How do you design and sequence proof activities that involve examples and counterexamples?

1 Why use examples and counterexamples?

Examples and counterexamples can enhance your proof activities by engaging students' curiosity, intuition, and creativity. They can also help students connect abstract concepts to concrete situations, clarify definitions and assumptions, and identify patterns and exceptions. By using examples and counterexamples, you can encourage students to generate, refine, and communicate their own conjectures and arguments, as well as to critique and evaluate others' reasoning.

2 How to select examples and counterexamples?

The quality and diversity of examples and counterexamples you use can significantly impact students' learning. When selecting these items, consider their relevance to the main goal or question of the activity and the mathematical content you want to emphasize. Additionally, strive for variety in terms of types, cases, or scenarios that illustrate or challenge the general statement or conjecture. Make sure to balance between simple and complex examples and counterexamples that can scaffold or extend students' learning. Avoid using only trivial, typical, or extreme examples that might limit students' thinking or generalize too quickly.

3 How to present examples and counterexamples?

Presenting examples and counterexamples in different formats, modes, or tools can affect how students perceive and interact with them. For instance, diagrams, graphs, tables, pictures, or animations are suitable for conveying geometric, spatial, or graphical information. Equations, expressions, formulas, or notations can be used to illustrate algebraic, numerical, or logical information. Additionally, words, sentences, stories, or scenarios can be helpful in demonstrating contextual, linguistic, or conceptual information. Finally, physical or virtual objects, models, or simulations are ideal for conveying tactile, kinesthetic, or dynamic information. When selecting the format and mode to use for presenting examples and counterexamples it is important to consider the nature and purpose of the examples and counterexamples as well as how it will facilitate students’ engagement and understanding.

4 How to sequence examples and counterexamples?

When introducing examples and counterexamples, the order and timing can influence the flow and effectiveness of your proof activities. You can use different strategies to sequence examples and counterexamples, such as the deductive approach, where you start with a general statement or conjecture and then provide examples and counterexamples to support or refute it. This can help students verify, justify, or revise their initial ideas and develop deductive reasoning skills. Alternatively, you can start with some examples and counterexamples and then ask students to infer a general statement or conjecture from them. This strategy can help them discover, formulate, or generalize their own ideas and develop inductive reasoning skills. Lastly, you can alternate between providing and eliciting examples and counterexamples and between stating and exploring general statements or conjectures. This strategy can help them compare, contrast, or synthesize different ideas and develop both deductive and inductive reasoning skills. Ultimately, you should choose the strategy that best matches the learning objectives and outcomes of the activity and that can foster students' curiosity and understanding.

5 How to facilitate examples and counterexamples?

As a teacher, you can shape how students use and learn from examples and counterexamples by providing role and guidance. Modeling is one technique to consider; you can demonstrate how to generate, test, or explain a general statement or conjecture with examples and counterexamples, as well as the appropriate language, notation, or representation to communicate reasoning. Questioning is another strategy you can use; open-ended, probing, or reflective questions can prompt students to explore, justify, or critique a general statement or conjecture. Additionally, feedback is an effective way to acknowledge, encourage, or challenge students' use of examples and counterexamples. Ultimately, the techniques you choose should suit your teaching style and goals while supporting students' confidence and competence in proof.

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?