Licentiate Thesis
Informatics with Specialisation in Work-Integrated Learning 2021 No. 32
Teachers’ tactics
when programming and mathematics converge
Ana Fuentes Martínez
SVANENMÄRKET
Licentiate Thesis
Informatics with Specialisation in Work-Integrated Learning 2021 No. 32
Teachers’ tactics
when programming and mathematics converge
Ana Fuentes Martínez
.
University West SE-46186 Trollh¨attan Sweden
+46 52022 30 00 www.hv.se
cAna Fuentes Mart´ınez 2021 ISBN 978-91-88847-88-1 (print) ISBN 978-91-88847-87-4 (electronic)
Acknowledgments
Two years had passed since I started this academic adventure and there are many people I need to thank for it. 2019 was for me a fast-paced year full of new places, new colleagues, new literature, and new ideas. 2020 became a much needed quiet year in which time for reflection and writing was gener- ously allowed for many and proved essential for this thesis.
Along the way, my supervisors have guided and encouraged me, safe and steady. Thank you, Lars Svensson, for your great visions and inspiring advice and, thank you, Thomas Winman, for all the times you asked me to ’write that down’ and elaborate the thoughts. It is nothing less than a privilege to have you both to show the way.
Being a Ph.D. student opens the gates of a wide research community, in which knowledgeable and open-minded thinkers magnanimously share their insights and help each other. I have the blessing of belonging to two spe- cialized research environments, Learning in and for the New Working Life (LINA) and the National Graduate school for Digital Technologies in Edu- cation -–abbreviated GRADE. I am grateful for all of you, junior and senior fellow researchers, for your support and constructive judgment, and for all the times you just dropped by, sent some words, or arranged a meet-up. My special gratitude to Senior Lecturer Peter Mozelius for his observant and chal- lenging questions at the outset of my investigations, and Professor Marcelo Milrad for contextualizing and sharpening the argument upon completion.
Many others have devoted their time to read and improve the multiple drafts of this work and they all deserve my gratitude. To Dr. Jennie Ryding, thank you for your genuine effort above what anyone could expect. Your perspec- tive on teaching and learning and your accurate remarks made a great impact in the final steps.
.
University West SE-46186 Trollh¨attan Sweden
+46 52022 30 00 www.hv.se
cAna Fuentes Mart´ınez 2021 ISBN 978-91-88847-88-1 (print) ISBN 978-91-88847-87-4 (electronic)
Acknowledgments
Two years had passed since I started this academic adventure and there are many people I need to thank for it. 2019 was for me a fast-paced year full of new places, new colleagues, new literature, and new ideas. 2020 became a much needed quiet year in which time for reflection and writing was gener- ously allowed for many and proved essential for this thesis.
Along the way, my supervisors have guided and encouraged me, safe and steady. Thank you, Lars Svensson, for your great visions and inspiring advice and, thank you, Thomas Winman, for all the times you asked me to ’write that down’ and elaborate the thoughts. It is nothing less than a privilege to have you both to show the way.
Being a Ph.D. student opens the gates of a wide research community, in which knowledgeable and open-minded thinkers magnanimously share their insights and help each other. I have the blessing of belonging to two spe- cialized research environments, Learning in and for the New Working Life (LINA) and the National Graduate school for Digital Technologies in Edu- cation -–abbreviated GRADE. I am grateful for all of you, junior and senior fellow researchers, for your support and constructive judgment, and for all the times you just dropped by, sent some words, or arranged a meet-up. My special gratitude to Senior Lecturer Peter Mozelius for his observant and chal- lenging questions at the outset of my investigations, and Professor Marcelo Milrad for contextualizing and sharpening the argument upon completion.
Many others have devoted their time to read and improve the multiple drafts of this work and they all deserve my gratitude. To Dr. Jennie Ryding, thank you for your genuine effort above what anyone could expect. Your perspec- tive on teaching and learning and your accurate remarks made a great impact in the final steps.
To the teachers that altruistically granted me access to their everyday prac- tices, their classrooms, and their plans, sincerely, thank you. Without your collaboration, your clever observations, and your pioneering work, this thesis would not have been possible. To all other mathematics teachers at my for- mer school and in the extended professional community online, thank you for sharing your reflections and ideas on the topic of programming.
I am fortunate to be surrounded not only by brilliant colleagues but also brilliant friends and family who made my journey joyful, the good moments better and the bad ones less so. To Carolina Mart´ınez, thank you, among many other things, for initiating me into the intricacies of de Certeau. To Birgitta, because you make me try to be that kindhearted person you see in me, thank you. To my sister, thank you for making the ’Face with Tears of Joy’–emoji, the most popular on my phone. To my parents, for inculcating the values of education and perseverance, opening a world of possibilities, and trusting my choices, gracias de todo coraz´on.
I want to thank my dearest children for their unconditional support. Jens, thank you for your time with LATEX-tables and python widgets, for your memes, and your delicious meals. Lycke, thank you for helping me with the graph- ics, the transcriptions, and the proofreading and for all the coffee-flavored conversations that make the evenings worth longing to. David, thank you for being there for swimming, reading, trip-planning, Pok´emon hunting, and binge-watching Friends, those are the moments that count.
To Peter, my beloved husband, because you are always the first person I want to tell the good news, the scary dreams, the crazy ideas. For all your love, my gratitude is incommensurable.
Popul¨arvetenskaplig Sammanfattning
L¨ararnas profession ut¨ovas i skolkontexter d¨ar deras autonomi i undervisnings be- gr¨ansas av regler, moraliska skyldigheter, fysiska och ekonomiska f¨oruts¨attningar och framf¨orallt officiella direktiv. Dessa f¨orh˚allanden ¨andrades i samband med de revide- rade matematikkursplaner f¨or gymnasieskolan, d¨ar programmering nu ska ing˚a. Hur l¨ararna anpassade sig till den nya l¨aroplanen och hur de navigerade de sp¨anningar och mots¨attningar som uppstod analyseras i denna licentiatuppsats i termer av taktiker och strategier. Studiens ¨overgripande m˚al ¨ar att bidra till en kritisk f¨orst˚aelse f¨or hur matematikl¨arare integrerar programmering i sin undervisning och hur denna integ- ration divergerar fr˚an intentionerna bakom reformen. Det empiriska materialet kom- mer fr˚an nio individuella intervjuer med programmeringskunniga matematikl¨arare.
L¨ararna bidrog med planeringar och lektionsmaterial d¨ar programmeringsaktiviteter ingick. Detta fungerade som utg˚angspunkt f¨or att leda samtalet mot ytterligare reflek- tioner ¨over den egna yrkesut¨ovningen. F¨or att f˚a en fullst¨andig bild av de nya villko- ren, unders¨oktes ocks˚a l¨aroplanen och andra relevanta policydokument. Dessa inklu- derade kurs- och ¨amnesplaner i matematik samt det st¨odmaterialet som publicerades i samband med reformen och en samling av programmerings¨ovningar fr˚an Skolverkets fortbildningsinsatser. Tv˚a taktiska tillv¨agag˚angss¨att blev tydliga n¨ar l¨arare b¨orjade in- tegrera programmering i matematik: Dual undervisning och instr¨ott programmering.
