## Friday, February 15, 2013

### what's the name of that graph

The Edge, with its deep thoughts and long interviews, seems at odds with typical Internet culture, and yet is representative of a particular type of discourse that could only exist with the Internet - or maybe the conversations like those at the Edge would exist without the net, but most of us would have no access to them.

Content aside, I noticed the graphic above in the promotional material for the new Edge book, This Explains Everything. At first I thought it was a complete graph on 12 vertices (K_12) - which you would think make a nice choice of graphic to represent the fully networked world that the Edge folk themselves exemplify (it has all possible edges).

But if you look again at the image on the graphic (sorry it is small), you'll see that this isn't quite right. It really looks more like this:

This graph is, I think, the sum of the complete graph on eight vertices and the empty graph on four vertices. The sum of two graphs is formed by taking both graphs and connecting the vertices from one graph to all the vertices of the other.

### Imperial and Alan

Almost every day as I walk to work and pass the Imperial Barber Shop I think of Alan Turing.

## Friday, February 8, 2013

### foiling understanding

A statement about mathematical understanding in the Common Core State Standards was recently pointed out to me:

One hallmark of mathematical understanding is the ability to justify, in a way appropriate to the student’s mathematical maturity, why a particular mathematical statement is true or where a mathematical rule comes from. There is a world of difference between a student who can summon a mnemonic device to expand a product such as (a + b)(x + y) and a student who can explain where the mnemonic comes from. The student who can explain the rule understands the mathematics, and may have a better chance to succeed at a less familiar task such as expanding (a + b + c)(x + y). Mathematical understanding and procedural skill are equally important, and both are assessable using mathematical tasks of sufficient richness. (CCSS for Mathematics, page 4).

No surprise that FOIL is singled out as the example of school mathematics where procedure trumps understanding. For this particular topic, I think that using generic rectangles to visualize the distributive law is better than relying on mnemonics at all, but I'm sure there are other equally good ways of avoiding the FOIL trap. Generally, whatever the topic, the challenge is to find representations that extend existing understanding rather than applying rules without comprehension.

Posts about generic rectangles are here and here.