Where it goes wrong
Traditional teaching
Fractions arrive as notation plus procedures: find common denominators, flip and multiply, cross-cancel. Pizza pictures appear for a week, then vanish, and fractions become symbol-shuffling with rules that seem arbitrary — which is exactly where most math trajectories quietly break.
The reteach
The Alice Method
Spend far longer than feels necessary on one idea: a fraction is a single amount — a point on a number line, a length of bar. Compare, order, and estimate fractions visually for weeks before any procedure appears. Then let procedures be discovered: a child who can see that halves are two-fourths has invented common denominators, and will never need to memorize them.
Mental model
One number, many names
1/2, 2/4, 3/6, and 0.5 are the same amount wearing different outfits. The denominator names the piece size; the numerator counts pieces. Bigger denominator = smaller pieces — the single most protective idea in elementary math.
Transfer
Where this shows up for the rest of their life
Renaming without changing value shows up everywhere: converting units, refactoring code so it's equivalent but clearer, restating an argument in the other person's terms. 'Change the representation, not the substance' is a lifelong thinking move.
Watch for these
Common misconceptions
The misconception
“Bigger denominator means bigger fraction.”
The repair
It's the opposite — more pieces means smaller pieces. This error is nearly universal and purely visual to fix: two bars, same length, different cuts.
The misconception
“Multiplying always makes things bigger.”
The repair
Multiplying by a fraction under 1 shrinks. 'Multiply' really means 'of': 1/2 × 8 is half of eight. That reading fixes the confusion permanently.
The misconception
“Add fractions by adding tops and bottoms.”
The repair
You can't count fourths and halves together for the same reason you can't add 3 apples and 2 oranges into 5 'appleoranges.' Same-size pieces first — that's all common denominators mean.
Seen, not said
The visual explanation
Two identical bars, cut differently, shaded differently. Comparison, equivalence, and the 'smaller pieces' insight all become visible in one image — no computation required.
Try it
Interactive example
Build two fractions and compare them by eye — then notice what bigger denominators do.
Which is bigger? Don't compute — look.
A fraction is an amount, not two stacked numbers. Same-length bars make comparison obvious.
1/2
2/3
2/3 is bigger — you can see it. Notice: a bigger bottom number means smaller pieces.