Where Are The Numbers
A recent comment on Twitter was talking about helping youngsters understand numbers. The idea was to write numbers on stickers in different ways, and then place the stickers on the number line. Then they could talk about their different representations.
So you have a number line. Where does the sticker with "1" on it go? Where does the sticker with "5" get placed? The whole numbers are, once past a certain age (although not!) fairly straight forward.
Then fractions. Where does "1/2" live? Where should we put "2/3"? What about "16/9"? And how about "23 3/4"?
Then decimals. Where should we put "0.1"? What about "3.25"? And you can see that this might be interesting. There are cetainly some possible misconceptions to be had here. Where is "0.025", and why is that further to the left than "0.1"? After all, "1" is smaller than "25". But teachers know of the pitfalls involving decimals, so this isn't such a problem.
So the exercise had the students putting these various items on the number line, and then observed that the same location had different representations. So the location "4" was also "20/5", and the location "3/4" was also "0.75". I think this is a brilliant idea. It's so important to realise that what we write is a representation of a place on the line, and the same place can have different representations. Perhaps we would have fewer problems with the idea that zero point nine recurring is equal to one if we just thought of them as different representations of the same point on the line.
I don't know how the lesson went, but it seems a reasonably fun activity to help understand numbers and their representations.
But then we get the question that brought me up short: Where is "15%"?
Now here's the thing. I've never really thought of "15%" as a number, or as a place on the number line. I've always thought of "15%" as an operation to perform. We take "15%" of a quantity. Certainly we can (and do) achieve that by the expediency of multiplying our quantity by the number "0.15", or by multiplying by "15" and then dividing by "100". But I've never thought of "15%" as being the number "0.15", or the number "15/100".
But clearly some people whose opinions I respect do think of "15%" as simply being "0.15". "What else can it be?", they ask.
And there is some logic to this. When we talk about "1/2" we happily place it on the number line, equi-spaced between "0" and "1". This, even though we usually talk about "Half of a quantity" in the same way as we talk about "15% of a quantity." There is a consistency there, to be sure. And if you ask "Where is 15% on the number line?" then there is nowhere else to put it. It's consistent, yes, so perhaps it's right to think of "15%" as a number.
But I'm still uneasy.
To equate "15%" with the number represented by the decimal "0.15" seems to me to require the same casual sliding between concepts as when I say that when I was trying to learn to juggle five balls there were "2 1/2" people in my local club who could do it.
Does that give you pause for thought? See, there were two who could, and one who was half way through learning. In a very real sense he could "half juggle" five balls, and since one times a half is the same as a half times one, one person who can half juggle five is the same as a half a person who can juggle five. Hence there was a half a person who could juggle five.
So when you say that 15% "is" the number 0.15, what exactly do you mean? Is this really as obvious as you think? Perhaps you have reached a level of sophistication where it is obviously "the same", but are you mixing concepts in a manner confusing to those not yet at your level?
Perhaps this comes back to my current understanding of multiplication not as repeated addition, but as an operation on the line. I don't think of "multiplying by three" as "add three copies together." I think of it as "Take the number line and stretch it by a factor of three." I freely admit that this is not the way to teach multiplication to five year olds, or seven year olds, or whenever multiplication gets taught. However, it is certainly the way I have come to think of it as I have to consider more and more sophisticated structures.
And are you comfortable thinking of multiplication not as repeated addition, but as a linear transformation on the structure under consideration? That's an operation in its own right, it extends the concept of repeated addition, but it's more general. So isn't that better?