- Title: Problem solving in the mathematics curriculum: From domain-general strategies to domain-specific tactics
- Authors: Colin Foster
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Paper summary
This article examines the effectiveness of teaching problem-solving in mathematics. It argues against the use of general problem-solving strategies, proposing instead the explicit teaching of specific, content-related tactics. The author contends that this approach, exemplified by techniques like “drawing an auxiliary line” in geometry, is more effective than broad, less applicable strategies and could improve problem-solving skills for all students. The article uses examples and research to support its claims, suggesting a structured teaching method to improve student outcomes. It also explores the difference between routine and non-routine tasks and their impact on learning.
What are the key implications for teachers in the classroom?
- Teachers should focus on teaching domain-specific problem-solving tactics rather than general problem-solving strategies. Domain-specific tactics are more concrete and actionable, and students are more likely to be able to apply them successfully. Teaching broad domain-general strategies such as “be systematic” or “draw a diagram” is not helpful for students.
- Teachers should select problems that are dramatically unlocked by using the tactic being taught. This will help students to see the power of the tactic and to understand when it is applicable.
- Teachers should allow students to struggle with problems initially, without explicitly teaching them the tactic. This will help students to develop their own problem-solving skills and to appreciate the value of the tactic when it is introduced.
- Teachers should explicitly teach students how to identify the features of a problem that make a particular tactic a good fit. This will help students to become more strategic in their problem solving.
- Teachers should provide students with opportunities to practice using the tactic on a variety of problems. This will help students to consolidate their understanding of the tactic and to develop fluency in using it.
- Teachers should interleave problems that require different tactics. This will help students to learn to discriminate between different tactics and to choose the most appropriate tactic for a given problem.
- Teachers should help students to understand the advantages and disadvantages of different problem-solving tactics. This will help students to make informed choices about which tactic to use in a given situation.
- Teachers should create a coherent problem-solving curriculum that systematically addresses important, high-leverage tactics. This will help students to develop a comprehensive set of problem-solving skills.
The sources emphasize that simply teaching mathematical content thoroughly and deeply is not sufficient for developing problem-solving skills. Students also need to be explicitly taught domain-specific problem-solving tactics. This approach has the potential to enable all students to become powerful problem solvers.
Quote
Breaking down some of the highest leverage of Polya’s strategies into detailed, actionable, domain-specific tactics, focused on well-defined content areas, seems to have the potential to mobilize students’ potentially inert content knowledge in the service of powerful problem solving of non-routine problems.
