3 mistakes learners make with proof, each one named and explained.
One counterexample disproves; no number of examples proves.
A value is given as a counterexample that in fact satisfies the statement.
Small values are the ones that come to mind, and they are usually the ones the statement was built on.
Question. n^2 + n + 1 is prime for all integers n
A common answer. n = 2, giving 7
The answer. n = 4, giving 21
Why. 7 is prime, so n = 2 supports the statement rather than disproving it.
Substitute your candidate and check it actually fails. A counterexample has to break the claim.
A general statement is argued by checking two or three particular cases.
Several cases working really is strong evidence, and it feels convincing.
Question. Prove the sum of three consecutive integers is divisible by 3
A common answer. 1 + 2 + 3 = 6 and 4 + 5 + 6 = 15, so it is true
The answer. n + (n+1) + (n+2) = 3n + 3 = 3(n+1)
Why. Examples cannot cover infinitely many cases; algebra covers all of them at once.
One counterexample disproves. Only a general argument proves.
Three consecutive integers are written in a form that is not consecutive, or the sum is mis-simplified.
Setting up the algebra is the step that decides everything, and it is done before any confidence builds.
Question. Three consecutive integers
A common answer. n, n + 1, n + 3
The answer. n, n + 1, n + 2
Why. Consecutive means each is one more than the last.
Write out three actual numbers first, then match your letters to them.
WAJD spots these patterns in your child's answers and names the one behind their wrong answers, instead of just marking them wrong.