Danish University Colleges
Developing reasoning competence in inquiry-based mathematics teaching
Larsen, Dorte Moeskær
Publication date:
2019
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Citation for pulished version (APA):
Larsen, D. M. (2019). Developing reasoning competence in inquiry-based mathematics teaching. Syddansk Universitetsforlag.
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1
Developing reasoning
competence in inquiry-based mathematics teaching
Prepared by Dorte Moeskær Larsen LSUL, IMADA, SDU
Submitted: 14th September 2019
Supervisor: Claus Michelsen Co-supervisor: Thomas Illum Hansen
2 Duration of this Ph.D.:
15th of September 2016 – 14th of September 2019
3
1 Publication by the author of this thesis
Peer reviewed and all written, submitted or published during this thesis.
The papers include in this thesis is marked with *
Dreyøe, J., Larsen, D. M., Hjelmborg, M. D., Michelsen, C., & Misfeldt, M. (2017).
Inquiry-based learning in mathematics education: Important themes in the literature.
Nordic Research in Mathematics Education, 329.
(Dreyøe, Larsen, Hjelmborg, Michelsen, & Misfeldt, 2017)
*Dreyøe, J., Larsen, D. M., & Misfeldt, M. (2018). From everyday problem to a mathematical solution-understanding student reasoning by identifying their chain of reference. In Bergqvist, E., Österholm, M., Grandberg, C., Sumpter, L. (Eds.) Proceedings of the 42nd Conference of the International Group for the Psychology of Mathematics Education. Umeå, Sweden
(Dreyøe, Larsen, & Misfeldt, 2018)
Gents, S., Christensen, M., Hjelmborg, M., Jensen, M., Larsen, D.& Hansen, R (2019).
How do mathematics teachers interact with the mathematic book and how does the use of the mathematics book and other resources influence the teaching? Abstract to Iartem 15 International Conference, Odense, Denmark
(Gents, Christensen, Hjelmborg, Jensen, Larsen & Hansen, 2019)
Gissel, S. T., Hjelmborg, M., Kristensen, B. T., & Larsen, D. M. (2019).
Kompetencedækning i analoge matematiksystemer til mellemtrinnet. MONA (3) p. 7- 27 Copenhagen, Danmark
(Gissel, Hjelmborg, Kristensen, & Larsen, 2019)
4
Larsen, D.M. (2017). Testing Inquiry-based Mathematic Competencies. In Ann Downton, Sharyn Livy, & Jennifer Hall (Eds) Proceedings of the 40th Annual Conference of the Mathematics Education Research Group of Australasia (MERGA), University of Monash
(Larsen, 2017)
*Larsen, D.M., Dreyøe, J., & Michelsen, C. (2019) How argumentation in teaching and testing of an inquiry-based intervention is aligned. Manuscript submitted for Eurasia Journal of Mathematics, Science and Technology Education. (Larsen, Dreyøe &
Michelsen, 2019)
Larsen, D. M., Hjelmborg, M. D., Lindhardt, B., Dreyøe, J., Michelsen, C., & Misfeldt, M. (2019). Designing inquiry-based teaching at scale: Central factors for implementation.
In U. T. Jankvist, M. Van den Heuvel-Panhuizen, & M. Veldhuis (Eds.), Proceedings of the Eleventh Congress of the European Society for Research in Mathematics Education (CERME11, February 6 – 10, 2019). Utricht: the Netherlands: Freudenthal Group & Freudenthal Institute, Utrecht University and ERME.
(Larsen, Hjelmborg, Lindhardt, Dreyøe, Michelsen & Misfeldt, 2019)
*Larsen, D. M., & Lindhardt, B. K. (2019). Undersøgende aktiviteter og ræsonnementer i matematikundervisningen på mellemtrinnet. MONA-Matematik-og Naturfagsdidaktik, (1). p. 7-21. Copenhagen, Danmark
(Larsen & Lindhardt, 2019)
*Larsen, D.M., & Puck, M.R. (2019). Developing a Validated Test to Measure Students’ Progression in Mathematical Reasoning in Primary School.Manuscript submitted for publication in International Journal of Education in Mathematics, Science and Technology
(Larsen & Rasmussen, 2019)
5
*Larsen, D.M., & Misfeldt, M. (2019). Fostering mathematical reasoning in inquiry- based teaching – the role of cognitive conflicts, Submitted to Nordic Studies in Mathematics Education (NOMAD) (Larsen & Misfeldt, 2019)
Larsen, D. M., Østergaard, C. H., & Skott, J. (2018). Prospective Teachers’ Approach to Reasoning and Proof: Affective and Cognitive Issues. In H. Palmér & J. Skott (Eds.) Students' and Teachers' Values, Attitudes, Feelings and Beliefs in Mathematics Classrooms (pp. 53-63): Springer.
(Larsen, Østergaard, & Skott, 2018)
Larsen, D. M., & Østergaard, C. H. (2019). Questions and answers but no reasoning...
In U. T. Jankvist, M. V. d. Heuvel-Panhuizen, & M. Veldhuis (Eds.), Proceedings of the Eleventh Congress of the European Society for Research in Mathematics Education (CERME 11, February 6 – 10, 2019). Utricht: the Netherlands: Freudenthal Group &
Freudenthal Institute, Utrecht University and ERME.
(Larsen & Østergaard, 2019)
Skott J., Larsen, D.M., & Østergaard, C.H. (2017) Reasoning and proving in mathematics teacher education. In Zehetmeier, S., Rösken-Winter B., Potari, D. &
Ribeiro, M., Berlin (Eds.), Proceedings of the Third ERME Topic Conference on Mathematics Teaching, Resources and Teacher Professional Development, European Society for Research in Mathematics Education p. 197-206
(Skott, Larsen, & Østergaard, 2017)
Skott, J., Larsen, D.M., & Østergaard, C.H. (in press). Learning to teach to reason:
Reasoning and proving in mathematics teacher education, In Zethetmeier, S. (Eds.), Routeledge.
(Skott, Larsen & Østergaard, in press)
6 Non-peer reviewed:
Dreyøe, J., Michelsen, C., Hjelmborg, M. D., Larsen, D. M., Lindhardt, B. K., &
Misfeldt, M. (2017). Hvad vi ved om undersøgelsesorienteret undervisning i matematik:
Forundersøgelse i projekt Kvalitet i Dansk og Matematik, delrapport 2.
(Dreyøe, Michelsen, et al., 2017)
Gissel, S. T., Hjelmborg, M. D., Skovmand, K., Kristensen, B. T., & Larsen, D. M.
(2017). Kompetencedækning i analoge læremidler til matematikfaget: Evaluering af analoge, didaktiske læremidler til matematikfaget i forhold til Fælles Mål med særligt fokus på kompetencedækning.
