Affect in Mathematical Modeling [1st ed.] 978-3-030-04431-2;978-3-030-04432-9

Table of contents :
Front Matter ....Pages i-vi
Front Matter ....Pages 1-1
Commentary on Affect, Cognition and Metacognition in Mathematical Modelling (Jonei Cerqueira Barbosa)....Pages 3-13
Chapter 1: The Construct of Affect in Mathematical Modeling (Scott A. Chamberlin)....Pages 15-28
Chapter 2: Metacognition in Mathematical Modeling – An Overview (Katrin Vorhölter, Alexandra Krüger, Lisa Wendt)....Pages 29-51
Chapter 3: Principles for Designing Research Settings to Study Spontaneous Metacognitive Activity (Marta T. Magiera, Judith S. Zawojewski)....Pages 53-66
Chapter 4: Engagement Structures and the Development of Mathematical Ideas (Lisa B. Warner, Roberta Y. Schorr)....Pages 67-85
Front Matter ....Pages 87-87
The What and the Why of Modeling (Alan H. Schoenfeld)....Pages 89-97
Engaging Students in Mathematical Modeling: Themes and Issues (Peter Kloosterman)....Pages 99-110
Chapter 5: Exploring a Conative Perspective on Mathematical Engagement (Gerald A. Goldin)....Pages 111-129
Chapter 6: Exploring Teacher’s Epistemic Beliefs and Emotions in Inquiry-Based Teaching of Mathematics (Inés M. Gómez-Chacón, Constantino De la Fuente)....Pages 131-157
Chapter 7: Mathematics Learning Experiences: The Practice of Happiness and the Happiness of Practice (Adi Wiezel, James A. Middleton, Amanda Jansen)....Pages 159-176
Chapter 8: Development of Modelling Routines and Its Relation to Identity Construction (Juhaina Awawdeh Shahbari, Michal Tabach, Einat Heyd-Metzuyanim)....Pages 177-199
Front Matter ....Pages 201-201
Commentary on Part III: Connections to Theory and Practice (Morten Blomhøj)....Pages 203-210
Commentary: Flow and Mathematical Modelling: Issues of Balance (Lyn D. English)....Pages 211-217
Chapter 9: The Complex Relationship Between Mathematical Modeling and Attitude Towards Mathematics (Pietro Di Martino)....Pages 219-234
Chapter 10: Teaching Modelling Problems and Its Effects on Students’ Engagement and Attitude Toward Mathematics (Zakieh Parhizgar, Peter Liljedahl)....Pages 235-256
Chapter 11: Affect and Mathematical Modeling Assessment: A Case Study on Engineering Students’ Experience of Challenge and Flow During a Compulsory Mathematical Modeling Task (Thomas Gjesteland, Pauline Vos)....Pages 257-272
Chapter 12: Flow and Modelling (Minnie Liu, Peter Liljedahl)....Pages 273-295
Chapter 13: A Coda on Affect (Bharath Sriraman)....Pages 297-301
Back Matter ....Pages 303-332

Citation preview

Advances in Mathematics Education

Scott A. Chamberlin Bharath Sriraman Editors

Affect in Mathematical Modeling

Advances in Mathematics Education Series Editors Gabriele Kaiser, University of Hamburg, Hamburg, Germany Bharath Sriraman, University of Montana, Missoula, MT, USA

Editorial Board Members Ubiratan d’Ambrosio (Sa˜o Paulo, Brazil) Jinfa Cai (Newark, NJ, USA) Helen Forgasz (Melbourne, VIC, Australia) Jeremy Kilpatrick (Athens, GA, USA) Christine Knipping (Bremen, Germany) Oh Nam Kwon (Seoul, Korea)

More information about this series at http://www.springer.com/series/8392

Scott A. Chamberlin • Bharath Sriraman Editors

Affect in Mathematical Modeling

Editors Scott A. Chamberlin School of Teacher Education University of Wyoming Laramie, WY, USA

Bharath Sriraman Department of Mathematical Sciences University of Montana Missoula, MT, USA

ISSN 1869-4918 ISSN 1869-4926 (electronic) Advances in Mathematics Education ISBN 978-3-030-04431-2 ISBN 978-3-030-04432-9 (eBook) https://doi.org/10.1007/978-3-030-04432-9 Library of Congress Control Number: 2019936163 © Springer Nature Switzerland AG 2019 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, express 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

Contents

Part I Commentary on Affect, Cognition and Metacognition in Mathematical Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jonei Cerqueira Barbosa Chapter 1: The Construct of Affect in Mathematical Modeling . . . . . . . Scott A. Chamberlin

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Chapter 2: Metacognition in Mathematical Modeling – An Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Katrin Vorhölter, Alexandra Krüger, and Lisa Wendt

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Chapter 3: Principles for Designing Research Settings to Study Spontaneous Metacognitive Activity . . . . . . . . . . . . . . . . . . . . . Marta T. Magiera and Judith S. Zawojewski

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Chapter 4: Engagement Structures and the Development of Mathematical Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lisa B. Warner and Roberta Y. Schorr

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Part II The What and the Why of Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . Alan H. Schoenfeld Engaging Students in Mathematical Modeling: Themes and Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Peter Kloosterman

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Chapter 5: Exploring a Conative Perspective on Mathematical Engagement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Gerald A. Goldin

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Contents

Chapter 6: Exploring Teacher’s Epistemic Beliefs and Emotions in Inquiry-Based Teaching of Mathematics . . . . . . . . . . . 131 Inés M. Gómez-Chacón and Constantino De la Fuente Chapter 7: Mathematics Learning Experiences: The Practice of Happiness and the Happiness of Practice . . . . . . . . . . . . 159 Adi Wiezel, James A. Middleton, and Amanda Jansen Chapter 8: Development of Modelling Routines and Its Relation to Identity Construction . . . . . . . . . . . . . . . . . . . . . . . . 177 Juhaina Awawdeh Shahbari, Michal Tabach, and Einat Heyd-Metzuyanim Part III Commentary on Part III: Connections to Theory and Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 Morten Blomhøj Commentary: Flow and Mathematical Modelling: Issues of Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 Lyn D. English Chapter 9: The Complex Relationship Between Mathematical Modeling and Attitude Towards Mathematics . . . . . . . . . . . . . . . . . . . . 219 Pietro Di Martino Chapter 10: Teaching Modelling Problems and Its Effects on Students’ Engagement and Attitude Toward Mathematics . . . . . . . . 235 Zakieh Parhizgar and Peter Liljedahl Chapter 11: Affect and Mathematical Modeling Assessment: A Case Study on Engineering Students’ Experience of Challenge and Flow During a Compulsory Mathematical Modeling Task . . . . . . . . 257 Thomas Gjesteland and Pauline Vos Chapter 12: Flow and Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 Minnie Liu and Peter Liljedahl Chapter 13: A Coda on Affect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 Bharath Sriraman References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

Part I

Commentary on Affect, Cognition and Metacognition in Mathematical Modelling Jonei Cerqueira Barbosa

Abstract In this chapter, I react to the texts by Chamberlin (2019); Vorhölter et al. (2019); Magiera and Zawojewski (2019); and Warner and Schorr (2019), which provide powerful insights to analyze mathematical modelling in terms of affect, cognition, and metacognition. Particularly, I use the sociocultural and sociocritical lens to discuss the ideas presented by the authors and to propose other questions. Keywords Affect · Cognition · Metacognition · Sociocultural perspective · Sociocritical perspective

Introduction The past decades have witnessed important theoretical insights on the practice of mathematical modelling in educational settings (Schukajlow et al. 2018). The community of researchers has given more attention to the analysis and theorising of what happens when mathematical modelling has been developed in educational environments. One of the focuses refers to students’ psychological processes, which I understand initially as dispositions for action, ways of thinking, and ways of communicating. In the case of the section one of this book Affect in Mathematical Modelling, the chapters by Chamberlin (2019); Vorhölter et al. (2019); Magiera and Zawojewski (2019); and Warner and Schorr (2019) address the concepts of affect, cognition, and metacognition. They present powerful ideas that provoke us into thinking about students’ doings in mathematical modelling activities. These concepts were originally formulated in the field of psychology, but they have been borrowed by those in the field of mathematics education. As Lerman (2010) underlines, our area has weak grammar, which means that it does not produce unambiguous empirical descriptions. Many conceptualizations compete in the fields, giving rise to what the sociologist J. C. Barbosa (*) Federal University of Bahia, Salvador, Brazil e-mail: [email protected] © Springer Nature Switzerland AG 2019 S. A. Chamberlin, B. Sriraman (eds.), Affect in Mathematical Modeling, Advances in Mathematics Education, https://doi.org/10.1007/978-3-030-04432-9_1

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B. Bernstein called horizontal structures of knowledge. As a result, I acknowledge that those descriptions presented by the authors of the book section have theoretical affiliations that compete in the field of mathematics education as well as others areas such as psychology and philosophy. The following commentary is more of a discussion based on my reflection of the texts than a summary. I cannot separate myself from the theoretical perspectives used in my research, which are based on the sociocultural theory (Barbosa 2010), the theory of codes by Basil Bernstein (Oliveira and Barbosa 2013), and Wittgenstein’s late philosophy (Souza and Barbosa 2014). Thus, what follows is a constructive dialogue with the authors on the concepts of affect, cognition, and metacognition and how they are used to analyse students’ doings in mathematical modelling practice. To help the discussion, I am going to present an illustrative episode of a classroom, which will be used as part of my arguments. I will clarify my perspective on mathematical modelling so that the reader might understand how it unfolds in the following discussion. Lastly, I will examine the notions of affect, cognition, and metacognition presented by the section authors.

A Classroom Episode The following episode was extracted from one of Teacher Marcelo’s first mathematical modelling-based lessons. It took place in a public school located in the Brazilian city of Salvador. Usually, mathematics lessons at the school followed this

Fig. 1 Bay of All Saints, where Salvador City and the island of Itaparica are located. (Source: https://www.google.com/maps)

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sequence: explanation, examples, and exercises. It is similar to what Alrø and Skovsmose (2002) called the tradition of school mathematics. Teacher Marcelo got interested in implementing mathematical modelling into his classes after taking part in an in-service programme. In the following, students are between 16 and 18 years old, and they enrolled in the third year of the secondary level. The teacher introduced the problem of building a bridge between the city of Salvador and Itaparica Island (see Fig. 1). In fact, the government was planning to build the bridge, and the issue had been discussed in the local press (Bochicchio 2017). After a short discussion about how the bridge will affect people’s lives, the teacher wrote the following question on the board: “What is the best location to build the bridge? And why?” The teacher allowed students to go online to seek information on the Internet (using smartphones connected to wireless). The question was open because assumptions and simplifications should be made, which might lead to different solutions. The students were organized into groups, and the teacher visited each group to follow students’ solutions. At a certain point, Teacher Marcelo approached the group formed by Ana, Lúcia, and Renato. [1] Renato: [2] Teacher: [3] Renato: [4] Lúcia: [5] Teacher: [6] Ana:

[7] Lúcia: [8] Teacher: [9] Ana: [10] Teacher: [11] Lúcia: [12] Renato: [13] Teacher: [14] Lucia: [15] Renato:

Teacher, we have no idea how to solve it. No idea? No! We were talking about what to find out the shortest distance [between the island and the mainland]. It is one possible answer. But the shortest distance is the best? There would be where the ferryboat is nowadays, but it is also very close to the port. . .and there is São Joaquim Market [the biggest outdoor market of the city]. You did not say any criterion. You want to know our opinion on the best location, is it? Yes, your opinion, but with some justification. I was reading here that the sea depth is a factor, depending on the technique to use. Great, as well as the distance between the mainland and the island. . .this is another factor. So, you want to know the cost. . .I mean the lowest cost? Teacher, how do you want indeed? Could you be more explicit? You who will decide. Let us then make a list of what influences the cost. It is becoming more difficult.

After that, the teacher left the group. Clearly, the students are struggling to deal with a realistic and complex situation. They do not understand how to solve the presented problem. Now I am going to use this episode to clarify my perspective on mathematical modelling in mathematics education and to build a dialogue with the chapter authors.

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Mathematical Modelling Before discussing the concepts of affect, cognition, and metacognition, I will clarify my understanding of mathematical modelling because it is going to unfold in the following discussion. In my view, mathematical modelling is a way to educate in which students are invited to question and investigate situations originally from daily life, sciences and/or vocational contexts through mathematics (Barbosa 2006). The episode above shows this concept in action: the situation comes from daily life. In fact, the population of the city and region were already discussing the building of the bridge and its environmental effects on the island. Therefore, the problem was not originally formulated in the context of school mathematics, but it was moved into classroom. It was an open problem for the students because there were not given clues about which mathematical strategies should be used. The context of school mathematics is evocative because the students easily recognize that the problem should be solved through school mathematics, though other arguments are allowed. This understanding of mathematical modelling has some similarities with the principles of model-eliciting activities developed by D. Lesh and colleagues, cited by Magiera and Zawojewski (2019), particularly regarding the reality principle. The so-called modelling cycles have been used as a theoretical model to describe the students’ actions, such as in Blum and Leiß (2006). It is well accepted that modelling involves those sets of actions described by the cycles, but they might not represent the sequencing developed by students (Barbosa 2006). Czocher (2014) shows that the cycles of modelling are highly idealized, artificial, and simplified, suggesting that the mathematical thinking involved is not sequential or quasiperiodic. The metacognitive activity itself links all supposed steps of the cycle. When the students of the aforementioned episode were discussing the criteria to be considered to decide the location of the bridge, they should have been given an idea about the solution. The student, Lúcia, said: “Let us then make a list of what influences the cost” [Turn 14]. This means that the solution must be in terms of the best cost. Therefore, I would consider the modelling phases pointed out by Vorhölter et al. (2019) more as actions involved in modelling than sequential steps. The separation between the mathematical world and the real world might be seen as problematic. For instance, suppose that the group of students mentioned in the episode are going to find the distance between any point belonging to the island border and any point belonging to the Salvador city border. Would they forget they are dealing with the problem introduced by the teacher that day? It seems that the line between mathematics and reality leads to the instrumental perspective on mathematics, as pointed out by Chamberlin (2019). As a consequence, mathematics might be seen as transparent (i.e., mathematics is a neutral instrument to describe the world itself). This idea has been challenged through the notion of mathematics in action discussed by Skovsmose (2011), who suggests that mathematics structures the empirical situations.

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In order to make this point clearer, I go to L. Wittgenstein (2001), who says that the mathematical propositions are normative (i.e., they provide standards, forms, and modes of conduct to organize our experiences). The mathematical propositions are never distorted by empirical experience. Look back to the group of students working on the bridge-building problem. They are invited to organize their experience with the issue through propositions that already pre-exist in mathematical grammar. The problem solution can only be produced from the norms provided by mathematics. Based on this idea, Souza and Barbosa (2014) have pointed out that mathematical modelling does not describe the reality but regulates it. Consequently, the fundamental educational task is to invite students to reflect on the ways mathematics regulates our experience. I have argued that a socio-critical perspective of mathematical modelling is in action when students engage in discussing the relationship between the assumptions and their resulting mathematical models as well as their uses in society (Barbosa 2010). This understanding might now be extended to include the discussion on how mathematical propositions shape the modelling process itself and its results. Since I have clarified my understanding on mathematical modelling, I am going to discuss students’ doings in mathematical modelling regarding the concepts of affect, cognition, and metacognition, seeking to build a dialogue with the authors of the section.

Affect Chamberlin (2019) provides a rich description of the conceptual field of affect, giving us an important map for new researches in the area. The author describes the concept regarding feelings, emotions, and beliefs; this is similar to one of the definitions discussed in Hannula et al. (2016), which is a synthesis on affect studies produced as a result of the 13th International Congress on Mathematical Education. Both Chamberlin (2019) and Hannula (2014) argue that the key issue is to understand the relationship between the affective components and the cognitive processes. In the classroom episode mentioned above, the student, Renato, showed impatience with a classroom situation based on a realistic problem. His discomfort or anxiety was explicit when he asked the teacher to give clear direction [Turn 12], leading to a state of apathy in relation to the resolution of the problem. A possible interpretation of Renato’s participation pattern in the episode might be understood in light of his previous experiences in the tradition of school mathematics or his attitudes towards mathematics built in that tradition. Warner and Schorr (2019) developed the concept of engagement structure, which refers to the affective dispositions present within the individual that become active according to the social circumstances. The authors mention the following structures: Check this out; I’m really into this; Get the job done; Stay out of trouble; Look how smart I am; Let me teach you; Do as I say or do as I want; Don’t let the group down. Warner and Schorr (2019) provide more details about each structure in their chapter.

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The empirical example shown by the authors illustrates how those constructs might be used to identify a shift in students’ forms of participation. As defined by the authors, the engagement structures belong to an individual’s internal instance and are activated by social circumstances. However, from a sociocultural perspective, the internal level is constituted from interactions with others. Therefore, the social context should not only be viewed as a kind of activator, but social settings are also seen as constituting individual dispositions (Lerman 2001). As a consequence, I would tend to re-interpret the engagement structures as forms of participation established in social contexts that were internalised by students as long as they have been taking part in pedagogical practices. Chamberlin (2019) sees the relations between affect, modelling, and cognition as reciprocal. The author presents and discusses a scheme in which the three components are interconnected, challenging the idea of affect as the primary cause. From this point of view, Renato’s impatience and apathy might be challenged in the practice of modelling since there is no cause-effect relationship between affect and cognition, but they are brought together. Also, Chamberlin (2019) sees affect, cognition, and mathematical modelling as processes, which, to my view, lead us to think of them as situated. As I suggested earlier, mathematical modelling is part of a pedagogical practice that we call school mathematics, and as such it provides what is more or less legitimate (or even not legitimate). The sociology of B. Bernstein (2000) talks about the concept of pedagogical practice as a code that regulates legitimate communication and establishes rules for acceptable meanings. Therefore, mathematical modelling seems to go beyond a process; it is a way of organizing the pedagogical practice, establishing rules for action in school mathematics. Referring back to the example in section 2 and considering the way the mathematical modelling environment was organized, someone might identify some rules: students are allowed to talk about a situation from daily life, control over students’ communication is weakened, and the students are required to make assumptions. The students (e.g., Renato) may not follow the rules, or they might struggle to address the rules, as Ana and Lúcia seem to do. This means that the practice of mathematical modelling provides positions, and those who participate in it react in different ways. These reactions might be seen as dispositions, attitudes, feelings, and so on. Therefore, it seems to be impossible to say that affective and cognitive processes are forms of dealing with social contexts, but I rather see them as shaped in contexts. When Renato asked for the teacher’s guidelines [Turn 12], perhaps he was acting similarly to other situations in school mathematics. His question would not be unusual in a context in which following examples provided by the teacher is the main rule. This illustrates what L. Wittgenstein (2001) called family resemblance. Renato acted by similarity. It leads us not to think of affective dispositions as internal structures that germinate in fertile conditions. On the contrary, they are shaped in social life; in such a way, I would emphasize the perspective that considers that feelings, emotions, beliefs, and dispositions are built and rebuilt in social practices.

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Cognition The concept of cognition is highly disputed in psychology, education, and mathematics education. Different foundational assumptions introduce different perspectives. For example, von Glaserfeld (2005) relates the concept to the process undertaken by the individual to organize their experiences; Lerman (2001), based on L. Vygotsky’s theory, sees cognition development as a result of internalization of social interactions; and Sfard (2007), based on the late philosophy of L. Wittgenstein (2001), sees cognition as communication. These three theoretical perspectives also suggest there are many ways of conceptualizing cognition in mathematical modelling. Chamberlin (2019) proposes to consider the categories of affect and cognition as inseparable. As I mentioned before, this challenges the idea of affect as a causal factor to cognitive processes, which seems to be a more dynamic way to describe students’ psychological processes. Earlier I argued seeing the affect being shaped in the positions made available by the pedagogical practice. Similarly, I am also going to understand cognition as shaped by rules of the pedagogical practice. From a sociocultural perspective, the students in the classroom episode discussed the best location to build a bridge between the city of Salvador and Itaparica Island. Over the years, they had been developed ways of thinking and speaking during the process of socializing in their (school) lives. To be more precise, according to Bernstein (2000), they learned legitimate forms of participation by taking part in social activities in school. However, social rules might change, as demonstrated in the episode above during which developing mathematical modelling introduced new rules. It implies a rearrangement of the legitimate ways of thinking and talking. In the classroom episode, some of the statements made by the students (e.g., asking the teacher for direction), can be interpreted in terms of following a traditional rule of school mathematics. The classroom episode shows that there is tension between different rules of communication between the colleagues and the teacher. Whereas students seem to ask for some direction, the teacher suggests they should make decisions. This suggests a broader idea of cognition that goes beyond the individual being. Magiera and Zawojewski (2019) suggest seeing individual cognition in terms of others’ regulation. In fact, regulating is an activity developed by all involved in pedagogical practices. It leads to the point made by Bernstein (2000): pedagogical practices address rules that set up the limits of legitimate communication. This understanding is in line with the perspective of thinking as situated in the social context. Also, we can go ahead and accept the formulation proposed by Sfard (2007) and Harré (2009): cognition itself is a communicative process. What we think is always related to a ruled social context. The only difference is in terms of communicating to yourself or to others. This understanding could lead us to review the construct of students’ modelling routes originally introduced by Borromeo Ferri (2006). The author states the concept of student’ modelling route as “the individual modelling process on an internal and external level” (p. 91). In light of seeing

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thinking as communicating, the internal level in modelling routes is not something beyond communication itself. Another import aspect is to consider modelling routes as relational to pedagogical contexts. Students’ strategies are not inseparable from the evocative contexts that address their rules; as a result, there is no separation between the students’ actions and the context.

Metacognition The concept of metacognition, generally speaking, refers to thinking about thinking and/or the monitoring of thinking. The subject has been part of the mathematics education agenda, particularly focusing on specific mathematical activities such as problem-solving (Schoenfeld 1992). Surely this tradition prompted us to research students’ metacognition when they are involved in producing mathematical models. Magiera and Zawojewski (2019) and Vorhölter et al. (2019) have drawn a detailed map on how the discussion has been developing in the field of mathematical modelling. The understandings presented in the chapters reflect the concept of metacognition by Flavell (1979), who highlights the role of monitoring and regulation on thinking. At this point, returning to the classroom episode mentioned earlier is useful. Consider the point in which the student, Lúcia, says that the problem question is to find the shortest distance between the island and the mainland. At that moment, would the student be only thinking about making the question clear? Would she be projecting the type of answer? Would she be starting to plan a strategy to approach the problem? Even though she did not anticipate a clear modelling route, Lúcia’s conversation is not isolated from the steps the student group is going to take. I am posing these questions to support the theoretical results presented by Vorhölter et al. (2019) about the difficulty of separating cognition and metacognition. I suggest that monitoring thinking and talking is always present in pedagogical practices. In the classroom episode, students do not forget that they are in a mathematics class in a school which has its own rules for legitimate communication. Bernstein (2000) introduced the notions of rules of recognition and rules of realization for respectively naming the regulation on what to say and how to say it in pedagogical contexts. Another construct developed by Bernstein (2000) was the rules of evaluation, which refers to the criteria for judging the legitimacy of the statements in pedagogical practice. We might analyse if the rules have stronger or weaker values, but they are operating in contexts. Magiera and Zawojewski’s (2019) chapter presents a sociocultural perspective on the concept of metacognition. Citing L. Vygotsky, the authors see metacognition as a process of internalization from the symbolic interactions in social contexts. Taking this into consideration, I would underline the inseparability among thought, metacognition, and pedagogical context (and its rules), which unfolds in the way people interact. Magiera and Zawojewski (2019) state that co-regulation or other-regulation

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takes place in solving problems collaboratively. The social organization of the classroom seems inviting for students to make forms of monitoring more visible. As a result, the educational task is that co-regulation and other-regulation should be visible parts of the pedagogical organization; in this case, the mathematical modelling. Magiera and Zawojewski (2019) present three principles to optimize the observation of metacognitive activity in modelling practices in school: organization of students for collaborative work, complex problems that require discussion, and working in groups. These characteristics for modelling practices seem to be useful for research purposes. However, they can also be seen as a way to give visibility to metacognitive activities in any mathematical modelling-based class. Metacognitive activities in modelling activities can refer to different scopes. For instance, from the socio-critical perspective (Barbosa 2010), one of the aims is to discuss how different assumptions made for the situation and mathematical descriptions generate different models and their possible uses in society. This refers to what I previously called reflective discussions (Barbosa 2010). For example, suppose that the student groups working on the bridge problem made different assumptions: one group considered the shortest distance between the mainland and the island; a second group assumed that the bridge should be built in the stretch of lesser depth; and, finally, a third group assumed that the starting point of the bridge in the city of Salvador should be a region of low economic development. Many of the arguments regarding the mathematical models and the answers differ, which could lead to a discussion about the close relationship between assumptions, the normative role of mathematics, and mathematical models and their uses in the societal debate. Perhaps these questions would not be valued in other approaches more focused on developing modelling competences or learning new mathematical contents. I therefore suggest that monitoring how to think and how to speak in the practice of mathematical modelling depends on the perspective in action. We can identify different purposes for the use of mathematical modelling in school (Kaiser and Sriraman 2006), which will put into operation different sets of rules. In this way, the students’ metacognitive activities cannot be seen as separated from the rules of the pedagogical practice in which modelling occurs.

Final Remarks The discussion about mathematical modelling in mathematics education in the light of the concepts of affect, cognition, and metacognition is fertile, as suggested by the theoretical and empirical results presented by the authors of the section one of the book Affect in Mathematical Modelling. The chapters are clear about their theoretical foundations, making them useful resources for the debate even if someone draws on different views. As noted at the beginning of the commentary, I have been working with social theories. In my task of discussing the chapters, I tried to challenge the conceptualization of affect, cognition, and metacognition in mathematical modelling in light of

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the social dimension. Instead of seeing the social context as a place to express affect, cognition, and metacognition, my main theoretical point is to see these psychological processes as inseparable from the context in which they occur. This does not imply that the rules of the pedagogical context determine how mathematical modelling and students’ psychological processes take place, but we can certainly say that contextual rules set limits on what students can think and speak. Furthermore, this does not mean that students, teachers, and other actors cannot challenge the rules altering their values. In addition, one question remains that needs further elaboration: How should the concepts of affect and metacognition be conceptualized if we assume there is no separation between internal and external instances? This understanding comes from the late writings of L. Wittgenstein (2011) and inspired Sfard’s (2007) commognitive approach. Harré (2009) says this Wittgensteinian view leads us to the second cognitive revolution, which is more radical than the first since it is abolishing the border between internal and external. This is an issue that the areas of mathematics education and mathematical modelling should put in their agendas.

References Alrø, H., & Skovsmose. (2002). Dialogue and learning in mathematics education: Intention, reflection, critique. Dordrecht: Kluwer. Barbosa, J. C. (2010). The students’ discussions in the modeling environment. In R. Lesh, P. L. Galbraith, C. R. Haines, & A. Hurford (Eds.), Modeling students’ mathematical modeling competencies (pp. 365–372). New York: Springer. Bernstein, B. (2000). Pedagogy, symbolic control and identity: Theory, research, critique (Rev ed.). London: Rowman & Littlefield. Blum, W., & Leiß, D. (2006). “Filling up” – The problem of independence-preserving teacher interventions in lessons with demanding modelling tasks. In Bosch, M. (Ed.), CERME-4 – Proceedings of the fourth conference of the European Society for Research in mathematics education, Guixol. Bochichhio, R. (2017, July 10). Ponte Salvador-Itaparica: edital sai em outubro. A Tarde. Retrieved from: http://atarde.uol.com.br/ Borromeo Ferri, R. (2006). Theoretical and empirical differentiations of phases in the modelling process. ZDM, 38(2), 86–95. Chamberlin, S. A. (2019). The construct of affect in mathematical modelling. In [to be completed]. Czocher, J. A. (2014). Towards building a theory of mathematical modelling. In P. Liljedahl, C. Nicol, S. Oesterle, & D. Allan (Eds.), Proceedings of the joint meeting of PME38 and PME-NA36 (Vol. 2, pp. 353–360). Vancouver: PME. Flavell, J. H. (1979). Metacognition and cognitive monitoring: A new area of cognitivedevelopmental inquiry. American Psychologist, 34(10), 906–911. Glasersfeld, E. (2005). Thirty years of radical constructivism. Constructivist Foundations, 1(1), 9–12. Hannula, M. S. (2014). Affect in mathematics education. In S. Lerman (Ed.), Encyclopedia of mathematics education. Dordrecht: Springer. Hannula, M. S., Di Martino, P., Pantziara, M., Zhang, Q., Morselli, F., Heyd-Metzuyanim, E., Lutovac, S., Kaasila, R., Middleton, J. A., Jansen, A., & Goldin, G. A. (2016). Attitudes, beliefs,

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motivation and identity in mathematics education: An overview of the field and future directions (ICME-13 Topical Surveys). Hamburg: SpringerOpen. Harré, R. (2009). The second cognitive revolution. In K. Leidlmair (Ed.), After cognitivism: A reassessment of cognitive science and philosophy (pp. 181–187). New York: Springer. Kaiser, G., & Sriraman, B. (2006). A global survey of international perspectives on modelling in mathematics education. ZDM, 38(3), 302–310. Lerman, S. (2001). Cultural, discursive psychology: A sociocultural approach to studying the teaching and learning of mathematics. Educational Studies in Mathematics, 46, 87–113. Lerman, S. (2010). Theories of mathematics education: Is plurality a problem? In B. Sriraman & L. English (Eds.), Theories of mathematics education: Seeking new frontiers (pp. 99–109). Heidelberg: Springer. Magiera, M. T., & Zawojewski, J. S. (2019). Designing research settings for the study of metacognitive activity: A case for small group mathematical modeling. In [to be completed]. Oliveira, A. M. P., & Barbosa, J. C. (2013). Mathematical modelling, mathematical content and tensions in discourses. In G. A. Stillman, G. Kaiser, W. Blum, & J. P. Brown (Eds.), Teaching mathematical modelling: Connecting to research and practice (pp. 67–76). Dordrecht: Springer. Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense-making in mathematics. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning. A project of the National Council of Teachers of Mathematics (pp. 334–370). New York: Macmillan. Schukajlow, S., Kaiser, G., & Stilman, G. (2018). Empirical research on teaching and learning of mathematical modelling: a survey on the current state-of-the-art. ZDM Mathematics Education. Advance online publication. https://doi.org/10.1007/s11858-018-0933-5. Sfard, A. (2007). Commognition: Thinking as communicating, the case of mathematics. New York: Cambridge University Press. Skovsmose, O. (2011). An invitation to critical mathematics education. Rotterdam: Sense Publishers. Souza, E. G., & Barbosa, J. C. (2014). Some implications of Wittgenstein’s idea of use for learning mathematics through mathematical modelling. International Journal for Research in Mathematics Education, 4, 114–138. Vorhölter, K., Krüger, A., & Wendt, L. (2019). Metacognition in mathematical modeling – An overview. In [to be completed]. Warner, L. B., & Schorr, R. Y. (2019). Exploring the connection between engagement structures and the development of mathematical ideas. In [to be completed]. Wittgenstein, L. (2001). Philosophical investigations (3rd ed.). Oxford: Blackwell.

Chapter 1: The Construct of Affect in Mathematical Modeling Scott A. Chamberlin

Abstract In this chapter, affect, mathematical modeling, and to a lesser degree cognition, are discussed in an attempt to provide readers with a fundamental understanding for the remainder of the book. Affect is described as a multifaceted construct that relates to beliefs, attitudes, and emotions (McLeod and Adams 1989). DeBellis and Goldin (2006) refer to meta-affect, which is affect about affect, though they later added the tetrahedral model, which included values. Affect, once considered a subset of cognition (Binet and Simon 1916) may now be considered a co-equal constituent with cognition. Mathematical modeling is considered a process or act, in which problem solvers seek to generate understanding of mathematical information through mathematizing in an iterative process. In this chapter, a model is provided in which the relationship between affect, cognition, and mathematical modeling is elucidated. Keywords Affect · Cognition · Mathematical modeling · Mathematical problem solving

The focus of this chapter is on explicating the relationship between affect in mathematics and mathematical modeling. The chapter is broken into three sections, with a priority on seminal research in the fields of (1) affect in mathematics, (2) mathematical modeling, and (3) their relationship. At the conclusion of the literature review, a model (found in Fig. 1) regarding the relationship between affect, modeling, and cognition is presented. This model is a theoretical one and some assumptions are made for its creation. The model was created based on empirical literature, but not per se on empirical data collected specifically to substantiate the model. The general field of affect is much older than many realize, dating to at least the mid-1700s (Smith 1759). Mathematical modeling gained notoriety circa the late 1800s with the work of Hertz (1894). Each field grew precipitously in the late 1960s S. A. Chamberlin (*) School of Teacher Education, University of Wyoming, Laramie, WY, USA e-mail: [email protected] © Springer Nature Switzerland AG 2019 S. A. Chamberlin, B. Sriraman (eds.), Affect in Mathematical Modeling, Advances in Mathematics Education, https://doi.org/10.1007/978-3-030-04432-9_2

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and early 1970s, as scholarly activity centered on the two foci and they thus became a concentration in the burgeoning field of mathematics education. Prior to the introduction of the discussion, it is critical to note three points. First, with respect to each focus (i.e., affect and mathematical modeling), each existed long before they were formally conceptualized in literature and the dates mentioned are the earliest known dates of well-recognized publications. Second, given the rather lengthy ascension of literature of affect in mathematics, the Smith publication may not formally be considered an area of mathematics education today. Instead, the Smith publication dealt formally with how people make moral judgments in the area of economics. Nevertheless, the publication is often credited with providing early theory about affective subconstructs such as interest, value, and emotions. Third, several notes about semantics may help readers interpret this chapter and the larger book. Affect is considered a noun and it relates to feelings, emotions, dispositions, and beliefs, in this case relative to mathematics (Anderson and Bourke 2000; McLeod 1994; McLeod and Adams 1989; Middleton and Spanias 1999). Mathematical modeling is considered an act or process and is therefore a verb. From the process of modeling, ideally mathematical models are created (Lesh and Zawojewski 2007) as modeling maintains a close relationship with mathematical problem solving. Cognition in the context of this model is generally considered a process (verb), though at times it could be considered a noun as well.

History of Affect It has become commonplace for educational researchers to establish their name among peers by identifying pre-existing concepts and constructs and coining new terms to describe them as Coleman (2006) suggested. Some individuals are sadly not aware that such psychological phenomena (e.g., affect) existed and had been formalized centuries ago. Affect was officially conceptualized nearly 200 years ago by Pinel. In the early 1800s, Pinel referred to, ‘Intellectual and affective factors’, as Binet and Simon did (1916), in relation to learning. This (affect) may be the construct that Binet discussed in 1916 when he referred to ‘intellectual manifestations’ as Pinel was regarded as one of Binet’s contemporaries. Almost simultaneously in 1829, Mill (1878) defined many components of the mind that were not identified as cognition per se, such as feelings, beliefs, anxiety, and attitude. One-hundred years later, Thurstone and Chave (1929) finalized a seminal book on attitudes, thus bringing attention to it among behavioral psychologists. Nevertheless, such non-cognitive constructs, as Messick (1979) referred to them, were not given much attention in mathematical psychology until the late 1960s. As an aside, Pinel, being a physician (formally trained as a psychiatrist) by trade, did have a high interest in mathematics, was a mathematics teacher, and therefore did matriculate many upper level mathematics courses (Pinel n.d.).

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Around the time of Binet and Simon (1916), Valentiner (1930) coined the term non-intellectual factors in relation to intelligence. More recently, the Educational Testing Service did refer to non-intellectual factors as early as 1951, regarding general intellect and not mathematics specifically. Throughout all of this theoretical writing, the construct of affect was difficult to disentangle from how individuals think. This is because much of the international emphasis in the early twentieth century pertained to the rather misunderstood construct called cognition (Jastrow 1901; Maher 1900). The sentiment among experts in the early twentieth century was that feelings, emotions, attitudes, dispositions, and beliefs were impossible to disentangle from the process of thought (cognition). They were thus considered somewhat distinct constructs, but affect may have been considered a subset of cognition. This rather ill-defined understanding of affect’s relation to cognition may not have helped (mathematical) psychologists make sense of the construct. Specific to mathematics, Feierabend (1960) may be credited with creating foci that led to subsequent research on affect in mathematics. Among her contributions were questions that provided direction regarding what areas to research in mathematics. In her article, she urged researchers to investigate: • If there was a prevailing negative attitude toward arithmetic and mathematics (p. 19) • What the relationship was between motivation and achievement (p. 19) • What the factors are underlying attitudes towards mathematics (p. 20) Within the decade, the School Mathematics Study Group (SMSG) convened the National Longitudinal Study of Mathematical Abilities (Higgins 1970; Romberg and Wilson 1969) to investigate such matters. The NLSMA may be considered the most comprehensive work of its era and provided significant direction for the nascent field of mathematical psychology. Specifically, NLSMA reports 4–7, 20, and 33 pertain to student affective factors as they were measured and as they relate to achievement. In this sense, the formal construct-domains of affect and mathematics had officially been conceived by the late 1960s and from there, the volume of literature grew almost exponentially with much of the work coming immediately after the NLSMA efforts. Hence, the work by the SMSG was instrumental in promulgating additional research in the field of affect in mathematics (Chamberlin 2010). Many studies proceeded the work of the SMSG, including a vast array of studies in which researchers investigated anxiety (Richardson and Suinn 1972; Suinn 1970) and attitude in mathematics (Aiken 1972, 1974). Fennema and Sherman’s work (1976) is often cited as seminal in the area of affect in mathematics because they assessed multiple subconstructs of affect with a single instrument. Prior to this date, instruments were typically created to assess one subcomponent of affect (Chamberlin). The Fennema-Sherman instrument was later revised by Lim and Chapman (2012). As additional research efforts were invested in investigating affect in mathematics, the field questioned what, precisely, constituted affect. The answer to this question may reside in a publication outside of the domain of mathematics. By 1982, then updated in the year 2000 with a second edition, Anderson and Bourke had clarified

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what affect was, at least in general educational settings, with their work entitled, Assessing affective characteristics in the schools. In being able to advertise that they could accurately assess affective characteristics in schools, they were tasked with conceptually establishing several foundational understandings such as (1) identifying effective approaches to assessing psychological constructs, including instrumentation, (2) providing a rationale for why affect should be assessed in schools, (3) ascertaining the specific subconstructs that constitute affect. It is the third piece of these foundational understandings that likely provided great direction to the domain of mathematical psychology. According to Anderson and Bourke, two tenets are critical. First, affect is synonymous with motivation. Second, affect is comprised of the following eight subconstructs: anxiety, aspiration, attitude, interest, locus of control, self-efficacy, self-esteem, and value. In the next section, additional conceptions of what affect in mathematics is will be presented. Similarly, in 1982, Hoyles sought empirical data to clarify whether hypotheses about negative affect in mathematics were apparent. According to Hoyles, high school students hold very emotional, predominately negative, responses to mathematics. In subsequent years, multiple researchers helped quantify students’ value of mathematics through monitoring and investigating course matriculation (e.g., Betz and Hackett 1983; Reyes 1984). Subsequently, an important step in the discussion of affect transpired in the late 1990s/early 2000s. During this time, the field realized a change from a normative to an interpretive perspective on the discussion of affect, regarding the approach to measurement and individual impact on mathematical problem solvers. In so doing, the focus in affect transferred from a process of explaining it to one of interpreting student affect. Practically speaking, this meant that researchers focused their efforts on understanding why individuals’ affective states were as they were through interpretation, rather than using general (or normative) guidelines to causally explain affective states (Hannula et al. 2004).

Multiple Conceptions of Affect in Mathematics A recurring issue with the psychological construct of affect is that it seems the longer experts discuss what, precisely it is, the more convoluted the conception of it becomes. Nevertheless, some clarification exists in theoretical writings by experts in mathematics education. Of particular note, are early writings by McLeod and colleagues (1989, 1992, 1994). In 1989, McLeod and Adams published a formative book that helped provide a foundation for researchers in mathematics education. The book, entitled Affect and mathematical problem solving, had a very similar focus and title to this book, Affect in mathematical modeling. Prior to this publication, the construct of affect in relation to mathematics was somewhat amorphous. That is to say, several of the aforementioned researchers had conducted investigations on various concepts of affect, but it may not have been formalized in one central publication quite as well as it was in the

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McLeod and Adams’ book. Hannula et al. (2004) referred to it as a ‘pioneer book’ (p. 132). Foci in the book included, the role of affect in problem solving, the conception of what constituted affect in mathematics, affect and learning, and affect and metacognition, to name a few. Subsequently, McLeod contributed to the International handbook of research on mathematics education, edited by Grouws in 1992. In this chapter, he again relied heavily on Mandler’s 1984 general publication about affect and emotions in the world of psychology, not the domain of mathematics per se. Throughout all of his work, he was steadfast in interpreting research in the general field of psychology and personalizing it to mathematics (education). In this 1992 publication, McLeod was famous for attributing actual constructs of affect to mathematics education including beliefs, attitudes, and emotions. Many scholars today, some 25 years later, still refer to affect in mathematics as comprised of beliefs, attitudes, and emotions, though some scholars reference a more comprehensive spectrum of affective components than the ones McLeod did. In addition, McLeod discussed the relationship between affect and cognition. These efforts may have been instrumental in helping affect become prominent among mathematics educators as its own field of inquiry, rather than being considered a subset of cognition, as has been postulated. In so doing, McLeod provided badly needed structure to affect and facilitated understanding among researchers for future work. In 1994, McLeod provided another major contribution to JRME in which he reviewed research on affect in mathematics from 1970 to 1994. This publication was instrumental in several respects. First, it identified the informal start date of research on affect in mathematics, circa 1970. Second, it provided a thorough overview of literature that researchers could utilize for several years. In essence, the publication afforded neophyte scholars the opportunity to access most research contributions relevant to affect and mathematics in one spot. A prospective shortcoming of the publication was that the research discussed was only that contained within JRME to date, not publications in other outlets. In the late 1990s, Ma published two meta-analyses on affect and mathematics achievement. These publications were instrumental in providing lucidity in interpreting the importance of affect, relative to achievement in mathematics. The first study was published in 1997 and in it Ma and Kishor explicated the effect of attitude in relation to achievement. After limiting studies to 107, an effect size of .12 was realized. Individual categories were also analyzed and they were gender, grade, ethnicity, sample selection, sample size, and date of publication with very similar results (minimally significant if at all). This study put to rest the notion that attitude can influence mathematics achievement. Though the finding(s) was/were statistically significant, the practical significance of attitude could be debated. Moreover, the results may have been negatively influenced by a shortsighted view of what constituted attitude since all of the studies preceded 1997. Subsequently (1999), Ma conducted a similar investigation on anxiety in relation to mathematics achievement. In this meta-analysis, Ma utilized 26 studies and found a correlation of .27 between anxiety and achievement. Though a negative relationship, as expected between anxiety and achievement, the .27 effect size was stronger than the rather modest .12. Practically speaking, anxiety correlates negatively, though nominally so, with

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achievement. In short, this means that low anxiety is better than high anxiety for high achievement and conversely high anxiety often correlates with low achievement. The results of the studies were certainly of importance to the field of mathematics education. Nearly as important was the collection of the corpus of literature in one study. Incidentally, as with the 1997 study, Ma investigated the relationship of anxiety and achievement in mathematics using several demographic variables, including gender, grade, and ethnicity and found each to be of no significant interaction with achievement in mathematics. It is important to note that Ma’s studies were correlational and therefore causation cannot be assumed. Within the decade, Middleton and Spanias provided another lens on affect in mathematics as they reviewed motivation and its effect on teaching and learning in mathematics. This publication was considered by many to be the most comprehensive discussion of motivation in mathematics at that time. Though not perhaps an original intent, the Middleton and Spanias publication brought to the fore the question of, “What precisely is the relationship between affect and motivation?” Within the year, Anderson and Bourke (2000), stated that motivation is the sum total of affect, though this statement came from the general field of educational psychology, and not specifically mathematical psychology. Malmivuori’s (2001) dissertation promulgated an emerging career in which she investigated affect in relation to such constructs as self-regulation (2006). McLeod later called Malmivuori’s dissertation far superior to his 1992 chapter in the Handbook of Research on Mathematics Education (Center for Research in Mathematics and Science Education n.d.). This comment was rendered likely for two reasons. First, her dissertation is considered by many to be widely influential and a model for high quality works. Second, when Malmivuori completed her dissertation, she had access to an additional 10 years of publications (both empirical and theoretical) on which to base her claims. Still, her analysis of literature prior to her dissertation was conducted at a very deep level, hence, McLeod’s comments on the high quality of her dissertation. The following year, 2002, Leder, Pehkonen, and Törner edited a book in which a comprehensive overview of affect was provided. They termed beliefs, what McLeod referred to as but one of three legs of affect, a ‘hidden variable’ in mathematics education. Various discussions were provided by researchers in this book relevant to topics such as a conceptualization of what beliefs are in mathematics, teacher beliefs and their relation to learning, student beliefs and their relation to learning, and beliefs in relation to other components of affect. Of great importance to this book was Goldin’s discussion of beliefs in relation to affect and meta-affect. In so discussing meta-affect, he referred to it as ‘affect about affect’ or a cognizance of emotions, beliefs, and attitudes. Interestingly, Goldin stated that affect is often downplayed in significance in mathematics because society believes it to be a purely intellectual endeavor and therefore fully devoid of an emotional component. The Goldin contribution was a theory developed from over a decade of work with colleague Valerie DeBellis (1991, 1993, 1997, 1999). Also of note is their tetrahedral model (Debellis and Goldin 2006) in which it is suggested that affect is comprised of four separate, but intricately intertwined categories. The category that DeBellis and Goldin added was values, which subsumed ethics, and morals, and were in addition to McLeod’s

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pre-existing three categories of affect, which are emotions, attitudes, and beliefs. The tetrahedral model provided a new perspective on affect as comprised of four components, rather than the previously agreed upon three. More recently, the 2009 publication by Maaß and Schlöglmann contains empirical results relevant to affect in mathematics. This book had an emphasis on teachers and their practical effect on classroom learning. As an example, Liljedahl’s chapter on teacher insights about the relationship between beliefs and practice and Sivunen and Pehkonen’s chapter on elementary teachers’ conceptions on problem solving are particularly salient in understanding the importance of beliefs and attitudes in mathematics.

The Conception of Modeling in Mathematics Modeling may be considered the process of creating mathematical models to explain and understand phenomena and concepts outside of mathematics in mathematical terms. Quarteroni calls mathematical modeling, “The third pillar of science and engineering, achieving the fulfillment of the two more traditional disciplines, theoretical and experimental.” (2009, p. 10). To generate mathematical models, the process of modeling must be engaged. The purpose of creating mathematical models is varied. As Lawson and Marion (2008) suggest, mathematical modeling may be used to create understanding of science (and mathematics), assess effects of change in systems, and facilitate decision-making. Lesh and colleagues (2000) stated that problem solvers create mathematical models to, “. . .reveal how they are interpreting mathematical situations that they encounter by disclosing how these situations are being mathematized (e.g., quantified, organized, coordinatized, dimensionalized) or interpreted (p. 593). Several points from this conception are important to note. Specifically, endemic to mathematical modeling are the processes of interpreting and mathematizing. First, to create mathematical models, some degree of interpretation must occur. That is to say, to create successfully a mathematical model, problem solvers must analyze some mathematical information (e.g., data), interpret the information, and then create a mathematical model to make sense of the information. Second, the process of mathematizing is instrumental. Mathematizing occurs when problem solvers analyze everyday information that may not ostensibly be mathematical and they make it mathematical (Presmeg 2003; Van Den HeuvelPanhuizen 2003). Treffers (1987) substantiates this point when he referred to mathematizing as, “Transferring a problem field into a mathematical problem” (p. 247). An example of mathematizing information to create a mathematical model may be defining and then quantifying factors. For instance, when most people go to a grocery store or market, an objective is to collect all desired products and then pay in as expeditious manner as possible. Though many people do not view this episode as an opportunistic one to create a mathematical model, listed below is a sample of factors that may lead to success in exiting the grocery store as quickly as possible. For instance:

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1. cashier speed in scanning items and bagging them, 2. amount of customer produce in basket as produce may require a special code that needs to be entered into the register manually while products with a Universal Product Code (UPC) are quickly scanned for information, 3. consumer form of payment (e.g., electronic forms of payment are nearly always quicker than checks or cash), 4. total number of items in the basket/cart, 5. speed of consumer in delivering items to the cashier It is important to reiterate that mathematical models may be used to understand mathematical situations and there are myriad situations in which a model could be created, but the general public may not realize a need for one. As an example, some may find it meaningless to create a mathematical model to quantify efficiency in shopping, but business experts may not see this example as trivial as (their) future success may depend on such models. Moreover, the need for the mathematical model may be of primary importance to certain constituencies, while not necessarily noticed by other parties, to which the model will perhaps not be utilized. Embedded in this example of creating a mathematical model is the process of interpreting information in the form of (1) identifying relevant factors that may affect speed in the cashier line, (2) mathematizing the factors through a process such as weighing the variables, (3) creating a mathematical model that is generalizable to as many shopping situations as possible, and (4) testing the model for efficiency. Two points are worth expanding here. That is, mathematizing occurs when problem solvers analyze information that has not been made mathematical and make it mathematical. It thus is easily considered in creating the final mathematical model. Second, generalizability in creating mathematical models is instrumental in assessing the value of the mathematical model. That is to say, if a model has greatly limited generalizability, then it has little applicability, with the exception of the current use of it. The best models, therefore, are often the most easily interpreted and the most generalizable to other contexts and domains. In addition, another process inherent in mathematical modeling is the creation of multiple iterations (Chamberlin 2008; Lawson and Marion 2008) of models in an attempt to create a highly refined mathematical model. This is the case because the first model that is created to respond to a situation is not always the most sophisticated one. The world of engineering and the design process embedded in it is perhaps most representative of this concept. As an example, the first cell phone did not resemble current cell phones at all. Initial prototypes of what is now known as a cell phone were far heavier, slower, and more cumbersome than today’s phones. Moreover, given few cell phone towers and an inability to pick up a signal in many cases, calls were often dropped or never even initiated. Today’s cell phones, several iterations after the first version in 1983 are lighter (Art Institute 2017), have far better reception, can connect to the internet wirelessly, and can execute hundreds of functions (often simultaneously) in relation to early prototypes. Similarly, the creation of mathematical models, whether created by a grade two student or a graduate student in mechanical engineering, require a process of creating

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a prototype (or first iteration) and then undergoing the creation of three to four additional models before the model can be considered a highly sophisticated one. Refinements, it may be postulated, are critical to highly efficient mathematical models, but are often quicker and easier alterations to the initial prototype. Throughout the creation of models, the influence of beliefs, attitudes, and emotions cannot be overstated.

Affect in Mathematical Modeling Thus far, a foundation in relation to extant literature has been provided regarding what the construct of affect is, what affect in mathematics is, and what mathematical modeling is. In this section, the three foci are combined to formulate one theory, presented in Fig. 1. In this section, the description of this theory is elucidated so that readers will have a basis on which to interpret discussion in the remaining chapters. The caveat with the theory provided is that it is simply that, theory. Theories, by definition, are unproven suppositions based roughly on previous evidence. In this case, the evidence is the empirical studies so explicated in literature. In Fig. 1, affect and mathematical modeling are larger circles because they represent the focus of this book. Cognition is a central component that links the two constructs in the field of mathematical psychology. It is important to note that the effect of each component is postulated to be a bi-directional relationship. That is to say, at any given time, and it may be theorized that these loops are perpetual. As importantly, affect influences (and the term affect as a verb is purposefully avoided in this chapter in lieu of influence) cognition, while cognition influences affect. Similarly, cognition influences modeling in mathematics, while the process of modeling (creating models as it was described earlier) influences the process of cognition. Finally, modeling influences affect, and vice versa, though it is significant to notice that the feedback loop that represents the interaction between modeling and affect influences cognition as well. To reiterate, all influences are predicated on feedback, which may come consciously or subconsciously. In the remaining

Cognition

Affect

Fig. 1 Theory of affect in mathematical modeling

Modelling

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sections, individual relationships will be explained, supplemented liberally with examples to support the theory. The Interrelationship of Affect and Cognition. As discussed, the relationship between affect and cognition is challenging to distinguish because the two are so intricately intertwined (Goldin 2017; Tuohilampi 2017). The two constructs are very much symbiotic in the respect that cognition directly influences and needs affect, but simultaneously, affect does not exist without cognition. In fact, early theorists and researchers in the field of educational psychology rarely mentioned affect without mention of cognition (Binet and Simon 1916; Bloom 1956). Affect certainly influences cognition in that one’s feelings, emotions, dispositions, attitude, and beliefs play an integral role in how one thinks. As an example, consider a problem solver that is engaged in a mathematical problem-solving task. Solving the problem naturally requires cognition. If the problem solver was involved in a tragic incident, such as a car accident the previous night, the influence of the incident most assuredly weighs on one’s ability to engage in cognition, without considerable distractions. Interestingly, a common misconception may be that only negative affect can serve as a distraction. However, inordinately positive affect can similarly serve as a distraction. If, for instance, an individual solving a mathematics problem had just won millions of dollars in a lottery, distraction or disengagement may also exist. When affective states detract from one’s ability to engage and concentrate, affect negatively influences cognition. Cognition also influences affect in that the degree of success that one is having and the level of engagement can often directly influence how effectively one is involved in cognition. The mechanism that apprises problem solvers of their progress, or lack thereof, is self-regulation. In some instances, individuals will receive feedback that adequate success is not being reached and thus be increasingly motivated, through persistence, to solve the problem. In any event, be it positive or negative feedback, cognition influences affect and affect directly influences cognition. The Relationship Between Cognition and Modeling. Similarly, the relationship between cognition and modeling is bi-directional and perpetual. That is to say, active engagement in thinking (cognition) is directly influenced by the modeling process. In fact, at times cognition and modeling are almost indistinguishable because the very notion of success in creating the first mathematical model for a problem and then refining it through the creation of subsequent iterations is contingent upon success in thinking. As an aside, one’s ability to mitigate the effect of interference when thinking (De Visscher and Noël 2014) can facilitate success in cognition, and ultimately success in mathematical model creation. This example speaks of the significance of thinking relative to mathematical modeling. However, it can be postulated that there is an influence of mathematical modeling on cognition. This is the case because the very demands of the individual problem may relate to the quality, level, and depth of cognition. It may be cliché to refer to background

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knowledge when considering cognition. However, it is most challenging to create a mathematical model without information that problem solvers have previously learned subconsciously entering into the formulation of the model. Though not formally studied, it can thus simply be hypothesized that if two groups of problem solvers each completed a modeling problem, one group with extensive background knowledge on a topic and the other group without such background knowledge, the first group may likely produce the more comprehensive model of the two groups. The exception to this rule may happen when the group with the more extensive background knowledge tries to over-formalize the model with pre-existing algorithms or formulae that are not relevant to the problem. The Relationship Between Affect and Modeling (and Cognition). Given the close relationship between cognition and modeling, and considering the proposed model (Fig. 1), affect influences modeling and modeling influences affect. One’s feelings, emotions, and dispositions could influence the type of model created. This is the case given the nature of creating multiple iterations to identify the most refined model. The subconstruct called persistence, not necessarily agreed upon by mathematical psychologists as a subconstruct of affect, is requisite for problem solvers to find success in identifying the most refined model. Persistence arguably has connections to value, interest, and locus of control. In fact, many theoretical mathematicians work on one problem for as long as a decade (personal communication with Zhuang Niu on 6 May, 2017). In viewing the relationship of modeling to affect, it also exists and suggests a bi-directional relationship in the two components. It may be safe to assume that mathematical modeling influences affect given the success one, or a group of problem solvers, attains in approximating an acceptable or highly refined product. As an example, if a group of problem solvers has worked on the creation of a mathematical model for a lengthy time, relative to the time estimated to identify a comprehensive model, the group may feel that identifying an acceptable model is impossible. Hence, the feedback loop suggesting that success in the creation of a mathematical model is not forthcoming may serve to demotivate the group to persist. In this sense, success in mathematical modeling is directly related to affect. The counter situation may likely be true as well. If, for instance, a group is achieving a high degree of success in finalizing a mathematical model, then the group may be highly motivated to persist with the creation.

Conclusion In this chapter, a review of affect, mathematical modeling, and cognition was provided. However, the theory presented was rather concise and will be supplemented by experts in this book. Given the importance of mathematical modeling and the applications to all levels of mathematics, providing the field of mathematical psychology with insight regarding affect during mathematical

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modeling is invaluable. Moreover, such insight might help ascertain the process that aspiring mathematicians use to facilitate understanding of mathematical principles and understanding affect in mathematical modeling episodes may have a direct effect on cognition.

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Chapter 2: Metacognition in Mathematical Modeling – An Overview Katrin Vorhölter, Alexandra Krüger, and Lisa Wendt

Abstract The importance of metacognition in modeling processes is not questioned in the international discussion on modeling. However, in contrast to the assumed importance, a relatively small number of studies were conducted focusing metacognition in modeling. One of the reasons may be the fact that metacognition is a rather vague concept developed in different domains and with different conceptualizations. Another reason could be challenges in measuring metacognition. In this article, we will first present concepts of metacognition, which are used by the studies presented afterwards. In addition, we will describe different methods and instruments for measuring students’ metacognition concerning modeling as well as important research results. We will close with a summary and open research questions as well as a description of a study that tries to tackle some of these open questions. Keywords Metacognition · Metacognitive knowledge · Metacognitive strategies · Mathematical modeling · Modeling competencies

Metacognition is a field of interest in different domains, reaching from the psychological perspective to educational sciences. For a long time, it was not clear, if there is an overall metacognition for different domains, or if metacognition in different domains is of a different nature. Veenman (2011) summarizes, that novices or younger learners develop their metacognitive skills and knowledge in different domains and task-related; later, metacognitive skills become increasingly overarching. In the international discussion on mathematical modeling, the importance of metacognition for modeling processes is widely accepted (Blum 2011). Research on mathematical modeling processes repeatedly reveals the possibility to overcome cognitive barriers while solving modeling problems by using metacognition (Stillman 2011). However, research concerning metacognitive modeling competencies is still at its beginning. K. Vorhölter (*) · A. Krüger · L. Wendt University of Hamburg, Hamburg, Germany e-mail: [email protected]; [email protected]; [email protected] © Springer Nature Switzerland AG 2019 S. A. Chamberlin, B. Sriraman (eds.), Affect in Mathematical Modeling, Advances in Mathematics Education, https://doi.org/10.1007/978-3-030-04432-9_3

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Concerning the fact, that metacognition has been raised as a topic of research in different domains with different aims and origins, the definition of metacognition is not explicit and therefore, often described as fuzzy. In the first part of this paper, we contrast different types of conceptualizations with respect to their aims and conclude metacognitive knowledge and skills for the modeling process. Afterwards, we specify metacognition necessary for modeling processes by means of a selected modeling problem. In the second part of the paper, methods for measuring different facets of metacognition used in the field of mathematical modeling are summarized. It becomes clear that there is not a single method for measuring students’ metacognitive modeling competencies, but several, that all have weaknesses and strengths. The third part of this paper aims at presenting empirical results about metacognition in modeling processes. Research results referring to the importance of metacognition in modeling processes and the influence of metacognition in students’ modeling processes are summarized. Furthermore, possibilities for promoting students’ metacognition as well as teachers’ requirements for fostering students’ metacognition are presented. The paper closes with research questions that still need to be answered in the future and the presentation of an ongoing study aiming at answering some of the open research questions mentioned before.

Conceptualizations of Metacognition in Research on Mathematical Modeling In recent years, the relevance of metacognition has increased in the context of mathematical modeling. According to Kaiser (2007), metacognition is part of global modeling competencies. Thus, it is quite important to involve metacognitive activities during modeling processes in order to support modeling competencies. Concerning this, Blum (2011) underlines: “There are many indications that metacognitive activities are not only helpful but even necessary for the development of modeling competency” (Blum 2011, p. 22). Yet, there is still no consistent definition of metacognition (e.g. Veenman et al. 2006). There is a huge variation of definitions of the concept of metacognition as it reaches from the knowledge about one’s own thinking to the field of self-regulated learning in problem solving processes (Schoenfeld 1992). The terms metacognition and self-regulated learning are often used synonymously, although they derive from different conceptual roots and theoretical perspectives. Whereas metacognition initially focused on people’s thinking about cognition, self-regulated learning strongly emphasizes the regulation of learning processes and learning outcomes. Nowadays, research on metacognition considers self-regulation as a part of metacognition, whereas researchers in the field of self-regulation consider metacognition as a part of self-regulation. Not only the distinction between metacognition and self-regulation, but also the distinction between metacognition and cognition is often criticized and named as

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vague and arbitrary (Veenman et al. 2006). Usually, metacognitive processes are seen as higher-order processes monitoring and regulating lower-order cognitive processes (Veenman 2011), for example, Flavell suggests “cognitive strategies are invoked to make cognitive progress, metacognitive strategies to monitor it.” (Flavell 1993, p. 154). Due to the different origins, theoretical conceptualizations of metacognition differ a lot. Thus, in the following section we will illuminate different leading concepts of metacognition to give an overview.

The Concept of Metacognition One of the first definitions of metacognition can be traced back to Flavell (1976): ‘Metacognition’ refers to one’s knowledge concerning one’s own cognitive processes and products or anything related to them (. . .). Metacognition refers, among other things, to the active monitoring and consequent regulation and orchestration of these processes in relation to the cognitive objects on which they bear, usually in the service of some concrete goal or objective (Flavell 1976, p. 232).

In the taxonomy developed by Flavell and Wellman (1977), the term metamemory was coined. It is characterized as (a) memory relevant characteristics of the individual itself, (b) memory relevant characteristics of the task, and (c) potential employable strategies. This aspect of metacognition is often called declarative metaknowledge. It is differentiated in firstly, the knowledge about a person’s variables, which contains the knowledge about certain personal characteristics, regardless if these characteristics are temporary or permanent. Moreover, the person’s variables can relate to the specific person as well as to others. Secondly, the knowledge about task variables describes any relevant task characteristics during task processing. In this case, the type of a task can exert influence how to work on it. At last, the knowledge about strategic variables contains certain experiences and recognitions, which were undergone using certain learning strategies. Therefore, the knowledge about strategic variables requires the activation of a person’s knowledge and task variables. Furthermore, the taxonomy of Flavell and Wellman (1977) is completed by the aspect of sensitivity. The authors are convinced that it is not enough to only know certain variables but rather develop a feeling for using certain strategic activities during the work on problem-solving: “It goes without saying that, like all of us, the young child is constantly learning and recalling things incidentally, i.e., without any deliberate intention to learn or recall.” (Flavell and Wellman 1977, p. 7). This is what the students shall get during task processes: They shall activate those strategies intuitively. Moreover, Flavell (1979) established the classification in (a) metacognitive knowledge (described before as meta-memory), (b) metacognitive experiences, (c) goals (or tasks), and (d) actions (or strategies). Another important concept of metacognition is the perception of Brown. She follows Favell’s concept by combining the understanding of knowledge with

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metacognition (Brown et al. 1983) referring to the psychological perspective on metacognition and focusing on the differentiation between knowledge about cognition and controlling as well as the regulation of cognition. Knowledge about cognition is described as something that is stable and one can talk about, although it should be considered that it is incorrect in many cases. On the other hand, the regulation of cognition is something that is instable and an executive controlling aspect of the metacognition. Furthermore, Brown classifies the regulation of cognition in three different processes: Planning activities, monitoring activities during learning and evaluating results. Planning strategies for example are (1) to forecast results, (2) to design strategies, (3) to play through different possibilities during the process. Brown describes monitoring activities as those, which students monitor, prove, or by which they change their learning strategies (if necessary). The evaluation of results includes an examination of whether the chosen strategies were effective and efficient in order to solve the problem or not. Brown observes that it is quite hard to differentiate between cognition and metacognition and further, that there are different traditions metacognition emerged from so that it is sometimes not quite clear, which one is referred to (Brown 1984). Finally, there is the concept presented by Sjuts (2003) who divides metacognition into a procedural, declarative, and a motivational component of metacognition. The procedural metacognition contains the functions of planning, monitoring, and proving (i.e. that you should imagine yourself looking over your own shoulder and checking your actions). These take place before, during, and after task processing. The declarative component of metacognition comprises three types of knowledge about cognition: Firstly, diagnostic knowledge, which is about one’s own and also about other persons’ thinking; secondly, evaluating knowledge about tasks and requirements; thirdly, strategic knowledge about solution plans and their chances of success. The last component of metacognition, the motivational aspect, says that motivation and the power of volition are necessary for the use of metacognitive strategies. These conditions must either be already given or need to be developed (Sjuts 2003). In comparison to other concepts, Sjuts extends the definition of metacognition by this component. In most concepts of metacognition, scholars differentiate between cognition and regulation of cognition (Brown 1987; Schraw and Dennison 1994) or also refer to metacognitive knowledge and the usage of metacognitive strategies or metacognitive skills. Knowledge of cognition comprises declarative knowledge about the process in general, procedural knowledge about the execution of cognitive strategies as well as conditional knowledge on how strategies can be implemented. This means, metacognitive knowledge comprises knowledge about what kind of metacognitive activities exist and further, knowledge about how metacognitive strategies can be implemented as well as the knowledge why and when those strategies can be used (Schraw and Moshman 1995). Moreover, regulation of cognition implies comprehension using metacognitive strategies, which help students to control and regulate their activities. At least, three different components of metacognitive strategies can be distinguished: planning, monitoring (and regulating), and evaluating.

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In the following, we would like to present, what is associated with the declarative and procedural component of metacognition in detail.

Metacognition in Modeling Processes Hattie et al. (1996) point out that metacognitive competencies become increasingly important in connection with the growing complexity of a task to be solved. Modeling tasks are complex problems, because neither the mathematics to be used nor the adequate understanding of the real-world situation is provided for the students. Further difficulties are created by the nature of the modeling problems, which stem from students’ actual or future everyday lives, from their environments or from sciences. Thus, metacognition is judged as increasingly important for solving modeling problems (e.g. Maaß 2006; Vorhölter and Kaiser 2016). Referring to the general concept of metacognition, metacognitive modeling competencies can be divided into declarative meta-knowledge and procedural metacognitive strategies. Declarative meta-knowledge contains (among others): • knowledge about the characteristics of a modeling problem, such as: requirement for developing an individual approach requirement for investigating the data as input or about making adequate assumptions requirement for using context knowledge requirement for developing a simple but adequate model. • knowledge about useful strategies for solving modeling problems knowledge about (heuristic) strategies knowledge about the different steps of a modeling cycle (so that it can be used as a metacognitive tool) • knowledge about the capabilities of oneself and other individuals involved knowledge about the mathematical competencies of oneself or other individuals involved information about the work specific preferences Metacognitive strategies for working on a modeling problem can be differentiated into: • strategies for planning the solution process considering the task that has to be worked on, the involved persons, specific circumstances

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• strategies for monitoring and, if necessary, regulating the working process, which can for example be done by using the modeling cycle as a tool, applying strategies systemically and goal-orientated realizing cognitive barriers • strategies for evaluating the modeling process in order to improve the modeling process An illustration of how these metacognitive aspects appear in the process of working on a modeling problem within a group is given in the next chapter.

Necessary Aspects of Metacognition to Work on the Modeling Problem “Uwe Seeler’s foot” In the following, we illustrate the occurrence and importance of metacognition while working on the modeling problem “Uwe Seeler’s foot” as an example (for a detailed presentation of useful metacognitive strategies while working on this problem see Vorhölter (2018)). The problem statement (see Fig. 1) shows pictures of a sculpture of a foot of Uwe Seeler, one of the famous players of the soccer club HSV from Hamburg, Germany. In the text, a citation from a well-known newspaper in Hamburg (called Abendblatt) is given. The writer explains that the real foot of Uwe Seeler fits 3980 times into the sculpture. Students must decide if this statement can be correct. In addition, Uwe Seeler’s shoe size is given. To find a solution there are at least two different ideas: on the one hand, the idea of a hollow sculpture, which can be filled up with shoes of the given size, can be considered; on the other hand, a comparison of the volumes may lead to a solution. Students usually chose one of the two following approaches1: • with the help of the scale of the sculpture (which has to be calculated by the values estimated or investigated), one can conclude how many times the real foot fits into the sculpture (concerning the volume). • one can simplify the sculpture as a prism or split it into several geometric bodies such as a cuboid and a prism (see Fig. 2) and the real foot in the same way. Then

1

Another approach, of course, is to identify how much water a foot of size 42 displaces. After that, the amount of water can be multiplied with 3,980. The calculated volume need to be compared with the real sculpture in Hamburg. Students need to think about whether their calculated volume can be realistic or not. In fact, this approach was only observed once due to difficulties in realizing this experiment.

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Fig. 1 Modeling problem “Uwe Seeler’s foot”

Fig. 2 Possible solutions for splitting the foot into geometric bodies

the volume of the geometric bodies of the sculpture and the real foot can be calculated and compared. Thus, the modeling problem “Uwe Seeler’s foot” is a rather complex one, as the students have to develop a mathematical model on their own based on their individual mathematical knowledge and have to identify necessary information from the text and investigate or estimate missing values, which are used for the model. Considering that working on such complex problems is usually done in groups (at least in schools), not only the metacognitive competence of single group members is important for a successful working process, but also the knowledge and strategies shared in the group. In the following, we will present useful and necessary

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metacognition for successfully working on the modeling problem in a goaloriented way.

Metacognitive Knowledge To find a solution, students must estimate the values by using comparable figures, have to be able to develop a model on their own, to judge the model and, if necessary, develop and work on another model. Concerning metacognitive knowledge, the following aspects are important: • students must be aware of the characteristics of the modeling task. In this particular case, they have to be aware that several approaches are possible and that they have to decide which one to use. They must also be aware of the fact that they are not only allowed to use context knowledge (such as the height of the person in the photo to estimate the size of the sculpture), but they have to make assumptions (for example, they could measure their own foot size or look up the size of a foot with size 10 ½). • students have to know useful strategies for solving modeling problems. To work on this problem successfully, useful strategies can be on how to get the necessary data and formula, how to organize the work in the group (for example worksharing), how to validate the received solution and how to use the modeling cycle as a tool. • students have to be aware of the capabilities of oneself and other group members, in order to choose one model to plan their approach, manage their time and in the end present the work process in class (e.g. Who knows how to calculate the volume of a chosen figure? Who is able to identify the necessary values? Who can look up a missing formula?). Thus, the work can be shared efficiently in the group. This leads to the procedural aspect of metacognitive modeling competencies, the use of metacognitive strategies.

Metacognitive Strategies Regarding metacognitive strategies, the strategies used by a student individually and the strategies shared in the group can be distinguished. To be able to develop a mathematical model, it is very important that students use strategies for getting to a common understanding (the volume of “Uwe Seeler’s foot” is asked for). Thus, the question that has to be answered (“Is it possible?”) has to be identified clearly. Based on this common understanding, students can begin to identify important and unimportant information in the task as well as missing information and items. Therefore, it is necessary to estimate or investigate missing information. Regarding our task, the shoe size is mentioned, so the volume of a foot

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of this size can be found out by investigating or by measuring a real foot of this size. The size of the sculpture can be estimated by comparing it to the height of the woman standing next to the sculpture. Although strategies for retrieving information are rather cognitive, the decision to use a particular strategy is a matter of metacognition and depends on conditional meta-knowledge. Thus, the interaction of cognitive and metacognitive strategies becomes obvious. Furthermore, a more or less explicit planning of the whole solution process or at least parts of the following working process is a matter of metacognitive strategies as well. Students should at least decide for one model to work with (as far as it does not cause a blockage) or split the group and work on two models separately at once or work on different models one after the other together in the group. With regard to the mentioned problem, it is useful as well to split the group so that students can measure or research different variables simultaneously; for example, part of the students measure a foot with shoe size 10 ½, others measure the proportions of the sculpture on the photo. Here again, the strategies used to find out the necessary values are cognitive, whereas the decision, which strategy to use, is a matter of metacognition. During the work on the modeling problem each student must monitor oneself. Additionally, at least one student of the group should monitor the group progess continuously. Usually, monitoring the group progress is not the task of an individual student, but several students are monitoring different aspects or at different times. Monitoring during the working process includes taking care of the time left as well as the progress of the work. The latter can be done by posing questions and explaining the procedure to each other. This way, a common failure (conversion of units) can be prevented or at least detected early. Furthermore, misleading assumptions as well as incorrect or too complex models can be identified quite early. If problems are identified, students must use strategies to regulate their work. Problems can be divided into those that hinder following a sense of direction and a goaloriented work and, those that cause great blockages in students’ further work. An example for a regulating strategy, which was used, while no blockages (i.e., stopping work/progress due to any number of factors) have been identified, is that students tell themselves to work on the tasks and do not speak too much about other things during working time or demand each other to decide for one approach. If blockages have occurred, a possibility for regulation is to try to identify, in which step of the modeling cycle these are and to find out thereby what to do next. Another strategy for regulating is deciding, whom to ask for help. Sometimes, regulating can lead to re-planning the further working process. After working on the modeling problem “Uwe Seeler’s foot” and finding out the dimensions of the feet, the students will see that their results are significantly larger than the values mentioned in the newspaper. So, their result should be surprising for them, which normally prompts them to search for explanations. In many cases, students do not trust their own calculations, so they try to detect errors in their calculations or in the whole modeling process. Both often result in going through the modeling cycle once more. If the students do not find any errors, they will ask themselves automatically, whether the result really matches the real problem as well as the real-world model or does not. This validation of the modeling process is an important part of the monitoring process.

