Emergent modelling to introduce the distributivity property of multiplication: a design research study in a primary school

References

• Bakker, A. (2004). Design research in statistic education: On symbolizing and computer tools. CD-Bèta Press.
   Google Scholar
• Bakker, A. (2018). Design research in education. A practical guide for early career researchers. Routledge.
   Google Scholar
• Barab, S. A., & Kirshner, D. (2001). Guest editor’s introduction: Rethinking methodology in the learning sciences. Journal of the Learning Sciences, 10(1/2), 5–15. https://doi.org/10.1207/S15327809JLS10-1-2_2
   Web of Science ®Google Scholar
• Barmby, P., Harries, T., Higgins, S., & Suggate, J. (2009). The array representation and primary children’s understanding and reasoning in multiplication. Educational Studies in Mathematics, 70(3), 217–241. https://doi.org/10.1007/s10649-008-9145-1
   Web of Science ®Google Scholar
• Blum, W. (2015). Quality teaching of mathematical modelling: What do we know, what can we do? The proceedings of the 12th international congress on mathematical education – intellectual and attitudinal challenges, S. J. Cho (pp. 73–96). New York: Springer.
   Google Scholar
• Blum, W., Galbraith, P. L., Henn, H.-W., & Niss, M. (2007). Modelling and applications in mathematics education, New ICMI Studies no. 14. Springer.
   Google Scholar
• Bonotto, C. (2005). How informal out-of-school mathematics can help students make sense of formal in-school mathematics: The case of multiplying by decimal numbers. Mathematical Thinking and Learning, 7(4), 313–344. https://doi.org/10.1207/s15327833mtl0704_3
   Google Scholar
• Bonotto, C. (2013). Artifacts as sources for problem-posing activities. Educational Studies in Mathematics, 83(1), 37–55. https://doi.org/10.1007/s10649-012-9441-7
   Web of Science ®Google Scholar
• Carpenter, T. P., Levi, L., Franke, M. L., & Zeringue, J. K. (2005). Algebra in elementary school: Developing relational thinking. ZDM, 37(1), 53–59.
   Google Scholar
• Chamberlin, S., Payne, A. M., & Kettler, T. (2020). Mathematical modelling: A positive learning approach to facilitate student sense making in mathematics. International Journal of Mathematical Education in Science and Technology. https://doi.org/10.1080/0020739X.2020.1788185
   Google Scholar
• Cobb, P., Confrey, J., diSessa, A., Lehrer, R., & Schauble, L. (2003). Design experiments in educational research. Educational Researcher, 32(1), 9–13. https://doi.org/10.3102/0013189X032001009
   Google Scholar
• Ding, M., & Li, X. (2010). A comparative analysis of the distributive property in U.S. And Chinese elementary mathematics textbooks. Cognition and Instruction, 28(2), 146–180. https://doi.org/10.1080/07370001003638553
   Web of Science ®Google Scholar
• Ding, M., & Li, X. (2014). Transition from concrete to abstract representation: The distributive property in a Chinese textbook series. Educational Studies in Mathematics, 87(1), 103–121. https://doi.org/10.1007/s10649-014-9558-y
   Web of Science ®Google Scholar
• Doorman, M. (2005). Modelling motion: From trace graph to instantaneous change. CD-Bèta Press.
   Google Scholar
• Doorman, M. (2019). Design and research for developing local instruction theories. Avances de Investigación en Educación Matemática, 15(15), 29–42. https://doi.org/10.35763/aiem.v0i15.266
   Web of Science ®Google Scholar
• Edelson, D. C. (2002). Design research: What we learn when we engage in design. Journal of the Learning Sciences, 11(1), 105–121. https://doi.org/10.1207/S15327809JLS1101_4
   Web of Science ®Google Scholar
• Fischbein, E., Deri, M., Nello, M., & Marino, M. (1985). The role of implicit models in solving verbal problems in multiplication and division. Journal for Research in Mathematics Education, 16(1), 3–17. https://doi.org/10.2307/748969
   Google Scholar
• Freudenthal, H. (1973). Mathematics as an educational task. Reidel.
