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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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Download date: 26. Sep. 2021

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

Duration of this Ph.D.:

15th of September 2016 – 14th of September 2019

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)

 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)

 *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)

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)

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

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

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

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

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) 11

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

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, [email protected]

Assistant Supervisor:

Thomas Illum Hansen, PhD., Research Director, Associate Professor, University College Lillebælt, [email protected]

The dissertation at hand was prepared in the Laboratory for Coherent Education and Learning (LSUL) department at the University of Southern Denmark.

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.

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.

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

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).

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

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.

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.

7 Keywords

Mathematical reasoning, argumentation, assessment, test, inquiry-based teaching, primary school, mixed method.

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.

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

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

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

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 26

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. 27

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. 28

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

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

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 31

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.

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.

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 34

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.

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 36

and not just a focus on content skills and concepts (Undervisningsministeriet, 2019b). Similar intentions have happened in the United States who introduced “The Common Core State Standards for Mathematics” (National Governors Association Center for Best Practices, 2010) which include “standards for mathematical practice” that contain, among others, problem solving, modelling and reasoning in mathematics. This trend also appears in the international studies on Trends in International Mathematics and Science Study (TIMSS) and the Program for International Students’ Assessment (PISA).

Mathematical reasoning is one of the eight mathematical competences described in the well- known Danish mathematical competence report (Niss & Jensen, 2002)5, which have had a major influence on the national curriculum in Denmark because as early as in 2003 it was part of the national teaching guide (Undervisningsministeriet, 2003) and in 2009 it was part of the national curriculum in Denmark (Undervisningsministeriet, 2009). Today, reasoning competence is seen as central to all students’ mathematical experiences throughout all the school years in Denmark (Undervisningsministeriet, 2019b).

Reasoning in mathematics is seen as fundamental for doing mathematics. It is seen as a basis of mathematical understanding and is essential for developing, establishing and communicating mathematical knowledge in order to articulate, but also the need for and appreciation of making convincing arguments (Ball & Bass, 2003; Carpenter, Franke, & Levi, 2003; Hanna & Jahnke, 1996; G. J. Stylianides & Stylianides, 2008). Ball and Bass (2003) contend that:“…the notion of mathematical understanding is meaningless without a serious emphasis on reasoning” (p. 28).

However, even with this major focus on competences in school mathematics there are still arguments for a requirement for more research about teaching and learning competences in school mathematics:

“Fostering, developing and furthering mathematical competencies with students by way of teaching is a crucial […] priority for the teaching and learning of mathematics in all countries [...] We now need to understand the specific nature of the contexts and other factors that help create such progress[… ]there is a huge task lying in front of us in making competency notion understood, embraced and owned by teachers and in empowering them to develop teaching

5 Translated into English in (Niss & Højgaard, 2011) 37

approaches and instruments that allow for implementation of conceptually and empirically sound versions of mathematical competencies and their relatives in mathematical teaching and learning all over the world” (Niss et al., 2016, p. 630).

Researchers claim different reasons for why we still face problems concerning students’ reasoning in school mathematics today. In the following, four reasons will be presented: 1) late introduction, 2) limited teacher knowledge, 3) the traditional practice of teaching reasoning and proof and 3) the lack of opportunities.

1) Late introduction: Several researchers (Ball, Hoyles, Jahnke, & Movshovitz-Hadar, 2002; Harel & Sowder, 1998; G. J. Stylianides & Stylianides, 2008) have identified students’ sudden introduction to reasoning and proof in secondary school levels as a possible explanation for the many difficulties that secondary school (Knuth, 2002) and university students (Harel & Sowder, 1998) face with reasoning and proof. The notion of reasoning and proof has traditionally been associated almost exclusively with high school and university mathematics (G. J. Stylianides & Stylianides, 2008). A. J. Stylianides (2007) suggests that the late introduction of reasoning and proofs may cause a disconnection in students’ mathematical experiences, that can contribute to later problems with reasoning and proof and reasoning in primary school needs to be more in focus. 2) Limited teacher knowledge: Secondly, teachers are seen as having limited knowledge about reasoning and proof in school (Schifter, 2009; A. J. Stylianides, 2016; A. J. Stylianides & Ball, 2008; A. J. Stylianides, Stylianides, & Philippou, 2004). Many teachers think of reasoning as one more piece of content to squeeze into an already full agenda (Schifter, 2009) or that reasoning and proofs are beyond students’ capabilities in primary school and that proving is only an appropriate goal for a few students who are “developmentally ready” for it (Bieda, 2010), or teachers think that the content only apply to the teaching and learning of geometry in the upper secondary school level (Knuth, 2002). A focus on teachers’ understanding of reasoning and proof position in primary school also needs to be stressed. Furthermore, studies have also been conducted on teacher-students' problems with the reasoning competence, explaining how important it is to explicit focus on how to teach this competence (Skott et al., 2017). 3) The traditional practice of teaching reasoning and proof: Yackel and Hanna (2003) argue that teachers’ current teaching in reasoning and proof does not allow students to develop understandings of reasoning, because they do not facilitate the students’ understanding of 38

the contents in question. Proof is typically taught as something to be memorised and proving comes to be understood by students as ready-made or a ritual that confirms what they already know to be true (validating obvious statements), rather than as a means of developing understanding. Teachers insist on formalities instead of doing reasoning in the classroom and, in this sense, they dissociate proof from sense-making. Brousseau and Gibel (2005) elaborate:

“If model proofs are still presented to students, they are meant to serve as “model reasoning” which the students could then use in producing their own original forms of reasoning. But there is always the risk of reducing problem solving to an application of recipes and algorithms, which eliminates the possibility of actual reasoning.”(p. 14).

Furthermore, in Harel and Rabin (2010a), the term arbiter of mathematical correctness is presented. This concerns who determines whether a particular assertion is true and how the truth is established. They presented some episodes where teachers were responding to students’ ideas and they gave examples of teachers serving as arbiters rather than a facilitator of debate. The debate was resolved when the teacher was satisfied with the outcome, or when students agreed with the teacher, but the agreement was often more socially rather than intellectually based. Therefore, a focus also needs to be on how to teach reasoning and proof in primary schools.

4) The lack of opportunities: Finally, a fourth reason is that students lack opportunities to develop reasoning and proof in school contexts (Bieda, 2010). In an analysis from G. J. Stylianides (2009) which included 4855 tasks from textbooks, only about 40% of these tasks (1852 tasks) were designed to offer students at least one opportunity for reasoning and proof. This is also found in a Danish research study (Gissel et al., 2019) where the eight most widely used mathematical systems from Copenhagen (lending statistics) were analysed for both explicit and implicit foci on different mathematical competences. For reasoning competence, this study found a great variance between the different systems, and a particularly little coverage on explicit competences. In the examination of implicit competences coverage (assignments where competences with small modifications could come in play), it showed far more occurrences, but this also requires that teachers themselves have both knowledge and understanding of how to address this.

Defining mathematical reasoning In mathematics education and probably also in other fields, some key concepts are often used without being defined at all (Niss, 2007). This is also the case of mathematical reasoning. Yackel and Hanna (2003) argue that reasoning in mathematics is complicated by the fact that the term is widely used with the implicit assumption that there is a universal agreement on its meaning. Such consensus about its meaning, however, does not exist (G. J. Stylianides, 2009). Jeannotte and Kieran (2017) define what they call “the conceptual blur” (p. 1) regarding mathematical reasoning and argue that “The discourse on mathematical reasoning is not monolithic; it does not consist of a single voice” (p. 1).

Some researchers problematise that the term is generally used by teachers of all subjects and by researchers with a variety of meanings (Brousseau & Gibel, 2005), because it makes it difficult to discuss and evaluate the term. Brousseau and Gibel (2005) claim that it is unrealistic to expect that teachers will be able to make good progress in teaching mathematical reasoning to students and evaluating whether they have learned what was intended to be taught, if there is no agreement about what is meant by the term.

Balacheff (2008) agree about the differences and tell us to be more cautious:

“The scientific challenge of research in mathematics education is not to develop opinions and beliefs about teaching and learning, but to shape a body of knowledge which should be robust (which means theoretically valid) and relevant (which means instrumental for practitioners and other stakeholders). Convergence should be the rule. It must be our priority. Divergences must be considered as symptoms of problems… by “convergence” I do not mean a unique system of thought. I mean that differences must be explained and related in a way which keeps coherent the overall understanding we have of process and phenomena related to the learning and teaching of mathematical proofs” (Balacheff, 2008, p. 501).

However, in the following sections, four major elements will be described to clear the “conceptual blur” of the concept to find some prototypical features of this flexible category with lots of grey areas. This is important to understand the concept of reasoning in this thesis and thus gain a better understanding of what this thesis is all about. Firstly, the role of reasoning needs to be elaborated. Secondly, the opposition of seeing reasoning as a relation, a process or a product will be explained. Thirdly, the association between proofs, arguments and reasoning

will be defined. And finally, the difference between different kinds of reasoning will be described.

9.1.2.1 Purposes of reasoning and definitions Hersh (2009) argues that the function and role of reasoning and proof in the classroom is different than their role in research. In research, the function of reasoning is to convince. In the classroom, convincing is no problem. Students are all too easily convinced (Hersh, 2009). What reasoning and proof also should do for the students is not only to help them understand the meaning of the theorem being proved (to see that it is true), but also to provide insight into why a theorem is true (Why is it that the sum of two odd numbers is an even number) and a general understanding of the function of reasoning and proof in mathematics (Hanna, 2000).

In a study of Zaslavsky and Shir (2005) they found that 60% of the students in their study did not even understand why proof was needed. Students did not see any need for mathematical proof because their need for certainty is personal and for many students empirical work is personally more convincing. Yackel and Hanna (2003) argued that reasoning and proofs are used for different purposes, but that the most powerful proofs in education are those which both explain and communicate at the same time. “chains of logical argument do not function as proofs unless they serve explanatory and communicative functions” (p. 228).

9.1.2.2 Reasoning as a product, process or relation In the earlier mentioned Danish mathematical competence report (Niss & Jensen, 2002), mathematical reasoning competency has a broad definition; Mathematical reasoning competence consists of the ability to create and carry out formal and informal arguments, as well as the ability to follow and evaluate the arguments made by others. It encompasses understanding what a mathematical proof is, the role of counterexamples, and the difference between a proof and other forms of mathematical reasoning such as explanations based on examples. In addition, it includes the ability to develop an argument based on heuristics into a formal proof. It states that the competence is not only about justification of mathematical theorems, but also about creating and justifying mathematical claims in general, such as answers to questions and problems (Niss & Jensen, 2002).

Reasoning competence is often seen as the ability to perform mathematical reasoning, where reasoning can be defined in many different ways. NCTM has developed a “reasoning and proof”

cycle (NCTM, 2008). A cycle that starts with an exploration then a conjecture or hypothesis and next a justification or argumentations. This approach is in line with both G. J. Stylianides (2008) and Lampert (1990) view about mathematics as a process and that reasoning is seen as part of this process. G. J. Stylianides (2008) argues that reasoning can be seen as a sequence of steps; making inquiry to make generalisations or conjectures and developing arguments for the truth or falsity of the generalisations, some of which may qualify as proofs.

In a study to analyse students’ reasoning, Lithner (2008) focuses on reasoning in connection to problem solving and defined reasoning as “the line of thought adopted to produce assertions and reach conclusions in task solving” (p. 257). He sees reasoning as a product of separate reasoning sequences. Each sequence includes a choice that defines the next sequence and the reasoning is the justification and conclusion for the choice that is made. This approach is leaning up against part of the definition from NCTM “to develop and evaluate mathematical arguments and proofs” (NCTM, 2000, p. 55). Lithner (2008) approach is not based on formal logic and not restricted to proof, and he argues that reasoning may even be incorrect as long as there are some kinds of sensible reasons backing it. In Brousseau and Gibel (2005), however, reasoning is defined as a relation between two elements, a condition or observed fact and a consequence, such that:

“A denotes a condition or an observed fact, which could be contingent upon particular circumstances; B is a consequence, a decision of a predicted fact; R is a relation, a rule or, generally, something considered as known and accepted… An actual reasoning contains moreover, an agent E (student or teacher) who uses the relation R; a project, determined by a situation S, which require the use of this relation” (Brousseau & Gibel, 2005, p. 18).

There are many different approaches to the reasoning concept as described. Reasoning in this thesis is defined with a broad perspective to include the line of thought (thinking process) but it is also seen as the product of these processes (like in Lithner (2008)) adopted to produce conjectures, argumentation and reach conclusions. In this perspective, argumentation becomes a substantiation part of reasoning and therefore an important aspect which will be in focus in the next section.

9.1.2.3 Argumentation as part of reasoning and the relation to proofs Learning mathematics can be seen as argumentative learning (Krummheuer, 2007). Argumentation is then seen as not only a teaching aim, in the sense that one must design mathematics instruction in a way that makes the students reach this goal and be able to argue at a sophisticated mathematical level (learning to argue), argumentation is also part of everyday mathematics classrooms as a precondition for the possibility to learn not only the outcome (Krummheuer, 2007).

Boero (1999) argues that argumentation is part of many aspects in doing mathematics and reasoning processes: it is part of the production of a conjecture, the formulation of statements, explorations of contents (and limits of validity) of the conjecture, selection and enchaining of coherent, theoretical arguments into a deductive chain, organisation of the enchained arguments into proof that is acceptable according to current mathematical standards, approaching formal proof. For Duval (2007), an argument is considered only to be anything which is advanced or used to justify or refute a proposition. This can be the statement of a fact, the results of an experiment, or even simply an example, a definition, the recall of a rule, a mutually held belief or else the presentation of a contradiction. It takes the value of a justification when someone uses them to say “why” he/she accepts or rejects a proposition. In Krummheuer (2007) descriptions of argumentation, he is interested in the process where argumentation leads to the verification of a statement which likely could include proving. Pedemonte (2007b) argues that argumentation activity might favour the construction of proving; “Experimental research… shows that proof is more accessible to students if an argumentation activity is developed for the construction of a conjecture.” (p. 4). She argues that because conjecturing constitutes a fundamental activity where argumentation and proof may be related, conjecturing can resolve the potential conflict between the two main functions of proof, namely validating within a theory and explanation. In contrast to seeing conjecturing and argumentation as having potential in developing proof, Duval (1999) is working with argumentation as an epistemological obstacle to the learning of mathematical proof. According to Duval (2007), deductive thinking does not work like argumentation, because both deductive thinking and argumentation use very different linguistic forms and propositional connectives, and he claims that general argumentation is one of the main reasons why most students do not understand the requirements of mathematical proofs. For Duval the:

“epistemic values do not matter within description or explanation, but they are in the foreground with every kind of reasoning. And between argument and valid deduction the difference lies in the role assigned to status. So, in any debate we can get convincing arguments without proving, that is to say without proving” (Duval, 2007, p. 143).

In the theories of Duval, argumentation is therefore part of reasoning, but it is opposed to proving where the status of a proposition is predominant rather than the content.

The term proof is also an important aspect in connection to reasoning: Hersh (2009) sees proofs as part of reasoning: “in facts proof is just reasoning but careful critical reasoning looking closely for gaps and expectations” (p. 19). A lot has been said about the “complex, productive and avoidable” relationship between argumentation and proof (Boero, 1999). But also the word “proof” or “proving” are used in a number of ways, even in academic disciplines like mathematics education, where the exact meaning of these words, however, are seen to be important. Reid and Knipping (2010) describe different ways of defining proof. They argue that, in general, they find both a narrow and a broad definition of the category of proof. A narrow view of proof requires that all proofs have three characteristics: they must be deductive, convincing, and at least semi-formal, whereas a broader category only requires proof to be deductive (Reid & Knipping, 2010). Hersh (2009) describes proofs loosely as a chain or tree of connected statements beginning from some that are taken as true and proceeds to a conclusion according to a few logical rules (e.g. modules ponens etc.)6. Duval (2007) sees proofs as the outcome of reasoning: “The outcome of any reasoning is not only to produce new information but also, above all, to chance the epistemic value of a proposition whose truth we want to prove or attempts to convince somebody else of.” (p. 139). In the broad description, proof can be seen as different things; concepts, objects, proof texts (in school books), convincing argument, personal verification, personal understanding, as a process or deductive reasoning (Reid & Knipping, 2010), while the roles of proof are also seen to be varied; proof can be used for verification, to change an epistemic value, for explanation, exploration, systematisation, communication; however, in school the most common role for proof seems to be verification (Reid & Knipping, 2010). Healy and Hoyles (2000) made a survey where they asked students

6 Modus ponens and modus tollens: In logic, two types of inference that can be drawn from a hypothetical proposition: Modus ponens refers to inferences of the form A B; A, therefore B. Modus tollens refers to inferences of the form A B; B, therefore, A ( signifies “not”)(Kjeldsen, 2011). ⊃ ⊃ ∼ ∼ ∼ 44

to describe proof and its purposes. The two roles mentioned most often were verification (50%) and explanations/communication (35%). Exploration/systematisation was mentioned by only 26 students (1%). 28% was unclear of the role.

Different kinds of reasoning We often see a distinction between different kinds of reasoning; abductive reasoning, inductive reasoning, deductive reasoning and reasoning by analogy (Reid & Knipping, 2010). Most of these terms are widely, but inconsistently, used in mathematics education. In this thesis, in order to be able to use the concepts interchangeably, a brief description of them will be given in the following section.

Abductive reasoning can be seen as the search for a general rule from which a specific case would follow (Reid & Knipping, 2010, p. 101). In general there is a growing literature on the importance of abductive reasoning in mathematics education, even though it is not new research, because although they did not name it abductive reasoning some of the reasoning considered by George Polya under the heading “guessing”, “conjecturing” and “hunches” also seems to be abductive in nature (Reid & Knipping, 2010).

In Reid and Knipping (2010) they refer to a quote from Sherlock Holmes that describes it in “A study in Scarlet”:

“Most people, if you describe a train of events to them, will tell you what the results would be. They can put those events together in their minds and argue from them that something will come to pass. There are few people, however, who, if you told them the results, would be able to evolve from their own inner consciousness what the steps were which led up to the results. This power is what I mean when I talk of reasoning backwards” (p. 100).

Inductive reasoning can be seen as proceeds from specific cases to conclude general rules. It uses what is known in order to conclude something previously unknown. Inductive reasoning is normally seen as only probable, and not certain. Instead of being valid or invalid, inductive arguments are either strong or weak, which describes how probable it is that the conclusion is true. Reid and Knipping (2010) introduce five different types of inductive reasoning: pattern observation, prediction, conjecturing, generalizing and testing. Where generalizing proceeds from a premise about a sample to a conclusion about the population and conjecturing refers to

making a general statement also from a sample, but the general statement requires additional verification, and a prediction draws a conclusion about a future individual from a past sample because the future is assumed to be like the past. Pattern observation is about observing a number of situations in which a pattern exists and concluding that that these patterns are true for all situations. Testing is about testing predictions and conjectures and conjecturing, e.g., by exhaustion.

Deductive reasoning is the process of reasoning from one or more statements (premises) to reach a logically certain conclusion. If all the statements (premises) are true, the terms are clear, and the rules of deductive logic are followed, then the conclusion reached is necessarily true. Deductive reasoning is often considered to be the basis of proof if it is seen with a narrow perspective of proof, because it is the only kind of reasoning which is thought to establish certainty and is often associated with verification. However, it can also be used to explain or explore as Hanna (2000) argues.

Reasoning by analogies can most easily be explained by giving an example. Reid and Knipping (2010, p. 111) give an example of two students who are asked to explain why the sum of two odd numbers is even. One student answer: “because a negative times a negative is a positive”. The student here takes another analogy and uses it in another task. Reasoning by analogy involves making a conjecture based on similarities between two cases, one well known, and another less well known.

Remarks about the social aspects in reasoning in mathematics classrooms Reasoning and proof are often defined from a cognitive perspective, focusing on arguments that help an individual gain conviction in a mathematical claim (Harel & Sowder, 2007), while other researchers (Balacheff, 1988; Ball & Bass, 2003; Krummheuer, 1995; Yackel & Cobb, 1996) see reasoning and proof with a social perspective, by focusing on how members of a mathematical community justify an argument as a proof. Balacheff (1988) defined proof as “an explanation which is accepted by a community at a given time” (p. 2). The argumentations are here seen as a social phenomenon, where what counts as proof depends on what arguments are accepted as true and what rules of inference are accepted as valid. The “rules of reasoning” that are accepted might change over time and from community to community. In this way it differs from one class to another; for example, what might be acknowledged in a primary school class

is likely to be insufficient evidence for an upper secondary school class, because the type of arguments that the students are expected to produce depends on the cognitive abilities of students, but also on the culture in the classroom. In this regard the culture in the mathematics classroom is very important. Yackel and Cobb (1996) argue that the classroom is a social context in which a particular mathematical culture is developed within and by interactions between teachers and students. The culture and the norms in a classroom are not pre-given but emerged in relation to social conditions. Another way to see the social perspective is seen in the competence report (Niss & Jensen, 2002), where the focus in the reasoning competence is not only on developing argumentations but also on being able to follow and judge arguments created by others, which is an aspect of the competence that requires social interaction.

Taxonomies of reasoning This section will focus on different kinds of reasoning and different levels and taxonomies of reasoning. Taxonomies are very important when assessing students or developing a test, as these measurements are about locating the student’s level of competencies. In the development of tests, there is often a need to place individual students at different levels. In this section, a more individual perspective will be given to reasoning, but the social aspects as described in Section 9.1.4 will always be present. Some theories propose stages of understanding; Skemp (1978) distinguishes between instrumental and relational understanding. Some define categories of cognitive achievement like the taxonomies of Bloom, Engelhart, Furst, Hill, and Krathwohl (1956) and Van Hiele (1999) or Biggs and Collis (1989). Other frameworks structure particular competencies, e.g., problem solving like Schoenfeld (2009). Taxonomies of reasoning are, however difficult to describe, because it does not make sense to say that the scope of the reasoning competence of a person who can reason for algebra, geometry and probability calculation is less than for a person who can reason within probability calculations, functions, infinitesimal billing and optimisation. Likewise, it does not equally make sense to compare the technical level of reasoning competence of a person who is an expert in treating expressions within, for example, trigonometry, with the technical level of a person who is good at reasoning within probability distributions (Niss & Jensen, 2002). In mathematics education, we however, find theories of proofs in a taxonomy. The purpose of proof is in this sense not necessarily deductive. Harel and Sowder (1998) view proof as a continuum and describe: “by proving we mean the processes employed by an individual to remove or create doubts about the truth of an

observation” (Harel & Sowder, 1998, p. 242). The main point in Harel and Sowder (1998) is how conjectures are rejected or rendered into facts. And they introduce the concept of “proof schemes” which are arguments that a person uses to convince themselves or others of the truth or falseness of a mathematical statement. The taxonomy of proof schemes consists of three classes: the external conviction proof schemes class, the empirical proof schemes class, and the deductive proof schemes class. To prove within the external conviction proof scheme depends on an authority such as a teacher or a book. It could also be strictly on the appearance of the arguments or on symbol manipulations. Proving within the empirical proof schemes is marked by its reliance on either evidence from examples or direct measurement of quantities, substitutions of specific numbers or perceptions. The third deductive proof scheme class consists of two subcategories, each consistenting of various proof schemes. The deductive proof schemes share different essential characteristics among them: generality and logical inference. The generality concerns a “for all” argument – not isolated cases and no exception is accepted, and the logical inference is that the argument must be based on the rules of logic and that the proving process must start from accepted principles/axioms (Harel, 2007). The distinction in different proof schemes from Harel and Sowder (1998) is based on an empirical study and the strength lies in the fact that it is not abstract nor a priori: “all [results] were derived from our observation of the actions taken by actual students in their process of proving”(p. 244). Balacheff (1988) made another differentiation concerning aspects of proof in practice of school mathematics, with the following categories: Naïve empiricism, the crucial experiment, the generic examples and the thought experiment. The naïve empiricism consists of asserting the truth of a result after verifying several cases. The crucial experiment refers to an experiment whose outcome allows a choice to be made between two hypotheses. For the generic example and the thought experiment, it is not enough to show that the result is true, just because ‘it works’; it is about establishing certainty by giving reasons/arguments. Generic examples include making explicit the reasoning for the certainty of an assertion by means of operations or transformations on an object that is not here in its own right, but as a characteristic representative of its class. The thought experiment is more detached from a specific representation and de-contextualised and formulated in generality.

In broad terms the empirical proof scheme from Harel and Sowder (1998) corresponds to Balacheff (1988) crucial experiment and the term deductive proof scheme deduced by Harel and Sowder (1998) corresponds to Balacheff (1988) thought experiment. There is, however, a

small difference in how they categorise the terms; e.g., Balacheff (1988) includes the generic examples in the more pragmatic justifications, while Harel and Sowder (1998) place the generic examples in the deductive proof scheme category.

Generally it is agreed that mathematical argumentation at times degenerates into authoritative/external or empirical proof schemes (EMS, 2011; Harel, 2007) because the concept of formal proof is “completely outside mainstream thinking” (EMS, 2011, p. 51) and it requires a new basis of beliefs which is distinct from the basis of other kinds of reasoning (Reid & Knipping, 2010, p. 157). In everyday life and in science, the means for justification available to humans are largely limited to empirical evidence (Harel & Rabin, 2010a, 2010b), but if during early school year our judgement of truth in mathematics continues to rely on empirical considerations then the empirical proof scheme will likely dominate our reasoning in later school years. Harel and Rabin (2010b) argue that it takes enormous instructional effort for students to recognise the limits and roles of empirical evidence in mathematics to begin to construct alternative deductively based proof schemes.

Analytical tools or frameworks used in the area of mathematical reasoning Due to the very extensive literature on reasoning in mathematics education, many different analysis frameworks and tools have been used in the research literature. To be able to analyse what is going on in the classrooms of the intervention, there is a need for an analytic framework. In the following, some of these entries will briefly be described.

As an analytic framework, both the concepts of ‘proof schemes’ from Harel and Sowder (1998) and the reasoning levels from Balacheff (1988) have been used previously, but Toulmin’s model (Toulmin, 1958) has especially been a popular analytic tool.

The Toulmin model for Argumentation Pattern (TAP) has been identified as a valuable analytical tool for reasoning in science (Erduran, Simon, & Osborne, 2004) and it has also been used by a large number of researchers in mathematics education (Krummheuer, 1995; Pedemonte, 2018). However, some lack of resolution for tracing teachers’ changing practices and children’s enhanced argumentation is described by using the TAP framework. For example, in Osborne, Erduran, and Simon (2004) they counted the amounts of rebuttals, an element in the TAP model, in student argumentation as an indicator of quality. As a result of their study,

Osborne et al. (2004) concluded that in order for the students to be able to form TAP-structured arguments, the students need to be explicitly taught to do so.

G. J. Stylianides (2008) made a conceptualisation of reasoning and proof in a table, which has been used as an analytical tool in different research, among other, in Wong and Sutherland (2018), who investigate both reasoning and proof in textbooks, but also as an analysis of a classroom episode. The table can be seen in Table 1.

Table 1: conceptualisation of reasoning and proving

Reasoning and proving Mathematical Making mathematical Providing support to mathematical claims component generalisations Identifying a Making a Providing a proof Providing a non-proof pattern conjecture argument Plausible Conjecture Generic example Empirical argument pattern Demonstration rationale Definite pattern Psychological What is the solver’s perception of the mathematical nature of a component pattern/conjecture/proof/non-proof argument Pedagogical How does the mathematical nature of a pattern/conjecture/non-proof component argument compare with the solver’s perception of this nature? How can the mathematical nature of a pattern/conjecture/proof/non-proof argument become transparent to the solver?

In Lithner (2006) a different framework is introduced; which includes a differentiation between imitative and creative reasoning (Figure 3). Imitative reasoning includes memorised or algorithmic reasoning where the strategies are founded on the recalling of answers or algorithms, and creative reasoning is where a new (to the reasoner) reasoning sequence is created or a forgotten one is re-created (Lithner, 2008). This framework has been used in previous studies to categorise tasks according to mathematical reasoning (Boesen, Lithner, & Palm, 2010; Palm, Boesen, & Lithner, 2011) or to categorise students’ mathematical reasoning

in problematic situations (Sumpter, 2013) or mathematical reasoning requirements in physic tests (Johansson, 2016).

Figure 3: Overview of different types of reasoning from Lithner (2006) The cognitive unity of theorems is another theoretical construct originally elaborated to interpret students’ behaviour in an open problem solving holistic approach to theorems (Pedemonte, 2007a) but it has also been used for interpreting and predicting students’ difficulties when they are engaged in providing statements of theorems. The cognitive unity of a theorem is based on the continuity existing between the production of a conjecture and the possible construction of its proof. The idea is that there is a continuity between the production of statements (the conjecturing) and the construction of their proof. This is in opposition to when a student gets the task “prove that…”as in this case the process of conjecturing is not demanded, and the unity is broken. Teachers can then use this construct of cognitive unity as a tool for predicting and analysing some difficulties met by the student when they have to construct a proof (Garuti, Boero, & Lemut, 1998).

To analyse the students’ reasoning processes in classrooms, a tool from the Anthropological Theory of Didactics (ATD), called Study and Research Paths (SRP) has also been used. Chevallard (2006) argues that SRP allow to model mathematical knowledge from a didactical perspective. To do the analysis, the research needs detailed information about context, contents, order of questions and answers. Several studies have focused on the potentials of SRP; Winsløw, Matheron, and Mercier (2013) examine how SRP and new diagrammatic representations can be used to analyse didactic processes; Barquero, Bosch, and Romo (2015) illustrate how SRP can be used in professional programmes for teachers; and Jessen (2017) studies how SRP can support the development of knowledge in bi-disciplinary settings. In

Larsen and Østergaard (2019) SRP is used to investigate whether the teacher and the students ask questions and answer questions in a way that allows the students to engage in mathematical reasoning related to statistics.

Finally in G. J. Stylianides (2008) four major elements in argumentation are described and used to analyse if an argument could count as proof at the primary school level: The four elements are: the arguments foundation - what constitutes its basis, the arguments formulation how it is developed, and the arguments representation - how it is expressed, and social dimension in the arguments - how it turns out in the social context of the community wherein it is shaped.

The different analytic models are used in different papers during this thesis. In Paper II, III and V, the concept of proof schemes (Harel & Sowder, 1998) have been used along with Balacheff (1988) levels of reasoning, and the four major elements from G. J. Stylianides (2008) were applied in Papers II and V. The idea of a cognitive unit from, among others, Garuti et al. (1998) has been used in the development of the test. This is also the case with the conceptualization of reasoning and proof in the table from G. J. Stylianides (2008) (sse Paper IV).

Both the TAP model and the analysis model from ATD have been tried in various analyses during the work process, but both have been rejected since the way they were used could not help with the analyses and developments that were needed in the papers. The ATD model, for example, did not include the context (media and milieu) thoroughly enough, and there were many aspects in the classroom that were not included, for example, teacher introduction, which is crucial for how the students reason in the process. For further elaboration on this see Larsen and Østergaard (2019).

The next section will discuss how the theoretical context is between inquiry-based teaching and the development of reasoning competence.

9.2 How can inquiry-based teaching support development of reasoning competence? Inquiry-based teaching is considered a potential and attractive approach to teaching mathematics (Blomhøj, 2013), but the question is whether it will strengthen students' reasoning skills to be taught from this approach. This is the focus in this Section 9.2. The section will start with a description of different approaches to teaching reasoning competence in primary school

(Section 9.2.1) followed by a Section (9.2.2) about inquiry-based teaching. The Section ends with a description of the relation between inquiry-based teaching and the reasoning competence and why it makes sense to study in which way inquiry-based teaching develops students’ reasoning competence (Section 9.2.3).

Teaching reasoning in school. Teaching mathematics in school often fails to provide students with reasoning competencies that make students able to reason on a sophisticated level, when they enter higher classes (Knuth, 2002; Yackel & Hanna, 2003). de Villiers (2010) argues that teachers often present theorems and their proofs, and afterwards the students are given exercises with assignments like “prove that”, where the students just have to copy what the teacher just presented to them. Goulding, Rowland, and Barber (2002) agree and argue that teaching reasoning often degenerates into ritualistic role-following approaches. This way of teaching in mathematics can easily create the false impression that teaching reasoning in mathematics is only about teaching deductively. However, as Polya (2014) and Lakatos (2015) have pointed out, mathematics must be learned in the making which is a more inductive approach. In G. J. Stylianides and Stylianides (2009) they claim that it is important not to devalue the role of empirical explorations when teaching reasoning in mathematics, because such explorations can help students organise mathematical observations into meaningful mathematical generalisations and gain insight into how the deductive approach might work out. In general, there has been an interest in students’ form of argumentation which appears in the course of resolving a problem. A. J. Stylianides (2007) argues that reasoning in mathematical classrooms needs to be both honest to mathematics as a discipline and honouring of the students as mathematical learners (the intellectual honesty principle), but also sufficiently “elastic” to allow description of reasoning across all levels of education (continuum principle) and pedagogically sensitive to support the study of the teacher’s role in engaging in reasoning.

Harel and Sowder (2007) problematise that students do not get convinced by the same type of evidence that could be used to convince mathematicians, but that the arguments that convince mathematicians are not necessarily deductive arguments but in the making can also be based upon empirical evidence and appeals to authority (Harel & Sowder, 1998). Students’ use of examples/empirical arguments in situations where deductive proof might be expected can at first sight be disappointing, however, the reason for this choice might be so simple, that research

mathematicians also make use of examples in the making for three purposes: understanding the statement, generating an argument, and checking the argument (Reid & Knipping, 2010). All mathematicians would, however, agree that it is a misconception that a few confirming cases are enough to establish the truth of a mathematical generalisation or that a single counterexample is not sufficient to refute a false mathematical generalization. These misconceptions are also expressed in the “solid finding” of research on mathematics teaching and learning identified by the Education Committee of the European Mathematical society … “… students provide examples when asked to prove a universal statement” (EMS, 2011, p. 50) and important not to disregard when teaching primary school classes mathematics. Healy and Hoyles (2000) note furthermore that students sometimes accept examples as verification in those cases where they already believe the results are true: “the data indicate that students were more likely to assess empirical arguments as general - to believe them to be proofs - if they were already convinced of the truth of the statement and so intuitively could extend the argument for themselves” (Healy & Hoyles, 2000, p. 412). Schoenfeld (1986) describes that reasoning also has a major part in problem-solving processes; student used both empirical reasoning and deductive reasoning to solve problems: “Contributions both from empirical explorations and from deductive proofs were essential to the solution… Had the class not embarked on empirical exploration… the class would have run out of ideas and failed in its attempt to solve the problem. On the other hand, an empirical approach by itself was insufficient.” (Schoenfeld, 1986, pp. 245-249). Duval (2007) argues in this direction that we need to separate general argumentation from proving and have a larger focus on proving in school mathematics because “anyone who does not understand how a local proof works cannot understand why a proof proves, just like someone who cannot understand any page or episode of a book cannot understand the whole book” (Duval, 2007, p. 140). Boero (1999) wants students to work on the relation between proving and augmenting by focusing on conjecturing and argues that this is the link to develop students’ understanding of proofs. And finally Rowland (1998) describes that generic evidence can contribute to a great understanding of other types of proofs and reasoning in general.

As described, there are many opinions and studies that point in many different directions to how to develop reasoning competence in primary school and the different approaches are likely to develop different aspects of the competence, but the question now is whether inquiry-based teaching will also be a way to develop and contribute to the students’ reasoning competence.

Inquiry-based teaching Inquiry-based learning and teaching have been terms which have emerged with increasing frequency in educational curriculum research and policy documents in the last two decades. Artigue and Blomhøj (2013) call it an educational trend. In general, there are many different interpretations of inquiry-based teaching as it does not only result from differences between scientific disciplines (mathematics or chemistry), but also the conviction for degree of autonomy to be given to the students and the teachers, for example, when posing questions or problems or in the process of finding strategies or the results. Moreover, there are different terms and concepts used synonymous with this term including inquiry-based learning, inquiry- based education, inquiry-based pedagogy or discovery-based learning, constructivist learning, problem-based learning (Maaß & Artigue, 2013). Artigue and Blomhøj (2013) define inquiry- based teaching loosely as a way of teaching in which students are invited to work in ways similar to how mathematicians and scientists work. Inquiry-based teaching is then seen as a student-centred way of learning and teaching, in which students learn to inquire and are introduced to mathematical and scientific ways of inquiry. Maaß and Artigue (2013) argue that in this approach students raise questions, explore situations, and develop their own ways towards solutions:

“Inquiry is a multifaceted activity that involves making observations; posing questions; examining books and other sources of information to see what is already known; planning investigations; reviewing what is already known in light of experimental evidence; using tools to gather, analyze, and interpret data; proposing answers, explanations and predictions; and communicating the results. Inquiry requires identification of assumptions, use of critical and logical thinking, and consideration of alternative explanations and scientific inquiry refers to the diverse ways in which scientists study the natural world and propose explanations based on the evidence derived from their work.” (p. 781).

Bruder and Prescott (2013) discriminate between three different ways to do inquiry: structured inquiry, where the teacher gives the students a problem or question to be solved as well as the appropriate methods and materials to solve it. Guided inquiry; where the teacher provides the students with the problems or questions and the necessary materials. The students then have to find the appropriate problem-solving strategies and methods. And finally, open inquiry where the students have to find the problem or question they would like to solve and answer. Moreover, they also decide the methods and the materials they would like to use. The goals in

inquiry-based teaching are often connected to an enlarged set of goals beyond learning mathematical content, including learning how scientists work and equipping students with strategies for further learning (Maaß & Artigue, 2013). The role of the teacher in such a setting is often described to differ from traditional teaching approaches, and the teacher’s role is described more as facilitators of students’ learning process:

“the teacher’s role includes: orienting students towards questions and problems of interest for them that contain interesting learning potential; making constructive use of students’ prior knowledge; supporting and guiding when necessary their autonomous work; managing small group and whole class discussions; encouraging the discussion of alternative viewpoints; and helping students to make connections between their ideas and relate these to important mathematical and scientific concepts and methods. In this setting, students are not left alone in their discovery but are guided by the teacher who supports them in learning to work independently.” (Maaß & Artigue, 2013, p. 782).

There has been a lot of research about inquiry-based teaching in Europe: large international projects like PRIMAS (PRIMAS, 2013) and Fibonacci ("The Fibonacci Project ", 2013) which all show positive results for this approach. The effect includes, among others, benefits for motivation, for better understanding of mathematics and for the development of belief about mathematics as well as for the relevance of mathematics for life and society. Bruder and Prescott (2013) describe many different research studies in this specific approach and write that the general understanding is positive. The question is, however, if one of these benefits also is the reasoning competence and if this competence will be developed with an inquiry-based approach. Moreover, this also raises the question of how this can be measured.

The development of reasoning competence in inquiry-based teaching. Teachers have an important role in the learning of reasoning in school mathematics (Ball & Bass, 2003) but also in general in an inquiry-based teaching approach (Maaß & Artigue, 2013).

In EMS (2011) it is argued that teachers often provide little space for students to reason independently: The teacher is often the one who often raises the questions, but also in the end answers them, because when a difference occurs among students, the teacher will serve as an arbiter rather than a facilitator for the debate among the students. If reasoning, however, is

learned in an inquiry-based approach the role of teacher must be different and instead the role must be to facilitate the students’ learning as described earlier.

Different studies have been conducted focusing on teaching reasoning competence. Mariotti and Goizueta (2018) studied teachers’ prompts to construction of validation of a mathematical problem. And they found that the teachers’ interventions did not always have the intended effect: If the teacher give no better reason for proof than “that’s maths” then the student will never find out why it is true. Instead Mariotti and Goizueta (2018) claim that the process needs to be more in focus. They argue that proving processes has two phases, the formulation of a conjecture and the construction of a proof. This is in line with NCTM cycles of reasoning and proof, where the students must first investigate the elements of a problem or situation in order to make a conjecture about it and secondly, arrange the arguments, so the conjecture can be proven. This fits very well with an inquiry-based approach which is often considered a process that starts with posing a problem, setting up a hypothesis, finding strategies to solve the problem and finding the result (R. Hansen & Hansen, 2013).

Overall, many of the approaches described as being positive for the development of students' reasoning competency, are also closely linked to working inquiry-based in mathematics teaching including both the teacher role and a greater focus on the students themselves as having to develop their skills by participating throughout the process.

One of the major problems described by EMS (2011) is for the students to go from reasoning empirically to have a more deductive approach. This problem is empirically in focus in relation to inquiry-based teaching in three of the papers (Papers I + II + III).

Papers IV and V have a stronger focus on testing the reasoning competence, which will be elaborated in the next section.

9.3 Measuring student’s development of reasoning competence “Measurement, in its broadest sense, is the process of assigning numbers to categories of observations in such a way as to represent quantities of attributes” (M. Wilson & Gochyyev, 2013, p. 3).

Measuring the students' development of a mathematical reasoning competence is not an easy task. In Denmark we have seen a significant empirical turn in education research after the 57

TIMSS and the PISA to focus more on evidenced-based results like tests and an orientation against more detectable effects in empirically-based research (von Oettingen, 2018). This is especially explicit for mathematics students (van den Heuvel-Panhuizen & Becker, 2003). In Denmark it is apparent by introducing the national test in 2010. Implementing and completing more tests can, on the one hand, be seen as positive, because it produces some data which will offer a kind of visual language to help the discussion between teachers, school leaders and/or researchers, but, on the other hand, data also generate a whole lot of new issues which are gradually getting clearer for all parts. One aspect is the interpreting of statistical data which requires specific data literacy; When, e.g., a researcher produces graphs of two different year 4 classes’ performances in mathematics, the difference between the graphs causes something to the two classes. Does the one teacher do a better job than the other? Are the students better in one of the classes than in the second class? It can put everyone in precarious situations which the teacher, leaders and researchers have to deal with carefully. The interpreting of data is therefore even more important because data are doing more than they show and it is important not to mistake statistical significance with educational significance (van den Heuvel-Panhuizen & Becker, 2003). Another aspect is that testing in teaching affects and influences the students. Already in the early 1970s researchers (Snyder, 1971) were engaged in studies of students’ learning and assessment and Snyder (1971) wrote in the “Hidden Curriculum” that what influenced students most was not the teaching but the assessment. Teachers are, however, also influenced by the assessment – this is called the backwash effect, which is when the assessment influences the way the students are taught either in a positive way or in a negative way. In this way the commonly accepted view – what is tested is what you get - can also be seen with an optimistic view like when de Lange (1992) argues: “…if the test is made according to our principles, this disadvantage (test-oriented teaching) will become an advantage. The problem then has shifted to the producers of test since it is very difficult and time consuming to produce appropriate ones” (p. 320).

With this in mind, there has of course also been a lot of negative focus on testing. Skovmand (2016) argues that one of the problems in testing is that it is only a special type of knowledge that tests are able to measure, and the question is if it is this type of knowledge, we want the students to develop in the classrooms today.

Based on this discussion it is important here to proclaim that the author of this thesis does not necessarily advocate the idea that the field of mathematics education should be transformed

into an evidence-based testing profession and that it is only this kind of experimental study with profound measurement of correlations that is important in the process. However, as a researcher you must not neglect and simply reject this approach, because still much more research is needed about testing in mathematics, so we will be better at measuring what we value instead of just throwing in the towel. “When assessments are based on teaching and curriculum content, testing can be an essential part of instruction rather than lost time for learning”(W. S. Wilson, 2009, p. 1).

In this thesis there will be a specific focus on the development of an achievement test, which must be able to measure reasoning competence in mathematics. This is not a well-written area in mathematics education and it is actually difficult to find didactic consideration about testing and measurement of competencies in mathematics. For instance, what could characterise different types of tests in mathematics? And what can these different kinds of measurements be used for? Including what do different types of items measure? The little description in the literature was surprising given that tests are likely to have a large impact on how and what is being taught in the classrooms today.

Measuring mathematical competencies Measuring the development of mathematical competences is generally not an easy task and many educators describe that “designers of tests overly use items which assess only the basic levels of mathematical content knowledge and rarely include items which assess complex levels of students understanding” (van den Heuvel-Panhuizen & Becker, 2003, p. 691). This is also seen in the national test in Denmark which focuses on three different profiles: number/algebra, geometry and statistic/probability (Undervisningsministeriet, 2019a), and it does not include any of the mathematical competences from the curriculum.

The concept of competence is, however, also a rather complicated concept. First of all, mathematical competencies are described in Niss and Jensen (2002) as the ability to understand, judge, do, and use mathematics in a variety of intra- and extra-mathematical contexts and situations in which mathematics plays or could play a role. The different mathematical competencies are not defined to be disjointed. On the contrary, they are intertwined and come together in large aggregate overlapping complexes, which is also shown in the visual representation by the intersecting petals in the flower diagram in Figure 4. 59

Figure 4: A visual representation of the eight mathematical competences from Niss & Højgaard (2011, p. 51)

So, even if there is a clear description of reasoning competence, its execution will typically draw in some of the other competencies as well, e.g., communication competencies or representation competencies.

In Niss et al. (2016) they give an example:

“…the competency of posing and solving mathematical problems will necessarily involve at least some basic aspect of dealing with mathematical representations, mathematical symbols and formalism, or mathematical reasoning. If each of these three competencies were absent there would simply be no mathematical problem solving”(Niss et al., 2016, p. 623).

Furthermore, to possess a mathematical competency is described by (Niss & Højgaard, 2011, p. 49) as: “a wellinformed readiness to act appropriately in situations involving a certain type of mathematical challenge”. A students’ possession of a given mathematical competency is in Niss and Jensen (2002) defined in three different dimensions: The radius of action specifies the range of contexts and situations in which the student can activate this competence. The technical level specifies how conceptually and technically advanced the entities and tools are with which the person can activate the competence. The degree of coverage is the extent to which the student masters the characteristic aspects of the competence. These dimensions are seen as non- quantitative, because a competene is not something which a student can possess in full, it is

something which a student can possess to a greater or lesser degree (Niss & Jensen, 2002). In contrast, other variables like e.g., “pregnancy” are clearly a dichotomy - one cannot be slightly pregnant or almost pregnant… mathematical competencies are here seen as a span with something in-between.

In Niss and Jensen (2002) they describe these three dimensions in the metaphor of a three- dimensional box. The volume of the box is the “product” of the degree of coverage, the radius of action, and the technical level. If one of the dimensions is measured to be zero, the volume then becomes zero!

The challenge is then how to measure all these dimensions at a given point in time. This will, however, probably never be possible with a single assessment instrument, and it will never be possible to satisfactorily express this in one score. In developing a competence test, one of the biggest challenges is therefore based on how to break the competence goals into operational units (for all dimensions). For example, by describing the development of a given dimension and progression in taxological levels, which can be used as the tool to assess a student's competence. And the disadvantage of even trying to do this is that a dilemma can arise between, on the one hand, having to work on operationalising competence goals (in order to also support the learning process) and, on the other hand, the risk that the assessment becomes detached from any context and just an instrumental approach.

Defining different terms and concepts in measuring In the development and the discussions about the quality of the achievement test, different kinds of terms need to be clarified and discussed. The terms ‘assessment’ and ‘test’ are from time to time used interchangeably and the distinction between them is often seen as overlapping, but there is an important distinction between them. Testing can be regarded as a method of collecting data for assessment, in the sense that assessment is a broader term, covering other methods of gathering and interpreting data as well as testing. In Johnson and Christensen (2014) testing is defined as “…the process of measuring variables by means of devices or procedures designed to obtain a sample behavior” (p. 164), while Assessment is defined as “the gathering and integration of…data for the purpose of making …an educational evaluation, accomplished through the use of tools such as a test, interview, case studies, behavioral observation, and specially designed apparatus and measurement procedures” (p. 164).

Harlen (2013) argues that assessment will encompass the following four areas a) students being engaged in some activity, b) the collection of data from that activity by some agent, c) the judgement of the data by comparing them with some standard and d) some means of describing and communicating the judgement. There are several forms that each of these areas in the assessment can take and it opens the possibility of many different ways of conducting assessment: a) Activities in which students are engaged can be, for example; their regular work, some written or practical tasks created by the teacher for the purpose of assessment, some written or practical tasks created externally. b) The data can be collected by, for example; the teacher, the students, the teacher and students together or an external agent (examination board, qualifications authority, test developer). c) The data can be judged in relation to; norms, criteria, or students’ previous performance. d) The judgements can be communicated; as a written or oral comment by the teacher, a mark or score or percentage, a profile of achievement, a level or grade, a ranking or percentile (Harlen, 2013).

The combination of these (a, b, c, d) creates different assessment tools depending on the situation and the aims. In c) data can be judged in relation to norms or to criteria. Criterion- referenced tests compare a person’s knowledge or skills in relation to a predetermined standard, learning goal, performance level, or other criterion - each student’s performance is then compared directly to the criteria, without considering how other students perform on the test. Norm-referenced measures on the contrary compare a person’s knowledge or skills to the knowledge or skills of the norm group. For student assessments, the norm group is often a nationally representative sample of several thousand students in the same school year. The difference is therefore in the scoring and interpretation and some tests can actually provide both criterion-referenced results and norm-referenced results.

In Denmark the results of the national test have, since the school year 2009/2010, been passed on a norm-based scale (1-100), which is a percentile scale. The percentage scale has been formed on the basis of the distribution of the students' earlier test results, where a percentile value of, for example, 40 corresponds to the students’ skills on the logit scale, where 40 per cent of the test results from earlier were below; however, the results have since 2014/2015 been presented on a criterion-referenced scale, where the students’ skills are placed on a logit scale. 62

A commission has then set criteria for which academic level a score corresponds to in each profile area. The levels are here defined as: very poor performance, inadequate performance, okay performance, good performance, really good performance and excellent performance (Undervisningsministeriet, 2019a).

Finally, the two terms validity and reliability need to be further explained in connection to assessment and testing. Validity is often defined as how well the assessment corresponds with the behaviour or learning outcomes that was the intention to be assessed. Various types of validity have been proposed depending on the kind of information used in judging the validity:

Content validity: refers to how adequately the assessment covers the subject domain being taught and is usually based on the judgement of experts in the subject.

Construct validity: refers to whether the assessment covers the full range of learning outcomes in a particular subject domain. Does the assessment sample all aspects of the intention? (Including the irrelevant aspects). A useful technique for examining the internal structure of tests is factor analysis (Johnson & Christensen, 2014).

The criterion validity: focuses on the usefulness of the test in predicting how people taking the test will perform on some criterion of interest. It is the extent to which a measure is related to the outcome (Johnson & Christensen, 2014).

The reliability of an assessment refers to the extent to which the results can be said to be of acceptable consistency or accuracy for a particular use. This may not be the case if, for instance, the results are influenced by those who conduct the assessment (e.g. different teachers) or they depend on the particular occasion or circumstances at a certain time. Like validity, different kinds of reliability exist, e. g. internal consistency reliability which refers to how consistently the items on a test measure a single construct or concept, or interscorer reliability which is the degree of agreement or consistency between two or more scorers or raters (Johnson & Christensen, 2014).

Reliability and validity are perhaps the two most important properties to consider in developing and using a test or assessment procedure and reliability is not only necessary but a sufficient condition for validity - so if an assessment should be valid then it first needs to be reliable (Johnson & Christensen, 2014).

The difference between summative assessment and formative assessment also needs to be clarified. In 1969 Benjamin Bloom applied the distinction in classroom tests between formative evaluation to provide feedback and correctives at each stage in the teaching-learning process (aids in the learning process) and summative evaluating which is the grading, judging and classification function of evaluation (Wiliam & Thompson, 2017).

Summative assessment is assessment that is carried out for the purpose of reporting achievement at a particular time, while formative assessment is to follow the student learning to provide ongoing feedback that can be used by the teacher to improve their teaching and by students to improve their learning. “Assessment for learning” and formative assessment are often used interchangeably, but there is a distinction: “Assessment for learning” is any assessment for which the purpose in its design is to promote students’ learning. Such assessment becomes formative assessment when the evidence is actually used to adapt the teaching and when the assessment forms the direction of the improvement. This can happen in three different instructional processes: by establishing where the learners are in their learning process, by establishing where they are going, by establishing what needs to be done to get them there (Wiliam & Thompson, 2017).

To develop an achievement test, whose aim it is to measure how much the students have developed in an intervention, there is a crucial need for the test to be connected to the focus of the intervention. The alignment between teaching and testing is therefore very important; Hattie (2009, p. 6)) argues that; “any course needs to be designed so that the learning activities and assessment tasks are aligned with the learning outcomes that are intended in the course.”. This alignment might be seen as common sense, but traditional practice and teaching, however, often ignore this alignment (Biggs, 2011). The alignment between testing and teaching in the KiDM intervention is described further in Paper V.

Finally, it is important to notice here that there are significant differences in how such information is used in the different classrooms. Assessment serves different functions: Wiliam and Thompson (2017) describe three different purposes: supporting learning (formative), certifying the achievements or potential of individuals (summative) and evaluating the quality of educational institutions or programmes (evaluative). These different functions it may serve are not easy to reconcile and Wiliam and Thompson (2017) argue that the same assessment cannot serve different functions adequately.

In the development of the KiDM test different things clearly need to be taken into consideration: Which kind of activities in the test? Who should carry out the data collection? Should it be norms or reference judged? Validity? Reliability? How to make sure there is alignment with classroom practice? And which function must the test have? All these aspects will be discussed in connection to the development of the KiDM test in Papers IV and V. In connection with developing the test, it was also important to look at what research is already available in the field. Therefore, a systematic review was retrieved, on which the KiDM test could be built. It is important here to note that the KiDM test does not only focus on the reasoning competence, and therefore the review has a broader approach. In the next section, this review can be found, followed by how the review was used in connection to the test development.

9.4 Developing a test – a systematic review In order to develop the best test possible, a systematic review was conducted to gain an overview of what existing research already knows about testing mathematical competences. In the following, this systematic review is presented.

The review had the following research question:

What issues and concerns are prevalent in the mathematics education literature with regard to developing a mathematic inquiry-based competences test?

The product should be some guidelines which could be used in the development of the test. After the review, a small section on how these guidelines are applied will follow.

Review Methods The framework in the review process is based on seven stages proposed by Petticrew and Roberts (2006):

1) Clearly defining the research question;

2) Determining the type of studies that need to be located;

3) Carrying out a comprehensive literature search to locate those studies;

4) Screening the results of that search;

5) Critically appraising the included studies; 65

6) Synthesising the studies and assessing heterogeneity among the study findings; and

7) Disseminating the findings of the review.

These seven stages form the basis of this review. The research studies included in the review were selected on the basis of a number of choices and criteria: The search period ranged from January 2006 to December 2016. This period was selected to ensure a certain breadth, while still focusing on the newest articles in the research. The language options were restricted to only English and Danish articles. This was necessary due to the lack of resources required to translate papers/articles from other languages. Only peer-reviewed articles were included and to restrict the search area further, the population was restricted to elementary/primary schools. The keywords were chosen by screening abstracts in special issues of journals which have a focus on inquiry-based learning, but also articles about inquiry-based teaching from each of the highest ranked journals in mathematics education - defined in Törner and Arzarello (2012) - were randomly selected and screened for relevant keywords to make sure all the most important journals were represented.

The keywords in addition to test*, assess*, evaluate* and mathematic*, included the following mathematical areas: inquiry, model*, problem*, discover*, reason* and metacognitive (the * indicates that all endings to these words are included). A comprehensive search strategy was used, and specific databases were selected to ensure that the results represented different scientific areas, such as psychology (Psyck info), science (Web of science), and pedagogics (Eric). Scorpus was also included because it is the largest abstract and citation database of peer- reviewed literature (Library, 2019).

Table 2: Criteria for the systematic review

Criteria for inclusion and exclusion

The search period From January 2006 to December 2016 (10 years) The selection of • Psychology (psyck info), databases • Science (Web of science), • Pedagogics (Eric) • Abstract and citation database (Scopus) Language English and Danish Population Elementary/primary schools

Keywords inquiry*, model*, problem*, discover*, reason*, metacognitive* and test*, assess*, evaluate* and mathematic* Other Only peer-reviewed

The search results were: Eric 961, Scorpus 412, Web of science 405, and Psyck info 437; the total was 2215 hits, and the outcome without dublix was 1280. To specify the search further, the inclusion and exclusion criteria were described, and used in four sequences.

1. The 1280 article headlines were analysed, resulting in 371 articles being included. 2. The abstracts of 371 articles were read and analysed and 115 articles were included. 3. The 115 articles were then read in full, and the most important findings and their research methods were described in an Excel sheet. 4. During the reading in full, more articles were rejected due to the exclusion/inclusion criteria, and the final results of the search were 51 articles out of 1280 articles.

Table 3: Number of articles in the review process

The process Number of articles/papers

1. Search Eric: 961 results Scorpus: 412 Web of science 405 Psyck info: 437

= 2215 hits and outcome without duplex was: 1280. After reading and analysing 371 headlines

After reading and analysing 115 Abstracts After reading and analysing 51 papers in full The references of the 51 articles can be seen in Appendix a.

Articles/papers In an overview of the research methods used in the 51 studies, mostly quantitative studies were found (41 studies), and of these 41 studies, 37 studies were experimental studies and four studies were non-experimental studies. Two of the 51 studies were theoretical reviews, five of the 51 studies were qualitative studies, only one was a mixed methods study and two were comments on other articles. This overweight of quantitative studies may be clearly related to the fact that it is about testing in mathematics teaching.

Table 4: overview of the methods used in the articles

Methods Quantitative Qualitative Theoretical Mixed Comment All studies studies reviews methods s on others Experimental Non- studies experimental studies Number 37 4 2 5 1 2 51 of articles/ papers

In an examination of the countries represented in the studies, most of them came from the USA, but Turkey, Holland and Australia were also well represented. The specific distribution can be seen in the Appendix b.

The issues and concerns raised and discussed in the 51 studies were condensed into six main themes related to testing by creating a mind map in an explorative study. The six themes were the following:

1) Assessing mathematical creativity,

2) Assessing problem-posing and problem-solving,

3) Assessing mathematical reasoning,

4) Assessing mathematical language and modelling,

5) Computer tests, and

6) Metacognition in assessments

These six themes will be presented in the following:

9.4.2.1 Assessment of Mathematical creativity Assessment of creativity in mathematics is in focus in several articles (Akgul & Kahveci, 2016; Arikan & Ünal, 2015; Bahar & Maker, 2015; Chang, Wu, Weng, & Sung, 2012; Kaosa-ard, Erawan, Damrongpanit, & Suksawang, 2015; Kar, 2015). Kaosa-ard et al. (2015) define some general “mathematical process skill” and include, among others, reasoning skills and creativity

skills. How to test mathematical creativity skills is the concern in Akgul and Kahveci (2016). Akgul and Kahveci (2016) claim that there are not enough valid and reliable scales to assess students' creativity in the domains of mathematics and science, but that it is very important, because if mathematical creativity potential can be measured, then the curriculum can be adapted to address the needs of creative students. For this reason, Akgul and Kahveci (2016) created a valid and reliable scale to measure the middle school students' mathematical creativity. The scale includes fluency, flexibility and originality. Kar (2015) states that problem posing promotes creativity skills and in Arikan and Ünal (2015) and Chang et al. (2012), they point out that problem-posing activities include creative activities for mathematics education. In Bahar and Maker (2015) they found that general creativity and verbal ability are the only variables that contributed significant variance to open-ended problem-solving performance.

In sum it seems important to measure the development of students’ mathematical creativity in an assessment test. A way to do this could be to make open-ended items with problem solving tasks or tasks where student pose problems by themselves. A coding scale could then include a taxonomy of fluency, flexibility and originality.

9.4.2.2 Assessment of Problem posing and problem solving Problem posing and problem solving are core themes in the research about testing in the field of mathematics (Arikan & Ünal, 2015; Bahar & Maker, 2015; Kar, 2015; Leh, Jitendra, Caskie, & Griffin, 2007; Lein et al., 2016; Limin, Van Dooren, & Verschaffel, 2013). Limin et al. (2013) use five instruments in a mathematics test: a problem-posing test, a problem-solving test, a problem-posing questionnaire, a problem-solving questionnaire, and a standard achievement test. In other articles, there are different issues concerning how to create problem- posing and problem-solving test items: Arikan and Unal (2014) use three types of problem- posing items: free type, semi-structured type and structured type. Chang et al. (2012) use a problem-posing system consisting of four phases: posing the problem, planning, solving the problem, and looking back. Lawson and Suurtamm (2006) state that a problem-solving item should contain four aspects: problem solving, calculation, understanding and communication. These four aspects will make it more authentic and congruent with the students’ present understanding of mathematics and mathematical knowledge. Including all four aspects will also support the shift from viewing mathematics as mechanistic and atomised to problem-solving in action. Munroe (2016) uses problem posing in an assessment by using a picture as a prompt. 70

Students are then asked to generate three arithmetic problems by using the picture and offer their solution to one of these problems. The assessment shows that more advanced students generate more complex questions and therefore this element can be used in a taxonomy.

Kim and Noh (2010) carry out a study to test and refine a framework for developing and grading descriptive assessment problems. The study shows that their structured approach to descriptive problem assessment was a powerful tool for improving mathematics education in primary school. To structure the problem posing, coding process is in focus in Limin et al. (2013) and they divide the codes into 4 dimensions: accuracy, complexity, originality and diversity. Kar (2015) analyses the problems posed in terms of their semantic structures and conceptual validity, and Charlesworth and Leali (2012) use an analytic scale to evaluate students’ problem- solving performance. Munroe (2016), however, problematises the posing of problems in tests; in his case, students were not willing to write and elaborate difficult questions, as they were required to solve the questions themselves. The students would maybe have written more difficult questions if their classmates or someone else were required to solve the questions.

All in all, there seems to be a general agreement among the researchers that both problem- solving and problem-posing tasks provide important information about students’ mathematical competences. However, Lawson and Suurtamm (2006) pointed out that, there is still a need to further investigate how to construct items, so they include more complex mathematics.

9.4.2.3 Assessing mathematical reasoning Mathematical reasoning is an important part of mathematics testing, e.g., test items in TiMSS and PISA, but how to assess this competence continues to present a challenge (Christou & Papageorgiou, 2007; Goetz, Preckel, Pekrun, & Hall, 2007; Hu et al., 2011; Hunsader, Thompson, & Zorin, 2013; Nunes et al., 2007).

Logan and Lowrie (2013) argue after testing students’ spatial reasoning that more attention should be paid to how to measure spatial reasoning in tests. In Nunes, Bryant, Evans, and Barros (2015) a framework for prescribing and assessing mathematics inductive reasoning of primary school students is formulated and validated. The major constructs incorporated in this framework are students' cognitive abilities of finding similarities and/or dissimilarities among attributes and relations of mathematical concepts. Also in Goetz et al. (2007) they examine reasoning in mathematics in connection to other aspects. In their study they examine test-related

experiences of enjoyment, anger, anxiety, and boredom and how this relates to students' abstract reasoning ability. Emotions were assessed immediately before, during, and after a mathematics achievement test and analysis of variance shows that emotions experienced during the test situation differed based on students' abstract reasoning ability level. Nunes et al. (2015) measure reasoning by using four subtests: a quantitative ability test, a verbal similarities test, a number skills test and matrices. In Hunsader et al. (2013) they present another framework to analyse the extent to which assessments (i.e., chapter tests) provide students with opportunities to engage with key mathematical processes. The framework uses five criteria to assess these processes: reasoning, communication, connections, representation (role of graphs) and representation (translation). Their results from coding over 100 tests clearly indicate the need for more efforts to ensure that assessments focus not only on what students learn (i.e., the content) but also on how they learn (i.e., mathematical processes).

Altogether, the term reasoning is broadly used in mathematics education and it is connected to many other areas like logic or spatial areas. To reason in mathematics is therefore not a unified concept, so to be able to assess reasoning in mathematics there needs to be some precision of the connection to different parts and definitions of reasoning.

9.4.2.4 Mathematical language and modelling problems In the development of test items, the use of language is very important. Mushin, Gardner, and Munro (2013) investigate the role that language plays in mathematics assessment. Demonstrating that understanding or non-understanding is not only a question of what students actually know about targeted concepts, but also about how well they have understood what has been asked of them and how well their response demonstrates (non)understanding of the concepts. Hickendorff (2013) argues that in the written test, students’ language levels have differential effects on the results, the effects being more pronounced on applied problem solving than on computational skills. Hickendorff (2013) asked the question: "to what extent are different abilities involved in solving standard computation problems versus solving contextual problems?”, while Kan and Bulut (2015) asked the question: “Can word problems and mathematically expressed items be used interchangeably regardless of their linguistic complexities?” Hickendorff (2013) finds that there is a big difference in terms of which abilities/competences the students use when they solve problems; a large part of the variance can be determined by contextual /non-contextual problems. Kan and Bulut (2015) find that 72

word problems are generally easier than mathematically expressed items. Hickendorff (2013) recommends mixed tests with both contextually and mathematically expressed items.

Boonen, van Wesel, Jolles, and van der Schoot (2014) examine the role of visual representation type, spatial ability, and reading comprehension in problem solving. They discover that students who produce an accurate visual-schematic representation increased their chances of solving a word problem correctly by almost six times compared to students who do not make a visual representation. Csikos, Szitanyi, and Kelemen (2012) main focus is also on the role of visual representations in modelling and problem solving. In their study, students themselves are invited to make drawings for each task. The experiment points to the importance of using visual representations in mathematical word problems.

Kinda (2010) presents a study where a new assessment technique is used, i.e. the story generation task, which involves examining students’ understanding of subtraction scenes. The students from four different school years generate stories under the constraints provided by a picture (representing a Change, Combine or Compare scene). In another study (Mousoulides, Christou, & Sriraman, 2008) it is argued that context and language play an important role in mathematic modelling; In the study they analyse the processes used by students when engaged in modelling activities and, to examine the students' modelling abilities, they proposed a three- dimensional theoretical model: structuring mathematising interpreting, solving real-world problems by creating models and working with mathematical models.

The conclusion here is that in a test development it is important to focus on how to make the written language clear and understandable for the students and, if possible then a mix of both contextual and mathematical expressed items would be preferred. Moreover, it would be beneficial to also include visual representations.

9.4.2.5 Computer tests and different types of feedback Technology today offers many new opportunities for innovation in educational assessment and feedback through rich assessment tasks, efficient scoring and reporting on a computer. In Kingston (2009) and Logan (2015) they argue that mathematics assessment and testing are increasingly situated within digital environments. Kingston (2009) describes how computerised tests differ from paper-and-pencil tests and finds that there are some small but perhaps significant cognitive differences related to responding to mathematics items on a computer,

compared to items on a piece of paper using a pencil. When working on a computer, students need to switch their focus while using scratch paper to work out answers. In a paper test, students can work out problems in the margins of the text booklet. In the latter case, the changes of focus are spatially much smaller since paper is essentially two dimensional, whereas switching attention from computer screen to paper and back to computer screen requires multiple changes of focus in three dimensions. Logan (2015) explores the influence the different modes of assessment, computer-based or paper-and-pencil-based, and students’ visuospatial ability have on students' mathematics test performance. The results reveal statistically significant differences between performance on computer-based tests and paper-and-pencil test modes across content areas concerning whole number algebraic patterns and data and chance.

Many computer assessment environments focus on grading and providing feedback on the final product of assessment tasks (Adesina, 2014; Chu, Hwang, & Huang, 2010; Fyfe, Rittle- Johnson, & DeCaro, 2012; Klinkenberg, Straatemeier, & van der Maas, 2011). Adesina (2014) uses a method that traces links on an interactive touch-based computing tool for the capture and analysis of solution steps in primary school mathematics. The approach yield performance scores similar to those from paper-and-pencil tests but also provides more explicit information on the problem-solving process. Klinkenberg et al. (2011) present a new model for computerised adaptive practice and monitoring. In the scoring guide, both accuracy and response time were accounted for. Results show better measurement precision and many interesting options for monitoring progress, diagnosing errors and analysing development. Fyfe (2016) wants to understand when and why feedback has positive effects on learning and to identify features of feedback that may improve its efficacy. In an experiment, children are assigned to receive no feedback, immediate feedback, or summative feedback from the computer. The results show that feedback results in higher scores relative to no feedback for children and that immediate feedback is particularly effective.

In sum, the review shows that research into computer tests in mathematics is growing and much new research is created in this area. In the development of an assessment test, there needs to be some consideration about if and why the test should be on a computer/iPad or not. There are differences in the test results depending on whether the test is a paper-and-pencil test or it is carried out on a computer. A computer test can, on the one hand, give immediate feedback which is seen as effective in the student’s results and it can reveal other aspects of the student’s

development, e.g., response time for problem solving, but a computer test can, on the other hand, be different from the intervention and therefore the alignment would be small.

9.4.2.6 Metacognition in tests Metacognitive monitoring and regulation are seen as playing an essential role in mathematical problem solving and test results, and 7 out of the 51 articles have this focus. The main focus is on how to assess the students’ metacognition and the metacognitive relations to achievement (Desoete, 2007, 2008, 2009; Jacobse & Harskamp, 2012; Neuenhaus, Artelt, Lingel, & Schneider, 2011; Ocak & Yamac, 2013; Roderer & Roebers, 2013), but emotions are also in focus (Childs, 2009; Mavilidi, Hoogerheide, & Paas, 2014; Nyroos & Wiklund-Hornqvist, 2011; Ocak & Yamac, 2013; Phelps & Price, 2016).

