10 Questions or Comments

1) From the text, 'The Art of Problem Posing', I like the line on page 3, '"The answers Gertrude, what are the answers?" - whereupon Gertrude allegedly responded, "The questions, what are the questions?"'

It reminds of a scene in the book 'The Hitchhikers Guide to the Galaxy'. Deep Thought, a super computer created by a pan-dimensional, hyper-intelligent race of beings, was faced with the task of figuring out the answer to Life, The Universe and Everything. 7 and a half million years later he eventually comes up with an answer, 42. Deep Thought then replies, "I think the problem, to be quite honest with you is that you've never actually known what the question was."

There are all these answers and facts out there but there isn't only one correct path to these answers. So what are the questions?

2) x^2 + y^2 = z^2. An interesting way to begin because it is so open ended. You could choose anything, there are an infinite number of possibilities. The book asks you "What are some answers?" and the only way to come up with any is to pose some questions your self. I simply chose some simple numbers so I could move on with the reading. After reading some of the answers written though it got me thinking about the vast number of responses, and the fact that the responses didn't have to do with math at all.

3) When I see x^2 + y^2 = z^2 I think of a circle first then Pythagoras and right angle triangles. It's a tool to get an answer. The triangles that come to mind are the standard 'special' triangles with ratios 1:1:root2 and 1:2:root3.

4) On page 14 they show a triangle with squares on each edge. I have seen this before, but I have never known the reasons, it seems pointless. How does it help? After writing this question I think I may have figured it out. Is it simply a reminder that to relate each edge length you must square each length first?

5) Do you ever give us any of the answers to the questions you pose? Some of the points brought up are interesting but I guess it isn't the purpose of the book to give answers. That's what the internet is for eh?

6) The hand shaking question on page 19. When I hear this problem I instantly think 'the number of people, choose 2'. nC2, where n is the number of people shaking hands. So, with 9 people, we get 36 handshakes.

Yes, I am one of those guys who came up with the question, "How many handshakes were there altogether."

7) It has been so long since the first time we actually learned the properties of isosceles triangles and Pythagoras, etc. It makes me wonder, how did I first learn these concepts? What went through my mind when it was introduced to me? Did I ever think of interesting questions such as "can I make a bicycle hub out of congruent isosceles triangles?" When I read this chapter today, I certainly didn't come up with any questions like that. I agree with what is written on page 22 whole heartedly, those who have been exposed to geometry "find great difficulty coming up with much more than observations relating the equality of the lengths to the equality of angles."

8) If I was given a Geoboard of 25 nails I would get an extremely stretchy rubber band and try to make the most complicated looking picture possible using all of the pins without the elastic ever crossing over itself.

9) Coming up with unique questions for these pythagoras, geoboard, isosceles and handshake problems is tough. I'm sure after teaching a few courses in math and seeing what students come up with I'll be more capable of thinking of questions, both in and out of context.

10) The handy list of questions on page 30. Add ons:
What is the origin of the formula? Is it continuous? Discrete? Is there an average? Are there visual representations? For whom is this useful to? Practical or theoretical? Can we prove it? Do we need to prove it?