(Gissel, Hjelmborg, Skovmand, Kristensen, & Larsen, 2017)
Michelsen, C., Dreyøe, J., Hjelmborg, M. D., Larsen, D. M., Lindhardt, B. K., &
Misfeldt, M. (2017). Forskningsbaseret viden om undersøgende matematikundervisning, Undervisningsministeriet, København.
(Michelsen et al., 2017)
Larsen, D. M. (2017b) Problemopstilling som vurdering for læring. MONA-Matematik- og Naturfagsdidaktik, (2) (pp. 84-87)
(Larsen, 2017b)
Hansen, N. J., Vejbæk, L., Lindhardt, B., Jensen, M., Larsen, D. M., Hjelmborg, M., Jørgensen, A. S. (2018). Matematikdidaktiske tanker - mod en mere undersøgende dialogisk anvendelsesorienteret matematik: KiDM.
(N. J. Hansen et al., 2018) Contributions:
Hansen, T.I., Elf, N., Misfeldt, M., Gissel, S.T., Lindhardt, B. (2019). KVALITET I DANSK OG MATEMATIK. Et lodtrækningsforsøg med fokus på undersøgelsesorienteret dansk- og matematikundervisning. Slutrapport.
(T. I. Hansen et al, 2019)
7
2 Content
1 Publication by the author of this thesis ... 3
2 Content... 7
3 Preface ... 11
4 Acknowledgement ... 14
5 Abbreviations & Acronyms ... 16
6 Abstract ... 17
6.1 English Abstract ... 17
6.2 Danish Abstract (Dansk resume) ... 18
7 Keywords ... 21
8 Introduction ... 22
8.1 Aim of the research - Theory into practice and practice into theory ... 24
8.2 Research questions ... 26
8.3 Reading guide ... 27
8.4 Short summary of the five papers ... 28
8.5 Theoretical considerations in this thesis ... 30
Networking the different theories ... 30
Theories in research, practice and problems ... 34
9 Clarifications and theoretical background ... 36
9.1 Reasoning in mathematics education ... 36
The importance of focusing on reasoning in mathematics education ... 36
Defining mathematical reasoning ... 40
Different kinds of reasoning ... 45
Remarks about the social aspects in reasoning in mathematics classrooms ... 46
Taxonomies of reasoning ... 47
Analytical tools or frameworks used in the area of mathematical reasoning .... 49
9.2 How can inquiry-based teaching support development of reasoning competence? .. 52
Teaching reasoning in school. ... 53
Inquiry-based teaching ... 55
The development of reasoning competence in inquiry-based teaching. ... 56
8
9.3 Measuring student’s development of reasoning competence ... 57
Measuring mathematical competencies ... 59
Defining different terms and concepts in measuring ... 61
9.4 Developing a test – a systematic review ... 65
Review Methods ... 65
Articles/papers ... 68
Summary of key findings and development of guidelines ... 76
9.5 Using the guidelines in the development of the KiDM test ... 77
10 The KiDM project ... 79
10.1 Overall description of the KiDM project ... 79
The preliminary study in the KiDM project ... 81
The intervention and test developed in a design-based research approach ... 82
10.2 The final KiDM intervention in mathematics... 87
10.3 The final KiDM test ... 90
11 Methodology and Methods ... 92
11.1 Pragmatism as the research paradigm ... 92
11.2 Methods as a research design ... 95
An experimental research design ... 95
11.3 Methods – Mixing the methods ... 99
11.4 The qualitative research ... 107
Case study ... 107
Sampling in the qualitative research ... 109
Observations in the classrooms ... 110
Transcription with NVivo and the coding ... 113
Reliability and quality in qualitative studies ... 114
Ethics in the qualitative methods ... 115
11.5 Quantitative methods ... 116
Sampling in the quantitative data ... 116
Survey in the KiDM project ... 117
Test ... 122
Thinking aloud - to qualify tests ... 126
Ethical considerations in quantitative methods ... 127
9
12 Results from the KiDM RCT study ... 128
12.1 The quantitative measurements ... 128
12.2 Results from the tests. ... 130
12.3 Results from the teacher survey... 134
12.4 Results from the student survey... 137
12.5 Discussing the quantitative results ... 139
13 Presenting the contributing papers ... 142
13.1 Summary of Paper I ... 143
Introduction: ... 143
Methods and analysis ... 144
Findings and discussion: ... 144
Relation to this thesis research and further perspectives ... 145
13.2 Summary of Paper II... 146
Introduction ... 146
Methods and analysis ... 147
Findings and discussion ... 147
Relation to this thesis research and further perspectives ... 149
13.3 Summary of Paper III ... 151
Introduction ... 151
Methods and analysis ... 152
Findings and discussion ... 152
Relation to this thesis research and further perspectives ... 153
Summary of Paper IV ... 154
Introduction ... 154
Methods and analysis ... 154
Findings and discussion ... 155
Relation to this thesis research and further perspectives ... 156
13.4 Summary of Paper V ... 157
Introduction ... 157
Methods and analysis ... 157
Findings and discussion ... 158
Relation to this thesis research and further perspectives ... 159
10
14 Contributing Papers ... 161
14.1 Paper I ... 162
14.2 Paper II ... 172
14.3 Paper III ... 188
14.4 Paper IV ... 209
14.5 Paper V ... 227
15 Discussions, findings, conclusions and implications ... 248
15.1 Discussion of the methods ... 248
15.2 Validity and reliability in the mixed study - the legitimation of the study ... 250
15.3 The overall findings ... 253
Findings in connection to: How it is possible to study the development of students’ reasoning competence in primary school mathematics classes – RQ1 ... 253
Developing reasoning competence in inquiry-based teaching - The transition from empirical argumentation to more abstract argumentation – answering RQ2 ... 257
The overall research question: How can an inquiry-based teaching approach impact students’ reasoning competence in primary school mathematics classes? ... 261
15.4 Implications for research and practice ... 262
15.5 Final conclusion and comments ... 264
16 References ... 266
17 Appendix ... 286
a. References from the literature review - testing competencies in mathematics.... 287
b. The authors' nationality for the various articles to the literature review ... 291
c. Item map ... 292
d. Observations guide 1 (in Danish) ... 293
e. Observation guide 2 (in Danish) ... 295
f. Letter of consent (in Danish) ... 297
g. Three central transcripts from the observed lessons (in Danish). ... 298
h. Transcription-guide to the KiDM project (in Danish) ... 347
i. Coding guide (in Danish) ... 349
j. Calculated results based on the regression model to the teacher’s survey ... 367
k. Calculated results based on the regression model to the students’ survey ... 372
l. Co-author Statement ... 374
11
3 Preface
“When it says ‘show’ in this assignment, do I then have to ‘prove’ it, or can I just calculate a few examples?”
