Re: Re: learning calculus through mathematica
- To: mathgroup at smc.vnet.net
- Subject: [mg108143] Re: [mg108082] Re: learning calculus through mathematica
- From: "David Park" <djmpark at comcast.net>
- Date: Tue, 9 Mar 2010 06:22:29 -0500 (EST)
- References: <3794357.1268049946227.JavaMail.root@n11>
To add something in the way of speculation: Some people might not be good at calculation, yet they might be good at mathematical thinking if they can get over the calculation hump. The Rainman was good at calculation but no good at mathematics. As an analogy I was never good at cursive writing and early on took to printing. I won't say how much time I've spent trying to master "&"! I've read exhortations by teachers claiming students must absolutely master cursive writing. So if they don't master it, does that mean they will never likely be good at expressing ideas in writing, say on a computer? Mathematica can open up mathematics to a wider range of students. But not out-of-the-box. It is too hierarchically thin, and too difficult to manipulate expressions to common forms. Approaches may need to be altered. There needs to be more of an axiomatic approach and a distinction between what is mathematics and what is "plug and chug". Almost any subject will have its axiom set and these should be implemented, either in the form of rules or definitions that carry out some transformation. But the axioms should not be automatically applied. For some subjects the higher level Mathematica operations may need to be bypassed. The student would then have to recognize when various axioms are needed and apply them. Perhaps the student could bring up a window that listed all the axioms. If a student were solving problems by this method wouldn't that be considered as doing mathematics? Also, wouldn't this keep the student concentrated on the foundations of the subject and give the right kind of practice? David Park djmpark at comcast.net http://home.comcast.net/~djmpark/ From: Murray Eisenberg [mailto:murray at math.umass.edu] One reason is very simple: by using a CAS to do many long symbolic calculations, students can focus on modeling and the resulting and relevant mathematical concepts and methods -- not the details of carrying out long chains of algorithmic, algebraic steps. My 45 years of teaching make perfectly clear that, for most students in calculus, e.g., they are so involved in trying to get the symbolic manipulations right, they have little or any idea of why they're doing them. They totally miss the forest for the trees. The other side of this situation, I regret to say from my experience, is that the lazier or intellectually weaker students are often incapable of rising above merely carrying out mechanically the symbolic manipulations -- many of which they get wrong anyway -- to have much of an understanding of the higher-level concepts involved. On 3/7/2010 4:06 AM, Andrzej Kozlowski wrote: > I have never seen or heard any convincing reason why using a CAS should > make it possible to understand and learn better those areas of > mathematics which are fully accessible to a student with only a pen and > paper. In fact I can see a few reasons why the opposite might be the > case. In many situations I can see clear advantages in performing > algebraic manipulations "by hand" or even "in the head", which is, in my > opinion, the only way to develop intuition. The same applies to > visualisation - while being able to look at complicated graphics can > often be a big advantage, I always insist on students developing the > ability to quickly sketch simple graphs by hand on the basis of > qualitative analysis of analytic or algebraic data. This is again > essential for developing intuition and I am not convinced that doing all > this by means of a computer will provide equivalent benefits.... -- 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