“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
• 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)