading a Mathematical Proof

An essential aspect of formal mathematics—both in a theoretical form relating to natural contexts in applications and in axiomatic formal pure mathematics—lies in giving meaning to mathematical proof. Many learners see this as a need to learn the proof by heart to reproduce it in examinations. However, it is possible to give a meaningful long-term interpretation of mathematical proof related to the manner in which the human eye and brain interpret text, symbolism and visual illustration. A written proof is laid out on the page so that the eye can look at various parts of it at will, seeing its structure as a whole, looking back at earlier details, focusing carefully on subtle key points. Using eye-tracking techniques, Inglis and Alcock (2012) confirmed that undergraduates devoted more of their attention to parts of proofs involving algebraic manipulation and less to logical statements than expert mathematicians. Hodds et al. (2014) developed materials to encourage ‘self-explanation’ by reading a proof line by line, to identify the main ideas, get into the habit of explaining to themselves why the definitions are phrased as they are and how each line of a proof follows from previous lines. They were counselled not to simply paraphrase the lines of the proof by saying the same thing in different words, but to focus on making connections to grasp the main argument and explain how the given assumptions and definitions in previous lines led to the current line and contribute to the following lines. Students who had worked through these materials before reading a proof scored 30% higher than a control group on making sense of a proof several weeks later. Language plays an essential role in formulating increasingly sophisticated ideas through naming concepts so that they can be called to mind as mental objects, describing and defining their properties and relationships. However, as contexts change, language that makes sense in one context may fail to make sense in another (such as the definition of a tangent in Euclidean geometry and in the calculus). This can act as an impediment to transition for the learner while the expert with a more experienced knowledge structure may see the change as an enlightenment.

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Experts from different communities of practice have differing views on the role played by language in mathematics. Einstein, as a theoretical physicist, used thought experiments to produce his theory of relativity and claimed that ‘words and language, whether written or spoken, do not seem to play any part in my thought processes’ (quoted from Hadamard, 1945, p. 142). The linguist Lakoff sees language and the use of metaphor as the foundation of how mathematics grows in human thinking (Lakoff & Johnson, 1980; Lakoff & Núñez, 2000). From a neurophysiological viewpoint, Amalric and Dehaene (2016, 2019) using fMRI scanning showed that specialists in algebra, analysis, topology and geometry used non-linguistic areas of the brain to respond to the truth or falsehood of mathematical statements and concluded that ‘Overall, these results support the existence of a distinct, non-linguistic cortical network for mathematical knowledge in the human brain’. Kahneman (2011), a psychologist and economist, who was awarded the Nobel Prize in Economics for his book Thinking Fast, Thinking Slow, formulated a distinction between immediate responses in a few seconds (as in Amalric and Dehaene’s use of fMRI) and the long-term deep thinking required for formal proof. In the centre of the brain is a diverse network of structures called the limbic system,3 which is involved in a range of aspects, including laying down and retrieving long-term memories, and emotional reactions to incoming sensory information (Fig. 12.14). These have a profound effect on the nature of mathematical thinking. Incoming sensory data pass quickly to the limbic system, which can take immediate ‘fight or flight’ decisions before the forebrain is able to think through conscious decisions. It also suffuses the whole of the brain with chemical neurotransmitters that enhance or suppress connections. Positive aspects enhance motivation and set the

Fig. 12.14 The limbic system

3

See: https://en.wikipedia.org/wiki/Limbic_system.

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brain on alert to solve problems, while negative aspects depress thinking processes along with anxiety and fear. In the latter case, it is not just that the individual is unwilling to tackle a problem, the lack of connections makes it very difficult to think about the problem at all. Creating an original mathematical proof is the final stage of presenting ideas in a coherent formal framework and is preceded by a varied range of activities, thinking about possibilities, formulating hypotheses which may be appropriate or erroneous, making and reformulating definitions and deductions (Byers, 2007). The differing interpretations available confirm the need for what I will term a meaningful long-term framework for mathematical thinking that encompasses the many different aspects that are essential in a complex society. This chapter supports the case for a long-term development incorporating the three-world framework of embodiment, symbolism and formalism through practical, theoretical and axiomatic formal mathematics as in Fig. 12.2. It also includes other social and personal aspects related to the nature of mathematics and how it is conceived and interpreted by us as human beings.

12.9

Discussion

The meaningful long-term framework formulated here is a response to the various approaches that have been followed in the last half century. Within a single lifetime, the changes in technological tools have radically changed how we think about mathematics. In the United States, the development of nuclear weapons in the 1940s and the Russian launch of Sputnik in 1957 led to massive federal funding for huge ‘New Math’ projects to enhance American international influence (Woodward, 2004; Dossey et al., 2016). Meanwhile, in colleges and universities, computers have changed the whole approach to mathematics, particularly in calculus (Bressoud et al., 2015). The New Math movement included the introduction of set theory to explain mathematical concepts in more precise language and other approaches, such as ‘guided discovery learning’ where learners are encouraged to develop their own ideas through practical problem-solving. These were promoted by a range of organisations, but also had a variety of opponents, from mathematicians who decried the lowering of technical facility (e.g. Kline, 1973: Why Johnny Can’t Add) to teachers and parents who could not make sense of the new ideas. In 1972, federal funding was withdrawn. Throughout the USA, different communities followed their own agendas with each state setting its own educational policies, often devolving decisions to local school boards. Meanwhile, in other countries in the world, communities were seeking their own solutions. Over the following decades, many different projects arose, some initially following a ‘new math’ approach, others, such as ‘realistic mathematics’ in the Netherlands, encouraged young children to construct their own mathematical ideas in real problem-solving situations. Some projects had a measure of success, but there

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were also problematic aspects. For example, in the Netherlands, it was found that students going to university lacked the fundamental skills previously required in university mathematics. Difficulties arose in dealing with whole number arithmetic, fractions and algebra (Gravemeijer et al., 2016), which could be explained in terms of problematic transitions between practical problem-solving and operational symbolism. The array of possibilities is further complicated by the amazing evolution of technology from simple numeric calculators in the early seventies through an annual cycle of new developments that radically affect not only the ways that we think about mathematics but also how we live our whole lives. It is impossible for any individual to grasp the full picture. An expert may have an insightful view of a wide range of aspects, but as we saw in §7, experts from differing communities may have radically different interpretations. Instead of attempting to deal with the complexity of competing approaches, I chose to seek ways that could be used by anyone to improve the possibilities for learners to make meaningful sense of mathematics relevant to their own lives in a complex society. This involved focusing on the long-term development of sophistication in mathematics and the underlying changes in meaning that can give insight to some and impediments to others. I wanted to encourage learners to develop a personal sense of confidence by realising that it is not their inadequacy that causes them to have difficulty in understanding more sophisticated ideas, it also relates to the changing nature of mathematics itself. I sought to help learners and teachers by finding general principles that remained consistent over several changes in context which could act as a secure basis to allow them to reflect on problematic aspects that needed to be resolved to take advantage of more sophisticated ideas. The plan is to develop successive levels of sophistication linking specific information in a particular context with the general principles to guide the long-term development. As an example, the principle of articulation can be used at every level to give meaning to mathematical expressions and the use of brackets. This is appropriate for young children in their early learning of mathematics, for older students who have difficulties with algebra and for teachers and experts to have a strategy to help learners develop meaningful mathematical thinking. It can be coupled with the conservation of number and the general principles of arithmetic to deal with the meaningful manipulation of expressions in arithmetic and algebra which may cause increasing difficulty over time if learned only by rote. The beauty of the principle of articulation is that it extends the notion of flexibility of expressions representing either process or concept, which Eddie Gray and I formulated as the notion of ‘procept’. This works in the interpretation of mathematical expressions throughout the whole of mathematics. Although mathematicians may not realise it explicitly, this flexibility is implicit in being able to look at an expression and ‘know’ how to break it down mentally into appropriate chunks to interpret its meaning. Mathematicians generally use the notion of ‘equivalence relation’ to explain how different processes represent the same mathematical object. For instance, equivalent fractions represent a single real number, and equivalent expressions represent the

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same function. In human terms, this is made meaningful by the embodied representation of equivalent fractions as a single point and equivalent algebraic expressions having the same visual graph. Embodiment also gives meaning to the natural idea of ‘constants’ and ‘variables’ as fixed and moving points. The process of ‘tending to a limit’ is then embodied in a practical sense by a variable point becoming visibly indistinguishable from a fixed point or a sequence of graphs becoming indistinguishable from the graph of a limit function. Now we have retinal display on computers and smart phones, we can imagine zooming in on a graph to see that a differentiable function is ‘locally straight’. It is then possible to look along the graph to see the changing slope and imagine the derivative as the changing slope of the graph itself, both visually and also translated into symbolism. Meanwhile, in the USA, calculus has become a high school subject, still firmly based on a traditional sequence of four ‘big ideas’ (College Board, 2016): • • • •

Limits and continuity. Derivatives. Integrals and the Fundamental Theorem of Calculus. Series.

These are all formulated in traditional language with no mention of the embodied meaning of ‘local straightness’, even though this can be seen and sensed by any learner using an interactive dynamic retina display. The language describes the derivative as ‘the instantaneous rate of change’ of the function, intimating how it is calculated as a limit, when the locally straight approach encourages the learner to look along the graph to see its changing slope to give it an embodied meaning. The calculus reform project at Harvard (Hughes-Hallett & Gleason, 1994) proposed that mathematics should be considered in three ways—graphical, numerical and algebraic—later extending to a ‘rule of four’ by adding ‘verbal’. However, this begins with the more sophisticated symbolic idea of ‘best linear approximation’ rather than the natural idea of ‘local linearity’, where the derivative is simply the slope of the graph itself. Instead of building up from the learner’s experience to the expert’s insight, the curriculum takes the expert’s knowledge and reformulates it in an informal manner. When university mathematicians reflect on their teaching, they may develop a range of different mathematical approaches in their course design. I did it myself when I first taught analysis and was given freedom to interpret the syllabus. I designed an approach from the physical drawing of graphs to the formal definitions of analysis but avoided the compactness proof of the link between pointwise and uniform continuity by only defining uniform continuity. This had some success but technical difficulties arose, such as the fact that the simple function f(x) = x2 is not uniformly continuous on the domain ℝ. My colleague Alan Weir decided to teach integration before differentiation but found that, whereas the derivative of xn was straightforward to calculate for specific n, the Riemann sum was difficult for n = 2 and almost impossible for n = 3, so it was not possible to begin with simple

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computational examples. The book by Moss and Roberts (1968) bases its whole development on the notion of continuity of a function f : D → ℝ at a cluster point of D. For instance, it defines a sequence (sn) to have the limit s if the function f(x) = sn for x = 1/n, f(0) = s is continuous on the domain D = {x| x = 0 or x = 1/n for n 2 ℕ}. Many other approaches are possible. This may involve reorganising the mathematical content in pure mathematics by taking a different starting point, but seeking to simplify ideas in one aspect may lead to a difficulty arising somewhere else. It may also involve seeking to focus on appropriate aspects in a different subject area involving specific applications. Different communities and individuals will take their own decisions as to how they teach mathematics, and that is their privilege. The main issue here is how each learner can be encouraged to develop their own mathematical thinking in a way that is appropriate for them as individuals and also as part of a wider complex society. This chapter offers a long-term framework for the meaningful development of mathematical thinking that takes into account the increasing sophistication of mathematical ideas and the cognitive and emotional growth of the individual. It also offers a contextual overview to encourage the comparison and cooperation of different communities of practice. It does not predict the future. It offers a framework for readers to challenge their own beliefs to make informed choices.

