- Title: Student-generated examples in the learning of mathematics
- Authors: Anne Watson and John Mason
- Access the original paper here
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Paper summary
This research article investigates common difficulties students face in mathematics, specifically their understanding of theorems and the use of examples. The authors propose that encouraging students to generate their own examples, especially boundary cases, improves comprehension and retention. They present various mathematical tasks designed to illustrate this approach and highlight the importance of constructing examples to explore the full range of possibilities within definitions and theorems. The study’s methodology involves creating and refining tasks based on observed student challenges, with validity determined by whether the approach informs future teaching practices. Ultimately, the paper advocates for a shift in teaching that emphasizes the active construction of mathematical objects, fostering a deeper understanding of concepts.
What are the key implications for teachers in the classroom?
The authors suggest that framing techniques for solving mathematical problems as processes of construction can help students understand mathematics more deeply. When students are taught techniques as a set of mechanical manipulations, they may not see the creative problem-solving aspects of the technique. If students understood that they were constructing objects that meet specified properties when they use techniques like solving differential equations or finding integrals, it might help them be more successful when asked to construct a mathematical object with specific properties.
The authors recommend that teachers help students develop an awareness of where techniques come from and how they are found. One way to do this is to show students the powerful and pervasive strategy of describing a general object, imposing constraints on the object, and using those constraints to determine the values of parameters in the general description. The authors explain that this strategy mirrors the shift from synthesis to analysis in early Greek geometry, which was at the heart of the development of algebra as a problem-solving tool.
The authors discuss several strategies for helping students appreciate a particular mathematical example, especially a boundary example that shows why constraints are required in a definition or theorem. Students sometimes dismiss these examples as pathologies. One strategy is to guide students to construct a family of objects with properties similar to the original example. Another approach is to use structured tasks that force students to consider a more general class of examples. A third strategy is to have students construct a simple example, then a peculiar example, and finally attempt to describe a general object of the class. A final suggestion is to present students with several definitions of a concept and have them construct examples to illustrate how the definitions differ and what is not covered by each definition. These strategies help students see how a particular example fits into a larger class of objects, and how that class of objects relates to the constraints in a definition or theorem.
Quote
Any technique which yields answers illustrates ways to construct mathematical objects. Solving a first order linear differential equation, finding an integral, disentangling eigenvalues and eigenvectors for a matrix, and changing basis are all examples of constructing an object with specified properties. Thus constructing mathematical objects is something students have seen and done a great deal of, yet for the most part they are unaware of it in these terms.
