This article describes a case study in mathematics instruction, focusing on the development of mathematical understandings that took place in a 10-grade geometry class. Two pictures of the instruction and its results emerged from the study. On the one hand, almost everything that took place in the classroom went as intended—both in terms of the curriculum and in terms of the quality of the instruction. The class was well managed and well taught, and the students did well on standard performance measures. Seen from this perspective, the class was quite successful. Yet from another perspective, the class was an important and illustrative failure. There were significant ways in which, from the mathematician's point of view, having taken the course may have done the students as much harm as good. Despite gaining proficiency at certain kinds of procedures, the students gained at best a fragmented sense of the subject matter and understood few if any of the connections that tie together the procedures that they had studied. More importantly, the students developed perspectives regarding the nature of mathematics that were not only inaccurate, but were likely to impede their acquisition and use of other mathematical knowledge. The implications of these findings for reseach on teaching and learning are discussed.
In particular, Schoenfeld's observations are largely predicated on "teaching to the test" of standardized finals (esp.: Regents testing in New York State), with students memorizing standardized procedures for each particular problem (including a rote repertoire of geometry proofs and constructions), and generally not being able to think through any problems outside those narrowly-formulated items.
Little did he dare imagine how much more corrosive standardized testing would be 30 years later! A colleague and I were just discussing this issue (narrow and fragile problem-solving knowledge of students) just yesterday.
Hat tip to Daniel Hast on StackExchange ME for the link.
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