Where it goes wrong
Traditional teaching
Rule lists: two negatives make a positive, keep-change-change, watch your signs. The rules are correct and completely unmotivated, so they scramble under exam pressure — is it minus times minus, or minus plus minus, that flips?
The reteach
The Alice Method
Ground every rule in one physical story: walking a number line. Positive numbers face right; the negative sign means turn around; subtraction also means turn around. Now 'subtract negative three' is two about-faces — you're facing forward again, so you walk forward. Every sign rule becomes a fact about turning, checkable with your body in a hallway.
Mental model
The about-face
A negative sign is an about-face. One turn: face backward. Two turns: face forward again. That's the entire mystery of 'minus a minus' — not a rule but a rotation, and rotations compose. (Later, this same model becomes multiplication by −1 as a 180° rotation — the doorway to complex numbers.)
Transfer
Where this shows up for the rest of their life
Signed quantities run accounting (debt), science (charge, temperature), engineering (direction, phase), and games (score swings). More deeply: 'reversing a reversal restores' is logic itself — the double negative in language, the inverse of an inverse in every system a child will ever study.
Watch for these
Common misconceptions
The misconception
“−7 is bigger than −2 because 7 is bigger than 2.”
The repair
On the line, −7 sits farther left — deeper in debt. Distinguish size of the number (position) from size of the absolute value (distance from zero) and the confusion dissolves.
The misconception
“Subtracting always makes smaller.”
The repair
Subtracting a negative moves you right. 5 − (−3) = 8. If the answer surprises a child, walk it: face right, turn (subtract), turn again (negative), step three.
Seen, not said
The visual explanation
A walker on a number line responding to each operation as a movement instruction. Sign rules become choreography — visible, repeatable, and impossible to misremember once danced.
Try it
Interactive example
Walk the line. Try '− (−3)' and watch the two turn-arounds cancel.
Negative numbers are directions, not mysteries
Adding faces you right. Subtracting turns you around. A negative sign turns you around again.
-10
-5
0
5
10
0