   Google Scholar
• Freudenthal, H. (1983). Didactical phenomenology of mathematical structures. Reidel.
   Google Scholar
• Freudenthal, H. (1991). Revisiting mathematics education. China lectures. Kluwer.
   Google Scholar
• Gravemeijer, K. (1994a). Developing realistic mathematics education. CD Bèta Press.
   Google Scholar
• Gravemeijer, K. (1994b). Educational development and developmental research in mathematics education. Journal for Research in Mathematics Education, 25(5), 443–471. https://doi.org/10.2307/749485
   Web of Science ®Google Scholar
• Gravemeijer, K. (1999). How emergent models may foster the construction of formal mathematics. Mathematical Thinking and Learning, 1(2), 155–177. https://doi.org/10.1207/s15327833mtl0102_4
   Google Scholar
• Gravemeijer, K., Bowers, J., & Stephan, M. (2003). Chapter 4: A hypothetical learning trajectory on measurement and flexible arithmetic. Journal for Research in Mathematics Education. Monograph, 12, 51–66. http://www.jstor.org/stable/30037721
   Google Scholar
• Gravemeijer, K. (2004). Learning trajectories and local instruction theories as means of support for teachers in reform mathematics education. Mathematical Thinking and Learning, 6(2), 105–128. https://doi.org/10.1207/s15327833mtl0602_3
   Google Scholar
• Gravemeijer, K., & Cobb, P. (2001). Designing classroom-learning environments that support mathematical learning. Paper presented at the Annual Conference of the AERA, Seattle.
   Google Scholar
• Gravemeijer, K., Cobb, P., Bowers, J., & Whitenack, J. (2000). Symbolizing, modeling, and instructional design. In P. Cobb, E. Yackel, & K. McClain (Eds.), Symbolizing and communicating in mathematics classrooms: Perspectives on discourse, tools, and instructional design (pp. 225–273). Lawrence Erlbaum Associates.
   Google Scholar
• Gravemeijer, K., & Doorman, M. (1999). Context problems in realistic mathematics education: A calculus course as an example. Educational Studies in Mathematics, 39(1-3), 111–129. https://doi.org/10.1023/A:1003749919816
   Google Scholar
• Greer, B. (1992). Multiplication and division as models of situations. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 276–295). Macmillan.
   Google Scholar
• Greer, B., Verschaffel, L., Mukhopadhyay, S., et al. (2007). Modelling for life: Mathematics and children’s experience. In W. Blum (Ed.), Modelling and applications in mathematics education (pp. 89–98). Springer.
   Google Scholar
• Hendriana, H., Prahmana, R. C. I., & Hidayat, W. (2019). The innovation of learning trajectory on multiplication operations for rural area students in Indonesia. Journal on Mathematics Education, 10(3), 397–408. https://doi.org/10.22342/jme.10.3.9257.397-408
   Google Scholar
• Izsak, A. (2004). Teaching and learning two-digit multiplication: Coordinating analyses of classroom practice and individual student learning. Mathematical Thinking and Learning, 6(1), 37–79. https://doi.org/10.1207/s15327833mtl0601_3
   Google Scholar
• Kaiser, G. (2017). The teaching and Learning of Mathematical Modeling, The Compendium for Research in Mathematics Education (stock no. 14330), NCTM, 267-291.
   Google Scholar
• Lampert, M. (1986). Knowing, doing, and teaching multiplication. Cognition and Instruction, 3(4), 305–342. https://doi.org/10.1207/s1532690xci0304_1
   Google Scholar
• Larsson, K. (2015). Sixth grade students’ explanations and justifications of distributivity. Proceedings of the 9th Conference of the European society for research in mathematics education, K. Krainer & N. Vondrova, (pp. 295-301). Praha: Czech Republic: ERME.
