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Re: if using Mathematica to solve an algebraic problem

• To: mathgroup at smc.vnet.net
• Subject: [mg109005] Re: if using Mathematica to solve an algebraic problem
• From: Helen Read <hpr at together.net>
• Date: Sat, 10 Apr 2010 06:52:27 -0400 (EDT)
• References: <hpml5n$9nu$1@smc.vnet.net>
• Reply-to: read at cems.uvm.edu

On 4/9/2010 3:32 AM, David Park wrote:
> Sometimes I find it difficult to understand these discussions.
>
> For example, Richard's: "There is of course the possibility that something
> really useful will be developed that will make it possible to teach all students
> everything they need to know." What kind of something would that be, and in
> what way would it make it possible? It seems like a rather vague but
> expansive goal.
>
> Similarly, David suggests that Mathematica might be "terribly dangerous"
> with the possibility of "becoming skilled in answering questions through
> Mathematica, rather than actually learning the subject!" (Not always bad. Is
> there anything wrong at becoming skilled at driving to various locations in
> your city without actually learning how the internal combustion engine
> works?)

I don't get this either. I teach my students to use Mathematica as part 
of a mathematics course -- mainly calculus, but other classes as well. 
We use Mathematica differently and for different purposes depending on 
the subject matter. Using Mathematica does not subtract anything from 
their learning of the course content. Rather, it adds to it.

As just one example, when we come to the topic of series in Calculus II, 
after a little chalk (well, whiteboard) and talk, I give the students 
some examples of series and have them make tables and plots of terms and 
partial sums in Mathematica, and try to guess whether the series 
converges (and if so, to what). I'll give them the following examples to 
work on.

(a) A series that converges very quickly and obviously. (Usually I'll 
use a geometric series for this, and I'll remind them of this example 
example later on when we discuss geometric series.)

(b) A series whose terms don't converge to 0 (and thus the series diverges).

(c) The Harmonic series.

They go through (a) and (b) pretty quickly. When they get to (c), they 
have a hard time deciding if the series converges or diverges. They will 
puzzle over it for quite a while. Some of them will make tables and 
plots of partial sums going out to thousands of terms in the series, and 
still they are unsure if the series converges or diverges.

After letting them work while I walk around and answer questions, we 
stop to discuss the examples. The class will tell me that Example (a) 
has terms that converge to 0, and the partial sums converge to (whatever 
I have rigged it to). For (b), they will explain that the terms converge 
to let's say 1/2, and so the partial sums increase approximately 
linearly with a slope of 1/2, and the series diverges. From this, they 
make the observation that if a series has terms that do not converge to 
0, there is no way the series could converge.

Then we get to (c). The terms converge to 0, and it's tough to tell 
whether the partial sums converge or diverge, but most of them will lean 
toward thinking that they diverge. So then I lower the boom. Let's 
*prove* that it diverges. This leads us into a discussion of the 
Integral Test. We continue with this interplay between concrete examples 
in Mathematica and analytical work on the board as we progress through 
the chapter.

I don't understand how anyone would think that this use of Mathematica 
is "dangerous" or "threatening" or somehow prevents my students from 
learning the subject. On the contrary, this use of Mathematica helps my 
students to gain some conceptual understanding of a topic that they 
otherwise find difficult and abstract.

--
Helen Read
University of Vermont