## Thursday, August 28, 2014

### Section 3-1: Angles and Their Measures (Day 16)

Chapter 2 of the U of Chicago text is complete. But I've decided to lump the first half of Chapter 3 with Chapter 2, since it focuses on angles. As I've mentioned before, Chapter 3 is awfully late to introduce angles, as most texts do so in Chapter 1. This blog compromises by including angles in Chapter 2.

I don't have much to say about the book's treatment of angles. This text is unusual in that it includes a zero angle -- an angle measured zero degrees. Then again, in Common Core we may need to discuss the rotation of zero degrees -- the identity function.

The key to angle measure is what this text calls the Angle Measure Postulate -- so this is the second major postulate included on this blog. Many texts call this the Protractor Postulate -- since protractors measure angles the same way that rulers measure length for the Ruler Postulate. The last part of the postulate, the Angle Addition Property, is often called the Angle Addition Postulate. Notice that unlike the Segment Addition Postulate -- which this text calls the Betweenness Theorem -- the text makes no attempt to prove the Angle Addition Postulate the same way. Notice that Dr. Franklin Mason's Protractor Postulate -- in his Section 1-6, has angle measures going up to 360 rather than 180.

So let me include a few more things in this post. First, here's another relevant video from Square One TV -- the "Angle Dance":

Now from time to time, I read articles commenting on how mathematics should be taught. Normally, I don't post these, but this one intrigued me because it was written by Dr. Hung-Hsi Wu -- one of the major inspirations for this blog.

http://www.sfgate.com/opinion/article/Americans-stink-at-math-but-we-can-fix-that-5704332.php

The topic of this article was not high school geometry, though, but elementary school arithmetic. Here Wu argues that elementary school math should be taught by math teachers -- as opposed to teachers who hold multiple subject credentials. Elementary school teachers typically choose that career because they like little children, not because they like math -- and if elementary teachers weak in math are unable to teach it well, the students undergo a domino effect where they can't learn middle math or high school geometry, and end up in dead-end remedial classes in college.

My feelings about Wu's proposal are mixed. I can see students in the higher elementary grades such as fifth grade having more than one teacher during the day. Indeed, my own elementary school did exactly this when I was a fifth-grader -- and it later extended to third graders. But I find it especially awkward to implement such a program in kindergarten. I prefer to see five-year-old children experiencing school for the first time in a single classroom the entire day -- the teacher and her aide are like mother figures during the day. And besides, the specific example mentioned in the article for having math specialists involves the multiplication and division of fractions -- and that's way beyond kindergarten.

So I'd say that Wu has a point with respect to the higher elementary grades, but this may be tough with the lower elementary grades. There may be a way to implement Wu's plan more gradually across the grades, and I may discuss this in future posts.