Financial Numeracy in Mathematics Education: Research and Practice [15, 1 ed.] 3030735877, 9783030735876

Table of contents :
Preface
Introduction: Financial Education in the Digital Era
Contents
Contributors
Epistemology and Curriculum
Financial Education and Mathematics Education: A Cross-Cutting Analysis of the Epistemological Intersection of Financial Numeracy
1 Introduction
2 Definition(s) of Financial Education: Moving from Financial Literacy to Financial Education
References
Financial Numeracy as Part of Mathematics Education
1 Introduction
2 Theoretical Foundations of Numeracy
2.1 Numeracy as Basic Arithmetic
2.2 Numeracy as Mathematical or Quantitative Literacy
2.3 Numeracy as Social Practice
3 Towards a Conceptualisation of Financial Numeracy
3.1 Three Dimensions of Teaching Financial Numeracy
4 Concluding Remarks
References
An Overview of Financial Numeracy in the Quebec Curriculum
1 The Quebec Education System
2 Broad Areas of Learning
3 Mathematics Curriculum (Financial Mathematics)
4 Financial Education Course
5 Discussion
References
Research Design of the Project
Background and Implementation of the Project
1 Introduction
2 The Quebec Context
3 A Qualitative Methodology
4 Implementation of the Methodology
5 Concluding Remarks
References
Building a Research Instrument on Financial Numeracy in Schools (Quebec and Romania)
1 Introduction
2 Sections of the Initial Questionnaire
2.1 Consent Form
2.2 Demographic Questions
2.3 Questions About Knowledge and Representations of FE
2.4 Questions About Teaching Practices and Needs
3 Conceptualising the Questionnaire in Romania
4 The Revised Questionnaire
4.1 Consent Form
4.2 Demographic Questions
5 Questions About Knowledge and Representations of FE
6 Comparing Implementation of the Questionnaire in Two Different Contexts
6.1 Consent Form
6.2 Demographic Questions
6.3 Questions About Knowledge and Representations of FE
6.4 Questions About Teaching Practices and Needs
7 Elements to Be Considered When Adapting a Research Instrument
8 Concluding Remarks
Using Tasks to Elicit Mathematics Teachers’ Thinking in Financial Numeracy
1 Introduction
2 The Importance of Task Design in Mathematics Classrooms
3 Different Stances in Focus Groups
3.1 Research Stance
3.2 Practice Stance
4 Concluding Remarks
References
Financial Numeracy Research in the Digital Era: Ethical Considerations
1 Introduction
2 Ethical Considerations When Planning a Research Project
2.1 Obtaining an Ethics Certificate
2.2 Ethical Considerations Specific to FE Research
3 Ethics During the Research
3.1 Ethical Dilemma: Mathematics Dimension
3.2 Ethical Dilemma: Personal Finances Dimension
3.3 Ethical Dilemma: Professional Dimension
4 Ethics After the Research: Implications for Researchers
5 Concluding Remarks
References
Results of the Project
Mathematics Teachers’ Financial Numeracy Representations and Practices
1 Introduction
2 Teachers’ Representations of Financial Education and Financial Numeracy
2.1 The Emphasis Given in Their Definitions
2.2 The Dimensions of Their Definitions
3 Their Financial Numeracy Teaching Practices
4 Their Needs for Teaching Financial Numeracy
5 Concluding Remarks
References
Making Sense of Mathematics: Two Case Studies of Financial Numeracy in Grade 11 Mathematics Classrooms
1 Introduction
2 Lesson 1: Science Stream Mathematics
2.1 Analysis
3 Lesson 2: Cultural, Social and Technical Stream Mathematics
3.1 Financial Dimension
3.2 Technological Dimension
3.3 Mathematical Dimension
3.4 Analysis
4 Discussion
5 Concluding Remarks
References
Implications for Teachers
Financial Numeracy in Secondary Schools in Quebec: Implications for Leadership
Some Financial Numeracy Tasks for Secondary-School Mathematics Classes
1 Introduction
2 Task 1: Margaret
3 Task 2: Cellphone
4 Task 3: Chicken Shack
5 Task 3: Chicken Shack Part 2
Conclusion: Financial Numeracy as an Emerging Field in Education
Financial Numeracy in Relation to the Community of Mathematics Education
References

Citation preview

Mathematics Education in the Digital Era

Annie Savard Alexandre Cavalcante   Editors

Financial Numeracy in Mathematics Education Research and Practice

Mathematics Education in the Digital Era Volume 15

Series Editors Dragana Martinovic, University of Windsor, Windsor, ON, Canada Viktor Freiman, Faculté des sciences de l’éducation, Université de Moncton, Moncton, NB, Canada Editorial Board Marcelo Borba, State University of São Paulo, São Paulo, Brazil Rosa Maria Bottino, CNR – Istituto Tecnologie Didattiche, Genova, Italy Paul Drijvers, Utrecht University, Utrecht, The Netherlands Celia Hoyles, University of London, London, UK Zekeriya Karadag, Giresun Üniversitesi, Giresun, Turkey Stephen Lerman, London South Bank University, London, UK Richard Lesh, Indiana University, Bloomington, USA Allen Leung, Hong Kong Baptist University, Kowloon Tong, Hong Kong Tom Lowrie, University of Canberra, Bruce, Australia John Mason, Open University, Buckinghamshire, UK Sergey Pozdnyakov, Saint-Petersburg State Electro Technical University, Saint-Petersburg, Russia Ornella Robutti, Università di Torino, Torino, Italy Anna Sfard, USA & University of Haifa, Michigan State University, Haifa, Israel Bharath Sriraman, University of Montana, Missoula, USA Eleonora Faggiano, Department of Mathematics, University of Bari Aldo Moro, Bari, Bari, Italy

The Mathematics Education in the Digital Era (MEDE) series explores ways in which digital technologies support mathematics teaching and the learning of Net Gen’ers, paying attention also to educational debates. Each volume will address one specific issue in mathematics education (e.g., visual mathematics and cyber-learning; inclusive and community based e-learning; teaching in the digital era), in an attempt to explore fundamental assumptions about teaching and learning mathematics in the presence of digital technologies. This series aims to attract diverse readers including: researchers in mathematics education, mathematicians, cognitive scientists and computer scientists, graduate students in education, policy-makers, educational software developers, administrators and teachers-practitioners. Among other things, the high quality scientific work published in this series will address questions related to the suitability of pedagogies and digital technologies for new generations of mathematics students. The series will also provide readers with deeper insight into how innovative teaching and assessment practices emerge, make their way into the classroom, and shape the learning of young students who have grown up with technology. The series will also look at how to bridge theory and practice to enhance the different learning styles of today’s students and turn their motivation and natural interest in technology into an additional support for meaningful mathematics learning. The series provides the opportunity for the dissemination of findings that address the effects of digital technologies on learning outcomes and their integration into effective teaching practices; the potential of mathematics educational software for the transformation of instruction and curricula; and the power of the e-learning of mathematics, as inclusive and community-based, yet personalized and hands-on. Submit your proposal: Book proposals for this series may be submitted per email to Springer or the Series Editors. - Springer: Natalie Rieborn at [email protected] (publishing editor) - Series Editors: Dragana Martinovic at [email protected] and Viktor Freiman at [email protected] Forthcoming titles: Expected 2021: 15 Years of Mathematics Education and it Connections to the Arts and Sciences Edited by Claus Michelsen, Astrid Berckman, Victor Freiman, Uffe Thomas Jankvist and Annie Savard Mathematics Education in the Age of Artificial Intelligence Edited by Philipe R. Richard, M. Pilar Velez and Steven van Vaerenbergh Mathematical Work in Educational Content Edited by Alain Kuzniak, Elizabeth MontoyaDelgadillo and Philippe R. Richard Expected 2022: Quantitative Reasoning in Mathematics and Science Education Edited by Gülseren Karagöz Akar, ˙Ismail Özgür Zembat, Selahattin Arslan and Patrick W. Thompson The Evolution of Research on Teaching Mathematics Edited by Agida Manizade, Nils Fredrik Buchholtz and Kim Beswick Mathematical Competencies in the Digital Era Edited by Uffe Thomas Jankvist and Eirini Geraniou

More information about this series at http://www.springer.com/series/10170

Annie Savard · Alexandre Cavalcante Editors

Financial Numeracy in Mathematics Education Research and Practice

Editors Annie Savard Faculty of Education McGill University Montreal, QC, Canada

Alexandre Cavalcante Ontario Institute for Studies in Education University of Toronto Toronto, ON, Canada

ISSN 2211-8136 ISSN 2211-8144 (electronic) Mathematics Education in the Digital Era ISBN 978-3-030-73587-6 ISBN 978-3-030-73588-3 (eBook) https://doi.org/10.1007/978-3-030-73588-3 © Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

This book was born through a collaborative research project from 2016 to 2019 in Quebec, Canada. Writing this book was a very rewarding journey for us as we consolidated our ideas regarding the place of Financial Education within mathematics teaching and learning. The process of writing helped us articulate some foundations of financial numeracy which we hope will support further debates and curriculum reforms. During the collaborative project, we engaged in debates with teachers, school board consultants, teacher educators and scholars invested in the field of Financial Education. Annie Savard recruited a team of graduate students and, together with them, founded the Lemonade Stand Lab at McGill University (https://www.mcg ill.ca/lemonadelab), a research group dedicated to developing educational perspectives on Financial Education in schools. Among those students, Alexandre Cavalcante stood out as a leader. Over the years, Annie and Alexandre collaborated in greater detail to generate the ideas shared in this book. What was once a relationship of supervisor–supervisee became one of colleagues throughout the writing of this book. Alexandre is now an Assistant Professor at the Ontario Institute for Studies of Education, University of Toronto. We hope this book will guide you in supporting the learning of financial numeracy teaching and learning, as well as developing collaborative projects in this emerging field of research. For us, this book was an important milestone in promoting financial numeracy worldwide. We have been able to reach out to colleagues in countries such as Romania and Australia, and have been developing further investigations of financial concepts in mathematics classrooms. Montreal, Canada Toronto, Canada

Annie Savard Alexandre Cavalcante

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Preface

Acknowledgements We are very grateful to the Autorité des Marchés Financiers du Québec (AMF)1 whose Education and Governance Fund funded the research project described in this book. Their contribution was essential and helped, among others, to support the doctoral research of Alexandre Cavalcante. The opinions and perspectives expressed here are solely those of Professor Annie Savard.

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https://lautorite.qc.ca/en/general-public.

Introduction: Financial Education in the Digital Era

The information era, also known as the digital age, has caused significant changes in the way people live their lives. These changes impact how people communicate, learn and work, and many of the books in the Mathematics Education in the Digital Era series have touched on these impacts. In this book, we also discuss one of the fields most impacted by the rise of information technology: financial practices. With the rise of financial technology, or fintech, in banking services has come more rapid consumption and increased consumer choice; with one click you can buy an item from a country on the other side of the world. While both information and technology are more widespread than ever, there is a growing concern regarding the quality of information available to consumers worldwide. In other words, the digital age has not necessarily led to an increased ability to make financial decisions on the part of consumers. For example, technological solutions have made investing so much easier that individual consumers can invest in the stock market without needing a financial adviser. Yet, this does not mean that consumers are making better investment decisions based on their risk tolerance and life goals. In fact, the amount of information available can lead to greater confusion and a false sense of security. Smartphones and other devices have enabled young people to purchase physical and digital products that they might not fully understand and for which they are not adequately prepared. Online fraud is also a major concern for many countries. With ever more sophisticated frauds accompanying the increase in digital financial services, recognising a financial scam is becoming an important aspect of digital literacy (Christian, Kamlesh & Dennis, 2020). These examples show that individuals’ ability to use technology is not enough for them to make sound financial decisions. Having the knowledge and skills to mobilise financial knowledge in these contexts is paramount. Financial education must therefore become an integral component of mathematics education in the digital era. Yet, there has been little development in this area among mathematics educators.

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Financial education (FE) has been receiving increasing attention from researchers and institutions worldwide since the 2008 global financial crisis. The COVID-19 pandemic has hit international and local economies hard. Individuals and their communities are facing challenging financial situations that expose current and previous financial practices and raise questions such as: Why are frontline and essential workers so poorly paid? Why have companies not built up a reserve fund for emergencies? How can we support local companies? The need for a better understanding of financial practices is even more essential in times of crisis. The Organisation for Economic Cooperation and Development (OECD) is also looking at FE. Since 2012, the OECD has been assessing the financial literacy of 15-year-old students via the financial literacy component of the three-yearly Program for International Student Assessment (PISA), the global national rankings of which are followed closely by stakeholders worldwide. Every three years the number of countries participating in the financial literacy component of PISA increases. With stakeholders requesting support with the implementation of school curricula, Education scholars are paying increased attention to Financial Education. Asian, European and North American colleagues have noted that, with a paucity of FE literature, they had been asked by their governments to write materials for schools and support teachers. The need for individuals to increasingly take charge of their own financial wellbeing has not been complemented by the necessary initiatives to educate and prepare students for the life awaiting them. Yet, multiple actors have debated how to shape future financial landscapes. Policy makers around the world have created national strategies to better support citizens in financial decision making (Grifoni & Messy, 2012; OECD, 2013). Education ministries have revised their curricula to incorporate FE under various disciplines (Aprea, 2016). Universities have implemented courses providing information about financial planning and student debt among their students (Grable, Law & Kaus, 2012). Scholars have intensified the production of FE knowledge from many different perspectives. The topic has been explored in detail by researchers in other fields such as economics, management, accounting, social work and psychology. These fields have produced a variety of theories and perspectives for understanding different aspects of FE, including consumer behaviour (how consumers respond and act according to the milieu, e.g., Dwiastanti, 2015; Suparti, 2016), financial choices and decision making (what factors impact the process of making a financial decision, e.g., Banaian, 2009; Garofalo & Kitchell, 2010), financial knowledge (theories, formulas, processes associated with finance as a discipline, e.g., Kim, Chatterjee, Young, & Moon, 2002; Walstad et al., 2017), debt and saving (similar to consumer behaviour, but focused less on consumption and more on money management, e.g., Brown & Ferguson, 2017; Reams-Johnson & Delker, 2016), and pensions and employability (topics that extend to public policy and social well-being, e.g., Fornero & Monticone, 2011; Lusardi & Mitchell, 2008, 2011; Lusardi & Mitchelli, 2007).

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However, this subject is yet to enter the mainstream of mathematics education. Few researchers have actually developed research programmes or lines of investigation on specific concepts and situations associated with formal Financial Education (e.g., Hamburg, 2009; Lucey, 2002; Pournara, 2013; Savard, 2008; Sawatzki, 2014). Financial Education, however, cannot be a mere reproduction of financial knowledge in classroom settings. The development of financially capable individuals, and the relationships between teachers, students and the content of FE needs to be understood specifically from a mathematics education perspective. Few scholars have investigated the emergent field of FE, especially at elementary and secondary levels. Most researchers come from other fields such as anthropology, economics, finance or sociology. We hope that more researchers in education will join this endeavour and find their place in this fascinating field.

Presentation of the Book Education scholars have only quite recently started to look at the field of financial numeracy, and this book aims to help researchers, teacher educators and stakeholders with a variety of motivations to become familiar with it. Some are interested because of the political need for people to be financially literate. This is especially true since the 2008 international financial crisis and the increasing need to reduce household debt and increase individual savings and pension plans. Around the world, educators and researchers are being expected to introduce Financial Education into their national curricula, deliver workshops for teachers or write pedagogical materials. Financial Education curricula are often imposed top down by policy makers, leaving teachers and principals at the lower end of the educational spectrum unsure of how to implement them. Some are interested in the cognitive development of financial concepts among students at different ages. They are motivated by how students learn and reason about money. Others have a genuine interest in finance and mathematics. Mathematics plays an important role when learning about finance, as does finance when learning mathematics. Some researchers and teachers include a substantial social-change component. These educators are strongly motivated to change social structures in order to provide their students with better conditions. For examples, researchers have investigated how gender inequality is reproduced in educational discourses. Some teachers have taught mathematics in culturally responsive ways to prepare their students to participate in political debate with better information and critical thinking skills. Those concerned with social justice and inequality may find that the role of financial literacy in developing citizenship competencies is paramount. Socio-politically motivated scholars and educators who want to build a more equitable society and develop a critical stance towards social reality among their students may find that financially literate citizens will stand a better chance in the dynamics of society.

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Introduction: Financial Education in the Digital Era

The book is arranged in four parts, each containing two or three chapters that address a specific theme. Epistemology and Curriculum introduces the theoretical foundations of financial numeracy, illustrated by the integration of financial numeracy in the school curricula of the Québec Education Program. Research Design of the Project explains the methodology used in our research project, examining the challenges of research with teachers on the sensitive topic of finances. Results of the Project presents the results of the project on teachers’ perceptions and on classroom practices. Implications for Teachers provides the rationale for teaching financial numeracy from the point of view of two teachers. Learning situations (lessons) on financial numeracy to be implemented in secondary-school mathematics classrooms are included, together with their implications. Annie Savard Alexandre Cavalcante

References Aprea, C., Wuttke, E., Breuer, K., Koh, N. K., Davies, P., Greimel-Fuhrmann, B. & Lopus, J. S. (Eds.) (2016). Financial literacy in the twenty-first century: an introduction to the international handbook of financial literacy. Springer. Banaian, K. (2009). Stimulating economics. Academic Questions, 22(4), 442–451. Brown, N. & Ferguson, K. (2017). Teaching financial literacy with Max and Ruby. Childhood Education, 93(1), 58–65. Christian, E., Kamlesh, K. & Dennis, P. (2020). Financial literacy and fraud detection. The European Journal of Finance, 26(4–5), 420–442, https://doi.org/10.1080/1351847X.2019.1646666 Dwiastanti, A. (2015). Financial literacy as the foundation for individual financial behavior. Journal of Education and Practice, 6(33), 99–105. Fornero, E. & Monticone, C. (2011). Financial literacy and pension plan participation in Italy*. Journal of Pension Economics & Finance, 10(4), 547–564. Garofalo, J. & Kitchell, B. A. (2010). Technology focus: Using technology to promote equity in financial decision making. NCSSSMST Journal, 16(1), 22–23. Grable, J. E., Law, R. & Kaus, J. (2012). An overview of university financial education programs. In D. Durband & S. Britt (Eds.), Student financial literacy. Boston, MA: Springer Grifoni, A. & Messy, F. (2012). Current status of national strategies for financial education: A comparative analysis and relevant practices. In OECD Working Papers on Finance, Insurance and Private Pensions, 16. Paris: OECD Publishing. Hamburg, M. P. (2009). Financial mathematical tasks in a middle school mathematics textbook series: A content analysis (Doctoral thesis). The University of Akron: United States. Kim, J., Chatterjee, S., Young, J., & Moon, U. J. (2002). The cost of access: Racial disparities in student loan burdens of young adults. College Student Journal, 51(1), 99–114. Lucey, T. A. (2002). The personal financial literacy of fourth grade students. (Doctoral thesis). The University of Memphis: United States. Lusardi, A. & Mitchell, O. S. (2008). Planning and financial literacy: How do women fare? American Economic Review, 98(2), 413–417.

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Lusardi, A. & Mitchelli, O. S. (2007). Financial literacy and retirement preparedness: evidence and implications for financial education. Business Economics, 42(1), 35–44. OECD (2013). OECD skills outlook 2013 first results from the survey of adult skills. OECD Publishing. Pournara, C. (2013). Mathematics-for-teaching in pre-service mathematics teacher education: the case of financial mathematics. (Doctoral thesis). University of the Witwatersrand: South Africa. Reams-Johnson, A. & Delker, S. (2016). The development of the financial literacy program at the Community College of Baltimore County. Community College Journal of Research and Practice, 40(7), 571–579. Savard, A. (2008). Le développement d’une pensée critique envers les jeux de hasard et d’argent par l’enseignement des probabilités à l’école primaire: vers une prise d décision. (Doctoral thesis). Université Laval: Canada. Sawatzki, C. (2014). Connecting social and mathematical thinking: the use of “real life” contexts. (Doctoral thesis). Monash University: Australia. Suparti (2016). Mitigating consumptive behavior: The analysis of learning experiences of housewives. International Education Studies, 9(3), 114–122. Walstad, W., Urban, C., Asarta, C. J., Breitbach, E., Bosshardt, W., Heath, J. & Xiao, J. J. (2017). Perspectives on evaluation in financial education: Landscape, issues, and studies. Journal of Economic Education, 48(2), 93–112.

Contents

Epistemology and Curriculum Financial Education and Mathematics Education: A Cross-Cutting Analysis of the Epistemological Intersection of Financial Numeracy . . . . Annie Savard and Alexandre Cavalcante

3

Financial Numeracy as Part of Mathematics Education . . . . . . . . . . . . . . . Annie Savard and Alexandre Cavalcante

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An Overview of Financial Numeracy in the Quebec Curriculum . . . . . . . Annie Savard, Alexandre Cavalcante, and Azadeh Javaherpour

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Research Design of the Project Background and Implementation of the Project . . . . . . . . . . . . . . . . . . . . . . Alexandre Cavalcante and Annie Savard

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Building a Research Instrument on Financial Numeracy in Schools (Quebec and Romania) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Annie Savard and Daniela Caprioara

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Using Tasks to Elicit Mathematics Teachers’ Thinking in Financial Numeracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Louis-Philippe Turineck and Alexandre Cavalcante

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Financial Numeracy Research in the Digital Era: Ethical Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Annie Savard and Alexandre Cavalcante

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Results of the Project Mathematics Teachers’ Financial Numeracy Representations and Practices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alexandre Cavalcante

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Making Sense of Mathematics: Two Case Studies of Financial Numeracy in Grade 11 Mathematics Classrooms . . . . . . . . . . . . . . . . . . . . . Alexandre Cavalcante and Annie Savard

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Implications for Teachers Financial Numeracy in Secondary Schools in Quebec: Implications for Leadership . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Benoit Brosseau and Jean-François Blanchet

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Some Financial Numeracy Tasks for Secondary-School Mathematics Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Louis-Philippe Turineck and Annie Savard

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Conclusion: Financial Numeracy as an Emerging Field in Education . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Contributors

Jean-François Blanchet Centre de Services Scolaire des Grandes-Seigneuries, Laprairie, QC, Canada Benoit Brosseau Centre de services scolaire des Hautes-Rivières, Saint-Jean-surRichelieu, QC, Canada Daniela Caprioara Universitatea Ovidius din Constanta, Constant, a, Romania Alexandre Cavalcante Ontario Institute for Studies in Education, University of Toronto, Toronto, ON, Canada Azadeh Javaherpour Department of Integrated Studies in Education, Faculty of Education, McGill University, Montreal, QC, Canada Annie Savard Department of Integrated Studies in Education, Faculty of Education, McGill University, Montreal, QC, Canada Louis-Philippe Turineck Department of Integrated Studies in Education, Faculty of Education, McGill University, Montreal, QC, Canada

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Epistemology and Curriculum

This part introduces the theoretical foundations that motivated our project and were refined throughout it. In these chapters we conceptualise a new term to fit into the field of both Financial Education and Mathematics Education. Hence, researchers from both areas will benefit from engaging with our ideas in this part. In the first chapter, we situate the construct of financial numeracy by first describing other terms that circulate in the Financial Education literature. We also situate it at the epistemological intersection between Financial Education and Mathematics Education. We conclude the chapter by introducing the term financial numeracy. In the second chapter, we delve deeper into our conceptualisation of financial numeracy with a definition of the meaning we give to numeracy, followed by our theoretical framework on the dimensional components of financial numeracy. In the third chapter, we show how financial numeracy is implemented in the Quebec secondary-school curriculum. This province is an interesting case study because it incorporates financial numeracy in three different ways: a standalone course in Financial Education, a mathematics sub-strand and a cross-curricular component. We hope these three chapters help the reader understand both our vision and the place of financial numeracy among education scholars. We also hope teachers and leaders can benefit from these chapters to develop their own epistemology for the teaching of financial numeracy.

Financial Education and Mathematics Education: A Cross-Cutting Analysis of the Epistemological Intersection of Financial Numeracy Annie Savard and Alexandre Cavalcante

1 Introduction This chapter examines the epistemological intersection between Financial Education (FE) and mathematics education (ME), and explores the importance of mathematics educators in the teaching of FE. The parallel development of mathematics and finance can be observed through historical examples. According to Ifrah (2000), thousands of years ago, shepherds needing to know the number of animals they were bringing to market used pebbles. The invention of money to replace the barter system established a standard measure for the exchange of goods and services (Ifrah, 2000). This shared development remains important, especially in the context of teaching and learning FE and ME. This chapter begins by briefly defining and discussing FE and ME, showing how the definitions are grounded in the literature. We then examine the intersection between FE and ME, which we refer to as financial numeracy. This intersection can be regarded as a developmental construct, derived from financial contexts, in which mathematics becomes the pragmatic measurement of financial practices.