References Amalric, M., & Dehaene, S. (2016). Origins of the brain networks for advanced mathematics in expert mathematicians. Proceedings of the National Academy of Sciences of the United States of America, 201603205. https://doi.org/10.1073/pnas.1603205113 Amalric, M., & Dehaene, S. (2019). A distinct cortical network for mathematical knowledge in the human brain. NeuroImage, 189(2019), 19–31. Argand, J. R. (1806). Essai sur une manière de représenter les quantités imaginaires dans les constructions géométriqonsues. Duminil-Lesueur. (2nd edition: Gauthier Villars, Paris, 1874). Bressoud, D., Mesa, V., & Rasmussen C. (Eds). (2015). Insights and recommendations from the MAA National study of college calculus. https://www.maa.org/sites/default/files/pdf/cspcc/ InsightsandRecommendations.pdf Byers, W. (2007). How mathematicians think. Princeton University Press. Cauchy, A. L. (1821). Cours d’Analyse de l’Ecole Royale Polytechnique. Imprimérie Royale. http:// gallica.bnf.fr/ark:/12148/bpt6k90195m/f12 College Board. (2016). AP calculus AB and AP calculus BC including the curriculum framework. College Board. https://apcentral.collegeboard.org/pdf/ap-calculus-ab-and-bc-course-and-examdescription.pdf Desmos. (2011). Desmos graphing calculator. https://www.desmos.com Dossey, J. A., McCrone, S. S., & Halvorsern, K. T. (2016). Mathematics education in the United States 2016. ICME-13. NCTM. https://www.nctm.org/uploadedFiles/About/MathEdInUS201 6.pdf Filloy, E., & Rojano, T. (1989). Solving equations, the transition from arithmetic to algebra. For the Learning of Mathematics, 9(2), 19–25. Gravemeijer, K., Bruin-Muurling, G., Kraemer, J. M., & van Stiphout, I. (2016). Shortcomings of mathematics education reform in The Netherlands: A paradigm case? Mathematical Thinking and Learning, 18(1), 25–44. https://doi.org/10.1080/10986065.2016.1107821

252

D. Tall

Gray, E. M. & Tall, D. O. (1991). Duality, ambiguity & flexibility in successful mathematical thinking. In Proceedings of the 15th conference for the international group for the psychology of mathematics education, 2, 72–79. Gray, E. M., & Tall, D. O. (1994). Duality, ambiguity and flexibility: A proceptual view of simple arithmetic. Journal for Research in Mathematics Education, 26(2), 115–141. Hadamard, J. (1945). The psychology of invention in the mathematical field. Princeton University Press. (Dover edition, New York 1954). Hodds, M., Alcock, L., & Inglis, M. (2014). Self-explanation training improves proof comprehension. Journal for Research in Mathematics Education, 45, 62–101. Hughes-Hallett, D., & Gleason, A. M. (1994). Calculus. Wiley. Inglis, M., & Alcock, L. (2012). Expert and novice approaches to reading mathematical proofs. Journal for Research in Mathematics Education, 43, 358–390. Kahneman, D. (2011). Thinking, fast and slow. Macmillan. Keisler, H. J. (1976). Foundations of infinitesimal calculus. Prindle, Weber & Schmidt. Kline, M. (1973). Why Johnny can’t add: The failure of the new math. St. Martin’s Press. Kolb, H. (2007). Webvision: The organization of the retina and visual system. Retrieved on, 26 Jan 2019 from https://www.ncbi.nlm.nih.gov/books/NBK11556/ Lakoff, G., & Johnson, M. (1980). Metaphors we live by. University of Chicago Press. Lakoff, G., & Núñez, R. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. Basic Books. Lave, J., & Wenger, E. (1991). Situated learning: Legitimate peripheral participation. Cambridge University Press. Mason, J., Burton, L., & Stacey, K. (1982). Thinking mathematically. Addison Wesley. Moss, R., & Roberts, G. (1968). A preliminary course in analysis. Chapman & Hall. Roberts, S. (2016). Mathematical beauty: A Q&A with fields medalist Michael Atiyah. https://www. scientificamerican.com/article/mathematical-beauty-a-q-a-with-fields-medalist-michael-atiyah/ Stewart, I. N., & Tall, D. O. (2014). Foundations of mathematics (2nd ed.). Oxford University Press. ISBN: 9780198531654. Stewart, I. N., & Tall, D. O. (2015). Algebraic number theory and Fermat’s last theorem (4th ed.). CRC. ISBN 9781498738392. Stewart, I. N., & Tall, D. O. (2018). Complex analysis (2nd ed.). Cambridge University Press. ISBN: 978-1108436793. Tall, D. O. (1980). Intuitive infinitesimals in the calculus. In Abstracts of short communications, fourth international congress on mathematical education. http://homepages.warwick.ac.uk/ staff/David.Tall/pdfs/dot1980c-intuitive-infls.pdf Tall, D. O. (1985). Understanding the calculus. Mathematics Teaching, 10, 49–53. Tall, D. O. (2013). How humans learn to think mathematically. Cambridge University Press. ISBN: 9781139565202. Tall, D. O. (2019). Complementing supportive and problematic aspects of mathematics to resolve transgressions in long-term sense making. In Fourth interdisciplinary scientific conference on mathematical transgressions, March 2019. Tall, D. O., & Katz, M. (2014). A cognitive analysis of Cauchy’s conceptions of function, continuity, limit, and infinitesimal, with implications for teaching the calculus. Educational Studies in Mathematics, 86(1), 97–124. Tall, D. O., Tall, N. D., & Tall, S. J. (2017). Problem posing in the long-term conceptual development of a gifted child. In M. Stein (Ed.), A life’s time for mathematics education and problem solving. On the occasion of Andràs Ambrus’ 75th birthday (pp. 445–457). WTM-Verlag. van Hiele, P. M. (1986). Structure and Insight. Academic Press. Woodward, J. (2004). Mathematics education in the United States: Past to present. Journal of Learning Disabilities, 37(1), 16–31.

References

Aguirre, J., & Speer, N. M. (2000). Examining the relationship between beliefs and goals in teacher practice. Journal of Mathematical Behavior, 18(3), 327–356. Akkerman, S., & Bakker, A. (2011). Boundary crossing and boundary objects. Review of Educational Research, 81, 132–169. Alcock, L. (2010). Mathematicians’ perspectives on the teaching and learning of proof. Research in Collegiate Mathematics Education, 7, 63–92. Alcock, L. (2018). Tilting the classroom. London Mathematical Society Newsletter, 474, 22–27. Amalric, M., & Dehaene, S. (2016). Origins of the brain networks for advanced mathematics in expert mathematicians. Proceedings of the National Academy of Sciences of the United States of America, 201603205. https://doi.org/10.1073/pnas.1603205113 Amalric, M., & Dehaene, S. (2019). A distinct cortical network for mathematical knowledge in the human brain. NeuroImage, 189(2019), 19–31. Argand, J. R. (1806). Essai sur une manière de représenter les quantités imaginaires dans les constructions géométriqonsues. Duminil-Lesueur. (2nd edition: Gauthier Villars, Paris, 1874). Arnold, I. J. M., & Straten, J. T. (2012). Motivation and math skills as determinants of first-year performance in economics. The Journal of Economic Education, 43(1), 33–47. Artemeva, N., & Fox, J. (2011). The writing’s on the board: The global and the local in teaching undergraduate mathematics through chalk talk. Written Communication, 28(4), 345–379. Artigue, M. (1998). Research in mathematics education through the eyes of mathematicians. In A. Sierpinska & J. Kilpatrick (Eds.), Mathematics Education as a research domain: A search for identity (pp. 477–490). Kluwer Academic Publishers. Artigue, M. (1999). The teaching and learning of mathematics at the university level. Crucial questions for contemporary research in education. Notices of the American Mathematical Society, 46(11), 1377–1385. Artigue, M. (2016). Mathematics education research at university level: Achievements and challenges. In E. Nardi, C. Winsløw, & T. Hausberger (Eds.), Proceedings of the 1st INDRUM (International Network for Didactic Research in University Mathematics) Conference: an ERME Topic Conference (pp. 11–27). Montpellier. Artigue, M., & Blomhøj, M. (2013). Conceptualizing inquiry-based education in mathematics. ZDM, 45(6), 797–810. Artigue, M., Batanero, C., & Kent, P. (2007). Mathematics thinking and learning at post-secondary level. In F. K. Lester (Ed.), The second handbook of research on mathematics teaching and learning (pp. 1011–1049). Information Age.

© Springer Nature Switzerland AG 2023 S. Stewart (ed.), Mathematicians’ Reflections on Teaching, Advances in Mathematics Education, https://doi.org/10.1007/978-3-031-34295-0

253

254

References

Asiala, M., Dubinsky, E., Mathews, D. M., Morics, S., & Oktac, A. (1997). Development of students’ understanding of cosets, normality, and quotient groups. The Journal of Mathematical Behavior, 16(3), 241–309. Atiyah, M. F. (1978). The unity of mathematics. Bulletin of the London Mathematical Society, 10, 69–79. Ausubel, D. P. (1968). Educational psychology: A cognitive view. Holt, Rinehart & Winston. Avigad, J. (2019). Learning logic and proof with an interactive theorem prover. In Proof technology in mathematics research and teaching (pp. 277–290). Springer. Bachelard, G. (1938). La formation de ‘lesprit scientifique. J Vrin. Ball, D. L., Hill, H. C., & Bass, H. (2005). Knowing mathematics for teaching: Who knows mathematics well enough to teach third grade, and how can we decide? American Educator, 29(1), 4–17, 20–22, 43–46. Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching. Journal of Teacher Education, 59(5), 389–407. Bandura, A. (1982). Self-efficacy mechanism in human agency. American Psychologist, 37(2), 122–147. Bardini, C., Bosch, M., Rasmussen, C., & Trigueros, M. (2021). Current interactions between mathematicians and researchers in university mathematics education. In V. Durand-Guerrier, R. Hochmuth, E. Nardi, & C. Winslow (Eds.), Research and Development in university mathematics education: Overview produced by the international network for didactic research in university mathematics. Routledge. Barone, J. (2019). Leonardo Da Vinci: A mind in motion. British Library. Barton, B. (2011). Growing understanding of undergraduate mathematics: A good frame produces better tomatoes. International Journal of Mathematical Education in Science and Technology, 42(7), 963–974. Barton, B., Oates, G., Paterson, J., & Thomas, M. O. J. (2014). A marriage of continuance: Professional development for mathematics lecturers. Mathematics Education Research Journal, 27(2), 147–164. Bass, H. (2005). Mathematics, mathematicians, and mathematics education. American Mathematical Society, 42(4), 417–430. https://doi.org/10.1090/S0273-0979-05-01072-4 Bass, H. (2015). Mathematics and teaching. Notices of the American Society, 62(6), 630–636. Bergqvist, E. (2007). Types of reasoning required in university exams in mathematics. The Journal of Mathematical Behavior, 26(4), 348–370. Bergstrom, A., Clark, R., Hogue, T., Iyechad, T., Miller, J., Mullen, S., et al. (1995). Collaboration framework: Addressing community capacity. The National Network for Collaboration. Retrieved from https://www.uvm.edu/sites/default/files/media/Collaboration_Framework_ pub.pdf Bills, L., Cooker, M., Huggins, R., Iannone, P., & Nardi, E. (2006). Promoting mathematics as a field of study: Events and activities for the sixth-form pupils visiting UEA’s Further Mathematics Centre (A UEA Teaching Fellowship Report). Available from Elena Nardi. Bishop, A. J. (1988). Mathematics education in its cultural context. Educational Studies in Mathematics, 19, 179–191. Biza, I., Jaworski, B., & Hemmi, K. (2014). Communities in university mathematics. Research in Mathematics Education, 16(2), 161–176. Bjørkestøl, K., Borge, I. C., Goodchild, S., Nilsen, H. K., & Tonheim, O. H. M. (2021). Educating to inspire active learning approaches in mathematics in Norwegian universities. Nordic Journal of STEM Education, 5(1) Online https://www.ntnu.no/ojs/index.php/njse/article/view/3972/3 620 . https://doi.org/10.5324/njsteme.v5i1.3972 Black, P., & Wiliam, D. (1998). Assessment and classroom learning. Assessment in Education: Principles, Policy & Practice, 5(1), 7–74. Bridges, J. (Ed.) (1897). The ‘Opus Majus’ of Roger Bacon with introduction and analytical tables. Clarendon Press. College Board. (2016). AP calculus AB and AP calculus BC including the curriculum framework. College Board. https://apcentral.collegeboard.org/pdf/ap-calculus-ab-and-bc-course-and-examdescription.pdf

