A conjecture is an educated guess — a statement believed to be true based on observation or reasoning but not yet proven. The process of mathematical discovery follows a cycle: observe patterns, form a conjecture, test it against examples, and then either find a counterexample (which disproves it) or attempt a proof (which confirms it). Testing a conjecture means deliberately trying to break it by checking diverse and extreme cases. A conjecture that survives rigorous testing is worth trying to prove; one that fails gets refined or discarded.
Give students numerical data and ask them to form conjectures. For example: compute 1+3, 1+3+5, 1+3+5+7 (getting 4, 9, 16) and conjecture a pattern. Then test with more cases. Discuss: "When have you tested enough?" Introduce famous conjectures (Goldbach's, Collatz) to show that even simple-looking conjectures can remain unproven for centuries. Emphasize the testing strategy: try small cases, boundary cases, and "weird" cases before attempting a proof.
Mathematics does not spring fully formed from axioms. It grows through a messy, creative, very human process: you look at examples, notice a pattern, guess that the pattern always holds, and then try to figure out whether you are right. That guess is a conjecture, and learning to form and test conjectures is how you start thinking like a mathematician.
The process has a natural rhythm. First, you compute examples and look for patterns. Computing 1 + 3 = 4, 1 + 3 + 5 = 9, 1 + 3 + 5 + 7 = 16, you notice the sums are perfect squares: 4, 9, 16. The conjecture writes itself: the sum of the first n odd numbers equals n². But noticing a pattern is only the beginning — you need to test it.
Testing means trying to break your conjecture, not just confirming it. It is tempting to check a few more cases (1+3+5+7+9 = 25 = 5², yes!), declare victory, and move on. Resist that temptation. Instead, ask: does it work for n = 1? (1 = 1², yes.) Does it work for large n? (Sum of first 10 odd numbers = 100 = 10², yes.) Does it work for n = 0? (The empty sum is 0 = 0², yes.) You are stress-testing the conjecture by deliberately choosing cases that might be tricky — small numbers, large numbers, boundary cases.
The reason testing cannot replace proof is the gap between "every case I checked" and "every case that exists." One of the most instructive examples is the polynomial n² + n + 41. Plug in n = 1, 2, 3, ..., 39, and every single result is prime. Forty consecutive prime outputs — surely this always works? But at n = 40, you get 40² + 40 + 41 = 1681 = 41², which is not prime. Forty confirming cases meant nothing against that forty-first failure. This is why mathematicians do not call something a theorem until there is a proof.
When a counterexample does surface, the productive response is refinement, not abandonment. The conjecture "the sum of two primes is always even" fails because of 2 + 3 = 5. But the counterexample reveals exactly what went wrong: the number 2 is the only even prime, and adding it to an odd prime gives an odd sum. The fix is simple — restrict to odd primes — and the refined conjecture ("the sum of two odd primes is even") is provable in one line. Counterexamples are not enemies of conjectures; they are editors that make conjectures sharper.