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William P. Berlinghoff

Posts: 21
Registered: 12/6/04
Posted: Jan 26, 1994 9:36 AM
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I can't resist replying to the 1/22/94 posting by Joe
Malkevitch about the geometry curriculum, in which he claims
that "the teaching of axiomatics makes little sense."

Without meaning to offend, I suggest that -- whatever the
merits of his conclusion -- his argument does not stand up
well to an (informal) axiomatic analysis.

He says:
"We must remember that very few students who study geometry
in high school go on in science, no less in mathematics.
The goal of high school mathematics should not be to locate
future mathematicians per se."
I agree, and accept that as Axiom 1.

Then he says:
"In any case, the teaching of axiomatics makes little sense.
Research geometers have relatively little interest in such
systems and they make even less sense as I stated for
non-math. types."
I have two problems with this.
(1) IF the goal of high school mathematics is NOT to locate
future mathematicians per se (Axiom 1), then why should we
be guided by the interests of research geometers?
(2) The claim that they "make even less sense...for non-math
types" has indeed been stated, but without a shred of
supporting evidence. I suggest that it is an implicit Axiom
2, which is by no means self-evident. I do not accept it.

He goes on to say:
"Modern geometry has some wonderful applications. Centering
a curriculum around these applications at the very least
shows how important math is to our current world...."
I agree with this, too. That is, I agree with the
self-evident statement (Axiom 3) that centering a curriculum
around applications shows how important math is to our
current world.

However, implicit in the subsequent discussion of examples
is an unstated (and undemonstrated) theorem which, I infer,
must be something like this:
'Because mathematics is useful in our current world,
therefore it must be interesting (or at least more
interesting than axiomatic geometry) to high school students
who will not go on to major in science or mathematics.'

I suggest that this statement is yet another axiom which has
been masquerading as a theorem in many discussions about the
geometry curriculum. Has there been a systematic research
study of student attitudes that has demonstrated this? I
don't know of one. I think, rather, that this is a case of
many mathematicians projecting their likes and dislikes,
"observing" what they have already decided they will find.

Prof. Malkevitch presumably finds the applications of graph
theory and the like fascinating and impressive, and hence
may tend to notice the students who share his enthusiasm
about these things. I, on the other hand, regard the fact
that mathematics has real-world applications only mildly
interesting and not at all amusing. (If that had been the
"pitch" for mathematics when I was in school, I would have
become an English major. When I was a college freshman, it
was not calculus, but a seminar on the axiomatic basis of
the number systems, that turned me on to mathematics.) In
my 30+ years of teaching first-year college students (most
of whom have been non-science majors), I have tended to
notice those who share my predilections about the subject.
I suspect that neither one of us has the whole answer; even
in mathematics, it may well be "diff'rent strokes for diff'rent

I am grateful to Prof. Malkevitch for the COMAP paper
reference; I've already requested a copy of it. I note,
however, that all the people he lists as contributors are
mathematicians, and -- given Axiom 1 -- I'm not completely
comfortable with using their opinions as our only guide.

Having said all this, let me concede that my position within
the writing team for MATH Connections is that of the
dissenting minority viewpoint. Within the next day or so,
I'll post for comment our team's tentative outline of 10th
grade mathematics, to the extent that we have it now.
You will notice that it is largely non-axiomatic and still
very fluid. Please comment freely on this proposal; your
timely comments can still have a significant impact on how
we handle the material.

Bill Berlinghoff, Senior Writer, MATH Connections

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