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The Framework for Posing Elementary Mathematics Problems (F-PosE): Supporting Teachers to Evaluate and Select Problems for Use in Elementary Mathematics

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Abstract

An essential task for mathematics teachers is posing problems. Selecting mathematics problems that develop mathematical proficiency and engage students in desirable mathematical practices is a critical decision-making process. We present the Framework for Posing Elementary Mathematics Problems (F-PosE) developed to focus prospective teacher noticing on desirable features of mathematics problems and inform decision-making processes around the selection of problems for use in elementary classrooms. Development of the framework was informed by a three-phase design research process consisting of an extensive review of the literature, document content analysis and successive testing of mathematics problems in elementary classrooms in partnership with teachers and children. Consequently, it draws from emergent practice informed by the collective endeavour of a community of educators. The framework consists of eight indicators: use of a motivating and engaging context, clarity in language and cultural context, curriculum coherence, attention to cognitive demand, an appropriate number of solution steps to support reasoning, a variety of solution strategies, facilitating multiple solutions and opportunity for success. This F-PosE provides a critical focusing lens for prospective teachers when creating and selecting mathematics problems specifically for use in elementary classrooms.

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References

  • Australian Curriculum, Assessment and Reporting Authority. (2012). The Australian Curriculum: Mathematics, Version 3.0, 23 January 2012. Author.

  • Bonotto, C. (2013). Artifacts as sources for problem-posing activities. Educational Studies in Mathematics, 83(1), 37–55.

    Article  Google Scholar 

  • Bonotto, C., & Dal Santo, L. (2015). On the Relationship Between Problem Posing, Problem Solving, and Creativity in the Primary School. In F. M. Singer, N. F. Ellerton, & J. Cai (Eds.), Mathematical Problem Posing: From Research to Effective Practice (pp. 103–123). https://doi.org/10.1007/978-1-4614-6258-3

  • Brenner, M. E., & Moschkovich, J. N. (Eds.). (2002). Everyday and academic mathematics in the classroom. National Council of Teachers of Mathematics.

    Google Scholar 

  • Brown, S. I., & Walter, M. I. (1990). The art of problem posing (2nd ed.). Erlbaum.

    Google Scholar 

  • Cai, J. (1998). An investigation of U.S. and Chinese students’ mathematical problem posing and problem solving. Mathematics Education Research Journal, 10(7), 37–50. https://doi.org/10.1007/BF03217121

    Article  Google Scholar 

  • Cai, J. (2003). What research tells us about teaching mathematics through problem solving. In F. Lester (Ed.), Research and issues in teaching mathematics through problem solving (pp. 241–254). National Council of Teachers of Mathematics.

    Google Scholar 

  • Cai, J., & Cifarelli, V. V. (2005). Exploring mathematical exploration: How two college students formulated and solved their own mathematical problems. Focus on Learning Problems in Mathematics, 27(3), 43–72.

    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. F. Ellerton, & J. Cai (Eds.), Mathematical Problem Posing: From Research to Effective Practice (pp. 3–34). https://doi.org/10.1007/978-1-4614-6258-3

  • 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. https://doi.org/10.1016/j.ijer.2019.01.001

  • Cai, J., & Hwang, S. (2002). Generalized and generative thinking in U.S. and Chinese students’ mathematical problem solving and problem posing. The Journal of Mathematical Behavior, 21, 401–421. https://doi.org/10.1016/S0732-3123(02)00142-6

    Article  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’ conceptions on problem posing and lesson design. International Journal of Educational Research, 102. https://doi.org/10.1016/j.ijer.2019.02.004

  • Cai, J., & Merlino, F. J. (2011). Metaphor: A powerful means for assessing students’ mathematical disposition. In D. J. Brahier, & W. Speer (Eds.), Motivation and disposition: Pathways to learning mathematics (pp.147–156). National Council of Teachers of Mathematics 2011 Yearbook. NCTM.

  • Carpenter, T. P., Franke, M. L., Jacobs, V. R., Fennema, E., & Empson, S. B. (1998). A Longitudinal Study of Invention and Understanding in Children’s Multidigit Addition and Subtraction. Journal for Research in Mathematics Education, 29, 3–20.