Quotes like this can be heard in many different arenas in mathematics education:
When I was a mathematics teacher in primary school classes, I often experienced that the students had difficulties justifying their claims and conjectures and it was quotes like the above during classroom teaching that intentionally made me aware of the importance of focusing specifically on how to validate in mathematics education as part of teaching in primary school classes.
From 2009 to 2016 I was an assistant and later an associate professor at the University College Capital1 teaching prospective teachers in mathematics. In this arena, I similarly experienced many students having trouble in reasoning and proof. Especially in their examination papers I saw that many of the students made empirical argumentation, for example, by using GeoGebra to prove or generalise instead of making more formal deductive argumentation. I frequently heard questions like “Are these argumentations enough to call it a proof” or “When I show in GeoGebra that it works in all cases, is that not a proof?”.
I have also often been amazed, when observing teachers in primary school classes, for example, as part of education programmes for mathematics teaching, teaching different subjects in mathematics, why the teachers never went into more depth when the students argued in different ways or why the teachers did not even get the students to argue for their claims.
Together with Professor Jeppe Skott2 and Associate Professor Camilla Hellsten Østergaard3 we therefore started a small pilot study named RaPiTE (Reasoning and Proving in Teacher Education). In this study we wanted to focus on students’ mathematical reasoning competence in teacher education and we found that many students actually think they are very good at mathematics, but, however still fail to argue mathematically for different claims. In an example
1 Now called University College Copenhagen
2 Jeppe Skott is a Professor at the Linnaeus University
3 Camilla Hellsten Østergaard is now a Ph.D. student at University of Copenhagen (IND)
12 where the students should prove why the sum of two even numbers gives an even number and why the sum of two odd numbers gives an even number, many students explained the claim by giving examples (Larsen et al., 2018). In another study in RaPiTE, we found that teacher- students, even after a specific course with a focus on reasoning, could not even pick out a videoclip from their in-service practice where students do mathematical reasoning, even though they had just completed a teaching programme about this particular topic (Skott, Larsen &
Østergaard, in press).
All these experiences and research made my interest grow in connection to improving reasoning in mathematics education.
My background as a teacher in primary school made it natural for me to make my focus in this thesis on mathematics teaching in primary school. At the same time, it is in these years that students develop a lot of basic understandings of and attitudes towards the subject.
During the writing of this thesis it has, however, also been a challenge to keep my feed on the pathway, because so many interesting things have been happening in both the Laboratory for Coherent Education and Learning (LSUL) and at the UCL University College, but, especially the project named ‘Kvalitet i Dansk og Matematik’ [Quality in Danish and Mathematics]
(KiDM), which is a big part of this thesis, has been an interesting project to follow. Sometimes it has been difficult not to work further into the project instead of pursuing the writing of this thesis, but on the other hand, this thesis has also been developing since the beginning along with the KiDM project and I am glad that I had all possibilities and practically no restrictions to pursue the paths and directions that the results and the project led me to, so I was able to become immersed in the topic in an exciting and evolving way.
In this connection, however, the challenge was to make an independent project that was not completely absorbed in the KiDM project, while I still participated in all phases of the KiDM project. I have participated in the pre-study of the KiDM project (Dreyøe, Larsen, et al., 2017;
Dreyøe, Michelsen, et al., 2017; Michelsen et al., 2017). I have taken part in the development of the intervention and the website, and I have participated in developing all the quantitative studies of the KiDM project in the mathematics part including the students’ test, the students’
questionnaire and the teachers’ questionnaires. Finally, I have also taken part in the writing of the background report as well as various articles (e.g. Larsen, Hjelmborg, Lindhardt, Dreyøe, Michelsen & Misfeldt, 2019). Nevertheless, I think that choosing to focus precisely on the
13 reasoning competence has been the strength of the separation between the KiDM project and this thesis, thereby my focus in this Ph.D. exclusively focused on developing and measuring this competence in an inquiry-based approach, which means that I can use the KiDM project in constructing my data. Meanwhile, the research on the reasoning competence in mathematics is solely part of this thesis.
Supervisor:
Claus Michelsen, Ph.D., Professor mso, LSUL, University of Southern Denmark, Clm@imada.sdu.dk
Assistant Supervisor:
Thomas Illum Hansen, PhD., Research Director, Associate Professor, University College Lillebælt, thih@ucl.dk
The dissertation at hand was prepared in the Laboratory for Coherent Education and Learning (LSUL) department at the University of Southern Denmark.
14
4 Acknowledgement
In order to complete this Ph.D. thesis, many people have helped me with different kinds of goodwill and cooperation which I now want to express my gratitude for.
First, I want to thank Professor and Centre Director for LSUL, Claus Michelsen, for being my supervisor and believing in me by giving me freedom to follow all the arising opportunities on my way and still encouraging me in new ideas. I also want to express my gratitude to second supervisor Ph.D. and Research Director for UCL – University College, Thomas Illum Hansen, for helping me with the writing of the thesis, for his patient guidance, enthusiastic encouragement and useful critiques of this research work.
I would like to thank the whole KiDM group, this includes the participating students, teachers, mathematics supervisors and researchers. For many reasons it has been a pleasure to work together with all of you during the whole project. Thanks to the teachers who have opened the doors to their classrooms so I was able to make observations, and thanks to all the students who have taken the KiDM test and done a lot of inquiry-based mathematics. A special thanks to Professor Morten Misfeldt for discussions, participation and feedback to different articles. Also a thanks to Associate Professor Mette Hjelmborg, who has been a stable support all the way.
She has supported the development of the KiDM test and been a co-operator on other papers on the way. Morten Rasmus Puck and Morten Petterson from UCL – University College must also be mentioned for their great help with the quantitative data – their help with the statistical analyses has been an important contribution in this thesis.
Special thanks should also be given to all the past and present employers at LSUL who all have giving me feedback and different kind of support on the way; including Marit Skou for discussions in mathematics education and Ph.D. student Stine Mariegaard for joyful and rewarding professional conversations, but also for sharing many time-measured working hours and Michael Fabrin Hjort for always giving me technical support or helping me with other problems on my way.
Finally, thanks to Associate Professor and Ph.D. student Camilla Hellsten Østergaard, who has been a stable support in both the happy and hard times during the different writing phases and for her important discussions and detailed reading of the thesis.
15 Last, but maybe most importantly, thank you to my family; to my husband Peter who always encouraged me to keep going and thanks to my three children - Alvilde, Molly and Vilfred.
Thank you.
