# Multiple Approaches to Problem Posing: Theoretical Considerations Regarding its Definition, Conceptualisation, and Implementation

### Abstract

The importance of mathematical problem posing has been acknowledged by many researchers. In this theoretical paper, we want to capture different meanings and aspects of problem posing by approaching it from three different levels: (1) by comparing definitions, (2) by relating it to other constructs, and (3) by referring to research and teaching settings. The first level is an attempt to organise existing definitions of problem posing. The result of this analysis are five categories, which shows that there is no consensus regarding the conceptualisations of problem posing. In the second level, we examine how problem posing is conceived by the research community compared to other mathematical constructs, such as problem solving, mathematical creativity, or modelling. Finally, in the third level, we summarise possible ways of implementing problem posing in research and teaching settings as they are depicted in the relevant literature. Given this broad variance regarding the conceptualisations of problem posing, we attempt to provide some arguments as to whether there is a need for consensus on a commonly accepted concept of problem posing.

### Downloads

### References

Abramovich, S., & Cho, E. K. (2006). Technology as a medium for elementary preteachers’ problem-posing experience in Mathematics. Journal of Computers in Mathematics and Science Teaching, 25(4), 309–323.

Barbosa, J. C. (2003) What is mathematical modelling? In S. J. Lamon, W. A. Parker, & S. K. Houston (Eds.), Mathematical modelling: A way of life. ICTMA11 (pp. 227–234). Horwood Publishing.

Bonotto, C. (2010a). Realistic mathematical modeling and problem posing. In R. Lesh, P. L. Galbraith, C. R. Haines, & A. Hurford (Eds.), Modeling students’ mathematical modeling competencies (pp. 399–408). Springer.

Bonotto, G. (2010b). Engaging students in mathematical modelling and problem posing activities. Journal of Mathematical Modelling and Application, 1(3), 18–32.

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. Ellerton, & J. Cai (Eds.), Mathematical problem posing. From research to effective practice (pp. 103–123). Springer.

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

Cai, J., & Hwang, S. (2002). Generalised and generative thinking in US and Chinese students’ mathematical problem solving and problem posing. The Journal of Mathematical Behavior, 21(4), 401–421.

Cai, J., & Hwang, S. (2003). A perspective for examining the link between problem posing and problem solving. In N. A. Pateman, B. J. Dougherty, & J. T. Zilliox (Eds.), Proceedings of the 2003 Joint Meeting of PME and PMENA (Vol. 3, pp. 103–110). University of Hawaii.

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.

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

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.

Cai, J., Moyer, J. C., Wang, N., Hwang, S., Nie, B., & Garber, T. (2013). Mathematical problem posing as a measure of curricular effect on students’ learning. Educational Studies in Mathematics, 83(1), 57–69.

Caniglia, J. (2016). Writing to Learn Mathematics through Formulating Problems. In J. K. Dowdy & Y. Gao (Eds.), Pump it up (pp. 105–110). Sense Publishers.

Christou, C., Mousoulides, N., Pittalis, M., Pitta-Pantazi, D., & Sriraman, B. (2005). An empirical taxonomy of problem posing processes. ZDM – The International Journal on Mathematics Education, 37(3), 149–158.

Cifarelli, V. V., & Sevim, V. (2015). Problem posing as re-formulation and sense-making within problem solving. In F. M. Singer, N. Ellerton, & J. Cai (Eds.), Mathematical problem posing. From research to effective practice (pp. 177–194). Springer.

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

Crespo, S. (2015). A collection of problem-posing experiences for prospective mathematics teachers that make a difference. In F. M. Singer, N. Ellerton, & J. Cai (Eds.), Mathematical problem posing. From research to effective practice (pp. 493–511). Springer.

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.

Dillon, J. T. (1982). Problem finding and solving. Journal of Creative Behavior, 16(2), 97–111.

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(3), 303–328.

Duncker, K. (1945). On problem solving (Psychological Monographs, 58 [5, Serial No. 270]). American Psychological Association.

Einstein, A., & Infeld, L. (1938). The evolution of physics. Cambridge University Press.

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(1), 87–101.

Ellerton, N. F., & Clarkson, P. C. (1996). Language factors in mathematics teaching and learning. In A. Bishop, K. Clements, C. Keitel, J. Kilpatrick, & C. Laborde (Eds.), International handbook of mathematics education (pp. 987–1033). Kluwer.

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

English, L. D., Fox, J. L., & Watters, J. J. (2005). Problem posing and solving with mathematical modeling. Teaching Children Mathematics, 12(3), 156–163.

