Tuesday, May 12, 2009

More Dispatches from the Front

More dispatches


When I taught Calculus I in 1957, my class had twenty students and all had taken college algebra, trig and plane geometry in high school. Their parents were paying a lot of money for their kids to go to this school.
The students in the Calculus I class I taught in 1995 had no coherent preparation. I was asked if ½ x was the same as x/2 by a student who picked up the idea of the derivative as a rate of change with no problem.
Well, you teach the class in front of you, you don’t wish they were some other class. There are a hundred students instead twenty; they didn’t get a good preparation in algebra, trig and geometry in high school and it doesn’t cost $40,000 a year to go to school.
I think that Communism doesn’t work because Marx designed his system for a race other than the human race. It was an intellectual design, no pun intended, based on how he wanted people to be.
Mathematics education seems to have the same design flaw. The “New Math” is an egregious example of other worldly educational design.
I must confess that when the “New Math” first appeared it seemed like a neat idea but I was a young and foolish graduate student. At that time in my career I thought that a topic that was clear to me was teachable.
I taught determinants and Cramer’s Rule in their full generality to a freshman calculus course. I taught Fourier Series from the point of view of Banach Algebras. It was so obvious; how could the students fail to understand. I eventually had the epiphany that what was obvious to me was not obvious to my students, probably because they didn’t have five years of thinking about mathematics behind them.

But I did finally realize that my students were not me and put some effort into seeing who they actually were.
I am not saying that students need to be talked down to; I’m saying that mathematics education has to be revamped so as to release the potential of our students.

I don’t think that traditionally mathematics was taught to 100 students at a time. Populations weren’t large enough to need 100 student classes. Throughout most of the educational history of the U.S. the country was agrarian; school houses and the classes held in them were small.
It is my contention that a conscientious teacher in a small class will turn out an educated student.
That’s why people say, “Why, I went to a one room school house and got a good education. I don’t see why these kids today can’t get a good education. Maybe they need standardized tests.” The reason the Senators from North Dakota got good educations was that the classes they attended were small.
My dad lived on a ranch in northern Montana when he was in grammar school. One winter on his way to school his horse fell on his foot and broke it badly. A high school girl from a nearby ranch was his teacher and an old man would bring over books for him to read; Dumas, Sabatini, Kenneth Roberts. He was a well educated man.
My grandmother, Rosie, had a high school education and taught school. She was at least as well read as many school teachers are today and could teach the three R’s with the best of them.


When towns grew into cities and school populations got bigger there were two ways to go; get more teachers and keep class size small or have the same number of teachers and let the class size grow. That part of government that finances education has generally taken the penny wise, pound foolish path. For reasons that are obscure to me it doesn’t seem to be universally recognized that viable economic and political systems are rooted in an effective education.

When I went to high school, the trades were a viable career option. One of my classmates in college figured that engineering and bricklaying had about the same financial future. I remember our family driving across the country in 1952 and passing by the U.S. Steel plant, its smokestacks belching smoke and declaring the health of American industry.
The copper pit in Bisbee, AZ and the smelter in nearby Douglas were going 24/7.
One of the effects on education was that high school became a reasonable terminus. A kid could finish high school and expect to get a good job at the mill or plant or factory or mine. I guess that in 1950 the girls looked forward to marrying the guys who went to work in the mill and raising their kids. Reading, writing and arithmetic were all that were necessary for a good life, sort of.
College Algebra was a college course and the elite course in high schools. In 1950 a College Algebra book covered Descartes’ Rule of Signs, Horner’s method of Extracting Square Roots and the solutions of the general cubic and quartic equations.
But that is the way it used to be. The jobs of society with an industrial base aren’t there anymore. An education that worked in the past doesn’t work in today’s America but we are trying to shoehorn the teaching techniques and educational philosophy of yesteryear into our brave new world.

And whatever changes are made, the most important is a “small teacher/student ratio”, the smaller the better.
I say “small teacher/student ratio” rather than “small class size” because classes may be one the artifacts that have to go. But I will talk about class size because I think that is where change has to start.

I asked a high school mathematics teacher in Tucson what the limit on class size was. She said that it was 46. I opined that I thought that was little large and she said, “I can handle a class of 46.” She didn’t mention teaching a class of 46.
A colleague told me that he could teach calculus to at least 60 students at a time. Yeah, right.
The idea seems to be that you lecture clearly and maybe work some problems; the student takes notes and goes home and studies them. If they have any questions they come to office hours and get them answered. There should be no limit to the number of students you can teach.
This sounds so reasonable but unfortunately it doesn’t work. Well, it worked for the student I was but that student isn’t in my class; most of my students aren’t planning for a career in higher mathematics.

So, what’s wrong with large classes?

If I am talking to one person, they pay attention to me; they try to understand what I’m saying. Say we’re sitting at a table in a bar. Somebody else joins us. Two people will now pay attention to what one of us is saying. If there are five people at the table, the attention of one of them might wander unless the speaker is quite forceful. The speaker has to hold their attention. As the crowd grows the conversation will split into two conversations; or maybe someone will pick up a newspaper left on a nearby table and read while the rest converse; or maybe one will take out her laptop.
In my calculus classes of 100 students, some of the students knew how to work the problem I was demonstrating and were bored, some were lost and had no idea what I was talking about, some were trying to pay attention but couldn’t stay awake. Some read the school newspaper, some took out their laptops.
When two people are talking to each other, I’ve never seen one of them fall asleep. I’ve never had a student come to office hours and start playing a game on their laptop in the middle of working a problem.
If I’m talking to two people, they both feel I’m talking to them. It is hard to make 100 students feel you are talking directly to them, especially if many of them are disinterested.

A bad teacher will get better if they teach small classes. In a small class teaching gets personal and you don’t want to look like a jerk in front of a bunch of kids you are getting to know.
In a large class, not only do the egos of the students get lost but so does the ego of the teacher.

Perhaps small classes would not solve all the problems but that change would be a giant step for education.

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