Where it goes wrong
Traditional teaching
Memorize facts in isolation, then stack them into multi-digit algorithms — carry the one, borrow from the tens — executed as rituals whose steps are never explained. Speed is graded; understanding is assumed.
The reteach
The Alice Method
Build number sense first: numbers are quantities that can be broken apart and reassembled. 8 + 5 isn't a fact to retrieve — it's 8 + 2 + 3, a jump to ten and a hop past it. Facts get memorized too (fluency frees working memory), but through two-minute daily retrieval — a Post-it wall, a car-ride game — never hour-long drills. Every algorithm is introduced only after the child can explain the quantity move it performs.
Mental model
Numbers are LEGO
Every number can be snapped apart and rebuilt: 47 is 40 + 7, or 45 + 2, or 50 − 3. Arithmetic is choosing the decomposition that makes the problem easy. A child who owns this model never freezes on 99 + 47 — they slide one over and compute 100 + 46.
Transfer
Where this shows up for the rest of their life
Decomposition is the master skill. Splitting 13 × 7 into (10 × 7) + (3 × 7) is the same move as splitting a big coding task into functions, a big essay into paragraphs, or a hard negotiation into separable issues. Children who learn 'make the problem friendlier before you attack it' use it everywhere.
Watch for these
Common misconceptions
The misconception
“Fast fact recall means my child is good at math.”
The repair
Fluency matters, but it's the floor, not the ceiling. A child can be a fast calculator and still have no idea whether an answer is reasonable. Ask 'about how big should the answer be?' before every computation.
The misconception
“The standard algorithm is the 'real' way; mental strategies are crutches.”
The repair
It's the reverse. Mental strategies expose the structure of number; the algorithm compresses it for efficiency. Teach structure first, compression second.
The misconception
“Counting on fingers should be discouraged.”
The repair
Finger counting is a healthy stage of building quantity sense. It fades on its own as facts become automatic through retrieval — banning it just adds shame.
Seen, not said
The visual explanation
Show 13 × 7 as a rectangle of dots, then slice it into a 10 × 7 block and a 3 × 7 block. The distributive property stops being a rule and becomes something the child can see: big multiplications are just small ones glued together.
Try it
Interactive example
Slide the factors and watch a 'hard' multiplication split into two easy ones.
Multiplication is area — and area explains the “tricks”
13 × 7 splits into (10 × 7) + (3 × 7). That's the distributive property — seen, not memorized.