    Article  Google Scholar 

  • Chapman, O. (2008). Helping pre-service elementary teachers develop flexibility in using word problems in their teaching. Paper presented at the Annual meeting of the North American Chapter of the International group for the Psychology of Mathematics Education, Toronto.

  • Chapman, O. (2012). Prospective elementary school teachers’ ways of making sense of mathematical problem posing. PNA, 6(4), 135–146.

    Article  Google Scholar 

  • Charmaz, K. (2006). Constructing grounded theory: A practical guide through qualitative analysis. SAGE.

    Google Scholar 

  • Christiansen, B., & Walther, G. (1986). Task and activity. In B. Christiansen, A. G. Howson, & M. Otte (Eds.), Perspectives on mathematics education (pp. 243–307). Reidel.

    Chapter  Google Scholar 

  • Cobb, P., Confrey, J., diSessa, A., Lehrer, R., & Schauble, L. (2003). Design experiments in education research. Educational Researcher, 32(1), 9–13.

    Article  Google Scholar 

  • Crespo, S. (2003). Learning to pose mathematical problems: Exploring changes in preservice teachers’ practices. Educational Studies in Mathematics, 52(3), 243–270.

    Article  Google Scholar 

  • Crespo, S., & Harper, F. K. (2020). Learning to pose collaborative mathematics problems with secondary prospective teachers. International Journal of Educational Research, 102. https://doi.org/10.1016/j.ijer.2019.05.003

  • Crespo, S., & Sinclair, N. (2008). What makes a problem mathematically interesting? Inviting prospective teachers to pose better problems. Journal of Mathematics Teacher Education, 11(5), 395–415.

    Article  Google Scholar 

  • Department for Education (2014). The national curriculum in England: Mathematics programme of study. https://www.gov.uk/government/publications/national-curriculum-in-england-mathematics-programmes-of-study/national-curriculum-in-england-mathematics-programmes-of-study. Accessed 13 October 2020.

  • Devlin, K. (2000). The four faces of mathematics. In M. J. Burke & F. R. Curcio (Eds.), Learning Mathematics for a New Century: 2000 Yearbook of the National Council of Teachers of Mathematics (pp. 16–27). NCTM.

    Google Scholar 

  • Doyle, W. (1988). Work in mathematics classes: The context of students’ thinking during instruction. Educational Psychologist, 23, 167–180.

    Article  Google Scholar 

  • Duncker, K. (1945). On problem solving. Psychological Monographs, 58(5), 270.

    Article  Google Scholar 

  • Ellerton, N. F. (2013). Engaging pre-service middle-school teacher-education students in mathematical problem posing: Development of an active learning framework. Educational Studies in Mathematics, 83, 87–101. https://doi.org/10.1007/s10649-012-9449-z

  • English, L. D. (2009). The changing realities of classroom mathematical problem solving. In L. Verschaffel, B. Greer, W. Van Dooren, & S. Mukhopadhyay (Eds.), Words and worlds: Modelling verbal descriptions of situations (pp. 351–362). Sense Publishers.

    Google Scholar 

  • English, L. D. (2020). Teaching and learning through mathematical problem posing: Commentary. International Journal of Educational Research, 102. https://doi.org/10.1016/j.ijer.2019.06.014

  • Gravemeijer, K., & Cobb, P. (2006). Design research from a learning design perspective. In J. Van den Akker, K. Gravemeijer, S. McKenney, & N. Nieveen (Eds.), Educational Design Research (pp. 45–85). Routledge.

    Google Scholar 

  • Grundmeier, T. A. (2015). Developing the Problem-Posing Abilities of Prospective Elementary and Middle School Teachers. In F. M. Singer, N. F. Ellerton, & J. Cai. (Eds.), Mathematical Problem Posing: From Research to Effective Practice (pp 411–431). https://doi.org/10.1007/978-1-4614-6258-3

  • Guberman, R., & Leikin, R. (2013). Interesting and difficult mathematical problems: Changing teachers’ views by employing multiple-solution tasks. Journal of Mathematics Teacher Education, 16, 33–56.

    Article  Google Scholar 

  • Guthrie, G. (1986). Current research in developing countries: The impact of curriculum reform on teaching. Teaching and Teacher Education, 2, 81–89.

