Most recent change of WhereAreTheNumbers

Edit made on May 18, 2013 by ColinWright at 17:10:41

Deleted text in red / Inserted text in green

WM
HEADERS_END
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.