References

255

Bonwell, C. C., & Eison, J. A. (1991). Active Learning: Creating excitement in the classroom. In Association for the study of higher education; ERIC clearinghouse on higher education. George Washington University, School of Education and Human Development. On-line https://files. eric.ed.gov/fulltext/ED336049.pdf Borasi, R. (1992). Learning mathematics through inquiry. Heinemann. Bosch, M., & Winsløw, C. (2016). Linking problem solving and mathematical contents: the challenge of sustainable study and research processes. Recherches en Didactique des Mathématiques, 35(3), 357–401. Bouligand, G. (1957). L’activité mathématique et son dualisme. Dialectica, 11, 121–139. Archives Bourbaki. (n.d.). Mode d’emploi de ce traité. Manuscript, located April 2, 2019, on.: http://sites.mathdoc.fr/archives-bourbaki/PDF/031ter_delr_007bis.pdf Bressoud, D., Mesa, V., & Rasmussen C. (Eds). (2015). Insights and recommendations from the MAA National study of college calculus. https://www.maa.org/sites/default/files/pdf/cspcc/ InsightsandRecommendations.pdf Brookes, B. (1976). Philosophy and action in education: When is a problem? ATM Supplement, 19, 11–13. Brousseau, G. (1984). The crucial role of the didactical contract in the analysis and construdion of situations in teaching and learning mathematics. In H. G. Steiner (Ed.), u.a.: Theory of mathematics education. Occasional Paper (Vol. Nr. 54, pp. 110–119). Institut für Didaktik der Mathematik. Brousseau, G. (1997). Theory of didactical situations in mathematics. Kluwer. Brousseau, G., & Warfield, V. (2014). Didactical contract and the teaching and learning of science. In Encyclopedia of science education. Springer Science+Business Media. https://doi.org/10. 1007/978-94-007-6165-0-93-2 Brown, S., Collins, A., & Duguid, P. (1989). Situated cognition and the culture of learning. Educational Researcher, 18(1), 32–41. Brown, A., DeVries, D. J., Dubinsky, E., & Thomas, K. (1997). Learning binary operations, groups, and subgroups. The Journal of Mathematical Behavior, 16(3), 187–239. Bruner, J. S. (1964). The course of cognitive growth. American Psychologist, 19(1), 1–15. Bruner, J. (1966). Towards a theory of instruction. Harvard University Press. Bruner, J. (1990). Acts of meaning. Harvard University Press. Burton, L. (2004). Mathematicians as enquirers: Learning about learning mathematics. Kluwer. Byers, B. (1984). Dilemmas in teaching and learning mathematics. For the Learning of Mathematics, 4(1), 35–39. Byers, W. (2007). How mathematicians think: Using ambiguity, contradiction and paradox to create mathematics. Princeton University Press. Carlson, D., Johnson, C. R., Lay, D. C., & Porter, A. (1993). The linear algebra curriculum study group recommendations for the first course in linear algebra. The College Mathematics Journal, 24(1), 41–46. Cauchy, A. L. (1821). Cours d’Analyse de l’Ecole Royale Polytechnique. Imprimérie Royale. http:// gallica.bnf.fr/ark:/12148/bpt6k90195m/f12 CBMS. (2012). The Mathematical Education of Teachers II. American Mathematical Society and Mathematical Association of America. Chevallard, Y. (1985). La Transposition Didactique. La Pensée Sauvage. Chevallard, Y. (1991). La transposition didactique – du savoir savant au savoir enseigné (first edition, 1985). La Pensée Sauvage. Chevallard, Y. (1992). Fundamental concepts in didactics: Perspectives provided by an anthropological approach. In R. Douady & A. Mercier (Eds.), Research in didactique of mathematics, selected papers (pp. 131–167). La Pensée Sauvage. Chevallard, Y. (1999). Didactique? Is it a plaisanterie? You must be joking! A critical comment on terminology. Instructional Science, 27(1), 5–7. Chi, M. T., De Leeuw, N., Chiu, M. H., & LaVancher, C. (1994). Eliciting self-explanations improves understanding. Cognitive Science, 18(3), 439–477.

256

References

Clandinin, D. J., & Connelly, F. M. (2000). Narrative inquiry: Experience and story in qualitative research. Jossey-Bass. Clarke, D., Emanuelsson, J., Jablonka, E., & Mok, I. A. C. (Eds.). (2006). Making connections: Comparing mathematics classrooms around the world. Sense Publishers. Cobb, P. (1994). Where is the mind? Constructivist and sociocultural perspectives on mathematical development. Educational Researcher, 23(7), 13–20. Collins, A. (1988). Different goals of inquiry teaching. Questioning Exchange, 2(1), 39–45. Conradie, J., & Frith, J. (2000). Comprehension tests in mathematics. Educational Studies in Mathematics, 42(3), 225–235. Cook, J. P., Pitale, A., Schmidt, R., & Stewart, S. (2013). Talking mathematics: An abstract algebra professor’s teaching diaries. In S. Brown, G. Karakok, K. H. RoH, & M. Oehrtman (Eds.), Proceedings of the 16th annual conference on research in undergraduate mathematics education (pp. 633–536). Denver. Cook, J. P., Pitale, A., Schmidt, R., & Stewart, S. (2014). Living it up in the formal world: An abstract algebraist’s teaching journey. In T. Fukawa-Connolly, G. Karakok, K. Keene, & M. Zandieh (Eds.), Proceedings of the 17th annual conference on research in undergraduate mathematics education (pp. 511–516). Denver. Courant, R. (1981). Reminiscences from Hilbert’s Gottingen. Mathematical Intelligencer, 3(4), 154–164. Courant, R., & Robbins, H. (1941). What is mathematics? Oxford University Press. Courant, R., & Robbins, H. (2001). Was ist mathematik? (5th ed.). Springer. Cowen, C. (1991). Teaching and testing mathematics reading. American Mathematical Monthly, 98(1), 50–53. Creswell, J. W. (2013). Qualitative inquiry and research design: Choosing among five approaches (3rd ed.). Sage. Croft, A. C., Grove, M. J., Kyle, J. & Lawson, D. A. (Eds.) (2015) Transitions in undergraduate mathematics education. Online Lulu.com. Cronin, A., & Stewart, S. (2022). Analysis of tutors’ responses to students’ queries in a second linear algebra course at a mathematics support center. The Journal of Mathematical Behavior, 67, 100987. https://doi.org/10.1016/j.jmathb.2022.100987 Crouch, C. H., & Mazur, E. (2001). Peer instruction: Ten years of experience and results. American Journal of Physics, 69(9), 970–977. Cuban, L. (1999). How scholars trumped teachers - Change without reform in university curriculum, teaching, and research, 1890-1990. Teachers College Press. Darlington, E. (2014). Contrasts in mathematical challenges in A-level Mathematics and Further Mathematics, and undergraduate mathematics examinations. Teaching Mathematics and its Applications: An International Journal of the IMA, 33(4), 213–229. Davis, E. (2006). Characterizing productive reflection among preservice elementary teachers: Seeing what matters. Teaching and Teacher Education, 22, 281–301. Davis, P. J., & Hersh, R. (1986). The ideal mathematician. In T. Tymoczko (Ed.), New directions in the philosophy of mathematics (pp. 177–184). Birkhäuser. Dawkins, P. C. (2012). Metaphor as a possible pathway to more formal understanding of the definition of sequence convergence. Journal of Mathematical Behavior, 31(3), 331–343. Dawkins, P. C. (2014). How students interpret and enact inquiry-oriented defining practices in undergraduate real analysis. The Journal of Mathematical Behavior, 33, 88–105. Dawkins, P. C., & Karunakaran, S. S. (2016). Why research on proof-oriented mathematical behavior should attend to the role of particular mathematical content. The Journal of Mathematical Behavior, 44, 65–75. Deci, E. L., & Ryan, R. M. (2000). Intrinsic and extrinsic motivations: Classic definitions and new directions. Contemporary Educational Psychology, 25, 54–67. Retrieved from http://selfdeterminationtheory.org/SDT/documents/2000_RyanDeci_IntExtDefs.pdf. November 23, 2020 Desmos. (2011). Desmos graphing calculator. https://www.desmos.com

References

257

Dewey, J. (1933). How we think: A restatement of the relation of reflective thinking to the educative process. D. C. Heath & Company. Dionne, J. J. (1984). The perception of mathematics among elementary school teachers. In J. M. Moser (Ed.), Proceedings of the 6th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (PME) (pp. 223–228). University of Wisconsin. Dossey, J. A., McCrone, S. S., & Halvorsern, K. T. (2016). Mathematics education in the United States 2016. ICME-13. NCTM. https://www.nctm.org/uploadedFiles/About/MathEdInUS201 6.pdf Dreyfus, T. (1991). Advanced mathematical thinking processes. In D. O. Tall (Ed.), Advanced mathematical thinking (pp. 25–41). Kluwer. Dreyfus, T. (2014). Mutual expectations between mathematicians and mathematics educators. In M. N. Fried & T. Dreyfus (Eds.), Mathematics & Mathematics Education: Searching for common ground (Advances in mathematics education) (pp. 57–70). Springer. Duah, F. (2017). Students as partners and students as change agents in the context of university mathematics. Unpublished PhD thesis, Loughborough University, UK. Duah, F., Croft, T., & Inglis, M. (2014). Can peer assisted learning be effective in undergraduate mathematics? International Journal of Mathematical Education in Science and Technology, 45(4), 552–565. Dubinsky, E., & Leron, U. (1994). Learning abstract algebra with ISETL. Springer. Dubinsky, E., & McDonald, M. (2001). APOS: A constructivist theory of learning. In D. Holton et al. (Eds.), The teaching and learning of mathematics at university level: An ICMI study (pp. 273–280). Kluwer. Dubinsky, E., Dautermann, J., Leron, U., & Zazkis, R. (1994). On learning fundamental concepts of group theory. Educational Studies in Mathematics, 27(3), 267–305. Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61, 103–131. Dweck, C. (1999). Self-theories: Their role in motivation, personality, and development. Essays in social psychology. Psychology Press. Elliott, J., & Adelman, C. (1975). The language and logic of informal teaching, in The Ford Teaching Project, Unit 1, Patterns of teaching. University of East Anglia, Centre for applied research in education. Epstein, J. J., Epstein, P. G., Koch, H., Burnett, M., Alison, J., Robinson, C., & (Writers). (1942). Casablanca [Motion picture]. Warner Bros. Pictures. Epstein, J., Powers, A., Stewart, S., & Troup J. (under review). Working in multiple worlds of embodied, symbolic, and formal thinking in linear algebra: The experiences of a mathematician and his linear algebra students. Erlwanger, S. H. (1973). Benny’s conceptions of rules and answers in IPI mathematics. Journal of Children’s Mathematical Behavior, 1(2), 7–26. Falk, B. (2006). A conversation with Lee Shulman - signature pedagogies for teacher education: Defining our practices and rethiniking our preparation. The New Educator, 2, 73–82. Fauconnier, G., & Turner, M. (2002). The way we think: Conceptual blending and the mind’s hidden complexities. Perseus Books Group. Fennema, E., Carpenter, T., Franke, M., Levi, L., Jacobs, V., & Empson, S. (1996). A longitudinal study of learning to use children’s thinking in mathematics instruction. Journal for Research in Mathematics Education, 27(4), 403–434. Filloy, E., & Rojano, T. (1989). Solving equations, the transition from arithmetic to algebra. For the Learning of Mathematics, 9(2), 19–25. Frank, M. L. (1990). What myths about mathematics are held and convoyed by teachers? The Arithmetic Teacher, 37(5), 10–12. Freeman, S., Eddy, S. L. McDonough, M., Smith, M. K., Okoroafor, N., Jordt, H., & Wenderoth, M. P. (2014). Active learning increases student performance in science, engineering, and mathematics. Proceedings of the National Academy of Sciences of the United States of America 111(23) June 10, 2014, 8410–8415. On-line. Retrieved May 14, 2018 from http://www.pnas. org/content/pnas/111/23/8410.full.pdf