Jacobse and Harskamp (2012) find that one proven valid, but time consuming, method to assess metacognition is by using think-aloud protocols, but they argue that although it is valuable, the practical drawbacks of this method necessitate a search for more convenient measurement instruments. Desoete (2007) evaluates students’ metacognitive skills through teacher ratings, think aloud protocols, prospective and retrospective child ratings and their results show that how you evaluate is what you get. Child questionnaires do not seem to reflect actual skills, but they were useful to evaluate the metacognitive "knowledge" and "beliefs" of young children. In Desoete (2008) and Desoete (2009) they find that students’ different metacognitive skills in mathematics are generally related, but that it is more appropriate to assess them separately. Neuenhaus et al. (2011) claim that in theory it is assumed that the development of metacognitive knowledge begins highly domain and situation-specific and becomes more flexible and domain- transcending with practice and experience, but they also pointed out a strong relation between general metacognitive knowledge and domain-specific metacognitive knowledge.

Nyroos and Wiklund-Hornqvist (2011) argue that a recognised possible adverse effect of testing is test anxiety among students, which has a negative impact on test performance. Mavilidi et al. (2014) investigate these negative thoughts that anxious children experience during tests and find that these negative thoughts exhaust a lot of the students’ working memory resources at the cost of resources for performing on the test. The results suggest that by looking ahead in a test, less working memory resources are consumed by intrusive thoughts, and consequently, more resources can be used for performing on the test.

Overall the seven articles’ point is to include students’ metacognitive skills in developing an assessment test by assessing the skills separately and in domain-specific situations. Furthermore, forcing the student to look ahead in the test will help the anxious children perform better in the test.

Summary of key findings and development of guidelines This review shows that how to accurately assess mathematical competences is still a current issue (Schoenfeld, 2015; van den Heuvel-Panhuizen, Robitzsch, Treffers, & Koller, 2009). Ferrara, Svetina, Skucha, and Davidson (2011) argue that developing items for a mathematics test is extremely difficult, and that previous studies have shown that coherent definitions of development are often a result of luck rather than design. Therefore, it makes sense to consider the implications that can be drawn from the overview generated by this literature review. The implications can be viewed in many different ways, but it is, however, very important to be careful about drawing the conclusions directly from these mostly empirical studies directly into practice in developing the test to the KiDM project.

In the following, the six themes are summarised into 10 guidelines:

• Guideline 1: Focus the items on how students learn, and mathematical processes which are: problem solving, problem posing, mathematical reasoning, mathematical modelling, communication and presentation skills, connection knowledge and creativity skills.

• Guideline 2: Focus the items on all aspects in the processes of problem solving and problem posing. Use open questions or semi-structured prompts.

• Guideline 3: Use a problem-posing item to measure mathematical creativity. The following categories: Fluency, flexibility and originality can be included in a creativity scale.

• Guideline 4: Use an item where the students find and argue for similarities and/or dissimilarities among attributes and relations of mathematical concepts to measure reasoning.

• Guideline 5: Incorporate visual representations in the test items and give the students the possibility to draw visual representations during the tests. 76

• Guideline 6: Use items with both contextual problems and computational problems. Be aware of the language in all items.

• Guideline 7: Use three-dimensional activities to examine students’ modelling competencies: structuring mathematising interpreting, solving real-world problems and working with mathematical models.

• Guideline 8: Consider that you will get different results and can give different types of feedback if you use a computer-test or if you use a paper-and-pencil test.

• Guideline 9: Use a multiple method design, e.g., teacher questionnaires, think-alouds, interviews and student/teacher ratings to measure students’ mathematical metacognition.

• Guideline 10: Students’ emotions affect the results. Let the students read the test in their heads.

9.5 Using the guidelines in the development of the KiDM test In the development of an inquiry-based competences test in KiDM (see also Section 10.3), several of the guidelines from this review were used; Problem posing items were used to measure mathematical creativity along with a coding scale similar to that of Akgul and Kahveci (2016) and Limin et al. (2013) which includes fluency, accuracy, complexity and originality (guideline 3). To create the problem posing items, semi-structured items were used, and like Munroe (2016), a picture was used as a prompt in the problem posing items. Students were asked to generate two problems and to offer their solutions to these problems, but it also included, as Chang et al. (2012) argued, a way of making the students look back, by asking the students to explain why the second posed problem was more difficult. The problem solving items in the KiDM test all contain, according to Lawson and Suurtamm (2006), 4 aspects: problem-solving, calculation, understanding and communication (guideline 1 + 2). To be aware of the language in the test, the students were able to get the questions read aloud, but an awareness of making context in the items with visual representations, and not always only by written words, was also in focus and both contextually and mathematically expressed items were included (guideline 5 +6). By using statistical diagrams in some items, the students had to 77

analyse and critically assess the statistical models and communicate the models’ results (guideline 7). As mathematical reasoning is an important part of mathematics, described in Section 9.1.1, test items with a focus on spatial reasoning, inspired by (Logan & Lowrie, 2013), were included, but also items with a focus on argumentation and where the students found and made arguments for similarities and/or dissimilarities among attributes and relations of mathematical concepts (guideline 4) were included. (Different items can be seen in Papers IV and V).

The mathematical part of KiDM includes students from more than 100 year 4 and year 5 classes, so a computer test would be necessary due to benefitting from the computer capability of coding many of the items. This was decided despite the knowledge from this review that there would be other results if a paper-and-pencil test was used (guideline 8). This, however, means that the alignment between the classroom practice and the test situation is not that obvious (see for further reflection about alignment in Paper V). Finally, to assess the students’ mathematical metacognition level in the test, a specific separate questionnaire was developed (guideline 9 + 10).

Developing a test is a very complicated assignment and of course there are many concerns to be aware of in this process, but this literature review, however, shows what have been the issues and concern in research in the last 10 years on testing in mathematics education and it gives guidelines and direction in the test development. In this sense, Schoenfeld (2015) argued that we are standing by a crossroad and that evaluation and assessment is an important lever for chance, because there will always be a need for testing students’ mathematical competences and it is now very important to take a new crossroad so that new ways of testing can be developed.

Having now described the concept of reasoning competence in mathematics from different angles, the next section will focus on the KiDM project – the intervention and the test development more specifically.

10 The KiDM project

‘Kvalitet i Dansk og Matematik’ (translated into: Quality in Danish and Mathematics), also called KiDM, is a three-year, research and development programme financed by the Danish government Department of Education in collaboration with the Danish school headmasters’ association and the Danish teachers’ association. The intention of the programme was to bring more quality into the subjects of both Danish and mathematics which at that time was increased by law with one lesson more per week in the Danish schools from year 4 to 97. The enhancement of quality in Danish and mathematics was defined in the KiDM project, to make the teaching both in the subject of mathematics and in the subject of Danish more inquiry-based. This Chapter 10 is a description of the overall KiDM project, which is important because of the close relationship to this thesis. The section includes first an overall description of who is involved in the project, the purposes and the overall process. Then follows a description of how the intervention and the test have been developed in a design-based research perspective and the justifications for this approach. The chapter ends with a section containing a description of the final mathematics inquiry-based intervention in the KiDM along with a section with a description of the KiDM test. In general, the whole chapter can be seen as an introduction to the next Chapter 11 - the method chapter, which will subsequently describe how the research has been carried out after the test and intervention has been developed.

10.1 Overall description of the KiDM project The KiDM project is a nationwide experiment that has involved 172 schools in an experimental and development programme with inquiry-based approaches to Danish in years 7 and 8 and mathematics in year 4 and 5. In the mathematics part, 83 schools or 4601 students have been involved in the experiment. The experiment is arranged as an experimental programme with multi-level interventions and experiments of various kinds. The focus in the different levels is on students, teachers and other educational staff, and the experiment involves technology and materials, didactics and teaching methods as well as organisation and team collaboration.

The KiDM project has the following general purpose in mathematics:

7 A description of the Danish school reform 2014 can be found in the document: https://www.uvm.dk/-/media/filer/uvm/udd/folke/pdf14/okt/141010-endelig-aftaletekst-7-6-2013.pdf?la=da 79

• To strengthen the students' mathematical competencies, especially in relation to problem solving/posing, modelling and reasoning, as well as strengthening the students' language development in mathematics. • Breaking the importance of social background for students' learning in mathematics teaching. • Strengthen the students' well-being and learning in mathematics teaching.

The overall intention was at the same time to develop new knowledge about how we can understand and contribute to quality in Danish and mathematics, by evaluating and researching with a combination of quantitative and qualitative methods, so that it is possible to elucidate various new aspects. The project was run in a collaboration between two universities (The University of Aalborg and the University of Southern Denmark) and four university colleges (University College North, University College Absalon, University College South and University College Lillebælt). Results and recommendations are described and elaborated in the final KiDM report (T. I. Hansen, Elf, Misfeldt, Gissel, & Lindhardt, 2019).

The framework and process in the KiDM project was written by the KiDM steering group and the process in the mathematical part involved the following:

Preliminary study Developing the Intervention The eksperiment: (I) and test in differnet Implementation and testing iterations (II) (III)

•Surveying the literature •Developing inquiry-based •Implement and test the on inquiry-based in teaching activities for a effect of the intervention mathematics four-month mathematics with a Random • Carrying out interviews teaching approach for Controlled Trial (RCT). with mathematical year 4 and/or 5 The control schools and supervisors in schools to •Developing content for intervention schools make a preliminary teacher meetings were randomly selected. investigation on the •Developing a Each school participated inquiry-based teaching competence and a with 2-4 classes each, approach in concept test and the intervention concerns all lessons in mathematics. •Developing surveys mathematics for a whole semester. The control schools only participated in the tests

Figure 5: KiDM project - the process in mathematics

In the following two Sections 10.1.1 and 10.1.2, the preliminary study and the development of the intervention and test will be described.

The preliminary study in the KiDM project The literature survey from the preliminary study (I) (Dreyøe, Larsen, et al., 2017) was conducted to gain insight into the most important issues and main concerns associated with inquiry-based teaching in mathematics. A systematic search of six of the highest-ranked journals led to five important themes/issues:

1) The literature stresses that communication in the mathematics classroom should be facilitated as open, inquiring, and related to the students’ activities with a starting point in the students’ prior knowledge.

2) Mathematical skills and competencies are critical to participation in inquiry-based teaching. The modelling view of problem activities especially holds the greatest learning potential for students in inquiry-based teaching. Inquiry-based teaching has a positive impact on students’ mathematical creativity.

3) The students should be allowed to move in and out of the mathematical domain, and hence it is important to use a wide spectrum of activities in teaching. This requires students to be flexible thinkers and prepares them to cope with situations outside of school.

Furthermore, the literature contained examples of and knowledge about both 4) tools and resources for planning and implementing inquiry-based learning and 5) professional development and collaboration.

Apart from surveying the literature, six supervisors8 in mathematics were also interviewed in mathematics prior to developing the specific designs in KiDM. The interviews revolved around similar themes including the need for supporting teacher-teacher and student-student collaboration.

From the above preliminary research, three principles were formulated whose implications were important for the didactical intervention of the design for KiDM:

• Principle 1: An exploratory, dialogical, and application-oriented teaching method with room for student participation increases the effect of the students’ understanding of mathematical concepts and develops appropriate ways of working.

8 Diplomas (60 ECTS) in mathematics supervision have been available since 2009, and many schools have a local supervisor in mathematics who initiates changes and supervises fellow teachers. The supervisors are organized in a national network. 81

• Principle 2: In order to enhance motivation and learning, we prioritise that the students’ experiences of the teaching and the content should be meaningful both from an internal mathematical perspective and from the perspective of the situation of application/inquiry. • Principle 3: An exploratory, dialogical, and application-oriented teaching approach with room for student participation increases the possibility of implementing mathematical competencies. (Dreyøe, Michelsen, et al., 2017; Michelsen et al., 2017). The main focus of these principles was to make the teaching explorative, dialogical, and make room for student participation, as well as make it meaningful both from an internal mathematical perspective and from the perspective of the situation of application/inquiry, but also to include the eight mathematical competences (Niss & Højgaard, 2011), which is a key concept in the Danish curriculum standards. These eight mathematical competencies include, among others, the reasoning competence where argumentation is a central aspect.

The intervention and test developed in a design-based research approach The inquiry-based mathematics intervention in the KiDM project was developed in collaboration with school teachers, supervisors in mathematics, teacher educators, researchers, and professors in an iterative process during which the intervention was tested in a number of cycles in special development schools and pilot schools. The intervention was created in a design-based research approach (DBR) (Paul Cobb, Confrey, diSessa, Lehrer, & Schauble, 2003). The DBR approach was developed as a way to carry out formative research to test and refine a specific design based on theoretical principles from prior research. “This approach of progressive refinement in design involves putting a first version of a design into the world to see how it works. Then the design is constantly revised based on experience, until the bugs are worked out” (Collins, Joseph, & Bielaczyc, 2004, p. 19).

In Poul Cobb, Jackson, and Sharpe (2017) they distinguish between classroom design studies, professional development design studies and organizational design studies. KiDM actually includes professional development design studies, where the focus is on developing the teachers' competencies to teach more inquiry-based in mathematics, but it also includes classroom design studies where the focus is on designing a mathematics teaching intervention including a mathematical concept and competence test.

However, in order to narrow the research area in this thesis, the choice has been not to include the teachers' professional development perspective (cf. the research questions). This is done even though the study will influence the classroom interactions in many ways because the teachers are important partners in relation to the students' development (elaboration on this can be seen in Larsen et al. (2019)).

10.1.2.1 The iterative process in the intervention and test development In this thesis the focus will be on both the classroom intervention and the KiDM test which were both developed together with other researchers in collaboration with mathematics teachers and supervisors in mathematics from specific development schools. It was important for the development of the intervention and the test design to make an effort to involve different participants in the design in order to bring their different expertise into producing and analysing the designs. The main group of the development of the KiDM intervention in mathematics was Bent Lindhardt and Niels Jacob Hansen, from UC Absalon, Flemming Ejdrup from UCN and the author of this thesis. The development of the KiDM test was mainly done by the author of this thesis, but with assistance of Mette Hjelmborg and Morten Puck from UCL – University College, and Bent Lindhardt from UC Absalon. Both the intervention and the test from the KiDM project are important for this thesis, but since the responsibility for the test was placed with the author of this thesis, extra focus will now be placed on the test development.

The test development in the KiDM project followed the three phases in DBR described by Brown (1992): The alpha phase, where the researcher has control of the context and provides ideal supportive conditions. The beta phase where try-outs take place in more, carefully chosen sites with less, but still considerable support. The gamma phase, where the design is widely adopted with minimal support.

In the following, the test design process will be described and elaborated on the basis of the three phases from Brown (1992) in which all three phases include one iterative cycle, and every iteration cycle includes three steps.

Figure 6: The design-based process In the alpha phase, there was only one iteration where the first step was part of the design phase, where theory from the systematic review from the preliminary study helped develop a first draft. Step two was a first try-out with only two students (one student from year 4 and one student from year 5). The two students did the test with a think-aloud approach (see Section 11.5.4), and step three was a reflection/evaluation of the result from the think-aloud test.

In the beta phase the second iteration started. The first step was again a new design (mostly the language and graphics were changed, but also a few items were changed in connection to their mathematical level). In step two the test was used in one of the special development schools named Mølleskolen in Ry, where a year 5 class tried the test. After that, three different students and their teacher were also interviewed about their experience with the test. In the subsequent step, reflection was made mostly about technical issues and language.

Finally, in the gamma phase a third design was made which was tried out in 14 different pilot schools. The pilot schools were selected so that they represented a wide range of different schools using different strata. The strata were created based on three conditions:

School size (400 students were cut-off. 400 is selected as it indicates whether the school has more than two tracks (2 classes with 20 students in each for 10 school years, equaling 400). Urban rural schools (urban schools are schools in municipalities from Aalborg, Aarhus, Odense and the Capital Area. They were included because of the mentality and more opportunities in these areas). Regions (to bring schools from all over the country). These strata were also used in the later experiments.

In this iteration there was lot of feedback from the teachers. Most of them were very positive, that the test was not “just like the national test”, but some of them were also a bit skeptical about the difficulty of the items, especially in connection to students who find mathematics

challenging. After this iteration the development of the coding guide also started. Which again made us make some changes in the final design.

If there had been more time, yet another interaction where the last changes were tried out would have strengthened the design further - it would have been an excessive strength for the design, if it had been possible to collect so much data that the statistical model could also have been tested better. Moreover, it might have strengthened the design by evaluating the design in more, diverse contexts (i.e. different types of classes and various settings (Gravemeijer & Cobb, 2006)) and making sure students with different special needs in mathematics had also tried the test.

The evaluations in the iterations showed that elements that needed to be changed in the design that simply were not possible within the framework, that were made available for this project (for example graphic limitations in the test programme and very small possibilities for making the items interactive in the computer program). Finally, the systematics of all the iterations could have been more consistent, for example, in the collection of feedback from the pilot schools. It was unfortunately not clearly stated what the teachers should give feedback on and who this should be sent to. Finally, the messiness of all the different kinds of data from the classroom observations and from the student and teacher feedback made the amounts of data very big to analyse systematically in the short time, which is not surprising because it is often seen as a challenge in DBR projects (Collins et al., 2004).

10.1.2.2 Real-world settings in DBR The overall ide by using DBR is that it allows the researcher to address theoretical questions or conjectures in context and study them in “real-world” settings, and use formative evaluation to derive findings (Collins, Joseph, and Bielaczyc, 2004). However, in this sense the concept of “real-world setting” needs to be elaborated, because DBR is an interventionist study that intervenes in what naturally happens (the real setting). This is also the case in the KiDM project where the KiDM group deliberately manipulates the condition in the development schools and got the teachers to teach according to some specific theoretical ideas of inquiry-based education. This was necessary in our case, because the type of learning that we wanted to investigate, and test was not immediately present in the real-world setting. The real-world context can, however, be seen as a challenge but also an advantage. A challenge, because it is not possible to isolate specific components when it is part of a complex context and the outcome will never be 85

independent of the complex and social context it is part of. An advantage, because contrary to many other approaches, in DBR the context has been well-thought-out in the various iterations. “…the whole really is more than the sum of its parts.” (Brown, 1992, p. 166). Bryman (2012) argues in this sense that an advantage of the DBR approach is that it has a high ecological validity.

10.1.2.3 Theory in DBR In DBR, theory plays quite a complex role. In general DBR is seen as both pragmatic theoretical in orientation (Poul Cobb et al., 2017). Pragmatic because it involves investigating and improving the test design and theoretical because it also involves the development of a theory of the revising of conjectures about testing in inquiry-based teaching. Design-based research is therefore claimed to have the potential to bridge the gap between educational practice and theory, because it aims both at developing theories about domain-specific learning and the means that are designed to support that learning (Bakker & van Eerde, 2015). The idea is that design-based research produces both useful products and accompanying scientific insight into how these products can be used in education.

“Educational design-based research (DBR) can be characterized as research in which the design of educational materials (e.g., computer tools, learning activities, or a professional development program) is crucial part of the research. That is, the design of learning environments is interwoven with the testing or developing of theory.” (Bakker & van Eerde, 2015, p. 430).

DBR can therefore be described as design which is based on research, but also research which is based on design (Bakker, 2018).

In many research approaches, changing and understanding a situation are separate aspects. However, in design-based research these are intertwined: “If you want to change something you have to understand it and if you want to understand something, you have to change it” (Bakker & van Eerde, 2015, p. 432). According to Paul Cobb et al. (2003) design-based experiments “are typically test-beds for innovation” with their intent being “to investigate the possibilities for educational improvement by bringing about new forms of learning in order to study them” (p. 10). An aim in this is therefore to theorise over the intervention’s implementation in the classrooms.

Design-based research can be seen as well suited to fit with making Random Controlled Trials, because the design is developed to be very robust with ecological validity, because it has been developed in multiple iterative cycles: “Design-based research has recently been described as a potentially fruitful methodology for generating causal accounts of learning and instruction that could form the basis for systematic, randomized clinical trials“ (DBR, 2003, p. 6). However, randomised trials are not necessarily the appropriate end goal of DBR, because RCT in some way can fail to account for phenomena that violate the idea in DBR, namely, to include context and other variables. Therefore, this thesis also focuses on qualitative studies on both the intervention and the test where it is possible to include more context-specific data.

10.2 The final KiDM intervention in mathematics The specific development of the inquiry-based intervention in mathematics in years 4 and 5 includes some theoretical and practical considerations which will shortly be discussed in this section.

The definition of inquiry-based teaching in KiDM is made very broad. Generally, there are many different understandings and definitions of inquiry-based teaching (Blomhøj, 2013; N. J. Hansen et al., 2018) which can be perceived as a kaleidoscope of definitions (Artigue & Blomhøj, 2013). In KiDM, to teach inquiry-based is defined in relation to including at least one of the following processes (translated from Danish):

• Studies based on observations and real physical objects. • To pursue and investigate issues that they perceive as the students' own. • Pupils' active participation in the planning of studies and formulation of specific questions. • That the students develop and use skills to organise and interpret data, reasons, suggestions. • Explanations and make predictions based on own investigations. • Collaboration on studies, communication of own ideas and positioning on other students’ ideas. • To express yourself in speech and writing using mathematical concepts and representations.

• Public discussions where they explain their investigations. • Self-critical reflections on both process and product aspects of their studies. • To apply the acquired skills and competences in authentic situations. (N. J. Hansen et al., 2018; T. I. Hansen et al., 2019)

The KiDM intervention consists of a Danish website (www.kidm.dk) that contains a detailed teacher's guide and accompanying student pages with various student activities that take into consideration students’ prior experiences, students’ opportunities for participation and appropriate ways of working with exploration, applications and mathematical competencies, especially communication, reasoning, and modelling. These inquiry-based activities and tasks were divided into 5 different categories of activities: ‘the brooder’, ‘the (re-)intervention’, ‘the modelling’, ‘the measuring’ and ‘the products’. These can be read more deeply in Paper II (Larsen & Lindhardt, 2019) where the different activities are described based on theoretical considerations about what defines inquiry-based teaching.

The activity called “product” from the KiDM is an activity that is about students producing a product where the investigative aspect becomes more aesthetic or concerns future perspectives (like how this product can be made even better or more beautiful). It can in many ways be discussed whether this kind of activity whatsoever can be defined as an inquiry-based activity, because the students on their ways through the activity just follow a guide. However, if the discussion afterwards makes the students reflect on new ways to construct the product and maybe even make new conjectures which they afterwards inquire and justify, this activity can be defined as an inquiry-based activity according to both R. Hansen and Hansen (2013) and Blomhøj (2017).

The intervention includes three different courses with a focus on, respectively, numbers and algebra, geometry, and statistics. The three courses all focus on the reasoning competence, but it is, however, not the main aim in all the tasks in the courses.

To make sure that the inquiry-based teaching approach focuses on processes in teaching, every activity was also divided into three different phases: introduction, activity and whole class discussion which have different aims in the students’ inquiry process (N. J. Hansen et al., 2018).

Already Dewey (1997) emphasised the need for a high degree of structure and systematics in an inquiry-based teaching practice. Dewey wrote that students in an inquiry approach compared to a traditional approach, consequently need “more, rather than less, guidance by 88

other”(Dewey, 1997, p. 21). It has since been supported by a number of effect studies and research studies aiming at achieving the best results when supplementing an open, investigative approach with scaffolding (Friesen & Scott, 2013), guidance (Decristan et al., 2015; Kirschner, Sweller, & Clark, 2006) and tutorials (Furtak, Seidel, Iverson, & Briggs, 2012).

As a consequence of this, the descriptions of the activities in KiDM include a very detailed teacher's guide and lesson plans with specific instructions on how the activities should be introduced, what hints can be given along the way, and how the collective discussion should be designed. The website also contains a 24-page long description of educational considerations about inquiry-based mathematics education, for example, how students with special needs in mathematics require special focus in this approach and the importance of communication in the classroom, including an overall question guide.

Figure 7: Screenshot from the KiDM.dk webpage (www.kidm.dk)

The figure to the left show four different boxes. In an activity from the KiDM intervention, the students were told that the boxes only were weight in pair to weigh respectively 8,10,12,14 or 16 kg. The students then have to find out what each box weighs.

Figure 8: Activity from the KiDM intervention

To assess the students’ development of mathematical concepts and mathematical competences related to inquiry-based teaching, the developed pre- and post-tests were used at all the intervention schools and all the control schools. Student and teacher surveys were also conducted focusing on changes in the students’ motivation and experiences and the teachers’ extent of implementation.

10.3 The final KiDM test Already established achievement tests were found unsuitable for assessing the development of the KiDM programme, because these externally created Danish tests for years 4 and 5 are often multiple-choice tests with a focus on testing the students’ procedures and calculating skills, so an assessment with a stronger focus on mathematical and inquiry-based competencies had to be developed. The developed achievement test had to be used in a large-scale study, and it was therefore decided that it should be a computer-based test conducted in Survey XACT which should help with the large amount of data. The content in the test had to be broad, going from conceptual understanding in algebra, geometry, and statistics, to the eight mathematical competencies (Niss & Højgaard, 2011).

In the KiDM test the intention was to measure many different aspects of what the students have achieved by working inquiry-based; however, these achievements are considered as the competence of performing various processes. Therefore, the test-specific focus on these

processes is; among others, to make observation, independently pose problems and plan to solve problems, interpreting of data, making suggestions, explaining and predicting.

In the development of an inquiry-based competences test in KiDM, several of the guidelines from a systematic review (see Section 9.5) was also used. This includes, among others:

• Items with problem-posing questions to measure mathematical creativity (Kar, 2015). • Coding-scale inspired by Akgul and Kahveci (2016) and Limin et al. (2013) (fluency, accuracy, complexity and originality). • Problem-posing items with semi-structured items like Munroe (2016). • Students were asked to generate two problems and to offer their solution to these problems, but it also included, as Chang et al. (2012) advocated for, a way of making the students look back by asking the students to explain why the second posed problem was more difficult. • The problem-solving items in the KiDM test all contain, according to Lawson and Suurtamm (2006), 4 aspects: problem-solving, calculation, understanding and communication. • Like Hickendorff (2013), mixed tests with both contextually and mathematically expressed items were used.

(See Paper IV for more description of the specific items).

KiDM includes many students, so a computer test would be necessary due to getting benefit from the computer capability of coding many of the items. This was decided despite the knowledge from an earlier review (ses Section 9.4) that the result would be different if the test was a paper-and-pencil test compared to a computer test. More statistical details of the development of the test can be seen in Paper IV.

11 Methodology and Methods

Methods can be seen to relate into three different levels (Dahler-Larsen, 2008): methods as a paradigm (epistemology and ontology), methods as design and methods as a technique for constructing and treating data material. All these levels are important in connection to answering the research question in this thesis and will now be the focus in this chapter.

The following methodology and methods chapter will consist of first a description of the paradigm in this thesis (Section 11.1). Then the experiment design will be described (Section 11.2) along with how the experiment will be explored in a mixed methods approach (Section 11.3). Afterwards, the method of the data construction technique will be split into qualitative (Section 11.4) and quantitative (Section 11.5) approaches where each section consists of constructing data procedures and how the data are analysed, but reliability and validity considerations are also included along with ethical considerations. Each section includes some methodology considerations along with descriptions of the methods used and some further reflection.

11.1 Pragmatism as the research paradigm Considerations of epistemology and ontology are important aspects to have clarified in connection with the overall methods of this thesis because this affects the whole approach and design of the project. In general, it is important to be clear about the scientific approach in all conducted research, because different scientific approaches have different ways of defining truth about nature; whether people actually invent new theories or people find the truth somewhere else, because this will affect the way we answer the research questions and how the research design is created. From the objective and measurable reality of positivism via the more contextualised understanding of realism to the subjective plurality of interpretivism, paradigms could be interpreted as prescriptive and as requiring particular research methods and excluding others (Feilzer, 2010). The paradigm can in this sense blind and limit researchers to some aspects, but it can also in other aspects open up to curiosity and creativity or new theories.

Argumentations and reflections about the paradigm are often not made explicit in short papers or small journal articles, but they will, however, always in some way be exposed in the

interpretations of the research findings. Therefore, this methods section will start with reflections about the overall research paradigm of this thesis.

In answering the research questions in this thesis, there is a need to generate new knowledge that hopefully can inform actions and activities in educational settings and researchers’ understanding. This new knowledge can, on the one hand, be seen to provide information about reality as the real “truth”, but it can, on the other hand, also be seen as subjective and process- oriented. In a pragmatic view, knowledge is viewed as being both constructed and based on the reality of the world we experience (Johnson & Christensen, 2014). In the theory of pragmatism, the argument is that cognition should not be understood as an objective reflection of reality, but as a useful tool that demonstrates its appropriateness in practice (Biesta, 2010). Pragmatists contend that most philosophical topics—such as the nature of knowledge, language, concepts, meaning, belief, and science—are all best viewed in terms of their practical use and successes (Brinkmann, 2017). The overall research paradigm in this thesis is pragmatism.

Pragmatism was founded by C.S. Peirce in the 1870s as a theory of meaning, but in the 20th century, Peirce and his students (William James, John Dewey, F.C. Schiller) developed it into a general theory of human realisation (Brinkmann, 2017). The intention in this section is not to review the evolution and long historic development of pragmatism as a philosophy, because this will be beyond the scope of this thesis; however, the intention is to explain how the pragmatic approach will be visible throughout this thesis.

To have a pragmatic approach to research is sometimes wrongly seen as a way where you can do whatever you find it necessary to answer the research questions, an approach we see insinuated in Johnson and Christensen (2014) when they write that pragmatism: “is to mix research components in a way that you believe will work for your research problem, research questions and research circumstances.” (p. 489). This is, however, not the case. In pragmatism, the idea is that the function of thought to describe, represent, or mirror reality is rejected (Dewey, 1958). Instead, knowledge must in some way be tested in practice to be valid and this makes experiences and actions an important part of research in the pragmatic paradigm.

“The only way to avoid a sharp separation between the mind which is the centre of the processes of experiencing and the natural world which is experienced is to acknowledge that all modes of experiencing are ways in which some genuine traits of nature come to manifest realization.” (Dewey, 1958, p. 24).

In pragmatism, reality therefore only reveals itself as a result of activities and reality is only experienced (Dewey, 1958).

In this thesis the ideas from John Dewey (1859-1952) will be the primary focus in the pragmatic approach. This is not only because Dewey was a world-renowned educator and philosopher. It is also because many of Dewey’s ideas are still relevant today (Biesta et al., 2003) and his thoughts have been a large part of the KiDM project already from the KiDM application to the development of the website.

For Dewey, the experience is the starting point for knowledge and thinking as well as the test of the validity of thought (Dewey, 1958). Pragmatists’ view of the measurable world relates more carefully to an ‘‘existential reality’’ where the reference is to an experiential world with different elements or layers, some which are objective, some which are subjective, and some which are a mixture of the two (Dewey, 1958). In Biesta et al. (2003) it is described that objectivism argues that knowledge is “of the world” and subjectivism that knowledge is of “the mind”, but that Dewey argues that knowledge is a construction. But it is not a construction of the human mind, it is a construction that is located in the organism-environment transaction itself. Dewey is also very clear that all experience is social - that is, experience requires communication and contact (Dewey, 1974). This pragmatic approach therefore forces researchers to be more flexible and open to the emergence of unexpected experienced data which is not the same as being able to do whatever they want.

At the level of translating these epistemological concerns into deciding on research methods, a pragmatic paradigm poses the methodological question: If phenomena have different layers, then how can these layers be measured or observed? Mixed methods research offers to plug this gap by using quantitative methods to measure some aspects of the phenomenon in question and qualitative methods for others. In the literature (Biesta, 2010), there are stronger and weaker versions of how mixed methods research fits a pragmatic approach; ranging from the suggestion that pragmatism provides the philosophical foundation for mixed methods research to the idea that pragmatism could provide philosophical support for mixed methods approaches.

However, mixing the methods and integrating the different methods—in the sense of looking at experiences from different perspectives and providing an enriched understanding is, however, not a simple approach. This will therefore be explicitly described in Section 11.3.

11.2 Methods as a research design In this section there will be a description of the methods used as the research design in this thesis. Research design refers to the outline, plan or strategy that is used to seek the answer to the research questions. The research design of the KiDM experiment was developed in the KiDM steering group (see Section 10), but the planning of how to answer the research questions in this thesis, which is closely linked to the KiDM project, will be explained in this section.

An experimental research design The KiDM project is designed as an experimental research project and because of the close connection between the KiDM project and the research questions in this thesis, the research design in this thesis is also defined as an overall experimental design. However, the focus of the output will be different.

An experiment is defined as: “An environment in which the researcher attempts to “objectively” observe phenomena that are made to occur in a strictly controlled situation in which one or more variables are varied and the others are kept constant” (Johnson & Christensen, 2014, p. 320).

The overall purpose of experimental research is to determine a cause-and-effect relationship (causality) in these controlled situations, because it allows the researcher to observe the effect of changing and manipulating one or more independent variables. This changing or manipulation is expected to cause a change in the dependent variable. This is, however, not an easy kind of research to do in education research, because of extraneous variables (Johnson & Christensen, 2014). Extraneous variables are variables other than the independent variable of interest that may be related to outcome and can be alternative explanations for the outcome. In Johnson and Christensen (2014) they use the term confounding variables to refer to extraneous variables that were not controlled for. Educational research often includes field experiments conducted in real life settings. The advantage of field experiments is that they are excellent for determining whether a manipulation works in real-world settings, but it also makes it difficult to exclude these extraneous variables.

The discussion about carrying out causal analyses in education research is an ongoing discussion. On the one hand, it argues for the importance of this type of research (Hattie, Biggs, & Purdie, 1996) while, for example, in Dahler-Larsen and Sylvest (2013), they describe that

the problem is not the causal analysis research in itself, but that it is more or less explicit in the ambition that all social sciences should be shaped as causal analyses and, as a result, most research today is primarily assessed on the basis of certain causal analytical calculations. However, it is not all research that poses causally-oriented questions, because we cannot then look at the unique things and the things that have not become regular.

In the KiDM project the experiment is an inquiry-based teaching intervention. The treatment group, the group that receive the experimental treatment, is in this thesis called the intervention schools, and the group who does not receive the treatment is the control schools. The control group serve as a comparison and a control for a rival hypothesis.

The experiment in KiDM is a random assignment, which is a procedure that makes random assignments to conditions on the basis of chance and in this way maximises the probability that comparison groups will be equated on all extraneous variables. Johnson and Christensen (2014) argue that “…random assignment is the best technique for equating the comparison groups on all variables at the start of an experiment” (Johnson & Christensen, 2014, p. 325). The idea is to equate all comparison groups on all variables and then systematically vary the independent variable, and then the researcher can claim that the changes in the dependent variable are caused by the independent variable that was systematically manipulated by the researcher.

To measure the changes to a variable in random controlled experiments a test instrument is often used, and then the research design will include a pre-test and a post-test for the two randomly assigned groups of participants. Random controlled experiments are repeatedly seen as the “gold standard” (Johnson & Christensen, 2014, p. 492) or as “true” experiments (Bakker & van Eerde, 2015, p. 433) for establishing evidence of cause and effect studies. However, some critiques of this approach are also described:

“…the practical usefulness and ecological validity of research on learning based on classical well-controlled experiments are more questionable than ever” (Engeström, 2011, p. 599).

Engeström (2011) argues that the RCT study which is taken from fields such as medicine and agriculture does not fit into educational research, for different reasons. Among other, because in an RCT study researchers expect the design of the intervention to be complete at the outset, so it does not recognise the complexity of the educational setting:

“…the process of design research is depicted in a linear fashion, starting with researchers determining the principles and goals and leading to completion or perfection. This view ignores 96

the agency of practitioners, students, and users. It seems blind to the crucial difference between finished mass products and open-ended social innovations, as well as between designer-led and user-led models of innovation process. Finally, in much of the literature on design experiments, a variable-oriented approach to research is tacitly endorsed, without questioning the underlying problematic notion of causality” (Engeström, 2011, p. 602).

In other words, experiments with human beings in social contexts are met with people with identities and agencies; they are not met with anonymous mechanical responses. The teachers and students in school in the RCT study are expected to execute the intervention without resistance and the experiment is expected to reliably generate the same desired outcome when transferred and implemented in new settings, which Engeström (2011) clearly disputes.

It is clear that there are many pros and cons with using a random controlled trial (RCT) study and this will of course also be taken into consideration in the discussions section (see Chapter 15). The overall view of RCT studies in this thesis is that the area cannot be seen as either black or white, but that it is more a barbell where the balance should not tilt over so that we do not look unilaterally at the random controlled experiments’ results, but that these are seen and interpreted in a larger context, which also means including other method approaches.

The RCT study used in the KiDM programme focuses on answering the question: Does inquiry- based teaching improve students’ achievement? The aim is to explore whether the treatment group learns more concepts of understanding and inquiry-based mathematical competencies in this inquiry-based treatment, than the control schools. The measurement in the KiDM project is, however, not so manifestly only through quantitative results, as the test measurement will be supplemented with observations and interviews. It should also be noted here that the intention of the KiDM project has a multi-strand approach and it also has a focus on whether the teachers change and on what, for example, the teacher group meetings mean to the intervention.

In this thesis, the focus will be to study whether the students develop their reasoning competence in this inquiry-based intervention more than in the control school and this will be measured in multiple ways.

1. By using an achievement test, but also 2. Surveys on teachers’ and students’ experiences will be used along with 3. Classroom observation 97

To be able to finally answer the research questions it is clear that this design shows caution about determining the effect only with a simple linear causality, but different approaches need to be included and interpreted.

The research design used in this thesis can be seen in the following Figure 9. The figure shows how the research is divided into two areas. To the left (blue square) it is shown that the design of the test and intervention took place in a design-based approach that ended with final versions (described in Chapter 10). To the right (red square) the RCT study is tried to be explained with tests and surveys (pre + post) and that the observations are done in the treatment group in the three trials. The combination of the DBR part of this thesis in combination with the RCT study can in some way be defined to have a development purpose where the methods of the design of the interventions and the test were used in the later experiment.

Figure 9: Overview of the research in the thesis

The timeline in this thesis can be seen in the following Figure 10. Where it is clear that the DBR part of the research is done in Autumn 2016 and Spring 2017, the three trials were done in

Autumn 2017, Spring 2018 and Autumn 2018. In 2019 the focus has been on analysing and interpreting the data followed by writing.

2016 Autumn Review of mathemati 2018 cal Autumn competen 3. trial ce test Pre- and Developme post-test 2017 2019 nt of test Analysing Autumn Autumn Review of the IBME 1. trial qualitative (until Developme Pre- and data from 15th of nt of post- the septemb IBME- test observatio er) interventio Observati ns Complete n ons in Analysing the Developme the the thesis as nt of the classroo qualitative well as mixed m with data from submissi study students the test on

2017 2018 2019 Spring Spring Spring Pilot test 2. trial Analysing of the Pre- and the test + post- test quantitati surveys Observatio ve data Pilot test n of Interpretati of classroom on of the intervent and mixed ion interviews study with students Analysing the qualitativ e data from the observati ons

Figure 10: Timeline in the Ph.D. project

11.3 Methods – Mixing the methods The experiment design in this thesis includes a mixed methods study. A mixed methods study involves the collection or analysis of both quantitative and qualitative data in a single study (Johnson & Christensen, 2014). This combination is, however, still an important topic within methodological debate in the field of mathematics education (Kelle & Buchholtz, 2015),

because “By now, only few empirical studies in mathematics education try to combine qualitative and quantitative research methods in a common design” (Kelle & Buchholtz, 2015, p. 322).

The combination of different methods in a single research design has, however, been used for several years, but it is only within the last decades that there has been a real discussion of basic principles of this type of combination (Frederiksen, Gundelach, & Skovgaard Nielsen, 2014).

In this section the design will be presented and the benefits and disadvantages of using this design will be discussed.

Mixed methods is often defined as a class of research studies in which a researcher mixes or combines quantitative and qualitative research approaches and techniques in a single research question for the broad purposes of breadth and depth of understanding and corroboration (Johnson & Christensen, 2014; Johnson, Onwuegbuzie, & Turner, 2007). The claim in mixed methods is that it is a strength to combine qualitative and quantitative research to answer the research questions in a more qualified manner. In Johnson and Christensen (2014) they argue, for example, that the RCT study called gold standard (see Section 11.2.1) will be even stronger if some qualitative aspects are added to it, and this is in line with T. I. Hansen (2018) who argues that mixed methods can instead be seen as the gold standard in educational research:

"With Mixed methods as a golden standard, we can combine the methods without hierarchizing them and making some types more inferior than others because the educational perspective must prevent the evidence from becoming blind and instrumental, while the evidence perspective, conversely, must oblige the formation in an empirical and perceptual correlate so that it does not become general and speculative” (T. I. Hansen, 2018, p. 258).

Different kinds of mixed methods are described in the literature and multiple researchers describe ways to categorise different kinds of mixing the methods (Brinkmann & Tanggaard, 2015; Creswell, Plano Clark, Gutmann, & Hanson, 2003; Johnson & Christensen, 2014; Johnson et al., 2007). In the field of education where the attempt is to solve manifold and complex problems there is, however, probably not one best way to mix the methods, but researchers advocating mixed methods research argue that it is important to use both an exploratory and a confirmatory method as part of the research (Johnson et al., 2007).

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The argument that mixed methods research is understood as a mix of quantitative and qualitative methods has in particular been that these methods each form a multiplicative paradigm with different complementary strengths. Frederiksen et al. (2014) argue, however, that the difference between qualitative and quantitative research can hardly be maintained in practice if it is subject to further investigation, because the vast majority of qualitative researchers comment on quantitatively and frequent performers structuring in their materials, and quantitative researchers will work with quantifications of something qualitative. Frederiksen et al. (2014) therefore argue that it is important to seek to understand mixed methods based on the method itself. This will, however, be the intention in this thesis.

In this thesis the different methods are used to expand the research, but in some ways, they are also used to complement each other. Purposes of mixing methods can according to Greene, Caracelli, and Graham (1989) have different approaches: Triangulation, Complementarity, Development, Initiation and Expansion (Greene et al., 1989, p. 259).

The purpose of traditional triangulation is often referred to as the strength in mixed methods. Method triangulation arose in qualitative research where the combination of more qualitative methods should provide a more accurate understanding of the phenomenon (Brewer & Hunter, 2006). Kelle and Buchholtz (2015) describe this with two different metaphors: “the use of different methods to investigate a certain domain or reality can be compared with the examination of physical objects from two different viewpoints or angles” (p. 332) and “empirical research results obtained with different methods are like the pieces of a jigsaw puzzle which provide a full image of a certain object if put together in the correct way.”(p. 332). Both metaphors provide pictures of triangulation as it assumes that there is actually a unique phenomenon that can be measured by combining the measurements like a positivistic approach. But the methods used may not be useful to validate each other and the different methods can also be seen as relating to different empirical experiences. However, in this thesis which has a pragmatic approach the different methods cannot help in understanding a phenomenon more objectively, but by combining different methods or research approaches we can hopefully get an extended (expansion) understanding of the experience in order to get elaborations and clarifications and to get a more complete picture of the research domain in the complex educational research. It is here important to acknowledge that the qualitative data are not seen as a lifesaving device to rescue the quantitative data or the other way around, but both the methods have their own importance in the overall research. Using the traditional 101

triangulation approach would also mean that if the research results from the different methods are dissimilar, then it would indicate validity problems; but if separate aspects of the investigated experience were examined with different methods then we would actually expect different results.

It is, however, not that easy to combine qualitative and quantitative methods. Quantitative and qualitative research can be seen as distinguished by different views of human behaviour. In quantitative research there is an assumption of determinism which means that all events are fully determined by one or more causes – (the cause-and-effect relationship). The qualitative research view, on the other hand, is more human and fluid, dynamic and changing over time where different people and groups are said to construct their own realities and perspectives. When mixing these methods, the idea is to value both the approaches. This is in line with Dewey's monotheistic philosophy because Dewey describes that research is about seeking holistic perceptions of life instead of starting from contradictions (Dewey, 1974).

The research in this study is defined as an experimental mixed methods (Creswell & Clark, 2017), because it, among other things, combines a test and surveys in a RCT study with some qualitative observations in the RCT study. The combination of both qualitative and quantitative methods is crucial in this thesis for answering the research questions in order to get a rich description of what happens in the complex classrooms under the experiments, but the quantitative research is also crucial to be able to assess the relationship between students’ development of the reasoning competence in control schools versus intervention schools.

The complementary strength by using mixed methods in this thesis is that the results will be more nuanced and extended because the sum of the parts is greater than only using one method.

In the following Figure 11 we can see an overview of all the data included in this thesis.

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Design Methods Data

Observations Qualitative data from intervention classrooms Mixed methods Test (pre and post) Quantitative data Survey (pre and post)

Figure 11: overview of the design, methods and data

The integration of the methods in mixed methods is in Johnson and Christensen (2014) defined in nine different designs (see Table 5) in a typology-matrix, where two dimensions are fundamental; time orientation and paradigm/research approach emphasis.

Table 5: Table from Johnson & Christensen 2014, p. 497 Time Order Decision Paradigm Concurrent Sequential Emphasis Decision Equal status QUAL+ QUAN QUALQUAN QUANQUAL Dominant status QUAL+quan QUALquan QUAN + qual QualQUAN QUANqual QualQUAL

To get an overview of and to understand the research in this thesis it is essential to be clear about whether the mixing is a combination of methods and techniques during data collection

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and analysis or it is about the integration of methodological approaches within the research design.

When using the matrix from Johnson and Christensen (2014) the research methods in this thesis can be defined as a QUAN (capital letters) + QUAL (capital letters) research because it relies on both the quantitative data (data from tests and surveys), and the qualitative data (data from observations). It is important that no one of the approaches has a dominant status.

Without the qualitative data it will be impossible to answer the “how” in the research questions, because the test item will not be able to measure this. The classroom observations, which are the qualitative data, are important in the explanation of what the impact of the intervention in the classroom is and to explore the actual experiment. For the quantitative data it is important to answer another aspect in the research question, because the quantitative test and surveys will focus more on the cause-and-effect aspect of the question and whether it is possible whatsoever to measure in a test a development in students’ reasoning competence.

The overall design of the mixed methods can in some way both be described as sequential, because the data from the DBR affect the later RCT study, and the general construction of the experiment. However after the design is developed (both the test and the intervention) the qualitative data do not affect the test data, and the test data do not affect the observations in the classroom, which means that the research can be seen as a concurrent design, where the notion concurrent refers to the qualitative and quantitative data as being collected at approximately the same time (in the different trials). The qualitative data are here all constructed during the experiment to explore the phenomena of reasoning competence in the classroom. And the quantitative test data and survey data are also constructed during the experiment.

In Creswell et al. (2003) different mixed methods designs are represented in different illustrations. An illustration of how the methods are mixed in this thesis can be seen in Figure 12. The method approaches in the five papers are very different, but all the approaches in the different papers will al together be able to answer the overall research question.

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Design Based Research (DBR)

sequentiel

Analysis of tests results QUAN IV

Papaer

Mixed concur rent Analysis of method

Quan classroom experimenta V observation + l study QUAL testanswers Papaer QUAN + Thesis QUAL

Analysis of classroom

QUAL I+II+III observation

Papaer

Figure 12: Overview of the mixing of methods in the thesis

The different designs in the different papers included in this thesis are not at all that straightforward because in Paper V the quantitative test data are analysed with a qualitative approach along with the qualitative data from the classroom observations. The quantitative data

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are therefore converted into data that can be analysed qualitatively – qualitizing (Onwuegbuzie & Johnson, 2006). We call this a nested relation (Creswell et al., 2003). A nested relation means that the participants selected for the quantitative phase of the study represent a subset of the qualitative phase of the study where the collected data from the classroom observations are connected to get a better understanding of the alignment (Paper V).

Multiple researchers (Brewer & Hunter, 2006; Frederiksen et al., 2014; Johnson & Christensen, 2014) argue that it is the way the methods are combined and integrated in the research that decides the quality of the outcome. Frederiksen et al. (2014) define 7 different ways of integrating different methods in mixed methods research: Theory-integration, design- integration, methods-integration and data-integration, analyze-integration, interpretation- integration and presentation-integration. The intention in this thesis is not that the study should be integrated in all these 7 ways. The point is to be explicit about which kind of integrating ways the research uses.

The integration of different methods in this research is seen in the design-integration, data integration, interpretations integration and in the presentations-integration.

Design-integration is conducted when the different designs are mixed in the overall research question. In this thesis the research questions can only be answered by using both qualitative and quantitative methods. The designs in Papers I, II, III are not mixed but in Paper V the methods are mixed in a nested design. Paper V can in some way be seen as a bridge-building between the quantitative test design and the qualitative classroom observations.

Data integration is conducted when the students from the intervention’s classes carry out tests and surveys, but it is also observed during the experiment trial. Data integration does also happen when the test answers were converted to qualitative data and analysed using qualitative methods to study the alignment (Paper V).

Finally, there is an interpretative integration when the different analyses from the different papers are put in relation to each other in order to answer the overall research question in this thesis. It is in the interpretation of the different results and in the presentation that the overall meaning in the separate and isolated results from the different papers is constructed.

In general, many advantages and disadvantages of mixed methods can be written, as in other methods. The biggest limitation is, however, that to do this study it requires expertise in

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designing and implementing both quantitative and qualitative phases and the most important thing is to be aware of and reduce the weaknesses of the different methods and still use the advantages of the different methods. The different methods will be discussed in the next section.

11.4 The qualitative research In Johnson and Christensen (2014) they have a list of the six most common methods of data collection used by educational researchers: tests, questionnaires, interviews, observations, constructed and secondary data. This dissertation contains almost all these methods of data collection, which also means that there is a lot of methodological issues to be aware of.

First, to make sure the qualitative methods are reliable and consistent, the methods need to have a high degree of transparency, which this section will now try to show.

Case study All the papers which use qualitative data (Papers I, II, III, V) employ a case study approach, wherein collected cases from the classroom observations are studied.

Case study research can be seen as a specific type of research (Stake, 1995; Yin, 2014) and be defined as research that provides a detailed account and analysis of one or more cases (Johnson & Christensen, 2014). Stake (1995) defines case as a “bounded system” (p. 2) to indicate that a case study is about figuring out what complex things go on within a specific system. The system is here a set of interrelated elements that form an organised whole (Johnson & Christensen, 2014). Cases are in general seen as holistic entities that have parts in the whole context environments. The idea is that the researchers provide a detailed account of one or more cases, but the focus is on each case as a whole unit.

Some researchers are very inclusive in what they call cases (Yin, 2014). For them, cases are not only an object or an entity with a clear identity, but they could also be an event, an activity or a process. In this thesis the cases described in the papers are all small bounded episodes from the classroom. In these small cases the interest is in the classroom interactions as a small system, and therefore, to get some holistic descriptions of the interactions, video observations were chosen to be a way to construct the classroom data. Video observations gave the opportunities

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to carefully examine the context of the cases to be able to describe and explain better the intervention in the classroom.

This is in contrast to the test studies and surveys in the KiDM experiment which deliberately separated the phenomenon from its context, attending only to the phenomenon of interest and only as represented by a variable - the context in these studies is entirely ignored because it is “controlled” by the statistical environment. Also, in the KiDM surveys the inclusion of the context is extremely limited. The idea in this thesis is that the case studies can instead offer insights not provided by the RCT studies. The main idea is that RCT can measure the effect of the intervention, but it is the case study which can help explain “how” this intervention worked (or not worked). The case studies in this thesis are therefore to explain the presumed causal links in real-world interventions that are too complex for both surveys or test methods (Yin, 2014). These cases are not seen as samples, but they are seen as a possibility to shed empirical light on the theoretical concepts of mathematical reasoning. “the short answer is that case studies, like experiments, are generalizable to theoretical propositions and not to populations or universes” (Yin, 2014). The role of generalisation in case studies can be characterised as analytical generalisation in contrast to the RCT study where the generalization results from statistical generalisation. Stake (1995) distinguishes between three different kinds of case studies: intrinsic case studies, instrumental case studies, and collective case studies. In the papers in this thesis all the cases can be seen as instrumental case studies, where the primary interest is in understanding something other than the particular case and to redraw generalisations or build a new theory. In the papers the cases are seen as important only as a means to learn something more in general. This also means that the goal seems to be less particularistic and more universalistic, and the conclusions are therefore also less specific to the case, but it applies more beyond the particular case. In Papers I and III the interest is in how the students go from reasoning in specific contexts to doing more mathematical specific reasoning and in Paper II the interest is in how students make argumentation or work with reasoning competence in different inquiry-based activities. The cases in Papers I, II and III can also be defined as collective case studies (or multiple-case design (Yin, 2014)) where several cases are studied in a relatively in-depth analysis to get more generalised results, because as Yin (2014) argues: there will be greater evidence with multiple cases if similar results would happen in a new case.

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Case study research has different kind of critiques. Case studies are often accused of being a research strategy in which the researcher can confirm his/her favorite hypothesis, although there is no reason to claim that this would happen more frequently in the case study than in other research strategies (Flyvbjerg 1994). In the process of doing the case study, a list from Ramian (2008, p. 105) was followed to be aware of as many biases as possible: This list includes among others, the researchers’ personal engagement and other personal values.

Sampling in the qualitative research In qualitative studies it is often not possible for researchers to collect data from everyone in a sample area or community. Therefore, the researcher needs to gather data from a sample, or subset, of the population in the study. In quantitative research it would be a goal to conduct a random sampling that is representative of the entire population to be able afterwards to generalise the results to the entire population. However, in qualitative research the goal is instead to provide a more in-depth understanding and therefore the researcher often targets a specific group or groups, or a special type of individuals to accomplish this goal.

In this thesis the qualitative sampling was purposeful, in which the participants were selected based on, to a small degree, pre-selected criteria; The overall idea was to get samples from classes which were representative of the population it came from (all year 4 and 5 classes participating as intervention schools). The process of drawing a sampling from all the participating classes from KiDM was done very pragmatically: At the first KiDM Kick-off meeting (trial 1) all the teachers were asked if a researcher could visit them three times. This was done in two (east and south) of the three regions in Denmark (east, south, north). One region was excluded because of the long distance to the researchers visiting the schools. Unfortunately, not many agreed to participate, but three teachers from the east region and two teachers from the south region volunteered. This means that five schools were visited three times. In trial two, one school volunteered, and this class was visited in two lessons and in trial three another school volunteered to the Kick-off meeting and this class was visited every Monday, as far as possible, throughout the whole intervention (7 Mondays).

The distribution of schools can be seen in Table 6.

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Table 6: Distribution of schools in the qualitative data

School Region School Village/city Observations In Denmark size /lessons School 1 East Large City 3 x 2 lessons School 2 East Large City 3 x 2 lessons School 3 East Small Village 3 x 2 lessons School 4 South Large City 3 x 2 lessons School 5 South Small Village 3 x 2 lessons School 6 South Small Village 2 lessons School 7 South Small village 7 x 2 lessons

In Table 6 it is obvious that the sample was not done in connection to sample criteria from the quantitative sample (school size and village/city) but it was volunteers. This was decided mostly because many teachers would not like to be videotaped and therefore this should take place as a voluntary option. The sampling of observing specific students in the classroom was done in cooperation with the teachers.

Observations in the classrooms Observations from intervention schools’ classrooms are the main data in the qualitative data. Observation is an important way of getting information about students, because students do not always do what they say they do in interviews (Johnson & Christensen, 2014, p. 236), so to answer the research questions, it was important to follow how students act in the different contexts during the KiDM intervention. The focus in all the observations was on the directly readable features of the situation, i.e. students' interaction with the material and social environment and the discussions in the classroom. To capture the complex situations in the classrooms, all the observations were video recorded. Video observations can be comprehended to be very complex. In this thesis this is, however, not considered as a barrier in the analysis, but on the contrary, as an advance in the analysis to get in depth as possible.

The purpose of the observations was to get as close as possible to the students’ actions, while they were happening in situ. The video recordings were all taken with a handheld camera, and it clearly shows how the videos are social constructions in themselves, as this only reflects what

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the researcher thinks in the situation is relevant to record, so some information was recorded on behalf of others. Most of the observations were done by the author of this thesis, but Jonas Meldgaard Dreyøe9 did the observation in the east region (see Table 6).

During the observations we were aware that our presence with a videorecorder influenced the behaviour of the students. The role in the observations was as “observer-as-participant” (Johnson & Christensen, 2014; Krogstrup & Kristensen, 1999) because we took the role as observer more seriously than the role as the participant. Furthermore, the participant was fully aware that they were part of a research study. The insider view can in these situations be difficult, and as we both are former primary school teachers the participant role was also tested in the observation situations, especially when the students needed help either socially in form of resolving conflicts, or when the students had problems in their mathematical reasoning process. In the appendix the observations guide (in Danish) are included. There are two different guides included; one for the first trial (Appendix d) and one used in the second and third trial (Appendix e). There are two reasons why two different guides were developed. A new observation guide was developed because the last observations had to have a slightly different focus, but also because the experience from observation in trial one led to improvements in guide 2.

Finally, there are some ethical considerations in connection to making observations: First, there are considerations that the observation in the classroom could make the participating teachers feel like being examined, mostly because one of the observers was also part of the development of the intervention, but secondly there was also the concern that the observation could have impact on the RCT study, which also must be taken into consideration.

All the parents of the observed students were informed about the video observation, and they were able to deselect their child from getting video recorded, because they had to give their informed consent. It was here very clear that there was a professional confidentiality and that the observations would only be used in research and not in any way made available to the public, all according to the Danish Data Protection Agency (GPDR).

In the Appendix f the letter for consent can be seen (in Danish). The consents from the parents were active and not passive which means that the parents and students had to actively fill the

9 Jonas Meldgaard Dreyøe was at that time a scientific assistant in the KiDM project. 111

consent form to participate and not the other way around. Only in one case did we have one student who did not want to participate in the video observations (even though his parents agreed he could participate). The student’s wish was of course fulfilled so the specific student was not video recorded.

The observations did not have a definite focus on specific activities from the KiDM intervention, so some activities were observed more than one time in different classes while others were not observed. In Table 7 the different activities can be seen.

Table 7: Different activities observed in the KiDM classrooms

Activities 1. Trial 2. Trial 3. Trial

Course 1: What do the boxes weigh 4 x 2 lessons

(numbers and fraction)

Course 1: Number board 2 x 2 lessons

(numbers and fraction)

Course 1: Rectangle numbers 1 x 2 lessons

(numbers and fraction)

Course 1: Division of pizza 4 x 2 lessons 1 x 2 lessons

(numbers and fraction)

Course 1: Island 2 x 2 lessons 1 x 2 lessons

(numbers and fraction)

Course 2: Rope triangle 1 x 2 lessons

(measuring and geometry)

Course 2: Chicken farm 1 x 2 lessons 1 x 2 lessons

(measuring and geometry)

Course 2: Square meter 2 x 2 lessons 1 x 2 lessons

(measuring and geometry)

Course 3: Categorisation 1 x 2 lessons

(statistic)

Course 3: Modelling 1 x 2 lessons

(statistic)

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Transcription with NVivo and the coding All the video observations made in this thesis were made into a transcript. The large amount of transcript is, however, not all included in this thesis. Because of the large amount only four central transcribed lessons are included in the Appendix g. The other transcripts are available to see on personal request. The transcriptions are all in Danish.

Transcriptions are, however, not an unproblematic reproduction of the complex video recordings because the spoken and acted language is very different than written language. When carrying out transcription of the observations a lot of information gets lost. It is, for example, difficult to make transcripts of body language, voice prompts, or other mimicking. Moreover, there will also always be an element of interpretation in transcripts because the transcriber makes some choices about what to write and not to write in the process. The transcripts made in this thesis were made by different teacher students or the author of this thesis and when using the transcripts in the analysis it was obvious that it contained minor discrepancies in the transcriptions mode, even though there was a detailed transcript manual to standardise the transcripts and to make sure which details were important to transcribe in order to answer the research questions. A thorough review of all the transcripts to make them more stringent was therefore required. See the transcription guide (in Danish) in the Appendix h.

All the transcripts were afterwards uploaded to the qualitative data analysis program NVivo10. The decision to use NVivo in the analysis of the qualitative data was done mostly because of the program’s ability to keep track of the data and give an overview of the different coding. This also gave us an easy way to make different coding comparisons, for example, to increase the intercoding reliability – like, for example, in Paper I where both Jonas Dreyer and the author of this thesis went through all the data and made the coding individually first before comparing the coding. The coding process is the process of marking segments of data with categories. The coding in the qualitative data was made in different ways in all of the papers.

In Paper I the coding started with an open explorative coding done both by the first and second author of the paper. Before the second coding it was decided to do a more theoretical coding because a specific theoretical model (Latour, 1999) was introduced and also applied in the analysis process.

10 https://www.qsrinternational.com/nvivo/ 113

In Paper II the coding started again with an open explorative coding by the first author. Afterwards this coding was discussed and a second coding was done in cooperation between the first and second author of the paper in an exchange between coding and analyses based on a self-developed theory of different activities in an inquiry-based teaching approach. This coding ended by selecting different cases to represent different elements in the analysis.

In Paper III the coding was done first with an open explorative approach to all the data, after which the focus was on specific concepts (cognitive conflicts). Specific cases were selected and as part of the coding the codes were made into a juxtaposition and it ended with different figures to represent the different coding.

In Paper V the coding was done using a theoretical approach, with a focus on different types of arguments and reasoning (G. J. Stylianides, 2008) in the observed lessons and in the test answers. The data were finally divided into these different categories and compared.

It is obvious here that the ways of coding in these four papers are different but very important for the results and this reinforces again that the results developed are constructions – we choose the way to code and analyse the data. The interpreting of the data is therefore important, and we need to be very cautious because data are doing much more than they show.

Reliability and quality in qualitative studies The goal of this section is to describe how errors and biases in the qualitative research were minimised. To overcome this, at least two things were done:

First, as described above, a double-coding was created in all the papers, which means that the same data were coded by two researchers. This double coding should result in some specific discussions about the codes boundaries. The purpose of this is that it should help to increase the reliability in the results and hopefully increase the consistency in the coding because it adds a little “objectivity” to the research.

Secondly, “Communication validity” was also applied. The concept is described by Dahler- Larsen (2008) who claims that it is about whether a statement from a study can hold in a verifiable dialogue. This can do as a "member check", but in the qualitative studies in this thesis 114

(Papers I and II) the statements from the study were tested through discussion with the KiDM community of researchers and teachers and the study durability was generally recognised. Unfortunately, communication validity was not introduced in Papers III and V as these articles were written and developed during a period when the KiDM group unfortunately did not have more meetings.

Ethics in the qualitative methods Qualitative research, like all other research, is often seen as a value-driven activity that contains some ethical issues, but it also contains some ethical potentials in contrast to quantitative research, where individual characteristics often are hidden behind numbers. Qualitative research works more with private and more subjective aspects of people. This can give rise to ethical issues, but it also allows for more complex issues to be addressed.

Concrete ethical issues can be divided into micro and macro problems (Brinkmann & Tanggaard, 2015). Micro-ethical issues are about taking care of the people who are immediately part of the research, e.g., the students and teachers and macro-ethnical issues are about the location of research in a social context.

In this work we followed 4 rules of thumb in research ethics described in Brinkmann and Tanggaard (2015):

1. Obtain consent from those involved in the research and inform the persons concerned that participation is voluntary: In the research done in connection to this thesis all the participants were informed about the purpose of the study and the teachers participated voluntarily. The persons observed and video recorded all had to give an active consent.

2. Confidentiality in the form of anonymity must always be an option if the participants do not want to be credited with their name mentioned so they get a voice in the research: All the participants were anonymous, and we were aware that the confidentiality also applies between information that was constructed when talking to the students and afterwards talking to the teachers.

3. The consequences of the research should be carefully considered: In relation to micro- ethical consequences, including any inappropriate consequences for the individual participant, we did not find any major issues. Macro-ethical consequences in relation to how the results in the research can be used by lawmakers and, managers in an 115

inappropriate manner can, however, be a problem in this thesis if, for example, the results are used in a way that is skewed in relation to the overall purpose of the study.

4. The role of the researcher is very important in qualitative research: it was very important for us to make the commitments clear in all the papers, what expectations we had of the teachers and the students, because we did not want any misunderstandings in relation to, for example, publications and the website, but also in relation to the fact that there may be elements in the descriptions that may appear as critical in relation to teaching. It is important here to point out that it has never been the purpose of the study to assess teachers, but that it is about understanding the teachers’ choices to focus on development of teaching.

11.5 Quantitative methods Quantitative research is used when the research field can be made measurable. Quantitative methods, in contrast to qualitative methods, weigh up the collection of larger amounts of "hard" data, for example, information that can be measured and quantified (almost) immediately. The quantitative data in this thesis are categorised using statistical methods, and the results are presented numerically. The use of quantitative methods often implies that the researcher considers the field of study as an object which can be examined on one or more variables. In this study the variable is “to be taught in an inquiry-based way” or “not to be taught in an inquiry-based way”, while the idea is that all the other variables have not been affected (see Section 11.2.1). The quantitative methods used in this study are surveying and testing.

Sampling in the quantitative data The quantitative data in this study are derived from the KiDM RCT study which means that the sampling is randomised. The randomising in the project was done in specific strata. Stratified sampling is a technique in which a population is divided into mutually elusive groups called strata and then samples are selected from each group. There were three different strata which were: city or countryside, big school/small school, north, east, or south region. The intention is that when a sample is random it will give a strong claim of cause-and-effect from the experiment.

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The three trials were not randomised in the same way, because there were some problems in the recruitment of schools in trial three. A more detailed description can be found in the KiDM report (T. I. Hansen et al., 2019).

All three experimental trials have been stratified in cluster-randomised school-level trials because the intention were capacity building and development of professional communities. This means that the students in the experiment were randomly divided into an effort and control group - not as independent units - but as groups of units clustered in classes and embedded in schools.

Participating schools received - in addition to the effort, understood as access to a web platform with student and teacher materials, participation in kick-off events, etc. - a compensation of DKK 20,000 per class, the participating control schools continued their usual business and received DKK 2,500 per class they participated in. If the intervention had only included a teaching effort - and should not also support building up professional communities - we would instead have been able to allocate the effort at class level, which would have increased the statistical power in the experiments markedly (see also the results in Chapter 12).

Survey in the KiDM project Survey research is a questionnaire of a particular type, namely a study that involves collecting systematic data across a sample of cases and then do statistics analysis of the results (Olsen, 2005). In questionnaires, questions can either be answered verbally or answered by a self- completion point. The idea is that a survey aims to collect standardised information for the same variable for everyone in the sample. “Researchers use questionnaires so that they can obtain information about thought, feelings, attitudes, beliefs, values, perceptions, personality and behavioral intentions of research participants” (Johnson & Christensen, 2014, p. 192).

Survey-based analysis results are achieved by putting the respondents' answers together, so that they act as indicators of a specific experience. It is important here to point out that survey data are not suitable for covering complex opinion structures, for which more in-depth qualitative methods are required.

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The questionnaires constructed for the overall KiDM project in mathematics were constructed by the author of this thesis in collaboration with Morten Rasmus Puck11. The questionnaires were developed for all the students and all the teachers in both the control and the intervention schools. However, in this thesis not all of these questions are relevant, and we will now focus on the questions that are interesting in relation to the research questions in this thesis.

In the teacher survey only seven questions out of 108 are relevant to this thesis. These questions are mostly about how the teacher experienced the interventions and if they think that the intervention helps students develop reasoning competence in mathematics.

An overview of the specific questions in the teacher survey is presented in the following Table 8.

11 Morten Rasmus Puck is scientific assistant at UCL- University College with specializing in statistical analyzes 118

Table 8: Questions from the teacher survey with a specific focus on reasoning in mathematics

Question Question in Danish Question – translated Different answer categories number into English (A) I hvilken grad arbejder du hen To what extent do you To a very high degree, to a imod, at eleverne: kan work towards the large extent, to some extent, to ræsonnere matematisk? students: can they reason a lesser extent, to a very mathematically? limited extent, not at all (B) I hvilken grad arbejder du To what extent do you To a very high degree, to a henimod at eleverne kan work towards students large extent, to some extent, to engagere sig i en dialog being able to engage in a lesser extent, to a very med andre elever om deres dialogue with other limited extent, not at all matematiske forståelse? students about their mathematical understanding? (C) I hvilken grad arbejder du hen To what extent do you To a very high degree, to a imod, at eleverne: selv kan work towards the large extent, to some extent, to opstille matematiske students: will they be able a lesser extent, to a very problemer/hypoteser? to set up mathematical limited extent, not at all problems / hypotheses themselves? (D) EN dialog mellem eleverne A dialogue between the Totally agree, very much opstår lettere når students arises more agree, agree, disagree, undervisningen er easily when the teaching disagree, disagree completely undersøgende is inquiry-based disagree (E) Hvor ofte gør du følgende i How often do you do the always, always - when matematikundervisningen? following in mathematics relevant, often, ever, rarely, Når I skal til at gå i gang med teaching? When you are never aktiviteterne: lader jeg eleverne about to start the opstille hypoteser eller gæt. activities: I let the students set up hypotheses or guesses/conjectures. (F) Hvor ofte gør du følgende i How often do you do the always, always - when matematikundervisningen? following in mathematics relevant, often, ever, rarely, Når I er i gang med teaching? When you are never aktiviteterne: handler det ofte doing the activities: it is om, at eleverne skal often about the students argumentere og opstille små having to argue and set ræsonnementer. up small reasoning (G) Hvor ofte gør du følgende i How often do you do the always, always -when matematikundervisningen? following in mathematics relevant, often, ever, rarely, Når I afslutter en aktivitet/ et teaching? When you never forløb: snakker vi om kvaliteten finish an activity / course: af de forskellige we talk about the quality løsningsforslag. of the different solution proposals.