L¨ararens anv¨andning av duala undervisningsmetoder eller intersidig programmering var olika taktiker som formades av och som respons p˚a villkoren i den nya l¨aroplanen samt deras egna preferenser och syn p˚a studenternas l¨arande. Dessa tv˚a taktiker av- sl¨ojar olika ontologiska ˚ataganden i f¨orh˚allande till de strategier som representeras i l¨aroplanen. Av relevans f¨or l¨arare och ¨amnesplansutvecklare ¨ar f¨orst˚aelsen f¨or (a) hur begreppet programmering och matematik som separata ¨amnen ger en f¨orekland bild av l¨ararnas faktiska integrationsmetoder, och (b) hur de val som g¨ors i kursplaner kan forma l¨ararnas undervisningstaktiker.
To the teachers that altruistically granted me access to their everyday prac- tices, their classrooms, and their plans, sincerely, thank you. Without your collaboration, your clever observations, and your pioneering work, this thesis would not have been possible. To all other mathematics teachers at my for- mer school and in the extended professional community online, thank you for sharing your reflections and ideas on the topic of programming.
I am fortunate to be surrounded not only by brilliant colleagues but also brilliant friends and family who made my journey joyful, the good moments better and the bad ones less so. To Carolina Mart´ınez, thank you, among many other things, for initiating me into the intricacies of de Certeau. To Birgitta, because you make me try to be that kindhearted person you see in me, thank you. To my sister, thank you for making the ’Face with Tears of Joy’–emoji, the most popular on my phone. To my parents, for inculcating the values of education and perseverance, opening a world of possibilities, and trusting my choices, gracias de todo coraz´on.
I want to thank my dearest children for their unconditional support. Jens, thank you for your time with LATEX-tables and python widgets, for your memes, and your delicious meals. Lycke, thank you for helping me with the graph- ics, the transcriptions, and the proofreading and for all the coffee-flavored conversations that make the evenings worth longing to. David, thank you for being there for swimming, reading, trip-planning, Pok´emon hunting, and binge-watching Friends, those are the moments that count.
To Peter, my beloved husband, because you are always the first person I want to tell the good news, the scary dreams, the crazy ideas. For all your love, my gratitude is incommensurable.
Popul¨arvetenskaplig Sammanfattning
L¨ararnas profession ut¨ovas i skolkontexter d¨ar deras autonomi i undervisnings be- gr¨ansas av regler, moraliska skyldigheter, fysiska och ekonomiska f¨oruts¨attningar och framf¨orallt officiella direktiv. Dessa f¨orh˚allanden ¨andrades i samband med de revide- rade matematikkursplaner f¨or gymnasieskolan, d¨ar programmering nu ska ing˚a. Hur l¨ararna anpassade sig till den nya l¨aroplanen och hur de navigerade de sp¨anningar och mots¨attningar som uppstod analyseras i denna licentiatuppsats i termer av taktiker och strategier. Studiens ¨overgripande m˚al ¨ar att bidra till en kritisk f¨orst˚aelse f¨or hur matematikl¨arare integrerar programmering i sin undervisning och hur denna integ- ration divergerar fr˚an intentionerna bakom reformen. Det empiriska materialet kom- mer fr˚an nio individuella intervjuer med programmeringskunniga matematikl¨arare.
L¨ararna bidrog med planeringar och lektionsmaterial d¨ar programmeringsaktiviteter ingick. Detta fungerade som utg˚angspunkt f¨or att leda samtalet mot ytterligare reflek- tioner ¨over den egna yrkesut¨ovningen. F¨or att f˚a en fullst¨andig bild av de nya villko- ren, unders¨oktes ocks˚a l¨aroplanen och andra relevanta policydokument. Dessa inklu- derade kurs- och ¨amnesplaner i matematik samt det st¨odmaterialet som publicerades i samband med reformen och en samling av programmerings¨ovningar fr˚an Skolverkets fortbildningsinsatser. Tv˚a taktiska tillv¨agag˚angss¨att blev tydliga n¨ar l¨arare b¨orjade in- tegrera programmering i matematik: Dual undervisning och instr¨ott programmering.
L¨ararens anv¨andning av duala undervisningsmetoder eller intersidig programmering var olika taktiker som formades av och som respons p˚a villkoren i den nya l¨aroplanen samt deras egna preferenser och syn p˚a studenternas l¨arande. Dessa tv˚a taktiker av- sl¨ojar olika ontologiska ˚ataganden i f¨orh˚allande till de strategier som representeras i l¨aroplanen. Av relevans f¨or l¨arare och ¨amnesplansutvecklare ¨ar f¨orst˚aelsen f¨or (a) hur begreppet programmering och matematik som separata ¨amnen ger en f¨orekland bild av l¨ararnas faktiska integrationsmetoder, och (b) hur de val som g¨ors i kursplaner kan forma l¨ararnas undervisningstaktiker.
Abstract
Title: Teachers’ tactics when programming and mathematics converge
Keywords: Programming; Mathematics; Curriculum; Strategies, Tactics, Michel de Certeau
ISBN (print): 978-91-88847-88-1 ISBN (e-print): 978-91-88847-87-4
Teachers’ everyday practices are embedded in school contexts in which their teaching autonomy is constrained by rules, moral obligations, physical set- tings, and official directives. When a curricular revision mandated that pro- gramming was to be a part of mathematics in upper secondary education, teachers’ conditions changed. How teachers adapted to the new curriculum and how they navigated the tensions and contradictions that they encountered is in this thesis analyzed in terms of teachers’ tactics and policy strategies. The overall goal of the investigation is to contribute to a critical understanding of how mathematics teachers integrate programming in their professional prac- tice and how this integration aligns and diverges from the intentions behind the reform. The empirical material is drawn from nine individual interviews with mathematics teachers that were already proficient in programming. The teachers’ unit plans and other lesson materials featuring programming activ- ities served as a trigger point to delve into further reflections upon their own professional practices. To complete the scene, the policy documents were also examined. These included the mathematics curriculum, as well as related of- ficial documents and a collection of institutionally sanctioned programming exercises and demonstrations. Two tactical approaches were made appar-
Abstract
Title: Teachers’ tactics when programming and mathematics converge
Keywords: Programming; Mathematics; Curriculum; Strategies, Tactics, Michel de Certeau
ISBN (print): 978-91-88847-88-1 ISBN (e-print): 978-91-88847-87-4
Teachers’ everyday practices are embedded in school contexts in which their teaching autonomy is constrained by rules, moral obligations, physical set- tings, and official directives. When a curricular revision mandated that pro- gramming was to be a part of mathematics in upper secondary education, teachers’ conditions changed. How teachers adapted to the new curriculum and how they navigated the tensions and contradictions that they encountered is in this thesis analyzed in terms of teachers’ tactics and policy strategies. The overall goal of the investigation is to contribute to a critical understanding of how mathematics teachers integrate programming in their professional prac- tice and how this integration aligns and diverges from the intentions behind the reform. The empirical material is drawn from nine individual interviews with mathematics teachers that were already proficient in programming. The teachers’ unit plans and other lesson materials featuring programming activ- ities served as a trigger point to delve into further reflections upon their own professional practices. To complete the scene, the policy documents were also examined. These included the mathematics curriculum, as well as related of- ficial documents and a collection of institutionally sanctioned programming exercises and demonstrations. Two tactical approaches were made appar-
ent when mathematics teachers began to integrate computer programming in their subject: Dual teaching and Interspersed programming. The teacher’s use of dual teaching practices or interspersed programming are tactics shaped by and in response to the conditions of the new curriculum and their own pref- erences and views on student learning. These two tactics disclose different ontological commitments in relation to the strategies dictated by the curricu- lum and reflect a cardinal distinction between planning mathematics activities with elements of programming and planning programming activities with elements of mathematics. Of relevance for teachers and curriculum designers is the un- derstanding of (a) how the notion of programming and mathematics as sepa- rate subjects oversimplifies teachers’ actual integration practices, and (b) how the curricular choices made by policy can shape the teaching tactics adopted by educators.