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After finding a solution for the given modeling problem, students have to evaluate the modeling process in order to find out, what can be improved next time. The evaluation should refer to their own behavior as well as to the group work. It should cover several aspects like working behavior, used strategies, time management, cooperative group work as well as methods applied to overcome blockages. The description of useful and necessary metacognitive knowledge and skills presented above is based on observations of several groups of students working on the problem. However, measuring the metacognitive knowledge and skills used is challenging and is still under discussion (see for example Schellings et al. 2013; Veenman 2005). Different methods for measuring metacognition used in the field of mathematical modeling are presented in the following chapter.

Measuring Students’ Metacognition Until now, several methods were (and still are) used for measuring metacognition in modeling. This is not astonishing, as metacognition comprises different facets (see chapter 2) and there has been a discussion in measuring metacognition as such. (Schellings et al. 2013; Veenman 2005). In general, online-methods like thinking aloud and observations are distinguished from offline methods like questionnaires and interviews. The latter can be divided further into pre- and post-methods. The validity of online and offline methods have been compared in several studies, many of these comparing thinking aloud protocols to questionnaires. (Schellings et al. 2013) Schellings et al. (2013) demonstrated that the correlations between both measuring methods are usually moderate to low. As students’ ability of reporting used strategies is doubted, questionnaires were seen as less valid instruments than online approaches. However, Schellings et al. (2013) developed a three-point-frequency questionnaire based directly on the taxonomy for coding think-aloud protocols. Twenty ninth-graders were asked to study a text and think aloud simultaneously. After studying the text, they were given the questionnaire. The overall correlation between the questionnaire and the think aloud protocols (r¼0.63) was promising. (Schellings et al. 2013). Most of the instruments for measuring metacognition in modeling were adapted from instrument of other domains, whereas others were developed by analyzing modeling processes. In the following, different methods for measuring various facets of metacognition are presented. Furthermore, their advantages as well as their limitations are displayed. Metacognitive knowledge about mathematical modeling processes (characteristics of modeling problems and their special demands, useful domain specific strategies as well as individual abilities with regard to working on modeling problems of all group members), is assumed to be crucial for working on modeling problems successfully. Thus, in a qualitative study with 42 students Maaß (2007) analyzed

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misconceptions concerning modeling as part of (inappropriate) meta-knowledge about modeling processes. Students’ meta-knowledge was measured by analyzing interviews and concept maps, which students had to create two times during the study (in the middle and at the end) on their own. To create the concept map, different terms related to the modeling process as well as to the teaching unit were given to the students, who were asked to develop a visual representation of these terms. The concept maps were analyzed, not the process of drawing them. In addition, the students were given a rather less complex modeling task together with an inappropriate solution of the task and they were asked to solve the task correctly. While doing so, they were observed and asked to assign their approach to the modeling cycle. The concept maps were analyzed with regard to questions such as in which way the terms were arranged, in which way the terms relating to the teaching unit were assigned to the theoretical terms, which terms could not be assigned at all and which appropriate conceptions and misconceptions could be reconstructed. To validate hints from the concepts maps, sections of the interviews were consulted. In contrast to this, Brand (2014) measured students’ meta-knowledge on modeling by using a paper-and-principle-test as part of a test measuring students’ modeling competencies three times (as a pre-, post- and followed-test of her study on fostering students’ modeling competencies). In one of the test-items, 377 students of grade 9 were asked to match 14 extracts of the problem-solving process to the appropriate part of a modeling cycle. Furthermore, she asked students to assess useful metacognitive strategies for modeling on a four-point Likert scale, focusing on planning, monitoring and regulating of modeling processes. Thereby, conditional knowledge about useful metacognitive strategies for modeling was measured (Brand 2014). These test-items were used in the ongoing study MeMo (Vorhölter 2018) as part of the modeling test used in this study. The items for measuring students’ conditional knowledge on useful metacognitive strategies were modified and supplemented by items for evaluating with reference to Rakoczy and Klieme (2005) and Ramm et al. (2006). Schukajlow and Leiss (2011) also measured students’ conditional knowledge of useful metacognitive strategies by using a questionnaire with a five-point Likert scale. They reduced the items on strategies for planning and monitoring, which were as well adapted to those of Rakoczy and Klieme (2005). Furthermore, students’ competence for solving modeling problems were tested by means of several modeling tasks. After working on the tasks, six of them were shown to the students again and students were asked how they would act if, they were asked to solve these tasks. Schukajlow used these items in several other studies (see for example Schukajlow and Krug 2013; Schukajlow and Leiss 2011). To measure students’ metacognitive skills and behavior during modeling, the research group of Stillman has used several methods like interviews with students after working on a modeling problem, videotapes of the working processes, audiorecording of the working processes, working sheets and observations (see for

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example Stillman 2011; Stillman et al. 2007; Stillman and Galbraith 1998). To analyze the data, a framework for identifying students’ blockages was developed based on the framework by Garofalo and Lester (1985) and used as a coding scheme. The underlying assumption of this framework is that metacognitive strategies are used especially for overcoming blockages, which occur in the modeling process. This framework was developed further and still is. It is subdivided into five steps, according to the steps of the modeling cycle used in this research group. Furthermore, Stillman and Galbraith observed cognitive and metacognitive strategies used during modeling processes and divided them into the different phases of metacognition (Stillman and Galbraith 1998). Stillman et al. used online-methods, where metacognition was mainly rated by researchers and not measured by students’ self-reports. In this way, they measured the metacognitive strategies used and not the conditional knowledge about useful metacognitive strategies. However, especially strategies for monitoring are often not verbalized by students. Thus, researchers cannot rate them. Furthermore, these methods are time- and cost-consuming, if used with a bigger sample. For evaluating those strategies and to measure metacognitive strategies of a bigger sample, a questionnaire was developed in the study MeMo (Vorhölter 2017) in several steps. Four hundred thirty-one students of grade 9 had to fill in this questionnaire just after finishing their work on a modeling problem at two points in time: after working on the first modeling problem as well as after working on the last of six modeling problems, which were part of an intervention study lasting ten months. The questionnaire comprises items on the individual strategy used (12 items) as well as on shared strategies in the group (11 items) before, while and after working on the problem. For the future, expert ratings are planned in relation to the students’ behavior in the videotapes by using the same questionnaire as the students. Thereby, students’ answers can be compared to raters’ judgements. Furthermore, 57 groups of three or four students each were videotaped, while working on the first and the last modeling problem. Excerpts of the videos were used as a stimulus for interviews with 57 students out of 17 of the videotaped groups and 14 teachers. Thus, students’ and teachers’ perception and attitude can be reconstructed from the interviews.

Empirical Findings In the last years, the importance of metacognition for learning processes has been proved several times (for a general overview see Veenman 2011; for an overview on metacognition in mathematics education see Schneider and Artelt 2010). Besides, the importance of metacognition for working on complex modeling problems successfully and goal-oriented has been proven several times as well. Blum (2011) even summarizes, that for developing modeling competencies, metacognition is not only helpful, but crucial. However, there have only been a few studies focusing on the influence of metacognition on students modeling processes. In the following, research results regarding the influence of metacognition in students’ modeling

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processes are presented. Afterwards, teaching units and arrangements aiming at fostering students’ metacognitive modeling competencies are described with respect to their potentials and limitations.

Metacognition in Students Working Processes As mentioned above, metacognition is assumed to be crucial for successful modeling processes. Studies focusing on the role of metacognition in modeling processes analyzed difficulties and barriers occurring due to a lack of metacognition as well as the productive usage of metacognitive strategies. Thus, in a qualitative study Maaß (2006) identified misconceptions regarding modeling as part of (inappropriate) meta-knowledge of modeling processes. Maaß differentiates between misconceptions relating to (a) setting up a real world model (e.g. simplifying and assuming being the same; it is possible to simplify so much that working mathematically can be reduced to a minimum), (b) setting up a mathematical model (e.g. no difference between real model and mathematical model is known), (c) the mathematical solution (e.g. numbers are always exact, solutions have to be in terms of a number), (d) the interpretation and validation (e.g. validating and interpreting are identical, validating is a degradation of modeling) as well as (e) general misconceptions (e.g. every approach is a correct one, so there are not failures; mathematics is not useful for solving real problems). In general, high metacognitive knowledge of students with high modeling competencies was reconstructed. Furthermore, Maaß revealed a relation between meta-knowledge about modeling and modeling competencies: misconceptions about real models were related to deficits in setting up a real model, misconceptions in validating to deficits in doing so. Furthermore, Maaß identified a parallel development of both meta-knowledge about modeling and modeling competencies, whereas the quality of meta-knowledge in most cases was related to the performance in modeling. Schukajlow and Leiss (2011) investigated the self-reported usage of cognitive and metacognitive strategies during modeling activities. They specify cognitive strategies as rehearsal, elaboration and organization strategies in contrast to the metacognitive strategies planning, monitoring and regulating. In their study, they could not find relations between students’ modeling competencies and the selfreported usage of different learning strategies. Therefore, they considered two factors. At first, it is possible that the use of strategies during modeling activities is less important than assumed so far. Based on this assumption, there could be a lack of essential competencies and knowledge instead of a lack of knowledge on strategies. Another possible explanation relates to the restricted value of self-reported strategies. Moreover, in the study of Schukajlow and Leiss (2011) the students used planning strategies less than other strategies during their working process consciously. A reason for this could be that students plan their working process automatically.

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The relevance of metacognition in modeling processes is emphasized by the substantial studies by Stillman together with Galbraith, Brown and Edwards (2007) (for an overview about the current state-of-the-art see Stillman 2011). One of their aims was to identify metacognitive triggers, i.e. special situations, which stimulate the use of metacognitive strategies during modeling activities. Considering that working on modeling tasks is difficult and complex for students, there are many barriers students have to overcome. Using metacognitive strategies helps students to eliminate cognitive barriers during their modeling process (Stillman 2011). Goos (1998) developed a model of how to deal with metacognitive barriers, which has been transferred and supplemented by Stillman (2011) to the area of mathematical modeling. Goos (1998) disposes so called red flag situations. At this point, students become aware of specific difficulties. Red flag situations can take place in three different ways: (a) There is a lack of progress: In this case, the students should reconsider their solution strategy and think about changing their strategy. (b) Detection of an error: In this case, they should check their calculations and correct them. (c) Anomalous result: In this case, they should check their calculations as well and should reassess their solution strategy. Relating to Goos (1998), metacognitive success is defined as the recognition of a red flag situation and the appropriate reaction on it. Stillman (2011) adds the following three steps on how to overcome a red flag situation: At first, students need to recognize the necessity of the implementation of a strategy. Secondly, the main focus is on the selection of a strategy (keeping in mind that the students should think about alternative strategies before specifying their model). Lastly, the chosen strategy is implemented successfully. Stillman notes that this depends on the

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Fig. 3 Design of the study MeMo with two comparison groups (TT Teacher Training) groups with focus on metacognitive modeling strategies (red), and used mathematics (blue)

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students’ individual resources (relating to strategies) as well as on the task. With the complexity of modeling activities in mind, it is comprehensible that the reaction on a red flag situation is not always successful. Figure 3 demonstrates possible reactions on red flag situations: At first, it occurs, students do not recognize a red flag situation: “If they fail to notice that something is amiss, for example, by persisting with the wrong strategy or overlooking a calculation error.” (Goos 1998, p. 226). This case is called metacognitive blindness. Furthermore, it is possible that students recognize a red flag situation but are not able to react in an appropriate way. Goos (1998) differentiates between the following: • metacognitive vandalism (a red flag situation is recognized but the students react in a destructive way, e.g., changing the problem in order to use available knowledge) • metacognitive mirage (students perceive a difficulty although it is none.) Stillman and Galbraith (2012) add two more possible reactions: • metacognitive misdirection (inadequate reaction on a red flag situation. Stillman & Galbraith (2012, p. 101) value the metacognitive misdirection as way more inappropriate as the metacognitive vandalism.) • metacognitive impasse (progress becomes stagnancy. Neither reflection nor strategic efforts can help to overcome the blockage.). As presented in the aforementioned chapter, Stillman et al. reconstructed several metacognitive strategies and differentiated them into different aspects of metacognition. However, this distinction is an analytical one and the interplay between the categories has not been investigated. Recently, within the study Memo (see section “Outlook” in this chapter for more information) the interplay of different strategies used and reported by students was investigated. A factor analysis of 23 items, divided into strategies used by an individual person and those shared in a group, revealed a structure of three different aspects: both items on an individual level and those on a group level could be divided into strategies used for evaluating the working process, strategies used if blockages occurred and strategies used for a (more or less) failure in the free working process (Vorhölter 2018).

Fostering students’ metacognition As Veenman et al. (2006) point out, a vast majority of students acquire metacognitive knowledge and skills spontaneously from their parents, peers or teachers. However, metacognition can be fostered by addressing it directly. Therefore, several teaching units as well as recommendations regarding classroom settings and teacher behavior were developed in the last years. In general, three principals for metacognitive instruction should be considered, that were referred to by Veenman as the WWW&H rule (What to do, When, Why, and How):

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(a) “embedding metacognitive instruction in the content matter to ensure connectivity, (b) informing learners about the usefulness of metacognitive activities to make them exert the initial extra effort, and (c) prolonged training to guarantee the smooth and maintained application of metacognitive activity.” (Veenman et al. 2006, p. 9) Successful programs for metacognitive instructions are for example Reciprocal Teaching (Brown and Palincsar 1987), cognitive apprenticeship (Collins et al. 1989), IMPROVE (Kramarski and Mevarech 2003), and Schoenfeld’s method for promoting problem-solving competencies (Schoenfeld 1992). The effects of using strategies have been researched several times, and most of the results are encouraging, though some of them are disappointing (Blum 2015). For example, Schukajlow and Leiss (2011) did not find any significant correlation between self-reported metacognitive strategies on the one hand and mathematical modeling competences on the other hand. In this study, 86 students (grade nine) participated. A test was used in order to evaluate modeling competencies as well as students’ use of strategies. A result of this study is that students plan their working process in much less cases than using other strategies. On the other hand, the study showed that students evaluate planning processes as relevant. Schukajlow and Leiss (2011) note that causally determined effects of planning strategies on students’ performance need to be surveyed. In most studies, planning strategies are never investigated separate from other strategies, so that there are no results in this area. To the contrary, Maaß (2006) used the following methods for imparting metacognition in modeling in her study: • • • •

metacognitive knowledge about the modeling process has been imparted different perceptions of students’ modeling processes were discussed. students’ mistakes were dealt with in a productive way and analyzed. based on a schema of the modeling process, students were demanded to plan, monitor and validate their working process. • different solutions were compared and discussed; reasons for different solutions were reflected. • positive examples of self-monitoring were pointed out. • the teacher monitored the students’ working process. The results of this study clearly indicate that most students developed metacognitive competencies: many students at least show basic knowledge about the modeling process and are able to relate steps of their working processes to modeling terms like “reality” or “mathematical model”. At the end of the study, a great part of the students show a deeper understanding of the modeling process and developed adequate insights into modeling processes, for example referring to the subjectivity of modeling processes (Maaß 2006). In addition to the development of teaching units and therefore the providing of learning opportunities, teachers are seen as crucial for fostering students learning of metacognitive strategies. However, Schukajlow and Krug (2013) identified a greater

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positive influence by the demand for developing multiple solutions on students’ planning and monitoring activities. With the learning environment DISUM as a basis, two experimental groups of students working on modeling tasks in different group compositions were surveyed. To focus on multiple solutions, each task was used in two different versions. The intervention group was asked to work out two solutions, whereas the other group did not need to do this. The metacognitive strategies planning and monitoring were measured before and after a teaching unit composed of five lessons. The scales used had already been established in former studies. The acquisition of procedural metacognition seems to be due to the task materials instead of respective teacher interventions (Krug and Schukajlow 2014; Schukajlow and Krug 2013). Stillman also focused on the role of teachers in modeling processes and demands metacognitive competencies for teachers. According to Stillman, treating red flag situations as metacognitive triggers can also be challenging for teachers. The teaching of modeling activities is always complex for teachers. They play a key role, because they have to monitor the working process of different individual learning groups during mathematical modeling activities (Stillman and Galbraith 2012). Strategic interventions can help to activate the students’ use of metacognitive strategies and allow them to work on modeling tasks as independent as possible (Stender and Kaiser 2015). Teachers have to reflect on a meta-meta level. They have to recognize whether their students use metacognitive strategies in an appropriate way or not. Thus, they monitor their students’ metacognition and act on a metacognitive level themselves by recognizing if the perceived metacognitive activities can be improved by using strategic interventions. (Stillman 2011, Stillman and Galbraith 2012) Stillman considers two different levels: The macro level observes how the teacher acts in general on this meta-meta level, whereas the micro level takes into account, how individual interventions can be used to implement metacognitive strategies in order to overcome possible barriers in the individual context. In the empirical discussion, there is the assumption that the use of a solution plan can be seen as a metacognitive aid, particularly in case of difficulties (Blum 2015). According to Maaß (2007), students perceive knowledge about the modeling process as helpful and think that the modeling cycle can help orientating. The study DISUM shows the importance of a solution plan composed of four stages: understanding the task, establishing the model, using mathematics, and explaining the results (Blum 2011; Blum 2015). Using this, the students reported to apply strategies more frequently than before the structure was learned. Moreover, the students’ usage of a solution plan caused higher achievements than those, which were reached by the control group (Blum and Schukajlow 2018, Schukajlow et al. 2015a, b). In addition, according to Adamek (2016) students use a solution plan in every step, which can help to monitor the modeling process step by step.

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Summary and Open Research Questions As mentioned above, the importance of metacognition in modeling processes is not questioned in the international discussion on modeling, but it is rather rarely in the focus of research. Outcomes of research sometimes do not come to the same results although the studies are conducted in a similar way, but in other domains than mathematical modeling. Furthermore, results cannot easily be transferred from or to other settings because of different reasons: First of all, the understanding of modeling and modeling competencies differ depending on the researchers’ perspective on modeling. Secondly, the concept of metacognition is rather vague, as it comprises different facets than traditional mathematical problems do. Earlier in the chapter, we introduced some of the most important concepts of metacognition and highlighted their similarities and differences. However, these conceptualizations are theoretical and have not been proved empirically in the domain of mathematical modeling sufficiently yet. Vorhölter (2018) made the first step by providing a three component model of metacognitive skills for modeling. However, this analysis did not aim at revealing a correlation between metacognitive knowledge and metacognitive skills as well as a connection between metacognition in modeling and other facets of modeling competencies. Thus, the following open questions remain: • How can the relation between metacognitive knowledge and metacognitive skills be modelled? • How are metacognitive modeling competencies and other facets of modeling competency linked? The second aspect of this paper was to summarize methods and instruments used for measuring facets of metacognition in different studies. As shown above, all have their strengths and weaknesses. In the domain of metacognition, there has not been a study comparing the outcome of different methods yet. Thus, the problem mentioned by Blum remains: • “[H]ow to measure strategy knowledge, on the one hand, and strategy use, on the other hand, and another problem is how to reliably link students’ activities to their strategies” (Blum 2015, p. 88) The third part of the paper dealt with empirical findings on metacognition used in the modeling process by students and teachers as well as the effect of teaching units on the promotion of students’ metacognitive modeling competencies. All studies either are case studies or are embedded into a specific setting. Therefore, most results cannot be transferred easily. Furthermore, the perception and evaluation of the ones, who have to deal with metacognition in classrooms, i.e. teachers and students, were not a focus. However, their attitude is crucial. Without being convinced of the importance and usefulness of metacognition for modeling, metacognitive modeling competencies cannot be provided and learned. Thus, the following questions still remain, although first indications do exist:

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• How can students’ metacognitive modeling competencies be promoted effectively? • What influences the effectiveness of teaching units on metacognitive modeling competencies? • In which way does the teacher influence the promotion of metacognitive modeling competencies? How should he or she behave, what competencies should he or she have? Answering these questions is crucial to get a better understanding of the influence of metacognition on modeling processes.

Outlook Some of the open research questions, mentioned above, are tackled by the research project MeMo (Metacognitive Modeling competencies). The study MeMo has been carried out at the University of Hamburg, Germany (Vorhölter, Krüger, Wendt) and aims at the evaluation of a learning environment, which was designed in order to stimulate students’ use of metacognitive strategies during modeling processes. It took place from October 2016 to July 2017 with 23 classes of grade nine and ten of Hamburger schools participating. The participating school classes were divided into two groups. One group focuses on the stimulation of metacognitive competencies whereas the second group deepens mathematical competencies during modeling processes. Both groups work on 8 modeling activities in a period of 10 months. The modeling activities include a pre- and a post-test right before and after working on the same six modeling tasks. The pre/post design of the study was chosen in order to evaluate the modeling competencies of the students. Additionally, the students execute another test to improve their metacognitive competencies. This test takes place immediately after the first and after the last modeling problem. The same questionnaires are used for both groups. The difference between both intervention groups relate to the last 15 minutes of every modeling activity. After working on a modeling task in small groups of 3 to 4 students, the modeling process is evaluated in a discussion with the teacher with respect to a special focus: deepening mathematical or metacognitive competencies. The modeling activities are supplemented by 3 teacher trainings in each group. For two modeling problems, a special teacher training was carried out with focus on these activities. The concept of the teacher trainings covers a theoretical background, a didactical preparation of the following two modeling activities as well as accompanying materials (every participating teacher gets a teacher handbook with theory, didactical analysis of every modeling task, course plans of every modeling activity, evidences of potential barriers and respective teacher interventions, references with respect to the consolidation of every intervention group, and much more). The

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teachers were pleased to orient themselves roughly at certain guidelines (noted in the handbooks) in order to guarantee comparability. Apart from the analysis of quantitative data, the study MeMo aims at the survey of qualitative data as well. In the context of two PhD works, the teachers’ perspective on the use of metacognitive competencies as well as the students‘ perspective is investigated. At two different times, right after the first and after working on the last modeling task, teachers and students were interviewed. For this purpose, the first and the last modeling process were videotaped. Following the three-step-design (Busse and Borromeo Ferri 2003), certain excerpts of the video were used in the interviews as stimulated recalls to ask focused questions towards the students’ use of metacognitive strategies or to inspire teachers to reflect on the use of metacognitive strategies. By now, the data evaluation of the study MeMo has been finished. In the following months, the qualitative and quantitative data will be analyzed with the intent of learning about the following foci: • teachers’ perception and evaluation of students’ use of metacognitive strategies during modeling processes and its’ stimulation • students’ perception and evaluation of their own use of metacognitive strategies during modeling processes • the influence of the teaching unit on the students’ metacognitive modeling competencies including a comparison of both intervention groups • conceptualization of the relation of metacognitive modeling competencies and other sub-competencies of modeling competence and of the correlation between different facets of metacognitive modeling competencies Thereby, some of the open research questions presented above may be answered.

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mathematisches Verständnis”: 1. Befragungsinstrumente. Materialien zur Bildungsforschung. Vol. 13. Frankfurt am Main: GFPF [u.a.]. Ramm, G. C., Prenzel, M., Baumert, J., Blum, W., Lehmann, R. H., Leutner, D., & Schiefele, U. (2006). PISA 2003: Dokumentation der Erhebungsinstrumente. Münster: Waxmann. Schellings, G. L. M., Hout-Wolters, v., Bernadette, H. A. M., Veenman, M. V. J., & Meijer, J. (2013). Assessing metacognitive activities: The in-depth comparison of a task-specific questionnaire with think-aloud protocols. European Journal of Psychology of Education, 28 (3), 963–990. Schneider, W., & Artelt, C. (2010). Metacognition and mathematics education. ZDM - The International Journal on Mathematics Education, 42(2), 149–161. Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense-making in mathematics. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning. A project of the National Council of Teachers of Mathematics (pp. 334–370). New York: Macmillan. Schraw, G., & Dennison, R. S. (1994). Assessing metacognitive awareness. Contemporary Educational Psychology, 19, 460–475. Schraw, G., & Moshman, D. (1995). Metacognitive theories. Educational Psychological Review, 7, 351–371. Schukajlow, S., & Krug, A. (2013). Planning, monitoring and multiple solutions while solving modeling problems. In A. M. Lindmeier & A. Heinze (Eds.), Proceedings of the 37th conference of the international group for the psychology of mathematics education (Vol. 4, pp. 177–184). Kiel: PME. Schukajlow, S., & Leiss, D. (2011). Selbstberichtete Strategienutzung und mathematische Modellierungskompetenz. Journal für Mathematikdidaktik, 32, 53–77. Schukajlow, S., Kolter, J., & Blum, W. (2015a). Scaffolding mathematical modeling with a solution plan. ZDM, 47(7), 1241–1254. Schukajlow, S., Krug, A., & Rakoczy, K. (2015b). Effects of prompting multiple solutions for modeling problems on students’ performance. Educational Studies in Mathematics, 89(3), 393–417. Sjuts, J. (2003). Metakognition per didaktisch-sozialem Vertrag. Journal für Mathedidaktik., 24(1), 18–40. Stender, P., & Kaiser, G. (2015). Scaffolding in complex modeling situations. ZDM, 47(7), 1255–1267. Stillman, G. (2011). Applying metacognitive knowledge and strategies in applications and modeling tasks at secondary school. In G. Kaiser, W. Blum, R. Borromeo Ferri, & G. Stillman (Eds.), Trends in teaching and learning of mathematical modeling. ICTMA14. International Conference on the Teaching of Mathematical Modeling and Applications (pp. 165–180). Dordrecht: Springer. Stillman, G. A., & Galbraith, P. L. (1998). Applying mathematics with realworld connections: Metacognitive characteristics of secondary students. Educational Studies in Mathematics, 36(2), 157–194. Stillman, G., & Galbraith, P. (2012). Mathematical modeling: Some issues and reflections. In W. Blum, R. Borromeo Ferri, & K. Maaß (Eds.), Mathematikunterricht im Kontext von Realität, Kultur und Lehrerprofessionalität (pp. 97–105). Wiesbaden: Vieweg+Teubner Verlag. Stillman, G. A., Galbraith, P. L., Brown, J., & Edwards, I. (2007). A framework for success in implementing mathematical modeling in the secondary classroom. In J. Watson & K. Beswick (Eds.), Mathematics: Essential research, essential practice. Proceedings of the 30th annual conference of the Mathematics Education Research Group of Australasia, [held at Wrest Point Hotel Casino, Hobart, Tasmania, 2–6 July 2007] (pp. 688–707). Adelaide, SA: MERGA. Veenman, M. V. J. (2005). The assessment of metacognitive skills: What can be learned from multimethod designs. In C. Artelt & B. Moschner (Eds.), Lernstrategien und Metakognition. Implikationen für Forschung und Praxis (pp. 77–99). Münster: Waxmann.

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Veenman, M. V. J. (2011). Alternative assessment of strategy use with self-report instruments: A discussion. Metacognition and Learning, 6(2), 205–211. Veenman, M. V. J., Hout-Wolters, B. H. A. M., & Afflerbach, P. (2006). Metacognition and learning: Conceptual and methodological considerations. Metacognition and Learning, 1(1), 3–14. Vorhölter, K. (2017). Measuring metacognitive modeling competencies. In G. Stillman, W. Blum, & G. Kaiser (Eds.), Mathematical modeling and applications: Crossing and researching boundaries in mathematics education (pp. 175–185). Springer. Vorhölter, K. (2018). Conceptualization and measuring of metacognitive modeling competencies – empirical verification of theoretical assumptions. ZDM - The International Journal on Mathematics Education, 16(3). https://doi.org/10.1007/s11858-017-0909-x. Vorhölter, K., & Kaiser, G. (2016). Theoretical and pedagogical considerations in promoting students’ metacognitive modeling competencies. In C. Hirsch (Ed.), Annual perspectives in mathematics education 2016: Mathematical modeling and modeling mathematics (pp. 273–280). Reston, VA: National Council of Teachers of Mathematics.

Chapter 3: Principles for Designing Research Settings to Study Spontaneous Metacognitive Activity Marta T. Magiera and Judith S. Zawojewski

Abstract Drawing on current theories about the development of individuals’ metacognitive ability, a framework is proposed for designing fruitful environments for research on spontaneous metacognitive activity. Using modeling problems as examples, we argue that research settings that facilitate the study of spontaneous metacognitive activity of problem solvers need to be designed with attention to problem complexity, small group diversity, and the authentic documentation of problem-solvers’ spontaneous metacognitive activity. Keywords Spontaneous metacognitive activity · Complex problem solving · Small group interactions · Research settings design · Modeling

Introduction This chapter proposes guidance for designing research settings that optimize the potential to observe and document spontaneous metacognitive activity. Given that research on metacognition suggests that complex activity is associated with greater frequency of metacognitive behaviors (Mokos and Kafoussi 2013), the first research setting design principle suggests engaging subjects in challenging problem-solving situations, such as mathematical modeling. The second principle promotes the use of small groups of diverse problem-solvers in the designed research settings. A reasonable level of diversity helps to ensure that during the collaborative problemsolving process individuals encounter, and need to interpret and reconcile, ideas different from their own. Further, the need to communicate among subjects externalizes thinking (both cognitive and metacognitive). Finally, since research on metacognition requires the observation and documentation of problem-solvers’ M. T. Magiera (*) Marquette University, Milwaukee, WI, USA e-mail: [email protected] J. S. Zawojewski Illinois Institute of Technology, Chicago, IL, USA © Springer Nature Switzerland AG 2019 S. A. Chamberlin, B. Sriraman (eds.), Affect in Mathematical Modeling, Advances in Mathematics Education, https://doi.org/10.1007/978-3-030-04432-9_4

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internal metacognitive activity, the third principle addresses the documentation of problem-solvers’ authentic thinking. The emphasis of this principle is on deliberately designing a research setting that is thought-revealing—i.e., rich in way(s) for the researcher to capture subjects’ naturally externalized metacognitive activity.

Spontaneous Metacognitive Activity A variety of current theoretical perspectives, including socio-cognitive, sociocultural and situative, link social engagement during problem-solving activity to the development of spontaneous metacognitive capabilities in individual students (e.g., Chiu and Kuo 2009, 2010; Larkin 2006; McCaslin and Hickey 2001; Volet et al. 2013; Zimmerman and Schunk 2001). Spontaneous metacognitive activity is executive or control behavior observed as naturally occurring—rather than in response to deliberate external prompting. These theoretical considerations are supported by recent research which shifts conceptualization of metacognition from solely a focus on and examination of individuals’ thinking about their own thinking to including individuals’ thinking about the thinking of others (Hadwin et al. 2011; Iiskala et al. 2004; Magiera and Zawojewski 2011; Siegel 2012; Vauras et al. 2003; Whitebread et al. 2007). The consideration of metacognitive functioning of individuals in social contexts is re-conceptualized as a product of interactions between an individual, or a group of individuals, and a surrounding context. When goals and solutions are collectively co-constructed and the desired product is socially shared cognition, group members regulate not only their own, but each others’ thinking and their collective problem-solving activity (Hadwin et al. 2011; Hadwin and Oshige 2011; Iiskala et al. 2004, 2011; Kim et al. 2013; Vauras et al. 2003). Researchers describe the development of spontaneous metacognitive abilities during social interaction in complex problem settings. For example, Larkin (2006) explained how the collaborative nature of small groups naturally supports individuals in learning the social skills of listening, contributing and sharing, which leads to learning to question oneself—a metacognitive behavior. Furthermore, interactions among individuals working together provide a natural context in which verbal tools are used to regulate the behavior of others, serving as a significant mechanism for activating one’s own metacognition. In that sense, metacognitive activity, initially directed toward other’s thinking—over a series of experiences— can be internalized within an individual. This Vygotskian developmental perspective suggests that essential to the acquisition of metacognitive capabilities are the processes of assimilation and the internalization of metacognitive activity initially directed at other’s thinking in social contexts. Individuals who spontaneously engage in monitoring and evaluating the thinking of others are positioned to internalize these social behaviors and self-monitor, self-evaluate, and self-adjust their own performance efforts (Chiu and Kuo 2009).

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Social aspects of metacognitive activity have been reported in a broad range of empirical studies that show individuals working collaboratively during problem solving who spontaneously (without any deliberate external prompting) try to regulate the behavior of others. For example, Iiskala et al. (2004) reported on middle school students working in pairs on a task who engaged in monitoring, controlling and regulating one’s own performance, partner’s performance, and collaborative joint performance. They described how these students spontaneously compensated, monitored, and evaluated one another’s thinking. They also described how students expressed awareness of their partner’s thinking and engaged in reciprocal monitoring and complementing each other in regulating a shared performance over the task episode. Other researchers made similar observations reporting how groups of high school students solving mathematical modeling problems spontaneously expressed metacognitive awareness, engaged in evaluation, and regulated their own thinking by inviting each other to analyze one other’s thinking and shared solutions (e.g., Goos et al. 2002; Kim et al. 2013; Magiera and Zawojewski 2011). The importance of complex cognitive activity for the development of metacognitive abilities is clear. Mokos and Kafoussi (2013) investigated students’ spontaneous metacognitive activity while working on mathematical tasks of varied levels of complexity (i.e., routine, open-ended, and authentic problems situated in real-life contexts). They found that while students spontaneously engaged in metacognitive planning in problems of all levels of complexity, spontaneous metacognitive evaluation and comprehension monitoring were most frequently occurring when students engaged in solving problems situated in authentic reallife contexts. Such problems required students to find, sort, define missing information, and formalize mathematical interpretations of a real life context. Mokos and Kafoussi’s result suggests that a high level of realistic problem complexity might provide a powerful research setting for observing and studying spontaneous metacognitive activity. An explanation for Mokos and Kafoussi’s results may be found in Funke’s (2010) description of the nature of realistic complex problems as requiring metacognitive coordination to manage a large knowledge base, and to flexibly govern that knowledge to reach the desired goal(s). Similarly, Gravemeijer and Stephan (2002) describe how problem solvers need to use higher-level abilities to effectively make sense of and express a complex real-life situation using the language of mathematics. Realistic applied problems, such as real world mathematical modeling problems, are complex, unique, and typically ill structured and ill defined (Funke 2010; Lesh and Doerr 2003; Lesh et al. 2000; Lesh and Zawojewski 1992, 2007). Research shows that the abilities required for solving such complex problems are different from those represented, emphasized, and assessed in traditional curriculum documents and tests (Kartal et al. 2016; Schraw et al. 1995). For example, in their recent study with college students, Kartal and colleagues found that conventional measures of mathematics performance (i.e., standardized tests that included “problem solving” portions) did not serve as valid predictors of student performance on mathematical

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modeling activities. Schraw and colleagues describe how performance on solving illdefined complex tasks is independent of performance on solving well-defined tasks, because ill-defined tasks engage a different set of skills when compared to problems with well-defined problem space. Constrained problem situations in which the problem space is clearly defined call for the application of a finite number of concepts, known solution strategies, and usually have single correct, or convergent answers. In contrast, ill-structured complex problems are typically based on realworld contexts that have divergent solutions and require the integration of several content domains, have multiple solutions, multiple solution paths, and several criteria for evaluating solutions.

Principles for Designing Research Settings The goal for the principles proposed in this section is to design research settings that optimize opportunities to observe and document students’ spontaneous metacognitive activity in problem-solving situations. Research suggests that problem solvers’ spontaneous metacognitive abilities develop over and within episodes of social interactions in complex problem settings (e.g., Iiskala et al. 2004; Whitebread et al. 2007). Therefore, the proposed principles encourage the creation of problem-solving environments as research sites that support the real-time development of metacognitive capabilities that facilitate managing complex problem situations while simultaneously dealing with many interacting factors and variables, (Lin et al. 2005). Lesh et al. (2003b) observed that productive higher-ordered mental activity of individuals varies greatly across problems and phases of problem-solving activity. For example, Lesh and Zawojewski (2007) described how metacognitive strategies “free” of evaluative stances, such as brainstorming, are needed early on in the problem-solving process to freely generate ideas for approaching a problem. Although monitoring might be needed to “keep an eye” on a proposed strategy, interpretation, or the general problem constraints, an early metacognitive goal during collaborative problem-solving is to avoid shutting down the idea-generating processes prematurely, compared to later stages of solution process when selecting or modifying ideas to better fit the problem situation at hand require more detailed monitoring. As a result, productive metacognitive activity and problem-solving processes must be studied together as a co-dependent activity, since they are entangled and exist as part of a complex cognitive system. The data gathered during such experiences is authentic and compelling, and can be used to develop deep understanding of how metacognitive capabilities evolve spontaneously. Overall, the challenge for researchers is to find optimal ways to elicit, capture, and document the data.

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Design Principle 1: Realistic Complex Activity The Realistic Complex Activity Principle ensures that a problem solver’s solution process will involve iterative cycles of expressing, testing and revising ways of thinking about the problem situation and the proposed solution. Task complexity has been identified in the existing literature as an essential factor that activates problem-solvers’ metacognitive functioning (e.g., An and Cao 2014; Baker and Cerro 2000; Efklides 2006, 2009; Kim et al. 2013; Lin et al. 2005; Prins et al. 2006). Thus effective research settings need to be designed in a way that amplifies opportunities for problem-solvers to use metacognition as they simultaneously attend to several highly interconnected variables, clarify vaguely defined or unclear goals and constraints, deal with multiple solution paths, or consider multiple criteria for evaluating solutions. Different levels of conceptual and cognitive engagement (e.g., analyzing, creating, selecting and reducing information, reformulating problem situations, making decisions, setting priorities) directly activate different types of metacognitive activity. A research setting designed to engage problem solvers in iterative cyclical processes of revising and evaluating one’s interpretation of the problem situation and evolving solution requires ongoing monitoring, evaluating, and judgment-making about one's own performance and product. Lin et al. (2005) pointed out that the nature of problems used in prior research on metacognition constitutes a primary weakness of early metacognitive research. The problem environments were designed as explicit teaching interventions, rather than to nurture spontaneous metacognitive activity. Therefore, limited opportunity was available for students to make adaptations to their understanding of the problem environment or range of expected solutions, since the tasks selected were typically well defined, value free, of limited duration, and seldom grounded in the complexities of real-world considerations. Instead, Lin and colleagues argue that to elicit the development of metacognitive abilities students need to work in realistic complex problem situations, which frequently involve making changes to one’s interpretation of the problem environment (e.g., considering only some, not all, involved variables, defining what constitutes a “quality” solution), and one’s initial and intermediate solutions. One example of realistic complex problems that engage students in iterative cycles of expressing, testing and revising ways of thinking are model-eliciting activities (MEAs) used by Lesh and colleagues (e.g., Lesh et al. 1983, 2000, 2003a; Lesh and Doerr 2003). To develop a solution to an MEA small groups of problem solvers are required to design a mathematical model (e.g., a sequence of steps, a representation, or an equation) and by doing so generate knowledge about unknown systems of interconnected variables that the problem solvers themselves identify and define. These problems are “realistic” in that they are open-ended and client-driven, and typically contain incomplete, ambiguous, or undefined information about the problem context. Such ill-structured and ill-defined characteristics are purposefully used to prompt groups to negotiate definitions, rationales and assumptions, enhancing opportunities for metacognitive activity spontaneously emerge,

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consistent with Kim’s et al. (2013) description of environments (e.g., grappling with complex problems) that can trigger metacognitive activity. Another example of complex problems comes from the Cognition and Technology Group at Vanderbilt (2010) and their Jasper Series, which was developed as an environment for testing the tenets of “anchored instruction.” Their technology-based problem environments were designed to engage students in reasoning about complex real-life situations (typically presented in a form of a video) closely linked to students’ experiences. Students needed to formulate problems for themselves using information about the problem situation embedded in the video, and draw on their problem-solving skills and multiple mathematics concepts as they develop a problem solution. In successive classroom trials with different versions of Jasper series, the researchers documented how Jasper Series problem environments spontaneously engaged students in metacognitive planning and comprehensive monitoring. While MEAs and the Jasper problems were designed using different sets of design principles, both represent the intent of the Realistic Complex Activity Principle. Both types of problems are complex, situated in a real life contexts. Both require students to go through iterative cycles of considering evolving or multiple solutions, and require ongoing reflection on both the problem interpretation and the current solutions’ strengths and weaknesses. An additional striking similarity of both sets of activities is the use of small groups in the problem-solving process, leading to the second research setting design principle.

Design Principle 2: Small Group Diversity When solving a complex problem in a small group, problem solvers spontaneously question each other, reflect aloud on their own, others, or shared goals, evaluate current ways of understanding, and plan solution-seeking activities together. Such externalized conversations often reveal spontaneous metacognitive activity. Opportunities to elicit metacognition are enhanced when group members compare and contrast different points of view, ways of thinking, assumptions about a real-world context, etc. Therefore, the Small Group Diversity Principle ensures that participants in a problem-solving group are selected to represent a reasonable range of mathematical and sociocultural backgrounds, experiences, or sets of values. Designing a research setting involves selecting reasonably diverse group members, setting expectations for members to respect and seriously consider each other’s perspectives, and monitoring and intervening when one member permanently overtakes the entire direction of group activity during the problem-solving episode. This principle is consistent with recent interpretations of metacognitive activity that have shifted from a sole focus on and examination of individuals’ thinking about their own thinking, to include individuals’ thinking about the thinking of others, and a group’s collective mental activity (Hadwin et al. 2011; Iiskala et al. 2004; Magiera and Zawojewski 2011; Siegel 2012; Vauras et al. 2003; Whitebread et al. 2007). Research settings for the study of metacognitive activity need to be designed in a

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way that provides opportunity for group members to engage in individual and interdependent and collectively shared regulatory processes as they work toward shared outcomes. Social interactions can amplify not only the frequency of spontaneous metacognition, but also the opportunity to observe the evolution of primitive problem-solving and metacognitive behaviors (e.g., group members arguing over problem interpretation) to more mature capabilities (e.g., analyzing different points of view, coordinating next steps) (Lesh and Zawojewski 2007). Research suggests that encountering diverse perspectives can prompt one to spontaneously engage in metacognitive activity (e.g., Hogan 2001; Magiera and Zawojewski 2011). For example, Hogan (2001) observed students spontaneously activating their metacognitive thinking and engaging in planning and regulating their thinking in, what she termed as, conceptual contexts. She described these situations as one in which the students “were in the midst of building an explanation, . . . needed to stop and think, go back and reconsider evidence,” (p. 210). She observed that students engaged in metacognitive activity during group or class discussions, and when they shared complex ideas or expressed confusion about a specific idea proposed for examination. Focusing on a small group of students solving a series of mathematical modeling problems, Magiera and Zawojewski (2011) identified a wide range of social-based and self-based contexts in which the students spontaneously engaged in metacognitive activity (awareness, regulation or evaluation). They found that most frequently students become metacognitive about their own thinking or the thinking of others when they were attempting to interpret diverse perspectives about the problem situation or a solution approach. In these situations, group members shared and considered different mathematical approaches or interpretations of the problem and its solution, examined new information proposed by peers, or cognitively struggled with information presented by their peers. Magiera and Zawojewski’s study uncovered two additional, albeit less prevalent, social situations in which students spontaneously engaged in metacognitive activity: when problem solvers sought to reconcile with other group members disagreements about the mathematical processes, interpretations, or results; and, when the students engaged in and shared explanations, such as when trying to explain to each other the effectiveness of a strategy or approach. The focus on small group diversity in planning the research setting requires the selection of participants that are not only reasonably diverse, but also likely to engage productively and communicate with each other. Stacey (1992) demonstrated that simply forming small groups does not necessarily promote metacognitive activity within or among individual group members. Goos and Galbraith (1996) linked group members’ metacognitive functioning in collaborative problem-solving situations to individuals’ ability to respect each other’s perspectives (no group member dominates group activity) and distribution of knowledge (members share common mathematical knowledge but also bring their unique experiences and perspectives into the collaborative activity). Thus, to optimize the potential to observe metacognitive actions and behaviors, participants with diverse perspectives who demonstrate promise for productive and externalized collaborative interactions need to be purposefully sought. Once the group members have been selected for

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reasonable diversity and propensity to function well as a group, the subsequent challenge is to design effective and efficient ways to document the emergence of spontaneous metacognitive activity.

Design Principle 3: Authentic Thought Documentation The Realistic Complex Activity Principle and the Small Group Diversity Principle work hand-in-hand to maximize opportunity to elicit and externalize students’ authentic metacognitive thinking through their conversations and written interactions as they engage in the multiple cycles of problem interpretations and developing problem solutions. While these two principles ensure environments in which students’ give voice to their spontaneously developing metacognitive capabilities, the challenge to researchers is to document these moments and episodes in practical ways that will inform the research question posed. The Authentic Thought Documentation Principle ensures that authentic data is captured that will inform the driving research question in a permanent form that lends itself well to analysis. Various approaches have been used to document students’ thinking during problem solving episodes and there is consensus among researchers that each individual method used to access metacognitive behaviors is fallible. Schoenfeld (1992) noted that any methodology used to assess cognitive-metacognitive functioning might provide an adequate picture of some behaviors, but a distorted view of others. For example, consider “think aloud” protocols commonly used in early studies of mathematical problem-solving strategies to identify instances of problem-solving strategies and metacognition. The data from think-aloud protocols might provide insights into the question about the frequency and type of problem solving strategies students apparently used. However, the same data from these protocols were more limited in terms of assessing the effectiveness, or spontaneity, of identified metacognitive strategies. van der Stel et al. (2010) argued “using metacognitive activity more frequently does not automatically mean that the metacognitive skills have the higher level of quality,” (p. 221) nor that problemsolving performance has been improved. Baker and Cerro (2000) criticized thinkaloud protocols for their potential to disturb a student’s work on task. They also noted that protocols in which a researcher prompts a student to think aloud during problem-solving episode might raise a question whether a metacognitive, or problem-solving strategy is a naturally occurring strategy, or perhaps, wouldn’t have occurred at all if the student wasn’t “probed” to articulate his or her thinking. The challenge, then, is to design a research setting in which spontaneous metacognitive activity can be documented as it naturally occurs. Lesh and colleagues (Lesh et al. 2000) explicitly designed MEAs as activities with multiple entry points, allowing problem-solvers to articulate diverse perspectives during the solution development. As problem solvers negotiate and evaluate

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the skills and abilities each individual brings to the table, the group externalizes their thinking, recognizes, evaluates and makes decisions regarding allocation of available cognitive resources. By design, embedded in MEAs problem statements are criteria which group members can use to judge the effectiveness of their thinking and their solution models, rather than requiring an external authority to assess the goodness of intermediate solution attempts. These criteria intensify the opportunity for spontaneous metacognitive activity to be externalized. Problem solvers typically engage in iterative modeling cycles, traversing back and forth between constraints of the applied context and solution criteria in the ongoing evaluation of their evolving solution model. Spontaneous metacognitive activity is typically externalized as a well-functioning small group goes through several iterations, testing and reviewing their models with respect to the articulated criteria and constraints. For example, consider an MEA problem statement that describes a client who needs to publish a procedure for her employees to use for cutting out pieces of material from a rectangular-shaped piece to be assembled together as a soccer ball (Soccer Ball MEA, https://unlvcoe.org/meas/). The requirement that problem solvers create a procedure (i.e., model) that is to be communicated to a clearly stated audience (the client’s employees) ensures that the group’s intermediate and final products, as written directions, reveal their mathematical ways of thinking. However, while such written products are rich for studying problemsolvers’ mathematical thinking (conceptual understandings), they might not always provide sufficient level of information about specific questions concerning spontaneous metacognitive activity. Designing a way to capture spontaneous metacognitive activity elicited by rich learning environments (i.e., settings that meets the first two principles) is a challenge. The third design principle requires simultaneously externalizing students’ cognitive and metacognitive evolution in permanent ways, and in a form that can be used to address the specific research questions. The research setting designed needs to be feasible and practical. While it might seem that the best way to accomplish these goals is to videotape complex problem-solving sessions with small diverse groups, careful consideration needs to be given to a number of issues. For example, perhaps student conversations alone would provide sufficient data to inform the specific research question; on the other hand perhaps facial expressions and gestures are important to the research question, which would require multiple cameras. The design process for documenting authentic thought necessarily involves iterative cycles between consideration of the research question and the planning of the documentation methods. One example of a study that examined spontaneous metacognitive activity during mathematical modeling illustrates the challenge of planning an authentic thought-revealing research setting. Magiera and Zawojewski (2011) sought to identify the specific characterizations of situations and contexts associated with the spontaneous elicitation of metacognitive activity of a small group of students collaboratively solving a series of mathematical modeling problems. Their challenge was to capture and document authentic student engagement in spontaneous monitoring, evaluating, controlling each other’s thinking and collective group

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activity, and in monitoring, evaluating, or controlling their own thinking. The researchers expected that metacognitive activity directed at the thinking of others would be readily seen in the videotapes of the problem-solving sessions, but they were concerned that metacognitive activity directed at one’s own thinking would be less apparent, if at all. The researchers were also concerned that their own interpretations of the students’ interactions in the videotaped sessions would be less authentic than students’ own interpretations. Therefore, Magiera and Zawojewski designed a two-phase thought-revealing research setting. In the first phase, a small group of students solved a series of MEA problems while the interactions were captured on a videotape (using one camera). The students’ externalized their thinking as they interpreted team members’ proposed scenarios for a solution model, and as the members of the group together reflected on, and refined their collaborative ideas towards a “best fit” model. To prepare for the second phase, the videotapes were analyzed by the researchers to identify “metacognitive moments” (i.e., students evaluating each other thinking, reflecting on their collaborative planning, or articulating their awareness of their own or groups’ developing understanding). Then, in the second phase of data collection, researcher-student interviews were videotaped as individual group members interpreted their own thinking and described the context and situation that gave rise to each presented and identified metacognitive moment. Magiera and Zawojewski finally used the videos (and accompanying transcripts) of the second phase interviews to generate, evaluate and iterate their own hypotheses about the social- and self-based nature of the identified metacognitive moments. Magiera and Zawojewski’s design of their research setting attempted to optimize occurrences of metacognitive activity by achieving the first two principles, and then optimize gathering data that was authentic and in a form that directly addressed their research question.

Reflection: The Classroom as a Setting for Research on Spontaneous Metacognitive Activity We have shared and provided motivation for designing research settings in which to study spontaneous metacognitive activity with three purposes in mind. The first purpose was to plan research settings that have the potential to maximize opportunity for spontaneous metacognitive activity to emerge (i.e., use of complex problems). The second purpose was to enhance the opportunity for students to engage in spontaneous metacognitive activity by establishing the need for the externalization of student thinking by using reasonably diverse small groups. The third goal was to enhance the authenticity and practicality of the data gathered. When students’ metacognitive activity emerges spontaneously as an inherent aspect of problemsolving activity, without deliberate external prompting by researchers, the data gathered is naturally occurring in real time, and therefore authentic. However,

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planning for what data to collect and how to collect that data is a challenge to the design of a research setting. An important benefit of the three proposed principles is the close link between the desired research setting and desired characteristics of classroom practice. The potential to conduct such research in real classrooms is enhanced when researchers work with teachers to use the first two principles to purposefully create classroom situations that enhance students’ opportunity to engage in metacognitive activity. Together they can work to create or select realistic complex problems that facilitate multiple solution approaches, are open to interpretations, and are accessible to students with diverse mathematical backgrounds and knowledge of problems’ realworld context. The selection of students to form reasonably diverse small groups can be facilitated by classroom teachers, who would have first-hand familiarity with individuals and a similar goal to enhance learning opportunities through student interaction. Classroom cultural norms that have been already established by teachers can greatly enhance the research setting, when students expect to respect each other’s points of view, contributions, and ways of thinking. The third principle, however, goes beyond normal classroom practice and is largely left to the researcher to design. The researcher, working in a classroom environment, is charged with finding practical ways to gather permanent authentic data that will not only efficiently and effectively inform the specific research question, but will be feasible for enactment in a classroom. When the researcher also acts as an agent for professional development, the third principle for authentic thought documentation can be co-designed by the teacher and researcher. Lin et al. (2005) stated that teaching has unique qualities and teachers need to recognize that classroom situations, even when they appear similar, have a number of hidden features that in fact make these situations different. They described a professional development goal for teachers to recognize and identify these features, and address them appropriately in the situation. Professional development that engages teachers in researching their own practice may provide an opportunity for parallel research activity between the teacher and the researcher. For example, perhaps a teacher wants to explore whether metacognitive activity varies across problems or across phases of problem solving, as hypothesized by Lesh et al. (2003b). Or perhaps a teacher wants to investigate whether a particular metacognitive activity is always productive, as questioned by Lesh and Zawojewski (2007). When a teacher has quest of his or her own alongside the researcher, then meaningful opportunities can emerge for the teachers and researchers to work together on designing the research setting. In such research settings, researchers and teachers can jointly plan for the enactment of the three principles; what problems to use with students, how to group students, and how to document students’ thinking in a way that is authentic and efficiently and effectively can inform each of their research questions.

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References An, Y., & Cao, L. (2014). Examining the effects of metacognitive scaffolding on students’ design problems solving and metacognitive skills in online environment. MERLOT Journal of Online Learning and Teaching, 10(4), 552–568. Baker, L., & Cerro, L. C. (2000). Assessing metacognition in children and adults. In G. Schraw & J. C. Impara (Eds.), Issues in the measurement of metacognition (pp. 99–145). Lincoln, NE: Buros Institute of Mental Measurements, University of Nebraska-Lincoln. Chiu, M. M., & Kuo, S. W. (2009). Social metacognition in groups: Benefits, difficulties, learning, and teaching. In C. L. Larson (Ed.), Metacognition. New research perspectives (pp. 117–136). New York: Nova Science Publishers. Chiu, M. M., & Kuo, S. W. (2010). From metacognition to social metacognition: Similarities, differences, and learning. Journal of Education Research, 3(4), 321–338. Cognition and Technology Group at Vanderbilt. (2010). The Jasper Series as an example of anchored instruction: Theory, program description, and assessment. Educational Psychologist, 27(3), 291–315. Efklides, A. (2006). Metacognition and affect: What can metacognitive experiences tell us about the learning process? Educational Research Review, 1(1), 3–14. Efklides, A. (2009). The new look in metacognition. In C. L. Larson (Ed.), Metacognition. New research perspectives (pp. 137–151). New York: Nova Science Publishers. Funke, J. (2010). Complex problem solving: A case for complex cognition. Cognitive Processing, 11(2), 133–142. Goos, M., & Galbraith, P. (1996). Do it this way. Metacognitive strategies in collaborative mathematical problem solving. Educational Studies in Mathematics, 30, 229–260 Goos, M., Galbraith, P., & Renshaw, P. (2002). Socially mediated metacognition: creating collaborative zones of proximal development in small group problem solving. Educational Studies in Mathematics, 49(2), 193–223. Gravemeijer, K., & Stephan, M. (2002). Emergent models as an instructional design heuristic. In K. Gravemeijer, R. Lehrer, B. Oers, & L. Verschaffel (Eds.), Symbolizing, modeling and tool use in mathematics education (pp. 145–169). Dordrecht, the Netherlands: Kluwer Academic Publishers. Hadwin, A. F., & Oshige, M. (2011). Self-regulation, co-regulation and socially-shared regulation: Exploring perspective of social in self-regulated learning theory. Teachers College Records, 113 (2), 240–264. Hadwin, A. F., Järvelä, S., & Miller, M. (2011). Self-regulated, co-regulated and socially shared regulation of learning. In B. J. Zimmerman & D. H. Schunk (Eds.), Handbook of self-regulation of learning and performance (pp. 65–84). New York: Routledge. Hogan, K. (2001). Collective metacognition: the interplay of individual, social, and cultural meanings in small groups’ reflective thinking. In F. Columbus (Ed.), Advances in psychology research (Vol. 7, pp. 199–239). Huntington, NY: Nova Science Publishers. Iiskala, T., Vauras, M., & Lehtinen, E. (2004). Socially-shared metacognition in peer learning? Hellenic Journal of Psychology, 1, 147–178. Iiskala, T., Vauras, M., Lehtinen, E., & Salonen, P. (2011). Socially shared metacognition within primary school pupil dyads’ collaborative processes. Learning and Instruction, 21(3), 379–393. Kartal, O., Dunya, B. A., Diefex-Dux, H., & Zawojewski, J. S. (2016). The relationship between students’ performance on conventional standardized mathematics assessments and complex mathematical modeling problems. International Journal of Research in Education and Science (IJRES), 2(1), 239–252. Kim, Y. R., Park, M. S., Moore, T. J., & Varma, S. (2013). Multiple levels of metacognition and their elicitation through complex problem-solving tasks. Journal of Mathematical Behavior, 32 (3), 377–396. Larkin, S. (2006). Collaborative group work and individual development of metacognition in the early years. Research in Science Education, 36(1-2), 7–27.

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Vauras, M., Iiskala, T., Kajamies, A., Kinnunen, R., & Lehtinen, E. (2003). Shared- regulation and motivation of collaborating peers: A case analysis. Psychologia: An International Journal of Psychology in the Orient, 46(1), 19–37. Volet, S., Vauras, M., Khosa, D., & Iisjala, T. (2013). Metacognitive regulation in collaborative learning: Conceptual developments and methodological contextualization’s. In S. Volet & M. Vauras (Eds.), Interpersonal regulation of learning and motivation: Methodological advances (New perspectives on learning and instruction) (pp. 67–101). New York: Routledge. Taylor & Francis Group. Whitebread, D., Bingham, S., Grau, V., Pino Pasternak, D. P., & Sangster, C. (2007). Development of metacognition and self-regulated learning in young children: Role of collaborative and peerassisted learning. Journal of Cognitive Education and Psychology, 6(3), 433–455. Zimmerman, B. J., & Schunk, D. H. (2001). Reflections on theories of self-regulated learning and academic achievement. In B. J. Zimmerman & D. H. Schunk (Eds.), Self-regulated learning and academic achievement: Theoretical perspectives (pp. 273–292). Mahwah, NJ: Lawrence Erlbaum Associates.

Chapter 4: Engagement Structures and the Development of Mathematical Ideas Lisa B. Warner and Roberta Y. Schorr

Abstract We describe the relationship that exists between shifts in engagement and shifts in mathematical thinking, using the construct of engagement structures. The engagement structure construct (Goldin et al. 2011) is a way to account for and describe the complex dynamical interactions that recur as students solve mathematical problems. Our research is focused on a group of eighth grade students solving a problem in a group setting in an urban district. Our analysis involves video-recorded episodes, retrospective interviews and comprehensive field notes. We also document the social conditions present in the classroom that surrounded the shifts. Our findings suggest a variety of changes that can occur within an individual student, and across students in the same classroom, depending upon the social context. At times, changes in mathematical ideas preceded shifts in engagement, and vice-versa. Aside from the within student differences, our research provides an example of how, within the same classroom, students can have very different engagement and mathematical experiences. Keywords Problem solving · Student engagement · Engagement structures · Motivation

Theoretical Perspectives and Constructs Engagement Structures: A Brief Overview Student engagement is a critical part of mathematical learning (Middleton and Jansen 2011; Middleton et al. 2017; Goldin et al. 2011). Engagement, as we refer to it here, involves affective, cognitive, social, and other important dimensions L. B. Warner (*) William Paterson University, Wayne, NJ, USA e-mail: [email protected] R. Y. Schorr Rutgers: The State University of New Jersey, New Brunswick, NJ, USA © Springer Nature Switzerland AG 2019 S. A. Chamberlin, B. Sriraman (eds.), Affect in Mathematical Modeling, Advances in Mathematics Education, https://doi.org/10.1007/978-3-030-04432-9_5

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(Fredricks et al. 2004; Goldin 2017; Goldin et al. 2011; Gómez-Chacón 2011; Middleton et al. 2016). These dimensions are highly complex, and inextricably dependent upon, and influenced by each other. This co-dependency and complexity is often ‘visible’ when students work together to solve cognitively challenging problems (Gómez-Chacón 2017; Schorr et al. 2010a, b). In such situations, mathematical understanding, social dynamics, and affect, for example, can shift, based upon the highly individualized experiences and perceptions of the students, as well as many other internal and external factors. For example, one student might feel (momentarily) challenged by a question posed by a peer, while another might consider it an opportunity to revise her thinking, another may choose to ignore it entirely. Regardless of the response, the question may impact the in-the-moment engagement of the questioner, the questioned, and those affirmed, annoyed, overlooked or disinterested. Research on engagement is sometimes focused on the longer term implications of attitudes, beliefs, motivations, etc.. However, many recent studies focus on what occurs ‘in-the-moment” (Goldin 2017; Goldin et al. 2011; Middleton et al. 2016, 2017). Middleton et al. (2016) note that: “Engagement in the moment is a place where educators may have some control over the eliciting conditions for the development of interest and goals, instrumentality and efficacy beliefs, prosocial behaviors, and productive affective structures” (pg. 25). Our own research, over the past decade, has focused on the types of engagement that occur, in-the-moment, in middle and high school classrooms, as students solve mathematical problems (e.g. Epstein et al. 2007; Sanchez Leal et al. 2013; Schorr et al. 2010a, b). As a result, we have introduced what we believe to be a necessary construct to account for, and better understand the complex, dynamical interactions that seemed to recur. We termed this construct an engagement structure. We now provide a very brief description of this construct. An engagement structure is a kind of behavioral/affective/social constellation, situated in the individual as a psychological construct, and becoming active in social contexts (see Goldin et al. 2011; Schorr et al. 2010b). Engagement structures can be found in all aspects of human activity, not just during mathematical problem solving (see Goldin et al. 2011 for a more complete discussion). Our conceptualization of engagement structures incorporate several components including: a characteristic motivating desire, one or more goals, implementation actions to achieve the motivating desire (sometimes involving social interactions and sometimes involving other patterns of behavior), “self-talk” (which refers to hypothetical internal speech), sequences of emotional states, strategies, and modes of interactions. Engagement structures are present within all individuals and become operative under given sets of circumstances. These do not occur in isolation, indeed, they may reference each other, and several may be active simultaneously. At times, as conditions or motivations change, one structure can ‘branch’ into another. We now summarize several examples of the engagement structures that will be discussed in this paper, and include brief examples to illustrate. (See also Goldin et al. 2011 and Schorr et al. 2010b for the first 6 engagement structures below; the last 2 structures are the result of current research by Warner et al. 2018):

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• Check this Out (CTO): In this structure, the individual realizes that solving the mathematical problem can have a payoff – either immediately, or at some future point. The motivation to engage mathematically can lead to (intrinsic) interest in the task itself, or heighten (extrinsic) interest in an external payoff. The student therefore decides to devote (at least) some attention to the problem. An example: a student realizes that if he solves the problem, he may get a good grade, which will be pleasing to his parents. This realization results in solving the math problem. • I’m Really Into This (IRIT): Here, the individual has an intrinsic interest in the problem or problem solving experience and develops deep concentration in the process. This concentration is generally so intense that it can result in the experience of “flow” (Csikszentmihalyi 1990). An example: A student finds the problem so intriguing that she “tunes out” her surroundings, not even aware that the class is about to end. • Get the Job Done (GTJD): This structure involves a person’s sense of obligation to fulfill his part of a work “contract.” Ultimate satisfaction comes from doing the work rather than enjoying the challenge of the task. As the student completes the task, the focus is on finishing. An example: A student knows that the teacher has told him to solve all three parts of a particular problem before moving on to something else. He is therefore motivated to finish the work and moves through all three parts (superficially in some cases), in order to finish and satisfy the teacher. • Stay Out of Trouble (SOOT): In this case, the person’s aversion to risk supersedes the mathematical aspects of the task. The person avoids trouble at all costs, with or without completing the work. An example: A student wishing to avoid conflict may shun interactions that have the possibility of leading to trouble – either with peers, or with an authority figure. The student may not contribute to his groups’ solution, even though he has valuable ideas, just to avoid conflict. • Look How Smart I Am (LHSIA): This structure involves the person’s desire to demonstrate to others that she knows something that the other person does not (or does not know as well). The person may express her ideas to peers in an effort to impress them with her superior knowledge. An example: A student is motivated by impressing a peer and explains something to a peer using deliberately complex terminology, simply to demonstrate her knowledge of the subject. (Note that the peer may still benefit, but the overarching motivation is different than in the structure that follows, LMTY.) • Let Me Teach You (LMTY): This structure involves the person’s desire to teach another person something that he knows that the other person does not know (or does not know well). The person may explain an idea to a peer with the genuine hope that the peer will learn something. An example: Unlike the student in the structure above (LHSIA), a student explains something to someone else for the purpose of genuinely helping the other person, without regard to showing off his own expertise. • Do as I say [DAIS] or Do as I want [DAIW]: This structure involves the person’s desire to control a situation perceived as requiring compliance or obedience and

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to have others do what he wants them to do. This may include communicating directions, offering incentives or punishments (carrot and stick), communicating a sense of obligation (what someone is supposed to do, or expected to do), and/or imposing rules. He may get annoyed, frustrated or angry if someone does not comply. An example: A student has a desire to control his fellow group members and distributes the workload amongst them, telling each exactly what to do. • Don’t Let the Group Down (DLTGD): In this structure, the motivating desire is to “carry one’s weight”. There is an underlying desire to contribute positively to the group, and not let others down. An example: A student may not be especially interested in the problem, but sees her fellow group members actively pursuing an idea. Not wanting to be viewed as a slacker, she explores her peers’ mathematical ideas further, in the hope of contributing productively. Our goal, for the purpose of this paper is not to provide an extensive overview of our work on engagement, but rather use case studies to highlight the relationship that engagement has to mathematical shifts in understanding.

Connection to Shifts in Understanding Mathematical Problems Engagement structures are highly complex, and can change at any given moment. We now discuss how engagement structures, particularly those mentioned above, can coincide with shifts in mathematical understanding throughout the problem solving process. We highlight these connections more specifically in the Results Section. Mathematical models involve the organization and development of knowledge and how it evolves over time. We believe that all knowledge is organized around situations and experiences. When a student is presented with a problem, ideas are mapped into previously existing internal descriptive or explanatory systems (models) that then guide actions. Models are not static; they can evolve as new information is considered and as the need arises. They develop over time as the learner refines, extends, tests and shares ideas (Schorr and Lesh 2003). (For a further description of mathematical models, please see: English et al. 2016; Lesh et al. 2013; Schorr and Lesh 2003; Schorr and Koellner Clark 2003; Schorr et al. 2010a). In this chapter, we look at shifts in mathematical ideas, and see how they coincide with shifts in engagement structures. We note that not all changes in mathematical thinking result in fundamental conceptual shifts. Nonetheless, such modifications can be indicative of, or precursors to, revisions in existing models. Our guiding questions are: How do shifts in mathematical ideas impact engagement, and similarly, how are mathematical ideas impacted by shifts in engagement? How does this evolve over the course of a problem solving experience? What other factors contribute to these shifts (i.e. teacher intervention, peer responses, etc.)? We answer these questions by providing a detailed description of the classroom context and social dynamic of the groups within which the students worked.

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Methods Subjects: For this study, we focus on two students who were part of a larger study involving an eighth grade classroom that consisted of 20 students, 93% African American and 7% Hispanic. The school is classified as low income in the largest city in the state of New Jersey. The class was homogeneous and designated as a low ability class (which in this school means that it was the lowest in terms of standardized test scores at the eighth grade level). Procedure: In the larger study, all classes, including the one that is the subject of this paper, were observed in each of four cycles, with each cycle spanning a period of two consecutive days. The first cycle occurred approximately one month into the school year and subsequent cycles occurred later on. Prior to the start of a cycle, an interview was conducted with the teacher to ascertain her plans for the lesson and what she expected to happen. A post-lesson interview (using a stimulated recall protocol) with the teacher and individual interviews with several pre-selected students designated by the teacher, took place after each cycle (for more details see Schorr et al. 2010b). Classroom interactions for each of the classes were videotaped using three separate cameras (one stationary camera capturing whole-class interactions and two roving cameras, primarily capturing pre-selected students and their interactions) and all student and teacher interviews were videotaped using one camera. Transcripts were created from videotapes and student work was collected and digitized. Subjects: For this paper, we chose 2 students. Both were chosen from the same class during the first cycle (about 43 minutes of instruction each day in 2 consecutive days). These 2 students were chosen because they were part of the group of pre-selected students who were interviewed and because their shifts in engagement and coinciding shifts in the problem solving process were different from each other. The first student discussed below is Shay. He is tall in stature, and is, according to his teacher, accorded a high social status amongst his peers. Will, the second subject of this paper, is an immigrant, who happens to be smaller in stature, and is, according to the teacher, lower in social status. Data: We transcribed all observation and interview data. At least two to three members of the research team viewed the data (alone and together) and chose excerpts that they all believed provided evidence for the activation of the engagement structures that had thus far been identified. Additional members of the research team reviewed the decisions. Only after full agreement were segments chosen for this study. Subsets of the larger research team further analyzed the mathematical shifts that occurred during these sessions. That analysis forms the basis for this research. During the class session, the students were working in groups of three to five on the following mathematical task1, chosen by the teacher: 1

The problem was adapted from Exemplars K-12 (2004) www.exemplars.com

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L. B. Warner and R. Y. Schorr Farmer Joe has a cow named Bessie. He bought 100 feet of fencing. He needs you to help him create a rectangular fenced in space with the maximum area for Bessie to graze. Bullet 1: Draw a diagram with the length and the width to show the maximum area. Bullet 2: Explain how you found the maximum area. Bullet 3: How many poles would you have for this area if you need 1 pole every 5 feet?

Results In this section, we begin by presenting each student’s problem solving experience, including mathematical shifts that coincide with shifts in engagement structures. Next we present data from each of their retrospective individual interviews to provide more evidence for the presence of specific engagement structures.

Shay’s Classroom Observation The teacher encouraged the students to work in groups of three or four. Shay, however, began by investigating the task alone, (for the first 13 minutes), as his group members spoke to each other. Our characterization of his engagement begins with the “Check This Out” structure. Initially Shay was creating a rectangle, adding the 4 sides to equal 100 and calling 100 the area (however, the 100 represents the perimeter). Shay said, “. . . it’s just got to be the area is a hundred. . . .see. (Begins to calculate again) 15 and 15, that’s thirty, and then two numbers that equals up to seventy, 35 and 35. There go one.” We infer that initially, his motivating desire was to understand how to construct a rectangle using the 100 feet of fencing. A mathematical shift in the problem solving process occurred when he realized that it was possible to create more than one rectangle with a perimeter (not area) of 100 feet. At this point, he moved to the floor to investigate this problem, by himself, in order to have more room and to further isolate himself from the others in his group (who were talking about other things). At this point, he appeared to be genuinely interested in solving the problem, indicating that the structure “I’m Really Into This” may have become active. He seemed intrigued by the mathematics, “tuning out” other elements of the environment, including his rather talkative group members. During the problem solving session, Shay only interacted with the other members of his group when the teacher encouraged him to share ideas with them, or when he found them to be useful. For example, when Shay created a 40 unit by 10 unit rectangle, the teacher encouraged him to work with a peer by saying, “Okay, so now show him”. Shay responded by saying, “Like umm, the height could be forty, and then this could be ten. . .”. During this time, we infer that “Get the Job Done” became active for him, fulfilling an implied obligation to do as the teacher asked (work with his peers). Shay experienced some difficulty in figuring out the length and width of potential rectangles. When the teacher walked over to his group, he told her that he was

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“struggling” to figure out a height for a given width. The teacher then asked him more about this, and he responded: “Because, if you trying to find the height. . .you pick two heights and it won’t work for the other one.” His verbal explanation of the struggle indicated a desire to understand the math. In Schorr et al. 2010b, we discussed Shay’s willingness to display certain vulnerability by expressing the difficulty he was experiencing in figuring out the problem. For the purpose of this paper, we would also like to highlight that a mathematical shift occurred when Shay became aware of the issue he was having in constructing the rectangles and was able to express it to another person (in this case the teacher). When Shay realized that many rectangles with a constant perimeter of 100 units (using only whole numbers) could be constructed, he told his group members, “You could do a lot of ‘em, . . .” He also said to a group member “You could do ummm, look watch this, look. . . .hold on. . . .twenty three. . .and then twenty seven, twenty seven. . .twenty seven, twenty seven equals to a hundred. I could do another one.” We infer that this breakthrough also reinforced the “Get the Job Done” structure as he quickly realized that constructing all possible rectangles with a constant perimeter of 100 units would take a lot of time. Consequently, enlisting his group members would help him complete this task more quickly. He said, “you can keep going but it can go all day”. He spontaneously asked his group members to provide him with supplies, such as a ruler and paper. He also began to direct his peers and organize the work by saying, “Move, I know what to do, look. . .what number, look. . .we gonna start, I’m gonna do the lower numbers, you do the higher numbers. . .” The other members of Shay’s group worked on constructing the rectangles that he had assigned them. It became obvious that he had a desire to complete the assigned mathematical task correctly and thought that he needed the members of his group to construct as many rectangles as possible to do this. As Shay continued to work, he began to develop some strategies for building rectangles with a perimeter of 100. Further, he found a method for constructing rectangles with a constant perimeter. When the teacher asked him to explain his method, he said, “Look, you do two sides and then you see what the remainder is and then you. . .” His method involved finding the width given a specific length. He added the side length to itself (for the length of two sides), subtracted the sum from 100, and divided the difference by two (to find the width). For any given length, he now had a method to find the width. At this point in time, we infer that this mathematical breakthrough coincided with a “Check This Out” structure. It appears that his motive was to deepen his understanding and solve the problem. He realized that finding the area of each rectangle by multiplying the length times width would allow him to see which rectangle had the most space inside. He continued with this strategy until he had another “aha” moment. Prior to this, Shay’s choices of rectangles were not systematic. He now began to find the area of the rectangle with dimensions of 23 by 27 feet, 24 by 26 feet, and then 25 by 25 feet (see Fig. 1). Another mathematical shift occurred when he found an efficient way to find the next rectangle by subtracting one from the length and adding 1 to the width of the previous rectangle. He continued with this method until he came to the conclusion that the rectangle with the maximum area had dimensions of 26 by

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Fig. 1 Shay’s 23  27, 24  26, & 25  25 rectangles

24 (because he didn’t consider the 25 by 25 square to be a rectangle). During this time period, it is possible that “Check This Out” and “Get the Job Done” were operating simultaneously. Next, we provide evidence that the “Do As I Say/Want” engagement structure also became active. As Shay and his group members were recording their solution on a large piece of chart paper, he directed the organization of their ideas. For example, he said, “Just put them in order, just keep going. . .” We infer that Shay wanted others to do what he told them to do, in this case, organize their ideas the way he wanted them on the chart paper. As Shay continued to direct his group members throughout the session, he occasionally spoke to his peers with some annoyance in his tone. For example, by the end of the first day’s session, Shay said to his group members, “That’s what we been trying to do all this time” in response to his peer stating something that Shay thought he should have already known (had he been listening to him). While it is entirely possible that his words and actions were indicative of “Get the Job Done”, we draw upon additional interview data to suggest that “Do As I Say/Want” became active as well. Our inferences are based upon his responses to the interviewer about these occasions and his tone of voice when talking about them, as will be discussed in the interview section below.