258

References

Freudenthal, H. (1973). Mathematik als pädagogische Aufgabe (2 Bd. ed.). Klett. Freudenthal, H. (1983). Didactical phenomenology of mathematical structures. Kluwer. Fried, M. (2011). Tangent powers. Private. communication. Fukawa-Connelly, T., Weber, K., & Mejía-Ramos, J. P. (2017). Informal content and student notetaking in advanced mathematics classes. Journal for Research in Mathematics Education, 48(5), 567–579. Fund, Z. (2010). Effects of communities of reflecting peers on student-teacher development, including in-depth case studies. Teachers and Teaching: Theory and Practice, 16, 679–701. Gabel, M. (2019). The flow of proof—Rhetorical aspects of proof presentation. Doctoral dissertation. Tel Aviv University. Gabel, M., & Dreyfus, T. (2017). Affecting the flow of a proof by creating presence—A case study in number theory. Educational Studies in Mathematics, 96(2), 187–205. Gardiner, A. (1987). Discovering mathematics: The art of investigation. Oxford Science Publications, Clarendon Press. Gattegno, C. (1970). What we owe children: The subordination of teaching to learning. Routledge & Kegan Paul. Glasersfeld, von E. (1987). Learning as a constructive activity. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics. Lawrence Erlbaum. Goldin, G. A. (2003). Developing complex understandings: On the relation of mathematics education research to mathematics. Educational Studies in Mathematics, 54(2/3), 171–202. Goodchild, S. (2001). Students’ goals. Caspar Forlag. Goodchild, S. (2008). A quest for ‘good’ research: The mathematics teacher educator as practitioner researcher in a community of inquiry. In B. Jaworski & T. Wood (Eds.), The mathematics teacher educator as a developing professional. Sense Publishers. Goos, M. (2015). Learning at the boundaries. In M. Marshman, V. Geiger, & A. Bennison (Eds.), Mathematics education in the margins (Proceedings of the 38th annual conference of the mathematics education research group of Australasia) (pp. 269–276). MERGA. Gouvêa, F. Q. (2004, October 1). Review of the book Elementary mathematics from an advanced standpoint: Arithmetic, algebra, and analysis (F. Kelin). MAA Reviews. https://www.maa.org/ press/maa-reviews/elementarymathematics-from-an-advanced-standpoint-arithmetic-algebraand-analysis Gravemeijer, K., Bruin-Muurling, G., Kraemer, J. M., & van Stiphout, I. (2016). Shortcomings of mathematics education reform in The Netherlands: A paradigm case? Mathematical Thinking and Learning, 18(1), 25–44. https://doi.org/10.1080/10986065.2016.1107821 Gray, E. M. & Tall, D. O. (1991). Duality, ambiguity & flexibility in successful mathematical thinking. In Proceedings of the 15th conference for the international group for the psychology of mathematics education, 2, 72–79. Gray, E. M., & Tall, D. O. (1994). Duality, ambiguity and flexibility: A proceptual view of simple arithmetic. Journal for Research in Mathematics Education, 26(2), 115–141. Gregorio, F., Di Martino, P., & Iannone, P. (2019). The Secondary-Tertiary Transition in mathematics. Successful students in crisis. EMS Newsletter, 9(113), 45–47. Hadamard, J. (1945). The psychology of invention in the mathematical field. Princeton University Press. (Dover edition, New York 1954). Halmos, P. (1985). I want to be a mathematician. An automathography. Springer. Halmos, P. R. (1994). What is teaching? The American Mathematical Monthly, 101(9), 848–854. Hannah, J., Stewart, S., & Thomas, M. O. J. (2011). Analysing lecturer practice: The role of orientations and goals. International Journal of Mathematical Education in Science and Technology, 42(7), 975–984. Hannah, J., Stewart, S., & Thomas, M. O. J. (2012). Student reactions to an approach to linear algebra emphasising embodiment and language. In Proceedings of the 12th international congress on mathematical education (ICME-12) topic study group, Seoul, Korea (Vol. 2, pp. 1386–1393).

References

259

Hannah, J., Stewart, S., & Thomas, M. O. J. (2013a). Conflicting goals and decision making: The deliberations of a new lecturer. In A. M. Lindmeier & A. Heinze (Eds.), Proceedings of the 37th conference of the international group for the psychology of mathematics education (Vol. 2, pp. 425–432). PME. Hannah, J., Stewart, S., & Thomas, M. O. J. (2013b). Emphasizing language and visualization in teaching linear algebra. International Journal of Mathematical Education in Science and Technology, 44(4), 475–489. https://doi.org/10.1080/0020739X.2012.756545 Hannah, J., Stewart, S., & Thomas, M. O. J. (2013c). Conflicting goals and decision making: The influences on a new lecturer. In A. M. Lindmeier & A. Heinze (Eds.), Proceedings of the 37th conference of the International Group for the Psychology of mathematics education (Vol. 2, pp. 425–432). PME. Hannah, J., Stewart, S., & Thomas, M. O. J. (2014). Teaching linear algebra in the embodied, symbolic, and formal worlds of mathematical thinking: Is there a preferred order? In S. Oesterle, P. Liljedahl, C. Nicol, & D. Allan (Eds.), Proceedings of the joint meeting of PME 38 and PME-NA 36 (Vol. 3, pp. 241–248). PME. Hannah, J., Stewart, S., & Thomas, M. (2015). Linear algebra in the three worlds of mathematical thinking: The effect of permuting worlds on students’ performance. In Proceedings of the 18th annual conference on research in undergraduate mathematics education (pp. 581–586). Pittsburgh. Hannah, J., Stewart, S., & Thomas, M. O. J. (2016). Developing conceptual understanding and definitional clarity in linear algebra through the three worlds of mathematical thinking, teaching. Mathematics and its Applications: An International Journal of the IMA, 35(4), 216–235. https:// doi.org/10.1093/teamat/hrw001 Hansen, M. T. (2009). Collaboration: How leaders avoid the traps, create unity, and reap big results. Harvard Business Press. Harel, G. (2008). What is mathematics? A pedagogical answer to a philosophical question. In B. Gold & R. Simons (Eds.), Proof and other dilemmas: Mathematics and philosophy (pp. 265–290). Mathematical Association of America. Harel, G. (2018). The learning and teaching of linear algebra through the lenses of intellectual need and epistemological justification and their constituents. In S. Stewart, C. Andrews-Larson, A. Berman, & M. Zandieh (Eds.), Challenges and strategies in teaching linear algebra (pp. 3–27). Springer. Hart, K. (Ed.). (1981). Children’s learning of mathematics 11–16. John Murray. Heron, J. (1985). The role of reflection in a co-operative inquiry. In D. Boud, R. Keogh, & D. Walker (Eds.), Reflection: Turning experience into learning. Kogan Page. Hersh, R. (1979). Some proposals for reviving the philosophy of mathematics. Advances in Mathematics, 31, 31–50. Hersh, R. (1990). Let’s teach philosophy of mathematics. The College Mathematics Journal, 21(2), 105–111. Hersh, R. (1997). What is mathematics, really? Oxford University Press. Hiebert, J., & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 1–27). Lawrence Erlbaum Associates, Inc. Hiller, G. (2000). Levels of classroom preparation. In I. Westbury, S. Hopmann, & K. Riquarts (Eds.), Teaching as a reflective practice: The German didaktik tradition (pp. 207–221). Psychology Press. HMSO. (1982). Mathematics counts: The Cockcroft report. HMSO. Hodds, M., Alcock, L., & Inglis, M. (2014). Self-explanation training improves proof comprehension. Journal for Research in Mathematics Education, 45(1), 62–101. Hodgson, B. R. (2012). Whither the mathematics/didactics interconnection? Evolution and challenges of a kaleidoscopic relationship as seen from an ICME perspective. In ICME conference, Plenary Presentation, Seoul. South Korea.

260

References

Hogue, T. (1993). Community-based collaboration: Community wellness multiplied. Oregon Center for Community Leadership, Oregon State University. Hospesova, A., Carrillo, J., & Santos, L. (2018). Mathematics teacher education and professional development. In T. Dreyfus, M. Artigue, D. Potari, S. Prediger, & K. Ruthven (Eds.), Developing research in mathematics education – Twenty years of communication, cooperation and collaboration in Europe (New perspectives on research in mathematics education series) (Vol. 1, pp. 181–195). Routledge. Howson, A. G., & Mellin-Olsen, S. (1986). Social norms and external evaluation. In B. Christiansen, A. G. Howson, & M. Otte (Eds.), Perspectives on mathematics education (pp. 1–48). Reidel. Hughes-Hallett, D., & Gleason, A. M. (1994). Calculus. Wiley. Iannone, P. (2014). Mathematics & mathematics education: Searching for common ground, edited by M. Fried and T. Dreyfus, New York, Springer, 2014, 402 pp., 90, ISBN978-94-007-7472-8. Research in Mathematics Education, 16(3), 330–334. Iannone, P., & Miller, D. (2019). Guided notes for university mathematics and their impact on students’ note-taking behaviour. Educational Studies in Mathematics, 1–18. Iannone, P., & Nardi, E. (2005). On the pedagogical insight of mathematicians: Interaction and transition from the concrete to the abstract. Journal of Mathematical Behavior, 24, 191–215. Iannone, P., & Simpson, A. (2015). Students’ views of oral performance assessment in mathematics: Straddling ‘assessment of’ and ‘assessment for’ learning divide. Assessment and Evaluation in Higher Education, 40, 971–987. Iannone, P., & Simpson, A. (2022). How we assess mathematics degrees: The summative assessment diet a decade on. Teaching Mathematics and its Applications: An International Journal of the IMA, 41(1), 22–31. Iannone, P., Czichowsky, C., & Ruf, J. (2020). The impact of high stakes oral performance assessment on students’ approaches to learning: A case study. Educational Studies in Mathematics, 103(3), 313–337. Inglis, M., & Alcock, L. (2012). Expert and novice approaches to reading mathematical proofs. Journal for Research in Mathematics Education, 43, 358–390. Inglis, M., & Foster, C. (2018). Five decades of mathematics education research. Journal for Research in Mathematics Education, 49(4), 462–500. Jaworski, B. (1991). Interpretations of a constructivist philosophy in mathematics teaching. Unpublished PhD thesis, Open University, UK. Jaworski, B. (1994). Investigating mathematics teaching: A constructivist enquiry. Falmer Press. Jaworski, B. (1998). Mathematics teacher research: Process practice and the development of teaching. Journal of Mathematics Teacher Education, 1(1), 3–31. Jaworski, B. (2001). Developing mathematics teaching: Teachers, teacher-educators and researchers as co-learners. In F.-L. Lin & T. J. Cooney (Eds.), Making sense of mathematics teacher education. Kluwer. Jaworski, B. (2003). Research practice into/influencing mathematics teaching and learning development: Towards a theoretical framework based on co-learning partnerships. Educational Studies in Mathematics, 54, 249–282. Jaworski, B. (2006). Theory and practice in mathematics teaching development: Critical inquiry as a mode of learning in teaching. Journal of Mathematics Teacher Education, 9(2), 187–211. Jaworski, B. (2008a). Building and sustaining inquiry communities in mathematics teaching development. In K. Krainer & T. Wood (Eds.), Participants in MTE (pp. 30–330). Sense Publishers. Jaworski, B. (2008b). Building and sustaining inquiry communities in mathematics teaching development. Teachers and didacticians in collaboration. In K. Krainer (Volume Ed.) & T. Wood (Series Ed.) International handbook of mathematics teacher education: Vol. 3. Participants in Mathematics Teacher Education: Individuals, teams, communities and networks (pp. 309–330). Sense Publishers.

