Research in didactics of mathematics

1. Research in didactics of mathematics

2. Faculty members
ass.prof. Madis Lepik
lect. Jüri Kurvits
researcher Kirsti Kislenko
lect. Tiiu Kaljas
Ph.D.students:
Indrek Kaldo
Regina Reinup

3. Problem areas to attract most interest and research attention
teachers’ , students’ and pupils’ beliefs about teaching and learning of mathematics (Lepik, Kislenko, Kaldo)
proof and proving in school mathematics (Lepik)
textbook research (Lepik)
development of mathematical knowledge (Kurvits, Reinup)
technology in mathematics education (Kurvits)

4. Teaching and learning of proof
Proof is a current issue in mathematics education and there is a renewed emphasis on proof and reasoning in many countries.
In this project we explore:
- What is the status/role of proof and reasoning in the school curricula in the countries involved in the study?
- How do secondary school teachers relate to proof and the teaching and learning of proof in these countries?
-How is proof dealt with in mathematics textbooks?
-How to develop proving skills?

5. Publications on proof
Hemmi, K.; Lepik, M.; Viholainen, A. (2013). Analysing proof-related competences in Estonian, Finnish and Swedish mathematics curricula—towards a framework of developmental proof. Journal of Curriculum Studies, 45(3), 354 - 378.
Lepik, M. (2012). The role of proof in Estonian curricula of lower and upper secondary mathematics . Teacher Education, 16(1), 56 - 61.
Hemmi, K.; Lepik, M.; Viholainen, A.; Raman, M. (2012). Proof and proving in Estonian, Finnish and Swedish upper secondary school curricula . In: Proceedings of Norma 11, The Sixth Nordic Conference on Mathematics Education : Norma 11, The Sixth Nordic Conference on Mathematics Education in Reykjavík, May 11-14, 2011 . (Toim.) G. H. Gunnarsdóttir, F. Hreinsdóttir, G. Pálsdóttir, M. Hannula, M. Hannula-Sormunen, E. Jablonka, U. Reykjavík, Iceland: University of Iceland Press , 2012, 309 - 318.
Hemmi, K.; Lepik, M.; Viholainen, A. 2011.Proof and proof related items in estonian, Finnish and Swedish compulsory school mathematics curricula. In: H. Silferberg, J. Joutsenlahti (Eds.). Integrated research into mathematics and science education in the 2010s. Tampere University Press, 132 – 150
Lepik, M. 2011. Tõestamisest koolimatemaatikas. Koolimatemaatika XXXVIII. Tartu: Tartu Ülikool, 58 – 64
Hemmi, K.; Lepik, M.; Viholainen, A. 2011. Upper secondary school teachers’ views of proof and proving- An explorative cross-cultural study. Current state of research on mathematical beliefs XVI. Tallinn: Tallinn University, 137 - 157

6. Textbook research (grant from NordForsk)
From many studies it is well documented that the textbook is one of the most influential element for pupils’ mathematical learning. In the Nordic and Baltic countries the mathematics textbook is dominating in the teaching and teachers are heavily dependent on textbooks.
The main aim of the network is to increase the Nordic and Baltic collaboration in research on mathematics textbooks. Some of aspects to be explored:
-how teachers use textbooks
-how pupils use textbooks
-how textbooks influence pupils’ learning of math
-how textbook facilitates teacher learning

7. Publications on textbook use
Bjarnadottir, K.; Christiansen, A.; Lepik, M. (2013). Arithmetic textbooks in Estonia, Iceland and Norway - similarities and differences during the ninetheenth century. Nordic Studies in Mathematics Education, 18(3), 27 - 58.
Lepik, M., Grevholm, B., Viholainen, A. (in press). Textbook use in mathematics classroom: the teachers’ view. Nordic Studies in Mathematics

8. Development of mathematical knowledge
Jüri focuses on students’ understandings of rational number different meanings and representations. The object of his longitudinal study (4 years) is transition from whole numbers to rational numbers, and misunderstandings that occur in this process. At the same time the development of students` proportional reasoning is also observed.
Publications:
Kurvits, Jüri; Kleemann, Kait (2012). Õpilaste lahendusstrateegiad proportsionaalse mõtlemise ülesannetes. Lepmann Lea, Lepmann Tiit, Kokk Katrin (Toim.). Koolimatemaatika XXXIX (23 - 36).Tartu Ülikooli Kirjastus
Kurvits, J. (2010). Operations with rational numbers in grades 5 to 7. Daugulis Peteris (Toim.). 11th International Conference Teaching Mathematics: Retrospective and Perspectives (61 - 65). Daugavpils, Latvia: Academic Press "Saule"