Estrada, J., & Santos, M. (1999). Eliciting students’ dilemmas through activities that involve posing questions or re-formulation of problems. In F. Hitt & M. Santos (Eds.), Proceedings of PME - NA 21 (Vol. 2, pp. 566–571). Cuernavaca, Morelos, Mexico.

Freire, P. (1970). Pedagogy of the oppressed (MB Ramos, Trans.). Seabury Press.

Freudenthal, H. (1991). Revisiting mathematics education. China lectures. Kluwer Academic Publishers.

Gonzales, N. A. (1996). Problem formulation: Insights from student generated questions. School Science and Mathematics, 96(3), 152–157.

Gonzales, N. A. (1998). A blueprint for problem posing. School Science and Mathematics, 98(8), 448¬–456.

Greer, B. (1992). Multiplication and division as models of situations. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 276–295). Macmillan.

Grundmeier, T. A. (2015). Developing the problem-posing abilities of prospective elementary and middle school teachers. In F. M. Singer, N. Ellerton, & J. Cai (Eds.), Mathematical problem posing. From research to effective practice (pp. 411–431). Springer.

Guilford, J. P. (1967). The nature of human intelligence. McGraw-Hill.

Hadamard, J. (1954). The psychology of invention in the mathematical field. Dover Publications.

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.

Jay, E., & Perkins, D. N. (1997). Creativity’s compass: A review of problem finding. In M. A. Runco (Ed.), Creativity research handbook (pp. 257–293). Hampton.

Joklitschke, J., Baumanns, L., & Rott, B., (2019). The intersection of problem posing and creativity: A review. In M. Nolte (Ed.) Proceedings of the 11th Conference MCG11 (pp. 59–67). WTM Verlag.

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.

Koichu, B. (2020). Problem posing in the context of teaching for advanced problem solving. International Journal of Educational Research, 102, 101428. https://doi.org/10.1016/j.ijer.2019.05.001

Kontorovich, I., Koichu, B., Leikin, R., & Berman, A. (2011). Indicators of creativity in mathematical problem posing: How indicative are they? In In M. Avotiņa, D. Bonka, H. Meissner, L. Ramāna, L. Sheffield, & E. Velikova (Eds.), Proceedings of the 6th International Conference Creativity in Mathematics Education and the Education of Gifted Students (pp. 120–125). University of Latvia.

Kwek, M. L. (2015). Using problem posing as a formative assessment tool. In F. M. Singer, N. Ellerton, & J. Cai (Eds.), Mathematical problem posing. From research to effective practice (pp. 273–292). Springer.

Lakatos, I. (1976). Proofs and refutations: The logic of mathematical discovery. Cambridge University Press.

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.

Leung, S. S. (1997). On the role of creative thinking in problem posing. ZDM – The International Journal on Mathematics Education, 29(3), 81–85.

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.

Lin, P. J. (2004). Supporting teachers on designing problem-posing tasks as a tool of assessment to understand students’ mathematical learning. In M. J. Hoines & A. B. Fuglestad (Eds.), Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education, (v. 3, pp. 257–264). Bergen University College.

Malaspina, U., Mallart, A., & Font, V. (2015). Development of teachers’ mathematical and didactic competencies by means of problem posing. In K. Krainer & N. Vondrová (Eds.), Proceedings of CERME 9 (pp. 2861–2866). ERME.

Mamona-Downs, J., & Downs, M. (2005). The identity of problem solving. The Journal of Mathematical Behavior, 24(3–4), 385–401.

Mamona-Downs, J., & Papadopoulos, I. (2017). Επίλυση Προβλήματος στα Μαθηματικά [Problem solving in mathematics]. Crete University Press.

Marquardt, M., & Waddill, D. (2004). The power of learning in action learning: A conceptual analysis of how the five schools of adult learning theories are incorporated within the practices of action learning. Action Learning: Research and Practice, 1(2), 185–202.

Mayring, P. (2014). Qualitative content analysis: Theoretical background and procedures. In A. Bikner-Ahsbahs, C. Knipping, & N. Presmeg (Eds.), Approaches to qualitative research in mathematics education (pp. 365–380). Springer Science & Business Media B.V.

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

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

Newell, A., & Simon, H. A. (1972). Human problem solving. Prentice-Hall.

Olson, J. C., & Knott, L. (2013). When a problem is more than a teacher’s question. Educational Studies in Mathematics, 83(1), 27–36.

O’Neil, J., & Marsick, V. J. (1994). Becoming critically reflective through action reflection learning™. New Directions for Adult and Continuing Education, 1994(63), 17–30.

