How do you use examples and counterexamples in your math classroom?

Examples are not helpful in a proof. Showing that one instance is true means nothing to the general statement. Instead examples can help to develop a general pattern or hypothesis you wish to prove. If we notice a pattern, we might think this is a larger truth that can be shown. Counterexamples are very important in proof as they have the ability to disprove. If we think there is an underlying pattern that is difficult to show, by simply finding a single instance in which it doesn’t work is enough to save us a lot of effort on trying to prove something that does not hold or is not valid.

Full class discourse is a great opportunity to highlight both strategies! Encourage to students to share different insights, then celebrate their contributions. Consider memorializing these strategies in a "problem-solving toolbox" for students.

Examples and counterexamples will helps to prove the given statement is true or false. For example, the statement "Sum of a positive and negative integers is always a positive integer" can be shown to students to check whether it is true or false. This type of questions will improve the critical thinking and problem solving skills of students. The student who check with the example, -5 + 3 = -2 will get a negative integer as answer whereas a student who check the example -5 +6 = 1 will get a positive integer. Also the one one who check -5 +5 =0 , will 0. So the answer depends on the operands. This activity helps students to come up with a conclusion that the sum of a positive and a negative integers can be positive, negative or 0.

The most natural counterexample is an incorrect solution! By using student missteps as counterexamples you can encourage a classroom culture where it is okay to make mistakes. It should be encouraged to try something new! If it doesn’t work, this is not a failure but instead a vital piece of information that furthers our understanding.

I think it’s important to choose relevant and engaging problems that align with your curriculum. Examples can be used to provide context and help students understand concepts, while counterexamples can challenge assumptions and promote critical thinking. It's also important to encourage students to generate their own examples and counterexamples, as this can help them develop a deeper understanding of the material. For example, in a geometry lesson on quadrilaterals, an example may be to give several examples of quadrilaterals (such as squares, rectangles, & trapezoids) and have students identify their properties. Then, ask them to come up with counterexamples (such as rhombuses & kites) whose properties don't fit in the other examples.