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During the problem solving process, the teacher encouraged each group of students to write their solution on chart paper. She also told them that the groups would be rotating and critiquing other groups’ work (as shown on the chart paper) by writing comments on small pieces of paper, near the work. After Shay thought he solved the problem, he became invested in seeing to it that all of his group members understood his method. We suggest that “Get the Job Done” was active at this moment, the goal shifting to a desire to get his group members ready to defend their mathematical ideas as they rotated amongst the groups. We could also suggest that “Let Me Teach You” became active and that Shay had a genuine desire to help his fellow group members understand or solve the problem, which went along with sharing his ideas with them. It is also possible that Shay wanted to look smart in front of his group members, indicating that “Look How Smart I Am” may have become active. We will provide further evidence for this when we present the interview transcript. The three engagement structures could have been operating simultaneously. As Shay and his fellow group members rotated to critique other groups’ solutions, Shay continued to try to help his peers understand the concepts. For example, one of his group members was showing the teacher a 40 unit by 10 unit rectangle written by another group. As the teacher asked questions, Shay attempted to help his group member understand what the other group had done. When a student from the other group (Dana) confronted Shay, he reacted quite firmly. This interaction is described, at length, in Schorr et al. (2010a, b). We now summarize the main shifts in Shay’s approach to the mathematics and hypothesized engagement structures, below (see Table 1):

Shay’s Retrospective Interview Retrospective interviews with careful examination of the classroom video revealed more information about Shay’s mathematical understanding, perceptions of his social interactions with peers, motivations, emotions and engagement. For example, the interview confirmed that Shay had several instances in which he tried to help his group members. However, as noted above and below, he avoided helping them because he felt that they did not value him and his work. Interviewer: Shay: Interviewer: Shay:

How did you feel? I felt. . . like, I was trying to help them. But they wouldn’t let me give any help. Do you feel like you try to help people a lot in class, or. . .? No, [be] cause . . . but everybody thinks my work would be wrong, because I play.

In the interview, Shay spoke about being confident in his mathematical ability but did not think the other students had confidence in his mathematical ideas, because he

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Table 1 Summary of Shay’s shifts in mathematical thinking and hypothesized engagement structures as noted above

Initial approach

Shift 1

Shift 2

Shift 3

Shift 4

Shift 5

Shifts in mathematical thinking Initially, Shay began by constructing one rectangle (and labeled his sides 15, 15, 35 and 35). He added 15, 15, 35 and 35 to equal 100, but was confused about the difference between area and perimeter, referring to the 100 feet as the area rather than the perimeter. He soon realized that he needed to focus on both area and perimeter, and that the 100 feet was a measure of perimeter. He noticed that it was possible to create more than one rectangle with a fixed perimeter of 100 feet. He endeavored to find several rectangles through a method of trial and error. He began to use a more systematic approach that involved adding a side length to itself (for the length of two sides), subtracting this sum from 100, and dividing the difference by two. This method, if used correctly could be used to find the width of a particular rectangle. For any given length, he now had a method to find the width. He realized that the rectangles were increasing in area in a particular manner. He now focused on how the areas were increasing (as the sides increased). He continued using this method until he came to the conclusion that the rectangle with the maximum area had dimensions of 26 by 24 (because he didn’t consider the 25 by 25 square to be a rectangle).

Hypothesized associated engagement structures Check This Out

Check This Out

I’m Really Into This

Check This Out, Get the Job Done and/or Do As I Say/Want

Check This Out, Get the Job Done, Let Me Teach You and/or Look How Smart I Am Check This Out, Get the Job Done, Let Me Teach You and/or Look How Smart I Am

Clearly, other structures may have been present. We opted to only highlight the ones for which we have reasonable evidence

“plays”. He also spoke about his peers’ desire to argue, instead of focusing on his mathematical explanations. Interviewer: Shay:

What about the people that you were working with in the group? They act like they didn’t want to do anything, like, they wanted to argue instead of like, seeing what I was talking about. But in the end, it came down that I would be right anyway.

As Shay continued to speak about the other students’ off task behavior, his facial expression and tone of voice indicated that he was annoyed with his group members. This was consistent with the annoyance we heard in the tone of his voice during the

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classroom session, when he issued directives to his peers. As noted above, this provided us with additional evidence that “Do As I Say/Do” was present during the problem solving session, as he was trying to “Get the Job Done”. Shay wanted to have others do what he wanted them to do. Shay also spoke about the way he contributed to his group by being the one to start his group off. Interviewer: Shay: Interviewer: Shay:

So, do you think you contributed to your group? Yes. Can you give me an example? Like, when we’re doing work, ain’t nobody ever start off. I had to tell them to get out, to make as many rectangles as you can to find out what the area is. And then we get the maximum area.

He spoke about working hard and feeling good about it. He said, “I play a lot. People don’t expect me to do work, but when I work, I would work a lot. I try to work hard at that moment.” He also spoke about others holding him back. He stated, “I felt like some people would try to hold me back. They would think just because I do one thing, I can’t do another. But it’s, it’s more than just playing.” Some of Shay’s responses during this interview validated the inferences made above. They also provide us with further insight regarding his motivations. For example, he stated that he did try to teach his group members at one point during the problem solving process, which validates that “Let Me Teach You” may have been active. He also spoke about his peers’ perceptions of him as someone who does not take the mathematics seriously. This supports our inference that it is likely that “Look How Smart I Am” was active, too, and it may explain why he may have felt the need to show others that he is smart. We are not sure if Shay’s action of helping his peers were motivated by a genuine desire to help others or a desire to impress them, or both. It is also likely that one or both of them were active along with “Get the Job Done”. Teaching the other students, and also directing them, may have been a part of getting the job done for Shay, which in the end may have been important for his group to defend their solution, if questioned by the other groups.

Will’s Classroom Observation Will was a member of a different group within the same class. He was often quiet during the discussions that took place. We, therefore, have very few excerpts in which he spoke. By all outward appearances, there were few opportunities for Will to share his own ideas. He did not assert himself at all. However, when the teacher walked over to his group and asked the group a question, he occasionally answered her questions, especially those directed specifically at him. We infer that “Stay out of Trouble” was

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the primary structure active for Will for most of the sessions, as his main focus was on avoiding interactions that might lead to conflict. We will provide evidence for this in excerpts of observation and interview data below. In the following excerpt, occurring at the beginning of the session, it is important to note that one of Will’s group members drew a 40 unit by 10 unit rectangle on chart paper before the teacher walked over. In response to the teacher’s questions about that rectangle, Will made one suggestion to try different numbers. Will’s group members ignored this idea. Teacher: Will: Teacher: Dana: Teacher:

How can you prove that this is the maximum area? You can try different numbers and see if [they] will work. Okay, so how would you try different numbers? Do you want to try that and I’ll be back? Why we can’t just leave our maximum length like this? I didn’t say you couldn’t leave it like this. I was just asking do you know. . . .how do you know that’s the maximum length?

The other students in the group continued to discuss this with the teacher as Will began to write and draw silently. We infer that Will’s silent drawing and writing may have been an attempt to fulfill a desire to understand how to construct a rectangle with a perimeter of 100 feet, indicating the activation of “Check this Out”. Teacher:

(asking Will) So what are you doing here?

Will didn’t answer her, but stared at the wall and ceiling. Over the next few minutes, Will silently continued to write his ideas on his piece of paper. It appears that a “Check This Out” structure became active for Will, as he drew a 30 unit by 20 unit rectangle on his piece of paper. Later in the session, the teacher walked back over to Will’s group. Dana: Will: Teacher: Ghee: Teacher: Dana:

We’re trying to figure out the, ummm. . . .well I know that we’re trying to figure out the length and the width [be] cause you said that You said you could do different numbers. . . Who said that? . . . No I said that. But she said we can make a uhh uhh, that we need to find the, uhh, area. I did say that There, maximum area just means that. . .

The teacher then asked Will a question directly. Will answered her. Teacher: Ghee: Will:

So [Will] just said that, what did you say? The areas the inside of the box. . . . .Width is 30 and 20. (Ghee is laughing)

When the teacher asked Will to draw a rectangle with those dimensions, the students in Will’s group told the teacher that he did draw it and crumpled it up and put it in his desk.

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So can you draw that and show them? Turn your paper over and. . . He did do it. He drew it. He balled it up.

Will went into his desk, began to unfold the balled up piece of paper in his hand and laid it down to his left. Teacher:

Ok, so can you bring it over into the group and see what happens?

Will brought the paper over to his group members and in a very low tone, shared his ideas with the teacher and students in his group. During the observation, it seemed as if “Get the Job Done” became active, with a motive to do what the teacher asked of him (bring over the crumpled up piece of paper to his group). We will later provide evidence, with the interview data below, that the activation of “Don’t Let My Group Down” is also possible if Will was acting on a desire to be a contributing member of his group. We are still unsure if, prior to this moment, his group members ever saw the dimensions of his rectangle, decided the rectangle they constructed was better, or didn’t see it at all. We do know that his group members were aware that Will crumpled up his mathematical work, put it in his desk, and did not use his mathematical ideas. Later in this session, his group members decided to use the 40 by 10 rectangle and share it with the rest of the class. This result is discussed at length in Schorr et al. 2010a, b. In sum, after silently working on the math problem by himself, Will created the 30 by 20 rectangle, crumbled up his work, put it in his desk and stopped working on the problem, out of fear of getting in trouble with his classmates (as noted in his interview). We suggest that “Stay Out of Trouble” was active for Will, during most of this classroom session. We infer that Will had a need to avoid interactions that may lead to trouble with his peers. His fear of conflict seemed to supersede the mathematical aspects of the task. We summarize Will’s approach to the mathematics and hypothesized engagement structures, below (see Table 2):

Will’s Retrospective Interview Retrospective interviews revealed a lot more information about Will’s mathematical understanding, reasons for his lack of social interactions with peers, motivations, emotions and engagement. In this case, the interview was especially important, since Will did not say much, and rarely interacted with peers, during the problem solving session. During the classroom observation, as noted above, Will’s group member, Dana, only drew one rectangle on his group’s chart paper. This was not the one that Will had drawn on his crumpled up sheet of paper. From Will’s retrospective interview, we found out that he realized that it is possible to construct rectangles with different dimensions for this problem, which indicates a shift in his mathematical thinking.

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Table 2 Summary of Will’s mathematical thinking noted above

Initial Approach

Shifts in mathematical thinking Will created a 30 by 20 rectangle.

Hypothesized associated engagement structures Primary Engagement Structure: Stay Out of Trouble Secondary Engagement Structures: Check This Out, Get the Job Done and/or Don’t Let My Group Down

Clearly, other structures may have been present. We opted to only highlight the ones for which we have reasonable evidence

We are not sure if this mathematical shift occurred during the problem solving session or some time after. Consequently we did not list it on Table 2. The interview data did reveal that he knew that his rectangle and the rectangle chosen by his group would both have a perimeter of 100. By his statements in the retrospective interview, we infer that he did not know that his 30 by 20 rectangle had a greater area than the 40 by 10 rectangle chosen by his group. He believed his solution and the solution chosen by the group were equally likely to be correct. However, he was reluctant for the group to choose his solution, for fear of backlash. Interviewer: Will: Interviewer: Will: Interviewer: Will: Interviewer: Will:

So what made you crumple it up? . . .I was just thinking not to use it anymore, just use [Ghee’s], just saying, we don’t need it anymore so we just tossed it away. How did that make you feel? . . .If they had chosen my wrong one, and the right one they tossed it away, they might’d get mad at me. Why did you take it back out? Cause Ms. B wants it, wanted to see... So how did that make you feel? That raised up my happiness back for me. Like I said, it, when they chose [Gee’s], my happiness came down a little bit, but when she wanted to see mine, it came up.

Will also expressed concern about arguments. Interviewer: Will: Interviewer: Will: Interviewer: Will:

So where’s your level of happiness during this? It was pretty much a little bit low, but I didn’t like, saying anything, or. . . Why not? Because, it might just cause an argument in the first place. And how do you feel when there’s an argument? I don’t like arguing with people, because mostly, they become more like a fight. . .and, if it comes to a fight, you just get suspended. . .and maybe, they just kick you out of school.

As we analyzed Will’s responses during his retrospective interview, it became clear to us that “Stay out of Trouble was the primary structure active for Will, during most

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of this session. He seemed motivated by fear of conflict with his classmates and his actions demonstrated that (e.g. crumpling up his work and putting it in his desk). Based upon our analysis, we suggest that Will felt that the situation could be dangerous and potentially lead to a fight. When visiting this class 6 months later, we learned that Will was, unfortunately, assaulted by several of his classmates, so his fear was not unfounded. His retrospective interview also revealed that Will felt happy when the teacher asked him to take out the crumpled up piece of paper and explain his ideas to her. We infer that the encouragement and support the teacher provided for him (as she asked him to take out his crumbled up piece of paper) may have contributed to a desire to contribute to his groups’ ideas (which we will later provide more evidence for), possibly triggering the brief activation of “Don’t Let My Group Down”. Those same actions could have also indicated the activation of “Get the Job Done”, simply following the teacher’s directions by taking out the crumpled up piece of paper and showing his solution. He seemed to have deep respect for the teacher, often answering her when she asked a question directly to him and doing what she asked, throughout the classroom session. We infer that both structures may have been operating alongside each other, with multiple motivating desires for the same action (having both the desire to comply with the teacher and be a contributing member of his group). Will also reported that he feels good when he is focused. Will: Interviewer: Will: Interviewer: Will: Interviewer: Will:

I don’t want to feel no no negative way or anything like that. Just focus of what we’re trying to do. So you said you weren’t negative? (shakes his head yes.) Hmmm What do you mean by that? Uh, I wasn’t thinking about any other thing. I was just trying to find out what’s the problem. Ok, and how does that make you feel? Mm, it makes me feel pretty good (smiling) because I’m more focused than some other children, cause some of them just waiting for something to come up, so they just stop.

He continued to say that he felt that way most of the time. Based upon the observation video data, it seems as if his group members ignored or dismissed his mathematical ideas. From the interview, it seems as if Will may not have perceived this as related to his mathematical ability and/or ability to focus. For example, he revealed in the excerpt above, that he views himself as being focused most of the time and more focused than some of the other children in his class. Also, in the excerpt below he said the students sometimes look up to him and ask him questions. Interviewer: Will:

Now in that group, is there a group leader anywhere? Mmmm..I think, mostly [Dana], because she is the one mostly being like, if I came up with an answer, she would try to figure it out to see if it was right.

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Interviewer: Will: Interviewer: Will: Interviewer: Will:

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Ahh interesting. So, did anyone choose her as a group leader? Um no (shakes his head) How did she get chosen? I don’t know. I don’t know if they wanted her to be the group leader because, [Gee] he maybe wanna be the group leader, [Dana] maybe wanna be the group leader, maybe even I would wanna be the group leader (pointing to himself with both hands) So do you want to be the group leader? Of course. Yeah. How does that make you feel to be group leader? It makes me feel a little bit proud, because everybody would look up to you. To ask you like questions, if this is right. . . So do they ever do that? Look up to you and ask you questions? Sometimes (saying it as if he means it).

This portion of the interview provided some evidence that it is possible that “Don’t Let My Group Down” may have briefly been active during the problem solving session (having a conflicting desire to contribute to his group, while fearing them). By what Will said in the excerpt above, it is evident that he had a desire to contribute to his group and even be the leader of his group. It is possible that he did briefly follow this desire with the action of taking out his crumpled up piece of paper and sharing his ideas with his group members during the session, when encouraged to do so by the teacher. This desire, reported in the interview, was not apparent, at all, from the classroom observation. We infer that even though Will did have a desire to carry his weight and be a contributing member of his group, his fear of getting in trouble with the other students was stronger, during most of the session (which was indicated by his actions). We remain confident that “Stay Out of Trouble” was the primary structure active for Will during this class session, even though it appeared as if a few additional engagement structures were briefly active, as well. He seemed very concerned about staying out of trouble with his fellow classmates. Will physically stayed back, and only shared his ideas quietly and unassertively, ideas which had he been willing to share more pointedly, might have productively changed the direction of the group’s solution.

Discussion and Conclusions We believe that the results presented above provide a clear link between shifts in mathematical ideas and shifts in engagement structures. In some cases, the changes in mathematical ideas preceded the changes in engagement, and in other cases, it was the other way around. We also document the social conditions present in the classrooms that surrounded the shifts. We note that Shay had several mathematical

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shifts, and these, coincided with shifts in engagement (see Table 1). Will, on the other hand, created only one rectangle and was engaged primarily in “Stay Out of Trouble”. Will may have experienced some brief instances of “Check This Out” (when drawing his own rectangle on the piece of paper that he ultimately crumpled up) and “Get the Job Done” and/or “Don’t Let My Group Down” (when he took the crumbled up piece of paper out of his desk and showed his rectangle to the teacher and group members). We believe, however, that he was primarily engaged in “Stay Out Of Trouble”, and his desire to avoid confrontation with his peers led to much of his mathematical stagnation, (see Table 2). Shay spent most of his time in “Check This Out” and “Get the Job Done”. If we only relied on classroom observations, it would seem as if “Let Me Teach You” was active when he interacted with his group members. The retrospective interviews, however, revealed that his motivating desire during some or all of those interactions, may be more indicative of “Get the Job Done”, “Look How Smart I Am” and/or “Do As I Say”. Shay felt that his group members did not take his mathematical ideas seriously, and perceived him as all “play”. However, he did not allow this to cause him to be deterred from solving the problem. This was not the case for Will, who was handicapped by his perception of his peers (and it would seem that this was appropriate and adaptive, given the conditions present). Both students seemed to have a deep respect for the teacher and responded positively to her when she asked them questions. It is unfortunate that at the time of the problem solving session, the teacher did not realize how unsafe Will felt. (She became aware of this after his interview.) We believe that by examining the role that engagement plays in mathematical shifts, and vice versa, researchers and educators can better understand how to help students who are affectively, socially or cognitively compromised, as well as enhance the experience for others. Encouraging Will to share his ideas may have been helpful, but it certainly was not a longer-term fix. Students in “Stay out of Trouble” need to be safe, intellectually, socially, and physically before they can solve and share ideas with ease. We also believe that connecting engagement (via the use of engagement structures) and mathematical breakthroughs is helpful in understanding the idiosyncrasies that occur during group problem solving. For instance, breakthroughs can coincide with a need to work alone and take the time to develop an idea more thoroughly before sharing with others. Moreover, there are times when a student’s knowledge can deepen by stopping to listen to another group’s strategy and times when this might take a student away from his/her own way of thinking and confuse him/her more. Aside from the within student differences experienced by Will and Shay, these case studies provide an example of how, within the same classroom, students can have very different engagement and mathematical experiences. These differences are critical to consider as students solve problems that are cognitively complex, require the development and evolution of mathematical models, and involve complex interactions with peers.

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References Csikszentmihalyi, M. (1990). Flow: The psychology of optimal experience. New York: Harper & Row. English, L.D., Bergman Arleback J., & Mousoulides, N. (2016). Reflections on progress in mathematical modelling research. The second handbook of research on the psychology of mathematics education: The journey continues pp. 383–413. Fredricks, J. A., Blumenfeld, P. C., & Paris, A. (2004). School engagement: Potential of the concept: State of the evidence. Review of Educational Research, 74, 59–119. https://doi.org/ 10.3102/00346543074001059. Goldin, G. A. (2017). Motivating desires for classroom engagement in the learning of mathematics. In C. Andrà, D. Brunetto, E. Levenson, & P. Liljedahl (Eds.), Teaching and learning in maths classrooms (pp. 219–229). New York: Springer. Goldin, G. A., Epstein, Y. M., Schorr, R. Y., & Warner, L. B. (2011). Beliefs and engagement structures: Behind the affective dimension of mathematical learning. ZDM – International Journal of Mathematics Education, 43, 547–560. https://doi.org/10.1007/s11858-011-0348-z. Gómez-Chacón, I. M. (2011). Beliefs and strategies of identity in mathematical learning. In Roesken, B. & Casper, M. (Eds.), Current state of research on mathematical beliefs XVII: Proceedings of the MAVI-18 conference, September 2012, Helsinki, Finland (pp. 74–84). Gómez-Chacón, I. M. (2017). Appraising emotion in mathematical knowledge: Reflections on methodology. In U. Xolocotzin (Ed.), Understanding emotions in mathematical thinking and learning (pp. 43–73). London: University of East Anglia. Lesh, R., English, L. D., Sevis, S., & Riggs, C. (2013). Modeling as a means for making powerful ideas accessible to children at an early age. In S. Hegedus & J. Roschelle (Eds.), The SimCalc vision and contributions: Democratizing access to important mathematics (pp. 419–436). New York: Springer. Middleton, J. A., & Jansen, A. (2011). Motivation matters, and interest counts: Fostering engagement in mathematics. Reston, VA: National Council of Teachers of Mathematics. Middleton, J. A., Jansen, A., & Goldin, G. E. (2016). Motivation in Springer open Attitudes, beliefs, motivation and identity in math education: An overview of the field and future directions, ICME 13, Hamburg, Germany, pp. 17–26. Middleton, J., Jansen, A., & Goldin, G. A. (2017). The complexities of mathematical engagement: Motivation, affect, and social interactions. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 667–699). Reston, VA: National Council of Teachers of Mathematics. Sanchez Leal, L., Schorr, R. Y., & Warner, L. B. (2013). Being challenged in an urban classroom: A case study documenting the engagement of a young male who wanted to “look smart”. Journal of Urban Learning, Teaching, and Research (JULTR), 9, 78–88. Schorr, R. Y., & Koellner Clark, K. (2003). Using a modeling approach to consider the ways in which teachers consider new ways in which to teach mathematics. Journal of Mathematical Thinking and Learning: An International Journal, 5(2), 191–210. Schorr, R. Y., & Lesh, R. (2003). A modeling approach to providing teacher development. In R. Lesh & H. Doerr (Eds.), Beyond constructivism: A models and modeling perspective on teaching, learning, and problem solving in mathematics education (pp. 141–157). Hillsdale, NJ: Lawrence Erlbaum. Schorr, R. Y., Epstein, Y. M., Warner, L. B., & Arias, C. C. (2010a). Chapter 27: Don’t disrespect me: Affect in an urban math class. In R. Lesh, P. L. Galbraith, C. R. Haines, & A. Hurford (Eds.), Modeling students’ mathematical modeling competencies: ICTMA 13 (pp. 313–325). New York: Springer. https://doi.org/10.1007/978-1-4419-0561-1_27.

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Schorr, R. Y., Epstein, Y. M., Warner, L. B., & Arias, C. C. (2010b). Mathematical truth and social consequences: The intersection of affect and cognition in a middle school classroom. Mediterranean Journal for Research in Mathematics Education, 9(1), 107–134. Warner, L. B., Schorr, R.Y., & Goldin, G. A. (2018). Analyzing prospective teachers’ motivating desires during mathematical problem solving. In L. Gomez Chovam, A. Lopez Martino, & I. Candel Torres (Ed.), ICERI2018 Proceedings (pp. 10436–10441). IATED Academy. ISBN 978-84-09-05948-5. Yakov M. Epstein, Roberta Y Schorr, Gerald Goldin, Lisa B. Warner, Cecilia C. Arias, Lina Sanchez-Leal, Margie Dunn & Tom Cain (2007). Studying the affective/social dimension of an inner-city mathematics class. Proceedings of the Twenty-Ninth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Lake Tahoe, NV, pp. 649–656.

Part II

The What and the Why of Modeling Alan H. Schoenfeld

Abstract I begin with reflections on the meanings and purposes of mathematical modeling. Central concerns are what it means to engage meaningfully in mathematical modeling and why some kinds of mathematical models – the ones that are explanatory in nature – are preferable for educational research to the kinds of taxonomic or statistical models that describe components of a system but fail to say how they work. I then turn to the Teaching for Robust Understanding (TRU) Framework, which highlights five essential dimensions of classroom practice. A key dimension of TRU, “Agency, Ownership, and Identity,” asks what opportunities students have to develop the sense that they are willing to and capable of engaging with the content (agency), to make the content their own, above and beyond “ingesting” content defined by others (ownership), and see themselves as “math people” (identity). This dimension in particular provides a way of re-examining issues of affect. The balance of the chapter is devoted to using the TRU framework to examine and in some cases reframe the inquiries into affect and modeling found in the chapters by Goldin, Gómez-Chacón and De La Fuente, Weizel, Middleton and Jansen, and Shahbari, Tabach, and Heyd-Meyzuyamim. Keywords Modeling · Affect · Teaching for robust understanding · TRU framework · Agency, ownership and identity

I am grateful for the opportunity to respond to the papers in Part II, because they provoked me to reflect on the nature of the modeling enterprise, at least as I practice it. There are, of course, multiple uses. As a researcher, I find modeling to be an essential part of my toolkit. If I’m to make claims about the nature of people’s thinking and learning, I need mechanisms for making and justifying those claims. As

A. H. Schoenfeld (*) Graduate School of Education, University of California, Berkeley, CA, USA e-mail: [email protected] © Springer Nature Switzerland AG 2019 S. A. Chamberlin, B. Sriraman (eds.), Affect in Mathematical Modeling, Advances in Mathematics Education, https://doi.org/10.1007/978-3-030-04432-9_6

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someone who wants to help students engage meaningfully in mathematics,1 I find (as some of the authors suggest) that modeling tasks can be useful tools. The question, if we want to be general and useful, is just what about such tasks makes them good tools. Let me expand on these two themes. First, the modeling enterprise. As I wrote in How we Think (Schoenfeld 2010), I suffered from the loss of certainty when I left pure mathematics for mathematics education. In mathematics, a proof is a proof is a proof: there is an if-then form of certainty. But even in mathematics, there are different kinds of proofs. Some (typically, for example, proofs by contradiction) show that: but others show how and why. For me, proofs of the latter type are desirable. In the physical and social sciences, there are no proofs; but, there are models. As in the case of mathematics, I look for models that help me understand why things happen the way they do. That means, for example, that I don’t find statistical models useful in explanatory terms, even if correlations may suggest a relationship or document that a relationship is strong. I want to know (as best as one can) how things work. This means that I find some things less useful than others. For one thing, I find knowledge taxonomies not to be terribly helpful. To give a particular example, I do find that the idea of pedagogical concept knowledge (called PCK, due to Shulman 1986, 1987) is useful. What I find useful about it is that Shulman identified a kind of knowledge that plays a fundamental role in how teachers act in the classroom – and, that we need to pay attention to. It means, in practical terms, that content knowledge is not enough: those who think that one can improve teaching by focusing professional development on content knowledge are missing a big part of what constitutes expert teaching. That is part of the beauty of Ma (1999), Knowing and teaching elementary mathematics: while expert teachers may have a small part of the content knowledge regarding fractions that Ph.D. mathematicians do, the mathematicians lack a significant part of the pedagogical content knowledge possessed by those teachers. Questions of how PCK develop, and what one can do to assist in the development of relevant PCK, are useful and important. I feel differently about taxonomic questions regarding what kinds of knowledge there are – PCK, CK, SMK, etc. Pretty pictures may denote the various kinds of knowledge – but to what end? Unless you can indicate that different kinds of knowledge are accessed differently (and I think it’s unlikely that CK and SMK reside in different parts of the brain and get accessed in different ways), then what’s the point? The same is the case with taxonomic differences between, say, the affective and cognitive domains – the heart, for example, of Bloom’s taxonomy (Bloom 1956; Krathwohl et al. 1964). If the key question of interest is “how and why do people act the ways they do when engaging mathematically”?, the separation of cognitive and affective does little good. What matters is what people are trying to achieve, and what they take to be take to be true, and useful in pursuing their goals.

1

Isn’t there a decent term for this? “Mathophilic pedagogue” doesn’t exactly do it.

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This has serious consequences for understanding what people do. If a student thinks that (a + b)2 ¼ a2 + b2, that misunderstanding is part of that student’s cognitive reality, and we’d better be aware of it if we are to understand how the student acts mathematically. Similarly, if a student’s experience has been that homework assignments can be solved quickly if one understands the relevant methods, he or she is likely to develop the understanding that “all problems can be solved in five minutes or less.” By the mid-1980s, I referred to such understandings as students’ “belief systems”; I argued (Schoenfeld 1985) that what students took to be true, both with regard to mathematical knowledge and with regard to the nature of mathematics and their engagement with it, were fundamental determinants of the students’ success or failure in mathematical problem solving. Again, to be explicit: taxonomic differences between the affective and the cognitive domain serve little purpose when one is trying to explain human behavior. I developed belief systems as an early attempt to explain what mattered in thinking and problem solving. It took another 25 years before I was able to put things together more generally, in How we Think (Schoenfeld 2010). Again, the key idea in trying to understand (preferably, to model) someone’s behavior is to see the world from their point of view. What are they trying to achieve? That’s the issue of goals. What resources (including knowledge, but also tools, etc.) do they have at their disposal? (Note that the resources include what they take to be true, not necessarily what is true.) And, what orientations (including beliefs, values, biases, dispositions, preferences, and tastes) shape their behavior? Behavior that seems irrational (or at least counterfactual or non-intuitive) from the outside may be the result of a coherent and internally consistent calculus, once it is seen (well, modeled) from the inside. For example, the student who spent more than 5 min carefully making a straightedge and compass construction that copied a geometric figure given in a problem statement would seem, from the outside, to be wasting a substantial amount of time. That’s if one assumes that her primary goal was to solve the given problem. But what if she thought she wouldn’t be able to solve it? Then, avoiding full engagement with the problem, while doing a construction, met multiple goals. It showed that she knew geometric constructions, demonstrating some knowledge to the person who posed the problem. It allowed her to avoid direct engagement with the problem, so she could avoid failure: “I didn’t try, so I didn’t really fail.” And it bought her some time, in case inspiration might strike. In sum, from an internal point of view, that behavior was quite reasonable.2 My first point, then, is that if you want your research to make a real difference in practice, it is essential to view (and model) the world from the student’s point of view. To be sure, there exists important fundamental research that does not make direct contributions to practice, as there is practical work whose contributions to the knowledge base are minimal. My personal preference, however, is to work on problems that sit squarely in “Pasteur’s Quadrant” (Stokes 1997) – “use-inspired basic research.”

2

The student became my research assistant, and she later told me what I have discussed here.

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This leads to my second point, which focuses on how we can think productively about learning environments. The key question about learning environments should be, “what does the learning experience look and feel like, from the learner’s point of view?” The challenge I have been working on for the past decade has been to find an “actionable” description of what matters in classrooms, as experienced by students. The key question driving the work was: “is it possible to identify a comparatively small number of dimensions of learning environments with the property that when instruction goes well along those dimensions, the students who emerge from those environments are knowledgeable and resourceful thinkers and problem solvers? The result of this work is the Teaching for Robust Understanding (TRU) Framework (Schoenfeld 2013, 2014, 2015, 2018). Documents relating to TRU and tools for enhancing classroom environments can be found at the TRU Framework web site https://truframework.org/ and the Mathematics Assessment Project web site http:// map.mathshell.org/. The key idea is that the following five dimensions are central to teaching for robust understanding: 1. The content – what opportunities do students have to engage with the key ideas and practices of the domain? 2. Cognitive Demand – what opportunities do students have for “productive struggle” and sense-making? 3. Equitable access – what opportunities does every student have to engage with central content and practices? 4. Agency, ownership, and identity – what opportunities do students have to develop the sense that they are willing and capable of engaging with the content (agency), that they can make the content their own, above and beyond “ingesting” content defined by others (ownership), and that they can see themselves as “math people” (identity)? 5. Formative assessment – the degree to which student thinking is made public, so instruction can respond in ways that support and enrich dimensions 1 through 4. In brief: if a classroom does well along all five dimensions, the students in it will become knowledgeable and resourceful thinkers and problem solvers. If there are significant challenges along any of the dimensions, they will not. Note that although the content (Dimension 1) is centrally important, four of the five TRU dimensions center on how students experience that content. This background, and my biases toward work that resides in Pasteur’s Quadrant, shape my responses to the papers in this section of the volume. My intention is to (very briefly) highlight the relationships between the papers and both kinds of modeling (modeling as a curriculum activity, and modeling as a research activity), with an eye toward the kind of “use-inspired basic research” highlighted by Stokes (1997). While I appreciate the heritage and substance of Goldin’s work on the conative perspective, the first part of his chapter sits squarely within the taxonomic tradition and is thus somewhat problematic for me. What does it mean for conation to interact

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with affect and cognition? As discussed earlier in this response, I was driven to build the notion of belief systems precisely because affect and cognition weren’t separate entities that interacted. The whole idea was to explain what people saw, what people felt, what people did. Ultimately, the goal was to build models that provided explanations, at a level of mechanism, of how and why people did what they did in problematic situations. The models I built in How we Think (Schoenfeld 2010) spanned problem solving, teaching, and medical decision making. Conation was not formally labeled, but the notion of orientations includes including beliefs, values, biases, dispositions, preferences, and tastes. Moreover, it includes them operationally, as part of a model: in a particular context, with a person with a particular constellation of Resources, Orientations, and Goals is likely to behave in certain ways.3 Thus, conation is built into the model, in operational form. Unless I’m missing something, this operationalizes and supersedes the aspects of modeling in the second part of Goldin’s chapter. Finally, although it would take far more unpacking than is possible here, the idea behind the TRU framework is to address the needs highlighted in the chapter. For example, a challenge within the TRU framework is to build learning environments that provide students with opportunities to develop a sense of personal and disciplinary agency, a sense that they participated in the creation of their own knowledge, and thus have ownership of it; and a positive disciplinary identity. As Goldin notes, these are inherently social activities. That means that productive learning environments need to attend to such issues. For tools that help support teachers in doing so, see Baldinger, Louie, and the Algebra Teaching Study and Mathematics Assessment Project 2016, and Schoenfeld and the Teaching for Robust Understanding Project 2016. Gomez-Chacon and De la Fuente raise a series of interesting modeling-related and epistemological issues. Their literature review is terse and to the point, highlighting many of the limitations of earlier views. My only caution is terminological: the philosophical overtones of the word “epistemology” are problematic in the educational realm. Philosophers are hung up on strict definitions of notions such as “knowledge,” which is typically characterized as “justified true belief.” On its own, that raises interesting dilemmas, which I will only hint at: what was the epistemological status of the Jordan Curve Theorem when the entire mathematical community believed an incorrect proof of it?4 But, the key point of understanding human behavior is that what really matters is what the individual takes to be true, whether or not it actually is. One has to be careful to recognize that personal epistemology is not simply about truth and what can be verified. I am happy that the authors are clear about this:

3 To be precise, the model can produce a probability distribution of likely responses. But even without that, the operationalization of decision making is explanatory in nature. 4 The error was found, as well as a correct proof. But the point is we can only rely on fallible human judgment to assess the “truth” of very complex mathematical ideas. So, what is “justified true belief?”

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A. H. Schoenfeld We consider that the concept of personal epistemology allows relating the creation of subjective and objective knowledge in mathematics. The way in which the individual acts to develop his structures of thought. Moreover, with regard to the teacher, the manner in which they act to favour the mathematical thinking structures of their students. This suggests that the contexts of “discovery” (creation) and justification cannot be completely separated, since justifications, like proofs, are the product of human creativity as concepts, conjectures and theories. (manuscript page 18.)

My friendly expansion to the paper is to be explicit about the meta-level implications of their findings. The mathematical task discussed in the paper is interesting. Importantly, the task embodies an important aspect of mathematical modeling: there is not one “right answer.” As the authors note, the teacher FC’s view of mathematics changed: his perception of knowledge moves from “something static to something that is constantly changing.” I agree, and I also see a change in the rules of the game. In pure mathematics, certainty is a goal. One writes proofs; one solves problems and gets “the” answer. The meta-level rules of the game that one develops are consistent with this. But in modelling, the idea is to be reasonable, to make sure you’ve included all the things that matter, to decide how “close” you want to be, and to make sure that your model is indeed within the bounds of tolerance you’d like. It’s a very different game than pure mathematics. As such, playing it will be facilitated if one is explicit about how the game is played. As the research indicates, beliefs and orientations are shaped by experience – but they are shaped more effectively if one reflects explicitly on those experiences. One can hardly disagree with the fundamental premise in the paper by Wiezel, Middleton, and Jansen, that more is entailed in learning mathematics than the mastery of facts, procedures, and concepts. As but one example, addressing “math anxiety” has been a major (and profitable!) industry for decades. As google search on that phrase alone yields about 1,360,000 hits in less than half a second. “Math avoidance” yields about 2,130,000 hits in under .36 s. We can go all the way back to Bloom’s taxonomy (Bloom 1956; Krathwohl et al. 1964) for documentation of the importance of affect and, as I discussed in the opening paragraphs of this essay, their unfortunate separation. So happiness is important. But, if one is to consider aspects of the non-totally cognitive, how are they to be conceptualized – especially if one is to think about ways to enhance them in classrooms? From my obviously biased perspective, that’s where the TRU framework comes in. Dimension 4, Agency, Ownership, and Identity, is the most relevant here. Think about the times you’re happy, or the arenas in which you tend to be happy. For me, yes, math is a source of pleasure; the other day I spent some time working (at first unsuccessfully) on a problem that intrigued me, partly because it seemed like I should be able to solve it, but I couldn’t see a solution. When I finally did, I wasn’t yet pleased with it; I sent a note to my research group asking how they might think about the problem. A number of solutions were shared (along with some enthusiasm). Both the mathematics and the exchanges about thinking about the problem (we are math educators, after all) were a source of pleasure. But, of course, there’s more to life than math, or even math ed. What about food, as another example? Or music, or reading and writing, or sports?

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I’d argue that if you take pleasure in it, you have a sense of agency. I worked on the math problem because I found it interesting and thought I could make progress on it. I love to cook (including the times when I’m confronted with random things in the refrigerator and have to figure out how to make a meal out of them). That love came with the enjoyment of food (which may be wired in, but also has a social component) but also with the fact that, after much time in the kitchen, the results reward my efforts. And so on. In all of the things I mentioned, and whichever happen to be sources of pleasure for you, there’s a willingness on your part to engage, with the sense that the engagement will bring positive results. Some of those positive results come in the form of “ownership” – what’s produced is yours. That’s not the case when you’re working routine exercises out of a text; it’s much more likely to be the case when you’re pursuing something of interest. And ultimately, engaging in these ways becomes a part of you – I am a math person, I am a foodie, I am reader, etc. Why is “Agency, Ownership, Identity,” along with the other four dimensions of TRU, a more useful framing than “Happiness?” Perhaps the most important reason is that the five dimensions of TRU are actionable. If you frame a question as “how can I arrange the learning environment so that students are happier?” it’s hard to get traction. But if you ask, “how can I provide more opportunities for students to see themselves as people who can do mathematics?”, things begin to open up. The math can be made richer and more accessible; the level of cognitive demand (Dimension 2 of TRU) can be monitored in ways that support students in being successful; the classroom environment can be arranged in ways that are safer for students, support generating ideas, building on each other’s work, etc. (See the TRU web site, https:// truframework.org/, for relevant tools.) The same is the case for other goals. Recently, a group of teachers I’ve been working with said that a major goal problem they’re facing is to get their students to persevere. This framing is a challenge. What do you do, whip them? Reward them? But if you reframe things: why do people persevere? In part because they have a sense of agency. How do they develop it? With legitimate success, over time. How can you make that more likely? By paying attention to the level of cognitive demand (Dimension 2), adjusting it via formative assessment (Dimension 5), etc. Note, by the way, that agency serves as a conceptual link between perseverance and happiness. Finally, we come to Shahbari, Tabach, and Heyd-Metzuyanim’s chapter, “Development of Modelling Routines and its Relation to Identity Construction.” Here, I note that the communicational framework (Sfard 2008), in its very framing, tears down the wall between the affective and cognitive domains: Visual mediators and routines may be seen as the aspects of mathematical content and practices (the cognitive aspects of the framework), which are inseparable from the words and narratives in which they are embedded. As the title of the chapter suggests, those words and narratives are fundamental aspects of identity. The paper itself describes the simultaneous evolution of both mathematical talk and subectifying talk – an indication of how content learning and identity development are intertwined. This points to the utility of dialogic analysis as a tool for unpacking aspects of the development of identity. And, as the authors

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note, modeling activities elicit subjectifying activity, affording a personal “way in” to the mathematics that is not provided by more common classroom activities – especially if those activities are of the traditional “show and practice” form (Lappan and Philips 2009). That brings us back to the second part of this article’s title, concerning the why of modeling. I have always conceived of mathematics as a sense making activity; it is a matter of personal dogma that every bit of formal mathematics, K-14, can and should be experienced as the codification of sense making activities. Whether it’s counting objects you care about in elementary school or trying to sort out the complexities of the Summer Camp activity or the Good Teacher activity, trying to make sense of something that can be profitably mathematized is a great starting point for both building mathematical ideas and, deeply intertwined, building a mathematical identity. I say “starting point” because there is much more. We need to unpack the pedagogies that support such activities – every time I read a paper about powerful learning that took place in a classroom, I wonder about how the context and norms were established that supported that learning – and to think about the kinds of curricular and pedagogical design that would enable more classroom activities to provide the opportunities for individual and collective sense making afforded by the best modeling tasks.

References Baldinger, E., Louie, N., & the Algebra Teaching Study and Mathematics Assessment Project. (2016). TRU conversation guide: A tool for teacher learning and growth. Berkeley/Lansing: Graduate School of Education, University of California, Berkeley/College of Education, Michigan State University. Retrieved from: https://truframework.org/tools and/or http://map. mathshell.org/materials/pd.php. Bloom, B. S. (Ed.). (1956). Taxonomy of educational objectives. Vol. 1: Cognitive domain. New York: McKay. Krathwohl, D. R., Bloom, B. S., & Masia, B. B. (1964). Taxonomy of educational objectives: The classification of educational goals. Handbook II: The affective domain. New York: David McKay. Lappan, G., & Phillips, E. (2009). Challenges in US mathematics education through a curriculum developer lens. Educational Designer, 1(3). http://www.educationaldesigner.org/ed/volume1/ issue3/article11/index.htm. Ma, L. (1999). Knowing and teaching elementary mathematics. Mahwah: Erlbaum. Schoenfeld, A. H. (1985). Mathematical problem solving. New York: Academic. Schoenfeld, A. H. (2010). How we think: A theory of goal-oriented decision making and its educational applications. New York: Routledge. Schoenfeld, A. H. (2013). Classroom observations in theory and practice. ZDM, The International Journal of Mathematics Education, 45, 607–621. https://doi.org/10.1007/s11858-012-0483-1. Schoenfeld, A. H. (2014, November). What makes for powerful classrooms, and how can we support teachers in creating them? Educational Researcher, 43(8), 404–412. https://doi.org/10. 3102/0013189X1455. Schoenfeld, A. H. (2015). Thoughts on scale. ZDM, The international journal of mathematics education, 47, 161–169. https://doi.org/10.1007/s11858-014-0662-3.

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Schoenfeld, A. H. (2018). Video analyses for research and professional development: The teaching for robust understanding (TRU) framework. In C. Y. Charalambous & A.-K. Praetorius (Eds.), Studying instructional quality in mathematics through different lenses: In search of common ground (An issue of ZDM). Manuscript available at: https://doi.org/10.1007/s11858-017-0908-y. Schoenfeld, A. H., & the Teaching for Robust Understanding Project. (2016). The teaching for robust understanding (TRU) observation guide: A tool for teachers, coaches, administrators, and professional learning communities. Berkeley: Graduate School of Education, University of California, Berkeley. Retrieved from: https://truframework.org/tools or http://map.mathshell.org. Sfard, A. (2008). Thinking as communicating: Human development, the growth of discourses, and mathematizing. New York: Cambridge University Press. Shulman, L. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4–14. Shulman, L. (1987). Knowledge and teaching: Foundations of the new reform. Harvard Educational Review, 57, 1–22. Stokes, D. E. (1997). Pasteur’s quadrant: Basic science and technical innovation. Washington, DC: Brookings.

Engaging Students in Mathematical Modeling: Themes and Issues Peter Kloosterman

Abstract This commentary begins with a brief synopsis of each of the four very different chapters in Part II. It follows with discussion of five themes related to affect and mathematical modeling that cut across the chapters: social engagement, shortterm vs. long-term engagement, the importance of context, the development of theory, and the extent to which the claims made in the chapters are specific to modeling as opposed to mathematics teaching in general. The commentary ends with key insights for researchers and practitioners. These include the novel ways in which the chapters address affect, the implications of the theoretical models and conceptual frameworks presented in the chapters, and the directions for additional research that can be derived from the questions raised in the chapters. Keywords Affect · Beliefs · Context · Emotion · Engagement · Mathematics · Models · Theory

Over the last 30 years, the study of affective factors in the learning of mathematics has evolved. The First Handbook of Research on Mathematics Teaching and Learning (Grouws 1992), for example, included two chapters that focused primarily on affective factors: one of those chapters was specific to research on teachers’ beliefs (Thompson 1992) and the other was specific to student affect (McLeod 1992). The remainder of the First Handbook focused on cognitive, instructional, and assessment issues with relatively minimal attention to affect. The Second Handbook of Research on Mathematics Teaching and Learning (Lester 2007) included a chapter on teachers’ beliefs and affect (Philipp 2007) but student affect, to the extent that it was covered, was addressed only in chapters focused on learning of specific mathematics topics. For example, in their chapter on problem solving and modeling, Lesh and Zawojeski (2007) included a relatively brief section on student beliefs and dispositions. Other affective issues were not specifically addressed by P. Kloosterman (*) Indiana University, Bloomington, IN, USA e-mail: [email protected] © Springer Nature Switzerland AG 2019 S. A. Chamberlin, B. Sriraman (eds.), Affect in Mathematical Modeling, Advances in Mathematics Education, https://doi.org/10.1007/978-3-030-04432-9_7

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Lesh and Zawojewski although some of the other topics they covered, including habits of mind, flow experiences, and communities of practice, have affective components. Most recently, the Compendium for Research in Mathematics Education (Cai 2017) includes 11 chapters in a section titled “Students, Teachers, and Learning Environments.” Although not specific to affect, the chapters deal with non-cognitive issues including race, gender, language, identity, classroom discourse, and professional development. The only chapter dealing specifically with affect, written by Middleton et al. (2017) and titled “The Complexities of Mathematical Engagement: Motivation, Affect, and Social Interactions,” argues that affect is an important aspect of student engagement with mathematics and that, as researchers, we need to continually consider engagement as we study learning. Because student engagement is an explicit or implicit affect-related theme in each of the four chapters in Part II, I have selected engagement as the primary lens to discuss those chapters.

Affect and Engagement in the Four Chapters The four chapters in Part II represent a diverse set of perspectives on affect and mathematical modeling. I begin by summarizing the key concepts in each chapter in relation to student engagement. These comments are followed with commentary on themes that appear across the chapters.

The Conative Perspective (Goldin) Engagement in learning is typically viewed as having both cognitive and affective components. Goldin argues that a third component, called “conation,” interacts with and to some extent overrides cognition and affect when trying to understand engagement and motivation in mathematics. According to Goldin, conation “refers to the dimension (or domain) of human needs and drives, desires and goals, choices, intentionality, and ‘will’–that is, the ‘why’ behind human behavior” (p. 2). Goldin sees emotions such as anxiety and satisfaction as different conative sensations of need and desire and argues that good mathematics instruction motivates student learning in mathematics by fulfilling basic needs such as social involvement in classroom environments and moving toward career goals. Learning mathematics can also help meet the human goal of belonging to a community of individuals who communicate using mathematics, and the need for understanding in general. In addition to providing a number of examples of the basic needs that learning mathematics fulfills, Goldin outlines the components of a preliminary model for mathematics instruction that takes into account conation. The model focuses on keeping the perspective of students’ needs and personality traits in mind when teaching mathematics. Although the chapter does not explicitly address the use of

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mathematical models as an aspect of mathematics learning, learning to use models is an implicit aspect of quantitative thinking (the cognitive dimension) and building useful models can promote good feelings (the affective dimension). In addition, successful mathematical modeling promotes engagement and helps to meet personal desires for knowledge and success (the conative dimension).

Mathematics Teaching and Personal Epistemology (GómezChacón and De la Fuente) In stark contrast to the Goldin chapter, which looks at student engagement from the perspective of basic human needs, the Gómez-Chacón and De la Fuente chapter focuses on teacher decision-making from the perspective of personal epistemology. More specifically, it focuses on why research in mathematics education should include study of teachers’ beliefs about the nature of mathematical knowledge and beliefs about what should be done to help students learn mathematics. The chapter includes data and findings from a study of a high school instructor using a single mathematical modeling task with a class of 17-year old students. A major tenet of the instructor’s epistemology is that analogies can be very helpful in guiding students toward potential solution paths and thus he uses analogies to help students understand and complete the modeling task. In essence, analogies are ways of making connections between what the student knows and what she or he needs to figure out to complete a mathematics task. Two types of analogies are described, (a) analogy to existing knowledge of mathematics concepts and solution processes, and (b) analogy between two mathematical concepts or processes. Gómez-Chacón and De la Fuente propose a model of how the instructor’s epistemology impacts his decision making as he uses analogies to help students understand and propose a solution for the modeling task. Although the model is based on data from one teacher in one setting, it provides a sense of how instructors in other settings might base their instructional decisions on their own epistemologies. Student engagement, while not a specific focus, is key to the success of instruction based on the model.

Happiness and Mathematics (Wiezel, Middleton, and Jansen) Wiezel, Middleton, and Jansen begin with a premise that readers of this volume will likely agree with–factors beyond achievement need to be considered when

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documenting success in mathematics. With this supposition in mind, they argue that the construct of student happiness when learning and doing mathematics is a key to student engagement and should be receiving more attention by teachers and researchers than is currently the case. Wiezel et al. state that “In an education sense . . . to be happy is to learn and grow in one or more domains. . ., to experience pleasure while learning, and to reflect with satisfaction upon one’s long term educational experiences” (p. 5). They credit Seligman et al. (2009) when they identify three components of happiness: engagement, meaningfulness, and pleasure. They are quick to point out that that not all mathematical experiences are going to be meaningful or pleasant but overall, making mathematics enjoyable enough that students are willing to learn the mathematics they need to have a productive and fulfilled life should be an instructional goal. Although Wiezel et al. do not go back to human needs in the way that Goldin does, one could argue that engagement, meaningfulness, and pleasure are in fact human needs and thus it makes sense that being happy while doing mathematics is desirable regardless of whether it is essential for learning. Like the first two chapters in this section, Wiezel et al. provide a conceptual model–in this case a model of happiness based on engagement, meaningfulness, and pleasure. As is the case in Goldin’s chapter, Wiezel et al.’s chapter is not very specific to mathematical modeling but given that challenging content such as mathematical modeling often brings out emotions in students, the claims made in the chapter are clearly relevant to the modeling aspect of the mathematics domain.

Modeling Routines and Identity (Shahbari, Tabach, and HeydMetzuyanim) The chapter by Shahbari, Tabach, and Heyd-Metzuyanim analyzes the types of problem-solving routines and the identity-related comments made by five preservice mathematics teachers when working together on two Model Eliciting Activities (MEAs). As explained by Lesh and Zawojewski (2007), MEAs are activities where the goal is to develop and use a mathematical problem-solving process more so than to find an answer to a single problem. Using Sfard’s (2007) commognitive framework, Shahbari et al. found that the group’s routines (problem-solving methods) became more systematic the longer they worked and that the initial comments that group members made about their own experiences (i.e., identity comments) gave way to comments more focused on the mathematics as they worked through the MEAs. In contrast to the other three chapters where affectrelated constructs were treated as important considerations in planning mathematics instruction, affect appears in this chapter only as an outcome–the identity-focused comments made while preservice teachers worked on the MEAs. The chapter overlaps somewhat with the Gómez-Chacón and De la Fuente chapter in the sense that identity can rest in part on personal epistemology and thus students’ decisions on

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how to proceed when solving a mathematics problem are based on personal experiences, at least until they fully analyze the mathematical complexities of the problem. MEAs are intended to teach modeling and like some of the research involving MEAs, the focus in this chapter is on what took place when students completed MEAs with little explicit discussion of how the research findings apply to other aspects of doing mathematics. In contrast to the other three chapters, explicit models of thinking are not proposed.

Themes and Issues Social Engagement All four of the chapters in Part II involve social engagement in that they deal specifically with interactions among students and between students and their teachers. Goldin argues that “most, but not all motivating desires are overtly social, in that they relate mainly to having or avoiding particular kinds of social interactions” (p. 12). Desires that fall into this category include “Look how smart I am,” “Focus on me,” “Don’t disrespect me,” and “Help me” (p. 12). Social interaction is key to satisfying all of these needs. The Wiezel et al. chapter is also specific about social engagement, arguing that it needs to be considered along with the three traditional aspects of engagement: cognitive, behavioral, and affective. Given that interaction is key to effective instruction in mathematical modeling, there is a strong case for considering social engagement in such instruction. Wiezel et al. argue that the four different forms of engagement are related in all mathematics instruction, including instruction in modeling where, for example, deciding on a process for solving some types of problem (cognitive engagement) is usually based on the input the modelers get from a teacher or peers (social engagement). The other two chapters do not explicitly mention social engagement although it is key to the perspectives on teaching and learning in the chapters. Gómez-Chacón and De la Fuente’s emphasis on a teacher’s epistemology assumes social engagement as a key aspect of the decision making during a modeling activity and their data support this assumption. Shahbari et al.’s focus on identity construction, and their findings about the impact of students’ identities on the processes they used when developing mathematical models also clearly involve social engagement. In short, all four chapters describe the importance of productive social engagement patterns in mathematics learning and the data reported in the chapters support such engagement.

“In-the-Moment” vs. “Long-Term” Engagement With respect to mathematical modeling, “In-the-Moment” engagement is normally thought of as a student’s interest and engagement when working on a specific task as

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compared to longer-term feelings and engagement about mathematical problem solving or mathematics in general. Goldin and Wiezel et al. are more explicit about in-the-moment engagement than Gómez-Chacón and De la Fuente or Shahbari et al., but all four chapters focus on ways to get students engaged in mathematics and modeling in the short term with the expectation that short-term success and the positive affect connected with that success leads to appreciation for and engagement in mathematics over periods of months and years. Goldin stresses the importance of short-term success with his comment that “in-the-moment engagement is highly malleable in a way that students’ motivational traits are not” (p. 8). Wiezel et al. provide a compatible perspective with the comment that “we can think about pleasure in the context of mathematics as not just an in-the-moment emotional response to the conditions of learning, but as meta-affect that stabilizes the students’ momentary frustrations and triumphs” (p. 14). Wiezel et al. go on to argue that over time, predominantly pleasurable experiences in mathematics increase the odds that students will be engaged and thus successful in the subject. In-the-moment engagement is not specifically addressed by Gómez-Chacón and De la Fuente or Shahbari et al. although both focus on student success in solving specific problems with the expectation that students who are engaged in the short term are becoming more confident problem solvers. The Gómez-Chacón and De la Fuente chapter keys on the notion that analogy is an effective tool for engaging students in specific tasks. Shahbari et al. found that subjectifying talk (relating mathematics problems to personal experience) supported short-term student engagement by relating problems to non-school experiences. In short, all four chapters support the premise that in-the-moment engagement in modeling tasks is important and over time, such engagement improves long-term affect and performance in mathematical modeling.

Context and Generalizability of Results None of the four chapters address the issue of generalizability of their work and while generalization is often not addressed in qualitative research, the question of the extent to which the claims made in the chapters apply to populations or settings beyond those used for studies is important. The Gómez-Chacón and De la Fuente chapter describes research on the decision making of one teacher in a Spanish high school mathematics class. The description of how the teacher builds and uses analogy is likely to be of interest to researchers with interest in different methods of teaching modeling at the high school level. The chapter describes the teacher’s self-reported epistemic beliefs and the relationship between those beliefs and decisions the teacher makes in designing and implementing the instruction. GómezChacón and De la Fuente do not comment on whether anything beyond belief in analogy as an instructional tool may have impacted the teacher’s actions but that would be an interesting topic to explore as teachers’ decision are often based on a multitude of competing goals and options. In particular, it would be interesting to look at the effectiveness of analogy with students of various ages and the extent to

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which students become better over an extended period of time at using analogy to help mathematize situations. Gómez-Chacón and De la Fuente provide evidence that epistemic beliefs are related to what a teacher does in the classroom although this is not necessarily new information (see Philipp 2007). Like Gómez-Chacón and De la Fuente, Shahbari et al. focus on what happened with a single group of students in a short-term study. In contrast to Gómez-Chacón and De la Fuente, Shahbari et al. focus on the students rather than the instructor and leave open the question of the extent to which the comments made by the instructor impacted the statements made by students as they worked through the two MEAs. As is common in research involving MEAs, it appears that the students were left alone to complete the tasks and thus the instructor may not have been a significant factor in the way the students proceeded. However, if the instructor spent class time discussing the first MEA before students started on the second–a practice that is common in the classroom but not typically part of studies involving MEAs–the instructor could have been a significant influence on the types of statements students made and thus impacted how students’ roles outside of school impacted their thinking (a key theme in the chapter). A broader question with respect to the findings of Shahbari et al. is the extent to which the backgrounds of the five prospective teachers in the study impacted the findings. Readers are not told very much about the five individuals although it is clear from the comments that at least a couple of them were parents, they had a common understanding of what happens at a summer camp (one of the MEAs was about a camp), and while they were expected to take the role of a school principal for the MEA that focused on how a principal selects new teachers, they did not clearly do that. In short, the question remains of which aspects of the participants’ backgrounds impacted how they made decisions. Related questions include whether (a) the mathematical talk of secondary students who have less varied backgrounds than the participants of this study would become more systematic as they worked through an MEA, and (b) the findings of this study have implications for how we prepare teachers. Had Shahbari et al. provided more background information on the participants and context of the study, speculating on these issues would be easier. Again, generalizability is not typically addressed in qualitative research using MEAs, although it is important to consider if one assumes that MEA-based research is going to have impact on how we teach students to develop and use mathematical models. Because the Goldin and Wiezel et al. chapters are strictly conceptual, the issue of when and where the theories in the chapters apply is at least as important as in the data-based chapters. The Wiezel et al. chapter includes what appears to be a hypothetical analysis of two women as they worked their way through high school and college. In the Goldin chapter, the age of the students who made the conative statements reported in the chapter is not specified but the statements are typical of middle and secondary school students. It is not clear whether either chapter is intended to apply to students or classrooms at the elementary level although students at that level clearly have feelings about mathematics. The two students described by Wiezel et al. were both high achievers. It would be interesting to know the extent to

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which the authors of any of the chapters feel that the claims made in their chapters apply to students across the performance spectrum. In short, it is important to keep in mind that context, which includes background and experience of students, is an important factor in the learning of mathematics. The nature of the mathematics tasks, the mathematical experiences of students, the beliefs of the teacher, the classroom assessment practices, and the out-of-school experiences of students are among the multitude of factors that impact what students think and feel about mathematics. It is not, obviously, possible to consider all of these factors in research on mathematics teaching and learning but reporting on as many contextual factors as possible in a study makes interpreting the results easier.

Development of Theory Theory development is the primary focus of the Goldin and Wiezel et al. chapters and is an implicit focus of the other two chapters. Goldin’s six-factor preliminary model for using the conative (human needs) perspective to improve engagement and motivation in mathematics leads to important research directions in the area of mathematical engagement. One such direction involves understanding engagement structures, which include “characteristic patterns occurring in mathematics classrooms” (p. 8). There is currently a lot of interest in research on classroom norms, discourse, and the notion of making classrooms into learning communities (e.g., Cobb et al. 2017). What would we find if such research was approached from the Goldin’s conative perspective? Are there reasons why the conative perspective, which is better known among psychologists than among mathematics educators, has rarely been applied to mathematics learning? Wiezel et al. focus on engagement as a linear combination of four factors and define happiness as an experience that is engaged, meaningful, and pleasant. How would this perspective impact the way we study student engagement and success in classrooms? How would a focus on happiness integrate with a focus on teaching modeling processes? Shahbari et al.’s focus on modeling routines and identity development also points the study of affect in mathematics learning in a new direction. In the chapter, we see evidence that the focus of discussion shifts to be more systematic and less focused on individual experience but readers are not provided with a sense of how this fits with other research on how students work together to solve mathematic problems outside of the MEA framework. Tying MEA-focused research to broader studies of how groups of students solve problems (e.g., Vig et al. 2015) would deepen the theoretical impact of MEA-based work. Shahbari et al. argue that personal experience is a key factor in what students try to do in a modeling situation until they develop routines that are specific to the problem in question. This notion makes sense although exactly how it drives thinking and action beyond providing an anchor point to begin discussion is not addressed. While Gómez-Chacón and De la Fuente provide the reader with background on personal epistemology and reasoning processes, the extent to which their findings

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are compatible with research on teacher decision making (e.g., Borko et al. 2008) is not addressed, limiting the theoretical significance of their findings. MEA-based studies are often not connected to broader research findings, but those studies focus on the modeling process rather than connecting modeling to psychological theories as was done by Gómez-Chacón and De la Fuente. In contrast to Goldin and Wiezel et al., Gómez-Chacón and De la Fuente do not provide a specific theoretical model but they do show that personal beliefs about mathematics teaching impact a teacher’s decision making in the context of building an instructional model.

Mathematical Modeling as an Aspect of “Doing Mathematics” Given that this volume focuses on mathematical modeling, which is usually a relatively modest part of what is normally taught or expected in school mathematics, it is also important to think about how the nature of mathematical modeling might impact the claims of these chapters. As explained in Kaiser (2017), the notion of mathematical modeling dates back to the nineteenth century although it was not until recent years that it has become a significant focus in mathematics instruction. As noted above, the Gómez-Chacón and De la Fuente chapter, and the Shahbari et al. chapter report on studies in which students were expected to build relatively sophisticated mathematical models. Students worked together on the models and thus social interaction was important part of the modeling process. Affect was a component of the analysis in the chapters, but mostly as a secondary factor in relation to the modeling process. In the Goldin and Wiezel et al. chapters, the affective, or at least non-cognitive, realm was the primary focus of the chapters. An assumption in these chapters was that students would be doing mathematical tasks that required substantial thinking, but the ideas appeared to be applicable to any non-procedural mathematical content. None of the four chapters were explicit about the extent to which the findings reported were more relevant to the teaching and learning of mathematical modeling than other aspects of mathematics or relevant to content beyond mathematics. I do not see this omission as a limitation but it is important to keep in mind that mathematics context makes a difference–mathematics pedagogy can vary depending on exactly what mathematics students are expected to learn and modeling is one of many emphases in mathematics instruction in schools today. What we learn from the Gómez-Chacón and De la Fuente, and Shahbari et al. chapters clearly applies directly to teaching modeling. The messages from the other two chapters apply to modeling, but mostly because they apply to the teaching of mathematics in general.

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What Can Be Learned from the Four Chapters in Part II? Looking across the four chapters, one can ask the “big question” of what new insights are provided to researchers and practitioners. To begin, the chapters all contribute new perspectives on engagement in mathematics and mathematical modeling. Conation is rarely discussed in the mathematics education literature as demonstrated by the fact that it does not appear in the indexes of any of the three major research handbooks of research in the field (Cai 2017; Grouws 1992; Lester 2007). Goldin’s chapter provides a nice introduction to those interested in looking at mathematics learning from this perspective. Happiness is a household word but in mathematics education the terms enjoyment and liking are much more common. Wiezel et al. apply the psychological literature on happiness to mathematics education and thus bring that view to the field. While epistemological beliefs have been considered in mathematics education research, Gómez-Chacón and De la Fuente connect epistemology with teaching through analogy and this is unique. And, while MEA-based research is common, studying modelling routines and personal identity in the context of MEAs also appears to be a first. But, what else can be said? Beginning with conation, Goldin notes that, “The fundamental challenge to mathematics educators . . . is to relate the learning of mathematics in a deep way to fundamental human needs as they manifest themselves in the personalities of learners” (p. 7). Teachers seldom think in terms of fundamental needs so, while it is hard to argue with this claim, the key question is whether thinking in terms of conation helps them to better meet such needs. There appears to be little research on this issue but it is an empirical question and Goldin provides thoughts on how to begin study in this area. Education, in any form, should help to meet needs for competence and self-actualization so understanding whether there are advantages to thinking about learning and engagement in relation to conation rather than just the traditional realms of affect and cognition is an area where research could be very helpful. The Gómez-Chacón and De la Fuente chapter reviews personal epistemology, epistemic reasoning, and analogy as a tool. All of these topics are reviewed in the context of building traditional mathematical models (i.e., mathematizing a complex problem situation) and thus the chapter integrates topics that are not usually considered together. Gómez-Chacón and De la Fuente indicate that their focus on analogy comes from Polya’s third book that grew out of his problem-solving heuristic of looking at the solution to a related problem. Polya’s work on analogy does not have the notoriety of his four-step problem solving process and thus Gómez-Chacón and De la Fuente bring the reader back to historically interesting ideas about problem solving heuristics. They use the actions of the teacher in their study to help build models of using analogy in teaching. The models are consistent with what they report about the decisions made by the teacher during instruction although it is sometimes a bit difficult to see exactly how the models are the result of specific actions by the teacher. From the perspective of student engagement, it would be ambitious but helpful to provide a better sense of the connection between the

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instructor’s use of analogy and the psychology of learning. In general, like many limited-scale studies, the chapter leads to as many questions as it does answers. With respect to the Wiezel et al. chapter, it is easy to agree with the authors’ premises that (a) success in mathematics must be more than high achievement scores, and (b) students who are happy when engaging in mathematics are more likely to study and use mathematics in the future and that their engagement leads to more learning. The authors stress that students should be happy about being able to complete challenging mathematics tasks–ability to complete procedural tasks is not enough. This is particularly important with respect to mathematical modeling, which needs to be much more than simply knowing which formula to use to solve a word problem. The authors mention their ongoing large-scale study of students’ affect in secondary mathematics classes. A key goal of the study is to build a model of students’ sense of efficacy and well-being in relation to the mathematics instruction they receive. It is not clear exactly how happiness fits into this work but given that mathematical modeling is a topic that seems to have the potential to impact students’ feelings about mathematics differently than more traditional mathematics topics, it would be interesting to get a sense of the extent to which engagement and happiness vary as mathematics topics and assignments change over time. Given the current emphasis on test scores in the United States, it would also be interesting to know what parents, teachers, school administrators, and policy makers feel about efforts to increase happiness in mathematics, with or without simultaneous emphasis on performance. As noted earlier, the Shahbari chapter is the only one that says very little about theories of affect early on. The chapter introduction focuses on mathematical modeling, especially research involving Model Eliciting Activities and Sfard’s commognitive framework for understanding classroom communication. The framework makes use of identity although the authors restrict most of their discussion of identity to that framework rather than the broader field of identity in mathematics (e.g., Langer-Osuna and Esmonde 2017). As sometimes occurs in MEA-based research, and as was the case in the Gómez-Chacón and De la Fuente chapter, readers are given only limited information about how the modeling tasks in the study were selected and whether there was anything unique to the study participants or the classroom settings that made those tasks particularly relevant (or, as is sometimes the case, intentionally irrelevant). The authors do a good job of showing what parts of the data led to their findings and thus provide an example of how research designed from the MEA perspective can be used to analyze topics that are not normally addressed in that research tradition. As noted earlier, more could have been done to give a sense of the extent to which the study findings have the potential to be generalizable to similar settings. In short, all four chapters in Part II look at affect in mathematical modeling from new and unique perspectives. All include conceptual development of those perspectives and the case can be made that the type of engagement described in the four chapters should be a goal for all mathematics instruction. Paralleling what we see in the mathematics education field as a whole, there is no consensus in the chapters on what is meant by mathematical modeling or whether affect should be treated differently when instruction is specific to modeling as opposed to other aspects of

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mathematics. Keeping in mind that the chapters were developed independently, it is not surprising that each provides different insights. I trust that readers will see connections between the chapters that I did not, and that is a good thing. Different insights and perspectives are, after all, what make academic life so interesting.

References Borko, H., Roberts, S. A., & Shavelson, R. (2008). Teachers’ decision making: From Alan J. Bishop to today. In P. Clarkson & N. Presmeg (Eds.), Critical issues in mathematics education (pp. 37–67). London: Springer. Cai, J. (Ed.). (2017). Compendium of research in mathematics education. Reston: National Council of Teachers of Mathematics. Cobb, P., Jackson, K., & Sharpe, C. D. (2017). Conducting design studies to investigate and support mathematics students’ and teachers’ learning. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 208–233). Reston: National Council of Teachers of Mathematics. Grouws, D. A. (Ed.). (1992). Handbook of research on mathematics teaching and learning. Reston: National Council of Teachers of Mathematics. Kaiser, G. (2017). The teaching and learning of mathematical modeling. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 267–291). Reston: National Council of Teachers of Mathematics. Langer-Osuna, J. M., & Esmonde, I. (2017). Identity in research on mathematics education. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 637–648). Reston: National Council of Teachers of Mathematics. Lesh, R., & Zawojewski, J. (2007). Problem solving and modeling. In F. K. Lester Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 763–804). Charlotte: Information Age. Lester, F. K., Jr. (Ed.). (2007). Second handbook of research on mathematics teaching and learning. Charlotte: Information Age. McLoed, D. B. (1992). Research on affect in mathematics education: A reconceptualization. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 575–596). Reston: National Council of Teachers of Mathematics. Middleton, J., Jansen, A., & Goldin, G. (2017). The complexities of mathematical engagement: Motivation, affect, and social interactions. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 667–699). Reston: National Council of Teachers of Mathematics. Philipp, R. A. (2007). Mathematics teachers’ beliefs and affect. In F. K. Lester Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 257–314). Charlotte: Information Age. Seligman, M. E., Ernst, R. M., Gillham, J., Reivich, K., & Linkins, M. (2009). Positive education: Positive psychology and classroom interventions. Oxford Review of Education, 35, 293–311. Sfard, A. (2007). When the rules of discourse change, but nobody tells you: Making sense of mathematics learning from a commognitive standpoint. Journal of the Learning Sciences, 16, 565–613. Thompson, A. (1992). Teachers’ beliefs and conceptions: A synthesis of the research. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 127–146). Reston: National Council of Teachers of Mathematics. Vig, R., Star, J. R., Dupuis, D. N., Lein, A. E., & Jitendra, A. K. (2015). Exploring the impact of knowledge of multiple strategies on students’ learning about proportions. In J. A. Middleton, J. Cai, & S. Hwang (Eds.), Large-scale studies in mathematics education (pp. 61–74). London: Springer.

