Problem-solving method
The before and after method, explained for parents
For every problem sum where something changes partway through. Here is why one drawing is never enough, the words that give it away, and what to ask once both pictures exist.
- Used from
- P3
- Taught at
- P3–P6
What this method is
The frame for any question where something changes partway through. Money is spent, stickers are given away, more water is poured in. You draw the situation twice, once before and once after, and the answer falls out of what the two pictures share.
It is less a technique than a habit: one bar cannot hold two moments in time. Nearly every hard question is a before and after underneath, which is why MOE lists it among the twelve heuristics.
When to reach for it
These are the words in a question that point at this method. You do not need to be able to solve it to spot them.
“at first”
There is a before state, and the question is almost certainly asking about it rather than about the change.
“gave 12 of his stamps to”
A transfer between two people. Two pictures, and the total across both of them has not moved.
“after spending”
The after state is the part you are told. The before state is the part you want, so draw both and label which is which.
“in the end”
You are being handed the finish line. Draw the after picture first, then work back towards the start.
“now has 3 times as many”
This describes the second drawing, not the first. Children who apply it to the before picture lose the question here.
Watch it work
Ben had 120 stamps and Sara had 48. Ben gave some of his stamps to Sara, and in the end they had the same number. How many stamps did Ben give to Sara?
Draw the before
Ben 120, Sara 48 → the gap is 72Two bars, lined up at the left, and the bit sticking out is the gap. The question never mentions 72, which is exactly why it has to be drawn rather than read.
Draw the after before you can fill it in
Ben = SaraYou know the shape of the second picture from the words "the same number", even with no values in it. Drawing an empty answer is allowed, and here it is the point.
See what moving one stamp costs
Ben − 1 and Sara + 1 → the gap shrinks by 2This is the step that only the two pictures give you. A stamp moved is not lost, it changes sides, so the gap closes twice as fast as it feels like it should.
Close the gap
72 ÷ 2 = 36 stampsCheck it against both drawings. 120 − 36 = 84 and 48 + 36 = 84, which is the equal after picture you drew in step 2 before you knew any of the numbers.
The one it gets confused with
Two pictures is the frame. What you do with them is a separate decision, and these are the three it gets confused with.
Does anything actually change? If the question is one moment in time, one picture is enough and a second only adds work.
Constant total, difference and part
Once both pictures exist, what stayed the same is the very next question, and it picks the method from there.
Is the change a chain of steps with only the ending known? Then undo them one at a time rather than drawing two states.
How can you tell it has clicked?
Give them a question with "at first" in it and watch what gets drawn. Two labelled pictures before any arithmetic means the habit is there. One bar with numbers crossed out and rewritten means it is not.
The other tell is the check at the end. A child who knows why the two totals match in a transfer will notice on their own when they do not.
One thing to try tonight
Whenever a question says at first or in the end, ask for two empty boxes on the page before anything else. One labelled before, one labelled after.
That is the whole exercise. Half of these questions come apart the moment the two moments stop sharing a drawing, and it takes ten seconds.
Where your child will meet it
Want the bigger picture on how these methods fit together? Maths heuristics, explained for parents
See which methods your child reaches for
StudyLah names the method in every worked solution, so you can see whether the problem was the method or the arithmetic.