Table of Contents
Acknowledgments i
Popul¨arvetenskaplig Sammanfattning iii
Abstract v
1 Introduction 1
1.1 Purpose and research questions. . . 3
2 Background 7 2.1 The revision of the mathematics curriculum . . . 7
2.2 Learning mathematics with technology . . . 9
2.3 Programming in mathematics education . . . 9
3 Tactics and Strategies 13 3.1 de Certeau and everyday practices . . . 13
3.1.1 Everyday practices . . . 14
3.1.2 Power and resistance . . . 16
3.1.3 Tactics and strategies . . . 17
3.1.4 Anti-disciplinary practices . . . 19
3.1.5 Methodological notes . . . 21
3.2 Ontological grounds . . . 21
3.3 Progression for programming . . . 27
3.4 The Knowledge Quartet . . . 28
4 Method 31 4.1 Curricular documents . . . 32
4.2 Empirical Data . . . 34
ent when mathematics teachers began to integrate computer programming in their subject: Dual teaching and Interspersed programming. The teacher’s use of dual teaching practices or interspersed programming are tactics shaped by and in response to the conditions of the new curriculum and their own pref- erences and views on student learning. These two tactics disclose different ontological commitments in relation to the strategies dictated by the curricu- lum and reflect a cardinal distinction between planning mathematics activities with elements of programming and planning programming activities with elements of mathematics. Of relevance for teachers and curriculum designers is the un- derstanding of (a) how the notion of programming and mathematics as sepa- rate subjects oversimplifies teachers’ actual integration practices, and (b) how the curricular choices made by policy can shape the teaching tactics adopted by educators.
Table of Contents
Acknowledgments i
Popul¨arvetenskaplig Sammanfattning iii
Abstract v
1 Introduction 1
1.1 Purpose and research questions. . . 3
2 Background 7 2.1 The revision of the mathematics curriculum . . . 7
2.2 Learning mathematics with technology . . . 9
2.3 Programming in mathematics education . . . 9
3 Tactics and Strategies 13 3.1 de Certeau and everyday practices . . . 13
3.1.1 Everyday practices . . . 14
3.1.2 Power and resistance . . . 16
3.1.3 Tactics and strategies . . . 17
3.1.4 Anti-disciplinary practices. . . 19
3.1.5 Methodological notes . . . 21
3.2 Ontological grounds . . . 21
3.3 Progression for programming . . . 27
3.4 The Knowledge Quartet . . . 28
4 Method 31 4.1 Curricular documents . . . 32
4.2 Empirical Data . . . 34
4.2.1 Selection of participants . . . 34
4.2.2 Interview design and implementation . . . 35
4.2.3 Unit plans and other teaching materials. . . 38
4.2.4 Classroom observations . . . 40
4.3 Compilation and analysis of data . . . 41
4.4 Trust and cumulative knowledge . . . 42
5 Analysis 47 5.1 Strategies behind the new curriculum . . . 48
5.1.1 Curriculum for the Non-Compulsory School System. . . 49
5.1.2 Programming activities and instruction examples . . . 51
5.2 From strategies to tactics . . . 54
5.3 Teaching programming and mathematics . . . 59
5.4 Dual teaching . . . 60
5.5 Interspersed programming . . . 64
5.6 Summary of findings . . . 68
6 Discussion 71 6.1 New mathematics teaching. . . 72
6.2 Teachers’ tactics . . . 75
6.3 Restrictions and limiting factors . . . 78
6.4 Strategies and tactics in times of change . . . 80
7 Conclusion 83 7.1 Outcomes of the investigation . . . 83
7.2 Further research . . . 84
References 89
Appendices 1
A Participants 3
B Programming activities 5
C Official records 9
D Swedish education system 13
First of all, nobody can be innocent and wise at the same time. That’s another one of those impossible combinations.
—Amy Schumer, The Girl with the Lower Back Tattoo
4.2.1 Selection of participants . . . 34
4.2.2 Interview design and implementation . . . 35
4.2.3 Unit plans and other teaching materials. . . 38
4.2.4 Classroom observations . . . 40
4.3 Compilation and analysis of data . . . 41
4.4 Trust and cumulative knowledge . . . 42
5 Analysis 47 5.1 Strategies behind the new curriculum. . . 48
5.1.1 Curriculum for the Non-Compulsory School System. . . 49
5.1.2 Programming activities and instruction examples . . . 51
5.2 From strategies to tactics . . . 54
5.3 Teaching programming and mathematics . . . 59
5.4 Dual teaching . . . 60
5.5 Interspersed programming. . . 64
5.6 Summary of findings . . . 68
6 Discussion 71 6.1 New mathematics teaching. . . 72
6.2 Teachers’ tactics . . . 75
6.3 Restrictions and limiting factors . . . 78
6.4 Strategies and tactics in times of change. . . 80
7 Conclusion 83 7.1 Outcomes of the investigation . . . 83
7.2 Further research. . . 84
References 89
Appendices 1
A Participants 3
B Programming activities 5
C Official records 9
D Swedish education system 13
First of all, nobody can be innocent and wise at the same time. That’s another one of those impossible combinations.
—Amy Schumer, The Girl with the Lower Back Tattoo
Chapter 1
Introduction
With Sweden recently revising its educational policy to increase the pace and quality of digitalization and with the burgeoning addition of computer pro- gramming to mathematics curricula, discussions around the objectives, struc- tures, policies, and pedagogical practices embedded in the reform are more important than ever. Scholars have long problematized the ways in which mathematics and computer programming complement each other (eg. Papert, 1980; Psycharis and Kallia,2017; Barker, Merryman, and Bracken,1988) and connections with computational thinking abound in the international literature (eg. Weintrop et al.,2016; Mannila, Dagiene, et al.,2014; Rodr´ıguez-Mart´ınez, Gonz´alez-Calero, and S´aez-L´opez, 2020; Kohen-Vacs, Kynigos, and Milrad, 2020). However, the actual structures and practices of the integration of math- ematics and computer programming in the context of non-compulsory mathe- matics education have not yet received much attention (eg. Fuentes-Mart´ınez, 2020b). The policy documents that underpin the curricular reform advocate for integration practices and expose the preferred strategies to achieve the ed- ucational goals but its implementation in teaching activities diverges among practitioners.