16
5 Abbreviations & Acronyms
ATD: The Anthropological Theory of the Didactics IBME: Inquiry based mathematic education
IRT: Item Response Theory
KiDM: Kvalitet i Dansk og Matematik (Quality in Danish and Mathematics)
LSUL: Laboratorium for sammenhængende undervisning og læring (Laboratory for Coherent Education and Learning)
RaPiTE: Reasoning and Proving in Teacher Education SRP: Study and Research Path
TAP: Toulmin model for Argumentation Pattern TDS: Theory of Didactical situations
17
6 Abstract
6.1 English Abstract
Learning to reason in mathematics education is often considered to be one of the most important competencies in mathematics education, but nevertheless it is also considered challenging, and how to get students to go from arguing empirically to arguing more mathematical deductively it is often problematised (EMS, 2011).
The purpose of this thesis is to investigate how an inquiry-based teaching approach affects students' development of reasoning competence in primary school mathematics teaching.
Furthermore, there is a focus on how to study students' development of reasoning competence and whether it is possible to develop a test that can measure this development.
The thesis is closely linked to the development and research project named Quality in Danish and Mathematics (KiDM), which is aimed at developing and studying inquiry-based teaching in mathematics teaching in 4th and 5th grade in a randomised controlled experiment over three experimental trials.
The thesis is designed as a mixed methods study, which uses both data from qualitative video observations from several KiDM classes, and data from quantitative teacher and student surveys. In addition, a competence test was developed and used in the KiDM experiment from which both quantitative and qualitative data are drawn.
The result of the thesis is that an inquiry-based teaching approach affects students' development of reasoning competence in several ways:
By analysing video observations from the KiDM intervention with a model developed by Latour (1999), the findings indicate that the students, through different representations, go in small steps, stage by stage, from arguing from the complexity of everyday objects to arguing based on more general and formal mathematical approaches, so the students’ representations in small steps lose locality, particularity, materiality and multiplicity. In general, Latour’s (1999) model can be said to have the potential to focus on and clarify the students' work processes and reasoning (Paper I).
In another analysis, the findings indicate that overall the difference between the students’
reasoning activities depends on which inquiry-based activity is in focus, and that this may have implications for how a teacher will grasp the class discussion (Paper II).
18 Finally, analyses from the qualitative observations also provide evidence that cognitive conflicts can drive the students reasoning process and that the environment has an important role of retaining the conflicting positioning by making them available for discussion and scrutiny. The process of resolving cognitive conflicts is a process stretched over time that involved taking different routes and exploring approached and understandings (Paper III).
The quantitative data from the student survey indicates that students who were part of the KiDM experiment in mathematics experienced that they were generally more often focused on dialogues and more often discussed the students’ different solutions.
The developed competencies test is described as both reliable and valid (Paper IV), and in some way aligned to the KiDM classroom teaching, although not in all aspects (Paper V), but the test does nevertheless not produce a significant total result for students' development of mathematical competencies after the three trials in the KiDM project.
The findings from the thesis indicate that an inquiry-based teaching approach in general can have a positive effect on the students' development of reasoning competence; however, the teacher's approach to the activities and the designs of the tasks have a major influence on this effect.
6.2 Danish Abstract (Dansk resume)
At lære at ræsonnere i matematik anses som en af de vigtigste kompetencer i matematikundervisningen, men ikke desto mindre anses det ikke som en let opgave at udvikle elevernes ræsonnementskompetence, og det problematiseres ofte hvordan det er muligt at få eleverne til at gå fra at argumentere empirisk til at argumentere mere deduktivt (EMS, 2011).
Formålet med denne afhandling er at undersøge hvordan en undersøgende undervisning påvirker elevers udvikling af ræsonnementskompetence i grundskolens matematikundervisning. Derindunder er der også fokus på, hvordan man overhovedet kan studere elevers udvikling af ræsonnementskompetencen og om det er muligt at udvikle en test der kan måle denne udvikling.
Afhandlingen er tæt knyttet til udviklings- og forskningsprojektet Kvalitet i Dansk og Matematik (KiDM), som netop har til formål at udvikle og afprøve undersøgende undervisning
19 i matematikundervisningen på 4. og 5. klassetrin i et randomiseret kontrolleret eksperiment i tre forsøgsrunder.
Afhandlingen er designet som et mixed methods studie, som både anvender data fra kvalitative video-observationer fra flere forskellige KiDM klasser, men som også anvender data fra kvantitative lærersurveys og elevsurveys. Der blev derudover også udviklet en kompetencetest der blev benyttet i KiDM eksperimentet, hvorfra der både er blevet undersøgt kvantitative og kvalitative data.
Resultaterne i afhandlingen er overordnet, at en undersøgende undervisning påvirker elevers udvikling af ræsonnementskompetencen positivt, men på flere forskellige måder:
De kvalitative data og analyser bidrager med forskellige resultater herunder, at ved at anvende en model af Bruno Latour (1999) tydeliggør de, at eleverne gennem forskellige repræsentationer går fra at argumentere ud fra den komplekse materialitet i hverdagsobjektet til at argumentere ud fra mere generelle og formelle matematiske tilgange. Generelt kan der siges om Latour’s (1999) model, at den har potentiale til at fokusere på og tydeliggøre elevernes arbejdsprocesser og ræsonnementer (Paper I).
Resultaterne tyder samtidig på, at der overordnet set er forskel på elevernes ræsonnerende virksomhed afhængig af, hvilken undersøgende aktivitet der arbejdes med, og at dette kan have implikationer for, hvordan en lærer skal gribe klassens opsamling an (Paper II).
Endelig er der i analyserne fra de kvalitative observationerne fra den undersøgende undervisning også tegn på at det netop er de kognitive konflikter der har en drivkraft til at kunne udvikle elevernes ræsonnements proces, mens miljøet og de tilhørende materialiteter og artefakter spiller en vigtig rolle i at fastholde disse konflikterne, således at de kan være udgangspunkt for diskussion og undersøgelse. At løse kognitive konflikter i undersøgende undervisning anses således som en proces udstrakt over tid, hvor elevernes skal afprøve og undersøge forskellige tilgange og forståelser for derved at udvikle deres argumentationskæder (Paper III).
Elev-surveyens kvantitative data peger på, at eleverne oplever, at der på interventionsskolerne generelt er mere fokus på at eleverne skal argumentere for deres løsninger samt at der oftere opstår diskussioner i klassen omkring de forskellige løsningsforslag.
20 Den udviklede kompetencetesten beskrives som både reliabel og valid (Paper IV), men dog ikke fuldstændig i alignment med undervisningen af ræsonnementskompetencen i KiDM projektet (Paper V), alligevel formår testen dog ikke at få et signifikant resultat for elevernes udvikling af matematiske kompetencer i et samlet resultat efter de tre forsøgsrunder i KiDM.
