Teaching mathematical definitions by examples and counterexamples reminds me of the game Zendo. I seriously think that playing this game will improve one’s ability to rapidly think of good examples and counterexamples.

I also recommend Jo Boaler’s book.

Here’s the stat (but no reference in the book) from p. 92:

In California in 2004, there was a staggering correlation of 0.932 between students’ scores on the mathematics and language arts sections of the tests used.

Just before that she had referenced the SAT-9, but it’s not clear if she’s talking about the same test.

You look at each place value in the bottom number and multiply it by each place value in the top number without carrying. Just write down the ones * ones, the ones * tens, the tens * ones, and the tens * tens. Then add down like normal.

So: (5 *5) + (5 * 30) + (20* 5) + (20 * 30)

Some kids really like this method but it’s not very efficient when you deal with larger numbers.

I even find myself using it occasionally, rather than the standard method, which is a great discussion to have with the students.

A would be answered correctly even by a student that doesn’t understand decimals (.01, .5, 7, 11.4) — just ignore the decimals and you get the right answer.

B puts everything in hundredths so the student can bypass some understanding.
Again, like A, if a student compared .60 and .45 they would get the answer right even if they didn’t understand decimals, whereas if they compared .6 and .45 they would need to know about decimal place value.

Regardless, I like her implication that getting Math teachers who know as much math as possible does not guarantee getting good teachers. Having a Master’s in math does not guarantee you would be a good teacher. You need to know how to manage the question -> response -> feedback/question cycle.

(Which is a good reminder that it’s useful to catch students doing this before it shows up on a test, so you can just ask them to explain their steps in person.)