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A prospective teacher’s conceptions about the nature of mathematical tasks and professional knowledge within task unfolding

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Abstract

This qualitative study aimed to identify the changes in a prospective’ conceptions about the nature of mathematical tasks and pedagogy in video cases and to investigate the characteristics of the prospective’ professional knowledge and views during task design/modification. The study used the mathematical task demand engagement process, which includes task-sorting activity, selecting tasks, reflecting on illustrative cases, and teaching and reflecting on tasks as enacted. The findings regarding one prospective, Kar, were reported. Focus group interviews, individual interviews, and two recorded lessons were used as data sources. The content analysis method was employed for the in-depth analysis of the data. The findings suggested a change in Kar’s ideas about the nature of mathematical tasks, particularly about what characteristics low and high-level tasks possess and the efficiency of the Task Analysis Guide. Kar also generated detailed ideas on the pedagogy of high and low tasks and realized the role of teachers as facilitators while enacting high-level tasks. Another finding is that Kar’s decisions regarding task adaptation and her reasons catalyze both situated and global knowledge and views. Our findings suggest that providing an opportunity for prospective to reflect on and discuss the nature of mathematical tasks as planned, implemented, and revisited in a professional context is beneficial.

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References

  • Nathan, M. J., & Petrosino, A. (2003). Expert blind spot among preservice teachers. American Educational Research Journal, 40(4), 905–928.

    Article  Google Scholar 

  • Ainley, J., & Margolinas, C. (2015). Accounting for student perspectives in task design. In A. Watson & M. Ohtani (Eds.), Task design in mathematics education (pp. 115–142). Springer International Publishing.

    Chapter  Google Scholar 

  • Amador, J. M., & Earnest, D. (2018). Launching forth: Preservice teachers translating elementary mathematics curriculum into lessons. Mathematics Education Research Journal, 31(3), 301–323.

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Ayalon, M., & Wilkie, K. (2020). Developing assessment literacy through approximations of practice: Exploring secondary mathematics prospective developing criteria for a rich quadratics task. Teaching and Teacher Education. https://doi.org/10.1016/j.tate.2019.103011

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Boston, M. D., & Smith, M. S. (2009). Transforming secondary mathematics teaching: Increasing the cognitive demands of instructional tasks used in teachers’ classrooms. Journal for Research in Mathematics Education, 40(2), 119–156.

    Google Scholar 

  • Byerley, C., & Thompson, P. W. (2017). Secondary mathematics teachers’ meanings for measure, slope, and rate of change. The Journal of Mathematical Behavior, 48, 168–193.

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Charalambous, C. Y. (2010). Mathematical knowledge for teaching and task unfolding: An exploratory study. Elementary School Journal, 110(3), 247–278.

    Article  Google Scholar 

  • Choppin, J. (2011). The impact of professional noticing on teachers’ adaptations of challenging tasks. Mathematical Thinking and Learning, 13(3), 175–197.

    Article  Google Scholar 

  • Choy, B. H. (2016). Snapshots of mathematics teacher noticing during task design. Mathematics Education Research Journal, 28(3), 421–440.

    Article  Google Scholar 

  • Cohen, L., Manion, L., & Morrison, K. (2011). Research methods in education (7th ed.). Routledge.

    Google Scholar 

  • Dreher, A., & Kuntze, S. (2015). Teachers’ professional knowledge and noticing: The case of multiple representations in the mathematics classroom. Educational Studies in Mathematics, 88(1), 89–114.

    Article  Google Scholar 

  • Dreher, A., Kuntze, S., & Lerman, S. (2016). Why use multiple representations in the mathematics classroom? Views of English and German preservice teachers. International Journal of Science and Mathematics Education, 14(2), 363–382.

    Article  Google Scholar 

  • El Mouhayar, R. R., & Jurdak, M. E. (2013). Teachers’ ability to identify and explain students’ actions in near and far figural pattern generalization tasks. Educational Studies in Mathematics, 82(3), 379–396.

