Conceptual and Procedural Knowledge in Mathematics

(Originally posted at http://joelnoche.multiply.com/journal/item/38/Conceptual-and-Procedural-Knowledge-in-Mathematics on February 9, 2011 8:40 AM)

Last night, I watched a nice talk by Eric Mazur about peer instruction.  (Click here for the video, which is around 1 hour and 20 minutes long.)  He focuses on the physics knowledge of undergraduates, but I believe that his statements also apply to undergraduates’ mathematics knowledge.  He makes some nice points, such as

The better you know something, the more difficult it becomes to teach because you’re no longer aware of the conceptual difficulties of a beginning learner. (at around 00:51:00)

My dissertation (which is still on-going) is about the relationship between conceptual knowledge and procedural knowledge in mathematics.  So I was very interested in what he was saying.  His statement that

Better understanding leads to better problem solving. (at around 01:02:00)

(gains in conceptual knowledge lead to gains in procedural knowledge) seems to be supported by many studies.

I do admire his honesty and humility.  While he says that

“Good” problem solving does not necessarily mean understanding. (at around 01:02:00)

note that he does not say that better problem solving does not lead to better understanding.  This is actually the main question that I want to answer:  do gains in procedural knowledge lead to gains in conceptual knowledge?  Many studies conclude that the answer is no.  But I think it’s because these studies use a narrow definition of procedural knowledge.

I believe that there is a way to teach mathematics that on the surface looks procedural but is actually procedural and conceptual at the same time.  I believe that the current dichotomy (conceptual vs. procedural) is producing more smoke than light.  Perhaps it’s time to see that focusing on both at the same time is better than focusing on one and not on the other.

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