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Re: Mathematica as a New Approach to Teaching Maths

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  • Subject: [mg127518] Re: Mathematica as a New Approach to Teaching Maths
  • From: David Bailey <dave at removedbailey.co.uk>
  • Date: Thu, 2 Aug 2012 04:46:21 -0400 (EDT)
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On 27/07/2012 09:57, djmpark 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
>
>
>
>
>
> From: amzoti [mailto:amzoti at gmail.com]
>
>
> Hi All,
>
> I just watched what is probably considered a hot button topic issue by some
> from "Conrad Wolfram's recent TED talk "Stop teaching calculating, start
> teaching math".
>
> I was wondering if any Mathematica users have ever explored this and how
> they may be approaching it.
>
> I love the idea of teaching students to use Mathematica as an exploratory
> tool which allows them to ask what if questions for learning to problem
> solve and to ask better questions.
>
> Has anyone developed or researched an approach to replace the traditional
> teaching methods (crank out silly answers) at any level?
>
> It would be great if Mathematica could even suggest such as approach!
>
> Anyway, would love to hear any feedback, pointers or ideas.
>
> Sorry if this is off-topic!
>
> Thanks
>
>

I hesitate to jump into this debate because although I leaned 
mathematics as part of my undergraduate and PhD work in chemistry, I 
have not used it in earnest throughout the rest of my career.

In general, I feel that Conrad's ideas go rather too far. Part of the 
value of obtaining symbolic solutions to integrals by hand (say), is 
that it somehow makes the student more comfortable with the concept of 
an integral. Something similar applies to other mathematical concepts. 
The mechanism is probably more psychological than logical, but no less 
real for that.

It is also said that some researchers will browse alternative 
statistical methods available in a software package - simply hoping to 
find one that will make their data significant! That can't be an 
approach to maths that is worth encouraging!

Nevertheless, there does seem to be a tendency to impose unnecessary 
mathematical rigor in some subjects, that just gets in the way of using 
the material. For example, how many people have picked up a book on 
wavelets, hoping to discover if they would help in their problem, only 
to find themselves drowning in theorems and lemmas. These things are 
meant to be practically useful, but often there is no guide as to which 
of the scores of methods work best, and are worth spending time to 
understand!

Other books and papers seem to impose unnecessary mathematical rigor, 
just to appear impressive. How many times have you read "x \[Element] R, 
where R is the field of real numbers", when x is self evidently a 
floating point number, and no use is made of field theory in the 
material. Pointless tricks like that serve only to intimidate.

David Bailey
http://www.dbaileyconsultancy.co.uk



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