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Teachers as redesigners of curriculum to teach mathematics through problem posing: conceptualization and initial findings of a problem‑posing project

A Publisher Correction to this article was published on 13 April 2021

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Abstract

Problem posing, the process of formulating problems based on a given situation, is an essential practice in mathematics and other disciplines. Although this is acknowledged in policy documents, problem posing is neither substantively nor consistently included in the school mathematics curriculum. In this paper, we first comment on the state of problem posing in school mathematics and discuss three recommendations to improve its integration into curriculum materials and classroom practice. These recommendations present a low barrier to entry for teachers and students and require only minor changes to common mathematics classroom activities and curriculum materials. Based on the three recommendations, as well as the features of effective teacher professional development, a program was created to investigate longitudinally the impact of problem-posing professional development on teachers’ conceptions of problem posing and their design of lessons to teach mathematics using problem posing, as well as on students’ learning. Initial findings are presented, including the significant changes in teachers’ conceptions of problem posing and teachers’ design of lessons to teach mathematics using problem posing, as well as the impact on students’ learning.

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References

  • Birman, B. F., Desimone, L., Porter, A. C., & Garet, M. S. (2000). Designing professional development that works. Educational Leadership, 57(8), 28–33.

    Google Scholar 

  • Bolam, R., McMahon, A., Stoll, L., Thomas, S., & Wallace, M. (with Greenwood, A., Hawkey, K., Ingram, M., Atkinson, A., & Smith, M). (2005). Creating and sustaining effective professional learning communities (Research Report No. RR637). Bristol, England: University of Bristol, Department for Education and Skills.

  • Bonotto, C. (2013). Artifacts as sources for problem-posing activities. Educational Studies in Mathematics, 83, 37–55.

    Article  Google Scholar 

  • Borko, H. (2004). Professional development and teacher learning: Mapping the terrain. Educational Researcher, 33(8), 3–15.

    Article  Google Scholar 

  • Bryk, A. S., Gomez, L. M., Grunow, A., & LeMahieu, P. G. (2005). Learning to improve: How American schools can get better at getting better. Harvard Education Press.

    Google Scholar 

  • Cai, J. (2004). Why do U.S. and Chinese students think differently in mathematical problem solving? Exploring the impact of early algebra learning and teachers’ beliefs. Journal of Mathematical Behavior, 23, 135–167.

    Google Scholar 

  • Cai, J., Ding, M., & Wang, T. (2014). How do exemplary Chinese and U.S. mathematics teachers view instructional coherence? Educational Studies in Mathematics, 85(2), 265–280.

    Article  Google Scholar 

  • Cai, J., & Howson, A. G. (2013). Toward an international mathematics curriculum. In M. A. Clements, A. Bishop, C. Keitel, J. Kilpatrick, & K. S. F. Leung (Eds.), Third international handbook of mathematics education research (pp. 949–974). Springer.

    Google Scholar 

  • Cai, J., & Hwang, S. (2002). Generalized and generative thinking in U.S. and Chinese students’ mathematical problem solving and problem posing. Journal of Mathematical Behavior, 21, 401–421.

    Article  Google Scholar 

  • Cai, J., & Hwang, S. (2020). Learning to teach through mathematical problem posing: Theoretical considerations, methodology, and directions for future research. International Journal of Educational Research, 102, 101391. https://doi.org/10.1016/j.ijer.2019.01.001

    Article  Google Scholar 

  • Cai, J., & Jiang, C. (2017). An analysis of problem-posing tasks in Chinese and US elementary mathematics textbooks. International Journal of Science and Mathematics Education, 15(8), 1521–1540.

    Article  Google Scholar 

  • Cai, J., Hwang, S., Jiang, C., & Silber, S. (2015). Problem-posing research in mathematics education: Some answered and unanswered questions. In F. M. Singer, N. Ellerton, & J. Cai (Eds.), Mathematical problem posing: From research to effective practice (pp. 3–34). Springer.