Afhandlingens samlede resultater tyder på, at en undersøgende undervisning kan have en positiv effekt på elevernes udvikling af ræsonnements kompetence, men at lærerens tilgang til aktiviteterne og designet af opgaverne, herunder anvendelsen af forskellige artefakter og materialer har en stor indflydelse på denne effekt.
21
7 Keywords
Mathematical reasoning, argumentation, assessment, test, inquiry-based teaching, primary school, mixed method.
22
8 Introduction
“If problem solving is the heart of mathematics, then proof is its soul…” (Schoenfeld, 2010, p.
xii).
This quote clearly reflects that the subject of this thesis - the reasoning competence - is not a small and insignificant area in mathematics education, but that the area is considered as one of the most central and significant areas in mathematics education. The quote uses the word proof and not the word reasoning, but proof is often seen as part of the much wider description of reasoning competence in mathematics. The verb reasoning is, on the other hand, regarded as the active part of working with reasoning competence (this will be further explained in Section 9.1.2.2). To reason is, however, similarly described as the most important competence in mathematics (Ball & Bass, 2003; Hanna & Jahnke, 1996) or it is even seen as doing mathematics (Krummheuer, 1995). On the other hand, there is no consensus about how to define reasoning in mathematics (see Section 9.1.2) and there is still a huge task in making this competence’s notion understood, embraced and preserved by not only teachers, as there is also a need to empower teachers or textbook writers to develop teaching approaches and instruments that will help implement sound versions of this competence (Gissel et al., 2019; Niss, Bruder, Planas, Turner, & Villa-Ochoa, 2016). In the literature there is, however, already a great number of well-developed theoretical frameworks focused on different aspects of teaching and learning of reasoning and proof (Reid & Knipping, 2010); there is literature about how students understand or typically misunderstand proof (Durand-Guerrier, Boero, Douek, Epp, &
Tanguay, 2011) or how largely marginal a place reasoning and proof have in the mathematical classroom practice (G. J. Stylianides & Stylianides, 2017), but there is a relative small number of research studies about intervention studies about how reasoning competence develops in primary school mathematics classrooms. G. J. Stylianides and Stylianides (2017) argue, in this sense, that it would be unrealistic to expect, that if left to the individual teachers, textbook authors or other stakeholders, they would be able to successfully navigate and design appropriate learning experiences to help students overcome significant difficulties they face in the area of the reasoning competence.
23 Furthermore, the measurement of the reasoning competence is not a well-researched area; Niss et al. (2016) argue that assessment of students’ mathematical competencies needs to become a primary priority in educational research:
“… the need for devising more varied as well as more focused modes and instruments of assessment of the competencies, both individually, in groups and in their entirety.” (Niss et al., 2016, p. 630).
“It is essential to get the assessment of students’ mathematical competencies more in focus, both from a holistic perspective which considers complexes of intertwined competencies in the enactment of mathematics but also in an atomistic perspective where the assessment zooms in on this specific competency” (Niss et al., 2016, p. 624).
Knowledge about testing and measurement of competences is important for many people who work with educational studies, because some studies have gradually shown that tests can have unintended and negative consequences (Kousholt, 2012; Nordenbo et al., 2009), for example, the backwash effect which is the effect that tests have on how the teacher teaches. It is important to acknowledge that as long as the test items are parallel with the objectives of the syllabus/curriculum, they will have potential positive backwash effects on the learners;
otherwise, they will influence their learning in a negative way. We therefore need to be very careful when introducing tests in schools. This also includes an achievement test, where the primary recipients are people at some distance who require an assessment of an overall effect and not detailed information on the individual students. These assessments can also have an powerful influence on teaching and learning: “we simply want to stress that accountability tests, by virtue of their place in a complex social system, exercise an important influence on the curriculum of the school” (Resnick & Resnick, 1992, p. 49). In the design of an achievement test for the KiDM project, it is therefore important to be aware of the kinds of communications about educational goals the test implicitly or explicitly shows and the kinds of instructional practices the test is likely to invoke.
In this thesis the measurement of students’ processes and results and its correlation with an educational input (an inquiry-based intervention) is central - and it is precisely in this respect that this thesis will focus on not only how the students’ development occurs, but also how it is possible to measure and study a development of reasoning competence. The intention is to bridge the gap of measuring students’ development of competencies in a quantitative way by
24 surveys and testing, and a more qualitative approach to identifying students’ development of reasoning competence in the classroom.
This thesis is therefore not only about measurement – it is also about learning – it is about how we can support worthwhile learning in mathematics and help students develop the important competence of reasoning in mathematics.
8.1 Aim of the research - Theory into practice and practice into theory
This thesis is written in the area of mathematics education research which is part of educational research. Educational research is often seen as not so much research about education as it is research for education (Biesta, William, & Edward, 2003). The purpose is that research in education often mostly focuses on more practical aims for example, by developing new strategies for teaching or new ways of evaluating. In mathematics education several researchers have, despite this, expressed concern that mathematics education research has not played a greater role in supporting improvement of classroom practice, especially improvement of students’ learning of mathematics (Sierpinska & Kilpatrick, 1998; Skott, 2009). Skott (2009) argues, e.g., that the field is often overly optimistic with regard to the potential impacts of research and that it concerns the interplay between theories and practice. Boaler (2008) describes this difference between research in mathematics education and practices with a large gab:
“An elusive and persistent gulf exists between research in mathematics education and the practices of mathematics classrooms […] Indeed, I would contend that mathematics is the subject with the largest gap between what we know works from research and what happens in most classrooms.” (p. 91).
While Cockburn (2008) agrees:“… is the apparent mismatch between the amount of research in mathematics education undertaken and the limited amount that filters down into teachers’
classroom practice.” (p. 344). Bishop (1998) also expressed concern over researchers’
difficulties of relating ideas from research with practice of teaching and learning mathematics and claims that researchers need to engage more with practitioners’ perspectives and what effectively supports learning. von Oettingen (2018, p. 37) describes the difference between theory and practice by saying that theory means "to view what is" while practice means “to act
25 meaningfully” (translated from Danish). In theory, a distance to the world is expressed because one observes something in its natural quality, whereas practice, on the other hand, expresses a participation in the world and, therefore, is about something that first comes about through concrete actions. von Oettingen (2018) further argues that the meaning of developed theory for practice is not just about the theoretical reflection, but more about the theory's relation to a concrete practice. “Without theory, practice will fall into custom and without routines and without practice the theory would lose its meaning.” (translated from Danish) (von Oettingen, 2018, p. 41). Theory cannot in this sense dictate its own reliance into practice, and whether a theory works or can find application in practice is a question about the specific practice. Theory can, however, help to develop experimental drafts and point out possible difficulties and consequences, although theory does not provide any guaranties. von Oettingen (2018) adds the concept of empirical to the theory/practice discussion when he argues that an educational experiment is not a meeting between theory and practice - what meets is an “action-oriented enlightenment of practice and a meaningful act in practice in an experimental play” (translated from Danish) (p. 43). The empirical experiment is therefore not an expression of "reality” but an engineered observation of reality with the purpose of being able to expose something evident and particular "about" reality. When the theory has difficulty getting practical, it needs empirical knowledge, and when the empirical study will investigate practice, it needs the theory.