References

261

Jaworski, B. (2019a). Inquiry-based practice in university mathematics teaching development. In D. Potari & O. Chapman (Eds.), International handbook of mathematics teacher education: Volume 1: Knowledge, beliefs, and identity in mathematics teaching and teaching development (2nd ed., pp. 275–302). Koninklijke Brill NV. Jaworski, B. (2019b). Preparation and professional development of university mathematics teachers. In S. Lerman (Ed.), Encyclopedia of mathematics education. Springer. https://doi. org/10.1007/978-3-319-77487-9_100027-1 Jaworski, B., & Potari, D. (2009). Bridging the macro-micro divide: Using an activity theory model to capture complexity in mathematics teaching and its development. Educational Studies in Mathematics, 72, 219–236. Jaworski, B., Fuglestad, A.-B., Bjuland, R., Breiteig, T., Grevholm, B., & Goodchild, S. (2007). Learning communities in mathematics. Caspar. Jaworski, B., Treffert-Thomas, S., & Bartsch, T. (2009). Characterising the teaching of university mathematics: A case of linear algebra. In M. Tzekaki, M. Kaldrimidou, & C. Sakonidis (Eds.), Proceedings of the 33rd Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 249–256). PME. Jaworski, B., Treffert-Thomas, S., Hewitt, D., Feeney, M., Shrish-Thapa, D., Conniffe, D., Dar, A., Vlaseros, N., & Anastasakis, M. (2018). Student Partners in Task Design in a computer medium to promote foundation students’ learning of mathematics. In Proceedings of INDRUM: Second conference of the International Network for Didactic Research in University Mathematics (pp. 316–325). University of Agder and INDRUM. Job, P., & Schneider, M. (2014). Empirical positivism, an epistemological obstacle in the learning of calculus. ZDM – The International Journal on Mathematics Education, 46(4), 635–646. Johnson, E., Caughman, J., Fredericks, J., & Gibson, L. (2013). Implementing inquiry-oriented curriculum: From the mathematicians’ perspective. The Journal of Mathematical Behavior, 32(4), 743–760. Johnson, E., Keller, R., & Fukawa-Connelly, T. (2018). Results from a survey of abstract algebra instructors across the United States: Understanding the choice to (not) lecture. International Journal of Research in Undergraduate Mathematics Education, 4(2), 254–285. Kahneman, D. (2011). Thinking, fast and slow. Macmillan. Keisler, H. J. (1976). Foundations of infinitesimal calculus. Prindle, Weber & Schmidt. Kember, D., & Kwan, K.-P. (2000). Lecturers’ approaches to teaching and their relationship to conceptions of good teaching. Instructional Science, 28, 469–490. Kensington-Miller, B., Yoon, C., Sneddon, J., & Stewart, S. (2013). Changing beliefs about teaching in large undergraduate mathematics classes. Mathematics Teacher Education and Development, 15(2), 52–69. Kensington-Miller, B., Sneddon, J., & Stewart, S. (2014). Crossing new uncharted territory: Shifts in academic identity as a result of modifying teaching practice in under-graduate mathematics. International Journal of Mathematics Education in Science and Technology, 45(6), 827–838. Killion, J., & Todnem, G. (1991). A process of personal theory building. Educational Leadership, 48(6), 14–17. Kilpatrick, J. (1981). The reasonable ineffectiveness of research in mathematics education. For the Learning of Mathematics, 2(2), 22–28. Kirsch, A. (1977a). Didaktik Math, 5(2), 87–101. Kirsch, A. (1977b). Westermanns Paedagog (Beitraege 29/1977/4, 151–157). ABDE Unterrichtliche Vereinfachung. Kirsch, A. (1978). Aspects of simplification in mathematics teaching. Aspekte des Vereinfachens im Mathematikunterricht (German). Kassel Univ. (Gesamthochschule) (West Germany). Kirsch, A. (2000). Aspects of simplification in mathematics teaching. In I. Westbury (Ed.), Teaching as a reflective practice: The German Didaktik tradition (pp. 267–284). Erlbaum. Klein, F. (1887). The arithmetizing of mathematics. In B. W. Ewald (Ed.), 1996, From Kant to Hilbert: A source book in the foundations of mathematics (pp. 965–971). Oxford University Press.

262

References

Klein, F. (1908a). Elementary mathematics from an advanced standpoint. Arithmetic, Algebra, Analysis (E.R. Hedrick & C.A. Noble, Trans.) (3rd edn., German edn.). Dover Publications. Klein, F. (1908b). Fundamental mathematics from a higher standpoint, I–III (G. Schubring, Trans.). Springer. Klein, D. (2007). A quarter century of US ‘math wars’ and political partisanship. BSHM Bulletin, 22(1), 22–33. https://doi.org/10.1080/17498430601148762 Kline, M. (1973). Why Johnny can’t add: The failure of the new math. St. Martin’s Press. Koichu, B., & Pinto, A. (2019). The secondary-tertiary transition in mathematics. What are our current challenges and what can we do about them? EMS Newsletter, 6(112), 34–35. Kolb, D. (1984). Experiential learning: Experience as the source of learning and development. Prentice Hall. Kolb, H. (2007). Webvision: The organization of the retina and visual system. Retrieved on, 26 Jan 2019 from https://www.ncbi.nlm.nih.gov/books/NBK11556/ Kondratieva, M., & Winsløw, C. (2018). Klein’s Plan B in the early teaching of analysis: Two theoretical cases of exploring mathematical links. International Journal of Research in Undergraduate Mathematics Education, 4(1), 119–138. Krupnik, V., Fukawa-Connelly, T., & Weber, K. (2018). Students’ epistemological frames and their interpretation of lectures in advanced mathematics. The Journal of Mathematical Behavior, 49, 174–183. Kuster, G., Johnson, E., Keene, K., & Andrews-Larson, C. (2018). Inquiry-oriented instruction: A conceptualization of the instructional principles. Primus, 28(1), 13–30. Lakatos, I. (1976). In J. Worral & E. Zahar (Eds.), Proofs and refutations: The logic of mathematical discovery. Cambridge University Press. Lakoff, G., & Johnson, M. (1980). Metaphors we live by. University of Chicago Press. Lakoff, G., & Núñez, R. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. Basic Books. Landgärds, I. M. (2019). Providing economics students opportunities to learn basic mathematics. Nordic Journal of STEM Education, 3(1), 185–189. Larsen, S. P. (2013). A local instructional theory for the guided reinvention of the group and isomorphism concepts. The Journal of Mathematical Behavior, 32(4), 712–725. Larsen, S. P., Johnson, E., & Weber, K. (2013). The teaching abstract algebra for understanding project: Designing and scaling up a curriculum innovation. Special Issue Journal of Mathematical Behavior, 32(4). Laursen, S. L., & Rasmussen, C. (2019). I on the prize: Inquiry approaches in undergraduate mathematics. International Journal of Research in Undergraduate Mathematics Education, 5(1), 129–146. Lave, J., & Wenger, E. (1991). Situated learning: Legitimate peripheral participation. Cambridge University Press. Leder, G. C., Pehkonen, E., & Törner, G. (2002). Beliefs: A hidden variable in mathematics education? Kluwer Academic Publishers. Lerman, S. (1996). Intersubjectivity in mathematics learning: A challenge to the radical constructivist paradigm? Journal for Research in Mathematics Education, 27(2), 133–150. Lerman, S. (2010). Theories of mathematics education: Is plurality a problem? In B. Sriraman & L. English (Eds.), Theories of mathematics education (pp. 99–109). Springer. Lerman, S. (Ed.). (2020). Encyclopedia of Mathematics Education. Springer. Leron, U., & Dubinsky, E. (1995). An abstract algebra story. The American Mathematical Monthly, 102(3), 227–242. Letiche, H. (1988). Interactive experiential learning in enquiry courses. In J. Nias & S. Groundwater-Smith (Eds.), The enquiring teacher: Supporting and sustaining teacher research (pp. 15–39). Falmer Press. Levenson, E., Swisa, R., & Tabach, M. (2018). Evaluating the potential of tasks to occasion mathematical creativity: Definitions and measurements. RME, 20(30), 273–294.

References

263

Lew, K., Fukawa-Connelly, T. P., Mejía-Ramos, J. P., & Weber, K. (2016). Lectures in advanced mathematics: Why students might not understand what the mathematics professor is trying to convey. Journal for Research in Mathematics Education, 47(2), 162–198. LMS. (2010). Mathematics degrees, their teaching and assessment. The London Mathematics Society. Retrieved October 14, 2020 from https://www.lms.ac.uk/sites/lms.ac.uk/files/ Mathematics/Policy_repors/2010%20teaching_position_statement.pdf Lockwood, E., Johnson, E., & Larsen, S. (2013). Developing instructor support materials for an inquiry-oriented curriculum. The Journal of Mathematical Behavior, 32(4), 776–790. Loos, A., & Ziegler, G. M. (2016). Was ist Mathematik lernen und lehren. Mathematische Semesterberichte, 63(2016), 155–169. Lortie, D. (1975). Schoolteacher: A sociological study. University of Chicago Press. Love, E., & Mason, J. (1992). Teaching mathematics: Action and awareness. Open University. Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States. Routledge. Madden, A., Stewart, S., & Meyer, J. (2023). A linear algebra instructor’s ways of thinking of moving between the three worlds of mathematical thinking within concepts of linear combination, span, and subspaces. In Proceedings of the 25th annual conference on research in undergraduate mathematics education. Madsen, L., & Winslow, C. (2009). Relations between teaching and research in physical geography and mathematics at research-intensive universities. International Journal of Science and Mathematics Education, 7(4), 741–763. Mali, A. (2016). Unpublished. PhD thesis, Loughborough University, UK. Mark, M., & Schramm, M. (2016). The College Mathematics Journal, 47(5), 334–339. Marton, F., & Säljö, R. (1976). On qualitative differences in learning I – Outome and process; II – Outcome as a function of the learner’s conception of the task. British Journal of Educational Psychology, 46(4–11), 115–127. Marton, F., & Säljö, R. (1997). Approaches to learning. In F. Marton, D. Hounsell, & N. Entwistle (Eds.), The experience of learning. Implications for teaching and studying in higher education (pp. 36–55). Scottish Academic Press. Mason, J. (1992). Developing Teaching through thinking mathematically and developing through reflecting on teaching. Draft SV, not yet published Mason, J. (1994). Only awareness is educable. In A. Bloomfield & T. Harries (Eds.), Teaching, learning and mathematics (pp. 28–29). Association of Teachers of Mathematics. Mason, J. (1998). Enabling teachers to be real teachers: necessary levels of awareness and structure of attention. Journal of Mathematics Teacher Education, 1(3), 243–267. Mason, J. (2002a). Reflection in and on practice. In P. Kahn & J. Kyle (Eds.), Effective learning & teaching in mathematics & its applications (pp. 117–128). ILT & Kogan Page. Mason, J. (2002b). Researching your own practice: The discipline of noticing. RoutledgeFalmer. Mason, J. (2002c). Mathematics teaching practice: A guide for university and college lecturers. Horwood Publishing. Mason, J. (2008a). From concept images to pedagogic structure for a mathematical topic. In C. Rasmussen & M. Carlson (Eds.), Making the connection: Research into practice in undergraduate mathematics education. MAA Notes (pp. 253–272). Mathematical Association of America. Mason, J. (2008b). Being mathematical with & in front of learners: Attention, awareness, and attitude as sources of differences between teacher educators, teachers & learners. In T. Wood (Series Ed.) & B. Jaworski (Vol. Ed.), International handbook of mathematics teacher education: vol.4. the mathematics teacher educator as a developing professional (pp. 31–56). Sense Publishers. Mason, J. (2009). Teaching as disciplined enquiry. Teachers and Teaching: Theory and Practice, 15(2-3), 205–223. Mason, J. (2011). Phenomenology of example construction. ZDM Mathematics Education, 2011(43), 195–204.

