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Topic: Example stories for K12 math classes
Replies: 4   Last Post: Feb 25, 2010 5:57 PM

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kirby urner

Posts: 3,690
Registered: 11/29/05
Example stories for K12 math classes
Posted: Feb 22, 2010 4:06 PM
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Keying off Disney's 1959 Donald Duck in math land movie,
I'm thinking of stories we might be using to contextualize
some traditional content.

Yes, I have some non-traditional content up my sleeve as well,
but the same technique applies: provide lore, ala Disney, and
you'll have a better experience learning (and teaching) the skills.

Lore: with polynomials up and going, thanks to the new Algebra
coming from Baghdad through Pisa, a high culture on the Italian
peninsula actually turned their factoring and solution (in terms
of roots) into a sport. Patrons would sponsor tournaments,
and those contestants with secret algorithms for finding solutions
could actually win bets for their compatriots.

Lore: when Newton and Leibniz introduced the ideas of Calculus,
there was some intense backlash from Bishop Berkeley, the same
one for whom UC Berkeley is named.[1] He decried use of
infinitesimals as elements in proofs and excoriated Newton and
Leibniz for doing so.[2] The next 100 years or so of Calculus
development might be characterized as a formalization aimed
at plugging the holes Bishop Berkeley (and other skeptics) had
been poking.

Lore: the European Renaissance had a lot to do with the rediscovery
of Greek contributions to civilization, much of that knowledge kept in
Arabic translations during the so-called "dark ages" interim. The
introduction of abacus-based positional notation and the algorithms
for bookkeeping into European cultures, made intelligent management
of money more doable and a merchant class was able to thrive and
patronize the arts and sciences. Much of our current mathematics
got started in this period, which led to the development of perspective,
the illusion of depth in a flat canvas. XY and XYZ coordinates come
down to us from these days of radical pioneering.

I admit right off the bat that none of these three stories seems to
directly answer the question "but why does that mean *I* should
be learning this material?" Rather, each story embeds aspects of
math's importance and the kinds of role it plays: in competitive
fields where one seeks advantage; in formalizing new techniques
in need of logical defense against skeptics; in empowering human
beings to take more control of their own affairs and enjoy higher
living standards as a consequence.

Moving closer to the present and maybe upping the controversy
level as a result, here's another story for sharing:

Lore: the rectilinear XYZ-based practices developed as a result
of the discovery of perspective techniques in the Renaissance Era,
provided a strong bias in favor of 90-degree angles as "normal"
and "orthodox". Perpendicularity became king. As microscopes
and other instruments brought in more data about naturally
occurring shapes and designs, a more 60-degree aesthetic
began to emerge, as symbolized by the hexane ring of organic
chemistry.[3] XYZ math may these days be taught side-by-side
with some contrasting alternative in some curriculum segments.
We will be doing a chapter entitled Tetrahedral Kites for the next
few meetings. You will find a syllabus at our web site.

This last bit is from a teacher training, not a direct to high
school students class, although those are also available in
some zip code areas.** There's a Tetrahedral Kites lesson
plan, using non-traditional volumes (tetrahedron as unit,
octahedron as 4x, same edge length) at the NCTM site, plus
a lot of Alexander Graham Bell sites to consult. The story
continues through the octet truss, with links to NASA, and
ends in basic chemistry, where the CCP and FCC lattice
amount to the same thing.

If you're an elementary school teacher and a member of a
faculty motivated to include some of the above pre middle
school, then you will be the one responsible for making the
lore accessible to your students, in ways you know they'll
appreciate. Collaborate with your peers. Find out what is
already out there why not? "Being responsible" doesn't
mean always having to work solo.

Ditto for middle school or high school versions: the teacher
translates to her or his local community in a way she or
he thinks will work best. This is not about dictating outcomes
so much as planting seeds. Shape how the plants grow.
Tend to your own garden. This is sometimes called a "place
based" curriculum and is highly respectable in international
circles (UNICEF etc) because it encourages keeping the
material localized and therefore relevant.

Now you might ask what, pray tell, the above stories could
possibly mean for some child in the hills of Borneo. Why
learn about Italians, Greeks, Arab scholars? Well, you
may not be up to date on how it's going in Borneo, but
some of those hill kids have Internet already, thanks to
micro-hydro power. Learning about the big world out there,
ala National Geographic, is what's on everyone's plate these
days. Developing a healthy respect for many cultures is
a feature many teachers look for, when considering what
curriculum to teach.

Remember: all mathematics is ethno-mathematics (comes
from somewhere, has subcultural roots).

Disney's 1959 movie was G rated and suitable for showing
in elementary schools, although some zip code areas might
have been scandalized by the inclusion of jazz music,
considered "of the devil" by some (I'm not the big ethno-
musicologist in the room, so feel free to chime in, anyone
more familiar with that lore -- math and music are conjoined
at many levels, as Disney well knew).

Getting back to XYZ, one might expect some raised eyebrows
regarding my tracing it as far back as the Italian Renaissance,
as we call these Cartesian or perhaps Fermatian coordinates
and these gentlemen lived quite a bit later in time. Please
consider all of the above lore as highly abbreviated, fine to
add all these details.

In my defense: you will find that painters were using a grid
system much earlier, to map the visual field to a canvas.[4]
The roots of XYZ thinking trace to polyhedral geometry and
to individual "solids" such as the cube. Crediting the Greeks
is not that far-fetched (we also like to tell that story about
how the discovery of irrational numbers was somewhat
crazy-making to the original Pythogoreans -- Carl Sagan
tells it well -- but that's for another day).


** in business, we speak of B2C vs. B2B. That's business
to consumer and business to business. I think of teacher
trainings as B2B. Imagine Intel a sponsor, and teachers
getting paid to be there (on the clock, serving the public).

[1] Founders' Rock: On this outcropping located on the
north side of the campus near the corner of Hearst Avenue
and Gayley Road, 12 trustees of the College of California
stood on April 16, 1860, to dedicate just-purchased property
as a future campus for their college. In 1866, a group of
College of California men stood at Founders' Rock,
watching two ships out at sea through the Golden Gate.
One of the men, Frederick Billings, was reminded of the
lines of Bishop Berkeley, "Westward the course of
empire takes its way." He suggested that the town
and college site be named for the 18th-century Irish
philosopher. On Charter Day, 1896, the senior class
commemorated the dedication of the campus by placing
a memorial tablet on Founders' Rock.




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