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Re: Mathematica and Education

• To: mathgroup at smc.vnet.net
• Subject: [mg65042] Re: Mathematica and Education
• From: Helen Read <hpr at together.net>
• Date: Sun, 12 Mar 2006 23:58:18 -0500 (EST)
• References: <duud4v$hon$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

David Park wrote:
> Peter,
> 
> I find your remarks very interesting and I think you state the principal
> reasons for NOT making the maximum use of Mathematica in education. It
> certainly helps to get the objections and perceived limitations on the
> table. However, I would like to try, to the best of my ability, to make the
> counter arguments.

[David's excellent counter arguments snipped.]

> As for preserving old skills, I'm not too sympathetic. Are students to be
> taught how to sharpen spears (no advanced bow and arrow technology allowed!)
> track animals and identify eatable grubs and berries, just in case we get
> thrown back into a hunter-gatherer society? It wasn't that many generations
> ago when almost all women knew how to weave or operate a spinning wheel.
> Should these skills be preserved? 

Colleagues only a little older than I am used slide rules in school. I 
never learned to use one; I had an early generation scientific 
calculator. I don't believe this has harmed me in any way. I did learn 
to interpolate off of trig tables, and probably my teachers were arguing 
at the time over whether they should still be teaching that. My brother, 
two years younger, never saw trig tables, and I don't think it hurt him 
any. Technology advances, and we should make full use of it.

> The problem of using Mathematica intelligently, and not blindly, is serious.
> Most students are not well enough prepared with Mathematica to use it to
> anywhere near its capability. Mathematica is not wide spread enough and
> students do not learn it early enough. Any student interested in a technical
> career could do nothing better than start learning it in high school.
> Furthermore, Mathematica is not optimized for students and researchers. When
> it comes to ease of use there are many gaps. I believe that Mathematica can
> truly effect a revolution in technical education. But it is not as simple as
> just installing it on a departmental server. A lot of preparation is needed.
> Additional packages geared to student use are needed. Educators have to
> learn how to take advantage of the resource. 

At my institution, we have a university wide site license allowing us to 
install Mathematica on all of our computers, not just those owned by the 
university, but also laptops and desktops owned by faculty, staff, and 
students. Mathematica is available to everyone literally 24/7.

I have been teaching calculus with Mathematica for 10 years, for the 
last 6 of those years in a classroom equipped with 31 networked PCs (one 
for each student, plus one for the instructor), and a printer. We now 
have two such rooms. The instructor's machine is connected to the 
overhead projector, and we have software allowing easy communication 
between the student PCs and the instructor. I can, for example, 
broadcast my screen or any of the students' onto the projector or onto 
everyone's monitor. Unlike the computer labs on campus, these rooms are 
designed for teaching, with clear lines of sight from every student to 
the teacher and whiteboard, enough space for the instructor to walk 
around and interact with the students, etc.

My students use Mathematica routinely. There are weekly assignments 
(notebooks that I post online) which they submit to me via e-mail. We 
use Mathematica for examples and exploration activities in class; on 
homework assignments that they do for practice and do not hand in; and 
on quizzes and tests. I prepare many examples for the students to do in 
class with Mathematica, and we fire up Mathematica any time we feel the 
urge -- e.g. if something comes up in the middle of a lecture that we 
want to see in Mathematica. I save these examples from class and post 
them on the web, so that students who didn't finish the examples in 
class, or couldn't get something to work, or missed class entirely, can 
download the solutions later. I also make up lots of worksheets for the 
students to do for homework with full use of Mathematica, as I have yet 
to find a textbook with the sorts of problems that I would like.

I'm haven't thrown out the baby with the bathwater, however. We still 
do, for example, the techniques of integration chapter, trigonometric 
substitution and all. But it is clear to me that the students benefit in 
all kinds of ways from using Mathematica, and I hope that *some* of the 
pencil-and-paper topics will eventually be toned down. (Right now there 
are constraints placed on us by the client departments.)

I give two-part tests. Part I is done with pencil and paper, no 
computers, no calculators. This part of the test consists of traditional 
skill questions that could have been on a calculus test 30 years ago -- 
e.g., using the chain rule, integrating by parts. I still teach these 
topics and expect my students to do them by hand (a) because it teaches 
some good mental skills, and (b) so that they have some understanding of 
what is going on when they use Mathematica to do these things for them.

When a student finishes Part I of the test, s/he turns it in and picks 
up Part II, which is done with the use of Mathematica. The students show 
all of their work for Part II in a Mathematica notebook, making a title 
with their name, section and subsection labels for the various 
exercises, text cells to insert their own comments and answers to 
questions, etc. They print their work and staple it to their test paper 
when they hand it in.

Here are just a few examples of the sorts of things that go on the 
Mathematica portion of the test.

* Calculus I: Here's some complicated function. Find the linear and 
quadratic approximations at some given x0. Plot f(x) with the linear 
approximation and with the quadratic approximation on an interval that 
gives a good view. On approximately what interval does the linear 
approximation give a "good" approximation of f(x)? Ditto for the 
quadratic approximation.

* Calculus II: similar to the above, but with say, a degree 5 Taylor 
polynomial.

* Plot a curve given parametrically. Determine when (what t) and where 
(x and y coordinates) the curve intersects itself. Make a new plot that 
shows only the "loop" of the curve (in between where it crosses itself). 
Find the length of the loop. Find the surface area of the resulting 
surface if the loop is revolved about (a) the x-axis; (b) the y-axis; 
and (c) various other axes. (Most students will include a 
SurfaceOfRevolution plot for (a) and (b) even if I don't ask for it. I 
always try to cook the examples so that the surface plots look cool.)

* Here's a series. Verify with a table / plot that the terms are 
positive, decreasing, and converge to 0. Make a table and plot of the 
first (say) 100 partial sums. [At this point they can *see* that the 
series is convergent.] Use the integral test to confirm that the series 
converges. Estimate the sum of the series numerically by finding (a) the 
partial sum from n=1 to (say) 500, and (b) using integrals to find upper 
and lower bounds on the error if the partial sum is used as an 
approximation of the sum of the series.

All of my students, even the weakest among them, are quite comfortable 
using Mathematica by the end of the semester. On quizzes/tests, they are 
permitted to use the Mathematica Help Browser and to raise their hands 
and ask me for help if they are having Mathematica issues; they need 
less and less assistance by the end of the semester.

The students almost always do better on Part II (the Mathematica 
portion) of the test than Part I, despite the fact that the problems on 
Part II are usually what I consider more difficult. Often they can catch 
mistakes when working on Part II, things they would have done 
incorrectly on a pencil and paper test without realizing it was wrong. 
For example, they will set up an arclength integral incorrectly, and get 
an answer from NIntegrate (of course with Mathematica, we are not 
limited to the small number of examples whose arclength can be 
calculated exactly)--and they can see the answer is off by an order of 
magnitude when they estimate the length of the curve that they plotted. 
Or, they'll find the equation of a tangent line and plot it with f(x), 
only to find that they messed something up -- the line is not tangent! 
On a pencil and paper test, they'd have done these things wrong, not 
known it was wrong, and moved on to the next question. With the use of 
Mathematica they can *see* something is wrong, and can very often fix 
it. They are much better at checking whether an answer is reasonable 
with all the graphical / numerical feedback that they get from 
Mathematica than without it.

-- 
Helen Read
University of Vermont