Recent reports indicate that mathematics teachers and school principals are concerned about the necessity of extensive training programs for teachers to learn programming themselves, and the inequalities that might arise dur- ing this period (Larsson,2017; Mannila, Nord´en, and Pears,2018; Misfeldt, Szabo, and Helenius,2019). Even before adding computer programming, les-
Chapter 1
Introduction
With Sweden recently revising its educational policy to increase the pace and quality of digitalization and with the burgeoning addition of computer pro- gramming to mathematics curricula, discussions around the objectives, struc- tures, policies, and pedagogical practices embedded in the reform are more important than ever. Scholars have long problematized the ways in which mathematics and computer programming complement each other (eg. Papert, 1980; Psycharis and Kallia,2017; Barker, Merryman, and Bracken, 1988) and connections with computational thinking abound in the international literature (eg. Weintrop et al.,2016; Mannila, Dagiene, et al.,2014; Rodr´ıguez-Mart´ınez, Gonz´alez-Calero, and S´aez-L´opez, 2020; Kohen-Vacs, Kynigos, and Milrad, 2020). However, the actual structures and practices of the integration of math- ematics and computer programming in the context of non-compulsory mathe- matics education have not yet received much attention (eg. Fuentes-Mart´ınez, 2020b). The policy documents that underpin the curricular reform advocate for integration practices and expose the preferred strategies to achieve the ed- ucational goals but its implementation in teaching activities diverges among practitioners.
Recent reports indicate that mathematics teachers and school principals are concerned about the necessity of extensive training programs for teachers to learn programming themselves, and the inequalities that might arise dur- ing this period (Larsson,2017; Mannila, Nord´en, and Pears, 2018; Misfeldt, Szabo, and Helenius,2019). Even before adding computer programming, les-
CHAPTER 1. INTRODUCTION
son time for covering the whole curriculum was already scarce and teachers had long seen their planning time being eroded with other chores (Nyroos, 2008). Therefore, scholars argue for a critical approach to the integration of programming to traditional mathematics teaching, focusing on the program- ming knowledge progression that the subject requires (eg. Zhang, Nouri, and Rolandsson,2020; Foerster,2016).
These scholars share a concern for issues of teacher readiness and edu- cational equity provided by mathematics teachers with different prior pro- gramming knowledge, particularly those whose teaching education did not include computer programming and relevant pedagogy in merging both sub- jects. Achieving equity and bridging the gender gap were some of the reasons behind many of the efforts of introducing programming at early ages (eg.
Regeringskansliet, 2017a; Fuentes-Mart´ınez,2019). While aspects of teacher training and students’ equal access to programming should not be disregarded, in the context of the revision in the Swedish upper secondary school mathe- matics curriculum, the focus of this work is on lessons to learn from mathe- matics teachers that are already implementing computer programming.
The thesis starts with an overview of the curriculum revision, its objectives, and the ideas supporting the decision of boosting digital skills by incorporat- ing computer programming in mathematics. Next, Michael de Certeau’s no- tion of strategies and tactics (de Certeau,1984) are operationalized to under- stand how the new curriculum and its conditions for implementation shape teachers’ classroom practices. The power of the governed is a recurrent theme in de Certeau’s theories of everyday practices. He called strategies to the in- stitutionally sanctioned plans whose objectives comply with the views of the power institutions that enforce them. Tactics, on the other hand, are sub- versive everyday actions characterized by a self-defined target. Tactics are therefore efforts that bypass or semi-accommodate the strategies of the given cultural economy, the non-regulated actions of resistance located in the every- day practices. The argument presented here implies that teachers engage in two types of tactics, dual teaching and interspersed programming” that in dif- ferent ways align with and diverge from the curricular strategies envisioned for mathematics education. Since mathematics teaching practices are influ- enced by many different contextual factors, including local training programs and school settings, data from interviews with teachers serve to exemplify the
1.1. PURPOSE AND RESEARCH QUESTIONS
argument of dual teaching and interspersed programming as tactics and to analyze the role of the new mathematics curriculum in shaping teachers’ prac- tices. Several implications for the current curriculum implementation are pre- sented as well as a discussion of how dual teaching and interspersed program- ming pedagogies offer principles for teachers and administrators to guide the incorporation of computer programming in mathematics.
1.1 Purpose and research questions
What is it that directs teachers’ choices when making instructional decisions to implement an integrated math and programming curriculum? Exploring this problem from the perspective of tactics and strategies as related to cur- riculum implementation is expected to raise teacher awareness of the capacity and responsibility for those decisions and inform the profession to provide effective support mechanisms.
It is generally understood that the decisions teachers make have an im- pact not only on student achievement but also on the overall success of larger interventions such as the present curricular reform. When achievement is at odds, many look for remedies in technology and economic resources, but the decisions teachers constantly make in their classroom instruction is a signif- icant factor that needs to be taken into account. Finding reliable evidence about how teachers adjust to the new curricular demands and the motives behind their positions will provide valuable knowledge for teachers and poli- cymakers to consciously embrace this and other upcoming reforms in the ever swinging pendulum of education.
The thesis aims at understanding the practices of teachers in their attempts to adapt their teaching to the new curricular requirements. The investigation is therefore framed within theoretical perspectives that are productive for ex- plaining the subjective actions surrounding learning activities. The thesis is positioned in the field of ICT and learning, with a special focus on the con- vergence of mathematics and computer programming. To critically analyze teachers’ everyday context with respect to the curricular and professional con- straints that govern their activities, and the social actors connected to this en- vironment, the thesis focuses on three dimensions: the curricular changes, the teachers as curriculum agents, and the subjects of mathematics and computer
CHAPTER 1. INTRODUCTION
son time for covering the whole curriculum was already scarce and teachers had long seen their planning time being eroded with other chores (Nyroos, 2008). Therefore, scholars argue for a critical approach to the integration of programming to traditional mathematics teaching, focusing on the program- ming knowledge progression that the subject requires (eg. Zhang, Nouri, and Rolandsson,2020; Foerster,2016).
These scholars share a concern for issues of teacher readiness and edu- cational equity provided by mathematics teachers with different prior pro- gramming knowledge, particularly those whose teaching education did not include computer programming and relevant pedagogy in merging both sub- jects. Achieving equity and bridging the gender gap were some of the reasons behind many of the efforts of introducing programming at early ages (eg.
Regeringskansliet, 2017a; Fuentes-Mart´ınez,2019). While aspects of teacher training and students’ equal access to programming should not be disregarded, in the context of the revision in the Swedish upper secondary school mathe- matics curriculum, the focus of this work is on lessons to learn from mathe- matics teachers that are already implementing computer programming.
The thesis starts with an overview of the curriculum revision, its objectives, and the ideas supporting the decision of boosting digital skills by incorporat- ing computer programming in mathematics. Next, Michael de Certeau’s no- tion of strategies and tactics (de Certeau,1984) are operationalized to under- stand how the new curriculum and its conditions for implementation shape teachers’ classroom practices. The power of the governed is a recurrent theme in de Certeau’s theories of everyday practices. He called strategies to the in- stitutionally sanctioned plans whose objectives comply with the views of the power institutions that enforce them. Tactics, on the other hand, are sub- versive everyday actions characterized by a self-defined target. Tactics are therefore efforts that bypass or semi-accommodate the strategies of the given cultural economy, the non-regulated actions of resistance located in the every- day practices. The argument presented here implies that teachers engage in two types of tactics, dual teaching and interspersed programming” that in dif- ferent ways align with and diverge from the curricular strategies envisioned for mathematics education. Since mathematics teaching practices are influ- enced by many different contextual factors, including local training programs and school settings, data from interviews with teachers serve to exemplify the
1.1. PURPOSE AND RESEARCH QUESTIONS
argument of dual teaching and interspersed programming as tactics and to analyze the role of the new mathematics curriculum in shaping teachers’ prac- tices. Several implications for the current curriculum implementation are pre- sented as well as a discussion of how dual teaching and interspersed program- ming pedagogies offer principles for teachers and administrators to guide the incorporation of computer programming in mathematics.
