Academic Success Skills: Problem-Solving and Critical Thinking

Developing productive problem-solving habits, and a disposition for critical thinking on which these habits rely, are essential components of mathematical proficiency. Additionally, problem solving and critical thinking provide a foundation for learning new mathematical concepts. A goal of entry-level mathematics instruction is to enhance students’ problem-solving ability while leveraging it as a foundation for their learning of central ideas.

Faculty at the MIP Academic Success Skills Initiation Workshop in May, 2019 recommended that instructional resources developed by CoRDs and ARCs addressing Problem-Solving and Critical Thinking for success in the Oklahoma entry-level college math pathways should:

1. Identify the cognitive habits and characteristics of effective problem solvers and critical thinkers.

2. Describe how the conceptual activity entailed in effective problem solving and critical thinking might develop.

3. Identify the mathematical ways of reasoning on which effective problem solving and critical thinking in algebra, precalculus, and calculus depends and describe how these ways of reasoning facilitate students’ problem solving in these areas.

4. Propose specific principles of curriculum design and instructional practice (including assessment) to enable students to develop the cognitive habits of effective problem solvers, and give at least one example of a task sequence that would make a good problem for each college algebra, precalculus, and Functions and Modeling.

5. Describe the potential affordances of the MIP focus on promoting students’ active engagement and leveraging meaningful applications for supporting students’ problem solving and critical thinking.

Suggested Resources

Carlson, M. P., & Bloom, I. (2005). The cyclic nature of problem solving: An emergent multidimensional problem-solving framework. Educational studies in Mathematics, 58(1), 45-75.

This resource details some of the cognitive habits and characteristics of effective problemsolvers. Specifically, the authors generalize the problem-solving behaviors of 12 mathematicians in a Multidimensional Problem-Solving Framework that includes the following four phases: orientation, planning, executing, and checking. Carlson and Bloom describe how progressing through these phases involves robust and well-connected knowledge of mathematical concepts, facts, and heuristics, as well as the ability to regulate one’s emotions during the problem-solving process.

Silver, E. A. (Ed.). (2013). Teaching and learning mathematical problem solving: Multiple research perspectives. Routledge.

Among other things, this comprehensive resource reviews the history of scholarship on the teaching and learning of mathematical problem solving and surveys its key insights (Chapters 1 and 13), identifies significant implications of research in cognitive psychology for instruction in mathematical problem solving (Chapters 7 and 8), argues for the necessity to attend to students’ experience when designing curricula to develop students’ mathematical problem solving (Chapters 11 and 12), describes various affective issues relevant to students’ problem solving (Chapter 13), discusses the role and significance of the teacher in promoting students’ problem solving (Chapters 14 and 15), identifies the cognitive habits and characteristics of effective and ineffective problem solvers (Chapter 16 and 21), and discusses features of social contexts that are most propitious to fostering students’ problem solving (Chapter 18).

Thompson, P. W. (1985). Experience, problem solving, and learning mathematics: Considerations in developing mathematics curricula. In E. A. Silver (Ed.), Teaching and learning mathematical problem solving: Multiple research perspectives (pp. 189–243). Hillsdale, NJ: Erlbaum.

Thompson’s purpose in this chapter is to describe efforts to develop curricula that are based in research on mathematical problem solving while also being informed by an image of a student progressing through a curricular sequence. Thompson characterizes mathematical problems solving as a constructive and reflective activity that is essential to learning mathematics generally. This chapter is most relevant to addressing foci (2), (3), and (4) above.