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An overview of the specific questions in the student survey is presented in the following Table 9.

Table 9: Questions from the student survey with a specific focus on reasoning in mathematics

Question number Question in Danish Question – translated into Different answer English categories

(A) Jeg deltager altid i I always participate in the always, always - klassediskussionerne i class discussions in when relevant, often, matematik. mathematics. ever, rarely, never

(B) Jeg begrunder altid mine I always justify my always, always - løsninger solutions when relevant, often, ever, rarely, never

(C) Var du ofte med til at diskutere Were you often involved in always, often, andre elevers løsningsforslag discussing other students' sometimes, rarely, solution proposals? very rarely, never

When using surveys in research, a number of different measurement problems can arise, e.g., in connection to validity.

The validity is about whether the measurements actually measure what is appropriate to measure - including whether the relationship between theoretical definitions and operational definitions is equal (content validity), but construct validity is also an important concept to discuss (Olsen, 2005).

The biggest concern here was that we used self-completed internet-based questionnaires in the teacher survey. This can in some way affect how many teachers actually fill out the survey, because some of the teachers are very busy and the challenge is to get the teachers to spend time answering the questions. This concerns the low response rate, even though the teachers actually get paid to do it, but it also concerns sloppy answering. To overcome this, we tried to make the survey very readable and the format and the design pleasant.

Another concern was that the students did not understand the questions at all, therefore we clearly described in the instructions that it was quite ok if the teacher helped the students answer the questions, and at the same time we ensured that all the questions could be read aloud by using a digital reading program.

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Both the teacher and student surveys in KiDM generally consisted of a mixture of open questions (responses formulated by respondents), closed questions (predefined response options) and closed items (statements with predefined response options). The open questions were primarily comment boxes.

In constructing the survey there were different concerns. The biggest was how the different concepts were defined and how they could be operationalised in the survey. That is, to make the measurable without losing the essence of the theoretical definition is very important; however, not something which is widely described in other research (Olsen, 2005). It is in this operationalisation process that content validity must be ensured. The meaning of different words like “reasoning competence” can actually be quite abstract or can be understood very differently from teacher to teacher. This means that apparently "similar" questions might not be answered equally by respondents.

In the construction process different choices were made to make the questionnaire the best as possible and to make sure the students and the teachers understand the questions and remain motivated to answer all the questions.

The rating scales were fully anchored. This was chosen in order to be very clear as to what the rating scale includes. The scales were created as balanced as possible in a 5-point or 6-point rating scale.

The structure and phrasing of the components followed everyday language. E.g., when phrasing the questions and responses, familiar and neutral words, simple syntax and exhaustive and mutually exclusive response options should all be used.

The number of open questions was generally minimised as it would take more time and effort for the teachers to formulate answers themselves than to choose predefined answers.

The length of the questions was also minimised. The response categories were formulated to be exhaustive and mutually exclusive as far as possible. Even though these rules were followed, there could, however, be a concern that a question was misunderstood by the respondents. To decrease this risk, a colleague and the author of this thesis went through the phrasing of all questions and the assisting text, and the “Danish group” in the KiDM project was also invited to discuss the phrasing and questions.

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Finally, the surveys were tested in the development schools and after multiple adjustments, the surveys were tested in the pilot schools, before the surveys were finally used in the three trials.

The biggest advantages of using surveys as methods of data collection is that large quantities of data from many respondents can be collected and analysed in a relatively short time compared to other more qualitative-oriented methods.

Test In Johnson and Christensen (2014, p. 164) testing is defined as “…the process of measuring variables by means of devices or procedures designed to obtain a sample behavior”.

The achievement test used in this thesis is the mathematical test that was used in the KiDM project. It was designed to measure the degree of learning that has taken place after the students have been exposed to the inquiry-based teaching interventions. A detailed description of the development process and the validity of the test can be found in Paper IV. This means that this section will mainly focus on the final test design and some more theoretical aspects, including reliability.

The test developed for the KiDM project includes two tests, a test that measures the fields of “Mathematic concepts understanding” and a test that measures “Inquiry-based competencies”.

There are 54 items in the “mathematical concept” test (concept test) and 23 items in the “Mathematical competence” test (competence test). The number of items were not equally divided between the two tests because the open-ended items from the competence test took the students a longer time to answer, and since the tests each had to take a maximum of 45 minutes, the difference in the number of items emerged.

The test was used in all three trials; first baseline and then endline and it was designed for students from year 4 or 5.

The format of the items was that it included multiple-choice items and close-constructed items which could be auto-scored because the students should calculate a specific number or a fraction, but it also included open-constructed items, where the students needed to write, e.g., a small paragraph with their arguments and their explanations or just make a statement. These were rated by teacher-students in cooperation with the author of this thesis by using a detailed coding guide. The coding guide can be seen in Appendix i in Danish. 122

In the multiple-choice items the students have the opportunity to answer: “I do not know the answer”. We chose this opportunity to make sure that the students have seen the items and reflected that he/she does not know the answer. The items have a mixture of dichotomous and polytomous items; however, the concept test includes only dichotomous items.

In the KiDM project there was a clear interest in having two dimensions in the test; a mathematical concept dimension and a mathematical competence dimension. This was, however, a big challenge, because we expected that there would be a high correlation between these dimensions, since there might be an underlying construct called mathematic understanding. In an item map (see Figure 13) the hardest items are at the top and the easiest items are at the bottom, we can see that the items from the concept test (the blue items) are in general easier than the items from the competence test (yellow items). This, however, supports the idea of keeping two different dimensions.

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Figure 13: Item map from the KiDM test – the higher the item is placed, the higher the difficulty of the item is Comparing the students’ development in an experiment requires the construction of a suitable measurement scale (Van Wyke & Andrich, 2006). This means that the students’ competences need to be recorded and mapped, and the performance of the students who have been tested, to be compared.

Classical test theory is based on the basic idea that the student has a “true score”, that is, a skill within the measured area, but that in a test you will always measure with a certain margin of error in relation to the result. One can say this in the formula: Observed score = true score + error. A good test will, of course, give an observed score close to the “true score”. The challenge is that you cannot know what the true score is, and you must therefore use other statistical methods, e.g., Cronbach's Alpha to estimate the quality of the test.

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However, the biggest problem with the classic test theory is that we cannot compare students' development over time. We cannot know how much a student has developed if he/she gets 10 out of 12 when he/she last got 7 out of 12. Either you do not know if the first task was just as difficult and if you choose to use the same tasks again, you do not know if the student can remember the answers or has talked about the tasks in the meantime. Item Response theory (IRT), however, provides another approach where the idea is that the student’s response to a particular test item is influenced by qualities of the individual but also by qualities of the item.

The KiDM test uses an Item Response Theory, which refers to a system that describes the relationship between an individual response to an item and the underlying trait being measured. In comparison to classical test theory, IRT has some unique properties and advantages for test construction, scoring, ability assessment etc. Fan and Sun (2013) describe four main advantages:

Item parameter estimates do not depend on the particular group of examinees of the population for which the test is developed; Examinee trait assessment does not depend on the particular set of administered items sampled for a population. Statistical information is provided about the precision of the trait estimates and Traditional reliability information is replaced by relevant statistics and their accompanying standard errors. The KiDM test used a Rasch model to analyse the data. The Rasch simple logistic model for dichotomous items specifies the way that the probability (p) of scoring a given category (k) at item (i), with the difficulty (δik) depending on the ability (θ) of the respondent and m is the maximal score in the item. The partial credit model can be written as the following:

( ) ( = k) = 1 + 𝑘𝑘 𝜃𝜃𝑛𝑛 −𝛿𝛿𝑖𝑖(k ) 𝑒𝑒 𝑝𝑝 𝑥𝑥𝑛𝑛𝑛𝑛 m 𝑘𝑘 𝜃𝜃𝑛𝑛−𝛿𝛿𝑖𝑖k ∑k=1 𝑒𝑒

As ability varies, the probability of a correct response to the item also varies. The probability that a person with low ability will respond correctly is correspondingly low, approaching 0 asymptotically as ability decreases. Symmetrically, the probability that a person with high ability will respond correctly is correspondingly high, and approaches 1 asymptotically as

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ability increases. The logit scale routinely set at 50% the probability of success for any person in an item located at the same point on the item-person logit scale (Bond & Fox, 2015).

Many testing programmes, however, require greater precision or more information than a simple right/wrong scoring system allows from any particular item. In these cases, polytomous scored items with several levels of performance may be required, which also were used in the KiDM test.

The whole design process of the test can be read further in Paper IV, e.g., how the test was analysed and checked for disordered thresholds, DIF and FIT.

The KiDM test is assessed on the basis of established criteria that are expressed in the coding manual. The idea in the coding manual is that not only is the correct answer different from the incorrect answer: it also is better than the incorrect answer in a crucially important way. We regard the correct answer as superior to the incorrect answer and we routinely regard children who give the correct answer as showing more ability than those who do not.

To keep the degree of interscorer reliability on a high level as possible we kept training and discussing the borders between the more open and difficult items in the hope of improving the interscore reliability. After every test (pre- or post-test) in the three trials we met and discussed the disagreements.

Thinking aloud - to qualify tests In the development of test items, two think-aloud protocols were used to qualify it further. Think-aloud protocols involve students thinking aloud as they are performing the test. Two students were asked to say whatever came into their mind as they completed the test. This might include what they are looking at, thinking, doing, and feeling. The idea is that this gives us an insight into the participant's cognitive processes (rather than only their final product). In the process, notes were taken of what participants said and did, without any attempting to interpret their actions and words.

The think-aloud process was in two steps: in the first step the interviewer probed the students as infrequently as possible, because students are easily distracted during problem-solving activities. When silent for several seconds, the interviewer merely probed the subject to “keep talking”. Neutral cues such as “keep talking” to encourage the student to think aloud, but not to

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bias the data by adding external ideas to the internal processes of subjects. Once the think aloud process was complete, the second step of the methods was to ask follow-up questions. Such questions may also be useful for the students who were unable to meet the cognitive demands of thinking aloud while problem solving (Branch, 2000).

The think aloud was important in the construction of the items, because it was important that the test was adapted to the year 4 and 5 level, but also to be sure that the items measured what we wanted them to measure. Of course, we also observed some limitations of this method. E.g. one of the students did not have good language competences which meant that we might not have clarified all of the student's thoughts.

Ethical considerations in quantitative methods Ethical considerations in quantitative methods are also based on both micro-ethical and macro- ethical levels. Results from statistical research are often considered to have a high degree of credibility so many politicians would like to exploit these kinds of results. It is therefore important to be aware if the results in anyway can be misinterpreted and used to the detriment of others. On the micro-ethical level there are different perspectives to consider.

First, there is the issue of letting children participate in the study. What does it mean for children to be exposed to both an endline and a baseline test? It is well known that children in Danish schools are already tested a lot - and perhaps too much, so what implication does it have that there are even more tests coming into their everyday life? The intention here has not been to give them yet another skill-based test like the national test, but to challenge the students by giving them a different test so that the students also experienced that tests can be in different forms, and thus should not provide too many negative washback effects.

Secondly, there is also a consideration about getting so many people involved in surveys and testing – to use their valuable time in this research was also part of the considerations before the research started.

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12 Results from the KiDM RCT study

In the KiDM final report (T. I. Hansen et al., 2019) the results from the quantitative study are thoroughly described. In this thesis some of these results are interesting in connection to the overall research question and in this section some of these results will be drawn directly from this report but new results will also be included. The chapter starts with a description of the quantitative measurements in general (Section 12.1) followed by three results’ sections; results from the test (Section 12.2), results from the teacher survey (Section 12.3) and results from the student survey (Section 12.4). The chapter ends with a discussion of these measurements (Section 12.5).

1) The results from the KiDM test are important because they will tell if the students develop their concept understanding and their mathematical competences more during the intervention compared to the control schools. The competence test includes items focused on different mathematical competences; problem-posing/-solving competence, modelling competence along with reasoning competence. It was not possible to separate the different competences to be able to focus only on the reasoning competence. It is, however, still important to see if the test was able to measure any development of the students’ mathematical competences whatsoever. 2) Specific results from the student survey are important because they give some indicators of the students’ experiences of getting taught with an inquiry-based approach and if the results show that the students experience that reasoning competence is more in focus in the intervention schools than in the control schools. 3) Specific results from the teacher survey are important because they show the teachers experiences of teaching inquiry-based and if the teachers experience that, by teaching inquiry-based, they automatically put more focus on the reasoning competence.

12.1 The quantitative measurements Recruitment of schools as well as the distribution of schools for intervention or control was performed according to a specific protocol in trial one and two: all Danish schools were first stratified on the following parameters: School size (number of students), rural or urban school and regional location. Secondly, the schools within each stratum were randomly selected to be

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invited as either intervention or control schools (T. I. Hansen et al., 2019). Trial three was, however, done differently because of recruitment challenges. Instead, Danish schools were invited to sign up, after which the same stratification parameters were used and then randomly assigned to either the intervention or the control school (T. I. Hansen et al., 2019) (sed Section 11.5.1).

The distribution of schools looks like the following:

Table 10: Overview of the participants in the mathematics trials

Trial 1 Trial 2 Trial 3 Pooled Inter- Control Inter- Control Inter- Control Inter- Control vention vention vention vention Schools 14 10 14 12 17 16 45 38 Classes 42 28 44 34 59 50 145 112 Students 725 458 837 630 1064 967 2626 2055 Note: The sum of students counts the number of students who completed the test (base and endline) and student survey.

As seen in Table 10, 4081 students completed the mathematics test and the student survey during the three trials. These students came from 83 different schools. When the randomisation is done at school level, the effect measurements must also be done on the school level and as a result of that, several of our RCT experimental trials are based on very small school samples, and the individual RCT trial themselves do not have sufficient statistical strength to identify small effect size. In trial 1 there were, for example, only 10 control schools and 14 intervention schools. The report of the effect estimates for the individual experimental trials is therefore not that interesting, so in order to be able to make a meaningful statistical test that has the possibility of identifying the effect of the intervention, it is necessary to pool the three trials to increase the precision of the effect estimator. Therefore, the pooled power estimate gives the statistically most accurate measure of the effect of the interventions.

Different kinds of tests were made to control and check the results. E.g., a balance test was made to check that the randomisation was fair, so the intervention schools and the control school did not differ systematically from one another. The balance test compares the intervention and control group to a number of background variables (e.g. age, language, socio-cultural background) and it showed that there was balance between the intervention and control schools 129

on the vast majority of characteristics. Only in relation to age was there a statistically significant difference between the two groups. However, the difference is relatively small and only significant at the 10 per cent level. Overall, there are therefore no significant problems with imbalances in the observed characteristics between the intervention and control group in the pooled mathematical experiment (T. I. Hansen et al., 2019).

Moreover, a fidelity analysis was made using a specific teacher survey which was sent and answered three times during the intervention. In this specific survey (including one survey to each course) the teachers were asked to answer which sub-courses of all 9 sub-courses they had started and finished in full. In total, 99 teachers answered all three surveys. On average, the 99 teachers started 8.0 sub-courses (SD = 1.46). None of the 99 teachers started less than 5 sub- courses and we can see that approximately 72% started 8 or 9 sub-courses. This must be seen in the light of the fact that there were many teachers who stated that there was too much material in the intervention. In the same teacher survey, the teachers were asked to what extent they had followed the teacher's guidance for the individual sub-courses. The teachers had the following options: "not at all", "to a lesser extent", "to some extent", "to a great extent", "to a very high degree" and "complete". The 99 teachers have, on average, completed and finished 6.8 (SD = 4.08) sub-courses out of the 9 sub-courses, where they, to a large extent, to a very high degree or completely followed the teacher's guidance (T. I. Hansen et al., 2019). On the basis of this fidelity analysis, it was concluded by the researcher in the KiDM project that “…the level of implementation of the efforts has been satisfactory” (translated from Danish) (T. I. Hansen et al., 2019, p. 122).

12.2 Results from the tests. The intention with the tests in the KiDM project (described in Paper IV) is to examine the effect of the inquiry-based mathematics intervention for students in years 4 - 5. The effect is tested on the basis of two developed tests. One test measuring the students’ concept understanding, and the other test measuring the students’ more complex competences (the competence test is described in Paper IV). The intention with this two-part effect measurement is to elucidate whether the intervention possibly affects both the students' complex mathematical competencies and their more basic understanding of concepts in mathematics.

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The final number of items in the test was 54 for the concept test and 23 for the competence test. In Figure 14 and Figure 15 we see the distribution of the students' skill (pink) on the respective competence scales and the distribution of the difficulty of the test item (blue). The more to the right you go the higher level the students have and the more difficult the items get.

Figure 14: The distribution of the students and the items in the concept test (T. I. Hansen et al., 2019, p. 116)

Figure 15: The distribution of the students and the items in the competence test (T. I. Hansen et al., 2019, p. 116) 131

In these figures we see that the competence test items (figure 15) are more difficult than the items in concept test because in Figure 14 the students and items are placed roughly opposite each other, while in Figure 15 there is a small group of items that are slightly moved to the right of the majority of the students. This shows that some items in the competence test may be too difficult for the majority of the students to answer correctly.

The results from the tests are shown in Figure 16, where the average effect of the intervention’s study on the competence test and concepts test can be seen. In the figure, both the competence test score and the concept test score are standardised, which means that each goal has an average of 0 and the standard deviation is 1. The effect estimates can therefore be interpreted directly as effect sizes.

The results show that the intervention has a small, positive and significant effect on the students' understanding of the concepts. The intervention leads to the students in the intervention group improving 0.09 standard deviations more on concept understanding compared to the students in the control group. The effect is significant at the 0.05 level. A relatively small effect size of the intervention is consistently found in the trials, where the effect estimate is marginally significant in round 1 (0.09, p <.1), significant in round 2 (0.16, p <.001), but insignificant (0.05, insignia) in round 3 (T. I. Hansen et al., 2019).

We did not find a significant positive effect of the intervention in the students' competence test, which can also be seen in the figure. The pooled effect estimates show that students from the intervention schools improved their competence with 0.09 standard deviations compared to the students in the control group, but the effect is not significantly different from 0. Thus, we cannot rule out that the difference between the intervention and control group’s level of investigation competence is due to pure coincidence. We find the same picture in the three individual trials. Here, the statistical uncertainty is considerably greater, as there are fewer schools in the individual experimental rounds. While the efficacy estimates are fairly consistent and positive, the effect sizes are at the same time small and insignificant in each trial.

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Concept-test

Competence-test

Note: The effect estimate for the experiments’ pooled is based on an OLS regression model with stratum indicator control, baseline competence targets and experimental indicator. Effect estimates for experiments 1-3 are based on OLS regression models with control for the stratum indicator and baseline competence targets. The thin lines show a 95% confidence interval. The default errors are corrected for clusters at school level. + p <.1, * p <.05, ** p <.01, *** p <.001.

Figure 16: The result from the concept test and the competence test in KiDM (T. I. Hansen et al., 2019, p. 123)

Furthermore, a Robustness Check was made, and the results are that the pooled power estimates are quite robust across model specifications. In the models for the students' competence, the

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power estimates in the range are between 0.082-0.142 - though without any of the effect estimates being significant at the 0.05 level. This range in the effect estimates reflects that there is a tendency for some bias in the balance between intervention and control, but without the bias being substantial or statistically significant.

The intention of the competence test in relation to measuring specificity in the reasoning competence was to divide the competencies into different subdimensions, so the reasoning competence had its own dimension. However, this was not possible as after removing many items that did not fit into the overall dimension (see Paper IV), there were basically too few items to be able to develop an independent reasoning competence dimension.

12.3 Results from the teacher survey In the mathematical part of the KiDM project, it was decided to make some quantitative measurements on whether the teachers express that their attitudes towards teaching inquiry- based changed significantly among the teachers who participated in KiDM project compared to teachers at the control schools. A pre- and post- survey was developed, and it included some questions connected to reasoning competence. It is the results of these questions that will be further elaborated and explained in this section. The results are interesting in this thesis because it could indicate whether teachers, after teaching inquiry-based, have changed their views and attitudes about how students develop reasoning competence.

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Table 11: Questions from the teacher survey Questions (response scale in parentheses)

Survey items Label

To what extent are you working towards the students being able to reason A mathematically?

(To a very high degree, to a great extent, to some extent, to a lesser extent, to a very small extent, not at all)

To what extent do you work towards students being able to engage in B dialogue with other students about their mathematical understanding?

(To a very high degree, to a great extent, to some extent, to a lesser extent, to a very small extent, not at all)

To what extent do you work towards the students themselves being able to C set up mathematical problems / hypotheses?

(To a very high degree, to a great extent, to some extent, to a lesser extent, to a very limited extent, not at all)

A dialogue between the students arises more easily when the teaching is D inquiry-based

(Totally agree, very much agree, agree, disagree, disagree, disagree completely disagree)

When you are about to start the activities, I let the students set up hypotheses E or guesses/conjectures

(Always, always - when relevant, often, sometimes, rarely, never)

When you are doing the activities, it is often about the students having to F argue for their process and their results

(Always, always - when relevant, often, sometimes, rarely, never)

When we finish an activity / course, I let students tell their arguments for G their solutions

(Always, always- when relevant, often, sometimes, rarely, never)

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The results from these specific questions can be seen from Figure 17 and the calculations can be found in Appendix j.

Note: The effect estimate for the experiments pooled is based on an OLS regression model with stratum indicator control, baseline competence targets and experimental indicator. Effect estimates for experiments 1-3 are based on OLS regression models with control for the stratum indicator and baseline competence targets. The thin lines show a 95% confidence interval. The default errors are corrected for clusters at school level. + p <.1, * p <.05, ** p <.01, *** p <.001.

Figure 17: Results from the teacher survey

What is obvious here is that no significant effects of the interventions are found in any of the questions, and is it is very clear in the graphical presentation of the results that the statistical uncertainty is enormous. This uncertainty is due to the fact that there very few teacher responses per school and 11 schools out of the 83 schools opted out because their teachers had not answered the survey (see the calculations in Appendices j and k). All in all only 79 teachers from the intervention schools and 50 teachers from the control school answered the survey.

Therefore, instead of discussing what can be derived form these results, the uncertainty is so high compared to these analyses that the results have limited value.

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12.4 Results from the student survey The KiDM project furthermore includes a student survey. The students completed the pre- and post-survey in connection with the completion of the concept and competence tests in both control and intervention schools, so, unlike the teacher survey, the response rate is fine (see Appendix k for further calculations).

In the following Table 12 the question in connection to developing reasoning competence is shown.

Table 12: Questions from the student survey: Questions (response scale in parentheses)

Survey items Label I always participate in the class discussions in mathematics. A

(totally agree, agree, agree, disagree, disagree or disagree)

I always justify my solutions B

(totally agree, agree, agree, disagree, disagree or disagree)

Were you often involved in discussing other students' solution proposals? C

(always, often, sometimes, rarely, very rarely, never)

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The results from these specific questions can be seen in Figure 18.

Figure 18: Results from the student survey

Unlike the teacher survey, here some interesting results are presented. As shown in Figure 18, all three questions have achieved significant results. All of these questions are formulated without directly applying concepts such as “reasoning competence”, as years 4 and 5 students typically do not understand such concepts. Therefore, the interpretation of these results requires a further interpretation.

The students’ experience is that, in a significant degree, the students at the intervention schools more often participate in the discussions in the class (question A). Participating in discussions can of course take place in many different ways, but it can, however, be seen as an indicator that the students have experienced more participation in the discussions and thus also gain more

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experience in discussing in mathematics. Discussion and argumentation are an important part of reasoning in mathematics (see Section 9.1.2.3).

To the question about whether the students experience that they more often justify their solution (question B), we also see a significant difference between the control and the intervention schools. This does not indicate anything about the quality of the justifications, but it does, however, indicate that the students who have participated in the intervention think that they more often justify their answers.

Finally, to the question about wheater the students from the intervention school experience that they are more often involved in discussing other students’ solutions proposals (Question C), a significant difference is found. When students are discussing other students’ solutions or proposals, the students are compelled to try to understand the arguments behind them in order to discuss them and maybe be able to argue against them. This can therefore in many ways also indicate that work is being done on the reasoning competence.

All in all, these three significant positive results could indicate that during the interventions the students work with the reasoning competence by discussing and justifying their results more often.

12.5 Discussing the quantitative results Despite different challenges in the quantitative results, they all suggest a positive outcome from the intervention.

Firstly, the insignificant results from the teacher survey do not in any way suggest that the teachers change in a negative way their views and attitudes towards students developing reasoning competence in an inquiry-based teaching approach. On the other hand, there is also no indication that the teachers change their views and experience in a positive direction. There may be several reasons for this result. Initially, the reason could be that the time interval is simply too brief to measure any development of the teachers’ views and attitudes; but it may also mean that the intervention simply has not affected any change for the teachers. From a retrospective perspective, it would have been desirable for the outcome that some things during the intervention would have been done differently. This is especially current for the teacher survey, where too few teachers have completed the survey and the KiDM group unfortunately

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has not been efficient enough to send out reminders to the teachers to increase the number of participants.

Secondly, the student survey is very interesting as it has a statistically significant result showing that it makes a difference for the students if they are taught in an inquiry-based way and the results indicate that the students experience a larger focus on reasoning competence compared to the control schools.

Thirdly, the developed KiDM test which include both a concept test and a competence test also together show a positive trend. The results from the concept test show a statistically significant development of the students’ concept understanding. However, the results from the competence test do not have significance, although they do indicate a positive development. There might be many reasons why the competence test does not have any significant difference between the intervention schools and the control schools. The most obvious reason might be, that the intervention simply did not change anything in this relatively short intervention. This does, however, contradict the results of the student survey. Another reason could be that the test does not work or is not good enough. However, even though the test is validated in different ways (see Paper IV), different challenges with the test are found and described in Paper IV; most importantly is perhaps that, after calibrating with the Rasch model, the test ended with few items and some of the included items were too difficult for the students, which led to far too many students getting 0 points on the test. Moreover, a challenge is found by using this test in years 4 and 5, because this test includes items where the students need to write down their arguments etc., which some students might find difficult at that age.

The relationship between the concept test and the competence test does also need a reflection, as there will always be a relation between the content of the reasoning process (like different concepts) and the reasoning. You will always have to reason about something. This also means that when the test shows that the students have developed their conceptual understanding, this might later affect their competence to reason in a longer perspective. This relation definitely deserves more research in the future.

Finally, another kind of challenge with the quantitative measurement is that the randomisation has taken place at the school level and the number of participating schools was relatively low in comparison to if the randomisation had occurred at the student level where over 4000 students participated (see further elaborations on these issues in Paper IV).

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All in all, the quantitative results provide different answers to the research questions of this thesis, but by using quantitative methods to measure a complex unit as the reasoning competence is very complicated, and the results clarify that it requires many considerations in the design of the method, but also in the implementation and analysing of processes.

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13 Presenting the contributing papers

The five papers constitute the main findings in this thesis. Three papers (I, II and III) concern only the classroom teaching and the qualitative data and the remaining two papers (IV and V) concern the test used in the KiDM project. All five papers are embedded in the KiDM project. In this chapter the five papers will be presented with an introduction, the methods and analysis and a summary of the main findings along with a discussion. Finally, a section is added, where the findings of the papers are discussed in connection to the overall research question of this thesis and future perspectives.

The five papers are all written in collaboration with different colleagues who are in different positions in their careers; scientific assistant, experienced teacher educators, university professors in mathematics education and from different professional fields; statistics and mathematics education. All these different collaborators affirm the complexity of answering a research question with a mixed approach.

The contributing papers can be found in full in Chapter 14.

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13.1 Summary of Paper I

Title From everyday problem to a mathematical solution—understanding student reasoning by identifying their chain of reference

Authors Jonas Dreyøe, Dorte Moeskær Larsen & Morten Misfeldt

Status Published 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 2018

Introduction: This paper investigates the process from reasoning in everyday problems to reasoning in a more formal and symbolic way. This focus has been central in many papers (EMS, 2011; Harel & Sowder, 1998; G. J. Stylianides & Stylianides, 2009).

The specific intention is to understand the students’ working process and thinking process in an activity where the students need to go from an ‘everyday problem’ to a more formal mathematical solution.

To do the analyses we use an approach that involves developing mathematical reasoning in a continuity, instead of the formalistic division of the world into a mathematical and an empirical world. We used a model called circulating reference developed by Latour (1999). In this model, Latour (1999) takes outset in the often described radical gap between “words and the world” and he argues that there is neither correspondence nor a significant gap. Latour argues that the distinction between words and the world is rather a chain of small fundamental distinctions between references that are both representative and material.

We ask the following question:

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What are the potentials and limitations of using the concept of circulating reference to study the manifestation of mathematical reasoning in student articulations, manipulatives, and representations when solving an everyday problem mathematically?

By using Latour’s model our aim is to combine mathematical reasoning with mathematical work processes and the use of representations in a single model, and in this way overcoming the distinction between deductive and empirical reasoning we see in, e.g., (Harel & Sowder, 1998, 2007) and the distinction between the mathematical the objects and semiotic representations we see in Duval (2006).

Methods and analysis The study is imbedded in the KiDM project and the data used in this article are specific cases from the teaching activity called “What do the boxes weigh?”. All in all, the data include 15 lessons of video observation from non-participant observations from the classroom, all transcribed in full. All observations were then analysed using Latour’s model of chains of references, but because of the scope in a conference paper, only one lesson is described in detail. In this lesson we found six operators which are thoroughly labelled in the paper, where the students go from 1) thinking in contextual narratives presented by the teacher. To 2) adapting the narrative and supplementing it with their own material structure. To 3) adding a written equation. To 4) stop using the boxes in their physical form, but to still remember and use the sizes of the boxes. To 5) reducing the complexity of the physical shape and amplifying the fact that they are entities assigned a weight. And to 6) finally a set of solutions that leads ultimately to the correct combinations of weights.

Figure 19: The chain of references in the case studied Findings and discussion: In the described case we observe that, stage by stage, the students’ representations lose locality, particularity, materiality, and multiplicity, such that in the end, there was almost nothing left but numbers on their paper; instead, however, we see a greater standardisation, calculations,

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relative universality, and formalised knowledge, which is an amplification of the desired properties.

By using Latour’s (1999) thinking, we aimed in this paper to coincide mathematical reasoning and proof with mathematical work in a single model, thus overcoming the distinction between deductive and empirical reasoning and mathematical objects and semiotic representations. Latour’s model has a focus instead on describing the process and provides us with a chain of small fundamental distinctions between the object outside of the mathematical domain and the representations inside the domain.

Overall, we found that Latour’s (1999) model had the potential to focus on the work process and thinking of students’ reasoning in the transformation from an everyday problem to a mathematical solution.

Relation to this thesis research and further perspectives Reasoning in this paper specifically focuses on students' systematic thought processes and not like in the other papers, where the focus is more on students’ different argumentations. The idea is that the students' thought processes cannot be observed directly but only interfered with based on some external manifestations, which therefore was in focus in the observations and analysis.

In relation to the research questions in this thesis, this article contributes with a model that can help with a more nuanced lens on how the students' line of thoughts and reasoning evolves throughout the different inquiry-based activities (RQ1), and it also contributes with a clear description of how the students in the described case reason in the different phases in the inquiry-based learning process. The students start to reason by using the concrete materials, but quietly move to reasoning with more focus on symbols and more formal mathematical aspects.

The idea that Latour’s model has potential to focus on the students’ thinking and reasoning processes is interesting in a future research perspective, because it now leaves us with a lot of further questions like what happens when/if the students go back and forth in the chain? Is it positive always to go forward in this chain or why do students go back? Furthermore, it is possible to think of some future implication in teaching; It might be ideal for the teachers to discuss the students’ chains in the whole class discussion and, for example, compare the different chains instead of just showing a new chain.

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13.2 Summary of Paper II

Title Undersøgende aktiviteter og ræsonnementer i matematikundervisningen i matematikundervisningen på mellemtrinnet

Translated into: Inquiry-based activities and reasoning in primary school mathematics classes

Authors Dorte Moeskær Larsen og Bent Lindhardt

Status Published in:

MONA, Matematik-og Naturfagsdidaktik -tidsskrift for undervisere, forskere og formidlere, 1- 2019, p. 7-21

Introduction Despite a great focus on inquiry-based teaching in mathematics teaching, our experience from dialogue with practice (Dreyøe, Michelsen, et al., 2017) is that many teachers still experience teaching inquiry-based as too complicated, risky and too unpredictable and therefore do not get involved (Michelsen et al., 2017). In the KiDM project the idea was therefore to try to dissect inquiry-based mathematics into smaller, more manageable units. A categorisation of five different inquiry-based activities was therefore developed in the KiDM project which is described in this paper. Two of these five inquiry-based activities are selected, nuanced and studied in relation to the students’ way of reasoning based on how the dialogue between students and teachers appears in the whole class discussion. Our hypothesis is: there is a difference between the students' way of reasoning depending on which inquiry-based activity is in focus in the teaching, and that this may have implications for how a teacher should do the final class's discussion.

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Methods and analysis To study the different kinds of inquiry-based activities, classroom observations were performed in five different classes that all worked with the KiDM material. The observations were video recorded and transcribed in full. The data coding began with an open and investigative approach to the entire dataset, focusing on students' argumentations in the whole class discussion. The data were studied and general trends were discussed. Then codes were developed on all the found arguments, first based on the content of an argument described by Stylianides (2007) (foundation, formulation, representation, social) and then based on descriptions of proof schemes (external, empirical or analytical) (Harel & Sowder, 1998, 2007). Selected lessons were coded together, and some cases were coded individually by the first author followed by a joint discussion of the found coding. The study uses case study as the overall research methods.

Findings and discussion The paper has a detailed description of how inquiry-based teaching is defined and implemented in the KiDM project and the five different inquiry-based activities are described and exemplified.

In Table 13 an overview of what is defined as open, closed or known in the five different activities can be found.

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Table 13: overview of the different types of activities in inquiry-based teaching from the KiDM project

Activity Inquiry- Perspective Problem Methods Results Inquiry- based idea based aspects

The Re- To test and Teacher Known Known Known intervention derive conceptual Student Closed Open Open To (in Danish: understanding investigate Opdagelsen)

The brooder To understand Teacher Known Known Known a problem and (In Danish: to find a Student Open Open Closed To Grubleren) solution investigate

The product Wondering Teacher Known Known Known about function (In Danish: or aesthetics of Student Closed/ Closed/open Closed/ To explore produktet) a product. Possible open open changes and personalisation

The A "scientific" Teacher Known Known Unknown Measurement study of something Student Closed Closed Open To (In Danish: through investigate measurement Målingen) and calculation

The Modelling Develop and Teacher Known Unknown Unknown test (unknown) (In Danish: mathematical Modelleringen) models and Students Open Open Open To describe and investigate analyse reality

The two cases selected – a case where an activity defined as a “Re-intervention study” is studied and a case where an activity defined as a “Brooder” is studied.

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In the cases of the “Re-intervention”, we found that the teacher always has an important goal/point that she/he wants the student to be convinced about. This is also very clear in the whole process in connection to students’ reasoning processes. This determination implies that the teacher's zeal for achieving the right answer/point can result in the teacher becoming the active argumentative part instead of the students. In the cases observed, we saw that it is often the external proof scheme (the teacher’s arguments) that determines the doubt and accepts the argument. The dialogue therefore gets a focus on "right and wrong" rather than that the students themselves being given the opportunity to build a chain of arguments that are tested and evaluated in the classroom.

In the case observed of the "Brooder", the whole class discussion focuses more on the students' explanation of their process. Students express many different types of arguments in this explanation, ranging from very subjective attitudes to more analytical approaches. The final answer to the task (the results), is, however, not argued for in any of the observed cases.

In general, we found that by dividing inquiry-based teaching into smaller more operational elements helped to study more nuances and detail in the understanding of which reasoning process happens in the classrooms. It might furthermore, if presented to teachers, have potentials to help teachers have a greater overview of how to teach in inquiry-based teaching, and it illustrates that the choice of activity in the inquiry-based teaching has consequences in terms of pitfalls and constraints and how to plan the lesson.

Relation to this thesis research and further perspectives The findings from this paper contribute to the overall research question in this thesis by delivering a study of students’ reasoning activities in different types of inquiry-based activities. The findings are that by differencing and nuancing the inquiry-based activities it helps to get a more nuanced picture of how the students work with reasoning competence in inquiry-based teaching. It is obvious that the students’ reasoning competence can develop in different directions depending on which activities are in focus and this paper indicates which direction it is possible to follow if the teacher uses two different types of activities (RQ1). It could be interesting in further research to analyse all these different types of activities empirically and compare with the developed scheme, so as to see if, in any way, an even larger pattern can be seen in what is “open” / “closed” / “known” / “unknown” in the different

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activities and which reasoning processes come into focus and perhaps arise to an even greater understanding of how an inquiry-based teaching approach can impact students’ reasoning competence in primary school mathematics classes.

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13.3 Summary of Paper III

Title Fostering mathematical reasoning in inquiry-based teaching – the role of cognitive conflicts

Authors Dorte Moeskær Larsen and Morten Misfeldt

Status Submitted to

Nordic Studies in Mathematics Education (NOMAD)

Introduction Students reasoning competence is often seen as fundamental for doing mathematics (Ball & Bass, 2003; Carpenter et al., 2003; Hanna & Jahnke, 1996; G. J. Stylianides & Stylianides, 2008).

In the literature, there is a clear distinction between reasoning in mathematics and reasoning in everyday life (Harel & Sowder, 1998) and most students in primary school argue by using empirical observations from everyday life. Research has shown that changing this way of making arguments to more formal deductive arguments is highly non-trivial (EMS, 2011). However, how to make the transition from empirical to more formal deductive argumentation is still open for further research (EMS, 2011). Additionally, inquiry-based teaching is often seen as potential in mathematics education (Artigue & Blomhøj, 2013), and in studying how the relation is between inquiry-processes and reasoning processes, we realised that cognitive conflicts play an important and sometimes positive role in bringing the students from open exploration towards more directed exploration and reasoning. We therefore ask the following question:

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What characterizes productive cognitive conflicts in students independent work with mathematics in inquiry-based teaching, and in which way does these productive cognitive conflicts relate to students’ mathematic reasoning process?

Methods and analysis Video observations from 14 lessons from one year 5 KiDM class was carried out. All the observed lessons were transcribed in full. The camera followed one group of students in each lesson. Each lesson is analysed focusing on finding important productive conflicts. The presented analysis consists of one selected episode from a double lesson where a group of students works with an activity called “rope triangles”. Figure 20 shows the students’ reasoning process from the “rope triangle” activity and the blue boxes indicate the environment and artefacts that advocates for the emerging cognitive conflicts. The cognitive conflict can in many ways here be seen as the explanation to why the students get further in their reasoning process.

Figure 20: The reasoning process in the "Rope triangle" activity

Findings and discussion The findings indicate that cognitive conflicts exist in the analysed inquiry-based 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 152

reasoning process, where the environment and the different artefacts have a role of retaining the conflicting positioning making them available for discussion and inquiry. In the “rope triangle” activity the process of resolving cognitive conflicts is stretched over time. 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 realize the need to resolve these conflicts.

Relation to this thesis research and further perspectives In relation to the overall research question, this paper shows how inquiry teaching by using different materialities and artefacts helps the student resolve different cognitive conflicts which has a great potential for developing the students reasoning competence.

In a future perspective, it could be interesting to develop and plan new inquiry-based teaching lessons where this potential is specifically considered already in the planning.

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Summary of Paper IV

Title Developing a Validated Test to Measure Students’ Progression in Mathematical Reasoning in Primary School

Authors Dorte Moeskær Larsen og Morten Rasmus Puck

Status Submitted for publication to:

International Journal of Education in Mathematics, Science and Technology

Introduction A hindrance for students developing their reasoning competence in schools is that if the students are assessed only with a focus on procedural knowledge and not on reasoning competencies, then the students will regard this competence as not crucial for the subject. This also concerns achievement tests (Resnick & Resnick, 1992), and therefore more research is needed in this area. The intention of the presented study is: to develop, present and discuss the reliability and validity of a new competence test which among others, should be able to measure the reasoning competence in primary school classes.

Methods and analysis The purpose of the developed test was to assess if the students developed their mathematical competencies related to the inquiry-based teaching intervention. These competences were not only reasoning competence, as they also included the competence of problem solving/posing and modelling competence. However, in this study the focus is only if the test is valid in connection to the part of the test connected to the reasoning competence. The test uses an item response theory (Rasch model) to analyse the empirical data, which is a test scale that allows the researchers to use the respondent’s scores and express the respondent’s performance on a scale that accounts for the unequal difficulties across all test items. The development process was divided into three phases; 1) design of the test, 2) developing the test and 3) testing the test. All phases are thoroughly described.

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The definition of reasoning competence which can be seen as the content part (what to measure) used in the test is a bringing together of definitions from NCTM standards, Niss and Jensen (2002) and G. J. Stylianides (2008). The items are a mix of multiple-choice and close- constructed response questions as well as open-constructed problems, where the open- constructed items need to be manually scored using developed item-specific coding taxonomies developed with inspiration from proof schemes (Harel & Sowder, 1998). 20% of the manually coded data was double coded in order to measure the intercoder reliability which was found to have an 82% consistency, which is considered reliable compared to, e.g., the TIMSS study.

The number of test items was relatively small because it was not possible to include too many different open-ended items, because ethically it is not acceptable to expose students in years 4 and 5 to overly lengthy tests. The person separation index will therefore be low in the test and could have been increased by expanding the number of items. To measure content validity, two think-alouds and an exhaustive study of the test answers from the pilot schools were done. To validate the construct, a monotonicity test, a test for unidimensionality, a test for response dependence and a measure about differential item functioning were conducted, but analysis of different coded item maps was also done. An item map shows the relation between the difficulty of the items and abilities of the students measured on the same scale. All in all, altogether these different aspects were found to be not completely optimal for the purpose, although not problematic.

Findings and discussion In general, the different measurements and calibrations of the developed test provided some evidence that the instrument was measuring in a way that matches the theory. However, some problems with the low number of items were also problematic for the results. In the final result when using the test in the KiDM project, the test was, however, not able to measure any positive effect on the students’ mathematical competences in the intervention school compared to the control schools, which is statistically significant at the 0.05 level. This can, however, be due to many different causes, among others, that the KiDM interventions did not cause any specific development in the mathematical competencies in the four-month intervention or because the randomisation was carried out at school level which means that cluster-corrected standard errors were applied, and the number of schools was only 38 intervention schools and 45 control schools. However, it might also be caused by the fact that the test during the validation ended 155

with only 23 items that fitted the model and that the items were relatively difficult for the majority of the students.

The final conclusion is that the test has some possibilities of measuring reasoning competence, but it, however, also has some limitations and challenges.

Relation to this thesis research and further perspectives This paper is relevant in connection to both the sub-research questions, but the focus is on answering how it is possible to measure students’ development of reasoning competence in an inquiry-based teaching approach. The final answer is, however, not simple, because the test does not measure any significant development in the KiDM intervention. The findings, however, indicate that no simple test can measure reasoning competence, and that there is a great need to carry out more research in this direction. The Rasch model seems to be of great value in this direction, because it in many ways helps to validate the test (construct validation) and to build on empirical mapping of the test items. This paper, however, gives some indication of how the students develop reasoning competence in an inquiry-based approach, when the theoretically developed taxonomy is compared to the empirically developed item map from the test. These analyses show that items including mathematical argumentation are placed on the item-map to be very difficult compared to other items without argumentations.

It would undoubtedly be interesting if the KiDM test here could be further developed – by first developing several more items and then be tested again in a context where the students do not first have to go through an hour-long multiple-choice test and an hour-long questionnaire before going through a competence test, so that it will be ethically reasonable for the students to be exposed to several more test items.

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13.4 Summary of Paper V Title How argumentation in teaching and testing of an inquiry-based intervention is aligned

Authors Dorte Moeskær Larsen, Jonas Dreyøe and Claus Michelsen

Status Submitted to:

Eurasia Journal of Mathematics, Science and Technology Education

Introduction Reasoning is an important aim in teaching mathematics today, but mathematical tests, however, often do not measure all aspects of mathematical reasoning, which have an influence on what is taught and what is learnt. Therefore, the alignment between the intended aim, what is taught in the classroom and what is tested, is very important.

An aim in KiDM was to develop students’ competences in mathematical argumentation by taking an inquiry-based approach. The focus in the study is on whether and how the alignment between the KiDM intervention and the KiDM test occurs with a focus on argumentation in mathematics, and we ask the following question: How is argumentation in teaching and testing of an inquiry-based intervention in the mathematics classroom aligned?

Methods and analysis In the study, mathematical argumentation is defined as a process of a connected sequence of assertions intended to verify or refute a mathematical claim (A. J. Stylianides, 2007). Arguments can be divided into non-proof arguments and deductive proof arguments and distinguish between two kinds of non-proof arguments: empirical arguments and rationales (G. J. Stylianides, 2008). A. J. Stylianides (2007) defines four elements of importance to consider when analysing whether arguments qualify as proof or non-proof arguments. This includes the four following aspects: the foundation of the arguments, the formulation of the arguments, the representation of the arguments and the social dimension. These four elements have been used

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to analyse both students’ arguments enacted in the classroom and students’ arguments enacted in the test answers.

Data applied in this study include video recordings from observations in KiDM classes, but they also include test answers from the open-constructed response items from the pre-/post-test.

In the analysis process all the arguments/episodes from the observed lessons and the different arguments from specific selected test items were analysed on the basis of the four major elements of an argument. In the paper not all the analysed classroom observations and test answers have been included, instead two test items and three classroom observations were selected. The selected items and the three classroom observations together represent all the various aspects of the different coded argumentations.

Findings and discussion When comparing the observed arguments in the test and in the classroom some differences have been observed but also some alignments. A summary of the observations can be seen in juxtaposition in Table 14.

Table 14: Juxtaposition of the argumentation in teaching and testing

ALIGNMENT Arguments in the classroom Arguments in tests Representation Verbal and body language Written on a computer/Tablet Everyday language with little use of Everyday language with little numbers use of numbers Formulation Feelings, rationales, empirical or Feelings, rationales, empirical or logical deduction logical deduction Calculational discourse + teacher- Calculational discourse induced conceptual discourse Foundation Definitions, concepts, observations Quantity of numbers, definitions, concepts Social Discussion and negotiation with Individual students dimension teachers and other students

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The classroom and the test also indicate a difference between whether the students’ argumentations are more generalised. In some of the classroom discussions we see that the teacher asks questions that makes the students present generalizations, whereas in the test situations all the outcomes are focused on calculations. This is based on Paul Cobb (1998) distinction between calculational and conceptual orientations in teaching.

In the end, the intellectual honesty principle and the continuum principle, described by A. J. Stylianides (2007), are also discussed in relation to these years 4 and 5 students’ performances along with the mutuality between the four major aspects from A. J. Stylianides (2007).

Finally, the article concludes with a discussion of whether the biggest problem about measuring alignment is that in the test the tests answers are products, while in the classroom teaching the arguments are stated in a process. The study points out recommendations to put more focus on how to align tests to teaching approaches, and to put more focus on how to teach argumentation in classroom settings.

Relation to this thesis research and further perspectives This paper contributes to answering the sub-research question (RQ1), because it problematises the possibility of testing developments in students’ reasoning competence in primary school mathematics classes with a focus on alignment. It is clear that it is not simple to test competencies in mathematics, because the analysis shows that there is not complete alignment in how argumentations in classrooms are enacted and how the argumentations are enacted in the test situations. Therefore, despite the validation of the test from Paper IV, the findings in this paper state the difference between how the reasoning competence is applied in the teaching and the way in which it is actually measured, which is problematic in different ways.

First, it is problematic for the outcome of the interventions project, because the test results do not give a completely correct picture of whether the students have actually developed their reasoning competence, since it is not that which is being tested.

Secondly, it is problematic because of the backwash effect in connection to the teachers’ understanding of the aim of the reasoning competence, but also in connection to the students’ understanding of what is important after the intervention.

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In future perspectives it would be interesting to see whether the model from Table 14 could have been used and involved in making revisions to the test.

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14 Contributing Papers

The following papers are included:

Paper Dreyøe, J., Larsen, D. M., & Misfeldt, M. (2018). From everyday problem to a 1 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

Paper Larsen, D. M., & Lindhardt, B. K. (2019). Undersøgende aktiviteter og 2 ræsonnementer i matematikundervisningen på mellemtrinnet. MONA- Matematik-og Naturfagsdidaktik, (1) p.7-21

Paper Larsen, D.M. & Misfeldt, M. (2019). Fostering mathematical reasoning in 3 inquiry-based teaching – the role of cognitive conflicts, Manuscript submitted for publication in Nordic Studies in Mathematics Education (NOMAD)

Paper Larsen, D.M., & Puck, M.R. (2019) Developing a Validated Test to Measure 4 Students’ Progression in Mathematical Reasoning in Primary School Manuscript submitted for publication in International Journal of Education in Mathematics, Science and Technology

Paper Larsen, D.M., Dreyøe, J., & Michelsen, C. (2019) How argumentation in teaching 5 and testing of an inquiry-based intervention is aligned. Manuscript submitted for Eurasia Journal of Mathematics, Science and Technology Education

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14.1 Paper I

Title From everyday problem to a mathematical solution—understanding student reasoning by identifying their chain of reference

Authors Jonas Dreyøe, Dorte Moeskær Larsen, & Morten Misfeldt

Status Published 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 2018

This version of the paper is the original document in the publication.

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FROM EVERYDAY PROBLEM TO A MATHEMATICAL SOLUTION — UNDERSTANDING STUDENT REASONING BY IDENTIFYING THEIR CHAIN OF REFERENCE  Jonas Dreyøe1, Dorte Moeskær Larsen2, & Morten Misfeldt3  Aalborg University1,3 & University of Southern Denmark2

This paper investigates a group of students’ reasoning in an inquiry-oriented and open mathematical investigation developed as a part of a large-scale intervention. We focus on the role of manipulatives, articulations, and representations in collaborative mathematical reasoning among grade 5 students. In this analysis, we apply the idea of the chain of reference from the studies of Bruno Latour (1999) to the exploration, generation, and formalization of scientific knowledge. This framework allows us to combine knowledge from mathematics education about language and representations, manipulatives, and reasoning in a way that allows us to follow the material traces of students’ mathematical reasoning and hence discuss the possibilities, limitations, and pedagogical consequences of the application of Latour’s (1999) framework. THE MATERIAL TRACES OF MATHEMATICAL REASONING This paper concerns how to understand students’ work process and thinking in the course of progressing from an everyday problem to a formal mathematical solution in the mathematics classroom by focusing on students’ material artefacts, articulations, and representations and the transformations between them. We aim to emphasize that the division of the world into a mathematical and an empirical world is problematic, too formalistic, and out of step with how students learn and make sense in the mathematics classroom. We propose an approach that involves developing mathematical reasoning in continuity with students’ everyday understanding and empirical inquiry, rather than as a categorical distinction between everyday knowledge and mathematical knowledge. To do so, we utilize Latour’s (1999) description of the process that generates discursive scientific facts from investigations of material reality: circulating reference. Latour (1999) takes outset in the radical gap between “words and the world” that he argues is hardwired into western philosophy and must be reduced. He suggests there is neither correspondence nor a significant gap, but instead he presents a new model: chain of reference (Latour, 1999). Using this model, he argues that the distinction between words and the world is not a fundamental gap, but rather

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a chain of small fundamental distinctions between references that are both representative and material. Materials, manipulatives, and utterances are all crucial elements for mathematical reasoning (e.g., Pimm, 1987; Duval, 2006; Defreitas & Sinclair, 2014). According to Duval (2006), mathematical objects cannot themselves be perceived or observed directly with instruments, but only by using signs and semiotic representations. His research therefore often focuses on the semiotic representations mobilized in mathematical processes. Though his research has given attention to troubles in the conversion of representations, he does not elaborate on the process of developing these mathematical representations. In the literature, we also find a distinction between reasoning in mathematics and reasoning in everyday life. An example of this is Harel and Sowder’s (1998, 2007) distinction between empirical proof schemes and formal proof schemes. The majority of students in elementary school have empirical proof schemes, and research has shown that changing students’ empirical schemes to formal proof schemes is highly non-trivial (Education Committee of the EMS, 2011). Nevertheless, the change is an important process in mathematics education. Unfortunately, this literature does not elaborate on how this change takes place. Using Latour’s (1999) thinking, we aim to combine mathematical reasoning and proof with mathematical work processes and the use of representations in a single model, thus overcoming the distinction between deductive and empirical reasoning and mathematical objects and semiotic representations. This leads us to the following research question: What are the potentials and limitations of using the concept of circulating reference to study the manifestation of mathematical reasoning in student articulations, manipulatives, and representations when solving an everyday problem mathematically? METHODOLOGY We employ a microethnographic design, as such an approach is well-suited for describing, analyzing, and interpreting a specific aspect of a group’s shared behavior, beliefs, and language in a specific setting (Creswell, 2014; Garcez, 1997). The study is embedded in a three-year design-based research, mixed methods, and randomized controlled trial program (n = 177 schools) that includes the development of a didactical design for inquiry-based learning over a four-month period. To achieve the aim, we employ a case study wherein we study one of the collected cases of the teaching activity called “What do the boxes weigh?” In the analysis, we conduct a within-case analysis and develop a case description (Yin, 2002). 164

Data collection and coding The data used for constructing this case include field notes from non-participant observations, interviews, and audiovisual materials. To broaden our understanding, we visited four schools three times to observe specific classrooms for 90 minutes, subsequently interviewing students. The corpus of this paper is one of the observations and interviews conducted. In collecting the data, we focused specifically on one student per visit. The students were chosen by the teacher as being industrious and emotionally robust. These selection criteria were chosen to ensure the students would work with the assignment and not be overwhelmed by the researcher following them and their work. The 15-to-35-minute unstructured interviews were conducted immediately after the observation as one-on-one interviews (Creswell, 2014) to allow the students to elaborate and reflect on their choices of materiality with their representations and reasoning fresh in their memory. They were asked questions, such as “Why did you express this in that way?” They were also shown alternative representations and asked in what ways they differ from their own. We also collected audiovisual materials by video recording the teaching activities in full, by following the students and taking pictures of their work, and by recording the interview. All of the recordings have been transcribed in full. Regarding data coding, we took an exploratory approach by reading the data end- to-end several times and discussing general trends. After, we developed codes based on the theoretical framework utilized in this paper (Latour, 1999). The first and second authors of the paper then coded the observation and interview individually and thereafter compared and discussed the coding with the last author. Individual “double” coding was conducted to avoid subjective bias in the analysis and to increase the inter-coding reliability (Russell, 2018). CHAIN OF REFERENCE In this section, we present the theoretical basis for the paper, which serves as a means to describe holistically the development of representations in a material form. Latour (1999) distinguishes between words and the world, but he does not perceive them as “...disjointed spheres separated by a unique and radical gap that must be reduced through the search for correspondence, for reference, between words and the world” (Latour, 1999, p. 69). On the other hand, in Latour’s (1999) framework, matter is at one end and form at the other. This is disjunct from the stage that follows by a gap, which he claims no resemblance can fill. This means that knowledge is not located in the mind when the subject is confronted with an object and that a reference in a language does not designate an object by means of words verifying the object.

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These elements, formed by matter, form, and a gap, are called operators. These operators are linked in a reversible chain. The essential property of a chain is its traceability, allowing you to travel in both directions. Latour (1999, p. 69) describes it as follows: “If the chain is interrupted at any point, it ceases to transport truth — ceases, that is, to produce, to construct, to trace, and to conduct it.”

Figure 1: Elements of an operator and chain of reference (Latour, 1999, p. 70). This conception of reference has no limit at either end, as opposed to the traditional understanding of the two disjunct spheres. This chain of reference can be extended at both ends by adding operators. However, it is not possible to cut the chain or skip any of the operators. To comprehend the chain of reference, Latour (1999) introduces two new terms: reduction and amplification. These two processes occur while moving from one operator to another. There is a reduction in locality, particularity, materiality, multiplicity, and continuity when moving from one operator to another. While reducing some parts of the operator’s complexity, other parts are amplified. The amplification regards compatibility, standardization, text, calculation, circulation, and relative universality. This means in the end, we are able to understand and explain the dynamics of a system, but we are also able to trace back to the complexity we meet at first. Latour (1999) depicts this reduction and amplification in a model in which two triangles overlap. In this tradition, it is never possible to understand a phenomenon fully, as it grows from the middle toward the ends, and there are no ends to the amplification and reduction we can perform (Latour, 1999). THE CASE OF AMY AND DAN In the following, we present the case of Amy and Dan working with the activity “What do the boxes weigh?” This task concerns six boxes that have been weighed in pairs. All the paired combinations of boxes weigh 6, 8, 10, 12, 14, and 16 kg, respectively. The students are to determine the individual weights of the boxes by inquiry. To understand the chain of reference and to grasp the dialectical processes of the amplification and reduction of the complexity of the representations that characterize each stage in the reasoning process, we will investigate and describe the chain of operators (Latour, 1999). First operator: Boxes and weights 166

In this case, the teacher begins his narrative by telling a contextual story in which he is in need of help because he has four boxes that fell out of his car earlier that morning. The problem to solve is the need to determine what each weighs separately. He brings four different-sized boxes into the classroom, which he showed the class. The boxes were labelled A, B, C, and D. He tells the class explicitly that these boxes are empty. Additionally, he writes on the board that the paired weights of the boxes are 6, 8, 10, 12, 14, and 16 kg, respectively. This denominates the first operator: the empty boxes in their physical form and the paired weights written on the board. Second operator: Boxes and drawings The two students, Amy and Dan, begin the task by drawing the boxes side by side onto Amy’s paper and writing the names of the actual boxes — A, B, C, and D — inside the drawn ones. At the bottom, Amy writes the paired weights. While drawing the boxes, Amy says, “Ehhh…It is just to be able to remember that two of the boxes’ weights combined is this; if not, we would be completely confused.” While trying to figure out how to solve the task, the students still use the actual boxes. Matilde keeps pointing at them, and Dan goes to the boxes and tries to weigh them by hand. By writing the names (A, B, C, and D) of the boxes and drawing them on her paper, the second operator begins, which seems an adaptation of the teacher’s presented narrative using the same names for the boxes and the same boxes. However, now they have added a new representation of the boxes on Amy’s paper. Third operator: Boxes, drawings, and equations Meanwhile, Amy sits at the table talking to Dan across the classroom. Amy says, “So, if we say that…if we take A and B — they do not look very big (...) [long pause] — but I do not think it’s those that weigh six; I would imagine four or something.” Subsequently, Amy notes the agreed-upon resulting weight, i.e., writing A + B = 4. This representation thus marks a shift in the operators, where the students write the agreed-upon paired weights as an equation onto their paper. Thus, the third operator is the boxes in their physical form, the weights on the board, the drawings of the boxes, and the addition of the boxes’ names equaling the paired weights assigned. Fourth operator: Drawings and equations The teacher approaches Dan, as he is still by the boxes and weighing them by hand.

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Teacher: (...) you can’t just weigh them by hand, I already told you. See if you can explain to me how. (...) But would it make sense to try and choose some number and then see if you can combine your way out of it? Dan: No [shakes his head]. (...) Teacher: Okay, maybe it could be so that one of them has a weight of one and another one with a weight of five. [Dan nods and looks at Amy over his shoulder]. Good, but then try to think more about that. (...) Now we need to know what the last two boxes weigh, but we should be able to make the combination to get all of these up here [the weight results on the blackboard]. Try to talk it over with Amy. They now stop using the boxes in their physical form; instead, they use the drawings of the boxes on Amy’s paper and write the paired weights on the paper as well. Amy: C and A are roughly the same size. Dan: Maybe A weighs 3 kg. Amy: And then C could weigh — well they are pretty much the same size. The situation when Dan speaks to the teacher leads the students to shift operators, as Amy and Dan discuss after what the teacher spoke about to Dan. They then exclude the boxes in the physical form from their reasoning. The fourth operator thus consists of the equations of the boxes, the drawings of the boxes, and the notation of the paired weights. However, even though the drawings of the boxes are the same size, it is still important to Amy that the large boxes weigh more than the small boxes. Thus, the sizes of the boxes are remembered, even though they are not depicted in the representations used in the operator. Fifth operator: An attempted solution While working with the drawings and equations, they stop mentioning the sizes of the boxes and start to think about them as numbers instead. Amy: Perhaps we have to think about which numbers we can get before we write them down. If I try, I will just make another one with A, B, and C instead of erasing. Okay? Dan: So, we need to swap [some of the numbers]. Amy: I think we need the number four... because we have already used it three times. Amy and Dan now try a solution with four different weights by writing them below the drawn boxes. However, the solution is incorrect. While working with this operator, the sizes of the boxes in their physical forms are still an obstacle for them in solving the task. The reduction of complexity offered by this operator, however, helps them to focus not on the sizes of the physical shapes, but instead on the amplified attributes: that they are each an entity 168

needing to be assigned a weight that can make up the paired weights, as identified by the task. The fifth operator is the concrete example that did not work. Sixth operator: A set of attempted solutions Amy and Dan try two additional sets of attempted solutions that differ from the first, and the fourth solution is one of the two correct solutions. The sixth operator evolves from the fifth operator by creating a set of attempted solutions as a structure for systematizing, enabling them to eliminate wrongful combinations. This systematization of more than one different set is thus the sixth and final operator. This systematization structure helps them to identify the result quickly, as they are not repeating wrong earlier sets. Amy and Dan’s chain of reference The focal point and first operator of the chain of reference is the contextual narrative presented by the teacher. In the second operator, the students adapt the narrative and supplement it with their own material structure. The third element in the chain involves an addition where the equations are written, thus enabling them to remember what they have done. The fourth operator requires the students to stop using the boxes in their physical form, but to still remember and use the sizes of the boxes. In the fifth operator, the sizes of the boxes are entirely uninvolved, thus reducing the complexity of the physical shape and amplifying the fact that they are entities assigned a weight. The fifth operator is thus the one attempted solution that did not work out. The sixth operator is a set of attempted solutions that enables them to eliminate already tried and unsuccessful solutions and that leads ultimately to the correct combinations of weights.

Figure 2: Amy and Dan’s chain of reference for the task: “What do the boxes weigh?”. Based on the case, it seems apparent that for Amy and Dan, it was imperative that they, disregard the importance of the boxes’ sizes, namely overcoming a gap in the chain of reference. This structured and transparent transgression to the more abstract seems necessary to be able to generate cognitively an overview of the problem. This insight would be hard to explain without the framework of circulating reference. DISCUSSION AND CONCLUSION Using Latour’s (1999) thinking, we aimed in this paper to coincide mathematical reasoning and proof with mathematical work in a single model, thus overcoming 169

the distinction between deductive and empirical reasoning and mathematical objects and semiotic representations. Research focusing on student reasoning (Harel & Sowder, 1998; Education Committee of the EMS, 2011) describes the importance of understanding deductive proof schemes. However, Latour’s (1999) model does not make any distinction between different proof schemes, instead focusing on describing the process. Latour’s (1999) model allows us to acknowledge the quality of the specific arguments and systematizations, without matching them to normative categories, such as deductive and inductive. Latour’s (1999) model provides us with a chain of small fundamental distinctions between the object outside of the mathematical domain and the representations inside the domain. By utilizing Latour’s (1999) model, it is thus possible to gain a broader insight into how the transformation from everyday object to mathematical solution takes place. In the described case of Amy and Dan, we found that by using manipulatives, articulations, and representations and by making transformations between them, the two students created a chain of reference to go from an everyday problem to constructing a mathematical solution. In our data, we have observations from three other classrooms working with the same problem, and in our analysis, we found that these students all followed a similar process; however, the students used other operators in the process, such as, e.g., colors as representations or descriptions of boxes as names. From the introduction of the everyday problem to the students’ mathematical solutions, we can observe that stage by stage, the students’ representations lose locality, particularity, materiality, and multiplicity, such that in the end, there is almost nothing left but numbers on their paper; thus, a reduction in complexity occurs. However, this reduction has not only removed complexity, but also yielded greater standardization, calculations, relative universality, and formalized knowledge, which is an amplification of the desired properties In general, Latour’s (1999) model has the potential to focus on the work process and thinking of students in the transformation from everyday problem to mathematical solution, as it makes the process noticeable. Teachers need to be aware that the construction of the chain is an important process, but it often needs some guidance, and the guidance should be aimed toward the point in the chain causing trouble, that is, students should be allowed to overcome gaps, such as the sizes of the boxes shown in the case. It is thus important to let the gaps in the chain remain, to reduce complexity, and to amplify particularity and generalizability, as well as to maintain the students’ ability to walk along the chain of reference to go from an everyday problem in mathematics to a more formal mathematical solution. REFERENCES 170

Creswell, J. W. (2014). Educational research: Planning, conducting and evaluating quantitative and qualitative research. Harlow, Essex: Pearson. De Freitas, E., & Sinclair, N. (2014). Mathematics and the body: Material entanglements in the classroom. Cambridge University Press. Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61, 103–131. Education Committee of the EMS. (2011). Do theorems admit exceptions? Solid findings in mathematics education on empirical proof schemes. Newsletter of the European Mathematical Society, 81, 50–53. Garcez, P. M. (1997). Microethnography. In N. H. Hornberger & D. Corson (Eds.), Encyclopedia of Language and Education (Vol. 8), (pp. 187–196). Dordrecht: Springer. Harel, G., & Sowder, L. (1998). Students’ proof schemes: Results from exploratory studies. Research in collegiate mathematics education III, 234–283. Harel, G., & Sowder, L. (2007). Toward comprehensive perspectives on the learning and teaching of proof. In F. K. Lester Jr. (Ed.), Second Handbook of Research on Mathematics Teaching and Learning (pp. 805–842). Charlotte, NC: Information Age Pub. Latour, B. (1999). Circulating reference — Sampling the soil in the Amazon forest. In Pandora’s hope essays on the reality of science studies (pp. 24–79). Harvard University Press. Pimm, D. (1987). Speaking mathematically—Communication in mathematics classrooms. London: Routledge. Russell, B. H. (2018). Research methods in anthropology: Qualitative and quantitative approaches (6th ed.). Lanham, Maryland: Rowman & Littlefield. Yin, R. (2002). Case Study Research - Design and Methods (3rd ed.). United States of America: SAGE.

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14.2 Paper II

Title Undersøgende aktiviteter og ræsonnementer i matematikundervisningen i matematikundervisningen på mellemtrinnet

Authors Dorte Moeskær Larsen & Bent Lindhardt

Status Published in:

MONA, Matematik-og Naturfagsdidaktik -tidsskrift for undervisere, forskere og formidlere, 1- 2019, p. 7-21

This version of the paper is the original document in the publication.

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ARTIKLER 7

Undersøgende aktiviteter og ræsonnementer i matematikundervisningen på mellemtrinnet

Dorte Moeskær Larsen, Bent Lindhardt, LSUL, Syddansk Universitet Professionshøjskolen Absalon

Abstract: I et dansk forsknings- og udviklingsprojekt ved navn KiDM blev der udviklet tre måneders un- dersøgelsesorienteret undervisning i matematik til 4. og 5. klasse. Undersøgelsesorienteret undervisning i matematik har dog en bred definition, og for at hjælpe lærerne blev der udviklet en kategorisering af forskellige typer af undersøgende aktiviteter i matematik. Denne artikel definerer og beskriver disse fem forskellige kategorier. Herefter udvælges to aktiviteter (“Opdagelsen” og “Grubleren”) som bliver undersøgt med fokus på hvilke ræsonnementer der kommer i spil i dialogen i opsamlingsfasen. Der bliver afslutningsvis reflekteret over forskellen på elevernes ræsonnerende virksomhed i de to forskellige aktiviteter.

Nærværende artikel skal ses som et foreløbigt udkomme af et større ministerielt projekt, Kvalitet i matematik og dansk (KiDM), som blev igangsat i et samarbejde mellem Undervisningsministeriet, Skolelederforeningen og Danmarks Lærerforening. Projektet har forløbet over perioden 2016-2018. Gennemførelsen af projektet blev lagt i hænderne på deltagere fra University College Syd, Professionshøjskolen Absalon, University College Lillebælt, University College Nord, Aalborg Universitet samt Syd- dansk Universitet og derudover en række matematiklærere fra folkeskolen i såvel udvikling, pilottest og afprøvning. I ansøgningen til projektet blev der argumenteret for at øget kvalitet i matematik kunne omhandle en øget undersøgende, dialogisk og anvendelsesorienteret undervisning. I projektet omsattes dette til en intervention af en varighed på ca. tre måneder udviklet af såvel forskere som praktikere. Interventionen skulle gennemføres over tre perioder af et halvt års varighed fra efterår 2017 til efterår 2018 med 45 forsøgsskoler med samlet 143 klasser på 4. og 5. klassetrin. Det er foreløbige overvejelser og resultater fra denne intervention som danner grundlaget for dette tematiske nedslag.

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Til trods for det mangeårige fagdidaktiske fokus på undersøgende matematikundervisning er vores erfaring i dialogen med praksis at mange lærere oplever det for kompliceret, risikofyldt og for uforudsigeligt og dermed undlader at inddrage under- søgende matematik i undervisningen (Michelsen et al., 2017). Det kan således synes hensigtsmæssigt at forsøge at dissekere undersøgende matematik ned i mindre, mere overskuelige enheder. I KiDM-projektet blev der udviklet en kategorisering af fem forskellige undersø- gende aktiviteter som vil blive fremstillet og beskrevet i denne artikel. Samtidig er en central del af den undersøgende matematik potentialet for elevernes ræsonne- rende virksomhed; derfor er der udvalgt to af de fem undersøgende aktiviteter som vil blive nuanceret ud fra hvordan elevers og lærers dialog fremstår i opsamlingen. Vores hypotese er at der overordnet set er forskel på elevernes ræsonnerende virksomhed afhængigt af hvilken undersøgende aktivitet der arbejdes med i undervisningen, og at dette kan have implikationer for hvordan en lærer skal gribe klassens opsamling an.

