Explaining Is Part of the Maths

Most of us were taught that Maths is about getting the answer. You had it or you did not, the tick settled it, and whatever happened in your head along the way was your own business.

That is not what the subject is, and it is not what your child is being assessed on.

The Australian Curriculum describes proficiency in Mathematics as four things rather than one: understanding, fluency, problem-solving, and reasoning. Reasoning is the one about explaining and justifying, saying why a step is allowed rather than only taking it. It is not an extra for the children who finish early. It is a quarter of what the subject is.

So instruction five on every BrainBike worksheet, the one asking you to get your child to explain their answer, is not a device for catching them out. It is the other half of the exercise.

The part that outlives the algebra

Most of what your child learns in Year 8 Maths they will never knowingly use again. This bit they will use constantly.

Someone who can take a thing they understand and lay it out so another person understands it has a skill that works everywhere: it is most of what a good engineer does, and a good GP, and a tradesperson explaining a quote. Being right is necessary and not sufficient. A person who cannot walk you through their reasoning gets overruled by people who are wrong and articulate, for the rest of their career.

Maths is an unusually good place to train it, because the ground truth is not up for argument. Nobody can tell your child their reasoning was a matter of taste. Either the steps hold or they do not.

Why the ticks alone will mislead you

The practical argument runs alongside the principled one.

Here is a Year 5 sheet. 5 × 10 = 50, correct. 0.5 × 10 = 0.50, wrong. Ask how, and you get "you add a zero."

That rule earned the tick on the first question. The tick is the problem. It recorded a right answer produced by a method that falls apart the moment a decimal turns up, and the marks alone would have had you file the first question as known and move on.

This cuts both ways, which is the point. A child who gets 24 ÷ 6 right by counting up in sixes on their fingers has handed you a correct answer and told you their times tables are not there. Both of those are things you can only learn by asking.

Why saying it out loud does anything

Drivers and guards on the Japanese rail network use a procedure called shisa kanko, or pointing-and-calling. Approaching a signal, the driver physically points a finger at it and announces out loud what it shows. Pointing at a green light and declaring it green feels ridiculous, and the figure usually quoted is a reduction in errors of up to 85%. James Clear tells the story in Atomic Habits, whose ideas I wrote about in Establishing a BrainBike Habit, and his summary of why it works is the best one sentence on this I know: it forces automatic behaviour back into conscious awareness.

That is the entire mechanism here. "Add a zero" survives precisely because it is never said out loud. Run silently it collects a tick and moves on. Said out loud it has to stand up in front of both of you, and most bad rules cannot.

Ask on the ones they got right

This is the part most households get backwards, including mine for a while.

"How did you get that?" tends to come out only after a wrong answer. It is a natural instinct and it ruins the question. A child who has only ever heard it after a cross learns that it is not a question at all, it is a verdict with a question mark on the end. Give it a month and they will reach for the eraser before they have said a word, because the asking already told them the answer was wrong.

Ask it when they are right. Same tone, same words, roughly as often. Once the question carries no information about whether they are in trouble, it stops being an interrogation and starts being a conversation, which is the only form in which anybody practises explaining anything.

Four questions

One or two of these, on one or two of the five questions. Not all five, and not every day.

"How did you get that?" The default.
"What do you think the answer will be?" Asked before they check. Committing to a prediction and then finding out makes the correction stick in a way that drifting into it does not.
"How would you check it?" A different skill from solving, and a child who can do it needs you less.
"Why is that step allowed?" Asked about the printed solution rather than their own working. This is the one for the questions they got wrong, where "how did you get that" has nowhere useful to go. Do not explain the solution to them. Put it in front of them and have them explain a line of it back to you.

Two things not to do. Do not lead: "so you multiplied, didn't you?" hands over the answer and buys you a nod. And do not fill the silence. The pause after you ask is the part that is working, and most of us jump into it inside three seconds.

You do not need to know the Maths

This is the reason a lot of parents decide the job is not theirs, and it is wrong.

You are not auditing the explanation, you are listening to its shape. An explanation somebody owns and one they are assembling on the spot to make you go away sound nothing alike, and you have been telling those apart your whole adult life in every other subject without being an expert in any of them. The worked solution is printed for you, so supplying the Maths was never your job.

Where to stop

If you ask how they got 7 × 8 and the answer is "I just knew it," that is the right answer and the goal. Recall is meant to become automatic, and digging for a method there teaches a child that knowing something instantly is suspect. Fluency and reasoning are two different proficiencies and they do not want the same treatment.

The whole thing is about a minute on top of the grading. An interrogation of all five questions every afternoon is the fastest way to make sure nobody wants to sit down tomorrow.

Five questions on paper, a minute of asking, five honest ratings. The middle part is the one the software cannot do, and it is also the only part your child will still be using at forty.

If you want that running in your house without planning it yourself, the free trial is the fastest way to see it. If you would rather read more of this sort of thing first, subscribe to the newsletter, it is where these land before anywhere else.

On the research: prompting learners to explain their reasoning has a solid evidence base of its own (Bisra and colleagues pooled 69 studies on it in Educational Psychology Review*, 2018), and even Craig Barton, who argues harder than most for explicit instruction over discussion, builds explanation into both of his books. The classroom routine known as a number talk is a different and much less well evidenced thing: Matney and colleagues went looking for the research behind that one in 2020 and found very little. Nothing above is a claim that talking will raise your child's marks.*