From: bxi@po.CWRU.Edu (Bruce Ikenaga)
Subject: Re: Calculus and Mathematica
Reply-To: bxi@po.CWRU.Edu (Bruce Ikenaga)
Organization: Case Western Reserve University, Cleveland, OH (USA)
Date: Thu, 20 Aug 92 18:10:52 GMT


In a previous article, goddard@NeXTwork.Rose-Hulman.Edu (Bart Goddard) says:

>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.

   In our case, the decision was made in sort of grass roots
fashion ... but it was constrained by the requirement that
the package run on Macs and DOS boxes. Mma did that, had nice
graphics, and the promising Notebook interface, so ...

>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).  

   What other techniques do people normally teach?

   I would say that if you're going to lean on a CAS to do integrals
(numerical integrals, for instance), perhaps the time to do it is
when you're looking at applications of integration. Arc length comes
to mind.

>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.)

   Mmmm ... I dunno about this. First, it seems to me that the
first things which get automated are the mechanical things. Hence,
it's important to teach people to decide what tool or technique
to use in a given situation.

   Moreover, the heuristics one uses are interesting because they
illustrate the idea of *control* --- keeping track of one's
progress in solving a problem. For example, suppose we have

      \int x e^x\, dx.

How do you decide what technique to use? Someone says "parts".
Why parts? Suggestion: "Try parts if the integrand contains
functions of different types." Now we can ask how well this
heuristic works --- can we think of any counterexamples? ...

   (An interesting question here is: How is Mma really doing
something like

      \int e^x \sin x\, dx?

It probably isn't using the do-parts-twice-and-solve-for-the-
integral trick ... so what is it doing? I don't know, but it
might be interesting to discuss in class --- Why does the
program do this differently than I would do it?)

   Suppose we decide we want to try parts. Do we do it this
way?

      u = e^x,   dv = x\, dx.

This doesn't work so well (How do you know this?). Why doesn't
it work so well? We might suggest another heuristic: "If you
have powers of x, it's better to put them into u rather than
dv." Are there are counterexamples? What is the scope of this
rule of thumb

   Likewise, you might ask why you try trig substitution in

      \int \sqrt{1 + x^2}\, dx

rather than (say) parts. (Or will parts work?) And if you
decide to do trig substitution, why use x = \tan \theta
(as opposed to x = \sin \theta)?

   One of the more amusing techniques is the one you use for
things like

      \int \sin x^(1/3)\, dx.

It is interesting to try to articulate in words *why* x = u^3
works.

   In all of this, I haven't said anything about actually
*doing* the computation. In a sense, we take that part as
"canned" --- concentrating on *what* to do, and *why* you
do it, rather than *how*. In fact, it's nice to have people
mess around and try to come up with a list of heuristics
--- together with examples which show when the heuristics
fail.

   One reason I like teaching integration techniques is that
it's a good place to talk about heuristics, control and
decision procedures. People come out of high school with a
"try the first thing that comes to mind" approach to problem
solving. (See some of Alan Schoenfeld's articles, for instance.)
I think that helping to refine their problem solving skills
is one of the more important things we can do.

>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.

   This sounds like a good idea.

   (Another thought --- parts does occur elsewhere. For example,
it's often used to find recursive formulas for integrals. The
initial value formula for Laplace transforms [which many students
will see when they take differential equations] is a conrete case.)

>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.

   Yep. Tools like Mma are used by many of the engineers around
here. We're starting to get questions about what we do with it
(and could we use it in our courses, so they can assume their
students have seen it). This can be an opportunity for math
departments, if they take advantage of it.

>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.   

   Hmmm. "A couple"? I would mention linearity, convergence,
continuity, rates of change, and adding-up-lots-of-little-
things-to-get-a-big-thing :-).  Which ones were you thinking of?

>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.

   Actually, I *like* problems that the software can't do or
that it screws up. I make a point of showing these to my
students.

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

   Sorry this is such a long reply. :-)

-- 
Bruce Ikenaga
---------------------------------------------------
US mail: Dept. of Math, CWRU, Cleveland, Ohio 44106
E-mail : bxi@po.CWRU.edu