264

References

Mason, J. (2019). Pre-parative and post-parative play as key components of mathematical problem solving. In P. Felmer, P. Lilljedahl, & B. Koichu (Eds.), Problem solving in Mathematics instruction and teacher professional development. Springer. Mason, J., & Davis, J. (1987). Only awareness is educable. Mathematics Teaching, 120, 30–31. Mason, J., & Johnston-Wilder, S. (2004). Fundamental constructs in mathematics education. RoutledgeFalmer. Mason, J., Burton, L., & Stacey, K. (1982). Thinking mathematically. Addison Wesley. Mason, J., Burton L. & Stacey K. (2010). Thinking mathematically (2nd edn). Addison Wesley. Mattessich, P. W., Murray-Close, M., & Monsey, B. R. (2001). Collaboration: What makes it work (2nd ed.) McDonald, J., Stewart, S., & Harel, G. (under review). A student-Centered lesson on eigenvalues and eigenvectors. Mejía-Ramos, J. P., & Weber, K. (2019). Mathematics Majors’ diagram usage when writing proofs in calculus. Journal for Research in Mathematics Education, 50(5), 478–488. Mejía-Ramos, J. P., Fuller, E., Weber, K., Rhoads, K., & Samkoff, A. (2012). An assessment model for proof comprehension in undergraduate mathematics. Educational Studies in Mathematics, 79(1), 3–18. Mejía-Ramos, J. P., Lew, K., de la Torre, J., & Weber, K. (2017). Developing and validating proof comprehension tests in undergraduate mathematics. Research in Mathematics Education, 19(2), 130–146. Melhuish, K., Fukawa-Connelly, T., Dawkins, P. C., Woods, C., & Weber, K. (2022). Collegiate mathematics teaching in proof-based courses: What we now know and what we have yet to learn. The Journal of Mathematical Behavior, 67, 100986. Mellin-Olsen, S. (1987). The politics of mathematics education. Reidel. Moon, J. (1999). Reflection in learning and professional development: Theory and practice. Kogan Page. Moore, R. C. (1994). Making the transition to formal proof. Educational Studies in Mathematics, 27(3), 249–266. Moore, S., Armstrong, C., & Pearson, J. (2008). Lecture absenteeism among students in higher education: A valuable route to understanding student motivation. Journal of Higher Education Policy and Management, 30(1), 15–24. Moore-Russo, D., & Wilsey, J. (2014). Delving into the meaning of productive reflection: A study of future teachers’ reflections on representations of teaching. Teaching and Teacher Education, 37, 76–90. Moslehian, M. E., Størmer, M., Thorbjørnsen, S., & Winsløw, C. (2017). Uffe Haagerup - His life and mathematics. Advances in Operator Theory, 3(1), 295–325. Moss, R., & Roberts, G. (1968). A preliminary course in analysis. Chapman & Hall. Nardi, E. (2000). Mathematics undergraduates’ responses to semantic abbreviations, ‘geometric’ images and multi-level abstractions in Group Theory. Educational Studies in Mathematics, 43(2), 169–189. Nardi, E. (2007). Amongst mathematicians: Teaching and learning mathematics at the university level. Springer Science & Business Media. Nardi, E. (2008). Amongst mathematicians: Teaching and learning mathematics at university level. Springer. Nardi, E. (2015, August). The many and varied crossing paths of mathematics and mathematics education. Mathematics Today (Special issue: Windows on advanced mathematics) pp. 212–215. Nardi, E. (2016). Where form and substance meet: using the narrative approach of re-storying to generate research findings and community rapprochement in (university) mathematics education. Educational Studies in Mathematics, 92(3), 361–377. Nardi, E. (2017). From advanced mathematical thinking to university mathematics education: A story of emancipation and enrichment. In T. Dooley & G. Gueudet (Eds.), Proceedings of the 10th conference of European research in mathematics education (CERME) (pp. 9–31). Ireland.

References

265

Living Edition | Editors: Steve Lerman Nardi, E. (1996). The novice mathematician’s encounter with mathematical abstraction: Tensions in concept-image construction and formalisation Unpublished doctoral thesis. University of Oxford. Available at http://www.uea.ac.uk/~m011 Nardi, E., & Iannone, P. (2004). On the fragile, yet crucial relationship between mathematicians and researchers in mathematics education. In Proceedings of the 28th annual conference of the international group for psychology in mathematics education (Vol. 3, pp. 401–408). ERIC Clearinghouse. Nardi, E., & Iannone, P. (2006). How to prove it: A brief guide for teaching proof to mathematics undergraduates (No. 978-0-9539983-8-8). Commissioned by the Higher Education Academy (Mathematics, Statistics and Operational Research branch). ISBN 978-0-9539983-8-8. Available at http://mathstore.ac.uk/publications/Proof%20Guide.pdf Nardi, E., Jaworski, B., & Hegedus, S. (2005). A spectrum of pedagogical awareness for undergraduate mathematics: From ‘tricks’ to ‘techniques’. Journal for Research in Mathematics Education, 36(4), 284–316. Nardi, E., Biza, I., González-Martin, A., Gueudet, G., & Winsløw, C. (2014a). Remarks on institutional, sociocultural and discursive approaches to research in (university) mathematics education: (Dis)connectivities, challenges and potentialities. In P. Barmby (Ed.), Proceedings of the British Society for Research into Learning Mathematics 34(1) (pp. 89–94). BSRLM. Nardi, E., Biza, I., González-Martin, A., Gueudet, G., & Winsløw, C. (2014b). Institutional, sociocultural and discursive approaches to research in university mathematics education. Research in Mathematics Education, 16(2), 91–94. Nardi, E., Ryve, A., Stadler, E., & Viirman, O. (2014c). Commognitive analyses of the learning and teaching of mathematics at university level: the case of discursive shifts in the study of Calculus. Research in Mathematics Education, 16(2), 182–198. Nardi, E., Winsløw, C., & Hausberger, T. (2016). Editorial. In Proceedings of the 1st INDRUM (International Network for Didactic Research in University Mathematics) conference: An ERME topic conference (pp. 6–9). Montpellier. National Research Council. (2013). The mathematical sciences in 2025. The National Academies Press. Neubrand, M. (2000). Reflecting as a Didaktik Construction: speaking about mathematics in the mathematics classroom. In I. Westbury, S. Hopmann, & K. Riquarts (Eds.), Teaching as a reflective practice: The German didaktik tradition (pp. 251–266). Psychology Press. Neumann, P. (1992). The future for honours degree courses in mathematics. Journal of the Royal Statistical Society. Series A (Statistics in Society), 155(2), 185–189. https://doi.org/10.2307/ 2982954 NOKUT. (2013). Standards and guidelines for centres and criteria for the assessment of applications. Norwegian Agency for Quality Assurance in Education. Oates, G., & Evans, T. (2017). Research mathematicians and mathematics educators: Collaborating for professional development. In K. Patterson (Ed.), Focus on mathematics education research (pp. 1–30). Nova Science Publishers, Inc. Ollerenshaw, J. A., & Creswell, J. W. (2002). Narrative research: A comparison of two restorying data analysis approaches. Qualitative Inquiry, 8(3), 329–347. Opstad, L., Bonesrønning, H., & Fallan, L. (2017). Tar vi opp de rette studentene ved økonomiskadministrative studier?: En analyse av matematikkbakgrunn og resultater ved NTNU Handelshøyskolen. [Do we enrol the right students on economics-administration studies?: An analysis of mathematics background and results at NTNU Business School]. Samfunnsøkonomen, 1, 21–29. Pais, A. (2013). An ideology critique of the use-value of mathematics. Educational Studies in Mathematics, 84, 15–34. Pajares, M. F. (1992). Teachers’ beliefs and educational research: Cleaning up a messy construct. Review of Educational Research, 62(3), 307–332.

266

References

Paoletti, T., Krupnik, V., Papadopoulos, D., Olsen, J., Fukawa-Connelly, T., & Weber, K. (2018). Teacher questioning and invitations to participate in advanced mathematics lectures. Educational Studies in Mathematics, 98(1), 1–17. Paterson, J., & Evans, T. (2013). Audience insights: Feed forward in professional development. In D. King, B. Loch, & L. Rylands (Eds.), Proceedings of Lighthouse Delta, the 9th Delta conference of teaching and learning of undergraduate mathematics and statistics through the fog (pp. 132–140). Delta. Paterson, J., Thomas, M. O. J., & Taylor, S. (2011a). Decisions, decisions, decisions: What determines the path taken in lectures? International Journal of Mathematical Education in Science and Technology, 42(7), 985–995. Paterson, J., Thomas, M. O. J., Postlethwaite, C., & Taylor, S. (2011b). The internal disciplinarian: Who is in control? In S. Brown, S. Larsen, K. Marrongelle, & M. Oehrtman (Eds.), Proceedings of the 14th annual conference on research in undergraduate mathematics education (Vol. 2, pp. 354–368). Oregon. Paterson, J., Thomas, M. O. J., & Taylor, S. (2011c). Reaching decisions via internal dialogue: Its role in a lecturer professional development model. In B. Ubuz (Ed.), Proceedings of the 35th conference of the international group for the psychology of mathematics education (Vol. 3, pp. 353–360). Middle East Technical University. Pawson, R., & Tilley, N. (1997). Realistic evaluation. Sage. Pawson, R., & Tilley, N. (2001). Realistic evaluation bloodlines. American Journal of Evaluation, 22(3), 317–324. Philips, C. (2015). The new math – A political history. University of Chicago Press. Pólya, G. (1945). How to solve it: A new aspect of mathematical method. Princeton University Press. Pólya, G. (1962). Mathematical discovery: On understanding, learning, and teaching problem solving (combined edition). Wiley. Pólya, G. (1965). Let us teach guessing (film). Mathematical Association of America. Postareff, L., Lindblom-Ylänne, S., & Nevgi, A. (2008). A follow-up study of the effect of pedagogical training on teaching in higher education. Higher Education, 56, 29–43. Potari, D., & Jaworski, B. (2002). Tackling Complexity in Mathematics Teacher Development: Using the teaching triad as a tool for reflection and enquiry. Journal of Mathematics Teacher Education, 5(4), 351–380. Putnam, A. L., Sungkhasettee, V. W., & Roediger, H. L. III. (2016). Optimizing learning in college: Tips from cognitive psychology. Perspectives on Psychological Science, 11(5), 652–660. Ralston, A. (2004). Research mathematicians and mathematics education: A critique. Notices of the American Mathematical Society, 51, 403–411. Rasmussen, C., & Wawro, M. (2017). Post-calculus research in undergraduate mathematics education. In J. Cai (Ed.), The compendium for research in mathematics education. National Council of Teachers of Mathematics. Rasmussen, C., Keene, K. A., Dunmyre, J., & Fortune, N. (2018). Inquiry oriented differential equations: Course materials. Available at https://iode.wordpress.ncsu.edu. This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. Reason, P., & Rowan, J. (Eds.). (1981). Human inquiry: A sourcebook of new paradigm research. Wiley. Reeder, S., Stewart, S., Raymond, K., Troup, J., & Melton, H. (2019). Analyzing the nature of university students’ difficulties with algebra in calculus: Students’ voices during problem solving. In A. Weinberg, D. Moore-Russo, M. Wawro, & H. Soto (Eds.), Proceedings of the 22nd annual conference on research in undergraduate mathematics education (pp. 501–508). Oklahoma. Rényi, A. (2006). A Socratic dialog on mathematics (Chapter 1). In R. Hersh (Ed.), 18 unconventional essays on the nature of mathematics (pp. 1–16). Springer.