    Chapter  Google Scholar 

  • Cai, J., Chen, T., Li, X., Xu, R., Zhang, S., Hu, Y., Zhang, L., & Song, N. (2020). Exploring the impact of a problem-posing workshop on elementary school mathematics teachers’ problem posing and lesson design. International Journal of Educational Research., 102, 101404. https://doi.org/10.1016/j.ijer.2019.02.004

    Article  Google Scholar 

  • Carpenter, T. P., Fennema, E., Peterson, P. L., Chiang, C. P., & Loef, M. (1989). Using knowledge of children’s mathematical thinking in classroom teaching: An experimental study. American Educational Research Journal, 26, 499–531.

    Article  Google Scholar 

  • Chinese Ministry of Education. (2001a). Curriculum standards for school mathematics of nine-year compulsory education (Trial version). Beijing Normal University Press.

    Google Scholar 

  • Chinese Ministry of Education. (2001b). Guidelines for curriculum reform of elementary education (Trial version). Beijing Normal University Press.

    Google Scholar 

  • Chinese Ministry of Education. (2003). Curriculum standards of high school mathematics (Trial version). People’s Education Press.

    Google Scholar 

  • Chinese Ministry of Education. (2011). Mathematics curriculum standard of compulsory education (2011 version). Beijing Normal University Press.

    Google Scholar 

  • Chinese Ministry of Education. (2017/2020). Curriculum standards of high school mathematics (2011 version and revised in 2020). People’s Education Press.

  • Cobb, P. (1994). Where is the mind? Constructivist and sociocultural perspectives on mathematical development. Educational Researcher, 23(7), 13–20.

    Article  Google Scholar 

  • Cochran-Smith, M., & Lytle, S. L. (1999). Relationships of knowledge and practice: Teacher learning in communities. Review of Research in Education, 24, 249–305.

    Google Scholar 

  • Crespo, S. (2003). Learning to pose mathematical problems: Exploring changes in preservice teachers’ practices. Educational Studies in Mathematics, 52, 243–270.

    Article  Google Scholar 

  • Darling-Hammond, L. (1994). Who will speak for the children? How “Teach for America” hurts urban schools and students. Phi Delta Kappan, 76(1), 21–34.

    Google Scholar 

  • Darling-Hammond, L., Wei, R. C., Andree, A., Richardson, N., & Orphanos, S. (2009). Professional learning in the learning profession: A status report on teacher development in the United States and abroad. Executive summary. Dallas, TX: National Staff Development Council.

  • Desimone, L. M., Porter, A. C., Garet, M. S., Yoon, K. S., & Birman, B. F. (2002). Effects of professional development on teachers’ instruction: Results from a three-year longitudinal study. Educational Evaluation and Policy Analysis, 24(2), 81–112.

    Article  Google Scholar 

  • Dieteker, L., Males, L. M., Amador, J. M., & Earnest, D. (2018). Curricular noticing: A framework to describe teachers’ interactions with curriculum materials. Journal for Research in Mathematics Education, 49(5), 521–532.

    Article  Google Scholar 

  • DuFour, R. (2004). What is a “professional learning community”? Educational Leadership, 61(8), 6–11.

    Google Scholar 

  • Einstein, A., & Infeld, L. (1938). The evolution of physics. Simon & Schuster.

    Google Scholar 

  • Fan, L., Chen, J., Zhu, Y., Qiu, X., & Hu, Q. (2004). Textbook use within and beyond Chinese mathematics classrooms: A study of 12 secondary schools in Kunming and Fuzhou of China. In L. Fan, N. Y. Wong, J. Cai, & S. Li (Eds.), How Chinese learn mathematics: Perspectives from insiders (pp. 228–261). World Scientific.

    Chapter  Google Scholar 

  • Garet, M. S., Porter, A. C., Desimone, L., Birman, B. F., & Yoon, K. S. (2001). What makes professional development effective? Results from a national sample of teachers. American Educational Research Journal, 38(4), 915–945.

    Article  Google Scholar 

  • Garet, M. S., Wayne, A. J., Stancavage, F., Taylor, J., Walters, K., Song, M., & Doolittle, F. (2010). Middle school mathematics professional development impact study: Findings after the first year of implementation (NCEE 2010–4009). Washington, DC: National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences, U.S. Department of Education.