Metode og empiri For at kunne studere de forskellige undersøgende aktiviteter i KiDM-undervisningen blev der udført klasserumsobservationer i fem forskellige klasser som alle arbejdede med KiDM-materialet. De fem klasser blev observeret og videofilmet 3 ¨ 90 minutter af bl.a. førsteforfatteren til denne artikel. De fem skoler havde alle deltaget i 1. runde og var udvalgt til at repræsentere by-/landskoler og store/små skoler og var geogra- fisk placeret på både Fyn og Sjælland. Ved indsamling af data fokuseredes specifikt på én elev pr. besøg når der blev arbejdet alene eller i grupper. Eleverne blev udvalgt af læreren som værende særligt arbejdsomme og følelsesmæssigt robuste. Disse ud- vælgelseskriterier blev valgt for at sikre at eleverne ville arbejde med opgaven og ikke blive overvældet af at observatørens kamera fulgte deres arbejde. Alle optagelserne blev efterfølgende transskriberet fuldt ud. Datakodning begyndte med en åben og undersøgende tilgang til hele datasættet med fokus på elevers argumentationer. Dataene blev læst fra start til slut flere gange, og generelle tendenser blev diskuteret. Derefter udvikledes koder på alle fundne argumenter, først på baggrund af et arguments indhold beskrevet af Stylianides (2007) (fundament, formulering, repræsentation, socialt) og derefter ud fra bl.a. beskrivelser af forskellige bevisskemaer (eksterne, empiriske eller analytiske bevisskemaer) (Harel & Sowder, 1998). Udvalgte lektioner blev kodet sammen hvorefter forskellige udvalgte cases blev kodet individuelt af førsteforfatteren efterfulgt af en fælles diskussion af disse kodninger. Denne kombination af individuelle og dobbelte kodninger blev udført for at undgå subjektiv bias i analysen og for at øge inter-kode reliabiliteten (Joh nson, 2014).

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Undersøgelsesstrategien lægger sig op ad casestudiet idet formålet er at opnå en detaljeret og specificeret beskrivelse af hvordan interventionen udfolder sig i forskellige cases. Begrundelsen for denne strategi er at et casestudie ses som en typisk strategi til empirisk udforskning af et udvalgt fænomen i den sammenhæng hvor fænomenet udspiller sig, hvorved også fænomenets kontekst kan inddrages i den videre argumentation (Robson, 2011).

Hvad er undersøgende matematikundervisning? Nogle lærere forestiller sig at for at lave undersøgende matematikundervisning skal eleverne starte fra bunden hvor de selv skal finde på den undersøgende problemstilling og selv skal gennemføre hele undersøgelsen, men dette er ikke den eneste måde at lave undersøgende matematik på. Ifølge Harlen og Allende (2006) findes der ikke en egentlig model for hvordan en undersøgelsesorienteret tilgang skal omsættes til undervisningspraksis. Implementering af metoden i undervisningspraksis vil variere med undervisningens tema, læreren, elevernes alder og ikke mindst hvilke ressourcer der er til rådighed:

“IBME [Inquiry-Based Mathematic Education] vil sandsynligvis tage en mangfoldighed af former i overensstemmelse med de institutionelle forhold og begrænsninger, hvor den udvikler sig.” (Artigue & Blomhøj, 2013, s. 809, vores oversættelse)

I Danmark taler man typisk om forskellige typer af forløb. Det kan være tematiske forløb (Blomhøj & Skånstrøm, 2006) eller matematiske modelleringsforløb (Blomhøj & Kjeldsen, 2006) eller forløb med undersøgelseslandskaber (Skovsmose, 1999). Artigue og Blomhøj (2013) argumenterer for at forskellige teoretiske tilgange kan støtte begrebsliggørelsen af IBME og dennes implementering. Yderligere beskriver de hvordan forskellige teoretiske tilgange som Realistic Mathematics Education, Theory of Didactic Situations, Anthropological Theory of Didactics, modellering og Problem- Based Learning alle har deres egen tilgang, men også er overlappende med IBME. IBME bliver således beskrevet som et kalejdoskop mere end en enstrenget struktur.

“Ligesom i IBSE [Inquiry-Based Science Education] involverer undersøgelsesbaserede metoder inden for matematik forskellige former for aktiviteter kombineret i undersøgel- sesprocesserne: uddybende spørgsmål; problemløsning; modellering og matematisering; søge ressourcer og idéer; udforske; analysere dokumenter og eksperimentere med data; opstille hypoteser, teste, forklare, begrunde, argumentere og bevise; definere og struk- turere forbindelser, repræsentere og kommunikere.” (Artigue & Blomhøj, 2013, s. 808, vores oversættelse)

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Blomhøj (2017) fremhæver at et undersøgende undervisningsforløb naturligt kan opdeles i en tredelt struktur med hver deres didaktiske fokuspunkt:

• Iscenesættelse hvor læreren igangsætter • Aktivitet hvor eleverne har frihedsgrader til at handle undersøgende • Opsamling og fællesgørelse.

Iscenesættelse I denne fase introduceres og tydeliggøres den undersøgende aktivitet med henblik på at igangsætte elevernes arbejde. Den didaktiske kontrakt afstemmes så eleverne kan og vil indgå i den rammesatte deltagelsesstyring af den fremlagte problemstilling/opgave.

Aktivitet I denne fase arbejder eleverne selvstændigt med en anvist problemstilling. I den igangsatte aktivitet kan der indgå forskellige grader af frihed og åbenhed som påvirker elevernes undersøgende arbejdsmåde. Det kan betyde arbejdsprocesser som indebærer en vis uforudsigelighed og usikkerhed, og som kræver fagligt vovemod hos eleverne (og læreren?). Man skal turde agere med risiko for at fejle.

Opsamling og fællesgørelse I denne fase opsummeres elevernes erfaringer, resultater og refleksioner som grundlag for opbygning af fælles faglig viden i klassen. Læreren er facilitator i processen for at sikre en rettethed mod en generalisering, præcisering, erkendelse osv. af elevernes arbejdsproces og produkter. I dette indgår dialogen som en central størrelse. Projektet valgte ovenstående tredelte struktur for undervisningen idet det tydelig- gjorde forskellen på elevernes eget undersøgende arbejde og den lærerstyrede klas- sesamtale. Vi har i projektet valgt at skelne mellem den undersøgende undervisning og den undersøgende aktivitet. Den første beskriver lærerens planlægning og struktur for undersøgende undervisningsforløb. Den anden berører elevernes undersøgende arbejdsmåde.

Hvad er undersøgende aktiviteter? Vi skelner mellem to principielle tilgange til det undersøgende: det eksplorative og det investigerende. Ordene er hentet fra engelsk, “exploration” og “investigation” – to udtryk vi ikke umiddelbart har præcise termer for på dansk.

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• Det eksplorative består i at være en udforskende, nysgerrig og observerende person som uden indledende problemstilling undersøger et begreb, et fænomen eller en genstand i situationen. Man er således opdagelsesrejsende i det ukendte hvor man undervejs justerer mål og arbejdsproces. • Det investigerende består i at forfølge nogle hensigtsmæssige strategier for at finde et kvalificeret svar. I det investigerende har man en problemstilling som er lede- tråden i arbejdsprocessen – en kurssætter som løbende skal sikre styringen mod et kvalificeret svar. Den indbefatter at eleverne etablerer en plan.

For bedre “at se” muligheder og nuancer i det undersøgende har vi forsøgt at klas- sificere og beskrive fem forskellige typer som på hver sin måde indeholder det un- dersøgende i form af det investigerende og eksplorative. Udvalget skal ikke opfattes som udtømmende for undersøgende aktiviteter, men have en eksemplarisk karakter. Derudover indgår der overvejelser om aktiviteters frihedsgrader – om graden af åbenhed knyttet til problemstilling (arbejdsopgave), metodiske valg for at løse problemet/opgaven og de mulige resultater/svar. Vi har også valgt at skelne mellem et lærerperspektiv og et elevperspektiv idet det tydeliggør forskellen i videnspositioner i den undersøgende aktivitet. Der er således forskel på at kende og guide elever mod en opdagelse af en bestemt begrebsmæssig sammenhæng som man må forvente læreren har et indgående kendskab til, og så at deltage som lærer i et forløb hvor ukendthedsfaktoren er betydelig højere. En systematisering af ovenstående parametre har resulteret i følgende fem forskellige aktivitetstyper:

Opdagelsen Hovedhensigten med “Opdagelsen” er at eleverne skaffer sig indsigt i og forståelse af udvalgte matematiske begreber. Åbenheden og det undersøgende består i at eleverne ikke kender de faglige pointer, men selv skal finde frem til dem i en form for erfarings- og eksperimenterende forløb. De skal således få øje på sammenhænge og systemer som kan lede dem mod en generaliseret viden inden for det udvalgte matematiske stofområde. Undersøgelsen er her mere et styret forløb – en afprøvning – hvor der stiles mod en ahaoplevelse hos eleverne. Det er bl.a. det Freudenthal omtaler som “guided reinvention” hvor idéen er at give eleverne mulighed for selv at genopfinde matematik ved at gøre det selv (Gravemeijer, 1999). Fra et elevperspektiv er problemet/opgaven ofte lukket; metoden og resultatet opleves med forskellige grader af åbenhed. Det åbne består i at eleverne selv arbejder sig undersøgende hen mod et nyt vidensniveau til forskel fra at øve sig på en viden som er formidlet af læreren. Fra et lærerperspektiv er aktiviteten typisk kendt i alle faser.

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Grubleren Hovedhensigten med denne aktivitet er at eleverne udvikler deres kreativt tænkende og ræsonnerende evner. Det centrale er ikke stoffet, men om eleverne kan og vil gå ind i “hvis … så”-relationer samt indgå i systematisk undersøgelse af muligheder. Det undersøgende fokus er således på elevens undersøgende metodik – måden man når frem til svaret. Her er svaret ikke det centrale, men de hensigtsmæssige arbejdsmåder der fører eleverne hen til svaret. “Grubleren” har i hverdagen mange navne som “nød- deknækker”, “kryptisk opgave”, “gåden”, “drillepinden” osv. Den findes både iklædt virkelighedens rammer og som rene matematiske problemstillinger. Fra et elevperspektiv er der en åbenhed i at forstå og tolke problemstillingen idet den ofte er atypisk og uvant. Det samme gør sig gældende for den metode man anvender for at nå et resultat. Det fremgår ikke umiddelbart hvilke løsningsmetoder der vil væ- re hensigtsmæssige, og der skal muligvis nyskabes eller kombineres kendte metoder. Som ved “Opdagelsen” er læreren bekendt med såvel problemstilling, mulige løs- ningsmetoder og svar. Der kan dog ligge uforudsigelige løsningsmetoder fra eleverne som læreren må forholde sig til.

Produktet Hovedhensigten er her at eleverne arbejder med at fremstille et produkt som “virker” – ud fra både funktionelle, men også æstetiske perspektiver. Er det en flyver, skal den kunne flyve ordentligt – er det fx et billede, skal det være smukt. I sådanne fremstillings- og forbedringsprocesser indgår der ofte matematik. Det undersøgende består i at eleverne “tager over” og går længere end til blot at følge en angivet frem- stillingsproces. De begynder at eksperimentere og forandre såvel proces som produkt. I dette kan indgå skabende, innovative processer. Der kan således indgå en “instruktion” til den praktiske udformning af produktet hvor det undersøgende kan opstå når man vil forandre, produktudvikle, forbedre, tydeliggøre, personliggøre m.m. produktet. Man kunne i denne sammenhæng tale om nysgerrighed som bærende element og dermed en mere eksplorativ tilgang til produktet. Instruktionen kan være mere eller mindre lukket – i en gradient fra op- findelse til håndværksmæssig ordentlighed.

Målingen Hovedhensigten med “Målingen” er at anvende matematik i en naturvidenskabelig ramme ved at foretage “en undersøgelse”. Man er således underlagt nogle “viden- skabelige krav og retningslinjer” for at gøre undersøgelsen tilstrækkelig pålidelig og gyldig. Det undersøgende består i at resultatet er ukendt for både lærer og elever. Det er således en måde at skabe sig ny viden på. Det kan fx være at man ønsker at undersøge indeklimaet på skolen ved at måle temperaturen på udvalgte steder over

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tid, det kan være trafiktælling for at undersøge trafiktætheden, det kan være en un- dersøgelse af elevernes løbepræstationer osv.

Modelleringen Hovedhensigten med “Modelleringen” er at fremme modelleringskompetencen. Eleverne skal forholde sig til en problemstilling i hverdagen som skal afgrænses og omsættes til en matematisk beskrivelse og analyse. På baggrund af det skal eleverne tolke de svar de får, og forholde sig kritisk til deres model. Her er der mange åbne, ukendte elementer for såvel lærer som elever der fordrer en undersøgende virksomhed. Til trods for at problemstillingen kan være kendt, er den ofte af en kompleks og åben karakter som kræver yderligere præcise spørgsmål eller hypoteser. Der indgår en åbenhed i hvilke variable og størrelser som er relevante for at skabe en anvendelig matematisk model til beskrivelse og analyse af problemet. Der er en åbenhed i mulige resultater som afhænger af de præmisser man har opstillet m.m. Fra et lærerperspektiv er der en stor grad af ukendthed – og dermed er elever og lærer ofte i samme undersøgende situation.

Aktivitet Undersøgende sigte Undersøgelses aspekt Perspektiv Problem Metode Resultat

Opdagel- Afprøve og udlede begrebs- Lærer Kendt Kendt Kendt sen mæssige sammenhænge Elever Lukket Åbent Åbent Investigerende

Gruble- Forstå problemstillingen og Lærer Kendt Kendt Kendt ren en mulig løsningsmetode Elever Åbent Åbent Lukket Investigerende

Produktet Undre sig over funktion Lærer Kendt Kendt Kendt eller æstetik ud fra produkt. Mulige ændringer og Elever Lukket Lukket Lukket Eksplorativt personliggørelse /åbent /åbent /åbent

Målingen En “videnskabelig” under- Lærer Kendt Kendt Ukendt søgelse af noget gennem måling og beregning Elever Lukket Lukket Åbent Investigerende

Modelle- Udvikle og afprøve Lærer Kendt Ukendt Ukendt ringen matematiske modeller og evt. beskrivelse og analyse af ukendt virkeligheden Elever Åbent Åbent Åbent Investigerende

Tabel 1. Overblik over hvad der tænkes som “åbent”, “lukket” eller “kendt” i de forskellige aktiviteter.

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En central del af undersøgende matematik er potentialet for elevernes ræsonnerende virksomhed, men at få eleverne til at ræsonnere i undervisningen kræver mere end blot at stille åbne opgaver eller blot at få dem til at forklare deres tænkning (Ball & Bass, 2003). I denne artikel har vi valgt at tage udgangspunkt i elevers ræsonnementer i opsamlingsfasen knyttet til to af de undersøgende aktiviteter. Det blev synligt i vores kodninger at langt de fleste ræsonnementer fra eleverne bliver synliggjort i denne fase. Det er ofte når læreren spørger ind at eleverne bliver tvunget til at synliggøre de ræsonnementer de har arbejdet med i løbet af aktiviteten. Når eleven arbejder alene, foregår dette ofte implicit i deres udregninger, og i gruppearbejdet afhænger det meget af gruppens arbejdsproces. Opsamlingsfasen kan netop afspejle elevernes systematisering og deres udvikling af forståelser i kraft af kommunikationen og de ræsonnementer eleverne fremfører. Det kan fx ske i forbindelse med at de i en opsamling skal give nogle forklaringer på hvad de har gjort, og retfærdiggørelse af hvorfor netop den valgte tilgang eller det udregnede resultat giver mening. Desuden tydeliggøres det når eleverne forsøger at forstå og udfordre andre elevers og lærerens forklaringer og spørgsmål. Ræsonnementerne i opsamlingen kan her ses både som et middel til at lære at forstå det matematikfaglige indhold (at lære af at ræsonnere), og det kan ses som et mål for læringen (lære at ræsonnere).

Elevers ræsonnementer ved undersøgende aktiviteter At kunne ræsonnere matematisk handler både om at kunne følge og bedømme en kæde af argumenter fremsat af andre samt selv at kunne udtænke og gennemføre argumentation (Niss & Jensen, 2002). I Whitenack og Yackel (2002) beskrives det at ræsonnere i matematik specielt handler om at eleverne skal udvikle matematiske argumenter for at kunne forklare deres idéer til andre. Hanna (2000) fremfører at argumenter kan have en overbevisende styrke og en forklaringsstyrke. Den overbevisende styrke kan fremstå absolut som ved deduktive bevisførelser eller relativ hvor den i så fald bliver mere personlig og subjektiv. Forklaringsstyrken i et argument ligger i at den kan bidrage med en indsigt i hvorfor noget er sandt. Elevernes udviklede argumenter kan have mange former. Det kan være faktaer- klæringer, resultaterne af et forsøg, et eksempel/eksempler fra praksis, en definition eller sætning, en tilbagekaldelse af en regel, en gensidig tro eller præsentationen af en modsætning. Stylianides (2007) beskriver at der indgår følgende fire elementer i brugen af argumenter: fundamentet, formuleringen, repræsentationen og den sociale dimension. Fundamentet er de forudgående definitioner, aksiomer, sætninger osv. Formuleringen

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handler om på hvilke måder argumentet bliver udviklet; er det fx udviklet som en generalisering, som en deduktion eller fra en case? Repræsentationen handler om hvordan argumentet bliver fremført; det kunne fx være mundtligt, skriftligt eller algebraisk. Det sidste element omhandler den sociale dimension, herunder hvem argumentet kommunikeres til, eller fx at lærere kan have forskellige tilgange til hvornår de anser en argumentationsrække for at være lødig i en matematikundervisning. Der er flere bud på hvordan man kan fjerne eller afvise tvivl og udvikle eller godtage sandheden af et argument (Balacheff, 1988; Harel & Sowder, 1998). I Harel og Sowder (1998) skelnes mellem tre forskellige typer af (over)bevisskemaer:

• Det eksterne overbevisende skema • Det empiriske skema • Det analytiske skema.

Argumentation i det eksterne skema anses som valid på baggrund af en autoritet, som når læreren eller facitlister blot godkender et argument for at være sandt uden yderligere begrundelser (Harel & Sowder, 1998). Argumentation i det empiriske skema valideres på baggrund af empiri hvor fokus er på at anvende fx konkrete eksempler. Det kan fx være induktivt formuleret (Harel & Sowder, 1998). Det analytiske bevisskema omhandler den deduktive argumentation som består af en række argumenter som følger af nogle gældende præmisser, love eller regler. Den deduktive argumentation tager dermed udgangspunkt i allerede bevidste påstande og teoremer og validerer påstande gennem logisk deduktion (Harel & Sowder, 1998). Matematisk argumentation er dog generelt kendetegnet ved at være et socialt fænomen (Krummheuer, 1995). Hvad der accepteres i klassen, afhænger af klassens normer, herunder både de sociomatematiske normer og sociale normer (Yackel & Cobb, 1996). En overbevisning eller en forklaringsstyrke afhænger af hvem det er der skal overbevises eller forstå en forklaring. En 4.-klasseselev har behov for en anden forklaringsstyrke end eksempelvis en gymnasieelev. Det er dog vigtigt at fremhæve at der i en matematisk problemløsningsproces ofte er en dynamisk relation mellem forskellige typer af argumenter. Der vil således indgå både bidrag fra empiriske undersøgelser, herunder fx “at prøve sig frem” og mere analytiske tilgange. Begge tilgange anses således som essentielle for at komme frem til løsninger af matematiske problemer (de Villiers, 2010), men det er dog samtidig vigtigt at understrege at det er de analytiske argumentationskæder der generelt anses som lødige i matematikundervisningen, og som samtidig beskrives som mest vanskelige for eleverne at udvikle (Education Committee of the EMS, 2011).

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Cases I det følgende har vi udvalgt to cases der illustrerer dialogen og elevernes argumentation i opsamlingerne for henholdsvis “Opdagelsen” og “Grubleren”. Begrundelsen for at det netop er disse to undersøgende aktiviteter der er udvalgt, er at der i empirien var flest observationer af netop disse to aktiviteter. De forskellige typer af aktiviteter er ikke repræsenteret lige meget i første tiltag i KiDM-materialet.

Case, “Opdagelsen” En af KiDM-aktiviteterne kategoriseret som “Opdagelsen” har navnet “En fjerdedel af hvad?”. Formålet med opgaven er at få eleverne til at “opdage” at en fjerdedel af noget er afhængig af helheden. Her indgår fjerdedeling af forskellige størrelser af pizzaer og lasagner (cirkler og rektangler). I en afsluttende opsamling fremlægger gruppen med Anders og Jens fra 4. a deres arbejde for resten af klassen. Det diskuteres hvilken deling af en rektangelformet lasagne der giver de største stykker, figur 1 eller figur 2:

Figur 1. Opdelingen af rektanglet Figur 2. Opdelingen af rektanglet med halvering af sidelinjerne. med diagonaler.

“Anders: Jeg vil helst have den med plustegn [figur 1], for så holder det lidt bedre sammen så det ikke falder fra hinanden. Lærer: Det er jo fint nok; den ligger pænest på tallerkenen. Men er det størst? Anders: Nej. Bo: Det er den der med diagonalerne der er størst [figur 2]. Lærer: Det er en meget spændende udlægning. Prøv lige at fortælle mig lidt om det; kig lige op (henvendt til klassen). Er det der stykke det største stykke? [En elev, Gustav, ryster på hovedet.] Er det større end det der? [Gustav ryster fortsat på hovedet.] Er det større end det der? Jens: Nej, kan I ikke bare se de er alle sammen lige store. Lærer: Hvorfor det? Jens: Fordi lige meget hvad, det er den samme plade fx, og den bliver stadig delt op i fire stykker lige meget hvad, så de er lige store alle sammen. Gustav: Nej, nej, nej. Lærer: Du skal ikke sige nej; nu hører vi hvad han siger, så kan du få lov at argumentere for noget andet.

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Lærer (hæver stemmen): Fordi det er den samme, det er i virkeligheden den samme plade. Vi skærer den ud i fire lige store stykker. Men vi er enige om at hvis vi havde haft sådan en her, og jeg havde gjort sådan her [læreren laver en meget skæv deling af lasagnen], så er vi ikke i tvivl om hvad for et stykke der var størst, vel? Jens: Nej.”

Som det fremgår, er det lærerens intention at få eleverne til at forstå at man kan dele et rektangel op i fjerdedele på forskellige måder. Typisk valgte eleverne at tegne diagonaler (se figur 2) eller at halvere siderne som et “kors” (se figur 1). Anders’ argument er kontekstrelateret og handler om hvordan man vil dele en lasagne derhjemme, men opfattes af læreren som et ikkelødigt matematisk argument da det straks affejes som ligegyldigt her i matematikundervisningen. Bo kommer derimod med en ny påstand om at stykkerne har forskellige størrelser. En påstand som læreren griber og gerne vil høre flere argumenter for. Jens får her lov til at komme med det endelige argument som har en slags ringslutning, og som i princippet ikke handler om om trekanterne i figur 2 er lige store. Alligevel verificerer læreren argumentet med et andet empirisk modargument, og diskussionen lukker.

Case, “Grubleren” I en anden KiDM-aktivitet, som er kategoriseret som “Grubler”, skal eleverne finde ud af hvad fire kasser vejer når de kun er blevet vejet parvis til at være henholdsvis 6 kg, 8 kg, 10 kg, 12 kg, 14 kg og 16 kg. Der findes to løsninger til opgaven: 2, 4, 6 og 10 kg og 1, 3, 5 og 9 kg. I nedenstående opsamling spørger læreren ind til processen:

“Lærer: Oplevede I nogle af de samme problemer som Karla havde med at få de store tal? Harbon: Jaaa. Lærer: Fordi 1 og 5, det rammer 6’eren, og så går jeg ud fra at når I så skal ramme 8’eren, så har I sat en 3’er på. Harbon: Nej. Lærer: Nej? Harbon: Der tog vi 7. Lærer: Der tog I 7 i stedet for 3, okay! Så tog I 7 og den næste … Harbon: Fordi så gav det 8, og så tog vi 9 i den, fordi så kunne den … Lærer: Er der nogen speciel grund til at I sprang 3 over? Harbon: Så kunne den komme højere op.”

I aktiviteten “Grubleren” er svaret ikke det centrale, men i højere grad mulige og hensigtsmæssige veje der kan føre eleverne hen til svaret. Opsamlingen bliver derfor

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en retfærdiggørelse af elevernes proces og ikke en overbevisning af resultatets vali- ditet. Processens retfærdiggørelse og argumentation har også forskellige niveauer. I den følgende samtale kan man konstatere hvordan læreren undlader at udfordre et meget subjektiv argument:

“Lærer: Men da I prøvede det, prøvede I så automatisk de ulige tal? Peter: Ja. Lærer: Hvorfor? Peter: Det føles bare bedst. Lærer: Det føles bare bedst … okay. Jeg kunne godt tænke mig, læg lige blyanterne fra jer, og kig herop … [læreren går videre].”

Udfordringen er her at eftersom læreren ikke afviser argumentet i matematikundervisning, men blot går videre, vil nogle elever måske efterfølgende tro at denne type af argumentation anses som tilfredsstillende validt i matematikundervisningen. På en anden skole med samme aktivitet ser vi følgende:

“Lærer: Hvorfor var I optagede af tallet 12? Caroline: Det var bare fordi 12, den havde vi bare haft med mange gange, og den kunne vi lave både 16 og sådan noget ud af …”

Caroline beskriver de argumenter hendes gruppe har udviklet i deres søgen efter et svar. Argumentet for at vælge tallet 12 understøttes af et empirisk argument. I en anden opsamling til samme “Grubler” diskuteres det hvad det betyder at lægge lige og ulige tal sammen da gruppen har fundet frem til at de to løsninger adskiller sig ved henholdsvis at indeholde lige tal og ulige tal:

“Lærer: Vilfred? Vilfred: Grunden til det er jo at et lige tal … hvis man fx har et lige tal, så vil det jo når man plusser en til, så bliver det et ulige tal, men hvis man plusser to ulige tal, så er der jo ligesom to tilovers fra det lige tal, og to er jo et lige tal så … [lidt uklart pga. han snakker lavt]. Lærer: Det er super godt forklaret, hvis vi fx tager vores 3’er og vores 5’er herover, så er det et lige tal, med en i overskud, og det her ovre er også et lige tal med en i overskud. Og hvis vi lægger de to sammen, så bliver de to overskydende til et lige tal. Vilfred: Så har du faktisk tre lige tal. Lærer: Det er så godt forklaret! Sindssygt godt! Er I med?”

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Det er tilsyneladende vigtigt for Vilfred at kunne argumentere for hvorfor alle tallene i én løsning enten skal være lige eller ulige. Læreren griber argumentet og prøver at udvide forklaringen og udtrykker samtidig begejstring for argumentet som måske kan siges at nærme sig det mere analytiske bevisskema (Harel & Sowder, 1998).

Diskussion I casen om “Opdagelsen” tydeliggøres lærerens målrettethed mod at konkludere ny viden om matematiske sammenhænge og begreber. Den tydeliggør også at denne målrettethed indebærer at lærerens iver for at nå det rigtige resultat kan medføre at han bliver den aktivt argumenterende part frem for eleverne. I casen med Jens ser vi at læreren ikke får italesat analytiske argumenter, men stopper ved de empiriske argumenter og via sin egen autoritet prøver at overbevise eleverne om den foreslåede påstand. Det bliver derfor det eksterne overbevisende skema (Harel & Sowder, 1998) der afgør tvivlen og godtager argumentet. Generelt i de øvrige observationer af “Opdagelsen” (fem andre klasser hvoraf kun tre indeholder en opsamling) ser vi ligeledes at læreren altid har en vigtig pointe som han/hun vil have overbevist eleverne om i den afsluttende opsamling således at der måske gås lidt på kompromis med de anvendte argumenters lødighed. Dialogen kan derfor få et fokus på “det rigtige og forkerte” frem for at eleverne selv får mulighed for at opbygge en kæde af argumenter som afprøves og vurderes. I aktiviteter med “Opdagelsen” bliver argumenterne fremført af både læreren og eleverne. Læreren inddrager ofte tavlen til at fremme visse argumenter. Elevernes argumenter bliver alle repræsenteret i et hverdagssprog, og de inddrager både deres empiriske erfaringer og deres fælles antagne viden til at argumentere for de forskellige påstande. Fundamentet i de forskellige argumenter bunder derfor ikke altid i en matematisk praksis. I casen “Grubleren” kan vi generelt se at der i opsamlingen bliver sat fokus på elevers forklaring af deres proces. Eleverne udtrykker mange forskellige typer af argumenter i denne forklaring, lige fra meget subjektive holdninger til mere analytiske tilgange. De endelige svar til opgaven bliver der derimod ikke argumenteret for i nogle af de observerede klasser. Overordnet kan vi se at de argumenter der bliver fremført, alle afholdes i et mundtligt hverdagssprog. Vi ser ingen steder et mere analytisk sprog anvendt. Argumenterne trækker alle på elevernes tidligere praksisser og deres fælles antagne viden, mens de inddrager deres egne empiriske erfaringer fra processen hvor fundamentet er placeret. I udsagn fra de deltagende lærere fra forsøgsskolerne i KIDM-projektet nævnes det ofte at der er behov for at elevens uformelle sprog formaliseres og gøres mere matematisk korrekt – med særlig fokus på korrekte matematiske termer. Man ser i

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observationerne meget forskel på hvilket fokus læreren har på det formelle sprog, og også her hvad der accepteres som lødigt i undervisningen. Den sociale dimension påvirker argumentationen i begge cases, og læreren bliver her også en vigtig spiller.

Opsamling Som det tidligere er omtalt, er vores hensigt med artiklen at se flere nuancer og de- taljer i forståelsen af undersøgende matematikundervisning. Vores tese er at ved at opdele billedet af undersøgende matematik til mindre, mere operationelle enheder er der mulighed for større overblik for læreren og dermed større gennemslagskraft i den daglige undervisning. Forskellige valg af fx typer af undersøgende aktiviteter har dog følgevirkninger som bør undersøges nærmere. Vi har her illustreret hvordan valg inden for de to undersøgende aktiviteter “Opdagelsen” og “Grubleren” kan have forskelle i det dialogisk argumenterende samspil der er mellem elever og lærer. En forskel man skal være bevidst om i valgsituationen og undervejs i undervisningen.

Referencer Artigue, M. & Blomhøj, M. (2013). Conceptualizing inquiry-based education in mathematics. ZDM, 45(6), 797-810. Balacheff, N. (1988). Aspects of proof in pupils’ practice of school mathematics. Mathematics, teachers and children, 216, 235. Ball, D.L. & Bass, H. (2003). Making mathematics reasonable in school. A research companion to principles and standards for school mathematics, 27-44. Blomhøj, M. (2017). Fagdidaktik i matematik. Frydenlund. Blomhøj, M. & Kjeldsen, T.H. (2006). Teaching mathematical modelling through project work. ZDM, 38(2), 163-177. Blomhøj, M. & Skånstrøm, M. (2006). Matematik Morgener – matematisk modellering i praksis. Kunne det tænkes, 7-23. de Villiers, M. (2010). Experimentation and proof in mathematics. I: Explanation and Proof in Mathematics (s. 205-221). Springer. Hanna, G. (2000). Proof, explanation and exploration: An overview. Educational Studies in Mathematics, 44(1-2), 5-23. Harel, G. & Sowder, L. (1998). Students’ proof schemes: Results from exploratory studies. Research in collegiate mathematics education III, 234-283. Harlen, W. & Allende, J. (2006). Report of the working group on international collaboration in the evaluation of Inquiry-Based Science Education (IBSE) programs. Santiago: FEBA. Krummheuer, G. (1995). The eth nography of argumentation: Lawrence Erlbaum Associates, Inc.

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Michelsen, C., Dreyøe, J., Hjelmborg, M. D., Larsen, D. M., Lindhart, B. K., & Misfeldt, M. (2017). Forskningsbaseret viden om undersøgende matematikundervisning. Undervisningsmi- nisteriet. Niss, M. & Jensen, T.H. (2002). Kompetencer og matematiklæring: Idéer og inspiration til udvikling af matematikundervisning i Danmark (Vol. 18). Undervisningsministeriet. Skovsmose, O. (1999). Undersøgelseslandskaber. Centre for Research in Learning Mathematics. Stylianides, A.J. (2007). The notion of proof in the context of elementary school mathematics. Educational Studies in Mathematics, 65(1), 1-20. Whitenack, J. & Yackel, E. (2002). Making mathematical arguments in the primary grades: The importance of explaining and justifying ideas. Teaching Children Mathematics, 8(9), 524. Yackel, E. & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics. Journal for Research in Mathematics Education, 458-477.

English abstract In a Danish development project named KiDM, a 3 months intervention of inquiry-based mathematics teaching was developed. Since inquiry-based teaching is broadly defined, to help the teachers focus their understanding of this concept a categorization of various investigative activities was developed; this article starts by defining and describing this categorization. Two of the mathematical activities – “The Brooder” and “The Discovery” – are described with special focus on students’ reasoning in whole-class discussion. In the conclusion some reflections are made about how the students reason in these two activities.

MONA 2019-1

14.3 Paper III

Title Fostering mathematical reasoning in inquiry-based teaching – the role of cognitive conflicts

Authors Dorte Moeskær Larsen & Morten Misfeldt

Status Manuscript submitted for publication in:

Nordic Studies in Mathematics Education (Nomad)

This version of the paper is the original document send in for publication.

188

Fostering mathematical reasoning in inquiry-based teaching – the

role of cognitive conflicts

Abstract:

Inquiry-based teaching is often endorsed as potential in mathematics education, where the student’s independent situations are considered important. In this article we scrutinise in what ways these situations entail for the students' development of mathematical reasoning. By studying cognitive conflict in one grade 5 class when participating in an inquiry-based intervention-study. The findings indicate 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 students' process of resolving cognitive conflicts is a process stretched over time that involve different routes, exploring approached and understandings.

Keywords

Reasoning, inquiry-based teaching, cognitive conflict

Students reasoning in inquiry-based teaching – the introduction

Students reasoning competence is very important in mathematics and is often seen as fundamental for doing mathematics. Ball and Bass (2003) even argue that: “…the notion of mathematical understanding is meaningless without a serious emphasis on reasoning” (p. 28). Reasoning can be seen as a basis for both mathematical understanding and communication, and as a critical part of

1 developing a mathematical approach that for instance appreciate convincing arguments (Ball &

Bass, 2003; Carpenter, Franke, & Levi, 2003; Hanna & Jahnke, 1996; Stylianides & Stylianides,

2008). In the literature, there is a clear distinction between reasoning in mathematics and reasoning in everyday life. An example of this is Harel and Sowder’s (1998, 2007) distinction between empirical proof schemes and formal proof schemes. The majority of students in primary school have empirical proof schemes, and research has shown that changing students’ empirical schemes to more formal proof schemes is highly non-trivial (Education Committee of the EMS, 2011) and how to make the transition from empirical to more formal deductive argumentation is still open for further research (Education Committee of the EMS, 2011).

Many important documents and initiatives point to the relevance of Inquiry Based Science,

Mathematics and Engineering education (IBSME) (Artigue & Blomhøj, 2013) in primary school classrooms, where the students’ independent formulation and exploration of mathematical problems are the focal point for the teaching processes. And indeed, such an approach has been explored and developed in a number of project and initiatives ("The Fibonacci Project ", 2013; PRIMAS, 2013).

The question is however, how the specific relations is between these independent situations in inquiry-processes and reasoning in such mathematical situations. Our interest in this relation led us to study the students’ independent reasoning situations, in an inquiry-based teaching approach.

When doing that, we realised that cognitive conflicts played an important and sometimes positive role in bringing the students from open exploration towards more directed exploration and reasoning.

By using the word independent situations, we refer to those situations where students either individually or often in groups perform an investigation or exploration and where the teacher is minimally present. In the terminology from Theory of Didactically Situations (TDS) (Brousseau,

2006) these situations will be defined as adidactical situations, but by using the term independent situations, we make clear that the situations sometimes are without the requirements that Brousseau sets for a situation to be defined as an adidactical situation for example that the student must be engaged in a game, "this game being such that a given piece of knowledge will appear as the means of producing winning strategies" (Brousseau, 2006, p. 57), which is not always the case in the

“independent situations” observed in inquiry-based teaching.

In this paper we will examine more closely the cognitive conflicts that one group of students experience when they engage in a mathematical inquiry. We follow how the students move from making meaning of a mathematical situation towards a more formal mathematical solution in the mathematical classroom. We focus particularly on which aspects of the environment, that are important to productive cognitive conflicts, and the subsequent reasoning processes and ask the following question:

What characterises productive cognitive conflicts when students work in independent situations

with mathematics in inquiry-based teaching and in which way does these productive cognitive

conflicts relate to students mathematic reasoning process?

Inquiry-based teaching

Inquiry-based teaching has been in focus in many empirical studies in mathematics where different things have been studied; the benefits of teaching inquiry-based (Bruder & Prescott, 2013); the difficulties of implementation (Dorier & García, 2013; Engeln, Euler, & Maass, 2013; Krainer &

Zehetmeier, 2013; Larsen et al., 2019; Maaß & Artigue, 2013; Maaß & Doorman, 2013; Schoenfeld

& Kilpatrick, 2013), inclusiv different approachedes to teaching inquiry-based (Kirschner, Sweller,

& Clark, 2006), parents involvement (Mousoulides, 2013), along with polity implications (Wake &

Burkhardt, 2013). However, research on the relations between inquiry-based teaching and students reasoning in mathematics is yet unexplored.

Inquiry-based teaching is an instructional theoretical model developed in several disciplines. The key idea is, that the students must solve “real” problems (Artigue & Blomhøj, 2013). Proponents of

Inquiry based mathematics education often take a constructivist approach suggesting that students construct knowledge following the lines of work of professional mathematicians themselves.

Mathematician often faces non-routine problems and make investigations where they search for information and develop conjectures, which they must justify and finally be able to communicate their results. Mathematics learning should follow a similar pattern (Artigue & Blomhøj, 2013). The progression is often defined as a process-cycle where students start by 1) identifying questions/new experiences, 2) making possible explanations (alternative ideas/existing ideas), 3) making prediction, 4) designing and conducting investigations, 5) interpreting data and 6) making a conclusion (Artigue, 2012). Within inquiry-based mathematics education, the teacher must lead students to experience the limitations of their knowledge and create the conditions for achieving the required cognitive evolution. (Artigue, 2012). Artigue and Blomhøj (2013) conclude, that inquiry- based teaching include that the autonomy and responsibility are given to students, but they also focus on the experimental dimension of mathematics (p. 809). The idea in inquiry-based teaching is that student learn more and better, when they can take control of their own learning by first defining their goals and monitoring their own progress in making the inquiry. Experiments in inquiry-based teaching can have many different approaches, but often when conducting investigation (phase 4 in the cycle) different kinds of manipulatives, representations or other resources are included in the experimentation, where also the context of the problem often is very essential (Baptist, 2012).

Cognitive Conflicts

In addition to inquiry-based teaching another perspective for learning suggests cognitive conflict as an instructional strategy in order to promote students conceptual change. Conceptual change is characterised as “the kind of learning required when the new information to be learned comes in conflict with the learners’ prior knowledge usually acquired on the basis of everyday experiences”

(Vosniadou & Verschaffel, 2004, p. 445). However conflicting ideas present a potential conflict and it only becomes a cognitive conflict when explicitly invoked by the students, because students may simple dismiss or treat it as an exception (Zazkis & Chernoff, 2008).

Tall (1977) described cognitive conflict in the learning of mathematics and proclaimed that understanding in mathematic often occurs in significant jumps accompanied by a clear sense of comprehension rather than a smooth steady process and he use cognitive conflict to explain that the learner after meeting these conflicts have to restructure his mathematical schema and restructure them in an appropriate manner. Implementing cognitive conflict has been reported in a variety of topics in mathematics education (Zazkis & Chernoff, 2008).

The KiDM project

The observed class participated in a design based, development and random controlled interventions experiment called Quality in Danish and Mathematics (KiDM). The aim in KiDM was to make the mathematics teaching in Danish compulsory schools more inquiry-based. The overall KiDM program had several stages starting by surveying the literature on inquiry-based teaching in mathematics and making some qualitative interviews with teachers and supervisors (Dreyøe,

Larsen, Hjelmborg, Michelsen, & Misfeldt, 2017; Michelsen et al., 2017), then developing an inquiry-based teaching program for a four-month mathematics teaching approach for both primary school year 4 and year 5, which afterwards was implemented in 145 classes. All the activities in the

5 intervention were all built around a simple 3-phase model, where the teacher 1) introduced a problem/investigation, after which 2) the students themselves made an inquiry with minimal guiding from the teacher and then 3) the activity ends with a whole class discussion. The control schools and intervention schools were randomly selected with respect to geography, size and ethnicity (T. I. Hansen, Elf, Misfeldt, Gissel, & Lindhardt, 2019). The study included some qualitative data which among others are observations from the intervention schools. It is these observations that form the basis of this study.

Observations from the KiDM experiment

The observations used in this study were collected in one year 5 intervention class. The class was randomly selected to be part of the KiDM project, and chosen to be observed, because their teacher volunteered at a kick-off meeting for the project. It was important that the teacher volunteered, because the intention was to observe the class once a week in four months (16 lessons). It is important to emphasise, that he therefore was not selected because of his specific abilities or special interest in inquiry-based mathematics. However, because of different practical issues in the class, the final number of lessons recorded in the class was 7 double lessons all together. All the observed lessons were transcribed in full by two preservice teachers and the first author of this article.

In the observed lessons the camera followed one groups of students and the group were chosen in collaboration with the teacher with the intention of the group being robust and not immediately disturbed by the camera.

This paper present one double-lesson. This double lesson can be seen as exemplary of how the other observed lessons proceed and the analysis presented later have been made in three different double lessons. The reason this specific lesson is presented here is, that the students independent work with

6 different representations and manipulatives visually shows the points clearly, as this lesson deals with geometry.

Presenting one double lesson

The observed lesson from the KiDM project follows the three-phased progression envisaged in the

KiDM project. The activity from the lesson can be found on the Danish webside: www.kidm.dk.

The lesson will first be described with specific quotes from the teachers and the students and afterwards analysed.

The case of “rope-triangles”

This activity is about conducting a systematic examination of different triangles. The activity is also used in other classes and analysed in the Danish part of PRIMAS (Artigue & Blomhøj, 2013). The intention of the activity is that the students must find all the possible triangles (whole number on the sides) with a rope with the circumference of 12 meter (and a knot tied each meter) and in this process learn about the triangle inequality.

The teacher introduces the double-lessons by talking about different kinds of triangles (right-angled triangles, obtuse triangles, acute triangles, equilateral triangles and isosceles triangles). The students then get handed out the ropes in groups of four or five and starts making different kinds of triangles outside their classroom. The students all have an expectation, that they can make many different triangles. An example of students doing this activity can be seen on figure 1.

Figure 1: Students investigating rope triangles

Picture from www.kidm.dk

The observed group, consisting of the students Ella, Alma and Nikolaj, quickly find by standing in different formations 6 different “triangles”: (4-4-4 + 2-6-4 + 2-5-5+ 6-1-5 + 6-3-3 + 5-4-3). In the process the group try to see if it is possible to make a triangle where one side is 7 meters, which they quickly reject, because the rope cannot in any way be outstretched. Finishing outside, they go back to the classroom and on the computer using the program GeoGebra they now must check if it is possible to draw the triangles in the programs. A student in the group (Ella) quickly draws all the triangles in GeoGebra, but when drawing the “triangle” 1-5-6 she, however, has some problems, but she anyway quickly announces “they all work”. Ella’s figure can be seen in Figure 2.

Figure 2: Ellas drawing of the "trianlge" 1-5-6

The teacher afterwards points at the 1-5-6 triangle and ask her: “Are you sure that this is a triangle?” Ella answer: “yes because it has three edges” When Nikolaj in the group tries to make the 1-5-6 triangle in GeoGebra he says: “that is not a triangle”. Ella answer him by saying “but it has three sides”. The third student (Alma) in the group announce: “it is very ugly – I am not cable of making this triangle” the discussion continues:

Nikolaj: Try to see mine!

Alma: That one is also very ugly.

Ella: [looking at her own triangle] Okay maybe now it fits better.

Nikolaj: [ask loudly to their teacher] We are finished what should we do now?

Ella: I would say that it is a triangle, because it has three edges, but wait a second. Wait a second.

It does not fit. No Nikolaj wait it is not a triangle, because these two (points at two sides) they are not long enough to reach together. That is why I don’t think it is a triangle. Those two together have the sum of 6 and that is why it is not a triangle, so Nikolaj it does not fit. There are only four triangles.

Nikolaj: But what about this drawing?

Ella: But it does not fit because this small space here shows that because 1+5 this is 6 and 6 + 6 is

12 or 5+1 this is 6 so it does not fit.

Nikolaj: Arhh okay!

Alma: But then we need to find new triangles.

Ella: Yes, and 7 does still not work. The number is not allowed to be bigger than 6 because then it does not fit at all, but I do not understand why that one fits [points at 6-4-2 triangle]?

Nikolaj: What?

Ella: I do not understand why that one fits – this one with 6-4-2 - when the other one does not fit.

[Ella put the sides from the triangle in GeoGebra again]. But Nikolaj try to see: this one does not fit either.

The groups afterwards try to find new triangles by using GeoGebra without any success.

Ella: If you started with 2 as the first line and then there could be one with 4, no it can only be 5 and 5 - we have that one. Okay then we go on and take 3 as the first line and then 4 and 5 and we also have that one. 3- 2 and then there must be 7 it cannot be done. 3 and 5 and then the last is 4 - we have made that one. Then we move on to 4 as the first line and then 6 and 2, but it can't be done either. Then there are 4 5 and 3 we have. And 4 and 2 it doesn't work. And 4 and 4 and 4 we have that one. Then 5 as the first line and 2 and 5 - yes and then 5 4 3 we have. We can't find any more.

And if we get to 6 then it doesn't fit anymore.

Different cognitive conflict in the students reasoning-process

After the thorough introduction, the students in the group believe that they can construct a lot of different triangles with the rope, but after the students experiments with the rope, they realise that this is not the case. This is done in a setting where the students work independently in a group, with no involvement from the teacher.

The activity with the rope shows the students that it is not possible to make triangle with a circumference of 12 with a side of 7 or larger, because then the rope cannot be stretch out. In this situation the rope visually helps the students to identify the cognitive conflict. A conflict between what the students think (all triangles are possible) and what the rope actually shows them (a side of

7 or larger is not possible). It is, however, still possible for the students to make a 1-5-6 “triangle”, a

6-4-2 “triangle” and a 6-3-3 “triangle” with the rope even though, it looks a little flat. The students work with the rope does in this case not give any useful constructive feedback and the students are still not aware of the triangle inequality. However, when the students afterwards work in the

GeoGebra-program, some of the students get more suspicious and skeptical – the 1-5-6-“triangle” does not fit very well, and “it looks ugly”, but they, however, still after some discussion conclude that the triangles with side 6 still works. In this situation the cognitive conflict intended to emerge by drawing the “triangles” in GeoGebra does not in the first case make the student realise, that they cannot make triangles where the sum of the two sides are not larger than the third side, because the feedback from the material is not constructive enough. The teacher, however, observes this error and intervene and pushes Ella a little, by asking her, if she is sure that the 1-5-6 triangle is a triangle. Initially this does not trigger any respond from Ella, but after a while Ella suddenly realise, by studying the drawings in GeoGebra again, that the triangles with side 6 does not works either and her arguments are this time not based on the rope or GeoGebra drawings, but on more formal

11 mathematics: the understanding of the triangle inequality. Afterwards the rest of the group now want to find more triangles, and these initiatives leads Ella to finally comes up with a systematic argumentation of how many triangles which is possible to find with this rope, where she explicitly articulates that she understands the triangles inequality.

In Figure 3 the steps in Ella’s reasoning process is illustrated with the different elements affected the process exemplified with blue arrows. Ella’s initial understanding ism that it is possible to make many different triangles and that there are no geometrical constraints apart from that the circumference should be 12. However, based on the interactions with the environment, Ella realises that the triangles sides should be less than seven. Later – due to the interactions with GeoGebra – the conflict becomes overt and the students discuss if 1-5-6 is a triangle. Pushed by the teacher Ella realises the triangle inequality and because she needs to explain her view to the group, she finally constructs a more or less deductive argument for the number of whole number triangles with 12 as circumference.

A systematic explanation of all the possible triangles - It is not possible to deductive make triangles with argumenting sides 6 - the triangles The drawings in get flat. Arguments geogebra do not from matheamtics matter - the triangles concepts It only work with with side 6 - still triangles with sides works. Arguments less than 7 - It is possible to make from empircal arguments from investigation many triangles - empirical arugments from the investigation students' rationale The students in the group The teacher want to find asks a The drawings The rope is not more triangles question in GeoGebra strected out

with sides greater than 7

Figure 3: The reasoning process in the Rope-triangle activity

Findings

In this paper we targeted to investigate the role of cognitive conflicts in students’ inquiry-based mathematical work by looking at one classroom episode of students reasoning process. Our overall findings are that a cognitive conflict exists in the analysed episode and that it is productive and important in relation to the students’ mathematical reasoning process. The cognitive conflict is in this sense the driving force for the students reasoning process, where the environment has a role of retaining the conflicting positioning and making them available for discussion and scrutiny. The process of resolving a cognitive conflict 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, but small steps towards resolving.

Instead the cognitive conflicts make the students involved in taking different routes and exploring different 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 short, we suggest that cognitive conflict can drive reasoning, and hence, that the mathematical environment plays a critical role to retain these cognitive conflicts. Finally, we argue that such an understanding of mathematical ideation as partly dependent on cognitive conflict, run counter to some of the typical understandings of that phenomenon.

Cognitive conflict can drive reasoning

In the analysed situation Ella experiences a cognitive conflict between the manifestation of the triangle inequality as ropes and as part of GeoGebra mathematical environment. This situation spurs

Ellas mathematical reasoning because her initial rationale (it is possible to create all sorts of triangles) is confronted with the reality, first in the situation with ropes, where she is able to accommodate her rationale to the empirical situation without creating cognitive conflict and later in relation to GeoGebra – where her rationale becomes difficult to maintain. In this specific situation

Ella has been able to experiment with and explore her (partly wrong) rationale (it is possible to create all sorts of triangles) and hence the conflict with the mathematical reality expressed in

GeoGebra is deeper, than if she was just confronted only with the triangle inequality or the

GeoGebra environment. This seems to increase Ella’s motivation for engaging in mathematical reasoning. The literature about reasoning in mathematics education has mainly focused on characterising the different types of argumentation and reasoning (Brousseau & Gibel, 2005; Harel

& Sowder, 1998, 2007), and less so about what makes students engage in reasoning processes and how to make the transition from making argumentation based on rationales and intuition to more deductive reasoning (EMS, 2011). We suggest, however, that Ella’s inclination towards more deductive reasoning is spurred by the cognitive conflict and that the deeper and more articulated the conflicting understandings are the larger the internal motivation for addressing this with mathematical reasoning.

A good environment retains cognitive conflict

One of the things that makes Ellas experience particularly strong is, that her wrong understanding/rationale is accommodated to the first exploration, and the work with the rope.

Therefore, Ella has a justified trust in her own rationale that makes it more difficult – we can

14 imagine – to give it up afterwards. By supporting that Ella retains and reinforces her rationale while also experiencing that it cannot be true, makes it pressing for Ella to address this. Student can typically easy dismiss conflicting views as exceptions and disturbances that does not need to be taken really into account (Zazkis & Chernoff, 2008) and the cognitive conflict must be explicitly invoked by the students. Hence the material and representations that support productive cognitive conflicts becomes more important, since students are more inclined to realise a conflict if the material and representations proponents for conflicting views are present at the same time and if the students independently have explored the conflicting views. In the analysed episode the conflicts are retained by material aspects of the environment and activities, such as the rope-exercise and the

GeoGebra exercise. The material aspects of the environment used in the lessons played an important role in producing the cognitive conflicts, because the materials in a visual way advocated for the conflict, the feedback form the materials, however, had different weight/power in the reasoning process. The feedback from the rope did not made it possible for the students to realise that the sides in the triangles cannot be 6 or larger, neither did the exercise in GeoGebra, but all together including the teachers hint made Ella resolve the cognitive conflict and understand the triangle inequality.

Resolving the cognitive conflicts will nevertheless take time. Ella experiences what can be described as an eureka moment when she realises the triangle inequality. This, however, does not happen as a response only to the environment she is engaged in at that time (the GeoGebra activity).

We suggest that this moment also owns credit to the work that Ella did with the ropes, where she experimented with and reinforced her ideas about that all sorts of triangles could be made. In that sense the fast realisation of how a conflict can be resolved owns it power to all the work that goes into exploring, articulating and reasoning about the different understandings that was in conflict. In

15 that sense the manifestation of conflicting views that can lead to student reasoning becomes a central design parameter in inquiry-based mathematic education.

Discussion and implications

Today, it is a common understanding that there must be an immediate learning benefit from all the activities students are working on in the classroom cf. the goal-oriented approach to teaching (R.

Hansen, 2015). This may entail, that if the students do not arrive immediately with a solution, the teacher will interrupt and give them the answer. However, it also entails that teachers will not allow complex and perhaps not immediately solvable tasks in the lessons, which, in addition to suppressing a natural urge to investigate, also deny the students to construct the answer themselves based on the productive cognitive conflicts, which can take time to resolve. During the KiDM interventions, we experienced teachers who would not apply the triangle-rope-activity, because the students were able to make “not-triangles” like the 3-3-6 triangle with the rope, which they found preventing for the students coming to understand the triangle inequality, since the students in a period of time have an incorrect understanding like Ella. It is therefore necessary, that mathematics teachers are aware that resolving a cognitive conflict can include small steps and that it will take time. The teachers need to be patient before they interact and interrupt the students otherwise it will destroy the students’ motivation for making their own inquiry and be part of the reasoning process and possibly preventing a deep understanding of the target concept.

Additionally, it is important that the teachers plan, that the material aspects of the environment advocate for resolving the conflicts. It is not necessary that the material aspects provide a definitive resolution, but they must advocate for a later resolution.

Conclusion

By analysing video observations from one year 5 primary school mathematics lessons where the teacher use an inquiry-based approach, we studied the students reasoning process, by finding and characterising the students’ cognitive conflicts. We found that the productive cognitive conflicts are important to mathematical reasoning, because they in some way can be seen as the motor that helps the students in the transition from empirical to more formal deductive argumentation in inquiry- based teaching. The environment will in these cases have an important role of retaining the conflicting positioning making them available for discussion and scrutiny.

References

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14.4 Paper IV

Title Developing a Validated Test to Measure Students’ Progression in Mathematical Reasoning in Primary School

Authors Dorte Moeskær Larsen & Morten Rasmus Puck

Status Manuscripts submitted for publication to: International Journal of Education in Mathematics, Science and Technology

This version of the paper is the original document send in for publication.

209

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Developing a Validated Test to Measure Students’ Progression in Mathematical Reasoning in Primary School

Dorte Moeskær Larsen1, Morten Rasmus Puck2

1LSUL, Southern University Denmark, [email protected], (Corresponding Author)

ISSN: 2147-611X 2University College Lillebælt, Denmark. [email protected]

To cite this article:

Larsen, D.M. & Puck, M.R. (2019). Developing a Validated Test to Measure Students’ Progression in Mathematical Reasoning in Primary School. International Journal of Education in Mathematics, Science and Technology (IJEMST), Vol(No), Page X-Page Y. DOI: 10.18404/ijemst.XXXXX

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International Journal of Education in Mathematics, Science and Technology

Volume X, Number X, Year DOI:10.18404/ijemst.XXXXX

Developing a Validated Test to Measure Students’ Progression in

Mathematical Reasoning in Primary School

Dorte Moeskær Larsen, Morten Rasmus Puck

Article Info Abstract Article History Not enough effort has been invested in developing reliable and valid assessment instruments to measure students’ development of reasoning Received: competences in mathematics. Previously developed tests rely mostly on 01 Month Year standardised multiple-choice assessments, which primarily focus on procedural knowledge and rote learning and not on how students argue for and Accepted: justify their results. This study reports on the process of developing a test to 01 Month Year assess students’ reasoning competence in primary school classes and on the

statistical results of a verification study. The test is based on item response Keywords theory (Rasch model) and was used in a large-scale mathematics intervention Assessment, project about inquiry-based teaching. The data was collected from 5,516 Item Response Theory, primary school students in Denmark. Data analysis indicates the test has Mathematical Reasoning, satisfactory validity and reliability, but the test was still unable to measure Primary School statistically significant progress at the intervention schools.

Introduction

Despite the importance of reasoning in school mathematics (Ball & Bass, 2003; Hanna & Jahnke, 1996), research shows that many students at all school levels face serious difficulties with reasoning and proving (EMS, 2011; Harel & Sowder, 1998). Consequently, children enter upper grades ill-equipped to develop their justifications and proofs (Knuth, 2002).

Multiple factors may have contributed to the fact that reasoning and proof have received a marginal role in primary and lower secondary school mathematics teaching (G. J. Stylianides & Stylianides, 2017). However, one particular hindrance could be that, if the way we assess students is focused only on procedural knowledge and not on reasoning competences, there is a risk that tests can obstruct the development of more complex mathematical competences by distorting the focus in teaching away from a competence orientation. As Biggs (2011, p. 197) argued, assessment determines what and how students learn more than a curriculum does. Even in achievement tests in which the primary recipients are people at some distance who require an assessment of an overall effect, rather than detailed information on individual students, assessments can have a 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). As a consequence, achievement tests need to be closely aligned with the target knowledge of the curriculum, but, although reasoning is well-known in mathematical teaching practice and has been the focal point of many studies (Lithner, 2008; Reid & Knipping, 2010; A. J. Stylianides & Harel, 2018), there is limited research on how to assess students’ competence with reasoning in mathematics education.

This paper introduces a newly developed large-scale achievement test that uses a Rasch model and is intended to measure students’ development of competence in mathematics. To determine the best way of conducting assessment in a particular case and context, it is necessary to consider the properties of possible tools in relation to the purposes and intended uses of the assessment results. Obvious important desirable properties of any assessment are its reliability and its validity for the intended purpose (Johnson & Christensen, 2014), which will be the focus in this paper.

2 Larsen & Puck

Theoretical Background

Developing a test that can measure reasoning competence requires both a study of how others have tried to measure it in the past and consideration of how to make the test valid and reliable. In the following, these two considerations will be elaborated.

Testing Reasoning in Mathematics Education

In both of the two major international large-scale assessments – the Trends in International Mathematics and Science Study (TIMSS) and the Programme for International Student Assessment (PISA) – we have seen some effort to measure reasoning competence, but, in general, the testing of competences in mathematics education is not a widespread nor well-documented area. Niss, Bruder, Planas, Turner, and Villa-Ochoa (2016) argued that assessment of students’ mathematical competences needs to become a priority in educational research from both holistic and atomistic perspectives, where a holistic perspective considers complexes of intertwined competences in the enactment of mathematics and ‘an atomistic perspective zooms in on the assessment of the individual competency in contexts stripped, as much as possible, of the presence of other competences’ (Niss et al., 2016, p. 624).

Through a literature review of assessment in mathematics (Larsen, 2017), created with a focus on developing such a test, different aspects of measuring reasoning in mathematics were found. Logan and Lowrie (2013) focused on testing students’ spatial reasoning in mathematics. In Nunes, Bryant, Evans, and Barros (2015), a framework for prescribing and assessing the inductive mathematics reasoning of primary school students was formulated and validated. The major constructs incorporated in this framework were students’ cognitive abilities for finding similarities and/or dissimilarities among the attributes and relationships of mathematical concepts. Nunes et al. (2007) focused on reasoning in logic, while Goetz, Preckel, Pekrun, and Hall (2007) examined reasoning in connection to another aspect – test-related experiences of enjoyment, anger, anxiety, and boredom and how they relate to students’ abstract reasoning ability.

Overall, we see that the term reasoning is broadly used in mathematics education and is connected to many different areas, such as logic, spatial, and cognitive abilities. Reasoning in mathematics is therefore not a unified concept, which makes an assessment of this competence a challenge. Yackel and Hanna (2003) argued that the reason for disagreement on the definition is, in fact, an implicit assumption of universal agreement on its meaning, yet there are many different conceptualisations of mathematical reasoning in the literature.

Brousseau and Gibel (2005) defined mathematical reasoning as a relationship between two elements: a condition or observed facts and a consequence. Duval (2007) described mathematical reasoning as ‘a logical linking of propositions’ (p. 140) that may change the epistemic value of a claim. G. J. Stylianides (2008) viewed mathematical reasoning as containing a broader concept; besides providing arguments (non-proof and proofs), it also encompasses investigating patterns and making conjectures. In Shaughnessy, Barrett, Billstein, Kranendonk, and Peck (2004), the National Council of Teachers of Mathematics (NCTM) described reasoning in mathematics as a cycle of exploration, conjecture, and justification. This is in line with G. J. Stylianides (2008), who focused on mathematical reasoning as a process of making inquiries to formulate a generalisation or conjecture and determine its truth value by developing arguments, where argument is understood to mean a connected sequence of assertions. In a Danish context, the definition of reasoning competence in the primary school mathematics curriculum is often based on the competence report (Niss & Jensen, 2002), which, among other factors, is very specific about students being able to distinguish proof from other kinds of argumentation and makes a clear distinction between ‘carrying out’ argumentations and ‘following’ the argumentations developed by others (e.g., other students or textbooks).

In the present paper, the definition of reasoning that will be tested in the developed testis a composite of definitions from the NCTM, the competence report (Niss & Jensen, 2002), and G. J. Stylianides (2008). This will be described in more detail in the Methods section.

Reliability and Validity in Tests

The term validity refers to whether or not a test measures what it claims to measure. On a test with high validity, the items will be closely linked to the test’s intended purpose. In 1955, Cronbach and Meehl wrote the classic article, Construct Validity in Psychological Tests, in which they divided test validity into three types: content, Int J Educ Math Sci Technol 3

construct, and criterion-oriented. For many years, the discussion has been about these different types of validity, but, in this paper, the validity issue will be focused more on obtaining evidence for unitary validity (Johnson & Christensen, 2014), which includes all three types of validity but in which the central question is whether all the accumulated evidence supports the intended interpretation of test scores for the proposed purpose.

The intention in this study is therefore to present and discuss whether the collected evidence supports the argument that the test can be seen as valid. Johnson and Christensen (2014) argued that, in this sense, complete validation is never fully attained – it is not a question of no validity versus complete validity. Validation should be viewed as a never-ending process, but, at the same time, the more validity evidence one has, the more confidence one can place in one’s interpretations.

The other aim in the development of this test is for it to provide reliable or trustworthy data. If a test provides reliable scores, the scores will be similar on every occasion, which is related to generalisation from the results (Black, 1998).

In summary, the aims of this study are to develop a test that, among other things, can measure reasoning competence in primary school classes and to examine the quality of that test.

Methods The methods section starts with at short description of the setting of the assessment. A description then follows of how the development process was divided into three different phases: design, development, and testing. The three phases are briefly reviewed, followed by information on how they relate to each other. The text then describes phase 1, which, in addition to a description of the definition of reasoning, also includes an example of an item from the test. In phase 2, the pilot study and the measurement model are described, including intercoder reliability, and, in phase 3, elaboration and consideration of content validity are described.

The Origin of and Reasons for Developing a Competence Test

This study is embedded in a large-scale, three-year, design-based, randomised-controlled-trial research and development programme in Denmark, called ‘Kvalitet i Dansk og Matematik’ (Quality in Danish and Mathematics; KiDM). The overall aim of KiDM is to make teaching in years 4 and 5 of Danish primary school more inquiry-based, and it includes the development of inquiry-based teaching activities for a four-month mathematics teaching plan implemented in 45 Danish schools (intervention schools). To assess students’ development of mathematical competences related to inquiry-based teaching, an achievement test, with a strong focus on mathematical competences, was essential to measure the difference between the 45 interventions schools and 37 control schools. A test was therefore developed specifically to measure whether students developed specific competences in mathematics. These competences included those of problem-solving/-posing, modelling, and reasoning.

In this study, we will focus on whether this test is valid and reliable in connection to the part of the test connected to reasoning competence. The intent in the test is to use a Rasch model analysis, which is a psychometric technique that can improve the precision with which researchers construct instruments, monitor instrument quality, and compute respondents’ performances (Wilson & Gochyyev, 2013). Rasch analysis allows researchers to construct alternative forms of measurement instrument because it does not treat a test scale as linear, with the raw scores of different respondents simply ‘added up’ to compare their levels of achievement. Rasch analysis instead allows researchers to use respondents’ scores to express their performance on a scale that accounts for the unequal difficulties across the test items. It is also a tool for finding items that measure the same phenomenon.

Designing, Developing, and Testing the Test

In this section, the development of the item design and the coding processes will be elaborated, together with the verification. The development was done in an iterative process comprising three different phases before the test was finally applied in three randomised controlled trials as part of KiDM.

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Phase 1: Designing the test: The components in the test (content and design) were based on theory identified in the established literature in a collaboration between the authors of this paper, two associate professors from two different mathematics teacher education colleges, and one professor in mathematics education.

Phase 2: Developing the test: Teachers from a primary school class and their supervisor in mathematics tested the test in their classroom, and interviews with three of their students were conducted. Think-alouds were also conducted with one student from year 5 and one student from year 4.

Phase 3: Testing the test: The test was piloted in 14 classes, with a focus on items appropriate to the intended pupils; the coding procedures were further developed, and the test administration was itself tested.

The three phases were not neatly divided because the various tests and trials required some corrections to the test, including the difficulty of the items, and the development of the coding process during the pilot test required changes to the content and the formulations, which needed consideration of the design and content, as will be described later.

Phase 1: Designing the test. In order for the test to measure the content of reasoning competence, which is aligned with the definition of reasoning competence in KiDM, it was decided that the definition should be process-oriented but also have a broad approach. Therefore, definitions from the NCTM, the competence report (Niss & Jensen, 2002), and G. J. Stylianides (2008) were all included. In Table 1, the three different broad definitions from the Introduction section are included in the left column to allow comparison of the definitions, and, in the right column is listed what is intended to be measured by different items in the test (only the final included items are listed).

Table 1 Content of items included in the final KiDM test

Standards Niss & Jensen G. J. The KiDM test Items with this from NCTM (2002) Stylianides focus (item (2008) number)* A Make and Making Can the students 1, 9, 10, 14, 15, investigate conjectures make inquiries to 17, 18, 19, 26 mathematical formulate conjectures assumptions/conjectu res in mathematics? B Investigating Can the students 13, 18 patterns and make inquiries to making formulate generalisations generalisations in mathematics? C Develop To devise informal Providing Can the students 1, 9, 17 mathematical and formal reasoning support to develop arguments (on the basis of mathematical argumentations in and proofs intuition), including claims; mathematics? transforming providing non- (proof/non-proof) heuristic reasoning proof into actual (valid) argumentations proof and providing proofs D Evaluate To understand and Can the students 15, 19, 29 mathematical judge a understand and judge arguments mathematical argumentations in and proofs argumentation mathematics? propounded by (proof/non-proof) others E Use various To devise and Providing Can the students 28, 29 types of implement informal support to select and use various reasoning and and formal reasoning mathematical types of reasoning? methods of (on the basis of claims; proof; intuition), including providing non- Int J Educ Math Sci Technol 5

select and use transforming proof various types heuristic reasoning argumentations of reasoning into actual (valid) and providing and methods proof proofs of proof F Select and To know and Can the students Items with this use various understand what a distinguish between content have types of mathematical proof different kinds of been removed reasoning is and how it is arguments – because they and methods different from other rationales, empirical were found too of proof argumentations arguments, and difficult for the deductive arguments? students in years 4 and 5

*Note: the remaining items focus on modelling competence or problem-solving/-posing competence.

The items are the substance of the test and have been developed with inspiration both from the research, such as A. J. Stylianides (2007), and from published TIMSS and PISA items as both tests explicitly state that they measure mathematical reasoning and, moreover, TIMSS is aimed at the same age range as our population.

The items included in the test are a mixture of multiple-choice and closed-constructed response questions as well as open-constructed problems that include, for example, making argumentation for one’s own conjecture or justifying one’s choice of which argumentation is correct in different problems. This specific mix was chosen because we wanted to use different types of item to accommodate different approaches to those items, but also because open items are very costly in the coding process and, therefore, it was not possible to have only open- constructed items. Multiple-choice items and closed-constructed items, in contrast, can be automatically scored by computer and are, therefore, almost cost-free. The open-constructed items were, in the KiDM project, manually scored by three (two in the last trial) preservice teachers. The open-constructed items include a mixture of dichotomous and polytomous questions; the dichotomous questions can only be coded 0 or 1 (incorrect or correct), while the polytomous questions have up to four different scoring categories. In Figure 1, an open-constructed item (item 9) from the KiDM test is presented that explores the similarities and differences between relationships in statistics in a close-to-reality task. In this task, the student must show, in connection to A and C in Table 1, that he or she is able to make inquiries to formulate a conjecture and develop argumentation to verify his or her conjecture.