1.1 Purpose and research questions
What is it that directs teachers’ choices when making instructional decisions to implement an integrated math and programming curriculum? Exploring this problem from the perspective of tactics and strategies as related to cur- riculum implementation is expected to raise teacher awareness of the capacity and responsibility for those decisions and inform the profession to provide effective support mechanisms.
It is generally understood that the decisions teachers make have an im- pact not only on student achievement but also on the overall success of larger interventions such as the present curricular reform. When achievement is at odds, many look for remedies in technology and economic resources, but the decisions teachers constantly make in their classroom instruction is a signif- icant factor that needs to be taken into account. Finding reliable evidence about how teachers adjust to the new curricular demands and the motives behind their positions will provide valuable knowledge for teachers and poli- cymakers to consciously embrace this and other upcoming reforms in the ever swinging pendulum of education.
The thesis aims at understanding the practices of teachers in their attempts to adapt their teaching to the new curricular requirements. The investigation is therefore framed within theoretical perspectives that are productive for ex- plaining the subjective actions surrounding learning activities. The thesis is positioned in the field of ICT and learning, with a special focus on the con- vergence of mathematics and computer programming. To critically analyze teachers’ everyday context with respect to the curricular and professional con- straints that govern their activities, and the social actors connected to this en- vironment, the thesis focuses on three dimensions: the curricular changes, the teachers as curriculum agents, and the subjects of mathematics and computer
CHAPTER 1. INTRODUCTION
programming. The idea behind this holistic approach is to enable deeper anal- ysis and a kaleidoscopic understanding of teachers’ practice by combining dif- ferent perspectives pertinent to the area of research.
Parting from Maxwell’s idea of a real world that exists independently of our thoughts and perspectives (Maxwell, 2012, p. vii), the investigation is grounded in a critical realist approach to knowledge, in the belief that our understanding does point to reality to some extent. The reality, in the par- ticular context of this work, refers to a new mathematics curriculum taking legal effect in Sweden from July 2018 (Skolverket,2019a) and teachers acting upon it with some grade of autonomy. The investigation asks therefore two questions:
How mathematics teachers adapt their practices in response to the addition of com- puter programming to the curriculum?,
and
How do teachers’ programming integration practices both align with and diverge from the types of programming integration practices encouraged in the new curricu- lum?
A valuable source in retrieving information about teachers’ integration practices are the teachers affected by the changes and whose responsibility is the pedagogical design that will guide classroom instruction. The abstract nature of inquiry about motives and autonomy delves into context-specific questions about how the addition of computer programming shapes math- ematics teachers’ practices and how those practices align with the intended outcomes of the reform. This was addressed by examining nine concrete ex- amples in which teachers share their decisions and insights in relation to their classroom instruction and unit plans1.
This thesis continues with the description of the necessary background in- formation that incorporates the results to the existing body of knowledge and positions its claims in the international, local and historical context in which they are to be recognized (chapter 2). The theoretical lenses presented inchap- ter 3provide direction to the research and grant a link between the observed micro-practices of individual teachers and the overarching structures of cur- riculum design and implementation. The method used to reach these micro-
1Unit plans comprise several lessons and are made to serve for a long period of time during which the topics in the unit are studied.
1.1. PURPOSE AND RESEARCH QUESTIONS
practices is comprehensively described inchapter 4followed by the formu- lation of two modes of integration that constitute the principal results: Dual teaching and Interspersed programming (chapter 5). The thesis ends with a discussion inchapter 6pointing at consequences for different stakeholders as well as contributions to theory and methodological paths.
CHAPTER 1. INTRODUCTION
programming. The idea behind this holistic approach is to enable deeper anal- ysis and a kaleidoscopic understanding of teachers’ practice by combining dif- ferent perspectives pertinent to the area of research.
Parting from Maxwell’s idea of a real world that exists independently of our thoughts and perspectives (Maxwell, 2012, p. vii), the investigation is grounded in a critical realist approach to knowledge, in the belief that our understanding does point to reality to some extent. The reality, in the par- ticular context of this work, refers to a new mathematics curriculum taking legal effect in Sweden from July 2018 (Skolverket,2019a) and teachers acting upon it with some grade of autonomy. The investigation asks therefore two questions:
How mathematics teachers adapt their practices in response to the addition of com- puter programming to the curriculum?,
and
How do teachers’ programming integration practices both align with and diverge from the types of programming integration practices encouraged in the new curricu- lum?
A valuable source in retrieving information about teachers’ integration practices are the teachers affected by the changes and whose responsibility is the pedagogical design that will guide classroom instruction. The abstract nature of inquiry about motives and autonomy delves into context-specific questions about how the addition of computer programming shapes math- ematics teachers’ practices and how those practices align with the intended outcomes of the reform. This was addressed by examining nine concrete ex- amples in which teachers share their decisions and insights in relation to their classroom instruction and unit plans1.
This thesis continues with the description of the necessary background in- formation that incorporates the results to the existing body of knowledge and positions its claims in the international, local and historical context in which they are to be recognized (chapter 2). The theoretical lenses presented inchap- ter 3provide direction to the research and grant a link between the observed micro-practices of individual teachers and the overarching structures of cur- riculum design and implementation. The method used to reach these micro-
1Unit plans comprise several lessons and are made to serve for a long period of time during which the topics in the unit are studied.
1.1. PURPOSE AND RESEARCH QUESTIONS
practices is comprehensively described inchapter 4 followed by the formu- lation of two modes of integration that constitute the principal results: Dual teaching and Interspersed programming (chapter 5). The thesis ends with a discussion inchapter 6pointing at consequences for different stakeholders as well as contributions to theory and methodological paths.
Chapter 2
Background
In this chapter, the particularities of the 2018 curriculum revision in Sweden are discussed and contextualized with its international counterparts and the historical circumstances that led to the reform. This background information is completed with a short overview of the role of technology and program- ming in mathematics.
2.1 The revision of the mathematics curriculum
Many countries are adjusting their education plans to develop students’ digi- tal competencies. These reforms ultimately seek to adapt to changes in orga- nizational and technological structures in the workplace as well as in research fields. As a response to the critiques questioning the adequacy of previous ICT curricula (eg. Wells,2012; Furber and Nurse,2012), computational sci- ence is now being incorporated into and even replacing many of the existing ICT initiatives. Allowing children to engage in computer programming in the scope of problem-solving activities is seen as a means of developing a wider range of computational thinking skills (eg. Zhang, Nouri, and Rolandsson, 2020; Brennan and Resnick,2012). While most countries have opted for sep- arate courses in computer science or programming alongside the traditional curriculum, Sweden has chosen the interdisciplinary path with the explicit purpose of clarifying the overall mission of education in strengthening stu- dents’ digital skills. This means that computer programming is introduced
Chapter 2
Background
In this chapter, the particularities of the 2018 curriculum revision in Sweden are discussed and contextualized with its international counterparts and the historical circumstances that led to the reform. This background information is completed with a short overview of the role of technology and program- ming in mathematics.
