Teaching against misconceptions

Telling a student the right rule rarely erases the wrong one — the old model has to be caught predicting, and failing.

The idea

A misconception isn’t an empty slot waiting for the right answer. It’s a working theory the student built and trusts, and it usually predicts a lot of the world correctly. So when you just tell them the rule, the new fact sits politely next to the old model — and under any pressure they reach for the model. The move that works is to make their own theory predict something, then meet that prediction with a case it plainly can’t explain. That collision is what loosens the grip. Then, and only then, you help them rebuild.

How it works

The reliable pattern has four beats. Skipping straight to beat three is why telling fails.

1. ELICIT     Surface the model. Ask them to predict, in their words.
              "Which is bigger, 1/8 or 1/2?"  ->  "1/8 — 8 is bigger."

2. CONFRONT   Choose a case their model predicts confidently AND
              reality refutes undeniably. Have them predict, THEN observe.
              "One pizza. Share it 2 ways or 8 ways — who gets more?"
              their model says: the 8 group
              their gut says:   the 2 group  <- collision = cognitive conflict

3. REBUILD    On top of the conflict, help them state the fix themselves.
              "More pieces means each piece is smaller,
               so a bigger bottom number makes a smaller fraction."

4. CHECK      Test transfer on a case they've never seen.
              "1/100 or 1/3?"   memorized -> "1/100, 100 is bigger"
                                understood -> "1/3, 100 pieces are tiny"

Telling does beat three with no beats one or two under it. The rule has nothing to attach to, so it stays a sentence, not a model.

When to use it

Watch out for

• Repeating the rule louder. If it didn’t land the first three times, a fourth telling won’t either. The grip bar has a floor that telling can’t get under.
• A counterexample they can explain away. Too small, too abstract, or one weird pair — and they file it as an exception and keep the rule. The case has to hit a prediction they made out loud.
• A case that doesn’t engage the belief. 3/8 vs 5/8 shares a denominator, so the “bigger bottom” rule makes no prediction there — nothing to be wrong about, no conflict.
• Confronting, then walking off. Cracking a model and not rebuilding leaves the student more confused than before. Beat three is not optional.
• Mistaking procedure for understanding. They can crank the algorithm and still predict a novel case with the old model. Only transfer tells you the model actually changed.
• Doing it in front of the class. If being wrong feels like being shamed, they defend the model instead of revising it. Make it safe to be caught.

Worked example

In a teaching-demo interview, the interviewer plays a student who insists 1/8 > 1/2. A weak candidate re-explains place value and denominators; the “student” nods and repeats it back, and the candidate calls it taught. A strong candidate elicits first — “predict it for me: 2 friends split a pizza, or 8 friends split the same pizza — who gets more?” The student, following their own rule, says the 8. Then they observe: everyone knows 8 sharers means thinner slices. The candidate names the collision gently — “so your rule just predicted the 8 group gets more, but we both know they get less” — and lets the student rebuild: “more pieces, smaller pieces.” Finally they check transfer with 1/100 vs 1/3. The difference between those two candidates isn’t warmth — it’s that one changed a sentence and the other changed a model.

Check yourself