ON THE WAY TO DEVELOP OPEN APPROACH TO MATHEMATICS IN FUTURE PRIMARY SCHOOL TEACHERS
In our contribution we focus on the possibility to develop open approach to mathematics in future primary school teachers during a university course on mathematics conducted in inquiry-based manner. In the first part of the research we analyse data obtained in the beginning and in the end of the mathematics course with respect to two main aspects related to open approach to mathematics: searching for all solutions of a task, and acceptance of different forms of notation of a given solution. Data analysis revealed in the participants three different shifts towards open approach to mathematics, and showed that after the active participation in the course each of the participants improved at least in one of the monitored aspects, and that none of the participants got worse in any of the aspects. In the second part of the research we analyse problems posed by participants several months after the end of the course, again with respect to the two aspects related to open approach to mathematics. As a special diagnostic instrument in our research we use an educational tool called Concept Cartoons.
- Artigue, M., Blomhøj, M. (2013) ‘Conceptualizing inquiry-based education in mathematics’, ZDM Mathematics Education, vol. 45, no. 6, pp. 797-810. http://dx.doi.org/ 10.1007/s11858-013-0506-6
- Boaler, J. (1998) ‘Open and closed mathematics: student experiences and understanding’, Journal for Research in Mathematics Education, vol. 29, no. 1, pp. 41–62. http://dx.doi.org/ 10.2307/749717
- Bruder, R., Prescott, A. (2013) ‘Research evidence on the benefits of IBL’, ZDM Mathematics Education, vol. 45, no. 6, pp. 811-822. http://dx.doi.org/10.1007/s11858-013-0542-2
- Brunner, J.S. (1966) Toward a theory of instruction, Cambridge: Belknap Press. http://dx.doi. org/10.1177/019263656605030929
- Dabell, J., Keogh, B., Naylor, S. (2008) Concept Cartoons in Mathematics Education (CD ROM), Sandbach: Millgate House Education.
- Dewey, J. (1938) Logic: The theory of inquiry, New York: Holt.
- Dorier, J.-L. and Maaß, K. (2014) ‘Inquiry-based mathematics education’, in Lerman, S. (ed.) Encyclopedia of Mathematics Education, Dordrecht: Springer. http://dx.doi.org/10.1007/ 978-94-007-4978-8_176
- Freudenthal, H. (1973) Mathematics as an educational task, Dordrecht: Reidel Publishing Company.
- Hellmig, L. (2010) ‘Effective ‘blended’ professional development for teachers of mathematics: design and evaluation of the "UPOLA"-program’, in Durand-Guerrier, V., Soury-Lavergne, S. and Arzarello, F. (eds.) Proceedings of CERME 6, Lyon, France: Institut National de Recherche Pédagogique, pp. 1694-1703.
- Hošpesová, A., Tichá, M. (2013) ‘One possible way of training teachers for inquiry based education’, in Ubuz, B., Haser, C. and Mariotti, M. A. (eds.) Proceedings of the Eighth Congress of European Research in Mathematics Education (CERME 8), Ankara, Turkey: Middle East Technical University, pp. 3105-3114.
- Hricová, D. (2015) ‘The formation of a performance evaluation model’, in Krejčí, I., Flégl, M. and Houška, M. (eds.) Proceedings of the 12th International Conference Efficiency and Responsibility in Education 2015 (ERIE 2015), Prague, Czech Republic: Czech University of Life Sciences, pp. 181-189.
- Jiang, F., McComas, W.F. (2015) ‘The effect of inquiry teaching on student science achievement and attitudes: evidence from propensity score analysis of PISA data’, International Journal of Science Education, vol. 37, no. 3, pp. 554-576. http://dx.doi.org/10. 1080/09500693.2014.1000426
- Keogh, B., Naylor, S. (1993) ‘Learning in science: another way in’, Primary Science Review, vol. 26, pp. 22-23.
- Koman, M., Tichá, M. (1988) Basic Research in Didactics of Mathematics in Czechoslovakia. Comment to the Development of the Functional and Algorithmic Approaches to Problem Solving, Praha: MÚ ČSAV, Preprint Series, no. 34.
- Koman, M., Tichá, M. (1998) ‘On travelling together and sharing expenses’, Teaching Mathematics and Its Applications, vol. 17, no. 3, pp. 117-122. http://dx.doi.org/10.1093/ teamat/17.3.117
- Kwon, O.N., Park, J.S. and Park, J.H. (2006) ‘Cultivating divergent thinking in mathematics through asn open-ended approach’, Asia Pacific Education Review, vol. 7, no. 1, pp. 51-61. http://dx.doi.org/10.1007/bf03036784
- Naylor, S., Keogh, B. (2013) ‘Concept Cartoons: What have we learnt?’, Journal of Turkish Science Education, vol. 10, no. 1, pp. 3-11.
- Naylor, S., Keogh, B., Downing, B. (2007) ‘Argumentation and primary science’, Research in Science Education, vol. 37, pp. 17-39. http://dx.doi.org/10.1007/s11165-005-9002-5
- Nohda, N. (1995) ‘Teaching and evaluating using "open-ended problems" in the classroom’, ZDM Mathematics Education, vol. 27, no. 2, pp. 57-61.
- Nohda, N. (2000) ‘Teaching by open-approach method in Japanese mathematics classroom’, in Nakahara, T., Koyama, M. (eds.) Proceedings of PME 24 (Vol. 1), Hiroshima, Japan: Hiroshima University, pp. 39-53.
