Understanding the Process of Mathematics Learning

I would like to set as the framing question for our discussion "How should basic research on Learning and Cognition impact the instruction of mathematics." This is clearly relevant to our group but I also propose it out of a selfish motive. I am on an NRC Panel that is suppose to inform an "out of the box" effort by the National Academy of Science to set up a (well funded) research program to impact education, the Strategic Educational Research Project (SERP). As they say provocatively at the beginning of their self description:

"Despite four decades of continuing effort, largely on the part of the Federal Government, this nation has not been able to build a system of scientific research that fuels far-reaching improvements in educational practice. This is not for lack of good minds or good work. There are pockets of significant research, just as there are excellent classrooms and schools. But the resulting intellectual capital has been too fragmented to have a marked effect on prevailing practice. We need to become better at accumulating our knowledge, extending it in promising areas, and incorporating the best of what we know in our teacher training programs and education curricula and materials. To give education research traction and to significantly enhance capacity will require new forms of organization that promote closer ties with practice; governance and management structures that create an environment for research planning that is protected from political influence; new kinds of partnerships and additional sources of funding."

They have commissioned two subpanels. The one that I am not on is devoted to the small matter of how this all is to be organized. The one I am on is concerned with "how can advances in research on human cognition, development, and learning be incorporated into educational practice." They have picked a number of areas for focus and one of these is (naturally) mathematics. In the area of mathematics they have identified a number of potential targets including whole numbers, rational or fractional numbers, and algebra. One view is being advanced that the work of Robbie Case (e.g., Griffin & Case, 1997) is a "poster child" for what can be done and the view more generally is that we know a lot about whole number from the work of developmental psychologists and can start to transition this knowledge to application. However, it is also observed that the real achievement problems for students are in rational numbers and algebra that at least in the case of algebra we need more basic research like that of the developmental psychologists on whole number.

So you can all ask yourselves how much of this analysis you believe and what the nature of that basic research would be. Ken has asked that I describe briefly my ACT-R architecture, including its brain connections, and how it might be relevant to these issues in mathematics education. My general views on the relevance of this architecture are developed in Anderson (2002). However, here I will briefly outline the model that we have developed for algebra equation solving and the imaging results obtained.

According to the ACT-R theory, cognition emerges as the result of manipulating information representations in various buffers. Essentially, there are production rules that recognize patterns of information in these buffers and request transformation of these patterns of information. We have looked at algebra equation solving by competent college students and have evidence that the critical buffers are a visual image buffer which holds a representation of the equation as it undergoes transformations (students prefer to solve these in their head) and a retrieval buffer which holds various arithmetic and declarative facts that are retrieved as part of the process of solving these equations. In this and other research we have successfully localized the visual image buffer as strongly represented in the left intraparietal sulcus and the retrieval buffer (for tasks like this) as strongly represented in the left prefrontal cortex (Brodmann areas 45/46).

A question for us all is how such research might inform us about mathematics. Dehaene (Dehaene, Spelke, Stanescu, Pinel, & Tsivkin, 1999) has argued that there are different components to mathematics thinking, some involving exact and some involving approximate reasoning, some involving visual representations and some involving verbal. We have shown in behavioral studies that, when novice students approach real-world problems that involve mathematics, they often use verbal, informal methods and with experience transition to more formal and (when equations are involved) more visual methods of solving these problems. One potential then is to diagnose how students are solving problems by what brain regions are involved.

Of course, brain imaging is perhaps an extreme of the importation of cognitive science research into mathematics educations. There are other methodologies and accompanying theoretical frameworks. The question is which are the ones that can provide the answers we want.

To provide slightly more specific discussion questions:

1. Do we know enough about the basic cognition underlying whole numbers to define a development program?
2. What do we know and what do we need to know about rational numbers?
3. What do we know and what do we need to know about algebra?
4. What can studies of brain function tell us about mathematics education?
5. What are the most promising cognitive science methods and theoretical frameworks for advancing the instruction of mathematics?
6. What would the qualities be of an adequate model/theoretical framework?

References

Anderson, J. R. (2002). Spanning seven orders of magnitude: A challenge for cognitive modeling. Cognitive Science, 26, 85-112

Griffin, S., & Case, R.. (1997) Rethinking the primary school math curriculum: An approach based on cognitive science. Issues in Education, 3, 1-65.

Dehaene, S., Spelke, E., Stanescu, R., Pinel, P., & Tsivkin, S. (1999). Sources of mathematical thinking: Behavioral and brain-imaging evidence. Science, 284, 970-974.