9. Web-based instruments in teaching/ learning of math
Jüri develops different teaching/learning strategies, materials and innovative teaching methods which help teachers to implement student-centered, collaborative based approaches to learning. He is very active in conducting different in-service courses.
Participated in Conrad Wolfram’s innovative project “Computer- Based Math”, developed innovative computer-based learning materials to teach data analyses.
 Publications:
Kurvits, J.; Kurvits, M. (2013). High School Students' Acquisition of Knowledge and Skills through Web-Based Collaboration. The International Journal for Technology in Mathematics Education, 20(3), 95 - 102.

10. Development of mathematical knowledge
Regina’s PhD project is in teaching and learning of percentages. It includes investigating students' conceptual understanding and attitudes towards this topic. The aim is to develop more understandable and emotionally gripping learning materialson this topic.
Publications
•Reinup, R. 2012. Teaching the topic of percentages in the secondary school. Proceedings of the Sixth Nordic Conference on Mathematics Education, University of Iceland Press, 707.
•Reinup, R. 2011. Teaching nimber lins, fractions, decimals and percentages as an integrated system. C. Winslow, R. Evans (Eds.). Didactics as Design Science. Copenhagen University Press, 71 – 81.
•Reinup, R. 2010. Developing of mathematics teachers’ community: five groups, five different ways. Proceedings of the Sixth Congress of the European Society for Research in Mathematics Education. Paris, Institut National de Recherche Pédagogique, 1831 - 1840
•Reinup, R. 2010. Mearuring pupils’ attitudes: an experience of a study. Proceedings of the conference MAVI-15: Ongoing research on beliefs in mathematics education. Department of Mathematics, University of Genoa, 193 – 203.
•Reinup, R. 2009. Emotional teaching methods in the elementary stage of percentage learning. In J. Maasz, W. Schloeglmann. Beliefs And Attitudes In Mathematics Education, Sense Publishers, 87 – 98.

11. Students’ view of mathematics
Indrek Kaldo explores university students’ view of mathematics across the disciplines having at least one compulsory mathematics course. More specifically, we want to answer the following questions:
a) What kind of structure can be identified to describe the construct view of mathematics?
b) What are the general tendencies in the Estonian university students’ mathematics related View of mathematics as measured through motivational orientation, Value of Mathematics, Competence beliefs, Perception of Teacher Role, and Cheating Behaviour?
c) Is there a difference between science and non-science students’ view of mathematics in Estonian universities?
Publications:
•Kaldo, I. (2011). Structure of students’ view of mathematics in an Estonian Business School. Nordic Studies in Mathematics Education, 16(1-2), 77 – 94.
•Kaldo, I., & Hannula, M. S. (2012). Structure of university students’ view of mathematics in Estonia. Nordic Studies in Mathematics Education, 17(2), 5 – 26.
•Kaldo, I., & Reiska, P. (2012). Estonian science and non-science students’ attitudes towards mathematics at the university level. Teaching Mathematics and Its Applications: International Journal of the IMA, 31(2), 95-105.
•Kaldo, I., & Hannula, M. S. (2014). Gender differences favouring females in university students’ views of mathematics in Estonia. Nordic Studies in Mathematics Education, 19(1), 3-22.
• Kaldo, I. (2014). View of mathematics – an investigation of Estonian university students. Nordic Studies in Mathematics Education, 19(2), 5-33.

12. Teachers’ beliefs
Belief research in mathematics education has focussed on how teachers view the nature of mathematics, its learning and teaching, and teaching in general (Dionne, 1984; Ernest, 1991; Liljedahl, Rösken, & Rolka, 2007).
Teachers’ beliefs concerning mathematics, its teaching and learning reflect a teacher’s priorities for the practices of mathematics classrooms and play a significant role in shaping teachers’ characteristic patterns of instructional behaviour (Thompson, 1992).
It is assumed, that what one beliefs influences what one does – beliefs act as teacher’s pedagogical predispositions. Beliefs are factors shaping teacher’s decisions, for example, about what teaching routines are apropriate, what goals should be accomplished and what should the learning look like (Schoenfeld, 1998).

