# 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.

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**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.

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**WHAT’S REALLY ESSENTIAL TO
PROBABILITY?**

In one group’s discussion of what’s essential to understanding
probability, the common thread became counting possibilities in an organized
and systematic manner. The group also discussed how students struggle
to see the difference between permutations and combinations –
particularly since many textbooks do one at a time, but the tests and
quizzes put them together and expect kids to know the difference. This
group talked about the importance of showing students both permutations
and combinations at the same time.

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