>I'm reasonably familiar with Derive, Mathematica, Mathcad, and Matlab, =
but=20
>I use a graphing calculator more often than all my math software =
packages=20
>combined. For most of my tasks, I can get answers from the calculator =
more=20
>quickly than the time it takes for my laptop to boot up (let alone the=20
>additional time it takes for Mathematica to load its kernel). If I =
wanted=20
>to keep a record of my work, I'd use something other than a calculator, =
but=20
>for quick and dirty answers, I find using a computer too slow and =
clumsy.
>
>Of course I work in the lofty ivory tower of a community college, so my =
use=20
>of calculators and computers won't reflect the real world. But I do =
know=20
>that even though I have a food processor essential for several of my=20
>recipes, I do most of my slicing, chopping, and mixing without it. The=20
>computer is unquestionably a wonderful tool with a multitude of=20
>applications, but it is not the only useful tool, nor is it always the =
best=20
>tool.
>
>Bruce Yoshiwara
>
I think these are good points -- the calculator remains a handy dandy=20
little artifact for many, and computers might just be overkill when=20
you need some quick and dirty graphs and or numeric outputs.
My own view of the place computers might play in a mathematics=20
curriculum is not particularly conventional, and should therefore
be presented with the necessary caveats.
My view is that students need more exposure to non-numeric operators,
which likewise follow strict rules and have a place in the underlying
processes which keep our civilization chugging along.
So I'm drawn to using computers because I can introduce notation=20
and syntax which calculators (and most pre-computer text books) do
not support.
I am NOT using Mathematica, Matlab or Derive. I do use some MathCad,
but often just to get some of the symbols cut and pasted to the web
(i.e. for typography e.g. big radical signs with greek letters,=20
sigmas and stuff like that).
My beef with the more conventional approach to mathematics is that
it is stuck in an era before we did a lot of electronic processing
with alphanumeric (not just numeric) data. So we don't expose=20
students to concepts of list, tuple, object, string in the context
of mathematics class -- we lump all that under the heading of=20
computer science (meaning many students don't get early exposure,
or none at all).
What I'm fighting is overspecialization and the acceptance of the
status quo, wherein "thinking about math" is over here in box A,=20
and learning to program your ideas on computer, explore logic and
algorithms using electronic means is over there in box B. I want
boxes A and B to smash together and integrate their content much
more successfully.
So, for example, I'll label the 4 vertices of a tetrahedron A-D,
and the inverted tetrahedron E-H. The two tetrahedra interpenetrate
to form a cube with vertices A-H. The dual to that cube is an=20
octahedron of 6 vertices: I-N. The cube and octahedron together
comprise a rhombic dodecahedron with vertices A-N. 12 more vertices
of the surrounding cuboctahedron (O-Z) give me an inventory of points
A-Z (26 vertices). That's an easy mnemonic -- something to carry
around in your head (along with the volumes table -- below).
So now I want to store the coordinates of all those points in a
dictionary of some kind. This is where math books go blank=20
(conventionally) because the idea of a "data dictionary" is=20
for computer people, even though math books customarily label
points with letters. We teach the paper and pencil way of=20
tracing a line from point A to point B, but what does this=20
look like on computer (which is the place you'll more likely
be needing to connect the dots in future)?
I want my students to become familiar with syntax like:
>>>datapoints =3D [ =
"A":(1,0,0,0),"B":(0,1,0,0),"C":(0,0,1,0),"D":(0,0,0,1)]
where A-D are "keys" and tuples (1,0,0,0)-(0,0,0,1) are "values".
Then I want them to write a function that traverses datapoints and
returns a new dictionary with all the tuples "inverted" e.g.
>>> newpoints =3D invert(datapoints)
>>> newpoints
[ "A":(0,1,1,1),"B":(1,0,1,1),"C":(1,1,0,1),"D":(1,1,1,0)]
And how does (1,0,0,0) get to be the vertex of a tetrahedron? =20
I'm not using XYZ coordinates, but something called "quadrays":
4 basis vectors from the center of a regular tetrahedron span
volume such that non-negative 4-tuples have a unique, lowest
terms mapping to XYZ's 3-tuples. So then I need functions=20
for converting XYZ->quadray and quadray->XYZ. Perhaps a=20
points dictionary (such as shown above) will be my input to
such functions.[1]
Note also that I'm not using the conventionally taught primitive
volume values for these polyhedra, preferring the following=20
table for its streamlined simplicity:
Volume
Tetrahedron ABCD: 1
Inverted EFGH: 1
Cube ABCDEFGH: 3
Octa IJKLMN: 4
Rh Dodec A-N: 6
Cuboct N-Z: 20
See animated GIF at: http://www.teleport.com/~pdx4d/intro.html
I get this table from "synergetic geometry", a frontier philosophy
which has by now spawned a rather large and growing research=20
community -- great for patching students into some recent=20
explorations (e.g. tensegrity, elastic interval geometry,=20
synergetic crystallography) that don't require years of preparation.