2.1 The revision of the mathematics curriculum
Many countries are adjusting their education plans to develop students’ digi- tal competencies. These reforms ultimately seek to adapt to changes in orga- nizational and technological structures in the workplace as well as in research fields. As a response to the critiques questioning the adequacy of previous ICT curricula (eg. Wells,2012; Furber and Nurse, 2012), computational sci- ence is now being incorporated into and even replacing many of the existing ICT initiatives. Allowing children to engage in computer programming in the scope of problem-solving activities is seen as a means of developing a wider range of computational thinking skills (eg. Zhang, Nouri, and Rolandsson, 2020; Brennan and Resnick,2012). While most countries have opted for sep- arate courses in computer science or programming alongside the traditional curriculum, Sweden has chosen the interdisciplinary path with the explicit purpose of clarifying the overall mission of education in strengthening stu- dents’ digital skills. This means that computer programming is introduced
CHAPTER 2. BACKGROUND
to pupils within already established subjects, such as mathematics and tech- nology (Skolverket, 2019a; Regeringskansliet, 2017a). For upper secondary school, the new national mathematics curriculum specifies that computer pro- gramming is to be incorporated by giving students methods “for mathemat- ical problem solving, including modeling of different situations”1but also to introduce varied representations of mathematical phenomena, describe think- ing processes and illustrate abstract concepts (Skolverket,2019c, p. 11). The mathematics curriculum for upper secondary school does not expect teachers to instruct students about how to program, nor does it dictate which kind of programming knowledge is necessary for each course. A separate course in computer programming is therefore still offered as an elective subject for all students in upper secondary school (see AppendixDfor a schematic descrip- tion of the Swedish school system).
The reform was introduced in 2018 simultaneously from kindergarten to upper secondary school, which proved challenging in several respects. First, it required that a large part of the mathematics and technology teachers in the country learned computer programming in a very short time, as well as pre- service teachers and teacher educators. Secondly, simultaneity also meant that the planned gradual progression in programming abstraction —from under- standing stepwise instructions to visual block programming and ultimately to text programming— would not be available for most students in the first years after the reform. This rapid and uneven implementation behooved teachers to cater for pupils with dissimilar skills regarding computer programming (Fuentes-Mart´ınez,2019). Alongside teacher professional development needs and disparity in students’ previous knowledge, the didactic issues of how and when to use programming in mathematics were not directly addressed in the policy documents which left a vacuum for pedagogues and textbook publish- ers to fill themselves. Therefore, the experiences of teachers that already feel comfortable with both mathematics and computer programming are seen as a valuable asset to learn about possible ways of merging learning activities that focalize the views of the curriculum.
1Swedish: Strategier f¨or matematisk probleml¨osning inklusive modellering av olika situa- tioner, s˚av¨al med som utan digitala verktyg och programmering (Skolverket,2017b, p. 10)
2.2. LEARNING MATHEMATICS WITH TECHNOLOGY
2.2 Learning mathematics with technology
Technological aids in mathematics education come from a long tradition of artifacts designed to facilitate calculations and as a consequence enabling in- teraction with a broader spectrum of applications (eg. Trouche,2005; S´eroul, 2000). From widely accepted mathematical formulas and numerical tables for standard functions to mechanical inventions such as the abacus or the slip- stick (slide rule), and later on electronic devices, the resources available have been used to simplify and accelerate tedious processes and for the sake of ad- vancing to more abstract and complex mathematics.
The legitimacy and integration of newer technologies into the mathemat- ical practice of schools was said to remain marginal (Guin, Ruthven, and Trouche,2005, p. 1) but more than 10 years later, most children in Sweden are familiar with the use of mathematical games in mobile devices (eg. H´erard and Carlsson,2017; Gunnarsdotter,2017; Helgen,2018). For students in sec- ondary education, the situation is probably similar. Applications showing the working alongside the solution of a given exercise are popular among pupils and they are, if not flagrantly promoted, in many cases sanctioned by their teachers. But as it is often said, quantity and quality may not come together.
This kind of games and shortcuts need to be used thoughtfully in order to advance learning which might not always be the case (H´erard and Carlsson, 2017).
In the scope of learning mathematics with technology, programmable de- vices have recently had a renaissance in accordance with the directives of the revised curriculum. The connection with the mathematics subject usually comes from spatial coordinates and geometry properties in the case of robots or drawing applications (eg. Bussi and Baccaglini-Frank,2015; Foerster,2016), but the variation increases when textual programming is available.
2.3 Programming in mathematics education
The essence of technology is using some phenomena for a purpose. Mathe- maticians resort to technological tools to achieve speed and precision in calcu- lations or even to handle fuzzy logic (Vojt´aˇs,2001). Computer programming has largely overtaken the practice of professional mathematicians in both re-
CHAPTER 2. BACKGROUND
to pupils within already established subjects, such as mathematics and tech- nology (Skolverket, 2019a; Regeringskansliet, 2017a). For upper secondary school, the new national mathematics curriculum specifies that computer pro- gramming is to be incorporated by giving students methods “for mathemat- ical problem solving, including modeling of different situations”1but also to introduce varied representations of mathematical phenomena, describe think- ing processes and illustrate abstract concepts (Skolverket,2019c, p. 11). The mathematics curriculum for upper secondary school does not expect teachers to instruct students about how to program, nor does it dictate which kind of programming knowledge is necessary for each course. A separate course in computer programming is therefore still offered as an elective subject for all students in upper secondary school (see AppendixDfor a schematic descrip- tion of the Swedish school system).
The reform was introduced in 2018 simultaneously from kindergarten to upper secondary school, which proved challenging in several respects. First, it required that a large part of the mathematics and technology teachers in the country learned computer programming in a very short time, as well as pre- service teachers and teacher educators. Secondly, simultaneity also meant that the planned gradual progression in programming abstraction —from under- standing stepwise instructions to visual block programming and ultimately to text programming— would not be available for most students in the first years after the reform. This rapid and uneven implementation behooved teachers to cater for pupils with dissimilar skills regarding computer programming (Fuentes-Mart´ınez,2019). Alongside teacher professional development needs and disparity in students’ previous knowledge, the didactic issues of how and when to use programming in mathematics were not directly addressed in the policy documents which left a vacuum for pedagogues and textbook publish- ers to fill themselves. Therefore, the experiences of teachers that already feel comfortable with both mathematics and computer programming are seen as a valuable asset to learn about possible ways of merging learning activities that focalize the views of the curriculum.
1Swedish: Strategier f¨or matematisk probleml¨osning inklusive modellering av olika situa- tioner, s˚av¨al med som utan digitala verktyg och programmering (Skolverket,2017b, p. 10)
2.2. LEARNING MATHEMATICS WITH TECHNOLOGY
2.2 Learning mathematics with technology
Technological aids in mathematics education come from a long tradition of artifacts designed to facilitate calculations and as a consequence enabling in- teraction with a broader spectrum of applications (eg. Trouche,2005; S´eroul, 2000). From widely accepted mathematical formulas and numerical tables for standard functions to mechanical inventions such as the abacus or the slip- stick (slide rule), and later on electronic devices, the resources available have been used to simplify and accelerate tedious processes and for the sake of ad- vancing to more abstract and complex mathematics.