A. Savard (B) Department of Integrated Studies in Education, Faculty of Education, McGill University, 3700 McTavish Street, Montreal, QC, Canada e-mail: [email protected] A. Cavalcante Ontario Institute for Studies in Education, University of Toronto, 252 Bloor Street West, Toronto, ON, Canada e-mail: [email protected] © Springer Nature Switzerland AG 2021 A. Savard and A. Cavalcante (eds.), Financial Numeracy in Mathematics Education, Mathematics Education in the Digital Era 15, https://doi.org/10.1007/978-3-030-73588-3_1

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2 Definition(s) of Financial Education: Moving from Financial Literacy to Financial Education This project began with current definitions of financial literacy (FL) grounded in the literature. The emerging field of FL is, it seems to us, moving from knowledge to competency. Researchers from a number of different fields have tried to define financial literacy in various ways, and over time the definition has come to mean a set of financial knowledge and its use in human behaviours (Huston, 2010; Johnson & Sherraden, 2007). Hogarth (2002) described three consistent themes emerging from the literature to define individuals who are financially literate: they (1) are knowledgeable, educated and informed on the issues of managing money and assets, including banking, investments, credit, insurance and taxes; (2) understand the basic concepts of managing money and assets; and (3) use that knowledge and understanding to plan and implement financial decisions (pp. 15–16). It is clear from this that knowledge is key. Knowledge of financial concepts and products includes, for example, knowing, what a profit is, how a credit card works and how you buy insurance. However, knowing is not enough: it is necessary to understand this knowledge in order to use it in life. The use of this knowledge and understanding can be seen in the conduct or actions of individuals in regard to personal finances, in other words, financial behaviours. Defining financial literacy in terms of knowledge and behaviours, and focusing on aspects of participation in economic life without considering competency to act and how personal decisions are linked to institutions, however, is quite limited and difficult to operationalise (Johnson & Sherraden, 2007). Financial literacy has therefore been reconceptualised as financial capability (Sherraden & Ansong, 2016), in which skills, behaviours and knowledge in four domains are articulated: managing money, planning ahead, choosing products and staying informed (Atkinson et al., 2007). In fact, financial capability can be understood as the ability to make informed judgements and decisions, using skills, behaviour and knowledge (Atkinson et al., 2007). According to Xiao and O’Neill (2016), financial capability is the ability to use financial knowledge to adopt financial behaviours that will achieve financial well-being. Such a definition focuses on specific aspects of participation in economic life, without taking into consideration competency to act and how personal decisions are linked to institutions (Johnson and Sherraden, 2007). In fact, as pointed out by Xiao and O’Neill (2016), financial capability can be regarded as a synonym for financial literacy. The term capability is largely used in different fields such as business, economics, and social development and social work. In a report on financial literacy in the US, Vitt and his colleagues (2000) suggested including complexity in the definition of competency: personal financial literacy is the ability to read, analyse, manage and communicate about the personal financial conditions that affect material well-being. It includes the ability to discern financial choices, discuss money and financial issues without (or despite) discomfort, plan for the future, and respond competently to life events that affect everyday financial decisions, including events in the economy generally. (Vitt et al., 2000, p. xii)

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The notion of competency seems more productive to us than capability, because financial capability is a generalisable behavioural outcome, a desirable aim in general. A competency, by contrast, is a situated aptitude or disposition held by a person. In education, a competency entails the development of a person. We agree with Perrenoud’s (2002) definition of competency as the mobilisation of a set of resources according to the context. These resources are knowledge, skills, tools, humans, everything to which an individual might have access in their environment. It could even be an online search or advice from a financial adviser. Mobilisation is not the application or transfer of knowledge, it is the adaptation or transformation of the knowledge in a particular situation. Thus, knowing how a financial product works does not in itself render you competent. Being competent means deciding whether this financial product is relevant and needed in your own situation, taking into account your personal needs, which includes your family’s or community’s needs. It also means using this financial product in a productive way for oneself and others. Making decisions is a complex process that requires the use of mathematics to be able to generate and assess alternatives (Halpern, 2003; Savard, 2008). In fact, it seems that being able to make decisions about personal financial choices cannot be done without strong mathematical knowledge (Agarwal & Mazumder, 2013; Cole et al., 2014). Recent work by Savard (2018) highlights the use of mathematical models by grade 4 students to make sense of financial concepts such as purchasing power, investment and risk. The complexity of financial competency was recognised and included in the definition of financial literacy for young people promoted by OECD in their PISA 2012 International Assessment Programme (OECD Programme for International Student Assessment 2014): Financial literacy is knowledge and understanding of financial concepts and risks, and the skills, motivation and confidence to apply such knowledge and understanding in order to make effective decisions across a range of financial contexts, to improve the financial wellbeing of individuals and society, and to enable participation in economic life. (p. 33)

Four areas of knowledge and understanding are thus defined as the content: money and transactions, planning and managing finances, risk and reward, and the financial landscape. Four processes and four contexts are also defined. Identifying financial information, analysing information in a financial context, evaluating financial issues, and applying financial knowledge and understanding are processes present in situations where financial knowledge, skills and understanding have to be used. The contexts range from personal to global: education and work; home and family; individual; and societal. Other public organisations have also defined the content to be taught in schools. In 2013, the Council for Economic Education in the United States established national standards of financial literacy education to be taught in schools in six important areas: earning income, buying goods and services, saving, using credit, financial investing, and protecting and insuring (Council for Economic Education, 2013). Each standard statement has “benchmarks” explaining content to be taught, as well as information and lifelong insight (Bosshardt & Walstad, 2014). Also in the United States, the

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Jump$tart Coalition for Personal Financial Literacy launched the fourth version of its educational programme in 2015. Six standards to be taught from kindergarten to grade 12 are presented: spending and saving, credit and debt, employment and income, investing, risk management and insurance, and financial decision making. In the content areas to be taught, there are consistent key concepts: for example, spending, saving and managing risk are part of all of them. It is also possible to combine them to cover a wider range of financial concepts. For instance, spending and saving is related to the concept of consumerism. However, as this project unfolded, we noticed that some important aspects of education, particularly mathematics education, were missing from these definitions. We changed our way of looking at FL to incorporate broader views on the subject. Instead of focusing only on personal finance (which is the current state of the art in the literature), we decided to embrace aspects of critical thinking, citizenship and community. For us, the focus on mathematics is also crucial, because without numeracy, finances and economics cannot be modelised. In this way we moved from financial literacy to Financial Education. Our working definition of FE for the purposes of this project is: Financial Education is the field of teaching and learning the financial dimension of the production and management of resources mediated by financial instruments (currency, models, concepts). The use of financial instruments can also lead to assigning a value to an action (service) or an object (good).

The cognitive and social aspects are considered, such as financial praxis in different contexts. Producing resources, for example, means developing both actual and new resources. For example, a student might produce new resources from existing available resources, such as recycling fabric to make pencil cases. The student can produce this resource for their own use, or they can sell it and obtain money. In this case, entrepreneurship education is included in our definition. Managing resources means managing individual and collective resources to develop well-being for oneself and one’s community. A student might decide to exchange a toy with a friend to get another one, instead of asking their parents to buy it. The cognitive and social aspects of the financial praxis should be understood in order to participate fully in the different communities. Financial Education should therefore aim to develop cognitive understanding of sociocultural practices about producing and managing individual and collective resources, thereby developing citizenship competencies in all students. In Financial Education, a learning situation should involve studying an object or phenomenon derived from financial praxis in a sociocultural context. The financial practices might be buying things in a store, or a situation involving financial concepts such as profit or compound interest. Mathematics is thus a privileged tool that models the situation to develop cognitive understanding. Knowledge and understanding developed in the cognitive context are about finance and mathematics, in other words, financial numeracy. FE is also about making thoughtful decisions about financial praxis for oneself and one’s community (Caprioara et al., 2020; Savard, 2017). Such decisions can also be regarded as the result of a critical thinking process (Lipman, 2003; Paul & Elder, 2001). Critically engaging students in financial praxis supports

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the development of citizenship competencies. Ten Dam and Volman’s (2004) definition of citizenship includes critical and responsible participation in social practices by a member of a community. This represents a democratic evolution of social practices and thus of the community itself. From this point of view, critical thinking and decision making can be considered citizenship competencies (Savard, 2008). Citizenship may have varied profiles among the members of a community. Westheimer and Kahne (2004) identify three kinds of citizens: the personally responsible citizen, the participatory citizen and the justice-oriented citizen. The personally responsible citizen acts in a responsible manner in their community, while the participatory citizen is active in the community at the local, state or national level. The justice-oriented citizen analyses and questions issues of social justice. For instance, the responsible citizen might donate money or food to a local charity, the participatory citizen volunteers at the fundraising activity and the justice-oriented citizen will try to find a way to eradicate poverty. Mathematics highlights the citizenship context that might be explicitly or implicitly present in the learning situation presented to students. For example, learning about buying goods in a store might lead to questions about fair trade, while learning about profit might lead to discussion of sustainability. Learning about compound interest might lead to saving money for education. Financial Education is thus broader than personal finances, it is about developing citizenship, and mathematics education plays a major role in this development. The next chapter identifies what we want for students: financial literacy using mathematics, in other words, financial numeracy.

References Agarwal, S., & Mazumder, B. (2013). Cognitive abilities and household financial decision making. American Economic Journal: Applied Economics, 5(1), 193–207. Atkinson, A., McKay, S., Collard, S., & Kempson, E. (2007). Levels of financial capability in the UK. Public Money & Management, 27(1), 29–39. Bosshardt, W., & Walstad, W. B. (2014). National standards for financial literacy: Rationale and content. Journal of Economic Education, 45(1), 63–70. Caprioara, D., Savard, A., & Cavalcante, A. (2020). Empowering future citizens in making financial decisions: A study of elementary school mathematics textbooks from Romania. In D. Flaut, S. Hoskova-Mayerova, C. Ispas, F. Maturo, & C. Flaut (Eds.), Decision making in social sciences: Between traditions and innovations. Springer. Cole, S. A., Paulson, A., & Shastry, G. K. (2014). Smart money? The effect of education on financial outcomes. Review of Financial Studies, 27(7), 2022–2051. Council for Economic Education. (2013). National standards for financial literacy. Halpern, D. F. (2003). Thought and knowledge: An introduction to critical thinking (4th ed.). Lawrence Erlbaum Associates. Hogarth, J. (2002). Financial literacy and family and consumer sciences. Journal of Family and Consumer Sciences, 94(1), 14–28. Huston, S. J. (2010). Measuring financial literacy. The Journal of Consumer Affairs, 44(2), 296–316. Ifrah, G. (2000). The universal history of numbers. From prehistory to the invention of the computer. Wiley.

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Johnson, E., & Sherraden, M. S. (2007). From financial literacy to financial capability among youth. Journal of Sociology & Social Welfare, XXXIV (3), 119–145. Lipman, M. (2003). Thinking in education (2nd ed.). Cambridge University Press. Organisation for Economic Cooperation and Development (OECD) (2013). OECD skills outlook 2013 first results from the survey of adult skills. OECD Publishing. Organisation for Economic Cooperation and Development [OECD]. (2014). PISA 2012 results: Students and money: Financial literacy skills for the 21st Century (Vol. VI). OECD Publishing. Paul, R., & Elder, L. (2001). Critical thinking: Tools for taking charge of your learning and your Life. Prentice Hall. Perrenoud, P. (2002). D’une métaphore à l’autre: transférer ou mobiliser ses connaissances? In J. Dolz & E. Ollagnier (Eds.), L’énigme de la compétence en éducation (pp. 45–60). Bruxelles: De Boeck. Savard, A. (2008). Le développement d’une pensée critique envers les jeux de hasard et d’argent par l’enseignement des probabilités à l’école primaire: Vers une prise de décision (Unpublished dissertation). Université Laval, Québec. Savard, A., Cavalcante, A., Turineck, L.-P., & Javaherpour, A. (forthcoming). Epistemological considerations towards the concept of money in school: Money as measurement. For the learning of mathematics. Savard, A. (2017). Making decisions in a complex world: Teaching how to navigate using mathematics. In C. Michelsen, A. Beckmann, V. Freiman, & U. T. Jankvist (Eds.), Mathematics as a bridge between the disciplines. Proceedings of mathematics and its connections to the arts and science international symposium (MACAS 2017), 27–29 June, Copenhagen, Denmark, 1–14. Savard, A. (2018). Teaching probability and learning financial concepts: How to empower elementary school students in citizenship. In K. S. Cooter & T. Lucey (Eds.), Financial literacy for children and youth (2nd ed., pp. 137–152). Peter Lang. Sherraden, M. S., Ansong, D. (2016). Financial literacy to financial capability: Building financial stability and security. In C. Aprea, K. Breuer, P. Davies et al. (Eds.), International Handbook of Financial Literacy (pp. 83–96). New York: Springer. ten Dam, G., & Volman, M. (2004). Critical thinking as a citizenship competence: Teaching strategies. Learning and Instruction, 14(4), 359–379. Vitt, L. A., Anderson, C., Kent, J., Lyster, D. M., Siegenthaler, J. K., & Watrd, J. (2000). Personal finance and the rush to competence: Personal literacy in the US. The Fannie Mae Foundation. Westheimer, J., & Kahne, J. (2004). What kind of citizen? The politics of educating for democracy. American Educational Research Journal, 41(2), 237–269. Xiao, J. J., & O’Neill, B. (2016). Consumer financial education and financial capability. International Journal of Consumer Studies, 40, 712–721. https://doi.org/10.1111/ijcs.12285

Financial Numeracy as Part of Mathematics Education Annie Savard and Alexandre Cavalcante

1 Introduction This chapter builds on chapter “Financial Education and Mathematics Education: A Cross-Cutting Analysis of the Epistemological Intersection of Financial Numeracy” to present how we conceptualise financial numeracy in mathematics education. We first define numeracy by laying the theoretical foundations underlying the construct. Next, we make a connection between numeracy and the school mathematics curriculum, highlighting how contemporary perspectives on numeracy can impact our understanding of teaching mathematics. We then present a three-dimensional model of teaching financial numeracy. The chapter concludes by discussing the implications of this model for mathematics education.

2 Theoretical Foundations of Numeracy The word numeracy is sometimes used as the name of a school subject or as a synonym for literacy in the context of numbers. Another meaning has recently emerged— numeracy as a social practice. In this section, we explore these three meanings of numeracy and discuss their contributions to or limitations on the teaching of financial concepts in mathematics. A. Savard (B) Department of Integrated Studies in Education, Faculty of Education, McGill University, 3700 McTavish Street, Montreal, QC, Canada e-mail: [email protected] A. Cavalcante Ontario Institute for Studies in Education, University of Toronto, 252 Bloor Street West, Toronto, ON, Canada e-mail: [email protected] © Springer Nature Switzerland AG 2021 A. Savard and A. Cavalcante (eds.), Financial Numeracy in Mathematics Education, Mathematics Education in the Digital Era 15, https://doi.org/10.1007/978-3-030-73588-3_2

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We believe the teaching and learning of mathematics should be focused on developing students’ ability to make sense of the world through mathematics and use this knowledge to make decisions and actively participate in society. These ideas broadly engender the notion of numeracy (Geiger et al., 2015). We position our work in this field as a way of emphasising how mathematical ideas are connected to real-life situations as opposed to simply reproducing a disciplinary epistemology (Goos et al., 2019).

2.1 Numeracy as Basic Arithmetic According to Yasukawa et al. (2018), in some popular understandings and in dictionary definitions, the concept of numeracy refers to basic arithmetic skills with numbers. It is sometimes confused with number sense or numeration. Aunio and Niemivirta (2010) define early numeracy as children’s understanding of numbers and their relationships. We define counting and comparing sets of objects as number sense. Using the term numeracy instead of number sense is confusing for teachers.

2.2 Numeracy as Mathematical or Quantitative Literacy Numeracy is also often identified as mathematical or quantitative literacy (Karaali et al., 2016). Thus, it refers to the ability and confidence to use, apply, interpret, and communicate mathematical information and ideas (OECD, 2013). This conceptualisation is particularly important for us because it expands the scope of the mathematical concepts and processes necessary for success in everyday life. This definition introduces multiple implications with important connections to financial education. Ability and confidence. Despite the traditional views of mathematics as a discipline detached from the real world, mathematics educators seek to develop numeracy as a way for students to be able to use such knowledge in their daily lives and professional practice. Financial situations are particularly rich in terms of the mathematical knowledge required to understand them; therefore they occupy a privileged place for the exercise (and development) of numeracy. This conceptualisation also includes a sense of being comfortable with situations that involve quantitative information. The issue of engagement in mathematics has been a source of much investigation among mathematics educators, and there has been a consensus that positive attitudes towards mathematics are important to the learning of mathematics. Research shows that financial situations can be a source of engagement in mathematics and students are generally highly motivated to learn more about financial matters (Attard, 2018). Using, applying, interpreting and communicating. This aspect broadens the scope of practices involved with the effective development of numeracy. Traditional mathematics teaching emphasises the memorising of mathematical facts and their applications to (mostly) solve problems or prove theorems. The notion of numeracy as

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mathematical (or quantitative) literacy emphasises the use of mathematics to make sense of the world. Consequently, other social practices such as interpretation and communication will be involved in this process. Financial situations require these and other practices by definition, since individuals have to understand financial products and services in order to engage in reasoning. Being well informed is a fundamental precondition for making sound financial decisions, hence being numerate is a sine qua non for financial education. Mathematical information and ideas. We have seen that this conceptualisation of numeracy extends beyond the strict limitations of knowledge. It also extends beyond the strict scope of arithmetic. In fact, being mathematically (or quantitatively) literate involves all strands of mathematics. Financial education scholars argue that mathematics is a critical aspect of being financially literate; however, they often reproduce the idea that mathematics is about performing basic operations. In fact, when we talk about financial situations, many mathematical elements emerge as important variables in the understanding of such situations. These elements include statistics, probability, measurement, geometry, etc. Hence, numeracy refers to the wider scope of mathematics presented in multiple ways instead of the traditional word problem containing only relevant information. In short, this conceptualisation seems to encompass a rich interpretation of the role played by mathematics in life. Not only does it incorporate a broader range of social practices, but also a broader range of mathematical knowledge. This broadening is important because financial situations and their related concepts are complex and require more than just basic arithmetic. However, conceptualising numeracy as mathematical (or quantitative) literacy still falls short when it comes to the role of contexts in its development. Some studies which have used this conceptualisation seem to imply that numeracy can be developed in class in isolation, as suggested by Street (2001). They propose that students can become numerate in class and that this ability can simply be applied in any context. We, however, agree with Baker (1998) in arguing that the context is a foundational aspect of developing numeracy. The next section introduces another conceptualisation which partly builds on the notion of mathematical literacy to understand numeracy as a social practice.

2.3 Numeracy as Social Practice Understanding numeracy as a social practice (NSP) entails looking at how people develop numeracy (meaning the skills and confidence to interact with mathematical knowledge, instruments and processes) in their daily lives. This perspective builds on the notion of mathematical literacy to emphasise not only its development, but most importantly its interaction with the social contexts in which it emerges. While the theories in which this perspective finds inspiration are multiple, four theoretical bodies have contributed significantly to its emergence: cultural-historical activity

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theory, ethnomathematics, New Literacy Studies, and situated cognition. Yasukawa et al. (2018) have described each of these bodies’ contribution to the establishment of the NSP perspective. Here, we discuss its overall characteristics as they apply to the teaching of financial concepts through mathematical knowledge, instruments and processes. NSP perspectives are mostly concerned with everyday life situations, focusing on what people do with numeracy rather than what they know in mathematical tests. Key to this characteristic is the importance of contexts. An NSP perspective emphasises the role of particular contexts for the development of numeracy skills, rather than developing mathematical knowledge in isolation. Moreover, focusing on the contexts in which numeracy is developed entails viewing numeracy as an activity that is culturally, historically and politically situated. Indeed, all numeracy activities convey the values and world visions of specific social groups. In the case of financial situations, it is paramount to unpack those values and ideologies in order to understand what is being conveyed to us and to make sound financial decisions. In this sense, it is important to be critical about numeracy practices in financial contexts. A third aspect of the NSP perspective refers to both explicit and implicit forms of mathematics. Particularly among ethnomathematics scholars (D’Ambrosio, 2016; Rosa & Orey, 2016), multiple studies have documented and analysed activities not traditionally recognised as numeracy, either by the social groups themselves or by researchers. In the case of financial activities, it is not as problematic because all money-related activities involve at least rudimentary arithmetic. It is difficult to conceptualise even a conversation about money which does not involve the use of numbers and some operations. However, the challenge here is to go beyond these rudimentary levels and comprehend that financial activities involve other aspects of numeracy such as estimation, comparison, interpretation, evaluations, etc. These are practices carrying a great many mathematical ideas that can be unpacked in the teaching and learning process. Finally, the fourth characteristic of the NSP perspective is serious attention to the power relations entailed in teaching, learning and researching numeracy. As Yasukawa et al. (2018) describe, “power relations are in play when a proficient numeracy user is reluctant to share information in case it threatens their own position. ‘Protecting their own patch’ could be viewed as a defensive social practice” (p. 13). This relates strongly to financial contexts, as it is often the case that the conditions and structures of financial products or services are not made explicit and clear for consumers, which in turn makes the relationship between them and financial institutions even more asymmetric. In that sense, it is important to empower students to develop the confidence and ability to question and challenge the situations in which they find themselves.

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3 Towards a Conceptualisation of Financial Numeracy Financial mathematics, it has been suggested by many, refers to the connections between the financial and mathematical worlds (Camiot & Jeannotte, 2016). Many scholars, policy makers and documents use this term when talking about the necessity of mathematics knowledge for learning financial concepts (NALA, 2015). However, financial mathematics refers to a very specific range of concepts that were developed by mathematicians interested in modelling the financial world. Within this field, concepts such as interest, compounding, annuities, and present and future value have been proposed for studying the behaviour of financial markets. We argue that the place of mathematics in the world of FE should be expanded beyond financial mathematics. According to the definition of numeracy described above, the entire scope of mathematical concepts, tools and procedures can be used to model financial situations in everyday life, and the term financial numeracy should be used to refer to this expanded understanding. Although the concept of financial numeracy is not new, it has been undervalued in the literature (Vacher & Wallace, 2013). Few studies have mobilised this term, only one of which was situated in mathematics education. It has been used to a limited extent to refer to the basic mathematical knowledge (mostly arithmetic) used in financial contexts. Lusardi (2008) distinguished financial numeracy from financial literacy by establishing that the former represents the most basic (and therefore weakest) knowledge required to develop the latter (strongest). Joo and Chatterjee (2012) conceptualised financial numeracy along the same lines, stating that “basic financial numeracy is the level of understanding basic financial math, such as interest rates and inflation (cognitive knowledge) while advanced financial literacy includes decision-making choices in the financial marketplace, such as risk diversification between company stock and mutual funds (behavioral aspects)” (p. 127). However, such a definition seems to draw on a superficial perception of numeracy. Not only does it confine numeracy to its cognitive dimension, but it also narrows the spectrum of mathematical domains that can emerge in financial contexts (typically basic arithmetical operations). In fact, this narrow understanding of numeracy seems to be reproduced in multiple studies (Skagerlund et al., 2018; Jayaraman et al., 2018; Huhmann & McQuitty, 2009). Building on the body of research in mathematics education, on the other hand, Camiot and Jeannotte (2016) propose a broader understanding of financial numeracy. The authors recognise that content knowledge (connaissances) is not sufficient to account for the complexity of numeracy experienced by individuals in financial everyday life. In fact, their conceptualisation of financial numeracy includes four different dimensions: cognitive, affective, behavioural and motivational. Therefore, financial numeracy goes beyond the mere application of basic arithmetic operations to solve financial questions. It encompasses the whole spectrum of mathematical concepts, procedures and instruments, as well as attitudes and the confidence to use and make sense of mathematics in financial contexts. Financial numeracy is situated at the intersection of financial education and mathematics education.

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However, such an intersection does not exist in a vacuum. Every scholar or educator arrives at this intersection from a specific background. Mathematics educators, for example, reach financial numeracy having their own epistemological stances. Financial contexts are frequently incorporated in a mathematics classroom by mathematics teachers, and these financial contexts have the potential to increase engagement in mathematics classes (Attard, 2018). Students are often interested in knowing more about real-life financial practices, products and services. By using them to teach mathematics we provide a real-life connection to abstract mathematical concepts, and that makes mathematics more meaningful. Discussing situations in which mathematics can be applied, such as financial situations, is an example of how mathematics can be perceived as situated knowledge, meaning that this form of knowledge is always produced within a specific context (Lave & Wenger, 1991). It is not detached from reality but embedded in social practices. By referencing specific everyday financial situations, we can not only study the application of mathematics (which implies learning it before applying it), but we can also investigate how mathematics can be extracted from them (learning mathematics from these practices). Another possible route for those interested in the concept of financial numeracy relates to understanding financial concepts (credit, debt, gambling), which requires us to engage in multiple mathematical practices such as modelling, estimating, measuring, counting, predicting, rounding, etc. In that sense, there is a general consensus that mathematics is an essential component of financial education. However, such consensus has been underexplored in the literature, which is why we believe financial numeracy can offer a powerful lens for us to reflect on the role played specifically by mathematics. Many ideas, such as profit and investments, acquire deeper layers of complexity once we explore them through a numeracy perspective. For instance, evaluating a profit according to absolute or percentage values (return on investment) relies on looking at profit as a mathematical structure (the difference between two quantities in relation to a third quantity). Hence, mathematical ideas are necessary to make sense of such financial concepts, empowering the individual to make better-informed decisions. A third way of approaching the field of financial numeracy concerns those interested in the social implications of financial products and services. While money is used in daily life as a unit to economically measure goods and services, questioning these measures—and thus revealing the social, cultural and political actors lying behind them that use mathematics to convey values—is, in our experience, not so common among mathematics educators. Take, for instance, the concept and measurement of inflation. Inflation is the general increase of prices that happens over time in a given economy, a phenomenon that is real and recognised in our daily lives. However, the calculation of this index is a social and economic convention that requires a “basket” of goods and services to be selected to represent the average increase. The weight with which the items enter the basket is also a matter of convention, and how the final index rate will be used by different institutions, companies and individuals depends on their understanding of this measure as well as the values possessed by each of

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Fig. 1 A conceptual framework for financial numeracy

the social actors. Mathematics plays an important role in empowering individuals to engage in critical debates about how financial concepts are mobilised and measured in daily financial life. We have created a pyramidal model to represent these different paths to financial numeracy, showing the intersection levels, along with the interactions between them. This model also represents three dimensions of teaching and learning financial numeracy in schools, particularly in mathematics classes. The three dimensions are equally important to implement in classrooms (Fig. 1).

