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AliceMethod

The Math Section

Every big idea in math, retaught.

Mathematics is the clearest demonstration of the whole method, because traditional teaching fails it so consistently: procedures before models, speed before sense, rules before reasons. Each topic below shows the traditional approach, the Alice approach, the mental model that makes it click, where it transfers, the misconceptions to expect — and something to play with.
01Ages 4–10

Arithmetic

Number sense before number facts

Mental model — Numbers are LEGO: Every number can be snapped apart and rebuilt: 47 is 40 + 7, or 45 + 2, or 50 − 3.

02Ages 7–12

Fractions

Amounts, not stacked numbers

Mental model — One number, many names: 1/2, 2/4, 3/6, and 0.

03Ages 9–13

Ratios & Proportion

The mathematics of recipes

Mental model — The recipe: A ratio is a recipe: the amounts can change, but the relationship can't.

04Ages 8–12

Decimals

Place value, continued

Mental model — The infinite zoom: The number line is infinitely zoomable.

05Ages 9–13

Percentages

A fair-comparison machine

Mental model — The common ruler: Percent rescales every comparison to the same 100-unit ruler.

06Ages 9–13

Negative Numbers

Directions, not mysteries

Mental model — The about-face: A negative sign is an about-face.

07Ages 10–15

Algebra

The grammar of patterns

Mental model — The balance scale: An equation is a statement that two pans balance.

08Ages 11–16

Functions

Machines, not formulas

Mental model — The input-output machine: Every function is a box with a slot in and a slot out.

09Ages 6–16

Geometry

Reasoning you can see

Mental model — Cut and rearrange: Area doesn't change when you slice a shape and slide the pieces.

10Ages 10–16

Probability

Thinking in long runs

Mental model — The long run: Probability is a statement about what happens over many trials, not a prophecy about the next one.

11Ages 10–17

Statistics

Detective work on data

Mental model — Compression with loss: An average is a lossy summary: one number standing in for many, chosen to be representative.

12Ages 11–17 (ideas from 8)

Exponents & Logarithms

The mathematics of growth

Mental model — Counting doublings: An exponent counts repeated multiplications the way ordinary numbers count repeated additions.

13Ages 14–18 (ideas from 10)

Calculus

The mathematics of change

Mental model — Zoom until straight: Every smooth curve is secretly straight, if you look closely enough.

“Traditional teaching asks: can the child perform the procedure?
The Alice Method asks: does the child own the idea?”