Math Musings: Lies My Teacher Told Me

I mentioned last time that the common phrase “Multiplication is repeated addition” is a mathematical lie we tell our children. And it’s not the only one.

Did you ever say, “Subtraction means take-away”? Or how about “Division is sharing”? I know I have, but both of those statements are also mathematical lies.

One of the reasons I like Cuisenaire rods so much is that they can help us avoid lying to our children about math.

Mathematical operations are not actions

We call addition, subtraction, multiplication, and division the four basic operations of arithmetic. But math operations are not actions, procedures, things we do to numbers.

Mathematical operations are relationships. They are part of the conceptual web that ties numbers together.

For example, addition is a relationship between two numbers which connects them to a third number we call the sum. Addition doesn’t combine two numbers to make the sum. All three numbers remain themselves, yet they are linked in the addition relationship

When we build an addition pattern with Cuisenaire rods, we can see all three numbers and how they relate.

Subtraction is not “take-away”

Subtraction does not remove anything from a number. When we say “10 − 6,” the 10 does not cease to be itself.

Subtraction is a relationship between two numbers along with a third number that names the difference between them.

With Cuisenaire rods, the pattern for this relationship looks like a missing-number puzzle.

Do you notice how similar this is to the pattern for addition? That’s what we mean when we say that addition and subtraction are inverse operations. They are two ways of looking at the same mathematical relationship.

Why does it matter what we say?

We want to introduce our children to the great ideas of mathematics. But when we tell children the lie that “subtraction means take-away,” we have only given them a procedure, a method for calculating by counting physical or mental beans.

We can use the procedures of combining, take-away, repeated addition, or sharing to solve early elementary problems in basic arithmetic. But if we tell our children that these procedures are what the symbols +, −, ×, and ÷ actually mean, we cripple their understanding and make it difficult for them to master later topics in math such as fractions, integers, and algebraic expressions.

The big ideas of math are not mere facts or procedures. To find true ideas, we must look to the principles of algebra.

What do we mean by a statement like “x – y = 4”?

Did y jump out of a dark alley to attack x, whacking him into pieces and shoving some of them into a bag, then run away leaving poor little 4 lying stunned in the street?

No, “x – y = 4” means this: “Find two numbers such that their difference is four.”

Swimming in the sea of ideas

How can we investigate a pattern like “x – y = 4”?

Perhaps we might think of the numbers 10 and 6. Are they different by 4?

Can we find some more? Take turns naming pairs of numbers that fit.

How can we tell whether we’ve found them all?

Do they have to be whole numbers? What if one of the numbers is a fraction, like 1/2? Can we find an “x – y = 4” partner for that?

Do the numbers have to be positive, or can we find a negative-number pair that works?

Can we find a number that doesn’t have a “x – y = 4” partner?

What else can we ask?

Can we create something from what we’ve learned? A graph, maybe, or another puzzle?

Taking a deeper dive

Once we start to explore a true idea in math, there is always something more to discover.

For example, given a pair of numbers that fits the “x – y = 4” pattern, how can we decide which one is x and which is y?

Notice that subtraction is different from addition or multiplication because it matters which number we put in which position. Subtraction is not commutative.

With addition, if we find two numbers that fit “x + y = 12,” it doesn’t matter which number we put first. The sum is the same whether we write “5 + 7” or “7 + 5.”

But with subtraction, “10 − 6” is not the same as “6 − 10.” NOT because subtraction is take-away. We aren’t trying to take 10 beans away from 6 beans.

Instead, it has to do with the meaning of the math symbols, with what we have all agreed (as a society) the minus sign will mean.

The subtraction equation:

“x – y = 4”

is the same number relationship as:

“x = y + 4.”

That means we can decide which number is x and which is y by seeing how they fit into the equivalent addition equation.

This can be especially helpful with negative numbers, where intuition often fails us.

If we know that −14 and −10 have a difference of four, then which of these addition equations makes sense:

−14 = −10 + 4?

or −10 = −14 + 4?

The second one is true, so we know our subtraction pattern must be:

−10 − (−14) = 4.

What about multiplication?

There was a huge kerfuffle in the online math education community several years back when Keith Devlin first pointed out that multiplication is not repeated addition.

If you’re interested, you can read all about it on my blog:

• What’s Wrong with “Repeated Addition”?

And check out a mathematician’s approach to multiplication in this video by James Tanton:

• What Is Multiplication, Anyway?