   Google Scholar
• Lesh, R., & Lehrer, R. (2003). Models and modeling perspectives on the development of students and teachers. Mathematical Thinking and Learning, 5(2&3), 109–129. https://doi.org/10.1080/10986065.2003.9679996
   Google Scholar
• Lesh, R., Post, T., & Behr, M. (1988). Proportional reasoning. In J. Hiebert, & M. Behr (Eds.), Number concepts and operations in the middle grades (pp. 93–118). Lawrence Erlbaum Associates.
   Google Scholar
• Maffia, A., & Mariotti, A. (2018). Intuitive and formal models of whole number multiplication: Relations and emerging structures. For the Learning of Mathematics, 38(3), 30–36.
   Google Scholar
• Maffia, A., & Mariotti, M. A. (2020). From action to symbols: Giving meaning to the symbolic representation of the distributive law in primary school. Educational Studies in Mathematics, 104(1), 25–40. https://doi.org/10.1007/s10649-020-09944-5
   Web of Science ®Google Scholar
• McKenney, S., & Reeves, T. C. (2012). Conducting educational design research. Routledge.
   Google Scholar
• Prahmana, R. C. I., Zulkardi, Z., & Hartono, Y. (2012). Learning multiplication using Indonesian traditional game in third grade. Journal on Mathematics Education, 3(2), 115–132.
   Google Scholar
• Schifter, D., Monk, S., Russel, S. J., & Bastable, V. (2008). Early algebra: What does understanding the laws of arithmetic mean in the elementary grades. In J. J. Kaput, D. W. Carraher, & M. L. Blanton (Eds.), Algebra in the early grades (pp. 413–447). Lawrence Erlbaum Associate/National Council of Teachers of Mathematics.
   Google Scholar
• Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26(2), 114–145. https://doi.org/10.2307/749205
   Web of Science ®Google Scholar
• Squire, S., Davies, C., & Bryant, P. (2004). Does the cue help? Children’s understanding of multiplicative concepts in different problem contexts. British Journal of Educational Psychology, 74(4), 515–532. https://doi.org/10.1348/0007099042376364
   PubMed Web of Science ®Google Scholar
• Steffe, L. P. (1983). The teaching experiment methodology in a constructivist research program. Proceedings of the fourth international congress on mathematics education, M. Zweng, T. Green, J. Kilpatrick, H. Pollak & M. Suydam, (pp. 469-471). Boston: Birkhäuser Inc..
   Google Scholar
• Streefland, L. (1985). Wiskunde als activiteit en de realiteit als bron (mathematics as an activity and the reality as a source). Tijdschrift Voor Nederlands Wiskundeonderwijs (Nieuwe Wiskrant), 5(1), 60–67.
   Google Scholar
• Treffers, A. (1987). Three dimensions. A model of goal and theory description in mathematics instruction-the wiskobas project. Reidel Publishing.
   Google Scholar
• Van den Heuvel-Panhuizen, M., & Drijvers, P. (2014). Realistic Mathematics education. In S. Lerman (Ed.), Encyclopedia of mathematics education (pp. 521–525). Springer.
   Google Scholar
• Van Dijke-Droogers, M., Drijvers, P., & Bakker, A. (2020). Repeated sampling with a black Box to make informal statistical inference accessible. Mathematical Thinking and Learning, 22(2), 116–138. https://doi.org/10.1080/10986065.2019.1617025
   Web of Science ®Google Scholar
• Van Dooren, W., De Bock, D., & Verschaffel, L. (2010). From addition to multiplication … and back: The development of students’ additive and multiplicative reasoning skills. Cognition and Instruction, 28(3), 360–381. https://doi.org/10.1080/07370008.2010.488306
   Web of Science ®Google Scholar
• Yackel, E. (2001). Perspectives on arithmetic from classroom-based research in the United States of america. In J. Anghileri (Ed.), Principles and practices in arithmetic teaching: Innovative approaches for the primary classroom (pp. 15–31). Open University Press.
   Google Scholar
• Young-Loveridge, J. (2005). Fostering multiplicative thinking using array-based materials. Australian Mathematics Teacher, 61(3), 34–40.
   Google Scholar