Cut The Knot!An interactive column using Java applets
by Alex Bogomolny
Once upon a time, at the first meeting of what was supposed to be a high school geometry course, the teacher surprised the students with the announcement:
In the ensuing discussion, students talked of the value of the award system, whether a teacher's salary was an award, how "school" was defined, and so on. They were offered an exercise,
An unorthodox beginning for a geometry course, isn't it? What followed was no less unusual. During the school year, only about half of the time was allotted to the geometric content, the other half was devoted to the general purpose discussions, like the above. In the spring, students in this and the control classes were offered a test in plane geometry, on which the students in "our" class performed as well if not better than students in other classes. Even more remarkably, "our" students exuded confidence that, given more time, they would have been able to solve more problems and improve their test scores. This is despite the fact that they were unfamiliar with much of the material covered by the test.
A remarkable achievement indeed. But there is more to the story. When interviewed decades later, the former students, now retired, not only all fondly remembered the course and the teacher, but claimed that taking the course was the single most important and influential event in their academic careers. Could there be a more potent argument? The course was an indisputable success.
For those who have not heard or read of the story before, the teacher was Harold F. Fawcett, mathematics professor at the Ohio State University and future NCTM President (1958-60), whose report of the experiment was published as the NCTM Thirteenth Yearbook in 1938 (a 1995 reprint is currently available.) The story was also presented in a talk by Frederick Flener of Northeastern Illinois University at the Annual NCTM Conference in Orlando, Florida on April 6, 2001. Copies of the presentation's write-up have been making rounds on the Web until one of them ended up in my inbox.
Flener's account tells us about the course, about meeting, interviewing and corresponding with the surviving students, Fawcett's children and friends, and adds a few strokes about Fawcett himself, his thoughtful and caring character. One of the correspondence remarked that One cannot separate the person and character of the man from his message. However, the course was taught by Eugene Smith from about 1945 to 1956. (It's to be regretted that later day students were not asked for their impressions.) Was Fawcett's success rooted in his personality or his approach? Would not one like to repeat his success story? Well, according to Flener, most of his colleagues have other ideas: better use of technology, more investigations, less emphasis on proof.
Still, let's take a closer look at Fawcett's philosophy and reasons for developing the course.
To quote from the book,
In Fawcett's view, geometry was the most suitable course in the secondary school to teach critical and reflective thinking. He provides a respectable selection of quotes to support his view and to explain the source of his dissatisfaction with the traditional courses. Traditionally,
As the result,
The worthy outcome for students taking a geometry course is not only proving and learning a set of theorems, but acquiring of mental habits that save them from floundering in the conduct of life. Not only students should learn to prove a number of theorems but also grasp the nature of proof, so that their analytic ability could be transferred to non-geometric situations. And how is this achieved? Fawcett cites R. H. Wheeler,
and W. Betz,
Fawcett concludes that transfer is secured only by training for transfer, which explains the unconventional opening of his course. Next he deals with methods and procedures suitable for such a study. His treatment is so pertinent to the modern day discussions (minding children's own logic and individual ways, group discussions, open ended approach, discovery and investigation) that Fawcett's experiment and the book deserve to be better known among math educators. The point of the opening discussion was to establish the need in agreed-upon definitions, which seemed foreign to the thinking of the pupils. For example, at the outset all students agreed that "Abraham Lincoln spent very little time in school" and no one raised the point that the truth of this statement depends on how "school" is defined. So, starting with the first meeting, students were led to recognize the importance of definitions and, later, the need for undefined terms. They were taught to recognize the presence of implicit assumptions even in the most elementary activities of life.
Flener interviewed Warren Mathews, a course graduate. Mathews' comments were:
To which Flener remarked
How true! And how sad! Except of course math educators have no particular reason to feel singled out in this respect. "In the field of education" should be considered as a generic designation.
But let's try to apply the course basics to the course itself. Is it correct to designate Fawcett's experiment the geometry course? What makes a school year long interaction of a group of students with one or more teachers a geometry course? Can you think of a suitable definition?
How does it jibe with the following remark [Fawcett, p. 102]?
I think that the description "Critical Thinking Course with Applications to Geometry" captures well the purpose, the proceedings, and the results of Fawcett's course. The skill transfer occurred in the direction opposite to the declared goal! What Fawcett's experiment demonstrates very convincingly is that development of critical thinking skills helps students master mathematics even when they feel no particular liking for the subject.
At the end of the course, Fawcett interviewed students' parents. In their parents view, the course helped 16 students improve their ability to think critically, but only 3 out of a little more than 20 students have learned to like mathematics.
And so what?
In an 1997 article Is Mathematics Necessary?, Underwood Dudley (see also this column, Jan. 2001) argues that the answer to the question in the title of the paper is a sound No. He ends the article with a pun,
This may or may not be so(1). But, in any event, some things seem to be more sufficient than others. (A discussion on what mathematics may be sufficient for, could have fit right in with Fawcett's geometry course.) We just saw how the critical thinking skills helped study mathematics. Flener too ties the success of the course to the fact that students who took the course were the University School veterans of three years and were used to open ended investigations.
Following is a more complete quote from Dudley's paper:
No doubt mathematician Fawcett knew about and could appreciate the glory and the beauty of mathematics. He was an outstanding teacher and could, if he wanted to, to do a better job passing on to his students this sense of beauty and amazement shared by all mathematicians(2). He apparently chose not to. His goal was to teach the students, via interaction with mathematics, critical and reflective thought. But the goals of education are many: acquisition of useful skills, absorption of the local and global cultures, development of the innate potential. Course offerings should match a variety of goals. It stands to reason that the manner in which a math course is planned and conducted should aim at a particular objective. There is no single right way to teach and study mathematics.
The definitions are important. To resolve the cross purpose discussions, it's no less important to accept a possibility that an approach may be as right, or as good, as another one perhaps for a different end.
(1) Is mathematics sufficient? Anecdotal evidence suggests that this may not be the case. M. Kline wrote [Kline, p. 325] about Jeremy Bentham (1748-1832)
(2) Upon discovering that , T. J. Osler of Rowan University wrote:
Alex Bogomolny has started and still maintains a popular Web site Interactive Mathematics Miscellany and Puzzles to which he brought more than 10 years of college instruction and, at least as much, programming experience. He holds M.S. degree in Mathematics from the Moscow State University and Ph.D. in Applied Mathematics from the Hebrew University of Jerusalem. He can be reached at firstname.lastname@example.org
Copyright © 1996-2001 Alexander Bogomolny