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On understanding mathematical problem-posing processes

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Abstract

Problem posing engages students in generating new problems based on given situations (including mathematical expressions or diagrams) or changing (i.e., reformulating) existing problems. Problem posing has been at the forefront of discussion over the past few decades. One of the important topics studied is the process of problem posing as experienced by students and teachers. This paper focuses on problem-posing processes and models thereof. We first provide an overview of previous research and then present the results of a scoping review regarding recent research on problem-posing processes. This review covers 75 papers published between 2017 and 2022 in top mathematics education research journals. We found that some of the prior research directly attempted to examine problem-posing processes, whereas others examined task variables related to problem-posing processes. We conclude this paper by proposing a model for problem-posing processes that encompasses four phases: orientation, connection, generation, and reflection. We also provide descriptions of the four phases of the model. The paper ends with suggestions for future research related to problem-posing processes in general and the problem-posing model proposed in particular.

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References

Papers that are part of the review are marked with an asterisk; annotated papers are marked with two asterisks.

  • * Armstrong, A. (2017). Bricolage in middle years school mathematics. For the Learning of Mathematics, 37(2), 19–24

  • Baumanns, L., & Rott, B. (2021). Rethinking problem-posing situations: A review. Investigations in Mathematics Learning, 13(2), 59–76.

    Article  Google Scholar 

  • *, ** Baumanns, L., & Rott, B. (2022a). The process of problem posing: Development of a descriptive phase model of problem posing. Educational Studies in Mathematics, 110, 251–269. https://doi.org/10.1007/s10649-021-10136-y. (Following Schoenfeld (1985), the authors analyzed processes of 64 preservice teachers to develop a descriptive model of problem-posing processes with five phases: Situation Analysis (analyzing situations and prompts), Variation (posing by altering givens), Generation (posing new problems), Problem Solving (solving self-posed problems), and Evaluation (assessing problems using individual criteria).)

    Article  Google Scholar 

  • Baumanns, L., & Rott, B. (2022). Identifying metacognitive behavior in problem-posing processes. International Journal of Science and Mathematics Education (online First). https://doi.org/10.1007/s10763-022-10297-z

    Article  Google Scholar 

  • Brown, S. I., & Walter, M. I. (1983). The art of problem posing. Lawrence Erlbaum Associates.

    Google Scholar 

  • Cai, J. (2022). What research says about teaching mathematics through problem posing. Education & Didactique, 16(3), 31–50.

    Article  Google Scholar 

  • 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.

    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. F. Ellerton, & J. Cai (Eds.), Mathematical problem posing: From research to effective practice (pp. 3–34). Springer.

    Chapter  Google Scholar 

  • ** Cai, J., Koichu, B., Rott, B., Zazkis, R., & Jiang, C. (2022). Mathematical problem posing: Task variables, processes, and products. In C. Fernandez, S. Llinares, A. Gutierrez, & N. Planas (Eds.), Proceedings of the 45th of the Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 119–145). International Group for the Psychology of Mathematics Education. (This is a paper written for a Research Forum at the 2022 PME international meeting. Both the paper and the Research Forum focused on task variable research on problem posing. This paper first reviewed previous efforts at understanding problem-posing processes and then examined the impact of task variables on both products and processes of problem posing at individual, group, and classroom levels).

  • Cai, J., & Leikin, R. (2020). Affect in mathematical problem posing: Conceptualization, advances, and future directions for research. Educational Studies in Mathematics, 105, 287–301.

    Article  Google Scholar 

  • Christou, C., Mousoulides, N., Pittalis, M., Pitta-Pantazi, D., & Sriraman, B. (2005). An empirical taxonomy of problem-posing process. ZDM—Mathematics Education, 37(3), 149–158. https://doi.org/10.1007/s11858-005-0004-6

  • Cruz, M. (2006). A mathematical problem–formulating strategy. International Journal for Mathematics Teaching and Learning, 79–90.

  • * Downton, A., & Sullivan, P. (2017). Posing complex problems requiring multiplicative thinking prompts students to use sophisticated strategies and build mathematical connections. Educational Studies in Mathematics, 95, 303–328. https://doi.org/10.1007/s10649-017-9751-x

  • Ellerton, N. F. (1986). Children’s made-up mathematics problems—A new perspective on talented mathematicians. Educational Studies in Mathematics, 17, 261–271.

    Article  Google Scholar 

  • English, L. D. (1998). Children’s problem posing within formal and informal contexts. Journal for Research in Mathematics Education, 29(1), 83–106.

    Article  Google Scholar 

  • English, L. D. (2020). Teaching and learning through mathematical problem posing: Commentary. International Journal of Educational Research, 102, 101451.

