Problems vs. exercises

[Horn, Peter]

I like the way you contrast problems and exercises, in fact I can say I wholeheartedly agree with your definition. so onto the discussion points you bring up in your original email. A good definition of a math problem is hard to come by. Because of the shear breadth of knowledge and topics available and the ways in which they differ, to try and hammer down one definition of a "Math Problem" would be foolhardy. Rather, we should attempt to discuss the depth and scope of questions posed. I feel that the higher on Bloom's taxonomy we go with our expectations as Math teacher the closer we come to challenging our students and giving them true "Problems."
Can a teacher know if a set of tasks are problems or exercises? with respect to my earlier answer, it is not whether or not a task is a problem or an exercise but rather, can the teacher know what the students will glean from performing the task? If the students are expected to strengthen their "toolset" with a set of exercises doesn't that help them with their "problem" solving? Which leads into the next topic, problem:exercise ratio and should it be the same for each student. In order to become good problem solvers, students obviously need skills which can only be acquired through the practice of skills (i.e. exercises) and since every student comes with a different "toolset" we must always look for ways to improve our students with fewer tool and enrich those with more tools. So, no the ration should not be the same for each student. This is where differentiated instruction becomes so crucial to the education of our children.

Peter Horn
Math Teacher
Dieruff High School
(484) 765-5500