    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(5), 524–549.

    Article  Google Scholar 

  • Hiebert, J., & Wearne, D. (1993). Instructional tasks, classroom discourse, and students’ learning in second-grade arithmetic. American Educational Research Journal, 30(2), 393–425.

    Article  Google Scholar 

  • Hošpesová, A. & Tichá, M. (2015). Problem Posing in Primary School Teacher Training In F. M. Singer, N. F. Ellerton, & J. Cai (Eds.), Mathematical Problem Posing: From Research to Effective Practice (pp 433–447). https://doi.org/10.1007/978-1-4614-6258-3

  • Kelly, A. E., Lesh, R. A., & Baek, J. Y. (Eds.). (2008). Handbook of design research methods in education: Innovations in science, technology, engineering, and mathematics learning and teaching. Routledge.

    Google Scholar 

  • Klein, S., & Leikin, R. (2020). Opening mathematical problems for posing open mathematical tasks: What do teachers do and feel? Educational Studies in Mathematics, 105, 349–365. https://doi.org/10.1007/s10649-020-09983-y

    Article  Google Scholar 

  • Koichu, B. (2020). Problem posing in the context of teaching for advanced problem solving. International Journal of Educational Research, 102. https://doi.org/10.1016/j.ijer.2019.05.001

  • Koichu, B., Harel, G., & Manaster, A. (2013). Ways of thinking associated with mathematics teachers’ problem posing in the context of division of fractions. Instructional Science, 41(4), 681–698.

    Article  Google Scholar 

  • Lappan, G., & Phillips, E. (1998). Teaching and Learning in the Connected Mathematics Project. In L. Leutzinger (Ed.), Mathematics in the Middle (pp. 83–92). National Council of Teachers of Mathematics.

    Google Scholar 

  • Leavy, A.M. & Frischemeier, D. (2022). Launching a statistical enquiry: Posing statistically worthwhile questions. Statistics Education Research Journal, 21(1). Article 10. https://doi.org/10.52041/serj.v21i1.226

  • Leavy, A. M., & Hourigan, M. (2020). Posing Mathematically Worthwhile Problems: Developing the Problem Posing Skills of prospective Teachers. Journal of Mathematics Teacher Education, 23, 341–361. https://doi.org/10.1007/s10857-018-09425-w

    Article  Google Scholar 

  • Leavy, A. M., & Hourigan, M. (2022). Enhancing the mathematical problem posing skills of prospective teachers through a mathematical letter writing initiative. Journal of Mathematics Teacher Education, 25, 293–320. https://doi.org/10.1007/s10857-021-09490-8

  • Lee, J. E. (2012). Prospective elementary teachers’ perceptions of real-life connections reflected in posing and evaluating story problems. Journal of Mathematics Teacher Education, 15, 429–452.

    Article  Google Scholar 

  • Leikin, R., & Elgrably, H. (2020). Problem posing through investigations for the development and evaluation of proof-related skills and creativity skills of prospective high school mathematics teachers. International Journal of Educational Research, 102. https://doi.org/10.1016/j.ijer.2019.04.00

  • Leung, S. S., & Silver, E. A. (1997). The role of task format, mathematics knowledge, and creative thinking on the arithmetic problem posing of prospective elementary school teachers. Mathematics Education Research Journal, 9(1), 5–24.

    Article  Google Scholar 

  • Liu, Q., Liu, J., Cai, J., & Zhang, Z. (2020). The relationship between domain- and task-specific self-efficacy and mathematical problem posing: A large-scale study of eighth-grade students in China. Educational Studies in Mathematics, 105, 407–431. https://doi.org/10.1007/s10649-020-09977-w

    Article  Google Scholar 

  • Ma, L. (1999). Knowing and teaching elementary mathematics. Lawrence Erlbaum Associates.

    Book  Google Scholar 

  • McKenney, S., Nieveen, N., & Van den Akker, J. (2006). Design research from a curriculum perspective. In J. Van den Akker, K. Gravemeijer, S. McKenney, & N. Nieveen (Eds.), Educational design research (pp. 62–90). Routledge.