    Article  Google Scholar 

  • * Ergene, Ö. (2021). Posing probability problems related to continuous and discrete sample space. International Journal of Mathematical Education in Science and Technology, 53(2), 311–336. https://doi.org/10.1080/0020739X.2021.2004464

  • *, ** Erkan, B., & Kar, T. (2022). Pre-service mathematics teachers’ problem-formulation processes: Development of the revised active learning framework. The Journal of Mathematical Behavior, 65. https://doi.org/10.1016/j.jmathb.2021.100918. (Using task-based interviews, the authors investigated how nine preservice teachers initiated problem posing. Most started by solving the given problem to understand its structure, find different contexts for the same mathematical structure, or determine its purpose. Using their insights, the authors extended the Active Learning Framework, adding details regarding cognitive, metacognitive, and instructional factors).

  • Felmer, P., Pehkonen, E., & Kilpatrick, J. (Eds.). (2016). Posing and solving mathematical problems: Advances and new perspectives. Springer.

  • * Fosse, T., & Meaney, T. (2020). Using problem posing to bring real-life into the mathematics classroom: Can it be too real? For the Learning of Mathematics, 40(3), 40–45

  • Garofalo, J., & Lester, F. K., Jr. (1985). Metacognition, cognitive monitoring, and mathematical performance. Journal for Research in Mathematics Education, 16(3), 163–176.

    Article  Google Scholar 

  • Goos, M., & Kaya, S. (2019). Understanding and promoting students’ mathematical thinking: A review of research published in ESM. Educational Studies in Mathematics, 103, 7–25. https://doi.org/10.1007/s10649-019-09921-7

    Article  Google Scholar 

  • Guo, M., Leung, F. K. S., & Hu, X. (2020). Affective determinants of mathematical problem posing: The case of Chinese Miao students. Educational Studies in Mathematics, 105, 367–387. https://doi.org/10.1007/s10649-020-09972-1

    Article  Google Scholar 

  • * Guo, Y., Yan, J., & Men, T. (2021). Chinese junior high school students’ mathematical problem-posing performance. ZDM—Mathematics Education, 53, 905–917. https://doi.org/10.1007/s11858-021-01240-7

  • Hartmann, L.-M. (in press). Prozesse beim Problem Posing zu gegebenen realweltlichen Situationen und die Verbindungen zum Modellieren [Processes of problem posing in given real-world situations and their relation to modeling]. Dissertation, University of Münster.

  • Horsley, T. (2019). Tips for improving the writing and reporting quality of systematic, scoping, and narrative reviews. Journal of Continuing Education in the Health Professions, 39(1), 54–57. https://doi.org/10.1097/CEH.0000000000000241

    Article  Google Scholar 

  • *, ** Jung, H., & Magiera, M. T. (2021). Connecting mathematical modeling and social justice through problem posing. Mathematical Thinking and Learning, 25(2), 232–251. https://doi.org/10.1080/10986065.2021.1966713. (This study uses a social justice lens to articulate decisions preservice teachers make while posing modeling problems and reflecting on their problems. While posing problems, the problem posers needed to consider whether their problems raised awareness of social justice issues on a broader (macro) or narrower (micro) level).

    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). Lawrence Erlbaum Associates.

    Google Scholar 

  • Koichu, B. (2020). Problem posing in the context of teaching for advanced problem solving. International Journal of Educational Research, 102, 101428.

    Article  Google Scholar 

  • Koichu, B., & Kontorovich, I. (2013). Dissecting success stories on mathematical problem posing: A case of the Billiard Task. Educational Studies in Mathematics, 83, 71–86.

    Article  Google Scholar 

  • Kontorovich, I., Koichu, B., Leikin, R., & Berman, A. (2012). An exploratory framework for handling the complexity of students’ mathematical problem posing in small groups. Journal of Mathematical Behavior, 31(1), 149–161.

    Article  Google Scholar 

  • Lavy, I., & Bershadsky, I. (2003). Problem posing via “what if not?” strategy in solid geometry: A case study. The Journal of Mathematical Behavior, 22(4), 369–387.

    Article  Google Scholar 

  • Leavy, A., & Hourigan, M. (2022). The Framework for Posing Elementary Mathematics Problems (F-PosE): Supporting teachers to evaluate and select problems for use in elementary mathematics. Educational Studies in Mathematics, 111, 147–176. https://doi.org/10.1007/s10649-022-10155-3

    Article  Google Scholar 

  • 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

  • * Nedaei, M., Radmehr, F., & Drake, M. (2021). Exploring undergraduate engineering students’ mathematical problem-posing: The case of integral-area relationships in integral calculus. Mathematical Thinking and Learning, 24(2), 149–175. https://doi.org/10.1080/10986065.2020.1858516

  • Page, M. J., McKenzie, J. E., Bossuyt, P. M., Boutron, I., Hoffmann, T. C., Mulrow, C.,... Moher, D. (2021). The PRISMA 2020 statement: An updated guideline for reporting systematic reviews. PLOS Medicine, 18(3). https://doi.org/10.1136/bmj.n71

  • *, ** Palmér, H., & van Bommel, J. (2020). Young students posing problem-solving tasks: What does posing a similar task imply to students? ZDM—Mathematics Education, 52, 743–752. https://doi.org/10.1007/s11858-020-01129-x. (This is one of the few papers addressing problem posing with young students, in this case with 6-year-olds. In this study, researchers first asked the children to solve a problem, following which these children were asked to pose a similar task to a friend. The researchers not only explored how the children interpreted the notion of similar but also explored how these children posed such similar problems).