Chapter 5: Exploring a Conative Perspective on Mathematical Engagement Gerald A. Goldin

Abstract Mathematical engagement is a complex, multidimensional, and dynamic construct. It involves giving attention to one or more objects of engagement – e.g. a mathematical concept, a problem to solve, a situation to be modeled, and/or a person or group in the immediate environment. For a student, engagement often entails social interactions with a teacher, a parent, a tutor, or peers. Sometimes it is characterized as involving interacting cognitive, affective, and behavioral aspects (although these do not constitute distinct types of engagement). This chapter explores briefly another of its dimensions – the conative, which encompasses individuals’ experienced needs, goals, desires, and meaningful purposes, and how these are (or are not) fulfilled. Relationships among mathematical engagement, fundamental human needs, conative feelings, motivating desires, and engagement structures are discussed. Then I outline a possible model describing students’ in-themoment mathematical engagement during challenging classroom activity such as mathematical modeling. The crucial question for educators becomes how immediate mathematical experiences can meet fundamental, universal needs. This points toward ways of removing barriers to motivation and productive engagement associated specifically with mathematics. Keywords Conation · Mathematical engagement · Motivation · Motivating desires

Introduction Students engage mathematically for diverse reasons. Moreover, a student’s immediate reasons for engagement vary considerably from one occasion to another, depending on features of the situation. But over time, far too many become unmotivated, and disengage with mathematics.

G. A. Goldin (*) Graduate School of Education, Rutgers University, New Brunswick, NJ, USA e-mail: [email protected] © Springer Nature Switzerland AG 2019 S. A. Chamberlin, B. Sriraman (eds.), Affect in Mathematical Modeling, Advances in Mathematics Education, https://doi.org/10.1007/978-3-030-04432-9_8

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In this chapter, I endeavor to develop a conative perspective on mathematical engagement. Conation encompasses individuals’ experienced needs, drives, desires, goals, choices, and meaningful purposes, and how these are (or are not) fulfilled. I hope to persuade the reader that to motivate students of mathematics in large numbers in the long term, it is important for educators to connect the study of mathematics “in the moment” far more explicitly with fundamental, universal human needs. To accomplish this, we need to understand conative processes in mathematical contexts, and their interactions with affect and cognition. The chapter is organized as follows. Section “The conative domain” outlines some aspects of conation, including what it is and why it is valuable for us to distinguish it from affect in mathematics education research. I discuss some interactions of conation with affect and cognition, and consider social interactions in relation to conation. Section “Needs or drives, hierarchies, and desires” addresses the topic of fundamental human needs, hierarchies of needs, and their relation to in-the-moment desires that can drive engagement. I raise again the basic, frequently-asked question, “Why study mathematics?” and propose that ultimately, the answer must relate mathematics to universal needs – not superficially, and not just to address one need, but in consideration of the full spectrum of needs suggested by research in the psychology of personality. In section “In-the-moment mathematical engagement”, I stress the complexity and importance for mathematics learning of in-the-moment engagement. This leads into the discussion of motivating desires and engagement structures – constructs introduced in some of my earlier work, but not previously examined from an overtly conative standpoint. I then consider a set of motivating desires in greater detail, including some of their conative, affective, and cognitive aspects. I discuss how motivating desires can arise in service of other motivating desires in classroom contexts, and some ways to classify common motivating desires for mathematics in relation to well-known theories of motivation. Then in section “Outline of a model for productive student engagement in mathematics”, I outline in a preliminary way a model for (productive) in-themoment engagement in mathematics based on fundamental needs, educational contexts, motivating desires, and engagement structures. This points toward ways to ensure that affective and motivational barriers commonly associated with mathematics do not arise, and suggests some future research directions. The scope of the chapter does not accommodate an extensive literature review, and much relevant research is omitted. I try to indicate relationships with some other models, and provide citations where the most closely-related literature is reviewed.

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The Conative Domain Description and Discussion Psychologists sometimes, but not always, distinguish conation (or volition) as a component of human mental activity that parallels cognition and affect. Conation refers to the dimension (or domain) of human needs and drives, desires and goals, choices, intentionality, and “will” – that is, the “why” behind human behavior (e.g., Snow et al. 1996 and references therein). It thus extends naturally to include a person’s planning, constructing, and/or organizing ways to meet her needs, achieve her goals, fulfill her desires, etc. “Subcomponents” of conation have been identified as (for example) direction, energizing, and persistence (e.g., Huitt 1999). Human mental activity can be regarded as involving complex and dynamic interactions among conation, cognition, and affect. The discussion here comes from the perspective of a mathematics educator, not an educational psychologist. Many of the ideas mentioned have been well known for some time, especially in the psychology of personality, and I necessarily omit much that is important to the study of conation and motivation. My main purpose is to highlight the importance and potential value to mathematics educators of giving serious attention to and elaborating on the conative dimension in fostering students’ mathematical engagement. A second purpose is to outline a preliminary model that may be of use to mathematics educators and researchers in mathematics education. In the considerable work on motivation in theoretical or empirical studies of mathematics teaching and learning, an explicit focus on conation (e.g., TaitMcCutcheon 2008) has in fact been relatively rare. Instead, the conative dimension is usually treated rather tacitly, with motivation most frequently studied by focusing on its cognitive, metacognitive, and affective aspects (e.g., Hannula 2006; Jansen and Middleton 2011; Middleton et al. 2017 and extensive references therein). More specifically, goals are typically identified. They are characterized and classified, and their emotional content is considered. Motivation is sometimes described along the dimensions of extrinsic to intrinsic (drawn from the nature of the goal) or non-self-determined vs. self-determined (e.g., Ryan and Deci 2000); absent or mild vs. intense; or other descriptors. Empirical studies identify descriptive variables correlated with various measures of mathematical engagement or success. But questions about why specific in-the-moment the goals exist and how they function remain comparatively unexplored. What are the conative feelings or drives behind students’ goals, and their relationship to fundamental needs of the personality on the one hand and mathematics on the other? How can students’ highly-variable, potentially malleable classroom engagement experiences with mathematics be shaped in ways that allow long-term interest and motivation to develop?

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In the extant literature, cognitive aspects of motivation pertain especially to the framing of goals based on the person’s prior knowledge, appraisal of their attainability, formulation of strategies for reaching them, attention, (metacognitive) selfregulation, relevant cognitive structures; and so forth. Affective aspects pertain especially to the person’s emotional feelings (states and traits), including outcome emotions and achievement emotions (Pekrun 2006; Pekrun et al. 2007), emotional aspects of attitudes and beliefs such as self-efficacy beliefs (Dweck 2000; Leder et al. 2002), meta-affect (DeBellis and Goldin 2006; Goldin 2000, 2002), affective structures, etc. The affective domain of motivation is often studied under assumptions for which there is considerable empirical evidence; for example: • That positive emotions (or alternatively, eventual positive emotions after some experiences of difficulty) enhance motivation, while negative emotions impede it; • That behavior is generally directed toward incentives or possible rewards, whose achievement leads to experiences of emotions such as elation, pride, or satisfaction. Considering both cognitive and affective aspects, it is often taken to be the case (again, with empirical support): • That the most important distinctions among goals students may have for mathematics include intrinsic vs. extrinsic, approach vs. avoidance, proximate vs. distal, or ego (performance) vs. mastery (Middleton et al. 2017 and references therein), • That beliefs (having both cognitive and affective aspects) influence both the appraisal of a goal and the choice of strategy for reaching it (Leder et al. 2002 and references therein). Why, then, is elaborating this kind of analysis insufficient? Why should we consider conation as distinct from affect and cognition in our approach to mathematical motivation?

Distinguishing Conation from Affect To understand the value of this distinction, I think it is useful to begin with the idea of conative feelings, which I take to be subjective sensations that may be termed wants. These are not typically included in proposed taxonomies of fundamental emotions. Thus, “basic” emotions are often taken to include joy, sadness, surprise, anger, fear, and disgust. More complex emotional feelings particularly important to mathematical activity, such as anxiety, boredom, hatred, frustration, satisfaction, disappointment, or pride, are (at least in principle) related to the more basic emotions occurring in combination in particular situations. Affective constructs such as attitudes, beliefs, and values in relation to mathematics certainly do involve strong emotional components (Goldin 2014; Hannula 2006; McLeod 1992, 1994; Pekrun and Linnenbrink-Garcia 2014 and references therein).

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However, emotional feelings are quite different from conative sensations of need or desire. These include such “basic” feelings as hunger, thirst, sleepiness, weariness, sexual desire or attraction, physical discomfort, or the desire to touch or be touched; and more complex feelings such as the desire to dominate or submit to domination, to be intimate, to belong or to be accepted, to communicate, to inspire, to know and understand, and so forth. Note too that we often use sensations of physical need as metaphors for sensations of higher needs: One may be said to “hunger” for companionship, to “hunger” or “thirst” for knowledge, or to have a “passion” for mathematics. And apart from such metaphorical usages (which may provide some hints as to underlying conative structure), I would conjecture here that such conative feelings connect strongly with mathematical motivation and engagement in ways that remain to be fully studied. Explicit consideration of conation allows us to explore the sources of student engagement in conative constructs: to address deeply the question of why what the student is doing matters (or does not matter) to him or her. Thus: • I would like to set aside the conjecture (often tacitly assumed) that in-the-moment goal formation and persistence is explained fully by anticipation of success, or by the positive emotions that will result from goal attainment – i.e., the conjecture that goals have an affective origin or can be fully understood through anticipated consequential affect. • I would like to replace this by the conjecture that every in-the-moment goal has a conative origin, distinct from affect and cognition. Thus, the goal may be framed and strategized cognitively, and affect may occur in anticipation of or as a consequence of the goal’s being attained or not, or the degree of progress toward the goal; but neither cognition nor affect is itself the source of the goal. For example, the expectancy-value theory developed by Wigfield and Eccles (2000) and their collaborators, focuses on cognitive and affective constructs such as expectancies of success, beliefs about ability, and subjective values, as explanatory of motivation. The self-determination theory developed by Ryan and Deci (2000) and their colleagues, identifies three “innate psychological needs” whose satisfaction leads to greater intrinsic or self-determined motivation: competence, autonomy, and relatedness. Characterization of these as needs points to the important role of conation. But if we take a conative perspective on motivation and engagement, we should understand not only the finer structure of these three needs (each of which may actually be categories embracing several more specifically-defined needs) but the role of many other fundamental needs of the personality. We should understand structures built up from such needs that pertain to mathematical activity, as we have come to understand cognitive structures and affective structures. We should likewise explore how extrinsic motivation relates to fundamental needs and to structures of conation. From this perspective, motivation “in the moment” may be regarded as a dynamic interaction of the conative with affective and cognitive dimensions. Likewise, the importance of social interactions to motivation can be understood from a conative perspective – through exploration of the needs that social interactions fulfill, can

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fulfill, or fail to fulfill in mathematical contexts. Then we may be able to see in greater detail the ways in which productive mathematical engagement relates to students’ needs, desires, and will, and lay a foundation for methods of fostering such engagement in classroom environments. I have mentioned only a small fraction of the relevant research literature. Additional references of importance having points of contact with this chapter include Sansone and Harackiewicz (2000), Midgley et al. (2001), Harackiewicz et al. (2002), Krapp (2005), Sansone and Thoman (2005), Hidi and Renninger (2006), Harackiewicz et al. (2008), Usher (2009), Renninger and Hidi (2011, 2016), Larson et al. (2014), and many others.

Conation Interacting with Affect and Cognition Some characterizations that pertain to the study of emotion are also relevant to conation. For example, as we do for emotions, we may distinguish conative states (“in the moment” desires or wants), from traits (longer-term conative characteristics of the individual). We may consider the architecture of conation (structures and relationships underlying or supporting various desires). Thus, to say a student desires recognition from the teacher of his correct problem solution (“in the moment”), may be descriptive of his state of conation. Such a desire may possibly be identified as seeking an affirmation of competence, or as seeking relatedness, or both. But there is much more to be said about this student’s state. To say the student has a high need for recognition or affirmation from those around him, may describe a personality trait – but this does not necessarily mean that on any specific occasion, he will experience this desire. Evocation of the desire is likely to depend on the context and circumstances of the mathematical activity. And to say that a student’s desire for recognition or affirmation can stem from a deeper need for nurturance or security is to assert something about the architecture of conation – a possible relation between her in-the-moment desire and a more fundamental need of the personality. Even if correct in general, such a relation may or may not be applicable to any specific individual’s traits, or to the individual’s desire as it occurs on a particular occasion. Problem solving (in mathematics, or more generally) is usually defined to occur in a situation where a person “has a goal” but does not immediately know how to reach it, or experiences some impasse in reaching it. Typically, the goal in a mathematics problem might be to find values of an unknown satisfying an equation, to prove a stated theorem, or to answer the question in a “story problem” situation. But what does it mean to “have” such a goal in the first place? What other goals, desires, or needs (of a “non-mathematical” nature) does the immediate problem goal serve? And why are these important to the student? These are conative questions. Let us consider briefly how conation is intertwined with affect and cognition in relation to mathematical activity. In earlier work, I have consistently considered in-

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the-moment or “local” affect occurring during problem solving to be representational (DeBellis and Goldin 2006; Goldin 2000). That is, emotional feelings encode essential information, some of which is strategic (i.e., cognitive). But let us look closely at what is encoded by some of the most commonly-occurring affect. Thus curiosity (often considered affective) typically encodes a desire (conative) to know (cognitive). Frustration (affective) likewise may encode unmet desire (conative) as well as unsuccessful strategy (cognitive). Loneliness (affective) may encode craving (conative) for the joy of companionship or acceptance (affective and cognitive). Fear (affective) encodes an urgent need (conative) for escape to safety (affective and cognitive). Attitudes, beliefs and belief structures likewise have important conative components, beyond their cognitive and affective aspects. Consider, for example, selfefficacy beliefs regarding mathematics, which may partially constitute a student’s mathematical identity (e.g. Dweck 2000). These involve propositions, imagery, etc. about the person’s abilities, held to be true, valid, or right in some sense (cognitive). The beliefs typically carry emotional charge for the believer (affective). But they also have powerful implications for what is desired or not desired in mathematical situations (conative) – e.g., whether possible objects of desire such as understanding a mathematical idea, persuading others through reasoning, achieving a high grade, impressing the teacher with one’s ability, or helping other students, are in actuality worthy of desire, or not. Beliefs influence not only what individuals desire when they engage with mathematics, but how they manage or control their desires – a metaconative aspect of their activity. It is plausible, then, to consider motivational structures or engagement structures (Goldin et al. 2011) pertaining to mathematics as fundamentally conative, involving needs, goals, and desires – whether long-term (e.g., a career goal) or immediate (e.g., a mathematical problem-solving goal). Needs and desires evoke cognitive appraisals of possible success and affective responses to failure or success. In the moment, students experience affective pathways (changing emotional states), meta-affect (affect about affect), and the longer-term influences of prior affective experience. But we cannot fully understand mathematical motivation without careful analysis of how the cognitive and affective domains interact with the conative. And as mentioned above, there is the important phenomenon of meta-conation (Snow 1996), which encompasses the internal monitoring and regulation of needs, goals, and desires, as well as the person’s desires about desires, or meta-desires – e.g., when one wants to want something but does not, or else has unwanted desires. Such monitoring, regulation, and meta-desire likewise involves both affect and cognition. These dimensions – the cognitive, affective, and conative – interact dynamically during mathematical engagement (in the moment), and combine into more enduring psychological structures. A theory which describes less than this is ultimately insufficient.

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Needs or Drives, Hierarchies, and Desires Many famous theories of personality begin with a characterization of universal human drives or needs. It is not necessary to make a specific commitment to any one theory in order to regard the following as unquestionably important to conation: • Sexuality, the “pleasure principle,” and “sublimation” (libido), and aggression (“death-wish”), fundamental to the psychoanalytic ideas of Sigmund Freud [1856–1939] (Freud 1990); • The “will to power” or “striving for recognition and superiority,” characterized by Alfred Adler [1870–1937] (Adler 1917, 1998); • The “will to meaning,” in the profound thinking of Victor Frankl [1905–1997] (Frankl 2014); • Needs of the personality as described by Henry Murray, including (for example) understanding, succorance, harm avoidance, deference to another, avoiding belittlement, achievement, avoiding blame, and nurturance (Murray 2008); • The “hierarchy of human needs” proposed by Abraham Maslow (1943), moving from “deficiency needs” to “growth needs”: (1) physiological needs, (2) safety needs, (3) social needs (love, belonging, intimacy), (4) needs for esteem and accomplishment, (5) the need for self-actualization and creativity. The above characterizations are overlapping, and of course they are described but sketchily in this rather inadequate summary. In my view, they may also be incomplete in important ways. But the basic idea that such needs exist, and that they drive behavior, is essential to conation. Let me remark that the concept of self-actualization (at least, as I interpret it here) need not imply a self-centered or “ego” focus. For many, the greatest selfactualization may occur through meeting the needs of others. Discussions of fundamental needs of the personality may seem quite far from mathematics education research. Historically, they have received little attention from our field. But at this point, let us consider the questions often asked by students. “Why study mathematics? What good is it to me? When will I ever use it?” The usual answers, “You need math for college. It is a prerequisite for many careers. You can use it in everyday life, when you go shopping or open a bank account. It has many other practical applications, which you will understand later in life,” or, “It helps you learn to think logically,” are usually unsatisfying, frustrating, and often untrue – what practical uses (now, or later in life) do most students ever find in trigonometric identities, the quadratic formula, or various criteria for establishing the congruence of triangles? The fundamental challenge to mathematics educators, in my view, is to relate the learning of mathematics in a deep way to fundamental human needs as they manifest themselves in the personalities of learners. It is especially important that we understand the immediate, in-the-moment desires that stimulate mathematical engagement, as well as the long-term goals and aspirations that motivating continuing mathematical study, as serving basic needs of the personality.

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Our goal should be for students to experience mathematics as conferring power, providing meanings, and offering possibilities for belonging, esteem, and selfactualization – not just for a few, but for everyone. And such possibilities should be offered not as far-off, intangible, potential benefits. They should occur in the hereand-now, lived experiences of students, beginning in kindergarten and extending throughout their education. When reading and writing are well taught, students can have such experiences. Why not in mathematics? To accomplish this, I think we must understand deeply the phenomenon of mathematical engagement – what drives it, what structures influence it or govern it, and what conditions foster or inhibit its occurrence and development. The main point of this chapter is to argue that the relevant structures are fundamentally conative. Personal empowerment can be enhanced through ways of thinking provided by mathematics, as well as by problem-solving skill as it develops and is applied with tangible results. Understanding increases with the pursuit of mathematical curiosity and the experience of reward in its awakening and fulfillment. Mathematics offers aesthetic experience, the beauty and fascination of exotic patterns, and many possibilities for creative thinking and artistic expression. Mathematics can fulfill social needs – it is a language for communication, opening possibilities to experience belonging to a local community and to a world community. It offers a way to “see” the world, a lens that is different from but complementary to ways of “seeing” that might be called emotional, literary, artistic, political, economic, spiritual, or other ways of experiencing life. Through mathematics, one acquires ways to contribute to others – to peers, to a local community, to humanity. Mathematics can contribute significantly to the formation of personal identity. And of course, it opens life opportunities – not only economic opportunities, but also social and recreational ones. Our challenge is to translate such potential ways in which mathematics can meet fundamental needs into students’ immediate school experiences. And as mathematics teachers, educators and educational researchers, we can also embrace the notion that inspiring the learning and appreciation of mathematics is a form of our own self-actualization.

In-the-Moment Mathematical Engagement Complexity and Importance Sometimes mathematical engagement is operationalized as a trait – a relatively stable characteristic of the individual that endures for months, a year or more, or a lifetime. Likewise, mathematical disengagement is seen as the opposite trait. Engagement is also sometimes interpreted as unidimensional: that is, a student may be more engaged with mathematics, or less. It is sometimes seen as tridimensional: researchers try to distinguish among a student’s cognitive engagement, affective engagement, and behavioral engagement – although these categories

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were originally introduced as convenient headings for a research review, rather than as distinct types of engagement (Fredricks et al. 2004). But to understand a student’s in-the-moment mathematical engagement as a dynamically evolving state, we need a different approach. First, one must note the complexity of the phenomenon. It entails an object of engagement (e.g., a math problem), but this object may change (and change back) from one moment to the next. There is an immediate focus of attention – that is, arousal in relation to specific feature(s) being attended to, related to the object of engagement. There is an interaction of the student with her immediate environment, and the interaction is one that matters to the student for reasons that may or may not be specific to mathematics. Engaged activity is highly variable, dependent on incentives, the task environment, the social press, and many other psychological and contextual factors. While complex and variable, and thus more difficult to study than students‘more stable characteristics such as attitudes and beliefs, mathematically engaged activity – as it takes place in classrooms and sometimes out of school – is recognized as centrally important to learning. And what takes place “in the moment,” as distinct from longer-term affective or motivational traits, is open in that same moment to influence by the teacher. That is, in-the-moment engagement is highly malleable in a way that students’ motivational traits are not.

Engagement Structures and Motivating Desires in Mathematics Classes To describe the dynamic complexity of in-the-moment engagement, my colleagues Yakov Epstein, Roberta Schorr, Lisa Warner, and I identified in earlier work some characteristic patterns occurring in mathematics classrooms, which we termed engagement structures (Goldin et al. 2011). An engagement structure is a kind of psychological/social/behavioral constellation. It is situated in the individual, and becomes active in a specific situation. It is archetypal, in the sense that similar engagement structures seem to develop nearly universally in human beings. It includes: • • • • • • • • • •

An in-the-moment goal or motivating desire (that defines the structure), Characteristic behaviors, including especially social interactions, Sequences of associated emotional states (affective pathways), Expressions of affect, Meanings and implications encoded in emotions, Meta-affect, Self-talk – that is, the person’s thoughts, or “inner voice,” Interactions with the person’s beliefs and values, Interactions with the person’s orientations and personality traits, Interactions with the individual’s (mathematical and non-mathematical) cognition.

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There is an analogy here with cognitive structures that have been characterized in describing the complexity of mathematical problem solving and development – e.g., conservation of number, volume, etc., proportional reasoning, coordination of conditions, and so forth. These, too, develop nearly universally in individuals as they develop, and become active in specific problem situations. They typically involve multiple systems of internal cognitive representation, characteristic cognitive processes, and observable patterns of behavior. They are what I regard as “mid-level structures” – far more complex than discrete skills or competencies, but not as “global” as overall mathematical ability or motivation. And they help us understand certain patterns in mathematical cognition and behavior that would otherwise be extremely difficult to analyse. For example, during problem solving, cognitive structures can “call on” or activate other cognitive structures, or one active structure can “branch” into another – for example, a student coordinating two mathematical conditions may make use of proportional reasoning in formulating one of the two conditions. But a student with a well-developed structure is not always using it. Its activation depends strongly on the context. Likewise, engagement structures as we conceive of them develop in most or all students, but are not always active. When they are, one may “call on” another, or “branch” into another structure. Engagement structures, too, are “mid-level structures” – more complex than goals, emotions, or behaviors, but less global than personality traits. In earlier work, we labeled engagement structures as “archetypal affective structures,” but my perspective now is that they are better considered as conative structures – the motivating desire is the pivotal element, in response to which the full engagement structure (behavior, thoughts, emotions, etc.) develops as a stable pattern. Indeed, we label the engagement structure by the motivating desire that initiates its activation in the moment. Examples of previously-identified motivating desires (and corresponding engagement structures) often active in mathematics classrooms are the following (Goldin et al. 2011): • To complete the assigned task, thus fulfilling a commitment (Get The Job Done) • To impress others (or oneself) with one’s mathematical ability (Look How Smart I Am) • To achieve a desired payoff, which may be “extrinsic” or “intrinsic” to the mathematics (Check This Out) • To experience the joy of understanding and/or deep immersion in the activity, or flow (Czikszentmihalyi 1990) (I’m Really Into This) • To confront a challenge to one’s status, or “save face” (Don’t Disrespect Me) • To avoid a possible interaction leading to trouble or conflict (Stay Out Of Trouble) • To address a perceived inequity and restore fairness (It’s Not Fair) • To help someone else understand the mathematics (Let Me Teach You) • To “look good” by giving a false impression of engagement (PseudoEngagement)

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• To have one’s culture or heritage acknowledged as it occurs in a multicultural mathematical context (Verner et al. 2013) (Value My Culture) In addition to the above, some subsequently proposed motivating desires occurring in mathematics classes include (Goldin 2017): • To escape from the current social environment (I Want Out) • To interrupt the ongoing mathematical activity of others in the class (Stop The Class) • To dominate another in the activity (Do As I Say) • To submit to another in the activity (Just Tell Me What To Do) • To have it acknowledged that one’s answer or solution is right, while a rival’s is not (I’m Right You’re Wrong) • To obtain assistance with the mathematics (Help Me) • To be held as worthy by the teacher or other students (Value Me) • To avoid notice by the teacher or other students (Don’t Notice Me) • To be the center of attention (Focus On Me) An ongoing exploratory study with preservice and practicing teachers of mathematics led to our identifying two additional motivating desires (Warner et al. in progress): • To make a meaningful contribution to one’s group, and • To avoid letting one’s group down. Note that the goal (the object of desire, or the situation to be avoided) may be tangible (e.g., a payoff), but more often is a situation likely to involve an emotional state. The above is more than a list of context-based goals resulting (if attained) in positive emotions. Each motivating desire stems from the possibility of meeting some more fundamental human need. Each commonly evokes some characteristic behavior patterns, strategies, thoughts or self-talk, affective pathways, etc., suggesting considerable structure. And as noted, when an engagement structure associated with a motivating desire become active, it may evoke other motivating desires in service of fulfilling the original one. For example, a student who wants to help another student understand a mathematical concept (“Let Me Teach You”) may find that his knowledge or mathematical expertise is not recognized, and his help is rejected. Responding to the frustration of his original goal, he may experience a desire to impress the other student with his ability – so that the patterns associated with “Look How Smart I Am” become active. It may then occur that the original goal becomes less salient or is forgotten. In that case, “Let Me Teach You” fully branches into “Look How Smart I Am.” Likewise, a student who wants her group to complete the assigned task (“Get The Job Done”) may observe that another student fails to understand some mathematical point. This may evoke the desire in her to help the other student, so that “Let Me Teach You” becomes active. Such branching of engagement structures is analogous to the way processes associated with well-established cognitive structures may call on or evoke other processes.

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Characterization of engagement structures associated with in-the-moment motivating desires allows a descriptive richness at the psychological level of the complex dynamics involved with mathematical engagement.

Some Ways to Classify Motivating Desires for Mathematical Activity The motivating desires identified may be classified along various dimensions that have been identified as important in models of mathematical motivation or engagement. Thus, some involve overtly mathematical goals, while others do not. Among those for which the object of attention most often relates explicitly to mathematical content, we have: • Get The Job Done • Look How Smart I Am • Check This Out (intrinsic payoff) • I’m Really Into This • Let Me Teach You • I’m Right You’re Wrong • Help Me An important distinction is drawn between mastery vs. performance (or ego) goals. The former are centered mainly on learning the mathematical content; the latter on comparison of oneself to others. Motivating desires mainly or most frequently involving mastery goals include: • Get The Job Done • I’m Really Into This • Check This Out (intrinsic payoff) • Let Me Teach You • Help Me Motivating desires mainly or most frequently involving performance or ego goals include: • Don’t Disrespect Me • Look How Smart I Am • It’s Not Fair • Pseudo-Engagement • Do As I Say

• Stay Out Of Trouble • Value Me • Just Tell Me What To Do (surrender of ego to another)

• Check This Out (extrinsic payoff) • Focus On Me • I’m Right You’re Wrong

Another important distinction is drawn between approach vs. avoidance goals. An approach goal involves reaching or attaining an objective; an avoidance goal involves preventing something from occurring, or staying away from a specific situation or outcome. Motivating desires typically involving an approach goal include: • Check This Out • Let Me Teach You • I’m Really Into This • Do As I Say • Stop The Class

• Look How Smart I Am • Get The Job Done • Value My Culture • Focus on Me • Value Me

• Don’t Disrespect Me • Focus On Me • I’m Right You’re Wrong • Help Me

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Those typically involving avoidance goals include: • Stay Out Of Trouble • Pseudo-Engagement • Don’t Notice Me • I Want Out • Just Tell Me What To Do Finally, most but not all motivating desires are overtly social, in that they relate mainly to having or avoiding particular kinds of social interactions. Here I would include: • Stay Out of Trouble • It’s Not Fair • Value My Culture • Do As I Say • Stop The Class

• Look How Smart I Am • Pseudo-Engagement • Focus On Me • Help Me • Value Me

• Don’t Disrespect Me • Let Me Teach You • I’m Right You’re Wrong • Don’t Notice Me • I Want Out

But merely classifying motivating desires does not itself address the profound role that conation plays in mathematical engagement. To better explore this role, I would like to propose in outline form a preliminary model that takes the conative dimension more deeply into account.

Outline of a Model for Productive Student Engagement in Mathematics Engagement structures as described above are complex constructs, organized around motivating desires but involving characteristic patterns that ensue. But the motivating desire or goal is itself already complex, with affective, cognitive, and conative aspects, and typically involves social interactions in an important way. The motivating desires that students experience in the moment, leading them to engage in mathematics, are diverse. How do they arise? How can we as educators understand them, encourage the most appropriate desires in various contexts, and make use of our understanding to promote long-term, fruitful relationships with mathematics in most or all of our students? Referring to Murray’s characterization of human needs, we suggested that each motivating desire could stem from an opportunity to fulfill such a need (Goldin et al. 2011). For example, “Get The Job Done” can stem from a need for deference, “to yield to the influence of an allied other” – in the mathematics classroom, the “allied other” is the teacher who describes the task to be completed. “Look How Smart I Am” stems from the need for achievement, “to increase self-regard by the exercise of talent.” Likewise, the need for understanding, “to represent in symbols the order of nature,” may underlie “I’m Really Into This;” while “Don’t Disrespect Me” may stem from a need for infravoidance, “to avoid conditions which may lead to belittlement,” and “Stay Out Of Trouble” from a need for harmavoidance, “to take precautionary measures.” Pseudo-Engagement stems from blame avoidance, “to avoid blame or rejection.”

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This point of view seems likewise compatible with the self-determination theory of Ryan and Deci (2000) mentioned above, where competence, autonomy, and relatedness needs (or types of needs) are identified. The present outline of a (preliminary) model does not commit to a specific theory or hierarchy of fundamental needs, nor does it posit some needs as superior to others in the kinds of motivating desires that can stem from them. Rather I propose several components that embrace the complexity of engagement with mathematics in classroom contexts. For each component, I suggest some directions for mathematics education research that – with attention to conation – can contribute to the advancement of engagement. The components of this model are: • (1) A sociocultural context conducive to meeting some fundamental needs through mathematical activity; • (2) Some ways offered by the teacher whereby such needs can be met; • (3) Students’ individual trait-dependent thresholds for various motivating desires to become active in the given situation(s); • (4) Engagement ensuing from the activation of motivating desires, with encouragement and monitoring; • (5) Characteristic patterns of emotions, cognition, social interactions and other behavior associated with engagement structures; and • (6) Engagement outcomes influencing the development of longer-term, powerful structures of affect and conation in relation to mathematics. Let us consider each of these briefly. (1) The school culture and the classroom culture set a context for mathematical activity. To foster engagement, the culture should be one in which students perceive opportunities to meet some fundamental need(s) through their mathematical activity, and can potentially experience a sense of fulfillment. Research is needed to investigate what constitutes such contexts, and why: e.g., expectations of meaningful mathematics, culturally relevant mathematics, classroom norms fostering respect for problem solving activity that includes valuing mistakes, expectations of success, belongingness and shared values, esteem by peers, etc. (2) The teacher of mathematics contributes (explicitly or implicitly) a variety of ways in which students’ need(s) can be met, together with possible or proposed objects of engagement (e.g., mathematical problems, modeling or other activities, team activity). This creates possibilities for one or more motivating desires. The immediate goal (e.g., recognition, helping another student, successful completion of a task) is perceived as desirable by students because, in this context, it meets some fundamental need. Research is needed on the conative aspects of various ways the teacher “sets the stage” for the mathematical activity – e.g., describes a task to be completed, appeals to students’ curiosity, offers an extrinsic reward, creates a teamwork situation, etc. – and what follows. The process of setting expectations and inviting possibilities may be influenced, for

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(4)

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example, by teachers’ characteristics – e.g., mathematical knowledge for teaching, attitudes toward mathematics etc. The students, of course, have a spectrum of personality traits, characteristics, and orientations toward mathematics, some more malleable (potentially) than others. These influence individual thresholds for various motivating desires of students to become active, in the context that has been set. Students’ cognitive and affective appraisals result in their identifying possible approach goals (objects or situations of desire) and/or avoidance goals. Research is suggested to investigate how student interests, self-efficacy and other beliefs, personality traits, mind-set, grit, and other characteristics, influence the activation and continuation of various motivating desires in different mathematical contexts that occur in schools or other environments. A variety of productive motivating desires (or possibly, less productive ones in the given context), as identified above, arise among the students. Each student’s motivating desire is oriented toward some object of engagement (or toward attaining some object or situation), and stimulates attention by the student directed toward that object or features related to the object. Engagement ensues, encouraged and monitored through group and peer interactions, and interactions with the teacher. Research is suggested to investigate the dynamics of conation, affect, cognition, and social interactions as activity proceeds, with special attention to the conative dimension – how desires and goals change during the activity, including branching from one motivating desire to another, and how they evolve over longer periods of time. Characteristic patterns of engagement then occur, associated with specific motivating desires. These can be understood as active engagement structures (as described in the literature) driving the students during their activity, including sequences of emotional feeling (affective pathways), self-talk, social interactions and other behaviors. Further research can identify additional engagement structures associated with mathematical activity, distinguish operationally among engagement structures, and characterize them further – particularly, their conative origins. Finally, outcomes of engagement fulfill (or do not fulfill) the multiple motivating desires initiating active engagement structures. When desires are fulfilled in ways that students experience as meeting fundamental needs over a sufficient length of time, what develops is long-term, productive mathematical engagement. When motivating desires appropriate to learning are frequently frustrated, mathematical activity itself becomes something to be avoided. Then we see the all-too-common development of low self-efficacy beliefs, “math anxiety,” and avoidance mechanisms. Further research is suggested to trace the conative dimension in the origins and development of mathematical motivation and mathematical aversion.

My conviction is that the greatest thing we can do as educators is inspire our students to discover who they are and to be all they can be. In the process of doing so, we

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come to know who we are and to be all we can be. Mathematics offers paths of discovery, beauty, understanding, ways of seeing patterns in the world, ways of thinking, means of communication with others, possibilities for expression, opportunities for intimate connection, and powerful means of accomplishment – all in service of full self-actualization. None of these are intrinsically limited to a small fraction of our students. Study of the conative dimension in mathematical learning, in interaction with cognition and affect, may be crucial to opening mathematical understanding as a way of being for all students in future generations.

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Tait-McCutcheon, S. L. (2008). Self-efficacy in mathematics: Affective, cognitive, and conative domains of functioning. In M. Goos, R. Brown, & K. Makar (Eds.), Proceedings of the 31st annual conference of the mathematics education research group of Australasia (pp. 507–514). Brisbane: MERGA. Retrieved December, 2017 from https://files.eric.ed.gov/fulltext/ ED503747.pdf. Usher, E. L. (2009). Sources of middle school students’ self-efficacy in mathematics: A qualitative investigation. American Educational Research Journal, 46(1), 275–314. Verner, I., Massarwe, K., & Bshouty, D. (2013). Constructs of engagement emerging in an ethnomathematically-based teacher education course. Journal of Mathematical Behavior, 32 (3), 494–507. Wigfield, A., & Eccles, J. S. (2000). Expectancy-value theory of achievement motivation. Contemporary Educational Psychology, 25(1), 68–81.

Chapter 6: Exploring Teacher’s Epistemic Beliefs and Emotions in Inquiry-Based Teaching of Mathematics Inés M. Gómez-Chacón and Constantino De la Fuente

Abstract In the present study, we describe the actions and decision-making of teachers in the teaching of modelling, based on their personal epistemology. The teacher’s personal epistemology (epistemic reasoning, beliefs and emotions) acts as a component of the cognitive and emotional conditions of a task required of students. The strategy of analogy as a tool to foster the students’ commitment and motivation in both real world and mathematical transitions, as well as the creation of multiple connections: both vertical (within the world of mathematics) and horizontal (within the real world, outside of mathematics) is prioritised. This paper proposes a conceptualisation of the term personal epistemology, from the current ontology and epistemology of mathematical knowledge. Keywords Inquiry-based teaching · Modelling · Epistemology · Decision-making · Problem solving · Analogy

Introduction Learning mathematics entails the development of an epistemological perspective about the content. It is recognised that the way in which mathematics is characterised in the classroom has much to do with the beliefs and epistemological views that the teacher holds. The subtle (explicit and implicit) messages communicated to students about mathematics and the nature of mathematical thinking affect, in turn, the way students grow in mathematical knowledge and the recognition they attribute to it. When we want to promote modelling in the classroom through inquiry-based teaching we consider mathematics as “a process and an experience”. Knowing maths I. M. Gómez-Chacón (*) Instituto de Matemática Interdisciplinar, Facultad de Ciencias Matemáticas, Universidad Complutese de Madrid, Madrid, Spain e-mail: [email protected] C. De la Fuente IES López de Mendoza Burgos (Spain), Madrid, Spain © Springer Nature Switzerland AG 2019 S. A. Chamberlin, B. Sriraman (eds.), Affect in Mathematical Modeling, Advances in Mathematics Education, https://doi.org/10.1007/978-3-030-04432-9_9

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is equated with doing maths. Research in mathematics education has focused on examining the characteristics of the context in which this “doing” is fostered. It is the “doing” – experimentation, abstraction, generalisation and specialisation – which constitutes mathematics, not a transmission through perfect communication by the teacher. The ‘Culture of inquiry’ models the activity of research mathematicians who expand mathematical knowledge and horizons through inquiry processes (Ernest 1990). Today, one of the challenges we have is the design and implementation of these types of activities that can be motivating for the students within the cultural and institutional constraints in which they operate (Artigue and Blomhoj 2013; Jaworski 2004, 2006, 2014; Maass and Doorman 2013). Designing research-based tasks requires deep analysis of the “mathematical experience” that is generated in both students and teachers. Thus, the creation of inquiry-based activity for their students is itself an inquiry process: teachers learn from the practices resulting from their teaching designs. This chapter proposes a reflection based both on a practical situation and on broader research (De la Fuente 2016; Gómez-Chacón and De la Fuente 2018). We approach the “mathematical experience generated” in a modelling activity focusing on the personal epistemology of the teacher (epistemic reasoning, epistemic beliefs and emotions). Addressing the personal epistemology of the teacher and the interaction between it and mathematical modelling can provide points of interest for the design of teaching modelling. The mathematical experience is something complex which must be understood from the cognitive and affective point of view. Here we will distinguish two ways of approaching the affective dimension: (1) the threshold of the interplay between cognition and affect, which from the heuristic point of view is of great interest, (2) the distinction in the way of thinking about the emotional aspect, which has to do with the subject who feels (personal epistemology and decision making according to the identification with mathematical objects) (for further information see Gómez-Chacón (2018)). The importance of an epistemological perspective in teaching and learning has been reviewed by different authors (Ernest 1991; Hersh 1986; Otte 1994). In this work the emphasis on epistemology is mainly presented as a meta-perspective, in how the teacher thinks about knowledge. Often a meta-perspective is expounded with a normative intent, driven by the assertion that a deeper reflection on epistemological issues would improve mathematical education. Research based on observational empirical studies of the epistemology of thinking in natural situations of interaction in the classroom (Schoenfeld 2010, 2016) is scarcer. Acquiring knowledge is not only a logical, sequential and standardised process, as the rationalists would say, but learning is considered ‘intuitive’. Often the teacher is faced with questions about how to deal with modelling teaching, and how to make decisions in the face of modelling processes with students, most of which involve epistemic cognition. This study tries to identify emergent configurations; some more routinised and others more spontaneous and intuitive, which involve the decision making of the teacher and penetrate the teaching of modelling.

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In what follows, sections “Personal Epistemology: Epistemic Beliefs and Emotions” and “Modelling and Epistemic Reasoning Processes” present antecedents on the topic and a theoretical frame of reference. These sections outline the concept of personal epistemology and the learning of modelling, in particular, the analogy as a heuristic tool for the transitions from real world modelling to the mathematical world. The methodology used (section “Objectives and Methodology”) is briefly described, giving rise to the presentation of the results from the development of a Mathematical Research Project (MRP) in the classroom, which details the epistemic mediation of the teacher through his actions and decision making (section “Personal Epistemology of Mathematical Knowledge and Decision Making in the Classroom”), and finally some conclusions are presented.

Personal Epistemology: Epistemic Beliefs and Emotions Personal epistemology is the study of people’s thinking about knowledge and about knowing. Its study is born in the field of educational and cognitive psychology (Barzilai and Zohar 2014; Hofer and Bendixen 2012; Hofer and Pintrich 1997). Even though in the last decades the field of personal epistemology has developed in several different directions there is convergence in some central descriptive dimensions of personal epistemology: the nature of knowledge, the certainty of knowledge, the simplicity of knowledge, the source of knowledge, and the justification of knowledge (Hofer and Pintrich 1997). Currently there is no single model guiding research on personal epistemology (Bendixen and Rule 2004). We briefly present some of the approaches to the concept of personal epistemology that the perspective of educational psychology poses: approaches to development, approaches to beliefs and approaches to resources.

Developmental Approaches Developmental models of personal epistemology generally view students as holding integrated epistemic positions or perspectives. These models describe students’ epistemic positions developing throughout the course of their life and studies, often following a typical trajectory (see Barzilai and Zohar 2014; Hofer and Pintrich 1997). Developmental approaches are concerned with identifying changes in students’ thinking. Thinking skills and theories about knowledge and knowing are deeply and intricately linked, capturing the close link between people’s views of knowledge and their reasoning processes by describing epistemic thinking as a “theory-in-action”. The developmental perspective encompasses both the epistemic reasoning processes and the epistemic beliefs and theories that underlie them. Beliefs reflect assumptions, expectations and attitudes that may affect reasoning processes.

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Leder et al. (2002), Maass and Schloeglmann (2009), and Schoenfeld (1985) have conducted extensive research in the field of beliefs in mathematical education. In this approach, personal epistemology is a term that refers to the beliefs that people hold about knowledge, both as to its nature and its acquisition and justification (Hofer 2002). Although different models of competence have been proposed, there is a consensus that epistemological beliefs refer to “belief about the nature of knowledge and knowledge processes” (Hofer and Pintrich 1997: 112) and in some cases learning (Op’t Eynde et al. 2006). Epistemological beliefs have sometimes been explicitly described as a type of metacognitive knowledge or as schemas (Muis et al. 2015; Schoenfeld 1985). These models of epistemic beliefs and self-regulated learning are mainly concerned with understanding how and why epistemic beliefs impact learning and how they are conditions that serve as inputs to metalevel learning standards.

Resources Approach A third important approach to the study of personal epistemology is the resources approach (Elby and Hammer 2010). This perspective emerges from the “knowledge in pieces” approach to the study and analysis of knowledge, and highlights the fragmented and contextual nature of students’ epistemologies. Epistemological resources are specific cognitive resources highly linked to the context that people use to understand and reflect on their epistemic knowledge, activities and positions. Epistemological resources may gradually advance into beliefs as they become entirely articulated and more stable. In the development of research, these approaches have often acted disjointedly, without taking into account, in an integrated way, that which each of them considers key components that underpin the concept of personal epistemology. One of the strongest criticisms of empirical studies on personal epistemology under these approaches to educational psychology is that they have little regard for the context and specific domains of knowledge (Bromme 2005). In the field of Mathematical Education we must highlight authors who have adopted a different perspective and have implemented this epistemic integration, although not under the coined denomination of personal epistemology (Schoenfeld 2010, 2016). For example, Schoenfeld has worked with metacognition as a central aspect of cognition and has related it to belief systems (Schoenfeld 1987). The author has developed a theory of decision making, centred around teachers (Schoenfeld 2010). This work is indicative of the fact that in the field of Mathematics Education there is a fundamental and productive dialectic between theory and practice; and contextual and knowledge components, allowing for the development of observational tools for reliable naturalistic interventions in which classrooms can serve as laboratories. The central idea of the Schoenfeld model is based on the claim that it is possible to describe, explain and predict teachers’ performance, decision making and actions

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during teaching based on their knowledge, beliefs and goals. In the last 20 years, the literature on teachers has identified and broadly described the knowledge, beliefs and goals of the teacher. Schoenfeld proposes to go some steps further, describing the ways in which these elements interact and result in teachers’ in-the-moment decision-making. In summary the Schoenfeld model is articulated in: resources (especially knowledge); goals; orientations (an abstraction of beliefs, including values, preferences, etc.); and decision-making (which can be modelled as a form of subjective cost-benefit analysis). Based on the Schoenfeld model, a deeper exploration of the construct that this author poses with Orientations would be interesting. We find it crucial to pinpoint epistemic beliefs and emotions -as an operative way of unpacking this category- as well as the interaction between epistemic reasoning and epistemic beliefs/emotions reflected in decision-making (Gómez-Chacón 2017). This interaction is understood as the application of heuristics and strategies in specific contexts and specific content in order to make judgments about right or wrong or true or false in mathematical knowledge. Thus, this study proposes the term personal epistemology as a multifaceted concept that operates at cognitive, metacognitive and affective levels. For an operational level of the term personal epistemology we have distinguished between epistemic reasoning, epistemic beliefs, epistemic emotions and decision-making. The term epistemic beliefs shall be used to refer to a person’s beliefs about the nature of human knowledge; its certainty and how it is conceptualised, and a person’s beliefs about the criteria for, and the process of, knowing. Likewise, the epistemic emotions are defined as emotions that arise when the object is the knowledge and the processes that involve the knowing are caused by cognitive qualities of task information and the processing of that information (Gómez-Chacón 2017; Pekrun and Linnenbink-Garcia 2012).

Modelling and Epistemic Reasoning Processes As mentioned in the preceding section, the analysis of personal epistemology will address the teacher’s resources, particularly their knowledge on modelling and heuristics in problem solving.

Recognising Modelling Skills Results from research on the learning of mathematical modelling from different perspectives (Blum and Leiß 2005; Haines and Crouch 2005), show that student learning in the transition from the real world to the mathematical model is hampered by the lack of knowledge and experience related to abstraction. This behaviour is less obvious when one moves from the mathematical model to the real world, in fact in this case the higher process level is more likely to be used.

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Several research studies on modelling have considered students’ mathematical modelling skills in terms of the expert-novice continuum (Crouch and Haines 2004). Students have difficulty keeping the real-world demands and the model in mind all at once. Novices tend to spend less time analysing the problem (Schoenfeld 1987), they have difficulty distinguishing between relevant and irrelevant aspects, and believe to have understood the problem sufficiently when they have not. Beginners immediately tend to start generating equations without recognising particular underlying abstract problem types or being able to access relevant concepts and procedures (Glaser and Chi 1988). In order to tackle these challenges, the experts attempt to find an answer to the following question: How could these students further increase their level of mathematical expertise? A certain level can be achieved by extended relevant motivated practice (with feedback) on all aspects of building models for a variety of problem types (Ericsson et al. 1993) or scaffolding of technical aspects to help beginning modellers through stages what can initially appear to be demanding and unfamiliar approach to problem solving and though metacognitive modelling competences (Stillman and Galbraith 1998; Stillman 2011). Such practice needs to be aimed towards developing recognition of underlying problem categories and formation of sufficiently and relevantly detailed problem representations that mediate between the abstract model and the real world problem context. We consider that students would also need to have extended practice to improve speed and accuracy in accessing and deploying appropriate mathematical procedures for particular model categories and with good heuristic-based strategic help supports. As it shows by Stender (2017), up to now, there is no empirical evidence of how good heuristic-based strategic help supports the modelling process and how the support must be adapted to groups of students of different ages, knowledge or culture. In this study is indicated that this “facilitators toolkit” might be a strong instrument to support students that are working on complex modelling problems and is exemplified by the use of analogy as heuristic tool.

Real-World: Mathematical World Transitions: Analogy as a Heuristic Tool Some aspects of analogy are specified in this section to support the understanding of the scope of the empirical data (section “Personal Epistemology of Mathematical Knowledge and Decision Making in the Classroom”). We will refer to the role of analogy in the search for patterns that allow the abstraction of contexts and generalisation of ideas, facilitating in the modelling processes the transitions from the real world to the mathematical world. In the problem-solving mathematical understanding, analogy may be considered the prior use of solution procedures to solve the problem. Solving problems by

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analogy involves the recognition of high-level relationships existing between two domains, although the two domains share very few similarities in their superficial characteristics (Gentner 1983, 1989). With respect to analogical transfer, the psychology of mathematics education distinguishes between two components of analogical thinking: access and use. Access relates to remembering the appropriate solution procedure (memory), while use refers to the correct implementation of the solution procedure. Researchers have noted that much of the inability of trainees to transfer the procedure of solving a new problem to an old one lies in access. People fail to remember the right memory solution. Once the student is told which solution procedure to use, the solution rates increase significantly. In the field of educational psychology a theoretical accessuse framework for the analogical understanding of problem-solving has been developed by Novick and Holyoak (1991) and in the mathematical framework a key model is Polya (1945, 1954). Authors in the field of psychology (Novick and Holyoak 1991) highlight two findings that are particularly relevant to mathematical learning: first, they find that students who best solve problems are those who abstract the structural characteristics of the problem; this is what cognitive psychologists call the “induction scheme”. Those who induce an appropriate scheme develop a better conceptual understanding of the type of problems represented in the experiment. Secondly, positive correlations are found between the transfer of solutions recognised and the qualification obtained in mathematics. This type of research suggests that the so-called “conditions of applicability” are critical to success in solving problems. In other words, being able to solve the problem is contingent on being able to recognise which solution is appropriate. For this purpose, two processes are important: understanding principles and executing procedures. According to Polya’s approach, “analogy” is not a method of solution; “looking for something analogous” or other variations of “looking for a related problem”. Polya presents it as a heuristic suggestion involving the articulation of processes of generalisation and specialisation. In Polya’s second book dedicated to solving problems (Polya 1954) -the first volume is entitled Induction and analogy in mathematics- the first thing he notes is that “Yet as we start discussing analogy we tread on a less solid ground” (p. 13) and he suggests that the only way to deal with the matter in a useful way is to conceptually specify analogy. This is what he calls “clarified analogy”, whereby: “two systems are analogous if they agree in clearly definable relations of their respective parts”. Thus a triangle in a plane can be said to be analogous to a tetrahedron in space, and the analogy is clarified in this case by specifying which are the relations in which they agree: two lines in a plane cannot enclose a part of it, whereas three can; likewise, three planes in space cannot enclose a part thereof, while four can: “The relation of the triangle to the plane is the same as that of the tetrahedron to space in so far as both the triangle and the tetrahedron are bounded by the minimum number of simple bounding elements” (p.14).

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Table 1 Uses of analogy Analogy. Uses in Maths class Use 1. Proceed by analogy (from a topic) Step 1 Step 2

Analysis of the topic Construction of a topic analogous to the initial one

2. Establishment of analogies (between two topics) Comparing the two topics Initial establishment of analogies between topics

This second book (Polya 1954) is devoted to the study of the formal structure of the reasoning made in the course of problem solving and that cannot be described with the classical deductive patterns of logic. In the Brief Dictionary of heuristics, Polya began by calling this “heuristic reasoning”, whereas in the second book he called it “plausible reasoning” (Polya 1945). This clarified analogy may already be more than a suggestion, inasmuch that not only does it imply that it would not be wrong to do something, but it also specifies the type of transformation to be performed.1 Finally, in his third book, Polya (1962–1965) attempts to advance towards what he calls a “general method” -which, although announced in the first volume, fails to appear in the second volume- He addresses the definition of general models that could encompass many ways of elaborating problem-solving plans. It reveals the two kinds of reasoning: demonstrative reasoning, being precise, final and “automatic”; and plausible reasoning, being vague, provisional and specifically “human”.2 In the process of conjecturing and justifying, complex chains of plausible reasoning are often elaborated, which may contain new nuances that enrich the patterns already known. A thorough analysis of these processes may make it easier to make them explicit and to model them, as this is usually done with known patterns. The use of analogy, presented in section “Personal Epistemology of Mathematical Knowledge and Decision Making in the Classroom”, which analyses a class session in the development of a MRP, will show us the epistemic type of mediations performed by the teacher. Two uses of analogy have been identified (De la Fuente 2016) (Table 1): Use 1: Proceed by analogy. This use is based on a topic or mathematical entity in a given context (a conceptual or procedural idea, a result, a model, a structure, . . .), analysing its qualities and proceeding by analogy building (designing or elaborating); in another context (more general or more concrete, generalising or particularising) another topic or analogous entity, which maintains the characteristics of the former, adapted to the new context. This process has two steps: (1) analysis of the mathematical topic; (2) construction of the analogue topic. Clarified analogy is a heuristic tool, the word “tool” meaning instrument of transformation. “To make a table” it is defined as a “skill” since it lacks the quality of “transformation” of the problem. 2 The emphasis was added by the author. 1

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Use 2: Establish analogies. This second use is based on two topics, entities or mathematical ideas (concepts, formulas, strategies or other procedures, demonstrations, results, models, structures, . . .) that are compared with each other. This is carried out through a comparative analysis using criteria to detect and establish possible similarities and common characteristics. The analogy is established when the common structure underlying both, independent of the contexts, is revealed; it is then said that, for those criteria, the two entities are analogous. In this case the process also has two steps: (1) comparing topics; (2) establishment of analogies. In a MRP these two uses can be sequenced: first use 2 and then use 1; That is to say, once the analogy between the two mathematical topics (use 2) is established by analogy (use 1), in order to construct another entity of the same type, situated in a more general or more concrete context, by generalising or particularising, depending on the case, the initial entities.

Objectives and Methodology Objectives The aim of the study is to explore the interrelationships between teacher’s personal epistemology and the knowledge of mathematical practice in the classroom. We show how the teacher’s decision-making in the classroom and the self-regulation of learning in modelling activities is supported by the teacher’s personal epistemology. In particular, the objectives of the study are: 1. To identify how the personal epistemology of the teacher (epistemic beliefs and emotions) act as a component of the cognitive and emotional conditions of a task required of students; 2. To explore what determines what that teacher does, on a moment-by-moment basis and what shaped the teachers’ decision-making in the teaching of modelling.

Methods The qualitative methodology used is based on methods of observation and case study (Bassey 1999). The criterion determining this case is that, on the one hand, this case is within a convenience sample, a selected sample category (Gliner et al. 2009) and on the other hand, the case (Teacher-FC) is a key informant, a secondary mathematics teacher considered of excellence, which allows to illustrate the interface between personal epistemology and inquiry-based teaching for modelling processes with a wealth of data.

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We focus on one modelling activity, where the inquiry based-learning is the mathematical activity involved in the process of transforming a problem-solving task (PST) in a Mathematical Research Project (MRP). The Mathematical research project (MRP) is the real investigation, based on the initial problem statement, the steps followed being completely analogous to those of scientific research, with the teacher acting as project advisor. The process of generating a Mathematics Research Project (MRP) from a particular problem will not only require the students’ creativity, but also the teacher’s mediation for the establishment of a suitable creative mathematical working space. The MRP: Functional models for modifying exam marks (see section “Personal Epistemology of Mathematical Knowledge and Decision Making in the Classroom”), was carried out with High School Students (17 years old), 25 students (15 boys and 10 girls) by the secondary mathematics teacher, denoted by TeacherFC. Section “Personal Epistemology of Mathematical Knowledge and Decision Making in the Classroom” presents the activity developed during four class sessions, each lasting 1 h. The research team includes two researchers and a secondary mathematics teacher (Teacher-FC). The method used for data collection has been the observation of classroom sessions, materials generated by the teacher and students in the development of the class and semi-structured interviews with the teacher. Although we focus here on a MRP, the study carried out with this teacher has been going on for more than 4 years with a diversity of projects and activities (De la Fuente 2016).The lesson was analysed by the research team. The analysis proceeded in stages. We decomposed the lesson into smaller and smaller “episodes,” noting for each episode which goals were present, and observing how transitions corresponded to changes in the goals and personal epistemology of the teacher. In this way, we decomposed the entire lesson – starting with the lesson as a whole, and ultimately characterising what happened on a line-by-line basis. The next step was to codify the material, taking as codes the indicators of actions and decisions made in the classroom by the teacher related to teach the analogy as a heuristic tool, together with a number of epistemological aspects (epistemic reasoning, epistemic beliefs and epistemic emotions) that had been registered in the teacher interviews. Encoding and analysing in detail each significant piece of every session permitted the understanding of the process of inquiry-based teaching and personal epistemology development and to report on the factors responsible for it: goals, modelling and epistemic reasoning, beliefs or strategic heuristic. Section “Epistemic Reasoning in Act: Actions and Decisions in the Classroom” shows the whole lesson and breaks it into major episodes (lesson segments), each of which has its own internal structure according to a main category: the analogy process as a heuristic tool for facilitating the processes of modelling (the transitions from the real world to the mathematical world). The analogy topic is selected because it is noted to be frequently used after a 2-year observation period of this teacher. It is a key tool in the actions and decisions taken in the classroom.

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Personal Epistemology of Mathematical Knowledge and Decision Making in the Classroom This section presents the results of the study. Before the study described, it was analysed based on observations that Teacher-FC used analogy quite often (emerges as a pattern of behaviour). The main result explained here is that, the teacher uses analogy in the teaching of Mathematical research project (MRP) as a heuristic tool to foster student engagement several times and that this frequency of use comes and is linked to the personal epistemology of the teacher. Firstly, we present the results that come from the lesson analysis, in natural situations of interaction in the classroom, where the teacher takes in the situation and adapts accordingly (section “Epistemic Reasoning in Act: Actions and Decisions in the Classroom”). Certain pieces of information and knowledge become salient and are activated and they show up individual’s resources, goals, and epistemic reasoning and beliefs, allowing us to model their behaviour. The sections “Epistemic Beliefs and Emotions” and “Epistemology and Ontology of Mathematical Knowledge”, based on the data from lesson analysis and the interviews with the teacher, show up the dimensions that support the teaching practice of Teacher-FC: the dimension of interconnectivity between epistemic beliefs and emotions and decision making in the classroom; the dimensions of process and of the coherence of the mathematical experience offered to students based on the epistemology and ontology of mathematical knowledge.

Epistemic Reasoning in Act: Actions and Decisions in the Classroom The mathematical modelling activity based on the study of functional models, which relates everyday life to the school environment was proposed to work in the class by Teacher-FC. The statement is as follows: Mathematical Research Project (MRP): Functional models for modifying exam marks3 A high school student returned home saying that his math teacher was dissatisfied with his students’ marks in a written test that they had done about functions, attributing it to perhaps the proposed questions had been rather difficult. The teacher decided to “fit” those marks using a correction factor: if the original mark was x (on a scale of 0 to 100), it would pffiffiffi be 10 x. That is, if the initial mark was 81, the corrected grade would be 90. Apparently, this factor is commonly used among teachers in Israel

The class sessions to carry out this project were analysed to attempt to answer the questions: What determines what that teacher does, on a moment-by-moment basis

3 The statement was adapted from Arcavi, A. (2007). El desarrollo y el uso del sentido de los símbolos. UNO. Revista de Didáctica de las Matemáticas, nº 44, 59–75.

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Fig. 1 Diagram of actions and decisions for discussing the MRP: “Functional models for modifying exam marks” (A1, A2, A3 . . . and D1, D2, D3 . . . coding refers to actions and decisions made in the classroom)

and what shaped teachers’ decision-making in the teaching of modelling? Figure 1 synthesises the process followed with respect to the actions and decisions carried out in the classroom development by the Teacher-FC. An in-depth analysis showed that the actions and decisions had an articulating axis: the use of the heuristic tool analogy. The figure represents the pattern (routine) that the teacher uses in the access and use 1 of the analogy (section “Modelling and Epistemic Reasoning Processes”). For the Teacher-FC an epistemic belief is that: The most important mathematical knowledge is the ways of doing and the specific methods of working in mathematics. Among this knowledge I highlight the search for patterns, regularities and mathematical laws, in changing processes, from particularisations and generic examples. (Teacher-FC Interview 2016)

Although Teacher-FC did not explicit in interviews, the repeated observation of his mathematical practice in class (example is the MRP presented here, see lesson episodes related to D1 and A3, A4, A6, A7, A8 in Fig. 1) has led us to claim that: the patterns to be developed further are the inductive and analogical patterns. The dominance of analogy is shown as an epistemic reasoning in act (tacit), moment by moment in the class.

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Following we illustrate this diagram describing the lesson episodes and pointing out the actions and decisions that are produced according to his goals and personal epistemology. [A14]: As the factor presented is common in Israel but it is totally unknown in Spain the teacher decided to present the previous paragraph in two phases: the first of the two points, asking the students to propose some appropriate correction factors if the exam was set by them. What correction factors can we propose to the teacher in order to modify the marks? Express them in algebraic form and represent them graphically. Analyse the advantages and disadvantages of each.

[A2]: Teacher-FC asks the class: What can be said about the subject? After a short discussion, several correction factors are presented by the students. No student agrees with the Israeli teacher’ proposed factor. Once formalized, we could describe them as follows: if x is a mark belonging to the interval [0, 10] and the mark obtained when correcting x, we can: Increase all of the marks by the same fixed amount c, y ¼ x +c rx r x Increase each mark by a percentage, r, y ¼ x þ 100 ¼ 1 þ 100 Round the mark to the nearest whole number, which is greater than or equal to the mark, y ¼ Ent[x] + 1 If the highest mark is the value a, transform this mark into 10 and transform the rest proportionally, y ¼ 10 a x As we can see they are unformalized factors: raise all of the marks by a quantity, raise it by a percentage, round, etc. Expressed like this. Then the teacher asks them to express them in a more rigorous, formalised form and to represent them graphically, with the aim, if they had not seen it yet, that they see that they are not analogous to the Israeli model. [D1-D3]: Once the advantages or disadvantages of each of them were discussed5 and facing up to the fact that the students’ answers do not formulate conjecture on some topic analogous to the Israeli’ model the second part is formulated by the teacher. [A4]: The teacher deliberately requests a factor analogous to the Israeli for the Spanish context. [A5]: The key is that the marks are between 0 and 10. The teacher asks to the students how to elaborate and obtain them.

4 [A1], [A2], [A3] . . . and [D1], [D2], [D3] . . . coding refers to actions and decisions made in the classroom, see Fig. 1. 5 It is omitted in order to not to lengthen the document, but they have a lot of didactic interest; for example, some images of interval values [0, 10] do not always remain in the interval, so new marks may be impossible values: for example 12, etc.

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Fig. 2 Factor representation y ¼ x; pffiffiffiffiffiffiffi y ¼ 10x

Adapt the correction factor of the teacher to our country, where the marks are between 0 and 10. Express it algebraically and make its graphical representation. Analyse its advantages and disadvantages with respect to the previous ones.

After a while, the students proposed to the teacher the desired factor (underlying [D1]) (Fig. 2). [D2]: As no further clarification is required, the teacher made the class work through the formulations made by students of the non-formalized analogue topic. Students used the Graph program for implementing the teacher’s proposal about comparison. [A3]: The teacher proposed the comparison to clarify the essence of the factor sought, indicating that the two functions can be compared, y ¼ x is the factor pffiffiffiffiffiffiffi that does not modify the initially obtained grades, and the factor y ¼ 10x, which is analogous to that of the Israeli teacher, but adapted to the Spanish context (underlying [D2]). [A3]: As we can notice in each of the two functions, the students transformed the interval [0, 10] into itself, so that the new marks remain values of the interval. This completes the analysis of the adequacy of the new factor to the Spanish context. [D4]: The teacher made the decision to go deeper into the subject, doing so after studying the new factor, he continued with the following question: Could we vary this factor to obtain similar ones? Test introducing some change in its algebraic expression: root index, exponent of 10 or exponent of x. Analyse the characteristics of each one and its suitability.

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Fig. 3 Synthesis of factor family graphs

[A2]: With this question the teacher led the class-group back to ask what can be said about the topic and he poses new questions that could provide the student with the model to follow in order to achieve it. Both teacher and students have passed through [D1] and [A3]. Then, although not described by the limits of the extension required in this article, students have proposed new correction factors. [A3]: As an expression of the work done by the students the teacher summarised the results as follows, indicating: the result of which we present in the following pffiffiffiffiffiffiffi graph, in which some of the functions analogous to y ¼ 10x appear (Fig. 3): [A6]: In the teacher’s proposal of the non-formalized analogue topic it is explained that: The expressions of all of them can be grouped into two, representing the two families of factors: F n ð xÞ ¼

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi n 10n1 x

Gn ðxÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi p n 10xn1

x 2 ½0; 10,

n2N

And if instead of having marks between 0 and 10, they were of the interval [0, N], it would have the families of factors:

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F n ð xÞ ¼

p ffiffiffiffiffiffiffiffiffiffiffiffi n N n1 x

G n ð xÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi p n N:xn1

x 2 ½0; N ,

n2N

After the study of the previous models, we can return to the idea of continuing to generalize and we can raise another question in the classroom. Here the teacher can decide whether or not to raise a new question, a new twist to the topic; that is, we can return from [D4] to [A2]. The teacher returned from [D4] to [A2]: Cyclically returning to [A2] the teacher did so by the following question: Reflecting on the expressions of the factors Fn and Gn, find an expression that encompasses the two, with the condition that all of them become part of a single family of correction factors.

[D4] and [A7]: In the teacher’s decision lies the idea that students become aware of how they can group them into a new generalisation of the model, as follows [D4] and [A7]: H ni ðxÞ ¼

p ffiffiffiffiffiffiffiffiffiffiffiffiffi n N i xni

n 2 N,

i 2 f0; 1; 2; . . . ; n  1g

In this new context, the teacher and students returned to the known factors so far, verifying that: – – – –

For i¼1, we obtain the factors Gn. For i¼n-1, we obtain the factors Fn. For i¼n/2, we obtain the factors F2¼G2. For i¼0, we obtain the factors Identity y¼x.

But the question can be further expressed if it is considered as the search for functional models that adapt to the situation and are of the trigonometric, logarithmic type. Here there would be successive cycles [D4], [A2], [A3], with different types of functional models. For each one of them the cycle is repeated in the diagram of actions and decisions. We present, by way of example, some of the factors proposed by the students (Figs. 4 and 5).6 Later, in other solutions given by students we can find other types of trigonometric models (Figs. 6 and 7).

In the first of the graphs a logarithmic function and the reciprocal exponential, which would be analogous to the logarithmic one, but for the reverse case, in which the marks had to be lowered. This last question can be worked simultaneously to the one of raising them. The second graph is the same approach with trigonometric functional models. The norm that is mentioned in the graph refers to that the functions transforming the interval [0, 10] into itself, a condition that they all must fulfill. 6

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Fig. 4 Exponential and logarithmic factors

[A7]: The teacher continued to encourage students to extend the research project. Let us note that the teacher’s types of orientations and commentaries, in the analysis of the student’s solutions, focused on suggestions of improvements for the attainment of the project objectives. The teacher and the students assessed the results, proposed essay improvements and suggestions for the elimination of errors and modification of results, suggested new paths, posed new questions, etc. For instance, in some of the models presented, the function was not strictly increasing as it had been demanded until then, this opens up new possibilities, analysing the meaning and interpreting the consequences that occur in the affected marks. One of the most interesting questions that one student posed  was thefollowing: what is the greatest value of p so that the function y ¼ x þ p: 1  cos πx 5 continues to transform the interval [0, 10] into itself. The same can be said for other functions that have maximum or minimum. This led the teacher and the students to consider the following model (Fig. 8). The teacher with the students verified this result. This solution seems incredible, but it is true: it leaves unchanged marks that are integers, raises the marks of the

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Fig. 5 Trigonometric factors adjusted to meet the norm

intervals (0, 1); (2. 3); . . . and lowers those belonging to the intervals (1, 2); (3. 4), ... In summary, through these episodes it can be verified that the actions and decision contexts, in which the teacher’s epistemic reasoning emerge, mark the epistemological norms that serve as an input for the self-regulated learning of the learner. The analogy is considered as a heuristic tool to foster student engagement and to analyse the complexity of a modelling problem. In the sections “Epistemic Beliefs and Emotions” and “Epistemology and Ontology of Mathematical Knowledge”, we will specify how this epistemic reasoning of the teacher is affected by his epistemic beliefs and emotions.

Epistemic Beliefs and Emotions Teacher-FC uses Polya-style heuristics as a decision-making mechanism for mediations with the students as well as for the establishment of knowledge domains ([D1], [D2], [D3] and [D4] in Fig. 1). In this teacher there is evidence of the

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Fig. 6 Sinusoidal factors to fix centralized and extreme marks

epistemic belief that “plausible patterns describe the logical form of mathematical reasoning” (e.g. [A4] in Fig. 1). As researchers, the repeated observation of this teacher’s action would lead us to affirm that “there is a model of competence in problem solving” based on the good management of interference that can occur in the coexistence of reasoning that responds to plausible patterns with reasoning that responds to deductive patterns. As in the way that plausible patterns are made explicit to students they are being given a tool for their reasoning in problem solving. Finally, note that the epistemic emotions that Teacher-FC emphasizes are intellectual courage, will, self-confidence and doubt. In his words: I think learning is really a process of mental change that normally requires someone to treat the content more than once, in a cyclical way (because the mathematical contents are very complex, polyhedral, so we cannot apprehend all their faces with only one contact with them). This implies that learning requires, to the student, intellectual effort and intention to do so; otherwise you can repeat it memorably, but it is not true learning. (Teacher-FC’s Interview 2016)

These epistemic beliefs and emotions not only constitute the framework for decision-making in their practice but also implicitly define which model of competence is related to “cognitive and affective aspects” in students. In most of the

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Fig. 7 Sinusoidal factors “by n” to fix centralized and extreme marks

development of the activity (section “Epistemic Reasoning in Act: Actions and Decisions in the Classroom”), Teacher-FC kept his students in three “meta-level” questions: What role does analogy play in modelling processes from the real world to the mathematical world? What do they learn by using analogy (reasoning, beliefs, epistemic emotions)? To what extent does mathematical trust depend on intellectual courage using these tools?

Epistemology and Ontology of Mathematical Knowledge As it was indicated at the beginning of the diagnosis described here, the personal epistemology of Teacher-FC has as initial point his behaviour in action, moment to moment (section “Epistemic Reasoning in Act: Actions and Decisions in the Classroom”). This analysis shows a unity between subjective and objective knowledge of mathematics when using inquiry-based teaching. In the creation of “mathematical

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Fig. 8 Sinusoidal factors, What does it get?

experience” in the classroom, the personal epistemology of the teacher is manifested in the ontological dimension of mathematics, in epistemic beliefs regarding knowledge structure, knowledge stability and sources of knowledge and justification. Teacher-FC, as a mathematician, has certain beliefs about the ontology of mathematics that influence his approach to teaching and learning. His position could be denoted as constructivist in the sense that both truths and mathematical objects are established by constructive methods. As Ernest (1991) points out the point of view of constructivists (intuitionists) is that “human mathematical activity is fundamental in the creation of new knowledge and that both mathematical truths and the existence of mathematical objects must be established by constructive methods” (Ernest 1991, p.29). Following the categories of epistemological beliefs indicated in section “Personal Epistemology: Epistemic Beliefs and Emotions”: structure, stability and source of knowledge in interviews this teacher points out: In relation to the knowledge structure, Teacher-FC indicates for the High School students that in the mathematical knowledge the most important thing is the knowledge of the mathematical practice and he indicates as specific knowledge of the teacher: search of patterns, regularities and mathematical laws, in changing process,

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from particularisations and generic examples; to mathematization of situations, contextualisation. In his own words: I started thinking that concepts were the gods of mathematical knowledge. After almost 35 years of professional experience as a math teacher, I believe that: – I have dethroned concepts like the kings of knowledge in my mind. This does not mean that I do not believe that concepts have no importance, they do, but in a teaching-learning context with teenagers, concepts are not the most important. – I have discovered that the most important mathematical knowledge is the ways of doing things and the specific methods of working in mathematics, as professional mathematicians do. Among this knowledge I think the following deserve to be highlighted: a) the search for patterns, regularities and mathematical laws, in changing processes, based on particularisations and generic examples; b) the mathematization of situations (from formal or academic mathematic to the real or everyday situations), through modelling (use of models, its construction and analysis of its adequacy and limitations); c) contextualization (from the academic, formal or mathematical to the real or everyday) as a reverse process to mathematization, through the search for new contexts, register of representation of ideas and the establishment of connections between contexts and different ways of representing or contextualizing ideas. (Teacher-FC’s Interview 2016)

In relation to the stability of knowledge, he indicates his evolution in the vision of mathematical knowledge: the passage of knowledge as something static to something that is constantly changing. In his expression: I also thought that what I knew, or thought I knew, was fully settled, but the passage of time has made me change my mind: – All knowledge, both emanating from established theories and that emanating from what we discover day by day, is moldable, modifiable, improvable. And this occurs at all levels, from elementary school to the areas of university research. And this should be experienced by our students, otherwise they will not know what mathematics really is. – It is better for the teacher to convey that knowledge is static, because that allows him to better control the classroom environment, there are fewer questions, less questioning and easier to finish a topic, start another, manage everything without leaving our field of security or our comfort space. The opposite involves living with the uncertainty of the changes, with the insecurity of uncomfortable questions, with the lack of total control of the class; and that is very complicated for a teacher to manage. (Teacher-FC’s Interview 2016)

Regarding to the source of mathematical knowledge, Teacher-FC also mentions its evolution. Going from the management of the same from authorities, whether moral or scientific, to the idea that this derives from empirical evidence and reasoning. In his words: I was always very obedient to my teachers and I believed in them and, in general, in their professional qualification, until in 3rd year of my degree. In that year I bought Proofs and Refutations, by I. Lakatos. With this book I realised that there were different conceptions of what mathematical knowledge is, its structure, its teaching, etc. Now I think that mathematical knowledge, in entailing discovery and creation, has a double aspect of “practical” experimentation (in the sense of Miguel de Guzman’s words) in the generation and elaboration phase, and theoretical idealism (in the sense of formalism and maximum rigor in reasoning) in the final presentation phase for fellow scientists. As for the sources of knowledge, I believe that knowledge, from authorities, is necessary, but to make it meaningful in oneself one has to “experience” it on a personal level: unravel it, break it

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down, analyse it, so that, in this process, we can internalise it, truly learn it, apprehend it and make it our own. I always tell students that they do some MRP, those that were worried that they would not discover something new: do not think that the goal is that you invent something new; if you take some existing mathematical knowledge, you study it thoroughly, you crumble it. If you analyse it and you become an expert in it, that for you already means discovering something new, because those ideas, new to you, that have rooted in your mind and have acquired life, form a living universe that will amaze and impel you to look for more, to know better . . . And that is part of the process of discovery and creation. (Teacher-FC’s Interview 2016).

