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Topic: Mathematica in Secondary Education

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Subject:   RE: Mathematica in Secondary Education
Author: Jon
Date: Feb 24 2005
On Feb 23 2005, Craig wrote:
> This posting is a bit like the "New Classroom" posting, but along a
> different line.

Mathematica is a very powerful, and somewhat
> expensive, tool for doing great exploratory mathematics and science,
> and for solving a variety of tough problems.  What if the cost
> weren't a factor?  What if it could be ubiquitous, almost (but not
> quite) like the graphing calculators?  How would having such a
> powerful tool change how and/or what math is taught?  What would you
> like to do with such a tool?  If you've never worked with
> Mathematica, you can liken it to almost any computer algebra system
> (but with many additional capabilities).

When similar questions
> were asked about the graphing calculator twenty years ago (was it
> that long?), there was a wide spectrum of responses; I doubt that
> either extreme of optimism or pessimism has been realized, but I
> firmly believe that the technology engendered a change in both
> pedagogy and curriculum.  Visualization became much more integral to
> math instruction.

Would "unleashing" Mathematica have such an
> impact?  I doubt that improvements to visualization alone would make
> a Mathematica explosion worthwhile--so many specialized (and less
> expensive) packages, in addition to the graphing calculators, do a
> pretty good job.  What about in understanding the mathematics
> underlying visualization, or the problem solving aspects?
> Mathematica appears to be a great tool for encouraging "what if"
> programming (and, once a student [or even teacher] overcomes a small
> syntactic hurdle, Mathematica programming is relatively
> straightforward).

I look forward to any comments, daydreams, or
> other thoughts...

I've been grappling with this question for several years now and have found that
t the most  vectors are very easy to use in Mathematica. I think we should do
far more in the secondary math curriculum. They are very useful in the study of
both plane and three dimensional geometry as well as being very important in the
study of physics. Mathematica makes it very easy to define vectors and then
calculate with them. The dot product and cross product are built in functions in
Mathematica and are very useful. Another feature of Mathematica is it allows you
to define a variety of different functions. For example, you can define a
funtion len[u]  which calculates the length of the vector u by entering
where Sqrt is a built in Mathematica function and "." is the standard dot
product. This can then be used to define the function ang[u,v] which calculates
the angle between two vectors  by entering
                         ang[u_,v_]:= (u.v)/(lng[u]lng[v])
Mathematica also makes it very easy to work with parametric equations. For
if the vectors OA and OB are defined where O is the origin then the line
A and B can be defined by
                       lineAB[t_]:=(1-t)OA + t OB
The advantage of this is that the parameter has some meaning now.  lineAB[0] is
lineAB[1] is OB, lineAB[.5] is the midpoint of AB. To find the intersection of
lines AB and CD, you'd type in
                            Solve[ {lineAB[t]==lineCD[s]},{s,t}]
Using only what I have outlined above you can do a lot of geometry in both two
and three dimensions because so far there has been no reference to dimension.

Enougn specifics. Mathematica does have a syntax that can be challenging and
takes a while to learn but once you get it, the effort is worth it. Mathematica
also has wonderful graphics capabilities and I have found that you can use it to
create many of the classic geometric diagrams like triangles with all the
centers plotted.
I'd love to hear from others. I'd also be willing to share anything I have.
Anyone interested in contributing to a Mathematica section in MathTools. Any
Derive users out there? Maybe we need a general computer algebra section.

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