If the rules never quite connected — invert-and-multiply, “two negatives make a positive” — that wasn’t you, and it wasn’t your teachers. The mathematician H. Wu calls the culprit Textbook School Mathematics: rules stripped of the definitions that give them meaning. Presented that way, math isn’t merely hard. It is, in Wu’s word, unlearnable.
These lessons rebuild the subject his way: one number line, precise definitions, and pictures that show each idea before any symbol asks you to trust it.
It all starts here. Choose 0. Choose 1 — the segment between them is the unit segment. Lay it down again and again: the whole numbers.
Every object is defined before it is used. A number is a point on the line. A fraction is a constructed point — not a pizza slice.
Each lesson is built around an interactive figure. You drag, slide, and reflect; the mathematics happens where you can see it.
Definitions and theorems are quoted verbatim from Understanding Numbers in Elementary School Mathematics, with the author’s permission — section and page.
One idea each, in order. Every lesson ends with a short note on why it’s taught this way.
The Number Line
A line, one point named 0, one point named 1. That single choice of unit builds every whole number.
Wu, §1.4 Lesson 02Addition, as Length
Lay one segment after another and measure where you land. Addition is concatenation — something you can watch.
Wu, §1.4 Lesson 03Fractions, as Points
Cut the unit into n equal parts, lay down m copies. A fraction is the exact point where you land.
Wu, §12.2 Lesson 04Equivalent Fractions
The Cancellation Law: one theorem, read in both directions, replaces every “rule” about reducing and expanding.
Wu, Thm. 13.1 Lesson 05The Rational Numbers
Reflect every fraction across 0. No negative signs yet — just mirror images, and a reason to want them.
Wu, Chs. 25–26 Lesson 06Adding Rational Numbers, as Vectors
Give each number a direction, slide, and read the endpoint. The “sign rules” become one provable observation.
Wu, Ch. 27The approach is Wu’s. The evidence for its spine — the number line — is broader.
“The fractions together with their reflected images will be seen to form a number system in the sense that we can perform the four arithmetic operations on them all, in a way that is consistent with the operations already defined on the fractions.” — Hung-Hsi Wu, Understanding Numbers in Elementary School Mathematics, p. 375