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Re: The audience for Mathematica (Was: Show doesn't work inside Do

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  • Subject: [mg102081] Re: The audience for Mathematica (Was: Show doesn't work inside Do
  • From: Helen Read <hpr at>
  • Date: Wed, 29 Jul 2009 05:07:48 -0400 (EDT)
  • References: <h4m4ca$ecg$>
  • Reply-to: HPR <read at>

Bill Rowe wrote:
> On 7/27/09 at 5:56 AM, siegman at (AES) wrote:
>> As a more specific definition of an expected audience, it seems to
>> me (and, I think, Helen Read) that Mathematica -- or at least a more
>> consistent and less perplexing form of Mathematica:
>> 1) Could be very accessible to bright high school students, maybe
>> with some hand holding;
> The problem here is most high school students are in the process
> of learning mathematics and have little knowledge of advanced
> mathematics. Is it really a good idea to give students learning
> say algebra access to a tool like Mathematica? If you do, what
> will they learn? Algebra or usage of the tool?

You can teach them both at the same time, using one to reinforce the other.

I teach my (university) calculus students *calculus*, using Mathematica

- as a tool for graphing and visualization
- as a tool for carrying out numerical calculations
- to check work done by hand (integrals, derivatives, algebra, etc.)
- to carry out such work (integrals, derivatives, algebra, etc.) after 
we have done the thinking / setting up
- for discovery learning, where the students can explore and learn by 
looking at examples, and build or re-inforce concepts

Here's an example from my class. We begin the study of series with a bit 
of discussion of the notation and terminology, and look at a specific 
example, writing out the first few terms and partial sums on the board. 
Then we fire up Mathematica, and make some plots (using DiscretePlot) 
and tables of (decimal values) of terms and partial sums, much farther 
out than we could ever do by hand. We use the tables and plots to make a 
conjecture as to whether or not the series converges.

Here's what I give them for the first few examples:
1. A convergent geometric series
2. A series whose terms converge to say 1/2
3. The Harmonic Series

I have them work on the examples (there is a computer for each student) 
while I walk around the room answering questions. The students go 
through the first two examples quickly. Then they get to the Harmonic 
Series, and are intrigued. They have trouble deciding if it converges or 
not. They start making plots and tables that go farther and farther out. 
I've seen some students carry out tables / plots to n = 10000 or even 
100000 or more (with a suitable increment!).

So then we stop and discuss the examples.

Example 1: The terms converge to 0 quickly, series looks obviously 
convergent. (We leave this one for now, but come back to it when we talk 
about geometric series another day.)

Example 2: The terms do not converge to 1/2, the partial sums increase 
without bound, the series diverges. This leads us to the N-th Term Test: 
if the terms of a series do not converge to 0, the series must diverge.

Example 3: So here the terms converge to 0, but rather slowly in 
comparison with the first example. I'll ask the class if they think the 
partial sums converge or not. Some of them will say that it's hard to 
tell, and they aren't completely sure. Then someone will say something 
like "I went out to 15000..."  -- at which point I'll put the student's 
plot up on the projector for all to see --  "...and it still just gets 
bigger. I think it diverges." And then I lower the boom: I ask the class 
how we can *prove* that the Harmonic Series diverges, and launch into a 
discussion of the Integral Test, which we carry out by hand on the board.

The next day, we discuss the Integral Test in more detail, and do some 
examples where we have to (a) prove that the series converges, (b) 
approximate (to some specified precision) what the sum. For (b), we use 
Mathematica to compute a partial sum (out to say n=100), and use 
integrals (which we might do by hand or in Mathematica) to find upper 
and lower bounds on the error in the approximation.

We continue this blend of mathematics and Mathematica as we progress 
through the chapter. I believe that it helps the students build 
conceptual knowledge of what it really means for a series to converge or 
diverge. We use Mathematica in similar ways when learning other topics.

Helen Read
University of Vermont

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