Mistakes Are Data

This class keeps a museum. Its exhibits are our best mistakes: the product differentiated factor by factor because sums split and surely products do too, the inner derivative that vanished from $\sin 3x$ leaving a lonely $\cos 3x$ behind, the vector “cancelled” from both sides of $\vec{u}\cdot \vec{v}=\vec{u}\cdot \vec{w}$ as if a dot product were ordinary multiplication — when $\vec{u}$ could simply be perpendicular to the difference and the equation true with $\vec{v}$ and $\vec{w}$ nothing alike. None of these are careless — each one is evidence that the notation invited a reasonable person to a wrong conclusion, and evidence is for studying, not shredding. Hence the standing rule at the whiteboards: first attempts are never erased. Cross them out, leave them legible, and let the room read what they have to say. Your first attempt is the only record of how you were thinking before you knew — destroy it and you cannot study the gap it was measuring.

Questions worth arguing about:

1. Nominate an exhibit: the error that got you this month. What makes it reasonable — what was the notation or the situation doing to invite it?
2. $\frac{dy}{dx}$ looks like a fraction. Sometimes it behaves like one — the chain rule reads like cancelling — and sometimes treating it as one leads straight off a cliff. Is trusting the notation a student mistake or a notation trap? Should the convention change, or should we?
3. The urge to erase gets stronger the smarter the person watching. What is that urge protecting — and what would make this room safe enough to leave wrong work up all period?
4. A slip ($7\times 8$ gone wrong inside a cross product) is not a misconception (believing $(fg{)}^{\prime }=f^{\prime }{g}^{\prime }$). Which deserves your study time, and how do you tell them apart in your own work?
5. Design a mistake worth making: what question would you have to attempt for the error to teach you the most?

Every practice set in this course carries a find the error item on purpose — reading a stranger’s slip trains the same eye that Checking Your Own Work turns on your own page. The norm protecting all of it lives in Our Classroom Norms: mistakes are data, and the museum only works if everyone donates.