I think there is great value in mathematics for its beauty and that not all
mathematics should be tied to applications that should be used. However, I
think it is time to find new ways to teach mathematics and mathematical
thinking for its beauty. Most of our students only see manipulation skills
that were historically important in order to support applied mathematics in
the era before technology. Now we still focus almost with tunnel vision on
the skills and contrived "applications" that justify teaching the skills.
There is beautiful mathematics that is neither routine nor is a perfect
model of the real world. Below are a couple of my favorites. I would like
to see other people's favorites.
Use one note card for each of the first three terms of a sequence. Put the
same number of red sticky dots on each of the three cards. Put 1 blue dot,
2 blue dots, 3 blue dots, respectively on cards 1, 2, and 3. Put 1 green
dot, a 2x2 square of green dots, a 3x3 square of green dots on cards 1, 2,
and 3. Have students create the 4th and 5th cards following the pattern,
then fill in a table generalizing to 6th, 7th, etc terms. The first column
in the table is the term of the sequence, second column number of red dots,
third is number of blue dots, fourth is number of green dots and fifth is
total number of dots on the card. Fill in the chart for sequence terms 1
through 10 ... n.
Have students plot points formed from 1st and last columns, graph the curve
y = whatever formula they came up with and see if they go through the
points.
This makes a good introduction to functions, sequences, leads to discussion
about input and output variables, etc. Students understand what constant
term means, linear and quadratic. Many other patterns, including
exponential, can be similarly made. Cubic functions are easy if you have
different colors of interlocking cubes instead of sticky dots and note
cards.
Here is calculus exploration that is strictly the study of patterns. This
takes 4 note cards. Have a group of 3 students divide 2 of the cards
evenly among them and record how much of a card (2/3) each student gets.
With the other 2 cards, they can only cut any one piece in half. They make
the cuts then divy out the as much as they can among the three people, then
again only making cuts of the left over pieces into halves, divy out as
much as they can with each successive cut. Keep track of how much of a
note card each student is getting each time. This is an interesting way to
see why 1/2 + 1/4 + 1/8 + 1/16 ....... = 2/3 and does attract student's
attention to a purely mathematical topic. (You do not have to be a
calculus student to understand the concept either.)
----------
> From: Martin KALMAR <Martin_KALMAR@CO.FREDERICK.MD.US>
> To: mathedcc@archives.math.utk.edu
> Subject: [MATHEDCC] Liberal Arts Math
> Date: Wednesday, August 27, 1997 3:00 PM
>
>
> Barbara Arementa wrote that her math department's criteria for the
contents
> of a liberal arts math course was "make it relevant...real life...usable"
>
> I wonder if anyone out there thinks that there is any value in teaching
> mathematics that isn't relevant or usuable. (When student's ask me why
> they need to take math, I like to tell them that I wish I could say that
> it is totally irrelevant and unusable.) Do English teachers choose the
> literture that liberal arts students must take, do they worry about
relevance
> or usability? Is there any value in teaching mathematics that is
> intellectually interesting or beautiful or fun?
>
>
> Martin Kalmar
> Frederick Community College
> Frederick, Md
>
> martin_kalmar@co.frederick.md.us
>
>
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