  • Garet, M. S., Wayne, A. J., Stancavage, F., Taylor, J., Eaton, M., Walters, K., Song, M., Brown, S., Hurlburt, S., Zhu, P., Sepanik, S., Doolittle, F., & Warner, E. (2011). Middle school mathematics professional development impact study: Findings after the second year of implementation (NCEE 2011–4024). National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences, U.S. Department of Education.

  • Guskey, T. R. (1999). Making the most of professional development. In J. H. Block, S. T. Everson, & T. R. Guskey (Eds.), Comprehensive school reform: A program perspective (pp. 417–430). Kendall/Hunt.

    Google Scholar 

  • Guskey, T. R., & Yoon, K. S. (2009). What works in professional development? Phi Delta Kappan, 90(7), 495–500.

    Article  Google Scholar 

  • Hadamard, J. W. (1945). Essay on the psychology of invention in the mathematical field. Princeton University Press.

    Google Scholar 

  • Hošpesová, A., & Tichá, M. (2015). Problem posing in primary school teacher training. In F. M. Singer, N. Ellerton, & J. Cai (Eds.), Mathematical problem posing: From research to effective practice (pp. 433–447). Springer.

    Chapter  Google Scholar 

  • Kennedy, M. M. (1998). Education reform and subject matter knowledge. Journal of Research in Science Teaching, 35, 249–263.

    Article  Google Scholar 

  • Kennedy, D. (2005). Editorial. Science, 310(5749), 787. https://doi.org/10.1126/science.310.5749.787b

    Article  Google Scholar 

  • Kilpatrick, J. (1987). Problem formulating: Where do good problems come from? In A. H. Schoenfeld (Ed.), Cognitive science and mathematics education (pp. 123–147). Erlbaum.

    Google Scholar 

  • Klamkin, M. S. (1968). On the teaching of mathematics so as to be useful. Educational Studies in Mathematics, 1(1–2), 126–160. https://doi.org/10.1007/BF00426240

    Article  Google Scholar 

  • Li, X., Song, N., Hwang, S., & Cai, J. (2020). Learning to teach mathematics through problem-posing: Teachers’ beliefs and performance on problem posing. Educational Studies in Mathematics, 105, 325–347.

    Article  Google Scholar 

  • Lloyd, G. M., Cai, J., & Tarr, J. E. (2017). Issues in curriculum studies: Evidence-based insights and future directions. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 824–852). National Council of Teachers of Mathematics.

    Google Scholar 

  • Morris, A. K., Hiebert, J., Sisofo, E., & Hwang, S. (2018). The continuous improvement of teaching in ordinary classrooms: Getting it right through testing and revising implementations of theories. Manuscript submitted for publication.

  • Mosteller, F. (1980). The next 100 years of science. Science, 209(4452), 21–23. https://doi.org/10.1126/science.7280662

    Article  Google Scholar 

  • National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Author.

    Google Scholar 

  • National Council of Teachers of Mathematics. (1991). Professional standards for teaching mathematics. Author.

    Google Scholar 

  • Newmann, F. (Ed.). (1996). Authentic achievement: Restructuring schools for intellectual quality. Jossey Bass.

    Google Scholar 

  • Olson, J. C., & Knott, L. (2013). When a problem is more than a teacher’s question. Educational Studies in Mathematics, 83, 27–36.

    Article  Google Scholar 

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

    Google Scholar 

  • Redding, C., & Viano, S. L. (2018). Co-creating school innovations: Should self-determination be a component of school improvement? Teachers College Record, 120, 110303.

    Google Scholar 

  • Remillard, J. T. (2005). Examining key concepts in research on teachers’ use of mathematics curricula. Review of Educational Research, 75(2), 211–246.

    Article  Google Scholar 

  • Resnick, L. B. (1987). Education and learning to think. National Academy Press.

    Google Scholar 

  • Richardson, V. (1996). The role of attitudes and beliefs in learning to teach. In J. Sikula (Ed.), Handbook of research on teacher education (2nd ed., pp. 102–119). Macmillan.

