Prospective Mathematics Teachers’ Task Modifications Utilizing Their Knowledge of Pattern Generalization

Year 2023, Volume: 16 Issue: 3, 596 - 616, 28.07.2023

Dilek Girit Yıldız

https://doi.org/10.30831/akukeg.1216428

Abstract

The purpose of the study is to evaluate how prospective mathematics teachers (PMTs) modify tasks to facilitate students' learning of pattern generalization through the use of their knowledge. The qualitative research method was used to determine the mathematical characteristics that PMTs use when modifying a mathematical task. In addition, the knowledge that PMTs draw from to modify the task has been outlined. Accordingly, data were collected from PMTs’ task modifications and reflection reports. When PMTs worked on two or more forms of modification, as compared to just one, they modified tasks more properly and comprehensively in a relevant manner. The PMTs who make condition modifications need to utilize specialized content knowledge and knowledge of content and student to help students understand through the use of models or tables. They also used their knowledge of content and teaching in organizing the questions in a way that encouraged inductive reasoning and problems based on real-world or familiar contexts in context modifications. Task modification activities can be a good way for future teachers to notice the role of tasks in mathematics teaching and demonstrate their use of knowledge.

Keywords

task modification, pattern generalization, prospective mathematics teachers, mathematical knowledge for teaching.

