Young students’ views on problem solving versus problem posing

Keywords: Problem posing, Problem solving, Early childhood, Mathematics

Abstract

For decades, problem solving has been of interest to researchers, and several studies have tried to capture the influence of students’ beliefs, attitudes and emotions towards mathematics and problem solving. However, problem posing as part of problem solving has not been investigated to the same extent. This article focuses on six-year-olds’ views on solving and posing problems. How do the students themselves describe their views on solving and posing problem-solving tasks, and what similarities and differences can be found? An educational design research study was conducted in three classes where the students first solved and then posed problem-solving tasks. Afterwards, the students were interviewed. In these interviews the students expressed positive views towards both solving and posing problem-solving tasks. The students expressed autonomy and challenge as positive when both solving and posing tasks. However, a posed task needed to be solved before being finished. Further, not all students considered problem posing to be a mathematical activity, and a plausible explanation for this is the students’ limited experience of problem posing.

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References

Anderson, T., & Shattuck, J. (2012). Design-based research: A decade of progress in education research? Educational Researcher, 41(1), 16–25. DOI: https://doi.org/10.3102/0013189X11428813

Antognazza, D., Di Martino, P., Pellandini, A., & Sbaragli, S. (2015). The flow of emotions in primary school problem solving. In K. Krainer, & N. Vondrová (Eds.), Proceedings of the Ninth Congress of the European Society for Research in Mathematics Education (pp. 1116–1122). Prague: Charles University

Batchelor, S., Torbeyns, J., & Verschaffel, L. (2019). Affect and mathematics in young children: An introduction. Educational Studies in Mathematics, 100(3), 201–209. DOI: https://doi.org/10.1007/s10649-018-9864-x

Boesen, J., Helenius, O., Bergqvist, E., Bergqvist, T., Lithner, J., Palm, T., & Palmberg, B. (2014). Developing mathematical competence: From the intended to the enacted curriculum. The Journal of Mathematical Behavior, 33(1), 72–87. DOI: https://doi.org/10.1016/j.jmathb.2013.10.001

Brown, S. I., & Walter, I. M. (2004). The art of problem posing (3rd edition). Hillsdale, NJ: Lawrence Erlbaum Associates.

Cai, J. (2010). Commentary on problem solving heuristics, affect, and discrete mathematics: A representational discussion. In B. Sriraman, & L. English (Eds.), Theories of mathematics education: Seeking new frontiers (pp. 251–258). Dordrecht: Springer.

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). Honolulu, HI: 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. DOI: https://doi.org/10.1016/j.ijer.2019.01.001

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). New York, NY: Springer.

Carrillo, J., Cruz, J. (2016). Problem-posing and questioning: Two tools to help solve problems. In P. Felmer, E. Pehkonen, & J. Kilpatrick (Eds.), Posing and solving mathematical problems: Advances and new perspectives (pp. 23–36). Cham, Switzerland: Springer.

Claessens, A., Engel, M., & Curran, C. F. (2014). Academic content, student learning, and the persistence of preschool effects. American Educational Research Journal, 51(2), 403–434. DOI: https://doi.org/10.3102/0002831213513634

Clements, D. H., & Sarama, J. (2016). Math, science, and technology in the early grades. The Future of Children, 26(2), 75–94. DOI: https://doi.org/10.1353/foc.2016.0013

Cobb, P., & Gravemeijer, K. (2008). Experimenting to support and understand learning processes. In A. E. Kelly, R. A. Lesh, & J. Y. Baek (Eds.), Handbook of design research methods in education: Innovations in science, technology, engineering, and mathematics learning and teaching (pp. 68–95). New York: Routledge.

Cross, C. T., Woods, T. A., & Schweingruber, H. (2009). Mathematics learning in early childhood: Paths toward excellence and equity. Washington, DC: National Research Council of the National Academics.

Csapó, B., & Funke, J. (Eds.). (2017). The nature of problem solving: Using research to inspire 21st century learning. Paris: OECD Publishing.

Debellis, V. A., & Goldin, G. A. (2006). Affect and meta-affect in mathematical problem solving: A representational perspective. Educational Studies in Mathematics, 63(2), 131–147. DOI: https://doi.org/10.1007/s10649-006-9026-4

Di Martino, P. (2019). Pupils’ view of problems: The evolution from kindergarten to the end of primary school. Educational Studies in Mathematics, 100(3), 291–307. DOI: https://doi.org/10.1007/s10649-018-9850-3

Dowker, A., Bennett, K., & Smith, L. (2012). Attitudes to mathematics in primary school children. Child Development Research. 2012, 124939. DOI: https://doi.org/10.1155/2012/124939

Dowker, A., Cheriton, O., Horton, R., & Mark, W. (2019). Relationships between attitudes and performance in young children’s mathematics. Educational Studies in Mathematics, 100(3), 211–230. DOI: https://doi.org/10.1007/s10649-019-9880-5

Ellerton, N. F., Singer, F. M., & Cai, J. (2015). Problem posing in mathematics: Reflecting on the past, energizing the present and foreshadowing the future. In F. M. Singer, N. F. Ellerton (Eds) Mathematical problem posing: From research to effective practice (pp. 547-556). New York, NY: Springer DOI: https://doi.org/10.1007/978-1-4614-6258-3_26

English, L. (1998). Children’s problem posing within formal and informal contexts. Journal for Research in Mathematics Education, 29(1), 83–106. DOI: https://doi.org/10.2307/749719

English, L., & Sriraman, B. (2010). Problem solving for the 21st century. In B. Sriraman, & L. English (Eds.), Theories of mathematics education: Advances in mathematics education (pp. 263–283). Berlin Heidelberg: Springer.

