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(This is the third in a series of articles from an ed school student working towards certification as a math teacher.  His first article is here, and his second here.  For reasons that will likely be quite obvious, he prefers to remain anonymous. -ed.)

For those of you who read my last missive regarding the highly agreeable professor, I’m sure it comes as no surprise that he would be the first to agree with the comments posted that many problems have only one answer—the right one.  I didn’t mean to pick on the guy so much.  He was merely echoing ed school wisdom about math education.  In ed school parlance, when more than one answer exists for a question, the thinking used to come up with answers is called “divergent”.  When only one answer exists, the thinking is called “convergent”.

In ed school, “divergence” is considered a good thing and “convergence” looked upon with disdain.  I think ed school teachers take an oath to uphold these beliefs as part of an attempt to turn math into a “divergent thinking” type of subject like social studies or English.  Such thinking reflects a significant and depressing lack of understanding of what math is about.  A math professor recently commented to me about this lack of understanding with respect to how it is taught:

“One problem I see that arises from how math is taught before college is that we get some math majors (who don’t all stay majors) who have a completely incorrect notion of what math is.  I’m not sure what they think it is, but when they have to take a course like abstract algebra and are asked to do proofs, they think this has nothing to do with math. Exposure to proofs in high school geometry would go a long way toward correcting such misconceptions early on.”

Unfortunately, the emphasis on proofs in geometry has been de-emphasized over the years, thanks in large part to the NCTM standards (see first letter in this series) and what ed schools think math is about.  The proofs that exist in today’s high school geometry courses are trivial; many textbooks have turned most theorems into postulates so that geometry has become a collection of “taken on faith” propositions with no proofs offered.  Geometry classes have become nothing more than memorization of formulae (areas, volumes, surface areas of volumes) and very few proofs. 

As part of my recent ed school course, I was required to log in 15 hours of field experience by observing math classes at the high school level.  I decided to visit a teacher who teaches honors geometry at the local high school.  I had tutored one of her students a few years ago.  I recall that she had supplemented the almost proof-less textbook by giving the students proofs to do, but even so, the number of proofs was far less than what previous generations had to do in non-honors courses.

I was surprised when I visited her class and saw how she “proved” the Pythagorean Theorem. (This is one theorem the textbook had not yet turned into a postulate.)   She handed out sheets of paper on which were drawn a right triangle with three squares extending from each of its three sides.  There is a famous proof in which the two smaller squares can be shown via congruence theorems to fit into the larger square of the hypotenuse.   Her version, however, was to have the students cut out pieces of the two smaller squares by cutting along lines marked within them and assemble the resulting pieces, like a jigsaw, into the big square of the hypotenuse.  This was how she proved the theorem.  (Oh, excuse me.  This is how she had the students prove the theorem.)  “Does this prove the theorem?” she asked.  The students said yes, because it showed that the areas of the squares of the two legs in a right triangle equal the area of the square of the hypotenuse.  Which is correct for the particular triangle on the sheet of paper she handed out.   There was no discussion of how the procedure could be generalized for right triangles for any size, nor why the teacher drew the lines within the two smaller squares where she did.

I asked her later if she offered any other proofs of the theorem.  “No,” she said. “We don’t spend much time on the Pythagorean Theorem in the Honors class simply because they’ve learned it before.” 

I visited another class as part of my assignment and observed an algebra 1 lesson.  The teacher of that class was actually quite good. He has a provisional license to teach and is getting his degree from the school I attend; even has the same advisor as I do it turned out.  One exchange in class caught my attention.  He announced they would be starting a new chapter in the book.  A student asked “Is there any math in this chapter?” to which the teacher, straight-faced, replied “Yes, but not too much.”  Since the algebra text itself is not very good (same authors as the proof-less geometry book) I wondered perhaps if the two of them were making a statement about the book. That wasn’t it.  He said the student was a wise-ass.  “I was just giving it back to him,” he said. 

I followed up with an email and asked what he thought of NCTM’s standards and whether our advisor liked them.  No reply.  I think I may have scared him.  That’s too bad; I’m only trying to help.  I hope he remains an excellent teacher.

I remain faithfully and sincerely yours in divergence of thought,
John Dewey