References

• Arbaugh, F., & Brown, C. A. (2005). Analyzing mathematical tasks: A catalyst for change? Journal of Mathematics Teacher Education, 8(6), 499-536. https://doi.org/10.1007/s10857-006-6585-3
• Ayalon, M., Naftaliev, E., Levenson, E. S., & Levy, S. (2021). Prospective and in-service mathematics teachers’ attention to a rich mathematics task while planning its implementation in the classroom. International Journal of Science and Mathematics Education, 19(2021), 1695-1716.
• Ball, D. L. (2000). Bridging practices: Intertwining content and pedagogy in teaching and learning to teach. Journal of Teacher Education, 51(3), 241-247.
• Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), 389- 407.
• Barbosa, A., & Vale, I. (2015). Visualization in pattern generalization: Potential and challenges. Journal of the European Teacher Education Network, 10, 57-70.
• Bartell, T. G., Webel, C., Bowen, B., & Dyson, N. (2013). Prospective teacher learning: Recognizing evidence of conceptual understanding. Journal of Mathematics Teacher Education, 16(1), 57–79.
• Basyal, D., Jones, D. L., & Thapa, M. (2023). Cognitive demand of mathematics tasks in Nepali middle school mathematics textbooks. International Journal of Science and Mathematics Education, 21, 863–879. https://doi.org/10.1007/s10763-022-10269-3
• Becker, J. R., & Rivera, F. (2005). Generalization schemes in algebra of beginning high school students. In H. Chick, & J. Vincent (Eds.), Proceedings of the 29th conference of the international group for psychology of mathematics education (Vol. 4) (pp. 121–128). Melbourne, Australia: University of Melbourne.
• Boston, M. D. (2013). Connecting changes in secondary mathematics teachers’ knowledge to their experiences in a professional development workshop. Journal of Mathematics Teacher Education, 16(1), 7–31. https://doi.org/10.1007/s10857-012-9211-6
• Brown, S. I., & Walter, M. I. (1990). The art of problem posing (2nd Ed.). Hillsdale, NJ: Lawrence Erlbaum Associates.
• Callejo, M. L., & Zapatera, A. (2017). Prospective primary teachers’ noticing of students’ understanding of pattern generalization. Journal of Mathematics Teacher Education, 20(4), 309-333.
• Chapman, O. (2013). Mathematical-task knowledge for teaching. Journal of Mathematics Teacher Education, 16(1), 1-6.
• Cheng, L. P., Leong, Y. H., & Toh, W. Y. K. (2021). Singapore secondary school mathematics teachers’ selection and modification of instructional materials for classroom use. In B. Kaur & Y. H. Leong (Eds.), Mathematics instructional practices in Singapore secondary schools (pp. 205-230). Singapore: Springer.
• Crespo, S. (2003). Learning to pose mathematical problems: Exploring changes in pre-service teachers’ practices. Educational Studies in Mathematics, 52(3), 243- 270.
• Doerfler, W. (2008). En route from patterns to algebra: Comments and reflections. ZDM Mathematics Education, 40(1), 143–160.
• Doyle, W. (1983). Academic work. Review of Educational Research, 53(2), 159–199.
• English, L., & Warren, E. (1998). Introducing the variable through pattern exploration. Mathematics Teacher, 91(2), 166–170.
• Girit Yildiz, D. G., & Akyuz, D. (2020). Mathematical knowledge of two middle school mathematics teachers in planning and teaching pattern generalization. Ilkogretim Online - Elementary Education Online, 19(4), 2098-2117.
• Guberman, R., & Leikin, R. (2013). Interesting and difficult mathematical problems: changing teachers’ views by employing multiple-solution tasks. Journal of Mathematics Teacher Education, 16(1), 33-56. https://doi.org/10.1007/s10857-012-9210-7
• Haggarty, L., & Pepin, B. (2002). An investigation of mathematics textbooks and their use in English, French and German classrooms: Who gets an opportunity to learn what? British Educational Research Journal, 28(4), 567-590.
• Healy, L., & Hoyles, C. (1999). Visual and symbolic reasoning in mathematics: Making connections with computers. Mathematical Thinking and Learning, 1(1), 59–84.
• Henningsen, M., & Stein, M. (1997). Mathematical tasks and student cognition: classroom-based factors that support and inhibit high-level mathematical thinking and reasoning. Journal for Research in Mathematics Education, 28(5), 524-549.
• Hidayah, M., & Forgasz, H. (2020). A comparison of mathematical tasks types used in Indonesian and Australian textbooks based on geometry contents. Journal on Mathematics Education, 11(3), 385-404. https://doi.org/10.22342/JME.11.3.11754.385-404
• Johnson, R., Severance, S., Penuel, W. R., & Leary, H. (2016). Teachers, tasks, and tensions lessons from a research practice partnership. Journal of Mathematics Teacher Education, 19(2), 169–185.