3.1 Three Dimensions of Teaching Financial Numeracy 3.1.1

Contextual Dimension

The contextual dimension is the study of mathematics in financial contexts. Learning about financial concepts and practices is not necessarily part of the goals of the contextual dimension of financial numeracy. The teaching of financial numeracy through this dimension tends to be limited to tasks that involve financial contexts. The context itself does not play a major role in justifying solutions to the task, but it can be a good way to generate motivation and engagement among students. In fact, the context is an opportunity to “stage” the task, which might be either attractive or unappealing to the students.

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Conceptual Dimension

The conceptual dimension refers to the teaching and learning of financial concepts which require mathematical activities such as representing, estimating, measuring, comparing, counting, predicting, rounding, etc. These activities are part of everyday financial life, hence they reflect the role of mathematics in the understanding of financial concepts, products and services. Many ideas, such as profit and investments, gain deeper layers of complexity once we incorporate the conceptual dimension. For instance, evaluating a profit according to absolute (dollar value) or percentage values (return on investment) is essentially a mathematical activity as it relates to comparing and using different number representations. This dimension also includes the mathematical domain of financial mathematics, which has been proposed by many as the appropriate term to refer to the connections between the financial and mathematical worlds (Camiot & Jeannotte, 2016). Many scholars, policy makers and documents refer to this term when talking about the need for mathematics knowledge in learning about financial concepts (NALA, 2015). However, financial mathematics represents a very specific scope of concepts, tools and procedures that were developed by mathematicians interested in modelling the financial world. Within this field, concepts such as interest, compounding, annuities, present and future value emerge in the study of the behaviour of financial markets. The conceptual dimension allows financial numeracy to expand its scope to other domains of mathematics, while keeping the same underlying goals. In fact, financial numeracy touches on all domains of mathematics in school (arithmetic, algebra, geometry, probability and statistics). Some teachers may perceive the potential of the financial concepts and propose complex learning situations to model, while others may not.

3.1.3

Systemic Dimension

The systemic dimension is the justification, or unpacking, of financial concepts in relation to their social implications. The systemic dimension entails investigating specific financial measures and how they are calculated, defined, modelled and portrayed in society. While money is used in daily life as a unit to measure goods and services, questioning these measures and unpacking what lies behind them is less common in mathematics curricula. Behind these measures are social, cultural and political actors that use mathematics to convey values. The concept of inflation is a good example: inflation as the general increase of prices over time in a given economy is a phenomenon that is real and recognised in our daily lives. However, the calculation of this index is a social and economic convention requiring a basket of goods and services to be selected to represent the average increase. The weighting of items in the basket is also a matter of convention. How the final index rate is used by different institutions, companies and individuals depends on each actor’s values and understanding of this measure. Mathematics plays an important role in empowering individuals to engage in critical debates about how financial concepts are mobilised

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and measured in daily financial practices, a role that we believe is incorporated in the systemic dimension of financial numeracy. In fact, the systemic dimension has strong potential in regard to financial education. However, this dimension is the most challenging to implement in classrooms, because the social, cultural and political factors it implies greatly increase the complexity of learning situations.

4 Concluding Remarks The model presented above has emerged from our research with secondary mathematics teachers. Our analysis of teachers’ practices revealed varying levels of engagement with financial education. Their practices touched mostly the first three levels of financial numeracy. The following chapters will use this conceptual framework on financial numeracy as a lens to view data related to financial education and mathematics, and also to look at teacher preparation for teaching FN.

References Attard, C. (2018). Financial literacy: Mathematics and money improving student engagement. Australian Primary Mathematics Classroom, 23(1), 9–12. Aunio, P., & Niemivirta, M. (2010). Predicting children’s mathematical performance in grade one by early numeracy. Learning and Individual Differences, 20, 427–435. Baker, D. (1998). Numeracy as social practice. Literacy and Numeracy Studies, 8(1), 37–50. Camiot, C., & Jeanotte, D. (2016). La numératie financière chez les jeunes adultes: une compétence à développer pour une meilleure gestion financière? In A. Adihou, J. Giroux, D. Guillemette, C. Lajoie, & K. Huy (Eds.), Actes du Colloque du Groupe de didactiques des mathématiques du Québec 2016 (pp. 55–67). D’Ambrosio U. (2016). An overview of the history of ethnomathematics. In M. Rosa, U. D’Ambrosio, D. Orey, L. Shirley, W. Alangui, P. Palhares, & M. Gavarrete (Eds.), Current and future perspectives of ethnomathematics as a program. ICME-13 topical surveys. Springer. Geiger, V., Goos, M., & Forgasz, H. (2015). A rich interpretation of numeracy for the 21st century: a survey of the state of the field. ZDM Mathematics Education, 47, 531–548. Goos, M., Geiger, V., Dole, S., Forgasz, H., & Bennison, A. (2019). Numeracy across the curriculum: Research-based strategies for enhancing teaching and learning (1st ed.). Allen & Unwin. Huhmann, B., & McQuitty, S. (2009). A model of consumer financial numeracy. International Journal of Bank Marketing, 27(4), 270–293. Jayaraman, J., Jambunathan, S., & Counselman, K. (2018). The Connection between Financial Literacy and Numeracy: A Case Study from India. Numeracy, 11(2), Article 5. Joo, S., Chatterjee, S. (2012). Financial Education Research Opportunities. In: D. Durband, S. Britt (Eds.), Student Financial Literacy. Boston, MA: Springer. https://doi.org/10.1007/978-1-46143505-1_10. Karaali, G., Villafane Hernandez, E. H., & Taylor, J. A. (2016). What’s in a name? A critical review of definitions of quantitative literacy, numeracy, and quantitative reasoning. Numeracy, 9(1), Article 2. http://dx.doi.org/10.5038/1936-4660.9.1.2 Lave, J., & Wenger, E. (1991). Situated learning: Legitimate peripheral participation. Cambridge University Press.

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Lusardi, A., Mitchell, O. S. (2008). Planning and Financial Literacy: How Do Women Fare? American Economic Review, 98(2): 413–17. NALA. (2015). A wealth of practice: Case studies of financial numeracy practice in Ireland. The National Adult Literacy Agency. www.nala.ie/publications Organisation for Economic Cooperation and Development (OECD). (2013). OECD skills outlook 2013 first results from the survey of adult skills. OECD Publishing. Rosa, M., & Orey D. C. (2016). State of the art in ethnomathematics. In: M. Rosa, U. D’Ambrosio, D. Orey, L. Shirley, W. Alangui, P. Palhares, & M. Gavarrete (Eds.), Current and future perspectives of ethnomathematics as a program. ICME-13 topical surveys. Springer. Skagerlund, K., Lind, T., Strömbäck, C., Tinghög, G., & Västfjäll, D. (2018). Financial literacy and the role of numeracy: How individuals’ attitude and affinity with numbers influence financial literacy. Journal of Behavioral and Experimental Economics (formerly The Journal of SocioEconomics), 74(C), 18–25. Street, B. V. (2001). Literacy and development: Ethnographic perspectives. Routledge. Vacher, H. L., Wallace, D. (2013). The Scope of Numeracy after Five Years. Numeracy, 6(1), Article 1. http://dx.doi.org/10.5038/1936-4660.6.1.1. Yasukawa, K., Rogers, A., Jackson, K., Street, B. (2018). Numeracy as a social practice: Global and local perspectives. London, UK: Routledge.

An Overview of Financial Numeracy in the Quebec Curriculum Annie Savard, Alexandre Cavalcante, and Azadeh Javaherpour

1 The Quebec Education System In Canada, there is no national curriculum. Each province manages its own education system and has its own provincial curriculum. In Quebec, the mandatory education system goes from kindergarten to grade 11. Students now start school at four and finish at around 17 years old. Kindergarten has two years: the first for 4-year-olds and the second for 5-year-olds. Elementary school encompasses grades 1–6, while secondary-school grades are from 1 to 5. After secondary school, students go to a CEGEP1 (Collège d’enseignement général et professionnel) to make the transition to university. At CEGEP level, students can do a 2-year programme in preparation for university or a 3-year technical programme in preparation for a job (Table 1). The Quebec Education Program, the provincial curriculum, states the competencies to be developed as well as the content to be learnt within each discipline through a number of broad areas of learning. It aims to prepare students to face the challenges of a pluralistic society which promotes diversity and to be ready to start work in a 1

For more information about CEGEP, see https://www.cegepsquebec.ca/en/cegeps/presentation/ what-is-a-cegep/.

A. Savard (B) · A. Javaherpour Department of Integrated Studies in Education, Faculty of Education, McGill University, 3700 McTavish Street, Montreal, QC, Canada e-mail: [email protected] A. Javaherpour e-mail: [email protected] A. Cavalcante Ontario Institute for Studies in Education, University of Toronto, 252 Bloor Street West, Toronto, ON, Canada e-mail: [email protected] © Springer Nature Switzerland AG 2021 A. Savard and A. Cavalcante (eds.), Financial Numeracy in Mathematics Education, Mathematics Education in the Digital Era 15, https://doi.org/10.1007/978-3-030-73588-3_3

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K-5

5–6 years old

K-4

4–5 years old

Kindergarden

6–7 years old

1

Cycle 1

7–8 years old

2

8–9 years old

3

9–10 years old

4

Elementary school

Cycle 2

Table 1 The education system in Quebec

10–11 years old

5

Cycle 3

11–12 years old

6

12–13 years old

Sec. 1 (7)

Cycle 1

13–14 years old

Sec. 2 (8)

14–15 years old

Sec. 3 (9)

CST 15–16 years old

16–17 years old

NS

Sec. 5 (11) TS

CST

NS

Sec. 4 (10)

Secondary school Cycle 2

TS

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constantly evolving market that is part of a globalized economy (Gouvernement du Québec, 2016). In its student-centred approach, teachers can teach beyond the classroom to develop competencies that will guide students through various areas of their daily lives.

2 Broad Areas of Learning Financial numeracy is indirectly mentioned in the document ‘Broad Areas of Learning’, which lists five topics that must be addressed by teachers in all disciplines from multiple angles. According to the document, these areas “deal with major contemporary issues young people will have to confront, both individually and collectively, in different areas of their lives. These issues also represent challenges to their talent and creativity” (Government of Quebec, n.d., p. 1). Each of the five areas includes important connections to financial numeracy: Health and well-being. The goal of this area is to “ensure that students develop a sense of responsibility for adopting good habits with respect to health, safety and sexuality” (p. 5). In that sense, teachers must develop students’ self-awareness and awareness of their basic needs; knowledge of the impact of their choices on health and well-being; active lifestyle and safe behaviours; and awareness of the consequences of collective choices on individual well-being. The connection to financial numeracy lies in its essential role in determining one’s wants and needs, and taking care of one’s own mental health. Career planning and entrepreneurship. The goal of this area is to “enable students to make and carry out plans designed to develop their potential and help them integrate into adult society” (p. 7). Within this goal, teachers should develop students’ selfknowledge and awareness of their potential and how to fulfil it; and strategies related to planning, familiarity with the world of work, social roles, and occupations and trades. The connection to financial numeracy is quite explicit here, since the world of work is permeated with financial issues. Planning one’s career and strategies related to financially supporting oneself require familiarity with financial concepts and processes. Environmental awareness and consumer rights and responsibilities. The goal of this area is to “encourage students to develop an active relationship with their environment while maintaining a critical attitude towards consumption and the exploitation of the environment” (p. 9). To achieve this goal, teachers must develop students’ knowledge of the environment; responsible use of goods and services; awareness of social, economic and ethical aspects of consumption; and attitudes towards a healthy environment based on sustainable development. Financial numeracy has an explicit connection to this area of learning as it plays an important role in understanding consumption practices and the consequences of our actions for the environment. Media literacy. The goal of this area is to “enable students to exercise critical, ethical and aesthetic judgment with respect to the media and produce media documents that respect individual and collective rights” (p. 11). To achieve this

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goal, teachers should seek to develop awareness of the place and influence of the different media in students’ daily lives and in society; understanding of media representations of reality; familiarity with methods of producing media documents; and respect for individual and collective rights and responsibilities regarding the different media. Financial numeracy is essential to developing an understanding of media representations of reality and their influence in everyday life. Citizenship and community life. The goal of this area is to “enable students to take part in the democratic life of the class or the school and develop an attitude of openness to the world and respect for diversity” (p. 13). Within this goal, teachers generally develop an understanding of the rules of social conduct and democratic institutions, participation, cooperation and solidarity, and the adoption of a culture of peace. It is also through this area that students develop an understanding that financial numeracy refers not only to personal financial matters, but also to the well-being of their communities and society.

3 Mathematics Curriculum (Financial Mathematics) In Quebec, secondary school is divided into two cycles. Cycle 1 covers the first two years of secondary school, grades 7 and 8. Cycle 2 is the last three years of secondary school, grades 9–11. In the last two years of cycle 2, grades 10 and 11, the mathematics curriculum offers three options: the Cultural, Social and Technical option (CST) (4 credits), the Technical and Scientific (TS) option (6 credits) and the Natural Sciences (NS) option (6 credits). Each option aims to develop students’ mathematical reasoning and mathematical knowledge so that they will be better prepared for life in society and vocational/technical training or pre-university education. The TS and NS options provide students with opportunities to engage in more complex mathematical content (particularly algebra and geometry). Students in these options usually pursue post-secondary studies which require more quantitative reasoning such as STEM, finance or business. Those who excel at mathematics in early secondary are encouraged to pursue the NS option as a way to keep their career options open. The CST option is typically regarded as the lower strand of mathematics as it focuses on the applications of mathematics in everyday life. Students in this strand often enrol in trade school or major in degree courses that are not heavily quantitative. In addition to financial mathematics, this strand also emphasises statistics and theories of social choice (e.g., voting systems). Financial mathematics concepts were added in 2016 to the last year of the CST option only. In this course, students are expected to use statistical tools and calculation of probabilities to assess risk, process data and draw conclusions (Government of Quebec, n.d.). Thus, the financial mathematics concepts reflect this orientation: interest rates (simple and compound), interest period, discounting (present value) and compounding (future value) (Government of Quebec, n.d.). The inclusion of financial mathematics in the CST curriculum does not imply the exclusion of financial numeracy in other grades or other options. In fact, many of the

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topics explored in this area can be explored in other domains of mathematics, as they involve the application of fundamental mathematics concepts such as exponentials, logarithms, proportions, percentage, number representation, etc. Furthermore, financial contexts are also used to teach learning situations. For instance, word problems requiring students to calculate using money will include budgeting, spending, saving, profit, sales, pricing, discount, chance, risk, commissions, salary, incomes and taxes. Thus, mathematics teachers present many different financial contexts to students and have the opportunity to help them learn mathematics in relation to Financial Education.

4 Financial Education Course In 2017, a new mandatory course, Financial Education (FE), was launched in Secondary 5 for all students. As it is a specific school discipline, students will get credits if they pass the FE course. Although a specific school discipline, FE in Quebec is not isolated among other scientific disciplines. According to Ministère de l’Education et de l’Enseignement Supérieur (MEES), FE as a school discipline (subject area) is a part of the broader area of social sciences (MEES, 2018). The programme focuses on personal finances as a required competency for achieving independence and living in society: In the Financial Education program, students acquire knowledge and know-how that enables them to assess situations that involve taking a position, considering the options available to them and, rigorously determining the consequences of their choices. They learn to recognize what influences them and to consider the legal aspects of the situations they encounter. In analysing financial issues that affect them, students exercise and develop their critical judgment. By exercising the competency developed in this program, they are also able to learn more about themselves, which helps them set their own goals and determine the degree to which they can tolerate the risks associated with the management of their personal finances. (MEES, 2018, p. 1)

In Quebec’s Financial Education programme, although students learn about some financial concepts such as credit cards, debt, mortgages and the stock market, mathematics does not play an explicit role in teaching those concepts. For example, on the topic of consumer credit, teachers will discuss the reasons why financial institutions give people credit, and how they can increase their credit score. They also discuss “the rights and responsibilities of consumers who enter into a variable credit contract (e.g. obtaining the good or service immediately, making monthly payments)” (MEES, 2018, p. 13). The programme presents financial concepts focusing on the social and citizenship aspects rather than the mathematical and numeracy aspects. The epistemology of designing an FE course in social sciences and humanities draws on the exploration of multiple perspectives on FE in social interactions with society. The course develops students’ cognitive and sociocultural learning, enabling them to make informed financial decisions. They will understand what questions need

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to be asked when gathering information, which websites are secure, what responsibilities and rights a consumer has, what considerations are necessary for entering the workforce, and the reasons for poverty and debt. This information is important to know from a social point of view. However, mathematics plays an implicit role in this programme by illustrating the financial concepts. Mathematics is not used as a cognitive tool to think critically about those concepts. We believe that lack of financial numeracy in developing FE as an individual discipline does not support students to mobilise mathematical reasoning in making financial decisions. To this end designing a programme that includes both the social sciences and mathematics aspects of FE alongside each other can be more effective than a course that focuses on one aspect of Financial Education.

5 Discussion Building curricula implies making epistemological choices. The Quebec Education Program (QEP) is an example of how financial numeracy can be implemented in different ways into a local (national or provincial) curriculum. Financial literacy is generally taught as a standalone subject in the Social Sciences field, while financial numeracy is present in the broad areas of learning and officially embedded into the mathematics curriculum. The QEP thus shows three different ways of including financial numeracy in curricula (Fig. 1). The first way would be for financial numeracy to have its own curriculum as a “stand-alone discipline” (Lenoir & Sauvé, 1998). For instance, the QEP proposes a mandatory Financial Education course for all students in grade 11. The second way might be to integrate financial numeracy within other curricula via the broad areas of learning, creating interactions between disciplines (Legendre, 1993). This interdisciplinary route mutually supports multiple disciplines, because the development of one discipline contributes to the development of the other (Fourez & Larochelle, 2003; Huntley, 1999). For instance, in the Quebec curriculum, financial numeracy

Fig. 1 Implementing financial numeracy in curricula

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might be used to support the construction of understanding of social phenomena and their complexity. Geography, history and citizenship education are the specific disciplines that support this understanding. The third way is to include financial numeracy within mathematics curricula as content of that discipline. This integration of contents within the same discipline is called intradisciplinarity (Huntley, 1999; Samson, 2014; Savard & Manuel, 2016). It can be a domain of a discipline, such as financial mathematics, which stands on its own, together with algebra, arithmetic or geometry. Intradisciplinarity can also refer to teaching two domains of the same discipline. In mathematics, some financial concepts (such as money and budgeting) could be integrated into the curriculum connected with all domains. Table 2 synthesises the three approaches to integrating financial numeracy in the curriculum. Implementing financial numeracy in a curriculum is not an easy task. Each curriculum is conceptualised and designed in unique ways that aim to reflect the needs of the society where it is embedded. In our experience, each of the three Table 2 Approaches to integrating financial numeracy in the curriculum Characteristics

Advantages

Disadvantages

Discipline

Taught for itself as a standalone course

Deeper understanding of the discipline: • Epistemology • Practices • Concepts • Processes • Instruments Guarantees that students will learn about financial matters

Might not be connected to other disciplines or to life Might be taught isolated or in silos Missing opportunities to have a broader understanding of a phenomenon

Interdisciplinary

Taught by two or more disciplines in such a way that one contributes explicitly to the other

Broader understanding of a phenomenon Connections between them show complexity of a phenomenon and thus requires strong thinking Develops a broader point of view on the disciplines

Might not go deeply enough into one discipline to understand all the facets of a particular concept or process Might present a narrow view of a discipline Takes time to teach

Intradisciplinary

Taught as a distinct domain (or in connection to other sub-domains) within a discipline

Deeper understanding of the two domains of the same discipline: • Epistemology • Practices • Concepts • Processes • Instruments Might save teaching time

Only one perspective of a phenomenon might be portrayed Some teachers might leave to the end of the academic year Some teachers might justify avoiding this topic based on lack of time

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types of integration serves specific purposes and tackles specific issues related to FE. For instance, while a standalone course might be beneficial for making financial numeracy more explicit and clarifying responsibilities, it runs the risk of conveying a single perspective (that of the teacher in question). On the other hand, while an emphasis on interdisciplinarity can address this risk, it may be the case that no teacher really integrates it for lack of clarity and expectations. Curriculum designers should consider Financial Education programmes in which knowledge and skills are integrated as a coherent and relevant subject from grade to grade (MEES, 2016). Although it is necessary to integrate FE as a core subject within K-12 school curricula, it does not have to be taught as a standalone subject (OECD, 2012). If FE aims to expose students to real-life contexts and develop their understanding about different dimensions of financial domains in different life situations beyond the classroom, it can be designed across different disciplines such as mathematics, social studies, economics, history, etc. (OECD, 2012). In both interdisciplinary and intradisciplinary designs, teachers require thorough pedagogical content knowledge of both the subject area and the FE domain. They need both educational and financial support to prepare relevant and meaningful materials for delivering curricular components in their practice. As Financial Education aims to foster students’ critical thinking and social skills, preparing them to make sound financial decisions and implementing these skills beyond the classroom environment, an FE class needs to include more student-centred materials and activities to foster thorough understanding of each component. Designing professional development programmes for teachers appears crucial to support these pedagogical approaches.

References Fourez, G., & Larochelle, M. (2003). Apprivoiser l’épistémologie. De Boeck. Huntley, M. A. (1999). Theoretical and empirical investigations of integrated mathematics and science education in the middle grades with implications for teacher education. Journal of Teacher Education, 50(1), 57–67. Javaherpour, A. (2017) A comparative study of Quebec & Ontario’s curriculum in financial literacy education: Presence, components and aims [Unpublished Master’s dissertation]. McGill Library, Montreal. Legendre, R. (1993). Dictionnaire actuel de l’éducation. Guérin. Lenoir, Y., & Sauvé, L. (1998). L’interdisciplinarité et la formation à l’enseignement primaire et secondaire: quelle interdisciplinarité pour quelle formation? Revue des Sciences de l’Éducation, 24(1), 3–29. Mandell, L. (1998). Our vulnerable youth: The financial literacy of American 12th graders. Jump$tart Coalition. Ministère de l’Education et de l’Enseignement Supérieur. (2016). Quebec education program: Mathematics, cycle 2. Gouvernement du Québec. Ministère de l’Education et de l’Enseignement Supérieur. (2018). Quebec education program: Financial education. Secondary V. Gouvernement du Québec. Organisation of Economic Cooperation and Development (OECD). (2012). Financial literacy: A core life skill. https://www.oecd.org/daf/fin/financial-education/FinEdSchool_web.pdf

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Samson, G. (2014). From writing to doing: The challenges of implementing integration (and interdisciplinarity) in the teaching of mathematics, science and technology. Canadian Journal of Science, Mathematics and Technology Education, 14(4), 346–358. Savard, A. (2015). Making decisions about gambling: The influence of risk on children’s arguments. The Mathematics enthusiast, 12 (1–3), 226–245. Savard, A. (2018). Teaching probability and learning financial concepts: How to empower elementary school students in citizenship. In K. S. Cooter & T. Lucey (Eds.), Financial literacy for children and youth (2nd ed., pp. 137–152). Peter Lang. Savard, A., & Manuel, D. (2016). Teaching statistics in middle school mathematics classrooms: Creating an intersection for intra and interdisciplinarity. Statistics Education Research Journal, 15(2), 239–256. Williams, J. et al. (2016). Interdisciplinary mathematics education: A state of the art. In: Interdisciplinary mathematics education (pp. 1–13). ICME. Springer.

Research Design of the Project

This part presents the research design of the project. The fourth chapter describes the context of the project in the province of Quebec, Canada and explains the place of Financial Education in the secondary-school curriculum before revealing the aims of the project and describing how it was implemented with teachers. The fifth chapter introduces our data-collection instrument: an online questionnaire. The rationale for using this research instrument is explored and each part of it explained. The authors also show how this research instrument was adapted for use in another country—Romania. The components of the research instrument in the two countries are compared, and some methodological considerations for adapting a research instrument are discussed. The sixth chapter presents the protocol used to collect focus group data. In this chapter, the authors discuss the importance of task design in mathematics classes and present a reflection on how to use tasks to generate discussions among teachers depending on the stance of those conducting focus groups (teacher educators vs. researchers). In the seventh chapter, we provide several considerations regarding the ethics of conducting a research project involving financial matters. We present the process taken at the beginning of the project to secure ethics approval from our university, and also share some situations that happened during the project in which we were confronted with ethical issues. We hope this chapter will be useful for those considering developing projects related to financial education in schools or among teachers.

Background and Implementation of the Project Alexandre Cavalcante and Annie Savard

1 Introduction This chapter presents the background to the research project. We start with contextual information about the province of Quebec before describing the qualitative methodology employed. We then explain the data collection process, highlighting the challenges of collecting a large amount of qualitative data in a short space of time (3 hours).

