10 Questions/Comments on "The Art of Problem Posing"

This response consists of 10 questions/comments regarding pages 1-32 of "The Art of Problem Posing" by Stephen I. Brown.

1. I valued most highly, the idea that problems are very much context-based. The difference between a lousy problem and a superb problem (on the same topic) lies almost solely in context. And I think a student can tell/appreciate when a truely good/unique/intriguing question has been asked (as opposed to a conventional one).
2. I often see my own contextual view of problem posing and solving as being quite narrow. What are some strategies or literature that might best help broaden my contextual view?
3. I like the notion of posing an idea and not a problem. The open-endedness of it is very freeing and allows the mind to explore new possibilities and solutions. I was amazed at how many answers there were to a given idea.
4. I also thought it was very interesting, the idea that the person being asked to respond to an idea like x^2 + y^2 = z^2, was, in a sense, posing and answering their own questions and that the problem-solver imposes their own views on an idea and becomes a unique problem-poser.
5. What do you do when the a student wants the proof to an equation or principle and the explanation of the proof would be too far above that student's current knowledge level? Often times, I think math teachers are compelled to have their students take their word for it, because the beautiful simplicity of an idea is too complex to explain to the student.
6. How strongly should I be emphasizing pattern recognition, rare sequences and phenomena in amongst the standard curriculum?
7. What license do I have as a teacher to assign value/marks to a student's answer that is a unique approach/solution to the problem for which they put the wrong answer? i.e. If I want the answer 5,12,13 and I get the answer -5,-12,-13?
8. I found it very funny in the hand-shake example of the jurors, because my mind immediately jumped to the problem of how many total handshakes. I guess that is my narrow contextual perspective.
9. How far can I probe into a simple concept before I run the risk of undoing the knowledge I taught and confusing my students?
10. What IS the use of examining phenomena?