Discussion and Conclusions This concluding section is based on two core elements: (1) the objectives of the study and the teaching of modelling and (2) open questions about the conceptualisation of personal epistemology. The present study takes a step forward to describe teacher action (decision making and actions) based on their personal epistemology (1 and 2 research objective). It was shown that the Teacher-FC, based on their personal epistemology, prioritises the strategy of analogy as a heuristic tool to foster student engagement and motivation in the real-world and mathematical world transitions, while also creating multiple connections, vertical (in mathematics) and horizontal (with the real world outside of mathematics). The use of the subjective valuations implicitly expressed by the teacher in the lesson, through the heuristic tool of the analogy, capture aspects of this teacher’s deep concern in teaching. Concern that has been explicitly expressed in the epistemic beliefs regarding the structure of knowledge, the stability of knowledge and the sources of knowledge alongside its justification, and the epistemic emotion of intellectual courage (as an emotional rudder to guide judgment and action). These epistemic beliefs and emotions are something that involve the nature of their rule of action, i.e., routines of action (in Schoenfeld’s terms). As noted, a detailed analysis reveals that the Teacher-FC, highly values students who do math and believe that these have the ability to point out questions as a result of student feedback. He has a style of teaching based on “routine of interrogation” that consists in asking questions and giving answers which integrate those given by the students. This routine is shown in Fig. 1. The routine seems to lead to the conscious level, since it acts as a component of the cognitive and emotional requirements of the task required of the students. It is also evident that the actions and decision contexts -in which the epistemic reasoning and the epistemic beliefs mark the epistemological norms in the classroom- are intended to achieve the self-regulated learning on the part of the student. In this regard, the process of epistemic cognition (reasoning and belief) fosters the establishment of a habit in the teacher’s teaching style: the use of analogy as a tool for modelling. Regarding the teaching of modelling, this study demonstrates that the use of heuristic strategies can be reconstructed within the modelling process. An example

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using an analogy presented here shows how strategic interventions can be created, based on heuristic strategies. The heuristic strategies can act as a conceptual toolkit for facilitators to analyse the complexity of a modelling problem, identify the important steps in the modelling process, and pre-formulate possible strategic support. Finally this study raises two open questions; one of them is how to verify the interaction between personal epistemology, reasoning and decision-making in teaching, “acting in the moment”. It was shown that it was not possible to perceive its presence as something static, yet rather something incardinated between the past and the future. The terms “reasoning” and “decision” often imply that the decision-maker has knowledge: (a) about the situation that requires a decision, (b) about the different options for action (answers), and (c) about the consequences of each of these options (results), immediately and the in future. The second question raised relates to the conceptualisation of personal epistemology, as a multifaceted and integrated perspective in which it is necessary to consider cognitive, metacognitive and affective aspects in contextualised and specific knowledge domains. Studies such as that by Schoenfeld (2010) or the one now presented involving teachers, make use of theoretical constructs as operational tools to analyse the decision-making and actions of teachers. Knowledge, goals and beliefs become resources for practice. These categories extend the traditional dimensions of the concept of personal epistemology (certainty of knowledge, simplicity of knowledge, source of knowledge, and justification of knowledge Hofer and Pintrich 1997). We consider that the concept of personal epistemology allows us to relate the creation of subjective and objective knowledge in mathematics to the way in which the individual acts to develop his structures of thought, while the teachers adapt the manner in which they act in order to favour the mathematical thinking structures of their students. This suggests that the contexts of “discovery” (creation) and justification cannot be completely separated, since justifications, like proofs, are the product of human creativity as concepts, conjectures and theories. Inquiry-based teaching identifies all mathematics students as mathematicians, and here we reflect on a spectrum of potential reasons behind the teacher’s decisions and actions aimed at developing this creation. Acknowledgements This study was funded by the research grant Visiting Scholar Fellowship, University of California in Berkeley, Scholarship “Becas Complutense Del Amo” 2015-16, Spain, project in collaboration with Dr. Alan Schoenfeld, Elizabeth and Edward Conner Professor of Education and Affiliated Professor of Mathematics at the University of California at Berkeley and by the Spanish Ministry of the Economy and Competitive Affairs under project EDU2013-44047-P.

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Chapter 7: Mathematics Learning Experiences: The Practice of Happiness and the Happiness of Practice Adi Wiezel, James A. Middleton, and Amanda Jansen

Abstract In this chapter, we present a framework for viewing happiness – both as a product and as a process – as a potentially desirable outcome in mathematics learning environments. Specifically, we explore how the components of engagement, meaningfulness, and pleasure may work together in the context of mathematics learners’ experiences to form happiness. We conclude by posing some questions for future directions, as well some recommendations for research tools that may be useful for continuing this line of inquiry. Keywords Positive psychology · Happiness · Engagement · Meaning · Mathematics learning

There are many valuable outcomes for learning mathematics in school. One that is often used in research on mathematics teaching, motivation, and affect is achievement (e.g., Hannula et al. 2014), the outward manifestation of how much mathematics a student has learned. Indeed, achievement, often measured in the form of grades, performance on standardized tests, and completion of higher coursework, is an important indicator of the success of mathematics learning experiences. It is associated with expectations for success and continued interest in, and career aspirations for, STEM and other mathematically intensive fields (Lazarides and Watt 2015). However, whereas mathematics achievement in school is a useful indicator, it is important to keep in mind that it is merely a proxy for success in mathematics, rather than an end in itself. It is a critical lesson we have learned in the past 30 years of research: when we consider achievement alone as our primary

A. Wiezel (*) · J. A. Middleton Arizona State University, Tempe, AZ, USA e-mail: [email protected] A. Jansen University of Delaware, Newark, DE, USA © Springer Nature Switzerland AG 2019 S. A. Chamberlin, B. Sriraman (eds.), Affect in Mathematical Modeling, Advances in Mathematics Education, https://doi.org/10.1007/978-3-030-04432-9_10

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outcome, we both miss the point of education (see Dewey 2007, reprinted from Dewey, 1938, for a good summation of the stance we take here), and we oversimplify the dynamics of students’ mathematical development, which includes factors such as their happiness and overall well-being. To illustrate, consider a hypothetical example of two high school freshmen, Claire and Ashley. At this stage in her young life, Claire is a set on becoming a physical therapist, and she fills her schedule with an intensive array of mathematics and science classes as is required of her future pre-medical major. She studies arduously for her mathematics exams, at times pulling all-nighters because she knows that strong grades in her math classes will keep her on track toward her goal of becoming a physical therapist. She earns high grades in Advanced Placement courses, and is successful in her first 2 years of college, earning straight A’s. She is relieved that her major requires her to take only two classes in mathematics—a subject that conjures up painful memories of the effort she had to put into late-night study sessions. Claire is not sure she would be able to continue doing well if she had to take more semesters of math. Now compare Claire’s experience to Ashley’s. Ashley doesn’t know what she wants to do professionally, but knows that college will help prepare her for many options. She takes the default courses in high school, because they are required of all students, and because she knows they are required for college entrance. As a junior, she tests into a calculus course, and soon she gets into a rhythm with the subject matter. As the year goes on, Ashley finds a certain satisfaction in gaining new skills in solving applied problems, and by the end of the semester, she ends up tutoring other students taking calculus. In college, Ashley continues to take more math and statistics classes than required for her major, and, like Claire, she earns straight A’s in these courses. But, unlike Claire, when Ashley goes to graduate school, she is excited to learn more about mathematics and statistics. Note that in each of these student’s cases, the criterion of achievement was met— both Claire and Ashley earned straight A’s in their math courses and each took courses beyond what was required in high school. However, we would say that Ashley had better and more productive mathematics engagement, because her longterm affective response to mathematics learning was an approach response as opposed to Claire’s avoidance response (see Middleton et al. 2017 for a discussion of productive engagement). The point we are making is that if two students can have the same level of achievement, but have qualitatively different approaches towards mathematics in their personal and professional lives, then achievement is only one limited indicator of what we mean by success in mathematics. The anecdote above suggests that success in mathematics also involves features like persistence—one’s continuation in mathematics despite encountering difficulty, intrinsic motivation—one’s desire to work on mathematics for internal reasons, perceived instrumentality of the subject matter—one’s sense that mathematics is useful to one’s goals, and self-efficacy—one’s belief that one is capable of succeeding. Moreover, much research shows that these variables interact significantly in predicting students’ achievement, continued mathematics course taking, and future career choices (Middleton et al. 2016). Therefore, considering the quality

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of the experience that students undergo as they build their mathematical selves, as well as the habits and dispositions toward mathematics that they learn, is fundamental to the understanding of what is meant by “success” at the individual, curricular, and policy level. We are tempted to add to the list of potential positive outcomes of mathematics education such variables as retention in mathematically-intensive subject matter, use of mathematics after graduation, and earning a satisfactory salary. These additional outcomes could lend a more complete assessment of the success of mathematics learning experiences. However, they too are only proxies, and lead to more questions—for example, how much mathematics usage after graduation is “enough,” for whom, and to what end(s)? How many mathematics courses should one take before we consider a person’s collective mathematical experiences successful? although these additional criteria are useful and important, we suggest a broader approach to thinking about mathematics learning experiences to help contextualize why we might care about metrics like these in the first place. Specifically, we take the view that the goal of education, including mathematics education, should be to give students the tools to lead productive and fulfilled lives. These, finally, are end goals. We require students to take mathematics as a part of compulsory education for nine or more years of their lives so that they can become productive and develop a sense of fulfillment. The variables we have already posited each point to one or both of these goals: productivity being both the ability and inclination of the person to learn and use mathematics for some greater good (Ashley’s tutoring for example); and fulfillment being the affective response to one’s collective experiences (Claire’s reticence to continue mathematics being an example of poor fulfillment, mathematically speaking). Because this line of reasoning echoes the work of Aristotle, who defined his term Eudaimonia (“happiness”) as both living life, and doing, well (Aristotle and Brown 2009), one useful way to think about the end goal of mathematics education is in terms of happiness—both in the instrumental sense (productivity), and in the personal sense (fulfillment). In the sections below, we detail the case for thinking about happiness in the context of mathematics education, discuss three relevant components of happiness and how they may relate to—and interact in—mathematics learning experiences, and we discuss exciting new directions in this line of thinking. As we progress, we address this fundamental question of mathematics education, “what exactly is the role of mathematics, and more specifically, the mathematical learning experiences one undergoes throughout compulsory education, in contributing to one’s happiness?”

Why Happiness? To situate our discussion of happiness and its role in thinking about mathematics learning experiences, we begin with a bit of history. The philosophy of happiness has a rich historical tradition, and has often been conceptualized in one of two different

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ways: Hedonism (literally, “pleasure-ism”, or “sweet-ism”), the philosophical stance that happiness is an intentional state that is directed toward the emotional experience of pleasure and away from the emotional experience of pain; and Eudaimonia (literally, “good spirit”), the stance that happiness is achieving the best conditions possible for a human being—not only pleasure, but also virtue, morality, and meaningfulness (Sidgwick 1907). More recently, Ryan and Deci (2001) described eudaimonic happiness as the development of human capabilities. In paraphrasing Waterman (1993), for example, Ryan and Deci describe eudaimonic happiness as “when people’s life activities are most congruent or meshing with deeply held values and are holistically or fully engaged,” and note that, compared to hedonic approaches, eudaimonic ones are “more strongly related to activities that afforded personal growth and development” (Ryan and Deci 2001, p. 146). Learning mathematics, therefore, may be described (for some people, in some circumstances) in these terms. But our contemporary view of happiness is more than potential for selfactualization. It also involves affective responses such as pleasure and satisfaction which arise in the moment of learning, which also reflect on the “hedonic” qualities of those moments (Ryan and Deci 2001). In an educational sense, then, to be happy is to learn and grow in one or more domains (such as mathematics), to experience pleasure while learning, and/or to reflect with satisfaction upon one’s long term educational experiences. Learners can have happy experiences while learning and doing mathematics and/or take happiness away from the process of learning and doing mathematics. Importantly, frustration at a math problem in the moment need not obviate the possibility of experiencing happiness in mathematics over the long haul—the very same math problems can be later construed as meaningful or pleasantly challenging in a way that closely aligns with the affective dimension of eudaimonic happiness. We consider happiness, with its self-actualization and affective dimensions, as valuable for describing mathematics learning experiences because it offers us two opportunities. First, it offers us the opportunity to leverage happiness in terms of the learning outcomes it gives us. For example, specific aspects of well-being, such as “positive” emotions, have been associated with better academic outcomes. Certain positive emotions, for example, encourage students to broaden the way they solve problems by widening the scope of their attention, which may, in turn, make them more successful in their academic pursuits (Fredrickson 1998, 2001; Valiente et al. 2012). Similarly, positive emotions such as joy and hope have been correlated with increased levels of self-efficacy, interest, and effort in testing settings, which can contribute to increased performance (Pekrun et al. 2004). Finally, happiness has also been linked with increased creativity and goal-relevant activity (Lyubomirsky et al. 2005). Thus, in this sense, happiness can be thought of as a product that gives us certain desirable outcomes. The benefit to considering happiness relative to, or in addition to, other outcomes such as achievement is that happiness can make us reconsider benchmarks for some of those other outcomes. So, returning to our earlier question about how to know how much mathematics persistence is “enough”—happiness can

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offer us a criterion against which to judge our answers: does this amount of mathematics experience contribute to this person’s happiness (e.g., Claire’s productivity and fulfillment)? In this respect, happiness can not only help us attain certain desirable outcomes, but also offer us a way to assess the benefit of those outcomes. The second opportunity that thinking about happiness in the context of school mathematics offers us is treating mathematics education as a sort of practice or process of happiness. If it is our interest to treat education as a means by which students can develop the tools to lead productive and fulfilled lives, then it makes sense to consider the ways in which mathematics education can also serve as a focused template for such happiness, one in which students can be exposed to what the process of happiness might look like, so that they may be able to experience it, extend it to future courses, and replicate it in other domains of their lives, such as in their careers and personal pursuits. This type of thinking thus treats the mathematics classroom as a purveyor of two kinds of content: the mathematical content itself, and a template for how to perform happiness while learning and doing mathematics. What does such a performance of happiness entail? We think a useful definition emerges from Seligman et al.’s (2009) characterization of happiness as an experience that is (1) engaged, (2) meaningful, and (3) pleasant. As suggested earlier, this is not to say that all mathematical experiences are meaningful or pleasant, nor that a person should be fully engaged in all mathematical experiences to be happy. Rather, reflection upon one’s collective experiences should convey an overall sense of connection with the experiences within which one is immersed. These experiences should somehow be greater as a whole, helping the person develop eudaimonic potential, and for the most part, this whole should also be enjoyable. We will address each of the three tenets of happiness (engagement, meaningfulness, and pleasure) in turn, and discuss how we think they may apply to mathematics learning environments specifically.

Engagement First, we focus on engagement. Seligman and colleagues treat the engagement component of happiness as complete absorption in a task that that requires the application of a high degree of one’s skills and talents (Seligman et al. 2009). This view is consistent with Csikszentmihalyi’s concept of flow, in which one feels a loss of time and emotion, and becomes one with the task at hand (Csikszentmihalyi 1990; Seligman et al. 2009). These characterizations consider the practice of happiness to be synonymous with optimal experience. The extent to which the psychological experience of the student embodies this absorption or flow is an indicator of the happiness they feel. Applying this to a curricular context, the activities that give rise to optimal experiences must be designed to promote active involvement and skill use. Indeed, Delle Fave and Massimini (2005) show that repetitive, passive, or unstructured tasks rarely promote optimal experience.

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However, engagement in the context of a classroom also includes components not directly linked to the task, such as attending to the ideas of other students, or to interactions with the teacher. Accordingly, we reconceptualize engagement as a relationship between the learner and her or his learning environment (see Middleton et al. 2017). Consider the example of a student named Connor, who is learning about polynomials in his algebra class. Connor’s teacher asks the class which terms in the expression “3x2 + 2x + 2 + 3x” can be combined. Connor raises his hand and suggests that “3x2 and 3x” can be combined because by the distributive property, 3x2 + 3x ¼ 3x(x + 1). The teacher asks if there are any other answers. This response is generally interpreted as meaning that a student’s answer is incorrect. Subsequently, Connor feels frustrated and confused as he realizes his answer is considered wrong, but not why it is wrong. He turns to his classmate, who points out that Connor should look at the degree of the terms, not the coefficients. Connor quickly says “oh,” raises his hand, and says “never mind, it’s 2x and 3x, since the exponents need to match, not the numbers before them.” Connor’s teacher nods and Connor beams with pride. We characterize such engagement as dynamic, for although environmental constraints direct students’ behavior (as in Connor needing to raise his hand before speaking, or being able to ask his peer for help after the teacher’s tacit evaluation of his response) and their affective appraisal of their experiences (as in Connor’s frustration at his initial response being wrong without meaningful feedback), these behaviors and feelings also change the environment by establishing social roles and norms. For example, Connor’s decision to correct his initial response is indicative of a social space in which self-correction is valued, transforming his initial emotional response from frustration to triumph. This dynamic process thus involves the emergence of practices that ultimately become “mathematics” for the student. Mathematical perseverance, for example, is shaped by such triumphant resolutions to challenging, even frustrating circumstances, and over time, it can become a habit of mind that is fully part and parcel of the student’s conception of what mathematics is and her or his role in it. Also, because it is a relationship, engagement can be similarly thought of as healthy and productive, like Ashley coming to enjoy mathematics and seek out more experiences with it, or counterproductive, where the student may become turned off by mathematics and seek to avoid it in the future. Connor’s situation could have gone either way, depending upon the norms in which his self-correction was handled. Lastly, we can see from this example that Connor’s engagement was simultaneously cognitive, behavioral, affective, and social. We now turn to the dynamics of the relationships among these aspects of engagement. The broad community of researchers focusing on school engagement typically divide engagement into three domains: behavioral, cognitive, and affective (Fredericks et al. 2004). To this triad, we add a fourth component: social engagement (Rimm-Kaufman et al. 2015). Behavioral engagement concerns the overt effort students put forth in mathematics activities, and the observable actions they proffer.

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For example, Connor showed behavioral engagement by raising his hand to respond to the teacher’s question. Cognitive engagement involves self-regulation strategies, and the direction of attention and memory resources to the activity at hand. Connor concentrated on his teacher’s question, and later mentally reworked his solution, consolidating it into an effective rule for combining like terms. Affective engagement involves the feelings individuals have about their interpretation of their involvement, and ranges from emotions as such curiosity and inspiration to frustration and anxiety. Connor experienced frustration at getting his answer wrong, and then pride when correcting his previous understanding. Finally, social engagement concerns the degree to which students attend to one another (and the teacher), coordinate their actions with others, build relationships with one another, and generally conform to the sociomathematical norms and practices of the classroom. Connor checking with his classmate, accepting proffered assistance, and volunteering a corrected response illustrate the influence that these social variables have on the cognitive, behavioral, and affective manifestations of his experience. Although Connor’s example showed one possible experience of engagement, a given moment of engagement can manifest itself in diverse mental activities and overt behaviors, such as attention, cognition, effort, and affect in varying combinations in each of the four domains of engagement. For example, Seligman’s and Czikszentmihalyi’s conceptualizations of flow relate to each of these factors in a particular way. Flow consists of focused attention and deep cognitive processing (cognitive), high effort (behavioral), and joy or elation (affective). Moreover, although social engagement has been under-researched in the theory of optimal experience, some researchers have posited that the use of complex and socially meaningful activities to encourage optimal experiences in education may be important for encouraging healthy social development (Delle Fave and Massimini 2005). For example, students’ willingness to engage in prosocial behaviors, such as helping peers, can serve as a hook to pull students into engaging more deeply with academic content (Jansen 2006). Take the case of Allen. Allen was a seventh-grade student who spoke of being anxious about sharing his thinking in front of his peers. He worried about being in front of everyone in class. When he was motivated to participate, it was at times when he noticed a peer could benefit from his help. One way he saw that he could help others was when he perceived that someone with a correct solution could also solve the problem correctly another way, and he was then willing to share his solution strategy. Additionally, Allen tended to notice if a friend was struggling, and he would feel a sense of responsibility to help him figure out the mathematics. These motivations to provide support to peers tended to override Allen’s initial nervousness about sharing his thinking with others. Through the process of articulating mathematical thinking to help a peer, Allen was able to reflect upon—and potentially improve—his own thinking as well. As such, mathematical engagement may then “spread” as students interact with each other, sharing and appropriating practices in the context of their social engagement.

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Meaningfulness Second, we address meaningfulness. Meaning(fulness) is conceptualized by Seligman and colleagues as the dual process of knowing one’s strengths and skills and applying them to serve something greater than the self, be it a cause, or a connection to others (Seligman et al. 2009; Seligman 2002). A teacher’s preparation of a lesson plan that will help his students learn important skills, a journalist’s final touches on a breaking news story vital to her community, and a clinician’s work on a treatment aimed to remedy throat cancer are each meaningful in this respect. In each of these cases, meaningfulness involves evaluating one’s capabilities and matching them both to one’s goals as well as to the environment at hand. As with our definition of engagement, we offer a more focused definition of meaningfulness as it relates to the mathematics learning environment. But here, we must first distinguish between meaning and meaningfulness. Meaning is defined as the gist one gets from an activity, and its place in the larger scheme of conceptions that one holds about the subject. As such, meaning can be thought of as a network of associations among ideas that one gains about mathematics content, one’s role in the social dynamics of the mathematics classroom, the normative practices in which one engages, and the skills one employs in developing some kind of gist as a takeaway for later recall. Meaning is individual in that it is unique for each student in the classroom community, yet it is framed and shaped by the social norms of the class, and by the material and historical norms that are seen as mathematically important in the greater community (Voigt 1994). By contrast, we think a useful definition of meaningfulness in the context of mathematics learning experiences is an individual’s subjective conception of the value or weight that they put on the meaning abstracted from their experience. Each of the different domains of engagement in the mathematics learning environment— cognitive, behavioral, social, and affective—contribute value (positive and/or negative) to this assessment. In other words, experiences can have meaning (e.g., one can get the gist of the math), and/or be meaningful (e.g., the experience can be seen as instrumental to the individual’s future aspirations). One helpful way to conceptualize meaningfulness of this sort is to begin by imagining of a normalized linear equation in which each variable (xi1, xi2, xi3, and xi4) corresponds to a different domain of engagement (cognitive, behavioral, social, and affective). The value of each of these variables ranges from 0 to 1, and together, they sum to 1 to capture the entire space of engagement. In this analogy, the beta weight for each of the xi variables (βi1, βi2, βi3, and βi4) corresponds to the unique proportion of value that a person ascribes to each domain of engagement. All of these beta weights can sum to 1, indicating that they work together to explain all of the value one can place on a given experience. Both the overall and the individual values can vary from student to student due to differences in prior experiences. So, for example, it is possible for Jill to place a higher subjective weight on cognitive engagement in a mathematics task, and a lower subjective weight on social engagement in the classroom. Jill may spend more

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time focusing on her bell work than chatting with her neighbors. However, it is possible for Jill’s classmate, Olivia, to instead place a higher subjective weight on social engagement, and a lower subjective weight on cognitive engagement. Olivia might find it to be more important to learn about the newest class gossip than attend closely to the morning bell work. Of course, because the process of assigning meaning is part of a dynamical system, it is unlikely that these variables form a perfect linear combination, however, they do contribute differential information, which may be of differential importance to the meaningfulness of an experience given prior learning and current goals. Meaningfulness also varies within individuals over time. For example, at the beginning of the semester, John may view cognitive and behavioral engagement as meaningful, but as he struggles to attain high marks in his class, he may revise his assessment and assign social engagement more value. Such adjustments can also occur across shorter periods of time, and are often contingent on the goals that students have and can attain in a classroom. For example, in his case study of goal development in a vocationally geared high school in the mid-Atlantic region of the U.S., Webel (2013) found that whereas engagement behaviors are relatively stable at the group level, they vary considerably at the individual level based on the personal goals of each student. Moreover, the expressions of each student’s goals, such as those related to self-worth preservation (which may be thought of as being related to affective engagement) and achievement, were displayed differently depending on whether they appeared to match the group’s goals. This suggests that individuals consider their individual goals in conjunction with the environmental constraints present, and evaluate the meanings and behaviors that are available to them in the situation. In addition, there can be a recursive relationship between engagement, outcomes, and meaning, which may stabilize over time within a class, but which may also become volatile under new rules and new roles afforded by different classes and different content. Meaningfulness can be in flux even within a given math classroom because instructors and students interactively develop an understanding of what constitutes a meaningful contribution. More specifically, the existence of meaningful engagement can be largely contingent on the subject matter being discussed. For example, if offering a “different approach” to a problem is generally considered valuable in a given math class (see Connor’s example above), this may require students to invoke different skills and competencies when solving a geometry problem than when solving an algebra problem later on in the semester (see Yackel and Cobb 1996 for how sociomathematical norms appear in a second grade classroom). Students can either view such a shift as a valuable opportunity as they are invited to solve mathematics in ways that make sense to them, or it can make students uncomfortable if they are used to waiting for an authority figure such as their teacher to demonstrate a solution path. Note that the first interpretation may lead students to view cognitive and behavioral engagement as more important in the class, whereas the second may lead students to reduce the perceived importance of cognitive and behavioral engagement. Accordingly, meaning can be unstable and recursive in math education environments.

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Pleasure Third, we discuss pleasure, a topic rarely thought of as pertaining to mathematics learning. It is particularly telling that, in a special issue of Educational Studies in Mathematics (Zan et al. 2006), devoted to the study of affect and meta-affect, the words pleasure and joy were mentioned only once—in the introductory article merely as examples of emotions. None of the other nine articles, by the most respected researchers in the field, even mentioned the words pleasure, joy, or happiness, let alone provided an analysis of their role in mathematics learning. This is puzzling given the powerful role these emotions play in human behavior, unless these emotions are somehow culturally divorced from our thinking as it relates to mathematics. This is a clear hole in our understanding, and a potentially rich area of inquiry, particularly if we are to leverage and promote positive emotions in the design of mathematics learning experiences. We begin to suggest ways to address this lacuna by focusing in on pleasure. Seligman and colleagues describe pleasure in terms of positive emotions, such as joy or contentment (Seligman et al. 2009). As with the previous two components of happiness, we specify our definition of pleasure so that it is useful within the context of mathematics education. These moments of pleasure in doing mathematics can be experienced as magical flashes of insight (Barnes 2000), experiences of making beautiful connections, and appreciation for the aesthetic of mathematics (Sinclair 2001). However, it is important to point out that not all mathematics experiences are immediately joyful or contentment-inducing. Indeed, oftentimes, learning in mathematics is marked by considerable effort and frustration (Goldin 2014). Nevertheless, as suggested previously, this does not eliminate the possibility that these very same frustrating experiences can be pleasurable when viewed from a meta-affective structure such as, “I am really into this.” When operating within a flow-like belief structure, effort can be seen as a challenge to overcome, and as an opportunity to test one’s abilities, rather than as a signal to disengage. These attributions of challenge, then, invoke positive emotions that result from an appraisal of the cognitive difficulty of a task, which in turn defines the whole of the experience as pleasureinducing (see Goldin et al. 2011). In this sense, we can think about pleasure in the context of mathematics as not just an in-the-moment emotional response to the conditions of learning, but as metaaffect that stabilizes students’ momentary frustrations and triumphs, and helps them evaluate the experience as positive or negative (DeBellis and Goldin 1997, 2006; Goldin 2002, 2014; Gomez-Chacon 2000). Meta-affect in this context is thought of as emotions (or more rightly, emotional attributions) about other emotions or cognitions. Just as hiking a mountain can be simultaneously painful and enjoyable, whereas a victory won through dishonesty can be shameful, meta-affect can transform an experience in a way that might not otherwise be expected. Of course, pleasure, as a meta-affective attribution is tied to all the previous experiences of the learner, many of them mathematical, as a kind of emotional

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summary—a means of evaluating the potential emotional content of future tasks. Thus, pleasure is projective, enabling the learner to approach situations similar to those in which they have previously experienced flow-like emotions. Meta-affect is also, therefore, closely tied to students’ mathematical self-concept (or, at the very least compared to it) as well as to their mathematical identity. We think another useful way of situating pleasure comes from Veenhoven (2011). Veenhoven’s model has two key variables that classify satisfaction into four types: Pleasure, Domain Satisfaction, Peak Experience (flow), and finally Life Satisfaction, which he equates with happiness. The key distinctions among these forms of satisfaction deal with, on the one hand, whether the satisfaction is focused on just one part of one’s life, or on one’s life as a whole. On the other hand, one’s satisfaction also depends upon whether the feeling is perceived as transitory or enduring. Thus, pleasure, to Veenhoven, is satisfaction about short-term events that are perceived to be transitory and not enduring—like the pleasure one receives in solving a particularly frustrating algebra problem, for example. Domain satisfaction pieces these pleasurable moments together, filtering out the non-pleasurable, over the whole of one’s experiences in a given domain. Thus, continually finding pleasure in solving particularly frustrating algebra problems might provide one with a feeling of satisfaction regarding one’s experiences in the domain of mathematics: an enduring disposition related to mathematical intimacy (DeBellis 1998). Peak Experience is encountered when a moment creates a lasting impact: an epiphany. Finally, Life Satisfaction or happiness, occurs when the whole of one’s experiences are evaluated as pleasurable in an enduring manner. This might be thought of as consonant with Goldin and colleagues’ (DeBellis and Goldin 1997, 2006) characterization of meta-affect: seeing the collective experience one has undergone and deriving pleasure from the feelings they have produced on the whole. Thus, we think of pleasure in terms of positive meta-affect related to mathematics learning experiences in a way that classifies the overall affect related to mathematics learning in a favorable light; this favorable disposition is then seen as an enduring quality of mathematics experiences. In the math classroom, pleasure as meta-affect can transform an otherwise frustrating and tedious problem set into a prideful illustration of what one has already learned. Note that pleasure in this sense is somewhat different than meaningfulness. Pleasure is an emotional attribution or evaluation, whereas meaningfulness is the subjective value we place on a specific aspect of engagement.

How Do the Three Components Interact to Form Happiness in Mathematics Learning Environments? We find it fitting to begin the discussion of how the components of happiness might interact by illustrating with a mathematical model. Earlier, we suggested that the relationship between engagement and meaningfulness for a given individual (e.g.,

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Joann) is akin to a normalized linear combination in which each of the four components of her engagement (cognitive, behavioral, social, and affective) serve as independent variables (xi1, xi2, xi3, and xi4) which can range from 0 to 1 (e.g., “how much does Joann, as a mathematics learner, engage cognitively or socially?”), and the coefficients (βi1, βi2, βi3, and βi4) which sum to 1, correspond to meaningfulness, or the subjective value a person places on each (e.g., “does Joann put more subjective weight on cognitive or affective forms of engagement?”). We now elaborate upon this model by including an individual’s pleasure (as quality of meta-affect) as well, in terms of a constant, pi, that can take a value between 0 and 1 (e.g., “how favorably does Joann evaluate the overall mathematics experience?”). Multiplying (βi1xi1 + βi2xi2 + βi3xi3 + β4ixi4) by the constant pi thus allows Joann’s meta-affect, or her overall degree of positive emotional attribution of a mathematics learning experience, to influence the meaningfulness (beta weights) that she applies to each of the four components of engagement (cognitive, behavioral, social, and affective). Notably, because this equation is normalized, happiness too takes a value from 0 to 1, reflecting the overall percentage of happiness present for a given person in a mathematics learning environment (“Is Joann unhappy (0% happy?) or fully happy (100% happy?”), or somewhere in between?”). hi ¼ pi ðβi1 Xi1 þβi2 Xi2 þβi3 Xi3 þβi4 Xi4 Þ where hi, happiness, is a scalar, referring to the degree of happiness a given individual (i) experiences in the mathematics learning environment; p is an individual (i’s) positive meta-affect; Xi refers to each of the four components of engagement for an individual (i) (Xi1 ¼ cognitive engagement, Xi2 ¼ behavioral engagement, Xi3 ¼ social engagement, and Xi4 ¼ affective engagement), representing vectors that correspond to complex thoughts, behaviors, social interactions, and feelings/emotions; and finally, the coefficients βi represent a given individual’s subjective meaning associated with each corresponding component of engagement. Of course, we do not think this model is sufficient to capture the whole story. For example, it is important to note that each of the vectors of values representing engagement interact with the other vectors in various combinations, which may themselves vary considerably over time. Moreover, in this linear analogy, correlations and multicollinearities undoubtedly must be present. But it is even more likely that the real relationship among these variables contains even more complexity— feedback loops, sensitivity to initial values, and emergent phenomena that are, as yet, not fully captured in the literature. For example, it appears likely that both the process of engagement in each of the four domains and the outcomes of that engagement may lead to evaluations which influence how much value (how much weight) learners give to each of those domains. In other words, engagement must influence meaningfulness. The more a student engages in mathematics learning, the more they are likely to see connections with other aspects of their life and learning, and to view mathematics as meaningful (either positively and negatively meaningful). Similarly, the meaning we place on the various domains of engagement may influence how much attention, and in turn, effort, learners devote to each of those domains. The four different forms of

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Fig. 1 Process model of some of proposed relationships between engagement, meaningfulness, and pleasure

engagement, too, may be related to one another; for example, cognitive and behavioral engagement are undoubtedly linked—the more a student pays attention in class, the more they may complete their assignments, and affective and social engagement too are likely connected—the more comfortable a student feels with the material and their classmates, the more likely they may be to help their peers. Moreover, it appears that pleasure—positive meta-affect in a mathematics learning environment—may serve to inform what kinds of things learners view as meaningful. When pleasure (as meta-affect) is high, it may reinforce the existing meaning structures; when pleasure is low, it may encourage learners to re-evaluate which aspects of engagement they should most value (and the literature on situational interest bears this out—see Rotgans and Schmidt 2014). Finally, it is unclear what role affective engagement may play in informing pleasure. For example, it is possible that this type of engagement may inform pleasure more directly than behavioral or cognitive engagement, because of the affective nature of the appraisal process that is involved in pleasure (see Fig. 1). Nevertheless, these relationships remain to be tested empirically. In examining happiness, we have conjectured that it has both process and outcome manifestations. It is important to note that we have emphasized the process manifestations here, because they are the levers by which we can influence the engagement of students to be more worthwhile in terms of the pleasure and meaning they may receive from their engagement. Through such leverage, we suspect that the cognitive engagement of students will become more productive, deep, and tied to their behavioral, social, and affective counterparts such that students can say (at least once and a while), “I was happy,” when referring to their mathematics experiences. Through this, we feel that the outcomes of students’ experiences, their sense of fulfillment, and their exposure to mathematical practices as an example of what the

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process of happiness might look like can lead to increased achievement and future engagement in mathematics, and to future life success.

Where Do We Go from Here? Just like considering happiness in the context of the mathematics classroom requires us to expand beyond traditional metrics of achievement, answering questions about it also requires us to broaden our methodological toolkit. Indeed, questions about happiness, engagement, meaningfulness, and pleasure in mathematics learning environments involve a variety of time scales, actors, and behavioral and affective dimensions. Addressing each of these components thus requires researchers to commit to measuring student learning experiences at various points in time, and to recording students’ overt behaviors and affect, as well as their internal interpretations of their experiences. Moreover, because of the critical role the social setting plays in “spreading” happiness, our methods must capture the roles of various influential persons in the mathematics environment (e.g., instructors, learners, classmates). This necessarily invites a mixed-methods approach (see Middleton et al. 2017 for a useful review). Methodologically, the arsenal that a researcher has at her or his disposal is immense. But a few techniques rise to the top in terms of providing exceptional utility to get at the process of engagement (what we have termed in-the-moment engagement). The first of these is Experience Sampling Methodology. This method focuses on experiences as the unit of analysis by sampling students’ experiences over time. This is typically done by paging students through some mobile device at random intervals during their mathematics experiences. The students would then, as soon as practicable, take a short survey about the experiences they are having right now. Over time, the characteristics of those experiences can then be determined empirically, both in an external frame (describing the tasks, social settings, and interactions students are having), and more importantly, in an internal frame (describing the growth of emotional responses, affective interpretations, and other interpretations of the lived experiences of the students). Shernoff and colleagues’ (2016) study of high school engagement used this method, along with classroom observation and other instruments. Using these combined methods, they were able to show that the presence of environmental challenge simultaneously with environmental support for motivational, relational, and social/emotional facets of students’ experiences improve their engagement and related affective outcomes. Observation of learning experiences is a second critical method that should be considered in any study of in-the-moment facets of engagement. Protocols which have been productive for capturing the characteristics of experiences that promote engagement have recorded (a) opportunities for belongingness, (b) the degree of competence and autonomy provided, and (c) meaningfulness of learning (e.g., Turner et al. 2014). These components have typically been compared with measures

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of student engagement: the cognitive, affective, social, and behavioral indices that assess long-term engagement with mathematics. In our current NSF-funded project, Secondary Mathematics in-the-moment Longitudinal Study, our team is beginning to approach some of these questions by taking both in-the moment and longitudinal approaches to collecting such data. For example, we use Experience Sampling Methods to measure, longitudinally, the different dimensions of ~5000 students’ engagement in high school-level mathematics tasks, and simultaneously, we record both their and their teachers’ views on the instructional practices that afford and constrain their engaged behavior at the time of learning (Middleton and Jansen 2017). ESM prompts are pushed to mobile devices at the end of experiences pre-identified by the teacher over the course of three consecutive days, at which time the students provide responses to prompts assessing facets of motivation, social and group behavior, affect, and cognition. Copies of the classroom tasks and student work are uploaded with participants’ responses. Additionally, project researchers video record classroom behaviors with which to situate and provide explanatory context for participants’ responses. But, moments taken out of the ongoing stream of experiences over time have little meaning for either the consolidation of concepts and skills for performance, or for the development of satisfaction and other indices of happiness (Veenhoven 2011). Accordingly, our project is examining the long-term effects of engaged mathematical experiences (and relatively-unengaged experiences, as the case may be) for 2 full years of high school. From these analyses, we will be building a model of the shortterm and long-term effects of instructional practices, as well as the relationship among motivational and engagement-related variables that impact the learner’s sense of efficacy and well-being, and using this to answer questions like the ones we posed above.

Conclusion In this chapter, we considered the benefits of moving beyond traditional indices of success in the classroom (such as achievement) to focus on a measure of something bigger: happiness. Although happiness may not be the first word to come to the mind of many in the context of mathematics classrooms, we think it useful both in the instrumental sense (productivity), and in the personal sense (fulfillment). As, incorporating happiness into mathematics learning environments would not only allow students to learn better, it would also provide them with a template for the process of happiness—a template they could take with them and apply to other domains of their lives. We view happiness in the mathematics context as consisting of three components: engagement, meaningfulness, and pleasure. In our view, engagement consists of a dynamic relationship between a learner and her or his learning environment. Engagement manifests itself in the form of mental activities and overt behaviors in four different domains: cognitive, behavioral, affective, and social. Meaningfulness,

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then, is the subjective weight or importance that a learner places on each of the four dimensions of engagement. Finally, pleasure is defined as meta-affect, or the learner’s affective appraisal of the whole mathematics learning experience. This framework poses a number of intriguing possibilities, including unpacking the relationships between various forms of engagement, and offering avenues for meaningful and long-term mathematics engagement. Unpacking these possibilities requires mixed methodological approaches, which involve studying mathematics classrooms across various dimensions, time frames, and persons. Ultimately, we hope to discover those aspects of the moment which provide (inter)personal connection, meaningfulness, and some modicum of pleasure such that interventions such as tasks, curricular sequences, teaching strategies, and tools can be designed to improve the probability that a student might begin to see mathematics as worthwhile in its own right, and instrumental to their future fulfillment. We invite others to join us in the exciting practice of learning more about how to add happiness into the equation of mathematics learning experiences.

References Aristotle, & Brown, L. (Ed). (2009). The Nicomachean ethics (D. Ross, Trans.). Oxford: Oxford University Press. Barnes, M. (2000). “Magical” moments in mathematics: Insights into the process of coming to know. For the Learning of Mathematics, 20(1), 33–43. Csikszentmihalyi, M. (1990). Flow: The psychology of optimal experience. New York: Harper & Row. DeBellis, V. A. (1998). Mathematical intimacy: Local affect in powerful problem solvers. In Proceedings of the twentieth annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 435–440). DeBellis, V. A., & Goldin, G. A. (1997). The affective domain in mathematical problem solving. In E. Pehkonen (Ed.), Proceedings of the 21st conference of the International Group for the Psychology of Mathematics Education (PME), Lahti, Finland (Vol. 2, pp. 209–216). Helsinki: University of Helsinki Department of Teacher Education. DeBellis, V. A., & Goldin, G. A. (2006). Affect and meta-affect in mathematical problem solving: A representational perspective. Educational Studies in Mathematics, 63, 131–147. Delle Fave, A., & Massimini, F. (2005). The investigation of optimal experience and apathy. European Psychologist, 10(4), 264–274. Dewey, J. (2007). Experience and education. New York: Simon and Schuster. Fredricks, J., Blumenfeld, P., & Paris, A. (2004). School engagement: Potential of the concept, state of the evidence. Review of Educational Research, 74(1), 59–109. Retrieved from http://www. jstor.org/stable/3516061. Fredrickson, B. L. (1998). What good are positive emotions? Review of General Psychology: Journal of Division 1, of the American Psychological Association, 2(3), 300–319. Fredrickson, B. L. (2001). The role of positive emotions in positive psychology. The American Psychologist, 56(3), 218–226. Goldin, G. A. (2002). Affect, meta-affect, and mathematical belief structures. In G. Leder, E. Pehkonen, & G. Torner (Eds.), Beliefs: A hidden variable in mathematics education? (pp. 59–72). Dordrecht: Kluwer.

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Chapter 8: Development of Modelling Routines and Its Relation to Identity Construction Juhaina Awawdeh Shahbari, Michal Tabach, and Einat Heyd-Metzuyanim

Abstract In this chapter, we link between modelling activity and affect through the concept of “identifying” or identity construction, as conceptualized within the communicational framework (Heyd-Metzuyanim and Sfard 2012; Sfard 2008). Our aim is to trace the development of modelling abilities through following the development of routines and the changes in identifying talk that co-occur along this development. For this aim, we follow a group of five prospective teachers as they worked on two model-eliciting tasks. Their working process was video recorded and transcribed. The participants’ discourse was analyzed to identify changes in routines while working on the two modelling tasks along with changes in their subjectifying talk (communication about themselves and others). We were able to trace changes in both these measures. Regarding the mathematical talk, we identify a change from a nonsystematic choosing-routine to systematic-choosing-routines and from routines that focus on choosing specific cases to routines that focus on eliciting criterions for choosing. Regarding their identifying activity, we show how participants initially build on their everyday roles in real life (such as mother, citizen and student), to justify their claims in the modelling activity. Later, when routines become more systematic and established, there is much less identifying talk, and claims are justified based on mathematical narratives. We link these findings to previous findings regarding the interaction of mathematizing and identifying activities in mathematical learning. Keywords Modelling · Model eliciting activities · Routine · Communicational framework · Identity · Subjectifying and mathematizing

J. A. Shahbari (*) Sakhnin College, Al-Qasemi Academy, Baqa El-Gharbiyye, Israel e-mail: [email protected] M. Tabach Tel-Aviv University, Tel Aviv, Israel E. Heyd-Metzuyanim Technion, Haifa, Israel © Springer Nature Switzerland AG 2019 S. A. Chamberlin, B. Sriraman (eds.), Affect in Mathematical Modeling, Advances in Mathematics Education, https://doi.org/10.1007/978-3-030-04432-9_11

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Introduction The modelling approach emphasizes the effectiveness of mathematics in real life (Vorhölter et al. 2014). Modelling activities are designed for small group work where the participants act as “a local community of practice” solving a real situation (Lesh and Zawojewski 2007). The participants are required to share responsibility in constructing their models through iterative cycles of translation, description, explanation, justification, and prediction of outcome data and solution paths (Lesh and Doerr 2003). This means, alongside the constructing of models and the mathematical development, that the participants have opportunities for social development, because they need to develop their argumentation skills while they assume, explain, and justify for each other (English 2003). Therefore, modelling activities are considered a rich experience for monitoring changes in both cognitive and affective domains. In the current study, we join the effort to study these aspects by focusing on the changes in modelling abilities with interrelation to the participants’ identity construction. A good candidate for such combined analysis is the commognitive framework (Heyd-Metzuyanim and Sfard 2012; Sfard 2008). Using the communicational perspective enabled us to closely monitor how learners’ mathematical and identity construction processes unfolded and changed while they engaged in sequence of two Model-Eliciting Activities (MEAs).

Framework Modelling Modelling offers learners opportunities to confront mathematical as well as everyday challenges (Lesh et al. 2000). MEAs are a relatively complex type of modelling task involving real situations with incomplete, ambiguous, or undefined information (English and Fox 2005). The learners are required to mathematize the situation in ways that are meaningful for them (Lesh and Doerr 2003). For the mathematization process, there is a need for type of quantities and operations in the realistic situation; the kinds of quantities that are needed include accumulations, probabilities, frequencies, ranks, and vectors. The operations include predicting, sorting, organizing, selecting, coordinating, quantifying, weighting, and representing data (English 2006). Therefore, engagement with MEAs fosters among learners modelling abilities, which is beyond the school mathematics textbook, and that are needed for our era (Doerr and English 2003). The product of engagement in MEAs are models for describing, explaining, or predicting the behavior of complex situations. These models are then extended, explored and applied in other situations. This process is considered as a central aim in model eliciting activities (English and Watters 2005).

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The Communicational Framework The communicational framework (Sfard 2008) is a socio-cultural perspective for studying learning processes. The framework suggests that mathematics is a type of discourse and that thinking is a certain form of self-communication. Sfard proposes four characteristics of mathematical discourse: (1) Words and their uses: Each discourse is characterized by its own keywords. Sometimes, the same words are used in different ways in colloquial and mathematical discourse. (2) Visual mediators: As mathematics is not about physical objects, in many cases communication is fostered by referring to visual realizations that are part of the communication (e.g., graphs or symbols). (3) Narratives: Narratives are sequences of utterances framed as descriptions of objects, relations between objects or processes with or by objects that can be endorsed or rejected (e.g., theorems and definitions). (4) Routines: Routines are repetitive discursive patterns characteristic of a specific discourse (e.g., solving a linear equation). According to the communicational framework, learning is a change in the individual’s discourse, that is, a change in words and how they are used, in narratives endorsed or in routines used. Sfard and Lavie (2005) divided routine to three subsets, opening, procedure and closing. The opening specifies the condition that routine may be evoked. The procedure is the how of the routine, it includes the subset that identifies the performance, and the closing describes the circumstances that indicate the completion of the routine performance. Sfard (2008) suggested two types of learning: learning at the object level and learning at the meta-level. Tabach and Nachlieli (2016) explained that object level learning involves expanding the existing discourse by expanding the range of word use, routines, visual mediators and endorsed narratives for a mathematical object. Meta-level learning involves changes in the meta-rules of the discourse. While learning and doing mathematics, participants not only talk about mathematical objects and preform mathematical routines, they also talk about themselves and about other participants. Following Heyd-Metzuyanim and Sfard (2012), we term this talk subjectifying and delineate it from the mathematical talk, which is termed mathematizing. Identity construction during mathematical learning can be first and foremost seen through subjectifying actions (whether verbal or non-verbal). These may include statements including personal pronouns such as “I don’t understand this” or “you can probably solve it” as well as more implicit messages that identify the speaker as a certain speaker. Heyd-Metzuyanim and Sfard (2012) termed these implicit subjectifications as indirect identifying, defined as actions intended to elicit a certain identifying narrative in the receiver of the message. As a whole, identifying activity, including emotional expressions and indirect subjectifying, constructs a set of narratives about participants that are significant (for the author of the identity story) and eventually reified into certain labels (such as “a mother”, “a gifted mathematician” or a “dutiful citizen”) (Sfard and Prusak 2005). This is what we term identity.

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Research Goal and Questions The aim of the current study is to monitor the development of modelling abilities among a group of prospective teachers through following the development of routines and the changes in subjectifying talk that co-occur along this development, by means of the communicational framework (Heyd-Metzuyanim and Sfard 2012; Sfard 2008). We ask the following research questions: 1. Which routines and participants’ identifying talk can be noted in the participants’ work on model-eliciting activities? 2. What changes in routines and participants’ identifying talk can be noted while they work on a sequence of two model-eliciting activities?

Method Research Participants This study is part of a larger research project aimed at examining the development of modelling abilities among practicing and prospective teachers while they engage in a sequence of modelling activities (e.g., Shahbari and Tabach 2016). In the current study, we followed and monitored one group with five participants [Areen, Fatena, Kaman, Raneen and Muhand – all pseudonyms]. All prospective mathematics teachers were in their 2nd year of studies in mathematics education track at a college of education in Israel. They participated in a problem-solving course taught by the first author as part of their studies. They had no previous experience with modelling activities.

The Model-Eliciting Activities In the current study we used two model-eliciting activities: one is “Summer camp activity” and the other “Good teacher” activity, both designed by Shahbari and Tabach (2017) based on the six design principles of Lesh et al. (2000).

Summer Camp Activity The Summer camp activity is about a mother who wants to choose a summer camp (s) for her two children. The mother organized the data about six camps (A, B, C, D, E and F) via four tables that provide information about each one, where each table refers to several components. We named the tables by using the letter C to refer to camp activity and Roman numerals to indicate the order of the table. For example,

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Table C.i is the first table in the camp activity. Table C.i provides the dates, transportation, food and cost of each camp. Table C.ii includes the types and number of entertainment activities at each camp. Table C.iii consists of data from the previous year about the number of participants and number of counselors at each camp. Table C.iv provides the parents’ evaluations and ranking of the camps for the previous year, with the ranking ranging from one to five stars. The participants need to write a letter to the mother explaining which camps are the most suitable based on her criteria, suggesting a model she can use every time she wants to choose a camp.

Good Teacher Activity The Good teacher activity is about a principal of an elementary school who is seeking a candidate for the position of math teacher at his school. He has a list of ten graduates who completed their B. Ed. The data about the candidates is described via four tables. We named the tables by using letter G to refer to the good teacher activity and Roman numerals to indicate the order of the tables. Table G.i includes the candidates’ ages and their average grades in their B. Ed. studies. Table G.ii includes the candidates’ ranking by their pedagogical instructors for their practice teaching, ranging from A+ to F over 3 years. Table G.iii includes the ranking of the candidates’ performance in the interview. Table G.iv includes the ranking for participation in social initiatives. The solvers need to write a letter to the principal that identifies the most suitable candidate, explain the choice and provide a model for choosing suitable candidates that he can use in the coming years.

Research Procedures and Data Sources The participants first worked on the Summer camp activity and after 1 week worked on the Good teacher activity. The instructor (the first author) introduced the camp and good teacher activities. After which the participants received the activity (hard copy) and were asked as a group to provide a solution. The main data sources were two video recordings of one group of five teachers working on the two model-eliciting activities. These recordings were transcribed verbatim: a 132-min video recording of the camp activity and an 80-min video recording of the good teacher activity. We also used the group’s reports and draft sheets as an additional data source.

Data Analyses The data derived from the video recordings were analyzed by the commognitive perspective in two phases: (a) the first phase included searching for routines, based

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on Sfard’s (2008) discourse analysis methodology. This search was conducted as follows: After transcribing the video recordings, we parsed the transcript into episodes according to participants’ working in each table or in the relation between the tables in the model-eliciting activities. If during one sub-task the participants enacted two routines, we separated the participants’ utterances into two episodes. In each episode, we searched for participants’ actions in order to trace the routines they employed while they worked on each model-eliciting activity. For each identified routine, we searched for two defining parts. The first was the when of the routine, comprising the routine’s opening and later its closing. The utterances between the routine’s opening and closing were classified as the how of the routine, i.e., the routine procedure. (b) The second phase included searching for identifying activity: in each routine we distinguished utterances that included subjectifying (HeydMetzuyanim and Sfard 2012); The subjectifying utterances included talk about people, for example “the interviews are important for me”; examining these subjectifying utterances, we found that an important theme rising from them, and that was highly connected with the context of the modeling problems, was the roles students were taking on while they were subjectifying. We delineated these subjectifying statements as role-taking identity statements. For example, if a student talked about “as a mother, I would. . .” or “I wouldn’t let my children go to. . .”, we classified this subjectifying statement as taking on the role of a mother.

Findings We first present the modelling routines that we identified in the two modelling activities. Then we focus on the closing condition for each routine.

Modelling Routines in the Two Modelling Activities We identified three main routines in the two modelling activities. The choosing specific cases routine was identified only in the Summer camp activity. The eliciting general model routine was identified in the two modelling activities. The implementing the elicited model routine was identified in the Good teacher activity only. We detail the sub-routines of these three main routines for each activity separately, and summarize all sub- routines in Table 1. Evidence from participants’ discourse for each sub-routine, are presented in the section about subjectifying activity in the two modelling activities to avoid repetition.

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Table 1 Routines identified in the two modelling activities Activity Summer camp activity- first phase

Routines Routine 1: Choosing specific cases (camps)

Features Non-systematic comparison Systematic comparison sub-set of cases

Summer camp activity- second phase

Good teacher activity

Routine 2: Eliciting general model

Routine 2: Eliciting general model

Routine 3: Implementing the elicited model

Integration between components Systematic comparison

Systematic comparison Integration between components Systematic comparison Systematic comparison Ranking according to elicited model

Sub-routines Looking at all cases [Episode 1] Looking at a sub-set of cases [Episode 2] Using average [Episode 3] Using estimation and ratio [Episodes 4 and 5] Assigning relative weighting [Episode 6] quantification of numerical data [Episode 7] defining range and scoring [Episode 8] quantification of qualitative data [Episode 9] Quantification of numerical data [Episode 10] Assigning relative weighting for components [Episode 11 and 12] Using average to assign values for quantitative quantities [Episode 13] Quantification of qualitative data [Episodes 14 and 15] Implementing the elicited model [Episode 16]

Modelling Routines in the Summer Camp Activity The participants’ work in the Summer camp activity can be separated into two phases. In the first phase, the participants tried to choose specific camps. The participants discussed the components of Table C.i and chose three camps. They then tried to check their choice based on the information provided in the other three tables without integrating the information from all four tables. The participants’ work during this first phase is denoted by Routine 1, which includes two main routines: non-systematic comparison and systematic comparison with focus on a sub-set of cases. Each of the two routines consists of two nested sub-routines: Looking at all cases; Looking at a sub-set of cases; Using average; and Using estimation and ratio. In the second phase, after about half of the total duration of the activity elapsed, the participants began thinking about the need to elicit general criteria for choosing between the cases rather than choosing individual cases. The need to write a letter about their considerations and decisions triggered this change. In the second phase of

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their work, we identified Routine 2 comprising two main routines: integration between the components, and systematic comparison, which consists of three sub-routines: quantification of numerical data; defining range and scoring; quantification of qualitative data. After working for 92 min, the participants could provide only a partial model – in which some components of the tables were still to be considered, without actual results and the participants were unable to write a letter about their recommendations.

Modelling Routines in the Good Teacher Activity The participants’ work in the Good teacher activity was different from their work in the camp activity, even though the two activities had similar external and internal designs and similar mathematical foci of weighing variables, which required similar modelling skills. In the Good teacher activity, the participants did not choose specific candidates. Rather, they elicited a model and then weighed all the candidates according to the model. The participants adapted most of the sub-routines of Routine 2 (quantification of qualitative data, quantification of numerical data and assigning relative weighting) and one sub-routine of Routine 1 (using average) from the camp activity. In addition, we identified a new routine in the Good teacher activity— implementing the elicited model derived by the participants. Table 1 presents the routines that were identified in the two modelling activities. Table 1 shows that all the sub-routines that we identified in the Good teachers activity while the participants elicited a general model, were also identified in Summer camp activity. Using average was identified in the two modelling activities. However, the use of average in Summer camp activity was identified in comparison sub-set of camps, while in the Good teacher activity it was identified in the systematic comparison between the candidates. Three of the sub-routines of Routine 2 were identified in the two modelling activities, yet in the Summer camp activity they were identified in the second phase. Routine 3 was identified in the Good teacher activity only because in the Summer camp activity, the participant get only partly model, as explained above.

Subjectifying Activity in the Two Modelling Activities Participants’ subjectifiyng talk was different in the two modelling activities. In the first activity- Summer camp, we identified more subjectifying talk, specifically in the first phase. While in the second activity- Good teachers, we identified less subjectifying talk.

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Subjectifying Talk in the Summer Camp Activity As we mentioned earlier, the participants’ work in the Summer camp activity separated into two phases. In the first phase, the closing sub-routines were all based only on personal authority condition, such as, the decision of choosing the number of the camps depended on participants’ personal experience as a mother. In the second phase, the closing condition of some sub-routines were personal authority and others were mathematical. Within sub-routines, participants took up different roles such as Mother, Father, Citizen, Consumer and Students. Table 2 presents the sub-routines and their structure (open, procedure and closing condition for each sub-routine). The episodes are detailed below. To demonstrate each sub-routine, we present the to participants’ discourse parsed into nine episodes. Episode 1 below shows the participants’ discussion immediately after they read the camp activity. Episode 1: Non-systematic comparing with focus on all cases [1] Fatena: [Reading the activity] [2] Areen: Let’s first consider the dates of the camp. The first one is from July 2 to 10 [3] Muhand: All the dates are suitable [7] Areen: Let’s compare camp A and camp C. What’s the difference between them? Both of them provide transportation. The first one provides food and costs 750. The third costs 900 [8] Kaman: The first is better [9] Areen: Right, C gives 1 day more but doesn’t provide food and is more expensive. So, A is more worthwhile than C [10] Kaman: We can choose both A and E [33] Areen I think it is enough. I am a mother. Enough [37] Kaman: So the camps are A, D and E

Episode 1 includes the participants’ discussion of the four components in Table C.i: dates, transportation, food and cost. The opening of the routine was triggered by the requirement to choose the most suitable camp or camps [1]. The procedure included direct and non-systematic comparison [2–36]. The routine closed with the choice of three camps—A, D and E [37]. As an example of the procedure, Muhand [3] determined that the dates of all the camps are suitable without explicitly relating to the features of each camp, such as number of days or other components. Kaman [8] determined that camp A is better without justification. She looked only at two components in Table C.i and did not consider the others. The closing of the routine was based on Areen’s personal authority [33]. She explicitly backed her statement with an identifying statement of herself as a mother (“It’s enough. I’m a mother. Enough”). After the participants worked with Table C.i, they discussed how to take Table C. ii into consideration. Episode 2 reproduces the participants’ discussions with respect to Table C.ii, which includes the number and types of entertainment activities at each camp.

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Table 2 Sub-routines in Summer camp activity, structure and their closing condition Sub-routines Looking at all case [Episode 1]

Looking at a sub-set of cases [Episode 2]

Using average [Episode 3]

Using estimation and ratio [Episodes 4 and 5]

Assigning relative weighting [Episode 6]

Quantification of numerical data [Episode 7]

Quantification of qualitative data [Episode 8]

Defining range and scoring [Episode 9]

Structure of routines opening, procedure and closing Opening: Consider Table C.i Procedure: Non-systematic comparison with focus all cases Closing: Choosing three camps Opening: Consider Table C.ii Procedure: Non-systematic comparing with focus the three chosen camps Closing: Endorsed the three chosen camps Opening: Consider Table C.ii Procedure: Using average to compare the three chosen camps Closing: Endorsed the three chosen camps Opening: Consider Tables C.iii and C.iv Procedure: Using ratio and estimation with focus on the three chosen camps Closing: Endorsed the three chosen camps Opening: The need for weighing tables by their importance Procedure: Assigning relative weighting Closing: Assigning values for the tables: 40%, 30%, 20% and 10% Opening: The need for assigning values for quantities Procedure: Quantification of numerical data Closing: Assigning quantity for each value Opening: The need for assigning values for quantities Procedure: Quantification of qualitative data Closing: Assigning quantity for each value Opening: The need for weighing numerical quantities Procedure: Defining range and scoring Closing: Assigning point for selected range: 15–20 get 18 points

Closing condition Personal authority

Personal authority

Personal authority

Mathematical and personal authority

Mathematical

Mathematical

Mathematical

Mathematical

Episode 2: Non-systematic comparison focusing on a sub-set of cases [39] Areen: Now, what about the other tables? Should we look at the data of all the camps or only the ones we choose? [43] Raneen: Let’s look only at the three camps [45] Areen: The first camp offers 8 h of sports, swimming . . . there is no scientific exploration [47] Fatena: At camp D there are more activities, one compensates for the lack of others (continued)

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Areen: Fatena: Kaman: Muhand: Muhand: Areen:

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It is strange! Without trips I will not send my son But look there are 1,1,2 trips, that’s means all the camps together 4 trips But you pay such amount [money], to send your son only for activities I mean, imagine yourself a child, what is the most important thing you want Also there are water games, we as children went for swimming, there is a swimming activities The three camps cover all the activities

Episode 2 reflects a non-systematic comparison between the components of the three camps chosen from Table C.i. The opening of the routine starts with Areen’s question [39]. She vacillated over whether to examine only the camps they chose in Table C.i or to discuss the data of all the camps. The procedure exposes their implicit decision to examine only the three camps. Their comparison between the camps was non-systematic [40–73]. The routine closed with Muhand’s decision [74] to stay with the three camps. The closing condition of the routine was, again, based on personal authority, with Areen’s [70] statement about her (and others) experience as children (“we as children went for swimming”). Areen’s statement follow other participants’ identifying statements. For example, Fatena [52] said “I will not send my son”, thus implicitly identifying herself as a mother and using this role to justify her statements. Muhand [56] co-constructed Fatena’s role by identifying her as a mother (“if you send your son”) who has to make a decision about the camp. Even though the participants began discussing Table C.i unsystematically, one of them later suggested using averages to discuss it systematically. Episode 3 shows the participants’ ongoing discussions with respect to Table C.ii. Episode 3: Systematic comparison based on average with focus on sub-set of cases [75] Fatena: [referring to computing the average number of entertainment activities at each camp] Maybe we should collect all of them and compute the average. Then we can see which one is better [76] Areen: Yes, let’s do that [82] Fatena: 9, 10, 8, 2, 7, 21 divided by 6 [83] Areen: 9 [the average]. . . [85] Kaman: A [camp A] is close to the average and F [camp F] too [89] Areen: Pay attention to the numbers. They are similar. We don’t need the average [93] Areen It is important that the child will have fun more than (that he) studies [95] Kaman: We’ll stay with A, D, and E

Episode 3 systematically compares the components of the three chosen camps from Table C.i. The routine starts with Fatena’s [75] sudden suggestion to compute the average of the number of entertainment activities at the camps. The procedure [76–94] includes computing the average, and the routine closes with the decision to stick with the same three camps [95]. The closing of the routine is based, yet again, on Areen’s [93] personal authority as a mother. Here, this identification is rather implicit, as Areen talks about a general “child” (not her son) that should have “fun

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more than studies”. However, given the previous explicit declarations of herself as a “mother”, such a value-laden statement can be directly linked to the previous role that Areen had taken up. After the participants finished working on Table C.ii, they discussed Tables C.iii and C.iv. They used the procedure—ratio and estimation—while handling data from the two tables. Episodes 4 and 5 show the participants working with these tables. Episode 4: Comparison by estimation and ratio of sub-cases in Table C.iii [115] Areen: Now let’s look at the third table [Table C.iii] [117] Fatena: Does it show the number of campers? [118] Areen: Ok, first let’s check the three camps we chose. If the numbers are right, we can compare them with the others [119] Kaman Camp A [120] Areen For five children, there is one counselor; the total number of children does not matter [meaning the total number of all children in the camp is not important] [121] Fatena What is important is how many children are in [each] group [126] Areen For 14 [referring to camp E] children there are two counselors. [127] Kaman: That means that each counselor has 7 children; they are almost the same

Episode 5: Examining the three chosen camps from Table C.iv using ratio [144] Muhand: Now let’s look at Table 4 (Table C.iv) [145] Areen: Showing how the parents evaluated the camps last year [146] Muhand: Let’s look at [camps] A, D and E [149] Areen: Let’s see, camp A is the highest, with 3 stars [150] Kaman: Look at the five stars [camp A] 159] Fatena: If we add all the categories we can get the total in each camp. No, we can get from the last table [Table C.iii] [161] Fatena: We can check the ratio [ratio between each category and number of children as appear in Table C.iii] [162] Areen: How? [163] Fatena: 68 [number of votes in the three star category] divided by 210 [number of all the children participate in camp A] [164] Areen: It is approximately [the number of parents], maybe parents have more than one child in the camp [174] Areen: I see that A is just not good. E and D good, D more than half gave 4 stars, right? 253 [number of votes in the three star category] from 570 [number of all the children participate in camp D]? [184] Fatena: But maybe a mother would say, no matter what others say, only the number of days is important [185] Areen: No, not to this level, they take into account other things, if in a camp they beat children, then because of the number of days we sent to it [sent the children to the camp] [187] Raneen: D and E. . . we agree about them [188] Areen: Yes [189] Raneen: Maybe we should replace A

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Episode 4 shows the participants’ discussion of Table C.iii, which includes data from the previous year about the number of participants and number of counselors at each camp. In the opening of routine using estimation and ratio the participants discussed whether or not to continue examining the three chosen camps [115–119]. The procedure [120–126] uses estimation of the ratio between the number of children and the number of counselors at each camp. The routine [127] closes with a confirmation of their choice of the three camps—A, D and E. Episode 5 shows the participants’ discussion of Table C.iv, which includes the parents’ assessments and ranking of the camps from the previous year, with the ranking ranging from one to five stars. As seen in Episode 5, the routine using estimation and ratio [144–146] opens with the participants checking only the three camps. The procedure [161–186] involves estimating and comparing the ratios of the rankings in each category. The routine [187–189] closes with the choice of camps D and E. The closing of the routine combined mathematical as well as personalauthority justifications. Thus, in [174] Areen decided which camps are good according to mathematical considerations (number of votes, proportion of 4 star ratings), but in [185] she based her justifications again on her role as a mother who is knowledgeable of parents’ considerations (“they take into account” and “we sent to it”). Episode 6 presents the participants’ initial attempt to search for general criteria. In this episode, the participants discuss how to rank criteria by assigning a different weight to each criterion. Episode 6: Assigning relative weights to the tables in the camp activity [216] Fatena: What do you say about the tables? Which one is more important? [225] Fatena: 4 [Table C.iv], it is about parents. They [parents] sorted or ranked [the camps], there are those who did not vote, perhaps they gave a high score but they did not vote. Like the local municipalities election, some [people] don’t vote in it, maybe they do support [the candidate] but they do not vote [226] Areen: I don’t think so. They [other people] drag them [people who don’t want to vote] [230] Areen: We understand that the first table is the most important, but what do we have to do to organize them (the tables)? [234] Muhand: We must arrive at a solution so that if you are a mother and get these lists of camps, you’ll know how to choose [235] Fatena: I’ll prepare a table. If the camp meets all the criteria in the table, it is suitable [249] Areen: But how can we express this? We have to be able to write it in a letter [256] Areen: We’ll assign a weight for the period of the camp [273] Areen Right, I agree with Fatima, it is right, we should consider the price [274] Muhand: So, you send your son without looking at the program [276] Fatena: We can’t leave anything out. We have to assign priorities. That means all of them are important [279] Areen: The activity reminded me..., you know if you want to buy food products or for example washing powder; if the price of 5 kg is like that. . . and for 6 kg the price like that. . ., so you need to check if it’s the same price per 1 kg [282] Fatena: We need to build a table with the more important criteria. (continued)

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[310]

Fatena

[320]

Kaman:

[339]

Areen:

[354] [376] [407]

Fatena: Kaman: Kaman:

Do you remember the feedback that we do about the lecturers? It is the same, we evaluated them, do they look at it If you want to choose a school for your son; at first, you will ask about the school I think we can say that table one is worth 50% and the other tables are 50%, then we can separate the 50% into the components; the whole is 100% We can say how much each table is worth, and then, how much in each table The first table is 50% 40%, 30%, 20% and 10%

The participants’ discussion in Episode 6 shows their attempts to find ways to organize and rank the tables by their importance. Episode 6 illustrates routine Assigning relative weighting, this routine opened with Fatena [216] asking her peers to rank the tables, Areen [230, 249] asking about how to organize the tables and Muhand [234] focusing on the need to find a solution. The procedure [339–406] uses weighting by assigning a percent to each table, according to Areen’s suggestion [339] to use percent where the whole is 100%. The routine closed by providing a percent for each component, as Kaman suggested [407]. The closing condition of the routine -Assigning relative weighting uses a mathematical justification. Within the participants assigning relative weighting for each table, they played different roles (mother, father, student, consumer and citizen). Fatina tried to explain that Table C.iv is not important by using her experience about assessment from her real life. Fatena [225] exemplified as a citizen the case of local council election, when supporter of one candidate did not go to vote for him. Areen [226] with the same role as citizen, rejected Fatena’s claim. With another role, Fatena continued to explain that feedback of parents in Table C.iii is not important. Fatena, [310] taking up a role of student exemplified the case of students’ feedback about the lecturers, which she claimed as not important because the college’s management did not consider it. The participants occupied the roles of mother and father; Mohand [274] turned to Areen and asked her about her choice as a mother, he used Areen’s role as a mother in order to emphasize the importance Table C.ii in the Summer camp activity. In similar way, Kaman [320] turned to Mohand, and asked him what he will do as a father in order to emphasize the importance of Table C.iii in the Summer camp activity. When the participants discussed the costs of the camps, we identified the role of consumer three times, as seen in Areen’s [279] explanation. Areen [279] explained about the need to check the cost of one unit to get the economical price, when there are products with different amounts and costs; Areen tried to explain the need to look at different things through comparing the costs. Episode 7 shows the participants’ discussion about assigning meaning to the quantities in the different tables. Episode 7: Quantification of numerical data in the camp activity [444] Areen: What is the meaning of 12? We need a key. What part of the 100% does it represent? [445] Kaman: A part of 40 (continued)

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Fatena:

[456]

Areen:

[457]

Fatena:

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What does a child need sciences research in camp for? He is bored in school (science, let alone at a camp) Wait a moment. I need to return to the first table. We ranked it from 1 to 40. So, what does this mean? For example, 10% of 40 is 4, so it is 4

Episode 7 illustrates routine- Quantification of numerical data, the opening of the routine was triggered by Areen’s [444] question about how to assign meaning to a quantity. The procedure [445–456] involved quantification of numerical data by defining the whole and giving a value for the quantity. The routine [457] closes by computing a value for the quantity. The closing condition is based on mathematical activity. However, through the process of participants’ quantifying numerical data in the different table, we identified subjectifying activity. For example, Fatena [452] subjectified the role of child through her rejection of the quantification of Table C.ii (“what does a child need science research in camp for?”). In trying to assign points to different quantities, the participants had to quantify qualitative data, such as providing food and transportation. Episode 8 is a short episode describing this quantification. Episode 8: Quantification of qualitative data in the camp activity [522] Fatena: Food and transportation? [523] Areen: There is or there isn’t, 4 or 0 [524] Fatena: yes is 4, no is 0

Episode 8 illustrates routine Quantification of qualitative data. The routine opens with Fatena’s question about the food and the transportation [522]. The procedure involves Areen’s suggestion to assign a numerical value [523], and the routine closes with assigning a numerical value to each variable [524]. The closing of the routine uses a mathematical justification, there is no evidence for personal authority. Episode 9 shows the participants’ discussions about how to quantify continuous quantities. Episode 9: Defining range and scoring in the camp activity [467] Fatena: For example, if one camp offers 20 days and another offers 8, she won’t choose the one offering fewer days [468] Areen: I understand (she looks at the researcher) but we need to tell her what a 12 or a 20 means. We must provide her a tool for calculating which is more important [469] Researcher: Here’s a chance to examine this by using a range. For example, certain scores will be assigned to all the prices within a certain range [478] Areen: The number of days? [479] Kaman: 1–20 [days] is the larger one [number of days] [480] Areen: From 1 to 20 [days] gets 20 [points] [504] Areen: A score equal to or greater than 20 is worth 20 points; between 15 and 20 [days] is worth 18 points; 10–15 [days] is worth 15 [points]. Note there is no need for addition here because we use ranking

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Episode 9 illustrates the routine Defining range and scoring. The routine [467] opens when Fatena asks about tools to help assign weights to the dates of the camps. The procedure begins with the researcher’s suggestion [469] to use a range. Then the participants [480–503] use the range by assigning points to several components. The routine closes by assigning the numerical values to the components, as reflected in several sentences, for example [504]. The closing of the routine is based on a mathematical justification.