- Novotná, J., Eisenmann, P., Přibyl, J., Ondrušová, J., Břehovský, J. (2014) ‘Problem solving in school mathematics based on heuristic strategies’, Journal of Efficiency and Responsibility in Education and Science, vol. 7, no. 1, pp. 1-6. http://dx.doi.org/ 10.7160/eriesj.2014.070101
- Novotná, J., Eisenmann, P., Přibyl, J. (2015) ‘Are heuristic strategies a domain only for gifted pupils?’, in Krejčí, I., Flégl, M., Houška, M. (eds.) Proceedings of the 12th International Conference Efficiency and Responsibility in Education 2015 (ERIE 2015), Prague, Czech Republic: Czech University of Life Sciences, pp. 406-413.
- Organization of Economic Cooperation and Development (2005) The selection and definition of key competencies: executive summary, Paris, France: OECD.
- Patáková, E. (2013) ‘The quality of mathematical problems – evaluation and self-evaluation’, Journal of Efficiency and Responsibility in Education and Science, vol. 6, no. 3, pp. 143-154. http://dx.doi.org/10.7160/eriesj.2013.060302
- Pehkonen, E. (1995) ‘Introduction: Use of open-ended problems’, International Reviews on Mathematical Education, vol. 27, no. 2, pp. 55-57.
- Ryan, J., Williams, J. (2011) Children's mathematics 4-15. Learning from errors and misconceptions, Berkshire: Open University Press.
- Samková, L., Hošpesová, A. (2015) ‘Using Concept Cartoons to investigate future teachers' knowledge’, in Krainer, K., Vondrová, N. (eds.) Proceedings of the Ninth Congress of the European Society for Research in Mathematics Education (CERME 9), Prague, Czech Republic: Charles University, Faculty of Education, pp. 3241-3247.
- Samková, L., Hošpesová, A., Roubíček, F., Tichá, M. (2015) ‘Badatelsky orientované vyučování matematice’, Scientia in educatione, vol. 6, no. 1, pp. 91–122.
- Samková, L., Naylor, S. (2015) ‘Concept Cartoons as educational and diagnostic tool’, in Novotná, J. and Moraová, H. (eds.) Proceedings of International Symposium Elementary Maths Teaching SEMT 15, Prague, Czech Republic: Charles University, Faculty of Education, pp. 374-376.
- Samková, L., Tichá, M. (2015) ‘Investigating future primary teachers' grasping of situations related to unequal partition word problems’, in Sabena, C. and Di Paola, B. (eds.) Quaderni di Ricerca in Didattica (Mathematics), n. 25, Supplemento n. 2. Proceedings CIEAEM 67, Teaching and learning mathematics: resources and obstacles, Palermo, Italy: G.R.I.M., pp. 295-303.
- Samková, L., Tichá, M. (2016a) ‘Developing views of proof of future primary school teachers’, in Balko, L'., Szarková, D., Richtáriková, D. (eds.) Proceedings, 15th Conference on Applied Mathematics Aplimat 2016, Bratislava, Slovak Republic: STU, pp. 987-998.
- Samková, L., Tichá, M. (2016b) ‘On the way to enhance future primary school teachers' beliefs about mathematics via inquiry based university courses’. Research report accepted for the conference 13th International Congress on Mathematical Education (ICME-13), July 2016, Hamburg, Germany.
- Samková, L., Tichá, M. (2016c) ‘Developing open approach to mathematics in future primary school teachers’, in Flégl, M., Houška, M., Krejčí, I. (eds.) Proceedings of the 13th International Conference on Efficiency and Responsibility in Education 2016 (ERIE 2016), Prague, Czech Republic: Czech University of Life Sciences, pp. 494-501.
- Samková, L., Tichá, M., Hošpesová, A. (2015) ‘Error patterns in computation in Concept Cartoons’, in Novotná, J., Moraová, H. (eds.) Proceedings of International Symposium Elementary Maths Teaching SEMT 15, Prague, Czech Republic: Charles University, Faculty of Education, pp. 390-391.
- Silver, E.A. (1997) ‘Fostering creativity through instruction rich in mathematical problem solving and problem posing’, ZDM, vol. 29, no. 3, pp. 75-80. http://dx.doi.org/10.1007/ s11858-997-0003-x
- Singer, F.M, Ellerton, N.F., Cai, J. (eds.) (2015) Mathematical Problem Posing. From Research to Effective Practice, New York: Springer. http://dx.doi.org/10.1007/978-1-4614-6258-3
- Sullivan, P., Clarke, D. (1992) ‘Problem solving with conventional mathematics content: responses of pupils to open mathematical tasks’, Mathematics Education Research Journal, vol. 4, no. 1, pp. 42–60. http://dx.doi.org/10.1007/bf03217231
- Tichá, M. (2009) ‘Problem posing in teacher training and as a research object in didactics: Two complementary perspectives’, in Spagnolo, F. (ed.) Quaderni di Ricerca in Didattica (Scienze Matematiche), n. 19, Supplemento n. 2. Proceedings CIEAEM 61, Palermo, Italy: G.R.I.M., pp. 445-449.
- Vyšín, J. (1976) ‘Genetická metoda ve vyučování matematice’, Matematika a fyzika ve škole, vol. 6, pp. 582-593.
- Wittmann, E.C. (1974) Grundfragen des Mathematikunterrichts, Stuttgart: Vieweg. http://dx. doi.org/10.1007/978-3-322-85668-5
- Wittmann, E.C. (2001) ‘Developing mathematics education in a systemic process’, Educational Studies in Mathematics, vol. 48, no. 1, pp. 1–20. http://dx.doi.org/10.1023/a:101 5538317850
- Zhou, C. (2012) ‘Teaching engineering students creativity: a review of applied strategies’, Journal of Efficiency and Responsibility in Education and Science, vol. 5, no. 2, pp. 99-114. http://dx.doi.org/10.7160/eriesj.2012.050205
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