Half this course happens out loud at the whiteboards; the other half happens when you put reasoning into words someone else can follow. Writing about mathematics has one rule that does most of the work:
Convince a sceptic, not a marker
Write for a classmate who missed today and doubts you — not for a teacher scanning for keywords. If your sceptic could ask “but why is that step allowed?”, answer it before they can.
The vocabulary is the toolkit
Mathematical claims come in strengths, and the words signal which one you are making:
| You are saying… | Say it like this |
|---|---|
| I see a pattern but haven’t tested it | ”I conjecture that…” |
| I tested cases and it held | ”It seems to hold because… (tried 3 cases)“ |
| I can say why it must be true | ”It must be true, because…” |
| I found where it breaks | ”Counterexample: and …” |
A conjecture that turns out wrong is not an error — it is the engine of the whole subject. Our class discussions run on them; so does Mistakes Are Data. In this course the top row of that table does the heaviest lifting it ever has: the power rule and the chain rule both enter the room as conjectures grown from data, and a finished optimization is nothing but “it must be true, because” sustained to the end — the candidates found, the endpoints checked, the winner defended. And when the writing is about a model, one more strength matters: saying where it stops. “This braking model holds until the car stops, and not one second past it” is not a weakness in your work — by the standard of What Makes a Model Good, it is the most professional sentence on the page.
Sentence stems that unlock a stuck reflection
- I noticed that… and I wonder whether…
- The step I trust least is… because…
- Another method that should give the same answer is…
- My estimate was… and the difference tells me…
The finished-solution standard
Before calling written work done, walk Showing Your Thinking’s checklist: the question restated, the reasoning visible, units carried through, and an answer sentence — is a value; “the car was doing 63 km/h when it passed the camera” is an answer.
The same standard applies to your Math Journal, with one addition: name what you did and thought, not just what the group did. “We found the maximum at the board” hides the learning; “I suggested checking the sign of the derivative on each side of the candidate, and the flat spot finally confessed it was a peak” shows it — and that is the evidence How Marks Work actually counts.