Subjectifying Talk in the Good Teacher Activity In the Good teacher activity we identified much less subjectifying utterances compared to the Summer camp activity, and the closing conditions of all of the sub-routines were mathematical. We identified only two turns in which some of the participants took up a role of a school principal and an expert on practices in the Ministry of Education. Table 3 presents the subroutines, their structure (open, procedure and closing and the closing condition of each sub-routine). Table 3 Subroutines, structure of each sub-routine and their closing condition in Good teach activity Sub-routines Quantification of numerical data [Episode 10]

Assigning relative weighting for components [Episode 11 and 12]

Using average to assign values for quantitative quantities [Episode 13]

Quantification of qualitative data [Episodes 14 and 15]

Implementing the elicited model [Episode 16]

Structure of routines opening, procedure and closing Opening: Consider Table G.i Procedure: Transforming average of each candidate to another value Closing: Assigned value for the each candidate Opening: Consider all the tables and components in Table G.ii Procedure: Assigning relative weighting Closing: Assigned percent for the each table and for each criterion in Table G.ii Opening: Consider Table G.ii Procedure: Using average to calculate the different scores for each candidate Closing: Assigned value for the each candidate Opening: Consider Tables Table G.iii and Table G.iv Procedure: Quantifying of qualitative data Assigned value for the different criteria in the two tables Opening: The need for implementation of the elicited model Procedure: Ranking candidates according to elicited model Closing: Choosing one candidate

Closing condition Mathematical

Mathematical

Mathematical

Mathematical

Mathematical

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To illustrate the sub-routines, we show the participants’ discourse in Episodes 10–16. Episode 10 depicts the participants’ discussion on how to scale the numerical quantities in Table G.i. Episode 10: Quantification of numerical data from Table G.i [1] Fatena: [Reads the good teacher activity] [5] Areen: We can start with their averages [9] Fatena: The lowest is 78, so the average is between 78 and 96. . . so we can say that whoever gets 78 will have low scores. Do you remember the use of scores? For example, 78 will get 5 points, and if the average is higher, it will get more points [26] Kaman: That means 70 is one point, 71 is 2 points, 72 is 3 points. . .100 is 30 points. Using range would be better [40] Fatena: There is no problem with counting, 95 get 26 points, 82 get 13, 85. . .16, 19 is 22. . .

Episode 10 represents routine Quantification of numerical data. The opening of the routine is the reading of the activity and the instruction to choose a good teacher [1] as well as Areen’s suggestion [5] to start with the average (Table G.i). The procedure of routine involves transforming the average to other quantities, as Fatena [9] explained to her classmates. She asked them if they remember a past routine: “Do you remember the use of scores?” Kaman [26] makes another suggestion to use routine Defining range and scoring by proposing to use range: “Using range would be better.” routine Quantification of numerical data closes by giving numerical values for the averages component [40]. The closing of routine based in mathematical consideration. Episodes 11 and 12 show the participants’ attempt to rank the components in Table G.ii and to rank all four tables given in the good teacher activity. The participants used routine Assigning relative weighting, a previous routine from the camp activity. Episode 11: Assigning relative weighting for components in Table G.ii in the good teacher activity [42] Muhand: Which is more important (in Table G.ii)? [43] Raneen: All of them [44] Fatena: There are five criteria [45] Raneen: Use average [74] Muhand: I think if we calculate the average, it means that the same importance [is given to all criteria in Table G.ii], but if I want an educator [teacher who is responsible for class management], so I want to emphasize the class management, I do not think it’s the same importance [like the other criteria] [81] Fatena: If you use average that means all of them will have the same importance [82] Muhand: So, you can order them by their importance [119] Areen: Let’s rank them [120] Raneen: To give an example so Fatena can understand, there are 5 criteria, we can compute from 100% [136] Kaman: 20, 25, 30, 5 and 20

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Episode 11 depicts routine Assigning relative weighting for components with respect to assigning relative weighting for the components in Table G.ii. Routine Assigning relative weighting for components opens with Muhand’s [42] question, “which is more important?” The procedure [119–134] is to give different values for the different criteria. The routine closes with Kaman [136] assigning a value for each criterion. The closing of the routine is based on mathematical justification. Within the routine of assigning values to the components in Table G.ii, we found subjectifying talk when Muhand [74] emphasized the importance of class management (“when you look for an educator”. He thus took up the role of school principal. However, this role was taken up only to justify the use of a certain routine (assigning different weights) rather than to end the routine. Episode 12: Assigning relative weighting for the tables in the good teacher activity [176] Areen: Which is more important? [177] Kaman The second (table) [148] Kaman: 30 [182] Fatena: No, even if you are now a “wow” student but with average 70, so what teacher you will be [what kind of teacher]? The first thing they look at in ministry of education is the Average [185] Fatena: No, it is low, the first (table) is 40 [187] Areen: The interview is important [188] Kaman: All of them are the same rate [189] Fatena: No [190] Kaman: 40, 30, 20 and 10

Episode 12 shows the participants using routine Assigning relative weighting. The opening is Areen’s [176] question about which table is more important. The procedure [177–189] includes assigning a percentage for each table, and Kaman’s [190] answer closes the routine. The closing of routine is based, again, on mathematical justification. Within this routine Fatena [182] took up the role of an expert on practices in the Ministry of Education. Again, this was that is not used as a closing argument. Rather, it was used for assigning a certain weight and choosing a routine (average). Episode 13 shows the participants’ discussion of how to assign values for different quantities in Table G.ii. Episode 13: Using average to assign values for quantitative quantities [77] Areen: It does not matter if the teacher was better during one particular year, so we can use the average [100] Raneen: You must build a model; these grades change [101] Areen: The average is the model [105] Areen We use average in order to consider all of them (the 3 years) [138] Areen: The grades are from 1 to 12. so what is 1 from 30 [percent]? [139] Kaman: Do the average first [173] Kaman: Adeam (one of the candidate teachers) gets 10.2

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Episode 13 depicts routine Using average about using average as a way of equating scales for different quantities. The opening of routine Using average is Areen’s [77] statement: “It does not matter if the teacher was better during one particular year.” The procedure [139–163] involves the use of average as a way to assign equal ranking to the 3 years. The closing of the routine is assigning a value [164–173] for each candidate. The closing of the routine based on mathematical condition. Episodes 14 and 15 depict the participants trying to give meaning to Table G.iii and Table G.iv, respectively. The two tables include qualitative criteria. Table G.iii shows the ranking of the participants’ performance in the interview, ranging from “not acceptable at all” to “broadly acceptable.” Table G.iv shows the ranking for participation in social initiatives, ranging from “did not participate at all” to “participated to a great extent.” Episode 14: Quantification of qualitative data in Table G.iii [186] Areen: To continue, the interview [197] Areen: How many columns (in the third table) [198] Muhand: 5 [199] Raneen: 20, 40, 60, 80, 100

Episode 15: Quantification of qualitative data in Table G.iv [209] Areen: The last table. . . we can separate into four [211] Raneen But why give points for non-participation [214] Areen: Ahh, yes . . .100 to divide by 3 [215] Raneen: Better without fraction [216] Areen: What do you want? 100, 60, 30 [217] kaman: Maybe here 65 [instead of 60%]

Episodes 14 and 15 depict routine Quantification of qualitative data. In each episode, the routine opens with a request to deal with the tables [186], [209]. The procedure of each [187–198], [210–215] involves assigning a numerical value to each criterion, depending on the number of columns. In each case, routine Quantification of qualitative data closes by providing the value— 20, 40, 60, 80 and 100 for Table G.iii [199] and 30, 65 and 100 for Table G.iv [216–217]. So, the closing condition of the routine is based on mathematical considerations, it depended on the calculating process. After the participants elicited a model for choosing a candidate, they began implementing this model. Episode 16 illustrates the participants’ discussion about the implementation of the elicited model. Episode 16: Ranking candidates according to the elicited model [231] Areen: Now we need to calculate the general score for each one [232] Fatena: Muhand (one of the participants) you calculate the scores for these two candidates [The group worked on these calculations in almost complete silence] (continued)

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Rawan gets 78.8 and the other 63.88 The highest score is for Adeam with 82.5, then Aram, Mayar, Nasreen, Nemreen

Episode 16 shows Routine 3 – which implements the elicited model. The opening of the routine is Areen’s [231] request to calculate the scores for each candidate. The procedure involves substituting in the model [232–238], and the routine closes [239] with participants’ answers about the scores for each candidate. The closing condition of the routine is mathematical; we identified a process of calculating the points for candidates, and then compering between them. The closing condition for each sub- routine in the two modelling activities and the role participants occupied within these sub-routines summarized in Table 4. Table 4 shows that in the Summer camp activity some sub-routines’ closing conditions were based on personal authority, and the roles that participants took up from their everyday life were central for their decisions. The same sub-routines in the Good teacher activity had mathematical closing conditions, and much less subjectifying activity was observed.

Table 4 Closing condition for each sub-routine and participants identifying activity Sub-routines Looking at all case Looking at a sub-set of cases Using average Using estimation and ratio Assigning relative weighting

Quantification of numerical data Quantification of qualitative data Defining range and scoring Implementing the elicited model

Closing condition and identifying roles Summer camp Good teachers Personal authority [Episode 1] – Occupied role: Mother Personal authority [Episode 2] – Occupied roles: Mother and child Personal authority [Episode 3] Mathematical [Episode 13] Occupied role: Mother Mathematical and personal – authority [Episodes 4 and 5] Occupied role: Mother Mathematical [Episode 6] Mathematical [Episode 11 and 12] Occupied roles: Mother, Father, Occupied roles: School principal and an Citizen, Consumer and Students expert on practices in the Ministry of Education Mathematical [Episode 7] Mathematical Occupied role: Child [Episode 10] Mathematical [Episode 8] Mathematical [Episodes 14 and 15] Mathematical [Episode 9]

–

–

Mathematical [Episode 16]

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Discussion The aim of the current study was to describe the development of modelling abilities among a group of prospective teachers through following the development of routines and the changes in subjectifying talk that co-occur along this development by means of the communicational framework (Heyd-Metzuyanim and Sfard 2012; Sfard 2008). The findings indicate that through engagement in the two model-eliciting activities there were changes in the mathematical talk and in the subectifying talk, and that these changes occurred simultaneously. In the first activity, the participants worked first with Routines 1 that included two sub-routines: non-systematic method and systematic method and Routine 2 that involved eliciting criteria for choosing, but when they had to elicit a model in the second activity they worked with Routine 2 and Routine 3—implementing the elicited model. The changes from a non-systematic choosing-routine to systematic-choosing-routines and from routines that focus on choosing specific cases to routines that focus on eliciting the criteria for choice are an indication that learning took place. We consider this learning to be at the meta-level, because the participants learned the rules of how to work on MEAs beyond the specific activity. The participants noticed that working with Routine 1 in the camp activity did not provide a solution. Then they began working with Routine 2 in response to the need to develop a tool for choosing appropriate cases. Routine 2 was part of a modelling discourse that the participants previously lacked. The new discourse focused on the following: eliciting models that involved interpreting and dealing with multiple tables of data; creating, using, modifying, quantifying and converting quantities; and coordinating, organizing data and representing findings in visual and textual forms. We consider this discourse to be an elaboration of modelling abilities. Regarding the subjectifying talk, we found that as the routines for modelling activities became established, so did the subjectifying activity decrease. In particular, participants took up less of their everyday roles (mother, child, citizen) and focused instead on more mathematical talk. In the first activity we identified more turns that include subjectifying talk of different roles, some of these roles such as citizen and consumer were not related directly to the situation. In the second activity, we found much less subjectifying talk, and the little everyday roles that were taken up were directly linked to the situation. These findings can indicate about alienation process of the routines; in the second activity routines were occurring of themselves, without the participation of human beings, so we identified less subjectifying talk and the closing condition of these routines were mathematical. However, when the routines were in progress, the closing condition was based in personal authority. Another important difference between the first and second activities was the status of role-related statements in the discussion. In the first activity, statements referring to being a mother, for example, were used as an ultimate arbitrator between the choices to be made. In the second activity, they were used only to assist in quantifying qualitative measures, such as the value of certain characteristic.

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Our analyses above emphasize the interaction of mathematizing and identifying processes in modeling activity. These findings continue previous findings, such as Heyd-Metzuyanim and Sfard (2012), who showed the adverse effects that excessive identifying activity can have on efficient mathematizing. However, our findings are also unique in that subjectifying activity is, in a sense, called for in modelling activities. In fact, modelling activities are designed to build upon students’ everyday experience (including their roles and identities out of the classroom), while helping them transform the everyday justifications for making a certain choice into mathematical justifications. Thus, we believe that the decrease in subjectifying and increase in mathematizing can be considered as an indicator of successful learning in modeling activities. Despite the importance of the current study and the interesting findings it yielded, we are aware that the study has two limitations. One limitation concerns the small number of participants, only one group of five prospective teachers. This limitation is related to the close observation needed in order to conduct this type of fine grained analysis. Second, the participants worked only in two model-eliciting activities, which were similar in their structure. We cannot know if similar findings would be observed if the activities’ structure was different. Future research is needed to address the two limitations pointed, in order to have the opportunity to generalize these findings. In addition, future studies may explore further the connection between the learning of modelling skills and subjectifying activity, particularly in relation to identity construction, positioning and affect.

References Doerr, H. M., & English, L. D. (2003). A modeling perspective on students’ mathematical reasoning about data. Journal for Research in Mathematics Education, 34(2), 110–137. English, L. D. (2003). Problem posing in the elementary curriculum. In F. Lester & R. Charles (Eds.), Teaching mathematics through problem solving (pp. 187–198). Reston: National Council of Teachers of Mathematics. English, L. D. (2006). Mathematical modeling in the primary school: Children’s construction of a consumer guide. Educational Studies in Mathematics, 62(3), 303–329. English, L. D., & Fox, J. L. (2005). Seventh-graders’ mathematical modelling on completion of a three-year program. In P. Clarkson et al. (Eds.), Building connections: Theory, research and practice (Vol. 1, pp. 321–328). Melbourne: Deakin University Press. English, L. D., & Watters, J. J. (2005). Mathematical modeling in the early school years. Mathematics Education Research Journal, 16(3), 58–79. Heyd-Metzuyanim, E., & Sfard, A. (2012). Identity struggles in the mathematics classroom: On learning mathematics as an interplay of mathematizing and identifying. International Journal of Educational Research, 51–52, 128–145. Lesh, R., & Doerr, H. (2003). Foundation of a models and modeling perspective on mathematics teaching and learning. In R. A. Lesh & H. Doerr (Eds.), Beyond constructivism: A models and modeling perspective on mathematics teaching, learning, and problem solving (pp. 9–34). Mahwah: Erlbaum.

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Lesh, R., & Zawojewski, J. S. (2007). Problem solving and modeling. In F. Lester (Ed.), The second handbook of research on mathematics teaching and learning (pp. 763–804). Charlotte: Information Age Publishing. Lesh, R., Hoover, M., Hole, B., Kelly, A., & Post, T. (2000). Principles for developing thoughtrevealing activities for students and teachers. In R. Lesh & A. Kelly (Eds.), Handbook of research design in mathematics and science education (pp. 591–644). Mahwah: Lawrence Erlbaum. Sfard, A. (2008). Thinking as communicating: Human development, the growth of discourses, and mathematizing. New York: Cambridge University Press. Sfard, A., & Lavie, I. (2005). Why cannot children see as the same what grown-ups cannot see as different?—early numerical thinking revisited. Cognition and Instruction, 23(2), 237–309. Sfard, A., & Prusak, A. (2005). Telling identities: In search of an analytic tool for investigating learning as a culturally shaped activity. Educational Researcher, 34(4), 14–22. Shahbari, J. A., & Tabach, M. (2016). Developing modelling lenses among practicing teachers. International Journal of Mathematical Education in Science and Technology, 47(5), 717–732. Shahbari, J. A., & Tabach, M. (2017). The commognitive framework lens to identify the development of modelling routines. In B. Kaur, W. Kin Ho, & B. Heng Choy (Eds.), Proceedings of the 41th conference of the international group for the psychology of mathematics education (Vol. 4, pp. 192–185). Singapore: PME. Tabach, M., & Nachlieli, T. (2016). Communicational perspectives on learning and teaching mathematics: Prologue. Educational Studies in Mathematics (ESM), 91(3), 299–306. https:// doi.org/10.1007/s10649-015-9638-7. Vorhölter, K., Kaiser, G., & Borromeo Ferri, R. (2014). Modelling in mathematics classroom instruction: An innovative approach for transforming mathematics education. In Y. Li, E. A. Silver, & S. Li (Eds.), Transforming mathematics instruction (pp. 21–36). Cham: Springer.

Part III

Commentary on Part III: Connections to Theory and Practice Morten Blomhøj

Abstract The four chapters forming Part III address two different but related issues concerning affective aspects of teaching and learning mathematical modelling. That is: (1) the possible effects of the students’ modelling activities on their attitudes towards mathematics dealt with in chapters “Chapter 9: The Complex Relationship Between Mathematical Modelling and Attitude Towards Mathematics” and “Chapter 10: Teaching Modelling Problems and Its Effects on Students’ Engagement and Attitude Toward Mathematics”; and (2) the affective aspects involved in the students’ modelling work addressed in chapters “Chapter 11: Affect and Mathematical Modelling Assessment-A Case Study on Students’ Experience of Challenge and Flow During a Compulsory Mathematical Modelling Task by Engineering Students” and “Chapter 12: Flow and Modelling”. The commentary discusses these chapters with regard to their connections to theory of the teaching and learning of mathematical modelling and to the practice of teaching modelling. For furthering and integrating research on affective aspects of mathematical modelling it is suggested to differentiate with regard to the two main educational goals for teaching mathematical modelling in general education. That is mathematical modelling as a didactical means for motivating, supporting and enhancing the students’ learning of mathematics or developing the students’ mathematical competency as an educational goal in it own right. Keywords Affects in modelling · Affects through modelling · Modelling as didactical means · Modelling competency

The four chapters forming Part III address two different but related issues concerning affective aspects of teaching and learning mathematical modelling. That is: (1) the possible effects of the students’ modelling activities on their attitudes towards mathematics dealt with in chapters “Chapter 9: The Complex Relationship Between Mathematical Modelling and Attitude Towards Mathematics” and “Chapter 10: M. Blomhøj (*) Roskilde University, Roskilde, Denmark e-mail: [email protected] © Springer Nature Switzerland AG 2019 S. A. Chamberlin, B. Sriraman (eds.), Affect in Mathematical Modeling, Advances in Mathematics Education, https://doi.org/10.1007/978-3-030-04432-9_12

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Teaching Modelling Problems and Its Effects on Students’ Engagement and Attitude Toward Mathematics”; and (2) the affective aspects involved in the students’ modelling work addressed in chapters “Chapter 11: Affect and Mathematical Modelling Assessment-A Case Study on Students’ Experience of Challenge and Flow During a Compulsory Mathematical Modelling Task by Engineering Students” and “Chapter 12: Flow and Modelling”. The affective aspects of the teaching and learning of mathematical modelling is not that well researched. Therefore, in general the book at hand is an important contribution to the field of research on the teaching and learning of mathematical modelling. In particular, the chapters in this section, focusing on the relationship between students’ perception of and attitudes towards mathematics on the one hand and affective aspects of the students’ modelling work on the other hand, contributes to the understanding of the potentials and challenges related to the integration of modelling in mathematics teaching in general and higher education. The chapters “Chapter 9: The Complex Relationship Between Mathematical Modelling and Attitude Towards Mathematics”, “Chapter 10: Teaching Modelling Problems and Its Effects on Students’ Engagement and Attitude Toward Mathematics” and “Chapter 12: Flow and Modelling” report research related to mathematics teaching from grade 2 to 13 with main focus on secondary level (grade 8–13), while chapter “Chapter 11: Affect and Mathematical Modelling Assessment-A Case Study on Students’ Experience of Challenge and Flow During a Compulsory Mathematical Modelling Task by Engineering Students” reports from a study on first year engineering students. The chapters are anchored in very different educational systems, namely Italy, Iran, Norway and Canada respectively. In all cases, the research reported has an empirical basis. The methods used variate from qualitative analyses of cases of individual students’ modelling activities over qualitative and quantitative analyses of essays from a large number of students to quantitative analyses of questionnaires on students’ affects and attitudes in relation to specific modelling activities. In chapter “Chapter 9: The Complex Relationship Between Mathematical Model ling and Attitude Towards Mathematics” the connections between mathematical modelling and attitudes towards mathematics is established theoretically with references to the literature in the two relatively detached fields of research. Nearly 1500 students’ essays on “Me and maths: my relationship with mathematics up to now” are analysed. From the theoretical analyses as well as from the essays, modelling has a potential for supporting both cognitive learning goals and for supporting the development of a positive attitudes towards mathematics. However, first of all the essays suggest a complex relationship between mathematical modeling and attitude towards mathematics. The personal history with mathematics and mathematics teachers and the degree of experienced personal success is key in the forming of the students’ attitudes towards mathematics. A general finding is that alignment between learning goals, teaching practice and form of assessment seems to be a prerequisite for developing positive attitudes towards mathematics in students. Accordingly, introduction of modelling at said secondary level in form of a few projects per year with no direct relevance for the formal assessment of the students’ learning cannot be expected to contribute to the forming of positive attitudes towards mathematics.

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In analyzing the essays, utterances about mathematics being useful in everyday contexts, future education or professional life as well as expressions about the experienced uselessness of mathematics, characterize to a high degree the two groups with very positive and very negative attitudes respectively. However, the picture is not symmetric. The usefulness of mathematics is expressed in many ways and is often connected to social aspects or mathematics competence as an instrument for education and carriers. While uselessness is expressed as a personal experienced phenomenon: “Mathematics is useless for me”. The statement might be generalized by the student with additions such as: “and for all other people how are not going to be engineers”. So for students with negative attitudes modelling activities will probably only change their attitude through personal experiences of being able to do something meaningful and successful in mathematical modelling. Moreover, the analysis uncovers a negative development in attitudes toward mathematics from primary to secondary level. The number of utterances expressing usefulness decreases and those expressing uselessness increases from primary to secondary level. That is interesting – and depressing – since there is an expected connection between students’ experienced usefulness of mathematics and the increased exposure to mathematical modelling and applications of models throughout secondary mathematics teaching. Such connection seems to be evident only for those students who have already developed positive attitudes towards mathematics and personalized the story about the usefulness of mathematics. Chapters “Chapter 10: Teaching Modelling Problems and Its Effects on Students’ Engagement and Attitude Toward Mathematics”, “Chapter 11: Affect and Mathematical Modelling Assessment-A Case Study on Students’ Experience of Challenge and Flow During a Compulsory Mathematical Modelling Task by Engineering Students” and “Chapter 12: Flow and Modelling” all use of the theory of flow developed in psychology by Csíkszentmihályi with the book from 1990 as the main reference. The theory is introduced in some detail in chapter “Chapter 10: Teaching Modelling Problems and Its Effects on Students’ Engagement and Attitude Toward Mathematics”. From the analyses in these three chapters it is evident that this theory and in particular the notion of balancing flow and challenge is a powerful theoretical tool for researching the affective aspects involved in students’ modelling activities. Chapter “Chapter 10: Teaching Modelling Problems and Its Effects on Students’ Engagement and Attitude Toward Mathematics” reports on an intervention study with 244 female students in grade 10 (aged 15–16) from three private high schools in Iran. Two groups of students participated in six sessions on modelling problem solving given in a teacher-centered instruction format or in a student-centered format respectively. A pre- and post-test instrument with questionnaires on the students’ attitudes towards mathematics and on the students’ experiences of flow during modelling and problem solving, and tests in modelling problems, word problems, and intra mathematical problems was developed and applied in the study. The three types of problems are illustrated with one example each. As example of a modelling problem a well-known problem is given: Find the optimal position in terms of distance from the backline for a football player to take a shot on goal when running parallel to sideline towards the goal. The way this problem is structured and

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formulated only challenge the students’ to work with some of sub-processes involved in mathematical modelling; namely the mathematization, the mathematical analysis, the interpretation, and possible the validation process. The quantitative analyses show that teaching modelling in this context has a positive effect on the students’ attitudes toward mathematics. This is supported by the teachers’ and the students’ reflections, which clearly show that working with modelling problems have brought about a change in the students’ attitude towards the mathematics course in question. Analyses of the flow questionnaire show that students found both intramathematical and modelling problems to be less engaging than word problems. Intra-mathematical problems were too easy and boring, while modelling problems were too hard and frustrating. In interpreting the last part of this finding, it should be taken into account that the students’ were not challenged to delimit and formulate problems for modelling, systematized or collect and analyze data as part of their modelling work. Working with full scale modelling would probably have led to even higher degree of frustration. Accordingly, it seems as if the choice of already simplified modelling problems in the design is adjusted to the teaching practice, which forms the context for the research. The findings are explained using the theory of flow. The word problems involved in the study were challenging for the students’ because they were different from the tasks in their normal teaching, and because they challenge the students to make sense out of the problem situations described in the tasks. These problems call for discussions in the group work, which was appreciated by students. Moreover, in this context the students’ experience a high degree of success solving this type of word-problems. However, in the study it is not really addressed to what degree the word-problems constitute mathematical problems for the students. That is to what degree the students need to draw on and develop their understanding of mathematical concepts and methods beyond the standard procedure in which they have been trained in order answer the word-problems. These findings are interesting but it is not really discussed what to draw from them in terms of recommendations for the development of the practice of teaching. If the development of modelling competency in students is seen as an important part of the justification of teaching mathematics, then modelling problems cannot be interchanged with word-problems. Moreover, in order to support the development of modelling competency the students need to work with full scale modelling on a regular basis. In the realization of this ambition, it is for sure relevant to research the possible effects of the students’ modelling work on their attitudes towards mathematics. In chapter “Chapter 11: Affect and Mathematical Modelling Assessment-A Case Study on Students’ Experience of Challenge and Flow During a Compulsory Mathematical Modelling Task by Engineering Students” we find a case study on students’ experience of challenge and flow during a group assignment on the modelling of a motion of the students own choice. As part of the research the task were given to 239 first year engineering students in groups of 2–3 as a compulsory assignment, which should be approved in order for the students to get access to the written exam.

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In the task description, it was stipulated that each group should document their motion in a video showing the actual movement of the object and that they should develop a mathematical model that describes the position of the moving object as a function of time. Some examples of motions were given in the task description as inspiration for the students: throwing a ball, jumping a skateboard, or driving a car. In addition, the students were encouraged to use their mobile phone for producing a video of the motion and a particular freeware for tracking a moving object in a video and generating coordinate representations of the position of the moving object as function of time. The requirement for the groups was to produce a poster documenting their modelling process with the given sections: Introduction, Observation, Model, and Discussion. The students’ experience of challenge and flow working with this task was measured by means of a questionnaire with 10 questions and some follow up qualitative interviews. In addition, the questionnaire contained questions on how the students’ have worked with the task including how they have used the digital tools to generate data describing their motion and if they have used other types of software for analyzing the data. The research question: “To what extent does an open task about video analysis of motion with mobile phones and free tracker software challenge and activate the students?” was answered confirmatively. The quantitative analyses of the questionnaire showed that 59% of the students experienced challenge and flow working with this task. This result was supported by observations of very active and engaged students at campus during the project and in working group sections. From the theory of flow, several features of the task and situation is lifted forward as possible explanations for the finding. The task description with the suggested use of digital tools make it easy for the group to get started. Probably, in most groups, at least some of the students were familiar with producing videos with their mobile phone and many students may have previous experiences with the use of tracker software. The fact that the students were allow to choose, which motion to work with, made it possible for the groups to take the ownership of the task and level the challenges in the task to their ambition and abilities. In addition, the clear requirements for the poster help the students to structure and report their modelling work. However, the most important feature of the task and situation for causing engagement and activity was that the assignment was compulsory with consequences for the students’ progress in the program. The assignment resulted in 100 posters. No list of the types of motions analyzed in the posters is given. However, two examples of the posters are presented in the chapter: Throwing a table tennis ball and A bouncing rubber ball. These examples indicate that some of the models developed can be at least partly based on physics theory. Hereby the posters could formed the basis for interesting discussions on the different status of mathematical models depending on their theoretical foundations. Chapter “Chapter 12: Flow and Modelling” present an in-depth analysis of the modelling work of two students at grade 8 in a Canadian developmental project. The point of departure is the students’ work with an open and rather complex modelling task on the design of a new school with associated facilities and surroundings on a

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predefine lot. In designing the school in terms of size and placement of buildings and facilities, the students need to take a number of requirement and restrictions into account, and that is what constitute the complexness of the task. The students’ modelling work is analyzed by means of their pathway through the modelling process and in accordance with the methodology developed by Borromeo Ferri (2006). Sixteen modelling actions are identified in the students’ work and these are associated with sub-processes in the modelling process. In addition, these actions are analyzed based on the transcript of the interactions among the students and with the teacher. The interactions are analyzed with respect to flow according to the model developed in Liljedahl (2018). By combining these two forms of analyses, it is possible to identify the modelling actions that caused challenges for the students and to pinpoint the dialogical interactions, which helped the students to overcome those challenges and stay in or getting back in to flow. As concluded in the chapter, these two theory based approaches for analyzing the students’ modelling activity work perfectly together. The combination constitutes a nice example of the interplay between research from mathematics education and research from psychology. There are good reasons to believe that this dual approach for analyzing the affective aspects of students’ pathways through the modelling process can produce new insides on how to help students overcome challenges related to the different sub-processes in mathematical modelling.

Connections to Theory of Teaching and Learning of Mathematical Modelling All four chapters related to the theoretical framework for researching on the teaching and learning of mathematical modelling. They all refer to the notions of a mathematical model, a modelling process and modelling competence, which are part of the basis of this framework. The main elements in this framework are: (1) the justifications for modelling in mathematics teaching at different levels and branches of educational systems – both the actual stated and the theoretical possible justifications; (2) suggestions for and reflections on how to implement modelling in curricula including forms of assessment; (3) didactical ideas and principles for designing and implementing modelling courses or activities in the practices of mathematics teaching at different levels; (4) empirical identified and theoretical supported learning potentials, as well as related cognitive and didactical challenges, connected to different types of modelling approaches. These elements developed in research on the teaching and learning of modelling and applications are summarized by Blum et al. (2007) in the ICMI-study 14. Taken together, they constitute a theoretical framework for research on the teaching and learning of mathematical modelling. As indicated in the chapters in this section research in the affective domain can contribute with new insides related to all four elements of the framework for research

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on the learning and teaching of modelling and application. Therefore, integrating the affective domain in this research field is a natural next step for further development.

Connections to the Practice of Teaching of Mathematical Modelling The framework for research on modelling and application has developed in close interplay with the development for mathematics teaching practices. In fact, the inclusion of modelling and applications is one of the major common trends worldwide in curricula developments during the latest decades in general mathematics teaching, particularly at secondary level. However, as pointed out by Blum (2015), modelling and applications are still not integrated element in mathematics teaching in general education. In the practices of mathematics teaching at secondary level (grade 8–13), modelling and applications are exotic and rare exceptions from the dominating teaching practice. Even in systems where modelling and application play some role in the curriculum, the practices of teaching particularly at secondary level is often determined by the formal assessment system – typically written exams without any room for modelling. Therefore, research in the teaching and learning of mathematical modelling suffer under the methodological challenge of researching something, which does not necessarily exist before and independently of the research. Accordingly, research in this field often includes the establishing of some experimental practice of teaching in which modelling and application are playing a role. When researching the possible effects of working with modelling on students’ attitudes towards mathematics this challenge is reinforced. Students’ beliefs about and attitudes towards mathematics and mathematics teaching are rather stable and based on their personal experiences over periods of years. So experimental teaching introducing modelling and application in a very limited period as part of a research study cannot be expected to influence the students’ attitudes toward mathematics significantly. Moreover, and more importantly, research on the possible effects in the affective domain of working with mathematical modelling most related to the educational justifications for integrating modelling and applications in the relevant mathematics teaching practice. The justification of mathematical modelling as an element of mathematics teaching in general education has been an issue in the mathematics education research for many years. Here I briefly state what I consider to be the three most important arguments in favor of mathematical modelling as a central element in general mathematics teaching (Blomhøj 2004): 1. Mathematical modelling bridges the gap between students’ real life experiences and mathematics. It motives the students learning of mathematics, gives direct cognitive support for the students’ conceptions, and it places mathematics in the culture as a means to describe and understand real life situations.

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2. In the development of highly technological societies competences for setting up, analysing and criticising mathematical models are of crucial importance. This is the case both from an individual perspective in relations to opportunities and challenges in education and work-life, and from the societal perspective in relation to the need for an adequately educated workforce. 3. Mathematical models of different kinds and complexity are playing important roles in the functioning and forming of societies based on high technologies. Therefore, the development of an expert as well as layman competence to critique mathematical models and the way they are used in decision making are becoming imperatives for the maintaining and further development of democracy. Of course mathematical modelling has different meanings at different educational levels and these arguments should not be given equal importance at all levels. It is clearly relevant to research the challenges and effects in the affective domain related to realisations of these justifications in the practice of mathematics teaching.

References Blomhøj, M. (2004). Mathematical modelling: A theory for practice. In International perspectives on learning and teaching mathematics (pp. 145–159). Gothenburg: National Center for Mathematics Education. Blum, W. (2015). Quality teaching of mathematical modelling: What do we know, what can we do? In S. J. Cho (Ed.), The proceedings of the 12th international congress on mathematical education – Intellectual and attitudinal challenges (pp. 73–96). New York: Springer. Blum, W., Galbraith, P. L., Henn, H., & Niss, M. (Eds.). (2007). Modelling and applications in mathematics education. The 14th ICMI study. New York: Springer. Borromeo Ferri, R. (2006). Theoretical and empirical differentiations of phases in the modelling process. ZDM, 38(2), 86–95. Csíkszentmihályi, M. (1990). Flow: The psychology of optimal experience. New York: Harper and Row. Liljedahl, P. (2018). On the edges of flow: Student problem solving behavior. In S. Carreira, N. Amado, & K. Jones (Eds.), Broadening the scope of research on mathematical problem solving: A focus on technology, creativity and affect. New York: Springer.

Commentary: Flow and Mathematical Modelling: Issues of Balance Lyn D. English

Abstract The four chapters in this third section provide an interesting and timely set of studies that primarily address students’ experiences of “flow” (Csíkszentmihályi 1990) during mathematical modelling. Factors pertaining to attitude and engagement are also featured in this section. Exploring affective components in students’ mathematical modelling is a challenging endeavour, especially when both constructs are complex with a long history. This commentary discusses key aspects of each of the chapters and highlights a few issues worthy of further research. The commentary examines the notions of “concreteness” of mathematical modelling, interpretations of flow in light of Liljedahl’s perspectives addressed in chapters “Chapter 10: Teaching Modelling Problems and Its Effects on Students’ Engagement and Attitude Toward Mathematics” and “Chapter 12: Flow and Modelling”, balancing context and model generation, the intrinsic appeal of modelling problems for generating flow, and the integration of modelling within other domains. Keywords Affect · Flow · Concreteness · Context · Model generation · Cross-domain modelling

The four chapters in this third section provide an interesting set of studies that primarily address students’ experiences of “flow” (Csíkszentmihályi 1990) during mathematical modelling. Factors pertaining to attitude and engagement are also of interest in this section. Examining affective components in students’ mathematical modelling is a challenging endeavour, especially when both constructs are complex with a long history (Chamberlin, chapter “Chapter 1: The Construct of Affect in Mathematical Modelling”, Part I). In Chamberlin’s informative overview of the history of affect and, more specifically, affect and mathematics education, it is clear that the construct has progressively become more “convoluted” (p. 23) over time. It is thus pleasing to see the chapters in this section confine their studies to just L. D. English (*) Queensland University of Technology, Brisbane, QLD, Australia e-mail: [email protected] © Springer Nature Switzerland AG 2019 S. A. Chamberlin, B. Sriraman (eds.), Affect in Mathematical Modeling, Advances in Mathematics Education, https://doi.org/10.1007/978-3-030-04432-9_13

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a few components of affect, despite the considerable overlap in ideas and frameworks across the chapters. Nevertheless, all four chapters provide a comprehensive account of how flow, engagement, and attitude impact on students’ mathematical modelling. In essence, the very nature of mathematical modelling is intimately connected to affect. As the authors point out, both modelling and affect are multifacted constructs that are interpreted variously in the literature. There is considerable overlap in definitions of models and modelling across the chapters, with Blum and Leiss’ (2007) framework and Borromeo Ferri’s (2006) representation of modelling processes referred to frequently. Chamberlin’s chapter, however, provides a comprehensive introduction to affect and modelling. In particular, for those not too familiar with models and modelling, Chamberlin presents real-world examples of the pervasiveness of mathematical modelling in everyday life. Using a real-world example of a shopper trying to exit a grocery store as soon as possible, Chamberlin illustrates how a shopper can apply informal modelling in determining the fastest checkout from which to exit. Taking such a meaningful example nicely illustrates some of the key features that are reported in diagrammatic form in Part III. In the remainder of this commentary I discuss interesting aspects of each of the chapters and highlight a few points that could be considered further. I review the notions of: (a) “concreteness” of mathematical modelling, (b) interpretations of flow with a focus on Liljedahl’s perspectives as featured in both chapters “Chapter 10: Teaching Modelling Problems and Its Effects on Students’ Engagement and Attitude Toward Mathematics” and “Chapter 12: Flow and Modelling”, (c) balancing context and model generation, (d) the intrinsic appeal of modelling problems for generating flow, and (e) the integration of modelling within other domains.

“Concreteness” of Mathematical Modelling Di Martino (chapter “Chapter 9: The Complex Relationship Between Mathematical Modelling and Attitude Towards Mathematics”) explores the complex relationship between mathematical modelling and attitude towards mathematics. In doing so, he presents a three-dimensional model for attitude, comprising “vision of mathematics”, emotions, and “perceived competence”. Although his chapter does not appear to give equal attention to all three components, Di Martino nevertheless presents interesting, although not completely unexpected, findings from his studies. Of relevance to modelling is what di Matrino refers to as “concreteness” (p. 223), that is, the degree to which the learner perceives the application of a mathematics situation to a realistic one. Perhaps not surprising, the degree of concreteness students found in mathematics problems declined with increasing grade level. This decline reinforces the need to give greater attention to how we contextualise the mathematical problems we present students—contextualisation and modelling go hand-in-hand. Although traditional school word problems can claim to use realistic contexts, the questions asked can often be unrealistic in the real world, as Di Martino points out: “The choice of the problems (or activities) is therefore crucial: the context has to be realistic and the posed questions significant” (p. 232). This is where mathematical modelling comes to the fore. One of the motivating features of

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modelling is its relevance to the real world, not just to familiar contexts such as school or neighbourhood environments, but also to challenging situations within these. Such situations must hold meaning for students, must be inviting, and must be sufficiently challenging yet within their capabilities. Liljedahl’s construct of “flow” in terms of challenge and skill provides a useful framework for exploring these modelling features.

“Flow” and Modelling Parhizgar and Liljedahl (chapter “Chapter 10: Teaching Modelling Problems and Its Effects on Students’ Engagement and Attitude Toward Mathematics”) investigated the impact of flow (specifically, engagement) and attitude in three types of problems, namely, modelling problems, word problems, and basic exercises. The authors also investigated the effects of teacher-centred and student-centred approaches on students’ attitude towards and engagement in each of these problems. The results of their questionnaires showed some surprising, and not so surprising results. With respect to the unexpected findings, it was the word problems that yielded the highest engagement before and after the intervention. The modelling problems were seen by the students as being too difficult, suggesting an imbalance between challenge and skill. Not surprisingly, the student-centred approach had a greater impact on students’ attitude towards modelling problems, with students appreciating the group work involved. What is rather troubling in Parhizgar and Liljedahl’s conclusions is the recommendation that students be “sufficiently taught how to solve real-world problems, especially how to do cycle modelling” (p. 251) so they can engage effectively with modelling problems. I question this advice, which appears to contradict the very nature and purpose of modelling problems. One of the many benefits of engaging students in these problems is fostering their independent application of mathematics knowledge and understanding to the solution of authentic problems that allow for various approaches and solutions. With their “low floor” and “high ceiling” features (English 2017, Gadanidis et al. 2018; Papert 1980) modelling problems are designed to be within reach of all students, a point to which I return. Furthermore, by implementing sequences of related modelling problems (Doerr and English 2003), students can apply and extend their learning to new situations without needing to be taught how to do so. One thus has to question the design of some of the problems in Parhizgar and Liljedahl’s study. Interestingly, in Liu and Liljedahl’s study (chapter “Chapter 12: Flow and Modelling”) a rebalancing of the challenges and skills in a modelling problem was undertaken to generate optimal flow. Quite rightly, the authors emphasize that flow is not static; rather, the balance between ability and challenge is dynamic. In line with my argument regarding the low floor/high ceiling feature of a modelling problem, students’ skills usually improve as they engage in a modelling activity. As Liu and Liljedahl point out, for students to remain in flow, the activity must be such that its challenge likewise increases. On the other hand, should an activity fail to be challenging, a “tolerance for the mundane and perseverance” (p. 277) can ensure students continue with a problem. Likewise, tolerance and perseverance are important when a problem’s challenge outstrips a student’s abilities at a particular

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point. I would argue that this latter characteristic is especially salient for innovative and creative thinking (Lucas and Nordgren 2015), a skill that is increasingly needed in an age of disruption (Rösel et al. 2016).

Balancing Context and Model Generation In illustrating the dynamic ways in which flow occurred during small group modelling involving designing a new school, Liu and Liljedahl highlight the important role of contextualized knowledge in such problems. In the interesting excerpts of students solving the problem, it was evident that the context was, at times, overshadowing their model generation. Although Liu and Liljedahl do not mention this point, it is worth commenting that, while a meaningful context is a core feature of modelling problems, there is the issue of another imbalance occurring within the dynamics of flow—that of contextual features versus model generation. It could be argued that with increased awareness and knowledge of contextual features comes a concomitant awareness of the need to balance context and modelling. On the other hand, insufficient contextual knowledge or experience can hinder students’ solution processes as was the case in Liu and Liljedahl’s problem where the students lacked driver experience of manoeuvring cars in parking lots. Such imbalances between context and modelling occur often in our lives. For example, many of us have experienced frustrations with architects (and vice versa) when our desires for the “perfect” home context need to be tempered with a consideration of core structural features required in the architect’s house design.

Intrinsic Appeal of Modelling Problems for Generating Flow Gjesteland and Vos (chapter “Chapter 11: Affect and Mathematical Modelling Assessment-A Case Study on Students’ Experience of Challenge and Flow During a Compulsory Mathematical Modelling Task by Engineering Students”) present an interesting variation in the nature of the modelling problem presented to students, in their case, undergraduate engineering students. As a compulsory outdoor assignment, with ample time for completion and accessible to all students, the modelling problem was designed to facilitate flow. The task required students to select a movement of an object of their choice, such as throwing a ball, jumping on their skate board, or even driving a car. They were to film this movement with their phones. Next, they were to “use free tracker software (http://physlets.org/tracker/) on their laptops to transform the movement into measurements, approximate the movement with a mathematical model, and then present their findings on a poster” (p. 257). Detailing their model, the poster was to contain an introduction, their observations (e.g., measurements), a mathematical model of the moving object’s trajectory, and a discussion of the model’s accuracy in comparison to the actual

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measurements. An important point made by Gjesteland and Vos, which is not emphasized as it should be in the other chapters, is that modelling lends itself to multidisciplinary contexts and content. In their study, the activity was set at the cross-roads of mathematics and physics. Interestingly, the authors chose not to use the term, mathematics, in their presentation of the task primarily due to the students’ frequently negative view of the discipline. In applying the flow construct of Csíkszentmihályi (1990), Gjesteland and Vos again highlight the core issue explored in the other chapters, namely, that studying affect when students are engaged in an activity can yield insights into different facets of affect, such as emotions, perception of utility etc. The authors’ goal in their study was to focus on students’ perceptions of “being challenged and activated by the activities” (p. 263). An important point made by Gjesteland and Vos is that flow can only occur if there is a degree of tension between challenge and skill. This feature of a learning activity can often be overlooked in mathematics curricula, but is a core element in maintaining students’ interest and thus motivation in the discipline. Modelling problems are ideal examples here because of their affordances for stimulating flow. One of the key characteristics of these problems is their intrinsic appeal to a diversity of learners. Because the problems have multiple entry and exit points, with more than one solution model possible, all students can achieve a sense of satisfaction in model creation. Invoking Papert’s (1980) notion of creating tasks that feature a low floor and high ceiling (although not citing this source), Gjesteland and Vos rightly highlight modeling problems as excellent examples of such tasks. Because modelling problems cater for a wide variety of achievement levels, all students can experience a degree of flow even if mathematics is not their preferred discipline. It is interesting how the authors deliberately avoided using the term, mathematics, in the task and follow-up questionnaire with the aim of not deterring students for whom mathematics had negative connotations. I would argue that such a decision might not have been necessary given the inviting nature of the modelling assignment, which many students found intrinsically worthwhile irrespective of it being an item of assessment. Although Gjesteland and Vos did not report on their students’ discipline learning, this is of course a core issue that cannot be overlooked. Elsewhere (e.g., English, November 2018) I have argued how modelling problems can foster learning innovation across STEM (science, technology, engineering, and mathematics), involving student generation of both discipline content knowledge and the adaptation and application of this knowledge to the solution of new problems.

Integration and Modelling An aspect that could have been emphasised more in this section is the integration of modelling within the other STEM disciplines and other areas of the curriculum. Although the focus of this book is on affect and mathematical modelling, the incorporation of other domains together with different cultural and social contexts (Anhalt et al. 2018; Verschaffel et al. 2009) can add to the intrinsic appeal of these modelling experiences. It is beyond the scope of this commentary to elaborate on

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these points but worthwhile examples include engineering-based modelling (e.g., English and Mousoulides 2009, 2011, 2015), modelling based on historical accounts involving first settlements and associated social issues (e.g., English 2016), and modelling involving ecosystems (e.g., Moore et al. 2014).

Concluding Points As the chapters in this section have demonstrated, issues pertaining to affect in mathematical modelling are manifold. The present chapters, and indeed the book as a whole, provide much needed research on affective components that remain underrepresented in studies on modelling. Yet the very nature of mathematical modelling invokes aspects of flow and positive attitude. Indeed, if a modelling problem creates a pleasing balance between challenge and skill, then flow should occur naturally for the student and use of the term, mathematics, should not be a deterrent. In exploring several foundational affective issues, the chapters raise further points needing attention. One aspect pertains to the design and implementation of modelling problems that capitalise on students’ natural talents and enable all students to experience the success that flow can bring. Modelling problems have several intrinsic features that are absent in many other mathematics learning experiences; the links between these features and various components of affect demand more investigation. Indeed, a subsequent volume on affect and mathematical modelling would be another welcome addition to the literature.

References Anhalt, C. O., Staats, S., & Cortez, R. (2018). Mathematical modeling and culturally relevant pedagogy. In Y. J. Dori, Z. R. Mevarech, & D. R. Baker (Eds.), Cognition, metacognition, and culture in STEM education (pp. 307–330). Springer. https://doi.org/10.1007/978-3-319-666594_14. Blum, W., & Leiss, D. (2007). How do students and teachers deal with mathematical modelling problems? The example sugarloaf and the DISUM project. In C. Haines, P. L. Galbraith, W. Blum, & S. Khan (Eds.), Mathematical modelling: Education, engineering and economics. ICTMA 12 (pp. 222–231). Chichester: Horwood. Borromeo Ferri, R. (2006). Theoretical and empirical differentiations of phases in the modelling process. ZDM, 38(2), 86–95. Csíkszentmihályi, M. (1990). Flow: The psychology of optimal experience. New York: Harper and Row. Doerr, H. M., & English, L. D. (2003). A modelling perspective on students’ mathematical reasoning about data. Journal for Research in Mathematics Education, 34(2), 110–136. English, L. D. (2016). Developing early foundations through modeling with data. In C. Hirsch (Ed.), Annual perspectives in mathematics education: Mathematical modeling and modeling mathematics (pp. 187–195). Reston: National Council of Teachers of Mathematics. English, L. D. (2017). Advancing elementary and middle school STEM education. International Journal of Science and Mathematics Education, 15(1), 5–24. 17, 347–365.

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English, L. D. (2018, forthcoming). Disruption and learning innovation cross STEM. Plenary paper to be presented at the 5th international STEM in education conference, 21st–23rd November, Brisbane. English, L. D., & Mousoulides, N. (2009). Integrating engineering education within the elementary and middle school mathematics curriculum. In B. Sriraman, V. Freiman, & N. Lirette-Pitre (Eds.), Interdisciplinarity, creativity, and learning: Mathematics with literature, paradoxes, history, technology, and modelling (pp. 165–175). Charlotte: Information Age Publishing. English, L. D., & Mousoulides, N. (2011). Engineering-based modelling experiences in the elementary classroom. In M. S. Khine & I. M. Saleh (Eds.), Models and modeling: Cognitive tools for scientific enquiry (Models and Modeling in Science Education Series) (pp. 173–194). Dordrecht: Springer. English, L. D., & Mousoulides, N. (2015). Bridging STEM in a real-world problem. Mathematics Teaching in the Middle School, 20(9), 532–539. Gadanidis, G., Clements, E., & Yiu, C. (2018). Group theory, computational thinking, and young mathematicians. Mathematical Thinking and Learning, 20(1), 32–53. Lucas, B. J., & Nordgren, L. F. (2015). People underestimate the value of persistence for creative performance. Journal of Personality and Social Psychology, 109, 232–243. Moore, T. J., Guzey, S. S., & Brown, A. (2014). Greenhouse design: An engineering unit. Science Scope, 37(7), 51–57. Papert, S. (1980). Mindstorms: Children, computers, and powerful ideas. New York: Basic Books. Rösel, A., Műnch, J., Richardson, I., Rausch, A., & Zhang, H. (Eds.). (2016). Are we ready for disruptive improvement? In M. Kuhrmann, et al. (Eds.), Managing software process evolution (pp. 77–91). Springer. https://doi.org/10.1007/978-3-319-31545-4_5. Verschaffel, L., Greer, B., Van Dooren, W., & Mukhopadhyay, S. (Eds.). (2009). Words and worlds: Modelling verbal descriptions of situations. Rotterdam: Sense Publishers.

Chapter 9: The Complex Relationship Between Mathematical Modeling and Attitude Towards Mathematics Pietro Di Martino

Abstract This chapter addresses the relationship between the promotion of mathematical modeling in the classroom – in particular linking mathematics and authentic real mathematics problem – and students’ attitude towards mathematics. This relationship has a twofold nature, based on two assumptions: on one hand mathematical modeling can help to construct or reinforce the belief concerning the utility and concreteness of mathematics, and therefore it can foster motivation in studying mathematics; on the other hand a positive attitude towards mathematics can strongly affect the way students approach real mathematics problems. Keywords Mathematical modeling · Problem solving · Modeling cycle · Attitude towards mathematics · Perceived utility of mathematics

Mathematical Modeling and Attitude Towards Mathematics: The Issue of Definition Mathematical modeling and attitude towards mathematics are two hot constructs in mathematics education: in the last two decades, a great amount of literature about them has been developed. Both constructs are also used in everyday language with a naïve and not welldefined meaning. For this reason, a lot of work in the literature has been devoted to clearly defining these constructs. Concerning attitude, we have addressed in depth the definition issue (Di Martino and Zan 2015): first developing a critical analysis of the existing literature (Di Martino and Zan 2001), then developing a definition of attitude towards mathematics strictly linked to students’ experience and emerging from students’ narratives (Di Martino and Zan 2010, 2011).

P. Di Martino (*) Dipartimento di Matematica, Università di Pisa, Pisa, Italy e-mail: [email protected] © Springer Nature Switzerland AG 2019 S. A. Chamberlin, B. Sriraman (eds.), Affect in Mathematical Modeling, Advances in Mathematics Education, https://doi.org/10.1007/978-3-030-04432-9_14

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In our study for the characterization of attitude, we used a narrative approach in order to investigate the dimensions students used to describe their relationship with mathematics. This choice is consistent with an interpretative approach: we wanted to give voice to the students and allow them to describe the aspects they themselves considered relevant for their experience with mathematics. In particular, we proposed the essay ‘Me and maths: my relationship with mathematics up to now’ in several Italian schools. According to Bruner (1990), we have assumed that – through autobiographical narratives – students would have introduced causal links between facts. These links do not have a logical perspective but rather a social, ethical and psychological one. Our goal was to construct what Spence (1982) calls narrative truth: this truth may be closely linked, loosely similar, or far removed from the objective truth. On the other hand, as Bruner (1990, pp. 119–120) claims: It does not matter whether the account conforms to what others might say who were witnesses, nor are we in pursuit of such ontologically obscure issues as whether the account is ‘self-deceptive’ or ‘true’. Our interest, rather, is only in what the person thought he did, what he thought he was in, and so on.

We established two main rules to collect the narrative data: essays were anonymous and they were assigned and collected in the class not by the class mathematics teacher. These two rules were given in order to leave the students free to describe even criticism and strong negative emotions towards either mathematics or teachers: We collected 1496 essays ranging from grade 2 to grade 13: 707 from primary school (grade 2–5), 369 from middle school (grade 6–8), 420 from high school (grade 9–13).

The analysis of the collected narratives suggests that almost all students describe their relationship with mathematics referring to one (or more) of the following three dimensions: • Their emotional disposition towards mathematics, • Their vision of mathematics, • Their perceived competence in mathematics. This analysis has also suggested the development of a three-dimensional model for characterizing attitude towards mathematics (TMA) represented in Fig. 1. Fig. 1 The threedimensional model for attitude

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Fig. 2 The mathematical modeling process

TMA takes explicitly into account the close relationship among the three dimensions. For what concerns mathematical modeling, the path towards a shared definition has been less rough, but not immediate. A first important aspect in the educational context is the distinction between model mathematics and mathematical modeling (Cirillo et al. 2016). The use of concrete objects to facilitate children’s understanding of abstract mathematical concepts (i.e. the operation of modeling mathematics) is different from “mathematizing authentic situations” (Yerushalmy 1997, p. 207). De Corte et al. (2000, p. 66) define mathematical modeling as the application of mathematics to solve problem situations in the real world. Using their words, mathematical modeling is: A complex process involving a number of phases (. . .). As several authors have stressed, this process of solving mathematical application problems has to be considered as cyclic, rather than as a linear progression from givens to goals.

We can characterize mathematical modeling starting by two elements: a problematic real-life situation (the authentic situation quoted by Yerushalmy) and a specific approach to the situation (the mathematical modeling cycle recalled by Verschaffel et al.). There are many schemas for the mathematical modeling cycle, but almost all essentially involve the following four phases proposed by Dossey et al. (2002): Formulation, Analysis, Interpretation, Test (see Fig. 2). If in classroom practice students are mainly engaged in the Analysis phase – that is, they are confined to the mathematical world – Formulation and Interpretation appear to be two key-phases in the mathematical modeling process. As a matter of fact, they are the two phases in which the mathematical world and the real world are interconnected. As Pollak highlights (2003, p. 649): What distinguishes modeling from other forms of applications of mathematics are (1) explicit attention at the beginning of the process of getting from the problem outside of mathematics to its mathematical formulation, and (2) an explicit reconciliation between the mathematics and the real-word situation at the end.

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Mathematical Modeling and Attitude Towards Mathematics in the School Standards In the NCTM News Bullettin of the September 1999 (Lappan 1999), the past President of NCTM recognized: We have not traditionally taken much responsibility for fostering students’ attitudes toward mathematics and their perceptions of their own role in learning the subject.

She underlined the need to consider affective goals in the teaching of mathematics, and in the new millennium, the attention to the affective goal in mathematics education has surely grown. For example, the OECD-PISA framework (OECD 2016) includes the attitudes in the parameters used to assess students’ performance. The promotion of mathematical modeling in the mathematics curriculum at all school levels is, by now, fostered in several official documents (Burkhardt 2006). Recently, a document called Guidelines for Assessment and Instruction in Mathematical Modeling Education (GAIMME) has been developed with input from the NCTM, in order to encourage teachers in incorporating the practice of mathematical modeling in their classrooms. The document discusses the reasons for this attention: Mathematical modelling can be used to motivate curricular requirements and can highlight the importance and relevance of mathematics in answering important questions. It can also help students gain transferable skills, such as habits of mind that are pervasive across subject matter. (VV.AA. 2016, p. 8)

In the Italian context, the development of a positive attitude has become an official goal in National Standards for the first cycle of instruction (from grade 1 to grade 8). In particular, one of the goals for the development of mathematical competence at the end of middle school is (MIUR 2012): “the student has strengthened a positive attitude toward mathematics through significant experiences and s/he has understood how mathematics is useful to operate in the real world”. This goal is interesting because it links attitude towards mathematics and mathematical modeling, evoking the two key elements for mathematical modeling: ‘to operate’ and the ‘real-world’. One reason to promote mathematical modeling is also the contribution modeling can give to a deeper comprehension of curricular mathematics (Blum and Borromeo 2009; Zbiek and Conner 2006). The GAIMME document reports other two main motivations for promoting mathematical modeling in the classroom: – To show the utility of mathematics skills in extra-mathematical contexts and for extra-mathematical purposes. This can help to develop a positive view of mathematics, to consolidate sense-making in and of mathematics and, definitely, to foster motivation for the study of mathematics (Niss 2012). This goal is clearly related to the assumed definition of attitude: it includes emotional aspects and the view of mathematics (what it is and how it can be used);

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– To prepare students to work professionally with mathematical modeling and to provide them with opportunities to link mathematics to other areas of the curriculum. In some sense, the development of skills related to mathematical modeling is a goal in itself (Zbiek and Conner 2006). In particular, the first goal highlights the supposed link between attitude towards mathematics (emotional disposition and view of mathematics) and mathematical modeling. This link becomes explicit in the development of the theoretical framework for the definition and assessment of mathematical literacy. Based on the seminal work of Niss (2003), OECD-PISA assumed the following definition of mathematical literacy: Mathematical literacy is an individual’s capacity to formulate, employ, and interpret mathematics in a variety of contexts. It includes reasoning mathematically and using mathematical concepts, procedures, facts and tools to describe, explain and predict phenomena. It assists individuals to recognise the role that mathematics plays in the world and to make the well-founded judgments and decisions needed by constructive, engaged and reflective citizens. (OECD 2016, p. 65)

Mathematical literacy includes the capacity to use mathematics in a variety of contexts but also the awareness of the role that mathematics plays in the world (view of math). The shared main assumption of the discussed documents is that mathematical modeling can help to construct or reinforce the belief concerning the utility and the concreteness of mathematics. It is also assumed that the promotion of this view of mathematics acts on the emotional disposition dimension of the TMA-model for attitude: fostering motivation in studying mathematics and developing positive emotions towards mathematics. In the following, we will critically approach these set of assumptions, discussing the complexity of the relationship between mathematical modeling and attitude towards mathematics on the basis of the educational literature and findings of different studies we have conducted. In particular, we will focus on two of three dimensions of the TMA-model for attitude: emotional disposition and vision of mathematics (with special reference to the perceived utility and concreteness of mathematics).

The Perceived Utility of Mathematics and Mathematical Modeling In the analysis of the narrative data collected in order to characterize the students’ attitude towards mathematics, the perceived utility of mathematics is one of the most recurrent themes in students’ narratives: it is therefore a main component of the students’ vision of mathematics.

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With respect to this issue, a difference between the primary level and the other levels stands out and appears to be particularly interesting. The evolution of opinions and views is clear: there is an evident shift from the belief that mathematics is essential – strongly present at the primary level and testified by the very high number of occurrences of expressions like “math is useful”, “math is important”, “math serves to” – to the idea that mathematics is useless. The word “useless”, practically absent in essays at the primary level, has almost twice as many occurrences as “useful” at the other school levels. What could be the reasons for this worrisome evolution? The analysis of the narratives highlights two main issues. The first one is the evolution in the students’ conception of utility of mathematics during the school period: particularly relevant appears the distinction between individual and social utility of mathematics. The second issue is the relationship between utility and concreteness of mathematics. In the following, we will discuss these two issues and the possible role of the promotion of mathematical modeling in this evolutionary phenomenon. An interesting phenomenon at the primary level is the identification between mathematics as a science and mathematics as a school subject: Mathematics is indispensable for “running the world”. It is necessary for everything. Without mathematics, we could not do many things; without mathematics, we could not know anything. Probably mathematics is the most important school subject. [4P.37]1

The distinction between mathematics as a science and mathematics as a school subject emerges in the following school levels. It is particularly relevant because it is at the basis of another distinction: that between social utility and individual utility of mathematics. Niss (2003) underlines how this distinction produces a sort of paradox (he calls it “the relevance paradox”): even there is a general agreement around the role that mathematics plays in the modern society, many, if not most students (and also adults) have increasing difficulty at seeing mathematics relevant to them, as individuals. In other words, the utility of mathematics for the development of the society does not imply that the knowledge of some elements of mathematics is considered essential for every single individual. In particular, the social relevance of mathematics does not automatically justify why the single individual should study mathematics. This consideration questions the link between the perception of social relevance of mathematics and motivation in studying it. This point of view was clearly stated by Vinner in his regular lecture at ICME 9 (Vinner 2007, p. 2): We live in a mathematical world, says the above Principles and Standards document, whenever we decide on a purchase, choose insurance or health plan, or use a spreadsheet, we rely on mathematical understanding... The level of mathematical thinking and problem solving needed in the workplace has increased dramatically. . . . Mathematical competence

Here and in the next excerpts the first number refers to the grade, the letter refers to the school level (Primary/Middle/High), the last number indicates the progressive numbering of the essay within the category. 1

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opens doors to productive future. A lack of mathematical competence closes those doors. This is not the place to elaborate in length how misleading are these claims. In short I will say only the following: No doubt mathematical knowledge is crucial to produce and maintain the most important aspects of our present life. This does not imply that the majority of people should know mathematics.

The clear distinction between the social and the individual conception of utility of mathematics is relevant also in the secondary-tertiary transition. In 2002, within a 3-year Italian Project on the evolution of students’ attitude towards mathematics, we developed a study on the relationship between secondary students’ attitude toward mathematics and their choices of how many and which mathematics courses to take in college (Di Martino and Morselli 2006). An open questionnaire was distributed to 1837 students attending 12th and 13th grades of secondary school: the students had half an hour to fill out the questionnaire in an anonymous way. We divided the sample into four groups on the basis of the declared emotional dispositions towards mathematics: very positive, positive, negative and very negative. Then, we characterized the four groups according to the answers to the questionnaires. In particular, one item was: “Choose 3 adjectives to describe mathematics”. The analysis of the answers to this item was particularly insightful for the issue of utility of mathematics. On the one hand, the adjective ‘useful’ is one of the most widely used in the answers to our questionnaire (376 occurrences). It was present in all groups and it did not characterize any group. On the other hand, the adjective ‘useless’ is recurrent in the very negative group and strongly characterises the group (chi2 ¼ 206,07). A possible interpretation of this is that the word useful is related to the social aspect while useless to the individual aspect. In any case, this result was significant for the research on affective factors. As a matter of fact, many traditional quantitative instruments (like validated Likert scales) use the item “mathematics is useful” to measure attitude. The implicit assumption is that the agreement with this sentence is related to a positive emotion, and consequently to a positive attitude toward mathematics. This assumption is strongly questionable: our results show how agreement and disagreement with respect to the belief “math is useful” are not symmetrical from the emotional viewpoint. The belief that mathematical utility is associated with a positive emotional disposition is also questioned by the analysis of students’ narratives: as a matter of fact, many students state “I do not like math” although they recognize its utility and sometimes even its indispensability. Another crucial difference between primary students and older students in the approach to the utility of mathematics is the fact that, at the primary level, the utility of mathematics is recognised in the (presumed) fact that school mathematics faces with concrete situations and learned mathematics is implementable immediately in the real life: In my opinion, mathematics is needed to do calculations faster, to calculate how many miles two cities are distant, how many square meters a field is extended; to draw a square, a circle

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or a triangle; to know how much you have to pay; to know how many grams or kilograms of strawberries mum has gathered in the field; to buy a liter of water. Mathematics is needed for everything. [4P.59]

This belief is probably also conveyed by the teacher in order to motivate children, but our narratives show as to identify the utility of mathematics in its immediate implementability can have a boomerang effect. As a matter of fact, many students wonder why they have to continue studying maths after learning the notorious “learning to calculate”: Mathematics is, of course, useful, but not always easy to apply. I wonder: when will I use an equation in my daily life?!? [3M.27] Another thing that contributes to my opinion about math is the fact that mathematical knowledge, apart from the elementary operations or the proportions that are used everywhere, is not used in everyday life. [3H.61] Mathematics is surely one of the most important subjects amongst those we have at school. Being able to write, read, and perform numeric operations has always been one of the primary needs of a man. However, most of the topics we learn in mathematics are not used in real life, unless you decide to undertake a well-defined job direction. [3H.23] Mathematics. . .a word which is as complex as its calculations, radical numbers, expressions and most of all lines, which, to me, are meaningless lines and also I don’t understand, except simple calculations, sums, divisions and multiplications, what the other things are for in one’s life if they do not decide to become a teacher or an engineer? [4H.14]

Sometimes this view appears as early as the elementary level. In this case the emotional engagement of the narrator is typically negative and very strong. In this respect, the following essay is particularly significant: To me mathematics is only a waste of time because once you have learned numbers, you can just stop, but no, we continue and lessons start to torture you slowly and it is an awful feeling when I write and don’t understand, and it seems to me I’m going down to hell: I start sweating from head to feet, I turn completely red and I feel like I’m exploding. [3P.28]

In the students’ narrative, the concreteness of mathematics is comprehensibly related to the perception of the degree of application of the learned mathematics to a realistic situation. The exposition to contextualized problems plays a crucial role in this perception and it is undoubted that mathematical word problems have a long tradition and an important role in mathematics teaching at primary school. If it is true that mathematical modeling and solving mathematical word problems are not the same thing (Erbas et al. 2014), it is also true that we recognize many similarities between them2 and that word problems appear to be the unique possible simulation of modeling in the first school levels. The most important reason for using

2 Our view is different from the one of Lingerfjard (2002): according to him it is unreasonable to compare the activity around word problems with that around modeling.

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word problems in primary school is exactly to train pupils to apply the formal mathematical knowledge to real-world situations (Verschaffel et al. 1994). According to this picture, a first clear difference between the primary level and the others emerging from our narratives is the growth of abstraction in school mathematics and the tendency to abandon word problems in favour of proposing problems in a purely mathematical context: My hate towards mathematics started from the year of high school when mathematics has become increasingly abstract and less applicable to everyday life. [3S.7] My doubt about the usefulness of mathematics grew over time during my period in middle school, reaching its peak at the high school where mathematics deals with completely abstract arguments and you ask yourself: what is it necessary to solve a derivative in my future? [5S.1]

Obviously, we can argue long about what being able to solve a derivate can be useful for, but, apart from that, this tendency to perceive mathematics as increasingly abstract is evident in our narratives: in some sense, it is even an educational choice. If therefore, contextualized problems have a role in the different assessments of the degree of concreteness of mathematics, it is also true that mathematics education literature has developed an extensive and meaningful criticism to school word problems, and in particular to their presumed realism. In particular, it has discussed that contextualization of traditional school word problems often appears to be artificial (Zan 2011) and that students tend to ignore contextual consideration when solving these problems (Lesh and Doerr 2003; Merseth 1993; Verschaffel et al. 1994). As Mellone et al. underline (2017, p. 1): The more pupils advance in their school path, the more we can witness a tendency to unthinkingly apply arithmetic operations when responding to word problems, without critically considering the reality that the word problem is actually referring to.

This does not mean that pupils in the early grades recognize the context described in word problem as artificial, rather that they act regardless of the specific context. As described by many authors (Greer 1997; Van Dooren et al. 2006) this alarming ‘disregard’ is strongly affected by some didactical choices. In particular, the fact that the typical school word problems can almost always be solved through the use of one or more arithmetic operations with the numerical data given in the text (Verschaffel et al. 1994). This implies that students do not need to consider and construct a situation model to get the right answer: to by-pass the text, without considering the context, becomes a winning strategy in order to get the right answer quickly (Zan 2011). The fact that this strategy is typically winning in the school context reinforces what D’Amore and Martini (1999) called formal delegation clause: when the student, computing some arithmetic operations, obtains the numerical result to the problem he just has to state it, whatever it is, no matter what it means in the problematic starting context. As is evident from the analysis of our narratives, in the early grades, the application of mathematics to trading situations appears to the pupils as proof of the concreteness, and consequently usefulness, of mathematics.

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This emerges clearly from our narrative data, the majority of occurrences of the expression “mathematics is useful” in primary students’ narratives is referred to trading situations: Mathematics is useful because people who want to work in the bakery, in the market or in a toy shop will have to do their calculations well. [3P.21] Mathematics is essential because when we do the shopping someone can cheat you with the change. [4P.1]

It is interesting that we found a strong and widespread criticism of the presumed utility of the mathematical word problems used in primary school almost only in the memories of the older students: Already when I was six years old, I did not understand why the teacher gave us the traditional problems regarding shopping in the everyday life. When she began to dictate the text, automatically a big smile would appear on my face. It was a ridiculous thing for me since it was my mother who was busy looking at foodstuff, paying a certain amount of money, etc. [2H.45]

This excerpt highlights how some situations – considered realistic at the age of elementary students – show all their limits in term of concreteness as students grow up. Sometimes, the alleged concreteness and realism of the primary word problems is (rightly so?) even ridiculed in secondary students’ narratives: We more or less wrote: “Mrs. Pina left the tap open. If 2 cubic meters of water per hour come out, how many hours are needed to flood her 400-cubic meter apartment?” Poor Mrs. Pina! I hope for her she has closed the tap before she gets submerged! [4P.1]

The reported excerpt is an example of how older students not only perceive the contexts described in mathematical word problems as different from the real world, but they also recognize the questions posed as artificial questions. Indeed, as Zan (2011) underlines, in the majority of the cases, the questions in mathematical word problems do not follow in a narrative way from the context, but they are artificial questions about the context. This observation is crucial for our discussion because it underlines the role of two components in the perception of the degree of concreteness of a mathematical word problem: the context and the posed question. If only one of these components is perceived as artificial, the problem will be perceived as artificial. The disconnection between mathematical word problems and concreteness has at least two worrisome consequences. On the one hand, this disconnection is one of the causes of the well-known phenomenon of children’s suspension of sense-making during mathematical problem solving (Greer 1997; Schoenfeld 1991). In particular, children answer mathematical word problems without considering the reality that the context of the problem could recall. This is obviously a big issue for the development of modeling competence because – as observed by Verschaffel et al. (2002) – it leads to an atrophied version of the modeling cycle (Fig. 1), where the two phases in which the mathematical world and the real world are interconnected (Formulation and Interpretation) are bypassed.

