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# Math Resources

## Thoughts on teaching

Here is a quick glance into some of the teaching discussions our groups have had:
- TEACHING CONGRUENCE THEOREMS – GOING UP V. GOING DOWN
- WHAT IS A POWER?
- WHAT’S REALLY ESSENTIAL TO PROBABILITY?

ABOUT CONGRUENCE THEOREMS – GOING UP V. GOING DOWN
In planning a lesson introducing students to the information necessary to determine a triangle, the group had planned a game that starts by giving all 6 parts (sides and angles) and asks students: could you make a matching triangle with only 5? Only 4? Only 3? This is the opposite direction than is often taught which usually starts with giving 1 and going up. Would 1 part determine a triangle? 2? 3? The group then discussed the difference between going up vs. going down, or advantages of one over the other in helping students understand congruence. They really thought and talked about this for awhile – finally deciding that going down had a lot of advantages – in that you probably would be more certain when you had enough.

WHAT IS A POWER?
The vast majority of books I look at are pretty poor about explaining to students why in 3^5, the power is not 5. I learned early on in my career to introduce a^b with three simultaneous definitions: a is the base, b is the exponent, the whole thing is the power. In fact, a^b is the bth power of a; but b is not the power. The more I examined this muddle, the more convinced I became that something else was protecting those of us who don’t make mistakes in matters like these. Kids are constantly encouraged to learn new vocabulary, usually defined by synonyms and usage examples. But in mathematics, it is extremely rare to come across one concept expressed by two or more words. In a^b, b is not the power because it is the exponent. Since this discovery, I started finding other examples of other ideas, let me call them meta-concepts, that underlie the very way in which mathematicians communicate. I have almost never seen these matters addressed in textbooks. Wise teachers engage a class in a discussion of these things. Lesson study may well be one of the most productive ways of finding these things.