… one of the most extraordinary experiments in mathematics instruction, based on Dewey’s work, was published in 1938 as an NCTM Yearbook called The Nature of Proof . The author, Harold Fawcett, taught a course in Geometry at the Ohio State Lab School that was arguably one of the greatest courses of all time. -Grant Wiggins
Well of course I had to read it. Here’s what I found:
Fawcett opens the book with the case that teaching geometry in high school “is no longer justified on the ground that it is necessary for the purpose of giving students control of useful geometric knowledge, since the facts of geometry which may at one time or another actually serve some useful purpose … can be learned in the junior high school” (p. 6) and that for sophomores instead, the point is widely assumed to be training students to think logically. “The degree of transfer of this logical training to situations outside geometry is a fair measure of the effectiveness of the instruction” (p. 8). And of course by that measure most geometry instruction is useless. Why? Because “If the kind of thinking which is to result from an understanding of the nature of proof is to be used in non-mathematical situations such situations must be considered during the learning process…. Transfer is secured only by training for transfer and teachers of mathematics can no longer expect that the careful study of ninety or more geometric theorems will alone enable their students to distinguish between a sound argument and a tissue of nonsense.” (p. 13)
So what did he do? He opened class with this: “There is no great hurry about beginning our regular work in geometry and since the problem of awards is one which is soon to be considered by the entire school body I suggest that we give some preliminary consideration to the proposition that ‘awards should be granted for outstanding achievement in the school.'” Students took sides, argued about whether a salary was an award and whether playing on the football team counted as “outstanding achievement.” Fawcett reports that “considerable time was spent in what might appear to have been useless discussion.” Finally one student offered, “Most of this trouble is caused by the fact that we don’t know what we mean by ‘awards’ or ‘outstanding achievement’.” (p. 31). He then introduced various topics and guiding students to analyze their own discussion. Over time, the students had determined the following principles (p. 34):
- Definition is helpful in all cases where precise thinking is to be done.
- Conclusions seem to depend on assumptions but often the assumptions are not recognized.
- It is difficult to agree on definitions and assumptions in situations which cause one to become excited.
He then reports that “there was general agreement that it would be interesting to make definitions and assumptions about concepts which did not stir the emotions and to proceed to investigate their implications”, but the students could not think of any such topic. So Fawcett proposed building a theory about “the space in which we live”, and students remembered that the course was about geometry. (p.35) This was four weeks into the year.
How to begin developing their theory? Students proposed defining things, but quickly hit quicksand. Fawcett gently guided them to start making propositions, defining some terms and leaving others undefined, and so on. He worked them to the point that he could give them a figure, say two intersecting lines, and instead of asking them to prove any particular theorem he just asked them to “state all the properties of the … figure that you are willing to accept.” Statements were listed on the board, discussed, critiqued, improved. Proven. Every student kept a notebook but no two were the same as different students found and proved different things which they shared with the class. By the end of the year, the students did better than ordinary classes on standardized tests despite not having been exposed to some of the material.
More importantly though, Fawcett was training for transfer. His exercises and tests throughout the year included items like asking students to analyze politicians’ platform statements, advertisements, supreme court decisions, and on and on. What are the key words that would have to be defined? What are the assumptions? What is the claim, and is it supported?
All this has left me wondering, and worrying, about what transferable skills I am hoping to teach in Algebra II and Precalc this year. My previous attempt is a good start but if I mean it, I need to defined observable behaviors that I am trying to train, and non-mathematical exercises for students to practice. I suspect I’ll be working on this question for a long time.
 Harold P. Fawcett, Ph.D. The Nature of Proof: The thirteenth annual yearbook of the National Council of Teachers of Mathematics, Teachers College, New York, 1938. 146pp.