14. NorBa study
It was agreed to focus on teachers’ beliefs about:
school microculture;
general pedagogical approach (conceptions of teaching/ learning in general);
effective/good teaching of mathematics;
their own classroom practice and textbook usage.
Research method: survey with Likert scale statements
Colleagues from Norway, Finland, Estonia, Latvia, Lithuania and Russia agreed to participate.
Overall sample size at the moment is approximately 1500 Nordic&Baltic teachers + 1000 Russian teachers.

15. General Pedagogical beliefs
Items of the questionnaire were subjected to Principal Component Analyses.
The number of factors extracted was determined by eigenvalues and scree diagrams. Based on these criteria it was decided to explore solutions of four, three and two factors.
The best solution was found in two-component structure.
The two-component solution explained a total of 32% of the variance.
The first factor (DF1) was labeled as
Reasoning and conceptual understanding (α=.73).
The second factor (DF2) was labeled as
Mastery of skills and facts (α=.68).

16. Good teaching means:
Mastery of skills and facts
- learning algorithms and drill of exercises
- learning facts
- quiet classrooms
Reasoning and conceptual understanding
- students own discoveries
- students work on practical and real-life problems
- students working, explaining and discussing in small
groups
- facilitating the students’ conceptual understanding

17. Two factors model
It is interesting that constructs described by factors 1 and 2 appeared as independent components and not as opposite extremes of one scale.
So in case of individual teacher they both may exist in parallel.
Teacher who emphasizes constructivist approach to teaching may value highly also practicing of routine procedures.

18. Belief profiles
According to the different degrees of agreement with ideas regarding these two factors typical belief profiles could be derived (DF1 x DF2) (9 possible profiles).
These belief profiles describe models of teachers’ conceptions of good mathematics teaching.

19. DF1= agree DF2= neutral
These teachers compromise both approaches.
Transmission of knowledge in combination with construction of knowledge. Rear use of discoveries and small group activities.
Instrumental aspect is not stressed. Neutral towards formal training of skills.
326 teachers (40%)
Estonia 42%
Latvia 37%
Finland 47%
Belief profiles: modest compromise

20. DF1= fully agree DF2= fully disagree, disagree
Teaching via discoveries, real-life problems; small group activities. Facilitating conceptual understanding.
Formal training of skills is not valued. Instrumental aspect is not stressed.
31 teachers (4%)
Estonia 1%
Latvia 6%
Finland 4%
Belief profiles: radical constructivists

21. DF1= disagree, neutral DF2= fully agree, agree
Teachers who tend to see the most important goal of mathematics instruction in formal training of skills. They value teaching through practicing of routines. Instrumental understanding is stressed.
Teaching is considered first and foremost the direct transmission of knowledge from the teacher to the pupil.
Teaching doesn’t use discoveries, nor real-life problems; nor small group activities. Transmission of knowledge.
16 teachers (2%)
Estonia 2.4%
Latvia 2.1%
Finland 2.1%
Belief profiles: radical traditionalists

22. DF1= fully agree DF2= fully agree, agree
Teachers believing into both approaches in parallel. They emphasize teaching activities aiming at developing conceptual understanding and at the same time value highly instrumental part of mathematical knowledge and stress training of routines and learning of facts and skills.
Teaching via discoveries, small group activities.
Teaching of skills, fluency through practicing of routines.
38 teachers (5%)
Estonia 5.1%
Latvia 5.1%
Finland 1.1%
Belief profiles: reconciliation of polarities

23. Contextual influences on beliefs
The implementation of teacher’s beliefs into the practice is influenced also by the context: norms and pedagogical traditions in the country, school culture, social background of the students, etc.
This makes the relationship between teachers’ beliefs and their teaching practice not linear; research often reports inconsistencies between teachers’ beliefs and their actions (Cooney, 1985; Skott, 2009).

24. Cross-cultural differences
So far, there have been few studies that compare teacher beliefs across countries (e.g., Andrews, 2007; Andrews & Hatch, 2000; Felbrich, Kaiser & Schmotz, 2012, OECD, 2009).
One commonly recognised finding is that beliefs are culturally informed and impact differentially on classroom practice (Andrews & Hatch, 2000; Felbrich, Kaiser & Schmotz, 2012).
Cross-cultural differences in teachers’ beliefs can provide important information regarding the scope of possible classroom practice and teachers’ inclination to different teaching approaches.
As such, beliefs held by mathematics teachers in different countries provide an interesting window through which to study mathematics teaching in those countries.

