Re: Re: The audience for Mathematica (Was: Show doesn't work inside Do
- To: mathgroup at smc.vnet.net
- Subject: [mg102131] Re: [mg102081] Re: The audience for Mathematica (Was: Show doesn't work inside Do
- From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
- Date: Thu, 30 Jul 2009 05:32:41 -0400 (EDT)
- References: <h4m4ca$ecg$1@smc.vnet.net> <200907290907.FAA19457@smc.vnet.net>
On 29 Jul 2009, at 18:07, Helen Read wrote: > Bill Rowe wrote: >> On 7/27/09 at 5:56 AM, siegman at stanford.edu (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
- References:
- Re: The audience for Mathematica (Was: Show doesn't work inside Do
- From: Helen Read <hpr@together.net>
- Re: The audience for Mathematica (Was: Show doesn't work inside Do