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 |---------------------------------------------------------------------------