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Re: if using Mathematica to solve an algebraic problem

My anecdotal experience has been that there is a high correlation 
between using Mathematica well and being able to do the corresponding 
paper-and-pencil calculations.  (Perhaps some third, underlying variable 
explaining the correlation?)

On 4/12/2010 6:54 AM, David Bailey wrote:
> Helen Read wrote:
>> 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
> Peace Helen!
> I am not in any way saying that how you teach is dangerous - I was more
> imagining a situation in which students had access to Mathematica at any
> time they wanted (student copy) and could use it to attack pencil and
> paper problems that you had set them. In that situation, I think you
> could end up with some students who became good at Mathematica, but
> didn't learn enough maths.
> One problem is that students may feel under time and grade pressure. If
> they see a way to cheat with Mathematica, they may not like the idea,
> but the competition might force it on them
> David Bailey

Murray Eisenberg                     murray at
Mathematics & Statistics Dept.
Lederle Graduate Research Tower      phone 413 549-1020 (H)
University of Massachusetts                413 545-2859 (W)
710 North Pleasant Street            fax   413 545-1801
Amherst, MA 01003-9305

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