But there is never a straight path from theory to empirical or vice versa (von Oettingen, 2018).
Many teacher educators and researchers entered the field of mathematics education with a commitment to make the learning of mathematics more successful for more students and, in essence, this is also the overall aim of this thesis and the way to reach this goal goes through different empirical data.
The idea in the KiDM project is through a design-based programme grounded on specific theories about inquiry-based teaching to develop a teaching design that, among others has a focus on students’ reasoning competence. The aim in this thesis is to make sure that the inquiry- based KiDM intervention has a focus on reasoning competence by using different theories in the design-based process, and it is to develop different ways to construct empirical data which will be able to measure if and how the students develop reasoning competence in the KiDM experiment. In this thesis the measurement of the reasoning competence will be conducted by constructing different video observations, constructing a test and different surveys. These different empirical data will then be used in an analysis to theorise over the KiDM project’s
26 inquiry-based intervention as an explicit empirical component in order to deepen the understanding of classroom practice that relates to developing reasoning competence in mathematics.
8.2 Research questions
The overall research question has the following wording:
How can an inquiry-based teaching approach impact students’ reasoning competence in primary school mathematics classes?
This question is about if and how the students develop their reasoning competence by having their teacher use an inquiry-based approach. To answer this overall research question, five papers, among others, are selected that answer the first two sub-research questions RQ1 and RQ2, but all together are able to answer the overall research question. All five papers give different perspectives into the two sub-research questions; however, Paper IV and Paper V answer sub-research question one in different ways more specifically:
RQ1: How is it possible to study the development of students’ reasoning competence in primary school mathematics classes?
The focus in RQ1 is on three different approaches. Can a specific developed KiDM test measure the students’ development of reasoning competence? And what is possible to study in video observations of the classroom interactions? Student and teacher surveys have also been developed and used.
In Paper I, Paper II and III there will be more specific answers in different ways to the sub- research question two:
RQ2: In what way can students’ reasoning competence develop within an inquiry-based teaching approach?
To answer this question the focus will mostly be on the qualitative classroom observations, but the quantitative results from surveys and the test will also be involved.
The overall research question is the unquestionable core of this thesis, but the five papers are essential for answering these questions. Therefore, the description of how these five papers together respond to the above research questions is a very important part of this thesis, including a discussion of the findings but also critical voices about these findings. However, the thesis
27 also includes additional details about the different theories used in the five papers, that could not fit into the short papers (for example a supplementary description of theoretical perspectives of reasoning in mathematics, and a more embracive description of the KiDM project). It also includes some quantitative results from the KiDM project which are not yet written into any articles. These results can be found in Chapter 12. Furthermore, it also includes a thorough description and justification of all the methods used in the research, including some criticisms of choosing these methods.
8.3 Reading guide
This thesis is based on a classic approach to academic paper writing: first, the thesis contains some theoretical descriptions (Chapter 9) both with regard to the reasoning competence and its relation to inquiry-based teaching, but also in relation to measuring and testing reasoning competence. Second, the KiDM project will be described more thoroughly along with a description of how both the KiDM intervention and the KiDM test have been developed in a design-based process (Chapter 10). Basically, the process in this thesis can be divided into a development phase - with a design-based approach (see Section 10.1) and an experimental phase (see Section 11.2.1) which include how the developed intervention has been tried out in intervention schools and the developed test and surveys have been used in both intervention schools and control schools.
In Chapter 12, some further results from the quantitative empirical data will be presented just before the five papers will be presented and discussed in Chapter 13 in connection to the overall research question in this thesis. The five specific papers can be found in Chapter 14 and finally, in Chapter 15 the two sub-questions will be answered based on the articles with related discussion and reflections along with answers to the overall research question followed by reflections on legitimation of the methods used.
The way in which this thesis is constructed offers the reader two choices of reading.
On the one hand, the reader can read the dissertation from the beginning to the end. In this way the thesis could be read almost like the way you read a monographic dissertation. Another way could be to read the 5 papers first and then read the rest of the theoretical perspectives and the general discussion. By doing this the reader would have an initial overview of the project and the results before going into the deeper details about the project.
28 8.4 Short summary of the five papers
In Paper I, (Dreyøe, Larsen & Misfeldt, 2018), we use a model from Latour (1999) to investigate how reasoning competence develops in a specific inquiry-based activity called “What do the boxes weigh?”. This activity is characterised by starting with an everyday-related problem and ending with a more formal mathematical answer. In the paper it is analysed how students develop their reasoning competence in this process by using a model from Latour (1999) called circulating references. The special interest in this paper is a specific focus on the students' actions and their use of materiality and embodiment in their development of their reasoning competence. The paper built on a qualitative case study - a specific video-recorded lesson from the KiDM schools, but references are made to other KiDM cases with similar findings. In the paper it is concluded that Latour’s model is an inspiring model to describe how the students, with small steps (operators), go from reasoning in everyday language, with the use of materials, artefacts and other representations with great focus on incorporating these into their reasoning, towards no longer constantly referring to the materialities and instead begin to write symbols (numbers or drawings or colour codes) against a more formal solution written with numbers. It is concluded that Latour’s model, which in principle has been developed to study how researchers go from concrete practice in other research fields to finally have a very specific and abstract theory, can also be used to analyse how students’ reasoning processes develop in an inquiry-based activity.
In Paper II (Larsen & Lindhardt, 2019), the focus is on how students reason in an inquiry-based teaching approach, but this article focuses not only on one activity, as it instead studies how the students reason in different teaching activities, comprising different categories of inquiry-based teaching. The categories of inquiry-based teaching are developed as part of the KiDM project.
In this paper, different cases from video-recorded observations from the KiDM project are analysed using a qualitative approach. The focus is on two of the different inquiry-based categories and, by using different theories about reasoning in mathematics education (Harel &
Sowder, 1998; G. J. Stylianides, 2008), the students' reasoning is analysed. To specify this, there is a special focus on how the students argue through the various activities. The results show that in the activity called "brooder4”, students mostly focus on arguing for their process and not the results, including explanations and descriptions of their empirical experiments.
4 Directly translated from the Danish word “grubleren” which is the verb ‘to brood’ made into a noun.
29 Whears, in the "discovery"activity, there is a larger focus on arguing for the understanding of the mathematical concept, which is the aim of the process, where the teacher often ends up, being the one who makes the argumentation because she/he has a specific focus on having the students understand a particular argumentation.