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Fig. 3 The three mountains test

On the other hand, from an affective point of view, the disconnection between word problems and real word reinforces the belief that mathematics is unrelated to the real world. In this case, problems that describe a realistic context posing an artificial question are particularly insidious because they foster the idea that mathematics is not related to any real human interest. About this, the pioneering study by Margaret Donaldson (1978) offers a very interesting definition of “concreteness” of a problem. She develops this definition to give an alternative interpretation of the children’s results in the famous Piagetian three mountains test (see Fig. 3). In the test, using a model of three mountains, where the mountains are distinguished from one another by some details (the snow on one, on house on the top of another, etc.), the experimenter puts a doll in another position with respect the child (Piaget and Inhelder 1967). The child is given a set of some pictures of the model taken from different angles and has to indicate the one representing what the doll sees. Usually, children below the age of seven choose the picture representing their own point of view. Piaget uses these results to conclude that children of this age are unable to enter in the point of view of others. Donaldson criticizes Piaget’s interpretation on the basis of the results of younger children (between the ages of 3 and 5 years) to a test developed by Martin Hughes (1975). In this test, the child is asked to hide a boy doll in a plastic model so two policemen placed in the model do not see it. To be successful the child had to consider and to coordinate two external points of view. Well, 90% of children’s responses were correct! Moreover, considering the differences between the requests of two tests, it is hard – if not impossible – to reconcile these findings with Piaget’s conclusions: children

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appear to be able to consider (and coordinate) others’ points of view. Donaldson’s interpretation is related to the authenticity of the tasks in their entirety (situation and question): The point is that the motives and intentions of the characters are entirely comprehensible, even to a child of three. The task [in Hughes’ test] requires the child to act in ways which are in line with certain very basic human purposes and interactions (escape and pursuit) —it makes human sense. Thus it is not at all hard to convey to the child what he is supposed to do: he apprehends it instantly [. . .] In respect of being humanly comprehensible, the “mountains” task is at the opposite extreme. Within this task itself, there is no play of interpersonal motives of such a kind as to make it instantly intelligible. (There is the question of the experimenter’s motives in asking the child to do it and of the child’s motives in responding, but that is quite another matter.) Thus the “mountains” task is abstract in a psychologically very important sense: in the sense that it is abstracted from all basic human purposes and feelings and endeavors. It is totally cold-blooded. In the veins of three-year-olds, the blood still runs warm. (Donaldson 1978, p. 17)

Donaldson clearly explains as this idea of concreteness/abstraction in a psychologically sense can affect children’s understanding of the task and so their behaviours. On the other hand, it seems by our narratives that this idea is unconscious in elementary students. Surely this conception of concreteness becomes more clear and explicit in older students: My relationship with math has not always been unhappy. At elementary school, it was one of the few subjects that I was passionate about, perhaps because the problems to solve seemed practical and the purpose was clear. I could not identify a precise date, but in the middle school, the first difficulties in mathematics began [. . .] The goal of learning certain strangeness was unknown for me. In my thoughts, mathematical contents appeared abstract and I did not see anything that could serve me in everyday life [. . .] I did not feel enriched by studying math and the desire to learn, know and deepen a so complex and wide matter did not grow in me. [5H.22]

This idea of abstraction in a psychological sense should always be kept in mind when we want to foster mathematical modeling in the classroom with the intention of promoting a positive attitude towards mathematics. It highlights the relevance of the choice of a concrete activity, in terms of the context involved and of the question posed. For example, in the Italian context, many controversies have been caused by the modeling problem proposed in the 2017 national graduation exam for scientific high schools (Fig. 4). The problem asked to study the motion of a square-wheeled tricycle. The text of the problem underlines the existence of a model of a strange such tricycle at New York’s Museum of Mathematics. In Italy, the modeling problem at the graduation exam was introduced – according to the new standard (MIUR 2010) – in the scholastic year 2014–2015. The explicit goals are to assess students’ modeling competence and to show that they are able to connect the learned mathematics and their everyday life.

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Fig. 4 The modeling problem in the 2017 Italian graduation exam

The square-wheeled tricycle problem is a good example of the difference between the literal meaning of the adverb concreteness and the affective meaning: the specific object exists, it is concrete in the literal meaning, but the problem (the context and the related question) is probably far from most students’ interests and everyday life and therefore it is abstract in the sense introduced by Donaldson. This kind of modeling is likely to obtain a negative effect: it can reinforce the opinion that mathematics deals only with abstract situations. We have discussed some critical aspects related to the presumed utility and concreteness of school mathematics (vision of mathematics), their consequences of students’ emotions (emotional disposition towards mathematics) and the role (and critical aspects) of promoting mathematical modeling in classroom. That is, we have discussed the relationship between two dimensions of the TMA-model for attitude towards mathematics and mathematical modeling.

Conclusions The relationship between mathematical modeling and attitude towards mathematics is more complex than it seems. It is very difficult to model any (even simple) real situation using basic mathematical knowledge developed in the school mathematics education. Therefore mathematical modeling is often declined in a simplified version: the resolution of mathematical word problems. This choice introduces some critical issues because of the differences between mathematical modeling and word problem solving.

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The first one is that mathematical modeling involves authentic real-life context while the context of the word problems is an idealized real-life situation (Lesh and Zawojewski 2007). The choice of the problems (or activities) is therefore crucial: the context has to be realistic and the posed questions significant. The second one is placed in the culture of the classroom wherein word problems are often presented in a stereotyped way (Greer 1997) developing automatic reactions in students rather than a strategic approach that should be the real goal of problem solving activity (it is the goal of competence). It is evident that the implementation of mathematical modeling in the classroom could have a great potential both for cognitive and for affective goals. As a matter of fact, it can promote the development of the specific competence related to mathematical modeling – that is an important goal of education in itself (Niss 2012) – and it can help to develop a positive attitude towards mathematics, showing the utility of learned math and therefore fostering motivation in studying it. On the other hand, the implementation of mathematical modeling in the classroom is often inhibited by its cognitive and didactical complexity. It emerges a sort of recurrent educational schema: mathematical modeling is considered too complex to be implemented in classroom; it is replaced (sometimes identified) by the resolution of simple word problems. These problems often are abstract in the sense of Donaldson, unrealistic, and their solutions involve only automatic and not strategic actions. In this way a corrupted idea of utility of mathematics and of mathematics education is developed since the primary school level. We believe that a completely different conception of utility of mathematics should be developed since the first school levels, repudiating its presumed immediate implementability in (equally presumed) concrete situations. The utility of mathematics education should be related to a crucial aspect also for mathematical modeling: the development of a critical approach towards the real world, fostering the desire to understand the reason for an observed phenomenon. This idea of utility is clearly related to the idea of active citizenship, therefore it has a universal value and it justifies the teaching of mathematics for all students, regardless of their specific future choices. On the other hand, the coherence between explicit goals of mathematics education and didactical choices is crucial. This appears clearly in the following narrative with which we want to close this chapter: Now I am OK, but not because I am able to reason with the formulas, but because I just apply them. I am sure that if I had to take a test that asks “whys” about the formulas I could not write a single word. Continuing along my path, linear equations, quadratic equations and radicals are not useful in the world of tourism, but these things we do to learn to reason, right. . .? But, if I do them because I know the rules but I don’t understand them, what do I need them for? There are people who spend their lives studying mathematics, but I ask myself how can they do that. If I could, math would be a subject that I would quit studying, since Ì hate it. I think that this “feeling” depends on the fact that I have always studied through rote memorization, mechanically, without worrying about really understanding the exercise that I was to solve. Is it my fault or the teachers’ fault? [2H.17]

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Chapter 10: Teaching Modelling Problems and Its Effects on Students’ Engagement and Attitude Toward Mathematics Zakieh Parhizgar and Peter Liljedahl

Abstract In this chapter, the engagement of 244 students is measured across three different types of mathematical problems (modelling problems, word problems, and mathematical exercises). We also investigate the potential of teaching modelling problems in changing students’ attitude towards mathematics. This research was conducted with a pre-test, followed by an educational intervention and a post-test. During the educational intervention, two groups of students with different instructional formats attended six sessions of modelling problem solving. The results of this study show that, although, there existed no significant difference in engagement existed between the three types of mathematical problems in the pre-test data, both quantitative and qualitative analysis of the post-test data showed that students generally experienced more engagement on word problems. Results also show that teaching modelling problems improved students’ attitudes towards mathematics in both groups. Keywords Flow · Engagement · Modelling problems · Word problems · Problem solving

Introduction Kaiser et al. (2011) argue that, because of the application of mathematics in science, technology, and in our daily lives, students’ ability to apply their mathematical knowledge to real-world situations has become an important issue in the twentyfirst century. Yet, as English and Sriraman (2010) argue, this application will require

This research has been funded by Iran cognitive sciences and technologies Council since 2013. Z. Parhizgar (*) Ferdowsi University of Mashhad, Mashhad, Iran e-mail: [email protected] P. Liljedahl Simon Fraser University, Burnaby, BC, Canada © Springer Nature Switzerland AG 2019 S. A. Chamberlin, B. Sriraman (eds.), Affect in Mathematical Modeling, Advances in Mathematics Education, https://doi.org/10.1007/978-3-030-04432-9_15

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a type of mathematical knowledge that is currently beyond their classroom experiences. As such, it is vital that we, as a field, begin to create opportunities which will enable students to improve their modelling skills. It is important for students to have practice in seeing situations in which mathematics might be helpful and in trying their hand at formulating useful problems, although every situation will not be formulated into a precise mathematical problem. (Pollak 1970, p. 318)

As a result, curricula all over the world are beginning to embrace mathematical modelling and the use of real-world problems as a necessary part of mathematics education. For example, the Iranian national curriculum documents (Research and Educational Planning Organization 2013) specify that the modelling of real-world problems and phenomena is considered to be within the territory of, “the teaching and learning of mathematics” (p. 25) further suggesting that, “the important mode of mathematics is to enable human beings to accurately describe complex situations and to predict and control possible natural, economical, and social situations” (p. 25). The ability to apply mathematics in solving problems in daily life is one of the most important aims of teaching mathematics. As such, “educational activities must be based on the mathematics of the environment and it must help students to observe and analyze mathematical concepts and prepositions in their environment and to obtain various interpretations for mathematical concepts in their surroundings” (p. 26). In addition, the incorporation of modelling into a mathematics curriculum can create the background for better understanding of mathematical concepts (Blum et al. 2002) as well as student motivation. . . . the more we can incorporate genuinely real-world problems within the curriculum, the better our chances of enhancing students’ motivation and competencies in mathematical problem solving. (English and Sriraman 2010, p. 268)

On the other hand, research shows that a perceived lack of connection to their real life is disenfranchising for some students, causing them to have negative attitudes about school mathematics (Alamolhodaei 2009). Generally, beliefs, attitudes and feelings, often called affect (McLeod 1992), play an important role in developing creative and critical thinking in mathematics and mathematical modelling can pave the way for students’ better understanding of mathematical concepts, thus leading to improvement on mathematical beliefs and developing creative thinking (Blum et al. 2002). Examining the modelling approach and its role in providing an environment in which students develop appropriate beliefs about, and positive attitude towards, mathematics is an important concept brought up in the International Congress on Mathematical Education (Blum et al. 2002). The research presented here investigates the potentials of teaching modelling problems in improving Iranian grade 10 students’ attitude toward mathematics.

Mathematics Attitude An individual’s attitude toward mathematics may refer to how s/he likes or dislikes mathematics, or to the extent to which s/he thinks understanding and learning mathematics are (un)important in their lives (Hannula 2002; Ma and Kishor 1997).

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Research shows that a variety of factors – including teacher’s support, interaction among students and behavioral and educational expectations from teachers – have a significant relationship with students’ attitudes and behaviors. Teaching methods which emphasizes active participation of students (student-centered) are contrary to traditional lectures in which the teacher provides information and students receive this information passively (teacher-centered) (Prince 2004). In educational environments in which teachers play a supportive role, students’ positive feelings and confidence in their abilities in attaining success are enhanced (Akey 2006). The teachers’ teaching method can, therefore, be an important factor in changing students’ attitude. Akinsola and Olowojaiye (2008) is a case in point for this kind of research; adopting two teaching methods, the authors learned that the teaching method implemented by teacher in the class exerts a strong influence on students’ attitude towards mathematics. In addition to the teaching method, the pertinence of mathematics subjects to students’ needs is also of great importance, playing a prominent role in motivating the students to learn mathematics. Attitudes towards mathematics as a part of the affective domain in mathematics are addressed in some research regarding the effect of real world problems on students’ mathematics attitude. Regarding the larger construct of affect and not merely attitude, Hardre (2011) proved that using various ways of getting students involved in the classroom as well as deliberately illustrating the uses of mathematics to students can enhance their efficacy and motivation in solving challenging problems. Adding to this, Bracke and Geiger (2011) found that incorporating modelling problems into mathematics lessons heightened student interest in these type of problems and even caused a change in attitude in mathematics in general. In the same vein, Eric (2011) investigated Singaporean sixth-grade students’ attitudes towards solving mathematical modelling problems in a problem-based learning environment. Their results indicate that the students’ interest, perseverance, and confidence increased after solving modelling problems. They concluded that providing students with a PBL environment is a good way to improve their attitude towards the learning of mathematics. Schukajlow et al. (2012) investigated the effects of teaching modelling problems on students’ enjoyment, interest, value and self-efficacy expectations concerning three types of mathematical problems: intra-mathematical problems, word problems, and modelling problems. The results of this work showed that teaching a unit on modelling problems had positive impacts on students’ affect with regard to all three types of problems. Taken together, it is expected that by providing activities which have a relationship to the real world, would strengthen the learning process, improve students’ interest and motivation, activate positive emotions, and increase the number of students who enjoy mathematics. We are interested in testing this idea by looking closely at student engagement—as a contextualization of these aforementioned affective variables—while solving mathematical modelling problems and comparing this with student solving more typical mathematical problems such as word problems and mathematical exercises. Hence, we have chosen to draw on the theory of flow (Csíkszentmihályi 1990, 1996, 1998), a theory developed specifically to explain the phenomenon of deep engagement.

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The Theory of Flow In the early 1970s Mihály Csíkszentmihályi became interested in studying, what he referred to as, the optimal experience (1990, 1996, 1998), . . . a state in which people are so involved in an activity that nothing else seems to matter; the experience is so enjoyable that people will continue to do it even at great cost, for the sheer sake of doing it. (Csíkszentmihályi 1990, p. 4)

The optimal experience is something with which most people are familiar. Flow is a period in which one is intensely focused and highly absorbed in an activity, that one loses track of time, is un-distractible, and is totally consumed by the enjoyment of the activity. As educators we have glimpses of this in our teaching and value it when we see it. In an effort to gain insight into this fleeting phenomenon, Csíkszentmihályi pursued it within contexts that he believed to be rich in optimal experiences and among the people working within these contexts – musicians, artists, mathematicians, scientists, and athletes. Emerging out of this work were a set of nine elements that turned out to have been common within every optimal experience – irrespective of it occurred for a musician, artist, mathematician, scientist, or athlete (Csíkszentmihályi 1990): 1. 2. 3. 4. 5. 6. 7. 8. 9.

There is a clear goal at every stage of the activity. There exists a mechanism for which feedback on one’s actions are provided. The doers ability is in balance with the challenge of the task. Action and awareness are merged. The doer becomes indistractible. The doer gives themselves to the task with no concern for failure. Self-consciousness disappears. Time becomes distorted. The goal of the activity shifts from getting done to doing the task.

The last six elements on this list are characteristics of the internal experience of the doer and manifest themselves as increase in focus, enjoyment, situated interest, and motivation. In contrast, the first three elements on this list are characteristics external to the doer, existing in the environment of the activity, and crucial to occasioning of the optimal experience. The doer must be in an environment wherein there are clear goals, immediate feedback, and there is a balance between the challenge of the activity and the abilities of the doer. This balance between challenge and ability is critical to Csíkszentmihályi’s (1990, 1996, 1998) analysis of the optimal experience and is articulated through the consequences of being in a state of imbalance. For example, if the challenge of the activity far exceeds the doer’s ability, they will experience a feeling of anxiety or frustration (see Fig. 1). Conversely, if the doer’s ability far exceeds the challenge of the task at hand, they will experience boredom (see Fig. 1). However, when there is a balance between challenge and skill, a state of flow is created (see Fig. 1) – which is the essence of optimal experience and the nine aforementioned elements into a single emotional-cognitive construct.

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Fig. 1 Graphical representation of the balance between challenge and skill

Flow is one of the only ways to talk productively about the phenomenon of engagement. The nine aforementioned elements of flow give us not only a vocabulary for talking about aspects of the subjective personal experience of engagement, but it also gives us a way to think about the potential environments that occasion engagement during mathematical activity.

Mathematical Problems Mathematical problems are often divided into three types: modelling problems, word problems, and intra-mathematical problems (Niss et al. 2007, p. 12). The distinction between the three types of problems lies in the degree of their connection to the real world. In what follows we summarize each of these type of problem along with the mental activities required for solving each type (Schukajlow et al. 2012).

Modelling Problems Simply stated, mathematical modelling is the application of mathematics in the solving of problems from real-life situations which do not have regular structure (Galbraith and Clatworthy 1990). Figure 2 shows an example of such a problem. One of the theories that can be used for describing modelling activities is the modelling cycle proposed by Blum and Leiss (2007). A commonly accepted solution process for a modelling problem can be characterized by a seven-step sequence of activities: (1) understanding the problem and constructing an individual “situation model”; (2) simplifying and structuring the situation model, thus constructing a “real model”; (3) mathematizing or translating the real model into a mathematical model; (4) applying mathematical procedures in order to derive a result; (5) interpreting this mathematical result with regard to the initial real situation, thus attaining a real result; (6) validating this result with reference to the original situation and, if the result is unsatisfactory, returning to step 2; and (7) exposing the whole solution process.

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Shot on Goal: You have become a strategy advisor to the new football recruits. Their field of dreams will be the FOOTBALL FIELD. Your task is to educate them about the positions on the field that maximise their chance of scoring. This means when they are taking the ball down the field, running parallel to the SIDELINE, where is the position that allows them to have the maximum amount of the goal exposed for their shot on the goal? Initially you will assume the player is running on the wing (that is, close to the side line) and is not running in the GOAL-to-GOAL corridor (that is, running from one goal mouth to the other). Find the position for the maximum goal opening if the run line is a given distance from the near post (Galbraith and Stillman 2006; Stillman 2011). Fig. 2 Modelling problem: Shot on Goal

It is important to note, however, that despite the linear nature of the aforementioned seven-step model students’ problem-solving processes are often not so linear. Rather, they frequently move back and forth between the real and mathematical aspects of the model (Borromeo Ferri 2007).

Word Problems Word problems are nothing more than unmasking a purely mathematical problem which is expressed in terms of the real world (Blum and Niss 1991). Figure 3 shows an example of a word problem. The process of solving these problems includes turning the words into mathematical world (Niss et al. 2007). Word problems can be related to the reality, although in this model mental activities related to reality are simpler in comparison to modelling problems and that is because the real model has been given in the problem from the beginning (Borromeo Ferri 2006; Chamberlin 2010; Schukajlow et al. 2012).

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Air plane take-off: An air plane taking off accelerates for 300 meters along the runway before taking off at a 45 degree angle to the ground. When the plane passes over the end of the runway it is already at a height of 150 meters. How long is the runway? (Bakhshalizade et al. 2013). Fig. 3 Word problem: Air Plane Take-off Fig. 4 Intra-mathematical problem: AB Length

B

A

C

The length of AB: In the Right triangle ABC, Tan B = √ and BC = √ . Calculate the length of side AB.

Intra-mathematical Problems The third type of the problems is the ones which does not have any connection to reality and are introduced using mathematical propositions (see Fig. 4). To solve these, only the appropriate mathematical procedures and concepts are needed. Intra-mathematical problems are not problems in the classic sense of problems. Many researchers would call them exercises. For consistency and parallelism, in this article we are referring to them as intra-mathematical problems.

Research Questions Taken together, we are interested in examining students’ attitude toward mathematics and the engagement of students while solving mathematical modelling problems and compare these to the engagement of students solving, the more common and more familiar, word problems and intra-mathematical problems. Such questions have not been adequately addressed (Schukajlow et al. 2012). At the same time, we know from prior research that a student-centered teaching approach can result in increased engagement and positive emotions (Gläser-Zikuda et al. 2005; Hänze and Berger 2007). However, there exists no research on the influence of educational environment and engagement. In addition to comparing engagement across different types of problem and investigating mathematics attitude, we are also interested in examining the role of teaching modelling on engagement and attitude. Blum (2011) partitions teaching into either teacher-centered or student-centered. According to Blum (2011), the teacher-centered approach is characterized by: The teacher writes the problem on the board, Students express their opinions and comments about the problem, The teacher asked them to think about the solution, The teacher brings the class to a common solution through teacher-students interactions, The final solution is written on the board.

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Meanwhile, the student-centered approach is characterized by (Blum 2011): The teacher writes the problem on the board, The students work in groups of three to five students to try to solve the problem, The teacher monitors the students and tries to guide them by posing appropriate questions.1 Taken together, this study has been designed and was conducted to answer the following questions: Question 1 What are the effects of teaching mathematical modelling problems on students’ attitude toward mathematics? Question 2 Are students’ attitude toward mathematics different in teacher-centered and student-centered lessons? Question 3 In which type of mathematical problems (modelling, intra-mathematical and word) do students experience the most engagement?

Method Because students did not have any previous experience with modelling activities in their mathematics classes, this research was conducted using a pre-test ! educational intervention ! post-test design.

Participants The participants of this study were 244 female students aged 15–16 from three private high schools in Iran. This study was done in nine grade 10 classes (the number of students in each class was between 29 and 33) which lasted approximately 4 months. The necessary sample size was estimated using PASS software, considering at least 80% power for t-test in pre-test and post-test independent sample t-test (its nonparametric equivalent, the Wilcoxon test) and a significance level of 5% based on the result of the Schukajlow et al. (2012) study. Seventy-two students were in a student-centered classroom during the course of this study, while 172 were in a teacher-centered classroom. Classes were categorized as ‘student-centered’ or ‘teacher-centered’ by systematic random sampling. This means that the students were randomly selected based on the estimated number and were divided into two groups.

This is done by using strategic interventions like ‘read the problem again’ or ‘draw a sketch’, comparing a groups’ solution to solutions from other groups, and having groups exchange ideas with other groups (Blum 2011). 1

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The majority of students in the schools in which the data were gathered had high scores in mathematics. The mean of the scores of classes (the average of students’ scores in each class ranged from 19 to 20 out of 20) were assimilated based on the students’ mathematics score in their previous year (grade 9).

Intervention Given the employment of two teaching methods (that is, teacher-centered and student-centered), students were divided into two groups. Six classes were run using the teacher-centered approach and three classes were run using the studentcentered approach (Blum 2011) with the same problems being used in both groups. In the intervention sessions students were exposed to modelling problems about the Pythagorean Theorem as well as Linear Functions. These lessons were designed based on aforementioned modelling cycle by Blum and Leiss (2007). The three students-centered classes were all taught by the lead author as well as four of the teacher-centered classes. The other two of the teacher-centered classes were taught by two teachers who normally taught using a teacher-centered approach. These two teachers were provided with all problems and solutions in a training session and delivered the lesson and the problems as intended.

Instruments The study consisted of the results from a pre-test, administered immediately prior to the aforementioned intervention, and a post-test, administered immediately after the intervention. The pre-test consisted of four separate sections. The first was a questionnaire asking about their attitudes towards mathematics designed by Tapia and Marsh (2004), which is used by many scholars (see for example Afari 2013). The scale2 is comprised of 40 items, the response to which range from ‘strongly disagree’ to ‘strongly agree’ based on five-point Likert scale (see Appendix A). The remaining three sections each consisted of mathematics problems relating to each of the three types of problems (modelling problems, word problems, and intra-mathematical problems) followed by an engagement questionnaire. The questionnaire about the students’ attitudes about mathematics. For the second section students first completed a modelling problem followed by a modified version of the Flow Perceptions Questionnaire (Egbert 2003) consisting

2

Some items in this scale were:

10. My mind goes blank and I am unable to think clearly when working with mathematics. 25. Mathematics is dull and boring. 40. I believe I am good at solving mathematics problems.

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of 15 items, using a 5 point Likert scale, to measure the participants‘engagement in solving the different types of problems. This questionnaire also had one written response question about their feelings that the student could fill out while solving the modelling problem. This same structure was used in section three and four with the exception that students were solving word problems and intra-mathematical problems respectively. The Flow Perceptions Questionnaire (Cronbach’s α ¼ .82) substantiated by Egbert (2003) with minor changes was used to examine student engagement. Egbert originally used this questionnaire to find out whether flow happens in foreign language classes (Egbert 2003). It has also been used in other studies (Azizi and Ghonsooly 2015; Mirlohi et al. 2011; Sedig 2007). This questionnaire is based on Likert format, having a 5-point scale from 5 (very strongly agree) to 1 (very strongly disagree). An example of one of the items is: This task excited my curiosity. The questionnaire originally consisted of 14 items. However, one of the items3 when translated to Persian, was not understandable to the students, so it was omitted from the questionnaire. Hence, the original psychometric properties of the instrument are technically in question (Huck 2012), though removing the item that could not be translated to Persian was the more preferable option to having an uninterpretable item. The Flow Perceptions Questionnaire was utilized to measure the quality of subjective experience such as level of interest, degree of concentration, enjoyment of the activity, and amount of perceived control of the activity. However, as there are no items directly related to perceived balance between challenge and skills, two items related to this quality were added to the questionnaire.4 As mentioned, an open question5 was also added to the questionnaire, resulting in a 16 item instrument (see Appendix B). After the intervention, the students again completed a four section test following the same structure as for the pre-test, but with different problems for the last three sections. See Table 1 for a summary of the instruments. Moreover, in order to study the students’ attitude towards modelling problems brought up during educational sessions, one open-ended question was also put to students along with the first questionnaire. The question includes: What effect has teaching modelling problems had on your attitude towards the mathematics course? The section for modelling problems included only one problem of mathematical modelling. The sections for word problems and intra-mathematical problems each contained two problems, one related to Pythagorean Theorem and the other related to Linear Functions. In what follows, samples of each type of problem are given, all relating to the Pythagorean Theorem.

3 During this task, I could make decisions about what to study, how to study it, and/or with whom to study? 4 My mathematical skills were on par with the provided challenges. I believe that my skills enabled me to overcome the challenges. 5 What were your feelings when you were solving this kind of mathematical problems? Write your comments about it.

Pre-test Mathematics attitude questionnaire Modelling problem Word problem section section Problem solving test Problem solving test Flow questionnaire Flow questionnaire Flow questionnaire

Intra-mathematics section Problem solving test

Table 1 Summary of instruments usedall tables are okay.> Intervention

Post-test Mathematics attitude questionnaire Modelling problem Word problem section section Problem solving test Problem solving test Flow questionnaire Flow questionnaire

Flow questionnaire

Intra-mathematics section Problem solving test

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Data The data consists of students’ attitude toward mathematics and flow questionnaire scores from the modelling, word, and intra-mathematical problems. The mean of these scores were computed for representing descriptive statistics and evaluating questionnaire by using repeated measures ANOVA through the SPSS Statistics 20 software. The reliability of the pre-test mathematics attitude and flow perception questionnaires were examined using the Cronbach’s alpha (see Table 2). In addition to the flow and attitude scores, there were also students’ answers to the two open questions about their feelings about these three kinds of mathematical problems and attitude toward mathematics. These were coded through the lenses of flow (Csíkszentmihályi 1990, 1996, 1998, ) as well as modelling (Blum and Leiss 2007). Although the student solutions for each of the problems were also collected, they are not being considered for the study presented here.

Result from the Mathematics Attitude Questionnaire The objective of the first research question was to examine the effect of teaching modelling problems on students’ attitude towards mathematics. In other words, can teaching modelling problems in six sessions improve the students’ attitude towards mathematics? In order to analyze their attitude the Two-Way Repeated Measures ANOVA was used. The mean of scores of students’ response to the attitude questionnaire is presented in Table 3. The results of this analysis indicate that teaching modelling problems had a positive effect on the students’ attitude towards mathematics F(1, 242) ¼ 22.86, p < .001. In the same vein, having taught modelling problems to 14–15 year-old students during a long-term (1 year-long) project, Bracke and Geiger (2011) claim that familiarity with, and working on, modelling problems improved students’ attitude in mathematics course. The main research question of their project was whether it was possible to incorporate the real-world modelling tasks into mathematics classes during the academic year. Their results indicated that incorporating the modelling into the curriculum in the long term would give the students the power to analyze these problems, thus yielding better results than short term projects. In regards to the second question, the examination of results of Two-Way Repeated Measures ANOVA showed that the effect of teaching method on the students’ attitude is not significant (F(1, 242) ¼ 0.25, P ¼ .617). As is shown in Table 3, the attitude of students in both groups changed for the better, but the effect of group was not significant as the modelling problems were new for students in both groups. Despite the lack of significance, after the project was completed, the students explicitly expressed that those problems needed to be solved in groups – something

Chapter 10: Teaching Modelling Problems and Its Effects on. . . Table 2 Reliability of the flow and mathematics attitude scales

Table 3 Students’ (n ¼ 244) mathematics attitude and engagement from pre-test and post-test (ANOVA type I test)

Reliability (Cronbach’s alpha) Mathematics attitude Intra-mathematical problems Word problems Modelling problems

Mathematics attitude Intra-mathematical problems Word problems Modelling problems

247 Pre-test 0.959 0.886 0.880 0.918

Post-test 0.958 0.893 0.901 0.935

Pre-test M (SD) 3.76 (0.62) 3.34 (0.67) 3.40 (0.69) 3.28 (0.79)

Post-test M (SD) 3.89 (0.60) 3.42 (0.68) 3.55 (0.68) 3.51 (0.80)

that may have increased the impact of the teaching method. This was nicely summarized by one of the teachers who employed the teacher-centered method in his/her class. Students tended to solve the problems in groups; the research methodology, however, required them to solve the problems individually.

In an effort to gain fuller information on the students’ attitudes, the students were asked to express the extent to which the teaching and learning of modelling problems had an effect on their attitudes, after the educational sessions ended. I am now more interested in mathematics and do not deem it a useless lesson. The current mathematics material included in the textbook is not interesting enough – just some mathematics formulas in which we insert values. The hardest task we have is find a formula related to some equations and solve problems. But when I got familiar with the modelling problems, I could see how well we were able to solve interesting problems by learning these formulas. I think this kind of problems help us live more easily and look at things closely. I became more interested in mathematics and am more excited about solving problems. Solving modelling problems has given us a wider perspective on mathematics and the lesson has become even lovelier than it used to be. I liked modelling problems a lot and learned that mathematics is of great importance for our lives. I used to think that mathematics was just used for calculating but now my view has changed – I know where mathematics can be applied and also study more eagerly in this field. I also thank you for changing my view toward math. We have been learning mathematics in the form of some formulas and through memorizing them since the elementary school. Thus, preparation for solving such problems must have been planned from the elementary school.

These data support the quantitative results of the research and clearly show that modelling problems have brought about a change in attitude towards the mathematics course. However, they feel that modelling problems should be solved in collaborative environment. All problems were fascinating and challenging; solving this kind of problems – especially in groups – causes all members to make their best effort to come up with a solution. I prefer

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solving diverse problems in more interesting subjects to repetitive, rote-learned problems about a single subject.

Results from the Flow Questionnaire In order to see if different types of problems generated a change in flow for the students, a repeated measures ANOVA Type I test was used. Statistical analysis shows that there is no significant difference in students’ engagement among these three types of problems (F(2, 486) ¼ 2.9, P ¼ .064). However, when we consider students’ flow score in pre-test and we compare them among the three types of mathematical problems, we see that the mean engagement for word problems are the highest, the intra-mathematical problems are second, and modelling problems are third (see Table 3). From Table 3 we see that students did not differentiate between the three types of problems. That is, there was no significant difference in the flow students experienced across the three problem types. Schukajlow et al. (2012) also reached the same conclusion in their study. In examining students’ motivation, interest, self-efficacy, and enjoyment across the same three types of mathematical problems, they found no significant difference between these variables. In the present study, similar results were obtained although students first solved the problems and then commented on them. A possible explanation for these results may be the lack of students’ experience with modelling problems that led them to underestimate the complexity of the problem leading to superficial or incorrect answers. When students’ responses to the open question about their feelings and comments on mathematical problems were analyzed, the comments clustered into 5, 6 and 7 comment types for intramathematical problems, word problems, and modelling problems respectively (see Table 4). From the data in Table 4, it can be seen that word problems were seen as being the easiest and most applicable of the three problem types. Intra-mathematical problems were seen as boring. However, because they know how to solve them and feel relaxed when they face them, they were sometimes seen as interesting. Students’ comments about modelling problems revealed that they found these types of mathematical problems applicable and valuable, but their skills did not allow them to face the challenges and gain more positive feeling about these problems. In general, the students had the most positive feelings about word problems. This was unexpected. We hypothesize that since word problems are practical compared to intra-mathematical problems and are simple in contrast with modelling problems, the students had more flow during solving word problems. However, investigating students’ comments on mathematical problems shows that the students found all kinds of mathematical problems interesting. This is

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Table 4 Students’ comments about mathematical problems in pre-test Problem type Intra-mathematical problems

Word problems

Modelling problems

Flow These problems were interesting and we are totally familiar with them I had control over the problem solving process I had enough skills for solving them This kind of problems was fun for me I had enough skills for solving them These problems were interesting I had control over the problem solving process They were interesting

Out of flow These kind of problems bored me

Others For solving this kind of problems, it’s just need to know mathematical formulas They are routine mathematical tasks

They were not so difficult

They are applicable

I do not possess sufficient mastery of solving them I could not understand the real situation

They were from our daily life

Long description of the real situation made me feel anxiety and irritated

They were thought provoking and needing creativity to solve them They were difficult and challenging

consistent with the statistical analysis above. It is difficult to explain this result, but it might be related to the fact that not all students answered the open question about their feelings. We examined the third research question again after training sessions of modelling problems. The results of the test shows that there is a significant difference in students’ flow score among the three types of mathematical problems (F (2, 486) ¼ 3.61, P ¼ 0.030). Students could not differentiate these three types of mathematical problems before the intervention, but after it they experienced more flow while solving word problems. Table 3 shows the mean of students’ flow experiences in post-test. After word problems, students experienced more flow in modelling and intra-mathematical problems respectively. From the analysis of students’ responses to the open item in the post-test questionnaire, it was clearly observed that students still have the most positive feelings about word problems (all the comment types were repeated). As mentioned

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before, however, unlike the comments after the pre-test, after the intervention students now also made some positive expressions about the modelling problems.

Conclusion This research attempted to assess students’ attitude towards mathematics before and after teaching modelling problems. The findings indicate that using educational sessions to get the students acquainted with modelling problems had positive effects on their attitude towards mathematics. The positive effect is in step with Bracke and Geiger (2011). Two educational forms (teacher-centered and student-centered) in teaching modelling problems are compared in the present study. Analyzing the data collected here supports that student-centered approach has a more useful effect on the students’ attitude. The student comments on group work in modelling problems indicate that cooperative learning environments can provide a better background in which to evoke students’ positive feelings. Previous research on student-centered environment (e.g., Schukajlow et al. 2012) has also especially emphasized the role of such environments. Although attitude toward mathematics is resistant to change, it can be improved using cooperative learning methods (Townsend and Wilton 2003). The other main goal of the current study was to determine for which kind of mathematical problems students experience the most engagement. Taken together, the quantitative and qualitative results suggest that modelling tasks were no more engaging to students than intra-mathematical and word problems. We conclude that students found both intra-mathematical and modelling problems to be less engaging than word problems – but for very different reasons. Intra-mathematical problems were too easy and boring while modelling problems were too hard and frustrating. A possible explanation might be that the modelling problems were not developmentally appropriate and using them with greater frequency might enable educators to alter the difficulty or wording of the problems to reach a delicate balance between challenge and developmental appropriateness. On the other hand, the students found word problems to be simultaneously challenging and approachable due to their comfort and familiarity with them. These results can be summarized by placing the various problems on Csíkszentmihályi’s (1990, 1996, 1998) flow diagram (see Fig. 5). Motivation plays an important role in learning. Flow is a kind of intrinsic motivation which is described as a desirable experience in which the person really enjoys doing the task. The aim of this theory is to create conditions under which students are highly involved in their educational activities. As such, it is a studentcentred theory (Whitson and Consoli 2009). In order to create the background for flow in students, we need to be aware of the teaching conditions in the classroom and

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Fig. 5 The modified flow diagram

more helpful problems for students. The evidence from this study suggests that word and mathematical modelling problems connected to the reality can be rich sources for creating engagement in students and improving their view towards mathematics. However, students should be sufficiently taught how to solve real-world problems, especially how to do cycle modelling in order to be able to engage in the problem solving process and gain necessary skills to overcome the challenges. Most participants were talented students who had high scores in their lessons especially in mathematics. Examining how talented students of mathematics acted in the modelling class, Hee Kim and Kim (2010) concluded that mathematical modelling is an appropriate curriculum to achieve the objectives specific to the talented students. According to this study, the objective of curriculum for talented students in South Korea is to nurture the capacity to creative thinking and expanding selfdirected learning; for them, talented students are those who possess exceptional abilities in solving mathematical problems and are committed to doing their tasks. A limitation of the study is therefore that most participants were students with high average and more motivated to study than average school students. It is therefore necessary to conduct the research on students with a different cultural and educational background. Additionally, the age level appropriateness of the modelling tasks may have been a limitation. More research would be needed in order to determine if this was, in fact, an issue. Finally, the study was limited by the fact that the problems were situated within only two mathematical concepts – linear functions and the Pythagorean theorem. A different set of contexts may have produced different results.

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Appendices Appendix A A selection of items from the Mathematics Attitude Questionnaire with high loadings on the attitude component. Participants responded to each of the following items on a scale from A (strongly disagree) to E (strongly agree). 6. 11. 14. 21. 24. 31. 36. 39.

Mathematics is one of the most important subjects for people to study Studying mathematic makes me feel nervous When I hear the word mathematics, I have a feeling of dislike I feel a sense of insecurity when attempting mathematics I have usually enjoyed studying mathematics in school Mathematics is a very interesting subject I believe studying mathematics helps me with problem solving in other areas A strong mathematics background could help me in my professional life

Appendix B Participants responded to each of the following items on a scale from 1 (strongly disagree) to 5 (strongly agree). Questions 3, 4, 11, and 12 were reverse-scored. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

This task excited my curiosity. This task was interesting in itself. I felt that I had no control over what was happening during this task. When doing this task I was aware of distractions. This task made me curious. This task was fun for me. I would do this task again. This task allowed me to control what I was doing. When doing this task, I was totally absorbed in what I was doing. I believe that my skills enabled me to overcome the challenges. This task bored me. When doing this task I thought about other things.

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13. 14. 15. 16.

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This task aroused my imagination. I would do this task even if it were not required. My mathematical skills were in par with the provided challenges. What were your feelings when you were solving this kind of mathematical problems? Write your comments about it. (open ended question) Note; you can answer this question with the help of questions above.

Appendix C (Figs. 6 and 7)

Giant's shoes: In a sports center on the Philippines, Florentino Anonuevo Jr. polishes a pair of shoes. They are, according to the Guinness Book of Records, the world's biggest, with a width of 2.37 m and a length of 5.29 m. Approximately how tall would a giant be for these shoes to fit? Explain your solution (Blum, 2011).

Fig. 6 Modelling problem: Giant Shoes (topic linear function)

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Cunning Running: In the Annual "KING OF THE COLLEGE" Orienteering event, competitor are asked to choose a course that will allow them to run the shortest possible distance, while visiting a prescribed number of checkpoint stations. In one stage of the race, the runners enter the top gate of a field, and leave by the bottom gate. During the race across the field, they must go to one of the stations on the bottom fence. Runners claim a station by reaching there first. They remove the ribbon on the station to say it has been used, and other runners need to go elsewhere. There are 18 stations along the fence line at 10 meter intervals, the station closest to Corner A is 50 meters from Corner A, and the distances of the gates from the fence with the stations are marked on the map.

Investigate the changes in the total path length travelled as a runner goes from gate 1 to gate 2 after visiting one of the checkpoint stations. To which station would the runner travel, if they wished to travel the shortest path length? For the station on the base line closest to Corner A, calculate the total path length for the runner going Gate 1 – Station 1 – Gate 2. Use Lists in your calculator to find the total distance across the field as 18 runners in the event go to one of the stations, and draw a graph that shows how the total distance run changes as you travel to the different stations. Observe the graph, then answer these questions. Where is the station that has the shortest run total distance? Could a 19th station be entered into the base line to achieve a smaller total run distance? Where would the position of the 19th station be? If you were the sixth runner to reach Gate 1, to which station would you probably need to travel? What is the algebraic equation that represents the graph pattern? Draw the graph of this equation on your plot of the points. If you could put in a 19th station where would you put it, and why? (Additional suggestions were provided as to how the work might be set out, and for intermediate calculations that provide some task scaffolding (Galbraith & Stillman, 2006). Fig. 7 Modelling problem: Cunning Runner (topic Pythagorean Theorem)

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Chapter 11: Affect and Mathematical Modeling Assessment: A Case Study on Engineering Students’ Experience of Challenge and Flow During a Compulsory Mathematical Modeling Task Thomas Gjesteland and Pauline Vos

Abstract This chapter describes a study on engineering students’ affect while working on the Tracker Project Task, a group assessment task that asks students (1) to use digital tools (the camera in their smart phones and free tracker software) to capture the movement of an object, (2) to mathematically model that movement, and (3) to create a poster reporting on the video analysis of the movement. We applied an activity-based conceptualization of affect in mathematics (“do you like this activity?”), which differs from a subject-based conceptualization of affect (“do you like mathematics?”). A subject-based conceptualization has two drawbacks: (1) it does not distinguish among different aspects of mathematics, and (2) it draws in students’ bias and beliefs from earlier, often bad experiences of poor mathematics teaching. We found an activity-based operationalization of affect by using the concepts of challenge and flow. Flow is a state of absorption, in which people forget about time and experience feelings of happiness. We assessed n ¼ 346 students through the Tracker Project Task. To study affect, we developed an instrument of 10 items (Likert-type) to measure students’ experience of challenge and flow. We administered the survey through a web-based platform yielding a high response rate (n ¼ 239, 69%) and good reliability (Cronbach’s Alpha: 0,795). The results revealed that three out of five students experienced challenge and flow, which expresses students’ positive affect regarding a mathematical assessment activity. This can be ascribed to, on the one hand, the activity and the instrument not clearly being related to mathematics, and thus not being tainted by students’ earlier negative experiences with mathematics. On the other hand the Tracker Project Task had characteristics that can bring about flow: being open, offering ample time to submit the product, being accessible to all students (low floor), but also enabling the better students to challenge themselves further (high ceiling). Such characteristics may be better feasible within mathematical modeling assessment than canonical mathematics assessment. T. Gjesteland (*) · P. Vos University of Agder, Kristiansand, Norway e-mail: [email protected] © Springer Nature Switzerland AG 2019 S. A. Chamberlin, B. Sriraman (eds.), Affect in Mathematical Modeling, Advances in Mathematics Education, https://doi.org/10.1007/978-3-030-04432-9_16

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Keywords Activity-based (conceptualization of affect) · Flow · Affect (and mathematical modelling assessment) · Project-based task · Tracker software

Introduction Introduction to Affect, Mathematics Education and Modeling Tasks A recently published book (Pepin and Roesken-Winter 2015) brings together recent research in the field of mathematics education and affect. In this book, the authors describe affect, values, emotions, beliefs, attitudes, and so forth, which they conceptualize in terms of complex, dynamic systems and participatory environments. However, while distinguishing between many aspects of affect, they hardly differentiate between aspects of mathematics education, whether this is instruction, curriculum or assessment methods. When taking mathematics in such a holistic way, students are asked to give their agreement or disagreement to items such as “mathematics is my favorite subject”, “I enjoy pondering over mathematics tasks” or “in mathematics there is always a reason for everything”. There is no room for how students experience separate phenomena in mathematics education, some of which maybe positive and some maybe negative, some temporary and some more lasting. There is evidence that teachers want the best for their students, but nevertheless knowingly offer inadequate instruction due to time pressure, examination demands, discipline problems, lack of confidence, and so forth (e.g. Nolan 2012). These limitations can hardly yield positive effects on students’ affect, as repetitive calculation exercises cause boredom, time-restricted tests cause stress, the distance between the teacher’s and the students’ discourses cause alienation, or the seemingly irrelevance and meaninglessness of algebraic expressions cause demotivation. Asking students holistically for their agreement or disagreement on a statement such as “mathematics is my favorite subject” gives little room for nuances and contexts. A student partly agreeing with this item might rather have said: “mathematics with this particular teacher is my favorite subject, but last year it was the opposite” or “mathematics is my favorite subject when we solve problems that I can relate to in my daily life”. Or the students partly agreeing with the statement “in mathematics there is always a reason for everything” might rather have expressed personal reasons: “in mathematics (classes) there is always a reason (for doing it), which is to get a pass”. So, when researchers of affect in mathematics education understand beliefs as “relatively stable, reified mental constructs” (Pepin and Roesken-Winter 2015, p. 4), the stability of these beliefs can well be related to the stable practices in mathematics classrooms with meaningless tasks, alienating symbols, stressful tests and, often, a friendly teacher, of which few students envy the job. To the general public, mathematics has a bad press, being the only subject linguistically associated to negative affect by the terms math anxienty (Tobias

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1978) and mathofobia (Hodges 1983). Paulos (1988) observed that incompetence with numbers is socially acceptable and many people have little shame saying “I always hated math”. Also, Brown et al. (2008) report of students saying that they would rather die than take mathematics. From our own experiences with mathematics education as student, parent, teacher, teacher trainer, researcher and colleague, we have observed that, indeed, many students experience moments of boredom, apathy, stress, alienation or demotivation in mathematics education. This means, in the first place, that there are good reasons for the poor image of mathematics in society and the stable beliefs of students. Second, it means that there is much room for improvement, whether it be in its instruction, curriculum, or assessment methods, all having consequences for students’ experiences, their curiosity, their creativity, their self-esteem, their beliefs, and of course not in the least, their knowing and understanding of mathematical concepts and their competencies to use mathematics flexibly for solving non-mathematical problems. Third, it means that even when changing educational practices robustly, this will not immediately or deeply change students’ affect towards mathematics at large. The weight of a social perception about mathematics as being hard, meaningless and only for nerds, cannot easily be countered by carefully coordinated reform practices. The subject of mathematics, being an institutionalized school subject with an elitist tradition of more than 2000 years, cannot easily be changed. However, it is possible to make small steps that break away from canonical mathematics education and design mathematical tasks, that many students experience as pleasant and meaningful challenges, even if these are part of institutionalized assessment. An important presumption for this is that mathematics does not necessarily need to be the context for mathematical activities. There are many non-mathematical areas, in which one needs mathematics to solve problems. One such area is physics, where the use of mathematical models is pertinent to describe phenomena. The study described in this chapter has an inter-disciplinary setting at the cross-road of physics and mathematics, whereby kinematics (the physics of movement) is the context for an assessment task, which requires a significant amount of mathematics. The task asks students to make a translation from the real world in which objects move into the mathematical world of graphs and formula. This translation is known as mathematization (Blum and Leiss 2005; Niss 2010). The task additionally asks students to reflect on the mathematization (e.g. precision, relation to laws of gravity). The task does not start from a real-world problem that needs to be solved, and students don’t make all mathematical modeling steps from the modeling cycle (Blum and Leiss 2005; Niss 2010). Nevertheless, we perceive the task as a modeling task, as mathematization is an essential activity within mathematical modeling. Another presumption is, that there exist a variety of task formats that challenge not only the typically gifted students, but also the more average student. Sullivan et al. (2011) and Sullivan and Mornane (2014) describe challenging tasks as requiring students to: • Plan their approach, especially sequencing more than one step;

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• Process multiple pieces of information, with an expectation that they make connections between those pieces, and see concepts in new ways; • Choose their own strategies, goals, and level of accessing the task; • Spend time on the task and record their thinking; • Explain their strategies and justify their thinking to the teacher and other students. The study described in this chapter centers on a task format that fits this description. It is called project: a task which cannot be completed within a limited time frame, which has a clear, but not straight-forward goal, and there is variety in the approaches to tackle it (Blomhøj and Kjeldsen 2006; Kaiser et al. 2011). However, unlike in other studies, in our case the task was an assessment task with a formal evaluation (pass or fail). Within the context of this situated project task requiring mathematizing, we studied students’ affect. To avoid the research being contaminated by students’ preconceptions of mathematics, we undertook this study without the use of the word mathematics in the instruments that measured students’ affect. To use the term mathematics could interfere with students’ earlier experiences of canonical mathematics education and their biases about mathematics as an institutionalized subject could interfere with their evaluation of their engagement with the task. The students in our study were from the engineering department; according to Harris et al. (2015) many of these students are disappointed by the mathematical demands in the 1st year of their studies and some would not have chosen the engineering direction if they had known about these demands. Not many of them have a positive stance towards mathematics, seeing it as a hurdle to get further in their studies. Only later on in their studies, they start to perceive the usefulness of mathematics for their future professional lives. Thus, we expected the participants in our study not to have very positive ideas about mathematics.

Introduction to the Research Setting At the Faculty of Engineering and Science of University of Agder (Norway), we deal with large student numbers (>300). Such numbers are a worldwide phenomenon as more and more students gain access to higher education. A few years ago, the large number of students made the faculty decide to abandon the laboratory training in the 1st-year Physics courses, because the laboratory facilities and its staff could no longer harbor the students. However, this policy only solved infrastructural problems on the short term. Abandoning lab training could lead to future problems when the graduates from our faculty have become engineers, managers, researchers, and so forth. In their future professional lives, our students will need skills to measure and model phenomena from the real world so they can describe and analyze these, and eventually, make predictions. For their proper training, it is insufficient to offer large-scale lectures, instructional videos or tutoring sessions to train for written examinations. They also need training in relating measurements to theoretical

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models. They need skills to practically handle instruments, calibrate these, measure as precisely as possible, work with error margins, and so forth. Therefore, we wanted to develop a task, in which lab training was combined with relating measurements to mathematical models, and the preferably outside of laboratory facilities. If such a task would be feasible with large student numbers, then laboratory training can be less dependent of university campuses, and even be feasible in less affluent regions, or within distance education. A second reason to develop a new task was to improve students’ motivation. The students in our engineering courses are no different from those described in Harris et al. (2015), who found that many 1st-year engineering students have a negative stance towards mathematics. Thus, we wanted a task, in which mathematics would serve engineering aims, for example by being related to technology and moving objects. We were inspired by Domínguez et al. (2015), who carried out research at a university in Mexico. They asked their students in an interdisciplinary mathematics/ physics course: a child is throwing a candy to another. Make a mathematical model of this movement. With such an open-ended, inquiry-based modeling task, students need to consider the what, how, and why themselves. Research in science education has demonstrated the advantages of such inquiry-based tasks over traditional lectures or teacher demonstrations (De Jong et al. 2013; Minner et al. 2010). We adapted this open-ended, inquire-based, kinematical modeling task from Dominguez et al. (2015) and added the use of smart phones for filming. Many students now have smart phones that contain cameras with the quality to film motion sufficiently precise for video analysis. To use equipment from students’ extrainstitutional, daily lives gave students more ownership over the task. Additionally, we added the use of free software that can capture the motion from videos. This software is based on pattern recognition through contrasts and is known as tracker software. Such democratic availability of digital equipment, both smart phone cameras and free software, opens new possibilities for inquiry-based laboratory training for which expensive laboratories are no longer needed.

The Task With support from the faculty administration, we gave the students an obligatory, inquiry-based laboratory task, which they had to fulfill to get access to the written examination. The task asked students to select a movement of an object; they could choose whatever: throwing a ball, jumping their skate board, driving a car. They had to film this movement with their phones. Thereafter, they had to use free tracker software (http://physlets.org/tracker/) on their laptops to transform the movement into measurements, approximate the movement with a mathematical model, and then present their findings on a poster. The poster had to contain an introduction, observations (measurements), a mathematical model of the moving object’s trajectory, and a

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discussion of the accuracy of the model in comparison to the measurements. The task had to be completed within the first 2 weeks of the course and to be done in groups of two or three. Collaboration was convenient, because one student alone cannot easily create and film a movement simultaneously. The delivered posters were assessed on their quality (pass or fail), whereby students had to obtain a pass to get access to the written Physics examinations at the end of the course. In our communication with the students, we indicated the task by the name “Tracker Task”. We limited the word mathematics, and used it only in sentences such as “you must find a mathematical model that describes the position of the object as a function of time”. It was our first time to implement such an open, practical task. Therefore, we did not want to focus on students’ learning effects in the first place. We considered it a pilot study with the aim to find out whether such an obligatory assessment was feasible with large numbers, without expensive laboratory equipment, and with students who have little experience with open-ended tasks. We felt that we – as lecturers and researchers – should first take the opportunity to learn and see whether the task activated students in a positive way, to the extent that the obligation and the assessment weren’t the instigators, but that the task in itself activated the students. Our research question was: to what extent does an open assessment task about video analysis of motion with smart phones and free tracker software challenge and activate the students?

An Activity-Based Conceptualization of Affect If one wants to study affect in mathematics education, there are many conceptualizations and instruments, but most of these address mathematics holistically. Instead, we wanted to study students’ affect when they engage in an activity, which is different from standard activities of canonical mathematics education. Thus, we sought an activity-based, and not a subject-based conceptualization. Looking at mathematics from an activity-based perspective goes back to, among others, Freudenthal (1973). He distinguished between mathematics as (1) a wellorganized deductive system, or as (2) a human activity which consists of organizing mathematical patterns when solving problems. The two perspectives are also known as “mathematics as a noun” and “mathematics as a verb”. The first perspective relates to mathematics as a holistic academic discipline, with its own symbols, and its procedures for establishing truth (by creating proofs), and so forth. The second perspective relates to mathematical activities, such as solving non-routine problems and using mathematics to solve problems within non-mathematical contexts, for example within kinematics. Within the first perspective, humans and their activities are less visible than within the second. An activity-based conceptualization of mathematics also aligns with a sociocultural perspective. One of its advocates, Lerman (2000), describes mathematics as a socio-cultural practice embedded within a community. If embedded within a school institution, mathematics is a practice embedded in a community of a teacher

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and a group of students. The activities consist, among others, of explanations by the teacher, and work on tasks by the students. This practice differs markedly from mathematics as a practice embedded within a research community, whereby the actors work on problems to which nobody knows an answer. Describing mathematics socio-culturally as a practice embedded within a community entails focusing on the activities undertaken by the actors, which are mediated by language, tools, and so forth. Using an activity-based conceptualization of mathematics enables us to (1) relate affect to activities and not to mathematics as a holistic entity, and (2) distinguish between different kind of activities within different contexts of mathematics education (doing repetitive exercises or an inquiry-based project). To study students’ affect while they engaged in an activity, one can still focus on different aspects, such as their emotions, their perception of the relevance or meaningfulness of a task, their boredom, their apathy, and so forth. We decided to focus on their perception of being challenged and activated by the activities. According to the Cambridge dictionary a challenge is: “(the situation of being faced with) something that needs great mental or physical effort in order to be done successfully and therefore tests a person’s ability” (http://dictionary.cambridge.org/ dictionary/english/challenge). Thus, in itself, a challenge is not necessarily activating, for example, because it is perceived as ‘too challenging’. However, a challenge can be related to activation through the use of the concept of flow, which is “a state in which people are so involved in an activity that nothing else seems to matter; the experience is so enjoyable that people will continue to do it even at great cost, for the sheer sake of doing it” (Csíkszentmihályi, 1990, p. 4). Flow is typically an activitybased concept. Further on in this paragraph, we will explain how this concept links challenge and activation to affect. However, we first present more background information on the concept of flow. Flow is a term connected to the Hungarian psychologist Mihaly Csikszentmihalyi, a researcher in the area of positive psychology, an area of psychology that studies causes for human’s happiness. In an overview of their studies, Csikszentmihalyi and Csikszentmihalyi (1988) describe how they observed artists, rock climbers, gamers and scientific researchers during their challenge, and how these people got fully absorbed in their activities and forgot about time, about basic needs such as eating and resting, or about simple responsibilities such as collecting the kids from kindergarten. However, once the activity came to an end and a product was finished (the paint was dry, so to speak), the observed people completely lost interest in the product. This implied that the process was considered more important than the end product. The state of this process was initially described as an autotelic experience, that is: as having a purpose in itself. Later, the autotelic experience was coined as flow, and nowadays it is also described as being in the zone. Csikszentmihalyi and his group at the University of Chicago studied many different groups, amongst which also adolescents. They observed that of all the places students hang out, the school is the one place they least wish to be were extended to other target groups (Csikszentmihalyi and Hunter 2003; Csikszentmihalyi and McCormack 1986). When they were in school, the classroom was the place they most strongly wished to avoid. They rather were in the cafeteria,

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the library, or the hallways. Interestingly, in secondary education, these researchers discovered that the people most likely to experience flow were teachers. According to the findings, teachers can experience flow if they have a sense of competency in their own work and a supportive work environment. Flow has also been studied in mathematics education (a.o. Armstrong 2008; Drakes 2012; Liljedahl 2006). However, the typical situation of most students in canonical mathematics classes is not to experience flow at all. Important aspects for flow to occur are: clear goals, during the activity one often gets feedback on the progress made, and one has a feeling of being in control. During the experience of flow, a person loses awareness of the self, of the larger environment, and loses awareness of time. Then, the activity is intrinsically rewarding irrespective of the outcome, or as Csikszentmihalyi and Csikszentmihalyi (1988, p. 33) write: “the mountaineer does not climb in order to reach the top of the mountain, but tries to reach the summit in order to climb”. According to Csikszentmihalyi and Csikszentmihalyi (1988), a precondition for experiencing flow is that a person should perceive that he/she is capable of doing it, that is: that one has sufficient skills for the activity, but that this activity is not perceived as easy. At this point, we see that skills bring challenge and flow together: a certain tension between skills and challenge brings dynamics into the activity, which can result in the actor experiencing flow. Flow forces people to stretch themselves in an activity, and improve on their abilities. However, if the skills are becoming better, and the activity does not become more challenging, the person will become bored. On the other hand, if the skills cannot meet the challenge, the person will become discouraged, alienated or, in the worst case, anxious. Thus, in relation to affect, flow is a technical term in the fields of intrinsic motivation and interest, describing “an optimal state of experience” (Csikszentmihalyi and Csikszentmihalyi 1988, p. 3). Flow is an activity-based concept (without activity, there cannot be an experience of flow), and it can only occur if there is a certain tension between challenge and skills. It remains important to note, that flow is an experience of a person, and that not all activities result in flow because of the above described tensions between skills and challenge. A task designer, thus, needs to consider this tension: if a task is too easy, the more gifted students will be bored, and if the task is too hard, the less talented students will not be able to start. Therefore, in our research we used a task with a low entry level to understanding the overall aims. In fact, being a group task, it was accessible to all students, irrespective of their initial skills. However, being accessible did not imply it was an easy task. As the task was open-ended and allowed for a variety of approaches, it invited the more gifted students to challenge themselves. This task characteristic is also known as low floor – high ceiling. Because of this type of task, we ignored the aspect of skills, and operationalized students’ affect in terms of students’ experience of challenge and flow.

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Methods In the Spring of 2017 we presented the task described on pages 262–263 to the students of the engineering department (Mechatronics, Electrical Engineering, Renewable Energy, Data Engineering, ICT, and others), as part of the 1st-year physics course. There were 346 students for whom the task was obligatory. The research design for studying students’ challenge and flow was a survey, whereby data were collected through a digital questionnaire (described below) within the university’s Virtual Learning System. Participation in the survey was voluntary and encouraged with prizes of NOK 500 (approx $60) for three randomly drawn participants. We removed irregular participation (e.g. participants who chose constantly a 3 as answer; 2nd-year students for whom the task wasn’t obligatory) and remained with n ¼ 239 students. The response rate of 69% can be considered high for a web-based survey (Bryman 2015). We developed the instrument, because a literature review did not yield any existing instrument that matched with the aims of our study. Reasons for discarding them were: too long questionnaires, or unsuitability to our Tracker Task. Therefore, we adapted items from instruments from earlier research and developed these in alignment with our needs. The items asked for (dis-)agreement to statements on a 5-point Likert scale. Ten items were designed to measure students’ perception of Challenge and Flow, see Table 1. It should be noted that the word mathematics does not appear in the instrument. While developing the items, we asked a few colleagues to review the items, and we organized a small pilot with a few 2nd-year students. The questionnaire contained six further questions about students’ collaboration and the ease to use the equipment or to find the mathematical formula. Those items were included to inform us about practical and technical issues, of which the results are irrelevant to the study presented here.

Table 1 The ten flow and challenge items in the questionnaire, with Cronbach’s Alpha ¼ 0.795 Statements q1 The “Modeling with Tracker Task” made me curious q2 (Inv.) This Tracker Task took too much of my time q6 Making a poster made me feel like a ‘real scientist’ q7 Time was flying when we worked in this task q8 (Inv.) This task is more suitable for Secondary Schools q9 This task helped me to better understand the theory q11 (Inv.) I was easily distracted when we worked on this task q13 During this task I started thinking about other movements (what if. . .?) q14 I would do this task even if it wasn’t obligatory q16 I would like to have more of such practical tasks

Cronbach‘s Alpha if item deleted 0.765 0.801 0.785 0.768 0.775 0.768 0.771 0.794 0.778 0.771

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In our research, we make a difference between challenge and flow as concepts (described in the previous paragraph), and the measurement scale of Challenge and Flow (with capital letters). The concepts of challenge and flow are subjective experiences of a person, and therefore these cannot be measured. However, we assume that they can be approximated by a score, which results from answering to the ten Challenge and Flow items from our questionnaire. A participant’s score on the Challenge and Flow items then is indicator of the extent to which he/she had positively experienced being challenged and/or activated. The score on the scale is calculated by adding the scores on the ten questions. As the score on one item ranges from 1 to 5, the score on the Challenge and Flow scale ranges from 10 to 50. To increase reliability, three items were inversely posed, and the scoring on these items was inverted, too. As measure of consistency between items (internal reliability), we calculated Cronbach ‘s Alpha for the ten items. If lower than 0.6, the consistency of a group of items is considered poor and unacceptable (Bryman 2015). It turned out that the ten items of the Challenge and Flow scale together had a good reliability, with Cronbach Alpha being 0.8. Additionally, we tested whether the consistency would improve if one of the ten items were deleted (it would mean that the item is inconsistent with the others). This analysis showed that nine items contributed positively to the scale and deleting them would lower the consistency. Only one item (“(Inv.) This Tracker Task took too much of my time”) did not show this, but deleting it would not significantly increase the consistency either, see Table 1.

Results As lecturers, we observed informally how enthusiastic students were everywhere on campus, throwing apples or golf balls, and even a cat was thrown (and fell on its feet). Students analyzed the flight of their skateboard, the turning of cars in the parking garages and the fall of a small parachute. Also, in the working groups where students come to practice examination tasks, we heard them discuss lively about the Tracker Task. We had offered office hours in case the students wanted clarification on the task, but not one student appeared. After 2 weeks we received more than 100 posters in our Virtual Learning System, of which we show two samples in Fig. 1 to give the reader an impression of students’ products. The format of the poster asked students to write an Introduction, give their Observation, give a Model, and write a Discussion. The poster on the left in Fig. 1 was made by students who threw a table tennis ball and mathematized its trajectory with a quadratic equation. On the right, students filmed the bouncing of a rubber ball and used MatLab to mathematize a sequence of parabolas, of which the height and width reduces with each bounce. In this chapter we don’t analyze the cognitive performance of the students in this assessment, such as their understanding of modeling, the depth of their analysis, the discussion of their measurements in relation to kinematical laws of gravitation, and so forth. Instead, we focus on students’ affect, which we operationalized in terms of

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Fig. 1 Two examples of products by the students

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Table 2 Mean scores on challenge and flow questions (n ¼ 239) q1 q2 q6 q7 q8 q9 q11 q13 q14 q16

Statements The “Modeling with Tracker Task” made me curious (Inv.) This Tracker Task took too much of my time Making a poster made me feel like a ‘real scientist’ Time was flying when we worked in this task (Inv.) This task is more suitable for Secondary Schools This task helped me to better understand the theory (Inv.) I was easily distracted when we worked on this task During this task I started thinking about other movements (what if. . .?) I would do this task even if it wasn’t obligatory I would like to have more of such practical tasks

Mean (sd.) 3.61 (0.75) 3.67 (0.88) 3.96 (1.03) 3.40 (0.92) 2.58 (0.95) 3.39 (0.88) 3.55 (0.91) 3.31 (1.12) 2.60 (1.13) 3.70 (1.02)

25

20

15

10

5

0 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50

Fig. 2 Frequencies of challenge and flow scores (middle score 30 in red)

challenge and flow and which we measured through the questionnaire. Table 2 shows the mean scores on each item (1 ¼ low, 3 ¼ middle, 5 ¼ high). The mean score on five of the ten items is higher than 3.5, being well on the positive affect side. This indicates that a majority of the students experienced challenge and flow to quite an extent. They largely agreed that the task made them curious (item 1), did not take too much of their time (item 2), or that they would like to have more of these tasks (item 16). The remaining five items obtained a score in the middle range (between 2.5 and 3.5). Not one item was answered below 2.5. When adding the students’ scores on the ten items, we obtain their total score on the scale Challenge and Flow. On this scale, the minimal score is 10 (not attained by any student), the middle score is 30 and the maximal score is 50 (not attained by any student). Figure 2 shows a histogram of the frequencies of the Challenge and Flow scores, with the bar for the middle score 30 in red.

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The histogram shows that the scores on the Challenge and Flow survey are skewed to the right, which means that students’ scores are, on average, on the high side of the scale. Out of 239 students, 54 students (22.6%) scored 28 points or lower, 42 students (17.6%) scored in the middle range of 29–31 points, and 143 students (59.8%) scored 32 points or higher. If we take 32 as a threshold score, it would mean that approximately three out of five students experienced a certain positive challenge and flow. Of course, the threshold is arbitrary; if we rather take 33 as a threshold score, then 129 students (54.0%) experienced challenge and flow.

Discussion and Conclusion Our research question was: to what extent does an open assessment task about video analysis of motion with smart phones and free tracker software challenge and activate the students? Based on the results from the survey, we find that a clear majority of the students (59%) experienced challenge and flow. They indicate that they forgot about time and wanted more of such activities. This result is anecdotally supported by our observations of buzzing students on campus, their discussions during workgroup sessions, and the high response rate to the survey. We searched the literature, but didn’t find earlier research in assessment of mathematics in which students expressed to want more of such assessment tasks. The data from the survey do not allow us to compare students’ appreciation of the Tracker Task to experiences of challenge and flow on other activities in the course (attending lectures, working on textbook problems). However, our 1st year engineering students aren’t in any way different from those described in Harris et al. (2015), avoiding additional mathematical tasks and seeking to minimize mathematical activities. In light of that, the high score on item q16 (“I would like to have more of such practical tasks”) can be interpreted as a comparison, whereby students express to favor the Tracker Task over other mathematics tasks. The results can be ascribed to a number of components. In the first place, we asked the student about an activity for which they could use their own smart phones. Thus, they had the laboratory equipment in their pockets. Being students from the engineering department, it could be expected that they liked using technological devices. These gave them ownership over the activity, and it made the activity different from prior experiences in mathematics education. Also, we were careful not to connect the activity to mathematics. In the questionnaire the word mathematics was not used once, and in the task the word mathematics was only used once when asking the students to create a mathematical formula. We did this, so the students would evaluate the activity in itself and not connect it to earlier, often negative experiences in mathematics classes. Harris et al. (2015) have clearly established that students in engineering, like the ones in our study, generally perceive mathematics not as their favorite. In the second place, the results can be explained in light of the task characteristics. Although the students were assessed on their product (the poster), the task was open,

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the students had ample time to submit the product, and the task was accessible to all students (low floor), yet enabling the better students to challenge themselves further (high ceiling). Such task characteristics may be better feasible within mathematical modeling education than canonical mathematics education. Also, the assessment being for groups may have added to students’ positive affect: unlike the large lectures that the students attended, the small groups offered them companions, informality and possibly even safety. We would like to highlight that we used an activity-based conceptualization for both mathematics and affect. Thus, we did not study affect in relation to mathematics holistically, but in relation to a certain mathematical activity. Central in the activity was the mathematization of the trajectory of a moving object, that is: the creation of a mathematical formula describing position of a moving object as a function of time, and additionally discussing to what extent that formula deviated from the actual measurements. We anticipated that students had not often been given such a task, in particular not in assessment. This newness enabled us to detach the task from mathematics education at large, which is dominated by explaining teachers and students practicing exercises (Nolan 2012). Also, we did not study affect holistically, but studied affect through an activitybased perspective. We studied whether the students were challenged and activated by a task, to the extent that they possibly got fully absorbed into the activity. In that case, the task was motivating in itself. For an activity-based conceptualization of challenge, we were able to build on Sullivan et al. (2011) and Sullivan and Mornane (2014). For an activity-based conceptualization of activation, we used the concept of flow, which is a state of happiness caused by an activity that is sufficiently challenging in relation to someone’s skills. In our research we contend that flow is an important aspect of affect, and recommend more research into students’ experiences of flow in mathematics classrooms. The concept of flow brings a new perspective on affect, not only being activitybased, but also carrying the possibility that flow can be experienced by students in mathematics classrooms, and even in assessment. How often do students experience flow in canonical mathematics classrooms, if at all? With the existence of math anxienty (Tobias 1978) and mathofobia (Hodges 1983), we contend that in many mathematics classrooms many students will hardly ever experience flow. However, our study demonstrates that mathematical activities can result in flow among a large group of students, even if these students are 1st year engineering students who, as reported by Harris et al. (2015), are not extremely good in mathematics. Of course, we do not know how the students’ responses would have been, if we had repeatedly included the word mathematics into the task or into the questionnaire. However, in general with modeling tasks, or in particular with a task to mathematize motion in the real-world, such tasks don’t look like the tasks from canonical mathematics education. Nevertheless, such tasks make students engage in mathematical activities. Thus, it remains a question whether affect research in mathematics education is tainted by the term mathematics, the looks of repetitive tasks in canonical mathematics education, and the lack of challenging tasks that intrinsically motivate students so they experience flow.

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In our study, we made 1st-year engineering students engage in mathematical activities, that made many of them experience challenge and flow. This was expressed by the high Challenge and Flow score, or expressed by the 3.7 score on item q16 (“I would like to have more of such practical tasks”). Therefore, we recommend mathematics education to include more open, easily accessible and inquiry-based tasks, in particular modeling tasks, and also we recommend more affect research into students’ experiences of challenge and flow, and into the lack thereof.

References Armstrong, A. C. (2008). The fragility of group flow: The experiences of two small groups in a middle school mathematics classroom. The Journal of Mathematical Behavior, 27(2), 101–115. Blomhøj, M., & Kjeldsen, T. H. (2006). Teaching mathematical modeling through project work. ZDM, 38(2), 163–177. Blum, W., & Leiß, D. (2005). “Filling up” – the problem of independence-preserving teacher interventions in lessons with demanding modelling tasks. In M. Bosch (Ed.), Proceedings of the 4th European Congress of Mathematics Education (pp. 1623–1633). Gerona: FUNDEMI IQS – Universitat Ramon Llull. Brown, M., Brown, P., & Bibby, T. (2008). “I would rather die”: Reasons given by 16-year-olds for not continuing their study of mathematics. Research in Mathematics Education, 10(1), 3–8. Bryman, A. (2015). Social research methods. Oxford: Oxford University Press. Csíkszentmihályi, M. (1990). Flow: The psychology of optimal experience. New York: Harper and Row. Csikszentmihalyi, M., & Csikszentmihalyi, I. S. (1988). Optimal experience: Psychological studies of flow in consciousness. Cambridge: Cambridge University Press. Csikszentmihalyi, M., & Hunter, J. (2003). Happiness in everyday life: The uses of experience sampling. Journal of Happiness Studies, 4(2), 185–199. Csikszentmihalyi, M., & McCormack, J. (1986). The influence of teachers. Phi Delta Kappan, 67 (6), 415–419. De Jong, T., Linn, M. C., & Zacharia, Z. C. (2013). Physical and virtual laboratories in science and engineering education. Science, 340(6130), 305–308. Domínguez, A., de la Garza, J., & Zavala, G. (2015). Models and modelling in an integrated physics and mathematics course. In Mathematical modelling in education research and practice (pp. 513–522). Cham: Springer. Drakes, C. I. (2012). Mathematical modelling: From novice to expert. Unpublished doctoral dissertation. Simon Fraser University, Burnaby. Freudenthal, H. (1973). Mathematics as an educational task. Dordrecht: Reid. Harris, D., Black, L., Hernandez-Martinez, P., Pepin, B., Williams, J., & the TransMaths Team. (2015). Mathematics and its value for engineering students: What are the implications for teaching? International Journal of Mathematical Education in Science and Technology, 46 (3), 321–336. Hodges, H. L. (1983). Learning styles: Rx for mathophobia. The Arithmetic Teacher, 30(7), 17–20. Kaiser, G., Schwarz, B., & Buchholtz, N. (2011). Authentic modelling problems in mathematics education. In Trends in teaching and learning of mathematical modelling (pp. 591–601). Dordrecht: Springer. Lerman, S. (2000). The social turn in mathematics education research. In J. Boaler (Ed.), Multiple perspectives on mathematics teaching and learning (pp. 19–44). Westport: Greenwood Publishing Group.

