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more on reform/response to Richard
Posted:
May 24, 1997 5:25 AM


I probably should have mentioned that my interest in mathematics came about in HS through the efforts of several HS math teachers.
When I was in 9th grade I was truant and in danger of being left back. Fortunately, at that time, NY had good assessment and it forced most kids to keep their acts together.
My parent's got me a math tutor who was an unemployed HS math teacher. At the first lesson he showed up with a copy of Birkoff and Ma Clane's Modern Algebra  I still have it. He told me he would show me how to excel in the rote stuff and that one needed to learn how to do that before learning more interesting things. After I started getting A's on 9th grade algebra exams I started teaching myself a little modern algebra. In HS I had excellent teachers who knew how to balance rote stuff with conceptual and theoretical stuff  but without a doubt the rote stuff was primary.
I was having a discussion with a colleague of mine about how in the "old days" we had to learn interpolation for evaluating logs. He was complaining about how dull this was. It was dull and it was very useful!! The way we evaluated logs was we used the three basic log properties to "beat a big number down to size" and then interpolate. As a result, we had a good understanding of the definition of log as well as it's fundamental properties.
These days the majority of calculus students don't know logs or exponential functions  the reform calculus books avoid the exponentials like the plague  I guess they know what isn't happening in HS's. Of course they don't know the properties either. Almost none of them know the difference between the log and ln buttons of their calculators and are clueless to what information these buttons give. The same holds for trig. and inverse trig. functions.
As a result two important and related concepts are a total mystery to most students  composition and inverse functions. Now I do not view the chain rule as symbol manipulation and anyone who teaches college math knows that one of the big stumbling blocks is composition of functions. Because these students have at most a calculator intro to the transcendental functions they really can't grasp composition or inverses in calculus. Most linear algebra texts are reform oriented these days and so composition is downplayed there too. The fact that matrix multiplication comes from composition of linear maps is unknown to most linear algebra students. Of course they may know what LU decomposition is  BIG DEAL.
So what we see is that an over use of calculators in HS and a deemphasis on rote properties of transcendental functions leads to major problems for math, science, engineering, CS and business students later on.
Again we see how reformers miss key issues. Rote stuff is not only good because the student is learning useful procedures, but much of it results in a deeper conceptual understanding of important ideas. As I said previously, rote stuff also gets the students in the habit of studying in a way which will serve them well in college and in their careers.
Richard mentioned that his student's used calculators and/or computers to generate conjectures which could be proved by induction. This is good. If his HS student's can do induction proofs in HS he has accomplished a very useful thing and my proverbial hat goes off to him  albeit, I don't see these student's at CSUN. None of my student's knows induction or the binomial theorem before I or someone else (David is another trouble maker who teaches these things to "virgin" minds).
However, I doubt that a machine is needed to generate enough conjectures to get a student started on induction proofs. In my 20 years teaching math and doing math research I have never needed a machine to come up with a conjecture for an induction proof. Of course, there are complicated conjectures where having a machine would be helpful. But I seriously doubt this is needed at the HS level. Later on, when a math, science, CS or engineering student has developed a certain amount of mathematical dexterity he/she will have little trouble using machines to help them formulate conjectures. But the sad reality of most (almost all) reform math is that it will never lead students to develop proof strategies and at best they will be OK at making guesses but at a loss to prove anything or to generalize or to explain to others.
There is nothing good about any reform math program I have seen. I suspect Richard, like my math tutor, is good at math, enjoys doing it and can teach it to others  I would conjecture he would be just effective, if not more, 20 years ago in the NYC school system which had standards and assessment and where it was taken for granted that students had to learn to progress.
Jerry



