10 Comments/Questions in Response to the first 32 pages of "The Art of Problem Posing"

How can I encourage the right line of questioning towards solving a problem? Assuming I can do so.

Does posing problems to students really allow them to explore "self-understanding"?

Was there a purpose in listing the reasons for repetition in 1,2,3,4,6,5 order?

I understand the importance of insistent and repetitive questions as they help to illuminate different nuances of problems but how does a teacher ask these questions of their students without boring or frustrating them?

How do I avoid annoying my students if I respond to a question with more questions?

It is clear that the underlying importance of education is to encourage independent thought and innovative thinking from an individual. In mathematics, the ideal would be to create lesson plans that elicit a deep-seated understanding, but how do teachers balance this depth of knowledge when there is a curriculum to follow and time constraints? Furthermore, some of the dialogues are between ONE teacher and ONE student. We expect as teachers, we will be working within a classroom with many students. What one student may question, has already been internalized by another. Asking or posing continuous questions will not benefit the student who has already come to terms with the material or the student who just wishes to complete the assignment/exam. That being said, I am totally excited to create "brain-teasers" problems for my students as challenges, I am also eager to try to push my students to take second looks at their own conceptions of the material.

After this article and some mentions of this in class, I realize that I was very interested in the personalities of the mathematicians whose discoveries I studied in math class. These stories of discovery or of the discoverers were only as an aside in post-secondary or from my own readings. It might make the topics a little more personable if the road to discovery was discussed in a highschool classroom. I will try to make a point in discussing the biographies of some of the mathematicians whose work is used in the curriculum.

It is true that given a list of tables, I find it difficult not to consider a pattern or a formula to describe the relationship between the numbers. Is this due to my mathematical training or an inherent curiosity that is just due to personality?

I agree that it is important to look at a problem and consider the possibilities prior to attempting to solve it. There is value in evaluating the approximations which will help to illuminate the strategies required for calculating a solution. It also serves to help to 'check' once a more precise solution is found. Should the solution be far away from the approximation, it could indicate that there is need for some more thought.

Is there such a thing as a stupid question?