2 The Quebec Context The research project emerged from a particular context and at a particular time. Financial Education began to attract increased attention in 2012, when the Organisation for Economic Cooperation and Development (OECD) started administering a Financial Literacy assessment via their triennial Programme for International Student Assessment (OECD, 2014). Some Canadian provinces participated for the first time in the programme’s second iteration in 2015. In Canada, each province and territory manages its own educational system. When Quebec participated in PISA in 2018, Financial Education had been added to the official secondary curriculum. A 2016 A. Cavalcante (B) Ontario Institute for Studies in Education, University of Toronto, 252 Bloor Street West, Toronto, ON, Canada e-mail: [email protected] A. Savard Department of Integrated Studies in Education, Faculty of Education, McGill University, 3700 McTavish Street, Montreal, QC, Canada e-mail: [email protected] © Springer Nature Switzerland AG 2021 A. Savard and A. Cavalcante (eds.), Financial Numeracy in Mathematics Education, Mathematics Education in the Digital Era 15, https://doi.org/10.1007/978-3-030-73588-3_4

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appendix to the Secondary 5 maths curriculum issued by the Ministry of Education included some financial mathematics concepts, and a new mandatory Financial Education course was introduced for all Secondary 5 students in 2017. When we collected the data for this project, the updated maths curriculum (CSTSecondary 5) had already been introduced, but teachers had few resources and textbooks had not been updated. The FE course, officially implemented in Fall 2017, was taught by social sciences teachers and maths teachers were not familiar with it. Not all schools implemented this new course in the same way (some had decided to postpone it to Winter 2018). It is important to highlight that the teachers we met in Spring and Fall 2017 were generally confused about the two new programmes and were reluctant to talk about FE for fear of encroaching on the territory of their social sciences colleagues.

3 A Qualitative Methodology The research team in this project comprised university professors, graduate students and school board consultants. The goal was to investigate the sources and needs of secondary maths teachers relating to financial numeracy. Its more specific aims were as follows: 1. 2. 3.

To identify the teaching practices that are used to teach Financial Education in mathematics classes. To identify information and training resources in Financial Education and assess their availability and efficiency. To identify teachers’ needs in terms of content and pedagogical methods in order to support them with future training. The research questions were:

1. 2. 3. 4.

How do the teachers teach financial numeracy in their class? What are the sources of information/training for the teachers in regard to financial numeracy? How do the teachers feel about FE and teaching FE? What are the needs of the teachers in terms of knowledge and pedagogy for teaching FE?

We decided that focus group discussions would be the most effective way to collect these data, as they would allow teachers to provide their own perspectives in contrast to those of their colleagues. Particularly in the context of financial numeracy, it is common for teachers to not be aware of what they do in their daily practices. A teacher might report not integrating financial numeracy in their classes when in fact much is done indirectly. Therefore, after the focus group discussion, we also asked some teachers to be video recorded in order to analyse their teaching practices. Only five teachers were willing to be video recorded.

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Focus group discussions allow teachers to see what others are doing and reflect on their own practice, increasing the likelihood of speaking up about their teaching. In addition to the research value of this approach, focus group discussions provide an opportunity to do professional development at the same time that data are collected from teachers. We were also concerned with the value that this research could provide to the participants directly, so we decided that the focus groups would also have a professional development aspect. During the planning meetings for the focus groups, we created a protocol to be used as a template for all groups regardless of who would be animating them. At the core of the protocol we established the four main questions to be posed to the teachers that should guide the discussions: 1. 2. 3. 4.

How do you teach Financial Education in your mathematics classes? What are your sources of information and training in financial education matters? How do you feel about the content and teaching of Financial Education? What are your needs in terms of content and pedagogy for teaching Financial Education?

These questions served as anchors to the discussions, especially because we wanted to guarantee that the research aspect would still be emphasised during the groups’ professional development sessions. The focus groups began with a questionnaire on personal knowledge of finance. This questionnaire, named The City, was developed by the Financial Consumer Agency of Canada (FCAC) and was available on their website.1 We expected teachers’ needs in respect of knowledge about personal finances to vary. Knowledge questions were also a way of avoiding discussion of their own financial situation. After the questionnaires, teachers were presented with mathematical tasks using financial contexts or financial mathematics concepts and processes. A discussion around the four research questions followed, which provided an opportunity to focus on some unanswered questions and points to be highlighted. Since it became clear in the focus group planning that time would be short, we designed an online questionnaire to obtain demographic information in advance of the focus groups. Participants were asked to complete the online questionnaire together with a consent form which, along with the entire research project, was approved by the McGill University Ethics and Research Committee (see chapter “Financial Numeracy Research in the Digital Era: Ethical Considerations”).

1

https://www.canada.ca/en/financial-consumer-agency/services/the-city.html.

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4 Implementation of the Methodology Between Spring and Fall 2017, data were collected from 36 teachers in focus groups of five to seven who reported integrating Financial Education in their mathematics classes. The groups were formed according to the different administrative regions of the province of Quebec. Teachers were recruited for the project through their school boards, especially through discussions with mathematics school board consultants (professionals providing pedagogical support to teachers). Most of the time, the school board consultant contacted teachers who were known to be interested in Financial Education and emailed other teachers on their school board to form the focus groups. Two of the groups asked for further clarification about our intentions, with a few school boards uncertain about the integration of mathematics and Financial Education. Given the context presented at the beginning of the chapter, some school boards thought we were referring to the newly approved Financial Education course. Since at the time the course had not yet been implemented, these school boards told us that we would not be able to collect data because they had not started teaching this course. Our response was to send them examples from the mathematics curriculum for each year of secondary school, clarifying that we were referring specifically to the mathematics curriculum and showing that Financial Education is already integrated, the extent depending on the teacher’s approach. This document was sent to the school boards and it proved to be effective since we were able to get their permission and recruit teachers to participate in the project. We wanted to question teachers working in the very different socio-economic contexts of urban and rural areas. For instance, in rural areas, the accessibility of public transportation is quite limited. Therefore, having a car is a necessity for many people, which is likely more expensive than using public transportation. In addition to that, the kinds of jobs available might differ too. Therefore, the needs in terms of FE might be different. The urban-area focus groups were held in Montreal, Longueuil and Quebec City, while those in rural areas were held in St-Joseph-de-Beauce and Trois-Rivières. It should be noted that two meetings were held in Longueuil, one for each high-school cycle. During these two meetings, teachers from a second Montérégie school board also participated. This school board has many schools in rural areas. All the meetings took place in French, except the one in Montreal which took place in English in a private school. The groups were facilitated by the principal researcher, the doctoral student who worked on the project, and the school board consultant of each region. All focus groups included men and women, except in St-Joseph-de-Beauce where no men participated. Before each focus group, we asked participants to complete an online questionnaire about their perceptions of financial education, their teaching practices, their sources of information, and their teaching needs (see chapter “Building a Research Instrument on Financial Numeracy in Schools (Quebec and Romania)”). The answers helped us analyse the focus group data and ask questions.

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Each focus group was facilitated by a member of the research team, and therefore had a unique dynamic in relation to the protocol that we designed. For instance, the focus groups from Longueuil were organised according to the secondary cycles taught by participants. One group had teachers from cycle 1 (Secondary 1 and 2), while the other group had teachers from cycle 3 (Secondary 3–5, from all mathematics options). Consequently, we used the situational problems designed for the respective cycle of the participants. For the other groups, we used a mix of situational problems at different levels, from Secondary 1 to Secondary 5. For more details on the tasks, see chapter “Using Tasks to Elicit Mathematics Teachers’ Thinking in Financial Numeracy”. At the end of each focus group, we asked if participants were interested in having one of their classes video recorded as part of our project. Participants were free to choose the topic of their lesson as the only requirement was to teach using any financial context. In each focus group, at least one teacher volunteered to receive team members in the classroom. The videos were recorded by the doctoral student who worked on this project as a research assistant. We asked teachers to tell us the time and the place, and the overall idea of their lesson. In some cases, the teachers recorded themselves because they lived in remote regions and had the equipment to do so. Two of those lessons are analysed in chapter “Making Sense of Mathematics: Two Case Studies of Financial Numeracy in Grade 11 Mathematics Classrooms”.

5 Concluding Remarks The focus groups aimed to collect information about teachers’ practices and needs. However, we wanted to give back to them, so that they gained something from us. The professional development part of the focus groups was really appreciated. In turn, it provided us with other insights about how teachers think about finances in relation to mathematics. Another important moment was when they filled out the questionnaire prior to the focus groups. The questionnaire was really helpful for collecting much demographical and contextual information about our participants’ practices and needs. This important research instrument took us in an alternative direction, which we explore in the next chapter.

References Government of Quebec. (n.d.). Secondary school education, cycle two: Mathematics, science and technology. Quebec Education Program. OECD. (2014). PISA 2012 results: Students and money: Financial literacy skills for the 21st century (Volume VI), PISA. OECD Publishing.

Building a Research Instrument on Financial Numeracy in Schools (Quebec and Romania) Annie Savard and Daniela Caprioara

1 Introduction As we saw in chapter “Background and Implementation of the Project”, the questionnaire developed to collect information about teachers’ practices, perceptions and needs was not part of the initial methodology plan for our research, which had been approved by the McGill Ethics Board. When the research team began to plan the questions and sub-questions to be asked during the focus groups, they soon realised that there were too many questions for the time allowed—3 h—as well as introducing the consent form. The solution they came up with was to have participants sign the consent form online prior to taking part in the focus group, and to add demographic and personal information questions not necessary for discussion in the focus group. The aim of the chapter is to introduce our research instrument—a questionnaire— and discuss methodological considerations for its adaptation in another country or context.

A. Savard (B) Department of Integrated Studies in Education, Faculty of Education, McGill University, 3700 McTavish Street, Montreal, QC, Canada e-mail: [email protected] D. Caprioara Universitatea Ovidius din Constanta, Constant, a, Romania © Springer Nature Switzerland AG 2021 A. Savard and A. Cavalcante (eds.), Financial Numeracy in Mathematics Education, Mathematics Education in the Digital Era 15, https://doi.org/10.1007/978-3-030-73588-3_5

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2 Sections of the Initial Questionnaire 2.1 Consent Form The questionnaire opened with a consent form briefly explaining the research project and the methodology for data collection. Participants were asked to agree to complete the online questionnaire, to participate in the focus groups, and to be video recorded during the focus group and—optionally—while teaching a lesson on FE in their classrooms.

2.2 Demographic Questions The demographic questions aimed to get a sense of the teachers’ professional experiences (see Fig. 1). Although it was important to associate teachers’ names with their experiences, pseudonyms were used. Gender was asked, in case any gender-related trends were found. Teachers’ backgrounds were important in order to know if they had been trained to teach FE in secondary school and to determine their needs. The level they currently taught and had previously done was also important in order to know the mathematical concepts taught, as well as to interpret their representations of their own FE teaching practices. Finally, information about the school and the school board enabled appropriate focus groups to be established.

2.3 Questions About Knowledge and Representations of FE The team was interested in teachers’ knowledge and representations of FE, and the meaning and importance they attached to teaching it. Any training they had received in the management of their own personal finances was also relevant (Fig. 2).

Name: ______________________________

Family name: ______________________________

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Fig. 1 Online questionnaire: demographic information

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1. To me, financial literacy is: 2. Do you teach financial education in your mathematics classes? Why? A. Do you usually discuss financial literacy with you colleagues? Yes No B. To what disciplines is financial literacy pertinent? C. Do teachers of other disciplines in you school approach financial literacy with students in their classes? D. Is their approach different from yours? How? 7. How have you learned to manage your personal finances? 8. Have you ever had any training (courses, conferences or workshops) about management of personal finances? What was its (their) content? A. Do you feel the need to have more training in this regard? B. When you need help, whom do you contact? 9. How do you keep yourself informed about managing your personal finances? Please indicate the resources you use:

Fig. 2 Online questionnaire: knowledge and representations of FE

2.4 Questions About Teaching Practices and Needs The team needed to know more about teachers’ initial teacher education, how they had learnt about teaching FE, and about their teaching practices, especially in regard to the concepts they do or do not teach, students’ motivation, parents’ reactions and their needs. These points were of particular importance for future research and professional development for teachers. The project funding agency was particularly interested in knowing whether the teachers used material created by different government agencies or by private-sector organisations interested in financial numeracy. Specific questions were therefore asked about different materials freely available for teachers (Fig. 3).

3 Conceptualising the Questionnaire in Romania The online questionnaire, specifically adapted for the context of the Romanian system and the individual respondents, was sent to primary-school teachers through the Romanian researcher’s professional networks. Factual data were placed at the end of the questionnaire to achieve greater objectivity. The Romanian version reflected the researchers’ interest in the teachers’ teaching practices, the financial concepts they taught and their degree of difficulty, students’ motivation for FE and an explanation of this. A very important objective of the Romanian questionnaire was also the identification of the needs of primary-school teachers in order to teach FE in school. The Romanian version sought to achieve at least two objectives:

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3. At each grade you teach, how many hours in a year do you spend in financial literacy? 4. What material(s) do you use in your mathematics classes to teach financial literacy? ● Financial Consumer Agency of Canada (FCAC) ● The Zone (FCAC) ● Financial Basics: A financial literacy workshop (FCAC) ● Your Financial Toolkit (FCAC) ● Make it Count: A resource for youth money management (Manitoba) ● Your Money (Canadian Bankers Association) ● Tesaffaires.com (AMF) ● FinÉcoLab (Cirano) ● Youth Blog (Desjardins) ● Charly et Max get involved! (Desjardins) ● Dollar sense (Société GRICS) ● Mes finances, mes choix (Cégep de Sherbrooke) ● Finance PGL (Fondation Paul Gérin-Lajoie) ● Parlons d'argent (Centre Ressources Jeunesse de l’Abitibi) ● Consommer sans illusion (ACEF Estrie) ● Treasure Academy ● Banking basics (Association des banquiers canadiens) ● Achète-toi une vie (TD) ● Fais ton budget (Université Laval) 5. Do you think such materials are appropriate for the grade(s) you teach? 6. Do you know the AMF, AMF's new toolkit, Bourstad, and Tes Affaires contest and website? AMF ( ) AMF's new kit ( ) Tes Affaires website ( )None ( ) 10. Which of the following financial concepts are you comfortable with? 1-Money and transaction: Consumerism, taxes, debit and credit card 2-Planning and managing finance: budget, saving, borrowing, investing (pension plan), credit, interest, earning, income taxes 3-Risk and reward: Investing (stock market), insurance, gambling 4-Financial landscape: Inflation, economy 5-Mathematical concepts: simple and compounded interest, period of interest, actualization and capitalization 11. Which ones would you use in your teaching? With which mathematical concepts? A. What topics do you cover during your financial literacy classes? B. Which financial concepts do you think are the most important to teach financial literacy at the secondary? Why? C. How do you select the concepts to teach in your classes? D. Are there concepts that are more difficult to teach than others? 12. Is financial literacy a motivating subject among the students in your mathematics classes? Why? A. Does the sociocultural context influence the interest and the choice of concepts to teach? B. What are the students' attitudes towards financial literacy? C. How do students react to financial literacy in comparison to other mathematics topics that you teach? 13. For you, does the socioeconomic background influence parents' reaction? 14. What are your needs to improve your teaching on financial literacy matters? Explain why. Financial knowledge ( ) mathematical knowledge ( ) didactical knowledge ( ) pedagogical material ( ) Other, explain ( ) Comments or questions:

Fig. 3 Online questionnaire: teaching materials and teaching practices

1. 2.

To identify teachers’ sources of information and training on financial education in order to assess accessibility and use. To identify their needs in terms of content and pedagogy for teaching financial education in order to support them by offering future training.

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4 The Revised Questionnaire 4.1 Consent Form Although institutional ethics boards exist in Romania, signed consent is not required for this type of online questionnaire. However, the opening of the questionnaire, in which the aims of the research project were briefly explained, was retained. Teachers volunteered to participate in the research project anonymously, so no questions about their identity were asked.

4.2 Demographic Questions The questionnaire, adapted specifically for the Romanian population, sought to interpret teachers’ practice and needs for teaching FE at primary-school level. Questions asked covered: Initial teacher education and years of teaching experience. Continuing professional development (in the Romanian system, teachers are required to attend training courses). Additional administrative responsibilities (to establish the extent of their influence on the staff). Nationality and gender. School location. Year group taught in 2017–2018. Population structure of the class.

Q II.1. Didactic degree (obtained by continuous training) and Q II. 2. Years of experience in the educational system. Q. II.3. Administrative responsibilities (No additional responsibilities for teaching, Responsibilities at the school level, Mentor of pedagogical practice, Methodist within the county school inspectorate, Director/Deputy Director, School inspector). Q II.4. Gender (Masculin; Feminin) and Q II.5. Nationality, Q II.6. Initial training (Liceul pedagogic, Colegiul pedagogic, Cursuri universitare de licenta, Cursuri universitare de masterat) and Q II.7. Continuous training. Q II.8. The class you taught in 2017–2018 year; Q II.9.School residence environment (urban/rural); Q II.10. County; Q II.11.. Estimate in percent (romani, romi, turci si tatari, maghiari si sasi, rusi si lipoveni, aromani, precizati daca sunt si altele).

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5 Questions About Knowledge and Representations of FE The questions in this section were structured into two categories of variables: knowledge and representations of FE and teaching practices and needs. Because financial education is not so well established in the Romanian educational system, especially at primary-school level, we first sought to understand primary-school teachers’ concepts of FE. It was also important to analyse teachers’ perceptions of the need for FE at primary level and their opinion on the person/institution that should deal with primary-school FE. We also wanted to know about their personal and professional training in the field of FE. Finally, we asked only those who do teach FE to their students about their teaching practices and needs Q I.1. What is financial education, in your view?). Q I.2.Do you consider that the financial education of students in primary classes is important / necessary? Explain why: Q I. 3. In your opinion, who do you think should be in charge of the financial education of primary school students? Q I.4. Why do you think it would be important to achieve financial education in school? Q I.5. Do you feel prepared enough to do financial education with your students? Justify: Q I.6. How did you learn to manage personal financial resources?; Q I.7. Have you attended any training (courses, conferences, workshops, etc.) on the topic of managing financial resources? If YES, what and on what topic?; Q I.8. Which of the following financial concepts are familiar to you: consumption, tax, debit card, credit card, budget, expenses, loan, investment, pension fund, simple interest, compound interest, tax, stock market, insurance, inflation, economy, capitalization? Q I.9. Do you do financial education with your students? If yes, please answer the following questions: Q I.9.1 How do you do financial education and in what context? Q I.9.2 What materials do you use to make it? Q I.9.3 Estimate the time allocated to financial education during a school year: Q I.9.4 What are your challenges / needs for teaching financial education? Q I.9.5 What financial concepts do you use in the classroom in financial education and why? Q I.9.6 Which of these concepts are easier to teach? Q I.9.7 Which of these concepts are more difficult to teach? Q I.9.8 Are your students motivated for financial education? Why? Q I.9.9 What would be useful for you to carry out financial education with your students? Justify the choices:

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6 Comparing Implementation of the Questionnaire in Two Different Contexts Throughout this process of adaptation, we noted elements to be considered. Some of them were learnt after the process, so this reflection is important for future collaboration.

6.1 Consent Form The two researchers discussed the consent form prior to the online launch. In terms of the research practices, the Romanian context was different from the Canadian one. Although verbal consent suffices in Romania, in Canada explicit and signed consent is mandatory even if the research is conducted in Romania. As a co-researcher on the project, the Canadian researcher had the same responsibilities within and outside Canada. The Romanian respondents were anonymous and their participation was voluntary, so the research could be conducted without signed consent.

6.2 Demographic Questions An important difference between the Canadian and Romanian research was the identity of the respondent. In the Canadian research project, the online questionnaire was a tool to collect more information on the participants of a focus group, while the online questionnaire was the only source of information in the Romanian research project. Anonymous questionnaires are the usual practice in research conducted in the Romanian system, in order to obtain more honest responses. For the same reason, the institutions in which they teach also remain anonymous. We asked only for the environment and the county, in order to have an idea of the national distribution of respondents. The structure of the population helped to define the profile of the school population that has access to financial education. Regarding the gender of respondents, in the Romanian questionnaire, two choices were given: female or male. It seemed appropriate at that time to the researchers and therefore they did not offer other choices. The Canadian questionnaire, from an inclusivity perspective, had a third choice, ‘other’, with a line to specify. Another aspect of inclusion is the language used. In the Canadian research project, the questionnaire was developed in French, because most of the participants were French speakers. For the one focus group that was conducted with English speakers, the questionnaire was translated into English. On the other hand, in Romania, the questionnaire was accessible only to Romanian speakers. In the Romanian education system there are programmes of study in the languages of minorities, but the teachers all speak the Romanian language.

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6.3 Questions About Knowledge and Representations of FE There are many questions common to the two research projects regarding initial and continuous training in FE. There is, however, a cultural difference between Canada, a capitalist country with rich experience of managing financial resources, and Romania, an emergent country in transition between communism and capitalism. The majority of the respondents were born and grew up in the communist era with no experience of finance management. They have had to adapt to the challenges of capitalism, including the concepts, tools and practices relevant to the management of personal or community financial resources. For this reason, the researchers were interested in how and when (both initially and ongoing) they did learn to manage personal resources and whether they felt sufficiently prepared to teach FE. The Romanian teachers’ responses were conclusive in the sense that almost all of them are self-taught in FE.

6.4 Questions About Teaching Practices and Needs The major difference between the two research tools consists in the questions about teaching materials used in the FE class. In Romania there are many commercially available games not necessarily for school use, but which can be used successfully in FE. The internet offers access to teaching resources produced in any country. The most important points are what is known, the interest of teachers and the actual possibility of acquiring these products. For this reason, the Canadian questionnaire mentions a list of teaching materials which was not included in the Romanian version. Because FE has been relatively recently introduced in the compulsory and optional Romanian curriculum and some financial concepts are not included, part of the questionnaire was restructured and simplified. Another difference between the Romanian and Canadian questionnaires consists in the reference to the role of mathematics in FE. There are two arguments for this distinction. First are the different levels of education – primary-school teachers in Romania and secondary-school teachers in Canada. The curricula are different in terms of content and vision. It was interesting to see whether financial mathematics concepts were present in the different mathematics curricula, because it shed lights on the intradisciplinary role of financial numeracy. Second, the students’ grade levels are different, with the elementary-school teachers trained as generalists, while the secondary-school teachers were trained as mathematics specialists. Their vision of mathematics is probably not the same, so their visions of the role played by mathematics in FE probably also differ. It was interesting for the researchers to find out whether the Romanian teachers consider that they are doing FE in mathematics lessons, considering that the measurement of value is mandatory in the primary mathematics curriculum.

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7 Elements to Be Considered When Adapting a Research Instrument Adapting a research instrument from one country to another implies the consideration of different elements. The choices we made were justified by the type of research undertaken, which was mainly qualitative methods in education, and the research theme, financial education. The two research projects were situated in different contexts, meaning that the adaptation process was contextualised in alignment with the characteristics of the participants targeted. The first element to consider is the participants: each of them is an individual. This seems obvious at first glance, but this element extends beyond the country they live in. The culture they live in is important, because it influences, among others, their willingness to participate, the wording of the questions, and how they respond. Trust in the researcher is an important element, because we asked them to give detailed information about their classroom practice. The respondents must feel comfortable and safe with the researcher. For instance, participants in Canada are used to speaking freely, because they have the right to do so. In Romania, the experience of about three decades of democracy is, on the whole, favourable for people in general and teachers in particular, to openly express their opinions. The fears of the communist era have largely been overcome. This also affects the language they are comfortable with, along with cultural references such as the name of their diploma: grade level, bachelor degree, bachelor in education or didactic degree. The second element to consider is the professional practice of the participants. The level of teaching (primary or secondary school) is a key point to consider, because the epistemology of the teachers is not the same. Teacher preparation is different, along with the content to be taught. Even if both teach mathematics, the concepts and processes covered are not the same, which leads to a different epistemology about mathematics. Professional practices are also influenced by cultural differences in schooling systems. In the Canadian province where the initial questionnaire was used, elementary school starts at grade 1 and finishes at grade 6. In Romania, elementary school starts at grade 1 and finishes at grade 5; curricula are different, and professional practices are tied to them. The third element is the language—the meaning conceptualised in one language that should be communicated in another one beyond mere translation. In this case, the questionnaire was developed in French and translated into Romanian using language related to the professional world of teaching.

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8 Concluding Remarks Adapting a research instrument is a complex process that goes beyond “copy and paste” questions or translation. It is a process that needs to take account of the cultural environment of the participants, their situated professional practices and the meaning given to the questions. In this sense, this adaptation of a research instrument is a re-conceptualisation of the research project.

Using Tasks to Elicit Mathematics Teachers’ Thinking in Financial Numeracy Louis-Philippe Turineck and Alexandre Cavalcante

1 Introduction As pointed out in chapter “Background and Implementation of the Project”, we used mathematical tasks during the focus groups in our project. These tasks touched on important financial numeracy concepts and had different degrees of depth and complexity. As a result of this project, one of the authors decided to investigate task design as part of his graduate thesis. In this chapter, we develop an argument for the importance of task design in mathematics classes and describe the use of such tasks in our focus group discussions. The chapter concludes by discussing some of the implications of these arguments for teaching and researching financial numeracy. For a review of the results of the focus group discussions, please refer to chapter “Financial Numeracy Research in the Digital Era: Ethical Considerations”.

