"There is no one way"

Tuesday, May 5, 2015

NCTM wrap-up

I only attended a few sessions at NCTM-Boston, because I spent a fair amount of time promoting the Lab Gear. I already posted my report on Geoff Krall's strategies to improve the problems we find in standard textbooks. In this post, I'll go over some of the other worthwhile ideas I came across.

- Scott Steketee gave a talk on making the connection between functions in geometry (e.g. the well-known transformations of the plane) and (linear) functions in algebra. His basic idea was to use geometric transformations in order to go from transformations of the number line, to function diagrams, to Cartesian graphs. One interesting point he made was that transformational geometry makes it possible to introduce function notation early on without the risk of confusion between f(x) and f·x. (In a strictly algebraic context, I prefer to wait until function notation is needed to talk about composition and inverse functions, so Algebra 2 or Precalculus.)

- Michael Pershan and Max Ray gave a talk on introducing complex numbers that was so compatible to my own pedagogy that I added my recollection of it to the Kinesthetics part of my Web site. Their approach also started with transformations of the number line. I was surprised and pleased when Max came up to me after the talk to say that they were familiar with my approach to complex numbers.

- Brent Ferguson gave a talk on compass and straightedge construction of numbers, given a starting line segment of length 1. Most interesting: new-to-me approaches to dividing a line segment into n equal parts. I was familiar with this traditional method, which I learned in high school:

To divide AB into three equal segments, draw a ray BC, mark a point D on it, use a compass to get points E and F so that BD = DE = EF. Join AF. Copy the angle ∠AFB at points D and E to get parallels to AF. Those lines intersect AB at G and H, and we have BG = GH = HA, which is what we wanted. Alas, while it's not hard to prove that this is entirely correct, it's a pain because copying angles with compass and straightedge is a hassle.

Brent's students came up with other methods. Here is one particularly elegant one:

 Construct perpendiculars to AB through A and B. Choose a point C on one of them, and use a compass to get D, E, and F. Join DE and CF, and you're done! No need to copy angles.

[This earlier post on geometric construction includes links to several more.]


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