“It is perhaps more important to be able to compose problems than to solve them.”
“Problem creation is the essence of mathematics.”
“The first effort is rarely if ever the final product.” (we must emphasize editing)
“Creating problems is theme and variations.”
“Determine what you are looking for and WRITE IT DOWN.”
“I don’t know what I want them to realize.”
“Beware of reinforced suspicion,” which is not a substitute for problem solving.
“If I’m 100% I will not forget it, I write it down.”
“There is nothing wrong with clarity by redundancy.”
Types of Problems
• Surprising
• Novel
• Fruitful
• Charming, imaginative, alluring
• Historically significant
• Haunting, musical
• Beautiful, elegant, sublime
Elements of Problems
1. a goal (construct, prove, maximize, minimize, classify, compare, compute)
2. given information
3. special rules (sometimes)
Some Big Ideas
- Plant a seed and then walk away until someone says something. Students and teachers struggle with silence, but it’s important and it saves time.
- Keep a notebook (both teachers and students) of interesting questions.
- We are well versed at turning a lot of words into symbols. Do the reverse.
- There are 3 varieties of equations: identities (always true), conditionals (sometimes true), and contradictions (never true)
“Problem creation is the essence of mathematics.”
ReplyDeleteThis is true at some level, but the question is which one.
Jim Yorke once told me that the big transition in high school mathematics is from specific calculations to calculation schemata. In college mathematics, it's from calculations (of any sort) to proving theorems. In graduate school, it's from proving theorems to stating theorems.
It's only at the last stage that one must gain skill at problem creation to be "good at math".
Hi Jesse!
ReplyDeleteWho's Larry Zimmerman? Much of this reminds me of George Polya (which is just fine...Polya's work should be loudly repeated, often.)
Sometimes these days I feel we are moving away from more traditional problem solving where all the required information is given up front. We're frequently encouraged to either : present a problem ambiguously and teach kids to ask clarifying questions, look stuff up, take measurements etc OR require them to sift through tons of data for the relevant, interesting gems. The approach is seen as more like encountering math 'in the real world.'
I wonder which way is better. Or if we need both.
Is this Larry Zimmerman of Brooklyn Tech fame? One of the best teachers that I've had in my entire educational career. Any ideas about how I can get in touch with him? Does he have a website? Just wanted to say hello... :)
ReplyDeleteThanks for the comment, Alex. Yes, indeed, Larry is a hero of mathematics education. I don't know if he has a website, but he often leads workshops for New York Math Circle (http://www.nymathcircle.org/faculty).
ReplyDeleteLong Live Mr. Zimmerman! Best teacher ever.
ReplyDeleteyes, Long Live Mr. Zimmerman, best math teacher ever, how do we contact him?
ReplyDelete