In Paper III (Larsen & Misfeldt, 2019) the target is to investigate the role of cognitive conflicts in students’ inquiry-based mathematical work by looking at one episode of the students reasoning process. The episode is part of the KiDM intervention, and it is one double lesson out of a total of seven videotaped lessons from a particular KiDM intervention school. The findings indicate that cognitive conflicts exist in the analysed episode and that it can be productive and important in relation to the students mathematical reasoning process. The cognitive conflict can in this sense be seen as the driving force for the students reasoning process, where the environment has a role of retaining the conflicting positioning making them available for discussion and scrutiny. The process of resolving cognitive conflicts is – at least in the examples provided here – a process stretched over time that do not necessarily entails large significant jumps in the students’ understandings. Instead the cognitive conflicts make the students involved in taking different routes and exploring approaches and understandings that are internally in conflict (and hence sometimes mathematically wrong) and build up to a situation where they call for reasoning in order to be resolved. This means that students can benefit from having conflicting understandings over extended timespan in order to realise the need to resolve these conflicts.
In Paper IV (Larsen & Puck, 2019), the focus is on how to measure mathematical competencies in a test. The paper describes how a competencies test is developed in the KiDM project, and how it is validated along with reflections and discussion on the constructs, the design, the outcome and the measurement model. The paper focuses specifically on the reasoning competence and how this competence is measured in the competence test. The validity and reliability are discussed thoroughly in relation to the measuring instrument by using, e.g., item maps. The final test results from the KiDM experiment are also discussed in connection to the test.
In Paper V (Larsen, Dreyøe & Michelsen, 2019), the focus is on how the KiDM test is aligned to the classroom teaching in the KiDM intervention. The study focuses more closely on the relationship between students’ reasoning in the classroom including which arguments come
30 into play in the teaching compared to how the reasoning competence is tested in the KiDM test.
The analyse is based on video recordings from the KiDM classrooms but also the many quantitative test answers are analysed using G. J. Stylianides (2008) definitions of argumentations. The paper is specifically looking at which arguments the students develop in the inquiry-based teaching approach and what arguments we see in the students’ answers from different items in the test. The result of this study clearly shows that, in some points, there is a clear alignment. For example, the fact, that the students use both empirical and a few more deductive approaches both in the teaching and in the test, whilst there are also examples of more emotional and rational arguments in both the test answers and the inquiry-based teaching. The arguments are also represented roughly in the same way in both cases, except that all the arguments in the classroom are oral and, in the test, they are all in writing.
The social aspects in the two situations are, however, very different; the teaching situation is characterised by the students talking to each other about their reasoning and the teacher's questions cause the students to develop their reasoning at a higher taxonomic level, while in the test the reasoning is carried out individually and no feedback is forwarded to the student.
8.5 Theoretical considerations in this thesis
The concept of theory has played an important part in this thesis as already mentioned in the aim of this study. Theory has played part in both designing and developing interventions and tests used in the KiDM project, but theories have also been used, for example, to evaluate the test in connection to alignment (Paper V), statistic results and validation (Paper IV) and to classroom observations of students’ reasoning processes (Papers I + II + III) all which have influenced the final results of this thesis.
In this section a more thorough discussion of the different ways of using theory in this thesis, which role theory has played, and how the theories are applied and connected, will be presented.
Networking the different theories
The many different theories used in this thesis is not exceptional. Sources and theories in mathematics education generally come from many different areas in the world and different cultures, different institutional settings and the complexity in the topics of mathematics
31 education is very high, because it involves many different research fields like psychology, pedagogy, philosophy and mathematics (Bikner-Ahsbahs & Prediger, 2010).
In (2007), the Danish researcher Mogens Niss described that the field of mathematics education had experienced a widening of research perspectives in different aspects which may have some consequences for the research in this field. Bikner-Ahsbahs & Prediger describe it very clear in the following sentence:
“Since mathematics learning and teaching is a multi-faceted phenomenon which cannot be described, understood or explained by one monolithic theory alone, a variety of theories is necessary to do justice to the complexity of the field” (2010, p. 1).
And since mathematics education includes practical, empirical and theoretical investigations, it is obvious that the field cannot draw on theories from one single approach alone (Eisenhart, 1991). Lester (2005) consequently argues that we need to act as bricoleurs:
“…by adapting ideas from a range of theoretical sources to suit our goals—goals that should aim not only to deepen our fundamental understanding of mathematics learning and teaching, but also to aid us in providing practical wisdom about problems practitioners care about”
(2005, p. 466).
The specific theories used in this thesis (see Chapters 9, 10 and 11) are from many different research areas like philosophy (theories about pragmatism – Section 11.1), psychometrics (theories about testing – section 9.4 ), mathematics (theories about deductive proving – Section 9.1.3), methodology (e.g. theories about design-based research – Section 9.1.2), but mostly the theories are from the specific domain of mathematics education (theories about reasoning in mathematics – Section 9.1).
This thesis has an overall empirical approach and it is therefore obvious, according to both Eisenhart (1991) and Lester (2005), that many different theories are used. However, according to Niss (2007) a consequence of this is that the thesis needs to present an account for all the different theories and their interplay. This is exactly what will be the focus in this section.
This thesis has an overall conceptual framework. Eisenhart (1991) distinguishes between three different types of frameworks where a framework is defined as a skeletal structure designed to support the investigation: 1) a theoretical framework, 2) a practical framework and 3) a conceptual framework. A theoretical framework is a structure that guides research by relying
32 on an already established coherent formal theory and in this framework the research problem would be derived from this theory and the results or findings would be used to support, extend or revise the theory (Eisenhart, 1991). A practical framework guides the research by using
“what works” in the experience (Eisenhart, 1991) and is not informed by formal theory but only accumulated by practical knowledge. The conceptual framework is based on previous research and represents the researcher’s synthesis/combination of theories on how to explain a specific phenomenon or problem. This thesis has a conceptual framework which means that in some way it can be seen as my understanding of how the different variables can and are able to be studie and connected with each other. The multiple theories used in the five different papers need to connect in a network to be able to answer the overall research question. To network different theories is typical for the conceptual frameworks, which do not necessarily aim at a coherent complete theory, but use different analytical tools for the sake of solving more practical problems or to analyse concrete empirical phenomena (Eisenhart, 1991).
