Problem Posing in Mathematics Education

Summary
The authors, Brown and Walter, highlight the importance of problem posing in math education to improve student learning. They begin by defining what appears to be a math problem is not really a problem. Their first example shows that finding Pythagorean triples in x^2 + y^2 = z^2 is not a problem per se because it is not contextualized in any way. In contrast, their next example is a simple triangle which lends itself to problem-posing more easily when there is no other information given. Students may pose problems about this triangle like finding its total area or dividing the triangle into 3 equal parts. Later, they suggest a strategy for posing a problem as in the famous Konigsberg bridge problem where one can keep all the original information of the problem and change only the question. In fact, historically, this original problem when re-posed repeatedly led to the invention of graph theory.

Next, the authors talk about the benefits of posing problems. By engaging with problem-posing, students can see the interrelationship between posing and solving. This activity gives students an opportunity to restructure the situation described in the original problem and allows them to find new connections that they don't expect to see by problem-solving. Finally, other benefits include promoting creativity, incorporating student learning into social settings, and reducing fears and anxieties caused by problem-solving.

Reflections
I do recognize problem-posing as an essential ingredient of both teaching and learning math. It seems to me that problem-solving (like an object) and problem-posing (like a mirror image) are complementary activities that act as "opposites". I associate posing/solving with "opposites" like  multiplication/division, squaring/square rooting, differentiation/integration, and so on. I think it is easier to do the former (like walking forward) in each pair than it is to do the latter (like walking backwards). When I tutor students privately in math, they may be struggling with a math concept which prevents them from solving the problem with this concept. I have to admit that in the past, I used to be afraid to take risks when posing problems for students to highlight certain aspects of a math concept. The reason is that I was not sure whether my posed problems would be solvable, yet some of these unsolvable problems offered good teaching moments. In addition, these moments also turned out to be good learning experiences for my students. Now, although I am not afraid to take risks anymore, I intend to give my students the opportunity to pose their own problems by modifying the original problem. Through posing, as they explore ideas, they reflect on every action they take and generate new questions to improve the flexibility of their thinking. On the other hand, most of my
low-achieving students struggle with posing more badly than they do with solving because they think posing a problem is like walking blind-folded.