Figure 1. Test item 9: Comparing temperatures in two different weeks

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In order to differentiate the quality of students’ answers to the different open-ended items, coding taxonomies were developed for each. These coding taxonomies (schemes) were based on a theoretical progression described in the literature; however, the concept of argumentation has very different definitions in the literature, and, to explain item 9’s coding scheme, we first look at theoretical descriptions of different approaches to taxonomies.

Arguments are often divided into non-proof arguments and deductive proof arguments (Reid & Knipping, 2010). A deductive argument is one that, if valid, has a conclusion that is entailed by its premises. In other words, the truth of the conclusion is a logical consequence of the premises and, if the premises are true, the conclusion must also be true. It would be self-contradictory to assert the premises and deny the conclusion, because the negation of the conclusion is contradictory to the truth of the premises. Therefore, deductive arguments may be either valid or invalid (Reid & Knipping, 2010).

G. J. Stylianides (2008) distinguished between two kinds of non-proof argument: empirical arguments and rationales. An argument counts as a rationale if it does not make explicit reference to some key accepted truths that it relies on or if it relies on statements that do not belong to the set of accepted truths of a community. Harel and Sowder (1998) introduced the concept of proof schemes, which classify what constitutes ascertaining and persuading for a person (or community). The taxonomy of proof schemes consists of three classes: external conviction, empirical conviction, and deductive conviction. Within the external conviction class of proof schemes, to prove depends on an authority, such as a teacher or a book. It could also be based strictly on the appearance of the arguments or on symbol manipulations. Proving, within the empirical proof scheme class, is marked by reliance on either evidence from examples or direct measurement of quantities, substitutions of specific numbers, or perceptions. The third, deductive proof scheme class consists of two subcategories, each consistent with different proof schemes: the transformational proof scheme and the axiomatic proof scheme.

In Table 2, the created theoretical taxonomy of items about ‘inquiry to make a conjecture’ and ‘developing argumentation’ (such as item 9 in Figure 1) is presented, based on theories from Harel and Sowder (1998) and G. J. Stylianides (2008). The scale is ordinal.

At Location I, students are only able to develop a conjecture, and there are no arguments at all. At Location II, the conjecture is explicit, but the argument is only a rationale (G. J. Stylianides, 2008) or from an external conviction proof scheme (Harel & Sowder, 1998). Location III includes students who can support a developed conjecture with an empirical argument. Locations IV and V indicate students who can make a conjecture, draw implications by linking pieces of information from aspects of the problem, and make arguments from one side (Location IV) or more than one side (Location V). This resembles what Harel and Sowder (1998) called a deductive proof scheme.

In the design phase, the research group’s main focus was on whether each textual passage and its associated items adhered to a theoretical model, but, in the development and testing phases, the main focus was on whether the students’ responses were associated with these theories.

Table 2 A theoretical model of the taxonomy of reasoning in mathematics for item 9

Location Taxonomy developed from theory Coding scheme for item 9 Codes for item 9 V Develop/follow/critique assumptions/claims/statements with two- sided comparative deductive/mathematical arguments. Draw implications by linking pieces of information from separate aspects of the problem and make a chain of argumentation using deduction/mathematical concepts. IV Develop/follow/critique ‘The mean is a good way of 3 assumptions/claims/statements supported comparing the temperature, and with one-sided mathematical/deductive the mean is higher in January than arguments. in December because, in Draw implications by linking information December, the mean is 3 and in Int J Educ Math Sci Technol 7

from the problem with argument(s). January it is 4. We can say that it is warmer in January.’ III Develop/follow/critique ‘In January, it’s a little warmer. 2 assumptions/claims/statements supported The highest temperature is 10 and with empirical argument. the lowest is 1.’ Draw implications from reasoning steps ‘Monday difference = 6, Tuesday within one aspect of the problem that difference = 1, Wednesday involves empirical entities. difference = 0, Thursday difference = 3, Friday difference = 1, Saturday difference = 4, Sunday difference = 2. Warmest in January.’ II Develop/critique an ‘December is snowy, so it is 1 assumption/claim/statement supported colder, and in January there is no without a sufficient argument. snow.’ Draw implications with only a simple rationale or external conviction arguments. I Only an assumption/claim/statement ‘December was the coldest week.’ 1 without argument. (no arguments – only an Draw implications without any assumption) argumentation. ‘January is warmer.’ 0 No claim or assumption in connection to Blank 0 the question – off track.

Phase 2: Developing the test. The coding guides were developed in an iterative process within the research group (the authors of this paper together with an associate professor) and teachers. Trainee teachers participated as raters in this process, and, to be sure of intercoder reliability, they double-coded 20% of all items in both the pilot study and the three trials in the KiDM project, to fine-tune the categories, all of which needed to be exhaustive. There was 82% consistency between the double-coded items in the KiDM project. Agreement between the scores awarded for TIMSS in 2011 and 2015 ranged, on average, from 92% in science to 98% in mathematics among the year 4 students (Foy et al., 2016), but it is important to note that most TIMSS items are dichotomic, while the current test has polytomous items with up to four different codes, and so the intercoder reliability will inevitably be lower.

The number of test items was relatively small because it was not possible to include many different open-ended items due to it taking a long time for students in years 4 and 5 to answer such questions, for which they had to write their arguments. For ethical reasons, no more items were developed, as the assignment would then be too extensive for these 9-to-11-year-old children to complete. The person separation index would therefore be low in the test and could have been increased by expanding the number of items.

The primary function of the measurement model is to bridge from the empirically found scores to the theoretically constructed model (Wilson & Gochyyev, 2013). The measurement model chosen in this test is the Rasch model. Georg Rasch (1960) introduced a probabilistic model for item response, which gives the probability p of scoring a given category k for item i, with difficulty δik and the ability of the respondent θ; m is the maximum score for the item. Here is the partial credit model:

( k ) ( = k) = 1 + m𝑘𝑘 𝜃𝜃𝑛𝑛 −𝛿𝛿𝑖𝑖( k ) 𝑒𝑒k=1 𝑝𝑝 𝑥𝑥𝑛𝑛𝑛𝑛 𝑘𝑘 𝜃𝜃𝑛𝑛 −𝛿𝛿𝑖𝑖 ∑ 𝑒𝑒 Rasch analysis allows us, for example, to compare the difficulty of mathematical reasoning problems while also locating the degree to which individual students have mastered the necessary skill set. This location of reasoning competence and students on the same unidimensional scale allows a fine-grained analysis of which aspects of the reasoning process being analysed makes one problem more difficult than another. The analysis was done using the RUMM2030 software (Andrich, 2009).

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In Figure 2, we see how the probabilities of the different scores (0, 1, 2, 3) develop as a function of the students’ ability on item 9 (from Figure 1). When we use a Rasch model as a measuring model, it is important to acknowledge that the answer at a higher location must not only differ from the answer at a lower location, but also be better in a significant way – it must be regarded as superior to the answer at a lower level because, in the Rasch model, we regard students who give an answer at a higher level as showing more ability than those who do not (Bond & Fox, 2015).

Figure 2. The probability curves for Item 9

Based on the students’ responses from the pilot test, each item was analysed using the Rasch model to calibrate the measurement tool, with the aim of eliminating obvious misfits from the test; six items were thus removed after the pilot test.

Phase 3: Content validity after the pilot test. To make sure the items were measuring the specific competence (content validity), two things were done. Together with a mathematical researcher, the first author of this paper worked through all the answers from the pilot schools to determine the extent to which the theoretical purposes were revealed in the students’ responses. This means that all the different types of student response were evaluated, and it was determined whether these responses were indicative of the student showing reasoning competence; whether the answers were more about being able to, for example, multiply; or whether they were non-rational guesses that did not show mathematical reasoning.

Furthermore, think-alouds were conducted with two students in which the students’ responses were audio recorded and afterwards analysed to ensure both that the items met the requirement of being understandable to the students’ age groups and that the students’ responses were desirable answers within the intended content areas. This was to verify that the students, in the process of answering the items, did mathematical reasoning – for example, whether they considered different arguments for their solutions or whether their answers were developed by a using simple skill algorithm or rote learning. As a result, many formulations of the items were changed, and more items were added in the different mathematical areas.

A good measurement process in education will provide for the estimation of just one ability at a time and will not intentionally or unintentionally confuse two or more human attributes into one measure. This is, however, difficult in close-to-reality items. For example, one item might rely on language comprehension to understand a context, while another might focus too much on students’ ability to interpret a complex geometrical drawing or algebraic reduction. The resulting score would then be uninterpretable, to the extent that these other abilities are manifested in the students’ responses. The important thing for us was that these aspects not overwhelm the role of the desired attributes. In the competence test, it was therefore decided that each of the items should contribute in a meaningful way to the construction of a mathematical competence and that each mathematical competence should have content from all areas – algebra, geometry, and statistics. Some competences had several items from each area.

Results Int J Educ Math Sci Technol 9

Construct Validity in the KiDM Test

To validate the construct of the KiDM test, we first conducted four important tests: a monotonicity test, a test for unidimensionality, a test for response dependence, and a measurement of differential item functioning (DIF). The analyses were performed using the test data collected at the baseline (in all three trials).

To investigate whether items met the requirement of monotonicity, two different conditions were examined. Monotonicity concerns whether the likelihood of a student responding correctly to an item increases the better the student is. First, we graphically inspected the empirical curve of each item, which shows the proportion of correct responses for student groups, broken down by their skill level and matched with the theoretically expected difficulty of the task. We also inspected the fit-residual statistics (as well as the chi-squared test), which tells how large the deviations are between an item on the empirical and theoretical curves. From a trade- off between the graphical inspection and the fit residual, four were excluded from the scale because the deviations between the empirical and theoretical curves were too large.

We also investigated whether the items fit to the dimension (a unidimensionality test) and did not overdiscriminate or underdiscriminate. With residuals smaller than −2.5, we found 11 items that overdiscriminated, while there were 22 items that underdiscriminated, with residuals larger than 2.5. For students with low ability, an item with a large discrimination would be the most difficult, but the same item would be the easiest for a person with high ability. Other items would overtake the difficulty of the high- discriminating item for the high-ability students, and the item difficulty would depend on the sample of students. We therefore deleted the 10 most extreme items.

To investigate whether there was response independence between the individual items, we looked at the residual correlations between each item pair. A residual correlation indicates whether the answer to a task affects the probability of responding correctly to another task. One item pair had a residual correlation above 0.3 and was eliminated from the test; this was a sign of a breach of the requirement for local independence between the items. The eliminated item was selected from a closer examination of which of the two items in the item pair had the highest residual correlation with other items.

Finally, to ensure that particular groups were not prejudiced, the DIF was tested. This involves controlling for respondents’ overall ability and identifying whether an item favours one group of respondents over another. Some groups might perform differently overall, but it is not desirable that students from one group find it harder or easier to answer an item correctly than students of equal ability from another group. We checked for the DIF according to grade level and whether the students attended an intervention school or a control school at the time the student took the test.

As assumed, we found no significant differences caused by the students’ belonging to the intervention or control group, because the schools were randomly selected, although we found some unproblematic differences in the grade level. Figure 3 shows that year 5 students (horizontal lines) had some advantages over year 4 students (sloping lines) because they were taught mathematics during their additional year of schooling. This did not, however, pose any DIF issues. We also saw some similarly unproblematic differences between the three trials: Trial 2 of the competence test had the highest mean (−.571), while the mean for Trial 1 was −.744 and the mean for Trial 3 was −.915. This is unsurprising, as Trial 2 took place in the spring, so the classes in that trial had attended school half a year longer than the classes in Trials 1 and 3.

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Figure 3. Person–item threshold in the competence test in school years 4 and 5

Validating through item maps.

By using the Rasch model to analyse the test answers, it was also possible to use an item difficulty map (Van Wyke & Andrich, 2006) (see Figure 4). This map shows the relationship between items and students, taking advantage of the fact that persons and items are measured on the same scale. The logit scale, which is the measurement unit common to both person ability and item difficulty, is displayed down the middle of the map. Because the logit scale is an interval-level measurement scale, equal distances at any point on the vertical scale represent equal amounts of cognitive development.

On the item map, the distance of the step from the bottom of the path represents its difficulty – items closer to the bottom are easier, those further up are more difficult. To the left, the students are indicated by small crosses. In Figure 4, one cross indicates 27 students. Students closer to the bottom have less ability than students at the top. SU is an abbreviation for the Danish word Scorede Undersøgende-opgave, which can be translated into scored inquiry-based items.

An item map can help researchers identify the strengths and weaknesses of an instrument, such as if some test items are measuring the same part of the variable or if there are areas of the tested variable that are missing from the test due to a lack of items with different levels of difficulty. In developing this test, our aim was to place enough stepping-stones along the path between little development and much development to represent all the points useful for our testing purposes; to do that, we needed to collect observations from enough suitable persons, but we also needed to have enough items.

Int J Educ Math Sci Technol 11

Figure 4. Item map for the items in the competence test

In the item map in Figure 4, we see that there are some gaps in the items along the vertical line, which may indicate that we lack some items with a specific level of difficulty. We can also use this item map to study the validity of the test by exploring whether there is consistency between the theoretically developed levels and the empirically developed item map. The idea of the item map is that each student will progress along the steps as far as their ability allows – a student will master the steps until the steps become too difficult. How far any student moves along the pathway will be our estimate of the student’s ability development. We must, however, remember that the Rasch model is probabilistic and therefore does not mean the students will correctly answer all the items up to the point at which they are placed and incorrectly answer all the items above it. Rather, the item map suggests that a person has a more than 50% probability of responding correctly to tasks below their skill level and a less than 50% probability for tasks above their skill level. The expected outcome will therefore be different from the actual outcome.

Such comparisons facilitate an assessment of construct validity by providing evidence that the instrument is measuring in a way that matches what theory would predict. In Figure 5, the item map from Figure 4 has been coded, which is shown with different colours. The items that do not have a specific focus on reasoning competence have been removed. The black boxes with white numbers are items for which the students made correct assumptions and claims but without any argumentation (Locations I and II in Table 2), and the grey boxes with black numbers are items for which the students made argumentations for their assumptions or for critiques (Locations III, IV, and V in Table 2). Figure 5 clearly shows that it is more difficult to produce arguments than to simply present assumptions or elicit claims in the test items.

The item at the top, SU9.3, is shown to be the most difficult item. In this item, code 3 (from Table 2) is where the students had the chance to show they could draw implications by linking pieces of information from separate 12 Larsen & Puck aspects of the problem to make arguments (Location V). In the beginning of the test-development phase (phase 2), more items in the test were possible to be answered in this way, but there simply were not enough students answering correctly at this difficulty level for this coding to be retained in the model, so we had to remove these codes from a few items and rethink the code boundaries or combine some codes. This might simply indicate that this is a very difficult way to reason for students in school years 4 and 5.

Figure 5. Item map with codes indicating argumentation

In Figure 6, a new coding has been made: the grey boxes with black numbers are complex items and the black boxes are non-complex items. A complex item is defined here as one in which the students need to calculate more than one result before being able to make an argument or claim. In the Figure, we see that most of the complex items are at the top.

Int J Educ Math Sci Technol 13

Figure 6. Item map with codes indicating complexity of items

Discussion of Validation

During the analysis, we coded for other aspects in the item map as well. For example, we coded items in which the students developed their own claims compared to items in which the students only had to understand existing claims. This coding, however, only showed that one was not significantly harder than the other, but we observed that some items can probably be solved using different methods and that this may call for the activation of the competences in different combinations or at different levels, which may have changed the item map in another direction. It is therefore recognised that some degree of dissimilarity in the outcomes is inevitable.

These kinds of comparison help to facilitate an assessment of validity by providing evidence that the instrument is measuring in a way that matches what the theory would predict. To test students’ ability in mathematical reasoning and to be aware of how it grows and becomes more sophisticated over time is interesting for multiple reasons. It is interesting for the students to be aware of where they are heading, but it is also important for researchers in the development of curricula, teaching materials, and tests and other evaluation tools.

Testing Mathematical Competences in the KiDM Project

The final number of items after calibrating the test was 23, of which 13 were focused on reasoning. This number may be considered large when one contemplates that open-ended items require students to write a short passage, but in the context of the Rasch model's requirements, the number is relatively small. After collecting baseline and endline data in the control and intervention schools in all three trials, the final results of the KiDM test indicate that it is not possible to measure any positive effect in student mathematical competences in the intervention schools compared to the control schools, with a statistical significance level of 0.05. This means 14 Larsen & Puck

that we cannot rule out that the differences between the intervention and control group levels of mathematical competence were coincidental.

There may be several reasons for this. The intervention, over a relatively short period, may not trigger any significant changes for the students at the intervention schools compared to the control schools, who may also have been taught competences in mathematics. Another issue is that, even though there were many students included in the trial, the randomisation was carried out at school level – cluster-corrected standard errors were used, taking into account that students who attended the same school were not independently drawn, but the students were clustered in schools. The number of intervention schools was 38, with 45 control schools, which is a relatively small number and may also have had an effect on the possibilities for finding a significant result.

However, we can identify two reasons why the test may not have worked as desired. Firstly, the fitting of the items to the construct meant that a relatively high percentage of the items were removed, resulting in relatively few items (23) that fitted the model; this relatively small number of items would affect the results. Secondly, many of the items were located at the difficult end of the scale, compared to the students, which means that many students did not respond correctly to any of the items.

Conclusion and Implications

We have shown that there is some consistency between our empirically developed item map and the theoretically developed map, but we also found some limitations. One disadvantage of the data is that, in this test, the items are separate components – it is unclear whether they can ever add up to higher-order thinking in which knowledge and skills can be detached from their contexts or their practice and use. Is it at all possible to measure a skill component in one setting that is learned or applied in another? This means that, perhaps, we cannot validly assess a competence in a context very different from the context in which it is practised or used (Resnick & Resnick, 1992); the test might have been improved if this had been even more in focus during development.

Moreover, we also see some limitations in connection with students who have difficulty writing. Although the tasks could be read out by a computer, the year 4 students especially may have had trouble arguing their answers in writing. Here, it could have been interesting for the students to be able to draw their answers, which, unfortunately, was not possible in our setting. Finally, a measure of concurrent validity is missing, and it could have been interesting to compare the students’ scores in this test to their scores in the national test, to see if the scores were predictive; however, this was also not possible in our setting.

The item maps are very interesting from a mathematical educational research perspective, because they can inform the discussion about what students find easier or more difficult using empirical findings and not just expert opinion. Many different codings could be made; for example, are items with context or without context more difficult? Do illustrative pictures make an item easier? However, to do studies like this, we need to be very aware of how the items are developed, to ensure that these aspects can be measured.

In conclusion, this paper presents a serious attempt to answer how an achievement test to measure reasoning competence in mathematics can be designed, developed, and tested, and the KiDM test is a serious suggestion for a more open-ended test that measures mathematical competence. Our purpose in this paper is to encourage mathematical education researchers to not only criticise achievement tests, but also to focus on how we can develop new alternatives to measuring other competences or to improve already developed tests, given that testing seems to be here to stay.

References

Andrich, D., Sheridan, B., & Luo, G. (2009). RUMM2030: Rasch unidimensional models for measurement. Perth, Western Australia: RUMM Laboratory. Ball, D. L., & Bass, H. (2003). Making mathematics reasonable in school. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.), A research companion to principles and standards for school mathematics (pp. 27–44): Reston, VA: National Council of Teachers of Mathematics. Biggs, J. (2011). Teaching for quality learning at university: What the student does. London, UK: McGraw-Hill Education. Int J Educ Math Sci Technol 15

Black, P. J. (1998). Testing, friend or foe? The theory and practice of testing and assessment. London, UK: Falmer Press. Bond, T. G., & Fox, C. M. (2015). Applying the Rasch model: Fundamental measurement in the human sciences (3rd ed.). New York, NY: Routledge. Brousseau, G., & Gibel, P. (2005). Didactical handling of students’ reasoning processes in problem solving situations. In C. Laborde, M.-J. Perrin-Glorian, & A. Sierpinska (Eds.), Beyond the apparent banality of the mathematics classroom (pp. 13–58). Boston, MA: Springer US. Cronbach, L. J., & Meehl, P. E. (1955). Construct validity in psychological tests. Psychological Bulletin, 52(4), 281–302. Duval, R. (2007). Cognitive functioning and the understanding of mathematical processes of proof. In P. Boero (Ed.), Theorems in school (pp. 137–161). Rotterdam, Netherlands: Sense. EMS. (2011). Do theorems admit exceptions? Solid findings in mathematics education on empirical proof schemes. Newsletter of the European Mathematical Society, 81, 50–53. Foy, P., Martin, M. O., Mullis, I. V. S., Yin, L., Centurino, V. A. S., & Reynolds, K. A. (2016). Reviewing the TIMSS 2015 achievement item statistics. In M. O. Martin, I. V. S. Mullis, & M. Hooper (Eds.), Methods and procedures in TIMSS 2015 (pp. 11.11–11.43). Retrieved from http://timss.bc.edu/publications/timss/2015-methods/chapter-11.html Goetz, T., Preckel, F., Pekrun, R., & Hall, N. C. (2007). Emotional experiences during test taking: Does cognitive ability make a difference? Learning and Individual Differences, 17(1), 3–16. Hanna, G., & Jahnke, H. N. (1996). Proof and proving. In K. C. A. Bishop, C. Keitel, J. Kilpatrick, & C. Laborde (Eds.), International handbook of mathematics education (pp. 877–908). Dordrecht, Netherlands: Kluwer Academic Publishers. Harel, G., & Sowder, L. (1998). Students’ proof schemes: Results from exploratory studies. In A. H. Schoenfeld, J. Kaput, & E. Dubinsky (Eds.), Research in collegiate mathematics education III (pp. 234–283): Providence, RI: AMS. Johnson, R. B., & Christensen, L. B. (2014). Educational research: Quantitative, qualitative, and mixed approaches (5th ed.). Thousand Oaks, CA: Sage. Knuth, E. J. (2002). Secondary school mathematics teachers’ conceptions of proof. Journal for Research in Mathematics Education, 33(5), 379–405. Larsen, D. M. (2017). Testing inquiry-based mathematic competencies. Short communication. Paper presented at the Merga Conference 40, University of Monash, Melbourne, Australia. Lithner, J. (2008). A research framework for creative and imitative reasoning. Educational Studies in Mathematics, 67(3), 255–276. Logan, T., & Lowrie, T. (2013). Visual processing on graphics task: The case of a street map. Australian Primary Mathematics Classroom, 18(4), 8–13. Niss, M., Bruder, R., Planas, N., Turner, R., & Villa-Ochoa, J. A. (2016). Survey team on: Conceptualisation of the role of competencies, knowing and knowledge in mathematics education research. ZDM, 48(5), 611–632. Niss, M., & Jensen, T. H. (2002). Kompetencer og matematiklæring: Idéer og inspiration til udvikling af matematikundervisning i Danmark [Competencies and mathematic learning: Ideas and inspiration to development of teaching in mathematics in Denmark] (Vol. 18). Copenhagen, Denmark: Danish Ministry of Education. Nunes, T., Bryant, P., Evans, D., & Barros, R. (2015). Assessing quantitative reasoning in young children. Mathematical Thinking and Learning: An International Journal, 17(2–3), 178–196. Nunes, T., Bryant, P., Evans, D., Bell, D., Gardner, S., Gardner, A., & Carraher, J. (2007). The contribution of logical reasoning to the learning of mathematics in primary school. British Journal of Developmental Psychology, 25(1), 147–166. Rasch, G. (1960). Studies in mathematical psychology: I. Probabilistic models for some intelligence and attainment tests. Copenhagen, Denmark: Danmarks Paedogogiske Institut (Chicago: University of Chicago Press, 1980). Reid, D. A., & Knipping, C. (2010). Proof in mathematics education, research, learning and teaching. Rotterdam, Netherlands: Sense Publishers. Resnick, L. B., & Resnick, D. P. (1992). Assessing the thinking curriculum: New tools for educational reform. In B. R. Gifford & M. C. O’Connor (Eds.), Changing assessments: Alternative views of aptitude, achievement and instruction (pp. 37–75). Dordrecht, Netherlands: Springer. Shaughnessy, M., Barrett, G., Billstein, R., Kranendonk, H., & Peck, R. (2004). Navigating through probability in grades 9-12. Reston, VA: National Council of Teachers of Mathematics. Stylianides, A. J. (2007). The notion of proof in the context of elementary school mathematics. Educational Studies in Mathematics, 65(1), 1–20. 16 Larsen & Puck

Stylianides, A. J., & Harel, G. (2018). Advances in mathematics education research on proof and proving: An international perspective. New York, NY: Springer. Stylianides, G. J. (2008). An analytic framework of reasoning-and-proving. For the Learning of Mathematics, 28(1), 9–16. Stylianides, G. J., & Stylianides, A. J. (2017). Research-based interventions in the area of proof: The past, the present, and the future. Educational Studies in Mathematics [Special Issue], 96(2), 119–274. Van Wyke, J., & Andrich, D. (2006). A typology of polytomously scored mathematics items disclosed by the Rasch model: Implications for constructing a continuum of achievement. Unpublished report, Perth, Australia. Wilson, M., & Gochyyev, P. (2013). Psychometrics. In T. Teo (Ed.), Handbook of quantitative methods for educational research (pp. 3–30). Rotterdam, Netherlands: Sense. Yackel, E., & Hanna, G. (2003). Reasoning and proof. In W. G. Martin & D. Schifter (Eds.), A research companion to principles and standards for school mathematics (pp. 227–236). Reston, VA: National Council of Teachers of Mathematics.

Author Information Dorte Moeskær Larsen Morten Rasmus Puck LSUL, University of Southern Denmark University college Lillebælt Campusvej 55, 5230 Odense, Denmark Niels Bohrs Alle 1, 5230 Odense, Denmark Contact e-mail: [email protected]

14.5 Paper V

Title How argumentation in teaching and testing of an inquiry-based intervention is aligned

Authors Dorte Moeskær Larsen, Jonas Dreyøe Meldgaard & Claus Michelsen

Status Manuscripts submitted for publication in:

Eurasia Journal of Mathematics, Science and Technology Education

This version of the paper is the original document submitted.

227

Alignment in teaching and testing

How argumentation in teaching and testing of an inquiry-based intervention is aligned

Contribution of this paper to the literature • This study fills in a research gap that focuses on the alignment between teaching and testing in mathematics classrooms with focus on argumentation. • By making a juxtaposition of how students argue in mathematics teaching situations and how they are subsequently tested in a specific research intervention the analysis is conducted. • Findings indicate a specific difference in how the students represent their arguments in test situations compared to teaching situations, while their arguments are based on the same fundament and have almost the same formulations.

Introduction and background This article is about how primary students’ learning activities with mathematical argumentation are mirrored in a written assessment test. The links between classroom activities and assessment test is investigated with the theory of constructive alignment developed by (Biggs, 1996), and with roots in curriculum theory and constructivism. At the classroom level, teachers use a variety of assessment strategies to determine whether their students are learning the mathematics they are supposed to learn. There is, however, broad agreement that most mathematical assessment tests do not measure all the wished aspects and since tests have a strong influence on what is taught, because they can influence students’ perception of what is important as well as how it should be learnt in mathematics, it is important to consider how to create alignment between classroom interaction and assessment. Surprisingly, the progress in aligning classroom activities and assessment has been limited since (Niss, 1993) in the introduction to the ICMI study about assessment in mathematics education concluded that the developments in content, working forms and student activities “have not, however, been matched by a parallel developments in assessment, where values, notion, and theory, practice, modes, and procedures are concerned. Consequently, an increasing mismatch and tension between the state of mathematics education and current assessment practices are materializing” (p. 4).

Alignment in teaching and testing

The Danish KOM report (Competencies and Mathematics Learning)(Niss & Jensen, 2002) refers to the problematic fact that many of the traditionally used forms of assessment permit only a limited evaluation of the mathematical knowledge and competencies that one wants to promote in mathematics teaching. Effective ways of evaluating competencies are crucial for a successful implementation of competency goals in teaching. A wide range of tools is required and there is a need to develop new tools for use in evaluating, not least in connection with various spectra of teaching and activity forms used in modern mathematics education. The tools developed should ensure that forms of assessment both provide for and exploit a sustainable coverage of students' mathematical acquisition and mastery (Niss & Højgaard 2011). In the article we address the problematic of creating alignment between classroom and assessment in the context of the Danish randomized controlled trial entitled ‘Kvalitet i Dansk og Matematik’ [Quality in Danish and mathematics] (KiDM), one of the main aims was to develop students’ competency in mathematical argumentation by taking an inquiry-based approach. Teaching inquiry- based is characterized by being dialogical, which puts the students’ argumentation to the fore (Engeln, Euler, & Maass, 2013). A test was developed (the KiDM test) to measure students’ mathematical development during the intervention including students’ development of more sophisticated ways of making arguments. This study focuses on whether and how the alignment between the KiDM intervention and the KiDM test occurs with focus on argumentation in mathematics, and we ask the following question: How is argumentation in teaching and in assessment by a written test of an inquiry-based intervention in the mathematics classroom aligned?

Alignment between the practice of teaching and assessment Empirical evidence indicates that thoughtfully implemented assessment practices improve students’ learning, increase students’ scores, and narrow achievement gaps between low-achieving students and others (Black & Wiliam, 1998). An important characteristic for a good educational system or any teaching for that matter, is the presence of alignment. Alignment means that the curriculum is designed to ensure that assessments and standards are addressed in the instructional process. If the tested content is not covered in instruction, then the students will not have the opportunity to understand the tested content. Hattie (2009, p. 6) argues: “… any course needs to be designed so that the learning activities and assessment tasks are aligned with the learning outcomes that are intended in the course. This means that the system is consistent.” This call for strategies means that

Alignment in teaching and testing

students’ mathematical reasoning serves as a method for eliciting and using evidence of students’ reasoning. In (Biggs, 1996, 2011) he uses the concept of constructive alignment which, he argues, has two premises: students construct meaning from what they do in order to learn, and the teacher aligns the planned learning activities with the learning outcomes. Thus, any course needs to be designed, so that the learning activities and assessment tasks are aligned with the learning outcomes that are intended in the course. This implies that students should not be expected to “work out” what is to be learnt and what it means to be successful in that learning (often they only learn this when they get the results of assessments back), but instead it is necessary to make these criteria for success clear before any teaching or assessments. Without such alignment, the powerful effects of assessment feedback and reporting, and self-regulated learning are less likely to occur:

From our student’s point of view, assessment always defines the actual curriculum… Students learn what they think they will be tested on. This is backwash…to refer to the effect assessment has on students’ learning. Assessment determines what and how students learn more than the curriculum does. (Biggs, 2011, p. 197).

The emergence of backwash has no negative outcomes for the learning of a subject, if and only if there is alignment between the aims of the teaching and the content of the assessment. The expectations of this would be that the learning activities, which are indicated in the intended outcome, are mirrored both in the teaching/learning activities undertaken by the students, and in the assessment tasks. To specify this, Biggs (2011) explains that the intended verb must be present both in the activity and in the assessment.

Constructive alignment may be common sense, however the traditional practice of teaching and assessment often ignores and forgets the importance of alignment (Squires, 2012).

Argumentation in mathematics education Learning mathematics can be seen as argumentative learning (Krummheuer, 2007). Argumentation is then seen as not only a teaching aim in the sense that one must design mathematics instruction in a way that make the students reach this goal and are able to argue in a sophisticated mathematical level (learning to argue), argumentation is also part of everyday mathematic classroom as a precondition for the possibility to learn not only the outcome. For Duval (2007) an argument is considered to be anything which is advanced or used to justify or refute a proposition. This can be the statement of a fact, the results of an experiments, or even simply an example, a definition, the

Alignment in teaching and testing

recall of a rule, a mutually held belief or else the presentation of a contradiction. Whitenack and Yackel (2002) distinguish between an argument and an explanation: “In some instances, a student may explain an idea to clarify his or her thinking for others. At other times, the student makes an argument to validate his thinking or to justify his activity” (p. 525). A. J. Stylianides (2007) defines mathematical argumentation as a process of a connected sequence of assertions intended to verify or refute a mathematical claim and describes some characteristics about argumentation; argumentation uses statements accepted by the classroom community that are true and available without further justification. It also employs modes of argumentation that are valid, known to, or within the conceptual reach of, the classroom community; and it is communicated with modes of argument representation that are appropriate and known to, or within the conceptual reach of, the classroom community.

G. J. Stylianides (2008) views proof as a way of argumentation and divides arguments into non-proof arguments and deductive proof arguments, distinguishing between two kinds of non-proof arguments: empirical arguments and rationales. The empirical argument is an argument that provides inconclusive evidence for the truth of a mathematical claim. The student may conclude that a claim is true after checking a suitable subset of all the possible cases covered by the claim, or after considering the full range of possible cases, but without showing that they did so. The empirical argument is in line with the empirical justification in Harel and Sowder (1998) description of proof schemes and the “naïve empiricism” in Balacheff (1988). EMS (2011) argues that students’ empirical proof schemes may be a consequence of students’ experience outside of mathematics classes. According to G. J. Stylianides (2008), an argument counts as a rationale if it does not make explicit reference to some key accepted truths that it uses or if it uses statements that do not belong to the set of accepted truths of a particular community. A. J. Stylianides (2007) defines four elements of importance to consider when analyzing whether arguments qualify as proof or non-proof arguments: the foundation of the arguments, the formulation of the arguments, the representation of the arguments and the social dimension. The foundation of the arguments concerns what constitutes the arguments’ basis. This can be definitions and axioms or mathematical attributes, but also specific observations and public knowledge can be the point of departure that defines the acceptable mathematical reasoning foundation within a given context.

The formulation of the arguments concerns how the arguments are developed. This can be a logical deduction or a generalization from a particular case, and it is not necessarily always rooted in the

Alignment in teaching and testing

mathematical discipline. According to G. J. Stylianides (2008), both the arguments’ foundation and formulation are important factors in their qualification as proof.

The representation of the arguments concerns how the argument is expressed. An argument can be expressed in everyday language, using body language or in more academic language, symbols and algebra. The everyday language is oriented towards the spoken language: specific, imprecise, partially unstructured and context-embedded, while the more academic language is more oriented towards the written language: abstract, generalizing and context-disembedded.

Finally, the social dimension concerns how the arguments are unfolded in the social context of the community in which they are created; how different social actors and practices impact the arguments’ creation and unfolding.

Argumentation is often studied by the TAP-model (Toulmin, 1958), however since this study is interested in a broader analysis in connection to alignment it must also include an analysis of the representations used and social aspects. A categorization of different types of argumentation (table 1) developed from G.J. Stylianides (2008) will therefore be the analytical model in this study. It is important to acknowledge that the table has not been part of the design of the KiDM teaching intervention nor the KiDM assessment-test.

Table 1: Classification of arguments

Analytical arguments: Empirical arguments: Rationale

deductive argumentation/proof Foundation definition, axioms, and Observations public knowledge / theorems intuition (feelings) in mathematics Formulation logic, deductive from one or multiple not logical, not deductive

cases or empirical Representation symbolic algebra / symbolic / everyday everyday language everyday language language verbal/written verbal/written/gestures verbal/written/gestures Social individually/ groups/ individually/ groups/ individually/ groups/ whole class whole class whole class

Alignment in teaching and testing

In Table 1 we distinguish between three categories: analytical argumentation, empirical arguments and rationale, which we have categorised to have different foundations and formulations, but all have the possibility to be represented in verbal or written language.

KiDM: An inquiry-based intervention project The KiDM programme include a researcher- and teacher-developed inquiry-based teaching intervention for a four-month long mathematics teaching approach implemented in 107 schools. The development of the intervention was created on the basis of a systematic review about inquiry-based learning in mathematics education (Dreyøe, Larsen, Hjelmborg, Michelsen, & Misfeldt, 2017), where three principles for inquiry-based teaching were presented. The main focus of these principles was to make the teaching explorative, dialogical and meaningful and to make room for student participation, but also to include the eight mathematical competencies (Niss & Højgaard, 2011), which is a key concept in the standard Danish curriculum. One of these eight mathematical competencies is the reasoning competency, where argumentation is central. The intervention includes three courses with a focus on numbers and algebra, geometry, and statistics respectively. The three courses all focus on the argumentation, however it is not the main aim in all the activities and tasks in the courses. The intervention is a web page1 which includes a detailed teacher manual that contains different tasks followed by specific questions and aims to enable the teacher to induce student reflection.

In order to assess student development of mathematical concepts and mathematical competencies related to inquiry-based teaching, a pre- and post-test were essential. A KiDM test with a strong focus on mathematical and inquiry-based competencies was developed. It was decided that the test should be computer-based. In terms of format, 50 of the items were multiple choice items, 31 were close- constructed-response items which were auto-scored and 22 were open-constructed-response items which were rated by the test setters and student assistants. The test content was broad, ranging from conceptual understanding in algebra, geometry and statistics, to the eight mathematical competencies (Niss & Højgaard, 2011).

The aim in this study is to examine the alignment between students’ enacted argumentation observed in the KiDM classrooms during the intervention and students’ enacted argumentation in the KiDM test and we use the theoretical classification of argumentation in Table 1 in the analyse.

1 www.kidm.dk

Alignment in teaching and testing

Method of constructing data and the coding process The constructed data used in this study includes video recordings from observations in KiDM classes and test answers from the open-constructed-response items from the KiDM test (pre-/post- test). Five intervention schools were visited three times each by the first and second author of this paper in order to observe classes for two lessons (90 minutes). The five schools were selected to represent different areas in Denmark and to represent big and small schools as well as rural and urban schools. The aim was to gain insight into the issue of the notion of argumentation in the classes through close examination of 4th and 5th grade students’ argumentations in the classroom context. Because of the complexity of this notion, all the lessons were video-recorded and transcribed in full. When observing, the focus was specifically on one student per visit. The students were chosen by the teacher for their industriousness and emotional robustness. These selection criteria were chosen in order to ensure that the students would work with the assignment and not be overwhelmed by the researcher following them and their work.

The different activities and tasks in the intervention have many different learning goals. Some of the activities have an explicit goal within reasoning while others did not. It is important to notice that all the observed lessons were selected not only for observation based on the specific content about reasoning, but for more practical reasons, including teacher consideration, but also continuity in the observations.

Regarding data coding, all the arguments/episodes from the observed lessons and the different arguments from specific selected test items were analysed on the basis of the four major elements of an argument according to our classification of arguments (Table 1). The first and second authors of the paper coded the observations and test answers individually and thereafter compared and discussed the coding with the last author. Individual “double” coding was conducted to avoid subjective bias in the analysis and to increase inter-coding reliability (Johnson & Christensen, 2014).

Due to the length of this article, it is unfortunately not possible to describe all the observed teaching lessons or to include all the different test items. Instead, two test items have been selected, which reflect different approaches to argumentation, while three separate classroom cases have been selected that show the breadth of the different arguments that were observed in the classroom teaching.

Alignment in teaching and testing

Presenting and analysing students’ arguments In analysing all the observed lessons, we found many different categories of argument. Below we present an activity from the KiDM project called “What do the boxes weigh?” This activity concerns six boxes that have been weighed in pairs. All the paired combinations of boxes weigh 6, 8, 10, 12, 14 and 16 kg, respectively. The students must determine the individual weights of the boxes by inquiry. There are two correct answers to this task. The reason why we have chosen to present this activity is that a main goal in this activity is explicitly argumentation and, we found more arguments than in any of the other observed activities.

In the following analysis we will present three episodes where three different categorized arguments are observed: Episode 1 In a 4th grade KiDM class a whole-class discussion about how to get the results has just begun:

Molly: We tried some different numbers and it did not fit, we kept having problems getting 16

Teacher: So, what did you do?

Molly: Then we tried some smaller numbers and used 10 as the biggest number and then it worked out.

Teacher: Ok. [the teacher asks another student]

In this episode, Molly uses an empirical argument [you find the answer by just trying different numbers] to explain their process. If we analyse Molly’s argument using table1, the foundation of the argument, which concerns how the argument is developed, is clearly done from observation - she found the answer by trying out different numbers. The formulation for the argument comes from multiple cases. It appears that Molly and her group did not try out 10 first, but after some experimentation they discovered that 10 was correct. The representation of the argument is verbal and in everyday language. Molly has not written down the arguments anywhere and the social element in this argument is important, because the only reason that Molly makes this argument explicit is because the teacher asks her to argue for their choice. This means that the articulated argument is produced in a negotiation process between her and the teacher in the whole-class discussion. At this point, the teacher seems to accept Molly’s argument, as the teacher does not immediately challenge this argument, and neither do the students oppose it.

Alignment in teaching and testing

Episode 2 Later in the same whole-class discussion, a student starts talking about the difference between the two specific results:

Teacher: When you tried all this, did you then automatically try all the odd numbers at one time or?

Peter: Yes

Teacher: Why did you do that?

Peter: It felt like the right way.

Teacher: It just felt the right way... [the teacher stops the discussion]

In this case the foundation of Peter’s argument [it felt the right way] is not based on any mathematical definition or axioms, but only on a feeling from the students. The formulation of Peter’s argument is difficult to define because it is implicit in the feeling. The feeling may have been developed in the process of making sums using odd and even numbers and hereby finding that the sum of four odd numbers will always be an even number, but the argument can also be completely random and not in any way systematic. Peter’s argument also has a verbal representation using everyday language. The social dimension of Peter’s argument is not clear either. The teacher’s repetition of Peter’s argument may mean different things; on the one hand it may mean that the teacher does not accept the argument, on the other hand it may also mean, that the teacher accepts the argument and is not interested in further clarification or elaboration. Episode 3 The last episode also comes from the whole-class discussion, where the class discusses whether all the numbers in one solution have to be either odd or even:

Teacher: …so can we find examples where they are the same [two odds or two even numbers] and where they are different [one odd and one even number]. So, we go from it's almost always, to always! Muhammad?

Muhammed: It's not always right because you can say 5 + 6

Teacher: But 6 is not an odd number. It is if we plus two numbers, both of which are odd.

Alignment in teaching and testing

Muhammed: Oh

Teacher: Vilfred

Vilfred: The reason for it is, that an odd number... If you have an odd number, for example... when you add one to an odd number it becomes an even number. So, if you sum two odd numbers, then it's like two left from the same number and two is an equal number so it is like two, two-by-two towers where even ones are level [He raises his arms and shows two towers with his hands and arms that are the same level] and the others are skew. [he shows two towers that have different levels with his arms/hands].

Teacher: It's super well explained, if we take 3 and 5, then it's an even number, with one extra [points at 3] and the above number [points at 5] is also an even number with one extra. [Draws the towers and then shows the skewed towers on the board] and if we put the two together, then the two extra become an even number and the tower is not skew.

In this episode the foundation of Vilfred’s argument is based on his definitions of even and odd numbers which is explicitly explained [when you add one to an odd number, it becomes an even number]. This definition is based on the deduction that the sum of two odd numbers or two even numbers is always even. The formulation of this argument can be seen as logical deduction from the definitions, but without using the formula of odd numbers [2n+1] and without expressing (or knowing) the formula. Instead, Vilfred’s argument is both verbally represented in everyday language but also in his body language using his arms and hands. The teacher clearly acknowledges the argument in a positive way and credits Vilfred’s argumentation. Summary: Three episodes from the classroom observations It is important to acknowledge that the foundation, the formulation and the representation that constitute the different arguments here are not predetermined but are negotiated by the participants as they interact in the classroom. Arguments in a mathematical community can be seen as a result of a social discourse in which participants agree to accept or reject particular theories. When the teacher and the other students accept an argument, it is the outcome of socially agreed sets of rules and mathematical axioms/theorems (Cobb, 2000). In the analysis of this classroom it is therefore important to be aware of both classroom social norms and sociomathematical norms. Among many of the observed lessons, we saw teachers who did not ask questions to clarify the student’s arguments, even though the KiDM material includes question guides to all the tasks.

Alignment in teaching and testing

A.J. Stylianides (2007) describes a theoretical framework that is comprised of two principles for conceptualizing the notion of proof in school mathematics: (1) The intellectual-honesty principle, which states that the notion of proof in school mathematics should be conceptualized so that it is honest to mathematics as a discipline and at the same time honours students as mathematical learners; and (2) The continuum principle, which states that there should be continuity in how the notion of proof is conceptualized at different grade levels so that students’ experiences with proof in school have coherence. The intellectual-honesty and continuum principles offer insight into what arguments might qualify, or not qualify as proof in the elementary school.

Vilfred’s arguments might have achieved a defensible balance between respecting his status as a mathematical learner but also honouring mathematics as a discipline. In a 4th grade classroom the construction of a more formal deductive argument is not expected because it might not be within a fourth grader’s conceptual reach.

The KiDM-test The KiDM test is an achievement test developed for the purpose of measuring students’ development after the KiDM intervention. The test is built from guidelines from a systematic review and several items have been inspired by the published items from TIMSS (Allerup, Belling, Kirkegaard, Stafseth, & Torre, 2016)

The test includes 8 items with specific focus on argumentation in mathematics. Below we will analyse two of these eight items from the test.

Alignment in teaching and testing

Figure 1: Items: U9 (Translated from Danish)

The students’ answers to this item are very diverse: There are answers: “There are no minus degree in January” where the argument can be seen as implicit in the answer, [therefore, it must be warmer than in December], however there are no minus degree temperatures in December either so the argument here does not make any sense. Another answer is:

Monday has 6 degrees of difference. Tuesday has 1 degree of difference. Wednesday has the same. Thursday there is 3 degrees difference, Friday has 1 degree of difference, Saturday has four degrees of difference and Sunday has two degrees of difference.

This answer uses mathematics by calculating the difference every day, but it does not make any claim or argument about which week (December or January) is the warmest. Other students have selected some specific days and calculated the difference: “Monday is much warmer in January than in December, therefore it is warmer in January”. This answer is based on the level of the temperature,

Alignment in teaching and testing

but the development is based on calculating only one day. The formulation can therefore be seen as empirical, because it provides inconclusive evidence for the truth of this claim, because the student concludes that the claim is true after checking only one subset of the days covered by the claim. Finally, there are answers such as:” In December, the average temperature is 3 degrees, in January the average temperature is 4 degrees, so it can be seen as warmer in January than in December”. This answer can be divided into two different arguments. Firstly, there is the argument that the average temperature in December is 3 degrees and in January 4 degrees, but there is also the argument that the average temperature can be used to tell which week is warmest. The first argument is founded on some mathematical calculations using the statistical descriptor referred to as mean/average, but the second argument is founded on the concept of average. The formulation which concerns how the arguments are developed can be seen as more mathematical and deductive. The average over the two weeks is calculated correctly, and the concept of average is chosen here as the model to answer the question. Basically the question could also have been answered by using other statistical models (e.g. mode or median).

All the different arguments in the test are represented as written answers. Formal mathematical symbols are not used, however in the students’ everyday language, some of the answers contain a little mathematical-language because they use symbols consisting of numbers to represent, for example, the temperature or the average. The students find and write the answer by themselves, they have no teacher or classmate to ask follow-up questions.

Another item shows two figures, P and Q:

P Q

Figure 1: Item from the KiDM test

Alignment in teaching and testing

The text in the item is (translated from Danish): “Describe how the two figures are similar - justify your answer” and “Describe how the two figures are different - justify your answer”

Like the other item, this item also has many different answers. The students’ argumentation for how the figures are similar ranges from. “they are alike” or “they are both white” to more sophisticated answers like: “the sum of all angles has the total of 180 degrees in both figures” or “they both have three sides and three angles” and the students’ argumentation for how the figures are different is, for example: “They are not the same triangle” or “They have different forms”.

But there are also more sophisticated answers such as: “Figure p has one right angle and figure q does not” or "The side-length of the triangles are not equal” If we try to analyse one of these answers, e.g. “figure P is right angled and figure q is acute”, the foundation is based on the mathematical concepts of acute angles and right angles and consequently different triangles. The formulation of this argument is not empirical, but it can be seen as a more deductive argument because the student begins with the statement about right-angled triangles and acute angles and reaches the logically certain conclusion that the triangles are therefore different.

The answers to these items are also represented in written form on the computer in an everyday language and only a few of the answers use symbols, but the numbers 180-degree, 90-degree or 3 are used several times.

All the students have written the answers themselves sitting in their usual classroom with a computer and iPad in front of them, and the students are not allowed to talk to each other and the teacher is not supposed to help the students if they get stuck. The students must do the test by themselves.

Alignment between arguments in the test and in the classroom If we align the arguments observed in the classroom with the arguments observed in the test, we see some differences, but also some similarities. Firstly, we see no alignment between the representation of the arguments - as the arguments observed in the classroom are all verbal or spoken in terms of students’ body language, while in the test the representation is always written on a computer/iPad. However, we see a minor use of mathematical symbols and always in the form of numbers in both situations. In the 15 lessons observed, we did not see any of the students write down their arguments. In the test, however, it was not possible to express the argument verbally only in writing. However, the arguments in the classroom and in the test were all represented in an everyday language.

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Looking at the basis for the foundation of the arguments, we can see that in both the test and the classroom the students’ arguments can be based on feelings or other subjective rationales (such as Peter, for example). We also see that students build their arguments on empirical and randomly selected numbers in a table in the test and in the classroom (such as Molly, for example). Finally, we see that students in the test and in the classroom base their arguments on more mathematical, deductive approaches, for example when pupils calculate the average temperature for the two weeks and match these numbers and argue accordingly or when Vilfred in the classroom argues why the sum of two odd number is always even.

Based on these observed lessons, however, we see that in the classroom, students' arguments sometimes appear more generalized. In some of the whole-class discussions the argumentation concerns not only an existing example, but becomes more generalized in the institutionalization phase or when the teacher interferes in the students’ discussions. Cobb (1998) made a distinction between calculational and conceptual orientations in teaching:

Calculational discourse refers to discussions in which the primary topic of conversation is any type of calculational process. This can be contrasted with conceptual discourse in which the reasons for calculating in particular ways can also become explicit topics of conversation. In this latter case, conversations encompass both students’ calculational processes and the task interpretations that underlie those ways of calculating. (p. 2)

In episode 3 we observe that the teacher in the classroom interactions asks generalizing questions [we go from it's almost always, to always!] and has a conceptual discourse. In the test however, all the questions posed require argumentation in the calculational discourse. All the items have an existing example with numbers or a specific figure (not general) from which the argumentation derives. Finally, the social elements in the argumentation are not aligned at all. The big difference is probably that the students in the test are developing their own arguments whilst sitting alone, while in the classroom the student and the teachers develop their arguments in a negotiation process, however in the classroom not all the observed arguments would have been articulated, if the teacher has not asked further questions. The table below (Table 2) shows a summary of the correlation between arguments in the classroom and in the test.

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Table 2: Juxtaposing the arguments in classroom teaching and the arguments in testing

ALIGNMENT Arguments in the classroom teachin Arguments in testing Representation Verbal and body language Written on a computer/Tablet Everyday language with little use of numbers Everyday language with little use of numbers Formulation Feelings, rationales, empirical or logical Feelings, rationales, empirical or deduction logical deduction Calculational discourse + teacher-induced Calculational discourse conceptual discourse Foundation Definitions, concepts, observations Quantity of numbers, definitions, concepts Social Discussion and negotiation with teachers and Individual students dimension other students

Discussion In general, both the arguments in the test and in the classroom need to be in line with A. J. Stylianides (2007) intellectual-honesty and continuum principle, this means among others that there needs to be a defensible balance between respecting fourth graders as mathematical learners and being honest to mathematics as a discipline. It might not be possible to construct formal deductive argumentation in mathematical tests as this might not be within a 4th or 5th grader’s conceptual reach, but we still need to honour mathematics as a discipline and develop some tasks, where the students can engage in making arguments in a logical, deductive way. The tasks must still also be in line with the KiDM project’s inquiry approach which here also is application-oriented and thus more in line with an inductive reasoning approach. These two concepts can therefore be seen to counteract each other, and this makes it important to maintain a balance between the two.

Moreover, the four elements in an argument defined by A. J. Stylianides (2007) are mutually dependent on each other. For example, in the test, students are required to respond in writing (representation), which of course also has some implication for the formulation of the argument, because of the students’ writing abilities in 4th grade, but also because of the social dimensions in the test situation; the students do not have anyone to ask follow-up questions or anyone to discuss

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the different arguments with. The students are consequently heavily restricted in their reasoning process in the test situation compared to the collaborative nature of the teaching situations.

Finally, it is important to acknowledge, that when we analyse the argumentation in the test we see the arguments as products (test answers), while argumentation in the classroom is more seen as part of a process where the students make an inquiry, which leads to an assumption and subsequently an argument to support this assumption. This means that there are in fact ontologically two different empirical elements that are not easily comparable. In addition, the products themselves do not necessarily give any insight into the process that leads to the student coming up with the answer. And in fact, the actual reasoning of the students in 4th grade is often only visible in the process of answering the question, not in the answer itself.

Does alignment in the social dimension mean that the test cannot be done individually, and what then happens in terms of the reliability of the measures of the achievements, if the test is done in pairs or groups? Is it possible that test answers can be represented verbally (or with body language) or does this require some impossible changes in the computer programmes or does classroom work need to focus more on the written answers because of the testing aspect? In general, there are some aspects which are difficult to change, if we choose a traditional testing situation.

Although there are still problems in aligning the tests with teaching, we should also focus on how to develop students’ mathematical argumentation competency in the teaching situation. The majority of students in elementary school validate their opinions by using empirical arguments, and research has shown that changing this to more formal and analytical argumentations is highly non- trivial (EMS, 2011). In many of the classroom episodes, the teachers were not explicit about why an argument was valid /not valid, and in all the students arguments they only used natural language, which does not encourage them to use more formal mathematical language.

Conclusion The alignment analysis showed that the arguments are represented in different ways. In the test, all the arguments were written on a computer, whilst in the classroom, the argumentations were observed verbally and/or through body language. Nevertheless, both in the test and in the classroom the arguments were represented in natural language using only numbers as a symbolic representation. However, the foundation and the formulation of the arguments were more aligned, but both the foundation and the formulation of the arguments in both classroom and test could be

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more honest to mathematics as a discipline by focusing more on analytical argumentation and could be closer to the continuum principle by bringing more continuity and coherence to how the notion of mathematical argumentation is conceptualized here in 4th grade and how the notion must be conceptualized in upper secondary school. According to the social dimension, there was no alignment at all, because all the tests arguments were made in a situation where the students sit alone without any possibilities to discuss the different arguments, while in the classroom almost all the arguments were expressed in a conversation with the teachers.

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References

Allerup, P., Belling, M. N., Kirkegaard, S. N., Stafseth, V. T., & Torre, A. (2016). Danske 4.- klasseelever i TIMSS 2015 : en international og national undersøgelse af matematik- og natur/teknologikompetence i 4. klasse. Fjerritslev: Forlag1.dk. Balacheff, N. (1988). Aspects of proof in pupils’ practice of school mathematics. In D. Pimm (Ed.), Mathematics, teachers and children (pp. 216-235). London Hodder & Stoughton. Biggs, J. (1996). Enhancing teaching through constructive alignment. Higher education, 32(3), 347- 364. Biggs, J. (2011). Teaching for quality learning at university: What the student does. UK: McGraw- Hill Education Black, P., & Wiliam, D. (1998). Assessment and classroom learning. Assessment in Education, Principles, Policy & Practice, 5(1), 7-74. doi:10.1080/0969595980050102 Cobb, P. (1998). Theorizing about mathematical conversations and learning from practice. For the learning of mathematics, 18(1), 46-48. Cobb, P. (2000). The importance of a situated view of learning to the design of research and instruction. Multiple perspectives on mathematics teaching and learning. Multiple perspectives on mathematics teaching and learning, 1, 45-82. 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, 329-337. Duval, R. (2007). Cognitive functioning and the understanding of mathematical processes of proof. In B. Paolo (Ed.), Theorems in school (Vol. 1, pp. 137-161). Rotterdam: Brill sense. EMS. (2011). “Solid findings” in mathematics education. Educational Administration Quarterly, Newsletter of the European Mathematical Society(81), 46-48. Engeln, K., Euler, M., & Maass, K. (2013). Inquiry-based learning in mathematics and science: A comparative baseline study of teachers’ beliefs and practices across 12 European countries. ZDM, 45(6), 823-836. Harel, G., & Sowder, L. (1998). Students’ proof schemes: Results from exploratory studies. In A. H. Schoenfeld, J. Kaput, & E. Dubinsky (Eds.), Research in collegiate mathematics education III (pp. 234-283): Providence, RI: AMS Hattie, J. (2009). The black box of tertiary assessment: An impending revolution. Tertiary assessment higher education student outcomes: Policy, practice research in Autism Spectrum Disorders, 259-275. Johnson, R. B., & Christensen, L. B. (2014). Educational research : quantitative, qualitative, and mixed approaches (5th ed.). Thousand Oaks, California: Sage. Krummheuer, G. (2007). Argumentation and participation in the primary mathematics classroom: Two episodes and related theoretical abductions. The Journal of Mathematical Behavior, 26(1), 60-82. Niss, M. (1993). Assessment in Mathematics Education and Its Effects: An Introduction. In M. Niss (Ed.), Investigations into Assessment in Mathematics Education: An ICMI Study (pp. 1-30). Dordrecht: Springer Netherlands. Niss, M., & Højgaard, T. J. (2011). Competencies and mathematical learningIdeas - Ideas and inspiration for the development of mathematics teachinglearning in Denmark: Roskilde Universitet. Niss, M., & Jensen, T. H. (2002). Kompetencer og matematiklæring: Idéer og inspiration til udvikling af matematikundervisning i Danmark [Competencies and mathematic learning: Ideas

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and inspiration to development of teaching in mathematics in Denmark] (Vol. 18). Danmark: Undervisningsministeriet. Squires, D. (2012). Curriculum alignment research suggests that alignment can improve student achievement. The Clearing House: A Journal of Educational Strategies, Issues Ideas, 85(4), 129-135. Stylianides, A. J. (2007). The notion of proof in the context of elementary school mathematics. Educational Studies in Mathematics, 65(1), 1-20. Stylianides, G. J. (2008). An analytic framework of reasoning-and-proving. For the learning of mathematics, 28(1), 9-16. Toulmin, S. E. (1958). The uses of argument. Cambridge: Cambridge university press. Whitenack, J., & Yackel, E. (2002). Making mathematical arguments in the primary grades: The importance of explaining and justifying ideas. Teaching Children Mathematics, 8(9), 524.

15 Discussions, findings, conclusions and implications

In this chapter the focus is a discussion of, first, the methods used (Section 15.1), followed by a discussion about the legitimation of the overall mixed project (Section 15.2). The findings in the five papers are afterwards discussed in connection to the research questions (Section 15.3), and thereafter follows a section with a focus on implications for research and practice (Section 15.4). The chapter ends with a final conclusion and some remarks (Section 15.5).

15.1 Discussion of the methods As described in the introduction, various theories and methods are combined and used in this thesis. The individual methods have each been presented and discussed individually in the method sections in Chapter 11, but in this section only some specific aspects will be pointed out as challenging and, as mixed methods research is often plagued by the problems of legitimation (Onwuegbuzie & Johnson, 2006), this aspect will be included.

Qualitative data can often be seen as interpretative research (Johnson & Christensen, 2014). To reduce the researcher's many subjective choices, various manuals have been explicitly designed (see Appendices e and f). However, many other choices had to be taken in the process of data construction; e.g., the aspect of only using handheld video observation from the different school classes. This was a choice made in connection to focusing on students’ reasoning processes. In fact, interviews were also conducted with all the teachers in the observed classes as well as during the pilot phase. These interviews might have given yet another perspective on the research question, but due to prioritising the focus on the students' viewpoint, this was opted out. Specifically using one handheld video camera in each observation class meant that there were many things in the classroom that were not recorded. This approach was chosen after testing different other approaches (e.g. audio recordings of all the groups and a specific camera on the teacher), because these extra recordings complicated the complexity of analyzing the data and gave a lot of non-useful data, so the data became unmanageable. Moreover, it should be emphasised that a logbook was used after each recorded lesson, which made it possible to add the extra observations that the camera did not capture.

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In the quantitative data (surveys and tests) there are things that could have been done differently. Firstly, the teacher survey reported in Section 12.3 can be criticised for having a small number of participants and more efforts should have been made to get teachers to answer the surveys. The teacher survey can in some way be considered as not so important compared to the fact that this thesis focuses on the students’ reasoning process. Secondly, when analysing the results from the student survey, it was obvious that the content in some of the questions could have been more thoroughly thought out in relation to the outcome. There were several questions which subsequently did not make sense at all - for example, when the students were asked whether they always did their homework, but in the KiDM project there was no homework included in the design. Thirdly, the test has different issues. Most important is probably the number of items which in retrospect should have been much higher. However, in this concern there is an ethical aspect here concerning what years 4 and 5 students should be exposed to. Furthermore, a different approach to the procedure in which the students conducted the tests should have been carried out, because both the concept test and the student survey were accomplished before the students lastly took the competence test, which probably affected the test results because the students will naturally have been more exhausted. In the design-based development of the test, a fourth iteration might have been rewarding, because some of the problems we ended up with in the test then might have been taken into account; e.g., the difficulty level of the items and the number of items (see Paper IV). Finally, some issues concerning applying RCT studies in educational research must be discussed, because if you try to isolate all other variables than the “inquiry-based teaching” variable (in the intervention schools) against “traditional teaching” (control schools), inquiry-based teaching will not necessarily be clearly separated, because the teachers in the control schools might similarly teach inquiry-based with a specific focus on the reasoning competence. This is because the reasoning competence is, as described earlier, part of the Danish curriculum (Undervisningsministeriet, 2019b), even though this is not particularly likely, cf. the background report from KiDM (Michelsen et al., 2017).

Moreover, when carrying out a causality study there might be an assumption that the intervention will be implemented in the intervention schools in an ideal manner, but the uncertainty about whether this implementation has actually taken place is likewise present and even though the fidelity analysis from the KiDM report indicated accepted fidelity (see Section 12.1), a concern can still be how the intervention is implemented. This is very important in

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connection to the reasoning competence, because the teachers need to be able to ask follow-up questions, create dialogues and get the students to make argumentations in order to develop the students’ reasoning competence. In the observed classes it was found that the enacted intervention was not always necessarily ideal (cf. the qualitative observations), which will affect the possibilities of measuring an effect in a test/survey. This is further discussed in the KiDM final report (T. I. Hansen et al., 2019) where it is argued that, on the one hand, the students and teachers in KiDM project initially experience a high degree of commitment to the intervention, which can be explained by them trying a new and open inquiry-based approach. On the other hand, during the trial both the teachers and the students, however, experience some repetition by having to work intensely and purposefully with this particular approach for four months. This means that no matter how much we have tried to vary and create space for engagement in the teacher’s guidance, there is a risk that the students will lose motivation. The challenge is perhaps that there has been too narrow a focus on the mathematical competencies opposed to their normal approaches, which typically include learning by rote and skills training. The loss of motivation will similarly affect the outcome of the test/survey results. This issue is connected to different implementation strategies which are included in the KiDM project – see also Larsen et al. (2019) for further elaborations about implementation strategies in the KiDM project.

15.2 Validity and reliability in the mixed study - the legitimation of the study In the methods chapter, a description of the validity and reliability of the different methods used in this thesis can be found. In this section the focus will be on legitimation of the mixed study. The intention is to argue to which extent the overall research question has been answered from the research design and the methods used. The focus will be on Onwuegbuzie and Johnson (2006) nine types of mixed research legitimation. Five of the nine types will be discussed, whilst the remaining four (Sequential validity, conversion validity, sample integration validity, multiple validities) are already included in the methods section.

The first type of legitimation is called inside-outside validity. Inside-outside validity is the extent to which the researcher presents the participants subjective insider views (the emic) and the researcher’s objective outsider view (the etic) (Onwuegbuzie & Johnson, 2006). In this mixed methods thesis, the intention has been on both aspects. Attempts as insider perspective have been made when trying to get close to the students' thoughts of perspective both in the 250

form of think-alouds from the test, and in the various approaches that have been applied to get closer to the students' understandings. This include both a survey approach where the students could express different understandings, and, especially in the video recordings, there was always a focus on a single student/group in order to be able to get as far into the student's perspective as possible. The test answers were also carefully studied to be able to understand the student's interpretations and results. The researcher outsider perspective has been an element built into the actual design of the various studies. For example, the various manuals designed to perform both observations (see Appendices d and e) and test coding (see Appendix i) have contributed to making the researcher's choice less subjective and to strengthening the transparency. In addition, there has always been another researcher into the various coding, so that intercoder reliability was strengthened - both in the qualitative and quantitative studies.

A second type of legitimation is called commensurability validity and it refers to the extent to which meta-inferences made in the study reflect a mixed world view (Onwuegbuzie & Johnson, 2006). As already described in Section 11.2, it is demanding to perform mixed methods studies as they require that you are fully trained to perform both qualitative and quantitative studies. In order to meet these requirements, help has been obtained to make the quantitative studies from two different statisticians12.

A third type of legitimation is paradigmatic mixing legitimation and it refers to the extent to which the researchers reflect on their philosophical paradigm (Onwuegbuzie & Johnson, 2006). In this thesis the research paradigm is described in Section 11.1, where the choice of pragmatism as the research paradigm is described. However, the pragmatic paradigm approach had an overall influence on many of the single methods used in the different papers in the sense of fitting into different epistemological and ontological standpoints, but most importantly it has had an impact on the possibilities of combining the different approaches in this thesis, since the combining of quantitative and qualitative approaches is sometimes considered to be tenuous because of competing dualisms.

Weakness minimization legitimation, the fourth type of legitimation, is the degree to which a mixed researcher combines the qualitative and quantitative approaches so that they have no overlapping weaknesses (Onwuegbuzie & Johnson, 2006). This aspect is also described in Section 11.3. However, it might be important in this connection to clarify that the mix of the

12 Morten Petterson and Morten Puch Rasmussen both form UCL – University College 251

different approaches could have been strengthened if the test had also been able to give a significant result from the RCT study about whether the students develop more reasoning competence in the inquiry-based teaching in the intervention school compared to the control school.

The last type of legitimation, which now will be discussed, is the Sociopolitical validity which refers to the extent to which a mixed researcher appropriately addresses the interest value and standpoint of multiple stakeholders in the research process (Onwuegbuzie & Johnson, 2006). This is an important area to enter regarding this thesis because there have been many different stakeholders present in the KiDM project; e.g., the KiDM steering group, Ministry of Education, mathematics supervisors and teachers. Fortunately, this thesis has been a deviation to the KiDM project and it has therefore been possible to create an independent project that has not been directly under the influence of these stakeholders. However different constraints have been made, e.g., in the development of the test and the development of the intervention which is directly part of the KiDM project, such as decisions about which computer programs to use and decisions about specific activities included in the intervention. However, the qualitative studies in this thesis have been carried out without interference by other stakeholders than the author of this thesis and the supervisor. The different stakeholders in KiDM project have also been discussed in connection to implementation in Larsen et al. (2019).

A last aspect, which will be included here, is the role of the author of this thesis. A potential conflict of interest lies in the fact that of acting as a part of designing the intervention, a designer of the test, but at the same time also being the observer in the classroom and analysing all the data. The author of this thesis has a natural desire for the intervention to work and for the test to reflect this result, which in some way may have affected the findings (Brown, 1992). However, that is why it has been very important to adhere as strictly as possible to scientific principles and make the methods and designs as transparent as possible, which hopefully was the case in this thesis.

Based on the above descriptions of meta reflections on the different challenges in the study’s legitimacy, the conclusion is, however, that this research study is truly mixed with a reasonable validity, reliability and legitimation.

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15.3 The overall findings The five separate papers in this thesis already include findings, discussions and individual contributions. In this section, however, the focus will be on the findings in a more coherent perspective and the intention is to discuss the final answers to the overall research questions and the two sub-research questions. First, the two sub-research questions: RQ1: How is it possible to study the development of students’ reasoning competence in primary school mathematics classes? And RQ2: In what way can students’ reasoning competence develop within an inquiry-based teaching approach? will be answered and then the focus will be to which extent the overall research question: How can an inquiry-based teaching approach impact students’ reasoning competence in primary school mathematics classes?, is answered.

Findings in connection to: How it is possible to study the development of students’ reasoning competence in primary school mathematics classes – RQ1 In this thesis, according to the methods section, many different approaches have been used to examine the students' development of reasoning competence. Two areas have been specifically in focus in this thesis; 1) The possibility to test students’ development in reasoning competence, but also 2) New analytical ways of studying classroom video observation in connection to understanding the students’ developments of reasoning competence. However, surveys have also been included which will be discussed in a third section.

15.3.1.1 Testing development in reasoning competence First of all, the not statistically significant result from the developed competence test (see Section 12.2) and Paper IV is not surprising in relation to the complexity of developing such a test.

Testing developments in competences in mathematics is unquestionably not easy. In the TIMMS study they argue that they measure different mathematical competencies (reasoning, modelling and problem-solving), but after developing the KiDM test and trying out other item types, there are still many challenges in connection to measuring development in mathematical competences in a test.

