hrubin@pop.stat.purdue.edu (Herman Rubin) (05/05/91)
In article <1991May2.195751.22316@psych.toronto.edu>, grant@psych.toronto.edu (Stuart Grant) writes: > In article <1991May02.171317.751@wimsey.bc.ca> balden@wimsey.bc.ca (Bruce Balden) writes: > >In article <1991May2.133856.8338@psych.toronto.edu> grant@psych.toronto.edu (Stuart Grant) writes: > >>>>I think you'll find that the majority of primary and secondary school > >>>>math teachers do not get their math education from a college's math > >>>>department in "regular" math courses but either from a regular college's > >>I agree that watered down courses in which students are not expected to learn > >>are not much use to anyone. However, I don't think that this is > >>the biggest problem with the math instruction in primary and secondary > >>schools. _Any_ math course taught at a college or university will be at > >>least as sophisticated as what teachers will be teaching in primary and > >>secondary schools. Not knowing how to do differential equations is not > >>the greatest problem math teachers have. This is too often true. Not knowing how to do differential equations, or even calculus is irrelevant. Not being able to do old-fashioned Euclidean geometry or to formulate word problems is relevant. Not knowing the structure of the integers or the real numbers is relevant. Until WWII, this material was not taught to undergraduates. In the 50s and 60s, it was taught to many undergraduates. Now, the high school students frequently do not get a proof-oriented geometry course, formulate few word problems, if any, have no idea how to argue by induction, all of which used to be "standard." Most students getting regular BAs in mathematics are little better. Even the kindergarten teacher should understand induction. Even the first grade teacher should be able to teach the very important use of symbols for precise expression. All junior high school and high school teaching of mathematics should be by those who understand proofs, and teach them, and can even occasionally produce them. > >Nevertheless, the good teacher of mathematics will have a deep appreciation > >of the way mathematics is actually used in the world at large and not just > >a good understanding of a traditional list of arithmetical and algebraic > >algorithms and formulas. The student who sees his mathematics teacher as > >inadequate, not only in the internal mechanics of the subject, but in > >success in making the subject relevant to the world at large, will correctly > >reason (YES, students are capable of reasoning) that this person has nothing > >of importance to tell him. The items I have mentioned above are the foundations. It does no good to know how to manipulate if you do not know when. This is what led to the "new math", which I do not think was done in the best manner, but which was the first real attempt to teach understanding. If failed because the teachers COULD NOT understand. At the present time, it is possible for someone to go through all the courses, and even get a MS in mathematics education, taking the special graduate courses on top of a standard BA, and never do other than manipulate. In days of old, the concepts were not taught well, and only a few courses, such as high school geometry and some parts of "college algebra", did more than present manipulations. However, the student who could not do word problems did not do well, and the concepts were reasonably tested. Teach the children to reason BEFORE they are brainwashed into believing that mathematics consists of manipulation. -- Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399 Phone: (317)494-6054 hrubin@l.cc.purdue.edu (Internet, bitnet) {purdue,pur-ee}!l.cc!hrubin(UUCP)
turpin@cs.utexas.edu (Russell Turpin) (05/06/91)
----- In article <11890@mentor.cc.purdue.edu> hrubin@pop.stat.purdue.edu (Herman Rubin) writes: > ... Not knowing how to do differential equations, or even > calculus is irrelevant. ... Somewhere along the way, we learned that the volume of a parallelepiped is the surface area of the bases times the perpendicular distance between them. An argument was made for this in terms of stacked plates shaped like the base, which could be made quite thin. (I think the teacher used a couple of decks of cards and pushed them askew against a rule to make the point that for the volume, it did not matter whether the stack was straight or slanted.) Later, we learned that the volume of a pyramid was one-third of the base times the perpendicular distance between it and the peak. Again, this has an intuitive argument. Later, we learned the volume of a sphere. But why is it *that*, I asked? The teacher had no easy argument. But, it was promissed, there is a later course, called calculus, where I could learn this, and how to calculate the volume inside many curved surfaces. Thus, long before I took calculus, I had some intuition about what it was partly about, and the study of limits, which is to many students largely unmotivated, I knew had ties to surfaces and volumes. Perhaps it would have been better had the teacher been able to lead me through a simple kind of argument for the volume of the sphere. But overall, the answer was not a bad one. What struck me was how different it was from the answers I had had from previous teachers. In elementary school, as soon as we learned squares, we were taught the Pythagorean theorem. But why is it so? To this question, the teacher at the time responded: things are just that way. I was left with no idea that it could be proven, or that some of the stuff we had learned about triangles and squares was the way to do it, nor even that its demonstration (as opposed to just memorizing it) was part of mathematics. (I suspect the teacher did not know enough about these things to pass on any useful information.) It may not be important for secondary math teachers to know how to do calculus or differential equations. But they should have a good idea of what these and other subjects are, because they pass on to students (or fail to do so) what mathematics is all about.