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:
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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.'
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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,
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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."
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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,
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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.
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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,
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... the major emphasis is placed on a body of theorems to be learned rather
than on the method by which these theorems are established.
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As the result,
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... 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.
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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,
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No transfer will occur unless the material is learned in connection with
the field to which the transfer is desired.
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and W. Betz,
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Transfer is not automatic. "We reap no more than we sow,"
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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:
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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.
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To which Flener remarked
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In the field of education we probably argue at cross purposes more because
we don't have the same definitions in mind.
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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]?
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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.
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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,
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Is mathematics necessary? No. But it is sufficient.
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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:
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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.
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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.
References
- W. Betz, The Transfer of Training with Particular Reference to Geometry, NCTM, 5th Yearbook, 1930
- U. Dudley, Is Mathematics Necessary?, The College Math. J., 28, 5, 1997, 360-364
- H. F. Fawcett, The Nature of Proof, NCTM, 13th Yearbook, Reprint 1995
- R. H. Wheeler, The New Psychology of Learning, NCTM, 10th Yearbook, 1935
- Necessary and Sufficient
(An attempt to draw conclusions from a remarkable experiment of more than
half a century ago and some more recent ideas.)
- Nature of Proof
(A report on the
above experiment.)
- 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.)
, T. J. Osler of Rowan University wrote:
is not
constructible. Yet 
, which
looks more complicated than