First of all, the test needs to be valid. This was the focus in Paper IV, and in many ways it could be argued that the developed KiDM test is valid (content validity and construct validity) in 253

connection to measuring reasoning competence. However, even though different definitions of reasoning competence (NCTM, 2000; Niss & Højgaard, 2011; G. J. Stylianides, 2008) are in some ways covered (see Paper IV), there is still a need for a reflection of whether the test also includes a dynamic measurement of the competence which involves measuring its “degree of coverage”, “radius of action” and “technical level” (Niss & Højgaard, 2011). Niss and Højgaard (2011) problematise testing mathematical competences with these notions:

“It is, therefore, meaningless to allege that the level of coverage of the problem tackling competency in a person who can only solve problems within algebra, geometry and probability theory, is less than in a person who can solve problems within probability theory, functions, calculus of inﬁnitesimals and optimisation. Similarly, it is also meaningless to compare the technical level of the symbol and formalism competency in a person who is a master at handling expressions within trigonometry, with the technical level of a person who is a master at calculations in probability distributions.” (p. 73)

Based on this quote, the assignment of making taxonomies for mathematical competencies that are necessary in designing a test is challenging, because both the content of the mathematical concept and the context are difficult to get around in the measurement of the mathematical competences. This corresponds to the close relationship between concept development and competence development, described in the preliminary study, where it says that it is necessary to have a conceptual understanding before working inquiry-based with different competencies (Dreyøe, Michelsen, et al., 2017). This is exactly why the development of empirical taxonomies, which is possible with the Rasch model, have been specifically interesting and useful in the test development. The empirically developed taxonomies were furthermore compared with theoretically developed taxonomies (see Paper IV) and found valid.

Secondly, the test must be aligned to the way reasoning is taught in the school context in such a way that the students experience that what they have learned in school – what has been validated in the mathematics lessons – is also what they must be held responsible for in a subsequent test. This issue is discussed in Paper V and the result shows, that in some aspects the KiDM test is satisfactory aligned to the teaching, but, however, not in all aspects. There are some areas that are very clearly not possible to align - it concerns specifically the social situation, which of course is very different in the two approaches according to the social aspect from G. J. Stylianides (2009). In the classroom context the teacher often plays a major role in

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the teaching, while the students in the test situation are not guided with follow-up questions and prompts etc. This has clearly affected the outcome. Concerning the content of the reasoning competence, the students make argumentation and justifications both in the test situation and in the classroom teaching based on their rationales, and empirical justification schemes, but also based on mathematical justification schemes. However, in the test situation this typically happens in a calculation discourse where in the classroom the most validated approach is to make argumentation in a conceptual discourse (Paul Cobb, 1998) which was not possible in the KiDM test.

An unexplored area in this thesis is also the age of the students in relation to their ability to handle tests, as well as the challenges that year 4 students may not yet be able to write down their calculations and arguments, which was part of many of the reasoning tasks; they have to argue for their own results / processes. The test might have had a significant result if the test was used in year 6.

In conclusion, it is important to note that the KiDM test never had the intention that this test could stand alone in the assessment of the students' development of mathematical competences, because the complexity of students’ development in mathematics will never be able to be satisfactorily expressed as one score on any one test. The first findings to answer RQ1 are therefore that the KiDM test was a serious attempt to measure the students’ reasoning competences, and although various challenges were found with this KiDM test, it was an attempt to get closer to a way of testing competencies which hopefully in the future can be used in new studies on measuring the reasoning competence.

15.3.1.2 Observing and analysing development in students’ reasoning competence In order to investigate the students' development of reasoning competence, several video observations have been carried out in the KiDM classrooms. If this approach is to be added to any cause and effect approach, then the control schools also ought to be observed, which, because of the scope of this thesis, was not included.

Video observers can focus either on the students' oral performance or/and the students’ action, including how the students use different materials and representations in their reasoning process. In Papers I and III there is a specific focus on how the different materialites and

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representations are advocates for the students’ reasoning developments, while in Paper II the focus is more specifically on the students’ oral reasoning developments.

These types of observations certainly provide some interesting new knowledge of whether the students develop their reasoning competence by working with the different KiDM activities, but it is not a direct measurement of the students' developments. In essence, one would never be able to claim that simply because we observe that students work with the competence in some lessons that the students have developed their competence, but this can be seen as an indication that when in a single lesson students make positive development from the beginning to the end of the lesson that it must have an effect on the students' reasoning competence.

In Papers I and II different ways of analysing the qualitative observations were used. Respectively, a model from Latour (1999) (circulating references) and a model developed in KiDM (categorising different inquiry-based activities) have been used to analyse the observations. Both models have been useful and given some interesting new perspectives on how the reasoning competence develops in inquiry-based teaching, and the model from Latour (1999) especially gives some new perspectives on how to analyse development in mathematics, which deserves further research in the future.

15.3.1.3 Measuring development – other approaches In this thesis, other approaches were also used to measure the students' development of reasoning competence. Both a student survey and a teacher survey were developed. These types of measurements are especially good in RCT studies, because by getting students and/or teachers from both the control school and the intervention school to fill them in, it is possible to find differences and similarities. However, it must be clearly emphasised here that these measurements indicate what the students think and feel at the specific time when filling out the survey (down strokes). These experiences do not necessarily indicate any sustainable development. Furthermore, these experiences are very subjective, but they unquestionably give a new answer to the research question about how to measure development of reasoning competence besides tests and direct observations.

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15.3.1.4 Summary of the answer to RQ1 The overall answer to RQ1 is that it is possible to measure the development of reasoning competence in different ways, but that the different approaches also give different types of answers, because the answers will always reflect the way you measure. Testing is challenging, but this study indicates that it is not impossible, but there, however, needs to be more research done in this direction. The research done in this thesis about testing reasoning competences can be seen as new steps along the way to be able to test students’ developments. To be able to use observation to measure students’ development of reasoning competence is maybe more well- reputed; however, the analyses of these observations can also be very complex and challenging. In this thesis the classroom observations have been analysed with different analytical tools and they all seem to have the potential to measure a development. Finally, the surveys also give an important and different perspective about the teachers’ and students’ experiences of the competence development. All in all, it can be said that the three measurement perspectives together give a broad picture of whether the students have developed reasoning competence.

It is not surprising, however, that it is complex to measure competence development, but through this thesis it has become clear that there are many different approaches and that it is certainly worth trying these to gain a greater insight into the students’ development of reasoning competence.

The studies that have been done in connection with this thesis are not exhaustive and therefore further studies in this field could be envisaged. For example, a focus on the students’ products such as portfolios or developed posters could be interesting - as well as other types of tests or more open evaluations. Think-alouds with students, for example, before and after certain courses (see also the literature review in Section 9.4), could most likely also contribute with some new perspectives in this connection.

Developing reasoning competence in inquiry-based teaching - The transition from empirical argumentation to more abstract argumentation – answering RQ2 As described in the theory Section 9.1.1 there is a current focus in the literature to propose forms of proof or activities that might support students in making the transition from making empirical argumentations to more formal proofs and to investigate how such progress might be achieved (EMS, 2011). This transition is explored in different ways in this thesis.

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The overall findings indicate that an inquiry-based teaching approach does help the students in this transition, but there are different factors to be aware of.

Firstly, the students’ way of reasoning does not develop in one big step (from everyday reasoning to reasoning mathematically), instead it develops in overcoming small steps in a chain of small gabs. In the KiDM intervention, the approach was that the students often started in an everyday context or with some artefacts or materials, whose purpose was to help the students find the solution to the problem posed. The idea was that students used these representations to work towards the more specific and formal mathematical results.

Latour's model show us that this process happens in small steps (operators) in a chain, which means that it is important that the students do not cut the chain or skip any of these operators, but they have to overcome all the small gabs by themselves. Latour (1999) describes that each operation is important in the process, which can have some future implication for teaching in inquiry-based teaching (see also Paper I).

Secondly, different types of activities (defined in the KiDM project) in the inquiry-based approach help to focus differently on the reasoning competence. In the KiDM project, inquiry- based teaching was divided into smaller units (different types of activities) and thereby studied in more detail. The findings indicate that the students’ work with these different activities brought different aspects of reasoning into play. This indicates that the choice of activity in inquiry-based teaching is not only about the content of the subject, but it also has an impact on how the students make their argumentations in inquiry-based teaching. If, for example, the goal of an inquiry-based teaching activity is that the students must understand or develop a specific mathematical concept, then the teacher more often compromises on the development of the students' reasoning chains - and instead the teacher make the argumentation and validation then the external proof scheme (Harel & Sowder, 1998) becomes crucial (see also Paper II).

Thirdly, inquiry-based teaching has potential to create cognitive conflicts which can be productive and important in relation to the students mathematical reasoning process. The cognitive conflict can 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 inquiry. The process of resolving the cognitive conflicts can be a process stretched over time that do not necessarily entails large significant jumps in the students’ understandings instead the conflicts make the students involved in taking different routes and

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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 (see also Paper III).

If we now turn to the test result for findings indicating whether the students develop their reasoning competence in inquiry-based teaching, the result shows a positive trend in the favor of the students developing their mathematical competencies (modelling, problem- solving/problem-posing, reasoning) compared to the control schools, but the result was not statistically significant.

In the student survey conducted in KiDM the quantitative results indicated that the students experienced that the teaching was focusing more on argumentation and dialogue in the inquiry- based teaching approach than in the control schools. This is because the students experienced that they participated to a greater extent in the class discussions, they argued more often for their solutions, and they more often discussed their classmates' solution proposals than the students at the control schools (see Section 12.4).

The teacher survey, which is also described in Chapter 12, did, however, not show any clear trend and it did not have any significant difference between the way the teacher in the intervention schools and the control schools experienced the teaching in connection to teaching reasoning competence. This result is actually quite interesting as it can signify different things: firstly, in many cases it would have been imagined that the teachers would respond positively after they had been involved in an intervention, but it could also indicate that the intervention was too brief, and that it simply is not possible to change the teachers' attitudes and ideas in such a short time.

In Figure 21 all the different perspectives are showed in one model, where the circles specify that the results come from different ways of measuring. At the same time, it must be made clear that the different results in many ways support each other from their individual perspective. The video-observed results support the results of the self-perceived surveys and the tested results and none of the results indicate a contradiction.

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Test results - a positive trend (not significant)

Classroom Student surveys - a observations - positive result analysing cognitive conflicts in paper III - - experince of more indicate a positive discussions and result dialogue

Students development of reasoning competence in inquiry-based teaching

Classroom observations - Teacher surveys - no analysing with a clear trend - (not model from KiMD - significant results) indicate a positive result

Classroom observation - analysing with the model of Latour (1999) - indication of development

Figure 21: Overview of different perspectives answering whether the students develop reasoning competence in an inquiry- based teaching approach. The colors indicate the different ways of measuring the student’s development. (red:tests, grey:surveys, blue:classroom observations)

All in all, when we study the students’ transition from empirical argumentation to more formal mathematical argumentation, some researchers have presented evidence that letting students come up with and formulate conjectures themselves may support proof production by creating a cognitive unity between conjecture and proof (Bussi, Boero, Ferri, Garuti, & Mariotti, 2007). Others have focused on generic proofs (Rowland, 2002), or how instructional sequences can help students overcome and realise the limitations of empirical arguments as methods for validating mathematical generalisations (G. J. Stylianides & Stylianides, 2009). However, the findings in this study indicate that by teaching inquiry-based in mathematics, the students develop reasoning competence. The above-mentioned aspects answer RQ2.

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The overall research question: How can an inquiry-based teaching approach impact students’ reasoning competence in primary school mathematics classes? In conclusion, the research question can be answered from the above findings. Inquiry-based teaching can impact students’ reasoning competence in primary school mathematics classes by putting the students in some situations where the students themselves have to do some inquiry, so that the students themselves get through small gabs in a chain and thus go from the everyday problems/context to more abstract mathematical solutions. Also, by using different types of inquiry-based activities it will help the students put different foci on which approaches to argumentation must be in focus. The situations where the students work independently in inquiry-based mathematics teaching also have great potentials for the students to experience cognitive conflicts that will cause the students to develop their reasoning competence. Finally, the students in the survey also give expression to the fact that an inquiry-based approach in mathematics teaching puts more focus on the dialogue and the discussions in the class, which thereby strengthens the students' competencies to argue for their own solutions.

It is not necessarily that the students develop formal, abstract mathematical proofs, but, if the students work with the inquiry-based activities it strengthens the students' understanding of mathematical validation and in all circumstances the inquiry-based mathematics teaching gets the students to work with different ways to make justifications. The overall findings are summarised here in 4 dots:

Inquiry-based teaching impacts students’ reasoning competence:

• by focusing to a higher degree on dialogue and discussions and by having the students argue for their results (results from the survey). • by including different categorised activities that are able to focus on different aspects of the reasoning competence (results from Paper II). • by creating cognitive conflict which can drive reasoning, and hence, that the mathematical environment plays a critical role to retain these cognitive conflicts (results from Paper III). • by letting the students go from “every-day problems” to more formal mathematical answers in small steps – resolving small gabs - and thereby making the transition from arguing with rationale and empirical situations to arguing more manageably with more formal mathematics (results from Paper I).

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These four findings have some implications for both practice and future research, which will be the focus in the next section.

15.4 Implications for research and practice On the basis of the presented findings, there are different kinds of implications for future research and practice.

The implications of using Latour's model for analysing students’ working processes is very interesting and might have future possibilities in other types of analyses that do not necessarily address the reasoning competence. The model might, for example, also have some possibilities to contribute to research about modelling in mathematics, as Latour’s model can be seen as contributing to the transition from everyday problems to a more abstract mathematical understanding, which also points into the didactics of modelling, but other areas in education could also benefit from using this model in analysing development. The results from using the model in the area of reasoning competence in Paper I also have implications for practice. The findings point out that teachers should be aware of how to guide the students to overcome all the small gabs in their working processes by themselves so the chain of reference (Latour, 1999) does not break. The cases from Paper I indicate that without the student overcoming the gabs by themselves, they will not understand the process completely, which can influence the final outcome for the students. It can also have implication for how the final whole class discussion must be carried out – must the teacher focus on all the students’ different chains of reference or what happens if the teacher present a new way of making the chain? However, more research in this area would benefit these implications.

Analysing the different types of activities in Paper II also has different implications for the teacher's practice. Firstly, it is important for the teachers to be aware of which parts of the reasoning competence are intended to be in focus since the different inquiry-based activities initiate different foci. Secondly, it might also entail that there can be pitfalls in connection to developing reasoning competence in the various activities that the teacher must be aware of.

Implications for understanding the artefacts and materialities as supportive of students experiencing and overcoming cognitive conflicts during their reasoning process is, that it is important that the materials support the cognitive conflict (e.g. by being part of the planning)

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and that the teacher is aware, if the cognitive conflict does not arise, so that the teachers know when and how to and are able to break into the situation and guide the students in the right way.

Developing the KiDM test has also denoted that new items have been tested and thus may play a new role in subsequent research on how to develop a test that is able to measure reasoning competence in mathematics. Papers IV and V and the results from Chapter 12 clarify what to consider in constructing a new version of a mathematical competence test.

As described, there are multiple new aspects to follow up in connection to the research done in this thesis, and if the future is auspicious, an inclusion of the teacher interviews conducted with the observed teachers would be very interesting. A process could be to revisit the teachers to carry out new observations and new interviews to be able to analyse how these teachers might have implemented the KiDM project and whether they possibly now focus more on reasoning competence?

Finally, an overall implication for primary school teachers’ practice is that, if they want to focus on the reasoning competence in mathematics teaching, an inquiry-based approach would be a good starting point, and the KiDM project can be a fine point of departure. However, there are also certain critical voices in the contemporary political educational research, in connection to a focus on competences in education that here, in the end, will get a single word. This is Gruschka (2016) who describes that the development of competence in schools is an orientation that mostly focuses on how the students presents their results, how they create mind maps, but where the content is all forgotten. The content is instead regarded as a means of developing competences. Gruschka (2016) claims that when students’ aims are to gain a certain measurable competence, it happens at the expense of knowledge and understanding of, for example, concepts. He argues that this is a wrong development and he contends that the most important aim in school is understanding and bildung13. When we examine the Danish curriculum, it is clear that the description of each competency in mathematics can be seen as a kind of recipe that just needs to be followed and trained, which does not necessarily lead to understanding as claimed by Gruschka. However, even though reasoning competence in some contexts can also be seen as a recipe with a list included (e.g. “to use reasoning”, “to develop and verify hypotheses” or “to have knowledge of simple mathematics proof”). These aims can, however,

13 Bildung is translated from the Danish word “dannelse” and there is no exact English word for it, but it is sometimes translated as the word “formation” 263

be seen as good points to focus on reasoning in mathematics, hence teaching in mathematics is not only focusing on training skills and rote learning, which is generally the tradition in mathematics. Gruschka (2016) argues moreover that the problem is if we change the evaluation or test of the teaching, so that it contains only competence descriptions without focusing on understanding, because then it will be psychometrics that set the agenda and not the pedagogy and didactics. This confirms the idea - as this thesis would like to reflect, that mathematics educators will need to get more into research about test development.

An opposite perspective is that of Bybee (2018), who points to the fact that context, content and competencies are inseparable and that when teaching competencies, it always has a content and it is always done in a context: “…an individual’s competence is directly influenced by the content of his or her knowledge of the natural, designed, and mathematical domains…” (Bybee, 2018, p. 88). The interpreting of this is therefore that this focusing on competences does not mean less focus should be placed on understanding different concepts in mathematics, but competencies are connected to both content and context. The intention in this thesis is therefore that the teaching of mathematics must stimulate both competence and understanding of the content, so that they interact with each other, because both aspects are important to develop in future primary school mathematics classrooms.

15.5 Final conclusion and comments The overall intention in this thesis was to deepen the theoretical understanding of classroom phenomena that relate to measuring and developing reasoning competence in mathematics by studying an inquiry-based teaching approach. According to Paul Cobb et al. (2003, p. 10) design-based research methods “are typically test-beds for innovation” with their intent being “to investigate the possibilities for educational improvement by bringing about new forms of learning in order to study them”. The idea was, after the development of the KiDM intervention and the KiDM test, to theorise over the developments as an explicit empirical component and draw on systematic methods of data collection (both qualitative and quantitative) for analysis in order to examine and ultimately offer supportive evidence for the potential and effectiveness of this large-scale intervention in connection to reasoning competence.

The hope is that the research done in this thesis will help teachers in the future to understand the concept of reasoning competence in mathematics better and, by making their teaching more

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inquiry-based, also contribute to focusing more on students’ development of reasoning competence in primary school mathematics classrooms.

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17 Appendix

Appendix Title of appendix included in the thesis

a References from the literature review about testing competencies in mathematics b The authors' nationality for the various articles to the litterature review c Item-map d Observations guide 1 (in Danish) e Observations guide 2 (in Danish) f Letter of consent (in Danish) g Central transcripts from the observed lessons h Transcription-guide to the KiDM project (in Danish) i Coding guide (in Danish) j Calculated results based on the regression model to the teacher survey k Calculated results based on the regression model to the students’ survey l Co-author Statement

286

a. References from the literature review - testing competencies in mathematics

1. Adesina, A., Stone, R., Batmaz, F., & Jones, I. (2014). Touch arithmetic: A process-based Computer-Aided Assessment approach for capture of problem solving steps in the context of elementary mathematics. Computers & Education ( 78), 333-343. doi:10.1016/j.compedu.2014.06.015 2. Akgul, S., & Kahveci, N. G. (2016). A Study on the Development of a Mathematics Creativity Scale. Eurasian Journal of Educational Research (62), 57-76. 3. Arikan, E. E., & Unal, H. (2014). Development of the Structured Problem Posing Skills and Using Metaphoric Perceptions. European Journal of Science and Mathematics Education, 2(3), 155-166. 4. Arikan, E. E., & Ünal, H. (2015). An Investigation of Eighth Grade Students' Problem Posing Skills (Turkey Sample). International Journal of Research in Education and Science, 1(1), 23-30. 5. Bahar, A., & Maker, C. J. (2015). Cognitive Backgrounds of Problem Solving: A Comparison of Open-Ended vs. Closed Mathematics Problems. EURASIA Journal of Mathematics, Science & Technology Education, 11(6), 1531-1546. 6. Boonen, A. J. H., van Wesel, F., Jolles, J., & van der Schoot, M. (2014). The role of visual representation type, spatial ability, and reading comprehension in word problem solving: An item-level analysis in elementary school children. International Journal of Educational Research, 68, 15-26. doi:10.1016/j.ijer.2014.08.001 7. Bostic, J. D., Pape, S. J., & Jacobbe, T. (2016). Encouraging sixth-grade students’ problem- solving performance by teaching through problem solving. Investigations in mathematics learning, 8(3), 30-58. 8. Burns, M. K., Codding, R. S., Boice, C. H., & Lukito, G. (2010). Meta-analysis of acquisition and fluency math interventions with instructional and frustration level skills: Evidence for a skill-by-treatment interaction. School Psychology Review, 39(1), 69. 9. Chang, K. E., Wu, L. J., Weng, S. E., & Sung, Y. T. (2012). Embedding game-based problem-solving phase into problem-posing system for mathematics learning. Computers & Education, 58(2), 775-786. doi:10.1016/j.compedu.2011.10.002 10. Charlesworth, R., & Leali, S. A. (2012). Using Problem Solving to Assess Young Children's Mathematics Knowledge. Early Childhood Education Journal, 39(6), 373-382. 11. Childs, R. A. (2009). "The First Year, They Cried": How Teachers Address Test Stress. Canadian Journal of Educational Administration and Policy (96), 1-14. 12. Christou, C., & Papageorgiou, E. (2007). A Framework of Mathematics Inductive Reasoning. Learning and Instruction, 17(1), 55-66.

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13. Chu, H.-C., Hwang, G.-J., & Huang, Y.-M. (2010). An Enhanced Learning Diagnosis Model Based on Concept-Effect Relationships with Multiple Knowledge Levels. Innovations in Education and Teaching International, 47(1), 53-67. 14. Csikos, C., Szitanyi, J., & Kelemen, R. (2012). The Effects of Using Drawings in Developing Young Children's Mathematical Word Problem Solving: A Design Experiment with Third-Grade Hungarian Students. Educational Studies in Mathematics, 81(1), 47-65. 15. Desoete, A. (2007). Evaluating and Improving the Mathematics Teaching-Learning Process through Metacognition. Electronic Journal of Research in Educational Psychology, 5(3), 705-730. 16. Desoete, A. (2008). Multi-Method Assessment of Metacognitive Skills in Elementary School Children: How You Test Is What You Get. Metacognition and Learning, 3(3), 189- 206. 17. Desoete, A. (2009). Metacognitive prediction and evaluation skills and mathematical learning in third-grade students. Educational Research and Evaluation, 15(5), 435-446. doi:10.1080/13803610903444485 18. Ferrara, S., Svetina, D., Skucha, S., & Davidson, A. H. (2011). Test Development with Performance Standards and Achievement Growth in Mind. Educational Measurement: Issues and Practice, 30(4), 3-15 19. Fyfe, E. R. (2016). Providing Feedback on Computer-Based Algebra Homework in Middle- School Classrooms. Computers in Human Behavior, 63, 568-574. 20. Fyfe, E. R., Rittle-Johnson, B., & DeCaro, M. S. (2012). The Effects of Feedback during Exploratory Mathematics Problem Solving: Prior Knowledge Matters. Journal of Educational Psychology, 104(4), 1094-1108. 21. Goetz, T., Preckel, F., Pekrun, R., & Hall, N. C. (2007). Emotional Experiences during Test Taking: Does Cognitive Ability Make a Difference? Learning and Individual Differences, 17(1), 3-16. 22. Hickendorff, M. (2013). The Language Factor in Elementary Mathematics Assessments: Computational Skills and Applied Problem Solving in a Multidimensional IRT Framework. Applied Measurement in Education, 26(4), 253-278. 23. Hu, W., Adey, P., Jia, X., Liu, J., Zhang, L., Li, J., & Dong, X. (2011). Effects of a 'Learn to Think' intervention programme on primary school students. British Journal of Educational Psychology, 81(4), 531-557. doi:10.1348/2044-8279.002007 24. Hunsader, P. D., Thompson, D. R., & Zorin, B. (2013). Engaging Elementary Students with Mathematical Processes during Assessment: What Opportunities Exist in Tests Accompanying Published Curricula? International Journal for Mathematics Teaching and Learning. 1-25 25. Jacobse, A. E., & Harskamp, E. G. (2012). Towards Efficient Measurement of Metacognition in Mathematical Problem Solving. Metacognition and Learning, 7(2), 133- 149. 26. Kan, A., & Bulut, O. (2015). Examining the Language Factor in Mathematics Assessments. Education Research and Perspectives, 42, 582-606.

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27. Kaosa-ard, C., Erawan, W., Damrongpanit, S., & Suksawang, P. (2015). How to Classify the Diversity of Seventh Grade Students' Mathematical Process Skills: An Application of Latent Profile Analysis. Educational Research and Reviews, 10(11), 1560-1568. 28. Kar, T. (2015). Analysis of Problems Posed by Sixth-Grade Middle School Students for the Addition of Fractions in Terms of Semantic Structures. International Journal of Mathematical Education in Science and Technology, 46(6), 879-894. 29. Kim, M. K., & Noh, S. (2010). Alternative Mathematics Assessment: A Case Study of the Development of Descriptive Problems for Elementary School in Korea. EURASIA Journal of Mathematics, Science & Technology Education, 6(3), 173-186. 30. Kinda, S. (2010). Assessment of Subtraction Scene Understanding Using a Story- Generation Task. Educational Psychology, 30(4), 449-464. 31. Kingston, N. M. (2009). Comparability of computer- and paper-administered multiple- choice tests for K-12 populations: A synthesis. Applied Measurement in Education 22(1) 22-37 [doi:10.1080/08957340802558326] 32. Klinkenberg, S., Straatemeier, M., & van der Maas, H. L. J. (2011). Computer Adaptive Practice of Maths Ability Using a New Item Response Model for on the Fly Ability and Difficulty Estimation. Computers & Education, 57(2), 1813-1824. 33. Lawson, A., & Suurtamm, C. (2006). The challenges and possibilities of aligning large- scale testing with mathematical reform: The case of Ontario. Assessment in Education: Principles, Policy & Practice, 13(3), 305-325. doi:10.1080/09695940601035486 34. Leh, J. M., Jitendra, A. K., Caskie, G. I. L., & Griffin, C. C. (2007). An Evaluation of Curriculum-Based Measurement of Mathematics Word Problem--Solving Measures for Monitoring Third-Grade Students' Mathematics Competence. Assessment for Effective Intervention, 32(2), 90-99. 35. Lein, A. E., Jitendra, A. K., Starosta, K. M., Dupuis, D. N., Hughes-Reid, C. L., & Star, J. R. (2016). Assessing the Relation between Seventh-Grade Students' Engagement and Mathematical Problem Solving Performance. Preventing School Failure, 60(2), 117-123. 36. Limin, C., Van Dooren, W., & Verschaffel, L. (2013). The Relationship between Students' Problem Posing and Problem Solving Abilities and Beliefs: A Small-Scale Study with Chinese Elementary School Children. Frontiers of Education in China, 8(1), 147-161. 37. Logan, T. (2015). The Influence of Test Mode and Visuospatial Ability on Mathematics Assessment Performance. Mathematics Education Research Journal, 27(4), 423-441. 38. Logan, T. & Lowrie, T. (2013). Visual Processing on Graphics Task: The Case of a Street Map. Australian Primary Mathematics Classroom, 18(4), 8-13. 39. Mavilidi, M. F., Hoogerheide, V., & Paas, F. (2014). A quick and easy strategy to reduce test anxiety and enhance test performance. Applied Cognitive Psychology, 28(5), 720-726. doi:10.1002/acp.3058 40. Mousoulides, N. G., Christou, C., & Sriraman, B. (2008). A Modeling Perspective on the Teaching and Learning of Mathematical Problem Solving. Mathematical Thinking and Learning: An International Journal, 10(3), 293-304. 41. Munroe, K. L. (2016). Assessment of a Problem Posing Task in a Jamaican Grade Four Mathematics Classroom. Journal of Mathematics Education at Teachers College, 7(1), 51- 58. 289

42. Mushin, I., Gardner, R., & Munro, J. M. (2013). Language Matters in Demonstrations of Understanding in Early Years Mathematics Assessment. Mathematics Education Research Journal, 25(3), 415-433. 43. Neuenhaus, N., Artelt, C., Lingel, K., & Schneider, W. (2011). Fifth Graders Metacognitive Knowledge: General or Domain-Specific? European Journal of Psychology of Education, 26(2), 163-178. 44. Nunes, T., Bryant, P., Evans, D., & Barros, R. (2015). Assessing Quantitative Reasoning in Young Children. Mathematical Thinking and Learning, 17(2-3), 178-196. 45. Nunes, T., Bryant, P., Evans, D., Bell, D., Gardner, S., Gardner, A., & Carraher, J. (2007). The contribution of logical reasoning to the learning of mathematics in primary school. British Journal of Developmental Psychology, 25(1), 147-166. doi:10.1348/026151006X153127 46. Nyroos, M., & Wiklund-Hornqvist, C. (2011). Introducing National Tests in Swedish Primary Education: Implications for Test Anxiety. Electronic Journal of Research in Educational Psychology, 9(3), 995-1022. 47. Ocak, G., & Yamac, A. (2013). Examination of the Relationships between Fifth Graders' Self-Regulated Learning Strategies, Motivational Beliefs, Attitudes, and Achievement. Educational Sciences: Theory and Practice, 13(1), 380-387. 48. Phelps, C., & Price, J. (2016). Slowing the hare: Quick finishers and class performance on standardized tests. Learning and Individual Differences, 51, 322-326. doi:10.1016/j.lindif.2016.08.005 49. Roderer, T., & Roebers, C. M. (2013). Children's Performance Estimation in Mathematics and Science Tests over a School Year: A Pilot Study. Electronic Journal of Research in Educational Psychology, 11(1), 5-24. 50. Schoenfeld, A. H. (2015). Summative and Formative Assessments in Mathematics Supporting the Goals of the Common Core Standards. Theory Into Practice, 54(3), 183- 194. 51. van den Heuvel-Panhuizen, M., Robitzsch, A., Treffers, A., & Koller, O. (2009). Large- Scale Assessment of Change in Student Achievement: Dutch Primary School Students' Results on Written Division in 1997 and 2004 as an Example. Psychometrika, 74(2), 351- 365.

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b. The authors' nationality for the various articles to the literature review

The nationality of the articles is based on the first author. In many of the articles the second, third or more authors are from another country, this is not included in this table.

Countries Articles Number of articles

USA 7,8,10,18,19,20,24,31,34,35,50 11 Turkey 2,3,4,5,26,28,47 7 Holland 6,22,25,32,39,51 6 Australia 37,38,42,48 4 Belgium 16,17 3 Germany 21,43,49 3 UK 1,44,45 3 Canada 11,33 2 China 23,36 2 Cypress 12,40 2 Japan 30,41 2 Taiwan 9,13 2 Hungary 14 1 Korea 29 1 Sweden 46 1 Thailand 27 1 all 51

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c. Item map

The yellow items are from the competence test and the blue items are from the concept test.

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d. Observations guide 1 (in Danish)

(Observationsguide to the first trial)

Plan:

- At følge én klasse i 3 dobbeltlektioner (gentages i 5-6 forskellige klasser)

Guide for feltbesøg i forskellige klasser – med 3 besøg i hver klasse

Der filmes direkte på en elev i hele lektionen. Inden undervisningens begyndelse udvælges i samarbejde med læreren en enkelt elev. Eleven vælges ud fra lærerens beskrivelse. Der ønskes en elev, der gerne både er arbejdsom i timerne, samt er robust til at kunne deltage i et efterfølgende interview, men som samtidig også gerne er forholdsvis stærk i matematik, således at der er stor sandsynlighed for at eleven får ræsonneret i lektionen.

Der indsamles kopi af dokumenter fra lærere og elever hvis relevant og muligt (fx elevproduktioner herunder fx plancher og tegninger; lærerens skriftliggjorte supplementer til undervisningen eller fx feedback på elevprodukter). Omfanget og relevansen af supplerende produkter baserer sig på en vurdering i situationen.

Der skal på forhånd laves en samtykkeerklæring så alle har lov til at deltage i videoen.

Alle aktiviteter i KiDM er bygget op omkring 3 faser: iscenesættelse, aktivitetsfasen og fællesgørelsen. Observationsguiden er også bygget op omkring disse tre faser.

Lektionen Fasen i lektionen Videoens fokus + Begrundelse

opmærksomhedspunkter

Iscenesættelsen Denne fase vil primært Eleven filmes hele tiden, der Eleven kan allerede i være lærerens lægges mærke til om eleven iscenesættelsen producere fremlæggelse af evt. skriver noget ned eller vigtige ræsonnementer som lektionens problemstilling andet, som derved skal filmes er afgørende for resten af eller aktivitet, men tæt på. aktiviteten. indimellem vil/kan eleverne bliver inddraget i drøftelser eller små Ved klasseundervisning er det Lærerens spørgsmål kan diskussioner eller der vil vigtigt også at få optaget hvad være vigtige informationer i kunne foregå små lærerne stiller af spørgsmål forhold til at få en større parøvelser. eller diverse hints. Ligeledes forståelse for hvilke ræsonnementer eleven

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hvis der står/skrives noget på producerer eller ikke tavlen. producerer.

Aktiviteten Aktivitetsfasen kan foregå Eleven filmes i alle Eleven kan producere meget forskelligt. henseender, både indendørs og ræsonnementer i mange udendørs. Det er vigtigt at være forskellige henseender i Dvs. der i denne fase opmærksom på, hvad eleven løbet af aktiviteten. Det kan typisk både vil være producerer - herunder hvad der både være i starten af adidaktiske og didaktiske bliver skrevet, tegnet, aktiviteten, hvor eleven får situationer, ligesom der bevægelser osv. nogle ideer til, hvordan kan foregå par-arbejde aktiviteten kan eller gruppearbejde. udføres/løses, men det kan også opstå undervejs, eller I gruppearbejde/pararbejde, er til sidst hvor der måske det vigtigt at få optaget/filmet efterrationaliseres og hvad hans/hendes evalueres i forhold til gruppe/makker siger udførelsen af aktiviteten undervejs. eller en generalisering.

Fællesgørelsen Fællesgørelsen kan foregå Eleven filmes hele tiden. Det er vigtigt at eleven meget forskelligt fra de filmes undervejs i forskellige aktiviteter. Det fællesgørelsen, da eleven i kan være fremlæggelser denne sidste fase måske Ved klasseundervisning er det eller fælles gennemgang stadig kan opnå at vigtigt også at få optaget hvad ved tavlen nedskrive/tegne/fortælle det er lærerne stiller af flere vigtige matematiske spørgsmål eller diverse hints, ræsonnementer. eller hvad der skrives/står på tavlen

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e. Observation guide 2 (in Danish)

For observation i runde 3, hvor samme klasse blev observeret igennem et helt KiDM intervention.

Plan:

- At følge én klasse igennem hele interventionen - besøge 1 gang om ugen - At udføre et interview med læreren før intervention og efter interventionen.

Guide for feltbesøg ca. 1 gang om ugen

Ved hvert besøg video optages (afhængig af adgang) hele klassen med håndholdt kamera ved introduktionen til aktiviteten samt fællesfølelsen af aktiviteten. Der udvælges én gruppe i hver lektion i samarbejde med læreren. Der filmes direkte på eleverne i aktiviteten. Der er hele tiden fokus på interaktion ml. lærer og elever og elever imellem. Både faglige og sociale interaktioner. Der udarbejdes umiddelbart efter besøget en feltkommentar i et fortløbende dokument med feltkommentarer. Feltkommentarer er en løbende logbog over observationen, som der skrives i efter hvert feltbesøg. Der indsamles kopi af dokumenter fra lærere og elever hvis relevant og muligt (fx elevproduktioner herunder fx plancher og tegninger; lærerens skriftliggjorte supplementer til undervisningen eller fx feedback på elevprodukter). Omfanget og relevansen af supplerende produkter baserer sig på en vurdering i situationen. Der noteres i feltnoter hvis der indsamles materialer. Der aftales tydeligt, hvordan jeg får adgang til disse materialer. Der nedskrives i feltnoterne, hvis der er sket ændringer i forhold til konteksten (fysiske omgivelser, faglige rammesætning, hvordan er eleverne placeret i rummet, herunder siddepladser). Der skal på forhånd laves en samtykkeerklæring så alle har lov til at deltage i videoen.

Alle aktiviteter i KiDM er bygget op omkring 3 faser: iscenesættelse, aktivitetsfasen og fællesgørelsen. Observationsguiden er også bygget op omkring disse tre faser.

Lektionen Fasen i lektionen Videoens fokus + Begrundelse

opmærksomhedspunkter

Iscenesættelsen Denne fase vil primært Ved klasseundervisning er det Eleverne i gruppen kan være lærerens vigtigt også at få optaget hvad allerede i iscensættelsen fremlæggelse af lærerne stiller af spørgsmål producere vigtige lektionens problemstilling eller diverse hints. Ligeledes ræsonnemeter som er

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eller aktivitet, men hvis der står/skrives noget på afgørende for resten af indimellem vil/kan tavlen. aktiviteten. eleverne bliver inddraget i drøftelser eller små Der fokuseres på interaktionen diskussioner eller der vil mellem lærer og elev/elever i Lærerens spørgsmål kan kunne foregå små gruppen. være vigtige informationer i parøvelser. forhold til at få en større forståelse for hvilke ræsonnementer eleverne producerer eller ikke producerer.

Aktiviteten Aktivitetsfasen kan foregå Der udvælges en gruppe som Eleverne kan producere meget forskelligt. filmes igennem hele ræsonnementer i mange aktiviteten. Herunder både forskellige henseender i Dvs. der i denne fase indendørs og udendørs. Det er løbet af aktiviteten. Det kan typisk både vil være vigtigt at filme elevernes både være i starten af adidaktiske og didaktiske interaktioner, men også være aktiviteten hvor eleverne får situationer, ligesom der opmærksom på, hvad der nogle ideer til, hvordan kan foregå par-arbejde skrives ned eller tegnes i aktiviteten kan eller gruppearbejde. gruppen. udføres/løses, men det kan også opstå undervejs, eller Hvis læreren henvender sig – til sidst hvor der måske skal interaktionen mellem efterrationaliseres og læreren og eleverne optages. evalueres i forhold til udførelsen af aktiviteten

eller en generalisering.

Fællesgørelsen Fællesgørelsen kan foregå Ved klasseundervisning er det Det er vigtigt at både læreren meget forskelligt fra de vigtigt at få optaget hvad det er og eleverne filmes undervejs forskellige aktiviteter. Det lærerne stiller af spørgsmål i fællesgørelsen, da eleverne kan være fremlæggelser eller diverse hints, eller hvad også i denne sidste fase eller fælles gennemgang der skrives/står på tavlen. stadig kan opnå at ved tavlen nedskrive/tegne/fortælle Elevernes svar og spørgsmål til flere vigtige matematiske læreren skal ligeledes optages. ræsonnementer.

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f. Letter of consent (in Danish) d.20/8 2018

Kære elever og forældre i 5. klasse på XXXXX

I forbindelse med klassens deltagelse i et ministerielt forsknings og udviklingsprojekt i matematikundervisningen ved navn ”Bedre Kvalitet i Dansk og Matematik” (KiDM), vil jeg komme på besøg i klassen og observere undervisningen flere gange i løbet af de 3-4 måneder forløbet varer. For at kunne indfange så mange aspekter af undervisningen som muligt, vil jeg meget gerne have mulighed for at undervisningen i disse lektioner blive videofilmet. Denne film vil på ingen måde blive offentlig tilgængeligt, men vil blot blive anvendt til intern forskning og evaluering i projektet, samt enkelte anonymiserede transskriberede uddrag vil blive anvendt i uddannelsesmæssige sammenhænge. Vi følger selvfølgelig de nuværende databeskyttelsesforordninger i forhold til opbevaring og brug af datamateriale.

Det er vigtigt at både I forældre giver samtykke til at lade jeres barn filme, men også at I elever selv giver samtykke til at blive filmet. Derfor vil jeg bede jer aflevere nedenstående seddel underskrevet til jeres matematiklærere XX inden d. 31. august.

Jeg glæder mig til at være en del af undervisningen på jeres skole i forbindelse med dette projekt.

Hvis I har nogle spørgsmål eller andre kommentarer er i velkommen til at kontakte mig.

Med venlig hilsen

Dorte Moeskær Larsen, ph.d. studerende, KiDM ([email protected]) eller tlf. xxxxxxxx ------Tilladelse:

Elevens navn (blokbogstaver):

Jeg og mindst en af mine forældre giver hermed tilladelse til, at de dele af ovennævnte datamateriale, hvori jeg optræder, må bruges på de ovennævnte betingelser.

Egen underskrift14:

Forældreunderskrift: ______

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g. Three central transcripts from the observed lessons (in Danish).

Title of the transcript - school Which paper are the transcript activity central

Hvad vejer kasserne School 1 Central in paper II and paper V [What do the boxes weigh]

Hvad vejer kasserne School 2 Central in paper I [What do the boxes weigh]

Rebtrekanten school 3 Central in paper III [Rope triangles]

All together there are 23 double lessons observed and transcribed. The remaining 20 which is not in the appendix can be sent if requested by contacting the author of this thesis.

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Transskribering fra skole 1 (hvad vejer kasserne)

I klassen sidder der ca. 20 elever fordelt på 4-mandsborde. Læreren står ved et smartboard.

Den elev der bliver observeret specifikt, har her fået navnet: Per

De andre elever i klassen er anonymiseret til Elev 1, elev 2 osv.

”Makker” er ham der sidder ved siden af Per og arbejder sammen med ham om opgaven

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1 Lærer: Vi blev lige lidt presset, der hvor i lige var ved at have fri, så i havde lidt svært ved at høre efter. Så jeg 2 kunne godt tænke mig, at vi lige fulgte op, hvad var det for nogle tal vi snakkede om i går. Og lige om et øjeblik 3 er jeg klar til at skrive det ned. Elev 1?

4 Elev: Primtal

5 Lærer: Primtal, primtal var en af de ting vi nåede frem til i går og hvad var det nu der var med primtal? Elev1?

6 Elev1: (Mumler)

7 Lærer: At der kun var et rektangel til primtal ikk? Og det var 1 og tallet selv det gik op i. Så det var den ene og den 8 fik vi skrevet en hel række op af.

9 Elev1: Den gider ikke.

10 Lærer: Jo den gider godt, den tænker bare helt vildt langsomt. [Begynder at skrive på smart boardet] Se. Så det 11 var primtallene. Så snakkede vi om en ting til.

12 [Eleverne rækker hånden i vejret]

13 Lærer: Elev2

14 Elev2: Lige tal

15 Lærer: Vi snakkede om lige tal og hvad blev vi enige om ved de lige tal?

16 Elev2: At de mindst havde 2.

17 Lærer: At de mindst havde 2. Når så snart vi nåede op over 5 ikke. Ja, fordi de havde den som primtal også har, 18 hvor 1 og tallet selv er med i og de har 2, den hvor 2 er med i os ikk? jaa, og så snakkede vi om en til. Per?

19 Per: Kvadrattallet

20 Lærer: Kvadrattallet, og hvad var det med kvadrattallet? [Skriver på tavlen]

21 Elev: mumler

22 Lærer: At de kunne lave…. Nu skriver jeg uden at kigge, nå sådan… At de kunne lave et kvadrat, hvad er det nu vi 23 laver et kvadrat, hvad er det nu et vil sige.

24 Per: At de er lige lange

25 Lærer: At de er lige lange ikk?

26 Per: Mm

27 Lærer: Så man kunne lave en hvor det var det samme tal på det ene og det andet led. De her er ret vigtige lige at 28 få skrevet ned bag øret, jeg sad jeg tænkte faktisk på da jeg kom hjem i går, at det kunne være vi skulle lave os 29 en mappe, hvor vi skriver nogle af de der matematikord op. Jeg plejer først at gøre det når vi rammer 6, men det 30 kan godt være vi skal starte allerede nu. Hvor vi får skrevet nogle af alle disse ord op og lige får skrevet med jeres 31 ord, hvad er det de betyder. Sådan så når vi om et år, lige pludseligt begynder at snakke om primtal, så har man 32 en opslagsmappe, som man kan slå det op i. Så det tror jeg vi gør, men det gør vi i næste uge. Så laver vi ligesom 33 en mappe. Nu skal vi nemlig til at snakke om noget andet. [Tegner et eller andet på tavlen] Nu har vi, nu har vi 34 talt bøger og fundet ud af, hvordan dælen det var. Så har vi store grupper, så har vi arbejdet med rektangler: 2, 35 2, 3, 3. I dag skal i også være 2 og 2 og i dag skal i også lave en lidt pudsig opgave. Som kræver at i tænker jeg 36 om. Åh nej, og så på en fredag. Så det der med at kigge rundt og sige: det ved jeg ikke. Det kan man ikke, det 37 kommer man ikke videre med vel? Den her opgave den handler om en mand

38 Elev: En mand?

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39 Lærer: som hedder, en mand, som hedder Svend Bent, Svend Bent Ejgild Nielsen. Svend Bent Ejgild Nielsen er i 40 gang med at pakke nogle ting ned, i sin lejlighed.

41 Elev: Kan vi ikke bare kalde ham Svend.

42 Lærer: Jo vi nøjes med at kalde ham for Svend, det er nemmere og Svend han er i gang med at pakke nogle ting 43 sammen og dem skal han sende med posten, eller sådan noget. Posten det er fint. Så ved i godt, at når man skal 44 sende noget med posten, så betyder det noget, hvor meget kassen den vejer. Så Svend han har fået vejet sine 45 kasser, han har 4 kasser. [Stiller 4 hvide kasser op på en reol] Han har 4 kasser. Han er bare kommet til at lave en 46 lille svipser, fordi han har vejet alle de her kasser, men han har kun fået af vide hvad de vejer parvis, altså 2 og 2. 47 Det vil sige, at han har fundet ud af [Roder med smartboardet]

48 Elev: Må jeg godt gå ud og tisse?

49 Lærer: Øhhh kun hvis det overhovedet ikke kan vente, fordi når vi er ved at gennemgå opgaverne så er det ikke 50 særligt smart. [Roder mere med smart boardet]

51 Tidskode 5 minutter

52 Lærer: Vi har vores kasser og øhhh de her kasser har han så fået vejet parvis og de her 2 kasser [Tegner 2 kasser] 53 De her 2 kasser til sammen, har han fundet ud af de vejer 6 kilo. Og så er der 2 kasser her, herover [Tegner 2 54 kasser], som til sammen vejer 8 kilo, nu har jeg tegnet dem lidt store, så nu er det lidt svært at få plads til dem 55 alle sammen. Og så er der 2 kasser her, som vejer 10 kilo, ja nogle af dem er samme størrelse, fordi det er lidt 56 nemmere, når jeg skal stå og tegne dem. Så er der dem her der vejer 12 kilo. Så er der 2 kasser dertil sammen 57 vejer 14 kilo. Og, 2 kasser herover, som står lige ved siden af hinanden, som dertil sammen vejer 16 kilo. Det vil 58 altså sige, det her det er de forskellige, nu blev det lidt noget rod, kan i godt fornemme det?

59 Elever: Jaa

60 Lærer: Jaaa, det står også på det papir i får lige om lidt. Det er de tal vi har, opgaven går ud på, hvor meget vejer 61 de enkelte af de 4 kasser?

62 Elever: Mumler og småsnakker

63 Lærer: Se og vi skal lige lave nogle aftaler på forvejen, fordi nu skal i arbejde sammen 2 og 2 om det her og så 64 skal i lægge en strategi, hvordan dælen vi lige finder ud af det her og så skal i have fundet frem til et resultat. 65 Derudover, derudover, så skal i love mig, at det ikke blive sådan noget med at løbe rundt til hinanden, altså til 66 nogle andre i ens gruppe. Og synes man, at man har en løsning eller et spørgsmål, så rækker man hånden i vejret. 67 Og det er ikke noget med åh, hvad har i fået og hvad har i fået? Stille og roligt selv, fordi det vi blandt andet lære 68 noget af den her, det er det arbejde der foregår herinde! [Peger op på sit hoved] Hvis der kommer en herude og 69 fortæller en, hvad man skal gøre hele tiden, så øver man sig ikke selv, så hvis i gerne vil give jer selv og jeres 70 klassekammerater en chance for at blive så gode som muligt, så giv folk en chance for at tænke sig om. Og det 71 er jo så fantastisk at i et klasselokale med 24-25 elever i. 25 elever i, så er der 25 mulige måder at tænke på og 72 der er ikke nogen af dem der er forkert. Okay? Elev3?

73 Elev3: Kan du ikke godt dele grupperne?

74 Lærer: Jo men som udgangspunkt så har I faktisk lige fået nye pladser, så nu arbejder i sammen med dem. Elev4 75 synes lige han havde vundet i lotteriet haha

76 Tidskode 8.54 Kildevældsskolen observation 1

77 Elev: Åh hvor er du heldig

78 Lærer: Schhhhh, husk nu lige, at det du siger ikke kun påvirker dem der over, men hvad tænker Elev5 nu siger 79 sådan?

80 Elev5: Hvad mener du?

301

81 Lærer: Jeg mener, at hvis der sad en ved siden af dig sad og tænkte: ”ahh fuck” og lige fik sagt det, hvad vil du så 82 tænke?

83 Elev5: (Mumler)

84 Lærer: Hvis Elev5 havde sagt, ej fuck den gruppe gider jeg ikke

85 Elev: Jaaa

86 Lærer: Og hun skal arbejde sammen med dig

87 Elev: (Mumler) så vil jeg bare tænke øh

88 Lærer: Nej, det er ikke fedt vel?

89 Elev: Nææ

90 Lærer: Nej…. Nå undskyld.

91 Elev: Mumler…

92 Lærer: Altså til sidst så har i jo selv fundet ud af det, vi følger op sammen til sidst og snakker om det. Jamen ved i 93 hvad, det fede er jo, at det kan i jo selv finde ud af, om hvad jeres svar er rigtigt, fordi hvis i finder frem til fire 94 kasser, som i tror vejer det og det og det. Så kan i jo tjekke det.

95 Elev: Nååå er der kun 4 kasser.

96 Lærer: Ja der er 4 kasser ikk, han har bare vejet dem sådan lidt på kryds og tværs, nå nu tager vi de to, nu tager 97 vi de to og i kan jo selv tjekke det, så hvis i tror de vejer henholdsvis 1, 2, 3 og 4 kilo. Så kan i jo ret hurtigt tjekke 98 og sige, at hvis de vejer 1, 2, 3 og 4 kilo, så vejer den der og den der 3 kilo tilsammen, ahhh, hvordan kan vi få 99 dem til at veje 8, hvis de vejer, 1, 2, 3 og 4 kilo. Ahh den bliver svær, så vejer de nok ikke 1, 2, 3 og 4 kilo.

100 Elev: Jamen du må ikke afsløre det hele.

101 Lærer: Nej, men jeg prøver også det bedste jeg har lært, at lade være. Med at sige noget om det. Elev2?

102 Elev: Det er bare mig og elev6 har været sammen i alle de der matematikgrupper, vi var også sammen i det der 103 rektangel

104 Lærer: Hvordan kan det være, der sad i da sådan her [Peger ud i begge sider af klassen]

105 Elev: Men der delte du jo holdene tilfældigt…

106 Lærer: Ja, ja, men i dag er i heldige, fordi i dag skal elev1 være med jer.

107 [Elever griner]

108 [Læreren går ned og klapper elev1 på ryggen]

109 Lærer: Jeg skal nok huske det. Men vi tager det lige sådan i dag. Okay? Jeg har.. jeg har nogle kasser som kan stå 110 rundt på bordene rundt omkring, som i kan bruge. Man må også godt få centicubes, hvis man vil det. Der er en 111 stak papir, hvis man skal bruge det til hovedregning. Øhmmm, det eneste man ikke må det er at pakke dem op. 112 Øh de fleste de er tomme, så de er ikke så spændende, men det er fordi de også skal bruges inde i parallelklassen.

113 Elev: Hvad er der inde i dem der ikke er tomme?

114 Lærer: Der er en, hvor der er sådan en dims i til at putte sukker i. Ligesom der står på cafeerne, dem pakker vi ud 115 når vi er færdige. Så er der en, hvor der er en sakseblok i. Det tror jeg er det, resten er tomme. Elev7?

116 Elev7: Mumler 302

117 Lærer: I finder ud af hvad i gør, så må i så finde ud af om i kan finde en løsning på dem. Okay? Jeg tror ikke der er 118 ret meget andet man ikke må, end at pakke pakkerne op. Elev2?

119 Elev2: Skal man ikke bare halvere dem?

120 Lærer: I finder ud af hvad i gør og hvis i tror det er det der løsningen, så prøver i det. Så må i tjekke om det bliver 121 rigtigt. Fordi man skal kunne lave alle de her kombinationer, [Peger op på smartboardet)

122 [Elever små snakker, imens læreren deler kasser ud]

123 Lærer: Jeg deler ud til bordene, fordi jeg har ikke nok til hvert par…. Nej lad den stå på midten af bordet. Der 124 kommer til at være fire kasser. 1, 2, 3, 4, 5, 6 [Tæller bordene i klassen] Jeg har til fem border. Så finder vi lige ud 125 af hvad vi gør med den sidste.

126 [Læreren beder en elev om, at udlevere papir, mens hun selv deler resten af kasserne ud]

127 Lærer: 4, og ved i hvad vi gør her? Det er jeres små kasser, det kan i godt ikke?

128 Tidskode 5 minutter

129 [Eleverne småsnakker 1 minut]

130 [Eleverne begynder at snakke om opgaverne]

131 Per: Det kan være 3, 4, 5, 6

132 Makker: eller, 5, 6, 7, 8

133 (Støj fra resten af klasse, så man kan ikke høre hvad de siger]

134 [Per + makker sidder bare og kigger på kasserne uden at snakke sammen]

135 Tidskode 8.54 Observation 2 færdig

136 [Per og makker sidder og snakker om opgaven, men de snakker meget lavt, så man kan ikke høre hvad de siger 137 pga. larmen der er i klassen: første minut)

138 Makker: Den vejer 6 kilo, den vejer 8 kilo [Peger på kasserne der står foran dem]

139 Per: 2 Kilo [Peger på den anden nederste]

140 Makker: 6 kilo og 8 kilo, det er 14 kilo!

141 [kigger lidt på hinanden]

142 Makker: Den vejer 6 kilo [Peger på den øverste kasse], den vejer 10 kilo [Peger på den anden øverste kasse] Den 143 vejer, 6 kilo, den vejer 8 kilo, den vejer 14 kilo [Peger på den anden nederste kasse] Den vejer 12 kilo [Peger på 144 den nederste kasse].

145 [Snakker sammen, men uklart hvad de siger pga. støj]

146 Makker: Det skal jo være 2 og 2, det må ikke bare være 1..

147 [25 sekunders tænkepause, hvor de begge sidder og kigger ned i hvert deres papir]

148 Per: Den der vejer 9 kilo [Peger på den øverste], den vejer 7 kilo [peger på den anden øverste], neej

149 [35 sekunders tænkepause]

150 Per: 9 [Peger på den øverste], 5 (Peger på den anden øverste), 7 [Peger på den anden nederste] 303

151 [Per sidder og mumler med sig selv imens hendes makker sidder og kigger ud i luften]

152 [Tænkepause 40 sekunder]

153 Per: Hvad var det nu?

154 Makker: Den vejer 6, den vejer 8, den vejer 4 og den vejer 2.

155 Per: 9, 7, 5, 1!

156 [15 sekunders tænkepause]

157 [De snakker sammen, men man kan ikke høre hvad de siger: 10 sekunder]

158 Per: 9 + 7 det giver 16.

159 Makker: Hvad så med 14?

160 Per: Giv mig lige den der. [Tager hans blyant]

161 Tidskode 5 minutter

162 [Per skriver noget ned på sit papir og rækker hånden op og venter i 20 sekunder, før hun rejser sig op, hendes 163 makker bliver siddende med hånden oppe, Per kommer tilbage og venter i 3 minutter på læreren kommer]

164 Lærer: Det der er ikke svaret, det der er et af svarene. [Peger ned på det Per har skrevet ned på sit stykke papir] I 165 kan jo starte med et til og så kan i overveje om der er flere. [Går igen]

166 Per: 5, 2 [Sidder og kigger ned i hendes papir]

167 Tidskode 8.54 observation 3 færdig

168 Per: jeg har det [Vil have blyanten igen] Prøv lige at giv mig den. [Makkeren beholder sin egen og Per finder en i 169 hendes penalhus.]

170 Per: 4 + 2 [Per sidder og regner selv på hendes papir og tænker 30 sekunder]

171 Makker: 4, 2, 6 og 8. [Viser sit papir]

172 [Per skriver det ned og rækker hånden op og de sidder og venter i 4,5 minut mens de kigger ud i luften]

173 Tidskode 5 minutter

174 [Per tager hånden ned og visker noget ud på hendes papir og skriver noget nyt ned. De små snakker imens 1 175 minut]

176 [Per rækker hånden op igen og venter 2 minutter]

177 Lærer: Jaa [Ser på hvad de har skrevet ned 20 sekunder] Godt, er der flere? Er der en måde vi kan finde ud af eller 178 forklare om der er flere. Hvis der er flere hvordan man finder dem, eller hvis der ikke er, hvorfor er der så ikke 179 flere. Prøv lige at snakke sammen. [går igen]

180 [De kigger opgivende på hinanden og løfter skuldrene] [Tænkepause 20 sekunder]

181 Tidskode 8.54 observation 4 færdig

182 Makker: Jeg tror ikke der er flere

183 Per: Nej det tror jeg heller ikke.

184 [De sidder og kigger lidt ud i klassen og lidt på deres papir, men snakker ikke med hinanden 2,5 minut] 304

185 Lærer: ja i lægger blyanterne og finder jeres hapser og går ud og holder pause.

186 Tidskode 2.56 – observation 5 færdig

187 [Eleverne er kommet tilbage fra lufter og læreren er i gang med at få folk i gang igen, eleverne småsnakker]

188 [En elev kommer over til Per og hendes makkers bord og begynder at røre på deres kasser.]

189 Elev: Jeg har allerede fundet begge løsninger

190 Makker: Jamen der er 3.

191 Elev: nej der er kun 2

192 [Per sidder og piller ved kasserne]

193 [Læreren hjælper en elev der sidder ved deres bord]

194 Lærer: Jeg tror jeg vil hjælpe dig så meget og sige, at vi arbejder med hele tal.

195 [Per og hendes makker sidder stadig bare og kigger ud i luften]

196 Lærer: Skal vi tage snakken fælles, så vi også får de andre med og ikke kun jer 3?

197 [Læreren går op til tavlen]

198 Lærer: vi har lige et par stykker der mangler øh, så derfor tillader jeg mig lige at give dem der er færdige noget 199 andet, de kan få lov til at rode med, imens de andre for lov til at blive færdige. Det er lidt i familie med det vi 200 lavede i går. Schhhh

201 [Per rækker hånden op og de får den nye opgave udleveret]

202 Tidskode 2.37 observation 6 færdig

203 Lærer: Schhh er i klar?

204 Elever: Yes

205 Lærer: Godt, vender i opmærksommen op mod mig? Elev 8 er du med?

206 [Læreren får alles opmærksomhed]

207 Lærer: Godt, jeg kunne tænkte mig at vide… Hvordan har i grebet det an? Fordi en ting er at finde svaret, som jeg 208 lige stod og snakkede med Ragna om, noget andet at sige er at vejen derhen faktisk også lidt vigtig, for hvad er 209 det for nogle tanker vi har gjort os undervejs. Så hvordan er i nået derhen. Elev 9?

210 Elev: Altså vi tog en af kasserne og så gav vi den bare et eller andet tal, men det skulle ligesom bare være over 2.

211 Lærer: Hvorfor skulle det være over 2?

212 Elev: Det ved jeg ikke… Vi synes bare det var lidt for lidt det første.

213 Lærer: Så i valgte et eller andet tilfældigt tal, hvad valgte i?

214 Elev: 4

215 Lærer: 4 og så gik i videre fra den og sagde okay, hvis vi bruger den som eksempel, hvad kan de andre så være? 216 Fik det jer i mål hele vejen?

217 Elev: Ja

305

218 Lærer: Var der nogle ting undervejs, hvor i tænkte: hov der var der lige noget der gik galt, fordi vi har glemt at 219 eller..

220 Elev: altså vi havde skrevet nogle tal og så passede de ikke rigtigt sammen med de andre vi skulle finde. Så 221 manglede vi nemlig 14 og 16.

222 Lærer: Så i startede med at have for lave tal til i kunne lave de største?

223 Elev: Ja

224 Lærer: Okay.

225 Elev: Så skrev vi 10 på den nederste, så gik det hele.

226 Lærer: Også passsede det?

227 Elev: Ja.

228 Lærer: Okay, ja! Per.

229 Per: (Lidt uklart pga. støj) Altså vi sagde, hvad skal dertil for at man kan få 6. Også tog vi så…

230 Lærer: Så i startede altså med at sige, vi ved at der er 2 kasser der til sammen skal give 6.

231 Per: Ja

232 Lærer: Så hvad kan de to kasser veje? Okay.

233 [Skriver de tal Per op på tavlen]

234 Per: 3+3

235 Lærer: 3+3

236 Per: 4 + 2

237 Lærer: Nu hørte jeg 2tallet, så jeg går ud fra at det andet var et 4 tal

238 Per: Og det sidste, 1 + 5

239 Lærer: og 1 + 5.. Okay, så når 2 af kasserne skal give 6, så ved vi at en af de her kombinationer skal med.

240 Per: Ja

241 Lærer: Ja

242 Per: Og så tog vi så bare og så tog vi så bare en af de tal og hvad mangler der så…

243 [Bliver afbrudt af læreren]

244 Lærer: Hvad startede i med for en? [Peger på de tal hun har skrevet på tavlen]

245 Per: øhhh 1 + 5

246 Lærer: 1+5 … hov, nu tager den 4-tallet. Så i startede med 1 + 5

247 Per: Ja.. også tog vi, hvad skal der til for at det kan give 8.

248 Lærer: Okay så gik i videre ligesom Karla, og så prøvede i tallene op ad og så hvad skal der så til for at få resten.

249 [Per nikker] 306

250 Lærer: Oplevede i nogle af de samme problemer som Karla havde med at få de store.

251 Per: Jaaa

252 Lærer: Fordi, 1 og 5 det rammer 6’eren, også går jeg ud fra, at når i så skal ramme 8’eren, så har i sat en 3’er på.

253 Per: Nej

254 Lærer: Nej

255 Per: Der tog vi 7

256 Lærer: Der tog i 7 i stedet for 3, okay! Så tog i 7 og den næste..

257 Per: fordi så gav det 8 og så tog vi 9 i den, fordi så kunne den..

258 Lærer: er der nogen speciel grund til at i sprang 3 over?

259 Per: Fordi så kunne det ikke give 16.

260 Lærer: Så allerede der, der havde i tænkt over, at i ville komme til at mangle i de store. Okay.. jaa. Andre måder? 261 Elev10

262 Elev10: øhmmmm, fordi 6 og 8 var de mindste tal, så var der ikke så mange af de store, så jeg vidste at for at få 263 et tal som kunne gå op i alle tal, så skulle det gå op i 6 som det mindste. Så alle de 6, kunne gå op i alle de andre 264 tal fordi de var så små (Lidt uklart)

265 Lærer: Jaa

266 Elev10: Så jeg tog 4 og 2

267 Lærer: Så det du fik ud af det når du siger 6, så siger du 4 og 2

268 Elev: Ja også det jeg havde gjort

269 Lærer: Fortæl mig lige resten at de du havde

270 Tidskode 5 minutter

271 Elev: Resten?

272 Lærer: af de du havde skrevet under 6 tallet, hvorfor nogle andre havde du hernede [Peger på det hun har skrevet 273 på tavlen]

274 Elev: Nååå 1 og 5 og

275 Lærer: 1 og 5

276 Elev: 2 og 4

277 Lærer: 2 og 4 det er det samme

278 Elev: Og så øhmm 3 og 3

279 Lærer: Jeg har en, men det er noget med dit sprogbrug at gøre, når vi plusser så er det jo er det jo ikke de store, 280 er du med på den? Fordi det er jo dem der går op i en når vi dividerer og når vi leger med tabeller ikke?

281 Elev: Jaa

307

282 Lærer: Jeg forstår godt hvad du mener og du har sagt, hvilke tal kan man plusse for at ramme. Du må bare ikke 283 bruge ordet …(uklart pga. larm fra kameraet) Men det er rigtigt, du har faktisk, det er lidt det samme som Per 284 ikke? Er det ikke rigtigt, at du gik hen og så sagde du 8 og så lavede du: Hvorfor nogle tal er der

285 Elev: [Mumler]

286 Lærer: Nej men du nåede at gøre det ret mange gange ikke? Altså du nåede at gøre det på flere af dem

287 Elev: [Mumler noget med alle sammen]

288 Lærer: Ja så fik du simpelhent skrevet op, 1 og 7 og 2 og 6 og 3 og 5

289 Elev: Ja jeg har bare skrevet 1, 2, 3, 4 og så tallene.

290 Lærer: Ja, var der noget med den her måde med at få dem alle sammen skrevet op du synes var okay, var det 291 meget fint, eller var det forvirrende eller var det..

292 Elev: øhh det var [.. stykke man ikke kan høre pga. larm fra kamera] .. hvad kan jeg bruge til at få 8, enten vil jeg 293 tage 2 og plusse med hinanden eller også 4. Så tænker jeg, hvis jeg tager 4 så har jeg jo to 4-taller og det kan man 294 ikke til at hjælpe mig særligt meget, jeg skal have et højere tal.

295 Lærer: Ja

296 Elev: Så tænkte jeg, okay så tager jeg 2 så jeg har 6 til det højere tal, så bliver det nemmere at få 16

297 Lærer: Så du gik herhen, fordi du siger okay 2 tallet har du allerede

298 Elev: Ja .. (Mumler noget uklart)

299 Lærer: ja ja

300 Elev: Og så puttede jeg 6 på

301 Lærer: Ja

302 Elev: og så puttede jeg 6 på, og så tjekkede jeg så, at jeg kunne få de mindste og så kunne jeg se, at for at få 16, 303 så blev jeg nødt til at have en 10’er.

304 Lærer: Ja.

305 Elev: 4, 2, 6 og 10

306 Lærer: Hov nu har jeg taget en rød, så den kom ligesom på som ekstra, fordi du kunne se: hov den mangler jeg 307 også – ja.

308 Elev: Vi blev ligesom bare ved med at prøve tallene, indtil vi ramte noget der gav mening.

309 Lærer: Så i startede med at vælge alle 4 - sådan tilfældigt? eller startede i med at vælge 1?

310 Elev: Nej vi startede sådan med at vælge nogle tal, som gav lidt mening

311 Lærer: Som for eksempel?

312 Elev: Som for eksempel 10, den synes jeg faktisk er god at have med

313 Lærer: Hvad var det første gæt i havde, gættede i ikke på dem alle 4 for at se om de passede.

314 Elev: Nåå jo

315 Lærer: Hvad var det første 4. I havde? 308

316 Elev: Det var 10, 4.. og 4 og 4 og 2

317 Lærer: Hvor mange 4taller?

318 Elev: 2.

319 Lærer: Og 2 [Skriver det op på tavlen] så det var jeres første gæt.

320 Elev: Ja

321 Lærer: Jaa

322 Elev: Den var bare ikke så god

323 Lærer: ved at prøve sig frem

324 Elev: Ja altså man kan sætte 4 og 2 sammen til en og så kan man sætte 4 og 4 sammen og 10

325 Lærer: Også mangler vi 10’eren, da i så kunne gå videre derfra, valgte i så 4 nye eller skiftede i dem så ud en efter 326 en eller?

327 Elev: øhmm vi skiftede lidt ud, så vi tog 10’eren, den beholdte vi.

328 Lærer: Mm

329 Elev: og så tog vi en 2’er, en 6’er og en 4’er

330 Lærer: Okay, jeg spørger lige igen – tog i 3 tilfældige, eller byttede i dem ud en efter en?

331 Elev: nej vi byttede bare en

332 Lærer: Ja det er det i reelt har gjort, men tænkte i sådan eller blev det en tilfældighed, at det var det i havnede i?

333 Elev: Neeeeej altså vi tænkte, hvad kan give 10 og det blev så 6

334 Lærer: Okay, så lavede i det næste. Så i virkeligheden, hvor mange af jer kender mastermind? Det der hvor man 335 har 4 farver og så skal modstanderen gætte, hvilke 4 farver man har.

336 [Ca. 6 elever rækker hånden op]

337 Lærer: Det er i virkeligheden lidt den leg vi har leget.

338 Elev: Jo

339 Lærer: at i siger, okay nu tager vi 4 tal og så ser vi hvad det cirka virker til vi kan bruge, så prøver vi at skifte en 340 og se om den virker og se kan vi hele tiden lige justere lidt, indtil vi rammer den rigtige kombination.

341 Elev: øhmm, jeg vil bare, eller vi startede med at vi lavede overslag, hvor vi ligesom tog nogle tal bare for at kunne 342 få et eksempel på det

343 Lærer: Lidt ligesom den her

344 Tidskode 10 minutter

345 Elev: og så ved en tilfældighed, så valgte vi så 9, 10, 4, 6 og 2 og så fandt vi ud af, nåå ja vi kan godt få dem alle 346 sammen også havde vi ligesom

347 Lærer: Så i ramte dem rigtigt i første hug

348 Elev: også fandt vi ud af at 2

309

349 Lærer: Jamen den venter jeg lige med et øjeblik. Elev 2?

350 Elev 2: Altså øhmmm, øhm i starten så forstod jeg den ikke helt, men så da jeg så forstod den, så sagde elev 1 4 351 tal og så var det rigtigt… og så ja.

352 Lærer: Så nu kunne jeg godt tænkte mig at høre Oskar. Var der nogle tanker bagved at du lige præcis ramte de 353 rigtige 4 eller var det lucky guess?

