4. Messages from Cognitive Psychology

Starting in 1985, Chickering and Gamson [4] led an extensive effort, with an exhaustive review of ''50 years of research on the way teachers teach and students learn'' and a conference of leading educators and researchers, to define what works in college education. The upshot was the publication of the following Seven Principles and supporting documentation (Appendix A in [4]):

Good practice in undergraduate education:

  1. Encourages student-faculty contact.
  2. Encourages cooperation among students.
  3. Encourages active learning.
  4. Gives prompt feedback.
  5. Emphasizes time on task.
  6. Communicates high expectations.
  7. Respects diverse talents and ways of learning.

Not long after, they published two detailed inventories -- for faculty ([4], Appendix B) and institutional administrators ([4], Appendix C) -- that could be used to assess the extent to which a school, its departments, and its faculty do or do not follow these principles. One does not need an inventory to see that much of the traditional teaching practice in mathematics is not in accord with these principles. But it doesn't have to be that way. Indeed, [4] is a straightforward handbook for implementing these principles.

Research in cognitive psychology has been sending us very consistent messages for a half-century or more, but it is clear that very few mathematicians have been listening until the current decade. As Chickering and Gamson summarize,

''While each practice can stand on its own, when they are all present, their effects multiply. Together, they employ six powerful forces in education:
  • Activity
  • Cooperation
  • Diversity
  • Expectations
  • Interaction
  • Responsibility.''

Another important message from cognitive psychology research is the Kolb learning cycle ([11], pp. 128-133). The four stages of this cycle are

The ideal learner cycles through these stages in each significant learning experience. The AE stage represents testing in new situations the implications of concepts formed at the AC stage. Depending on the results of that testing, the cycle starts over with a new learning experience or with a revision of the current one. The ideal learning environment is designed to lead the learner through these stages and not allow ''settling'' in a preferred stage.

In fact, there are very few ideal learners. Most of us (and our students) have preferred learning activities and styles, and they are not all alike. This is one reason why learning experiences work better for everyone in a diverse, cooperative, interactive group. (The preceding sentence mentioned three of Chickering and Gamson's ''powerful forces,'' and the other three are there implicitly.)

In the plane of the Kolb cycle there are two primary axes, the action-reflection axis (AE-RO), and the concrete-abstract axis (CE-AC). Those axes divide the cycle into four quadrants associated with the four dominant learning styles, which Kolb describes as follows ([11], pp. 131-132):

Most people are not rooted at a single point in the learning style plane, but rather move around in some subset of this plane, depending on the task at hand. However, it is safe to say that most mathematicians spend most of their time in the Assimilator quadrant, whereas the students in a calculus class are likely to come from at least three and maybe all four quadrants. If our pedagogical strategies address only the students who are ''like us,'' we are not likely to succeed in reaching all the students who expect to learn from us.


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| Brain Research |
| Technology and Learning | Technology and Curriculum |
| Renewal in Calculus Courses | References |