Giaconi, V., Varas, M. L., Tuohilampi, L., & Hannula, M. (2016). Affective factors and beliefs about mathematics of young Chilean children: Understanding cultural characteristics. In P. Felmer, E. Pehkonen, & J. Kilpatrick (Eds.), Posing and solving mathematical problems. Advances and new perspectives. London: Springer.

Hannula, M. S. (2011). The structure and dynamics of affect in mathematical thinking and learning. In M. Pytlak, T. Rowland, & E. Swoboda (Eds.), Proceeding of the Seventh Congress of the European Society for Research in Mathematics Education (pp. 34-60). Rzeszów, Poland: University of Rzeszów and ERME.

Hannula, M. S. (2016). Introduction. In G Kaiser (Ed.), Attitudes, beliefs, motivation and identity in mathematics education: An overview of the field and future directions (pp. 1-2). Cham, Switzerland: Springer.

Hannula, M., Pantziara, M., & Di Martino, P. (2018). Affect and mathematical thinking. Exploring developments, trends, and future directions. In T. Drefys, M. Artigue, D. Potari, S. Prediger, & K. Tuthven (Eds.), Developing research in mathematics education. Twenty years of communication, cooperation and collaboration in Europe. London: Routledge.

Jakobsson, A. (2012). Sociokulturella perspektiv på̊ lärande och utveckling

Lärande som begreppsmässig precisering och koordinering [Sociocultural perspectives on learning and development: Learning as conceptual clarification and coordination]. Pedagogisk Forskning i Sverige, 17(3–4), 152-170.

Kadir, Adelina, R., & Fatma, M. (2018). Enhancing students’ mathematical problem posing skill through writing in performance tasks strategy. Journal of Physics: Conference Series, 948, 1–7. DOI: https://doi.org/10.1088/1742-6596/948/1/012022

Lee, Y., Capraro, R. M., & Capraro, M. M. (2018). Mathematics teachers’ subject matter knowledge and pedagogical content knowledge in problem posing. International Electronic Journal of Mathematics Education, 13(2), 75–90. DOI: https://doi.org/10.12973/iejme/2698

Lesh, R., & Zawojewski, J. (2007). Problem solving and modelling. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 763–799). Charlotte, NC: National Council of Teachers of Mathematics and Information Age Publishing.

Leung, S. S. (1997). On the role of creative thinking in problem posing. ZDM - The International Journal on Mathematics Education, 97(2), 48–52. DOI: https://doi.org/10.1007/s11858-997-0004-9

Leung, S. S. (2013) Teachers implementing mathematical problem posing in the classroom: Challenges and strategies. Educational studies in Mathematics, 83(1), 103-116. DOI: https://doi.org/10.1007/s10649-012-9436-4

Liljedahl, P. (2017). On the edges of flow: Students engagement in problem solving. In T. Dooley, & G. Gueudet. (Eds.), Proceeding of the Tenth Congress of the European Society for Research in Mathematics Education (pp. 1146-1153). Dublin, Ireland: DCU Institute of Education and ERME.

Lowrie, T. (2002). Designing a framework for problem posing: Young children generating open-ended tasks. Contemporary Issues in Early Childhood, 3(3), 354–363. DOI: https://doi.org/10.2304/ciec.2002.3.3.4

Mason, J., & Johnston-Wilder, S. (2006). Designing and using mathematical tasks. St. Albans, UK: Tarquin.

McLeod, D. B. (1992). Research on affect in mathematics education: A reconceptualization. In D. A. Grouws (Ed.), Handbook of research on mathematics learning and teaching (pp. 575–596). New York, NY: MacMillan.

National Agency for Education. (2017). Curriculum for the primary school, preschool class and leisure time centre 2011. Stockholm: National Agency for Education.

National Agency for Education. (2018a). Curriculum for the compulsory school, preschool class and school-age educare. Stockholm: National Agency for Education.

National Agency for Education. (2018b). Curriculum for the preschool. Stockholm: National Agency for Education.

Niss, M., & Højgaard, T. (2019). Mathematical competencies revisited. Educational Studies in Mathematics, 102, 9-28. DOI: https://doi.org/10.1007/s10649-019-09903-9

Op’t Eynde, P., De Corte, E., & Verschaffel, L. (2001). “What to learn from what we feel?”: The role of children’s emotions in the mathematics classroom. In S. Volet, & S. Järvelä (Eds.), Motivation in learning contexts: Theoretical advances and methodological implications (pp. 149–167) (Advances in Learning and Instruction Series.). Oxford, UK: Elsevier Science.

