Re: if using Mathematica to solve an algebraic problem

*To*: mathgroup at smc.vnet.net*Subject*: [mg109106] Re: if using Mathematica to solve an algebraic problem*From*: Murray Eisenberg <murray at math.umass.edu>*Date*: Mon, 12 Apr 2010 23:03:02 -0400 (EDT)

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 > http://www.dbaileyconsultancy.co.uk > > > -- Murray Eisenberg murray at math.umass.edu 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