What are some strategies to help students generate and test their own examples and counterexamples in proof?

1 Why are examples and counterexamples important?

The use of examples and counterexamples is essential in the process of proof. It helps students comprehend the implications of definitions, theorems, and concepts, while also allowing them to identify and avoid mistakes. Additionally, it enables them to develop intuition and motivation for generalization and abstraction, as well as discover new patterns, relationships, and questions. Furthermore, it assists them in communicating and justifying their arguments and critiques. Consequently, it is necessary for students to learn how to generate and test their own examples and counterexamples, rather than merely relying on given ones or memorizing facts.

2 How to generate examples and counterexamples?

When attempting to generate examples or counterexamples for a proof, there are various strategies to consider depending on the context and the goal of the proof. For example, one could start with simple or familiar cases and then vary or extend them, or utilize diagrams, tables, graphs, or other visual representations to explore patterns and properties. Additionally, extreme, boundary, or special cases can be used to uncover exceptions or contradictions. Algebraic or numerical methods can be used to manipulate or simplify expressions or equations, while logical operators or quantifiers can be used to modify or negate statements or hypotheses. Finally, counterexamples can be used to generate examples, or vice versa by changing or reversing some conditions or assumptions.

3 How to test examples and counterexamples?

Testing examples and counterexamples goes beyond just plugging in values or checking calculations; one must also understand the scope and limitations of the examples and counterexamples, compare and contrast them to identify similarities and differences, evaluate their validity and relevance for the proof, explain how they support or refute the claim or conjecture, and generalize or refine them to make them more representative or specific.

4 How to scaffold examples and counterexamples?

Scaffolding is a way of providing guidance and support for students to develop their skills and confidence in proof. To do this, one can model how to generate and test examples and counterexamples by thinking aloud and asking questions, provide feedback and hints that encourage reflection, use open-ended or multiple-choice questions to prompt students to generate and test examples, use collaborative activities that let students compare their examples with peers, and use rubrics or criteria to help them assess the quality of their examples. All of these strategies can help students become more confident in their proofs.

5 How to challenge examples and counterexamples?

Challenging students is a great way to stimulate and extend their thinking and learning in proof. Questions such as asking for more or different examples and counterexamples to illustrate or contradict a claim, finding the smallest, largest, or most unusual example that satisfies or violates a condition, finding a general rule or formula that generates or excludes all possible examples of a type or class, finding a counterexample to a counterexample, or critiquing and improving the examples given by others are all great ways to challenge students.

6 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?