The legitimacy and integration of newer technologies into the mathemat- ical practice of schools was said to remain marginal (Guin, Ruthven, and Trouche,2005, p. 1) but more than 10 years later, most children in Sweden are familiar with the use of mathematical games in mobile devices (eg. H´erard and Carlsson,2017; Gunnarsdotter,2017; Helgen,2018). For students in sec- ondary education, the situation is probably similar. Applications showing the working alongside the solution of a given exercise are popular among pupils and they are, if not flagrantly promoted, in many cases sanctioned by their teachers. But as it is often said, quantity and quality may not come together.
This kind of games and shortcuts need to be used thoughtfully in order to advance learning which might not always be the case (H´erard and Carlsson, 2017).
In the scope of learning mathematics with technology, programmable de- vices have recently had a renaissance in accordance with the directives of the revised curriculum. The connection with the mathematics subject usually comes from spatial coordinates and geometry properties in the case of robots or drawing applications (eg. Bussi and Baccaglini-Frank,2015; Foerster,2016), but the variation increases when textual programming is available.
2.3 Programming in mathematics education
The essence of technology is using some phenomena for a purpose. Mathe- maticians resort to technological tools to achieve speed and precision in calcu- lations or even to handle fuzzy logic (Vojt´aˇs,2001). Computer programming has largely overtaken the practice of professional mathematicians in both re-
CHAPTER 2. BACKGROUND
search and applied fields (Broley, Caron, and Saint-Aubin, 2018). The rela- tion between programming and mathematics has been present from the early days of computer science. Programming started as a branch of mathemat- ics in higher education that soon grew too large and became a core subject in the field of computer science (Campbell-Kelly, 2018). Its wide range of applications and the availability of manageable programming environments made programming a part of many other higher education programs as well as curricula in pre-university courses (Campbell-Kelly,2018; Rolandsson and Skogh, 2014). For these new arenas, the technical aspects of programming and its affordances were essential whereas the connection with mathematics became less prominent. A survey about computer science in secondary ed- ucation showed that the majority of countries did not expect programming to support mathematics, despite many authors claiming cognitive benefits for learning mathematics with programming (eg. Foerster,2016; Putri,2018; Pa- pert,1980). This is nevertheless a debated issue, with many pitfalls around the assumptions of what qualifies as mathematical knowledge and what are the transfer effects regarding higher-order reasoning. Whilst those factors are undeniably important, they call for a student-centered research framework, which falls outside the scope of this study.
Already in 1997, the Swedish National Agency for Education included technology in mathematics among the goals to work toward (Skolverket,2000, English version). In that document it was regulated that the school in its teach- ing of mathematics should aim at ensuring that pupils
[. . .] develop their knowledge of how mathematics is used in in- formation technology, as well as how information technology can be used for solving problems in order to observe mathematical re- lationships and to investigate mathematical models (ibid., p. 61).
By introducing the wording ”computer programming” in the later amend- ment (Skolverket,2017a), information technology was shaped into a very con- crete set of activities with its own idiosyncratic ways of operating and rep- resenting knowledge. programming in mathematics is further narrowed to be a digital tool for problem-solving. This limited view is nevertheless chal- lenged in practice. Also in a Swedish setting, the study by Bergsten and Frejd demonstrated how pre-service teachers used programming “for the purpose
2.3. PROGRAMMING IN MATHEMATICS EDUCATION
of generalising students’ conceptual knowledge in mathematics” (Bergsten and Frejd,2019, p. 941)
Over the years, many scholars have devoted investigations to the role of programming in mathematics. Pea and Kurland (1984) studied the cogni- tive effects of such an endeavor and concluded with doubts about whether programming promoted mathematical rigor or even meaningful understand- ing of concepts. On the bright side, they too saw possibilities for program- ming to improve problem-solving skills in mathematics (ibid., pp. 159-160).
In line with current ideas in the field of Work Integrated Learning (WIL), John Monaghan calls for employing computer programming to make school math- ematics relevant to activities beyond mathematics classrooms and to connect with out-of-school mathematical practices (Monaghan, Trouche, and Borwein, 2016, p. 333). This particular view was further explored in Fuentes-Mart´ınez (2020b), where the work experience of computer programming profession- als was merged with mathematics teaching by means of pre-service practice.
David Berlinski claimed that “mathematics, like physics, may yet become an empirical discipline, a place where things are discovered because they are seen2”. Hopes are undoubtedly being renewed on a fruitful convergence of mathematics and programming in primary and secondary education but the pitfalls are still plenty, particularly regarding shortage of teachers who are able to teach an integrated programming and mathematics curriculum.
2Gound Zero: The Pleasures of counting (Berlinski,1997)
CHAPTER 2. BACKGROUND
search and applied fields (Broley, Caron, and Saint-Aubin, 2018). The rela- tion between programming and mathematics has been present from the early days of computer science. Programming started as a branch of mathemat- ics in higher education that soon grew too large and became a core subject in the field of computer science (Campbell-Kelly, 2018). Its wide range of applications and the availability of manageable programming environments made programming a part of many other higher education programs as well as curricula in pre-university courses (Campbell-Kelly,2018; Rolandsson and Skogh, 2014). For these new arenas, the technical aspects of programming and its affordances were essential whereas the connection with mathematics became less prominent. A survey about computer science in secondary ed- ucation showed that the majority of countries did not expect programming to support mathematics, despite many authors claiming cognitive benefits for learning mathematics with programming (eg. Foerster,2016; Putri,2018; Pa- pert,1980). This is nevertheless a debated issue, with many pitfalls around the assumptions of what qualifies as mathematical knowledge and what are the transfer effects regarding higher-order reasoning. Whilst those factors are undeniably important, they call for a student-centered research framework, which falls outside the scope of this study.
Already in 1997, the Swedish National Agency for Education included technology in mathematics among the goals to work toward (Skolverket,2000, English version). In that document it was regulated that the school in its teach- ing of mathematics should aim at ensuring that pupils
[. . .] develop their knowledge of how mathematics is used in in- formation technology, as well as how information technology can be used for solving problems in order to observe mathematical re- lationships and to investigate mathematical models (ibid., p. 61).
By introducing the wording ”computer programming” in the later amend- ment (Skolverket,2017a), information technology was shaped into a very con- crete set of activities with its own idiosyncratic ways of operating and rep- resenting knowledge. programming in mathematics is further narrowed to be a digital tool for problem-solving. This limited view is nevertheless chal- lenged in practice. Also in a Swedish setting, the study by Bergsten and Frejd demonstrated how pre-service teachers used programming “for the purpose
2.3. PROGRAMMING IN MATHEMATICS EDUCATION
of generalising students’ conceptual knowledge in mathematics” (Bergsten and Frejd,2019, p. 941)
Over the years, many scholars have devoted investigations to the role of programming in mathematics. Pea and Kurland (1984) studied the cogni- tive effects of such an endeavor and concluded with doubts about whether programming promoted mathematical rigor or even meaningful understand- ing of concepts. On the bright side, they too saw possibilities for program- ming to improve problem-solving skills in mathematics (ibid., pp. 159-160).