L.-P. Turineck (B) Department of Integrated Studies in Education, Faculty of Education, McGill University, 3700 McTavish Street, Montreal, QC, Canada e-mail: [email protected] A. Cavalcante Ontario Institute for Studies in Education, University of Toronto, 252 Bloor Street West, Toronto, ON, Canada e-mail: [email protected] © Springer Nature Switzerland AG 2021 A. Savard and A. Cavalcante (eds.), Financial Numeracy in Mathematics Education, Mathematics Education in the Digital Era 15, https://doi.org/10.1007/978-3-030-73588-3_6

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2 The Importance of Task Design in Mathematics Classrooms The idea of classroom mathematics not being useful in everyday life is a notion shared by many who have been through the education system. Boaler (1998) identified a growing concern in mathematics education that students are not able to use the mathematics they have learned in school in real-world mathematical situations. Students often feel mathematics, as traditionally taught, has no connection with real life, and there has been a call by curriculum developers (Bosshardt & Walstad, 2014) for the mathematics curriculum to solve more real-life and authentic problems (Wiest & Vega, 2016). The goal of incorporating financial numeracy can be seen as joining up the disconnect that students face during traditional mathematics teaching. The goal of financial numeracy may be to present students with situations that are relevant to their real-world lived experience and show them the value of learning this type of content for real-world applications (Attard, 2018; Batty et al., 2015; Wiest & Vega, 2016; Sawatzki, 2016). Specific recommendations by Attard (2018) and Sawatzki (2016) call for more mathematical tasks using money and financial mathematics to be embedded in realistic and authentic problems. If they are to improve their engagement with mathematics, financial numeracy tasks must be driven by students’ interest and be authentic. Sawatzki (2016) suggests that financial numeracy tasks should be authentic, imaginable and useful. Sawatzki’s (2016) study, in which data were collected from 14 teachers and more than 300 grade 5 and 6 students, used financial tasks as an educational intervention in the mathematics classroom. The results showed that students viewed those tasks as challenging but they saw realistic contexts as useful “when you grow up” (p. 15). The students reported the process of learning through problem solving as valuable and they preferred tasks that were relevant to their everyday observations. The findings from the students reveal that they enjoyed the challenge presented in the tasks and were able to see the merit of these tasks beyond school. The implications for practice are that teachers are best suited to modifying existing tasks to their local setting to promote more authentic and accessible tasks. The one-size-fit-all model of task design is not ideal when incorporating financial numeracy into mathematics. The tasks must be authentic and situated in local settings to be effective. Sawatzki (2016) illustrated these intervention tasks with a visit to local shopping areas in Darwin, Australia. She researched the pricing at the two locations offering laser tag and designed a mathematical task using the real prices of these two locations. The tasks compared the price per game as well as the rebates, discounts and memberships offered for multiple games. The task design sought to understand local life, and create an authentic task that is situated in reality and is relatable to students’ everyday life and lived experiences. Thus, financial numeracy can act as knowledge connecting the real world with in-class mathematics and can provide students with the opportunity to use acquired knowledge about the real world in school, fundamentally empowering them to know how everyday transactions work. Consequently, we argue that financial numeracy

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can shape their identities by developing their ability to critically reflect on financial situations. This critical reflection is done through everyday financial choices with which they are presented in the form of mathematical tasks. Criticism of the existing mathematics curriculum points out that students are going through their education without basic knowledge of taxes, mortgages, insurance, cell phone plans and financial contracts. Financial numeracy can also become imbued in students’ critical consciousness by framing and shaping tasks that reflect real-world inequalities or realities. This type of discourse surrounding citizenship can be seen throughout the literature on financial education more broadly (Arthur, 2012; Blue and Pinto, 2017; Henning, 2017; Masnan & Curugan, 2016; Silva & Valero, 2018). The word “citizen” is very often cited in the literature as a goal of financial numeracy aimed at creating financially educated citizens. In line with the sociopolitical turn in mathematics education (Gutiérrez, 2013), some financial numeracy authors have sought to transform mathematics into a social justice discipline. Their emphasis on real-world applications and illuminating existing injustices helps promote the notion of emancipatory mathematics. For example, Pinto and Coulson (2012) argue that without social justice, financial numeracy would be “reduced to replicating inequities and continuing to marginalise already vulnerable low socioeconomic populations” (p. 26). Silva and Valero (2018) explored Brazilian high-school textbooks, the power of situational problems, and how tasks are framed to promote “good citizen behaviour” and develop caring citizens. One task that was analysed in these textbooks presented a mathematics situation about water use but included information such as “do not leave the faucet leaking, it can cause water waste of approximately 50 litres of water per day” (Silva & Valero, 2018). They conclude that tasks can extend beyond mathematics and have the power to govern people. The tasks that were analysed promoted good citizenship values and were in line with government regulations that outline what constitutes good behaviour. These textbook tasks were about politics, culture, and power, and were designed in such a way as to highlight and reflect current government regulation on citizenship, morality and ways of life. This highlights the political power that textbooks possess. The literature includes many recommendations for task design and implementation. Here we summarise these as a set of guidelines for teachers who are answering the call to create their own tasks (Bush et al., 2012; Sawatzki, 2017). Bush et al. (2012) calls for more teacher training and professional development on task design for teachers. Teachers can create their own tasks that are better suited to the local setting, culture and students, and are culturally relevant. This is in line with the ideas of Sawatzki (2017) who dismisses the idea of the “one-size-fits-all task” and illustrates efforts by teachers to create their own tasks within the students’ local setting. Sawatzki (2016) also provides recommendations for creating authentic, imaginable and useful tasks that students can connect to meaningfully because the financial situations provided are drawn from students’ observations and experiences with money (Sawatzki, 2017). Bush et al. (2012) suggests that financially situated tasks foster reasoning skills and decision making, and promote mathematical practices in middleschool mathematics. This will make mathematics relevant and applicable and help

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students prepare for their future (Bush et al., 2012).Tasks can reflect citizenship, moral issues and the responsibility to promote order and progress (Silva & Valero, 2018). Arthur (2012) calls for a more open-ended approach, with tasks that allow students to think for themselves, and suggests a focus on the collective aspects of financial literacy as opposed to individual consumer and personal finance (Arthur, 2012). To summarise, teachers and other educational stakeholders will benefit from the following five orientations with regard to financial numeracy task design: (1) (2) (3) (4) (5)

Tasks should be designed based on the local environment and setting (Sawatzki, 2017). Tasks should be closely connected to students’ experiences with money (Sawatzki, 2017). Tasks should be relevant and applicable to real life (Attard, 2018; Bush et al., 2012). Tasks can reflect aspects of responsible citizenship and morality (Silva & Valero, 2018). Tasks should focus on the collective aspects of financial literacy rather than merely individual consumer and personal finance (Arthur, 2012).

With these guidelines in mind, our research team constructed a series of situational problems to use during the focus groups with secondary mathematics teachers. Not only did these tasks present promising situations for them to use in their own classes, they also helped us to pose questions and collect data about the participants’ ideas, attitudes and practices regarding financial numeracy. The next section discusses the outcomes of this methodological strategy in more detail.

3 Different Stances in Focus Groups The research was conducted collaboratively by academics, school board consultants and graduate students from diverse academic backgrounds, so naturally, each focus group was conducted in a unique manner. In order to maintain consistency and ensure the collection of data that could be compared and could answer the research questions, the project team had several meetings to align their goals and their perceptions of the use of rich tasks in the discussions. Nevertheless, team members took different approaches to the collection of data through the focus groups, particularly as regards the role of the mathematical tasks within these groups. Two broad stances emerged from the focus groups: research and practice.

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3.1 Research Stance Three focus groups were conducted by team members primarily concerned with the research aspect of the project: Montreal, Quebec City and St-Joseph-de-Beauce. For example, during the discussion on the questionnaire The City, the team members scanned each participant’s response before sharing the answers. This was intended to ensure that the original response was kept for later analysis. Also, the discussion of this questionnaire mostly involved teachers agreeing or disagreeing with the answers. The team members did not impose any specific view on the answers and provided as much space as possible for the participants to share their understanding. With regard to the mathematical tasks, these members mobilised them as a way to have teachers reflect on their practices and provide insights into their perceptions and attitudes towards financial numeracy. Consequently, the team members integrated the research questions (which were meant to be posed directly to participants) during the discussion of the tasks, as opposed to asking them at a different time. Also, mathematically solving the tasks was not the main concern for these members. In fact, they often introduced the task, had a brief discussion on how to go about solving it, and then presented the solution. In these focus groups, the emphasis was on knowing if teachers were implementing similar situations, tasks or problems in their classrooms and if so, what their teaching approach was. If teachers responded negatively, i.e., they did not introduce similar tasks in classrooms, the researchers investigated what exactly they were doing in class and why. In both situations, though, the mindset was to generate and collect sufficient data to understand critically what happens in the secondary mathematics classrooms in Quebec with regard to financial numeracy. It was only at the end of the discussions that the team members suggested to the teachers that these tasks could be used in their own classrooms. They provided links to the online repository in which the tasks were available and invited teachers to take a chance and integrate financial numeracy in their mathematics classrooms. The advantage of such an approach to our project was that it generated richer data for analysis. The teachers spent more time discussing their perceptions and practices instead of trying to figure out the solution or the teaching of the mathematical tasks. Consequently, we were able to delve deeper into their needs and how to provide recommendations to policy makers. The disadvantage of this approach was the lack of support provided to the teachers after the focus group. We wanted the teachers to be excited about our ideas and construct a long-lasting relationship that could develop into new partnerships and investigations, and had we provided more support in that sense, more teachers could have volunteered to receive us in their classrooms.

3.2 Practice Stance The other three focus groups (two in Longueuil and one in Trois-Rivières) were conducted by team members that were primarily concerned with the professional

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development of teachers in terms of financial numeracy. These members work on teacher education in different institutions (school boards and university) and took part in the project due to its applied nature and potential to change financial numeracy teaching practices. In fact, two of these members were responsible for changes in the Quebec curriculum and its introduction of financial mathematics. The third member worked for several years as a school board consultant before becoming a university professor and supervising pre-service teachers in their Bachelor of Education field placements. When discussing the answers to the questionnaire The City, the team members mainly focused on getting teachers to understand the financial and mathematics concepts related to each question as opposed, for example, to discussing whether teachers feel comfortable teaching such concepts. They also kept the questionnaires together with the teachers while showing the answer key, a strategy that allowed the participants to check their answers and make sense of eventual wrong answers. These members used mathematical tasks to support the teachers in integrating financial numeracy in their classes. It was an opportunity for teachers to have concrete ideas of what to do and how to approach in their teaching practices, and this opportunity was valued by these team members because of the lack of specific support for mathematics teachers. Consequently, as the tasks were introduced, the discussion started with the possible solutions and strategies to calculate them. The group then highlighted which mathematics concepts could be taught through the tasks, and debriefed eventual questions or concerns about the teaching of financial numeracy. It was only at the end that the team members posed the research questions to the participants. They did so in order to ensure that the questions were answered, but the discussions did not last long. The advantage of this approach was that teachers felt better supported in terms of trying some of the tasks in their own class. This is evidenced by the fact that we conducted subsequent projects with these teachers, which included collaboration with social sciences. The disadvantage, however, was that the data directly related to our research questions were not as rich, and although we were able to analyse the participants’ perceptions indirectly based on their reactions to the tasks presented in the focus groups.

4 Concluding Remarks Our research team’s experiences with incorporating mathematics tasks in the focus group discussions highlight the importance of clear goals and alignment among team members. Any task can be used in multiple ways in a research project, and it is important to share the same vision. Despite our efforts to keep the data collection as consistent as possible, variations in the way the discussions were conducted emerged. In that sense, the mathematics tasks had differing roles according to who was in charge of the focus groups. We believe these differences occur naturally in a project which is the result of a collaboration between individuals with different backgrounds

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and motivations. It is also expected, in qualitative research, for data collection to be flexible in response to the dynamics of participants and researchers. Hence, we do not argue that there is one single approach that fits all needs. As in any situation in life, collaborative groups need to make choices according to the needs of each member, and we should be mindful of the limitations and benefits of each approach. In our case, having more emphasis on the research or practice stances had different consequences. While groups conducted with a research stance provided richer data for analysis, those conducted with a practice stance provided better opportunities for further collaboration. Our conclusion is that we should strive for a balance, especially when doing research with diversity of epistemologies. It is by having balance that our project found its strength.

References Arthur, C. (2012). Consumers or critical citizens? Financial literacy education and freedom. Critical Education. Critical Education, 3(25). Attard, C. (2018). Financial literacy: Mathematics and money improving student engagement. Australian Primary Mathematics Classroom, 23(1), 9–12. Batty, M., et al. (2015). Experimental evidence on the effects of financial education on elementary school students’ knowledge, behavior, and attitudes. Journal of Consumer Affairs, 49(1), 69–96. Blue, L. E., & Pinto, L. E. (2017). Other ways of being: Challenging dominant financial literacy discourses in aboriginal context. Australian Educational Researcher, 44(1), 55–70. Boaler, J. (1998). Open and closed mathematics: Student experiences and understandings. Journal for Research in Mathematics Education, 29(1), 41–62. Bosshardt, W., & Walstad, W. (2014). National standard for financial literacy: Rationale content. The Journal of Economic Education, 45(1), 63–70. https://doi.org/10.1080/00220485.2014.859963 Bush, S. B., McGatha, M. B., & Bay-Williams, J. M. (2012). Invest in financial literacy. Mathematics Teaching in the Middle School, 17(6), 358–365. Gutiérrez, R. (2013). The sociopolitical turn in mathematics education. Journal for Research in Mathematics Education, 44(1), 37–68. Henning, M. B. (2017). Elementary preservice teachers’ and teacher educators’ perceptions of financial literacy education. Social Studies, 108(4), 163–173. Masnan, A. H., & Curugan, A. A. (2016). Financial education program for early childhood education. International Journal of Academic Research in Business and Social Sciences, 6(12), 113–120. Pinto, L., & Coulson, E. (2012). Social justice and the gender politics of financial literacy education (vol. 9). Sawatzki, C. (2016). Insights from a financial literacy task designer: The curious case of problem context. Mathematics Education Research Group of Australasia. Sawatzki, C. (2017). Lessons in financial literacy task design: Authentic, imaginable, useful. Mathematics Education Research Journal, 29(1), 25–43. Silva, M., & Valero, P. (2018). Brazilian high school textbooks: Mathematics and students’ subjectivity. In E. Bergqvist, M. Österholm, C. Granberg, & L. Sumpter (Eds.), Proceedings of the 42nd Conference of the International Group for the Psychology of Mathematics Education (vol. 5, pp. 187–194). PME. Wiest, L. R., & Vega, S. (2016). Teaching middle-grades mathematics through financial literacy AU—Crawford-Ferre, Heather Glynn. Kappa Delta Pi Record, 52(2), 79–82.

Financial Numeracy Research in the Digital Era: Ethical Considerations Annie Savard and Alexandre Cavalcante

1 Introduction It is important to address ethical considerations when conducting research about financial numeracy. In Canada, there are certain responsibilities to protect participants involved in people-centred research. Discussions around money and financial praxis may also give rise to a variety of perceptions according to different cultural beliefs or schemas.

2 Ethical Considerations When Planning a Research Project In Canada, a research project at university level that involves humans or animals must be officially approved by an Ethics Board. These boards are present in every Canadian university and apply the benchmark for ethical conduct involving humans given by the Tri-Council Policy Statement.1 The goal of these policies is to respect humans participating in research projects. Since the Nuremberg trials of 1945–1946, special attention has been paid to 1

http://www.pre.ethics.gc.ca/eng/policy-politique/initiatives/tcps2-eptc2/Default/.

A. Savard (B) Department of Integrated Studies in Education, Faculty of Education, McGill University, 3700 McTavish Street, Montreal, QC, Canada e-mail: [email protected] A. Cavalcante Ontario Institute for Studies in Education, University of Toronto, 252 Bloor Street West, Toronto, ON, Canada e-mail: [email protected] © Springer Nature Switzerland AG 2021 A. Savard and A. Cavalcante (eds.), Financial Numeracy in Mathematics Education, Mathematics Education in the Digital Era 15, https://doi.org/10.1007/978-3-030-73588-3_7

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conducting research with humans, and Canada has accordingly raised the bar for human rights in research projects. Protecting all citizens is thus a priority, especially in regard to fragile or at-risk populations such as children or homeless people. The policy is designed around three components: respect for persons; concern for welfare; and justice. A course on research ethics2 is also required for anyone seeking to do research involving humans in Canada, i.e., researchers and graduate students. All researchers and graduate students should also obtain an ethics certificate in order to conduct a research project. Funding from the three major Canadian grant agencies depends on delivery of the certificate by the relevant academic institution.

2.1 Obtaining an Ethics Certificate In Canadian universities, researchers applying for an ethics certificate generally fill in a form giving an overview of the rationale and the expected value of the research, along with its goals, the research questions and the anticipated reporting of the results. The methodology of the project is explained, in particular the collection of data. The data collection instruments should be described in detail, which means setting out the questions participants will be asked. Data capture methods, such as video recording, should be stated and justified. Recruitment of participants and location of the research must be carefully detailed, including approval from other relevant institutions (such as a school board) and the compensation for participation in the research. Important information about what the project entails in terms of action, time and commitment must be clearly presented to participants. The risks and the actions to reduce those risks should be described, as well as how the privacy and confidentiality of participants will be respected at all times. The process of gaining consent from participants is detailed, including obtaining consent from authorities such as parents or employers. The researcher should provide copies of all consent forms that will be used.

2.2 Ethical Considerations Specific to FE Research There are a number of specific ethical considerations when designing and conducting an FE research project. It is very important to clarify for participants the aims and the intentions of the research project: in our case it was about developing our knowledge in regard to a particular situation among a situated population. We believe researchers should state, when this is the case, that they are not funded by banks or other financial agencies. They should also make it clear that they are not trying to sell services or provide financial advice. This clarification is important because participants might

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http://www.pre.ethics.gc.ca/eng/education/tutorial-didacticiel/.

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be concerned that sensitive information about their personal finances might be used without their consent. Talking about finance and money is perceived very differently among people and is taboo in some cultures. This might prove a major obstacle when advertising the research project and recruiting participants. A survey conducted in 2015 showed that 35% of Quebecers are not comfortable talking about personal finances and money in general.3 Moreover, 53% of the 1023 participants said that they are not comfortable discussing money and finance with colleagues. Reasons for this—which might also be reasons for not participating in a research project—could include not have enough information, having difficulty managing their money, or not wanting people to judge them for having (or not having) money. Other concerns are related to people’s attitudes towards children and teenagers. Some adults believe it is not appropriate to talk about money and personal finances with children and teenagers, arguing that they are too young to really understand. Others think it is the role of parents and family members, not the school, to discuss money and finances with them. For participants in a low-income environment, people may feel it is inappropriate to talk about money and finances when children have no money of their own (and indeed may not have enough to eat). During the focus group discussions, we noticed that the teachers had distinct perceptions of finances and ways of dealing with them. Some seemed confident when talking about finances, others were hesitant. The fact that most of them were not familiar with talking about money in an official setting with colleagues and a pedagogical school board consultant highlighted their comfort level. These concerns are important to consider when designing and conducting research projects. Beyond the administrative process required to obtain ethics approval (also called procedural ethics), it is also important to address ethics in practice – the difficult and unpredictable situations that arise in the course of research, described by Guillemin and Gillam (2004) as the ethically important moments.

3 Ethics During the Research Having conceptualised the research and received ethics board approval, we conducted our project with the focus groups, questionnaires and classroom recordings. During this phase of the research (data collection), we also encountered some situations in which ethical concerns were raised due to the nature of our investigation. In this section we report some of these situations and the ethical considerations for researchers in the field of financial education.

3 Survey entitled “Les québecois, l’argent et les finances personnelles”, conducted by Léger Recherche Stratégie Conseil. Available on: lautorite.qc.ca/grand-public/specialistes-en-educationfinanciere.

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3.1 Ethical Dilemma: Mathematics Dimension One of the classes that we video recorded was a Secondary 4 mathematics class (grade 10) in the SN option (natural sciences, typically regarded as the higher strand of mathematics). The researcher positioned the camera at the back of the classroom in order to capture the teacher and the students who had agreed to be video recorded. Students who did not agree were outside the camera’s reach, but still participated in the lesson. The topic chosen by the teacher was statistics, specifically the Mayer Line linear regression method. In line with our prompt of using a financial context to teach a mathematics lesson, the teacher introduced the context of insurance companies and how they use mathematics to define prices of insurance policies. The first part of the lesson used the context of car accidents, while the second used the context of mortgage prices in Canadian cities. In the first part of the lesson, the teacher explained how the Mayer Line method works and which processes are conducted with a dataset in order to produce the linear regression. In the second part, he separated students into groups and asked them to carry out the regression using different datasets (represented by different Canadian cities’ mortgage prices over time). According to the teacher, the main point in this lesson was to clearly define outliers and exclude them from the linear model. The students had difficulty defining the outliers in their new dataset, so they produced different linear regression models. In other words, the linear functions that they created to represent the behaviour of mortgage prices in a given Canadian city differed significantly. In addition to that, part of the task was to compare their results with those of “experts” predicting future prices in a news article. In the face of this apparent inconsistency, the students started discussing what to do. The teacher was circulating among the groups to answer questions and prompt their thinking having regard to the objective of the lesson. In an informal conversation with the researcher after the lesson, he stated that he had expected students’ models to diverge. He also mentioned that he intended the selection of the outliers to be challenging, as it reflects the subjective nature of this activity. Although there are statistical methods to define whether an observation is an outlier or not, the underlying principle is a subjective one, and students should understand the decision-making process behind it. At some point during the class, the students who were sitting at the back of the classroom, near the camera and the researcher, turned round and asked the researcher whether he understood the topic of the lesson and if he could check their work to see what was wrong. It is worth mentioning that the researcher in question had sufficient knowledge of the subject. He was a PhD student in mathematics education at the time, and had experience teaching mathematics in grade 10. The subject was familiar to him, which posed an ethical dilemma of how to respond to the students. Several responses would be possible in this situation, each presenting specific ethical concerns.

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First, the researcher could lie and simply reply that he did not know how to solve the problem. This approach would probably be the least disruptive to the lesson dynamics and would limit the interaction between students and the researcher. The lie is problematic, however, since students could easily figure it out (by asking about the researcher’s work, his background, looking him up online, or listening to conversations between him and the teacher on different occasions). The trusting relationship between teacher, researchers and students would be damaged by this approach. In addition, the idea that it is possible to be impartial during video recordings is an illusion. The very fact that there is an outsider video recording the lesson disrupts the dynamics in multiple ways. Our position is that we acknowledge this disruption as a dialogic process that generates insights about the research object, and account for it in any analysis and report. The second possible response is the opposite approach. The researcher could have revealed his knowledge of the subject and provided help with the task (either by assessing their work, or by giving the correct answer regarding the outliers). However, this approach brings the disruption of the classroom dynamics into question, since one of the teacher’s objectives was to let students experience the subjective nature of defining outliers and he had anticipated that they would struggle with this issue. Had the researcher provided any sort of direction to the students, the teacher’s lesson plan would have been disrupted. A further professional ethical consideration here concerns the depth of orientation the teacher would deem acceptable. Since the researcher and the teacher were not in fact collaborating in the lesson plan (the goal was to record a lesson as prepared by the teacher), the researcher could not know how much feedback the teacher himself would provide to students during the task. In the end, the researcher took a third approach to the students’ question. He told them that he was familiar with he subject, and that he understood their struggle with the outliers and the overall linear regression. He then mentioned that they could tell the teacher about their concerns, and that other students might be having the same difficulties. In other words, the researcher was honest with the students, acknowledged their task and their difficulty, and directed them to the appropriate resources in class to solve this issue. While we understand that this approach might not work every time (students can be more persistent, especially if they figure out that the researcher knows the subject), it was sufficient to have an honest, yet less disruptive, response to the students. The experience described here highlights an important practice that can help researchers prevent these kinds of ethical considerations: establish a protocol with the teacher on how to respond or interact with students during the class. Some teachers might prefer the researcher to be at the back of the class and to not interact at all with the students. Others might feel more comfortable including the researcher in the classroom dynamics. By establishing a proper protocol, both teacher and researcher clarify their expectations and prevent ethical considerations in situations like the one described. Regardless of the protocol established by the teacher, they should tell students more about the researcher’s role in the classroom. If the researcher is not allowed to help students, this should be clearly stated by the teacher. The researcher can then remind students that they are not allowed to help.

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3.2 Ethical Dilemma: Personal Finances Dimension During the focus group discussions, we faced some ethical dilemmas regarding financial concepts that students might or might not know about. For instance, some teachers mentioned that some of their students did not have a savings account, which makes talking about saving and managing money challenging. Other teachers and the researcher pointed out that teenagers should know about savings and having a savings account was a useful tool. One of the teachers then said that her 13-year-old son did not have a savings account and realised that he should have one at his age. It was an embarrassing moment, one that should not occur while conducting a research project. The teacher concluded that she would help him to start a savings account and the discussion moved on to other financial concepts. On another occasion, teachers were discussing compound interest. The discussion led on to whether or not they should discuss pension plans with students. Some teachers and the school boards consultants mentioned the benefits of 16- and 17year-old students learning about it. They would be building a model of investing at a young age, which would be important, especially for workers whose employers did not provide a good pension plan. One of the teachers said to the researcher that his daughter, a hairstylist, has no pension plan and asked for advice. Given the direction of the discussion, it would have been easy for the researcher to say yes. She realised, however, that this teacher saw her as a knowledgeable person in terms of personal finances. Even if she had some knowledge, she was not a certified financial adviser and did not want to be seen as usurping such a role or providing advice. She responded that his daughter should consult a certified financial adviser and check whether they were registered on the AMF website.

3.3 Ethical Dilemma: Professional Dimension A professional ethical dilemma arose in a focus group discussion in a remote rural area where teachers live and work in smaller communities. When discussing compound interest, investment and pension plans, one of the teachers mentioned that she was not on a permanent contract and her status with the school board was precarious, even though she had worked full time for the last nine years. The other teachers, all tenured, were surprised and said they did not know that. They then looked at her differently when she said that she was not putting money aside for her pension plan because she had just bought a house. This moment was a bit uncomfortable for this teacher and she said that she would start to save money. It was also embarrassing for the researcher, who did not want her to be perceived differently in her small community. We understand that it is almost impossible to prevent situations like that from happening. Teachers are bound to share their perspectives on whatever issue is being discussed. However, the ethical consideration here is about how the researcher

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manages the situation. In our understanding, the best approach is to reorient the discussion back to the topic in question (their practices, sources and needs). It is interesting to note that in the three situations presented, it was the teachers who said something embarrassing. They were not asked by the researcher about the management of their personal finances: that was not the aim of the research project. The researchers wanted to know more about the knowledge they had about finances and their teaching practices about it, not about what they do in regard to personal finance in their personal life. Another interesting point is that those three situations arose when talking about saving and investment, about making money, not about debt and not having enough money. This reminded us that for Quebecers, having a lot of money is a taboo subject. We do not know if a cultural schema (Nishida, 1999) lies behind these ethical dilemmas, but it highlights the importance of considering the culture of the participants not only when they are recruited, but also when collecting data.

