Missing-number problems are not really about addition

1Subtraction wearing the face of addition

To fill in □ + 5 = 12, a child has to work out 12 − 5. The sign on the page says plus; the operation needed is minus. That is what an inverse problem is: the symbol and the action come apart.

A child who knows addition and subtraction are two views of one relationship gets there by themselves. A child who does not adds 12 and 5 and writes 17. The wrong answer tells you which of the two you have.

2How the equals sign is being read

There is a second problem underneath, and it runs deeper. A child raised on 3 + 4 = □ learns to read "=" as an instruction: write the answer next. Left to right, answer at the end. That is what it has always meant.

Read that way, □ + 5 = 12 looks malformed: the answer slot is on the wrong side. Show the same child 8 + 4 = □ + 5 and a common response is 12 - the answer, placed to the right of the equals sign, as always.

"=" says only that the two sides are the same size. It carries no direction and no order of writing. Missing-number problems ask whether that reading has arrived, and they ask it while looking like ordinary arithmetic.

3Memorised facts are not enough

Number facts can be memorised whole, and there is nothing wrong with that. But a forward-facing problem cannot tell memorised from understood. Both children answer, and at much the same speed.

Put a box in and they separate. If the relationship is there, the same remembered fact can be entered from either end. If it is not, the child stops. Same material, approached from another side.

4This is also where algebra starts

Replace the box in □ + 5 = 12 with an x and you have x + 5 = 12. The work is already algebra. Secondary school changes the notation; the thinking has been met once already, in primary.

English-speaking schools call them missing number problems; Japanese textbooks use an empty box and the name says just that. Both arrive in the middle primary years, and both are after the same thing: making the equals sign mean a relationship.

5Not mixed in with the rest

In this drill the box appears on either side - □ + 5 = 12 and 7 + □ = 12. The same equation, but not the same step to get there.

They are left out when the app is choosing questions for you. Mixing a different kind of thinking into a timed run makes both the speed and the record harder to read. The boxes are there for when the practice is chosen deliberately. The full breakdown is in the 42 kinds.