354 Elev: øhh jeg tænkte bare over, hvad der gav bedst mening og så…

355 Lærer: Men, men hvad er det så der gør, at du tænker, okay det her giver mening og det her giver ikke mening. 356 Altså hvad er det for en tanke der gør, at du kan se, at nogle tal giver mening?

357 Elev: [Mumler noget om det højeste tal, men man kan ikke høre det på grund af larm fra kameraet]

358 Lærer: Okay, så i stedet for at starte med 6tallet, som der var flere af de andre der gjorde, så startede du med 16 359 og sagde, hvad skal vi have fat i for, at det kan give 16.

360 Elev: Nikker

361 Lærer: Ja okay.. Øhhh er der flere vi mangler at høre fra? Altså jeg ved at der var en del, der var nogle der, der 362 var nogle som der havde lidt svært ved at gå i gang og finde ud af, hvordan dælen får jeg hul på den der. Og så 363 gik jeg hen og så sagde jeg, hvad nu hvis vi bare beslutter, at en af kasserne vejer et eller andet. Og så spurgte 364 jeg, hvad nu hvis vi bare bestemmer, at den her kasse den vejer 4. [Skriver 4 i en kasse hun har tegnet på tavlen] 365 og så simpelhent tager tallene en af gangen og så siger, jamen vi skal have 2 kasser der tilsammen giver 6. Den 366 ene vejer 4, så bliver vi også nødt til at have en der vejer 2, fordi eller kan vi ikke få den til at give 6. Så skal der 367 være noget der vejer 8, hvad kan vi få til at give.. jeg lavede den lidt sammen med elev herover, jeg kan lige vise 368 jer opstillingen herover. [går over til Per’s bord og tager de kasser de brugte] Nu må jeg se om jeg kan få dem 369 herop uden de vælter. [Tager kasserne med op foran klassen] Godt vi startede her, og sagde, hvad nu hvis den 370 ene vejede 2, så må der også være en der vejer 4, fordi ellers kan vi ikke få det til at give 6. Det næste vi skal have 371 til at ramme, det er at en skal give 8. Det vil altså sige, at der må være en af kasserne der enten vejer 4 fordi de 2 372 tilsammen giver 8, eller 6 og 2 tilsammen giver 8. Så den her nede har lige nu 2 muligheder [Peger på en af 373 kasserne] Så gik vi videre og sagde, okay så skal der være en der vejer 10, men det kan de to gøre tilsammen 374 [Peger på kassen med 4 og 6] Så fjerner vi lige den her [Fjerner hendes 4 tal] Nu har vi 3 kasser, kan vi så arbejde 375 videre herfra, kan i se systemet?

376 Elever: mmmm

377 Lærer: Så man starter simpelhent et eller andet sted og så udelukker man hen af vejen. Ved at hive de sedler fra 378 der ikke passer og fjerner de tal der ikke passer. Der er mange af jer der så er nået i mål med den ene løsning, 379 som hed 2, 4, 6 og 10 ikke? [Skriver tallene på tavlen igen i nye kasser]

380 Elever: Ja

381 Lærer: Ja, så var jeg pisse irriterende og sagde, hvaaaa’ er der flere?

382 [5 sekunders tænkepause]

383 Lærer: Ragna

384 Elev: Altså, vi kom først lige i tanke om, at hvis det var sådan der, at der var de der 6, eller de der 4 andre tal de 385 gav 22 i alt, så det andet måtte give 22, fordi ellers ville det jo ikke passe sammen. Og så, vil du sige det [Kigger 386 over på sin sidemakker, som så snakker videre]

387 Elev: Så prøvede vi sådan, at se om vi kunne tage nogle ulige tal med

388 L: Ja!

310

389 Elev: Og så prøvede jeg mig frem og tænkte, om man kunne tage 2 lige og 2 ulige, men det kunne man ikke rigtigt, 390 så det endte med at vi, at vi tog 4 ulige tal.

391 Lærer: Ja, var der noget, den der med at det ikke lykkes at blande 4 ulige tal.

392 [10 sekunders tænkepause]

393 Lærer: Det er fordi, jeg vil gerne høre hvad du siger, men en mumler og det er derfor jeg går lidt i stå, fordi jeg 394 prøver at høre hvad han siger, men jeg kan ikke. Var der noget med den, hvor i tænkte, ej det passer ikke, fordi…

395 Elev: Ja vi tænkte, at vi ville prøve ikke at bruge de lige tal, fordi dem havde vi brugt i den første

396 Lærer: okay, så fordi i havde brugt mange af de lige tal, så tænkte i, ahh vi skal ikke bruge de samme tal engang 397 til. Så er der ikke så mange lige tilbage at tage af.

398 Tidskode 15 minutter

399 Elev: og så var det at vi kom frem til, at 5, 1, 3 og 9 og det fandt vi så ud af, at det ikke virkede

400 Lærer: Så i testede, nu skriver jeg dem hernede [Skriver på tavlen] 5, 1, 3 og 9

401 Elev: Og de gik så op i alle, undtagen 16

402 Lærer: Så i strandede på den samme, som mange af jer gjorde med de lige tal, at man ret hurtigt kunne se, at 403 det passede i hvert fald ikke med 16

404 Elev: Også fortalte Niklas mig en teknik, jeg tror gerne Niklas gerne vil fortælle den

405 Lærer: Ja det tror jeg også gerne han selv vil, for elev 11 du startede faktisk med de ulige gjorde du ikke?

406 Elev 11: Jo så altså det er fordi jeg vidste at det ville være svært at lave 6 og 8, fordi der var ikke så mange tal jeg 407 ligesom kunne bruge, og jeg vidste også godt, at 6 og 8, er meget lave tal. Så jeg tænkte, hvis 16 det er svært at 408 lave, så tænkte jeg, hvad går op i 6 og 8, altså så jeg kunne bruge de samme tal med 6 og 8. Også bagefter da jeg 409 havde gjort det, nu skal jeg lige kigge [Kigger i sin mappe] Nåå ja, jeg tog 1 og 5, fordi så fik jeg et lidt højt tal med 410 og 1 kan nemt bruges

411 Lærer: då man kan sige, fordi 5 i denne sammenhæng er højest til at give 6, så valgte du den der til at starte med, 412 ja

413 Elev 11: Jo fordi så kunne jeg nemmere prøve at få 16. Så tænkte jeg at okay, øhm det bliver stadig rimelig svært 414 at få 16, hvis jeg ikke tog 11, og 11 kan man nærmest ikke bruge, fordi det er så højt!

415 Lærer: Ja

416 Elev 11: Øh og så tænkte jeg

417 Lærer: fordi de ikke kan bruges til ret mange af dem, ja

418 Elev 11: Så så jeg at 9 og 7 også går op i 16 og de kan bruges til lidt mere, fordi de begge to er lidt lavere.

419 Lærer: Så, den der overvejelse kan jeg meget godt lide, den der med at sige, det nytter ikke noget med at sige, 420 den her kasse den vejer 16 kg fordi så er den dækket ind. Fordi de 3 andre skal kunne kombineres på rigtig mange 421 måder for at give resten, så vi bliver nødt til at holde os til nogle tal der bruges i flere, ja…

422 Elev 11: [mumler noget med 4 kasser, uklart pga. støj fra kameraet]

423 Lærer: Ja, elev 3

424 Elev 3: Vi brugte også 11

311

425 Lærer: I brugte også 11, men har i fundet en løsning hvor i kan bruge 11

426 Elev: Ja

427 Lærer: Hvad er jeres kasser, hvis den ene skal være 11?

428 Tidskode 17.43 observation 7 færdig

429 Lærer: Hvad skal de 3 andre tal så være

430 Elev: 1, 5 og 7

431 Lærer: 1 og 5 og 7, men så kan vi jo tjekke igennem og så siger de to tilsammen giver 6 og de to sammen giver 8, 432 hvorfor nogle tilsammen giver 10?

433 Elev: 11 – 1

434 Lærer: Men du må jo ikke minus vel, fordi du kan jo ikke minus på en vægt

435 Elev: Nå nej.

436 Lærer: Så du kan godt se, at du mangler en?

437 Elev: ja

438 Lærer: Vi kan ikke stille 2 kasser på en vægt og så trække den ene vægten fra den anden kasse.

439 Elev: [Elev mumler]

440 Lærer: Men det gør vi ikke, vi stiller dem på en vægt, det er jo ligesom det der er det hele … for denne opgave 441 ikke? 442 Hvad med de her ulige, er der andre der har gjort sig nogle tanker omkring de ulige.

443 [Elev mumler]

444 Lærer: Vent den tager vi til sidst, Per.

445 Per: [Mumler]

446 Lærer: Du snakker helt vildt lavt

447 Per: Vi gjorde også lidt det samme som før det var den vi startede med

448 Lærer: Ja

449 [Per svarer læreren, men man kan ikke høre hvad hun siger pga hun snakker lavt og der er larm fra kameraet]

450 Lærer: Ja, nu rettede jeg lige den her ikke, for vi er enige om der skal stå noget andet her ikke? [Retter noget på 451 tavlen] Så vi har …. Anton

452 Elev: Øhm altså i den der så kiggede vi lidt på størrelserne af kasserne og tænkte sådan, at den store måtte veje 453 lidt mere end den mindre, den mindste

454 Lærer: Den store måtte veje lidt mere end den mindste [Skriver det op på tavlen]

455 Elev: Ja altså der er jo en stor kasse og en lille kasse og så

456 Lærer: Altså vægtmæssigt?

457 Elev: Ja

312

458 Lærer: fordi i havde jo lige præcis 4 kasser, der så fuldstændigt ens ud

459 Elev: Ja

460 Lærer: Vægtmæssigt måtte der være

461 Elev: Ja

462 Lærer: Jaa

463 Elev: Og så kastede vi bare nogle tal ind, og så talte vi i lidt tal og så var der et tal der ikke virkede og så puttede 464 vi et andet tal ind og så virkede det.

465 Lærer: Ja så der gjorde i også, at i testede et eller andet og så tog i en ud, tænkte i over det?

466 Elev: Mm, eller nej vi tænkte ikke over det, men det var bare ..

467 Lærer: men da i prøvede det, prøvede i så automatisk de ulige tal?

468 Elev: Ja

469 Lærer: Hvorfor?

470 Elev: Det føles bare bedst

471 Lærer: Det føles bare bedst.. okay. Jeg kunne godt tænkte mig, læg lige blyanterne fra jer og kig herop. Det vil 472 være rigtig dejligt, hvis I ikke lavede påskeklip og alt muligt mærkeligt ud af opgaverne, fordi vi skal bruge dem 473 igen til nogle andre opgaver, så det ville være rart, hvis de ikke var brugt op.

474 Elev: Jaaa, undskyld

475 Lærer: Hvis vi nu, kigger på dem her, kun 3 fra dig, og kun 3 fra Kian og kun 3 fra Jakob, som i øvrigt ikke må tegne 476 på bordet. Så bliver det alligevel ret mange ikke.

477 [En elev fortæller læreren noget med at han/hun har glemt at smide noget ud i skraldespanden ude i gården]

478 Lærer: Det gør du lige om lidt… der er 2 løsninger [Peger op på tavlen] er der flere løsninger end de 2?

479 Elev: Det er der ikke,

480 Lærer: Fordi?

481 Elev: Fordi det skal jo gå op i 6 …. I de to definitioner [Meget uklart]

482 Lærer: Så hvis vi går tilbage til det du startede med, det med at vi skal have 2 kasser der tilsammen skal give 6. 483 Det kan vi enten gøre med 1 og 5 eller med 4 og 2,

484 Elev: Ja og når vi bruger de to, så er der jo kun 3 og 3 tilbage.

485 Lærer: Ja

486 Elev: Så kunne vi se, at vi blev nødt til at have en 5’er så vi kunne lave 8….

487 Lærer: Så man kan sige, at du har udelukket det ved at sige, at den sidste løsning det ville være den her og så ville 488 du ikke kunne få det til at fungere.

489 Elev: Ja

490 Lærer: Okay

491 Elev: så den eneste måde man kan få den til at gå op i 6, 5, og 16, det er ved at lave 3, 3, 5, 8.. 313

492 Lærer: Kunne man have forestillet sig, at det var 4 og 2 og så i stedet for 6 og 10, så var det 2 andre tal.

493 Elev: Hmmm [5 sekunders tænkepause]

494 Lærer: Den kan du lige simre lidt over. Hvad siger Per?

495 Per: Altså jeg prøvede [Uklart pga. larm fra kameraet] men det kunne man ikke..

496 Tidskode 5 minutter

497 Per: Jeg kunne kun komme frem til, at det var de tal der skulle bruges.

498 Lærer: Så det han sidder og prøver og lege med nu og siger, hvis det er 4 og 2, vi skal bruge med 2 andre, det har 499 du testet, og du har ikke kunne finde nogle der virkede.

500 Per: nej

501 Lærer: Okay, så reelt burde vi ikke kunne finde flere. Nej og det passer også. Hehhehhehe, det passer også. Æhhh 502 der er en sjov lille matematisk ting, med det her med de lige og ulige tal. Ragna sagde på et tidspunkt, når jeg 503 plusser 2 ulige tal…

504 Elev: Den var en fejl…

505 Lærer: Ja, ja men hvad var, jeg ved det godt, men det lære man noget af. Hvad var det du sagde?

506 Elev: At hvis jeg plusser 2 lige tal,

507 Lærer: Så kan jeg ikke få et lige tal, var det sådan du fik sagt det?

508 Elev: nej men det er kun, hvis det ikke er to ulige tal, at det bliver et lige tal.

509 Lærer: Hvis vi plusser 2 ulige tal,

510 [Elev mumler]

511 Lærer: vent vent vent, ikke så hurtigt [er i gang med at skrive det op på tavlen] Hvis vi plusser 2 ulige tal, hvad får 512 vi så? Elev12 siger, hvis det er 2 forskellige, hvad var det du sagde?

513 Elev 12: Altså jeg sagde, at hvis det var 2 forskellige, så blev det jo ikke et lige tal,

514 Lærer: Såå, kan vi lave en regel, Jakob læg blyanten, Anna Liv læg blyanten, er der flere, så læg blyanterne. Så 515 kan vi lave en regel, når vi plusser 2 ulige tal, hvad sker der så? Asta?

516 Elev: Så bliver det et lige tal,

517 Lærer: Så bliver det et lige tal – altid!

518 [Elever mumler, hvordan det?]

519 Lærer: elev 7

520 Elev 7: Neeej ikke altid, det skal være 2 af de samme tal

521 Lærer: Hvad giver 1 + 5?

522 Elev 7: 6

523 Lærer: Er det lige eller ulige?

524 Elev 7: Lige

314

525 Lærer: Okay, hvad så med 3+3?

526 Elev 7: Det giver 6

527 Lærer: Det giver også lige, så vi kan finde eksempler på, hvor de både er ens og hvor de er forskellige. Så vi går ud 528 fra at det er det næsten altid, til det er det altid! Hvad siger elev3?

529 Elev 3: Det er da ikke altid lige, fordi man kan da sige 5+6

530 Lærer: Men 6 er jo ikke ulige. Det er hvis vi plusser to tal der begge er ulige.

531 Elever: Nååå

532 Lærer: elev 10

533 Elev 10: Grunden til det er jo, at et ulige tal.. hvis man for eksempel har et lige tal, så vil det jo når man plusser 534 en til, så bliver det et lige tal. Så hvis man plusser 2 lige tal, så er der jo ligesom 2 tilovers fra det lige tal og 2 er 535 jo et lige tal så.. [Lidt uklart pga han snakker lavt]

536 Lærer: Det er super godt forklaret, hvis vi for eksempel tager vores 3’er og vores 5’er herover, så er det et lige tal, 537 med en i overskud og det her over er også et lige tal med en i overskud. [Viser det på tavlen] Og hvis vi lægger de 538 to sammen, så bliver de 2 overskydende til et lige tal.

539 Elev 10: Så har du faktisk 3 lige tal

540 Lærer: Det er så godt forklaret! Sindssygt godt! Er i med?

541 Elever: Jaa

542 Lærer: Så hvis man ser det for sig, for eksempel med centicubes, jamen når vi bygger tallene, hvis vi bygger tallene 543 op af 2 og 2 og 2, så snart vi kommer til en ny, så kommer der til at lægge en ny ovenpå. Og hvis vi så tager en 544 ulige med, som er bygget op af 2 og 2 og 2 og 2, så vil der ligge en enkel lige ovenpå. Og så vil de to enere jo så 545 sætte sig sammen til to, SUPER godt forklaret! Jep, så 2 ulige tal plusset sammen, må give et lige tal. Hvad siger 546 elev 7?

547 Elev 7: Jeg har et ret svært spørgsmål

548 Lærer: Prøv!

549 Elev 7: Er 0 et ulige tal?

550 Lærer: Er 0 et ulige tal?

551 [Elever mumler] Neeej – det er det begge dele –

552 Lærer: 0 er, 0 er en skævert og det er 1 i virkeligheden også og det er 2 på nogen måder også. Kan i huske vi 553 snakkede om primtal i går?

554 Elever: Jaa

555 Lærer: Også snakkede vi om, at lige tal ikke kan være primtal, bortset fra 2.

556 Elev: Jeg vil godt bare lige sige, at elev 10 siger at hver gang man plusser en, så bliver det et lige tal.

557 Lærer: Ja, den der ene ekstra der ligger ovenpå

558 Tidskode 10 minutter

559 Elev: Hvis du plusser 1 med 0, så giver det 1

315

560 Lærer: Jaa og det der er, det der er med 0, 1 og 2, der er faktisk at der er hen af vejen nogle lidt specielle regler 561 omkring de 3, fordi de er lidt særlige, da vi lavede, da vi lavede rektangel opgaven i går, da startede vi med, at vi 562 skulle undersøge fra 5 til 29, og det er fordi at i de små der sker der nogle ting der er lidt sværere. Der sad engang 563 en masse kloge matematikere, og lavede lange beregninger om for at bevise det ene og det andet

564 Elev: Var det dig?

565 Lærer: Nej, så klog er jeg ikke.. Men så derfor, er der noget med de tal, som vi bare må sige – sådan er det. Indtil 566 vi bliver ældre og klogere og kan forstå beviserne ikke.

567 Lærer: jaa jeg kan godt forstå dig, og det er super godt at du tænker sådan og det er en super god tanke. Øhm 568 fordi vi plejer jo at sige, at de lige tal det er dem vi kan skære over og dele i to lige store grupper. Men 0, hvad 569 sker der med den?

570 Små snak i klassen

571 Lærer: Schh, schh, schhh hvad siger du?

572 Elev: jeg føler lidt at 0 det er et lige tal fordi, altså det lige tal det kommer først, 30 der er der også et 0 bagpå, 573 også er det et lige tal og så går vi til 31 og det er ulige. Så jeg føler lidt sådan at

574 Lærer: Ja, det føles lige, det føles lige ja, men det er jo også sådan, at hvert andet tal det er lige. [Støj fra kamera] 575 Men det der er, det der er jo, at det som de der meget kloge matematikere har gjort engang, er at de siger, at vi 576 bliver nødt til at bevise, at sådan er det! Så vi kan sige med overbevisning, at det føles sådan her og det er sådan 577 her fordi.. også har de lavet et eller andet bevis.

578 [støj fra kamera 10 sekunder]

579 Lærer:… at de har sagt, det føles som om, at det her må være rigtigt, men nu skal vi lige finde ud af helt præcis, 580 hvorfor det er sådan her. Ja?

581 [Peger på Per]

582 Per: Jeg finder det ulige, fordi hvis vi nu skriver 0 på et stykke papir og så vil du dele det op, det kan du jo ikke. Så 583 det er bare, at den ene får – så laver vi to sider, så skriver vi 0 på det ene og på den anden side der står der jo 584 ikke noget

585 Lærer: Neej, så hvis du var matematiker, så ville du sætte dig for at bevise, hvorfor det ikke er et lige tal! Også vil 586 man sige, at så vil det enten lykkes dig eller også vil det ikke lykkes dig! Ja.. Vi skal lige videre med den her. [Peger 587 op på tavlen] Fordi, hvad nu hvis vi siger to lige tal. Hvis man plusser dem. Hvis vi plusser de 2 lige tal, hvad så?

588 Elev: Så får man også et lige tal

589 Lærer: Så får man også et lige tal fordi..

590 Elev: Fordi hvis man har 2 + 2, så bliver det 4.

591 Elev: Altså 2 er et ulige og et lige tal..

592 Lærer: Men lige nu har vi fat i 2 lige tal. Den der tager vi til sidst. Elev 13?

593 Elev 13: Altså 2 lige tal vil altid gå op i et lige tal fordi ligesom Sebastian sagde, så er 2+2 4, uanset hvor meget 594 eller hvorfor nogle forskellige nogle der er med, vil det altid give et lige tal.

595 Lærer: såå

596 Elev 13: Jeg tænker sådan lidt (mumler i 7 sekunder)

597 Lærer: Nej det er lige meget, kom indtil kernen af det du gerne vil sige

316

598 Elev 13: mumler igen

599 Lærer: Ja det er på grund af, at ….

600 Elev 13: Det er på grund af, at alle tallene i 2 tabellen det er lige tal, så det er ligesom at plusse 2 +2 når du siger 601 6 + 6, fordi ligesom 2 tabellen.

602 Lærer: Ja, ja men i har forklaret det, men ikke med eksempler, jeg ved godt du kan [Snakker til Vetus] Anton

603 Elev 2: Altså ligesom det elev 10 sagde, som der var så godt, at så vil der ikke være nogle overskydende

604 Tidskode 15 minutter

605 Lærer: Så hvis vi holder fast i tanken fra før med centicubes, der ligger 2 og 2 og 2 og 2

606 Elev: Ja, så vil der aldrig komme en overskydende.

607 Lærer: Nej der vil ikke ligge en alene ovenpå vel, fordi begge tårne de vil være fine og det vil de blive ved med at 608 være, også når man sætter dem ovenpå hinanden. Så 2 lige tal, giver også et lige tal. Det er lige præcis derfor, at 609 det er et super godt billede, fordi vi kan jo bruge det igen ikke. Så hvis jeg nu siger, at vi plusser et lige og et ulige, 610 hvad sker der så? Hvad sker der så? Elev 6

611 Elev 6: Så bliver det ulige

612 Lærer: Fordi?

613 Elev6: Det ved jeg ikke

614 Lærer: Prøv lige at hold fast i elev 10’s ide med tårnet af centicubes, hvis vi har et lige tal, hvordan ser tårnet så 615 ud elev 6?

616 Elev 6: Det ved jeg ikke..

617 Lærer: Så er det pænt og lige ikke, fordi de ligger 2 og 2 og 2 og 2. Hvis vi så tager det ulige tårn, så ligger der en 618 centicubes alene foroven ik’. Så når du sætter de to sammen, så vil den ene centicubes for oven, blive ved med at 619 ligge der. Så når vi tager den her, så giver det et ulige tal. Også kan man sige, når vi vender tilbage til vores kasse 620 opgave, alle de tal, som vi fik af vide at kasserne kunne veje til sammen. De er jo lige.. så vi kan ikke finde en 621 løsning, hvor vi blander de lige og de ulige tal. Fordi de vil altid få ulige resultater. Giver det mening?

622 Elever: jaaa

623 Lærer: Jeg sagde at de 4 kasser, de 4 ulige blandet, så får vi problemer, for når vi så lægger dem sammen, så giver 624 det jo ulige tal. Men på den her opgave der er de alle sammen lige. Så ligesom du sagde, at hvis vi valgte en kasse 625 der var 16, så ville den kun kunne bruges til en ting. Hvis vi valgte en kasse der var ulige og så de andre var ..

626 Tidskode 17.43 Observation 8 færdig

627 Lærer: lige, så ville vi få svært ved at ramme alle tallene fordi, vi ikke kunne lave kombinations mulighederne om.

628 [Støj fra kamera imens en elev siger noget til læreren]

629 Lærer: Nej.. elev 3

630 Elev 3: Altså der hvor vi sagde, at et lige tal plus et ulige tal, så er det jo at det kommer an på hvilke tal det er

631 Lærer: Nej

632 Elev 3: Jo

633 Lærer: Nej, hvis du tager et lige tal og et ulige tal og plusser, så giver det altid ulige tal. 317

634 Elev 3: Hvad nu hvis 4 + 3

635 Lærer: Prøv at se, prøv at se, hvis vi tager fat i den her med centicubesene [Begynder at tegne på tavlen]

636 Elev 3: Nååååå

637 Lærer: Alle tal kan du bygge op af centicubes par, alle lige tal – alle ulige tal bygger du op af centicubes, hvor der 638 så kommer til at sidde en ovenpå.

639 Elev 3: nååå

640 Lærer: Hvis du kombinerer 2 tårne,

641 Elev mumler noget uklart

642 Lærer: men lige nu snakker vi om lige og hele tal ikke halve. Når du kombinere to tårne med ingen på toppen, så 643 bliver det ved med at være et tårn med ingenting på toppen, men hvis du kombinere to tårne, som har en liggende 644 ovenpå, så vil de passe sammen og blive til en med ingen ovenpå, men hvis du kombinere en med en ovenpå og 645 en uden, så vil den der bliver ved med at være der [Illustrerer det oppe på tavlen]

646 Elev: Men så hvis, neej øhm,

647 Lærer: elev 7?

648 Elev 7: De der to lige tal ikke, de kan godt blive til et lige (uklart pga. han snakker så lavt)

649 Lærer: Hvordan?

650 [Elev mumler igen]

651 Lærer: Ja men jeg siger hele tal, så du må ikke have de halve med vel. Okay?

652 Elever: Ja

653 Lærer: Godt! Ikke så tosset vel, i har været mega seje, godt gået! Godt eksempel. Kasserne pænt ned i kassen 654 heroppe og papiret ind i matematikmappen! 655 Tidskode 2.31 Observation 9 færdig

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Transskribering fra skole 2 (hvad vejer kasserne)

I klassen sidder der ca. 25 elever i en hestesko. Læreren står ved et smartboard.

De to elever der bliver observeret specifik, har her fået navnene: Bo og Ane

Dorte er observatøren som på et tidspunkt stiller et par spørgsmål.

De andre elever i klassen er anonymiseret til Elev 1, elev 2 osv.

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1 10-10-2017 Optagelse 00023

2 Elev: (mumler) nej lige meget 3 Lærer: Hvorfor det? 4 Elev: Fordi de har samme størrelse 5 Lærer: Okay. Kan man altid være sikker på det? 6 Elev 2: ...nej. 7 Lærer: ..Jeg spørger bare..?? Hvis jeg har sådan en, hvis jeg har en bærepose. Hvis jeg tager to bæreposer og 8 stiller op foran jer; den ene er fyldt med sand, den anden er fyldt med fjer. Vil I så umiddelbart tro de vejer det 9 samme? 10 Flere elever: neeej. (tøvende, i kor) 11 Lærer: Men de har samme størrelse.. Okay, så man kan godt forestille sig at selvom to kasser de har samme 12 størrelse, så er det ikke nødvendigvis at de vejer det samme. [flere elever rækker hånden i vejret for at sige noget] 13 ..men vi kan faktisk ikke vide det. 14 Lærer: De kunne jo faktisk godt veje det samme. Elev1? 15 Elev 1: hvad er der i kassen? 16 Lærer: Hvad’ der i? 17 Elev 1: Ja. De er helt fyldt med luft. 18 Elev 2: Hvordan kan de så veje 16 kg? 19 (et barn der sidder tæt ved læreren udbryder noget utydeligt og gestikulerer med henvisning til kassen.) 20 Lærer: Så skal du så prøve at tænke med i historien. Ikke også. Jeg har jo ikke noget i kasserne. 21 Elev 4: Jamen så vejer de jo ikke seks kg – eller noget som helst. [slår opgivende ud med armene] 22 Lærer: Det er rigtigt! 23 Elev 2: det var jo det jeg sagde! 24 Elev 4: Jah, er jeg ikke bare go’! [eleven rækker triumferende armen i vejret.] 25 Lærer: Men det er jo det med, at nu prøver vi at forestille os det, som.. 26 Elev: næh.. [forsigtig fnisen..] 27 Lærer: ..nå ok. Men hvis jeg tager to af kasserne, så kan de veje seks kg. Hvis jeg tager to andre kasser så (afbryder 28 sig selv) – hvad ville I umiddelbart tro det var for to kasser hvis de skulle veje 6 kg? 29 Elev: (mumler, uden at være spurgt) A og C. 30 Lærer: bare et gæt hvis man skal kigge på størrelsen af den, hvad ville I så tro? [flere elever vifter med hænderne 31 for at få lov at svare.] 32 Lærer: elev2? 33 Elev 2: B og A 34 Andre elever: nej.. 35 Elev 2: Jo. 36 Elev 5: Hvorfor?

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37 Elev: Det kan jeg se! 38 Lærer: Det ved vi da ikke. Det kunne godt være den her [lyden af en kasse der bliver slået på] – at der ikke var 39 noget inden i den her, men der var beton i den her kasse. 40 Elev2: Jamen lærer, hvis vi tager A og C og det vejer seks kg, og så du tager A og D, altså.. [tager sig til hovedet, 41 synligt i tvivl om sit eget ræsonnement.] 42 Lærer: Du er allerede gået i gang med at tænke over hvad det kunne være at det var, det var. Men nu skal I høre 43 hvad opgaven den består i nu. Om et øjeblik, så sætter I jer sammen to og to. [aajj, lyder det klagende fra en 44 elev...] – og så skal I prøve at finde ud af hvad kasserne vejer. Der er forskellige muligheder man kan gøre det på. 45 Som jeg sagde, så ligger de her sedler herovre, som jeg øh… som jeg har klippet ud, hvor der står alle mulige 46 forskellige vægte på. Dem kan man jo tage nogle af og så kan man jo prøve at sidde og sige: ’hvis jeg skal vælge 47 fire vægte, så kunne jeg gå over og tage fire forskellige sedler, så skulle jeg sige kan jeg lave de heroppe.’ 48 Det kan også være der er nogen der kan tænke sig til det. Der er også nogle andre her, så hvis man har brug for 49 dem så kan man klippe dem ud – der står nogle sakse herovre. Så er der også, afslutningsvis er der faktisk også 50 sådan nogle centicubes. 51 Elev: Kan vi ikke bruge de nye sakse? 52 Lærer: Ved du hvad, de der sakse, de virker helt fint til det her. Her! – der er der, hvad hedder det, de der 53 centicubes. Og det kunne godt være der var nogen der ville synes at det var en fed idé at sige, ’nå men hvis den 54 vejer fire kg, den ene kasse, så tager jeg fire centicubes og sætter sammen, og så tror jeg egentlig den vejer noget 55 andet (lidt utydeligt, underviseren virker distraheret af noget andet) og så kunne jeg, så kan man på den måde 56 måske... det kan godt være der var nogen der synes, det kunne være rart. Det der er jeres opgave, det er, om lidt, 57 to og to, at prøve at finde ud af hvad I tror de vejer de forskellige kasser. Så er der nogle spørgsmål? 58 Elev 6: Det er dig der deler grupperne ud? 59 Lærer: det er rigtigt. 60 Bo: Hvad… [synligt utilfreds, lægger han sig hen over bordet...] 61 Elev 7: ahvad? 62 Elev 2: Det er ham der sætter grupperne… [Elev 7 slår opgivende i bordet] 63 Elev: Ja elev 5? 64 Elev 5 [der har siddet med hånden oppe siden seancens start]: Må man gerne prøve at tage dem i hånden og se 65 hvor tunge de er? 66 Lærer: Ja det må du godt… 67 Elev 2: elev 5, der er ikke noget... 68 Lærer: men det er optisk bedrag. 69 Elev 2 [henvendt til elev 5]: Der er ikke noget indeni. 70 Lærer: der er ikke noget i nogen af kasserne. 71 Bo: Hvordan skal vi så finde ud af hvad de vejer. (Andre elever taler i munden på hinanden) 72 Lærer: prøv at høre, man kunne jo gøre det at hvis man havde centicubes på samme måde, så kunne man jo gøre 73 det at.. føles de som om de vejer lige meget – nu er det lidt lidt – de vejer jo kun et gram stykket, det er måske 74 lidt svært at forestille sig (halvt henvendt til sig selv), men ellers kunne man godt forestille sig det var på den 75 måde. Det kunne måske være meget sjovt, for jeg har selvfølgelig en idé om hvad det kunne være de ku’ veje. Så 76 hvis jeg nu havde puttet lige meget i kasserne, så kunne det måske godt give mening. [Flere elever gaber, ser 77 ukoncentrerede eller forvirrede ud.] MEN ikke lige i den her sammenhæng. Godt nu skal I høre, jeg har lavet nogle 78 grupper og til forskel for sidste gang, så sidder man to og to sammen og arbejder.

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79 Elev: hvem er de to og to? 80 Lærer: det hører du lige om lidt. Du er en lille smule utålmodig; du glæder dig til at komme i gang kan jeg 81 fornemme, det er godt at høre. Det vi gør (hæver stemmen lidt), måden vi arbejder med den her opgave, det er 82 ved at I bruger jeres hæfte, det hæfte som I får udleveret af mig lige om lidt, som I har liggende nede i skabet. Og 83 så får I øh... så skal i prøve at se om I kan finde ud af det i det hæfte. De der materialer derovre, dem må I tage 84 alt det af I kan bruge. Og hvis der er nogle andre materialer I tænker ville give mening for jer for at prøve at se 85 om I kan løse opgaven, så er det også muligt. 86 Tidskode 5 minutter 87 Materialer: Sakse, centicubes, sedler med vægtangivelser, papirer med flere vægtangivelser. 88 På den hvide tavle er skrevet øverst; 6, 8, 10, 12, 14, 16. 89 Elev: hvornår hører man hvem man er sammen med? 90 Lærer: det hører du lige om to sekunder. Godt. Og det er jo super fedt I er her alle sammen, for jeg har lavet nogle 91 grupper ud fra at det var alle der var her. 92 Elev: Hvor mange er vi? [flere elever begynder at tælle halvhøjt for dem selv]. 93 Lærer: Hvor mange er det vi er? 21. Går det op i 2? 94 Elev: nej. 95 Lærer: så derfor er der en gruppe der er tre. Ooogg nu skal I høre – shh. Vi har en gruppe der hedder: xxx 96 Lærer: Var der noget af det jeg sagde I ikke forstod? [flere elever fniser til hinanden) Nå ok... Og så har vi en 97 gruppe der hedder øh… elev, elev og elev – I arbejdede så fint sammen sidst. 98 Elev: nej!! [flere griner] 99 (Flere taler i munden på hinanden) 100 101 Lærer: Og I arbejdede fint sammen... (gentager overbærende/afvisende) nej nej... Og den sidste (hæver stemmen 102 for at overkomme larm fra eleverne) det er elev og elev. [elev slår i bordet og læner sig demonstrativt tilbage] Og 103 det kræver lidt, ved jeg godt, af jer to, men jeg har fuld til... Jeg har fuld tillid til. 104 Elev: Hvorfor skal jeg være i den samme gruppe igen!!? (opgivende) 105 Lærer: Fordi han har brug for dig. 106 Elev: Jamen det er irriterende! 107 (Høj småsnakken) 108 Elev: eej, lærer.. 109 Lærer: Jeg har tillid til jer! [går ned langs hesteskoen og klapper eleven på skulderen, før han fortsætter] 110 Elev: Du har jo bare taget (utydeligt) .. og sat dig selv i den. 111 Lærer: nej det har jeg ikke. 112 (småsnakken intensiveres) 113 [Bo lægger sig på maven på bordet og begynder at skubbe sig ned langs hesteskoen] 114 Tidskode 7:00 minutter (Optagelse 10-10-2017 00023, slut) 115

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116 Bo: Vi lige lavet en (utydeligt).. 117 Ane: øhmm… det bare for lige at huske os på at to af dem vejer det der til sammen, for ellers så kunne vi – så 118 ville vi blive helt forvirret. 119 Bo: sådan der. 120 Ane: Så, så hvis vi siger at.. hvis vi tager A og B – de ser jo ikke særligt store ud… [lang pause]... men jeg tror 121 alligevel ikke det er dem der giver seks; jeg kunne forestille mig fire eller sådan noget. [kigger på elev 4] [Bo rejser 122 sig]. Jeg ved det ikke (delvist henvendt til sig selv.) 123 [Bo demonstrerer noget for Ane, som nikker, hvorefter han trækker tilbage til bordet] 124 Ane: …så hvis man lægger A og C sammen, så må det jo give seks. [griner] [hun fatter sig og ser opmærksomt 125 frem for sig hvor hendes samarbejdspartner er ved at demonstrere noget uden for kameraets rækkevidde.] 126 Ane: er det det? Okay..men så kom herned. [Hun signalere han skal komme tilbage til hæfterne.] Så skriver vi 127 øhm.. A, B, C. Det ved jeg egentlig ikke hvorfor jeg skriver, men nu har vi fundet en der giver seks. (Ane virker, 128 om ikke tynget, så meget bevidst om kameraets tilstedeværelse.] ..Så øh skal vi skrive, vi skriver A + øh hvad er 129 det nu det hedder.. A + C er lig… 130 Ane: (mumler) jeg ved ikke hvor meget det skal veje… 131 Bo: var det ikke C? 132 Ane: nårh jo. [finder sit viskelæder og retter fejlen.] 133 Elev 2: HEJ! [Adresserer kameraet lige foran linser, mens han smilende vinker.] 134 Dorte: Shh.. du må ikke forstyrre. 135 Bo: hm… så er det her måske forkert. [peger ned i sit hæfte.] [Ane nikker forbeholdent.] 136 Ane: Så ved vi i hvert fald at A og C må ca. være tre kg hver. 137 Bo: Måske er C fire [peger ned i Ane’s hæfte] … og så er det to… 138 Ane: ja. Men okay, men så kan vi i hvert fald lige lave en let streg over seks [taber sin blyant på gulvet] for så ved 139 vi, den har vi sådan nogenlunde, sådan... 140 Ane: Hvad kunne så give otte, det kunne være… 141 Bo: øjeblik [forlader sin stol og billedet…] 142 Ane: Bo jeg har en idé.. ..det kunne være D + B.. [henvendt til Bo der står ved tavlen uden for billedet] – nej jeg 143 mener den lille kasse Bo, jeg mener den lille kass’ [kameraet panorerer og viser en masse børn der står ved 144 kateteret og flytter rundt på forskellige kasser. Bo løfter en af dem på plads og går derefter tilbage mod 8.] 145 Ane: og det var D .. + B. 146 Bo: Skriver du det ved C? 147 Ane: Jaa... jeg ved det ikke, jeg skriver det lidt alle steder... [bege skriver i deres hæfter.] 148 Bo: Hvilket tal var det nu? 149 Ane: otte. 150 [Bo går op til kateteret igen for at løfte på kasserne.] [Han spørger læreren om noget.] 151 Lærer: Ja men du kan ikke bare løfte på dem, det har jeg fortalt dig. Prøv at se om du kan forklare mig hvordan. 152 [Lærerne sætter alle kasserne op på kateteret foran Bo.] Men kunne det give mening, kunne det give mening for 153 jer – og du må snakke med Ane om det – men kunne det give mening i stedet for at stå og løfte på de kasser så

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154 her, som i udmærket ved I ikke kan bruge til noget det har jeg jo fortalt jer. Men kunne det give mening at prøve 155 at vælge nogle tal og så se om i kan kombinere jer ud af det? 156 Tidskode 5 minutter

157 158 Bo: …nej. [ryster på hovedet.] 159 Lærer: Hvad er det du ikke forstår? 160 Bo: Det der med tallene. 161 Lærer: Okay. Hvis jeg nu siger til dig at den [peger] vejer 5 kg, hvad tror du så den anden vejer? 162 Bo: To. 163 Lærer: Prøv lige : fem plus to.. 164 Bo: Seks. 165 Lærer: Hvad? Fem plus to, er det seks? 166 Bo: Nej syv. 167 Lærer: Godt. [Peger op på tavlen.] Så den kombination, den er her ikke i hvert fald. 168 Bo: neeej. 169 Lærer: Okay, så det kunne være der var en der vejede en og en der vejer fem. [Bo nikker forstående og kigger sig 170 over skulderen.] Godt. Men så prøv at tænke lidt videre ud af den. Kunne man gå videre ud af den vej? 171 Bo: Ja. 172 Lærer: Så skal vi finde ud af hvad de sidste to kasser vejer. Men vi skal kunne lave kombinationen så vi får alle 173 dem herop. Prøv og gå ned og snak med Ane om det. 174 [Sætter sig ned ved siden af Ane igen.] 175 Bo: Altså det er fordi, hvis nu D den vejer fem, så B den vejer en.. så giver det jo seks. 176 Ane: Ja men hvis D nu vejer fem, så hvis det skal gi otte så skal B’eren veje 3..?.. 177 Bo: Ja men den er vel lille. 178 Ane: Jamen så siger vi B er lig med en. Jeg visker bare det hele ud. [Der viskes ud.] 179 Ane: Jeg har en idé. 180 Bo: Hvad. 324

181 Ane: Ikke det hele, hele, men under B – den er jo meget lille – der kunne vi sige en til to kg. 182 Bo: hm... 183 Ane: Det kan vi skrive nedenunder. Og hvad kunne D’eren være? Det kunne være noget fem-seks. 184 Bo: hm... [Begge noterer ned.] 185 Ane: øhh den vejer vist kun et tal. 186 Bo: okay - Fem eller seks, hvaffor en af dem? [Henvendt til Ane.] 187 Ane: Det er fordi du… Altså hvordan skal vi komme… hvis vi skal lave den der med seksten, hvordan kan vi så 188 nøjes med en kasse der kun vejer fem kg? Er det så ikke bedre at tage seks kg. Jeg ved det ikke, det er bare noget 189 jeg tænker. Hvad synes du? 190 Bo: Joh… det er faktisk rigtigt. 191 Ane: Skal vi sige at D er seks kg? 192 Bo: Ja. 193 Ane: C og A de er jo nogenlunde lige store. 194 Bo: Måske vejer A tre kg. 195 Ane: Og så C’eren det kunne være – de er jo egentlig lige store… 196 Bo: Eller... det kan være der er beton i den ene, og så bare fjer i den anden… 197 Ane: Men bare lige for at have en masse tal at arbejde med, så... 198 Bo: Okay, fire. 199 Ane: Bare lige så vi har noget at forholde os til. Okay. 200 Bo: ...Og D den vejer en eller to kg. 201 Ane: Skal vi ikke bare, - jeg har bare taget en. Okay. Så laver jeg bare lige en streg. 202 Bo: Hvis det er man lægger alle dem samme, så gir det jo stadig ikke seksten. [Viser med sin blyant på papiret.] 203 Ane: Nej men det giver fjorten. Men man kunne, man kunne, man kunne plusse to… (Utydeligt) D. en...? 204 Tidskode 10 minutter 205 Ane: Hvordan kan man så få fjorten? 206 Bo: Hva? 207 Ane: Hvordan kan vi så, øhm… hvordan kan vi så få fjorten og seksten til at, øhh… Vi kan jo ikke lægge det her 208 sammen. 209 Bo: Vi kan ikke engang få tolv, hvis vi kun lægger de der tre tal sammen… 210 Ane: Jah... Øhh måske skal vi prøve at vælge nogle tal som det hele kan gå op i. Fx, hvis vi kan få seksten, så tror 211 jeg også vi kan få mange af de andre. [Bo nikker indforstået] 212 Ane: Måske kunne B være 2 kg. [De noterer begge.] Fordi så lille er den jo heller ikke. Og så er A tre igen. – Nej 213 så det femten. [Griber ud efter viskelæderet.] Det giver femten hvis vi lægger de normale tal tilbage. 214 [Bo peger til noget han har skrevet øverst i sæt hæfte.] 215 Ane: Jaer. Og så tre ved C også. 216 Bo: Ja. Fordiii..?? 325

217 Ane: Det giver jo seksten. 218 Bo: Ja okay. 219 Ane: Men så kan vi ikke få fjorten. [Smiler til Bo.] 220 Lærer: I skal lige huske det kun var to kasser, der til sammen vejede... Nu lægger I alle sammen, er det rigtigt? 221 Ane: Nårh ja… 222 Ane: Men så kan vi heller ikke få otte. Så det er nogle meget mærkelige tal vi har valgt. 223 Bo: Og fem så kan vi ikke (Mumler) [Begge visker ud.] Jeg beholder altså ottetallet. 224 Ane: Okay. 225 Ane: Men D kan jo ikke kun veje otte, fordi de andre kan jo ikke veje otte, og otte plus otte giver seksten. Så jeg 226 tror D må være tungere. Så det skal nok være noget... hvad ved jeg, 12 måske. [Bo slår ud med armene.] 227 Bo: Jaja, lad os prøve. (Siger noget, tilsyneladende henvendt til sig selv, men Ane: opfanger det og svarer:) 228 Bo: Jojo, men jeg kom til at skrive seksten. Men så kunne C være fire kg. Og så bare tre og en. 229 Bo: Ahh der ja. [peger ned i Anes kladdehæfte.] 230 Ane: Vi kan ikke få det til at gi... Skal vi prøve med at to med B, for det gælder jo om at prøve noget. Ja. Skal vi 231 sætte et kryds over seksten? 232 Bo: Ja. 233 Ane: Og så skal vi skrive C + D er lig med seksten? 234 Bo: Så C + D det gir seksten. Og vi kan også – B + C det er seks. 235 Ane: B + C? 236 Bo: Hmm.. (Affirmativt.) 237 Tidskode 15 minutter 238 Bo: Øhh.. vi kan ikke få otte..? [tænkepause] Vi kan ikke bruge tre for det er uligt. Eller det kan vi godt… 239 Ane: Åhh lige tal, lige tal... Skal vi så tage to? 240 Bo: Øhh ja, for den er jo meget... 241 Ane: – men så skulle C måske være seks? Skal vi prøve det. 242 Bo: Den er så fire ved A. 243 Ane: Men så kan vi jo ikke bruge de her regnestykker til noget. [visker ud.] Hvad var det nu C skulle veje? 244 Bo: Øhh.. seks. 245 Ane: Okay. Og A? Det var fire, var det ikke? 246 Bo: Jo. Så kører vi på, A + B det er seks. Og vi kan få otte, for B + C det er otte. 247 Ane:Og. Og A + D giver seksten. 248 Bo: Og A + C det er ti. 249 Bo: Vi kan ikke få tolv?! 250 Ane: Naaj… Men D + B, skal vi ikke bare skrive det? 251 Bo: D + B 326

252 Bo: Og så tolv det er det eneste vi mangler. 253 Ane: Ja som vi ikke kan tage, uden at lægge tre sammen i hvert fald. 254 Bo: Måske vejer… nej. 255 Ane: Nej fordi hvis vi – hvis vi ændrer et af tallene… 256 Bo: – så vil alt det her bliver ødelagt. 257 Lærer: (Besked til hele klassen.) Hvis man finder en løsning, skal man holde den for sig selv. 258 Ane: Bare man måtte lægge tre sammen. [Smiler til Bo.] Vi kan jo heller ikke bare tage D. 259 Bo: Neej, vi skal jo lægge de to sammen. Jeg visker det ud. 260 Ane: Jeg gemmer det for en sikkerheds skyld. 261 Bo: Ja okay, det kunne vi godt have tænkt på lidt før. 262 Ane: Måske vi skal tænke over hvaffor nogle tal vi kan få inden vi skriver ned. Hvis jeg prøver, - jeg laver bare lige 263 en til med A, B og C i stedet for at visk’. Okay. 264 Bo: Såå.. vi skal have byttet… 265 Ane: Jeg synes vi skal have tallet fire… fordi det har vi brugt tre gange. 266 Bo: Øh, ja. Ved A eller C? 267 Ane: Bare ved A. Hm… Så hvis vi også vil have otte... ved C [Kameraføreren undskylder til en pige der kommer og 268 vil sætte sig.] 269 Bo: Og så D. 270 Ane: Ja D. 271 Bo: …det kan være.. 272 Ane: Skal det være tolv igen? 273 Bo: Ja. Jo. 274 Ane: Og så to ved B. Ja, for så kan vi både få seksten. Jeg laver lige en ekstra streg hvor vi kan skrive det. [Tegner] 275 Tidskode 20 minutter 276 Ane: Okay D ..+ A er lig med seksten. [Begge skriver] Og A, ej vent lige… Og tolv, og D... 277 Bo: A +... A + B 278 Ane: Nej men, nej men jeg mener når vi starter med seksten, så skal vi tage dem bagfra – det er lettere, er det 279 ikke? 280 Bo: hm… 2 +… B + B. 281 Ane: Ja. Og så. Og så A + C. 282 Bo: A + C… Nårh det er ti ... eller otte? 283 Ane: Nej det er ti nu. 284 Bo: A + C? 285 Ane: Nej det er tolv. A + C det er jo tolv. 286 Bo: Nårh... 327

287 Ane: Og så B + C. 288 Bo: B + C. (Gentager) Men vi kan jo ikke få otte. 289 Ane: Jo… hov [smiler] 290 Bo: Jeg laver en ny en. 291 Ane: Ej, vi er så tæt på hver gang. Måske skal vi vælge et tal som er med, som vi skal bruge. Men der har vi jo 292 valgt tolv, men det gik jo fint nok. 293 Bo: Hm… det er det eneste tal vi kan bruge. 294 Bo: Kan D ikke være… hm. Måske vi skal prøve kun med ulige? 295 Ane: Hvordan har du tænkt dig med tolv? Hm… Skal vi prøve? 296 Bo: Ja. D det kan være elleve. 297 Ane: Okay. D er elleve. 298 Bo: A eller C kan være fem. 299 Ane: Så tager vi C 5. Og B det kan være et et-tal. Og A et tre-tal. 300 Bo: Nej, nej: B skal være tre. 301 Ane: Okay. Og hvad skal A så være? 302 Bo: Det… ska.l.. være... tre. 303 Ane: Jamen så har vi jo to tre’ere. 304 Bo: Men A + B det er seks. 305 Ane: Hvad med A og en? 306 Bo: Hvad? 307 Ane: A og en. 308 Bo: Nårh ja. 309 Ane: Men så kan vi jo ikke få... 310 Bo: Seksten, vi starter bagfra igen. 311 Ane: Er det ikke A, A + D? 312 Bo: Nej det er jo ikke seksten. 313 Ane: Nårh ja. [Begge skriver.] Så er det fjorten. Det er D +… 314 Bo: D + B. 315 Ane: Og så D + A. 316 Ane: Ti? 317 Bo: Prøv lige at vente. 318 Ane: Vi kan ikke få ti.. [Begge tager sig til hovedet.] 319 Bo: Måske vi skal ikke have A som et. Det burde jo være B. 320 Tidskode 25 minutter

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321 Ane: Jamen det har jo ikke noget at gøre med… – det er lige meget hvor den står henne. Jeg, jeg har snart ikke 322 mere plads på min side. 323 Bo: Hm... jeg ved ikke… [En anden elev kommer til syne i billedet og kigger drilsk ind i kameraet. Bo smiler til ham 324 og vifter ham væk.] 325 Ane: Fire og tolv, dem er vi kommet langt med. Og så måske også… ej det kunne vi ikke. Ok C, hvad kunne C 326 være? 327 Bo: C? 328 Ane: Jeg tror ikke vi skal bruge de samme tal igen og igen og igen. 329 Bo: Nej. 330 Dorte: Prøv lige at overveje den der igen. I holder hele tiden fast i jeres tolv’er; måske I skulle prøve noget andet 331 der. 332 En anden elev : Ej det er mega irriterende, vi mangler kun et tal. 333 Ane: Okay, hvis vi prøver et andet tal end tolv.. ti? 334 Bo: ...Ja! 335 Ane: Så B er en, eller hvad? 336 Bo: Tror B er to. 337 Ane: Ja. 338 Bo: Måske skal vi prøve bare… 339 Ane: – Og så fire os. 340 Bo: Jamen den har du brugt mange gange 341 Ane: Ja, men så vi kan... vi kan...vi kan ikke lave seksten uden en fire. Eller hvad? [Bo peger ned i Anes hæfte 342 hvorpå de begge begynder at skrive.] 343 Bo: ...Det der det er ti… 344 Ane: Jaer. Jeg tror faktisk... Hvis vi nu prøver at skrive op. Hvad var det nu for nogle tal, det var.. [kigger på tavlen, 345 og skriver ned.] Okay hvis du skal tage den forfra : A + B er lig med seks. 346 Bo: Og så er det B + C. 347 Ane: Okay. Og så ti? Det er A + C. 348 Bo: Ja A + C. 349 Ane: Så er det D. 350 Bo: Hvad var det nu? 351 Ane: A + C. 352 Bo: Det var ti? 353 Ane: Ja. 354 Bo: Og så er det tolv, så det er B + D. 355 Ane: Og så D + A. Og så D + C. [Spjætter da det går op for hende at tallene passer sammen, og kigger smilende 356 over på Bo, der nikker. Ane rækker begge hænder i vejret og Bo rejser sig hurtigt fra stolen.] 357 Tidskode 29.52 minutter 10.10.2017 (00024 slut) 329

358 359 10.10.2017 Optagelse 00025 360 Kontekst. Bo og Ane præsenterer deres løsningsforslag til læreren. Omkring dem står også en anden gruppe 361 og lytter med. De har taget deres hæfter med for at demonstrere deres løsning. 362 Lærer: Hvad har I gjort, prøv at fortæl hvad I har gjort? 363 Bo: A + B giver seks, det er jo seks’eren der. Så tager vi B + C det giver otte – så streger vi otte ud. Så tager vi A + 364 C det giver ti. Og B + D giver tolv og så A + D det er... fjorten. Og D + C det er seksten. [Bo peger ned på 365 udregningerne på papiret mens han forklarer.] 366 Lærer: Men hvad vejer A? 367 Bo: Den vejer fireogtyve. 368 Lærer: Den har I skrevet deroppe. 369 Ane: Ja. 370 Lærer: Okay. Men hvordan kommer I herfra og derned? Det forstår jeg ikke helt. Gættede I bare? 371 Ane: Altså først – vi holdt meget fast i tolv. 372 Bo: men Dorte hun hjalp os lidt. 373 Ane: Ja så hun sagde måske, måske skulle vi prøve… noget… så minusset’ vi med noget to og så fandt vi ud af at 374 ti det ville være meget godt. 375 Lærer: Hvorfor var I optagede af tolv? 376 Ane: Det var bare fordi tolv, den havde vi bare haft med mange gange og den kunne vi lave både seksten og 377 sådan noget ud af… 378 Lærer: Okay. Så I havde en vægt der var tolv fordi det havde I fornemmet at det gav god mening... 379 Ane: – Også fire den har vi også haft meget med. 380 Lærer: Okay. Og så prøvede I og så landede I der. [Peger på udregningen.] 381 Lærer: Okay, prøv at se – nu slår jeg en streg her – og så siger jeg det er et super(t) bud det der; kunne man, kan 382 man finde et andet bud? Jeg kan give jer et hint – prøv at gå ned – [peger på resultaterne] hvad har alle de tal 383 der til fælles? 384 Ane: Okay. 385 Lærer: Hvad har de der tal tilfælles? 386 Ane: Skal vi sige det? 387 Lærer: Ja. 388 Ane: Alle sammen i to-tabellen. 389 Lærer: Ne-ja, men hvad er de også? 390 Ane: Alle sammen lige. 391 Lærer: Godt. Hvad nu hvis vi opererede i.. 392 Ane: Ulige. 393 Lærer: Det kunne man måske overveje. Prøv lige at gøre jer en overvejelse med det. 394 Tidskode 1:49 (slut 00025) 330

395 396 10.10.2017 Optagelse 00026 397 Kontekst. Tilbage på plads, forsynet med hæfter, blyanter og viskelæder. 398 Bo: Hvem har skrevet det?! [Ser på sit hæfte der tilsyneladende er blevet vandaliseret] 399 Ane: Okay, hvilke talskal vi vælge. 400 Bo: Øhm... fem! 401 Ane: Fem? 402 Bo: A fem. 403 Ane: Og så B (i kor) en. 404 Bo: Og så C det kan være syv. 405 Ane: Ja. 406 Ane: ni.. nej. Skal vi vælge ni? 407 Bo: Jaja. 408 Ane: Ja for sidste gang tog vi elleve. Okay, A + B. A + B er seks. 409 Ane: A + B er lig med tre, seks. [Griner] Ikke noget. Se så det B + C det lig med syv. 410 Bo: Øh… Jo det er tolv. 411 Ane: - Nej, er lig med otte mener jeg. 412 Bo: Nej… [Peger på en udregning på Anes papir.] 413 Ane: Jo… fordi jeg mener B + C. Og det er otte. 414 Bo: hva? 415 Ane: Men så siger vi så, vi kan ikke få tolv. 416 Bo: Øh… jo. [Peger ned i Anes udregninger.] Hvis det er man trækker dem fra så har man to tilbage. 417 Ane: Må man det? 418 Bo: Ja du skal jo bare plusse to. Fem plus syv det tolv. 419 Ane: Nårh… Okay. Det er A + C. 420 Ane: Og så er det… 421 Bo: - Det er C plus D. 422 Ane: Okay. 423 Bo: Seksten. 424 Ane: Nårh, men lige nu fjorten. Det er... A + D. 425 Bo: A + D? 426 Ane: Ja det er fjorten. 427 Bo: Og så det C, D. (halvt råbende) CD! 428 Ane: (Gentager) CD! 331

429 Bo: Så er vi færdige igen. 430 Ane: Jaja. 431 Tidskode 3.21 (Slut 00026) 432 433 10.10.2017 Optagelse 00027 434 Kontekst. Gruppen (Bo og Ane) har opsøgt læreren for at vise deres nye resultater. 435 Lærer: Shh... [vifter nogle andre elever væk.] Hvad gjorde I her? 436 Bo: Du forklarer det, den her gang. 437 Ane: Okay. Det var fordi, de her tal, de er lige. Så tog vi den ulige version af dem. Altså ti-ni, seks-syv... på den 438 måde. 439 Lærer: Hvorfor gik I en ned der? Hvorfor gik I ikke en op? 440 Bo: Fordi vi havde haft elleve en gang, og den virkede ikke. 441 Ane: Det vir- det syntes vi ikke virkede så godt. 442 Lærer: I havde prøvet med elleve, eller hvad? 443 Bo: Vi havde også prøvet med ulige en anden gang. [Han bladrer og fremviser siden i kladdehæftet.] 444 Lærer: Må jeg spørge jer; hvad giver de der tal til sammen? [Peger på de resultater gruppen skal nå.] 445 Ane: [tænker] toogtyve. 446 Lærer: Hvad giver de tal hernede til sammen? [Peger på værdierne gruppen har stillet op for at ramme 447 resultaterne.] 448 Bo: Det er toogtyve! [Ane griner til Bo.] 449 Ane og Bo: Jo det er toogtyve. 450 Lærer: Okay... det er da egentlig mærkeligt, tænker jeg…? 451 Bo: Ja. 452 Ane: Det er ligesom det der med de der hjørner vi skulle lægge sammen. 453 Lærer: Kunne vi overveje noget ift det? Tror I at I ville kunne vælge fire andre tal der også gav toogtyve til sammen, 454 og så ville I kunne løse opgaven. 455 Ane: Ja. (Halvt overrasket.) 456 Bo: Orv, nu har jeg fundet tre... 457 Lærer: Gå lige ned og prøv det.. 458 Tidskode 1.52 (00027 slut) 459 460 10.10.2017 Optagelse 00028 461 Kontekst. Tilbage ved bordet, for at forsøge at udforske den nyvundne indsigt: summen af både resultattallene 462 såvel som de anslåede værdier, er den samme, toogtyve. 463 (Utydeligt) Bo repeterer deres

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464 Ane: Okay, skal vi tage de ulige eller lige? 465 Bo: Vi starter med ulige den her gang. Skal vi prøve med elleve? 466 Ane: Elleve og ni. 467 Bo: [Mens der skrives.] Elleve. Og C ni? 468 Ane: Ja. 469 Bo: Nu gider jeg ikke trykke fire mere… 470 Ane: -ej nu er vi altså allerede oppe på tyve. Vi skal have et mindre tal; syv! Fordi B det bliver jo ni... eller et. Og 471 så skal A være tre. Og det giver jo toogtyve igen. 472 Bo: Surprise.. [Smiler] 473 Bo: Men vi kan ikke få tre. Det var ikke et godt trick, elleve virker ikke! 474 Ane: Nej elleve virker ikke. 475 Elev: Jeg gider ikke det her pis mere... 476 Bo: Vi har løst den to gange. [Mens han bogstaveligt talt holder resultaterne tæt til kroppen.] 477 Elev: Tillykke med det Bo! 478 Bo: Tak. [Bukker og smiler.] 479 Ane: Vi har brugt fem mange gange i den sidste... 480 Bo: Ja.. men vi må jo ikke bruge de samme tal mere… 481 Ane: Jamen hvordan skulle vi så få seks? 482 Bo: Fordi vi skulle lave dem med fire andre. [Peger ned i Anes hæfte.] Så vi kan ikke have ni, syv, et eller fem. 483 Ane: Tag et ettal, så. 484 Bo: Lad os prøve med tre. 485 Ane: Ved B’eren. 486 Bo: Okay. Så skal jeg lige have hvisket det ud ved A’eren. Og ved D’eren der kan der være 13..? (eftertænksomt) 487 13! (Konstaterende) 488 Ane: Ved D? 489 Bo: Hmm... (Godkendende) 490 Ane: Okay. Så har vi i hvert fald seksten der. 491 Ane: Elleve ved C, eller er det lidt dårligt? Nej, vi tager ni ved C. 492 Bo: Nej den har vi allerede. 493 Ane: Nårh… Okay skal vi prøve elleve igen? 494 Bo: Argh.. Ok, man kan ikke have elleve. 495 Ane: Okay, det her giver over firetyve. Toogtyve. Hvorfor kan jeg ikk’ snakke dansk! 496 Dorte: Det kan jo også være han snyder jer lidt. Det kan jo være der ikke er flere, har I tænkt over det? 497 Bo: Ja jeg tror ikke der er flere. [Ryster på hovedet.] 498 Ane: Det tror jeg virkelig heller ikke.. 333

499 Dorte: Hvorfor tror I ikke det? 500 Bo: Fordi der ikke er flere der kan give toogtyve. 501 Dorte: Der er simpelthen ikke mere der kan give toogtyve, siger I? 502 Bo: Jo der var nogen, men der kan man ikke få de seks. 503 [To andre børn diskuterer med hinanden på hver sin side af gruppen.] 504 Bo: Hva?! (Henvendt til en af de diskuterende parter.] 505 Ane: Skal vi gå op til Kim og sige at han… 506 Bo: Har snydt os! 507 Tidskode 4.56 (00028, slut) 508 509 10.10.2017 Optagelse 00029 510 Kontekst. Til evaluering hos læreren. Omkring står også tre andre børn. 511 LÆRER: Var det det jeg sagde I skulle gøre, prøve? 512 Bo: Næh… 513 Lærer: Nej, hvad var det I skulle prøve? 514 Ane: Det samme mål. 515 Lærer: Ja det var det. 516 Bo: Vi skulle blande dem sammen. 517 Lærer: Neej. Jeg sagde ikke I skulle, - det har jeg ikke sagt. Jeg sagde bare at de til sammen skulle gi..? 518 Bo: Toogtyve. 519 Lærer: Det var det, det var det vi opdagede eller.. 520 Ane: – men man må gerne blande ulige og lige sammen..? 521 Lærer: Det kunne jo godt være.. 522 Bo: Men vi fandt – 523 Lærer: – Jamen jeg ved det ikke.. 524 Bo: Vi fandt en mere ulige der gav toogtyve, men den kunne ikke være seks. 525 Lærer: Nej, godt – hør her, (hæver stemmen for at tale til hele klassen) femte?!? 526 Elev: A! 527 Tidskode 00.30 (00029 slut) 528

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Transskribering fra skole 3 (Rebtrekanten)

I klassen sidder der ca. 20 elever fordelt på 4-mandsborde. Læreren står ved et smartboard.

Den gruppe der bliver observeret specifik, har her fået navnene: Vera, Alvilde, Molly og Christian

De andre elever i klassen er anonymiseret til Elev 1, elev 2 osv.

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1 Transkribering af d. 8. oktober 2 Lærer: Vi skal til at arbejde lidt med nogle trekanter – nogle figurere. Og det I skal lave i dag det er noget med 3 nogle trekanter I skal danne. Og for at danne det, så får I på et tidspunkt sådanne nogle reb her. (viser nogle reb 4 frem) og der er nogle knuder på, og hvis I slår op i KiDM så vil I se at der er nogle opgaver de kalder for 5 knudetrekanter. Og det er det her (viser rebet frem) og grunden til at de kalder det knudetrekanter det er at man 6 skal bruge de her knuder til at tælle og til at finde ud af hvilke trekanter man kan danne, så derfor kalder man det 7 knudetrekant, men der er lige nogle ting vi skal have lidt styr på inden vi går i gang med de der knudetrekanter, 8 men der findes nogle forskellige trekanter. Kan I nævne mig nogle trekanter? Vera

9 Elev 1: Ligebenede trekanter 10 Lærer: Ligebenede trekanter – ja jeg tror jeg bruger den her nede i stedet for (går bagerst i klassen hvor der er en 11 klassisk tavle) Hvad er en ligebenet trekant?

12 Elev 2: En ligebenet trekant det er en trekant hvor to af siderne er lige lange. 13 Lærer: to af siderne er lige lange ja. Husk det på jeres ben. De er forhåbentlig 14 sådan nogenlunde lige lange – for nogen af os (hæ hæ). Så ja ligebenet så er 15 to af siderne lige lange. Det vil jo sige det kunne jo være sådan en som – kan I 16 se for de der stole der? – [tegner en ligebenet trekant ved tavlen] det kunne jo 17 være sådan en som den – forestil jer at de der to er lige lange. Så de der to de 18 er lige lange (peger på benene i trekanten) (kort pause) så findes der nogle 19 andre? Skal vi lige fjerne dem her [fjerne fire stole der er placeret på bordet 20 foran tavlen], men der findes også nogle andre. Hvad er det Elev 3?

21 Elev 3: En ligesidet trekant. 22 Lærer: En ligesidet trekant. Hvad er det så?

23 Elev 3: En trekant hvor alle tre sider er lige lange. 24 Lærer: En trekant hvor alle tre sider er lige lange – uhh – nu skal jeg se hvad 25 mine egenskaber rækker til – sådan cirka. (tegner nu en trekant med alle tre 26 sider lige lange). Det vil være sådan en der, hvor alle tre sider er lige lange. Vi 27 har altså en ligebenet trekant – det er det der med vores egne ben, og så har vi en ligesidet og så…

28 Elev: en retvinklet trekant. 29 Lærer: en retvinklet trekant, hvad er det?

30 Elev: det er sådan en (utydeligt) 31 Lærer: ja og den der (viser en ret vinkel med hænderne) den er halvfems grader – må jeg ikke godt viske det der 32 ud… (tegner nu en ret vinklet trekant på tavlen) 33 Lærer: det kunne fx være sådan en der. En ret vinkel, hvordan markerer vi denne her vinkel.

34 Elev: man sætter sådan en lille firkant der.

35 Lærer: man sætter sådan en lille firkant der. (Læreren tegner en firkant på trekanten)

36 Elev: ja ude i hjørnet 37 Lærer: så viser man at den der er halvfems grader. Den er ikke 89 den er heller ikke 91 den er 90 grader. Præcis – 38 det er den der vinkel. Hvis man nu i stedet for skulle markere hjørnet på den her (peger på den ligebenede trekant) 39 med et eller andet, hvad gør jeg så Elev 4.

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40 Elev 4: Du laver sådan en lille bue. 41 Lærer: Ja jeg laver sådan en lille bue for at markere at den der vinkel den måler jeg nu eller 42 den skal jeg måle. Eller jeg skriver at den der vinkel den er så og så stor. Så viser jeg det 43 med sådan en bue i stedet for. Så den eneste gang man anvender en firkant det er altså 44 ved 90 grader. For at vise at den er sådan skarp ligesom et vindue eller – vi har jo rigtig 45 mange 90 graders vinkler hvis vi sådan kigger os rundt. Elev5 hvad har vi så mere?

46 Elev 5: en spids trekant og en stumb trekant…

47 Lærer: spids - ja vi har spidse (tegner en spidstrekant)… og hvad er en spids?

48 Elev 5: det er når den er under 90 grader. 49 Lærer: ja det vil sige – den der (peger på den ligebenet trekant øverste vinkel) det er faktisk 50 en spids vinkel. Fordi den er mindre end 90 grader. Det er sådan en man kan stikke sig på. 51 Så I princippet kan man sige at en 89 graders vinkel det er faktisk en spids vinkel, selvom jo den er jo næsten, altså 52 hvis vi kigger på den så er den jo måske retvinklet at se på men i princippet så er den spids vinkel fordi den er 53 under 90 grader – og hvad sagde du så – en stumb vinkel hvad er det så?

54 Elev 5: en stumb vinkel det er når den er over… 55 Lærer: Så det vil være (tegner en stump vinkel) sådan en der fx – over 90 grader. Yes (kort pause) så altså 56 ligebenet, ligesidet og retvinklet, spidsvinklet og stumpvinklet (nævner dem mens han peger på trekanterne efter 57 tur) Det er nogle af dem I skal ud og eksperimentere med i dag, og se om I kan danne trekanter ved hjælp af det 58 tov. Det går ikke så meget ud på at se om det nu er en retvinklet eller spidsvinklet trekant eller – det går det ikke 59 ud på. Det går ud på at danne nogle trekanter og kan jeg egentlig danne de her trekanter. Og så i første omgang 60 da er det sådan set ligegyldig om det er den ene eller den anden form. Og når I kommer ud – og nu siger jeg ud 61 fordi I skal udenfor at gøre det

62 Flere elever: yes 63 Lærer: så skal I lave en skitse over de der trekanter I nu får dannet derude. Og hvad er en skitse for noget? Hvad 64 er en skitse? Når I kommer ud og skal tegne det?

