Re: Re: Re: Re: learning calculus through mathematica
- To: mathgroup at smc.vnet.net
- Subject: [mg108083] Re: [mg108058] Re: [mg108015] Re: [mg107971] Re: learning calculus through mathematica
- From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
- Date: Mon, 8 Mar 2010 06:11:36 -0500 (EST)
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I don't see any real contradiction between this and what I wrote. "Long symbolic computations" can well qualify as something that isn't "fully accessible" to many students without a computer. Also, I don't think you would dispute that all students should be able to do "short symbolic computations" by hand. If they don't, then, in my opinion, they will never fully understand what it is that Mathematica is doing for them. Ultimately it is a question of finding the right balance. There are lots of things in mathematics that one needs to do just once completely by oneself in order to develop an intuitive understanding of what is involved. Once this understanding has been developed, there is no need to perform ever again these often tedious computations and manipulations by hand. It seems to me that showing students that all such things can be done by a CAS before they have understood the basic concepts can sometimes be seriously harmful (of course mostly to the "the lazier or intellectually weaker" ones - the others would probably not be satisfied with mere "button pushing"). On the other hand, there are enormously many fascinating things in calculus and other areas of mathematics that can't be done by an average student without a CAS either at all or within a reasonably time period. I think all computer aided calculus courses should include such examples (and even perhaps concentrate on them) because it is only such examples that can really convince both students and sceptical academics that CAS can be seriously useful in mathematics. Andrzej Kozlowski On 7 Mar 2010, at 16:15, Murray Eisenberg wrote: > 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
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- Re: Re: Re: learning calculus through mathematica
- From: Andrzej Kozlowski <akoz@mimuw.edu.pl>
- Re: Re: learning calculus through mathematica