    Google Scholar 

  • Roschelle, J., Shechtman, N., Tatar, D., Hegedus, S., Hopkins, B., Empson, S., Gallagher, L. P., et al. (2010). Integration of technology, curriculum, and professional development for advancing middle school mathematics: Three large-scale studies. American Educational Research Journal, 47(4), 833–878.

    Article  Google Scholar 

  • Rosenshine, B., Meister, C., & Chapman, S. (1996). Teaching students to generate questions: A review of the intervention studies. Review of Educational Research, 66(2), 181–221.

    Article  Google Scholar 

  • Senk, S. L., & Thompson, D. R. (Eds.). (2003). Standards-based school mathematics curricula: What are they? What do students learn? Erlbaum.

    Google Scholar 

  • Silver, E. A. (1994). On mathematical problem posing. For the Learning of Mathematics, 14(1), 19–28.

    Google Scholar 

  • Silver, E. A., & Cai, J. (1996). An analysis of arithmetic problem posing by middle school students. Journal for Research in Mathematics Education, 27(5), 521–539.

    Article  Google Scholar 

  • Singer, F. M., Ellerton, N. F., & Cai, J. (Eds.). (2015). Mathematical problem posing: From research to effective practice. Springer.

    Google Scholar 

  • Stigler, J. W., Hiebert, J., & Givvin, K. B. (2018). Does VAM + MET = improved teaching? In R. P. Ferretti & J. Hiebert (Eds.), Teachers, teaching, and reform: Perspectives on efforts to improve educational outcomes (pp. 56–74). Routledge.

    Google Scholar 

  • Thompson, A. G. (1992). Teachers’ beliefs and conceptions: A synthesis of the research. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 127–146). Macmillan.

    Google Scholar 

  • Timperley, H., & Alton-Lee, A. (2008). Reframing teacher professional learning: An alternative policy approach to strengthening valued outcomes for diverse learners. Review of Research in Education, 32(1), 328–369. https://doi.org/10.3102/0091732X07308968

    Article  Google Scholar 

  • Toluk-Uçar, Z. (2009). Developing pre-service teachers understanding of fractions through problem posing. Teaching and Teacher Education, 25(1), 166–175.

    Article  Google Scholar 

  • Van Harpen, X. Y., & Sriraman, B. (2013). Creativity and mathematical problem posing: An analysis of high school students’ mathematical problem posing in China and USA. Educational Studies in Mathematics, 82, 201–221.

    Article  Google Scholar 

  • Vescio, V., Ross, D., & Adams, A. (2008). A review of research on the impact of professional learning communities on teaching practice and student learning. Teacher and Teaching Evaluation: An International Journal of Research and Studies, 24(1), 80–91.

    Article  Google Scholar 

  • Yoon, K. S., Garet, M., Birman, B., & Jacobson, R. (2006). Examining the effects of mathematics and science professional development on teachers’ instructional practice: Using professional development activity log. Council of Chief State School Officers.

    Google Scholar 

  • Yoon, K. S., Duncan T., Lee, S. W., Scarloss, B., & Shapley, K. L. (2007). Reviewing the evidence on how teacher professional development affects student achievement (Issues and Answers Report REL 2007, No. 033). Washington, DC: U.S. Department of Education, Institute of Education Sciences, National Center for Education Evaluation and Regional Assistance, Regional Educational Laboratory Southwest. Retrieved from http://ies.ed.gov/ncee/edlabs.

  • Zhang, H., & Cai, J. (2020). Teaching mathematics through problem posing: Insights from an analysis of teaching cases [Manuscript submitted for publication].

  • Zhang, D., Yao, Y., & Cai, J. (2021). Elementary mathematics teachers learning to teach through problem posing: Initial findings of a longitudinal study [Paper presentation]. In The 14th International Congress on Mathematical Education, Shanghai, China.

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Correspondence to Jinfa Cai.

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Cai, J., Hwang, S. Teachers as redesigners of curriculum to teach mathematics through problem posing: conceptualization and initial findings of a problem‑posing project. ZDM Mathematics Education 53, 1403–1416 (2021). https://doi.org/10.1007/s11858-021-01252-3

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Keywords

  • Teacher as curriculum redesigners
  • Problem posing
  • Teaching through problem posing
  • Teacher learning
  • Longitudinal study