• Kaput, J. J. (1999). Teaching and learning a new algebra. In E. Fennema & T. Romberg (Eds.), Mathematics classrooms that promote understanding (pp. 133–155). Mahwah, NJ: Erlbaum.
• Kaur, B., & Lam, T. T. (2012). Reasoning, communication and connections in mathematics: An introduction. In B. Kaur & T. T. Lam (Eds.), Reasoning, communication and connections in mathematics: Yearbook 2012 association of mathematics educators (pp. 1–10). Singapore: World Scientific Publishing.
• Kusaeri, K., Arrifadah, Y., & Asmiyah, S. (2022). Enhancing creative reasoning through mathematical task: The quest for an ideal design. International Journal of Evaluation and Research in Education (IJERE), 11(2), 482-490.
• Lannin, J. K., Barker, D. D., & Townsend, B. E. (2006). Recursive and explicit rules: How can we build student algebraic understanding? Journal of Mathematical Behavior, 25(4), 299–317.
• Lee, E.J., Lee, K.H. & Park, M. (2019). Developing pre-service teachers’ abilities to modify mathematical tasks: Using noticing-oriented activities. International Journal of Science and Mathematics Education, 17(5), 965-985.
• Lee, K. H., Lee, E. J., & Park, M. S. (2016). Task modification and knowledge utilization by Korean prospective mathematics teachers. Pedagogical Research, 1(2), 54.
• Lee, M. Y., & Lee, J. E. (2021). An analysis of elementary prospective teachers’ noticing of student pattern generalization strategies in mathematics. Journal of Mathematics Teacher Education, 26, 155-177. https://doi.org/10.1007/s10857-021-09520-5
• Liljedahl, P., Chernoff, E., & Zazkis, R. (2007). Interweaving mathematics and pedagogy in task design: A tale of one task. Journal of Mathematics Teacher Education, 10(4), 239–249.
• Llinares, S., & Krainer, K. (2006). Mathematics (student) teachers and teacher educator as learners. In A. Gutie’rrez & P. Boero (Eds.), Handbook of research on the psychology of mathematics education: Past, present and future (pp. 429–459). Rotterdam: Sense Publishers.
• MacGregor, M., & Stacey, K. (1995). The effect of different approaches to algebra on students’ perceptions of functional relationships. Mathematics Education Research Journal, 7(1), 69–85.
• Magiera, M. T., van den Kieboom, L. A., & Moyer, J. C. (2013). An exploratory study of pre-service middle school teachers’ knowledge of algebraic thinking. Educational Studies in Mathematics, 84(1), 93–113.
• Malara, N. A., & Navarra, G. (2009). The analysis of classroom-based processes as a key task in teacher training for the approach to early algebra. In B. Clarke, B. Grevholm, & R. Millman (Eds.), Tasks in primary mathematics teacher education (pp. 235–262). Berlin: Springer.
• Markworth, K. A. (2010). Growing and growing: Promoting functional thinking with geometric growing patterns [Doctoral dissertation, University of North Carolina]. ProQuest Dissertations Publishing.
• Merriam, S. B. (2009). Qualitative research: A guide to design and implementation. San Francisco, CA: Jossey-Bass.
• Moss, J., Beatty, R., Barkin, S., & Shillolo, G. (2008). "What is your theory? what is your rule?" fourth graders build an understanding of functions through patterns and generalizing problems. In C. Greenes, & R. Rubenstein (Eds.), Algebra and algebraic thinking in school mathematics: 70th NCTM yearbook (pp. 155–168). Reston, VA: National Council of Teachers of Mathematics.
• Mosvold, R., Jakobsen, A., & Jankvist, U. T. (2014). How mathematical knowledge for teaching may profit from the study of history of mathematics. Science & Education, 23(1), 47-60.
• Papatistodemou, E., Potari, D., & Potta-Pantazi, D. (2014). Prospective teachers’ attention on geometrical tasks. Educational Studies in Mathematics, 86(1), 1–18.
• Pepin, B. E. U. (2015). Enhancing mathematics / STEM education: a ‘‘resourceful’’ approach. Technische Universiteit Eindhoven.
• Prestage, S., & Perks, P. (2007). Developing teacher knowledge using a tool for creating tasks for the classroom. Journal of Mathematics Teacher Education, 10(4), 381-390.
• Radford, L. (2008). Iconicity and contraction: a semiotic investigation of forms of algebraic generalizations of patterns in different contexts. ZDM, 40(1), 83–96.
• Radford, L. (2011). Embodiment, perception and symbols in the development of early algebraic thinking. In B. Ubuz (Ed.), Proceedings of the 35th Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 17–24). Ankara, Turkey: PME.
• Rivera, F. D. (2010). Visual templates in pattern generalization activity. Educational Studies in Mathematics, 73(3), 297-328.
