Two Educational Views
Research on mathematical cognition indicates that learning the basics — that is, mastering the facts and procedures of the discipline — is only a small part of what learning to think mathematically is about. Other vitally important aspects of mathematical thinking and problem solving are heuristic problem solving strategies (rules of thumb for making progress when you’re “stuck”); “control” skills (having a degree of self-awareness during problem solving that keeps you on the right track, and keeps you from squandering problem solving resources on wild goose chases); and “having a sense of what mathematics is all about” — developing a mathematician’s point of view and being able to engage in mathematics rather than merely knowing about it. There is nothing special about mathematics, at least in this regard: the same is true of all problem-solving domains, including the sciences, and even writing. A mistaken focus on subject matter “mastery” alone can have some disastrous consequences; our teaching must be much broader. This talk outlined these principles in mathematics, with a few examples from other fields. Professor Schoenfeld described what can go wrong and provided a few examples of what can go right if we attend to the broad spectrum of problem solving competencies in all of our instruction.
Professor Metz’s talk explored the reasons that students have trouble succeeding in science courses: teaching scientific knowledge as a series of memorizable facts decontextualizes it and prevents students from learning scientific critical thinking skills. Taking a social constructivist perspective on learning, Metz addressed this problem by describing the relationship between content knowledge (“conceptual understanding”) and active participation in the norms of scientific culture (“engagement in disciplinary practices”). Following Brown, Collins, and Duguid, who argued in “Situated Cognition and the Culture of Learning” (1989) that knowledge is inextricable from its context and that learning is a process of enculturation in the social and intellectual norms of a discipline, Metz explained that thinking about learning in this way helps students to become independent scholars. The social cognitive approach to learning is reflected in the National Research Council’s new “Four-Strand Model of Scientific Literacy,” advanced in 2007 by the Committee on Science Learning, Kindergarten through Eighth Grade, of which Metz was a member. Metz concluded her talk by describing her own current research project, in which she is exploring the extent to which second- and third-graders can develop an understanding of natural selection, variation, and other concepts underlying evolutionary theory. Ultimately, Metz’s research demonstrates that explicitly teaching students how to participate in scientific culture as an enterprise of collaborative discovery can scaffold a far more advanced understanding of complex scientific theories at a young age than was previously thought.
Students, particularly undergraduate students, are not simply amassing facts; they are acquiring “disciplinary habits of mind” that provide them with the analytical framework and context in which to use their knowledge.
Most college disciplines involve some problem solving activity — attempts to confront situations that do not have obvious answers or procedures to find answers. Students’ abilities to handle problem solving are dependent on their knowledge base, their problem-solving strategies, their metacognition or self-regulation, and their beliefs about how the discipline is conducted.
Social cognitive theory holds that “students actively construct their knowledge; that social interaction, collaboration, and participation in learning communities are key to students learning; that academic learning is part of a process of enculturation into a “community of practice”; [and that there is] an integral connection between knowing and doing” (see Metz above).
Teaching denotative knowledge and focusing exclusively on content result in less substantial learning for students at all levels. Teaching complex thought processes and the social norms of a discipline results in more robust learning and critical thinking skills and leads students toward becoming independent scholars.
Schoenfeld, Alan. How We Think: A Theory of Goal-Oriented Decision Making and Its Educational Applications. New York: Routledge, 2010.
Key learning principles:
- Students learn more effectively when they are taught not only content but also problem-solving strategies.
Metacognition, the ability to analyze one’s own critical thinking process and to adjust it when necessary, is an essential element of robust learning.
- Students learn most effectively when they are in a continual process of actively constructing their knowledge.
Learning is a communal endeavor that depends on enculturation into the norms of a discipline.
Inspired by these talks:
Activities for the college classroom designed and implemented by members of the How Students Learn Working Group, Spring 2011. See what’s already under way at UC Berkeley to address the learning principles described in Professor Schoenfeld’s and Professor Metz’s talks.
Luke Segars, GSI
|Challenge Problem Set||Computer Science 10|
|Michael Hutchings||Understanding Limits||Math 1A|
Activities to try:
Resources for post-secondary instructors looking to implement some of these learning principles in the classroom.
Teaching Metacognition from Carleton College’s Geoscience Department (applicable to all disciplines)
Exam Wrappers from Carnegie Mellon’s Eberly Center for Teaching and Learning
Teaching Problem Solving from Vanderbilt University’s Center for Teaching
These articles explore the current state of educational research as it relates to higher education. Some articles may require access through the UC Berkeley library or proxy server.
Brown, John Seely, Allan Collins, and Paul Duguid. “Situated Cognition and the Culture of Learning.” Educational Researcher 18.1 (Jan.-Feb. 1989): 32–42.
Clauss, Jon and Kevin Geedey. “Knowledge Surveys: Students Ability to Self-Assess.” Journal of the Scholarship of Teaching and Learning 10.2 (June 2010): 14–24.
Halpern, Diane F. “Teaching for Critical Thinking: Helping College Students Develop the Skills and Dispositions of a Critical Thinker.” New Directions for Teaching and Learning 80 (Winter 1999): 69–74.