Contextualized Mathematics

Should mathematics be learned within the context of a situation that creates (or at least, shows) the need for the mathematics? Do students learn better when we do? What do we want our students to gain?

Some of my recent posts might leave a person with a belief that I think mathematics should be highly contextualized. However, I definitely do not think this is true. My reasons deal with two discrete issues that connect within our classrooms — the impact of high context on learning, and identifying the goals (what we want students to gain).

The research on contextualized learning is not particularly strong at this point. Certainly, the general researchers in learning & cognition conclude that context can actually interfere with learning; this is simply a corollary to the principle that learning is improved by making the target (the thing to be learned) as visible as possible. Context can hide the big ideas. For a good summary, see this article about cognitive psychology in mathematics http://act-r.psy.cmu.edu/papers/misapplied.html — one of the best single sources I’ve ever seen.

A second component of the learning dimension is language and culture. As soon as we present a context, we make demands of our students about other knowledge … sometimes fairly unrealistic. One example I saw recently involves ‘shooting free throws’, rebounds, and ‘dunks’; another dealt with a baseball infield. Sports are not uniformly followed by our students. The same difficulty arises when we talk about projects around the house (whether it is sewing or woodworking). These language and cultural factors affect both native speakers and those who learn English as a second language, and the problems cut across economic standing as well.

The other dimension of my concern deals with our goals … what do we want our students to gain? Some people bring in contextualized learning so that students can experience ‘doing mathematics’ like a mathematician does. Other people use high-context because it motivates students. To me, both of these goals are important … however, they are not the whole story. A major goal of any math class should be to provide general tools that can be used, especially in future classes that the student needs to take, and this suggests a need to be able to transfer learning to other situations. This transfer is inhibited when a learner has not practiced a skill or process repeatedly; meeting this threshold is very difficult if we contextualize most problems. See http://jackrotman.devmathrevival.net/sabbatical2006/3%20Life%20in%20The%20Grey%20Zone.pdf, which deals with ‘how much practice is enough’.

I am especially concerned about preparing students to cope with the quantitative needs in their science and technology classes. These needs vary from the very specific context to quite broad conceptualizations, and we seldom know which mixture of needs a particular student will have. Developmental mathematics needs to deal primarily with broad sets of needs. I do not think we can limit the mathematics to that which there are contexts that the student will understand.

It would be simple to say that we need ‘balance’ in our curriculum, and this would be true. However, we should talk about what students should gain. Some of their future classes are actually after some very specific skills (such as equivalence of different forms, or dimensional analysis); others are general … almost theoretical (such as behavior of types of functions for biology). For a particular college, the needs might shift the ‘balance’ more strongly in one direction or the other.

The New Life reference material (at http://dm-live.wikispaces.com/) was developed to summarize many of these needs. I encourage you to look at the sound mathematics described, most of which can be well served by combining learning about the process along with dealing with various contexts. Needs exist for which contexts may not exist; some needs deal with theory, where context is a temporary step off the trail.

Yes, there are “what works” studies that conclude a high-context approach improves math classes. These are not proofs of a result. Like other ‘what works’ studies, there are many factors involved and only a few measured for inclusion in analysis. From a learning standpoint, the best to be expected is “no significant difference” with high context … and this would reflect a great deal of effort to avoid the known difficulties with high context.

We are preparing students for success, and their success involves a multitude of needs. Our math classes should focus primarily on general concepts, with a limited role for contextualized learning.

Developmental Mathematics Revival!

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