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Liljedahl, P. (2006). Learning elementary number theory through a chain of discovery: Preservice teachers’ encounters with pentominoes. In R. Zazkis & S. Campbell (Eds.), Number theory in mathematics education: Perspectives and prospects (pp. 141–172). Mahwah: Lawrence Erlbaum Associates. Minner, D. D., Levy, A. J., & Century, J. (2010). Inquiry-based science instruction – what is it and does it matter? Results from a research synthesis 1984 – 2002. Journal of Research in Science Teaching, 47(4), 474–496. Niss, M. (2010). Modeling a crucial aspect of students’ mathematical modeling. In R. Lesh, P. L. Galbraith, C. R. Haines, & A. Hurford (Eds.), Modeling students’ mathematical modeling competencies: ICTMA 13 (pp. 43–59). New York: Springer. Nolan, K. (2012). Dispositions in the field: Viewing mathematics teacher education through the lens of Bourdieu’s social field theory. Educational Studies in Mathematics, 80(1), 201–215. Paulos, J. A. (1988). Innumeracy: Mathematical Illiteracy and its consequences. New York: Hill & Wang. Pepin, B., & Roesken-Winter, B. (2015). From beliefs to dynamic affect systems in mathematics education. Zürich, Switzerland: Springer. ISBN 978-3-319-06808-4 Sullivan, P., & Mornane, A. (2014). Exploring teachers’ use of, and students’ reactions to, challenging mathematics tasks. Mathematics Education Research Journal, 26(2), 193. Sullivan, P., Cheeseman, J., Michels, D., Mornane, A., Clarke, D., Roche, A., et al. (2011). Challenging mathematics tasks: What they are and how to use them. In L. Bragg (Ed.), Maths is multi-dimensional (pp. 33–46). Melbourne: Mathematical Association of Victoria. Tobias, S. (1978). Overcoming math anxiety. New York: Norton.

Chapter 12: Flow and Modelling Minnie Liu and Peter Liljedahl

Abstract In this chapter we look at student engagement while doing a modelling task. We present the case of two grade 8 students (age 12–13) and look at their modelling behavior through the double lenses of flow and modelling. During their modelling process, these students experienced an imbalance between the challenges and skills presented to them. Results show that the rebalancing of the challenges and skills were facilitated by the teacher, the students, and the naturally evolving complexities of the task. Keywords Flow · Engagement · Modelling · Extra-mathematical knowledge · Tolerance · Perseverance

The grade 8 mathematics classroom was noisy and charged with energy, as grade 8 classrooms are apt to be. But this was a different kind of noise – a different kind of energy. Instead of the usual teenage banter heard among students of this age, the noise was a din of discussion, arguments, calculations, discussion about calculations, and punctuations of outbursts of emotions. “Argh! This is not working!”, a girl complained as she leaned over to chat with a friend working in another group. Across the classroom, a boy shouted with joy, “We got it!”, and high-fived his group mates. Immediately, three students ran from one end of the classroom to see what they had done and then dashing quickly back to their groups with a sliver of an idea. In the middle of all this the bell rang, but not much changed. The students kept working, occasionally sharing ideas and celebrations with other groups until they were satisfied that they had cracked the problem. Only then did they submit their work and shuffle off to their lockers and then to lunch.

Scenes like this are a rare occurrence in any classroom, and even more rare in a high school mathematics classroom. What could be motivating this behavior, this energy, this engagement? Maybe it was a test – a type of group problem solving test – a whole class problem solving test – worth a lot of marks – with very complex questions. What else could explain the intra-group work, the inter-group work, the M. Liu (*) · P. Liljedahl Simon Fraser University, Burnaby, BC, Canada e-mail: [email protected]; [email protected] © Springer Nature Switzerland AG 2019 S. A. Chamberlin, B. Sriraman (eds.), Affect in Mathematical Modeling, Advances in Mathematics Education, https://doi.org/10.1007/978-3-030-04432-9_17

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running around, the overt celebrations, and the willingness to stay well into their lunch break? But this was not a test. There were no marks involved. There wasn’t even a requirement to finish. It was just a class of students working in groups to solve a modelling task – a particular type of modelling task. But there was energy. And there was engagement. And there was commitment – to finish, to produce an answer with which the group was satisfied. Not the right answer, but a reasonable answer. In this chapter we will look more closely at a case such as the one described above and unpack it through the double lenses of engagement and modelling behavior. Along the way we will introduce a unique type of modelling task that seems effective in invoking the type of frenzy described above.

Engagement, the Optimal Experience, and Flow Within the field of mathematics education one of the few ways to discuss engagement through a theoretical lens is through Mihály Csíkszentmihályi’s (1975, 1990, 1996, 1998) notion of an optimal experience – a state in which people are so involved in an activity that nothing else seems to matter; the experience is so enjoyable that people will continue to do it even at great cost, for the sheer sake of doing it. (Csíkszentmihályi 1990, p. 4)

In the anecdote above, the students being described were having an optimal experience. They were so focused on and absorbed in the task they were working on that they lost track of time. Their focus was so great that they did not hear the bell – they were un-distractible. Csíkszentmihályi, in his pursuit to understand the optimal experience, studied this phenomenon across a wide and diverse set of contexts (1975, 1990, 1996, 1998). In particular, he looked at the phenomenon among musicians, artists, mathematicians, scientists, and athletes. Out of this research emerged a set of elements common to every such experience (Csíkszentmihályi 1990): 1. 2. 3. 4. 5. 6. 7. 8. 9.

There are clear goals every step of the way There is immediate feedback to one’s actions There is a balance between challenges and skills Action and awareness are merged Distractions are excluded from consciousness There is no worry of failure Self-consciousness disappears The sense of time becomes distorted The activity was autotelic – a reward unto itself

The last six elements on this list are characteristics of how the doer experiences the phenomenon of an optimal experience. In the anecdote above the students would claim that they had lost track of time, that they did not hear the bell, that they were

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Fig. 1 Graphical representation of the balance between challenge and skill (Liljedahl 2018, p. 507)

not worried about failure and that they were not doing the task because it was for marks. In contrast, the first three elements on this list can be seen as characteristics external to the doer, existing in the environment of the activity, and crucial to occasioning the optimal experience. The doer must be in an environment wherein there are clear goals, immediate feedback, and there is a balance between the challenge of the activity and the abilities of the doer. This balance between challenge and ability is central to Csíkszentmihályi’s (1975, 1990, 1996, 1998) early work on the optimal experience and can best be understood when looking at imbalances between challenge and ability. For example, if the challenge of the activity far exceeds a person’s ability they are likely to experience a feeling of anxiety or frustration. Conversely, if their ability far exceeds the challenge offered by the activity they are apt to become bored. When there is a balance in this system a state of, what Csíkszentmihályi refers to as, flow is created (see Fig. 1). Flow is, in brief, the term Csíkszentmihályi used to encapsulate the essence of optimal experience and the nine aforementioned elements into a single emotional-cognitive construct.

Flow in the Mathematics Classroom Thinking about flow as existing in that balance between skill and challenge, as represented in Fig. 1, obfuscates the fact that this is not a static relationship (Liljedahl 2018). Flow is not the range of fixed ability-challenge pairings wherein the difference between skill and challenge are within some acceptable range. Flow is, in fact, a dynamic process. As students engage in an activity their skills will, invariably, improve. In order for these students to stay in flow the challenge of the task must similarly increase (see Fig. 2). In a mathematics classroom, these timely increases of challenge often fall to the teacher. But this is not without obstacles. For example, if a student’s skill increases either too quickly or too covertly for the teacher to notice, Csíkszentmihályi’s theory of flow predicts that student may slip into a state of boredom (see Fig. 3). Likewise,

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Fig. 2 Graphical representation of the balance between challenge and skill as a dynamic process (Liljedahl 2018, p. 508)

Fig. 3 Too fast an increase in skill (Liljedahl 2018, p. 508)

Fig. 4 Too great an increase in challenge (Liljedahl 2018, p. 508)

when the teacher does increase the challenge and if that increase is too great flow predicts that the student may become frustrated (see Fig. 4). Csíkszentmihályi (1975, 1990, 1996, 1998) predicts that if either of these states occur that a student is apt to quit. However, Liljedahl (2018) found that, even in cases of extreme imbalance, students did not always quit. Looking more closely at such cases, Liljedahl (2018)

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Fig. 5 Modified representation of the balance between challenge and skill (Liljedahl 2018, p. 522)

Fig. 6 Reaction to too great an ability (Liljedahl 2018, p. 522)

showed that the boundaries between flow and boredom and flow and frustration actually contained within them two previously unknown intermediary states – tolerance for the mundane and perseverance (see Fig. 5). While the first of these states accounts for cases where students worked at repetitive tasks without getting bored and without quitting, the second accounts for situations where students worked on a task where the challenge far outpaced their abilities without getting frustrated and quitting. Liljedahl (2018) further found that, in some cases, these states acted as buffers between flow and quitting by delaying the transition to boredom or frustration long enough for the imbalance between ability and challenge to be rebalanced. In the case of tolerance, this rebalancing was the result of an increase in complexity (see Fig. 6) while in the case of perseverance, rebalancing could happen as a result of either a decrease in challenge or an increase in ability (see Fig. 7).

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Fig. 7 Reaction to too great a challenge (Liljedahl 2018, p. 522)

Modelling, Modelling Tasks, and the Modelling Cycle Csíkszentmihályi (1975, 1990, 1996, 1998) and Liljedahl (2018) found flow across a wide variety of activities, classroom contexts, and mathematical problems. Common to all of these activities, contexts, and problems was the characteristic of evolving complexity. That is, as students’ abilities improved there had to be a commensurate increase in the challenge of the task at hand. In this chapter we look at flow within the context of modelling tasks.

Modelling Tasks Modelling tasks are messy problem solving questions that are situated in reality. Barbosa (2006) describes modelling tasks as problematic activities that, although not purely mathematical, require modellers to investigate the situation “with reference to reality via mathematics” (p. 294), and relate what happens in reality to mathematics by applying their mathematical knowledge and skills to determine a possible solution to the problem. As such, modellers need to combine their knowledge of mathematics with their knowledge from outside of mathematics (such as their lived-experiences) to produce a realistic and meaningful solution. Modelling tasks can be classified based on a variety of classification schemes (Maaβ 2010): 1. Modelling activity, where tasks may focus on the entire or a part of the modelling process and can promote modellers’ understanding of the steps taken to solve a modelling task; 2. Data, where tasks require modellers to investigate and critically decide the information relevant to solve the task, and/or to acquire the missing information required to solve the task; 3. Nature of relationship to reality, where tasks range from fantasies to genuine context only;

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4. Situations, where tasks are classified based on their relationship between the modeller’s life and the context of the task. This may range from personal (closest to a modeller’s life), to educational/occupational, public, and scientific (furthest away from a modeller’s life); 5. Type of model used, including descriptive models, which describe the situation as close to reality as possible; and normative models, which only describe certain aspects of reality while ignoring other aspects; 6. Type of representation, where tasks are presented as text; pictures; text and pictures; printed materials; and situations; 7. Openness, where tasks are classified based on the final product produced by modellers; 8. Cognitive demand, which include tasks that focus on extra-mathematical modelling, inner-mathematical working, “Grundvorstellungen” (mental objects that help with the transitions between reality and the world of mathematics), dealing with texts containing mathematics, mathematical reasoning, and dealing with mathematical representations; 9. And mathematical content, where tasks are classified based on their mathematical demands and their audience.

Modelling Cycles The process to solve modelling tasks can be referred to as a modelling cycle. Modelling cycles describe general modelling behaviors and can be illustrated through the use of various diagrammatic representations (Blum and Leiβ 2005; Borromeo Ferri 2006; Kaiser 2005; Mason and Davis 1991; Pollak 1979). These diagrams “illustrate key stages in an iterative process that commences with a real world problem and ends with the report of a successful solution, or a decision to revisit the model to achieve a better outcome” (Galbraith 2012, p. 8). For the purpose of this study, we use Borromeo Ferri’s modelling cycle as a unit of analysis. Borromeo Ferri (2006, 2010) describes the modelling cycle from a cognitive perspective, and focuses on modellers’ individual thinking processes, where these processes are made explicit through modellers’ actions during their modelling processes (see Fig. 8). Borromeo Ferri’s (2006) modelling cycle begins with a scenario situated within reality. As modellers interpret the real situation, they build a “visual, verbal, auditive, or formal” (Borromeo Ferri 2010, p.104) mental representation of the situation (MRS). MRS highlights the mental process modellers go through to understand the task, which happens on an implicit level, and is dependent on the modeller’s thinking style and experiences (Borromeo Ferri 2010). In the formation of a MRS, modellers may make connections with and simplify the original situation and discuss with each other the approach of the problem. Therefore, a MRS is a representation of modellers’ thinking process, a demonstration of modellers’ understanding and interpretation of the situation, and an illustration of the direction in which modellers take to solve the problem. After modellers

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Fig. 8 Representation of Borromeo Ferri’s (2006, p.92) modelling cycle

create a MRS, they make further assumptions of the situation and draw on their lived-experiences to create a real model to represent the situation. Borromeo Ferri (2006) refers to these lived-experiences as extra-mathematical knowledge (EMK). EMK includes any knowledge or experiences that originate from outside of a modellers’ mathematical experiences. EMK may have little or no clear connections to mathematics, but plays an essential role in the modelling process, as it enables modellers to consider the problem situation from a real world perspective, and allows them to produce a reasonable real model and mathematical model to represent the situation (Blum and Borromeo Ferri 2009; Borromeo Ferri 2006). In the process of building a real model, modellers make decisions on their assumptions about and their approach to the problem situation. A real model may use some mathematics to represent the situation, but also contains the original context. It is strongly connected to the MRS and “contains essential features of the original situation, but is on the other hand already so schematized that (if at all possible) it allows for an approach with mathematical means” (Blum and Niss 1991, p.38). Next, modellers mathematize the real model into a mathematical one. The process of mathematization can be described as “the transition from real model to mathematical model . . . [where] extra-mathematical knowledge (depends on the task) is strongly demanded by the individuals and used to build a mathematical model” (Borromeo Ferri 2006, p.92). Modellers then use the mathematical model along with their mathematical skills to produce mathematical results, apply their EMK once again to interpret these results in terms of the context of the problem and then validate these results by comparing them to the original situation. If modellers decide the real solutions are acceptable, this concludes their modelling cycle. If not, modellers re-enter the modelling cycle and make adjustments and modifications to their work.

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Research Questions Whereas the modelling cycle can give insights into students’ modelling behavior, the theory of flow give insights into student engagement. The question is, can these theories help explain the frenzied and energetic display of student engagement described in the opening paragraph of this chapter? More specifically, can Borromeo Ferri’s (2006) modelling cycle and Liljedahl’s (2018) modified theory of flow be used in symphony to understand the phenomenon of student engagement while working on a modelling task?

Methodology In order to answer this question we conducted a study in which we looked at student engagement while working on modelling tasks. In what follows we describe the setting of this research as well as the methods used to capture and analyze the data.

Setting Data for the research presented here were collected in a Mathematics 8 (age 12–13) classroom in a high school in western Canada. Within this jurisdiction mathematics is a required program of study until grade 11. With few exceptions, Mathematics 8 and Mathematics 9 are considered part of the common stream in that all students, irrespective of ability, take the same course and sit in the same classrooms. In grade 10 students may choose to stay in the common stream or shunt themselves into the Workplace and Apprenticeship stream if they do not see themselves as continuing onto post-secondary education or are aiming to enter into a trades program. In grade 11 the common stream splits again into a Pre-Calculus stream and a Foundations stream. Although both of these streams are eligible for admission into academic post-secondary programs, only the pre-calculus stream is eligible for entrance into the sciences. The class in which the research was conducted was an exception to this common stream in that it belonged to an enriched and accelerated program for students who are motivated and eager to learn and who have been recommended by their grade 7 teacher. The program covers the regular Mathematics 8, 9, and 10 curricula in 2 years as opposed to 3. This allows the students to finish the pre-calculus stream by the end of grade 11 and frees them up to take additional elective mathematics and science courses in grade 12.

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DESIGNING A NEW SCHOOL Your city is getting a new 11000 m2 middle school. It is going to be built on a lot (200 m x 130 m) just outside of town. Besides the school, there will also be an all-weather soccer field (100m x 75m), two tennis courts (each 15m x 27.5m), and a 30 car parking lot on the grounds. The following requirements must be met: all fields, courts, buildings, and parking lots must be no closer than 12.5m to any of the property lines. any leftover property will be used as green space – grass, trees,shrubs. good use of green space is an important part of making the school grounds attractive. To help you with your design and layout you have been provided with a scaled map of the property (every square is 10m x 10m). Present your final design on a copy of this map. Label all structures and shade the green space.

Task The task that the students worked on comes from a genre of tasks called Numeracy Tasks. These tasks have been designed specifically to meet the numeracy goals of the local curriculum and are crafted around “an aggregate of skills, knowledge, beliefs, dispositions, habits of mind, communication abilities, and problem solving skills that people need in order to engage effectively in quantitative situations arising in life and work” (Steen 2001). The particular task that the students were asked to work on is called the Design a New School task (see Fig. 9). Although this can broadly be considered a planning task (Liljedahl 2010) it is, more specifically, a spatial planning task. Although not requiring the use of advanced mathematics, the Design a New School task contains characteristics from the aforementioned classification schemes. It can be described as an open and realistic modelling task that focuses on the modelling activity. It is also cognitively demanding in the sense that it contains an extra-mathematical modelling focus which requires modellers to acquire and apply extra-mathematical knowledge to generate a reasonable solution.

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Data and Analysis Data for the research presented here were collected in the first author’s grade 8 (n ¼ 28) class while students worked on the aforementioned task. Although it is not possible to know if the grade 8’s had seen similar tasks in their previous years, this was one of the first of such tasks this group had been given in their grade 8 school year. Students were randomly assigned to groups of 2–4 and worked on the task during a 75 min class. There were no instructions provided other than what can be seen in Fig. 9. While the students worked the teacher (first author) circulated naturally through the room and engaged in conversations with the students – sometimes prompted by her and sometimes prompted by the students. These conversations were audio recorded and transcribed. At the same time photographs of student work were taken and students’ finished work was collected. These, coupled with field notes summarizing the interactions as well as observed student activity, allowed us to build cases for each group of students. Each of these cases is a narrative of their modelling experience punctuated by significant moments of activity and emotive expression. These cases constitute the data. Given that natural and unscripted nature of the teacher's movement through the room, not all of the cases are equally well documented. Regardless, each of these cases were analyzed separately through the lenses of modelling and flow. More specifically, the cases were analyzed using Borromeo Ferri’s (2006) modelling cycle as well as through Liljedahl’s (2018) modified theory of flow. The results of these disparate analyses were then combined and compared on an event by event basis to see if there were relationships between student engagement and various aspects of their modelling activities intersected. In what follows we present one of the more complete and comprehensive of the aforementioned cases – the case of Amy and Angela. This is followed by the modelling cycle analysis, the flow analysis, and finally the joint modelling-flow analysis.

Results and Discussion Amy and Angela reacted to the modelling task by first asking questions about the parking lot. Amy believed that one of the key factors of the parking lot is the dimensions of a vehicle, and suggested to go outside to the staff parking lot to take some measurements. Amy:

How big is a car [talks to herself]? Can I go outside for a second [asks teacher]?

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Fig. 10 Representation of Amy and Angela’s work to identify the usable space (red)

Fig. 11 Representation of Amy and Angela’s work with the soccer field (green)

When she came back, Amy discussed these measurements with Angela, and suggested that they should increase these measurements to accommodate for large vehicles. Amy: Angela: Amy: Angela:

The car I measured was 2.5 metres by 1.5 metres, but it was a slightly smaller car so probably make it a bit bigger? ‘Cause there are bigger cars in the parking lot? Like a Chevy. What’s that? It’s a truck.

After this conversation, Amy and Angela decided to put the parking lot on hold and investigated the possible locations and orientation of some of the building structures on the grid. First, they re-read the instructions provided, and paid attention to the areas on the grid which they were allowed to put buildings and the actual length each square represents. They divided 12.5 (distance between the border and all buildings) by 10 (each square represents 10m) and got “one and one-fourth”, and outlined a rectangle one and a quarter squares inside the border of the grid to represent the space they could put the buildings (see Fig. 10 for details). Angela:

So, all fields, courts, buildings, and parking lots must be no closer than 12.5 metres to any of the property lines. So one and one-fourth.

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Fig. 12 Representation of Amy and Angela’s work to include the tennis courts (purple)

Amy and Angela then used the measurements given in the instruction to work on the soccer field. They divided the length and width of the soccer field and those of the tennis courts by 10, and drew a rectangle that is 10 by 7.5 squares to represent the soccer field (see Fig. 11). They then realized they could not fit the tennis courts beside the short edge of the soccer field, and rotated the soccer field 90 to allow the tennis courts (two rectangles that are 2.75 by 1.5 squares long) to fit beside the soccer field (see Fig. 12). Amy: Angela: Amy: Teacher: Amy:

Now it’s just the tennis fields. Aw. Man. We can flip it. Ah, like, rotate it. So rotate this sideways, instead of placing it lengthwise. So you can fit the tennis courts . . . . . . beside it.

After drawing four rectangles to represent the usable space, the soccer field, and two tennis courts on the grid, the girls went back to the instructions and read the information given on the parking lot. They drew a quick sketch of a few parking spots and tried to visualize what the parking lot might look like based on their drawing. However, they had difficulties visualizing the parking lot and the things they needed to consider other than the areas taken up by parked vehicles. A brief discussion with the teacher (first author) led Amy and Angela to realize that there was more to consider than just the area each parked vehicle takes up. Amy: Teacher: Amy: Teacher: Amy: Teacher: Angela: Teacher: Amy:

The car space. . . 4 by 2. Because, there are bigger cars. But that’s the size of a car? That’s the car I measured.1 Okay. So that’s the size of a car. But once I parked the car. . . You can’t get out. Uhuh. I need to get out. So. . . would I need. . . what does that mean? You need some extra space! You need some extra space! So, how much is that extra space? Like. . . 0.5 meters?

1 The vehicle Amy measured was 1.5m by 2.5m. She has already increased the dimension to accommodate for vehicles larger than the one she measured.

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Fig. 13 Representation of Amy and Angela’s work to include the parking lot (blue)

Angela: Amy: Angela: Amy: Teacher: Amy/Angela: Teacher: Amy/Angela:

A car can’t fit! Like, 0.5 meters between each car. Oh okay. You need to be able to go behind a car You need to go behind the cars and to. . . Drive Drive into the lot. Oh!

In this conversation, Amy pointed out that cars in parking lots do not park right next to each other. Rather, there is a gap between each car which allows drivers and passengers to enter and to exit their vehicles. Other than a gap between the vehicles, Amy and Angela also pointed out the parking lot needs a driveway for the vehicles to drive into and out of the parking spaces. After further discussion with each other, the girls created an outline of the parking lot. It is rectangular in shape, 60m long, and 5m wide. All 30 parking spaces are lined up along the long edge of the parking lot, and each parking space is 2m wide and 4m long. The driveway, which runs along the long edge of the parking lot, is 1m wide and 60m long. It is a single direction driveway, where drivers enter the parking lot through an entrance on one end of the parking lot and leave the parking lot using the exit on the opposite end. They did not specify how they arrived at a 1m wide driveway. They divided these measurements by 10, and drew a rectangle that is 6 squares by 0.5 squares on the grid (see Fig. 13). Afterwards, Amy and Angela moved on to the final requirement, the school building. Based on the instructions, the girls first decided that the length of the school building was 110m and the width was 100m to accommodate for an area of 11,000m2. Similar to what they previously did, Amy and Angela divided 110m by 10 and 100m by 10 and determined that they could represent the school building on the grid using a rectangle that is 11 squares long and 10 squares wide. Very soon, they were stuck. Amy and Angela could not fit a 11  10 rectangle on the grid. They then recognized that each square is 100m2 and 11,000m2 means 110 squares, and looked for 110 squares on the grid for the school building. They were stuck again. Angela complained that she couldn’t fit the school building on the grid because there was not enough space to do so. Amy described Angela’s frustration as a “mental breakdown”, and called the teacher over for help. During their discussion with the

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teacher, Amy had an “AHA!” moment (Liljedahl 2005) and realized that she could “stack” the school building because in reality, it is possible to have buildings taller than one floor. Teacher: Amy: Teacher: Angela: Teacher: Angela: Teacher: Angela: Teacher:

Angela: Teacher: Angela: Teacher: Angela: Teacher: Angela: Teacher: Amy: Teacher: Amy:

What’s wrong? [Angela] is having a mental breakdown. Yea. . . looks that way. Why is that? I can’t fit it in? Okay. Explain to me what you mean by you can’t fit it in. I have no more space left! You have no more space left to fit what? Um. . . the rest of the school. The rest of the school. So, tell me, tell me what you have so far. So on the left, you have your soccer field, okay, and next to your soccer field you have your two tennis courts I can’t fit it in. Oh you can’t fit it in? Is the school big enough? The school is big enough. Okay. . . No. The school is not big enough and you don’t have any more space. Oh my. . . Oh no. . . Oh no. . . so we need more space. But, if we squish it in, it would be ugly. So don’t squish it in. But there is no space! Oh there is always space. Stack them! What do you mean stack them? Two floors!

Amy’s “AHA!” moment (Liljedahl 2005) happened about 30 min into their modelling process. While they realized that they could have a school building that is taller than one floor, they have not quite grasped what the building might look like and how much space they wanted for each floor. As they recognized the possibility to build a multi-floored school building, they also joked about creating a “110 floors” building, to which the teacher took the opportunity to discuss with them building shapes and floor area. Angela:

Amy: Angela: Teacher: Amy: Angela:

Oh! Can we do those lab rooms thingy? . . . Oh windows! [and draws an “S” shape school] . . . Oh Oh Oh, how about this [points at Amy’s representation of the school building]? There. It’s a circle! [draws 2 rectangles, one inside another] Can we do 3 floors? Yea you can do 3 floors. [Our school] has 3 floors. The basement the main floor and our floor. [Our neighbor school] has four though. Oh, you know what, in New York, my school has 5 floors, I eat breakfast at school, on the fifth floor. [laughs]

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As Amy and Angela discussed the shape of the school building and the number of floors they wanted, they also explored the idea of having a two floor tennis court building, and the idea of in-cooperating the two tennis courts into the school building by putting them on the roof of the school building. Angela: Teacher: Angela:

Wait so tennis court can be two floors. Um, you can, but I don’t know if that’s cool, though. On the roof, the tennis courts!

After discussing the possibility to include the tennis courts as a part of the school building, Amy and Angela decided to not make changes to their designs and explored the idea of a school building with three equal-area floors that totals to 11,000m2, and decided that each floor would take up 3670m2, or 36.7 squares.2 Angela: Teacher: Angela: Amy: Angela: Amy:

We got this! What’s that? What’s that 70? Two floors? Yea. Three floors! Wait. . . . 1-1-0 divided by . . . 36.66667 so I round this to 7. 36.7?

Upon further discussion, Amy and Angela interpreted 36.7 as the need to draw a rectangle that has an area of 36.7 squares on the grid. When they discussed the possible length and width of a rectangle with an area of 36.7 squares, they quickly dismissed their solution, and further chatted with each other the school building they were in and came to the realization that the floors do not need to have equal floor areas. They eventually settled on a building with two floors where the main floor is larger than the second floor. They then drew two rectangles, one inside the other, to represent the school building, and assigned the remaining space on the grid as green space (see Fig. 14). Although they had a solution that satisfied all of the task requirements, Amy and Angela were not satisfied. They began to make modifications to their design to improve students’ school life (see Fig. 15). First they turned the all-weather soccer field into a soccer stadium, and then added a garden next to the parking lot for additional green space. The garden is 60m by 20m with lots of trees and hedges. There is a gate (approximately 3m wide) on one side of the garden, and a bench inside the garden. As a finishing touch, Amy and Angela added a path to the front entrance of the school, and two 20m wide front and back doors to the school. Unfortunately, the two doors were placed on the second floor of the school building rather than the first, and the path connects the edge of the school grounds to the second floor of the school building rather than the ground floor. This could simply be a mistake as they overlooked where the outline of the first floor of the school building was. After installing these additional features, Amy and Angela concluded their solution was satifactory and submitted their solution.

Three floors of 3670m2 per total up to 11,010m2, which is slightly larger than the required 11,000m2.

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Fig. 14 Representation of Amy and Angela’s work to include the school (orange)

Fig. 15 Amy and Angela’s submitted solution

Modelling Analysis Amy and Angela’s modelling behavior can be charted through Borromeo Ferri’s (2006) modelling cycle (see Fig. 16). This begins when they identified the parking lot as something ambiguous, as no information was provided other than that it needs to fit 30 vehicles (real situation (RS13). Drawing from her EMK, Amy identified the length and width of vehicles as important factors in the designs (MRS2) but were not certain what to make of these measurements within their design. These actions helped reduce some ambiguity, but were not sufficient to help them move forward.

3 RS1 means that students were at the real situation (RS) stage of the modelling cycle, step 1 of their overall modelling process. MRS stands for mental representation of the situation, RM stands for real model, MM stands for mathematical model, MS stands for mathematical solution, RSoln stands for real solution, and MC stands for modelling cycle.

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Fig. 16 A diagrammatic representation of Amy and Angela’s entire modelling process. The green arrows (3–8, MC1) represent their work on the usable space; the orange arrows (9–14, MC2) represent their work on the soccer field; the purple arrows (15–20, MC3) represent their work on the tennis courts; the blue arrows (1–2, 21–26, MC4) represent their work on the parking lot; and the red arrows (27–41, MC5–7) represent their work on the school building

Instead of spending more time on the parking lot, Amy and Angela put it on hold and worked on other aspects of the problem. As the girls re-read the instructions they focused on the distance between the buildings and the property lines (RS3). They interpreted this as a restriction to the “usable space” on the grid (MRS4), made plans to create an outline on the grid (RM5), drew from the instructions the information they needed, and applied their mathematical skills to convert the distance away from the property line into number of squares on the grid (MM6, MS7). They interpreted their solution as an outline on the grid and used it to represent the space they could use (RSoln8). This is Amy and Angela’s first complete modelling cycle (MC1, steps 3–8). After the outline, the girls moved on to the soccer field (MC2, steps 9–14) and the two tennis courts (MC3, steps 15–20). They took a similar approach to the usable space and outlined these structures on the grid. One thing that is worth mentioning here is Amy and Angela’s rotation of the soccer field after they generated a real solution for the tennis courts, as they recognized the overall relationships between the locations and placements of the buildings, and that the rotation allows them to have a better use of space. Moving on to the parking lot (MC4, steps 21–26), the girls re-read the instructions (RS 21), and focused on creating a 30 car parking lot (MRS22). They drew some sketches of parking spaces based on the measurements they took, but

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experienced difficulties in visualizing the relationship between the vehicles and the parking spaces and the relationship between the parking spaces and the parking lot (RM23). Although they have both visited various parking lots as passengers, neither had reached the legal driving age at the time of the study, and had never experienced parking lots from a driver’s perspective. Their lack of EMK became a hindrance to their modelling process. A discussion with the teacher led them to deepen their understanding of parking lots, to update their real model, and eventually to generate their mathematical model (MM24), mathematical solution (MS25), and real solution (RSoln26). Unfortunately, despite all their hard work, the students failed to create a reasonable parking lot design: there was no indication of an entrance or an exit to the parking lot in this design; they did not include any driveways that connect the edge of the school property line with the parking lot; and their driveway inside the parking lot is too narrow for most if not all vehicles. It seems that their discussion with the teacher and with each other was not sufficient to expand their EMK for the purpose to generate a reasonable parking lot design. Finally, the girls proceeded to work on the school building (MC5, steps 27–32). They re-read the instructions (RS27) and aimed to determine the amount of floor space the school building needed on the grid (MRS28). They proceeded to make decisions about the shape of the school building (RM29), built a mathematical model by looking for the factors of 11,000, converted these into number of squares (MM30, MS31), and interpreted 11 and 10 as the length and width of the school in terms of squares (RSoln32). This is when they got stuck, as they could not find the space they needed on the grid. Amy’s “AHA!” moment (Liljedahl 2005) during their discussion with the teacher helped them to recognize the possibility to extend the vertical height of the school building in order to satisfy the floor area and to decrease the construction area required (MC6, steps 33–37). This realization (MRS33) allowed the girls to draw from their past experiences to think about the various shapes and possible height of the school building, and to possibly make changes to the designs of the tennis courts. As they moved forward, they toyed with the idea of a three-floored school building (RM34), built a mathematical model (35) to determine the area occupied by the building on the grid (MS36), but became dissatisfied with their solution (RSoln37). They abandoned their solution as they realized that the floor areas of the school building did not need to be equal, and they re-entered the modelling cycle once more (MC7, steps 38–41), made modifications to their real model (38), and eventually settled on a two floor building where the main floor is larger than the second floor (MM39, MS40, RSoln41). Finally, Amy and Angela read over the instructions again to verify that they had satisfied all the requirements. However, they felt they could further improve their work and therefore made modifications to their designs to make the school grounds aesthetically pleasing. To make their school grounds more appealing, they created a garden next to the parking lot, added a path to the front entrance of the school, and added a front and a back door to their school building.

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Flow Analysis Amy and Angela’s progress through the Design the New School task can also be analyzed through the lens of flow. More specifically, their progress on the task can be charted on Liljedahl’s (2018) modified flow diagram (see Fig. 17). Amy and Angela began the task as presented in Fig. 9 (1). Their first choice was to work on designing the parking lot (2). Although this proved to be too challenging for them at the time they did make some immediate progress around how big a single car could be (3). From here they decided to shift to what they thought was the easiest of the aspects to work on – determining the usable space around the perimeter of the lot (4). This was easily achieved so they used the same strategy to determine the size and placement of the soccer field (5). This shift from the boundary to the soccer field did not represent an increase in challenge, but allowed for the sequential development of the skill of scaling dimensions in the problem into the diagram (6). Amy and Angela then moved onto trying to figure out how to place the tennis courts (7). This proved to be a little bit more challenging as they needed to reposition the soccer field and scale and place the tennis courts (8). After this they shifted back to figuring out the needed dimensions and placement of the parking lot (9). As before, this was a significant increase in challenge for the two girls requiring them to, for the first time in the task, make estimates based on Amy’s earlier measurements (10). The only aspect left for them to consider was the dimensions and placement of the actual school building. At first, this seemed like it was going to only be a matter of doing some scaling (11), but it actually turned out to be much more challenging (12) and was not resolved until they realized that they could make a two story structure (13). At this point they needed only to figure out how to divide up the area of the two floors (14). Once they had satisfied all of the requirements of the task they then challenged themselves to make it more aesthetically pleasing (15) which resulted in the recalculation of the area of the two floors of the school and adding some realistic elements (16). The overall time that Amy and Angela spent on the task was 75 min, during which they were highly engaged the whole time. When looking at the flow analysis Fig. 17 Amy and Angela’s flow analysis

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of their activity it is easy to see why. Although they got off to a rough start by deciding to first solve the parking lot issue, their perseverance provided a buffer until they autonomously opted for something more manageable. From there, they progressed in a staircase fashion wisely selecting, in turn, progressively more challenging aspects of the task to work on as their abilities increased (4–12). At that point, they got stuck. But, again, their perseverance kept them working until they had their break through. From there it was routine work until they placed the school (12–14). At that they again exercised their autonomy and chose to increase the challenge by improving the aesthetics of their solution.

Modelling-Flow Analysis Putting the two aforementioned analyses together we begin to see a synchronicity between the nature of the tasks itself, Amy and Angela’s modelling behavior, and their engagement. Most obvious in this is the way in which the task, with its naturally evolving complexity, increases the challenge of the task at hand as the girls move through the problem. This evolving complexity is due, in part, to the degrees of freedom the students have in solving this problem. More significant, in this regard, however, was the demand for extra-mathematical knowledge that was just beyond the students’ grasp. Knowing how wide a car was, although easy to measure, still left them with the task of figuring out the size of a parking spot and how wide the access lane should be. Although more accessible, the extra-mathematical knowledge also came into play when Amy and Angela struggled to fit the school building into the space that was left. This synchronicity between the modelling task and flow also worked the other way – in keeping the modelling process going. The fact that the students were in flow motivated the students to keep going around and around the modelling cycle over and over again. The wrong turn at the beginning of this cycle was buffered both by Amy and Angela’s perseverance as well as their autonomous actions to pull the challenge into balance with their skill level. Without either of these it is easy to imagine the modelling cycle grinding to a halt before it really got going.

Conclusion It is clear from the aforementioned analysis that Borromeo Ferri’s (2006) modelling cycle and Liljedahl’s (2018) modified theory of flow worked in perfect synchronicity in analyzing the experiences of Amy and Angela as they worked through the Design a New School task. The charting of Amy and Angela’s dynamic state of flow demonstrated the interplay between the evolving nature of the modelling task and their emerging abilities. Sometimes this interplay resulted in an imbalance between

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challenge and skill. But, in each of these cases, the girls were able to remain in the state of flow through a rebalancing of skill and challenge. This rebalancing was sometimes facilitated by Amy and Angela, themselves. Liljedahl (2018) found that changes in challenge and ability were often selffacilitated through students’ autonomous actions to seek challenge and help from their peers and teacher. As we see in the case of Amy and Angela, the girls first attempted to design the parking lot. But, their abilities were not up for this challenge. They lacked the required extra-mathematical knowledge (EMK) to proceed. As such, they moved away from the challenge and worked on things that were less demanding to stay engaged with the problem. As their skills improved, they came back and took on the challenge they once shied away from. During their designs of the school building, the challenges they faced also exceed their ability. Conversely, when they had satisfied the conditions of the task they increased the challenge by now considering the effect of their design on student life. However, changes in the balance between challenge and ability may also be facilitated by the evolving nature of the task. The Design a New School task consists of elements with various complexities – while some are not very cognitively demanding (such as the usable area), others require students to acquire and combine their EMK along with their mathematical skills to produce a reasonable solution (such as the parking lot and the school building). As we see in the case of Amy and Angela, the girls successfully solved the task by moving forward from the least complicated element of the task to the most complicated. As such, the interplay between extra-mathematical knowledge and the task contribute to the maintenance of the students’ engagement.

References Barbosa, J. C. (2006). Mathematical modelling in classroom: A critical and discursive perspective. ZDM, 38(3), 293–301. Blum, W., & Borromeo Ferri, R. (2009). Mathematical modelling: Can it be taught and learnt? Journal of Mathematical Modelling and Application, 1(1), 45–58. Blum, W., & Leiß, D. (2005). How do students and teachers deal with modelling problems? The example “Filling up”. In Mathematical modelling (ICTMA 12): Education, engineering and economics. Chichester: Horwood Publishing Limited. Blum, W., & Niss, M. (1991). Applied mathematical problem solving, modelling, applications, and links to other subjects—State, trends and issues in mathematics instruction. Educational Studies in Mathematics, 22(1), 37–68. Borromeo Ferri, R. (2006). Theoretical and empirical differentiations of phases in the modelling process. ZDM, 38(2), 86–95. Borromeo Ferri, R. (2010). On the influence of mathematical thinking styles on learners’ modeling behavior. JMD, 31, 99–118. Csíkszentmihályi, M. (1975). Beyond boredom and anxiety (1st ed.). San Francisco: Jossey-Bass Publishers. Csíkszentmihályi, M. (1990). Flow: The psychology of optimal experience. New York: Harper and Row.

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Csíkszentmihályi, M. (1996). Creativity: Flow and the psychology of discovery and invention. New York: Harper Perennial. Csíkszentmihályi, M. (1998). Finding flow: The psychology of engagement with everyday life. New York: Basic Books. Galbraith, P. (2012). Models of modelling: Genres, purposes or perspectives. Journal of Mathematical Modelling and Application, 1(5), 3–16. Kaiser, G. (2005). Mathematical modelling in school–examples and experiences. Mathematikunterricht im Spannungsfeld von Evolution und Evaluation. Festband für Werner Blum, 99–108. Liljedahl, P. (2005). Mathematical discovery and affect: The effect of AHA! experiences on undergraduate mathematics students. International Journal of Mathematical Education in Science and Technology, 36(2–3), 219–236. Liljedahl, P. (2010, September 30). Numeracy tasks. Retrieved October 2017, from http://www. peterliljedahl.com/teachers/numeracy-t. Liljedahl, P. (2018). On the edges of flow: Student problem solving behavior. In S. Carreira, N. Amado, & K. Jones (Eds.), Broadening the scope of research on mathematical problem solving: A focus on technology, creativity and affect (pp. 505–524). New York: Springer. Maaβ, K. (2010). Classification scheme for modelling tasks. JMD, 31, 285–311. Mason, J., & Davis, J. (1991). Modelling with mathematics in primary and secondary schools. Geelong: Deakin University. Pollak, H. O. (1979). The interaction between mathematics and other school subjects. In New trends in mathematics teaching (pp. 232–248). Paris: International Commission on Mathematical Instruction (ICMI). Steen, L. A. (Ed.). (2001). Mathematics and democracy: The case for quantitative literacy. Washington, DC: The National Council on Education and the Disciplines.

Chapter 13: A Coda on Affect Bharath Sriraman

Abstract A coda to the different chapters on affect in mathematical modeling is offered in relation to the psychological literature on affect. Keywords Coda · Affect · Mathematical modeling · Psychological theories

Mathematical modeling is by its very nature inexplicably linked to what mathematicians find interesting and worthy of investigation, whether it be natural phenomena, or phenomena occurring in the social and digital worlds. The mathematics invented by Newton to solve problems in mechanics and optics is arguably an exemplary case of viewing differential Calculus as the language of modeling the laws of the universe. Today modeling is essential to understand problems in biology, epidemiology, information sciences, business and phenomena occurring in social media. Unlike the canned “word” problems that occur in differential equations textbooks that require recipe driven approaches that involve the application of canonical functions (exponential, logistic etc.), problems that warrant mathematical modeling in the real world lie at the heart of human interest. Human interest in turn is intertwined with the notion of affect. It has become common to encounter books in the popular mathematics or science sections of book stores with titles such as Loving and Hating Mathematics (Hersh and John-Steiner 2010) or Love and Math: The heart of hidden reality (Frenkel 2013), which suggest that there is more to the discipline of mathematics than its assumed objectivity and neutrality in the sciences. Seemingly there are dispositions that one takes to the subject, and “comments about liking (or hating) mathematics are as common as reports of instructional activities” (Mcleod 1992, p. 575). There are

B. Sriraman (*) Department of Mathematical Sciences, University of Montana, Missoula, MT, USA e-mail: [email protected] © Springer Nature Switzerland AG 2019 S. A. Chamberlin, B. Sriraman (eds.), Affect in Mathematical Modeling, Advances in Mathematics Education, https://doi.org/10.1007/978-3-030-04432-9_18

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also other ways in which the subject has been anthropomorphized, in references to its beauty or coldness. Bertrand Russell once remarked: Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, without the gorgeous trappings of painting or music.

Given these allusions to the discipline of mathematics per se, and consequently to its teaching, learning and doing, it should come as no surprise that affect is an important dimension that needs to be studied and understood if the goal of mathematics education is to stimulate learners’ interest in the subject. The chapters in this book on Affect in Mathematical Modeling invoke a cornucopia of terms such as (in alphabetical order) anxiety, beliefs, challenges, emotions, engagement, feelings, flow, frustration, happiness, etc., and numerous others that can be subsumed under the umbrella of affect. With a little effort one could conceivably use up the entire alphabet to coin terms that can fit within the study of affect. With this skepticism in mind, the goal of this coda is twofold: 1. To fix several arbitrary starting points to research on affect in mathematics education. 2. To re-examine terms borrowed from psychology that form the basis of much of the research on affect in mathematics education. The opening chapter by Chamberlin provides us with a detailed introduction on the origins of affect research in psychology in the study of cognition and its subsequent development in mathematics education in relation to understanding problem solving and modeling. The relationship between the three constructs of affect, cognition and mathematical modeling is also developed. Given the lengthy history of affect in Chamberlin’s chapter, can a circumspect reader point to a different starting point for mathematics education? One could point to Alba Thompson’s seminal dissertation on the beliefs and practices of middle school teachers completed in 1982. Thirty six years later, have we made any progress in relation to that the starting point? The findings of the three case studies that constituted this dissertation were reported in Educational Studies in Mathematics (Thompson 1984), which delineated the correspondence between beliefs on the nature of mathematics and mathematics teaching and its correspondence to practice in the classroom. Thompson’s (1984) paper has been cited more than 2000 times to date, and the findings of her study have been well validated for the population of both pre-service elementary school mathematics teachers as well as in-service teachers. In a nutshell beliefs held by an individual are a stable construct and difficult to “perturb” or change and shape their instructional behavior in the classroom. The reason for using the term “beliefs” in a coda that purports to address affect is because beliefs as Pajares (1992) noted travels under the guise of “attitudes, values, judgements, perceptions, conceptions, . . .”, in other words the entire gamut of the affective domain. Leder and Grootenboer (2005) proposed a conceptual model to help us understand the affective domain of research in mathematics education. In this model one end of the spectrum contains increased cognition whereas the other end contains increased affect. A reciprocal relationship is suggested between cognition

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and affect, i.e. higher cognition is linked with lower affect, and higher affect is linked with lower cognition. Beliefs, attitudes, emotions (and values) are the variables that determine where one lands in the spectrum. These three terms need further unpacking. What if any is the difference between beliefs, attitudes and emotions? The investigation of affect in mathematics education has no definite starting point per se. One could look for its origins in psychology, although earlier in the coda the dissertation by Thompson was pointed to as a reasonable starting point. Some might argue that among the earliest educators to note the importance of affective as well as cognitive aspects of learning was John Dewey. In The Psychological and The Logical in Teaching Geometry (1903) he pointed out that a geometry teacher cannot just be concerned with “development into the most orderly intellectual system possible. He or she must take into account that learning develops out of the present habits and experiences of emotion, thought, and action of the students” (p. 399).“What a given statement means to a pupil depends absolutely on the interaction set up between the topic presented and the habits which the pupil brings with him to it” (p. 391). However Dewey did not elaborate on the emotions or habits that might indicate whether or not the student was prepared to study a geometry course that emphasized proof. In mathematics education the term beliefs is often conflated with attitudes and these two terms are conflated with emotions and/or feelings. Beliefs are typically defined as the mental acceptance of, or conviction in the truth or actuality of something. According to Mcleod (1991), beliefs are largely cognitive in nature, and are developed over a relatively long period of time. D’Andrade (1981) suggested that beliefs develop gradually through a process much like “guided discovery” where children respond to the situations in which they find themselves by developing beliefs that are consistent with their experience. The development of beliefs about mathematics is also heavily influenced by the cultural setting of the classroom (Schoenfeld 1989). These factors make beliefs very hard to change. Hence educational beliefs of students play a part in their knowledge base and subsequent behaviors (teaching and learning of mathematics). According to Hiebert (1999) “change does not happen automatically; it requires learning.” (p. 15). Namelylearning alternatives to traditional modes of teaching and teacher-student norms and expectations. Without changing or causing conflict in students’ beliefs, status quo becomes the outcome. Hence cognitive conflict is needed to cause changes in beliefs. Some neuroscientists posit that neural pathways are more or less hardwired over time and reconfiguring them is difficult. Mandler (1984) claimed that most affective factors arise out of emotional responses to the interruption of plans or planned behavior leading to “blockage”. In Mandler’s theory, these interruptions result in a physiological response and the individual attempts to evaluate the meaning of this blockage. Finally evaluation of this blockage could be either positive or negative, reinforcing or contradicting the original belief. Research on beliefs was considered important because beliefs (about mathematics, about self, about mathematics teaching and learning, among others) directly influence the development of attitudes and emotional responses to mathematics.

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Here is also conflation of the terms “beliefs” and “attitudes.” Students often have a certain state of mind (or disposition) regarding mathematics. For example a student may either like geometric proofs or dislike it. Numerous studies have indicated that changing attitudes is difficult even under good conditions since attitudes have to first be brought to the surface of one’s cognitive thoughts and then discussed in detail (Bassarear 1986). Attitudes often develop out of emotional responses and as a teacher one can influence students’ attitudes regarding mathematics to a certain extent, by providing positive experiences. In comparison to research on beliefs and attitudes, research on emotional reactions to mathematics has only recently come to the forefront of affective research. Buxton’s (1981) research dealt with adults who reported that their emotional reaction to mathematics was panic. This panic led to a high degree of physical arousal, which was so difficult to control that it disrupted their ability to concentrate on a task. Anxious people process information in a highly selective way: they attend to the most threatening elements of the information presented. This selective attention may cause math-anxious students to focus on irrelevant parts of a math problem. This drains cognitive resources and lessens performance (Ashcraft and Faust 1994). This type of anxiety is also felt by mathematicians working on long-standing problems, but many are able to persevere in spite of setbacks because of their meaningful engagement with problems. Some commentaries for the three parts in the book discuss the interaction of affect and cognition in the context of solving problems (modeling) and examine the notion of “belief systems” in which affect and cognition are not treated as separate entities (e.g., Schoenfeld). Goldin views student engagement with mathematics as a basic human need or creating a sense of belonging in the classroom setting, which is further elaborated in the context of “happiness” as a basic human need in contrast to anxiety (Wiezel et al.). Ideally solving mathematical problems should result in a sense of pleasure or satisfaction. Jindřich Nečas who used to run advanced seminars on partial differential equations when I was in graduate school often remarked on the need to “feel” natural phenomena such as the turbulence of water and current flows near rocks and dams before beginning the process of modeling such phenomena. As one of the foremost applied mathematicians of his generation he devoted a major portion of his life to the general blow-up solution of the Navier-Stokes equation in R3 which is one of the seven (unsolved) millennium problems. In spite of setbacks he derived pleasure and satisfaction at being able to immerse himself into a problem that went back to the discovery of Newton’s second law. We don’t expect students to engage in problems the way some professional mathematicians do, but modeling offers both context and meaning that can fuel affect for persistence. This is the hope for investigating the interaction of affect with mathematical modeling in this book.

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Index

A Ability, 24, 38, 59, 71, 75, 81, 109, 115, 117, 121, 122, 153, 161, 213, 235, 236, 238, 263, 275, 277, 281, 294, 300 Abstract, 136, 137, 221, 227, 230–232 Abstraction, 132, 135, 136, 227, 230 Achievement, 17, 19, 45, 101, 109, 114, 118, 124, 159, 160, 162, 167, 172, 173, 215 Activity, 3, 16, 53, 68, 113, 132, 162, 178, 212, 236, 259, 274, 297 Aesthetic, 119, 168, 293 Affect, 3, 15, 68, 112, 131, 165, 198, 211, 236, 258, 297 Affective, 7, 8, 16–19, 24, 67, 68, 83, 90, 91, 95, 99–101, 103, 107, 112–115, 117, 119–122, 124, 126, 132, 135, 149, 154, 160–162, 164–167, 170–173, 178, 203, 205, 208–211, 216, 222, 225, 229, 231, 232, 237, 298–300 Agency, 92–95 Algebra, 93, 164, 167, 169 Analogies, 101, 104, 108, 109, 121, 133, 136–142, 148, 150, 153, 166, 170 Analyses, 172, 173, 178, 181, 182, 198, 204–208, 219–221, 223–225, 227, 238, 246, 248, 249, 261, 262, 266, 269, 279, 283, 289–294 ANOVA, 246–248 Anxiety, 7, 16–19, 94, 100, 114, 126, 165, 238, 249, 275, 298, 300 Apathy, 7, 8, 259, 263 Approaches, 5, 10–12, 18, 33, 34, 36–39, 41, 59, 60, 63, 75, 76, 79, 80, 114, 120, 123,

126, 132–137, 151, 160–162, 167, 169, 172–174, 178, 208, 213, 220, 221, 223, 225, 232, 236, 241, 243, 250, 259, 264, 279, 280, 290, 297 Aspirations, 18, 118, 159, 166 Assess, 17, 18, 21, 39, 60, 61, 163, 173, 222, 230, 250, 262, 269 Assessments, 92, 93, 95, 99, 106, 161, 166, 167, 189, 190, 204, 208, 209, 215, 222, 223, 227, 258–271 Attitudes, 7, 8, 16–20, 23, 24, 40, 46, 68, 114, 117, 120, 126, 133, 203–206, 209, 211–213, 216, 219–223, 225, 230–232, 235–252, 258, 298, 299 Autonomy, 115, 125, 172, 293 Autotelic, 263, 274

B Behaviors, 16, 38–40, 43, 53–55, 59, 60, 68, 76, 91, 93, 100, 113, 114, 118, 120–122, 125, 126, 164, 165, 167, 168, 170, 172, 173, 178, 237, 273, 274, 279, 281, 289, 293, 298, 299 Beliefs, 7, 8, 16, 17, 19–21, 23, 24, 68, 91, 93, 94, 99, 101, 104, 106–108, 114, 115, 117, 120, 126, 131–154, 160, 168, 209, 223–226, 229, 236, 258, 259, 282, 298–300 Binet, A., 16, 17, 24 Boredom, 114, 238, 258, 259, 263, 275, 277 Boring, 206, 243, 248, 250 Bruner, J., 220

© Springer Nature Switzerland AG 2019 S. A. Chamberlin, B. Sriraman (eds.), Affect in Mathematical Modeling, Advances in Mathematics Education, https://doi.org/10.1007/978-3-030-04432-9

327

328 C Calculus, 91, 160, 297 Challenges, 6, 8, 9, 11, 56, 60, 61, 63, 68, 69, 92, 93, 95, 108, 118, 119, 121, 132, 136, 168, 172, 178, 204–210, 213, 215, 216, 238, 239, 244, 248, 250–252, 258–271, 274–278, 292–294, 298 Challenging, 8, 24, 25, 38, 45, 53, 68, 102, 109, 162, 164, 206, 211, 213, 237, 247, 249, 250, 259, 263, 264, 270, 292, 293 Classrooms, 4–7, 9–11, 21, 43, 46, 58, 62, 63, 68, 70–79, 81–83, 90, 92, 94–96, 100, 105, 106, 109, 112, 113, 116, 120, 121, 124, 125, 131, 132, 134, 139–148, 151, 152, 163–167, 169, 172–174, 198, 221, 222, 230–232, 236, 237, 242, 250, 258, 263, 270, 275, 277, 278, 281, 298–300 Cognition, 3–12, 15–17, 19, 23–25, 32, 54, 58, 93, 108, 112–117, 121, 125–127, 132, 134, 153, 165, 168, 173, 298–300 Cognitive, 7–9, 12, 16, 29–32, 34, 37, 40–42, 44, 54–57, 60, 61, 67, 90, 91, 94, 95, 99–101, 103, 107, 112–115, 117, 119, 121, 122, 124, 126, 132–135, 137, 139, 149, 153, 154, 164–168, 170, 171, 173, 178, 204, 208, 209, 232, 238, 266, 275, 279, 299, 300 Collaboration, 262, 265 Commognitive, 12, 102, 109, 178, 181 Communication, 8–10, 25, 109, 119, 127, 132, 179, 262, 282 Complex, 5, 11, 33, 35, 37, 39, 40, 42, 45, 53–59, 61–63, 68–70, 83, 93, 108, 113–115, 120, 121, 123, 124, 132, 136, 138, 149, 165, 170, 178, 204, 207, 211, 212, 221, 226, 230–232, 236, 258, 273 Conation, 92, 93, 100, 108, 112–118, 124–126 Conative, 92, 100, 101, 105, 106, 111–127 Concept, 3, 4, 6–12, 16, 18, 21, 22, 30–33, 39, 46, 47, 56, 75, 90, 94, 100, 101, 118, 122, 133–136, 139, 152, 154, 163, 169, 173, 206, 221, 223, 236, 241, 251, 259, 260, 263, 264, 266, 270 Conception, 18–21, 39, 152, 164, 166, 209, 224, 225, 230, 232, 298 Conceptualizations, 3, 11, 20, 30, 31, 46, 48, 54, 68, 165, 262–264, 270 Conclusion, 15, 73, 76, 133, 153, 213, 229, 248 Concrete, 31, 138, 139, 221, 225, 230–232 Confidence, 75, 237, 258 Conjectures, 94, 115, 143, 154, 171 Consistency, 241, 266

Index Construct, 8–10, 15–26, 68, 72, 73, 78, 79, 102, 112, 114, 115, 124, 135, 139, 154, 179, 211–213, 215, 219, 220, 223, 227, 237, 238, 258, 275, 298 Contextualize, 154, 212 Contextualization, 152, 227, 237 Create, 15–17, 21, 22, 25, 33, 39, 61, 63, 71, 72, 76, 79, 83, 125, 154, 169, 216, 236, 238, 250, 262, 269, 280, 286, 290, 291 Creating, 17, 21–25, 57, 72, 153, 197, 215, 251, 262, 287, 290, 300 Creativity, 94, 118, 140, 154, 162, 249, 259 Cronbach, 244, 246, 247, 265, 266 Csikszentmihalyi, M., 69, 163, 263, 264 Curiosity, 117, 119, 125, 165, 244, 252, 259

D Descriptive statistics, 246 Design, 22, 32, 47, 48, 53, 56–58, 60–63, 96, 132, 168, 180, 206, 207, 213, 214, 216, 242, 259, 265, 288, 289, 291, 294 Development, 9, 11, 30, 41, 44, 54–57, 60, 63, 68, 70, 83, 90, 95, 100, 106, 109, 119, 121, 125, 126, 131, 133, 134, 138, 140, 142, 149, 160, 162, 165, 167, 173, 178, 180, 197, 204–206, 209, 210, 220, 222–224, 228, 232, 292, 298, 299 Dimension, 12, 37, 67, 68, 73, 76, 78, 79, 92, 95, 100, 101, 113, 115, 117, 123, 124, 126, 127, 132, 133, 141, 151, 154, 162, 172–174, 220, 221, 223, 231, 283, 285, 292, 298 Discourse, 100, 106, 179, 182, 185, 193, 197 Discussion, 4–7, 10, 11, 16, 18, 20, 23, 29, 38, 45–47, 59, 68, 77, 103, 106, 109, 112–114, 143, 160, 161, 169, 185, 187, 189–191, 193–195, 197, 206, 207, 214, 228, 261, 266, 269, 285, 286, 288, 291 Dispositions, 3, 7, 8, 16, 17, 24, 25, 91, 93, 99, 161, 169, 220, 223, 225, 231, 282, 297, 300 Domain, 17–19, 38, 46, 91, 92, 100, 102, 113, 114, 166, 169, 208–210, 237, 298 Dynamic, 9, 70, 113, 115, 120, 164, 173, 213, 214, 258, 275, 276, 293

E Education, 3, 5, 9–12, 16, 18–20, 40, 90, 101, 102, 108, 109, 112, 113, 118, 119, 125, 132, 134, 137, 160, 161, 163, 167, 168, 180, 194, 204, 205, 208–211, 219, 222, 227, 231, 232, 236, 258–264, 269–271, 274, 281, 298, 299

Index Educational psychology, 20, 24, 133, 134, 137 Elaboration, 12, 41, 152, 197 Elucidate, 23 Emotions, 7, 8, 16, 17, 19–21, 23–25, 75, 79, 100, 102, 114–116, 120–122, 125, 132, 135, 139–141, 148–150, 153, 162, 165, 168–170, 212, 215, 220, 223, 231, 237, 241, 258, 263, 298, 299 Empirical, 3, 6, 8, 11, 15, 18, 20, 21, 23, 30, 40, 45, 46, 55, 108, 113, 114, 132, 134, 136, 152, 204, 208 Engage, 7, 24, 54, 56–61, 63, 69, 90, 92, 95, 111, 117, 124, 165, 170, 180, 213, 251, 262, 270, 271, 275, 282, 300 Engineering, 21, 22, 204, 206, 214–216, 258–271 Epistemic, 104, 105, 108, 132–151, 153 Epistemology, 93, 94, 101–103, 106, 108, 132–135, 139–141, 143, 150, 151, 153, 154 Evaluate, 38, 44, 47, 54, 58, 60, 62, 167, 168, 170, 171, 269, 299 Expectancy-value, 115 Experience, 7, 9, 31, 54, 56, 58, 59, 68–70, 72, 83, 91, 92, 94, 100, 102–104, 106, 113, 114, 116, 117, 119, 121, 122, 124–126, 132, 135, 141, 150, 152, 159–174, 178, 180, 185, 187, 190, 198, 205–207, 209, 211, 214–216, 219, 220, 222, 236, 238, 239, 242, 244, 248–250, 258–260, 262–264, 266, 269–271, 274, 275, 278–280, 283, 291, 293, 299, 300 Externalization, 62 Externalize, 60, 61 Extrinsic, 69, 113–115, 121, 123, 125

F Feelings, 7, 8, 16, 17, 24, 25, 101, 104, 105, 109, 113–115, 117, 164, 165, 169, 170, 230, 236, 237, 244, 246, 248–250, 253, 298, 299 Flavell, J.H., 10, 31 Flow, 100, 121, 163, 165, 168, 169, 205–208, 211–216, 237–239, 244–251, 263–266, 268–271, 275–278, 281, 283, 292–294, 298 Formula, 36, 109, 118, 247, 259, 265, 269, 270 Frustrations, 104, 114, 117, 122, 162, 164, 165, 168, 206, 214, 238, 275, 277, 286, 298 Function, 22, 32, 60, 113, 141, 144–147, 207, 251, 253, 262, 270, 297

329 G Generalization, 104 Goal, 31, 34, 36, 37, 40, 55, 56, 62, 63, 70, 75, 91, 93–95, 100, 102, 109, 113–117, 119, 120, 122–125, 153, 160–162, 167, 180, 205, 215, 220, 222, 223, 230, 232, 238, 250, 260, 298 Goldin, G.A., 111–127, 160, 164, 168, 169, 172, 300 Graph, 144–146

H Happiness, 80, 94, 95, 102, 106, 108, 109, 160–163, 168–174, 263, 270, 298, 300 High-ceiling, 213, 215, 264, 270 Hypothesis, 18, 62 Hypothesize, 63, 75, 76, 248

I Identity, 92, 93, 95, 96, 100, 102, 103, 106, 108, 109, 117, 119, 169, 178, 179, 182, 198 Ill-defined, 17, 56, 57 Implement, 45, 208, 262 Inquiry, 19, 131, 132, 139, 140, 150, 168, 261, 263, 271 Instrument, 6, 17, 38, 136, 138, 205, 244, 265 Interest, 16, 18, 25, 29, 68, 69, 90, 95, 103, 104, 106, 113, 132, 143, 159, 162, 163, 171, 211, 215, 220, 229, 237, 238, 244, 248, 263, 264, 297, 298 Interpret, 8, 16, 21, 23, 59, 118, 223, 279, 280 Interpretation, 7, 18, 21, 41, 56, 57, 59, 165, 167, 206, 225, 229, 230, 279 Interview, 38–40, 48, 62, 71, 72, 74, 75, 77–83, 140–142, 151, 181, 182, 194, 195, 207 Intra-mathematical problems, 205, 237, 239, 241, 243, 244, 246, 248, 250 Intrinsic, 69, 113–115, 121, 123, 160, 212, 215, 216, 250, 264

J Joy, 114, 117, 121, 162, 165, 168

K Knowledge, 4, 24, 29–33, 35–41, 43–46, 55, 57, 59, 63, 69, 70, 83, 90, 91, 93, 94, 101, 114, 115, 122, 126, 131–136, 139, 141, 142, 148, 150–154, 213–215, 224–227, 231, 235, 278, 280, 282, 293, 294, 299

330 L Learning, 11, 16, 19–21, 30–32, 40, 41, 44, 45, 47, 48, 54, 61, 63, 67, 89, 92–96, 99, 100, 102–104, 106–109, 112, 113, 118–120, 123, 126, 132–135, 137, 139, 140, 148, 149, 151–153, 159–164, 166–170, 172–174, 179, 197, 198, 203, 204, 208, 209, 213, 215, 216, 222, 226, 230, 236, 237, 247, 250, 251, 262, 298, 299 Lesh, P.L., 6 Lesh, R., 16, 21, 56, 57, 59, 60, 63, 70, 99, 178, 180, 227, 232 Literature, 15–17, 19, 20, 23, 57, 93, 108, 112, 114, 116, 126, 135, 170, 171, 204, 212, 216, 219, 223, 227, 265, 269 Locus of control, 18, 25 Low floor, 213, 215, 264, 270

M Mathematical, 3, 15, 29, 53, 91, 112, 131, 160, 178, 221, 259, 278, 298 Mathematical modelling, 3–12, 132, 135, 136, 141, 203–206, 208–212, 215, 216, 236, 237, 239, 241, 242, 244, 251, 279, 282 Mathematical problem solving, 16, 18, 68, 91, 104, 121, 228, 236 Mathematical psychology, 16–18, 20, 23, 25 Mathematics, 3, 15, 33, 55, 90, 111, 131, 159, 178, 219, 258, 274, 297 Mathematics education, 16, 18–20, 40, 90, 100, 101, 108, 109, 112, 113, 118, 125, 132, 137, 161, 163, 168, 180, 208, 209, 211, 219, 227, 231, 232, 236, 258–260, 262, 269, 270, 274, 298, 299 Mathematize, 21, 96, 105, 178, 266, 270, 280 McLeod, D.B., 16, 18–20, 99, 114, 236, 297, 299 Meaningful, 63, 102, 106, 112, 122, 125, 152, 162–167, 170, 171, 174, 178, 205, 212, 214, 227, 259, 278, 300 Meaningfulness, 102, 162, 163, 166, 167, 169–174, 263 Meta-affect, 20, 104, 114, 117, 120, 168–171, 174 Meta-affective, 168 Metacognition, 32, 33, 35, 37, 38, 40–47, 53, 54, 57–60, 134 Meta-conative, 117 Method, 30, 38–40, 44, 46, 61, 91, 102, 104, 116, 139, 142, 151, 152, 172, 204, 206, 237, 243, 250, 258, 259, 281 Misconceptions, 24, 38, 41

Index Model, 6, 15, 33, 57, 90, 112, 132, 169, 178, 220, 259, 279, 298 Model-eliciting activity (MEA), 57, 60–62, 102, 103, 105–109, 178, 197 Modeling/modelling, 3–11, 21, 33, 35, 47, 94, 108, 131–133, 136, 139, 140, 142, 148, 150, 152, 153, 178, 180, 182–184, 196–198, 203–209, 212–216, 222, 236, 237, 239–251, 265, 268, 274, 278, 279, 281–283, 287, 289–291, 293, 298 Modifying, 56, 140–142, 197 Monitor, 37 Motivation, 100, 113, 250

N National Longitudinal Study of Mathematical Abilities (NLSMA), 17 NCTM, 222

O Objective, 139 Ontology, 141, 151–153 Open-ended, 55, 57, 244, 261, 262, 264 Organizing, 8, 113, 178, 197, 262 Ownership, 92–95, 207, 261, 269

P Participants, 58, 59, 105, 109, 173, 178–185, 187–198, 242, 244, 251, 252, 260, 265, 266 PBL, 237 Pedagogies, 96, 107 Persevere, 95, 300 Perspectives, 4–6, 8–11, 18, 21, 29, 30, 32, 46, 48, 54, 59, 60, 67–83, 92, 94, 100, 101, 103, 104, 106, 108, 109, 111–127, 131–135, 154, 178, 179, 181, 210, 212, 220, 247, 262, 270, 279, 280, 291 Philosophy, 4, 9, 161 Piaget, J., 229 Pleasure, 94, 102, 104, 118, 162, 163, 168–174, 300 Polya, G., 108, 137, 138 Principles, 6, 11, 26, 53–63, 114, 118, 137, 180, 208, 224 Problem solving, 16, 18, 19, 21, 30, 55, 60, 63, 68–72, 75, 77, 79, 80, 82, 83, 91, 93, 99, 104, 108, 116, 117, 121, 125, 135, 136, 138, 149, 205, 224, 228, 231, 236, 245, 249, 251, 252, 273, 278, 282, 298

Index Problem statements, 34, 61, 91, 140 Procedures, 37, 61, 71, 94, 136, 137, 139, 179, 181, 182, 185–187, 189–196, 206, 223, 239, 241, 262 Process, 16, 29, 69, 101, 112, 131, 178, 221, 260, 275, 299 Proofs, 90, 93, 94, 152, 154, 227, 262, 299, 300 Psychology, 3, 9, 16–20, 23–25, 109, 112, 113, 133, 134, 137, 205, 208, 263, 298, 299 Pythagorean theorem, 243, 244, 251, 254

Q Questionnaires, 38–40, 47, 204, 205, 207, 213, 215, 225, 243–250, 265, 266, 268–270 Questions, 5, 6, 8, 10–12, 17, 20, 30, 36, 37, 39, 46–48, 54, 58, 60–63, 68, 70, 75, 77, 78, 81–83, 90, 92, 95, 104–106, 108, 109, 112, 113, 116, 118, 132, 136, 141, 144–147, 150, 152–154, 161, 162, 165, 172, 173, 180, 187, 191, 194, 212, 213, 222, 224, 228–230, 232, 241, 242, 244, 246, 248, 249, 253, 262, 265, 266, 268–270, 273, 281, 283

R Reasoning, 58, 106, 108, 117, 121, 132, 133, 135–142, 148–150, 152–154, 161, 223, 279 Reconceptualize, 164 Regulation, 9, 10, 30–32, 37, 59, 117 Reliability, 246, 247, 266 Representations, 39, 57, 121, 136, 144, 152, 207, 212, 239, 275, 277, 279, 280, 284, 285, 287, 289, 290 Research, 4, 15, 29, 68, 99, 112, 132, 159, 180, 225, 258, 281, 298 Results, 4, 7, 9–11, 19–21, 30, 32, 37, 40, 42, 44–46, 55, 56, 59, 68, 70, 79, 82, 91, 92, 95, 104–106, 108, 115, 119, 126, 133, 135, 138, 141, 145, 147, 153, 154, 168, 184, 207, 213, 225, 227, 229, 236, 237, 239, 241–243, 246–251, 264–266, 269, 270, 277, 280, 283, 299, 300 Revising, 57 Routines, 55, 95, 102, 106, 108, 142, 153, 178–198, 249, 262, 293

S School Mathematics Study Group (SMSG), 17

331 Schukajlow, S., 3, 39, 41, 44, 45, 237, 239–242, 248, 250 Self-determination, 115, 125 Self-efficacy, 18, 114, 117, 126, 160, 162, 237, 248 Self-esteem, 18, 259 Self-regulation, 20, 24, 30, 114, 139, 165 Simon, T., 16, 17, 24 Sociocultural, 4, 8–10, 54, 58, 125 Sociomathematical, 165, 167 Solutions, 5–7, 32–39, 41, 42, 44, 45, 54, 56, 57, 59, 60, 62, 63, 69, 74, 75, 77, 80–82, 94, 101, 108, 116, 122, 136, 137, 146, 147, 165, 167, 178, 181, 189, 190, 197, 213–215, 232, 239, 241, 243, 246, 247, 278–280, 282, 288–291, 293, 294, 300 Spontaneous, 43, 53–63, 73, 132 Standardized, 55, 71, 159 Strategies, 6, 10, 31–48, 56, 59, 60, 68, 73, 83, 114, 117, 122, 135, 139, 153, 165, 174, 227, 260, 292 Stresses, 104, 109, 112, 221, 258, 259 Student, 17, 30, 67, 99, 111, 131, 159, 182, 219, 258, 281 Student-centered, 205, 237, 241–243, 250 Study, 19, 30, 71, 100, 112, 132, 160, 178, 220, 259, 279, 298 Subjectifying, 96, 179, 180, 182, 191, 194, 196–198

T Talented, 251, 264 Taxonomies, 31, 38, 90, 94, 114 Teacher, 4, 20, 40, 69, 101, 116, 132, 164, 180, 220, 258, 275, 298 Teacher-centered, 205, 237, 241–243, 247, 250 Teaching, 20, 41, 77, 100, 126, 131, 159, 181, 222, 298 Tetrahedral model, 20, 21 Theoretical, 3, 4, 6, 9–11, 15, 17, 18, 20, 21, 25, 30, 39, 46, 47, 54, 67–83, 106, 113, 133, 137, 152, 154, 204, 205, 207, 208, 223, 260, 274 Theory, 4, 16, 47, 106, 112, 134, 165, 237, 265, 275, 299 Thinking, 3, 6, 9, 10, 24, 30, 32, 38, 54, 55, 57–63, 68, 70, 76, 79–81, 83, 89, 91, 92, 94, 101, 103, 105–108, 118, 119, 127, 131–133, 137, 152, 154, 161, 163, 165, 179, 183, 214, 224, 236, 251, 260, 265, 268, 275, 279 Traits, 100, 104, 114, 116, 119–121, 126

332 U Understandings, 6, 7, 9, 10, 12, 17–19, 21, 26, 31, 33, 36, 44–47, 56–58, 61, 62, 68, 70, 73, 75, 79, 83, 91–93, 100, 105, 106, 108, 109, 117–119, 121, 124, 127, 134, 136, 137, 140, 161, 165, 167, 168, 204, 206, 213, 221, 224, 230, 232, 236, 239, 259, 264, 266, 278, 279, 291, 298

V Values, 10, 12, 16, 18, 20, 22, 25, 34–37, 41, 43, 57, 58, 75, 91, 93, 113–116, 120, 123–125, 135, 143, 144, 147, 153, 162,

Index 164, 166, 167, 169–171, 183, 186, 191–195, 197, 232, 237, 238, 247, 258, 298 Variables, 20, 22, 31, 37, 56, 57, 113, 120, 160, 161, 165–167, 169, 170, 173, 184, 191, 237, 248, 299 Vygotsky, 9, 10

W Word problems, 109, 205, 212, 213, 226–229, 231, 232, 237, 239–241, 243–245, 247–250