This chapter followed the learner: the child transcribing numbers, the pupil reading a word problem, the student writing a solution. The guide below gathers the habits that help at all these moments. Six habits Read with a pencil in hand. A mathematics text is dense: one line may be worth a paragraph. Paul Halmos advised: “Don’t just read it; fight it!” Redo the calculations, invent examples, ask why each step holds. Read aloud. Saying $\sum_{k=1}^{n} k^2$ as “the sum of $k$ squared, for $k$ from one to $n$” forces you to recognise each part. Where speech stalls, reading stalled too. Look at the shape before calculating. What is the last operation? Are there repetitions, symmetries, a number that appears twice? Bogdanov-Belsky’s blackboard is solved by looking. Translate both ways. From text to formula, giving a name and a unit to each unknown; and from formula back to text, to check that it says what the problem says. Test with numbers. Before trusting a rule, substitute small values. A counterexample refutes a false rule; many examples do not prove a true one, but they give the confidence to look for a proof. Write for someone. Complete sentences, words between formulas, the goal announced before the calculations. If a classmate cannot follow the solution, it is not finished yet. A complete reading of a badly written solution A student hands in the following solution to “solve $2x + 3 = 11$”:
Ordinary mathematical writing is two-dimensional. In $\frac{x+1}{2}$, what is above the bar is divided by what is below it; in $x^2$, the $2$ is up high and therefore an exponent; in $\sqrt{x+1}$, the stroke of the root covers exactly what is inside. Position is part of the sign, like the place of a digit in a numeral. The eye reads all this at once. But what about someone who does not read with the eye?
All mathematical writing has a reader. Sometimes the reader is another person; sometimes it is the author, days or years later. The two call for very different kinds of writing, and two famous notebooks show what happens when a text written for oneself falls into other people’s hands. Gauss’s diary On 30 March 1796, an 18-year-old student in Göttingen opened a small notebook and noted, in Latin, that he had discovered how to divide the circle into 17 equal parts with straightedge and compass, a problem open since the Greeks. It was Carl Friedrich Gauss, and it was this result that made him choose mathematics over philology. Over the next eighteen years he wrote 146 short entries in the notebook, almost always a single sentence, sometimes a formula.