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Re: Mathematica for High School Students

  • To: mathgroup at
  • Subject: [mg24110] Re: [mg24097] Mathematica for High School Students
  • From: Andrzej Kozlowski <andrzej at>
  • Date: Wed, 28 Jun 2000 02:11:36 -0400 (EDT)
  • Sender: owner-wri-mathgroup at

Since you are inviting comments I can't resist adding a few of my own,
although I suspect they will not be very helpful.

It seems to me that the very fact that this issue is being discussed in the
context of High School is a terrible indictment of the standards of
mathematics education in the US. (I would have been more shocked if I had
not eperienced this myself in the days when I taught at a large state
university in the Midwest.)

Let me just say that this year my daughter finished her high school
(International Baccalaureate)  in Britain. In her final year she did three
mathematics projects: one on the Plateau Problem (soap bubbles), one on
Fibonacci numbers and one on Statistics. She used Mathematica in all of
them, for a variety of purposes, ranging  from using Solve to solve
equations which would be too complicated to solve by hand even for a
professional mathematician, to writing her own simple programs. As for the
Gauss trick: she learned it in the first year of her (Japanese) Junior High

I do not know if there is an age when one is too young to be introduced to
computer mathematics (though I suspect it depends on the student). But if
there is such a boundary it certainly should not be in High School!

Andrzej Kozlowski
Toyama International University

on 6/27/00 1:51 PM, David Park at djmp at wrote:

> Hi Ted,
> Well, I can see that my high school notebooks didn't come off too favorably
> with you; but I would like to continue the discussion because it interests
> me; because we do have a difference of opinion here; and because I do value
> your opinion. I hope you don't mind if I post this to MathGroup because I
> tried to get a discussion going on the topic about a month ago. Maybe I can
> stimulate more discussion from the larger group.
> (There were two notebooks: one on solving simple equations step by step
> using pure functions and Map, and one centered around Gauss' feat of adding
> the numbers 1 to 100.)
> ____________________________________________
> Ted: The way Gauss added the numbers from 1 to 100 is very easy to explain
> without using Mathematica.
> ______________________
> Perhaps, but most mathematical ideas can be explained without the aid of
> Mathematica. The vast majority of mathematical books don't even mention
> Mathematica. The purpose of the notebooks was to teach something about
> Mathematica using an idea that might appeal to students. So, it doesn't seem
> to be a reasonable argument to say that we can teach Mathematica without
> using it.
> ____________________________________________
> Ted: The way I see one shouldn't learn about Mathematica features that are
> not
> fully intuitive until their 3rd year of college (except in a Computer
> Science class about Mathematica). Even in the 3rd and 4th years of college I
> would only encourage gradual introduction of less than intuitive features in
> Mathematica.
> _____________________
> And what features of Mathematica are not "fully intuitive" so that the
> student "shouldn't learn" them? Would you hazard to put forward a list of
> Mathematica statements to which young students would be restricted? Or a
> list that would be banned from their use?
> Are the functional programming statements ones that you would call not
> "fully intuitive"? Is it easier to understand the For statement, than to
> understand the Map statement? I can't remember exactly, but it seems that it
> was in junior high school that I learned that an equation remains true if we
> do equal operations on each side of the equation (other than dividing by
> zero). So if you are going to ban pure functions and Map, how would you
> implement that simple concept with Mathematica, which presumably the student
> already knows?
> Would you allow the young student to use Solve?
> I believe that there is much work to do in learning how to apply tools such
> as Mathematica to secondary education. It is not something that I am
> directly involved in, and have no expertise in; but I am still interested.
> I would like to put forward several arguments for allowing and encouraging
> the full use of Mathematica for the INTERESTED student.
> First, there is the great advantage to the student of just gathering sheer
> mathematical experience. For most students this is not easy to get. Don't
> get in his way, but help him. It is much easier to see a new idea, when one
> has the mathematical experience that the idea fits on to.
> There are mathematical ideas that are far easier to teach and learn with
> Mathematica than out of a text book, such as some of the functional
> programming ideas. You can actually see them in action. It is not that the
> ideas are difficult or abstruse; they just don't fit into the rigid
> curriculum. There is no reason why a bright student shouldn't be exposed to
> these ideas.
> I don't think that any young student interested in Mathematica should be
> shackled in his use of it, or even worse, admonished for using it. I have a
> sister fifteen years younger than me. One day when I was home from college,
> and reading a book, she said: "I wish I had a book to read." "You don't have
> any books?" "They gave us one book, but I've read it." "Why don't you take
> out some books from the library." "O No! We're not allowed to do that."
> Later I related this experience to a primary school teacher, expressing my
> disapproval of such restrictions. Did I ever get a piece of her mind! There
> was a very "scientific" method of teaching students how to read. If they
> jumped ahead and read something out of order, the whole plan was ruined!
> How do we know that the young student isn't another Gauss? We shouldn't
> erect any road blocks. We should help as far as we can and see how far he
> goes. Richard Feynman's high school teacher put him on to doing variational
> calculations. Would you say that he shouldn't have been permitted such
> things until his third or fourth year in college? Don't worry, there is
> little chance that a student will charge ahead of his or her capabilities; a
> far greater danger is that he will fall short of his capability.
> David Park
> djmp at

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