Activity design for improving mathematical understanding

Abstract

This study aims to present a conceptual framework on the design of activities to improve comprehension and evaluate the impact of activities developed on students' mathematical understanding. Conducted as a case study in the theoretical dimension, the activities developed within the scope of the study were applied to 21 8th grade students determined by convenience sampling method. The data obtained were analysed according to whether the students used algorithms correctly, came up with multiple solutions, made logical inferences and discoveries, made connections with daily life, made use of multiple representations, established relationships between concepts, and were able to recognize the need for a concept in history and produce solutions for it. The results of the study showed that the designed activities were effective in improving students' mathematical understanding.

Keywords

• Activity design
• mathematical understanding
• mathematical understanding activities
• understanding.

Matematiksel anlamayı geliştirmeye yönelik etkinlik tasarımı

Abstract

This study aims to present a conceptual framework on the design of activities to improve comprehension and evaluate the impact of activities developed on students' mathematical understanding. Conducted as a case study in the theoretical dimension, the activities developed within the scope of the study were applied to 21 8th grade students determined by convenience sampling method. The data obtained were analysed according to whether the students used algorithms correctly, came up with multiple solutions, made logical inferences and discoveries, made connections with daily life, made use of multiple representations, established relationships between concepts, and were able to recognize the need for a concept in history and produce solutions for it. The results of the study showed that the designed activities were effective in improving students' mathematical understanding.

Keywords

• Anlama
• etkinlik tasarlama
• matematiksel anlama
• matematiksel anlama etkinlikleri

References

1. Berry, J. S., & Nyman, M. A. (2003). Promoting students’ graphical understanding of the calculus. The Journal of Mathematical Behavior, 22(4), 479–495. https://doi.org/10.1016/j.jmathb.2003.09.006
2. Bingölbali, E., & Özmantar, M. F. (2014). Matematiksel zorluklar ve çözüm önerileri. Pegem Akademi.
3. Boaler, J. (2011). Changing students’ lives through the de-tracking of urban mathematics classrooms. Journal of Urban Mathematics Education, 4(1), 7–14.
4. Boesen, J., Lithner, J., & Palm, T. (2010). The relation between types of assessment tasks and the mathematical reasoning students use. Educational Studies in Mathematics, 75(1), 89–105. https://doi.org/10.1007/s10649-010-9242-9
5. Breen, S., & O’Shea, A. (2019). Designing mathematical thinking tasks. PRIMUS, 29(1), 9–20. https://doi.org/10.1080/10511970.2017.1396567
6. Cai, J., & Ding, M. (2017). On mathematical understanding: Perspectives of experienced Chinese mathematics teachers. Journal of Mathematics Teacher Education, 20(1), 5–29. https://doi.org/10.1007/s10857-015-9325-8
7. Chapman, O. (2013). Mathematical-task knowledge for teaching. Journal of Mathematics Teacher Education, 16, 1–6. https://doi.org/10.1007/s10857-013-9234-7
8. Chen, Q., & Weng, K. Q. (2003). On understanding learning in mathematics learning. Journal of Mathematics Education, 12(1), 17–19.