• Smith, E. (2008). Representational thinking as a framework for introducing functions in the elementary curriculum. In J. L. Kaput, D. W. Carraher, & M. L. Blanton (Eds.), Algebra in the early grades (pp. 133–160). New York: Taylor & Francis Group.
• Smith, M. S., Bill, V., & Hughes, E. K. (2008). Thinking through a lesson: Successfully implementing high-level tasks. Mathematics Teaching in the Middle School, 14(3), 132-138.
• Son, J. W., & Kim, O. K. (2015). Teachers’ selection and enactment of mathematical problems from textbooks. Mathematics Educational Research Journal, 27(4), 491–518.
• Steele, D. F., & Johanning, D. J. (2004). A schematic-theoretic view of problem solving and development of algebraic thinking. Educational Studies in Mathematics, 57(1), 65-90.
• Stein, M. K., & Smith, M. S. (1998). Mathematical tasks as a framework for reflection: From research to practice. Mathematics Teaching in the Middle School, 3(4), 268-275.
• Stein, M. K., Smith M. S., Henningsen, M. A. & Silver, E. A. (2000). Implementing standards-based mathematics instruction: A casebook for professional development. Teachers College Press.
• Stephens, A. C. (2006). Equivalence and relational thinking: Pre-service elementary teachers’ awareness of opportunities and misconceptions. Journal of Mathematics Teacher Education, 9(3), 249–278.
• Sullivan, P. A., Clarke, D. M., & Clarke, B. A. (2009). Converting mathematics tasks to learning opportunities: An important aspect of knowledge for mathematics teaching. Mathematics Education Research Journal, 21(1), 85-105.
• Sullivan, P. A., Clarke, D. M., Clarke, B. A., & O’Shea, H. F. (2010). Exploring the relationship between tasks, teacher actions, and student learning. PNA (Revista de Investigación en Didáctica de la Matemática), 4(4), 133- 142.
• Sullivan, P., & Mousley, J. (2001). Thinking teaching: Seeing mathematics teachers as active decision makers. In F.-L. Lin & T. Cooney (Eds.), Making sense of mathematics teacher education (pp. 147–163). Dordrecht: Springer.
• Strauss, A. & Corbin, J. (1998). Basics of qualitative research: Techniques and procedures for developing grounded theory. Sage.
• Swan, M. (2008). Designing a multiple representation learning experience in secondary algebra. Educational Designer, 1(1), 1-17.
• Thanheiser, E. (2015). Developing prospective teachers’ conceptions with well-designed tasks: Explaining successes and analyzing conceptual difficulties. Journal of Mathematics Teacher Education, 18(2), 141-172. https://doi.org/10.1007/s10857-014- 9272-9
• Thompson, D. R. (2012). Modifying textbook exercises to incorporate reasoning and communication into the primary mathematics classroom. In B. Kaur & T. Lam (Eds.), Reasoning, communication and connections in mathematics (pp. 57–74). Singapore: World Scientific Publishing Company.
• Thomson, S. & Fleming, N. (2004). Summing it up: Mathematics achievement in Australian schools in TIMMS 2002. Melbourne: Australian Council for Educational Research.
• Ubuz, B., Erbaş, A. K., Çetinkaya, B., & Özgeldi, M. (2010). Exploring the quality of the mathematical tasks in the new Turkish elementary school mathematics curriculum guidebook: The case of algebra. ZDM-International Journal on Mathematics Education, 42(5), 483-491.
• Walkowiak, T. A. (2014). Elementary and middle school students’ analyses of pictorial growth patterns. Journal of Mathematical Behavior, 33(2014), 56-71.
• Warren, E. (2000). Visualisation and the development of early understanding in algebra. Paper presented at the 24th Conference of the International Group for the Psychology of Mathematics Education (PME), Hiroshima, Japan.
• Warren, E., & Cooper, T. (2008). Patterns that support early algebraic thinking in the elementary school. In C. Greenes, & R. Rubenstein (Eds.), Algebra and algebraic thinking in school mathematics: 70th NCTM yearbook (pp. 113–126). Reston, VA: National Council of Teachers of Mathematics. Watson, A., & Mason, J. (2007). Taken-as-shared: a review of common assumptions about mathematical tasks in teacher education. Journal of Mathematics Teacher Education, 10(4), 205-215.
• Wilkie, K. J. (2014). Upper primary school teachers’ mathematical knowledge for teaching functional thinking in algebra. Journal of Mathematics Teacher Education, 17(5), 397-428.
• Yıldırım, A. & Şimşek, H. (2013). Sosyal bilimlerde nitel araştırma yöntemleri (9. baskı). Ankara: Seçkin Yayıncılık.
• Zaslavsky, O. (2008). Meeting the challenges of mathematics teacher education through design and use of tasks that facilitate teacher learning. In B. Jaworski & T. Wood (Eds.), The mathematics teacher education as a developing professional (pp. 93-114). Rotterdam: Sense Publishers.