References

267

Resnick, L. B., & Resnick, D. P. (1992). Assessing the thinking curriculum: New tools for educational reform. In B. R. Gifford & M. C. O’Connor (Eds.), Changing assessments: Alternative views of aptitude, achievement, and instruction (pp. 37–75). Kluwer. Richardson, K. (1998). Models of cognitive development. Psychology Press. Roberts, S. (2016). Mathematical beauty: A Q&A with fields medalist Michael Atiyah. https://www. scientificamerican.com/article/mathematical-beauty-a-q-a-with-fields-medalist-michael-atiyah/ Roh, K. H. (2008). Students’ images and their understanding of definitions of the limit of a sequence. Educational Studies in Mathematics, 69(3), 217–233. Roh, K. H. (2010). How to help students conceptualize the rigorous definition of the limit of a sequence. Primus, 20(6), 473–487. Schneider, M. (2014). Epistemological obstacles in mathematics education. In S. Lerman (Ed.), Encyclopedia of mathematics education (pp. 214–217). Springer. Schoenfeld, A. H. (1985). Mathematical problem solving. Academic. Schoenfeld, A. (1992). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics. In D. A. Grouws (Ed.), Handbook of research in mathematics teaching and learning (pp. 334–370). MacMillan/National Council of Teachers of Mathematics. Schoenfeld, A. H. (2004). The math wars. Educational Policy, 18, 253–286. Schoenfeld, A. (2008a). The math wars. Educational Policy, 18(1), 253–286. Schoenfeld, A. H. (2008b). On modeling teachers’ in-the-moment decision-making. In A. H. Schoenfeld (Ed.), A study of teaching: Multiple lenses, multiple views (Journal for research in mathematics education monograph 14, pp. 45–96). NCTM. Schoenfeld, A. H. (2010). How we think: A theory of goal-oriented decision making and its educational applications. Routledge. Schoenfeld, A. H. (2011). How we think: A theory of goal-oriented decision making and its educational applications. Routledge. Schoenfeld, A., Thomas, M. O. J., & Barton, B. (2016). On understanding and improving the teaching of university mathematics. International Journal of STEM Education, 3(4), 17. Available from https://stemeducationjournal.springeropen.com/articles/10.1186/s40594-0160038-z Schön, D. (1983). The reflective practitioner: How professionals think in action. Temple Smith. Selden, A. (2012). A home for RUME: The story of the formation of the mathematical association of America’s special interest group on research in mathematics education. https://www.tntech. edu/cas/pdf/math/techreports/TR-2012-6.pdf Selden, A., & Selden, J. (2007). Overcoming students’ difficulties in learning to understand and construct proofs (Tech. Rep. No. 2007-1). Tennessee Technological University, Department of Mathematics. Sfard, A. (1994). Reification as the birth of metaphor. For the Learning of Mathematics, 14(1), 44–55. Sfard, A. (1998). A mathematician’s view of research in mathematics education: An interview with Shimshon A. Amitsur. In A. Sierpinska & J. Kilpatrick (Eds.), Mathematics Education as a research domain: A search for identity (pp. 445–458). Kluwer Academic Publishers. Sfard, A. (2008). Thinking as communicating. Human development, the growth of discourse, and mathematizing. Cambridge University Press. Shipman, B. A. (2012). Determining definitions for comparing cardinalities. Primus, 22(3), 239–254. Shipman, B. A. (2013). Convergence and the Cauchy property of sequences in the setting of actual infinity. Primus, 23(5), 441–458. Shulman, L. S. (2005a). Signature pedagogies in the profession. Daedalus Summer, 134(3), 52–59. Shulman, L. S. (2005b). The signature pedagogies of the profession of law, medicine, engineering, and the clergy: potential lessons for the education of teachers. In Manuscript Delivered at the Math Science Partnerships (MSP) Workshop ‘Teacher Education for Effective Teaching and Learning’ Hosted by the National Research Council’s Center for Education, February 6–8, 2005, Irvine, California.

268

References

Shulman, L. (2007). Signature Pedagogies in the professions. In D. Lemmermöhle (Hrsg.), Professionell lehren – erfolgreich lernen (pp. 13–22). Münster: Waxmann. Sierpinska, A. (1987). Humanities students and epistomological obstacles related to limits. Educational Studies in Mathematics, 18, 371–397. Simon, H. A. (1972). Theories of bounded rationality. Decision and organization, 1(1), 161–176. Simpson, A. (2018). Princesses are bigger than elephants: Effect size as a category error in evidence based education. British Educational Research Journal, 44(5), 897–913. Skemp, R. R. (1976). Relational understanding and instrumental understanding. Mathematics Teaching, 77, 20–26. Skemp, R. R. (1982). Communicating mathematics: Surface structures and deep structures. Visible Language, 16(3), 281–288. Solomon, Y., Croft, T., & Lawson, D. (2010). Safety in numbers: Mathematics support centres and their derivatives as social learning spaces. Studies in Higher Education, 35(4), 421–431. ISSN: 0307-5079. Speer, N. M. (2008). Connecting beliefs and practices: A fine-grained analysis of a college mathematics teacher’s collections of beliefs and their relationship to his instructional practices. Cognition and Instruction, 26(2), 218–267. Speer, N. M., Smith, J. P., & Horvath, A. (2010). Collegiate mathematics teaching: An unexamined practice. Journal of Mathematical Behavior, 29, 99–114. Sriraman, B. (2022). Uncertainty as a catalyst and condition for creativity: The case of mathematics. ZDM – Mathematics Education, 54(1), 19–33. Sriraman, B., & Törner, G. (2008). Political union/mathematical education disunion: Building bridges in European didactic traditions. In L. English (Ed.), The handbook of international research in mathematics education (2nd ed., pp. 660–694). Routledge, Taylor & Francis. Stake, R. E. (2000). Qualitative case studies. In N. K. Denzin & Y. S. Lincoln (Eds.), Handbook of qualitative research (pp. 443–466). Sage Publications. Steen, L. A. (1999). Review: Theories that Gyre and Gimble in the Wabe reviewed work(s): Mathematics education as a research domain: A search for identity by Anna Sierpinska; Jeremy Kilpatrick. Journal for Research in Mathematics Education, 30(2), 235–241. Steffe, L. P., & Thompson, P. (2000). Interaction or intersubjectivity? A reply to Lerman. Journal for Research in Mathematics Education, 31(2), 191–209. Stein, M. K., Grover, B. W., & Henningsen, M. (1996). Building student capacity for mathematical thinking and reasoning: An analysis of mathematical tasks used in reform classrooms. American Educational Research Journal, 33(2), 455–488. Stenhouse, L. (1984). Evaluating curriculum evaluation. In C. Adelman (Ed.), The politics and ethics of evaluation. Croom Helm. Stewart, S. (Ed.). (2017). And the rest is just algebra. Springer. Stewart, S. (2018). Moving between the embodied, symbolic and formal worlds of mathematical thinking in linear algebra. In S. Stewart, C. Andrews-Larson, A. Berman, & M. Zandieh (Eds.), Challenges and strategies in teaching linear algebra (pp. 51–67). Springer. Stewart, S., & Epstein, J. (2020). Linear algebra thinking in the embodied, symbolic and formal worlds: Students’ reasoning behind preferring certain worlds. In S. S. Karunakaran, Z. Reed, & A. Higgins (Eds.), Proceedings of the 23rd annual conference on research in undergraduate mathematics education (pp. 546–553). Boston. Stewart, S., & Schmidt, R. (2017). Accommodation in the formal world of mathematical thinking. International Journal of Mathematics Education in Science and Technology, 48(1), 40–49. https://doi.org/10.1080/0020739X.2017.1360527 Stewart, I. N., & Tall, D. O. (2014). Foundations of mathematics (2nd ed.). Oxford University Press. ISBN: 9780198531654. Stewart, I. N., & Tall, D. O. (2015). Algebraic number theory and Fermat’s last theorem (4th ed.). CRC. ISBN 9781498738392. Stewart, I. N., & Tall, D. O. (2018). Complex analysis (2nd ed.). Cambridge University Press. ISBN: 978-1108436793.

References

269

Stewart, S., & Tran, T. (2022). Linear algebra proofs: Ways of understanding and ways of thinking in the formal world. In C. Fernández, S. Llinares, A. Gutiérrez, & N. Planas (Eds.), Proceedings of the 45th conference of the International Group for the Psychology of mathematics education (Vol. 4, pp. 43–50). Alicante. Stewart, S., Thomas, M. O. J., & Hannah, J. (2005). Towards student instrumentation of computerbased algebra systems in university courses. International Journal of Mathematical Education in Science and Technology, 36(7), 741–750. Stewart, S., Thompson, C. A., Kornelson, K., Lifschitz, L., & Brady, N. (2015a). Balancing formal, symbolic, and embodied world thinking in first year calculus lectures. In T. Fukawa-Connolly, N. Engelke Infante, K. Keene, & M. Zandieh (Eds.), Proceedings of the 18th annual conference on research in undergraduate mathematics education (pp. 970–976). Pittsburgh. Stewart, S., Schmidt, R., Cook, J. P., & Pitale, A. (2015b). Living in the formal world of mathematical thinking. In K. Beswick, T. Muir, & J. Fielding-Wells (Eds.), Proceedings of 39th psychology of mathematics education conference (Vol. 1, p. 201). PME. Stewart, S., Schmidt, R., Cook, J. P., & Pitale, A. (2015c). Pedagogical challenges of communicating mathematics with students: Living in the formal world of mathematical thinking. In Proceedings of the 18th annual conference on research in undergraduate mathematics education (pp. 964–969). Pittsburgh. Stewart, S., Thompson, C., & Brady, N. (2017). Navigating through the mathematical world: Uncovering a geometer’s thought processes through his handouts and teaching journals. In T. Dooley & G. Gueudet (Eds.), Proceedings of the Tenth Congress of the European Society for Research in Mathematics Education (CERME10, February 1–5) (pp. 2258–2265). DCU Institute of Education and ERME. Stewart, S., Troup, J., & Plaxco, D. (2018a). Teaching linear algebra: Modeling one instructor’s decisions to move between the worlds of mathematical thinking. In A. Weinberg, C. Rasmussen, J. Rabin, M. Wawro, & S. Brown (Eds.), Proceedings of the 21st annual conference on research in undergraduate mathematics education (pp. 1014–1022). San Diego. Stewart, S., Thompson, C., & Brady, N. (2018b). Examining a mathematician’s goals and beliefs about course handouts. In A. Weinberg, C. Rasmussen, J. Rabin, M. Wawro, & S. Brown (Eds.), Proceedings of the 21st annual conference on research in undergraduate mathematics education (pp. 1069–1075). San Diego. Stewart, S., Andrews-Larson, C., Berman, A., Zandieh, M., & (Eds.). (2018c). Challenges and strategies in teaching linear algebra. Springer. Stewart, S., Reeder, S., Raymond, K., & Troup, J. (2018d). Could algebra be the root of problems in calculus courses? In A. Weinberg, C. Rasmussen, J. Rabin, M. Wawro, & S. Brown (Eds.), Proceedings of the 21st annual conference on research in undergraduate mathematics education (pp. 1023–1030). San Diego. Stewart, S., Epstein, J., Troup, J., & McKnight, D. (2019a). An analysis of a Mathematician’s reflections on teaching eigenvalues and eigenvectors: Moving between embodied, symbolic and formal worlds of mathematical thinking. In A. Weinberg, D. Moore-Russo, M. Wawro, & H. Soto (Eds.), Proceedings of the 22nd annual conference on research in undergraduate mathematics education (pp. 586–593) Oklahoma. Stewart, S., Epstein, J., Troup, J., & McKnight, D. (2019b). A mathematician’s deliberation in reaching the formal world and students’ world views of the eigentheory. In U. T. Jankvist, M. Van den Heuvel-Panhuizen, & M. Veldhuis (Eds.), Proceedings of the eleventh congress of the European Society for Research in mathematics education (CERME11, February 6–10, 2019) (pp. 4835–4842). Freudenthal group \& Freudenthal institute, Utrecht University and ERME. Stewart, S., Troup, J., & Plaxco, D. (2019c). Reflection on teaching linear algebra: Examining one instructor’s movements between the three worlds of mathematical thinking. ZDM Mathematics Education, 51(7), 1253–1266. Springer. https://doi.org/10.1007/s11858-019-01086-0

