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

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  • Subject: [mg102131] Re: [mg102081] Re: The audience for Mathematica (Was: Show doesn't work inside Do
  • From: Andrzej Kozlowski <akoz at>
  • Date: Thu, 30 Jul 2009 05:32:41 -0400 (EDT)
  • References: <h4m4ca$ecg$> <>

On 29 Jul 2009, at 18:07, Helen Read wrote:

> 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

I think this is a beautiful way to teach mathematics and provides a  
superb answer to Richard Fateman's question, to which I just posted my  
own, much less impressive, answer. If I had seen this post first I  
would not have bothered.

Andrzej Kozlowski

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