Mathematics of Juggling

This is an excellent, in-depth video on the mathematics of juggling, from Cornell University professor (and world-class juggler) Allen Knutson.

http://www.youtube.com/watch?v=38rf9FLhl-8

This hour-long video covers the mathematical notation developed to classify and communicate juggling patterns, and Knutson explains how that inherent mathematical structure can be used to create new patterns.

There’s a lot of very sophisticated math here, which may surprise some people.  But as Knutson says at the beginning, “Anything that is sufficiently understood … there should be a mathematics of that thing.”

There’s also some good juggling in here, too!

The Saccheri Quadrilateral

We’ve been exploring non-Euclidean geometries lately, and the Saccheri Quadrilateral plays a pivotal role in this particular mathematical history.

The Saccheri quadrilateral is a biperpendicular quadrilateral with two congruent legs.  It’s an object that is “obviously” a rectangle in Euclidean geometry, but proving that without the aid of the parallel postulate turns out to be rather tricky.

In fact, just proving that the measure of the green angle is less than the measure of the orange angle is pretty tough!

Bilingualism, Math, and the Brain

This is a short summary of recent research into how bilingualism strengthens the brain.

The details are a bit fuzzy, as is always the case in social science, but these results would not surprise me.  In fact, it’s surprising to me that people ever thought bilingualism would be a hindrance to cognitive function, as opposed to an aid.

I think about bilingualism often as a math teacher.  I try to consistently preach mathematical bilingualism, imploring students to train themselves to see problems both algebraically and geometrically.  Being able to see a situation in two (or more) ways not only creates great flexibility in problem solving, but it strengthens the overall inter-connections between mathematical ideas.

It seems only natural to me that traditional bilingualism would do the same.  Not only does it offer multiple filters for processing the world, but the development and maintenance of the cognitive systems is a real intellectual workout.

CTL 2012: The Write Angle for Teaching Math

I had the pleasure of running a workshop at this weekend’s Celebration of Teaching and Learning in New York City.  The workshop explored ways to inspire and structure student writing in math class, and was sponsored by Math for America.

In addition to the workshop, I wrote a companion piece for the CTL’s Edblog, titled  “The Write Angle for Teaching Math“:

http://thirteencelebration.org/blog/edblog/edblog-the-write-angle-for-teaching-math/3193/

The incorporation of writing into math class has played a pivotal role in my development and evolution as a math teacher.  This article discusses some of the ways I get students writing about math, both technically and creatively.

Writing in math class opens up a whole new world for both teachers and students.  Give it a try!  You might be surprised at where it takes you.

Spherical Geometry Explorer

This is a simple and powerful tool for exploring some basic ideas of spherical geometry.

http://www.math.psu.edu/dlittle/java/geometry/spherical/toolbox.html

In addition to investigating points, lines, and line segments on the surface of sphere, you can construct triangles, antipodes, and even circles in spherical geometry!

A picture may be worth a thousand words, but a good interactive spherical geometry applet is worth a thousand well-drawn diagrams!

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