Math Points: Research-Based Pedagogy

by Jack Rotman

Note: The name reflects a normal paradox in education – we must work with objects that have no real dimensions, like a point in space. The name also reflects my belief that every reader needs to start off assuming that a writer has no real dimension; that is, assume that the writer does not know anything, and judge by the validity of the presentation whether there is something “real” to be learned.

One of the challenges each of us faces is “Over-Reacting”; an example is the reaction sometimes felt when another driver cuts right in front of us, with a clearance of 1.75 inches between their vehicle and ours. Professionally, we face a similar challenge, and this note deals with one of mine.

Over the last ten years or so, mathematics education has had an influx of pedagogies that are “based on research”. Some examples are constructivism, contextual learning, and cooperative learning; the adoption of such methods is often advocated by reference to research. In addition, methods – including these – are often suggested based simply on a theoretical basis, without reference to any evidence of their validity. My initial reaction to almost all of these has been an over-reaction that is similar to the rude-driver scenario. Even in calmer moments, the concern remains: Are these methods truly supported by research?

I’d encourage all of you to read an article written by researchers at Carnegie Mellon University; the authors are John Anderson, Lynne Reder, and Herbert Simon. The document is available on the web at http://sands.psy.cmu.edu/personal/ja/misapplied.html, and is entitled Applications and Misapplications of Cognitive Psychology to Mathematics Education. The authors (based on numerous primary research sources) review how cognitive psychology has been applied to mathematics education; they include many helpful summaries of what the research does and does not support.

Personally, I think that the research supports the use of more active learning, that provides more context for learning, and imbeds more learning in a social situation, when compared to the ‘traditional’ myth of passive, abstract, and individual (psychological) learning. I label this a myth because learning never has occurred in such an extreme environment as is often depicted in the current movements. When learning occurred in our classrooms, it involved some activity, some combination of abstract and concrete, and some combination of social and psychological environments.

The “new pedagogies” present a dichotomy that does not really exist. These methods suggest that we need to have students be “active”, as if other methods have not done that. (The programmed learning methods were fantastic at keeping students “active”; however, that was not sufficient for the method to thrive.) These methods suggest that students need a “context” to develop their understanding, as if “context” was an issue of only being relevant to that particular student. (Should our students see no connections when they do not apply to themselves? What value does this reinforce?) These methods suggest that we need to have students involved in collaborative groups during class time, as if no social support system was ever found in other methods. (Have college students studied in isolation with the “traditional” methods?)

Actually, I still have not talked about the part of these methods that bothers me the most, the reason that they elicit such a strong response: I believe that these methods, as they are being presented to practitioners, are demeaning to the abilities present in human beings. When somebody makes this suggestion:

“transfer from textbook to one’s own performance doesn’t – and can’t – work.” (D. Hestwood, of Minneapolis Community Technical College, in a presentation at the NCTM meeting, 1997.)

I get very upset. This statement is patently false; every semester, I observe students (some of whom have learning disabilities) make this transfer. It is true that the transfer requires appropriate conditions, but the transfer is, in fact, quite possible.

Another claim made is that “Schemas aren’t transferable”. Our species would be extinct if we could not transfer learning – survival depends on being able to apply old knowledge to new but similar situations. There are certainly factors in how a schema is acquired that effect its ease of transference, but it is not a ‘never’ situation.

I believe that claims are being made about “learning” that are false and that are an insult to the wondrous ability of human beings to learn. Each member of our species comes into the world with a great learning capacity, and we should not let people convince us that learning is so dependent upon certain methods and conditions. We can learn much about improving learning, if we are willing to critically review the suggestions we receive “based on research”.