Cognitive Science and Mathematics Education

By Alan H. Schoenfeld | Go to book overview

7
New Knowledge about Errors and New Views about Learners: What They Mean to Educators and More Educators Would Like to Know

Stephen B. Maurer Department of Mathematics Swarthmore College

One of the insights of the cognitive science approach to learning theory is that many of the mathematics errors students make are systematic. These errors are bugs, like bugs in computer programs, not slips.

So what? Astute teachers have known this for years (without the computer terminology). Also, astute researchers in traditional learning theory had documented this back in the 1920s.

The purpose of this paper is to answer that question--so what?--from the perspective of some mathematics educators who have had the opportunity to learn a bit about what the cognitive learning theorists are doing.

There is something new in today's statement that students make systematic errors. Briefly put (we say more later), researchers are now able to predict a large number of the arithmetic mistakes that individual students will make--before the students work the assigned problems! Hundreds of systematic error patterns have been identified. (On the other hand, more questions have been raised than answered. For instance, very little is known about the stability of these bugs; sometimes they are lasting, sometimes they self-correct and sometimes they reoccur spontaneously.)

In any event, it is not the specifics of this new knowledge which is really important to educators, not at least until the specifics are more complete. What is important is the general insight these studies give into how students learn. Indeed, "learn" may not be the best word to describe what happens; "interpret" may be better.

All told, the research brings Good News and Bad News. The Good News is that, basically, students are acting like creative young scientists,

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