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Re: if using Mathematica to solve an algebraic problem

  • To: mathgroup at smc.vnet.net
  • Subject: [mg109042] Re: if using Mathematica to solve an algebraic problem
  • From: Helen Read <hpr at together.net>
  • Date: Sun, 11 Apr 2010 04:32:42 -0400 (EDT)
  • References: <hpmlcd$9v0$1@smc.vnet.net> <hpplf0$m8t$1@smc.vnet.net>
  • Reply-to: HPR <read at math.uvm.edu>

On 4/10/2010 6:55 AM, Richard Fateman wrote:
> Bill Rowe wrote:
> <snip>
>
> If one becomes very skilled at using Mathematica to
>> solve problems correctly wouldn't there have to be some
>> corresponding gain in understanding of how the same problems
>> would be solved without Mathematica?
>
> Not necessarily.
>
> If you use programs like Factor[], regardless of how many times you use
> them, why would you have the slightest idea of how you would factor
> polynomials without Mathematica?

Because you learned to factor polynomials by hand _first_, and were 
required to practice it, and have been tested on it. Once you understand 
how to do it for reasonable polynomials, why on earth would you not use 
a tool like Mathematica to save time when faced with something that's 
just too big to factor by hand?

>> I wonder if it is really possible to become highly skilled at
>> getting good results from Mathematica without also learning the subject.
>
> It may work for you, since you have a certain level of curiosity about
> the subject and about Mathematica.  For students who have no curiosity
> about either, they will learn as little as possible.

That's true with or without Mathematica. The students who don't want to 
learn, won't. They will memorize as little as they can get away to eke 
by, without any real understanding, and won't retain a thing after they 
walk out the door.

What I do with Mathematica in my classes helps to engage the students. 
Yesterday, on a Friday afternoon late in the day, we did some 
chalk-and-talk on finding slope and area for parametric curves, and did 
an example on the board. (The example involved finding the area inside 
an ellipse, and finding the slope at a particular point on the ellipse.)

Then I gave the class the following problem to work on.

(a) Plot the curve {x(t)=t^2 - 3t - 18, y(t)=-2t+5}, determine its 
orientation, and use the DrawingTools to mark the orientation with arrows.

Note that the ParametricPlot is not going to reveal the orientation; the 
students had to think about it and figure it out.

(b) Find the equation of the tangent line when t=5. Plot the tangent 
line together with the curve.

Note that Mathematica isn't going to come up with the equation for them. 
The students had to work it out.

(c) Find the exact coordinates of the leftmost point on the curve.

This is a question they had never been asked before. I hadn't done an 
example of it, and I didn't tell them how to do it. They had to think 
about it, discuss with their neighbors, and figure out what to do.

(d) Find the area of the region bounded by the curve and the y-axis (on 
the left of the y-axis).

To set up the integral, they had to figure out the appropriate domain 
for t, which could not be done by mimicking the ellipse example that we 
had done.

(e)  Take the region from (d) and revolve it around the y-axis, and find 
the resulting volume.

We did volumes for solids of revolutions two months ago, for Cartesian 
functions; they had never been asked this for a parametric curve, and 
had to reason it out themselves.

Now, again, this was late in the day on a Friday afternoon, late in the 
semester. Every one of the students was engaged and interested in the 
work. The students used Mathematica for some of the work (derivatives, 
integrals, etc.), and did some of it on paper, as they saw fit. There 
was a lot teamwork and discussion and thinking. I walked around and 
answered questions and gave hints where needed, but didn't outright tell 
them how to do anything. They were so involved in what they were working 
on that they lost track of time and stayed nearly 10 minutes past the 
end of class time, at 4:00 p.m. on a Friday.

I don't understand how anyone would think this a bad thing.

-- 
Helen Read
University of Vermont


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