Larry Zimmerman is an extraordinary man, teacher, mind; one of those math teachers that infectiously inspires creativity and enthusiasm. As a teacher, he is alert, industrious, sensitive, clear, direct. He presents an entirely different model of teaching than anything I’ve seen before, and it made me wonder, if a little desperately, how my life would change if he had been my coach. He has recently retired from a long career at Brooklyn Tech and is pretty involved in the New York Math Circle (http://www.nymathcircle.org/)
My notes and quotes on him came from a workshop he taught on problem creation yesterday. My take on this part of his teaching is that it taps into students' (& teachers) metacognition (thinking about thinking, thinking about the fact that they are doing math while they're doing math), pattern seeking abilities and natural curiosity. I'm trying to approach planning by simply stating the questions I want to be prepared to ask during a lesson, as well as return to through the course of a unit. Here's an example of the list of questions that came from one initial problem (which I didn't initially even think was all that interesting.) I found the questions we came up with fascinating and pretty surprising.
“Theme”: How many distinct positive integer factors has the number 36?
“Variations” in no particular order:
- How many distinct positive integer factors has the number 37?
- List the factors of 36.
- What is the smallest positive integer which has the same number of factors as 36 (including 1 and itself)?
- How many distinct positive integer factors has the number 40?
- How many distinct positive integer factors has the number 49?
- How many distinct positive integer factors has the number 944?
- If x < y, and x divides y, can the number of factors of x be greater than or equal to the number of factors of y?
- Is there a positive integer for which the integer is less than the number of its factors?
- Can you find the number of factors of a number without enumerating them?
- Which positive integers have an even number of factors?
- Which positive integers have an odd number of factors?
- How many distinct positive integer factors of 36 are even/odd/perfect squares/multiples of 6/etc.?
Other inspired questions, not particularly related to the theme:
- Is zero an even integer?
- Is zero a factor of 5?
- Is 5 a factor of zero?
- Is 2 prime?
- What is the next even prime?
- Define a definition.
- What is the purpose of a definition?
Why do even numbers, like 8, have an even number of factors (1, 2, 4, 8) while odd numbers, like 9, have an odd number of factors? (1, 3, 9)
ReplyDeleteI like using this to get kids talking.
Jonathan
Jonathan, you mean you lie to them?
ReplyDeleteCome to think of it, no. I pose this as a true/false. The misdirection is real, though. And they do end up arguing.
ReplyDeleteOf course I follow with "Why?" and their counterexamples usually are 4 and something else, which combined with 8 and 9 usually lead to a quick conjecture, some more thought...
It's a good one.
On the other hand, lying? Most common conversations in my class, not related to math:
Me: [Something outrageously false]
Student: Really?
Me: No.
Jonathan