Attack on the Teaching of Mathematics

An Attack on the Teaching of Mathematics

Why is it that the United States ranks so low internationally in mathematics?
The reason that occurs to me is that the United States ranks low in respect for intellectual activity. The United States ranks high in belief of creationism or intelligent design. The United States elected Bush as President twice and regardless of how one rates his policy, I don’t think anyone ever accused him of being intellectual.
I think that every country can have its stars. A country can put together ten basketball players to win the Olympics but that country can’t staff an NBA. I would guess that every school in America from middle school on up has a basketball team. Everybody the U.S. knows who Shaq is but how many know who Smale or Stein is.
The United States can produce great mathematicians but can’t produce the players to support an infrastructure. We are missing the intellectual minor leagues.
Mathematics is nothing if not an intellectual pursuit.

The majority of students in the United States seem to feel that school is a social institution, not an intellectual institution. When I was involved in home schooling kids I was told that I was depriving them of the socializing influence of going to school. (This was a lame excuse for going to school in my opinion, an opinion that I am willing to defend but not here.)
The United States, until well into the 20th Century, was 90% rural. Kids went to school when the farm work was done. There was no centuries old tradition of scholarship.
This does not say that there were no American intellectuals but America was born as a Hog Butcher and Steel Maker, not a scholar. Intellectuality in America has not reached the "tipping point" nor does it appear that it will soon.
Not all that long ago people could build their own house and fix their own car. In the country it is still valuable to have the skills that built this country and made it great. A hundred years ago people were needed who raised hogs and poured steel. Intellectuality wasn’t needed, know-how was. With a high school diploma a person was prepared for their future life.
Now steel is poured overseas and hogs are raised on a mega-farms. A high school diploma gets you a minimum wage job.
Unfortunately teachers have grown up in an environment of diminishing interest in the intellectual. In fact, my dad’s grammar school in Deer Lodge, Montana in 1910 was more intellectual than the grammar school my daughter went to in 1998. “School to work” is a catch phrase of education. As the number of graduating mathematicians decreases, the Business School grows.
The teaching establishment looks for new ways to teach the same old stuff apparently thinking that this time it will work but doing the same thing and expecting a different result is a definition of insanity.
Every calculus book I have ever taught from asks the student to maximize the volume of an open top box with a square base with a given area of sides and base. They had the same list, more or less, of functions preceded by “Take the derivative of the following.” or “Evaluate the Integrals” etc. The even numbered are assigned because the back of the book has the answers to the odd numbered problems.
I think it is insane to expect better test results this semester than last.
Another example of this kind of insanity is thinking that armed force will change the minds of an indigenous population.

Instead of making the school experience intellectual, teachers in the lower grades tell me they try to make it fun.
Now my definition of fun is meeting a challenge but I get the impression that the definition the teachers have in mind is riding a merry-go-round. I understand that a merry-go-round can be used to demonstrate the coriolis force but I don’t think that was the fun they had in mind.

The first calculus text I used was Sherwood and Taylor, a reasonable book of reasonable size. The first calculus book I taught from was the first edition of Thomas. It was a bit thicker than Sherwood and Taylor but I thought it was a good book.
Thomas went through a sequence of new editions which evolved into new editions of Thomas and Finney. Every edition was thicker than the last until the last edition I taught from was so heavy that I had to cut it into two pieces so I could carry it.
Each edition, and God only knows what edition they are on now, was more unteachable than the previous edition, continuing a monotone sequence of increasing obscurity.
When I read a section (a section a day was the drill) that I had assigned, I would see that in three hours I couldn’t read it and work the assigned problems. The literary style sucked. The section was not self-contained and referred to previous sections scattered throughout the book. The shaded formulae were unhelpful. (I would suggest to my students that they take a black magic marker and blot out shaded formulae.)
I am not going to give a section by section critique of Thomas’ legacy but I would point out that the evolution of Thomas is typical of the evolution of solutions to any serious problem in this country, maybe in all countries; by making things worse.
To solve traffic problems a city will add lanes to existing roads and put more curlicues in highway interchanges asymptotically approaching perpetual gridlock.
The tax code is made thicker and more indecipherable each year as are laws generally, asymptotically approaching the livelihood of an infinite number of lawyers. Simplification is an unknown art.

As in many areas of human endeavor a lot of dead ends are followed. The New Math, Piaget rods, the Harvard Program are a few in teaching mathematics come to mind. (Surely the students know, deep down, that they are being fed crap.)
This procedure has many names: the quick fix; too little, too late; a day late and a dollar short; looking under the street light for your lost keys because the light is better there.
The surge of troops in Iraq falls in this category

Personally I have no hope that the teaching of mathematics will change for the better in the next several decades and for the general student population it will get steadily more irrelevant.

One of the evolutionary trends that I noticed over thirty-seven years of teaching was the evolution of the normal distribution of grades to a bi-modal distribution; students got it or they didn’t. The class average became a meaningless statistic. It isn’t as though we have no good students in the sciences and mathematics but the gap between the haves and have not’s is widening. Where is the B Student of yesteryear?