The Forgotten Art of Decomposition

I have a belief that all of us are curious beings who have an innate desire to see how things work: to some extent or another, we will take something apart to figure out the inner mechanisms then reassemble it. Being able to take an object apart, and to then put it back together again tells us a lot about that object: its features, what it can do, what it can't do. This ability to decompose (and recompose) objects is fundamental to problem solving.

However, I think it is something that is often overlooked in Maths.
For example, look at this number, and think about how many different ways you see it.

40

How did you see it?
Forty?
Four tens?
Eight fives or five eights?
Half of eighty? Eighty halves? 
Ten less than fifty? Sixty less than one hundred?
Two twenty pence pieces (in the UK)?
Four dimes? A quarter, a dime, a nickel (Canada and the U.S.)?

Or do you get more dangerous and think:
Four hundred tenths?
120 thirds?
The sum of two squares?
The sum of two prime numbers? A different sum of two prime numbers?
The sum of three prime numbers?
2×2×2×5?

To me, being good at Math necessarily involves the ability to decompose and recompose objects. In number sense, this is much, much more than simply memorising facts. Now don't get me wrong: kids (and adults) do need to learn and understand their additive and multiplicative facts. But I worry that when students memorise these without making connections then we create situations where kids 'know' 7×5 but don't know 5×7. Or they know both of these but cannot do 35÷?=5. Or they cannot look at a number like 40 and figure out its factors.

We can help students learn how to decompose numbers by beginning with some concrete representations. Here, grade 1 students work with deciblocks to see 10 decomposed in a variety of ways.

This in turn can be backed up with diagrammatic representations:

We can also share and model decomposition strategies that we use. I ran a Math Night for parents in one of our schools recently and shared with them the following:

• knowing that 9 is one less than ten means that adding nine is more simply done as 'add 10-1'
• multiplying by 9 is more simply done as 'times by 10 subtract the number' so 9×27 is 270-27=243

Think how powerful this is: by decomposition, we have connected multiplying by ten to multiplying by nine. We can then connect these to multiplying by 11, or twelve (ten times plus a double) or multiplying by 20 (times by ten then double). The smiles on the parents' faces after I shared these strategies were great to see and we agreed that there is a big difference between memorising a fact and understanding a fact.

Even when learning facts, it is powerful to see four sixes as the same as two twelves hence 24. In fact whenever you multiply by four knowing this as doubling twice is very powerful. Arrays are fantastic tools to help students understand why facts. Consider this from Jordan Ellenberg in his book How Not To Be Wrong:

"Here's my earliest mathematical memory. I'm lying on the floor in my parents' house...looking at the stereo. Maybe I'm six. This is the seventies and therefore the stereo is encased in a pressed wood panel, which has a rectangular array of airholes punched into one side. Eight holes across, six holes up and down. So I'm lying there, looking at the airholes. The six rows of holes. The eight columns of holes...Six rows with eight holes each. Eight columns with six holes each.
And then I had it-eight groups of six were the same as six groups of eight."

Finally, a little puzzle: Suppose you are being paid mileage at 45 cents per km and you have driven 218 km. How could you work out your expenses without using a calculator or an algorithm? Scroll down to see my method.










50c/km would pay $109, 5c/km would pay $10.90.
Thus 45c/km would pay $109 - $10.90 = $98.10.