Why carrying is where children get stuck
A child who answers 2 + 5 can still stop at 7 + 5, and it is not about the size of the numbers. Carrying, broken into the three steps it really is.
A child who answers 7 + 5 can stall at □ + 5 = 12. Same numbers,
same answer. It is a different problem because it asks a different question.
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.
"Undo addition with subtraction" sounds like the whole rule. It is not: for some operations, moving the box from one side to the other changes the step needed. Four operations and two positions make eight cases:
□ + 5 = 12 → 12 − 5 ／ 7 + □ = 12 → 12 − 7□ − 5 = 7 → 7 + 5 ／ 12 − □ = 7 → 12 − 7□ × 3 = 6 → 6 ÷ 3 ／ 2 × □ = 6 → 6 ÷ 2□ ÷ 3 = 2 → 2 × 3 ／ 6 ÷ □ = 2 → 6 ÷ 2
Addition and multiplication undo the same way whichever side the box is on, because the
order does not matter (7 + 5 is 5 + 7).
Subtraction and division are different.
12 − □ = 7 is solved by subtracting (12 − 7), and
6 ÷ □ = 2 by dividing (6 ÷ 2). A child who learned
"do the opposite" as a rule writes 12 + 7 = 19 and 6 × 2 = 12.
Of the eight cases, these two are the ones the rule gets wrong.
What gets a child through these two is not a better rule but picturing the quantities the equation describes. "Twelve, take some away, seven left - how many went?" makes it plain that the step is subtraction. When a child is stuck here, that rewording is what helps.
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.
This is not a thought experiment. 8 + 4 = □ + 5 is the very item Falkner,
Levi and Carpenter gave to children in grades 1 to 6 in a 1999 study in the United States.
Besides the correct 7, they report answers of 12 (work out the
left side and stop) and 17 (add every number in sight). Both come from
reading "=" as an instruction.
"=" 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.
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.
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.
The eight cases in section 2 are not equally hard. Once the forward facts come easily, taking them in this order keeps each stumble separate from the next:
□ + 5 = 12□ − 5 = 7 (undo by adding)12 − □ = 7 and 6 ÷ □ = 2A child who stalls at step four does not need another rule. Going back to quantities - "twelve, some taken, seven left" - gets there sooner.
In this drill the box appears on either side - □ + 5 = 12 and
7 + □ = 12 - because, as section 2 showed, the step needed can change.
Any of the 42 kinds can be set to use a box, and each of the
9,417 problems has a left and a right version, so there are
18,834 box problems in all, including the 12 − □ = 7 and
6 ÷ □ = 2 shapes.
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.
Pick a kind by hand and the boxes appear. No sign-in needed.
For a walk-through with screenshots, see the full introduction.
More reading
A child who answers 2 + 5 can still stop at 7 + 5, and it is not about the size of the numbers. Carrying, broken into the three steps it really is.
Treat a x b and b x a as one fact and the pile halves. Which ones are last to stick, and what is gained by taking one table at a time.
Wrong, slow, or not seen for a while - three faces of weakness weighed into one number, plus what has to be added so that practice stays bearable.
Where the four- and ten-second marks came from, and what to do about the case that timing children makes them afraid of maths.
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