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Michael Paul Goldenberg

Posts: 7,041
From: Ann Arbor, MI
Registered: 12/3/04
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Posted: Jan 23, 1997 3:47 PM
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At the invitation of a previous poster to find "reason" about mathematics
education on the Internet, I started up my browser with great anticipation.
Having jumped to the Web page suggested, I found myself back at that
bastion of collective self-assurance: the Mathematically Correct site.
Rather than relieving my sense of the vast ambiguity surrounding the debate
about mathematics education reform, it made me recall something I've been
reading from another period of educational turmoil, the '60's.

In his essay collection SCIENCE, CURRICULUM, and LIBERAL EDUCATION, Joseph
Schwab has a wonderful chapter about John Dewey entitled "The 'Impossible'
Role of the Teacher in Progressive Education." There is a section that I
think sheds vast amounts of light on our struggles with reform vs.
traditional mathematics classrooms.

Schwab writes: "Dewey seeks to persuade men to teach a mode of learning and
knowing which they themselves do not know and which they cannot grasp by
their habitual ways of learning. It is the same problem of breaking the
apparently unbreakable circle which Plato faces in MENO and Augustine in
his treatise, ON THE TEACHER.

"To appreciate this seeming paradox more fully, let us turn to an analogous
but simpler situation. Suppose a man wishes to show that the classic logic
can lead to error. The one thing he cannot do is to give a classic argument
which leads to this conclusion. For, if the classic argument does so lead,
the conclusion for that very reason, is suspect. Nor can he hope to succeed
by arguing for his new logic by means of the new. For then he follows rules
which his hearers do not know, much less agree to. This is a true
dilemma--the live equivalent of a paradox.

"We can begin to see Dewey's solution to this problem by continuing the
analogy. Suppose that the man with the new logic changes his intention. He
proposes, not to 'prove' the fallibility of the old logic but to persuade
men to try the new system which he conceives to be sounder. By this change
of intention, he opens for himself a new route. He points out many
assertions which his hearers will agree to be erroneous. He then points out
that these errors were conclusions arrived at by the old logic. He does not
(for he cannot) show that the fault lay in the old logic itself rather than
in its faulty application. Hence, by these 'pointings' he PROVES nothing.
But he has, perhaps, raised a reasonable doubt in the minds of reasonable
men. He has created a situation in which some men of good will may be moved
to try the new logic and thereby submit it to the test of practice--provide
they can understand it.

"This last is the remaining great stumbling block. If the enterprise is to
be successful, it is the new logic and not some radically mistaken version
of it which must be tried. Yet this is the unlikeliest outcome of all. For,
if the new logic be described in its own terms, its hearers must struggle
hard for understanding by whatever means they have. These means, however,
are the old modes of understanding, stemming from the old logic.
Inevitably, the new will be altered and distorted in this process of
communication, converted into some semblance of the old."

I find many valuable parallels between Schwab's analogy about old and new
logic and our disputes about traditional and new approaches to mathematics
education (curricula, classroom cultures, discourse, assessment, etc.). Of
the many things I'd like to comment upon, a key one is the impersonal tone
Schwab uses: the problems he describes are attributed to modes of thought
rather than to evil intentions on the part of individual thinkers or groups
of like-minded thinkers. If that tone were present on the pages of
Mathematically Correct, I'd find it far less annoying and useless to wander
over them. (And, no doubt, many folks wish I and a few others on this list
would take some of that medicine ourselves.)

Other of Schwab's points I'd like to stress by inserting the appropriate
terms into his writing:

1) the reform educator will not be PROVING that traditional methods are
faulty. Doing so is neither possible nor desirable; attempting to do so
leads to dilemmas best left alone.

2) a goal of the reformer instead is to point out problems which advocates
of the traditional approaches can agree have their sources rooted in the
old ways. Examples might be anything, from SAT scores to anecdotal
evidence, that raises reasonable doubt in the minds of reasonable people
who consider themselves traditionalists about the adequacy of the old ways.


3) succeeding in the second point, the reformer would hope that people of
good will who are loyal to the traditional methods will be moved to try
some of what is emerging from the new approaches. This hope, I suspect, is
for traditionalists to approach the new ideas with open minds.

4) the open-mindedness issue connects with Schwab's concern that what is
tried is not some radically mistaken version of the new approaches. If we
look only at the worst, most poorly developed examples of reform thinking,
not much is likely to be gained. So open-mindedness suggests that some
restraint must be used in deciding what examples are 'fair samples.'

I'd like to think that I can justify the length of this post by seeing it
as a way to help define the terms of the debate a little better, both to
help reformers and supporters of reform clarify what they might reasonably
hope to achieve; and to help opponents/critics focus upon what kinds of
counter-arguments are likely to resonate effectively with reformers. If
that can be achieved at all, perhaps we'll see a drop in the bile level and
a rise in substantive conversation.

|---------------------------------------------------------------------------
|Michael Paul Goldenberg
|University of Michigan 310 E. Cross St.
|School of Education 4002 Ypsilanti, MI 48198
|Ann Arbor, MI 48109-1259 (313) 482-9585
|(313) 763-9683
|
|"I wish I knew as much about ANYTHING now
| as I knew about EVERYTHING when I was twenty."
| Bill Ayers
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