Jonathan Groves
Posts:
2,068
From:
Kaplan University, Argosy University, Florida Institute of Technology
Registered:
8/18/05
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Re: Presenting Numbers To Adults
Posted:
Sep 18, 2010 9:40 AM
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On 9/17/2010 at 12:04 pm, Alain Schremmer wrote (in part):
> Are fractions really necessary--other than code for
> division? If so,
> for what? (Aside from appeasing ignorant
> administrations.)
Alain and others,
You may recall that my post on the table of contents for my book got
copied over to Math-Teach by accident. Bill Marsh on Math-Teach
had written to me, and his response is found in the post
http://mathforum.org/kb/message.jspa?messageID=7208301&tstart=0.
My response to him can be found at
http://mathforum.org/kb/message.jspa?messageID=7208551&tstart=0.
I had explained to him that what I had planned so far for the
book is only undergoing development, not a final plan, and in
fact will almost certainly need to change anyway and that my reasons
for thinking this were motivated by what Alain had said. Actually,
I was wondering about that development even before Alain had said
something, but he had encouraged me to think about other issues
as well in deciding what goes into the book.
Robert Hansen had replied to me last night, and I just wrote him
a reply. Because the comments I had said are of interest on this
list, I will include my reply below (since the post has not appeared
yet, there is no link I can give):
On 9/17/2010 at 9:31 pm, Robert Hansen wrote:
> Jonathan wrote...
>
> "Are fractions really necessary--other than code for
> division? If so, for what?"
>
> How can you not include fractions? They are the basis
> of ratios, percentages and all other rational
> expressions. Can you be more specific by what you
> mean when you say "not include fractions"? Is this
> primarily a book for remediation?
Robert,
First, that was a question Alain Schremmer asked, and I think it is a
good question to ask and to consider. He does at least make a good
point in that perhaps we beat fraction arithmetic to death too much, that
many support a heavy emphasis of fractions in math courses simply because
of conventional wisdom or mere tradition. If there is indeed a good
reason for fraction arithmetic receiving the emphasis it does, I don't
see that reason. And here I'm talking about the general student, not
those interested in STEM careers or other specialized careers where a
strong knowledge of how to perform fraction arithmetic is essential.
I don't see that many careers where a strong knowledge of fraction
arithmetic is essential.
Adding and subtracting fractional measurements seems necessary to
a lot of people but only because some people need that and only because
the customary system of measurement uses fractions. The metric system
does not, and if America goes completely metric someday--and I'm sure
it will happen, but when that will happen is anyone's guess--then
all our measurements will be expressed in decimal form. I really
do wonder how many people will truly need to know how to add or
subtract measurements expressed as fractions, and much of that need
will disappear when we go metric. I also wonder how many people
can even read rulers for themselves well enough to express lengths
as fractions. I would guess that relatively few know how to do that.
Yes, I agree that fractions are the basis of ratios and percentages,
but not any significant knowledge of how to perform fraction arithmetic
is required. We will need, of course, to know how to take a fractional
or percentage amount of a quantity and need to know that nx to ny is the
same ratio as x to y, but even that last statement can be proven by noting
that multiplying the dividend and divisor of a division by a common number
does not change the quotient.
I would not argue that fractions be omitted entirely because they do appear
in the real world. But an understanding of what fractions mean in terms of
partial units and in terms of division (which is actually a more general
interpretation of the "parts of a whole" idea) and what it means to take
a fractional amount of a quantity are the most important ideas. Many
students don't get even that. And if they don't, then all this fraction
arithmetic makes no sense to them. And I have observed huge numbers of
students who are greatly confused enough about fraction arithmetic that
they can't learn even to do fraction arithmetic correctly. Many others
who do learn it quickly forget it, especially if they had struggled
to learn it or that their work clearly shows that they "get" it
somewhat but that their understanding of how to do the fraction
arithmetic is still shakey.
Of course, education is not really career training or at least it
shouldn't be reduced to nothing but career training, so we can
rightfully justify teaching some ideas in mathematics if they at
least help students learn to reason mathematically, even if those
ideas are ideas they will never use later on. So if fraction
arithmetic is indeed not going to be very useful for students later
on in terms of the mathematics they will use, then can we really
justify placing the emphasis on fraction arithmetic that we do in
terms of what I just said at the beginning of this paragraph?
Perhaps so, but there are probably better choices we could make to
accomplish such goals. That is, we can help students learn to
make sense of mathematics and learn to reason mathematically
via other ways.
Even if we can truly justify the emphasis with fraction arithmetic
in these courses, we cannot justify the current ways that we teach
it because it ends up being a huge waste of time for everyone and
ends up becoming a major headache for the teacher to see huge
numbers of students not getting it.
I do plan to keep the material on fraction arithmetic in my book,
regardless of the answers to these questions. Those who will
need it later on and can learn to reason mathematically can learn
the fraction arithmetic they need from the book. And I will
include it also because many teachers will feel the need to teach it
and because some students will be interested in learning fraction
arithmetic, especially if I motivate it by the fact that nonzero
terminating decimal numbers are not closed under division and
also by the fact that fractions are generalizations of terminating
decimal numbers in that fractions allow us to give the sizes
of partial units where the unit is split into any number of equal-sized
parts rather than a number of equal-sized parts that is a power of 10.
In short, Alain Schremmer has raised a good question that is worth
considering, and that has motivated me to ponder it further. It has
reminded me that we should not justify that this idea here or that
idea there is essential to a math curriculum or should receive the
amount of emphasis it typically does just because lots of people
say so.
Jonathan Groves
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