Counterexamples can be used to dispel misconceptions, deepen conceptual understanding, and help students construct viable arguments. Standard for Mathematical Practice 3 states:
Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples.
In the traditional sense, a counterexample is a specific case that shows that a statement is false. First, let’s take a look at the impact of using a counterexample to a student’s misconception. Consider the following scenario.
Mr. Gebra is teaching a lesson on simplifying radical expressions. He models the following examples before assigning students some problems to complete independently.
Mr. Gebra notices that several students have the following response to the first problem on their assignment.
Several questions come to mind…
- What misconception could possibly be leading students to this incorrect response?
- How might our choice of model problems contribute to students’ misconceptions?
- Could we avoid misconceptions by anticipating students’ thinking and strategically using counterexamples?
Secondly, we can also incorporate the use of counterexamples in learning tasks to help students explore the truth of their conjectures. Students benefit from generating counterexamples to a false statement. Consider the following statement:
Division makes numbers smaller.
In determining weather this statement is true or false, students might generate counterexamples. Presented with a falsehood, students can then “fix” the statement to make it their own conjecture. How would you revise the statement to make it true? Students can test the validity of their conjectures and critique each other’s reasoning. It is through this process, that opportunities for learning emerge.
Benefits of using counterexamples include:
- Deepens conceptual understanding
- Deemphasizes the reliance on tips and tricks
- Reduces or eliminates misconceptions
- Useful tool for constructing viable arguments
- Provides a nonthreatening method for critiquing the reasoning of others
- Encourages students to practice using the language of mathematics
- Discourages the use of mathematically incorrect overgeneralizations or imprecise vocabulary
- Advance students’ mathematical understanding
- Enhances critical thinking skills
- Increases students’ openness to new or atypical problems
- Makes learning more active and creative
- Helps students avoid repeated mistakes
- Increases self-confidence
- Promotes mathematical habits of mind
- Allows students to search for truth when faced with falsehoods
- Opportunities for learning emerge