Cut The Knot!

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by Alex Bogomolny

Necessary and Sufficient

October 2001

There is certain philosophia prima on which all other philosophy ought to depend; and consisteth principally in right limiting of the significations of such appellations, or names, as are of all others the most universal; which limitations serve to avoid ambiguity and equivocation in reasoning, and commonly called definitions ...
Thomas Hobbes (1588-1679)
Leviathan, ch. 46
Penguin Classics, 1982
  1. Why require the studying of math?

  2. I guess kids wouldn't like the comment that has been echoing in my head through this thread...

    "Because it's fun!"

  3. Hey, I'm a kid and I think maths is fun!


From a k12.ed.math discussion

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:

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 we give some preliminary consideration to the proposition that 'awards should be granted for outstanding achievement in the school.'

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,

Accepting the definition of "school" as "Any experience from which one learns" indicate your agreement or disagreement with the proposition: "Abraham Lincoln spent very little time in school."

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 has been 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 correspondents remarked that One cannot separate the person and character of the man from his message. However, the course has been taught by Eugene Smith from about 1945 to 1956. (It's to be regretted that later day students were not queried 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,

There has probably never been a time in the history of American education when the development of critical and reflective thought was not recognized as desirable outcome of the secondary school.

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,

... the major emphasis is placed on a body of theorems to be learned rather than on the method by which these theorems are established.

As the result,

... there is little evidence to show that pupils who have studied demonstrative geometry are less gullible, more logical and more critical in their thinking than those who did not follow such a course.

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,

No transfer will occur unless the material is learned in connection with the field to which the transfer is desired.

and W. Betz,

Transfer is not automatic. "We reap no more than we sow,"

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 much 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 in 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,

I remember all our work with definitions. When I was a vice president at Hughes, and now in my work with my church, I realize how important definitions are. It is amazing that when we can agree on our definitions most of the conflict ends.

To which Flener remarked

In the field of education we probably argue at cross purposes more because we don't have the same definitions in mind.

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]?

While the control of geometric subject matter was not one of the major purposes to be accomplished by the pupils in class A, nevertheless it seemed desirable to compare their achievement in this respect with that of pupils who had followed the usual course in geometry.

I think that the Description "Critical Thinking Course with Applications to Geometry" serves 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,

Is mathematics necessary? No. But it is sufficient.

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:

Can you recall why you fell in love with mathematics? It was not, I think, because of its usefulness in controlling inventories. Was it not because of the delight, the feeling of power and satisfaction it gave; the theorems that inspired awe, or jubilation, or amazement; the wonder and glory of what I think is the human race's supreme intellectual achievement? Mathematics is more important than jobs. It transcends them, it does not need them.

Is mathematics necessary? No. But it is sufficient.

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, 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 could and 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. W. Betz, The Transfer of Training with Particular Reference to Geometry, NCTM, 5th Yearbook, 1930
  2. U. Dudley, Is Mathematics Necessary?, The College Math. J., 28, 5, 1997, 360-364
  3. H. F. Fawcett, The Nature of Proof, NCTM, 13th Yearbook, Reprint 1995
  4. R. H. Wheeler, The New Psychology of Learning, NCTM, 10th Yearbook, 1935
  1. Necessary and Sufficient
    (An attempt to draw conclusions from a remarkable experiment of more than half a century ago and some more recent ideas.)

  2. The Nature of Proof
    (A report on the above experiment.)

  3. Simson Line
    (A sequence of nice geometric facts with the word define emphasized. Just imagine what would happen if we did not agree on the definitions or did not use them altogether.)

(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)

If there is such a thing as a mathematical mind, Bentham possessed one. ... Even his deficiencies, notably in the field of romance, were those associated with mathematicians. After fifty-seven years of remoteness from the society of women he decided to marry and carefully reasoned to his choice. He then proposed by letter to a woman he had not seen in sixteen years. He was refused. But the logic of his proposal remained the same, and so after twenty-two more years, during which he carefully re-examined its impeccability, he again offered himself to the same woman, hoping, possibly, that she had learned some mathematics in the meantime and would see the force of his case. Apparently she was equally sure of her logic, or intuition, for she again refused.


  1. M. Kline, Mathematics In Western Culture, Oxford University Press, 1953

(2) Upon discovering that , T. J. Osler of Rowan University wrote:

It is known that is not constructible. Yet , which looks more complicated than , is constructible because it can be expressed by the golden section. This is an unexpected result. It is surprises of this type that make it a joy to be a mathematician.

Osler's identity can be verified by direct inspection, i.e., by taking the cube of both sides. Similarly surprising identities can be found elsewhere.)


  1. T. J. Osler, Cardan Polynomials and the Reduction of Radicals, Am Math Monthly, v 74, n 1, 2001, 26-32.

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