Osana, H. P., & Pelczer, I. (2015). A review on problem posing in teacher education? In F. M. Singer, N. Ellerton, & J. Cai (Eds.), Mathematical problem posing. From research to effective practice (pp. 469–492). Springer.

Pólya, G. (1957, 2nd ed.). How to Solve It. Princeton University Press.

Pólya, G. (1981). Mathematical discovery (combined ed.). Wiley.

Ruthven, K. (2020). Problematising learning to teach through mathematical problem posing. International Journal of Educational Research, 102, 101455. https://doi.org/10.1016/j.ijer.2019.07.004

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.

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

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

Silver, E. A. (1995). The nature and use of open problems in mathematics education: mathematical and pedagogical perspectives. ZDM – The International Journal on Mathematics Education, 27(2), 67–72.

Silver, E. A. (1997). Fostering creativity through instruction rich in mathematical problem solving and problem posing. ZDM – The International Journal on Mathematics Education, 29(3), 75–80.

Silver, E.A., Mamona-Downs, J., Leung, S., & Kenny, P.A. (1996). Posing mathematical Problems in a complex environment: An exploratory study. Journal for Research in Mathematics Education, 27(3), 293–309.

Singer, F. M., & Moscovici, H. (2008). Teaching and learning cycles in a constructivist approach to instruction. Teaching and Teacher Education, 24(6), 1613–1634.

Singer, F. M., & Voica, C. (2015). Is problem posing a tool for identifying and developing mathematical creativity? In F. M. Singer, N. Ellerton, & J. Cai (Eds.), Mathematical problem posing. From research to effective practice (pp. 141–174). Springer.

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.

Singer, F. M., Pelczer, I., & Voica, C. (2011). Problem posing and modification as a criterion of mathematical creativity. In M. Pytlak, T. Rowland, & E. Swoboda (Eds.), Proceedings of CERME7 (pp. 1133–1142). University Rzeszów.

Singer, F.M., Ellerton, N., Cai, J., & Leung, E. (2011). Problem posing in mathematics learning and teaching: A research agenda. In B. Ubuz (Ed.), Proceedings of the 35th PME Conference, (v. 1, pp. 137–166). PME.

Stanic, G., & Kilpatrick, J. (1988). Historical perspectives on problem solving in the mathematics curriculum. In R. Charles & E. Silver (Eds.), The teaching and assessing of mathematical problem solving (pp. 1–22). National Council of Teachers of Mathematics.

Stickles, P. (2011). An analysis of secondary and middle school teachers’ mathematical problem posing. Investigations in Mathematics Learning, 3(2), 1–34.

Stillman, G. (2015). Problem finding and problem posing for mathematical modelling. In K. E. D. Ng & N. H. Lee (Eds.), Mathematical modelling. From theory to practice (pp. 41–56). World Scientific.

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

Tichá, M., & Hošpesová, A. (2009). Problem posing and development of pedagogical content knowledge in pre-service teacher training. In V. Durand-Guerrier, S. Soury-Lavergne, & F. Arzarello (Eds.), Proceedings of CERME 6 (pp. 1941–1950). Lyon, France.

Tichá, M., & Hošpesová, A. (2013). Developing teachers’ subject didactic competence through problem posing. Educational Studies in Mathematics, 83(1), 133–143.

Van Harpen, X. Y., & Sriraman, B. (2013). Creativity and mathematical problem posing: an analysis of high school student’s mathematical posing in China and the USA. Educational Studies in Mathematics, 82(2), 201–221.

Verschaffel, L., De Corte, E., & Lasure, S. (1994). Realistic considerations in mathematical modeling of school arithmetic word problems. Learning and Instruction, 4, 273–294.

Verschaffel, L., Van Dooren, W., Chen, L., & Stessens, K. (2009). The relationship between posing and solving division-with-remainder problems among Flemish upper elementary school children. In L. Verschaffel, B. Greer, W. van Dooren, & S. Mukhopadhyay (Eds.), Words and worlds: Modelling verbal descriptions of situations (pp. 143–160). Sense Publishers.

Wilson, J. W., Fernandez, M. L., & Hadaway, N. (1993). Mathematical problem solving. In P. S. Wilson (Ed.), Research ideas for the classroom: High school mathematics (pp. 57–78). Macmillan.

In order to ensure both the widest dissemination and protection of material published in CEPS Journal, we ask Authors to transfer to the Publisher (Faculty of Education, University of Ljubljana) the rights of copyright in the Articles they contribute. This enables the Publisher to ensure protection against infringement.