25. Two levels of contextual factors We suggest an overall theoretical frame for the role of culture, school micro-culture, and teacher beliefs in the formation of actual classroom practices:
CULTURE Nationality Language SCHOOL MICROCULTURE TEACHER BELIEFS General teaching beliefs Mathematics teaching beliefs TEACHING PRACTICES

26. Publications: belief studies
Lepik, M.; Kislenko, K. (2014). Estonian Mathematics Teachers' Beliefs about Teaching and Their Self-Reported Practices. A.Liimets, M. Veisson (Eds.). Teachers and Youth in Educational Reality. Frankfurt: P. Lang, 23 – 41.
Lepik, Madis; Elvisto, Tiina; Oder, Tuuli; Talts, Leida (2014). Õpetajate üldpedagoogiliste uskumuste struktuur ja tüüpprofiilid. Krull, E.; Leijen, Ä.; Lepik, M.; Mikk, J.; Talts, L.; Õun, T. (Toim.). Õpetajate professionaalne areng ja selle toetamine. Tartu: Eesti Ülikoolide Kirjastus, 248 - 273.
Pipere, A.; Lepik, M. (2013). Job satisfaction, beliefs and instructional practice: The case of Latvian and Estonian mathematics teachers. Electronic Journal of Research in Educational Psychology, 11(1), 162 - 192.
Lepik, M.; Pipere, A.; Hannula, M.S. (2013). Comparing mathematics teachers’ beliefs about good teaching: the cases of Estonia, Latvia and Finland. Nordic Studies in Mathematics Education, 17(3-4), 177 - 198.
 Hannula, M.; Pipere, A.; Lepik, M.; Kislenko, K. (2013). Mathematics teachers' beliefs and schools' micro-culture as predictors of constructivist practices in Estonia, Latvia and Finland. A.Lindmeier; A.Heinze (Toim.). Proceedings of the 37th conference of the International Group for the Psychology of Mathematics Education (433 - 440). Kiel, Germany: PME
Lepik, M.; Pipere, A.; Hannula, M. (2013). Mathematics teachers' beliefs about good teaching: A comparision between Estonia, Latvia and Finland. M. Hannula, P. Portaankorva- Koivisto, A. Laine & L. Näveri (Eds.). Current State of Research on Mathematical Beliefs. Helsinki: University of Helsinki Press, 327 – 340.
Hannula, M.S.; Lepik, M.; Pipere, A.; Tuohilampi, L. (2013). Mathematics Teachers' Beliefs in Estonia, Latvia and Finland. Proceedings of the Eighth Congress of the European Society for Research in Mathematics Education. Middle East Technical University, Ankara: ERME, 1865 – 1875.
Lepik, M.; Pipere, A. (2012). Baltic- Nordic comparative study on mathematics teachers' beliefs: Designing research instrument to describe the beliefs and practices of mathematics teachers. Acta Paedagogica Vilnensia, 27, 115 - 123.

27. Metaphors …
Metaphors enable people to understand one phenomenon by comparing it to something else
Metaphors reflect teachers’ unconscious beliefs about teaching and the teacher’s role
Metaphors are seen as a “blueprint” of professional knowledge of teachers’ thinking (Martinez, 2001).
Metaphors are the “master swich” to change teachers’ beliefs (Tobin, 1990)
Metaphor study

28. “Teacher is like ... My brief explanation of the metaphor is as follow.”

29. What?
-Didactics expert
-Pedagogigal expert
-Subject expert
Who?
-Self-referential
Where?
-Contextual
(Beijaard, Verloop and Vermunt 2000,
Löfström, Poom-Valickis and Hannula, 2011)
Categorazing the metaphors – The extended Beijaard model

30. The second mother- who teaches, helps, cares, supports... all of my students, all the time and they know that. They can always talk to me, discuss whatever problems they want and share all their joys with me as well.
Gardener - who sows seeds, weeds, waters, cuts off branches when needed. And so it goes round and round every year, whatever flood or drought there might be, the gardener should always guarantee the blossoming of his/her garden with whatever effort it takes!
Guide - whose task it is to prepare his/her students for real life, guiding the child carefully and gradually into the grown-ups’ world.
Book – offering knowledge and concrete help whenever students ask a question, they will either get an answer or are guided to find the answer themselves.

31. Categorazing the metaphors – emotional connotation (positive, neutral, negative)
A teacher is like a ray of sun that makes others happy creating a friendly and motivated atmosphere
Teacher is like a fool, everyone can call her names, she needs not to be listened to and it is better to disrupt the class