The reason that this thesis does not have a theoretical framework is that a theoretical framework in some way forces the researcher to explain that their results are given by one specific theory - the data need to fit that theory, rather than be evidence from the empirical data. Moreover, by having a theoretical framework, there is, however, a specific risk of ignoring important information in the process, for example, the concept of context, which is a very important element in the analysis in two of the papers. The thesis, however, has an overall scientific paradigm (pragmatism, se Section 11.1) which is seen as a way to be able to combine the different theories in a conceptual framework without limiting the access in a single theoretical framework. The different theories applied in the different papers are then combined in individual conceptual frameworks in each of the five papers. These conceptual frameworks in each paper are, however, not alike in the different papers, because each paper answers different empirical problems. The argumentations for how these different theories are connected in each paper are more or less explicitly explained in the separate papers, but in this section, we elaborate on the overall conceptual framework in this thesis.
Bikner-Ahsbahs and Prediger (2010, p. 492) made a landscape of strategies for connecting theoretical approaches seen in Figure 1.
33
Figure 1: Bikner-Ahsbahs and Prediger 2010, p.492
Network theories go in this sense from just understanding other theories to synthesizing the different theories (see Figure 1). In all the papers included in this thesis the network of the theories can be seen as a combination of theories. This follows Prediger and Bikner-Ahsbahs (2014) description that strategies of combining theories are mostly used for a networked understanding of an empirical phenomenon or a piece of data which is typical for conceptual frameworks because they do not necessarily aim at a coherent complete theory but use different analytical tools (Prediger & Bikner-Ahsbahs, 2014, p. 119). This is very clear in Paper V where theories from G. J. Stylianides (2008) are combined with theories from, among others, Harel and Sowder (1998). The process of combining theories also has its limits and Radford (2008, p. 323) argues that in networking theories, theories will have an “edge” that a theory cannot cross without a substantial loss of its own identity and beyond such an edge, the theory conflicts with its own principles. (Radford, 2008, p. 323). This had the consequence that every time a new theory was included and combined with the other theories, considerable considerations about the theoretical background of these theories had to be discussed and a decision had to be made about whether it was possible whatsoever to combine this theory to the other theories.
Basically, this had an impact on the choice of the overall scientific paradigm of this thesis, because there was a need in this thesis to combine both theories with social perspectives (like the social perspectives on reasoning in mathematics classrooms), to theories with a more so- called “positivistic” background (theories about causality), where the social factors have less influence. However, the choice of having a pragmatic scientific approach provides the opportunity to include both these perspectives (ses Section 11.1 for further explanation on this).
All the different theories used in this thesis do not play the same role or purpose in the different papers. This will be the focus in the next section.
34 Theories in research, practice and problems
Niss (1999) suggests that a theory in mathematics education entails both a descriptive purpose, aimed at increasing understanding of the phenomena studied, and a normative purpose, aimed at developing instructional design.
To show that the theories in this thesis are used in many different ways, Figure 2 is constructed.
Figure 2: Models of theories used in the thesis
First, the scientific paradigm seen in the blue circle indicates that the theory of the chosen scientific paradigm of pragmatism refers to the epistemological and ontological positions or orientations which can be seen as the guide that informs the research and analysis processes in the overall research approach.
In addition, it has been a deliberate choice that most theories used in the research are from mathematics education. If, for example, scientific aspects had also been included, for example, in defining inquiry-based teaching, the definitions would have been significantly different. The theories used within mathematics education are more specifically, mainly focused on the
35 reasoning competence. The theories about reasoning competence have been used for various purposes. There are theories used normatively. For example, theories that have been used to develop didactical elements, by giving inspiration to the activities in the intervention and the test items. This mainly concerns different theories about reasoning in inquiry-based mathematics teaching and testing, but it also includes other theoretical implications used to make prescriptive statements to guide practice. Theories used in the development process also include theories found in two literature reviews made in the beginning of the process. Some of the same theories, but also other new theories, are then used later in the process in a descriptive way, e.g., like a lens (Silver, Herbst, & learning, 2007). This is done when different theories have been used to study classroom interaction, like in Papers I, II and V where respectively theories from Latour (1999), Harel and Sowder (1998) and G. J. Stylianides (2008) were used in analysing video observations and test answers. Different theories have also been used to describe gabs and problems in earlier research and to confirm that gabs found in the empirical data are not already described and resolved theoretically. Finally, the thesis also includes different theories, whose role is to describe or even explain practice. In this thesis, theories about social aspects like Yackel and Cobb (1996) are included to explain some of the findings from the analysis. Another particular case of theory that provides rational description is that of statistical theory (item response theories) whose aim is to understand practice by demonstrating a correlation between the two variables; the control school and the intervention school (Paper IV). The last aspect which is also indicated in Figure 2 by a yellow triangle at the bottom is the methods used and the methodology behind these methods. These theories can be seen as a mediating connection between the theory of reasoning competence, the research question and practice. These theories are often used in the practice of doing research specially to justify the choices in the design and how to construct and analyse the data. This concern, among other, theories about design methods, and theories about methods to construct empirical data. As can be seen, the triangle goes through all the other circles, because the methods depend very much on what scientific paradigm the overall orientation is, but at the same time the research methods also depend on which area is being studied and which methods are able to measure mathematical reasoning competence. By making these few elaborations it should be clear that the roles and purposes of theory in this thesis have been many and different.
36
9 Clarifications and theoretical background
Before going deeper into the research done in this thesis, there is a need for elaboration in connection to justification of why studying this competence (Section 9.1.1), but also consideration about defining reasoning competence in mathematics education (Section 9.1.2- 9.1.4), are important. The taxonomies of the competence will also be defined in this chapter (Section 9.1.5) along with description of different analytical tools used in educational research to study the reasoning competence (Section 9.1.6). The connection between inquiry-based teaching and reasoning competence is described in Section 9.2. Additionally, there will be a section with a description of the measuring and testing of competencies in mathematics education (Section 9.3). The section also includes a systematic review made as a background for developing some guidelines to be used in developing the competence test used in the KiDM project (Section 9.4 + 9.5).
9.1 Reasoning in mathematics education
A fast-increasing amount of research has presented important insights and understanding into the area of reasoning, which has led to an escalation of publications about different aspects of reasoning in mathematics (A. J. Stylianides & Harel, 2018). The intention in this section is not to try to give an exhaustive review of all research conducted in this field, rather the focus is to present theories and research results that have a particular influence or have the potential to shed light on the critical issues in connection to the research in this thesis.
In the chapter below, proof is often included as a part of reasoning, since these two concepts are often described side by side and often referred to collectively in the theoretical literature.
Proofs are therefore also included here in this approach. This does not mean that there is no difference between the two concepts, and the relation between them will be further explained in Section 9.1.2.
The importance of focusing on reasoning in mathematics education
Developments in education have in the recent decades been directed towards mathematical competencies by researchers, organisations or national levels. In Denmark, the national frameworks in mathematics now have a main focus on developing mathematical competencies
