For February 1 - Brown and Walter - Problem Posing in Math Education

The authors write about problem in posing in mathematics by focusing on what they define as "sensitivities" or categories to consider when working in a problem-solving setting and asking problems. The five sensitivities they pose (titles directly copied) are:

1) An Irresistible Problem Solving Drive: We have a (trained?) sense of wanting to find an answer to a problem, even if the problem is poorly posed and may be interpreted in multiple ways.

2) Problems and Their Educational Potential: Posing problems can lead students to pose new problems and may require (and breed) creativity in solving problems.

3) Interconnectedness of Posing and Solving: Problems often require “restructuring” (pp. 21), resulting in the ability to connect different content to a seemingly unrelated problem.

4) Coming Up With Problems: Rather than accepting a set of conditions and finding only one answer, more successful and meaningful problems can be created when students are asked to make generalizations, broadening their scope.

5) The Social Context of Learning: Collaborating solutions can help students explore personal understanding and extend their knowledge of problem solving.

I appreciated the authors’ acknowledgement of social interaction and the view of mathematics as an independent venture. While a great deal of mathematics struggling can happen on a student’s independent time, a Vygotskian method of problem solving can expose students to a variety of solving methods as well as unite students in a common effort. It makes me wonder, however, if problem posing can be open-ended without losing students along the way. For example, I posed the following problem to a grade twelve student in a physics class which had been assigned a work period while the teacher was away:

In a long hallway are 1000 closed doors. People begin to run through the hallway and to do the following: if a door is open, they close it. If a door is closed, they open it.

These people run in a very specific manner:

Person #1 changes every door.

Person #2 changes every second door.

Person #3 changes every third door.

This pattern continues until 1000 people have run through the building. How many doors remain open after the last person has run through the hallway?

This is a problem that some of my grade 8 and 9 students have successfully solved, yet they did not need the pre-requisites that the grade 12 student today thought. In fact, his response when I suggested that he use a different approach was, “Oh, so don’t do math, just solve it?” While he still did not provide me with a solution (nor did I provide him with hints beyond sketching the first few cases), he gave me some insight into his mind. I wonder if he meant that calculations are mathematics and problem-solving is a procedure, or if solving without the use of numbers is no longer mathematical? Perhaps his comment represents a hybrid of the two? I would have liked to listen to him work on the problem with a classmate. The single interaction I managed to observe was when his colleague asked him of the problem “Is it a riddle or an actual math problem?” to which the student responded, “A math problem.”

            Perhaps students decide on the nature of a question first before deciding how to answer it – and, perhaps, whether or not it is worth their time to answer it. This makes problem 4, which discussed more generalized questions, seem to be even more important. If students cannot generalize a math problem beyond the immediate scope of the question and immediately related content they are solving, perhaps it is not worth solving at all.