4 Ethics After the Research: Implications for Researchers Having obtained ethics approval from all necessary parties and collected all the data in our research, we moved on to the analysis, and later, the publication of results. Ethical considerations also emerge in these two phases, although they are not as intensive (or unexpected) as other moments described previously. The first consideration is how to store all the data that was collected. In this project, we collected two types of data: responses to an online questionnaire and videos from focus groups and classrooms. With regard to video data, we decided to keep everything in an encrypted external hard drive to ensure data confidentiality. The online questionnaire data (teachers’ responses) were downloaded and saved in the same hard drive. One important consideration in this regard is the use of cloud services. The rise of information and communication technologies has made it possible for researchers to use online, collaborative tools in their work. However, it has also introduced problems of confidentiality and security in data storage. Using an online platform for questionnaires (popular ones include Google Forms, Survey Monkey, etc.) means that the data collected are not necessarily stored in the same country as the researchers and participants. On top of that, the data are kept by a private company that has no connections to the university issuing the ethics certificate. Any data leaks or change in legislation might affect the confidentiality of such data. In light of this issue, our university created its own server for online surveys, cloud storage, and collaborative platforms, among other tools that comply with its ethics boards. This means that all data stored in its servers are kept in the university premisses and are covered by Canadian legislation. Although we never asked about our participants’ personal finances, their perceptions of finance and financial education must be treated carefully and all data should be kept confidential since it

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might be seen as sensitive. In this project, we decided that all data would be stored for seven years, and after that would be destroyed. The second ethical consideration refers to how we analyse the data in our project. In this phase of the research, all identifiable data had to be removed in order for participants not to have their information shared inappropriately. Identifiable data means not only their names, but any sort of information (or a combination of different pieces of information) that hints at who that person might be. For example, the location of teachers’ work, their experience, level of education and university might provide clues to who that teacher is. This principle informed our analytical method and led us to choose to analyse the data by group instead of by individual. In other words, to preserve anonymity, we decided that the analytical unity of our research would fall within the focus groups (and their respective questionnaire responses and classroom recordings). In addition to not identifying our participants, we also attempted to remain neutral during the analysis. Our focus was on the teachers’ practices, sources of information and need to implement financial education, therefore we paid particular attention to not being judgemental about their opinions, experiences and perspectives. To enable us to remain as neutral as possible, the analysis of each focus group was conducted by at least two graduate students and principal researchers. We also had regular meetings to share the progress of the analysis, when we could discuss how to analyse certain moments or certain perspectives. The third and final ethical consideration emerges during the publication of the results of our research, both in the final report to the funding agency and as scholarly writings. In these publications, we kept in mind that all information shared could potentially be used by financial institutions as a marketing tool. For example, this research was conducted with teachers, a professional category with specific purchasing power. Therefore, financial institutions or companies could potentially use our publications to produce particular products or services targeted at them. Any information that could lead these institutions to create and sell products to teachers, however unlikely, was not published. Thus we protected our participants’ right of privacy, anonymity and neutrality.

5 Concluding Remarks Ethical considerations in research are not over when an ethics certificate has been obtained. As we have demonstrated, they permeate and inform all phases of the research process, and often lead to methodological and analytical choices in order to protect the participants (who have generously agreed to contribute to the project) and the researchers (who need to be careful in dealing with data). We hope the insights we have shared in this chapter can help others in their research, particularly in the realm of financial education. Since this is a relatively new field within education, certain sensitive aspects may emerge, and knowing how to manage them is key to successful project development.

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References Guillemin, M., & Gillam, L. (2004). Ethics, reflexivity and “ethically important moments”. Research. Qualitative Inquiry, 10(2), 261–280. Nishida, H. (1999). A cognitive approach to intercultural communication based on schema theory. International Journal of Intercultural Relations, 23(5), 753–777.

Results of the Project

This part presents the main empirical results of our project. Its two chapters describe and analyse data from distinct sources (focus groups, online survey and video-recorded lessons). In the eighth chapter, we introduce the representations of financial numeracy shared by secondary mathematics teachers from Quebec. We make connections between their perceptions and the stances they showed when discussing the teaching and learning of financial numeracy. We also describe their self-reported teaching practices in this area and their need to improve the integration of financial numeracy in mathematics. The ninth chapter focuses on the classroom data collected from participating teachers. We analyse two cases of mathematics classes in which the teachers used financial situations to teach specific mathematical concepts. These cases highlight how the teaching of financial numeracy can happen in multiple ways: from a contextual, conceptual or systemic perspective. Together, these chapters provide evidence of the state of financial numeracy in the Canadian province of Quebec. We hope to shed light on what teachers think and do, as well as to highlight a mathematics education perspective related to the broader field of Financial Education.

Mathematics Teachers’ Financial Numeracy Representations and Practices Alexandre Cavalcante

Abstract So far in this book, we have suggested that financial numeracy can be integrated in mathematics classrooms in multiple ways. Regardless of how this idea is introduced and developed, preparing teachers to approach it effectively is paramount. Research communities in mathematics education have proposed frameworks to understand the specific knowledge that mathematics teachers possess and to support them in developing this kind of knowledge to improve the quality of their instruction. In this chapter, I analyze the representations of secondary mathematics teachers when discussing their views of financial numeracy education and how they mobilize financial concepts in their teaching practices. Overall, the teachers feel insecure about their content knowledge of financial concepts as well as their teaching practices in class.

1 Introduction So far in this book, we have suggested that financial numeracy can be integrated in mathematics classrooms in multiple ways. Regardless of how this idea is introduced and developed, preparing teachers to approach it effectively is paramount. Research communities in mathematics education have proposed frameworks to understand the specific knowledge that mathematics teachers possess and to support them in developing this kind of knowledge to improve the quality of their instruction. In general, while European traditions of Didactics of Mathematics have studied the didactical transposition (Chevallard, 2006) of content, the rise in interest in teachers’ knowledge among North American scholars can be linked to Shulman’s work (1986, 1987), in which the author conceptualised subject matter knowledge (SMK) and pedagogical content knowledge (PCK) in an attempt to distinguish knowledge of the discipline from the knowledge required to transform that disciplinary knowledge for A. Cavalcante (B) Ontario Institute for Studies in Education, University of Toronto, 252 Bloor Street West, Toronto, ON, Canada e-mail: [email protected] © Springer Nature Switzerland AG 2021 A. Savard and A. Cavalcante (eds.), Financial Numeracy in Mathematics Education, Mathematics Education in the Digital Era 15, https://doi.org/10.1007/978-3-030-73588-3_8

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teaching. In my doctoral work, I have used this conceptualisation to reflect on what secondary mathematics teachers know in the area of financial numeracy. While SMK refers to formal, disciplinary mathematical knowledge, PCK is primarily concerned with the transformation of knowledge into forms that make it comprehensible to others. Shulman (1986) attempted to define it as “the most useful forms of representations …, the most powerful analogies, illustrations, explanations, and demonstrations—in a word, the ways of representing and formulating the subject that make it comprehensible to others” (p. 9). Ball and her colleagues (Ball et al., 2005, 2008) built on Shulman’s ideas to develop a practice-based theory of mathematical knowledge for teaching based on the mathematical work that teachers actually do on a regular basis. Following from this analysis, they propose a typology of teachers’ knowledge which elaborates on Shulman’s SMK–PCK distinction (Fig. 1). Within the category of SMK, Ball et al. (2008) distinguish common content knowledge (CCK) from specialised content knowledge (SCK) and horizon content knowledge (HCK). CCK is mathematical knowledge used “in a wide variety of settings” (p. 400) and not unique to teaching. By contrast, SCK “is mathematical knowledge not typically needed for purposes other than teaching” (p. 400). HCK concerns the trajectory of a particular topic through the curriculum and ultimately its relationship to advanced mathematics. Ball et al. (2008) also propose three sub-categories of PCK: knowledge of content and students (KCS), knowledge of content and teaching (KCT), and knowledge of content and curriculum (KCC). These three categories reflect the intertwined nature of mathematics and pedagogy in teachers’ knowledge. The authors identify four components of mathematics teachers’ knowledge which they suggest cut across the

Fig. 1 Framework of mathematical knowledge for teaching (Ball et al., 2008)

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different categories of their typography. This knowledge gives teachers the ability to: (1)

(2) (3)

(4)

Unpack or decompress mathematics, making it accessible to learners. In fact, they suggest that the ability to unpack mathematics may be one of the distinctive features of teachers’ mathematical knowledge. Connect mathematical content to other topics within a grade, and to related content in previous or higher grades. Retain the integrity of mathematics while still providing grade-appropriate explanations and examples that minimise the chance of learners developing misconceptions. Pay attention to the features of mathematical practice, such as representing, defining, reasoning and proving.

In this chapter, I discuss the main findings of the project upon which this book is based. Our focus was on secondary mathematics teachers from Quebec, Canada. We investigated their ideas, practices and needs with regard to financial numeracy. The chapter discusses three topics: (1) teachers’ representations of financial numeracy; (2) practices reported by the teachers in the questionnaire and during the focus groups; and (3) their needs to develop financial numeracy in their mathematics classes. The chapter concludes with a discussion on the connections between these three topics.

2 Teachers’ Representations of Financial Education and Financial Numeracy Ahead of their participation in the focus groups, the teachers answered the questionnaire described in chapter “Building a Research Instrument on Financial Numeracy in Schools (Quebec and Romania)”. Among the questions were some that prompted them to reflect on how they would define Financial Education (FE) in relation to mathematics. We decided not to use the word financial numeracy with the teachers for two reasons. First, this concept is not yet widespread among practitioners, and we thought it could cause confusion among the participants as they would not be sure of what it meant. Second and more importantly, this is a term that we propose as a way to conceptualise the role of mathematics for the development of FE (given the recognised importance of the discipline); it does not stand on its own in the mathematics curriculum, so it needs to be unpacked in order to become explicit. My goal here is not to assess teachers’ ability to define financial numeracy or to construct a final definition of financial numeracy based on their responses. It is, rather, a way for us to understand their perspectives and how these definitions inform what they do in class and how they perceive the practices that they bring to their mathematics classes. The table below summarises their responses according to their focus and what content they made reference to in their definitions of financial numeracy (Table 1).

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Table 1 Focus of the teachers’ definitions of financial numeracy Emphasis given in their definitions

Dimensions reported by the participants Personal finance

Citizenship

Mathematics

5

6

2

4

1

1

2

3

0

1

Total

11

12

7

4

General comments

9

General

Specific

Teacher focused

6

Student focused

3

Content focused

2.1 The Emphasis Given in Their Definitions Within their definitions, teachers emphasised a variety of aspects that relate to their professional stances. While some teachers defined financial numeracy through what is meant to be taught and by whom, others emphasised the end goals of financial numeracy. In order to make sense of their responses and understand how they might connect to their practices, I organised the diverse definitions into three emerging themes: teacher-focused definitions, student-focused definitions and content-focused definitions. Finally, I also accounted for some responses that represent general comments with no specific mention of the participants’ professional stance. Teacher-focused definitions: in 19 definitions, participants mobilised their knowledge of content and teaching (KCT) to emphasise the role of a teacher in developing financially numerate students. Among these definitions, which encompassed a variety of dimensions and concepts, the expressions used by the teachers engendered two main ideas: deliver content and develop consciousness. With regard to delivering content, some of the expressions involved ideas such as inform, teach, educate or equip. The other responses related to developing conscientious students; they used expressions such as raise awareness, prepare and familiarise. Overall, these two types of responses reveal our participants’ educational stances and the specific nature of their definition of FE as something that related to their own professional identity. Student-focused definitions: nine definitions drew on our participants’ knowledge of content and students (KCS). Participants who defined financial numeracy with a focus on the students provided answers that touched on the understanding and abilities that students should have of financial concepts and practices, but also their learning of these concepts. Those definitions that made reference to students’ learning on top of their understanding show that these teachers are also mobilising their professional stance in defining financial numeracy. Content-focused definitions: a minority of participants (six) defined financial numeracy according to the content it refers to. Those participants typically listed a range of different financial concepts (or practices) that are not necessarily connected to their curriculum requirements. In that sense, this definition can be understood as

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only indirectly related to their knowledge of content and curriculum (KCC). At the same time, these definitions speak directly to the teachers’ common content knowledge (CKK) stemming mostly from personal financial experiences with consumption, credit and investment. However, the participants did not specify what should be done in relation to such content in specific classroom settings, which reveals some of their orientations to financial numeracy and the challenges posed by the lack of proper support for teachers in this area. General comments: a total of nine answers given by the teachers did not provide a definition of financial numeracy in itself, but rather only recognised the importance and relevance of financial numeracy in schools and for individuals’ personal lives. Answers such as “important, practical, increasingly important, interesting” reveal that, although participants perhaps were not able to construct a definition (or interpreted the question in a different way), they do regard financial numeracy positively, ultimately revealing a positive attitude to potentially integrating it in classrooms.

2.2 The Dimensions of Their Definitions In addition to specific emphases, the teachers’ definitions of financial numeracy made reference to a variety of content: while some connected their definitions to the mathematics content necessary to be financially numerate, others touched on the financial content or even some attitudes associated with the concept. Personal finance: the majority of teachers constructed their definitions by drawing on personal finance content that must be taught. Their answers varied in terms of the specific concepts related to financial numeracy. The most general definitions made reference to the importance of financial concepts for understanding the world and managing our own finances: “to equip students to understand basic concepts related to the world of finance”; “to give students knowledge to enable them to manage their personal finances well”. These ideas highlight that participants recognise the epistemological and practical value of learning financial concepts. Some more specific concepts were provided by other teachers. Their examples touched on a myriad of different financial concepts, which reveals not only the richness of this topic, but also the complexity of integrating it into the mathematics class. Some definitions mentioned everyday elements such as salary, credit cards and budgeting, while others mentioned more complex concepts such as stocks, investments and income tax. Overall, however, teachers mixed different levels of complexities in their definitions, with responses such as: “To inform young people about how wages, credit cards, taxes, duties, investments, interest, RRSPs, stocks, donations, etc. work…”; “All that concerns a person’s finances: Consumption, investments, credit, budget…”; “It’s learning to manage a budget, a credit card, being able to compare offers and find the best one for yourself…”. Citizenship: a second category of definitions provided by the mathematics teachers connected financial numeracy to attitudes towards Financial Education. In other words, their definitions highlighted the importance of being financially numerate in

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order to become a critical citizen (although not all the answers mentioned citizenship). Some responses were solely connected to being critical: for example, one teacher defined financial numeracy as “preparing students for the future”. Others incorporated critical elements into other content aspects: “Ability to decode financial terms; ability to apply financial concepts; ability to make decisions within the context of financial situations”; “To bring the student to understand the basic financial concepts and certain economic principles important for the future”. These definitions provide an opportunity to see financial numeracy beyond the scope of personal finance, but also connect it to broader issues faced in our society. Most importantly, decision making seems to be a key aspect of understanding financial numeracy in multiple contexts: these teachers seem to be aware of the importance of financial numeracy as a way to make better—more informed—decisions. Mathematics: a minority of four teachers defined financial numeracy with specific references to the mathematical content involved with the concept. Their references positioned mathematics as a “context for teaching mathematics”. They also mentioned that financial numeracy might “allow students to understand certain calculations related to finance and to take a critical look”. Finally, these participants also argued that financial numeracy has important intersections while acknowledging the important in the mathematics class: “Knowledge of finance in math teaching and applicable to youth in secondary classes”.

3 Their Financial Numeracy Teaching Practices Financial numeracy seems to be integrated into our participants’ mathematics courses in Quebec. Out of the 36 teachers who participated in this project, 31 reported developing financial numeracy to a certain extent in their mathematics classes. Five teachers said they did not teach financial numeracy. Among the reasons for not teaching it, we found that the curriculum was the primary one: apart from compound interest, the teachers mentioned that financial concepts/topics are not in the secondary mathematics curriculum, therefore they “do not teach to any degree”. Other reasons were also mentioned “the curriculum, my competencies in this domain, the level of students, the time”. Interestingly, one teacher initially mentioned that she does not teach financial numeracy; however, in her response she acknowledged that “I make connections when concepts apply”, so we decided to include that response together with those of other teachers who currently approach FE. In fact, this type of response was reaffirmed by participants in the focus groups. They argued that they teach “in examples or problems because it is important for the development of my students”. Most participants referred to their teaching of financial numeracy either as a means to test mathematical concepts concretely (like perimeters and percentages) or as the content itself (especially in Secondary 5, CST option). Responses varied in terms of the intensity with which the teachers incorporate financial numeracy, however. When we asked about their approach to integrating financial numeracy,

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four teachers emphasised the limited extent of their practices, with responses such as “implicitly when the context is of a financial type, for example the calculation of a tax, a discount”. Two other participants argued that they teach FE indirectly. They went as far as to say that they teach “without using the word financial, we study the percentages (and therefore taxes, interest, deductions…) and I answer questions when the subject is suitable or questions arise”. One of them mentioned that she teaches “a few basics, because this knowledge is important in everyday life”. Nonetheless, even those participants who believe their practices are limited in terms of integrating financial numeracy seem to resonate with the other teachers who feel more confident in the way they approach financial numeracy. They emphasised the importance of teaching it—with responses such as “this is important for the student’s future”; “Certain notions of the program give me the opportunity to do so, I consider it essential that the students have certain notions in this regard”. For these teachers, their rationale for selecting financial numeracy stems from the curriculum, and it goes beyond the CST curriculum. For instance, one teacher mentioned that “I consider that this notion is relevant and fits well into the curriculum for Secondary II”. In other words, the general mathematics content is well suited to financial numeracy teaching for participants like this one. According to them, finance seems to provide a good context for mathematics, with one teacher recording that “it was the most useful thing they (the students) learned all year!” Overall, the majority of teachers agree upon the importance of integrating financial numeracy in their classes. However, there is no consensus as to the extent to which they currently do it. Consequently, we notice that the time allocated to this topic varies widely across the participants. First, financial numeracy seems to be embedded in the mathematics classes mostly in the upper grades of secondary school. For instance, when asked how many hours were devoted to financial numeracy in a year, the teachers reported an increased number in cycle 2 (Sec 3, Sec 4 and Sec 5) as opposed to cycle 1 (Sec 1 and Sec 2). However, there does not seem to be a consensus among the teachers as to how many hours of financial numeracy are being taught. Due to the disparities between CST and other streams of mathematics in cycle 2, the time devoted to the teaching of FE varies a lot. When controlled for the math streams, two main categories emerge in our data: those who teach mathematics CST, reporting an average of 25 to 40 h, and those who teach other streams of math, reporting zero to five hours in a year. In the specific case of the CST math option—the only stream that includes financial mathematics as a mandatory topic—one teacher responded by mentioning the chapter dedicated to financial mathematics, as well as the differences between cycles 1 and 2 of secondary school: “Sec 5 (1 whole chapter) therefore about twenty hours. In Sec 2 (especially during the chapter on % and scenarios-skills exam 1), around ten hours.” In fact, one participant believes that it is CST students that need financial numeracy the most. He argued that “for students in Secondary 5, young people lack culture in finance”. While students in the CST option have access to financial numeracy, the SN and TS options do not seem to provide the same opportunities: one teacher argued that “in maths SN5, [I teach] very little only when we talk about exponential functions.

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When I was teaching math in CST 5, we approached financial math in 20–25 lessons”. Another teacher who teaches SN said in the focus group that she explores investment and interest rates when teaching exponential equations. In terms of content, when teaching financial numeracy, teachers seem to explore both mathematical and financial concepts. Percentage was the mathematics concept most mentioned by the teachers of lower secondary grades, whereas in upper secondary they mentioned exponential functions and simple and compound interest. As with the time distribution, the content in upper secondary was heavily influenced by the math option. One teacher even mentioned that “in math CST 2 years ago I tackled all subjects except games of chance without insisting too much on formulas. In the math SN compulsory program, I only deal with interest, taxes and savings”. Other concepts included taxes and budgets. During the focus group, one teacher went into more detail, explaining that “the reality is that I do not have too much space and in the Sec 3 program I do not stray too far from the curriculum”. One teacher during the focus groups mentioned the importance of learning financial vocabulary, specifically for the students in the higher-level math (SN/TS): “in Secondary 5, we do optimization, so we do profit, but it’s basic vocabulary. In the exponential function, I found that I did not make enough vocabulary and then there, I wanted to embark there”. The financial concepts involved in their classes were much more varied: taxes, credit cards, discounts, profit, interest, savings, income tax, revenue, expenses, profit. During the focus groups, some mentioned that when approaching financial concepts in Secondary 1, they focus more on the mathematical concepts and calculations instead of teaching students a life skill: “we approach them in Sec 1, more how to do calculations, we don’t really approach them like I am teaching you a life skill, it is more how to do percentage calculations, but it should be!” Another teacher during the focus group mentioned that she uses her personal Excel budget spreadsheets and demonstrates how to do calculations within Excel: “that’s one thing I actually did in my CST class though, was I shared my Excel budget file with them so they could use it for themselves”. Throughout the focus groups, our participants discussed the rationale for their choices of financial numeracy content. Across all 36 teachers, it was noticeable that they relied basically on two criteria: the provincial curriculum and the real-life needs of students. The curriculum criteria were acknowledged by all teachers as a guide to what they had to teach in their classes, and therefore for most teachers it both enabled and imposed limitations on the teaching of financial numeracy. One teacher went as far as to say that she teaches “ONLY what’s in the provincial curriculum”. Two other teachers used the curriculum to select the depth they can go into with certain concepts. However, in the focus groups, some teachers expressed concerns with the teaching of financial numeracy. One of the participants was concerned with how much time is available in math classes and states her concern by saying that “in math class we need to do math, we can’t just talk for the sake of talking”. Others seemed to agree and another teacher mentioned: “I don’t talk about it much. Me just a little bit for a while, I talked a little bit but there in our math class, you have

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to do math, not just chat to chat”. The participants’ concerns with the curriculum were also revealed by the comments that they made about provincial exams. In fact, the pressure of the mandatory ministry exams was emphasised just as much as the content that must be converted from the curriculum. These exams, according to the teachers, limit their flexibility in class because they must cover all the content for the ministry exam. Consequently, solidifying the mathematics content first is a priority. Here is an excerpt from one of the focus groups which exemplifies this point: Yes but for me, when the program tells me that, I prepare the students to finish the course. I can go with general education, but before I go towards general education, I have to make sure that the student finishes his course. I can’t go and talk about, for example, actions, or credit cards when I know that the exponential function is not very well understood, so I master the program for the student to succeed.

All in all, we can see the tensions imposed by the curriculum when it comes to teaching financial numeracy. More clarity in the document, and more intentionality in educational policy would be beneficial in this context in order to support teachers in their practices. Finally, teachers were also mindful of their students’ needs regarding financial numeracy, leading to the second of their criteria for selecting concepts to teach. One teacher, for instance, selects the concepts according to the benefits that financial knowledge can bring to the students, arguing that she teaches “anything that helps prepare students for the real world”. In that sense, the teachers usually tailor their classes by adding new elements to the traditional textbooks. During the focus groups, most of them collectively agreed that when teaching financial numeracy, they create their own problems since these are not included in their textbooks. They use real-world resources such as old receipts and flyers from stores. Three participants provided specific examples such as: “Credit, the 2008 crisis, investments are all essential subjects for me because it is very concrete for the students.” During one focus group, however, a teacher was more sceptical. She explained that she uses contexts with money but it’s far from teaching financial numeracy: “there are not many situations, you know, there are contexts with functions in which we see with the salary context, the number of hours they work but it’s never, uh, it’s just a context where there is money. But, from there to do financial education, with a similar problem, my reality is not there at all”.