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Pelczer, I., & Gamboa, F. (2009). Problem posing: Comparison between experts and novices. In M. Tzekaki, M. Kaldrimidou, & H. Sakonidis (Eds.), Proceedings of the 33th Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 353–360). International Group for the Psychology of Mathematics Education.

  • Pittalis, M., Christou, C., Mousoulides, N., & Pitta-Pantazi, D. (2004). A structural model for problem posing. In M. J. Hoines & A. B. Fuglestad (Eds.), Proceedings of the 28th annual meeting of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 49–56). Bergen University College.

  • Pólya, G. (1945). How to solve it. Princeton University Press.

    Book  Google Scholar 

  • * Radmehr, F., & Drake, M. (2017). Exploring students’ mathematical performance, metacognitive experiences and skills in relation to fundamental theorem of calculus. International Journal of Mathematical Education in Science and Technology, 48(7), 1043–1071. https://doi.org/10.1080/0020739X.2017.1305129

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

    Article  Google Scholar 

  • Schoenfeld, A. H. (1985b). Mathematical problem solving. Academic Press.

    Google Scholar 

  • Segal, R., Stupel, M., Sigler, A., & Jahangiril, J. (2018). The effectiveness of ‘what if not’ strategy coupled with dynamic geometry software in an inquiry-based geometry classroom. International Journal of Mathematical Education in Science and Technology, 49(7), 1099–1109. https://doi.org/10.1080/0020739X.2018.1452302

    Article  Google Scholar 

  • * Silber, S., & Cai, J. (2017). Pre-service teachers’ free and structured mathematical problem posing. International Journal of Mathematical Education in Science and Technology, 48(2), 163–184. https://doi.org/10.1080/0020739X.2016.1232843

  • *, ** Silber, S., & Cai, J. (2021). Exploring underprepared undergraduate students’ mathematical problem posing. ZDM—Mathematics Education, 53, 877–889. https://doi.org/10.1007/s11858-021-01272-z. (This is one of the few studies involving mathematically underprepared undergraduate students as subjects. The authors gave 45 undergraduate students four problem-posing tasks. The findings indicated that many of the students were able to identify key ideas of the given situations and use them in their posed problems. The study suggests the role of knowledge in the problem-posing process).

    Article  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., & Cai, J. (2013). Problem-posing research in mathematics education: New questions and directions. Educational Studies in Mathematics, 83(1), 1–7.

    Article  Google Scholar 

  • Song, S. H., Yim, J. H., Shin, E. J., & Lee, H. H. (2007). Posing problems with use the ‘What if not?’ strategy in NIM game. In J. H. Woo, H. C. Lew, K. S. Park, & D. Y. Seo (Eds.), Proceedings of the 31st Conference of the International Group for the Psychology of Mathematics Education, Vol. 4 (pp. 193–200). International Group for the Psychology of Mathematics Education.

  • Törner, G., & Azarello, F. (2012). Grading mathematics education research journals. EMS Newsletter, 86, 52–54.

    Google Scholar 

  • Wessman-Enzinger, N. M., & Mooney, E. S. (2021). Conceptual models for integer addition and subtraction. International Journal of Mathematical Education in Science and Technology, 52(3), 349–376. https://doi.org/10.1080/0020739X.2019.1685136

    Article  Google Scholar 

  • Williams, S. R., & Leatham, K. R. (2017). Journal quality in mathematics education. Journal for Research in Mathematics Education, 48(4), 369–396.

    Article  Google Scholar 

  • Xie, J., & Masingila, J. O. (2017). Examining interactions between problem posing and problem solving with prospective primary teachers: A case of using fractions. Educational Studies in Mathematics, 96, 101–118. https://doi.org/10.1007/s10649-017-9760-9

    Article  Google Scholar 

  • * Yao, X., & Manouchehri, A. (2019). Middle school students’ generalizations about properties of geometric transformations in a dynamic geometry environment. The Journal of Mathematical Behavior, 55, 1–19. https://doi.org/10.1016/j.jmathb.2019.04.002

  • * Zhang, L., Cai, J., Song, N., Zhang, H., Chen, T., Zhang, Z., & Guo, F. (2022). Mathematical problem posing of elementary school students: the impact of task format and its relationship to problem solving. ZDM—Mathematics Education, 54(3), 497–512. https://doi.org/10.1007/s11858-021-01324-4

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Funding

During the preparation of this paper, Jinfa Cai is supported by a grant from the National Science Foundation (DRL- 2101552). Any opinions expressed herein are those of the authors and do not necessarily represent the views of the National Science Foundation.

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  1. University of Delaware, Newark, USA

    Jinfa Cai

  2. University of Cologne, Cologne, Germany

    Benjamin Rott

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Cai, J., Rott, B. On understanding mathematical problem-posing processes. ZDM Mathematics Education 56, 61–71 (2024). https://doi.org/10.1007/s11858-023-01536-w

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