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Can Mathematical Problem Solving Be Taught? Preliminary Answers from 30 Years of Research

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Posing and Solving Mathematical Problems

Part of the book series: Research in Mathematics Education ((RME))

Abstract

In this chapter, the authors note that during the past 30 years there have been significant advances in our understanding of the affective, cognitive, and metacognitive aspects of problem solving in mathematics and there also has been considerable research on teaching mathematical problem solving in classrooms. However, the authors point out that there remain far more questions than answers about this complex form of activity. The chapter is organized around six questions: (1) Should problem solving be taught as a separate topic in the mathematics curriculum or should it be integrated throughout the curriculum? (2) Doesn’t teaching mathematics through problem require more time than more traditional approaches? (3) What kinds of instructional activities should be used in teaching through problems? (4) How can teachers orchestrate pedagogically sound, problem solving in the classroom? (5) How can productive beliefs toward mathematical problem solving be nurtured? (6) Will students sacrifice basic skills if they are taught mathematics through problem solving?

During the preparation of this article, Jinfa Cai was supported by a National Science Foundation grant (DRL-1008536). He is grateful for the support, but any opinions expressed herein are those of the author and do not necessarily represent the views of the NSF.

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Notes

  1. 1.

    We wish to emphasize that due to the complex nature of problem solving, there are no hard and fast rules concerning what students can learn about problem solving or how it should be taught. Indeed, the main theme of this analysis is that the suggestions we provide are meant as guidelines for teachers’ to consider seriously not directives that should be rigidly followed.

  2. 2.

    Hiebert et al. (1997) describe “mathematical tools” as the collection of language, materials, and symbols that students have available when they engage in mathematical activity.

  3. 3.

    LieCal Project (Longitudinal Investigation of the Effect of Curricula on Algebra Learning) was funded by the National Science Foundation. It investigated whether the Connected Mathematics Program (CMP) can effectively enhance student learning of algebra. The LieCal Project investigated not only the ways and circumstances under which the CMP curriculum can or cannot enhance student learning, but it also looked at the characteristics of the curriculum and implementation that lead to student achievement gains.

  4. 4.

    A process-constrained problem requires a student to carry out a procedure or a set of routine procedures to solve the problem. In other words, the problem is set in such a way that it constrains a student’s solution to a rather limited process. Usually, a process-constrained problem can be solved by applying a “standard algorithm.” On the other hand, a task that is process open may not require an execution of a procedure or a set of procedures; instead it requires an exploration of the problem situation and then finding the solution to the problem. Therefore, the task is set in such a way that it allows students to use alternative, acceptable solution strategies. Usually, a process-open task cannot be solved by following a “standard algorithm.” See Cai (2000) for details.

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Authors and Affiliations

  1. Indiana University, Bloomington, IN, 47405, USA

    Frank K. Lester Jr.

  2. University of Delaware, Newark, DE, USA

    Jinfa Cai

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  1. University of Chile, Santiago, Chile

    Patricio Felmer

  2. University of Helskini, Helsinki, Finland

    Erkki Pehkonen

  3. University of Georgia, Athens, USA

    Jeremy Kilpatrick

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Lester, F.K., Cai, J. (2016). Can Mathematical Problem Solving Be Taught? Preliminary Answers from 30 Years of Research. In: Felmer, P., Pehkonen, E., Kilpatrick, J. (eds) Posing and Solving Mathematical Problems. Research in Mathematics Education. Springer, Cham. https://doi.org/10.1007/978-3-319-28023-3_8

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