If Not Methods: Reasoning About Subtraction

We’ve been examining the fact that, while there may be only one right answer to a math problem, but there’s never only one right way to get that answer.

What matters in math is the journey. How do your children make sense of the problem and reason their way to that answer?

As always, real math is not about the answers but the thinking.

But if we don’t want to give our children a method, how can we teach? What if we pose a problem and the child doesn’t know how to solve it?

What if our children get stumped on a subtraction calculation like 431 – 86?

When Kids Say, “I Don’t Know How”

There is a way to teach that doesn’t cripple a child’s understanding, an age-old technique of encouraging curiousity that leads to learning. I call it the Notice-Wonder-Create cycle.

Ask your children to Notice everything they can about the problem. Take turns listing details about the numbers, shapes, or story problem situation.

Ask your children to Wonder about the possibilities. Take turns posing questions related to the things you’ve noticed.

Ask your children to Create something new, based on the ideas they’ve noticed and wondered about. They may create a solution to the problem, an explanation of how to solve it. Or they may create a similar problem of their own, or perhaps a journal entry about their wonderings.

“Notice, Wonder, Create” is not a three-step method for solving math problems. It’s the natural, spiraling cycle by which our minds learn anything.

For Example: 431 – 86

What if our children get stumped on a subtraction calculation like 431 – 86? As we take turns noticing and wondering, we might say things like…

• I notice the first number is 431.
• I notice we’re subtracting, so the answer will be smaller.
• I wonder, does subtraction always make things smaller?
• I notice that 86 is more than 31, so when we subtract, the number will go under 400.
• I wonder, will it go under 300?
• I notice that 86 is less than 100.
• I wonder, how much less?
• I notice that because 86 is less than 100, I know the answer must be more than 431 – 100 = 331.

And so on, until the child’s mind sparks with an “Aha!” that leads to creating a solution.

(Children will also notice or wonder things that go radically off-topic. A little of that is fine, but don’t hesitate to direct their attention back to the problem at hand.)

How To Solve It

So how might a student stuck in counting mode (or just beginning to move into additive thinking) reason about subtraction?

We have two main ways to think about subtraction: take-away, and distance.

Take-Away: Counting Down

With numbers too big for our fingers, we need to count in chunks.

For 431 – 86, we can begin by taking away one (85 to go). Then subtracting 30 more will bring us down to 400.

How much farther do we need to go? We still need to take away 55. Removing 50 gets us down to 350, and taking away five more makes the answer 345.

431 – 1 – 30 – 50 – 5 = 345

As children develop their number skills, they will be able to reason in larger chunks. Perhaps they will move from 431 to 400 in one step, and from there directly to the answer:

431 – 31 – 55 = 345

Distance: Counting Up

How far apart are 86 and 431? How much would we need to add to go from one number to the other?

We can count up four, from 86 to 90. Then add 10 more to get 100. We’ve added fourteen so far. Plus 300 will bring us up to 400, and then we still need 31 more:

4 + 10 + 300 + 31 = 345

As children develop their additive thinking, they will be able to reason in larger chunks. Perhaps they will move from 86 to 100 in one step, and from there directly to 431:

14 + 331 = 345

Take-Away: Subtracting Over

A child who has grown into a more sophisticated way of thinking might look at the problem and reason this way:

“431 – 100 is easy. That’s 331, but I’ve taken away too much. How much extra do I need to add back? 86 is fourteen less than one hundred, so…”

431 – 100 + 14 = 345

Distance: Constant Difference

Another sophisticated thinking strategy: If we add (or subtract) the same amount from both numbers, the distance between them stays the same.

So we can add four to both numbers, creating the easier calculation 435 – 90. Then we can add ten more, making it even better:

431 – 86 = 445 – 100 = 345

Two Rules of Mathematics

Your children don’t have to solve math problems by the same method you would use, nor by following the procedure in the book. They can use any method that makes sense to them, as long as they can explain why it’s true.

As master teacher W.W. Sawyer said:

“The main purpose is to get him into the habit of thinking, and to show him that he can think the problems out for himself.

“Pupils ask ‘Am I allowed to do this?’ as if we were playing a game with certain rules.

“A pupil is allowed to write anything that is true, and not allowed to write anything untrue!

“These are the only rules of mathematics.”

—W. W. Sawyer,
Vision in Elementary Mathematics