Art of Problem Posing - Questions and Comments on Pgs 1-32.

Goal: Write 10 questions or comments to the author regarding pages 1-32.



(1) "We believe that is is necessary first to get "caught up in" (and sometimes even 'caught,' in the sense of 'trapped by') the activity in order to appreciate where we are headed." - Pg 2

I believe this is a fantastic idea for capturing the attention and imagination of the curious learner, but should we be concerned that we may frustrate the less interested students as well?



(2) "Coming to know something is not a 'spectator sport'... As Dewey asserted many years ago, and the constructivist school of thought has vigorously argued more recently, to claim that 'coming to know' is a participant sport is to require that we operate on and even modify the things we are trying to understand." - Pg 2

Thank you very much for clearly writing this statement. After reading this, I am finally beginning to understand the Brent Davis lectures, and I feel less stressed out over the first assignment.



(3) "It is worth appreciating, however, that not everything we experience comes as a problem. Imagine being given a situation in which no problem has been posed at all. One possibility of course is that we merely appreciate the situation, and do not attempt to act on it in any way... Another reasonable response for some situations, however,might be to generate a problem or to ask a question, not for the purpose of solving the original situation, but in order to uncover or to create a problem or problems that derive from the situation." - Pg 4-5

I find this statement acts as a reflection of my own life. There was a time when I questioned and thought critically about everything I experienced. From my questioning came a thirst for knowledge, but more questions arose than answers. Although I have already trained my mind to think critically and question everything, I found it wasn't until years later that I was simply able to appreciate the simplicity of certain situations, and let my mind relax. While I believe it is important to train the minds of students to question, it is equally important to train them how to be at peace. As for how to accomplish this second part, I am still unsure.



(4) "For example, exploration of the perimeters of various rectangles with area 24cm^2 by means of models and drawings, with data as recorded in [the table below], could lead to student recognition and formulation of such problems as the following: Is there a rectangle of minimum perimeter with the specified area? What are it's dimensions?" - Pg 9

This would also make a great launching activity for discussing which three-dimensional geometric shapes have the least surface area to volume ratio, and tie this into nature where soap bubbles minimize their surface area (and surface tension) by choosing a spherical shape. The student wouldn't need to know the complex mathematics and physics of the principle, just understand why this works.



(5) "x^2 + y^2 = z^2; What are some answers?" - Pg 12

Perhaps because of my physics background, I mentally solved for the f(x), f(y), and f(z) algebraically. Although I like the idea of an open ended mathematics question, I feel that further constraints are required before posing such a problem to the students... Half a page later, and I realize that this example was meant to illustrate that exact point. :P



(6) "x^2 + y^2 = z^2; What are some questions?" - Pg 14

My questions:


How do you solve for each individual variable?


Where is this formula used in mathematics?


When is this formula used in real life?

Again, I notice that my answers are different from those in the book. Because I view this formula as a tool, my first thought is "when do I use this tool?" I feel that if students can also see the relevance of this knowledge, they are more likely to be encouraged to learn it.



(7) "[Pythagorean Theorem] has to do with squares on the sides of a right triangle" - Pg 16

I am completely lost on this one. Is it a paper folding exercise?



(8) "Does the [Supreme Court Judges] handshake have an effect on subsequent cordiality? - Pg 20

I think this is a fantastic question to ask the students, but how can it be graded, since I do not know the answer myself?



(9) "If one angle of an isosceles triangle is twice another, is the shape of the triangle determined?" - Pg 21

From this question I immediately thought "Can you solve for every angle in the triangle?" Additionally, when I saw the figures below, I thought "If a triangle is made up of four identical isosceles triangles, what is the relationship between the angles of the larger triangle and the smaller ones?" This is a pretty self-evident question, but may help to make the struggling students in the class feel that they are developing a deeper understanding, and therefore open them up to further exploration in the subject.



(10) The geoboard has an appeal to the uninitiated student as well as to the sophisticated one... What would you do with it? - Pg 22

I would love to ask students how many squares can be made using the board. I recall a fellow that I did my physics undergrad with that had trouble with traditional homework questions, but excelled at puzzles like this. I think this would be a great opportunity to "level the playing field" in the classroom between the mathematically-inclined students and those that struggle with the subject.