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: <firstname.lastname@example.org>
- Sender: owner-wri-mathgroup at wolfram.com
David Park wrote:
> 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
* Calculus II: similar to the above, but with say, a degree 5 Taylor
* 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.
University of Vermont
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