    Article  Google Scholar 

  • Estrella, S., Zakaryan, D., Olfos, R., & Espinoza, G. (2020). How teachers learn to maintain the cognitive demand of tasks through Lesson Study. Journal of Mathematics Teacher Education, 23(3), 293–310. https://doi.org/10.1007/s10857-018-09423-y

    Article  Google Scholar 

  • Gün, Ö., & Taş, F. (2021). An evaluation of mathematical tasks designed by prospective within the framework of task design principles. International Journal for Mathematics Teaching and Learning, 22(2), 17–31.

    Article  Google Scholar 

  • Henningsen, M., & Stein, M. K. (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, 524–549.

    Article  Google Scholar 

  • Johnson, H. L., Coles, A., & Clarke, D. (2017). Mathematical tasks and the student: Navigating “tensions of intentions” between designers, teachers, and students. ZDM, 49, 813–822. https://doi.org/10.1007/s11858-017-0894-0

    Article  Google Scholar 

  • Kaput, J. J. (2000). Transforming algebra from an engine of inequity to an engine of mathematical power by “algebrafying” the K-12 curriculum. US Department of Education, Office of Educational Research and Improvement, Educational Resources Information Center.

  • Kisa, M. T., & Stein, M. K. (2015). Learning to see teaching in new ways: A foundation for maintaining cognitive demand. American Educational Research Journal, 52(1), 105–136. https://doi.org/10.3102/0002831214549452

    Article  Google Scholar 

  • Kuntze, S. (2012). Pedagogical content beliefs: global, content domain-related and situation-specific components. Educational Studies in Mathematics, 79(2), 273–292. https://doi.org/10.1007/s10649-011-9347-9

    Article  Google Scholar 

  • Lannin, J., Webb, M., Chval, K., Arbaugh, F., Hicks, S., Taylor, C., & Bruton, R. (2013). The development of beginning mathematics teacher pedagogical content knowledge. Journal of Math Teacher Education, 16, 403–426. https://doi.org/10.1007/s10857-013-9244-5

    Article  Google Scholar 

  • Lee, E., Lee, K., & Park, M. (2019a). Developing preservice teacher’s abilities to modify mathematical tasks: Using noticing-oriented activities. International Journal of Science and Mathematics Education, 17(5), 965–985. https://doi.org/10.1007/s10763-018-9891-1

    Article  Google Scholar 

  • Lee, H. S., Coomes, J., & Yim, J. (2019b). Teachers’ conceptions of prior knowledge and the potential of a task in teaching practice. Journal of Mathematics Teacher Education, 22(2), 129–151.

    Article  Google Scholar 

  • 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(4–6), 239–249. https://doi.org/10.1007/s10857-007-9047-7

    Article  Google Scholar 

  • Mason, J. (2002). Researching your own practice: The discipline of noticing. Routledge.

    Book  Google Scholar 

  • Mason, J. (2008). Being mathematical with and in front of learners: Attention, awareness, and attitude as sources of differences between teacher educators, teachers and learners. In T. Wood & B. Jaworski (Eds.), International handbook of mathematics teacher education The mathematics teacher educator as a developing professional (pp. 31–56). Sense.

    Google Scholar 

  • Miles, M. B., & Huberman, M. A. (1994). Qualitative data analysis: An expanded sourcebook (2nd ed.). SAGE Publications Inc.

    Google Scholar 

  • National Council of Teachers of Mathematics (NCTM). (2000). Principles and standards for school mathematics. NCTM.

    Google Scholar 

  • Olson, J., White, P., & Sparrow, L. (2011). Influence of lesson study on teachers’ mathematics pedagogy. In L. Hart, A. Alston, & A. Murata (Eds.), Lesson-study research and practice in mathematics: Learning together (pp. 39–57). Springer.