4 Their Needs for Teaching Financial Numeracy In order to introduce our inquiry into our participants’ needs to integrate financial numeracy in their practice, we asked all 36 teachers about the financial concepts that they are comfortable with. The results are shown in Fig. 2. The financial concepts are divided into four categories using the OECD (2017) framework: daily transactions, financial planning, long-term financial topics, and issues not related to personal finance (described in the figure as “society”). We found that the majority of teachers

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Fig. 2 Answers to the question “Which financial concepts are you comfortable with?” (N = 36)

feel comfortable with concepts related to daily financial transactions or financial planning activities (which occur repeatedly for adults). Since these are math teachers, each financial concept was also categorised according to the three most important mathematics topics required to understand it, and the average of teachers who responded to this content was calculated (see Fig. 3).The categories were drawn from the Quebec Education Program for secondary mathematics (Government of Quebec, n.d.). Each of the five branches of mathematics (arithmetic, algebra, probability, statistics and financial mathematics) is subdivided into three areas from which the mathematics concepts stem. The findings suggest that teachers are more comfortable teaching financial concepts which require a simpler application of mathematical knowledge. Unsurprisingly, teachers were less comfortable with teaching financial concepts that require a more complex mathematical understanding, generally those related to actuarial sciences and stochastic thinking. These results show that teachers need more opportunities to develop their knowledge of the content (financial concepts). Financial numeracy can be incorporated into basic mathematics at the secondary level, but if we are aiming to deepen students’ understanding, it is important that teachers feel comfortable with less trivial aspects, which also involve more complex mathematics. Further professional development could tackle this gap in their knowledge by addressing specific financial context concepts. Finally, in this project we also encouraged teachers to discuss how much they would appreciate training in Financial Education. We asked specifically whether they would like to have more training in order to incorporate financial numeracy in mathematics classes. A strong sense of eagerness to learn more about FE emerged from

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Fig. 3 An analysis of the mathematics behind the financial concepts with which teachers feel comfortable (N = 36)

the participants, with the vast majority, 30 of the 36 teachers, responding positively to the question.. We asked participants to provide more details about the nature of their needs. The majority (40%) said they would benefit from pedagogical materials to incorporate FE into their classes. It is important to note that pedagogical materials not only refers to textbooks or worksheets, but can often be physical materials (manipulatives or software) that facilitate the understanding of abstract concepts (like compounding). Another 32.5% of the teachers understandably feel the need to improve their own financial knowledge, on the basis that current teacher education programmes do not offer a component related to finance or its teaching. The last two responses, more relevant to the field in which the teachers are supposedly more comfortable (and have been trained): 15% and 12.5% of them reported the need for didactical and mathematical knowledge, respectively. During the focus groups, much time was devoted to discussing the teachers’ needs with regard to financial numeracy, which enabled us to develop a more nuanced understanding of their needs. The need for pedagogical materials was a repeated theme. Teachers claimed that the textbooks do not include adequate financial numeracy situations and they often resort to creating problems themselves. Teachers were impressed with the resources presented in the workshop1 ; however they expressed concern over their own ability to create such rich resources. One of them argued: “But developing 1

For more details of the tasks used during the focus groups, please refer to chapters “Using Tasks to Elicit Mathematics Teachers’ Thinking in Financial Numeracy” and “Financial Numeracy in Secondary Schools in Quebec: Implications for Leadership”.

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activities like that, I think it comes from knowing people who share. Producing it all ourselves is difficult. We don’t have time.” This highlights the overall sentiment from the focus group that teachers often create their own material and wish to have access to better materials. This implies that teachers are unaware of the resources available to them. The teacher quoted above emphasised the importance of sharing resources among educators. Teachers do not create their own resources because they do not like what is out there, they do so because they are unaware of what is available. Some teachers who were aware of the materials mentioned that online resources are hard to navigate. They wanted a more mathematics-friendly search process where teachers who plan to teach a certain mathematical topic can find resources that integrate financial numeracy components in that specific math content. For example, a participant mentioned that “I start exponential [functions] with groups of, uh, strong groups. Will there be something that can be exploited right away with exponentials? In terms of math financial for SM, because we touch it at the exponential level but I would like to touch it more, something to finance, something concrete immediately”. Most of the focus groups emphasised that teachers must cover all of the required math content before they can introduce financial numeracy lessons in their classes. One teacher even mentioned that students need to master theoretical mathematics components before being able to apply them to a new “fun” context: “What I see in high school 5 a lot is that I see teachers who have logs [students] and the rules for financial math go further, so you need to have seen all the logs and exponential functions before you can start doing business fun”. In other focus groups we saw a similar trend of teachers feeling they need to cover all the content for the ministry exams or that is mandated by the curriculum before being able to teach financial numeracy. This implies a disconnect between the math content needing to be taught and financial numeracy. The teachers do not see a clear relation between the two, prompting a need to present teachers with these materials with an emphasis on their connection to the curriculum. Overall, the results show that integrating financial numeracy in mathematics is more than simply introducing new content; it requires different aspects of mathematics, pedagogy (didactics) and finance to be mobilised.

5 Concluding Remarks In summary, the teachers who participated in this project recognised the importance of financial numeracy education, but believed it can be challenging to integrate it in their own classes due to external constraints (lack of materials and obstacles in the curriculum, for example). They also recognised their own needs and seemed to be open to learning more and refining their teaching practices accordingly. In this chapter, I have presented the representations, practices and needs as reported by the teachers themselves.

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Their definitions of financial numeracy are mostly based on the teaching– learning process, which provides some evidence that teachers have a specific epistemology towards Financial Education. Overall, these teachers seem to value financial numeracy as something important to be developed in secondary school. They recognise that teaching mathematics in financial contexts can be not only a tool to improve the quality of mathematics instruction, but also an important asset in the development of citizenship competencies, which include knowledge of financial concepts and decision-making skills that impact our everyday life. Their practices reveal that this epistemology might be heavily informed by direct factors, and not so much by external factors. Consequently, their needs refer to the specifics of the teaching of financial numeracy. They also recognise the limitations of their qualifications for fostering financial numeracy to make it explicit and systematised in secondary mathematics (something that the provincial government attempted to do in their 2016 curriculum revision). A major finding is that, despite the teachers’ recognition that financial numeracy is important and motivating to students, they do not seem to have enough support to improve their teaching practices in this respect. A significant proportion of the teachers feel insecure about their knowledge as well as their practices in class, and they do not seem to be satisfied with the current state of things. They acknowledge their need for more training, educational materials and knowledge to support students in developing financial numeracy. These data must be interpreted with caution for a number of reasons. First, teachers might already be integrating some financial numeracy aspects in their classes without noticing or acknowledging it. On the other hand, some might also be overestimating the impact of their practices in their responses to the online questionnaire. Second, their needs might also be under- or over-represented in the way they talk about training and materials needs or even their own ability to teach financial numeracy. However, the results described in this chapter point to the need for more research and development in the field of Financial Education in general, and financial numeracy in particular. More specifically, we believe teachers need more resources tailored to their reality (and that is one of the reasons why we decided to write this book). Part III of this book discusses some implications of this project for practitioners and teacher educators.

References Ball, D. L., Hill, H. H., & Bass, H. (2005). Knowing mathematics for teaching: Who knows mathematics well enough to teach third grade, and how can we decide? American Educator, 14–46. Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59, 389–407. Chevallard, Y. (2006). Steps towards a new epistemology in mathematics education. In M. Bosch (Ed.), Proceedings of the 4th Conference of the European Society for Research in Mathematics Education (CERME 4) (pp. 21–30). FUNDEMI-IQS.

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Government of Quebec. (n.d). Secondary school education, cycle two: Mathematics, science and technology. Quebec Education Program. Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15, 4–14. Shulman, L. S. (1987). Knowledge and teaching: Foundations of the new reform. Harvard Educational Review, 57, 1–21.

Making Sense of Mathematics: Two Case Studies of Financial Numeracy in Grade 11 Mathematics Classrooms Alexandre Cavalcante and Annie Savard

1 Introduction This chapter describes two situations or lessons, created by teachers participating in this research project, that show two ways of incorporating financial numeracy in mathematics classrooms. The first was conducted with students from an Englishspeaking school in Montreal. The students were enrolled in the science stream for mathematics (SN), therefore they did not have financial mathematics in their curriculum. The second lesson was recorded with French-speaking students from the cultural, social and technical stream (CST). These students have financial mathematics as part of their curriculum, and it was within this framework that the teacher situated his lesson (Government of Quebec, 2011). We video recorded and transcribed the lessons and analysed them according to the framework presented in chapter “Financial Numeracy as Part of Mathematics Education”. Here, we provide a description of the overall flow of each lesson, highlight important aspects, briefly analyse how these lessons fit with our model of financial numeracy, and finally discuss student engagement in the lesson.

A. Cavalcante (B) Ontario Institute for Studies in Education, University of Toronto, 252 Bloor Street West, Toronto, ON, Canada e-mail: [email protected] A. Savard Department of Integrated Studies in Education, Faculty of Education, McGill University, 3700, McTavish Street, Montreal, QC, Canada e-mail: [email protected] © Springer Nature Switzerland AG 2021 A. Savard and A. Cavalcante (eds.), Financial Numeracy in Mathematics Education, Mathematics Education in the Digital Era 15, https://doi.org/10.1007/978-3-030-73588-3_9

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2 Lesson 1: Science Stream Mathematics Title of the lesson

Linear Regression and Financial Numeracy: Insurance and Real Estate

Length

70 min

Grade and age

Grade 11–16/17 years

Mathematical concepts

Inference statistics Linear regression Mayer Line Method [Appendix 1 provides an overview of how this method works] Outliers

Financial concepts

Insurance Car insurance policies Real estate Evolution of real-estate prices

Materials used

Data sets in the form of graphs: Car accidents based on average distance traveled Real-estate prices in different cities over the year Scientific calculator

The lesson aimed to develop a linear regression using the Mayer Line (when the data are divided into two groups and the averages of each group are taken as the parameters for a linear function). The teacher decided to use the context of insurance companies and real-estate prices to teach the students how to create linear regressions. The main motivation, according to the teacher, was to show the students how insurance companies determine the prices that they charge. This lesson consisted of two tasks in which the students were presented with a set of data (in the form of graphs and tables) based on which they had to create a linear regression model. To do so, the students had to first decide which observations were relevant and which were not. In other words, they had to spot outliers and disregard them in their calculations. The first task served to introduce the idea of linear regression, so the teacher explained that insurance companies use statistical data to determine car insurance prices. He explained what a linear regression is and how students can easily create one based on their knowledge of linear functions. To do so, he provided the students with a scatterplot graph, with car accidents on the Y-axis and the number of people driving on the X-axis. He also provided the same data in the form of a table with each of the 17 observations (Fig. 1). The main discussion during this class was on the process of identifying the outliers. The teacher had the students observe the scatterplot graph and notice that one of the observations seemed to be way off the general tendency (which, for the Mayer Line Method, had to be a straight line). He explicitly showed the students which point was the outlier and what it meant. He connected students’ thinking with their real-life experience by explaining what an outlier means:

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Fig. 1 Car accidents task graph and table

Did you notice that (point)? I want you to circle that and write the word “Outlier” next to that point! … We are going to shun that point! That point is not representative of the data. It was either mistaken in data collection or something weird happened that day. Maybe there was a terrible snow storm and there was no accident! Maybe there was police and everyone on their best behavior! It is an outlier we will not include an outlier in our analysis.

We can see that, by describing the meaning of this outlier, the teacher provided the opportunity to connect the maths with the specific context of car accidents, which is important for students to have access to the task. The idea of outlier is rather complex when we situate it in a mathematical classroom, where students generally work with deterministic calculations and exact answers. Defining an outlier is a mathematical practice that requires a judgement about what is relevant or not to data analysis. Such practice cannot be implemented without understanding the meaning behind the data. This is the boundary between the mathematics and the financial context (in this case, car accidents). What we observe from this task is that this teacher was confident about the social context and how mathematics is connected to it. By clarifying the meaning behind the different statistical concepts related to linear regression, the teacher allowed students to deepen their understanding of the purpose and analysis of a linear regression. However, there was little discussion about insurance company practices and how they use this knowledge to determine the prices of insurance, or even what insurance means to the average consumer.

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Fig. 2 Real-estate task

The second task was introduced with a bar graph containing real-estate prices of different cities over a period of about 30 years with different time gaps (Fig. 2). The task required students to find the linear regression, make predictions about real-estate prices for the following three years and check whether their predictions match with those of specialists. Unlike in the car accident task, in this task the teacher did not emphasise the meaning of an outlier. Given that the students were also required to develop a linear regression model, defining the outlier(s) was important if they were to have the best model possible.

2.1 Analysis In this lesson, financial contexts are used to study explicitly mathematical concepts and processes. In the first part of the lesson and with the help of a scientific calculator, students were asked to calculate a linear regression to modelise how insurance companies determine prices for car insurance. In the second part of the lesson, still with the help of a scientific calculator, students were asked to calculate a linear regression to modelise real-estate prices. In this lesson, mathematics was used to make sense of financial numeracy. In both parts of the lesson, a scientific calculator was used by students to help them with arithmetic calculations, such as multiplication and division to find the linear

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regressions. The technology was used to decrease the cognitive demand on students. Instead of focusing on performing calculations, the tasks focused on modelisation and interpretation of two financial contexts. In this lesson, technology was used to support students to understand the mathematics needed to modelise and interpret two financial contexts. Students were engaged in the lesson. In fact, calculating car insurance is a particularly interesting financial context for them, because many students take driving lessons when they are 16 (the age at which young people can start driving in Quebec). Many of them knew about car insurance (mandatory in Quebec), because it is part of the driving lesson or because their parents had talked about it (car insurance for young drivers usually costs more than for drivers over 25). The second financial context, real-estate prices, might appeal to students in terms of understanding society and its economic complexity. During the lesson, students were mostly curious about the mathematics mobilised rather than the financial aspect. One of the reasons might be that they had some knowledge of the financial context, enough to be engaged in the modelisation process. Another reason might be that students wanted to modelise the measurement aspect of a financial situation. They were paying more attention to the mathematics first in order to understand the phenomenon. A third reason might be an effect of the didactical contract (Brousseau, 1997). These students might think that questions on mathematics are the ones that will be assessed, so they wanted to make sure they understood the mathematical concepts and processes first.

3 Lesson 2: Cultural, Social and Technical Stream Mathematics Title of the lesson

My Choice of Financing: Credit Cards and Interest Period

Length

75 min

Grade and age

Grade 11–16/17 years

Mathematical concepts

Financial mathematics Compound interest Interest period

Financial concepts

Borrowing and debt Credit cards and credit instruments Minimum payments

Materials used

Questionnaire about how credit cards work in Canada Laptops with internet access Spreadsheet of credit card payments Product prices as found on the internet Information on different credit cards

The lesson aimed to develop students’ practical understanding of how compound interest works in the context of debt repayment. The teacher decided to use the

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context of credit cards because it is an upcoming reality for the majority of students in grade 11, who are about to go to college or enter the job market. In addition, credit cards work in unique ways that are not necessarily understood by consumers, and the teacher wanted to explore not only the mathematical modelling aspect, but also the financial practice aspect (credit card rules, the importance of paying on time, etc.). This context also provided a rich opportunity to understand the impact of interest rates over time and under different circumstances. The teacher started with a few questions about how credit cards work, such as: What is the common interest rate, that is to say the maximum permitted by law that could be used by credit card-issuing banks? When you receive an account, do you have to make the payment immediately within a week, two weeks or three weeks? When you don’t pay your card in full, who pays? The merchant, Visa-Mastercard or the bank? If your account is overdue, are you eligible for the grace period? Are you entitled to the three weeks? Yes, no or does it depend on sales?

At this stage, students mostly participated by providing short answers with their ideas about how credit cards work in Canada. For the majority, it was a guessing game since they did not have much experience of the rules of such instruments. The questions served to establish their attention and engage them in the context of credit cards, particularly by showing how much they did not know about an apparently common product. In the second part of the lesson, which took up 75% of the time, the teacher introduced and monitored the task. Students were supposed to find the “products of their dreams” and check how much they cost. They were also asked to find two different credit card companies and collectinformation (interest rate, annual fees, etc.). The task asked students to calculate how much time it would take for them to pay off their credit card debt if they decided to only make minimum payments every month. Three aspects in particular reflected students’ engagement with different dimensions of this lesson: financial practice dimension, technology dimension and mathematical dimension.

3.1 Financial Dimension During the lesson, the teacher circulated among the students, checking their work and discussing issues that emerged. Most of the students’ questions were about financial practices related to credit cards. For example, having collected data about the time to pay off the debt, some students questioned what would happen if it took too much time to get rid of such payments. The teacher responded: You have seen everyone, in the simulations that you have done, there are some that it takes more than, you started with a large amount it takes 45 years, 50 years to do that. I received

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the question: what happens if we don’t pay and we die? We don’t stop paying and we die. It goes to our heirs unfortunately. Well if they don’t want anything, I think it can go to court.

Students then asked how inheritance works, to which the teacher replied: “I think that the Curateur public [Public Guardian] has the right to administer inheritances on behalf of others that have refused their debts, I am not sure.” One student shared something that happened in his personal life, and the teacher provided a more nuanced explanation: Your parents have life insurance huh? Life insurance usually covers the amount of debt okay. Okay, I’m not saying in all cases but someone who has no life insurance and who has debts, debts go to heirs or life insurance. The amount of life insurance, you have to pay the debts first. Okay, it can happen. But these are questions that at 15, 16 years old you do not ask yourself yet.

Another group was debating the purchase of a car using credit cards: Student:

Sir, is it okay that it costs $45,000?

Teacher:

It’s expensive [laughs]. No no, do your business… Take the number of years and calculate on the amount…

A few minutes later, the teacher turns to the whole class and says: Be realistic. By the way, what you pay $10,000, you don’t put that on a credit card. A car on a credit card… it is usually financing, it’s reimbursement - you pay for cars over three years, five years, four years, five years, but not on a credit card.

These examples, and others throughout the lesson, show the students’ level of engagement with the financial practice dimension of the activity. By questioning inheritance and the possibility of purchasing something apparently too expensive, not only were they posing authentic questions, but they were also connecting their personal experiences with the interpretations of the task. The teacher, on the other hand, also mediated their experience by allowing them to continue with “their business” and mentioning that it was not common. This kind of adaptation is part of what Chevallard and Bosch (2014) conceptualises as “didactic transposition”, or the adaptations that need to be performed in order to provide appropriate conditions for learning.

3.2 Technological Dimension In order for them to solve the tasks without carrying out every calculation, the teacher provided the students with a pre-programmed spreadsheet for credit card minimum payments (see Fig. 3). Students were able to customise the spreadsheet with the cost of their product, the interest rate charged by the credit card company (annual rate) and the policy regarding minimum monthly payments. The students also had questions about how to use the spreadsheet provided by the teacher, which highlight the importance of the technological dimension in this

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Fig. 3 Credit card task

lesson. For example, when students downloaded the spreadsheet, some of them were not sure what to do with the document as it was already pre-filled. The teacher then had to explain that “this is the whole document; you add it once here” and pointed to the cells in grey. During the exploration time, a group pointed out a mistake with the formatting of percentages in the Excel sheet. Some of the cells were not formatted to show numbers in percentage format, and this group was not sure if there was a mistake or if they had to do something about it. The teacher replied: Oh yes, I forgot to change the percentages. Question two, please change the percentages because card one is 12.9% and card two is 19.9%.

What we noticed is that the integration of technology in this class outweighed the shortcomings of the lesson. Despite the challenges that some groups faced, students were able to make the necessary calculations to anticipate what would happen if they only made minimum payments on a credit card. In responding to students’ questions about the format of the spreadsheet, the teacher showed the importance of being familiar with the technology used in class, a growing concern in the field of mathematics education (Koehler & Mishra, 2008).

3.3 Mathematical Dimension The interpretation of the results from the spreadsheet highlights the importance of attending to the mathematical aspect of this task. Because the first payment on the spreadsheet was year one, month one, the teacher had to explain that these dates should be understood as ordinal numbers, not cardinal. They represented the moment

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in time that the payment would be made, not the number of years and months that it would take to make that payment. The following excerpt illustrates this discussion: Teacher:

So look how long it takes you to pay it all that $2000 there.

Many groups:

16 years.

Teacher:

I have 16 years and two months. But be careful when I say year 16 month two there, because here I start at year 1 month 1. Fact that in, year 16 month 2 means that it is in the second month that I finished paying it okay. Just be careful, I’m coming to the file, we start at year one, month one, so I say year 16, month two, that’s how it’s been for 15 years, you understand? When I say - look here, year one, month one, I’m not starting at year zero, I’m starting at year one, so when you get to year 16, month two, year 16, month one is not for 16 years, it is for 15 years. Just be careful because I’m starting at year one.

These results align with those of Pournara (2015) in his work with pre-service teachers. When working with financial mathematics tasks, time is an essential component in the interpretation of results. Time can be represented in many ways in tasks like the one presented here, and it is important to know if payments are made at the beginning or at the end of the period, how dates are recorded and how much time has elapsed since the last payment. These are important aspects that are not necessarily at the core of the lesson, but which provide opportunities to learn mathematics meaningfully.

3.4 Analysis In this lesson, mathematical models are used to explicitly study how financial concepts work. With the help of a spreadsheet, students were asked to calculate compound interest and interest periods to modelise how credit cards charge interest. In this lesson, mathematics is used as a tool to understand financial mathematics concepts. In fact, the maths often seems implicit. If someone had looked at this classroom at any point, it would not resemble a traditional mathematics class, but rather a discussion about everyday contexts with the teacher sharing his experiences and providing some insights into the financial system in Canada. The spreadsheet used by students helped them with the compound interest calculations. The technology was used to decrease the cognitive demand on students. Instead of focusing on performing calculations, the tasks focused on modelisation and interpretation of two financial concepts: compound interest and interest period. In this lesson, technology was used to support students to understand the mathematics needed to modelise and interpret two financial concepts. It is interesting to note how this class was tailored to the needs of students in the CST mathematics stream (typically regarded as the lower maths option). The students could approach the problem of calculating compound interest and interest period in two ways: executing arithmetic calculations over and over until they found the point when the debt would be paid; or using financial mathematical formulae

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for compound interest (which, despite being structurally exponential functions, are usually memorised based on financial terms). Instead, this teacher established a middle ground for students to work on. The spreadsheet shows the dynamics of the credit card debt over each month, but it is automated in a way that allows students to focus on the interpretation of results. The students could therefore focus on higher cognitive aspects of the lesson, without the need to memorise formulae that would not necessarily be applied in real-life situations. Students were engaged in the lesson. Discussing credit cards interests them, because they know about them. In fact, we believe that most of the students have seen their parents using credit cards and other credit instruments. We also think that credit cards might be used by many of them when downloading games, music or videos. Furthermore, the need to pay online is a reason why young people might want a credit card. Buying the product of their dreams using a credit card is also about using credit instead of saving to buy a product they really want. This is an aspect that we noticed emerging at the end of the lesson when the teacher asked students about their conclusions. Most of the answers were negative about the use of credit cards, and it seems that it was a moment of awakening: while some said buying with credit cards should not be an option, others concluded that debts should be paid off as soon as possible, and it is not worth being burdened with an overdraft. During the lesson, students were curious, mostly about about the financial praxes of buying a product with a credit card. Some of them wanted something so expensive that a loan would be a better way of getting credit than a credit card, indicating that it is more important to discuss credit in general than mathematical models of it. For instance, when discussing interest period, students connected this concept to life expectancy in general. Calculating the time to pay down a credit card debt, one student asked what would happen if someone dies before the debt was paid, which allowed the teacher to discuss briefly rights of inheritance and how wealth is treated. While we realise that this kind of discussion is not strictly mathematical (and we are not arguing that it is the teacher’s duty to teach such things), it gave meaning to the mathematics in the classroom.

4 Discussion The two lessons presented in this chapter can be situated squarely within our model of financial numeracy. The first lesson, on car insurance and real-estate prices, could be situated as the modelling level. In both parts of the lesson, the students were asked to modelise mathematically the financial contexts presented using representational measurements. Since the SN curriculum does not include any financial component, using financial contexts to modelise is an interesting way of incorporating financial numeracy in class, as well as providing a deep and meaningful understanding of linear regressions (i.e., how they are used in real life). The second lesson, on credit cards, could be situated as financial mathematics, moving towards pragmatic measurement. In fact, since the lesson orientation was

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more towards numeracy than mathematics, it was difficult at some points to find the mathematics. Furthermore, the pragmatic aspect of the lesson was clear during the discussions, with students questioning the functioning of credit instruments, and discussing how interest rates are established, what happens with debt over time, how much time is enough, and how to interpret the results. Understanding how financial numeracy can be integrated in mathematics education makes aspects of these two lessons uniquely interesting. First, the lesson materials were created either by the teacher or by the school board consultant. The materials are not part of a textbook, so the teachers had to plan the lesson in addition to the regular schedule and materials they are used to. Both lessons used an inquiry-based approach in which students actively worked on problems that did not have a final answer. Instead, they exercised their judgement to verify their work and interpret their results. Second, both teachers have educational backgrounds related to finance (economics and engineering), so they are comfortable with this financial content. The project asked for volunteer teachers who were happy to have their classes video recorded, so we understand that not every mathematics teacher will be familiar or comfortable with the approaches described here, or with managing discussions of that nature. Chapters 10 and 11 in this book explore the implications of our findings for teacher education and professional development. These examples are representative of the possibilities that our analytical model poses for mathematics teachers. While sometimes teachers can use financial contexts as a means to clarify or shed light on the mathematics being explored (contextual dimension), at other times they might choose to delve deeper into these contexts and discuss with students how the mathematics emerges in real-life situations (conceptual dimension). Finally, teachers can also sometimes choose to unpack financial concepts and how they are constructed in relation to other epistemologies (systemic dimension).

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5 Concluding Remarks The two lessons described in this chapter represent two possibilities for teaching financial numeracy in mathematics classrooms: the modelling level using representational measurement, and pragmatic measurement. These different possible ways have implications for teachers and students. Even if the conceptual framework on financial numeracy is hierarchical, we consider that it is equally important to address each component in mathematics classrooms. In fact, we recognise that, alongside other institutional constraints, teachers have obligations in respect of a particular programme or curriculum. In many cases, it may be hard to find an opportunity to teach financial numeracy, which is why it is so important to seize those opportunities when they do arise. On the other hand, teachers may need support to identify them. For instance, how to link a mathematical concept to a financial praxis or a financial context is not always obvious at first sight. Our research project highlighted the fact that teachers, with their different backgrounds and experiences, have a variety of different understandings of financial education.

References Brousseau, G. (1997). Theory of didactical situations in mathematics 1970–1990. Dordrecht: Kluwer. Chevallard, Y., & Bosch, M. (2014). Didactic transposition in mathematics education. In S. Lerman (Ed.), Encyclopedia of mathematics education. Dordrecht: Springer. Government of Quebec. (2011). Secondary school education, cycle two: Mathematics, science and technology. Quebec Education Program. Koehler, M. J., & Mishra, P. (2008). Introducing TPCK. In AACTE Committee on Innovation and Technology (Ed.), The handbook of technological pedagogical content knowledge (TPCK) for educators (pp. 3–29). Mahwah, NJ: Lawrence Erlbaum Associates. Pournara, C. (2015). Talking time, seeing time: the importance of attending to time in financial mathematics. African Journal of Research in Mathematics, Science and Technology Education, 19(1), 82–94.