In line with current ideas in the field of Work Integrated Learning (WIL), John Monaghan calls for employing computer programming to make school math- ematics relevant to activities beyond mathematics classrooms and to connect with out-of-school mathematical practices (Monaghan, Trouche, and Borwein, 2016, p. 333). This particular view was further explored in Fuentes-Mart´ınez (2020b), where the work experience of computer programming profession- als was merged with mathematics teaching by means of pre-service practice.
David Berlinski claimed that “mathematics, like physics, may yet become an empirical discipline, a place where things are discovered because they are seen2”. Hopes are undoubtedly being renewed on a fruitful convergence of mathematics and programming in primary and secondary education but the pitfalls are still plenty, particularly regarding shortage of teachers who are able to teach an integrated programming and mathematics curriculum.
2Gound Zero: The Pleasures of counting (Berlinski,1997)
Chapter 3
Theoretical framework:
Tactics and Strategies
A unique empirical material on mathematics teachers’ everyday practices and their choices regarding teaching with programming was collected at the outset of the reform. It guides the central theoretical decisions of the present work toward the elucidation of the Research Questions. The curricular changes, the teachers as curriculum agents, and the subjects of mathematics and com- puter programming are the central dimensions that delimit the field of inter- est. This leads to an overarching approach informed by the work of Michael de Certeau in which tactics and strategies are fundamental terms in the analy- sis of everyday practices. Consequently, the chapter is organized to expound on the significance of this theory (section 3.1) and relate it to other chief con- cepts necessary for the analysis (section 3.2). To finalize, a common learning progression regarding programming is presented (section 3.3) together with a brief overview of another theoretical framework, the Knowledge Quartet (section 3.4), which was an auxiliary tool for the initial data analysis.
3.1 de Certeau and the theory of everyday practices
Michel de Certeau (1925-1986) was a French cultural theorist with academic roots in history, psychoanalysis, philosophy, and social sciences. His mul- tidisciplinary background is unmistakable present both in his ideas and in
Chapter 3
Theoretical framework:
Tactics and Strategies
A unique empirical material on mathematics teachers’ everyday practices and their choices regarding teaching with programming was collected at the outset of the reform. It guides the central theoretical decisions of the present work toward the elucidation of the Research Questions. The curricular changes, the teachers as curriculum agents, and the subjects of mathematics and com- puter programming are the central dimensions that delimit the field of inter- est. This leads to an overarching approach informed by the work of Michael de Certeau in which tactics and strategies are fundamental terms in the analy- sis of everyday practices. Consequently, the chapter is organized to expound on the significance of this theory (section 3.1) and relate it to other chief con- cepts necessary for the analysis (section 3.2). To finalize, a common learning progression regarding programming is presented (section 3.3) together with a brief overview of another theoretical framework, the Knowledge Quartet (section 3.4), which was an auxiliary tool for the initial data analysis.
3.1 de Certeau and the theory of everyday practices
Michel de Certeau (1925-1986) was a French cultural theorist with academic roots in history, psychoanalysis, philosophy, and social sciences. His mul- tidisciplinary background is unmistakable present both in his ideas and in
CHAPTER 3. TACTICS AND STRATEGIES
his eclectic ways of approaching knowledge. For example, the return of the repressed, with its Freudian and historical reminiscences, is a central idea in de Certeau’s most influential piece, The Practice of Everyday Life. This notion was first developed in his earlier publications, particularly those related to the events of May ’681. These political revolts mark Certeau’s research inter- ests toward contemporary social issues and his later contributions framed in everyday practices.
de Certeau’s theories have impacted a wide range of modern social re- search, including policy studies in education (eg. Saltmarsh, 2015; Brewer and Werts,2017). It is in this framework of contemporary knowledge that de Certeau’s body of theoretical questions, methods, categories, and perspectives are examined to critically analyze teachers’ everyday practices with respect to the curricular and professional constraints that regulate their activities. In the next sections, de Certeau’s theoretical framework is introduced and dis- cussed, highlighting the notions that are considered to be most relevant for understanding how teachers adapt their practices in response to the addition of computer programming to the curriculum.
3.1.1 Everyday practices
In The Practice of Everyday Life (de Certeau,1984)2and its subsequent volume, The art of living and cooking (de Certeau et al.,1998)3, de Certeau seeks to an- alyze the humblest concerns of the ordinary people as they are reflected in their everyday practices. He examines and formalizes the means by which the dominant culture is enacted in expectations —social norms, a product’s instructions-of-use, explicit laws— and how these rules are re-appropriated in everyday situations by the same people upon which expectations are built.
In that analysis, de Certeau resorts to the term culture in its wider sense as the spectrum of uses and practices of a social group.
de Certeau explains that humans make “innumerable and infinitesimal transformations of and within the dominant cultural economy in order to adapt it to their own interests and their own rules” (de Certeau, 1984, pp.
xiii–xiv). This process gives intrinsic meaning and agency to the activities of
1The capture of speech and other political writings. de Certeau (1997)
2First published as l’invention du quotidien, de Certeau (1980)
3L’Invention du quotidien - Volume 2, Habiter, cuisiner
3.1. DE CERTEAU AND EVERYDAY PRACTICES
individuals. It creates a private space in which those, otherwise assumed to be passive receivers, establish their way of operating within the arena organized by external techniques of sociocultural production.
Everyday life is thus the life of ordinary people at the microlevel, defined by its inherent messiness and the complex relationship between human ac- tions and ‘the system’. It encapsulates the signs, tools, products, and ideas that are readily available and part of the daily routine. In this system, every- day actions constitute a means of manipulating the common environments in which men and women conduct their lives. Therefore, everyday life also com- prises speech, walks, meals, and other expressions of a heterogeneous society in all their diversity.
de Certeau is primarily concerned with describing the emancipatory resis- tance mechanisms of ’the ordinary man’ rather than pointing out class strug- gles and inequalities. The ordinary man is the common hero, the ubiquitous character whose goal is to get by and who does so by making-do. Unlike the superheroes of popular fiction, these common heroes are not guided by moral or aesthetic ideals, but by their personal circumstances, needs, and prefer- ences. They are anonymous to the statistics that try to categorize them and to the cultural structures that subdue them but not mere spectators. Instead, they engage in tacit daily practices to pursue their own projects and to evade disciplinary boundaries. Those practices emerge as the unit of analysis in de Certeau’s investigations. Hence, his primary concern lies in the particular schemata of action that appear within a culture disseminated and imposed by the elites. The study of the subjects conducting those actions is for de Certeau, subsidiary to the actual types of operations and it is approached only indi- rectly.
The theoretical framework defined to examine these infinitely diverse arts of living differentiates styles of action according to matter, form, time, place, situations and circumstances (Rico de Sotelo,2006). It leads to an analysis of the practices organized at two levels: the modalities of action and types of opera- tions. The modalities of action correspond to the formalities of practices (tac- tics and strategies, see3.1.3) and those can be characterized by their different types of operations i.e. the different role of spaces (de Certeau,1984, pp. 29- 30). The same tactical modality could have different meanings and require a different type of operation depending on whether it takes place at home or at
