From: goddard@NeXTwork.Rose-Hulman.Edu (Bart Goddard)
Subject: Re: Calculus and Mathematica
Organization: Rose-Hulman Institute of Technology
Date: Wed, 19 Aug 1992 16:38:05 GMT


In article <347@moene.indiv.nluug.nl> toon@moene.indiv.nluug.nl (Toon  
Moene) writes:
> My advise: Unless you're absolutely sure MMA (or the other symbolic  
> algebra package you're looking at) does what you want it to do, or  
you  
> have enormous amounts of time doing it the hard way (i.e. programming  
it  
> in MMA's language), don't buy into symbolic algebra - it is no  
Artificial  
> Intelligence.
> 
> --
> Toon Moene (toon@moene.indiv.nluug.nl)

This is good advise.  It applies to anything: Don't hire a contractor  
unless he's going to do what you want.  Don't by a car unless it  
does/has all the things you want.  Don't marry that woman unless you're  
sure things will "work out",.....

I have been using MMa in the classroom, but the choice of package was
not mine.  I would have chosen Maple, since it is more powerful and  
flexible, runs on a smaller machine (so more students could have it  
running in their dorm rooms and wouldn't have to come to the labs to do  
their homework), and runs faster (at least on our network) than MMa.

The incorporation of Computer Algebra Systems into math classroom  
changes the game quite a bit.  Two years ago, I taught all my calc  
students several integration techniques: parts, substitution, partial  
fractions, trig substitutions, etc.  Last year, with the introduction  
of MMa, I was forced to think hard about whether integration by parts  
was a worthwhile topic.  After all, the machine will do most of the  
book problems, and in real life, one can "always" do an integral  
numerically.  So not doing parts would free up some time to spend  
studying more theoretical things (When CAN'T you do an integral  
numerically?).  On the other hand, parts is derived from the product  
rule for differentiation, and students who understand this relationship  
have learned something about how functions behave, and I think this  
might be important, whether any of them ever actually evaluates an  
integral with this technique. So I ended up teaching the techniques I  
named above, (but no others).  However, I didn't give an assignment  
that asked them to evaluate a "mixed bag" of integrals, wherein they  
would have to choose a technique and hope it worked.  This sort of  
assignment is a "by hand" skill which (I think) is largely obsolete  
except for it's pedagical value (one learns about funtions by playing  
with them.)

Having decided that my calc course would include the topic of  
integration by parts, I went to the next step:  How to make them see  
the topic the way I see it (this is what I think teaching is). Two  
years ago, I had them do several integrals that required parts.  They  
were able, then, to do integration by parts.  Several of them  
understood the connection with the product rule.  Last year, I gave the  
new class roughly the same list (but longer) and had them integrate  
them on the machine.  (I had to check that the maching could do all of  
them before I made the assignment.  If the machine can't, the problem  
isn't a good example for this exercise.)  Then they were to find a  
pattern in the  answers they were getting.  We discovered the  
integration by parts relationship, and it was more fun this way.  (By  
fun, I mean that instead of 30 students (with glazed eyes and drool  
running down the pencils they've been chewing on) taking notes and  
dissappearing, I had 30 students with whom I was interacting (something  
we don't get to do much of in mathematics.))  I think that one year  
later, these students are just as able to integrate by parts as the  
students from the year before (which isn't saying much). And they  
understand the relationship to the product rule better than the earlier  
class.  PLUS, almost all of our students are engineering majors, and  
they have the added bonus of learning mathematics in the environment in  
which they will use it in their upper-level courses and in their  
careers.

Calculus consists of just a couple easy ideas, upon which we build in a  
a straightforward manner.  Once a student finishes the course, he  
wonders what was so hard.  Now that his brain has abstracted and sorted  
everything out, all of calculus fits into just a couple brain cells.   
My job is to make my students see things this way. I can do it with or  
without a computer algebra system, but it's more interactive WITH. It  
doesn't matter if there is an occasional problem which stumps MMa, it  
still helps in other areas.  Besides, by the time you figure out  
exactly what you want to do and how to do it, the software has been  
updated and it WILL do those persnickety problems.

Sorry this is such a long tirade.  My "friends" say that talking to me  
is like drinking from a fire hydrant.

Bart Goddard
Rose-Hulman Inst. of Tech
goddard@nextwork.rose-hulman.edu