Elizabeth Fennema - Mathematics Classrooms That Promote Understanding

Mathematics is perceived by most people as a fixed, static body of knowledge. Its subject matter includes the mechanistic manipulation of a variety of numbers and algebraic symbols and the proving of geometric deductions. This perception has determined the scope of the content to be covered and the pedagogy of the school mathematics curriculum. Specific objectives, which students are to master, have been stated; the teachers role has been to demonstrate how a manipulation is to be carried out or to explain how a concept is defined; and students have been expected to memorize facts and to practice procedures until they have been mastered.

The arithmetic of whole numbers, fractions, decimals, and percents represents the majority of the first 7 of 8 years of the school mathematics curriculum for all students. For those students who successfully complete the hurdles along the arithmetic path, yearlong courses in algebra and geometry follow. Then, for many college-bound students, what follows is a second-year course of algebra and perhaps a year course of precalculus mathematics. Finally, for a few students, a course in calculus is often available. In each course, the emphasis is on mastering a collection of fixed concepts and skills in a certain order. This layer-cake structure that we have inherited is deeply embedded not only in our curricular structure but in our larger societys expectations regarding schools. This approach to mathematics education reinforces the tendency to design each course primarily to meet the prerequisites of the next course. This layer-cake filtering system is responsible for the unacceptably high attrition from mathematics that plagues our schools.

The traditional three-segment lesson, which has been observed in many classes, involves an initial segment where the previous days work is corrected. Next, the teacher presents new material, often working one or two new problems followed by a few students working similar problems at the chalkboard. The final segment involves students working on an assignment for the following day.

This mechanistic approach to instruction of basic skills and concepts isolates mathematics from its uses and from other disciplines. The traditional process of symbol manipulation involves only the deployment of a set routine with no room for ingenuity or flair, no place for guesswork or surprise, no chance for discovery; in fact, no need for the human being. Furthermore, the pedagogy of traditional instruction includes a basal text (which is a repository of problem lists), a mass of paper-and-pencil worksheets, and a set of performance tests. There is very little that is interesting to read. Workbook mathematics thus gives students little reason to connect ideas in todays lesson with those of past lessons or with the real world. The tests currently used ask for answers that are judged right or wrong, but the strategies and reasoning used to derive answers are not evaluated. This portrayal of school mathematicsa tedious, uninteresting path to follow, with lots of hurdles to clearbears little resemblance to what a mathematician or user of mathematics does. What is clear is that students do not do mathematics in traditional school lessons. Instead, they learn a collection of techniques that are useful for certain purposes. Because mastery of techniques is defined as knowledge, the acquisition of those techniques becomes an end in itself, and the student spends his or her time absorbing what other people have done.

Traditional school mathematics has failed to provide students with any sense of the importance of the disciplines historical or cultural importance, nor any sense of its usefulness. Is it any wonder that many students dislike mathematics and fail to learn it? The premise of this book is that traditional teaching and learning of mathematics has not enabled students to learn mathematics with understanding and that our first step must be to redefine mathematics.