65 Elev: er det ikke hvor du tegner som det skal se ud? 66 Lærer: jo det kan man godt sige

67 Elev: ja sådan lidt hurtigt. 68 Lærer: ja sådan lidt hurtigt. Skal det være præcist? Nej det behøver det ikke. Så når I får papir – eller jeg har 69 printet de her opgaver ud, så I kan tegne på dem. Det vil sige at nu har I dannet en trekant så tegner I lige sådan 70 en (tegner nu endnu en trekant på tavlen) og den kom til at se sådan her ud. Og hvad er der så mere med en 71 skitse? Når I kommer ind, så kan I ikke huske hvor stor var siderne.

72 Elev: du skal skrive mål på. 73 Lærer: du skriver mål på. Du skriver fx at den der er 4 meter, men det er den jo ikke på tegningen. (læreren skriver 74 4 på bundlinjen) En skitse der tegner man sådan lidt en hurtig tegning. Det var den form den havde og den side 75 der den var 4 meter og den side der den var 3 meter eller hvad den nu var (skriver 4 på en side i trekanten på 76 tavlen). Så har vi en skitse for så kan man huske det når man kommer ind. Så I skal tegne en skitse over de 77 trekanter I får dannet når I kommer ud. (læreren går ned til den anden tavle) Den næste opgave – når I nu har 78 fået dannet de her trekanter det er at I skal bruge GeoGebra. Den er lidt – det er lidt besværligt at komme derind. 79 Øhh jeg kan lige vise jer (vil åbne GeoGebra på computeren)

80 Elev: du skal bare …(utydeligt) 81 Lærere: der var den… kan I huske at vi arbejdede med GeoGebra for noget tid siden. 337

82 Flere elever: ja 83 Lærer: og I søger på G-E-O--G-E-B-R-A. Så kommer den her side frem (i google) og så skal I bruge den øverste 84 (klikker på den øverste) Det er derfor jeg har taget computeren med her ind, og selvom I har arbejdet sammen i 85 grupper, så skal I alle sammen prøve at gå ind på GeoGebra og sidde med det alle sammen. I skal også alle 86 sammen tegne de skitser ned med de mål, det vil sige når I kommer ind i klassen igen, så har alle i gruppen de 87 samme mål. De samme skitser på deres papir. Så vi har det alle sammen. Så kigger I der, men når I kommer dertil 88 så skal I bruge noget helt bestemt. Et lille program. Det finder I ved at I søger på KiDM. K-I-D-M

89 (læreren logger ind på KiDM) 90 Lærer: vi er nu nået til indsats 2, den ligger der. Vi er nået til det første forløb der hedder knudetrekanter. (han 91 klikker på computeren) Så er der tre forskellige herinde, og det øverste elevark, det har jeg printet ud til jer. Det 92 får I, det er bare sådan en beskrivelse af hvad I skal lave. Og på den næste side der er der sådan nogle firkanter I 93 kan bruge til de der skitser. Det vil sige at I hver trekant der laver I en skitse af den trekant I nu har dannet med 94 det reb der og skriver mål på. Så tager I det med ind. Og så skal I prøve det i GeoGebra og se om I kan lave de der 95 trekanter i GeoGebra. Der ligger sådan en lille en (åber en GeoGebra fil) den synes jeg vi skal se på først.

96 (eleverne ser filmen af omkring trekanter fra KiDM hjemmesiden – læreren kommentere lidt undervejs) 97 Lærer: det er det I skal gøre når I kommer ind, så skal I prøve at taste jeres mål ind i de her gule firkanter. Og så 98 prøve at danne en trekant med de tal I nu kommer ind med og se om det I det hele taget kan lade sig gøre. Det 99 er sådan lidt svært at få den røde og blå plet til at ramme lige oven i hinanden, men det kan altså godt lade sig 100 gøre og hvis man sådan lige kan se at de overlapper lidt så er det også i orden fordi så er de jo nået sammen. 101 Men det I skal gøre det er at på den her side (går til KiDM siden med rebtrekantens forside) her ligger den her der 102 hedder GeoGebra, og når I klikker på den, så henter I sådan noget ned – nu håber jeg det sker- ja der kan I se den 103 ligger sig herned. Så henter I den ned på computeren. Når I så går ind i den der – hvor har vi den der GeoGebra 104 henne?

105 Elev: det er den første. 106 Lærer: den ligger der ja. Når I så går ind i GeoGebra, så vil I hurtigt kunne hente den frem. Det håber jeg I kan på 107 denne her computer også. Jeg kunne i hver fald på min egen computer. Den skal lige have oversat det der… bum 108 bum (computeren arbejder)

109 Heropppe, kan I bruge den der hedder åben (læreren klikker på ”åben”) og så mappen der – så burde den ligge 110 derinde…

111 (læreren viser dem fortsat hvordan man helt præcis kommer derind med lidt hjælp fra eleverne)

112 Elev 1: hvad er grupperne?

113 Læreren: det får I lige at vide. Prøv at se her (tager rebet frem)

114 Elev: det er nogle nye du har lavet! 115 Lærer: ja det er nogle nye jeg har lavet. Det er frygteligt langt. Og I kan se at de er bundet sammen, så lige nu er 116 det faktisk en cirkel. Så er fidusen at der imellem de her (peger på to knuder) hvor langt tror I der er imellem dem?

117 Elev: 1 meter. 118 Lærer: ja der er en meter. Sådan ret præcist er der en meter mellem hver knude. Hvis så Elev6 og mig vi var i en 119 gruppe, så tog Elev 6 fat der (giver Elev 6 rebet) Så går jeg herned og tager fat i den her knude. Og Elev 6 bliver 120 ved med at holde i knuden. Og det gør jeg også – og så strammer vi ud.

121 122 Og så tager den tredje person fast et eller andet sted og går ud. Kan I forestille jer det? Så får vi en trekant. 338

123 Flere elever: mmmnnn 124 Lærer: med Elev 6 i det ene hjørne og mig i det andet hjørne og … (peget lidt rundt) Elev 7 i det 125 tredje hjørne. Vi gør det ikke lige herinde, for vi kan simpelthen ikke være der. Men I skal holde 126 på knuderne, ikke noget med at holde på midten og sige nu skal min være 3 ½ meter på den 127 ene. Vi snakker hele meter. Det vil sige hvor lang er den her linje hen til Elev 6? Hvor lang er 128 den?

129 Elev: 3 meter 130 Lærer: 1-2-3, så det er forholdvis nemt at tælle op. Når I så har den figur I vil danne og som I 131 skal tegne, så vil den ene side på jeres skite jo være 3 meter og den anden 5 meter og så den 132 trejde – hvad den så vil være. Så har I en skitse hvor I har en trekant. Er I med på hvad det går 133 ud på?

134 Flere elever: ja ja 135 Lærer: Så var der de der grupper. Og når jeg nu har nævnt dem, så skal I altså udenfor og I skal 136 have en blyant og et reb med. Og jeg har også taget et skriveunderlag med hvis I får brug for det. Sådan at I kan 137 sætte papirerne fast. Det vil sige når I kommer ud så er I jo …. Ja der skal jo være 3 til at lave trekanten og den 138 fjerde kan jo så være skriver Elev 8. Det vil sige at når I så kommer ind så starter I lige med hvis I ikke har noget 139 på jeres papir at skrive det op fra den fjerde – så I alle sammen har tallene. Så tager I en computer og prøver at 140 ligge det ind i GeoGebra

141 [Grupperne bliver oplæst] 142 Lærer: I skal have sådan et ark med der ligger her også.

143

144 (videoklip nummer 2: foregår i skolegården)

145 (2 piger (Alvilde, Molly og en anden er ved at vikle et reb ud)

146 Alvilde: sådan – prøv at give slip – så går jeg herover.

147 (rebet er ikke en cirkel)

148 Alvilde: jeg tror bare vi skal have bundet dem sammen. Så hvis du giver mig din ende, så holder jeg dem bare 149 sammen hele tiden. 150 Iærer : mangler I ikke nogen?

151 Alvilde: jo vi mangler 2. Vi mangler hende der kommer der.

152 (en tredje piger ankommer)

153 (De tre har fat med begge hænder (ikke ved en knude)

154 Alvilde: nej I skal give slip for den kan ikke være en halv. Hvor 155 meget er den? Den er fire der og fire der – det er jo – Hov 156 Molly – hvor er dit ark hende?

157 (den sidste pige samler noget op)

158 Alvilde: du skal tegne en trekant og skrive 4 på alle siderne. 159 (Alvilde forklarer Molly) Det er lige meget hvordan den ser ud 160 du skal bare tegne en trekant og skrive 4 på hver side. Det er lige meget bare vi ved at den er 4 på alle siderne.

161 Molly: okay men det bliver rigtig grimt. 339

162 Alvilde: okay jeg prøver lige at læse igen. Okay hvis jeg nu rykker mig – nej Christian hvis du rykker en knude op 163 (Christian rykker sig) ja sådan ja. Okay hvor mange er der der? Der er 1-2-3-4-5 og 4 og 3

164 Molly: 3-4 og 5

165 Alvilde: ja 3 4 og 5

166 Molly tegner endnu en trekant og skrive 3, 4 og 5 på skitsen – imens rykker gruppen rundt igen.

167 Alvilde: og så skal du tegne en ny en, hvor der skal stå 2, 6 og 4

168 (gruppen rykker igen)

169 Alvilde: nej vi skal bare finde ud af hvad vi gør nu. For jeg ved ikke helt – for vi har jo lavet en lige en. Hvor mange 170 har vi (Christian rykker igen samme tur som sidst) ja og så kan vi ikke lave andre her – der kan ikke være en her…

171 172 Alvilde: men er der flere? måske skal Vera prøve at rykke? Du stiller dig bare et eller andet sted Christian.

173 Hvad er det? Det er 3-4-5

174 Vera: Det er et bjerg

175 Alvilde: den har vi lavet. Jeg kan ikke finde flere. Prøv at stil jer forskellige steder, så ser vi om vi ikke har lavet 176 den? Der burde jo være flere. (de prøver igen at stille sig i en ny kombination) Den er 3 og 4 og 5 – den har vi 177 allerede lavet.

178 Vera: så prøver jeg at stå her.

179 Alvilde: hvad er det? Det er 2 … er det ikke 2- 5 og 5

180 Christian: jo

181 182 Alvilde: 2- 5 og 5 – og så rykker I (de laver ny formation) Det er sku også en trekant. (Christian slipper) Nej Christian 183 hvad laver du. Du skal holde den. Hvis du tæller den der (peget på den ene side) så tæller jeg den der. Er det 5 184 og 6. Vera var der ikke 6 på din?

185 Vera: jeg ved det ikke.

186 Alvilde: 6-1 og 5 ja 6-1 og 5

187 Christian: der var 7

340

188 Beneikte: nej _ tæller efter – nej der er kun 6.

189 Christian og Vera – nåhh ja.

190 Alvilde: okay så en ny trekant – er vi i gang med en ny en?

191 Vera: det ved jeg ikke.

192 Alvilde: den er 2 og

193 Molly: hvad skulle jeg skrive ved den ene side?

194 Alvilde: øhh 2-6 og 5 (de prøver at lave en ny formation) Vi skal prøve at lave en på 7 men jeg tror ikke det går 195 op? Vera prøv at gå til den næste knude – ja. (de lykkes ikke rigtig og de laver en helt ny) Prøve lige og se om vi 196 har 4-3 og 5

197 Christian: ja det tror jeg vi har

198 Alvilde: ja okay – vi skal finde nogle flere – har vi egentlig en hvor den er 1? Prøv lige at stil dig. Og hold i en. 199 (Christian er lidt forvirret) nej du skal holde i to knuder. Og Vera skal give slip. Christian prøver forskellige 200 formationer) Nej Christian du skal videre. Sådan ja.

201 Christian: er det en trekant?

202 Alvilde: jeg tror ikke rigtig det går op fordi…

203 Alam: det er en trekant

204 Daivd: det er en trekant.

205 Alvilde: ja men det er fordi den skal jo være stram og den er overhovedet ikke stram. Så giv Vera det ene hjørne 206 (Vera tager over) Se det går ikke rigtig helt op. (turen smutter ud fra hænderne). Nu prøver I bare – så holder jeg 207 her og tæller – så gør I noget – det skal nok blive godt. (de stiller sig igen) sådan det er 3-4-5

208 Vera: så rykker jeg.

209 Alvilde: så er det 2 og 5 og 5. Har vi ikke haft den?

210 Molly: øhh jo

211 Alvilde: så rykker Christian (Vera rykker igen) sådan det er 1 og 6 og 5, den har vi haft.

212 Vera: så rykker Christian derover ad. (Christian rykker)

213 Alvilde: men det går jo ikke op Vera (de har en 4 1 og 7 trekant ) (Vera rykker nu runndt igen) den har vi haft

214 Christian: så er det jo det samme igen.

215 Alvilde: ja det er jo det – vi har jo lavet dem alle sammen.

216 Christian: så har vi lavet dem alle sammen.

217 Alvilde: nu har vi gået hele tovet rundt. Så skal vi ind og tegne det over fra Molly. Så I skal tage jeres ark og gå 218 indenfor.

219 (video 3 starter)

220 Lærer: så skal I ikke bruge rebet – den tager jeg lige og folder sammen. (gruppen går indenfor)

221 Molly: det er megagrimt lavet bare så I lige ved det.

222 Alvilde: det er lige meget. 341

223 (video 4 starter)

224 (gruppen har nu hentet en computer hver – der er lidt uro, Christian peger på Alvildes computer – for at hun kan 225 hente filen)

226 Christian: ok så tryk åben og så tryk bare kryds der. (Christian hjælper også Vera – han tager bare hendes 227 computer og henter det)

228 (der bliver filmet ganske kort på en anden gruppe)

229 Alvilde går i gang med at tegne trekanterne i GeoGebra.

230 Alvilde: sådan, så har jeg lave denne. (Alvilde tegner en ny)

231 Christian: nu skal vi tjekke om alt det her er rigtig. 4-4-4 det er én okay

232 Alvilde: ja og 2-4-5 er også okay. Jeg mangler kun 6-1-5.

233 Alam: skal man reset?

234 Alvilde: du kan bare skrive det andet tal – jeg har slet ikke lavet reset.

235 Christian: og 4-5-3 – tjek

236 Alvilde: alle passer – jeg er bare lige ved at tjekke den sidste

237 Christian: 5-5-2 den burde også passe

238 Alvilde: (klapper) ja jeg har fået dem alle sammen til at passe. Så done.

239 Christian: ja nu venter du lige.

240 Alvilde: jeg venter bare lige på jer alle sammen.

241 (Alvilde kigger stadig på 5-1-6, mens Christian sætter fluetegn på endnu en trekant)

242 Alvilde: nu kan jeg ikke få dem til at passe bedre.

243

244 lærer: er du sikker på det er en trekant?

245 Alvilde: ja for der er jo trekanter jo. Er der en der er 6-4-2?

246 Elev: det må der være.

247 Alvilde: ja det passer også

248 Christian: okay den sidste det er 6-5-1

249 Alvilde: ja det giver 12.

250 (Christian tegner ind)

251 Christian: Det er sku da ikke en trekant.

252 Vera: ej den kan jo rykkes

253 Christian: Alvilde kan det her godt være en trekant?

254 Alvilde: det var jo det – jeg synes jo godt det kunne være en trekant. For der er jo 3 kanter. For den kan jo godt 255 være en trekant for der er 3 kanter.

342

256 Vera: Ej den her den irriterer mig.

257 Alvilde: Christian hvordan kan vi sige at der ikke er flere trekanter?

258 Vera: fik du din til at se sådan her ud? Den har jo tre kanter.

259 Alvilde: jeg vil også sige at det er en trekant for der er jo tre kanter.

260 Christian: vi har kørt hele vejen rundt

261 Alvilde: vi har kørt hele vejen rundt og du har rykket dig en 262 tak og jeg har rykket mig en tak.

263 Vera: det er faktisk rigtig grimt – den her trekant kan jeg ikke 264 lave.

265 Christian: så prøv at se den her.

266 Vera: den er jo også grim.

267 Christian: det er sådan noget de er vant til.

268 Alvilde: (pusler igen med trekanten) sådan nu kan jeg ikke få 269 den til at passe bedre.

270 Christian: kigger på kameraet og siger – hvad skal vi nu lave? 271 Lærer: er du sikker på det der er en trekant?

272 Alvilde: Jeg vil sige at det er en trekant for der er tre kanter. Men prøv at vent. Wait a second. Det går jo ikke op. 273 Nej Christian vent det er jo ikke en trekant. Fordi de der to de går jo ikke sammen, men vi har jo fået dem til at 274 gå sammen. Det er bare fordi den passer ikke før at vi gør den helt lige. Derfor tror jeg ikke det er en trekant. De 275 to tilsammen er jo 6. Derfor er det ikke en trekant. Det passer ikke. Christian det passer ikke. Der er kun 4 276 trekanter af hvad vi kan finde.

277 Christian: men hvad med den her tegning?

278 Alvilde: men det passer jo ikke fordi den der lille mellemrum den viser at – fordi 1 + 5 det er 6 og og 6 + 6 det er 279 12, eller 5 +1 det er jo 6 – så det går ikke op. Den sidste passer ikke.

280 Christian: nåhh. Den er en dååå

281 Alvilde: Okay skal vi prøve at finde en ny en? Vi skal prøve at lave en ny en og de skal give 12 tilsammen.

282 Christian: 7… nårhh det skal give 12 tilsammen….

283 Alvilde: jahh 7 det kan ikke gå op. Tallet det må ikke være højere end 6. fordi så går det slet ikke op…. Jeg forstår 284 dog ikke hvorfor den der går op når den der ikke går op. (peger på en anden trekant med 6 -4-2)

285 Christian: hvad?

286 Alvilde: jeg forstår ikke hvorfor den der går op – den der med 6-4 og 2 når den anden ikke går op. (Hun indsætter 287 tallene igen)

288 Christian: vent 7-3-2

289 Alvilde: jo det fik vi til at gå op men det – prøv at se her. Men det her det går jo op. Prøv at se den her går jo 290 heller ikke op før vi gør den helt flad.

291 Christian: 0 og 12 – hæ hæ

343

292 Alvilde: men Christian prøv at se den her går jo heller ikke op før end at vi gør den helt lige.

293 (Christian flytter på QR-koden)

294 Alvilde: prøv at hør vi har lavet noget forkert.

295 (Alvilde rykker nu også på QR-koden)

296 Christian: hvad med hvis alle sammen er 0.

297 Alvilde: og den der går heller ikke op mere.

298 Christian: hvorfor?

299 Alvilde: fordi det gør den bare ikke – fordi de der to tal de giver jo også 6 tilsammen. Den går ikke op den er også 300 bare en lige streg.

301 Christian: hallo vi skal finde 2 andre fordi der er også den med 6-4-2 den går også ikke op.

302 Vera: det tror jeg måske at jeg forstod.

303 Molly: hvilen var det. Er der andre der har 6?

304 Alvilde: ja det var dem med 6 tallet der ikke virker. Nu starter vi forfra med 1 nåhh nej – så vi starter med 2. 305 (Bendikte prøver sig frem med først 2)

306 Christian: hør Alvilde – hvad med den her kan den gå op (viser en 2-4-6 figur på hans computer)

307 Alvilde: nej fordi hvis du viser lægger dem der to sammen så er det en lige streg. Så et tal med 6 det duer ikke.

308 Christian: alt med 6 går ikke op.

309 Vera: Christian jeg har lavet en med 6

310 Alvilde: må jeg se Vera (lidt nedgørende) (Vera vender hendes computer så de kan se) men det skal give 12 Vera.

311 Vera: det kan da også bare give 12 et par gange.

312 Christian: hvad med 4-5-3

313 Alvilde: den har vi allerede lavet.

314 Christian: har vi?

315 Alvilde: ja det er nummer 4 trekant.

316 Alvilde: vi har 3-4-5 og 2-5-5 og 4-4-4, fordi den der oppe går heller ikke op.

317 Molly: mig og Vera vi fatter ikke noget af det I har lavet.

318 Alvilde: I skal slette alle dem med 6 fordi sådan er det.

319 Vera: har vi en 5-5 og 4?

320 Molly: nej den har vi ikke.

321 (lidt uro)

322 Vera: hallo har vi den her 5-5-4

323 Alvilde: 5-5-4 – den giver ikke 12

324 Vera: nej – jeg kan vist ikke pludselig…

344

325 Christian: den giver 14.

326 Molly: 5-5 og 2

327 Alam 5-5 og 3

328 Alvilde: 5-5-3 Vera det giver 13.

329 Christian: 5 5 2 den har vi.

330 Alvilde: 5-5-2 den har vi og den er rigtig

331 Vera: ohh my god – det giver ikke mening…(strækker sig)

332 Alvilde: kan det her være en trekant fordi man kan jo også bare lægge den ned.

333 Vera: 8-3 og 1

334 Alvilde: (prøver et nyt forsøg) sådan nu går det op.

335 Christian (har slettet noget af siden) ej jeg har slettet det.

336 Alvilde: ej det gjorde du bare ikke. Det var ligesom dengang hvor vi lavede en eller anden test og så bare gik ind 337 bagefter og rettede og lavede sjovt. Ej jeg kan ikke få det til at passe.

338

339 (Kameraet går over til en anden gruppe som har lavet 5 trekanter)

340 (Læreren bryder ind) 341 Lærer: prøv at find ud af det sidste spørgsmål. Hvad står der. Hvordan kan I sige at der ikke er andre trekanter 342 end de trekanter I har tegnet? Hvordan kan I vide det? Tag en snak om det i de der grupper. Hvorfor er der ikke 343 bare uendelig mange trekanter.

344

345 Elev: så skulle man have et større reb hvis man skulle lave flere trekanter.

346

347 Video 5:

348 Alvilde: 6 eller over kan ikke være med. Vi kan kun finde 3. En ligebenet og et spidst nej jeg kan ikke huske hvad 349 de hedder. Hey hvad hedder den der spids trekant ting?

350 Christian: hvad den der under?

351 Alvilde: en spids trekant. Sådan en har vi fundet, ej det var ikke helt sådan en. Vi kan kun finde en ligebenet og 352 ja sådan en her. 353 lærer: hvordan kan I vide at der ikke er flere?

354 Alvilde: det ved vi ikke, Molly prøver sig lige nu frem. Enten så har vi lavet dem -fordi 6 den duer ikke eller så….

355 Lærer: hvordan kunne I være systematiske?

356 Alvilde: ja hvis man startede med 2 og så kunne der være en med 4, nej det kan kun være 5 og 5 – den har vi. 357 Okay så går vi videre og tager 3 i den første kasse og så 4 og 5 og den har vi også. 3 2 og så skal der stå 7 det kan 358 heller ikke lade sig gøre. 3 og 5 det er så 4 til sidst det har vi lavet. Så går vi videre til 4 og 4 og 6 og 2, men det 359 kan heller ikke lade sig gøre. Så er der 4-5 og 3 den har vi. Og 4 og 2 den går heller ikke. Og 4 og 4 og 4 den har

345

360 vi. Ja vi kan ikke finde flere. 5 og 2 og 2 – ja og så 5 4 3 den har vi. Vi kan ikke finde flere. Og hvis vi kommer til 6 361 så går det ikke op.

362

363 Kameraet går hen til en ny gruppe: 364 Lærer: dvs. I påstår der er 5 trekanter. Men hvad nu hvis jeg påstår at det er der ikke der er 6. Kan I så vise mig at 365 det er der ikke. Det vil sige at hvis jeg nu sagde til jer at I mangler en. Jeg siger ikke I gør det, men hvis nu jeg siger 366 til jer at I mangler en. Hvordan kan I så sige til mig at det gør vi ikke fordi…dum dum dum det kan ikke lade sig 367 gøre fordi…. Prøv lige at snakke om det.

368

369 Elev: det skal jo give 12 – og så skal de samtidig også kunne gå op.

370 Elev: 3 tal der går op i 12.

371 Elev: så skal vi bare finde ud af hvor mange der gør det. Den der gør (peger på en trekant) er der flere? 372 Lærere: 5 klasse prøv lige at høre her – næste gang det bliver på onsdag – da skal jeg prøve at høre jeres 373 argumentation for hvorfor der kun findes det antal trekanter som I har stående på jeres papir. Det vil sige I skal 374 kunne sige til mig at der findes kun 5 fx fordi når rebet er 12 meter så du du – det skal vi høre på onsdag. Vi skal 375 også lige høre om de der trekanter I har fundet, da er nogle jeg ikke er helt enige i. Det snakker vi om på onsdag. 376 I skal gemme papirerne. 377

346

h. Transcription-guide to the KiDM project (in Danish)

(this appendix was given to the students who helped doing the transcriptions, they also received an example of a lesson which already was transcripted)

• I Indledningen af dokumentet beskrives hvad der ses i klassen, evt. hvordan de sidder, gruppeborde eller lignende. (Der skal være tidskodning ca. hvert 5. minut. o Det markeres med minuttal, eksempelvis ved minut 40 og 12 sekunder: [00:40:12] • Der sættes nummerering på linjerne. (Det kan gøres automatisk i Word) • I undervisningstransskriptionen sættes læreren og lærerens ord i kursiv og i interviewene sættes intervieweren og interviewerens ord i kursiv. • Alle uklarheder og pauser sættes i parentes () og ved tænkepause præciseres det, hvor lang tids pause der er. Eks: (5 sek. Tænkepause) • Hvis noget er uklart, beskrives dette (uklart) Eksempelvis hvis eleverne mumler, taler i munden på hinanden, støj eller lign. • Hvis der siges noget, som eksempelvis meget tydeligt er ironisk - tydeliggøres dette ved en parentes efterfølgende (ironisk). Eller hvis der siges noget meget højt, skal dette markeres med kapitæler. • Skift linje når en ny person begynder at tale.

Den kvalitative undersøgelse i KiDM handler om ræsonnementer og materialitet, dvs. det eleverne gør, siger, manipulerer med og de artefakter eller materialer de arbejder med. Det betyder at der i transskriberingen skal være ekstra fokus på nogle bestemte ting undervejs: • De udvalgte elevers kropssprog skal beskrives tydeligt. Udfører eleven nogle fagter med armene, eller peger på bestemte ting skrives dette i klammer undervejs []. Hvis det ikke kan beskrives tydeligt nok med ord, så tilføj da et screenshot fra videoen. • Alle de materialer der anvendes skal beskrives - eksempelvis hvis der anvendes centicubes, taltavler osv. Det er vigtigt, at det tydeliggøres, hvad eleverne gør med disse materialer. Peger på dem, tegner, deler dem op osv. Dette skrives også i klammer []. Eksempelvis [Eleverne deler centicubesene op i fem grupper med tre centicubes i hver]. Hvis det ikke kan beskrives tydeligt med ord, så tilføj da et screenshot fra videoen. • Hvis det er svært at forstå indholdet i samtalerne så anvend gerne billeder fra videoen, så det giver mening for den kommende læser. Eksempelvis hvis eleverne tegner eller skriver noget. Hellere et billede for meget end for lidt! • Generelt handler det om at det skal være tydeligt for os efterfølgende, at forstå hvad der sker uden at skulle kigge på videoen igen.

347

Vi anbefaler at bruge ”Express scribe” som kan downloades på https://goo.gl/cE17qE og vejledningen på https://goo.gl/rPJjrD, men det er også helt ok at anvende word.

Hvis der undervejs opstår nogle spørgsmål så ring eller skriv til enten:

Jonas Dreyøe tlf.: xxxxxxx eller [email protected]

Dorte Moeskær Larsen tlf.: xxxxxxx eller [email protected]

348

i. Coding guide (in Danish)

The open-ended test items need to be coded manually. This was done by 3 different teacher students by using this guide.

B11:

Skriv det regnestykke som giver det rigtige svar til spørgsmålet herunder: Prisen på 1 kg. Pærer er 12 kr. hvor meget koster 2,5 kg.

Kodning Svar Eksempler

0 Ikke korrekt 10+12+8

(eller blot 15+15 resultatet uden 14+16 regnestykke) 30

1 Korrekte 18 + 12 eller 12 + 12 + 6 eller 12:2+12+12= eller 12*2,5 eller 12*2+6 svar 12*2=24+6=30 eller 2*12+12:2=30 eller 12+12=24+6= eller 2*2+halvdelen af tolv

12 gange 2,5

24 + 6

først skal man gange 12 med 2 og så skal man finde haldelen af 12 ogbså leger man det man har regnet sammen

24+6

12+12=24 så de halve af 12 er 6 24+6=30

12.2,5

2 gange 12 = 24 så plus den halve som er 12-6=6 så det er 24 plus 6 som er lige med 30

1 kg+1 kg=2 kg+ 0,5 kg altså 30 kroner

12 gange 2 plus 6

tag to kilo som bliver 24 og tage det halve af tolv når man kun skal have et halvt kilo

2*12+6=36

12+12=24 12-6=6=30

2*12+12:2

jeg vil regne sådan at jeg siger at et kg pærer er 12 kr så siger jeg så er 2 kg jo 24 kr. Også til sist regner det der fem tal det er et halv siger jeg at hvad er så halvdelen af tolv og så ligger jeg det sammen

12.kr*2 6*1

349

12*2+12/2

6*5

B51:Se på de to figurere herunder: Beskriv på hvilken måde de to figurere er ens:

Kodning Svar eksempler

0 Ikke korrekte ”samme slags”

(de har samme form, de har begge lige sider) ”Samme form”

1 Korrekte svar ”de er begge to trekante”

(begge er trekanter/begge har 3 sider/begge har ”de er begge to trekanter og har tre hjørner” samme antal sider/begge har 3 vinker/begge har 3 hjørner) ”de har begge to tre ganter”

”de er bege to trekanter figur p liger barer ned”

(hvis der er flere begrundelser skal der minimum ”trekant” være en der er korrekt - også selvom de andre ikke er korrekte) ”lige bunden”

”de har begge 3 spidser i kanterne”

”De er begge spidse”

"De er lige høje og de er begge to spidse

de har 3 sider

de har en spis

de er begge 180 grader. For en trekant skal være 180 grader for at være en trekant

Spids

Alle vinklerne giver tilsammen 180 grader i begge

B52

Se de to figurere: Beskriv på hvilken måde de to figurere er forskellig:

Kodning Svar Eksempler

350

0 Ikke korrekte svar De har ikke den samme form

(de har/har ikke den samme form) De er ikke den samme trekant

Der skrives ofte om placering - den ene er væltet De er skæve osv. - det giver ikke point. den ene ligger ned og den anden stor op Der skrives at den er ”ligesidet” (hvilket ingen af dem er) de er vendt på hver sin måde

den ene er bred den anden er lang

de har forskellig form

de er to forskellige slags trekanter

den ene er længere end den anden

Den ene er ligesidet den anden er ikke ligesidet

Den ene trekant er højere end den anden

1 Korrekte svar de er trekanter men de er ikke vent på samme måde

Generelt hvis der står mere end en begrundelse, så figur p har en ret vinkel og det har figur q ikke er det ok hvis bare en af tingene er korrekte. Figur P er stør end figur Q Typisk ligebenet omtales at den ene er ligebenet den anden er ikke eller at den ene er retvinklet den "den ene har en tre spidse vinkler og den anden har kun to" anden er stump. "De har ikke lige lange sider" hvis der blot står ”ret” og ikke vinkel ”Den ene er længere på siden nederst”

Mette, Louise og Per gik i tivoli. Mette betalte et lodt i tambolen til dem hver. Hunbetalte 2 kr. Louise betalte karusellen. Hun betalte 8. kr. for hver af dem. De spiste alle 3 en lille is. Per betalte alle is. En is kostede 4. kr. Sidst på dagen skulle de finde ud af hvor meget de skulle betale til hinanden. Prøv at regne dette ud

Kodning Svar Eksempler

0 Ikke korrekte svar 42

6 24 12

1 Næsten korrekt svar 14

Dog gives svaret ikke tydeligt og reduceret 6+12+24=42 42/3=14 de skal betale 14kr hver

Der står blot hvor meget hver skal betale (14 kr) og mette skylder de andre 12 kr i alt. Louise skylder de ikke hvor meget de skal give til hver andre 6 kr i alt. Per skylder de andre 10 kr i alt

De løsninger hvor de blot skriver hvor meget de skal ”Louise og per skal betale Mette 2 kr. hver. Louise og give til hinanden uden at det er reduceret eksempelvis Mette skal betale per 4 kr. hver. Per og Mette skal 351

når Mette har betalt 6 kr - så skylder de andre hende 2 betale Louise 8 kr. hver = Mette for 4 kr. tilbage Per kr. hver osv. for 8 kr. tilbage og Louise for 16 kr. tilbage.

Mette skylder 12 kroner Louise skylder 6 kroner Per skylder 10 kroner”

2 Korrekt svar. ”14 kroner 2+4+8 Mette skal betale 8 kr, Louise skal have 10 kr Per skal betale 2 kr Når regnestykket bliver reduceret således at det Louise skal have 2kr af per og Mette skal give hende fremgår tydeligt hvor meget der skal betales imellem 8 kr” børnene - giver det 2 . ”Per skulle give 2 kr til louise og mette skulle give 8 kr til louise.

Den ene har betalt 8 kroner for lidt. Den anden har betalt 2 kroner for lidt. Den tredje har betalt 10 kroner for meget. De to første skal betale 10 kroner til den tredje.

U2:

Antal elever i gymnastiksalen. Din skoleleder vil gerne undersøge, hvor mange elever der kan stå i skolens gymnastiksal, da der skal holdes en koncert. Hvordan kan du hjælpe ham med at undersøge dette?

Dette er et FERMI-spørgsmål, hvilket betyder at de skal kunne finde nogle strategier der er realistiske for at kunne svare på opgaven.

For at få fuld point skal elev-svarene indeholder følgende 3 områder: dele gymnastiksalen op i enheder (kvadratmeter eller blot firkanter) finde ud af hvor mange elever der kan være på en enhed/kvadratmeter gange eller gentagen addition (ikke nok blot at regne det ud!)

Hvis der kun er 2 af disse områder med så giver det kun 2 point.

Mindre end 2 områder betyder =0 point.

Husk her ikke at tænke videre og gætte på hvad de mon forestiller sig de ville have skrevet, hvis de ikke skriver det selv.

Kode Svar Eksempler

352

0 ikke tilstrækkelig eller ikke identificeret "først ville jeg sætte en madrase og lave som om de er scenen og så kan alle børn sidde og så kan nogen side i en madrase og nogen -når der ikke står noget om at tælle eller ikke på vipperne og nogen på bænken og på gulvet så er der en koncert" forstår svaret ”Tag alle elever ind i En veldefineret strategi kan ikke blive identificeret gymnsatiksalen”

Der antydes at der skal søges information Jeg vil først tage 1. klasse ind så vil jeg tage 2. klasse ind udefra… ”vi spørger pedellen”

1 urealistisk strategi Jeg vil tage hele klasen ind og tælle dem

(det handler om at tælle alt/prøve sig frem på Jeg vil først tage 1. klasse ind så vil jeg tage 2. klasse ind - og så en urealistisk måde) vil jeg tælle dem alle.

"trin 1 først vil jeg gerne mål hvor stort der er inde i gymnastiksalen trin 2 bagefter vil jeg gerne tage så mange mensker så der kan være trin 3 og til sidst vil jeg tale hvor mange mensker der kan være der"

2 Realistisk - dog med fejl og mangler. "1: Jeg vil gerne dele det op i flere Grupper 2: Jeg vil Delle det op efter 10 i hver gruppe 3: så vil jeg se hvor mange der kan At eleverne bliver delt op i små enheder, men være der" ikke tydeligt hvordan man kommer videre herfra. "1. jeg vil finde ud af hvor mange der er på skolen. 2. så vil jeg tage gennemsnits tykkelsen og måle gymnastik salen. 3. der efter (minimun 2 dele beskrevet - herunder: regner jeg det sammen"

dele gymnastiksale op i enheder "hvor stor er gymastiksallen stille alle på en rakke og så vil jeg (kvadratmeter!) sige at der fx skulle stå 25 på en rakke og så gange det med ti og så ved man hvor mange der kan være" finde ud af hvor mange elever der kan være på en enhed "1 Hvor meget fylder et barn 2 arealet på gymnastiksalen"

gange eller gentagen addition) "jeg ville først finde ud af hvor maget gulet kan holde til så ville jeg finde ud af genem snits vegten af dem der kommer og så ville jeg divider det"

"jeg finde ud af hvor stor gymnastiksalen er altså i kvadratmeter også ville jeg sige en person hver Â½ kvadratmeter"

3 God og realistisk strategi "1. Først vil jeg dele halen op i lige bider. 2. Så vil jeg måle et normalt barns fødder. 3. Så vil jeg dele bidderne op i striber. 4. Reducere problemet – løser et mindre problem så skal børn stille sig ind i en stribe. 5. Så vil jeg tælle hvor mange – laver en forholdsmæssig opskalering børn kan stå i en stribe. 6. Også vil jeg gange."

Her skal både være opdeling af arealet/rummet "Trin 1. først ville jeg finde ud af hvor mange der kan sidde på og derefter en tælling af antal børn/elever pr. hver række. Dernæst ville jeg finde ude ud af hvor mange rækker enheden samt en multiplikation eller divison der kan være. Og så ville jeg gange de to tal. Og så ville du finde af dette. svaret."

(ikke nok blot at regne det ud!)

U4 + U6

353

Skriv en matematikopgave (let og svær)

Nb: hvis der skrives ”længe af figuren” går vi ud fra at det er omkreds!

Nb: Hvis facit står der betyder det ikke noget for point-givningen - også selvom det er forkert!

Kode Svar eksempler

0 Ikke forståelig/ikke identificerbar Jeg forstod det ikke!

Når det ikke er en matematisk opgave som kan Hvor er den 12 cm? besvares med et matematisk svar. "hvilken farve har den" Hvis der ikke står noget regnetegn mellem tallene ”10”

”20”

Peg på siden….

"hvor stort er det største sted"

”Hvad er figuren tilsammen”

"hvad giver det i alt"

2 4 10

”hvor lang er arealet?”

1 Hvis regnestykke er uden kobling til 100*10 figuren/billedet: hvad er 11+2" Når det er ren tal regnestykker - og tallene ikke forekommer på billedet! (13, 199 osv) 10+9

Hvis der fremgår bare et tal i regnestykket som 4*2*10-4-2-12*4*3 ikke er med på figuren fx 6 4*6 Eller hvis regnestykket er helt sort hvis det er koblet til billedet.

Hvis tallene anvendes men i en anden kontekst

2 6cm+3cm+6cm+3cm

Regnestykket er koblet direkte til 2+4+10 figuren/billedet, men kræver ikke komplekse udregninger "12+10+4+10+12"

Udregningerne er finde en side på figuren "hvor stor er arialet på den blå figur" (minus den blå figur - hvor siden skal udregnes) plus alle tal

Aflæse og tælle sider hvor mange vinkler er der?

hvor mange sidder er der er i alt ?"

Hvis der kommer enheder på tallene i form af "hvor mange cm er i venstre side" (også højreside) cm eller kvadratcentimeter er det koblet til 354

figurerne eller at tallene kommer fra billedet ”Hvor lang er den mindste side på dukkehuset?" 12+10 +4) "man skal plusse alle centimeterne."

"hvor lang er figuren" Direkte aflæsnings opgaveer (hvor mange vinkler/sider osv) "Hvad er summen af alle talen"

Når der ikke skal regnes noget ud - blot skrives ”hvor lang er toppen af figuren?” et tal fra billedet

3 Regnestykke koblet til figuren/billedet og ”Du skal plusse den blå side den øverste side og den yderste side kræver komplekse udregninger - dvs. der skal til højre” foretages mere end én simpel udregning/aflæsning for at kunne få 3 point. "Hvor mange sider er 10 cm ?" Der skal minimum være 2 regnestykker indeholdt i opgaven. ”hvad er omkredsene af dukkehuset?”

Her findes også de ”gode” matematik opgaver ”Hvad er længde på hele dukkehuset?" som eleven godt nok ikke kan svare på (dog "hvor stor er arealet på figuren? relevant til 4-5 klasse niveu) ”hvad er omkredsen”

"Her ser du et dukke hus du skal finde ud af hvad omkredsen er

Du siger 12+16+10+4+10+2+10

Hvad er længden på det blå areal

Hvor er midten?

Hvor lang er det længste sted på figuren?

"Hvad er højden af figuren?"

"hvor lang er hele figuren"

"hvor stor er toppen"

"hvor høj er toppen"

"Hvor lang er siden til venstre gange med den der er oppe over"

"hvor lang er den længeste linje?"

"Hvor mange cm er der på siden af spidsen?"

"hvor mange af siderne er 10 cm"

"Hvor lang er venstre side at skorstenen?"

”Hvilke figurere er der og hvad hedder de?”

”Hvor mange firkanter kan man lave i figuren?”

U09

Sammenlign de to tabeller for temperaturen i december og januar

355

NB: Varmere er lig ”bedst” (vejret er bedre i januar = vejret er varmere i janaur)

Kode Svar eksempler

0 Ikke korrekt "det er da godt for så kan man tænke sig om at du kan få frisk luft"

"den er forskellige"

Noget der ikke giver mening eller irrelevant "de andre dage er lidt mere varmer"

"mandagen i december sammenligner med lørdagen i januar"

Blot aflæsning af tabellen uden ”21 og 28” sammenlignings tilgang. ”det er næsten det samme” /”det er næsten samme temperatur”

1 En sammenligning uden matematiske "temperaturen har virkelig rykket sig op og ned den er varm nogle deskriptorere måske en sproglig… dage i december og kold andre dage i januar."

"Mandag er der 6 grader forskel. Tirsdag er der 1 grade forskel. Onsdag er det det samme. Torsdag er der 3 grader forskel. -sammenlinger fx dag for dag "at tempertuepå en på en tirsdag er den nul grader derfor er det - enkelte dage fremstilles koldest"

-udsagn om at det er varmest i januar - uden ”Der er ikke frost i januar” begrundelser "der er en lille forskel på dem og det er varmest i januar"

"mandag.6 tirsdag.1 onsdag.0 torsdag.3 fredag.1 lørdag.4 søndag.2 ---- 16"

2 Begyndende deskriptoere - dog ikke helt "Januar er 7 grader varmer end December." korrekt "det hart været varmer i januar det højeste temperatur er 10" Største og mindsteværdi "at det i gennemsnitet er blevet varmest i januar" Gennemsnit - begyndende - evt- blot summen! ”december har den laveste temperatur og januar den højeste temeratur”

3 Deskriptorere tages i brug korrekt I december er gennemsnits temerature 3 grader i januar er gennemsnitstemperaturen 4 grader

U12

Anders vil gerne undersøge hvor mange vanddråber er der i en hel spand – hvordan kan han gøre det?

Dette er et FERMI-spørgsmål, hvilket betyder at de skal kunne finde nogle strategier der er realistiske for at kunne svare på opgaven.

356

For at få fuld point skal elev-svarene indeholder følgende 3 områder:

1. dele vandet i spanden op i mindre enheder (dl-mål, kopper, sprøjter osv.) - antal dl i spand!

2. finde ud af hvor mange dråber der kan være i en enhed (antal dråber i en dl)

3. gange eller gentagen addition (ikke nok blot at regne det ud!)

Hvis der kun er 2 af disse områder med så giver det kun 2 point.

Mindre end 2 områder betyder =0 point.

I denne kan man også tænke vægt på følgende måde:

1. veje hvor meget spanden vejer

2. finde ud af hvor meget en dråber vejer

3. division (ikke nok blot at regne det ud!)

Kode Svar Eksempler

0 ikke tilstrækkelig eller ikke identificeret strategi Undersøge hvor meget der er i spanden.

Der antydes at der skal søges information udefra… ”Jeg gætter på at der er 1000 dråber”

Gæt - uden udregninger

1 urealistisk strategi (men dog en strategi der kan lade sig ”Tælle alle dråberne” gøre hvis vi havde uendelig tid) "hæl det over i noget og så hel det tilbage og se hvor – at tælle alt/prøve sig frem på en urealistisk måde mange dråber der er" (Dermed tælle alle)

De anvender areal/længer i stedet for rumfang

2 Realistisk - dog uden tydelig proces "Først kan jeg for at gøre det nemt tage et lille bitte glas bruge en sprøjte putte vand i sprøjten og helde små Det antydes at mængden skal deles op i mindre vanddråber ud i det lille bitte glas (jeg tæller hvor mange enhender og gange op… men dog med fejl. banddråber jeg putter i nu siger vi at der er 20 dråber i glasset) indtil glasset er fyldt op til kanten. Så helder jeg Dråberne skal deles op, men det beskrives ikke klart vand .” hvad man så skal gøre efterfølgende - evt. blot at gange eller ”plusse” men ikke tydelig med hvad! "først vil jeg tele hvor meget det vejer dernæst vil jeg gerne veje en dråbe så vil jeg rejene" Generelt skal der her fremgå minimum 2 korrekte trin i processen for at den kan give 2 point. (opdele, opmåle, "Først ville jeg regne ud hvor meget en spand kan udregne) indeholde. Bagefter hvor stor en regndråbe er. Så ville jeg regne det ud."

357

3 God strategi finde en lille skål og så tælle hvor mange dråber der kan være i den. Efter det vil jeg se hvor mange af de små Reducere problemet – løser et mindre problem – laver skåle der kan være i spanden. Så vil jeg gange de to en forholdsmæssig opskalering tal."

Alle 3 områder er med: "Man skal først regne ud hvor mange dråber der skal til for at få en dl det kan du bare tele. så kan du måle hvor 1. dele vandet i spanden op i mindre lang spanden er og så kan du sige spanden er 3 dl og der enheder (dl-mål, kopper, sprøjter osv.) - antal dl i skal 200 dråber for 1 dl så der kan være 600 dråber" spand!

2. finde ud af hvor mange dråber der kan være i en enhed (antal dråber i en dl)

3. gange eller gentagen addition (ikke nok blot at regne det ud!)

Eller evt. alle 3 områder omkring vægt er med:

Veje dråber, veje vand, division

U 14

Tyggegummi: Du skal købe 6 pakker tyggegummi. På hvor mange måde kan du købe 6 pakker tyggegummi? Skriv de forskellige måder herunder

Kode Svar Eksempler

0 Ikke forståeligt "jeg getter at de få lige mange"

fejl "4 måder"

"tag den der ik koster meget, og køb flere."

1 En ufuldstændig løsningen "jeg kan sige 3kr plus 3kr =6"

De skriver ikke alle 3 svar, men blot en af dem "4 af dem er koster 6kr eller 2 af dem er koster 8kr eller 1 der koster 9 og 1 der koster 6" De skriver blot 3

Der kan også være en fejl i de 3 svar

Hvis de har mere end 3 måder (også selvom de har 3 rigtige svar) - for så har de jo forkerte måder med!

2 Korrekt svar "jeg køber to 8kr jeg kan også købet tre 6kr jeg kan også købet en 9kr og en 6kr" Alle svarene er noteret

U15

358

Legetøjsbil: Gert siger "I Norge er bilen billigere ”Har Gert ret? – Begrund dit svar”

Kode Svar Eksempler

0 Ikke korrekt ”han har ikke ret”

Fejl udregninger og dermed et forkert svar. ”1 doller er 7 kr”

Volapyk ”fordi USA dollars er syv kroner værd i norge”

Forkert svar, men korrekt udregning ”man skal bare gange med syv”

”nej fordi i Norge betaler man ikke skat SÃ… ALLT ER DYRET”

”nej”

1 Utilstrækkeligt ”fordi at 7 kroner er 1 dollar så hvis 28 kroner er det biglliger” Der står noget omkring dollars og kroner men ingen tydelig begrundelse men tallene bliver brugt korrekt. ”i Norge er der billigere”

Der står tydeligt (Sprogligt) at Gert har ret, men der ”ja den i Norge er 15kr billiger end i USA det er for de at er ingen begrundelse eller der er regnefejl i USA er lidt dyer en Norge” begrundelsen. ”hen har ret fordi at det 28 kr er mindre end 5 dollar”

Gert har ret

Korrekt ”fordi at 5 dollar er cirka 35 kr vis det var 4 dollar var det cirka det samme” 2 Dollars bliver omregnet til kroner eller modsat. ”USA dollar 5x7 =35 Norge 28kr 4x7 =28” Den indeholder en udregning.

U16

Du har fået denne kasse med elastikker og vil gerne undersøge hvor mange elastikker der i alt, men du gider helt sikkert ikke tælle dem alle sammen. Hvordan finder du ud af hvor mange der er i alt?

Dette er et FERMI-spørgsmål, hvilket betyder at de skal kunne finde nogle strategier der er realistiske for at kunne svare på opgaven.

For at få fuld point skal elev-svarene indeholder følgende 3 områder: tælle hvor mange eleastikker der kan være i en lille enhed /kasse

359

tælle hvor mange kasser der er (15) gange eller gentagen addition (ikke nok blot at regne det ud!)

Hvis der kun er 2 af disse områder med så giver det kun 2 point.

Mindre end 2 områder betyder =0 point.

I denne kan man også på samme måde tænke vægt på følgende måde:

1. veje hvor meget en elastik vejer

2. veje hele kassen

3. division (ikke nok blot at regne det ud!)

Kode Svar Eksempler

0 ikke tilstrækkelig eller ikke identificeret strategi Jeg vil læse på pakken

(når der ikke står noget om at tælle eller volapyk) Jeg vil dele dem op i grupper

Rene gæt

En veldefineret strategi kan ikke blive identificeret

Der antydes at der skal søges information udefra…

1 urealistisk strategi (men dog en strategi) Jeg vil tælle alle elastikkerne

Det handler om at tælle alt/prøve sig frem på en Jeg vil tælle 10 20 30 osv. urealistisk måde Jeg vil dele dem op i 10’er bunker og så bruge 10 tabellen

2 Realistisk strategi men dog med mangler Først vil jeg telle hvor mange der er i en bøtte og så vil jeg gange det Elastikkerne skal deles op, men det beskrives ikke klart hvad man så skal gøre efterfølgende - evt. blot at Tæller et lille rum, og så kan man finde resten, fordi der gange eller ”plusse” men ikke tydelig med hvad eller nok er lige mange i de andre. modsat at der står at bunkerne skal tælles og hvor mange bunker der er skal tælles, men det fremgår ikke "Først vil jeg tælle en af rummene Så vil jeg gange alle at de skal ganges osv. rummene"

(der er kun 1 mangel jf Ferm-spørgsmålene nedrest) "1. jeg vil telle den ene box og så gange 2. og så vil jeg gange det med de andre box 3. og så vil jeg finde =."

Jeg vil tælle hvor meget der er i et rum derefter gange det med 13

360

3 God strategi Jeg vil tælle hvor meget der er i et rum derefter gange det med 15 Reducere problemet – løser et mindre problem – laver en forholdsmæssig opskalering. jeg tæller hvor mange elastikker der er i en af rummene og så tælle hvor mange rum der er på begge led og så gange det helle

(alle 3 punkter er med jf. Fermi-spørgsmålene) "Først vil jeg tælle et felt Så ville jeg gange det med hvor mange felter der er Og så var jeg færdig Eller jeg ville se på pakken"

hvad er den inderste omkreds af figur a /figur b

U17:

En Butiksindehaver ville gerne vide, hvilke snacks han skal sælge i sin forretning. Han foretog en undersøgelse og lavede en liste over de snacks, han solgte. Dette gjorde han to uger i træk. Hvilke to snacks ville du sælge

Kode Svar

0 Ikke korrekt "bøf. fordi man putter det i bøger og fordi bøger smager godt" Ikke tilstrækkelig begrundelse "jeg vil gerne sælge dem m. ost/bacon og den almindelige." Her kan der vælges kun en korrekt snack. "jeg vil gerne sælge dem m. ost/bacon og den almindelige."

"ost/bacon røget bacon skaldyrsmag"

"Jeg vil sælge bøf, dem kan jeg godt lidt. røget bacon, det kan jeg godt lide"

1 2 Korrekte uden begrundelser Skaldyr og røget bacon

Skaldyr og røget bacon fordi det smager godt!

Skaldyr fordi det er sundt og røget bacon fordi det er usundt

2 Her kommer de besvarelser der er korrekte, men Skaldyr og røget bacon fordi det var dem der solgte godt som har en begrundelser.

U 19

Morten mener, at Stjerne købmanden har den billigste tube lim. Har Morten ret? Begrund dit svar.

361

Kode Svar Eksempel

0 Ikke korrekt begrundelse "han har ret fordi at det er en meget billig lim så folk gerne vil købe en." Kun svar - ingen begrundelse "fordi stjerne købmandens er billgist" Volapyk! "Moden har radt fodi at den koster minder" Hvis de kun har regnet de 2 ud uden tydeligt at sige at den anden er dyrere….

1 Korrekt begrundelse "6kr:3=2kr pr tube lim 9kr:4=2,25kr pr tube lim"

Udrenget som 2 og 2,25 kr "morten har ret fordi der er 4 tube lim for 9 kroner hos lillykøbmanden og 3 tuber lim for 6 kroner ved Sterne Den ene regnet ud - og at den anden er købmanden man kan så regne ud at det ville koste 2 kr per dyrere/billiger! (uden udregninger på den anden) lim ved stjerne købmanden

"morten har ret fordi at stjerne købmanden har tilbud 2 for hver tube."

"hos stjerne købmanden koster en tube lim 2 kr. hos lily købmanden koster den sidste tube lim 3kr."

"han har ret fordi han sparer 1 kr"

"s. købmanden er 1kr. billigere"

"hvis du tagger to af dem der koster 9 er det 18 og hvis du køber 3 af de andre gir det også 18 men du for fler tuber"

U21 + U23

billedrammer - lave en let/svær opgave

Kode Svar Eksempler

0 Ikke forståelig/ikke identificerbar "hvor mange centimeter er b laver ind a nor den ligger ned"

Ingen spørgsmål/ evt. blot svaret på et spørgsmål "hvad er a+B"

volapyk "det er en firkant og det er 6 cm på den ene side og på den anden side er den 10 cm" herunder også hvis det er uklart hvad det er der skal findes eksempelvis - Hvad er arealet? Eller hvad er "du ganger bredden 2 gange som så ville give 18 og så omkredsen? længden 2 gange som så ville give 26 og så plusser du bare 18 og 26 det giver så 44" Hvad er størst? "vilken trekant er størst a er 2 på den ene og 5 på den Hvis det har noget med pigerne (på billedet) anden side. b er 3 på den ene og 7 på den anden."

"hvilken en er mindst a) b b) a C) de er lige store"

"du har nu 1 ramme omkredsen er 12 og 10"

"tand dianater i en figurer"

"ramme a og ramme b plus dem sammen ___ ?"

362

”Hvad er arealet”

”hvad er der på billedet?”

1 Regnestykke uden kobling til figuren/billedet: ”47+332

(det gør ikke noget at de er udregnet forkert- da det ”Du har en masse 7-taller hvordan kan du få 50?” drejer sig om opgaven) Der er 9 snore og 12 elastikker…” (tallene må ikke være 9,12,13,10,1 dog hvis der er decideret skevet en anden kontekst til tallene fra 9+9+9+9+9+9+9+9+9+9+9+9+9+9+9+9+9+9 figuren)

Hvis det bare er et tal (der godt nok er fra billedet) der bliver brugt til ligegyldigt udregning fx 9+9+9+9+..

2 Regnestykke koblet direkte til figuren/billedet, men "10+12" kræver ikke komplekse udregninger. ”26+18” (omkreds) Kan afgøres ved blot at se på figurerne, eller blot en udregning (plus, minus, gange, dividere eller "1*13=13 1*9=9 9*13=117" lignende) "1cm + 1cm" Vær opmærksom på om vi skal kode således at hvis tallene fremgår på billedet/figuren så er det koblet til Højden i figuren figuren! Længden i figuren Hvis det er tydeligt at se at tallene er lagt sammen fra "hvor mande mm er b vandrat" billedet fx 26, 18, 20, 24 "hvor høj er billedramme b"

"hvor mange mm er der fra figr a yderste streg til den enderste"

"er de to figurer paralle"

"hvilken slags figur er figur A?"

"Hvad er det for en figur"

Hvor mange diagonaler er der?

"hvad giver de to længste side til sammen"

"hvor høj er det indrest af figur b"

"hvor mange fire kanter er der"

"hvor bred er a og b tilsammen?"

"hvad er arealet på figur A." (flyttet hertil da det kun er et regnestykke)

3 Regnestykke koblet til figuren/billedet som kræver Er figur A større end figur B?/ Er billederne lige store? komplekse udregninger - dvs. kan ikke aflæses direkte eller findes ved blot et simpelt regnestykke "hvor langt er der fra hjørne til hjørne" (plus, minus, division eller multiplikation) der skal minimum 2 udregninger til "hvordan finder du midten af figurerne"

363

Herunder også sammenligninger mellem figurerne - ”har figur a og b den samme omkreds” herunder udregninger af areal/omkreds for begge figurere og sammenligninger af disse. ”er arealet af figur B større end arealet af figur A?”

4 Genial - original "Der skal 5 af ramme A op at hænge. De skal hænge med 10 cm i mellem. Der skal være 5 cm ud til væggens ende. Hvor meget plads skal man bruge for at hænge billederne op ?"

U22 + U24

Svar på opgaverne til de opstillede opgaver

SKAL IKKE KODES

U25

En vandhane hjemme hos vilfed … hvordan vil du undersøge det?

Dette er et FERMI-spørgsmål, hvilket betyder at de skal kunne finde nogle strategier der er realistiske for at kunne svare på opgaven.

For at få fuld point skal elev-svarene indeholder følgende 3 områder: tælle antal dråber pr. minut/time/enhed ELLER antal minutter for 1 dl/kop/enhed. finde ud af hvor mange minutter/enhed der er på 1 uge gange eller gentagen addition (ikke nok blot at regne det ud!) eller division

Hvis der kun er 2 af disse områder med så giver det kun 2 point.

Mindre end 2 områder betyder =0 point.

Kode Svar Eksempler

0 ikke tilstrækkelig eller ikke identificeret "du kan åbne vandhanden og slukke den åbne den og slukke den" når der ikke står noget om at tælle dråberne/veje "set en kop under." En veldefineret strategi kan ikke blive identificeret

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Der antydes at der skal søges information "1 jeg vil vide hvor mange litter der kunne vere i 2 jeg ville udefra… regne på lommeregner 3 jeg ville se om hvor mange dl og milidl" Hvis det tager mere end 1 dag at undersøge! (fx en hel dag!) "få en ting til at tælle hver dråbe der falder ned fra vandhanen"

"undersørge hvor mange dråber det falder ned også vil jeg tælle hvor mange der er falde ned"

Bruge tabellen og tælle 10 af gangen og sætte en streg

1 urealistisk strategi (men dog muligt - fx tælle en "Først vil jeg se hvor meget den har dryppet på en dag. Så vil dag og gange med 7) gange det med syv."

– at tælle alt/prøve sig frem på en urealistisk måde

Det må her godt tage en dag at undersøge…

2 Realistisk "han kunne veje en dråbe vand. Dernæst helle den tilbage i spanden, og så veje spanden. Så ser han hvor mange gange Del op i dele og gange op… men dog med fejl. vandåbens vægt kan gå op i spandens vægt."

Der antydes at der skal deles op i små enheder - "Først vil jeg se hvor mange dråber der kommer på et min Der ofte er den dog ikke helt gennemtænkt da der efter hvil jeg se hvor mange der kommer i en time Der efter mangler en information (det kan være gange, hvil jeg regne det ud" dele op, eller måle i dl eller ligne) "fÃ¸rst vil jeg finde ud af hvor mange gange den drypper pÃ¥ Der mangler ofte at stå noget med at der også en time, hvis nu det var 10 drÃ¥ber pÃ¥ en time sÃ¥ vil jeg skal tælles hvor mange dråber der hører til i en gange 24 med 7 = 168" enhed. Spand eller lignende

Jf. Fermi-spørgsmål skal der være to områder med i besvarelsen

3 God strategi "1 finde ud af hvor meget vand der er dryppet på en time stil et krus under der hvor det drypper 2 mål hvor meget vand der er Reducere problemet – løser et mindre problem – i koppen regn ud af hvor meget vand der er faldet i løbet af 24 laver en forholdsmæssig opskalering timer brug gange 3 gang resultatet med 7 fordi der er 7 dage i en uge nu skulle Det kan typisk være ved at fylde en mindre enhed - rumfangsmæssigt på en bestemt Trin 1 find ud af hvor hurtigt den drøbber i timen. Trin 2 Find tidsenhed eller måle hvor lang tid det tager at så ud af hvor mange gange den drypper i timen. På en dag er fylde en enhed. der 24 timer. Trin 3 find så ud af hvor mange gange den drypper om ugen ved at gange det med 7. Der skal her også være en slags måleenhed (kop, deciliter eller lignende) "Trin 1. Først ville jeg finde ud af hvor meget vand der drypper på en time. Ved at sætte en spand under hvor jeg også kan hvor mange deciliter der er. Trin 2. Bagefter ville jeg så gange det antal deciliter pr time med timerne på et døgn. Og så gange det med 7"

U29

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Cirkus. Julie siger at der vil være 11 forestillinger i alt og Mette siger at der vil være 10 forestillinger i alt. Hvem har ret og begrund dit svar?

Kode Svar Eksempler

0 Ikke korrekt Mette har ret.

Ikke tydeligt Har ret

eller forkert

(hvis svaret er forkert men udregningerne er korrekte)

1 - Utilstrækkelig ”Har ret”

Uden begrundelse - fx blot skrives 11 dage Hun tænkte sig om

Hvis der skrives at Julie har ret men dog er "hun også tager den 25 med" begrundelsen ikke helt god "hun har ret"

”der er 11 dage”

"fordi det godt kan være der r en ekstra forstilling"

2 Korrekt "julie har ret fordi at der er en d. 25 dec. og der er 31 dage i december fra den 25 til den 31 er der 7 dage med den 25. og så er Med begrundelse eksempelvis at det er fordi fire dege i januar hvor de holder forstilling og 7+4 giver 11" der er 31 dage i december. "Julie har ret Der er 11 forestillinger fordi at i december er der 31 dage"

"Julie har ret og det er fordi at vis man tæller fra 25 og op ad til 4 januar så får man svaret."

"Julie har ret fordi der er 31 dage i december mÃ¥ned og ikke 30."

"25 26 27 28 29 30 31 1 2 3 4 det er julie fordi det jeg lige skrev"

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j. Calculated results based on the regression model to the teacher’s survey

Tabel A: The interventions effect on the teachers attitudes and experience

To what extent are you working towards the students can reason mathematically? (1) (2) (3) (4)

Intervention -0.047 -0.031 0.007 0.009

(0.180) (0.170) (0.159) (0.160)

Constant 0.029 -0.136 -0.156 -0.110

(0.156) (0.217) (0.180) (0.212)

Adjusted R2 0.007 0.016 0.117 0.104

Stratum indicator Nej Ja Ja Ja

Baseline Nej Nej Ja Ja

Intervention indicator Nej Nej Nej Ja

nschools 72 72 72 72

n 128 128 128 128

Note: Each column shows the estimates of the intervention from a separate OLS regression model. The teachers' perception (the dependent variable) is standardized. Cluster-corrected standard errors (at school level) are reported in parentheses. + p <.1, * p <.05, ** p <.01, *** p <.001.

Tabel B: The interventions effect on the teachers attitudes and experience

To what extent do you work towards students being able to engage in dialogue with other students about their mathematical understanding? (1) (2) (3) (4)

Intervention -0.256 -0.217 -0.187 -0.181

(0.172) (0.173) (0.169) (0.167)

Constant 0.158 -0.143 -0.162 -0.074

(0.138) (0.309) (0.302) (0.313)

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Adjusted R2 0.008 0.011 0.133 0.133

Stratum indicator Nej Ja Ja Ja

Baseline Nej Nej Ja Ja

Intervention indicator Nej Nej Nej Ja

nschools 72 72 72 72

n 128 128 128 128

Tabel C: The interventions effect on the teachers attitudes and experience

To what extent do you work towards the students themselves being able to set up mathematical problems / hypotheses? (1) (2) (3) (4)

Intervention -0.077 -0.075 -0.065 -0.059

(0.186) (0.191) (0.177) (0.169)

Constant 0.048 0.150 0.083 0.050

(0.152) (0.265) (0.249) (0.285)

Adjusted R2 0.007 0.017 0.194 0.210

Stratum indicator Nej Ja Ja Ja

Baseline Nej Nej Ja Ja

Intervention indicator Nej Nej Nej Ja

nschools 72 72 72 72

n 128 128 128 128

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Tabel D: The interventions effect on the teachers attitudes and experience

A dialogue between the students arises more easily when the teaching is inquiry-based (1) (2) (3) (4)

Intervention 0.409* 0.366+ 0.196 0.188

(0.177) (0.184) (0.185) (0.180)

Constant -0.253+ 0.166 0.036 0.006

(0.147) (0.164) (0.143) (0.180)

Adjusted R2 0.032 0.006 0.120 0.118

Stratum indicator Nej Ja Ja Ja

Baseline Nej Nej Ja Ja

Intervention indicator Nej Nej Nej Ja

nschools 72 72 72 72

n 128 128 128 128

Tabel E: The interventions effect on the teachers attitudes and experience

When you are about to start the activities, I let the students set up hypotheses or guesses/conjectures (1) (2) (3) (4)

Intervention 0.132 0.143 0.208 0.204

(0.187) (0.183) (0.170) (0.174)

Constant -0.082 0.334 0.198 0.139

369

(0.145) (0.367) (0.415) (0.452)

Adjusted R2 0.004 0.008 0.103 0.093

Stratum indicator Nej Ja Ja Ja

Baseline Nej Nej Ja Ja

Intervention indicator Nej Nej Nej Ja

nschools 72 72 72 72

n 128 128 128 128

Tabel F: The interventions effect on the teachers attitudes and experience

When you are doing the activities, it is often about the students having to argue for their process and their results

(1) (2) (3) (4)

Intervention 0.007 0.082 0.090 0.090

(0.178) (0.168) (0.166) (0.170)

Constant -0.004 -0.546 -0.582+ -0.654+

(0.142) (0.332) (0.306) (0.337)

Adjusted R2 0.008 0.034 0.129 0.115

Stratum indicator Nej Ja Ja Ja

Baseline Nej Nej Ja Ja

Intervention indicator Nej Nej Nej Ja

nschools 72 72 72 72

370

n 128 128 128 128

Tabel G: The interventions effect on the teachers attitudes and experience

When we finish an activity / course, I let students tell their arguments for their solutions

(1) (2) (3) (4)

Intervention 0.205 0.222 0.226 0.220

(0.164) (0.156) (0.154) (0.154)

Constant -0.127 -0.039 -0.065 -0.071

(0.115) (0.278) (0.267) (0.225)

Adjusted R2 0.002 0.038 0.084 0.103

Stratum indikator Nej Ja Ja Ja

Baseline Nej Nej Ja Ja

Intervention indicator Nej Nej Nej Ja

nschools 72 72 72 72

n 128 128 128 128

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k. Calculated results based on the regression model to the students’ survey

Tabel A Effect of the intervention on perception of own participation I always participate in the class discussions in mathematics.

(1) (2) (3) (4)

Intervention 0.097** 0.099** 0.091*** 0.091***

(0.031) (0.030) (0.026) (0.025)

Constant -0.055* -0.087* -0.035 -0.004

(0.021) (0.037) (0.038) (0.048)

Adjusted R2 0.002 0.002 0.185 0.185

Stratum indicator Nej Ja Ja Ja

Baseline Nej Nej Ja Ja

Intervention indicator Nej Nej Nej Ja

nschools 83 83 83 83

nstudents 4809 4809 4809 4809

Tabel B Effect of the intervention on perception of own participation

I always justify my solutions

(1) (2) (3) (4)

Intervention 0.097** 0.099** 0.097*** 0.097***

(0.031) (0.029) (0.026) (0.026)

Constant -0.054* -0.064 -0.056 -0.032

372

(0.021) (0.043) (0.049) (0.062)

Adjusted R2 0.002 0.004 0.128 0.128

Stratum indicator Nej Ja Ja Ja

Baseline Nej Nej Ja Ja

Intervention indicator Nej Nej Nej Ja

nschools 83 83 83 83

nstudents 4809 4809 4809 4809

Tabel C Effect of the intervention on perception of own participation – POOLED Were you often involved in discussing other students' solution proposals?

(1) (2) (3) (4)

Intervention 0.186*** 0.191*** 0.183*** 0.183***

(0.042) (0.037) (0.031) (0.031)

Constant -0.104** -0.061* -0.052* -0.068*

(0.035) (0.030) (0.026) (0.033)

Adjusted R2 0.008 0.013 0.092 0.092

Stratum indicator Nej Ja Ja Ja

Baseline Nej Nej Ja Ja

Intervention indicator Nej Nej Nej Ja

nschools 83 83 83 83

nstudents 4809 4809 4809 4809

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l. Co-author Statement

Co-author statements from Morten Misfeldt to paper I and paper III,

374

375

Co-author statement from Bent Lindhardt to paper II,

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377

Co-author statement from Claus Michelsen to paper V:

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