Testing and refining conjectures helps students move from “it worked in my examples” towards arguments that are both mathematically sound and accessible to their classroom community.
In mathematics, a conjecture is a claim that appears to be true based on observations or examples but has not yet been established. Moving from “this seems to work” to “this always works, and here is why” requires students to test conjectures systematically, search for cases in which they fail, and refine the conditions or wording of a claim when needed.
Students often see a claim hold true in a few examples and conclude it must be true across the board. Finding three examples that work isn’t the same as knowing the claim will hold in every case: what’s missing is an explanation of why. The shift students need to make is from collecting confirming examples to building a case for why the pattern would hold generally, not just in the cases they have checked.
This doesn’t mean that every lesson needs to end in a formal proof. A conjecture only needs enough testing to justify it as likely true. Proof is the further step of explaining why a claim must always hold, not just that it has held so far. Most classroom work can stay at the conjecturing stage: testing cases, hunting for exceptions, and refining the claim when an exception is found.
Students generating, testing, refining, and justifying conjectures is a core part of learning to reason mathematically. In Widjaja and Vale’s (2021) study of a conjecture about perimeter and area, many students generated examples that supported a false conjecture but struggled to search for counterexamples or to recognise that a single counterexample was sufficient to refute an “always” claim. The authors argue that students need explicit opportunities to search systematically for counterexamples. Doing so can challenge persistent misconceptions, develop conceptual understanding and strengthen students’ capacity to justify claims and disagree mathematically.
One practical way to support this is through Always–Sometimes–Never statements. Buchbinder and McCrone (2023) describe these tasks as a way of making explicit the different roles of examples and counterexamples in proving and disproving claims. A teacher can present a mathematical claim and ask students to decide whether it is always true, sometimes true or never true. Students then test examples, search deliberately for counterexamples, and refine the wording of the claim if needed. For example, the statement When the perimeter of a rectangle increases, its area also increases invites students to test cases, find exceptions and discuss what conditions would be needed for a more accurate claim.
The teacher’s role is to help students move from noticing patterns to testing claims systematically. This includes prompting students to ask:
- Does this always work?
- What cases have been tested?
- Can a counterexample be found?
- What conditions need to be added?
- What would make the argument convincing to the class community?
Discuss with your colleagues
For pre-service teachers
- If a group of students in front of you had just found three examples that confirmed their idea, what would you say to push them toward looking for a counterexample instead?
- Pick a claim from a topic you’ll teach (e.g. “multiplying two numbers always makes a bigger number”). Is it always true, sometimes true, or never true? What would a student need to test to find out?
For classroom teachers
- Where do students already notice patterns or make claims in your lessons, and how are these claims currently tested?
- Try an Always–Sometimes–Never statement with your class this week. What did students do when asked to find a counterexample, and where did they get stuck?
For leaders
- Where in your school’s scope and sequence do students first meet the idea that examples alone don’t prove a claim, and is this idea revisited enough as they move toward more formal reasoning?
References
Buchbinder, O., & McCrone, S. M. S. (2023). Preparing prospective secondary teachers to teach mathematical reasoning and proof: the case of the role of examples in proving. ZDM – Mathematics Education, 55, 779-792. https://doi.org/10.1007/s11858-023-01493-4
Widjaja, W., & Vale, C. (2021). Counterexamples: Challenges faced by elementary students when testing a conjecture about the relationship between perimeter and area. Journal on Mathematics Education, 12(3), 487–506. https://files.eric.ed.gov/fulltext/EJ1339966.pdf