    Chapter  Google Scholar 

  • Osana, H. P., Lacroix, G. L., Tucker, B. J., & Desrosiers, C. (2006). The role of content knowledge and problem features on preservice teachers’ appraisal of elementary mathematical tasks. Journal of Mathematics Teacher Education, 9(4), 347–380. https://doi.org/10.1007/s10857-006-4084-1

    Article  Google Scholar 

  • Otten, S., & Soria, V. M. (2014). Relationships between students’ learning and their participation during enactment of middle school algebra tasks. ZDM, 46(5), 815–827.

  • Papatistodemou, E., Potari, D., & Potta-Pantazi, D. (2014). Prospective teachers’ attention on geometrical tasks. Educational Studies in Mathematics, 86, 1–18. https://doi.org/10.1007/s10649-013-9518-y

    Article  Google Scholar 

  • Parrish, C. W., Snider, R. B., & Creager, M. A. (2022). Investigating how secondary prospective teachers plan to launch cognitively demanding tasks. Journal of Mathematics Teacher Education. https://doi.org/10.1007/s10857-022-09534-7

    Article  Google Scholar 

  • Pettersen, A., & Nortvedt, G. A. (2017). Identifying competency demands in mathematical tasks: Recognising what matters. International Journal of Science and Mathematics Education. https://doi.org/10.1007/s10763-017-9807-5

    Article  Google Scholar 

  • Philipp, R. A. (2007). Mathematics teachers’ beliefs and affect. In F. K. Lesters (Ed.), Second handbook of research on mathematics teaching and learning (pp. 257–315). Information Age Publishing.

    Google Scholar 

  • Remillard, J. T. (1999). Curriculum materials in mathematics education reform: A framework for examining teachers’ curriculum development. Curriculum Inquiry, 29(3), 315–342.

    Article  Google Scholar 

  • Remillard, J. T., & Bryans, M. B. (2004). Teachers’ orientation toward mathematics curriculum materials. Implications for teacher learning. Journal for Research in Mathematics Education, 35(5), 352–388.

    Article  Google Scholar 

  • Rule, A., & Hallagan, J. (2007). Using hands-on materials to write algebraic generalizations (grades 5–8). In: Paper presented at the Annual Conference of the Association of Mathematics Teachers of New York State.

  • Sapkota, B. (2022). Preservice teachers’ conceptualizations of mathematical tasks. The Mathematics Educator, 30(2), 3–32.

    Google Scholar 

  • Senk, S., Tatto, M., Reckase, M., Rowley, G., Peck, R., & Bankov, K. (2012). Knowledge of future primary teachers for teaching mathematics: An international comparative study. ZDM (Online first), (pp. 1–18).

  • 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, 132–138.

    Article  Google Scholar 

  • Stake, R. (1995). The art of case study research. Sage Publications Inc.

    Google Scholar 

  • Stake, R. (2005). Qualitative case studies. In N. K. Denzin & Y. S. Lincoln (Eds.), The Sage handbook of qualitative research (3rd ed., pp. 443–466). Sage.

    Google Scholar 

  • Steele, M. D., Hillen, A. F., & Smith, M. S. (2013). Developing mathematical knowledge for teaching in a methods course: The case of function. Journal of Mathematics Teacher Education, 16, 451–482.

  • Stein, M. K., Grover, B. W., & Henningsen, M. (1996). Building student capacity for mathematical thinking and reasoning: An analysis of mathematical tasks used in reform classrooms. American Educational Research Journal, 33(2), 455–488.

    Article  Google Scholar 

  • Stein, M. K., Remillard, J., & Smith, M. S. (2007a). How curriculum influences student learning. Information Age Publishing In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (Vol. 1, pp. 319–369). Charlotte.

    Google Scholar 

  • Stein, M. K., Remillard, J., & Smith, M. S. (2007b). How curriculum influences student learning. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 319–370). Information Age Publishing.

    Google Scholar 

  • 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, 268–275.

    Article  Google Scholar 

  • Stein, M. K., Smith, M. S., Henningsen, M., & Silver, E. A. (2000). Implementing standards-based mathematics instruction: A casebook for professional development. Teachers College Press.

    Google Scholar 

  • Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2009). Implementing standards-based mathematics instruction: A casebook for professional development (2nd ed.). Teachers College Press.

