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Students often solve mathematics problems by executing known procedures to generate answers. As long as problems are familiar to those done in the classroom, then doing mathematics continues to be a matter of applying familiar formulas and procedures. However, difficulties often arise when students are challenged to justify their solutions or solve novel problems. This paper proposes vee diagrams as a means of facilitating students' critical thinking and reasoning to systematically analyse the mathematical content of problems as a means of identifying relevant principles and concepts that can inform the identification of plausible solutions. Constructing a vee diagram requires solvers, not only to show how they solved the problem (methods), but in addition, they make explicit why their methods worked (relevant theoretical principles). Doing so routinely encourages the view of mathematical problems as the application context and site for the critical synthesis of the conceptual structure of mathematics and its methods and procedures. The paper presents examples of vee diagrams constructed by mathematics teachers, student-teachers and school students from a number of studies conducted to investigate how mathematics learning could be made more meaningful for students. Implications for teaching school mathematics and teacher education are provided. |
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