    Google Scholar 

  • National Council for Curriculum and Assessment (2018). Draft specification: Primary Mathematics, junior infants to second class. Available at: https://ncca.ie/media/3148/primary_mathsspec_en.pdf

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

  • National Governors Association Center for Best Practices. (2010). Common core state standards for mathematics. National Governors Association Center for Best Practices, Council of Chief State School Officers.

  • Nicol, C. C., & Crespo, S. M. (2006). Learning to teach with mathematics textbooks: How preservice teachers interpret and use curriculum materials. Educational Studies in Mathematics, 62(3), 331–355.

    Article  Google Scholar 

  • Norton, A., & Kastberg, S. (2012). Learning to pose cognitively demanding tasks through letter writing. Journal of Mathematics Teacher Education, 15(2), 109–130.

    Article  Google Scholar 

  • O’Shea, J., & Leavy, A. M. (2013). Teaching mathematical problem-solving from an emergent constructivist perspective: The experiences of Irish primary teachers. Journal of Mathematics Teacher Education, 16(4), 293–318. https://doi.org/10.1007/s10857-013-9235-6

    Article  Google Scholar 

  • Phillips, E., & Crespo, S. (1996). Developing Written Communication in Mathematics through Math Penpals. For the Learning of Mathematics, 16(1), 15–22.

    Google Scholar 

  • Plomp, T. (2007). Educational Design Research: An Introduction. In T. Plomp & N. Nieveen (Eds.) An Introduction to Educational Design Research (pp. 9–35). Proceedings of the seminar conducted at the East China Normal University, Shanghai (PR China), November 23–26, 2007.

  • Polya, G. (1954). Mathematics and plausible reasoning. Princeton University Press.

    Book  Google Scholar 

  • Polya, G. (1973). How to solve it: A new aspect of mathematics method. Princeton University Press.

    Google Scholar 

  • Ralph, N., Birks, M., & Chapman, Y. (2015). The methodological dynamism of grounded theory. International Journal of Qualitative Methods, 14(4), 1–6.

    Article  Google Scholar 

  • Richey, R., & Klein, J. (2005). Developmental research methods: Creating knowledge from instructional design and development practice. Journal of Computing in Higher Education, 16(2), 23–38.

    Article  Google Scholar 

  • Rosli, R., Capraro, M. M., Goldsby, D., Gonzalez, E., Onwuegbuzie, A. J. & Capraro, R. M. (2015). Middle-Grade Preservice Teachers’ Mathematical Problem Solving and Problem Posing. In F.M. Singer, N.F. Ellerton, & J. Cai (Eds.), Mathematical Problem Posing: From Research to Effective Practice (pp. 334–355). https://doi.org/10.1007/978-1-4614-6258-3

  • Ruthven, K. (2020). Problematising learning to teach through mathematical problem posing. International Journal of Educational Research, 102. https://doi.org/10.1016/j.ijer.2019.07.004

  • Schindler, M., & Bakker, A. (2020). Affective field during collaborative problem posing and problem solving: A case study. Educational Studies in Mathematics, 105, 303–324 (2020). https://doi.org/10.1007/s10649-020-09973-0

  • Schoenfeld, A. H. (1983). Problem solving in the mathematics curriculum: A report, recommendations, and an annotated bibliography. The Mathematical Association of America.

    Google Scholar 

  • Schoenfeld, A. H. (1988). When good teaching leads to bad results: The disasters of “well-taught” mathematics courses. Educational Psychologist, 23(2), 145–166.

    Article  Google Scholar 

  • Schoenfeld, A. H. (1989). Exploration of students’ mathematical beliefs and behaviours. Journal of Research in Mathematics Education, 20(4), 338–355.

    Article  Google Scholar 

  • Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning. Macmillan.

    Google Scholar 

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

    Google Scholar 

  • Silver, E. A. (1997). Fostering creativity through instruction rich in mathematical problem solving and problem posing. ZDM-Mathematics Education, 3, 75–80.

    Article  Google Scholar 

  • Silver, E. A., & Cai, J. (2005). Assessing students’ mathematical problem posing. Teaching Children Mathematics, 12(3), 129–135.

