Knowledge 9. Understanding the Process of Mathematics Learning

The framing question for the discussion was "How should basic research on Learning and Cognition impact the instruction of mathematics." There were two parts to this discussion and three discussion questions associated with each part.

Part 1. SERP Proposal Background

The first part of the discussion surrounded the research agenda associated with mathematics that is being developed by a NRC Panel on Learning and Instruction. This panel is supposed 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. One is devoted to how to organize a new research effort. The other panel, the panel on Learning and Instruction, 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 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.

Part 1. SERP Proposal Discussion

There were three questions suggested the goals of the panel with respect to mathematics instruction:

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?

One of the issues that raised a lot of discussion concerned how we should defined what the content of mathematics education should be. It was noted that algebra has become a civil right because of indication of its connection to future earnings. However, the evidence is unclear how important algebra skills are to job performance and how much it is just a matter of credentialing. Some argued that it is a training ground for generalization skills that go beyond mathematics per se and others argued that it should be a valued part of our culture independent of its economic importance. We need more research on the mathematics that people use as adults and the degree to which mathematics generalizes to other competences.

There was also some discussion of the underlying concepts of number and the degree to which things like the number line are acquired or innate (as implied by Dehaene). What are the instructional implications of various conceptions of the number line.

Part 2. Cognitive Science Background

The second part of the discussion concerned what the role was of different types of cognitive theories and studies of brain function. As part of this discussion Anderson discussed his ACT-R architecture, including its brain connections, and how it might be relevant to these issues in mathematics education. His general views on the relevance of this architecture are developed in Anderson (2002). According to the ACT-R theory, cognition emerges as the result of manipulating information representations in various cortical buffers. Essentially, there are production rules that recognize patterns of information in these buffers and request transformation of these patterns of information. Anderson has looked at algebra equation solving by competent college students and found 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 equations 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 he has 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's areas 45/46).

There is research indicating that there are different ways of solving mathematical problems. For instance, 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. Koedinger & Nathan (in press) 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.

Part 2. Cognitive Science Discussion

The three initial discussion questions for this part of our meeting were:

1. What can studies of brain function tell us about mathematics education?
2. What are the most promising cognitive science methods and theoretical frameworks for advancing the instruction of mathematics?
3. What would the qualities be of an adequate model/theoretical framework?

These questions evoked some discussion about just what was meant by Cognitive Science. Some felt that it implied a central belief in computational machinery and denied certain methods such as those in cultural anthropology. On the other hand, others pointed out that Cognitive Science in its original conception was intended to be much broader than this although it in fact is somewhat dominated by traditional cognitive psychologists.

There was some discussion of the need to shed light on what it means for someone to understand mathematics. Many students do not appreciate the level of understanding involved in mathematics and tend to think of it as routine. They do not even recognize certain conceptual discussions as involving mathematics. Cognitive science does a great service when it parses complex mathematics done in workplace or by professionals into terms or pieces that we can look at.

Another point of discussion was the importance of including motivation in a theoretical analysis of mathematics. David Geary's (1994) view was discussed that to learn advanced mathematics, for which we are not biologically prepared, requires a supporting cultural structure.

Finally, there was a discussion of what we can learn from the study of brain function. One suggestion was that there are different types of mathematical reasoning as indicated, for instance, by Dehaene's research indicating that there are different brain regions subserving sharp calculations versus estimation. Some questioned whether this research provided evidence for innate mathematical faculties as claimed by Dehaene.

Somewhat differently, brain imaging research might indicate that different students are approaching problems in different ways. Another way to look at this is to think of brain studies as refining our view of what resources students bring to the task.

Recommendations

While it would not be accurate to say that we emerged from the discussion in total agreement the following are points on which there would be a fair degree of consensus:

1. Cognitive science, broadly construed, is making mathematics learning analyzable and visible, in part by breaking a complex process down into tractable subparts.
2. The process cannot be fully understood in isolation of its context: bridges to motivation, economics, and sociocultural context need development and must be incorporated in models of mathematical education.
3. Converging brain and cognitive evidence point to multiple strategies and subsystems that can be engaged by a given mathematical task. Data on brain activity, as well as externally observable activity such as eye movements, reveal individual strategy differences in real time. Evidence of separate representations used in exact and approximate arithmetic suggests that early mathematics education needs to target both competencies.
4. Theory, practice, and policy need to be informed by a clearer view of what it means to understand mathematics.

References

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

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

Geary, D. C. (1994). Children's mathematical development: Research and practical applications. Washington, DC: American Psychological Association.

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

Koedinger, K. R. & Nathan, M. J. (in press). The real story behind story problems: Effects of representations on quantitative reasoning. The International Journal of Learning Sciences.