Reflections on Doing and Teaching Mathematics

This chapter in the "Mathematical Thinking and Problem Solving" compilation tackles the epistemological issues questions of "what does it mean to know mathematics" and "what does it mean to do mathematics." There is a questioning of mathematical certainty and a response that mathematics, like other sciences, is actually a theory in progress; new discoveries upset old theories and are reconstructed to match our "reality."

As a result, there is a suggestion that it is foolish and socially unjust to let mathematical authority reside in the hands of any teacher. Instead, classrooms should resemble microcosms of the larger mathematical community, one in which students are doing mathematics and are the arbiters of mathematical correctness.

QUOTES
"Our classrooms are the primary source of mathematical experiences (as they perceive them) for our students, the experiential base from which they abstract their sense of what mathematics is all about." (p. 53)

"When mathematics is taught as received knowledge rather than as something that (a) should fit together meaningfully, and (b) should be shared, students neither try to use it for sense-making nor develop a means of communicating with it." (p. 57)

"The issue is the character of mathematical knowing: whether mathematicians can always be absolutely confident of the truth of certain complex mathematical results, or whether, in some cases, what is accepted as mathematical truth is in fact the best collective judgement of the community of mathematicians, which may turn out to be in error." (p. 59)

"The activities in our mathematics classrooms can and must reflect and foster the understandings that we want students to develop with and about mathematics." (p. 60)

"The means are social, for the approach is grounded in the assumption that people develop their values and beliefs largely as a result of social interactions." (p. 61)

"They have little idea, much less confidence, that they can serve as arbiters of mathematical correctness, either individually or collectively. Indeed, for most students, arguments (or proposed solutions) are merely proposed by themselves. Those arguments are then judged by experts, who determine their correctness. Authority and the means of implementing it are external to the students." (p. 62)

"I hope to make it plain to the students that the mathematics speaks through ll who have learned to employ it properly, and not just through the authority figure in front of the classroom. More explicitly, a goal of instruction is that the class becomes a community of mathematical judgement which, to the best of its ability, employs appropriate mathematical standards to jedge the claims made before it." (p. 62)

"The implicit but widespread presumption in the mathematical community is that an extensive background is required before one can do mathematics." (p. 65)

Schoenfeld, A. (1994). Reflections on doing and teaching mathematics. In A. Schoenfeld & A. Sloane (Eds.), Mathematical Thinking and Problem Solving (pp. 53-70). Hillsdale, NJ: Lawrence Erlbaum Associates, Inc.