    Google Scholar 

  • Strauss, A., & Corbin, J. (1998). Basics of qualitative research: Techniques and procedures for developing grounded theory (2nd ed.). Sage.

    Google Scholar 

  • Stylianides, A. J., & Stylianides, G. J. (2008). Studying the classroom implementation of tasks: High-level mathematical tasks embedded in “real-life” contexts. Teacher and Teacher Education, 24(4), 859–875. https://doi.org/10.1016/j.tate.2007.11.015

    Article  Google Scholar 

  • Sullivan, P., & Mornane, A. (2014). Exploring teachers’ use of, and students’ reactions to, challenging mathematics tasks. Mathematics Education Research Journal, 26, 193–213. https://doi.org/10.1007/s13394-013-0089-0

    Article  Google Scholar 

  • Tarr, J. E., Reys, R. E., Reys, B. J., Chavez, O., Shih, J., & Osterlind, S. J. (2008). The impact of middle-grades mathematics curricula and the classroom learning environment on student achievement. Journal for Research in Mathematics Education, 39(3), 247–280.

    Google Scholar 

  • Tekkumru-Kisa, M., & Stein, M. K. (2014). Using contrasting video cases of enactment of cognitively demanding science tasks in professional development. In J. L. Polman, E. A. Kyza, D. K. O ’Neill, I. Tabak, W. R. Penuel, A. S. Jurow, K. O ’Connor, T. Lee, & L. D ’Amico (Eds.), 11th International Conference of the Learning Sciences (Vol. 2, pp. 808–815). Boulder, CO: International Society of the Learning Sciences, Inc.

  • Tekkumru-Kısa, M., Stein, M. K., & Doyle, W. (2020). Theory and research on tasks revisited: Task as a context for students’ thinking in the era of ambitious reforms in mathematics and science. Educational Researcher. https://doi.org/10.3102/0013189X20932480p.1-12

    Article  Google Scholar 

  • 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). World Scientific Publishing Company.

    Chapter  Google Scholar 

  • Wilhelm, A. G. (2014). Mathematics teachers’ enactment of cognitively demanding tasks: Investigating links to teachers’ knowledge and conceptions. Journal for Research in Mathematics Education, 45, 636–674.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Wilkie, K. J. (2016). Learning to teach upper primary school algebra: Changes to teachers’ mathematical knowledge for teaching functional thinking. Mathematics Education Research Journal, 28(2), 245–275.

    Article  Google Scholar 

  • Yeo, J. B. (2017). Development of a framework to characterise the openness of mathematical tasks. International Journal of Science and Mathematics Education, 15, 175–191.

  • Yin, R. K. (2014). Case study research: Design and methods. Sage.

    Google Scholar 

  • Yurekli, B., Bostan, M. I., & Cakiroglu, E. (2020). Sources of preservice teachers’ self-efficacy in the context of a mathematics teaching methods course. Journal of Education for Teaching, 46(5), 631–645. https://doi.org/10.1080/02607476.2020.1777068

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No funding was received for conducting this study. The authors have no financial or proprietary interests in any material discussed in this article.

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  1. Department of Mathematics and Science Education, Faculty of Education, Düzce University, 81620, Düzce, Turkey

    Emine Aytekin-Kazanç

  2. Department of Mathematics and Science Education, Middle East Technical University, Ankara, Turkey

    Mine Işıksal-Bostan

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  1. Emine Aytekin-Kazanç
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  2. Mine Işıksal-Bostan
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All authors contributed to the study’s conception and design. Material preparation, data collection, and analysis were performed by all authors. The first draft of the manuscript was written by all authors. All authors read and approved the final manuscript.

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Correspondence to Emine Aytekin-Kazanç.

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Aytekin-Kazanç, E., Işıksal-Bostan, M. A prospective teacher’s conceptions about the nature of mathematical tasks and professional knowledge within task unfolding. J Math Teacher Educ (2023). https://doi.org/10.1007/s10857-023-09584-5

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