9. Creswell, J. W., & Poth, C. N. (2018). Qualitative inquiry and research design: Choosing among five approaches (4th ed.). Sage.
10. Dempsey, M., & O’Shea, A. (2017). Critical evaluation and design of mathematics tasks: Pre-service teachers. In CERME 10.
11. Fauvel, J. (1991). Using history in mathematics education. For the Learning of Mathematics, 11(2), 3–6.
12. Fujita, T., & Yamamoto, S. (2011). The development of children's understanding of mathematical patterns through mathematical activities. Research in Mathematics Education, 13(3), 249–267. https://doi.org/10.1080/14794802.2011.624730
13. Garner, M. (2007). An alternative theory: Deep understanding of mathematics. Measurement: Interdisciplinary Research and Perspectives, 5(2–3), 170–173. https://doi.org/10.1080/15366360701487583
14. Goldin, G. A. (2018). Discrete mathematics and the affective dimension of mathematical learning and engagement. In E. W. Hart & J. Sandefur (Eds.), Teaching and learning discrete mathematics worldwide: Curriculum and research (pp. 53–65). Springer. https://doi.org/10.1007/978-3-319-70308-4_4
15. Goos, M., Geiger, V., & Dole, S. (2013). Designing rich numeracy tasks. ICMI Study 22: Task Design in Mathematics, 589–597.
16. Gulikers, I., & Blom, K. (2001). A historical angle: A survey of recent literature on the use and value of history in geometrical education. Educational Studies in Mathematics, 47(2), 223–258.
17. Hatisaru, V. (2020). Exploring evidence of mathematical tasks and representations in the drawings of middle school students. International Electronic Journal of Mathematics Education, 15(3), em0609. https://doi.org/10.29333/iejme/8482
18. Huinker, D. (2015). Representational competence: A renewed focus for classroom practice in mathematics. Wisconsin Teacher of Mathematics, 67(2), 4–8.
19. İdikut, N. (2007). The effect of benefiting from history in education of mathematics on the student's attitudes towards mathematics and their success on it (Master's thesis). Yüzüncü Yıl University, Institute of Social Sciences.
20. Kadijevich, D. (2018). Relating procedural and conceptual knowledge. Teaching of Mathematics, 21(1), 15–28.
21. Kieran, C., Doorman, M., & Ohtani, M. (2015). Frameworks and principles for task design. In Task design in mathematics education (pp. 19–81). Springer. https://doi.org/10.1007/978-3-319-09629-2_13
22. Kilpatrick, J., Swafford, J., & Findell, B. (2001). The strands of mathematical proficiency. In Adding it up: Helping children learn mathematics (pp. 115–118).
23. Koç, M. H. (2019). Evaluation of class teachers’ activity preparation and implementation processes. Journal of Education for Life, 33(1), 69–84. https://doi.org/10.33308/26674874.201933193
24. Krathwohl, D. R. (2002). A revision of Bloom’s taxonomy: An overview. Theory into Practice, 41(4), 212–218. https://doi.org/10.1207/s15430421tip4104_2
25. Lai, M. Y., Kinnear, V., & Fung, C. I. (2019). Teaching mathematics for understanding in primary schools: Could teaching for mathematising be a solution? International Journal for Mathematics Teaching and Learning, 20(1), 1–17. http://dx.doi.org/10.4256/ijmtl.v20i1.111
26. Lauritzen, P. Å. L. (2012). Conceptual and procedural knowledge of mathematical functions [Doctoral dissertation, Itä-Suomen yliopisto].
27. Lesh, R., Cramer, K., Doerr, H., Post, T., & Zawojewski, J. (2003). Using a translation model for curriculum development and classroom instruction. In R. Lesh & H. Doerr (Eds.), Beyond constructivism: Models and modeling perspectives on mathematics problem solving, learning, and teaching (pp. 35–58). Lawrence Erlbaum Associates. https://doi.org/10.4324/9781410607713
28. Lincoln, Y. S., & Guba, E. G. (1985). Naturalistic inquiry. Sage.
29. Liu, L. H. (2009). Thoughts on mathematical understanding learning. Journal of Xianning College, 29(3), 18–20.
30. Liu, P. (2003). Do teachers need to incorporate the history of mathematics in their teaching? The Mathematics Teacher, 96(6), 416. https://doi.org/10.5951/MT.96.6.0416
31. Ministry of National Education [MoNE]. (2013). Secondary school mathematics course curriculum (9, 10, 11 and 12. grades). Boards of Education.
32. Ministry of National Education [MoNE]. (2018). Mathematics curriculum (primary and secondary school grades 1–8). Boards of Education.
33. Mi, S., Lu, S., & Bi, H. (2020). Trends and foundations in research on students’ conceptual understanding in science education: A method based on the structural topic model. Journal of Baltic Science Education, 19(4), 551–568. https://doi.org/10.33225/jbse/20.19.551
34. National Council of Teachers of Mathematics (NCTM). (2000). Principles and standards for school mathematics. NCTM.
35. O’Shea, A., & Breen, S. (2021). Students’ views on transition to university: The role of mathematical tasks. Canadian Journal of Science, Mathematics and Technology Education, 21(1), 29–43. https://doi.org/10.1007/s42330-021-00140-y
36. Özdemir, A., & Yıldız, S. G. (2015). Using history of mathematics in the classroom: Babylonian number system. Amasya Education Journal, 4(1), 26–49.
37. Paolucci, C., & Wessels, H. (2017). An examination of preservice teachers’ capacity to create mathematical modeling problems for children. Journal of Teacher Education, 68(3), 330–344. https://doi.org/10.1177/0022487117697636
38. Powell, A. B., Borge, I. C., Fioriti, G. I., Kondratieva, M., Koublanova, E., & Sukthankar, N. (2009). Challenging tasks and mathematics learning. In Challenging mathematics in and beyond the classroom (pp. 133–170). Springer.
39. Rasmussen, C., Zandieh, M., King, K., & Teppo, A. (2005). Advancing mathematical activity: A practice-oriented view of advanced mathematical thinking. Mathematical Thinking and Learning, 7(1), 51–73.
40. Sierpinska, A. (2013). Understanding in mathematics. Routledge.
41. Stein, M., Smith, M., Henningsen, M., & Silver, E. (2000). Implementing standards-based mathematics instruction: A casebook for professional development. Teachers College Press.
42. Stylianides, A. J., & Stylianides, G. J. (2008). Studying the classroom implementation of tasks: High-level mathematical tasks embedded in ‘real-life’ contexts. Teaching and Teacher Education, 24(4), 859–875. https://doi.org/10.1016/j.tate.2007.11.015
43. Suzuki, K., & Harnisch, D. L. (1995, April). Measuring cognitive complexity: An analysis of performance-based assessment in mathematics. Paper presented at the Annual Meeting of the American Educational Research Association, San Francisco, CA.
44. Swan, M. (2014). Designing tasks and lessons that develop conceptual understanding, strategic competence and critical awareness. In Tarefas matemáticas: Livro de Atas do Encontro de Investigação em Educação Matemática (pp. 9–28).
45. Usiskin, Z. (2012). What does it mean to understand some mathematics? In Selected regular lectures from the 12th international congress on mathematical education (pp. 821–841). Springer. https://doi.org/10.1007/978-3-319-17187-6_46
46. Yang, H. J. (2016). In-depth understanding and flexible application: Experimental study of junior middle school mathematics comprehension teaching. New Curriculum: Middle School.
47. Yao, Y., Hwang, S., & Cai, J. (2021). Preservice teachers’ mathematical understanding exhibited in problem posing and problem solving. ZDM–Mathematics Education, 53(4), 937–949. https://doi.org/10.1007/s11858-021-01277-8
48. Yatim, S. S. K. M., Saleh, S., Zulnaidi, H., Yew, W. T., & Yatim, S. A. M. (2022). Effects of brain-based teaching approach integrated with GeoGebra (B-Geo Module) on students' conceptual understanding. International Journal of Instruction, 15(1), 327–346. http://dx.doi.org/10.29333/iji.2022.15119a