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Replies: 0

 Ed Wall Posts: 845 Registered: 12/3/04
Posted: Mar 15, 2001 11:22 PM

Professor Zhang

Before I reply to this question, I notice logical thinking has
been put forward again as a reason for doing something. I am not
convinced that pure mathematics develops logical thinking; however,
perhaps I misunderstand your use of the word logical. Logical covers
a bit of territory - there is for example, mathematical logic or
symbolic logic (Hegel is known for his Science of Logic (Wissenschaft
der Logik) - which is quite different from either of these) and so
forth. I would even venture to say that if what you indicate is true,
then practitioners of pure mathematics must be among the most logical
since they have had the most practice. Unfortunately, taking logical
in the larger sense, there is substantial evidence that this is not
true.
So, and I am open to argument, perhaps one could say justly that
pure mathematics is useful in developing a logic useful for doing
pure mathematics. Then a question arises: What else is this
particular logic useful for? Let us say, for example, it is useful
for physics. Could one argue that all the logic a top notch physicist
needs is the logic of a pure mathematician? Let us say, for example,
it is useful for biology? Could one argue that all the logic a top
notch biologist needs is the logic of a pure mathematician? Let us
say, for example, it is useful for carpentry? Could one argue that
all the logic a top notch carpenter needs is the logic of a pure
mathematician?
So, and I am open to argument, perhaps a basic course, one in
which one only experiences pure mathematics, is insufficient
logically (as taken in a larger sense). Perhaps something that
incorporates 'real life mathematics' might offset that insufficiency.

Now 'real life mathematics.' If one wants to provide a certain
kind of motivation to students at a certain point in their studies
this might make sense. However, it is unclear that a problem must be
real life in order for students to be mathematically interested and
curious. And it is very unclear what makes a problem 'real life' -
sometimes what is passed off as this is anything but 'real life.' It
might, for example, a fairly standard piece of basic mathematics
dressed up in the vocabulary of 'real life.' Some of the best 'real
life' mathematics problems that I have seen were done in physics
classrooms because often only there can you take the time to make it
real. That is not to say that there aren't some possibilities and
that takes me to the next topic.

Again, keep in mind that I do know some about the Chinese
curriculum, but I know little about the detailed facts of enactment.
What are needed, I think, are 'real interesting' mathematical
problems (and by this I mean something that addresses substantial
mathematics). From what I can tell, your curriculum does not always
provide such (and ours doesn't either!) - however, how your teachers
use the curriculum may very well be another matter (there are good
math teachers no matter what the curriculum). If a 'real interesting'
math problem looks 'real life' that is great. If it doesn't, that is
fine also. Again, this depends on what you want mathematics to open
up for students and, probably, some of what is called applied
mathematics - although it is not strictly pure mathematics - has all
the rigor one might desire (actually, at one time many meant by pure
mathematics number theory).

Finally, I agree that 'real life' mathematics is not important at
the abstract level if you mean that in the sense of a tautology; that
is a play on abstract and real. However, it seems fairly clear, that
human children, at often very different rates, tend to roughly move
through stages of abstraction. Students in college may be at very
different places than they were in high school. It seems, hence, that
a teacher (and a curriculum writer) might want to pragmatically think
about 'real' in a relative sense. That is, by 'real' I mean an
indication of one's mathematical experience at the moment of
teaching. So for a small child 'real' might be fingers and toes, and
several years later the number line. And, it seems to me, going from
one's fingers to the the number line is a substantial feat of
abstraction.

Oh, V. A. Krutetskii (1976) The Psychology of Mathematical
Abilities in Schoolchildren might provide some useful information as
regards abstraction.

Ed Wall

>Question 2: What role does "real life mathematics" have in the classroom?
Some would say that Chinese mathematics education is too pure. Others
would say that basic mathematics must be pure.
[Background] Chinese mathematics teaching emphasizes knowledge of
procedures and algorithms. Real life mathematics is important, but
not at the abstract level. Pure mathematics consists of algorithms.
Developing logical thinking is important too.