    Article  Google Scholar 

  • Silver, E. A., Ghousseini, H., Gosen, D., Charalambous, C., & Strawhun, B. T. (2005). Moving from rhetoric to praxis: Issues faced by teachers in having students consider multiple solutions for problems in the mathematics classroom. Journal of Mathematical Behavior, 24, 287–301.

    Article  Google Scholar 

  • Silver, E. A., Mamona-Downs, J., Leung, S. S., & Kenney, P. A. (1996). Posing mathematical problems: An exploratory study. Journal for Research in Mathematics Education, 27, 293–309.

    Article  Google Scholar 

  • Silver, E. A., & Stein, M. K. (1996). The QUASAR Project: The ‘revolution of the possible’ in mathematics instructional reform in urban middle schools. Urban Education, 30, 476–521.

    Article  Google Scholar 

  • Singer, F. M., & Voica, C. (2013). A problem-solving conceptual framework and its implications in designing problem-posing tasks. Educational Studies in Mathematics, 83(1), 9–26.

    Article  Google Scholar 

  • Smith, M., Bill, V., & Hughes, E. (2008). Thinking through a Lesson Protocol: A Key for Successfully Implementing High-Level Tasks. Mathematics Teaching in the Middle School, 14(3), 132–138.

    Article  Google Scholar 

  • Smith, M. S., & Stein, M. K. (1998). Selecting and creating mathematical tasks: From research to practice. Mathematics Teaching in the Middle School, 3, 344–350.

    Article  Google Scholar 

  • Star, J. R., & Newton, K. J. (2009). The nature and development of experts’ strategy flexibility for solving equations. ZDM-Mathematics Education, 41, 557–567.

    Article  Google Scholar 

  • 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, 455–488.

    Article  Google Scholar 

  • Stein, M. K., & Lane, S. (1996). Instructional tasks and the development of student capacity to think and reason: An analysis of the relationship between teaching and learning in a reform mathematics project. Educational Research and Evaluation, 2, 50–80.

    Article  Google Scholar 

  • Stein, M. K., Remillard, J. T., & Smith, M. S. (2007). How Curriculum Influences Student Learning. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 319–369). Information Age Publishing.

    Google Scholar 

  • Stoyanova, E., & Ellerton, N. F. (1996). A framework for research into students’ problem posing in school mathematics. In P. C. Clarkson (Ed.), Technology in mathematics education (pp. 518–525). Mathematics Education Research Group of Australasia.

    Google Scholar 

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

    Google Scholar 

  • Vacc, N. (1993). Questioning in the mathematics classroom. Arithmetic Teacher, 41(2), 88–91.

    Article  Google Scholar 

  • van den Akker, J. (1999). Principles and methods of development research. In J. van den Akker, R. Branch, K. Gustafson, N. Nieveen, & T. Plomp (Eds.), Design approaches and tools in education and training (pp. 1–14). Kluwer Academic Publishers.

    Chapter  Google Scholar 

  • Verschaffel, L., De Corte, E., & Lasure, S. (1994). Realistic considerations in mathematical modeling of school arithmetic word problems. Learning and Instruction, 4, 273–294. https://doi.org/10.1016/0959-4752(94)90002-7

    Article  Google Scholar 

  • Verschaffel, L., Greer, B., & De Corte, E. (2000). Making sense of word problems. Swets and Zeiglinger.

    Google Scholar 

  • Voica, C., Singer, F. M., & Stan, E. (2020). How are motivation and self-efficacy interacting in problem-solving and problem-posing? Educational Studies in Mathematics, 105, 487–517. https://doi.org/10.1007/s10649-020-10005-0

    Article  Google Scholar 

  • Watson, A., & Ohtani, M. (2015). Task design in mathematics education: An ICMI study 22. Springer International.

    Book  Google Scholar 

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  1. Department of STEM Education, Mary Immaculate College - University of Limerick, South Circular Road, Limerick, Ireland

    Aisling Leavy & Mairéad Hourigan

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Leavy, A., Hourigan, M. The Framework for Posing Elementary Mathematics Problems (F-PosE): Supporting Teachers to Evaluate and Select Problems for Use in Elementary Mathematics. Educ Stud Math 111, 147–176 (2022). https://doi.org/10.1007/s10649-022-10155-3

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