[Of course it irks me that so many mathematics teachers choose=20
to ignore the simple advantages of this volume table, and the=20
boost it gives, especially to a younger set just starting out,
when attempting to wrap one's minds around spatial geometry --
but I've diatribed enough on that in other contexts and needn't
bore readers here with any tirades or lectures in this regard]
So instead of Mathematica and Matlab, I'd rather have my studends
studying Java and Python, backending these into a ray tracer like
Povray (for visuals), or into a VRML world. All three of these=20
languages and platforms are freeware and very capable. =20
So some of the elements of my non-traditional approach are:
(a) use some synergetic geometry and it's streamlined approach=20
to polyhedra and sphere packing to get "beyond flatland"
(b) converge geometric visualization w/ algorithmic processes=20
known and proved by mathematicians (e.g. Pythagorean=20
theorem), but with the syntax/notation of computer languages=20
to explore these concepts interactively
(c) in particular, study coordinate geometry and polyhedra in=20
conjunction with matrix transformations of shapes (rotate,=20
scale, translate). I even introduce quaternions at this=20
point, in the context Hamilton intended for them: as objects=20
that rotate vectors in space.[2]
Visualization is at the core, so even though students are manipulating
symbols ala algebraic texts, they have rendered output in a lot of
cases. =20
But this doesn't mean I avoid real algebra either. My website gets=20
into Pascal's triangle + Binomial Theorem (a standard link) and=20
uses sphere packing to get to power series (series in general) and=20
Bernoulli numbers.[3] =20
Lots of probability and statistics notions can fit in at this point=20
(Pascal's triangle defines a Gaussian distribution, as those science=20
museum "falling ball" demos so appropriately demonstrate). Plus you=20
need some of the permutation and combination concepts to adequately=20
define these series (Pascal, Bernoulli). =20
Sigma notation goes here -- prelude to the Riemann sum -- along with=20
a seamless integration with do-loops (by the time students hit sigma=20
notation, they should already have fluency enough in some computer=20
language to implement them -- a basic premise of my curriculum).
=46rom such a discrete math background, I move towards the calculus by=20
"increasing the frequency" (frequency is a key term, defined elsewhere=20
in the curriculum)[4] -- same as going to the limit, in terms of=20
positing a continuum (at which point we finally get to the "real=20
numbers" -- until now we've been using floating points, integers and=20
other sets which have specific meaning in the electronified world of=20
computers and calculators, but which are not the "reals" until we get=20
to the metaphysics of "infinity" (something computers don't deal with=20
except by over and under flowing)). =20
That's right: I don't start with the "reals" -- I get to them later,=20
as abstractions involving "infinity" need not clutter our investigations=20
right out of the starting gate (I think we hit kids too early with a=20
lot of "infinity talk" -- before they have any philosophical training,=20
ergo have no defenses against sloppy philosophies (not that all philo-
sophers who use "infinity" are sloppy -- but many of them are)).
Anyway, that's where I'm heading in a nutshell. I need computers=20
because calculators are too difficult to program adequately,=20
especially in the realm of alphanumeric (not just numeric)=20
algorithms.
Students like this mix of computerese and conventional mathematics
because having some exposure to Linux, HTML, Java, Python, XML/XSL,=20
and SQL is what looks great on a resume. These are hot button topics.
No reason to divorce mathematics from what it takes to get a job
these days. On the contrary, I side with mathematician Keith Devlin: =20
math is about making the invisible visible, and if you have no insight
into what all these computers are doing all around you, then your
math class has failed to lift the veil of mystery and ignorance
from in front of your face.
Kirby
Curriculum writer
Oregon Curriculum Network
http://www.inetarena.com/~pdx4d/ocn/
[1] http://www.teleport.com/~pdx4d/quadrays.html
also "Success Story: Using VFP to Teach Object-Oriented
Programming" by K. Urner, FoxPro Advisor (Advisor Media, Inc.,=20
March 1999), pg 48
also http://www.inetarena.com/~pdx4d/oop.html
[2] http://www.inetarena.com/~pdx4d/ocn/oopalgebra.html
[3] http://www.teleport.com/~pdx4d/sphpack.html
[4] http://www.teleport.com/~pdx4d/omnihalo.html
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