Implications for Teachers

This book would not be complete without a dedicated space for teachers. This part aims to speak directly to teachers and act as a tool for both teachers and stakeholders. Ultimately, it is teachers who implement financial numeracy in their classrooms. Therefore, it is important to include their thoughts and ideas about why financial numeracy should be part of mathematics classrooms in secondary (junior and high) schools. The fourth part presents, among others, material from two former secondaryschool mathematics teachers—Benoit and Jean-François—who went on to become mathematics pedagogical consultants into two different school boards (districts). In the tenth chapter, they explain their motivations, as mathematics teachers, for teaching financial numeracy. Over the last 20 years, they have shared with Quebec teachers the learning situations (lessons) they have developed on financial numeracy, grounded in mathematics. Each learning situations modelises a financial phenomenon. The learning situations were originally developed and shared in French: this is the first time they have been accessible in English. The leadership of Benoit and Jean-François and their work as consultants was so influential that some financial mathematics concepts have been incorporated into the CST Secondary 5 curriculum. The concluding remarks of the chapter contain comments and suggestions, with implications for leadership that are rooted in the field.

Financial Numeracy in Secondary Schools in Quebec: Implications for Leadership Benoit Brosseau and Jean-François Blanchet

Before presenting our recommendations about teaching financial mathematics and Financial Education, let us take a brief look at the origins of our interest in this subject. During a course on the theory of interest as part of an Actuarial Science degree in the early 1990s, the discovery of the effect over time on the cumulative capital of a sequence of equal instalments (annuity) at compound interest left us speechless. The longer the investment period (number of years), the faster the interest on payments and interest accrues. This cumulative capital ($) situation can be expressed as a function of time (years) by an exponential model. The long-term forecast of the capital accumulated by a series of equal instalments presents figures spectacular enough to fuel the interest of anyone wishing to build a nest-egg for their old age. This is a situation that is of great interest to students in upper secondary school. As educators, we thought it was not normal for students to acquire this financial knowledge only at university and only on specific courses. We believe that students need to be aware of personal finances right from high school onwards. It is not only compound interest investments that show such impressive results. There are many other domains, such as credit cards, lines of credit, personal loans and mortgages. Buying a home normally results in the most important debt that most people could have in their lifetime. However, it is also the most important investment that many people will make. Understanding the effect of compound interest when borrowing is paramount for all consumers and citizens. It is therefore important to dwell on it. For example, it takes about 18 years to repay a loan of $3000 on a credit card with an interest rate of 19.9% if you only pay the minimum each month (3% B. Brosseau Centre de services scolaire des Hautes-Rivières, Saint-Jean-sur-Richelieu, QC, Canada e-mail: [email protected] J.-F. Blanchet (B) Centre de Services Scolaire des Grandes-Seigneuries, 1325 rue Industrielle, Laprairie, QC, Canada e-mail: [email protected] © Springer Nature Switzerland AG 2021 A. Savard and A. Cavalcante (eds.), Financial Numeracy in Mathematics Education, Mathematics Education in the Digital Era 15, https://doi.org/10.1007/978-3-030-73588-3_10

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of the amount or $10). The total amount reimbursed is slightly more than twice the amount borrowed. When they become aware of this, almost everyone says: “Why did not I have access to this kind of knowledge during my studies?” As Quebec high-school mathematics teachers since the 1990s, we were certainly not alone in using financial contexts to address certain mathematical notions, such as decimals and operations on them, percentages, linear functions, exponential functions, and so on. However, we noticed that colleagues were afraid to use this kind of context because they did not feel comfortable with the subject. When we became mathematics school board consultants, we wanted to share this interest in financial mathematics and support teachers to integrate financial numeracy in secondary school. It seems clear to us that we need to include some financial concepts in our financial mathematics learning sequence because it helps to make sense of learning. In addition, for schools that wanted this essential learning for good citizenship to be recognised, we developed a local course aimed at developing specific financial skills that would allow students to obtain additional credits. In addition to building situations, we experimented with them in several classes (as teachers in the 1990s and as school board consultants in the 2000s), having held these positions for more than 10 years. This local initiative on financial mathematics led to workshops in several school boards, and to the annual conference of the Groupe des Responsables des Mathématiques au Secondaire (GRMS) in Quebec for several years. During our training sessions, we realised that some teachers shared our passion for financial numeracy. Besides this local initiative, continued to approach provincial ministerial bodies. We met with mathematics course managers at the Quebec Ministry of Education and shared our vision and our record on financial numeracy. In the wake of the reorganisation of the Cultural, Social and Technical (CST) stream, the decision was made in 2016 to add financial mathematics concepts to the Secondary 5 CST course, a move widely supported by teachers and professionals in the field. We had won our bet to have financial mathematics officially included in the high-school curriculum. Our classroom experiments have been very favourably received by students. The key is to offer them concrete and meaningful situations. Young people enjoy situations that address topics such as the operation of credit cards, investments with compound interest or loans of all kinds (car loan, personal loan, mortgage, etc.). The teachers we have worked with are unanimous on this point: real-life situations that deal with financial mathematics are of much greater interest to students than other mathematical notions in the curriculum. Some students who are not motivated by other aspects of the mathematics course suddenly become very attentive. We have even heard students say, “Finally, a few things that will serve us concretely in everyday life”. Many positive comments have come from parents who are happy to see these topics discussed in class with their child. Some financial situations even evoked some parent–child discussions at home. In our opinion, it is essential to foster young people’s curiosity about certain notions of personal finance and thus develop skills in this area. As the foundation of

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high-school mathematics teachers, we first developed financial mathematics situations and then expanded our field of expertise to include elements of personal finance (hence moving toward financial numeracy). This chapter brings together some observations, reflections and recommendations regarding financial numeracy. Before 2016, no financial mathematics content was prescribed in the Quebec Education Program (QEP). Only limited financial contexts were available for teachers to address certain mathematical concepts. As these were only proposed contexts and not mathematical content elements, the students only came into contact with them if the teacher had an interest in financial mathematics, felt competent in the field and/or considered these contexts relevant to education in general. Prior to 2016 the teaching of financial mathematics remained a local initiative, although some teachers were able to benefit from the dissemination of material during GRMS training. Another finding is that the majority of teachers feel that they are not well equipped with this dimension of mathematics, which means that few of them are tempted to approach financial contexts in the classroom. Teachers lack training in Financial Education and of course financial mathematics. The initial training of high-school mathematics teachers (baccalaureate) does not include these ideas. Moreover, since these are only contexts to be mobilised and not elements of the prescribed course, unless the teacher shows an interest in FE, no training is offered. The degree to which they feel comfortable will depend on their interest and experience with personal finances. The needs of teachers in terms of training are at the mathematical and didactic levels, but especially at the financial level (financial knowledge). The latter are currently available in the school boards through educational advisers or training at, among others, the GRMS annual conference. We therefore recommend, first, that future initial teacher education should cover financial mathematics. and second, that school boards, through their pedagogical advisers, provide support to teachers with respect to financial mathematics and FE. Another finding is that the amount of time spent in the classroom depends on two factors: the level of education and whether or not the teacher has been trained in financial mathematics. Secondary 5 CST teachers give more time to financial mathematics than other teachers because content elements are now prescribed. In addition, teachers who have been trained give more time to financial mathematics and financial numeracy than untrained colleagues. The latter often separate the mathematical content from the financial content, whereas they should develop in synergy. Pedagogical advisers play an important role in training and sharing resources. Most teachers who cover financial numeracy in the classroom use materials that they prepare themselves, as these situations are rarely found in published materials. Material from all walks of life can be used to address different contexts: promotional blurb, invoices, receipts, payroll, personal budget, tax grid, etc. Few materials and resources produced by the AMF (Autorité des marchés financiers du Québec) or other organisations are widely available in the education community, so teachers are not familiar with them. Materials developed by institutions such as the AMF must be accessible and circulate among teachers. This material should also be presented with clear and concise explanations related to the course.

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Concluding remarks. We recommend a greater degree of collaboration between school boards and organisations such as the AMF to promote financial numeracy in Quebec. By partnering with educational advisers or teachers, the AMF would ensure that the material produced is better suited to teaching and the Quebec Education Program would benefit from greater visibility in the education network. The AMF would also benefit from using the well established GRMS network to disseminate the resources available to teachers. We also recommend the use of technology, such as amortisation schedules in Excel solver finance, as they are excellent tools for addressing more complex and non-prescribed concepts. For example, the repayment of a debt (concept of annuity: continuation of equal payments) can be tackled with the help of an Excel spreadsheet, in order to eliminate tedious and repetitive calculations. In the Technical Sciences (TS) and Natural Sciences (NS) streams, this same technology makes it possible to validate calculations made using the calculator. We recommend that Secondary 5 mathematics and Financial Education teachers take the time to standardise their practices and define each others’ roles. For example, for some situations such as the use of a credit card, mathematics could focus more on the calculation aspect while the concept of credit in a broader sense requires a different approach. We also recommend that a mathematical field be given to financial numeracy (or financial mathematics) in mathematics courses in Quebec, similarly to arithmetic and geometry (Jean-François Blanchet and Benoit Brosseau, GRMS training, October 2011). Having a mathematical field would bring importance to this essential knowledge that will help students better manage their personal finances throughout their lives. The great interest of students in financial mathematics is explained by the relevance of addressing these notions. The students feel that these notions will really serve them in real life. This is the main reason for their interest in financial numeracy and financial mathematics.

Some Financial Numeracy Tasks for Secondary-School Mathematics Classes Louis-Philippe Turineck and Annie Savard

1 Introduction The tasks presented in this chapter were devised by Louis-Philippe Turineck as part of his master’s degree project, for which Annie Savard supervised him. These tasks, which can be used in mathematics classrooms from grade 7 to grade 9, need to be contextualised for the country in which they will be used. For instance, in many countries, taxes are included in the price, but this is not the case in Canada. The most important thing is that the tasks should engage students in modelising meaningful financial praxis.

2 Task 1: Margaret In this task, Margaret, a student who wants to buy basketball shoes, has various options for earning money by doing good deeds for her neighbours. The task values saving, working towards a goal, helping the community, and understanding the time and value of work. It touches on percentages, arithmetic, division and decimal numbers, and can be used to explore topics such as tables of values, graphs and linear equations. The task can be approached in many different ways and is openended enough for students to engage in critical thinking about the situation and use an appropriate mathematical method or model to solve it. L.-P. Turineck (B) · A. Savard Department of Integrated Studies in Education, Faculty of Education, McGill University, 3700 McTavish Street, Montreal, QC, Canada e-mail: [email protected] A. Savard e-mail: [email protected] © Springer Nature Switzerland AG 2021 A. Savard and A. Cavalcante (eds.), Financial Numeracy in Mathematics Education, Mathematics Education in the Digital Era 15, https://doi.org/10.1007/978-3-030-73588-3_11

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Margaret Margaret is a grade 6 student on the school’s basketball team. She has been saving up for new shoes since last Christmas. They cost $200 plus taxes (15% in Quebec) but she only has $15.50 in her piggy bank. Her mom tells her that she can earn money by helping out the neighbours. She has a few choices: – Paul says he will pay Margaret $8.50 to shovel the snow for an hour once a week. – Sally says she will pay Margaret $14.50 if she can babysit her son on Friday nights for 2 h. – Ali says he will pay Margaret $35.50 to walk his dogs for the month. This task takes 20 min per day, five days a week. – Jimmy says he will pay Margaret $1.50 every day, if she comes on weekdays to feed his cats after school. This task takes 5 min per day, repeatable up to five times per week. – A local company says they can pay Margeret $7.50 per day to deliver the weekly newspaper. This task takes 30 min and is repeatable up to twice a week. Questions to answer: 1. 2. 3. 4. 5.

How long will it take Margaret to save enough to buy her shoes? Which tasks offer the best price per hour? Which task offers the worst price per hour? Is it reasonable to assume Margaret can agree to do all the tasks above? Why or why not? If you were Margaret, how would you approach this situation? What tasks would you choose to do and why?

3 Task 2: Cellphone This situation presents students with the opportunity to explore the different options consumers have when choosing a cellphone. The prices indicated in this task were taken directly from telecommunications company websites, and reflect the actual costs associated with each service and each device. As consumers, we can either buy a phone ourselves and pay cheaper monthly fees on a “bring-your-own-device” plan, or we can pay upfront and sign up for a contract with a company (usually for 1–3 years) with significantly higher monthly fees. The tasks allow students to decide the parameters of the problem, represent the situation with a mathematical model of their choosing and justify their choice of cell phone plan. This type of mathematical situation reflects real-life choices faced by consumers choosing a specific device and cell phone plan. The task also prompts students to identify the benefits of each cell phone plan, and to question why companies can charge more for similar

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service/similar devices. The financial literacy goals of this task are in line with the PISA (Program for International Student Assessment) Financial Literacy Framework (OECD 2015) under the heading of Money and Transaction: Explore and analyse a financial market. Cell Phone Problem You have decided to purchase a new phone but aren’t sure about which phone and plan you should go for. For now, you know that you want a plan with at least 4 GB of data and Canada-wide free calls. After consulting different providers’ websites, you have gathered all the information from different providers. The carriers offer you two options: you can either buy the phone outright and pay for a “bring-your-own-phone” plan monthØly; or you can sign up for a contract with an upfront cost and monthly total. Each phone has an associated upfront cost and a monthly cost.

Questions to answer: 1. 2. 3.

For your phone, which plan is better, the contract or the bring-your-own-device plan? Represent your solution using a mathematical model. What kind of conclusions can you draw from the information given? Justify your reasoning mathematically. Discuss your results with another group. What similarities and differences did you notice about the prices for different devices?

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What conclusions can you draw about cell phone prices with each company? Is it better to go with a contract option or with a bring-your-own-device plan? Name two benefits of the bring-your-own-device option. Name two benefits of the contract option. Why can a company offer the same service but at different prices? Why do you think they can charge different prices for phones with similar performance? Why do certain phones cost more? Which phone is the most expensive and which the least expensive?

4 Task 3: Chicken Shack This task is inspired by a chapter in Freakonomics by Stephen J. Dubner and Steven Levitt (2005), economists who explore “the hidden side of everything”. In their book, they recount a story about visiting a local chicken shop in New York City. As economists, they were baffled by the pricing model. This chicken shop offered specials which include a medium fries and small coleslaw. If you look at the first option, the 2-wing special costs $3.03 and the 3-wing special costs $4.50. The only difference between the two options is an extra wing, which costs $1.47 more. If you assume that each wing cost $1.47, then that would bring the implied price of the fries and coleslaw to $0.09. Further, the difference between the 5-wing and 6-wing deals is significantly more than the difference between the 4-wing and 5-wing deals. This is odd since usually economics of scale suggest you would be offered better value for ordering a greater quantity. This situation presents students with the opportunity to note pricing model patterns by graphing the pricing structure using a broken-line graph, and to explore any potential flaws in the pricing structure. It also encourages them to create their own pricing structure and justify their prices. YuJin’s Chicken Shack YuJin decides to open her very own chicken store in her hometown. She calls it YuJin’s Chicken Shack. She is designing her menu and wants to offer her customers chicken wing deals. All of her specials include a medium fries and coleslaw.The prices for YuJin’s Chicken Shack are as follows:

YuJin’s Chicken Shack Chicken Wing Deal: Includes medium fries and a small coleslaw 2-Wing Deal

$3.03

3-Wing Deal

$4.50 (continued)

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(continued) 4-Wing Deal

$5.40

5-Wing Deal

$5.95

6-Wing Deal

$7.00

Questions to answer: 1. 2. 3. 4.

Graph this pricing structure on a broken-line graph. What do you notice? Compare the 2-wing deal and the 3-wing deal. What do you estimate is the price of the fries and coleslaw? What offers the best value? Justify your answer mathematically. YuJin asks for your services as a menu consultant. She asks you to make recommendations and to re-do her pricing model for the special. Fill out the chart below with a new pricing model, providing a brief justification for your choice of prices.

YuJin’s Chicken Shack Chicken Wing Deal: Includes medium fries and a small coleslaw 2-Wing Deal

—

3-Wing Deal

—

4-Wing Deal

—

5-Wing Deal

—

6-Wing Deal

—

5 Task 3: Chicken Shack Part 2 This task is an extension of the previous Chicken Shack task. I wanted to incorporate an element of Microsoft Excel into one of my tasks. Excel is a very useful tool for organising data, completing complex maths functions, turning data into helpful graphs and charts, and analysing data. This tool is under-utilised in schools, especially during elementary and early years of high school. Thus, I wanted to create an extension of one of the tasks that would allow students to use Excel and develop their competencies with the software. The task can be delivered following a few introductory lessons on Excel. Previous knowledge required includes how to input data,

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how to do basic mathematical calculations using functions, and how to create graphs and charts to represent data. This situation presents students with the opportunity to graph the data on the restaurant menu and choose the best graph to represent the pricing model. Then students will be asked to analyse this graph and draw conclusions. They will go on to create a new menu with new prices and choose the same graph to compare their new pricing model with the old one. This will allow them to use graphs to analyse, make mathematical claims and justify their reasoning while developing their competencies with Excel spreadsheets. Excel Task YuJin decides to open her very own chicken store in her hometown. She calls it YuJin’s Chicken Shack. She is designing her menu and wants to offer her customers chicken wing deals. All of her specials include a medium fries and coleslaw.

YuJin’s Chicken Shack Chicken Wing Deal: Includes medium fries and a small coleslaw 2-Wing Deal

$3.03

3-Wing Deal

$4.50

4-Wing Deal

$5.40

5-Wing Deal

$5.95

6-Wing Deal

$7.00

Questions to answer: 1. 2. 3. 4.

5. 6. 7.

Input the data of this pricing model in Excel. Choose an appropriate graphic or chart that would represent the pricing model of YuJin’s Chicken Shack. What do you notice about the chart? What conclusions can be drawn by examining the chart? YuJin asks for your services as a menu consultant. She asks you to make recommendations and to re-do her pricing model for the special. Using an Excel function, create a new pricing model. Create a new chart using your pricing model and compare it to her original pricing chart. What conclusions can be drawn? Use both charts as evidence to justify your claims. Using your own pricing model, use an Excel function to map out your pricing model for 7-, 8-, 9- and 10-wing deals.

Conclusion: Financial Numeracy as an Emerging Field in Education

This chapter presents a short recapitulation of the main ideas developed in this book. We propose the launch of new directions in research in mathematics education in the digital area, and we provide some recommendations for teacher preparation. Financial Education is an emerging field in education that has strong connections with mathematics education. To illustrate those strong connections, we have developed a model of financial numeracy that highlights not only the role of mathematics in developing understanding of financial concepts and practices, but also the potential benefits of introducing financial practices in the teaching and learning of mathematics. These ideas are aligned with contemporary research in mathematics education that reveal higher student motivation, engagement and understanding in mathematics classes when financial contexts are introduced (Althauser & Harter, 2016; Bansilal et al., 2012; Pournara, 2013; Wilburne et al., 2007). These ideas are also aligned with curricula worldwide which have been increasingly promoting financial numeracy in multiple ways (as a standalone course, integrated across different disciplines, or integrated within the mathematics curriculum). As we developed the epistemological foundations of financial numeracy using data from our research project with secondary mathematics teachers (grade 7 to grade 11), we were struck by the tensions those teachers faced in regard to teaching this area. For instance, they recognised the importance of educating students about financial issues, but at the same time they were not sure about how far to go, depending on the students’ age. The greatest tensions that concerned their own knowledge and the time allocated to the teaching of mathematics. These tensions can be addressed by training mathematics teachers about financial numeracy and showing how it can be seamlessly integrated into the broader curriculum. Suitable initial and in-service teacher preparation programmes could support teachers in developing the knowledge and competency to teach financial numeracy.

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Pre-service teacher preparation for mathematics teachers should offer at least one entire course on financial numeracy. This course will aim for novice teachers to develop: Their knowledge in regard to financial mathematics concepts; Their knowledge in regard to students thinking about financial numeracy; Their knowledge to design and adapt tasks according to the financial numeracy model; A critical stance in regard to the production and management of resources in our society. This course should also be offered in pre-service teacher training for kindergarten and elementary-school teachers. While the content will be adapted for younger students, the aims will be the same. In-service teacher training for mathematics teachers and for kindergarten and elementary-school teachers should consist in a series of workshops over two or three years. The aims of the series of workshops will be the same, but the way of reaching them will be grounded in individual teaching practices. Thus, each workshop will present some theoretical and practical content. It could be, for example, that teachers develop learning situations or lessons for their students using the new knowledge learnt. The teachers will have to implement some of the situations or lessons developed in the workshop in their classroom and then share their experiences at the next workshop. This iterative process will support teachers to make connections between the epistemology of mathematics in regard to financial numeracy and their teaching practices. As pointed out by Savard and Polotskaia (2014), one activity presented in a professional development workshop for teachers is not enough to effect profound changes in the teaching practices of elementary-school teachers. A series of different activities is necessary in order to truly support teachers to implement new teaching practices in regard to financial numeracy.

Financial Numeracy in Relation to the Community of Mathematics Education The ideas shared in this book, while unique and innovative, are not disconnected from recent advancements in the field of mathematics education. They are part of various movements that aim to reconceptualise the role and structure of what it means to teach and learn mathematics, particularly in formal school. These multiple movements have proposed innovations in mathematics classrooms, teaching practices and the curriculum that point in distinct directions. For example, algebraic thinking is one trend that has emerged in response to challenges faced by students in the transition from elementary to secondary mathematics. Scholars who endorse this approach state that algebraic thinking can ease the introduction of formal algebra in secondary school by promoting mathematical reasoning

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in elementary school related to generalisations, representations, functional thinking and abstractions (Kieran, 2018; Polotskaia & Savard, 2018; Radford, 2010). This approach entails both alignments and tensions in relation to financial numeracy. It seems aligned in the sense that, like financial numeracy, we aim at improving the experience of students with mathematics and strive to move from rote memorisation to a critical understanding of how mathematical concepts can be applied. Tensions emerge, however, once we compare the role of real-life contexts in the classroom: while algebraic thinking scholars seek to develop whatever is generalisable in a context (with the goal of abstracting), financial numeracy scholars focus specifically on each context and how mathematics is constructed in real-life situations. The same applies to computational thinking, an influential trend in the books from the MEDA series. Scholars concerned with computational thinking propose that learning about programming, automatisation and other processes are fundamentally the business of mathematics educators. They propose that the reasoning behind these processes is mathematical as it involves dealing with variables, abstraction and generalisations. In this sense, computational thinking can be developed both as part of the mathematics curriculum, or as a way to help students learn other mathematics concepts (Kotsopoulos et al., 2017; Gover & Pea, 2013; Weintrop et al., 2016). This discussion, as has been shown throughout the book, is very similar to financial numeracy. It is about finding connections between mathematics and other practices (either financial or computational) and investigating the possibilities of integrating them into formal mathematics. In doing so, we believe mathematics can become more dynamic, interesting and relevant for all students. Overall, these trends reveal the dynamics of mathematics education research communities around the world and their effort to change the status quo of teaching and learning mathematics in the digital era. In that sense, teaching financial numeracy in mathematics classrooms has the ultimate goal of developing citizenship among young learners—which entails learning beyond personal finances. It means preparing the 21st-century citizen. This ultimate goal will be achieved only with the support of the schooling and research communities. References Althauser, K., & Harter, C. (2016). Math and economics: implementing authentic instruction in grades K-5. Journal of Education and Training Studies, 4(4), 111– 122. Bansilal, S., Mkhwanazi, T., & Mahlabela, P. (2012). Mathematical literacy teachers’ engagement with contextual tasks based on personal finance. Perspectives in Education, 30(3), 98–109. Grover, S., & Pea, R. (2013). Computational thinking in K–12: A review of the state of the field. Educational Researcher, 42(1), 38–43. https://doi.org/10.3102/ 0013189X12463051 Kieran, C. (ed.). (2018). Teaching and learning algebraic thinking with 5- to 12year-olds: The global evolution of an emerging field of research and practice. Springer.

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Kotsopoulos, D., Floyd, L., Khan, S. et al. (2017). A pedagogical framework for computational thinking. Digital Experiences in Mathematics Education, 3, 154–171. https://doi.org/10.1007/s40751-017-0031-2 Polotskaia, E., & Savard, A. (2018). Using the relational paradigm: effects on pupils’ reasoning in solving additive word problems. Research in Mathematics Education, 20(1), 70–90. https://doi.org/10.1080/14794802.2018.1442740 Pournara, C. (2013). Teachers’ knowledge for teaching compound interest. Pythagoras, 34(2). https://doi.org/10.4102/pythagoras.v34i2.238 Radford, L. (2010). Algebraic thinking from a cultural semiotic perspective. Research in Mathematics Education, 12(1), 1–19. https://doi.org/10.1080/147948 00903569741 Savard, A., & Polotskaia. E. (2014). Gérer l’accès aux mathématiques dans la résolution de problèmes textuels : une exploration du côté de l’enseignement primaire. Éducation et Francophonie, XLII(2), 140–159. Weintrop, D., Beheshti, E., Horn, M. et al. (2016). Defining Computational Thinking for Mathematics and Science Classrooms. Journal of Science Education and Technology, 25, 127–147. https://doi.org/10.1007/s10956-015-9581-5 Wilburne, J. M., Napoli, M., Keat, J. B., Dile, K., Trout, M., & Decker, S. (2007). Journeying into mathematics through storybooks: A kindergarten story. Teaching Children Mathematics, 14(4), 232–237.