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WM

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This is a Work In Progress ...

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This is a pre-draft - it's not even at the stage of being

a draft yet. It's putting down some of the ideas. Not all

the ideas are here yet, and they certainly aren't in the right

shape. Even */I/* don't agree with some of what follows, but

I thought it might be interesting to some people.

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A recent comment on Twitter was talking about helping youngsters

understand numbers. The idea was to place numbers on the number line,

and then talk about their different representations.

So you have a number line. Where does "1" go? Where does "5" get

placed? The whole numbers are, once past a certain age (and not

always then!) fairly straight forward.

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 is interesting. There are cetainly some

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".

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".

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.

But then we get the interesting question: Where is "15%"?

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

(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. But is

it right to think of "15%" as a number?

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.

I'm still uneasy.

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.

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 positions 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.

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.

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?

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Still to add:

We sometimes say that "1/2" is "1/2 of something", and then we refer

to "1/2" as a number, a place on the line. Surely just because we usually

call a percentage a percentage *of*something* that shouldn't then stop us

from putting it on the line.

Well, maybe it should. And maybe it should give us pause about how we

talk about fractions with children. Maybe some of them can't add

fractions exactly because we casually confuse "2/3 of something" with

the number "2/3" (whatever that means). Maybe we are seriously doing

them a disservice, and this question about percentages is highlighting

the casual confusion we take for granted.