Palmér, H., & van Bommel, J. (2018a). Problem solving in early mathematics teaching: A way to promote creativity? Creative Education, 9(10), 1775–1793. DOI: https://doi.org/10.4236/ce.2018.912129

Palmér, H., & van Bommel, J. (2018b). The role of and connection between systematization and representation when young children work on a combinatorial task. European Early Childhood Education Research Journal, 26(4), 562–573. DOI: https://doi.org/10.1080/1350293X.2018.1487141

Palmér, H., & van Bommel, J. (2018c). Young children’s feelings towards problem-solving tasks: What does “success” imply? In B. Rott, G. Törner, J. Peters-Dasdemir, A. Möller, & Safrudiannur (Eds.), Views and beliefs in mathematics education: The role of beliefs in the classroom (pp. 69–78). Cham: Springer. DOI: https://doi.org/10.1007/978-3-030-01273-1_7

Palmér, H., & van Bommel, J. (2020). Young students posing problem-solving tasks: What does posing a similar task imply to students? ZDM - the International Journal on Mathematics Education, 52(4), 743–752. DOI: https://doi.org/10.1007/s11858-020-01129-x

Passolunghi, M., Cargnelutti E., & Pellizzoni, S. (2019). The relation between cognitive and emotional factors and arithmetic problem-solving. Educational Studies in Mathematics, 100(3), 271–290. DOI: https://doi.org/10.1007/s10649-018-9863-y

Peng, A., Li, J., Nie, B., & Li, Y. (2017). Characteristics of teaching mathematical problem solving in China. In R. Huang, & Y. Li (Eds.), Teaching and learning mathematics through variation. Mathematics teaching and learning (pp. 111-125). Rotterdam: Sense Publishers.

Perry, B., & Dockett, S. (2008). Young children’s access to powerful mathematical ideas. In L. D. English (Ed.), Handbook of international research in mathematics education (pp. 75–108). New York, NY: Routledge.

Pinxten, M., Marsh, H. W., De Fraine, B., Van Den Noortgate, W., & Van Damme, J. (2013). Enjoying mathematics or feeling competent in mathematics? Reciprocal effects on mathematics achievement and perceived math effort expenditure. British Journal of Educational Psychology, 84(1), 152-174. DOI: https://doi.org/10.1111/bjep.12028

Prediger, S., Gravemeijer, K., & Confrey. J. (2015). Design research with a focus on learning processes: An overview on achievements and challenges. ZDM Mathematics Education, 47(6), 877–891. DOI: https://doi.org/10.1007/s11858-015-0722-3

Rogoff, B. (2003). The cultural nature of human development. Oxford: Oxford University Press.

Schoenfeld, A. (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 (pp. 165–197). New York: MacMillan.

Stickles, P. (2011). An analysis of secondary and middle school teachers’ mathematical problem posing. Investigations in Mathematics Learning, 3(2), 1–34. DOI: https://doi.org/10.1080/24727466.2011.11790301

Stoyanova, E. (2005). Problem-posing strategies used by years 8 and 9 students. Australian Mathematics Teacher, 61(3), 162–175.

Swedish Research Council. (2017). Good custom in research. Stockholm: Vetenskapsrådet.

Swedish Schools Inspectorate. (2009). The teaching of mathematics: The content and the effectiveness of the education. Kvalitetsgranskning rapport 2009:5. Stockholm: Skolinspektionen.

The Stationery Office. (1999). Primary school curriculum: Mathematics. Dublin: The Stationery Office.

Utdanningsdirektoratet. (2013). Læreplan i matematikk fellesfag. Retrieved from http://data.udir.no/kl06/MAT1-04.pdf

van Bommel, J., & Palmér, H. (2018). Enhancing young children’s understanding of a combinatorial task by using a duo of digital and physical artefacts. Early Years. DOI: https://doi.org/10.1080/09575146.2018.1501553

Watson, A., & Mason, J. (2002). Student-generated examples in the learning of mathematics. Canadian Journal of Science, Mathematics, and Technology Education, 2(2), 237–249. DOI: https://doi.org/10.1080/14926150209556516

Wedege, T., & Skott, J. (2006). Changing views and practices? A study of the KappAbel mathematics competition. Trondheim: Matematikcentret.

Wyndhamn, J., Riesbeck, E., & Schoultz, J. (2000). Problemlösning som metafor och praktik [Problem solving as a metaphor and practice]. Institutionen för tillämpad lärarkunskap: Linköpings Universitet.

Yackel, T., & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics. Journal for Research in Mathematics Education, 27(4), 458–477. DOI: https://doi.org/10.2307/749877

Published
2021-02-19
How to Cite
van Bommel, J., & Palmér, H. (2021). Young students’ views on problem solving versus problem posing. Journal of Childhood, Education & Society, 2(1), 1-13. https://doi.org/10.37291/2717638X.20212165