Connecting information, not learning new rules

One of the things I enjoy most about teaching mathematics is showing students that most mathematical rules aren't invented out of thin air.

Unfortunately, that's often how mathematics is presented in school. Students learn one topic, memorise a handful of rules, complete a worksheet, and then move on to the next topic. Before long, mathematics can begin to feel like an enormous list of unrelated facts that simply have to be remembered.

But that's not what mathematics really is.

One of the beautiful features of mathematics is that new ideas almost always grow out of old ones. Concepts aren't replaced, they're extended. Each new topic builds on the foundations that came before it, creating a logical structure in which everything fits together. Mathematics isn't a collection of isolated topics. It's a single connected story. Each new idea grows naturally from ideas we've already developed, allowing us to solve increasingly sophisticated problems. Once students begin to see those connections something changes, mathematics becomes far easier to understand, and much harder to forget. Mathematics becomes less about remembering rules and more about understanding why those rules have to be true. I want to expand on this with an example:

A negative multiplied by a negative equals a positive.

Who remembers this from year 7 or year 8? I thought about this very recently when tutoring a (wonderfully) curious student who said that her teacher had told them this, but she didn’t know why.

Many students can recite this rule, but far fewer understand where it comes from. Let us look at the steps in the development of this concept, starting much earlier in our school lives:

To understand why this rule exists, we don't start with negative numbers. We start somewhere much more familiar: the idea of multiplication itself.

Step 1: Multiplication begins as repeated addition

When we first learn multiplication in primary school, it isn't introduced as a completely new operation. Instead, it is presented as a simpler and more efficient way of writing repeated addition.

Suppose we have four groups of three objects. We could write

3 + 3 + 3 + 3

and then calculate the answer by adding each group together.

Eventually we're shown a much shorter way to express exactly the same idea:

4 × 3 = 12

Nothing about the mathematics has changed. Multiplication is simply acting as shorthand for repeatedly adding the same quantity.

This interpretation works beautifully whenever the first number is a positive whole number.

For example,

5 × 2

means adding 2 together five times.

Similarly,

8 × 10

means adding 10 together eight times.

Even something like

100 × 4

can be understood as adding 4 one hundred times—although we're certainly grateful that multiplication allows us to avoid writing such a long sum.

Notice what's happening here. Multiplication hasn't appeared as a completely new idea. Instead, it has grown naturally out of something we already understand: addition. We would then connect this understanding to rectangular arrays: in school we soon discover another interpretation of multiplication through rectangular arrays. Four rows of three objects produce exactly the same calculation as repeated addition, but they also introduce multiplication as an area model. Rather than replacing the repeated-addition idea, this simply gives us another way of thinking about the same mathematics.

This pattern appears throughout mathematics. Rather than inventing entirely new concepts, mathematics continually extends existing ones so they can solve bigger and more interesting problems.

So what happens when we introduce negative numbers?

Can multiplication still be understood using repeated addition?

The answer is yes, at least to begin with, and that simple observation explains much more than most students realise.

Step 2: Adding and subtracting negative numbers

It will definitely help students to understand multiplication with negative numbers if an effort has been made to help them understand addition and subtraction of positive and negative numbers. It’s helpful to remember that negative numbers aren’t really a concrete thing students experience in life, like the counting numbers. If you want to focus on understanding that students don’t find this intuitive, you just need to look at history. Historically, negative numbers were not always accepted. Some early mathematicians considered them absurd because they could not imagine a quantity that was "less than nothing." Even mathematicians had trouble making ‘sense’ of a negative number. (I find it nice to refer back to this when introducing the unfortunately named imaginary numbers in later years also!)

Yet merchants and accountants in ancient cultures, including China and India, found negative numbers extremely useful for recording debts and credits: a positive number could represent money owned, while a negative number could represent money owed. Over time, mathematicians realised that treating these ideas consistently led to a powerful and elegant number system that extended the mathematics people already understood. I found it a very useful paradigm in which to teach negative numbers – suppose you have a certain amount of money, and some is added or subtracted. This provides a really nice context for taking away someone’s debt and leaving them better of than they were. Subtracting a negative number increases the total you have.

Step 3: Repeated addition with negative numbers

So now we can easily deal with, say, 4 × -3. We are repeatedly subtracting 3, or adding negative 3 4 times.

-3 + -3 + -3 + -3 = -12. The multiplication rules make sense it terms of the ideas we already understand.

Step 4: Multiplying a negative number by a positive one

How do we interpret -5 × 2 ? Can we think about adding 2 minus 5 times? We have some choices here, and it’s nice to potentially show all of the connections to students. We can use our understanding of multiplication as commutative (e.g. 6 × 7 = 7 × 6) to infer that this must be equal to 2 × -5, which we understand how to do, or we could investigate the interpretation found in Step 5.

Step 5: Then what about (-4) × (-3)?

The problem we have here is that our idea of multiplication as repeated addition was developed in the context of positive numbers. How do we interpret (-4) × (-3)? Here's where repeated addition reaches its limit. What would "negative four groups of negative three" even mean?

Repeated addition no longer gives us a direct interpretation.

Instead, mathematics asks a different question:

What answer keeps all the patterns we've already established true? (This is a key idea; mathematics often tries to generalise as far as possible).

We know

4 × (-3) = -12

3 × (-3) = -9

2 × (-3) = -6

1 × (-3) = -3

0 × (-3) = 0

Notice that every time the first number decreases by 1, the answer increases by 3.

To keep that same pattern going, the next answer must be

(-1) × (-3) = 3.

Then

(-2) × (-3) = 6

and eventually

(-4) × (-3) = 12.

If negative times negative were anything else, multiplication would stop following the same logical patterns that work everywhere else. This is the way it must work for negative numbers. Notice what's happened so far. Every new idea has grown out of an earlier one. We haven't memorised any new rules, we've simply extended ideas that already made sense

The bigger lesson

One of the beautiful things about mathematics is that very few rules are completely arbitrary.

Most of them grow naturally from ideas you've already learned.

Multiplication begins as repeated addition.

Negative numbers extend addition to include opposites.

Eventually, these ideas meet, and the familiar integer rules emerge because they're the only rules that keep mathematics consistent.

That's why I encourage students to ask "Why?" as often as possible.

Understanding always lasts longer than memorisation.