270

References

Stewart, S., Axler, S., Beezer, R., Boman, E., Catral, M., Harel, G., McDonald, J., Strong, D., & Wawro, M. (2022a). The linear algebra curriculum study group (LACSG 2.0) recommendations. The Notices of American Mathematical Society, 69(5), 813–819. https://doi.org/10.1090/ noti2479 Stewart, S., Cronin, A., Tran, T., & Powers, A. (2022b). Ways of thinking and ways of understanding in the formal world: Students’ perspectives on nature of proofs in a second course in linear algebra. In S. Karunakaran & A. Higgins (Eds.), Proceedings of the 24th Annual Conference on Research in Undergraduate Mathematics Education (pp. 1152–1157). Boston. Strang, G. (2006). Linear algebra and its applications (4th ed.). Thomson Higher Education. Strauss, A. L., & Corbin, J. (1998). Basics of qualitative research: Grounded theory procedures and techniques (2nd ed.). Sage. Stray, C. (2001). The shift from oral to written examination: Cambridge and Oxford 1700–1900. Assessment in Education: Principles, Policy & Practice, 8(1), 33–50. Stylianides, G. J., Stylianides, A. J., & Weber, K. (2017). Research on the teaching and learning of proof: Taking stock and moving forward. In J. Cai (Ed.), The compendium for research in mathematics education. National Council of Teachers of Mathematics. Subroto, T., & Suryadi, D. (2019). Epistemological obstacles in mathematical abstraction on abstract algebra. In 3rd International Conference on Mathematical Sciences and Statistics. IOP Conference Series: Journal of Physics: Conference Series (Vol. 1132). https://doi.org/10. 1088/1742-6596/1132/1/012032 Suchman, L. (1994). Working relations of technology production and use. Computer Supported Cooperative Work, 2, 21–39. Swinyard, C., & Larsen, S. (2012). Coming to understand the formal definition of limit: Insights gained from engaging students in reinvention. Journal for Research in Mathematics Education, 43(4), 465–493. Tall, D. O. (1980). Intuitive infinitesimals in the calculus. In Abstracts of short communications, fourth international congress on mathematical education. http://homepages.warwick.ac.uk/ staff/David.Tall/pdfs/dot1980c-intuitive-infls.pdf Tall, D. O. (1985). Understanding the calculus. Mathematics Teaching, 10, 49–53. Tall, D. (1987). Constructing the concept image of a tangent. In Proceedings of the 11th conference of the international group for the psychology of mathematics education (Vol. 3, pp. 69–75). Tall, D. O. (Ed.) (1991). Advanced mathematical thinking. Kluwer. Tall, D. O. (1993). Mathematicians thinking about students thinking about mathematics. Newsletter of the London Mathematical Society, 202, 12–13. Tall, D. (Ed.). (1994a). Advanced mathematical thinking. Kluwer Academic Publishers. Tall, D. O. (1994b). Understanding the processes of advanced mathematical thinking. In Abstracts of invited talks, International Congress of Mathematicians, Zurich, August 1994, pp. 182–183. Tall, D. O. (1994c). Understanding the processes of advanced mathematical thinking. In International Congress of Mathematicians, Zurich, August 1994. Full Lecture at: http://homepages. warwick.ac.uk/staff/David.Tall/pdfs/dot1996i-amt-icm.pdf Tall, D. (2004a). Thinking through three worlds of mathematics. In M. Johnsen Joines & A. Fuglestad (Eds.), Proceedings of the 28th conference of the international group for the psychology of mathematics education (Vol. 4, pp. 281–288). Tall, D. O. (2004b). Building theories: The three worlds of mathematics. For the Learning of Mathematics, 24(1), 29–32. Tall, D. O. (2008). The transition to formal thinking in mathematics. Mathematics Education Research Journal, 20(2), 5–24. Tall, D. O. (2010). Perceptions operations and proof in undergraduate mathematics, community for undergraduate learning in the mathematical sciences (CULMS). Newsletter, 2, 21–28. Tall, D. O. (2013). How humans earn to think mathematically: Exploring the three worlds of mathematics. Cambridge University Press. ISBN: 9781139565202. Tall, D. (2019a). Long-term principles form meaningful teaching and learning of mathematics. To be published.

References

271

Tall, D. O. (2019b). Complementing supportive and problematic aspects of mathematics to resolve transgressions in long-term sense making. In Fourth interdisciplinary scientific conference on mathematical transgressions, (March 2019). Tall, D. O., & Katz, M. (2014). A cognitive analysis of Cauchy’s conceptions of function, continuity, limit, and infinitesimal, with implications for teaching the calculus. Educational Studies in Mathematics, 86(1), 97–124. Tall, D. O., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12(2), 151–169. Tall, D. O., Tall, N. D., & Tall, S. J. (2017). Problem posing in the long-term conceptual development of a gifted child. In M. Stein (Ed.), A life’s time for mathematics education and problem solving. On the occasion of Andràs Ambrus’ 75th birthday (pp. 445–457). WTM-Verlag. Thom, R. (1973). Modern mathematics. Does it exist? In A.G. Howson (Ed.), Developments on Mathematical Education. Proceedings of the Second International Congress on Mathematical Education (pp. 194–209). Cambridge: Cambridge University Press. Thoma, A. (2019). Transition to university mathematical discourses: A commognitive analysis of first year examination tasks, lecturers’ perspectives on assessment and students’ examination scripts. Doctoral thesis:. University of East Anglia, UK. Thoma, A., & Nardi, E. (2018). Transition from school to university mathematics: Manifestations of unresolved commognitive conflict in first year students’ examination scripts. International Journal for Research in Undergraduate Mathematics Education, 4(1), 161–180. Thomas, S. (2012). An activity theory analysis of linear algebra teaching within university mathematics. Unpublished doctoral dissertation. Loughborough University, UK. Thomas, M., de Freitas Druck, O., Huillet, D., Ju, M. K., Nardi, E., Rasmussen, C., & Xie, J. (2014). Key mathematical concepts in the transition from secondary school to university. In S. J. Cho (Ed.), Proceedings of the 12th International Congress on Mathematical Education (pp. 265–284). Springer. Thompson, P. W. (2014). Reflections on collaboration between mathematics and mathematics education. In M. N. Fried & T. Dreyfus (Eds.), Mathematics and mathematics education: Searching for common ground, advances in mathematics education (pp. 313–333). Springer. Thompson, C. A., Stewart, S., & Mason, B. (2016). Physics: Bridging the embodied and symbolic worlds of mathematical thinking. In T. Fukawa-Connolly, N. E. Infante, M. Wawro, & S. Brown (Eds.), Proceedings of the 19th annual conference on research in undergraduate mathematics education (pp. 1340–1347). Pittsburgh. Thurston, W. P. (1990). Mathematical education. Notices of the American Mathematical Society, 37, 7, 844–850. https://arxiv.org/pdf/math/0503081.pdf (Note: page numbers in this text refer to the available pdf, not the original pagination.) Thurston, W. P. (1994). On proof and progress in mathematics. Bulletin of the American Mathematical Society, 30(2), 161–177. https://arxiv.org/pdf/math/9404236.pdf (page numbers in the text refer to the latter pdf.) Tilley, N. (2000). Realistic evaluation: An overview. In Founding conference of the Danish evaluation society (8). Retrieved on the 04.06.2019 from http://healthimpactassessment. pbworks.com/f/Realistic+evaluation+an+overview+-+UoNT+England+-+2000.pdf Toeplitz, O. (1909). Über die Auflösung unendlich vieler linearer Gleichungen mit unendlich vielen Unbekannten. Rendiconti del Circolo Matematico di Palermo, 28(1909), 88–96. Törner, G. & Pehkonen (1996). On the structure of mathematical belief systems. International Reviews on Mathematical Education, 28(4), 109–112. Törner, G., Rolke, K., Rösken, B., & Sririman, B. (2010). Understanding a teacher’s actions in the classroom by applying Schoenfeld’s theory teaching-in-context: Reflecting on goals and beliefs. In B. Sriraman & L. English (Eds.), Theories of mathematics education, advances in mathematics education (pp. 401–420). Springer.

272

References

Van den Heuvel-Panhuizen, M., & Drijvers, P. (2014). Realistic mathematics education. In S. Lerman (Ed.), Encyclopedia of mathematics education (pp. 521–525). Springer. van Hiele, P. M. (1986). Structure and Insight. Academic. Verret, M. (1975). Le temps des études. Paris. Vincent, B., LaRue, R., Sealey, V., & Engelke, N. (2015). Calculus students’ early concept images of tangent lines. International Journal of Mathematical Education in Science and Technology, 46(5), 641–657. https://doi.org/10.1080/0020739X.2015.1005700 Vygotsky, L. (1978a). Mind in Society: The development of the higher psychological processes. Harvard University Press. Vygotsky, L. S. (1978b). Mind in society. Harvard University Press. Wasserman, N. H., Fukawa-Connelly, T., Villanueva, M., Mejia-Ramos, J. P., & Weber, K. (2017). Making real analysis relevant to secondary teachers: Building up from and stepping down to practice. Primus, 27(6), 559–578. Watson, A., & Mason, J. (2002). Extending example spaces as a learning/teaching strategy. In A. Cockburn & E. Nardi (Eds.), Mathematics. Proceedings of PME 26 (Vol. 4, pp. 377–385). University of East Anglia. Watson, A., & Mason, J. (2005). Mathematics as a constructive activity: Learners generating examples. Erlbaum Weber, K. (2004). Traditional instruction in advanced mathematics courses: A case study of one professor’s lectures and proofs in an introductory real analysis course. The Journal of Mathematical Behavior, 23(2), 115–133. Weber, K. (2012). Mathematicians’ perspectives on their pedagogical practice with respect to proof. International Journal of Mathematical Education in Science and Technology, 43(4), 463–482. Weber, K., & Mejia-Ramos, J. P. (2011). Why and how mathematicians read proofs: An exploratory study. Educational Studies in Mathematics, 76, 329–344. Wells, G. (1999). Dialogic inquiry: Toward a sociocultural practice and theory of education. Cambridge University Press. Wenger, E. (1998). Communities of practice, learning, meaning and identity. Cambridge University Press. Wenger, E., McDermott, R., & Snyder, W. M. (2002). Cultivating communities of practice: A guide to managing knowledge. Harvard Business School Press. Wertsch, J. V. (1991). Voices of the mind: A sociocultural approach to mediated action. Harvard University Press. Westbury, I., Hopmann, S., & Riquarts., K. (Eds.). (2000). Teaching as a reflective practice: The German Didactics tradition. Erlbaum. Wilson, N. (2006). Encyclopedia of ancient Greece. Routledge. Winsløw, C., Biehler, R., Jaworski, B., Rønning, F., & Wawro, M. (2018a). Education and professional development of university mathematics teachers. In V. Durand-Guerrier, R. Hochmuth, S. Goodchild, & N. M. Hogstad (Eds.), Proceedings of the 2nd INDRUM (International Network for Didactic Research in University Mathematics) Conference: an ERME Topic Conference (p. 12). Kristiansand. Winsløw, C., Gueudet, G., Hochmut, R., & Nardi, E. (2018b). Research on university mathematics education. In T. Dreyfus, M. Artigue, D. Potari, S. Prediger, & K. Ruthven (Eds.), Developing research in mathematics education – Twenty years of communication, cooperation and collaboration in Europe (New perspectives on research in mathematics education series) (Vol. 1, pp. 60–74). Routledge. Winsløw. (1993). Strongly free actions on subfactors. International Journal of Mathematics, 4, 675–688. Woods, C., & Weber, K. (2020). The relationship between mathematicians’ pedagogical goals, orientations, and common teaching practices in advanced mathematics. The Journal of Mathematical Behavior, 59, 100792. Woodward, J. (2004). Mathematics education in the United States: Past to present. Journal of Learning Disabilities, 37(1), 16–31.

References

273

Zakariya, Y. F. (2021a). Undergraduate students’ performance in mathematics: Individual and combined effects of approaches to learning, self-efficacy, and prior mathematics knowledge. Doctoral Dissertation,. University of Agder. Zakariya, Y. F. (2021b). Self-efficacy between previous and current mathematics performance of undergraduate students: An instrumental variable approach to exposing a causal relationship. Frontiers in Psychology, Section Educational Psychology, 11, 556607. https://doi.org/10.3389/ fpsyg.2020.556607 Zammattio, C., Marinoni, A., & Brizio, A. (1980). Leonardo the scientist. McGraw-Hill. Ziegler, G. M., & Loos, A. (2017). What is mathematics? and why we should ask, where one should experience and learn that, and how to teach it. In Proceedings of the 13th International Congress on Mathematical Education (pp. 63–77). SpringerLink.