Re: Mathematica as a New Approach to Teaching Maths
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- Subject: [mg127484] Re: Mathematica as a New Approach to Teaching Maths
- From: Ralph Dratman <ralph.dratman at gmail.com>
- Date: Sat, 28 Jul 2012 02:41:20 -0400 (EDT)
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David, I agree that Mathematica is quite difficult to learn, but I also think a good teacher could make the learning process significantly easier by going carefully over certain key points which are not emphasized in the documentation. For example, one of the most fascinating features of the language -- its ability to mix symbolic and numeric quantities -- can also be a source of great confusion for the neophyte. I would spend a lot of time talking about that and showing how useful and interesting the results can be. I would also go very carefully over what one must do to keep the front end and/or kernel from going out of control, and how to recover when that does happen. I have a number of related ideas to help students get the feel of the Mathematica environment before plunging in to accomplish serious work. I would love an opportunity to teach Mathematica to a small group of smart high schoolers or beginning undergraduates. Ralph Dratman On Fri, Jul 27, 2012 at 4:56 AM, djmpark <djmpark at comcast.net> wrote: > Thanks for mentioning the Conrad Wolfram Ted Talk. > > http://www.ted.com/talks/lang/en/conrad_wolfram_teaching_kids_real_math_with > _computers.html > > I think Conrad's talk was correct in the main points, glib about the > difficulties, correct about the potential. > > 1) The first point is that Mathematica is difficult. There is a long > learning curve. Therefore students headed for careers with technical content > should start learning Mathematica early, long before they have to tackle > difficult mathematical material. The students would learn some mathematics > as they are learning Mathematica but it is a question as to what subjects > should be treated when the main objective is to learn Mathematica. (If the > aim is to learn real mathematics John Stillwell's "Numbers and Geometry" > might be a good source of material.) > > 2) The test of having skill with Mathematica is the ability to turn real > world problems into Mathematica specifications - one of Conrad's points. I > call it flying solo, as opposed to copying someone else's code. > > 3) Conrad disparages learning math by hand with paper and pencil. I would > rather co-opt the paradigm. Don't think of Mathematica as a super graphical > calculator, or as a programming language (although it is in part these > things) but think of it as a piece of paper on which you are writing your > ideas, exploring them and presenting them. It is indeed a magic piece of > paper with its active calculation, memory and dynamics - but still a piece > of paper. > > 4) This means that students should also learn how to use the Sectional > structure of notebook and discuss their material and its development in Text > cells. It means that material will often have to be presented in stages with > multiple definitions and derivations, graphical presentations and dynamic > presentations. One cannot often present coherent material in a single > Manipulate statement with lots of Sliders. > > 5) I didn't quite follow the example that Conrad presented in place of a > test. I'm not certain if the student was just to use the dynamic > presentation or if he was to design and implement it. The first might not > be especially instructive and the second might be too difficult. In any > case, I would say that the tests or homework should be in the form of essay > questions using the techniques of the preceding point. If a student writes > an essay notebook he has something to keep, refer to in the future, and show > off. > > 6) The analogy of jumping over a chasm is one I presented on MathGroup in > 2008. http://forums.wolfram.com/mathgroup/archive/2008/Nov/msg00714.html > > 7) In general, the entire topic of using Mathematica in education is quite > difficult because the capabilities that Mathematica brings are so > revolutionary. It is all too easy to be unconsciously mired in old paradigms > or to fall into the pit of "computer junk". In some cases full-fledged > Applications will be necessary, done in Workbench with documentation and > examples or course material. Students must learn how to write routines (and > their usage messages), but maybe not every extra routine convenient for some > subject matter. Such applications should not put the student into a box but > rather provide a set of routines that supplement and extend regular > Mathematica. An example might be a set of axioms. in the form of rules or > routines that apply the rules, for some field of mathematics. Then a student > could do derivations or proofs using the axioms. What better way to become > familiar with them using them and seeing them in action. > > 8) Some things along these lines are in the Presentations Application. One > of the things students have most difficulty with is custom graphics because > the WRI paradigm is really convoluted when it comes to combining things or > making geometrical diagrams. Presentations tries to fix that. There is a > section on single variable integrals that allows a student to do various > manipulations on the integrals such as change of variable, integration by > parts or trigonometric substitution so they can see what is happening. There > is a Students Linear Equations section that allows matrices to be > manipulated with primitive commands and see the results. The matrices also > have row and column labels to give them context. I've been working with John > Browne's GrassmannAlgebra Application, primarily trying to learn it but also > helping with interface and the writing of some introductory examples. This > Application would be great for teaching plane geometry because one can > easily define points, lines, triangles and other objects algebraically; do > things such as calculate lengths, areas and angles; rotate and translate > objects; calculate perpendiculars and find intersections; or determine if a > point is inside or outside a triangle - all with algebra. One can also draw > the geometric diagrams directly using the Grassmann algebra expressions for > the coordinates. > > It's the kind of thing that can be done but it's more than regular > Mathematica and it takes development. > > > David Park > djmpark at comcast.net > http://home.comcast.net/~djmpark/index.html