Reflecting on Sputnik:  Linking the Past, Present, and Future of Educational Reform
A symposium hosted by the Center for Science, Mathematics, and Engineering Education

Symposium Main Page

 

Current Paper Sections Introduction
Lesson One
Lesson Two
Lesson Three
Lesson Four
Lesson Five

 

J. Myron Atkin
Rodger W. Bybee
George DeBoer
Peter Dow
Marye Anne Fox
John Goodlad
(Jeremy Kilpatrick)
Glenda T. Lappan
Thomas T. Liao
F. James Rutherford

 

Symposium Agenda

 

 

 

 

 

 

 

 

Current Paper Sections Introduction
Lesson One
Lesson Two
Lesson Three
Lesson Four
Lesson Five

 

Symposium Main Page

 

  Email questions or comments to csmeeinq@nas.edu

Five Lessons from the New Math Era (continued)
Jeremy Kilpatrick, University of Georgia

Lesson Two: Mathematical thinking is not bookable.

"Bookable" is a term used by publishers to describe the capability of a concept or mental process to be captured in print in a form that teachers will accept and can use. Many of the new math projects concentrated on bringing about reform primarily through the production of innovative materials. In particular, by providing sample textbooks for mathematics courses, the SMSG attempted to influence the commercial textbook publishing process, which would presumably then change how mathematics was taught. The SMSG author teams, by providing fewer problems for students to work, expected that those problems would be treated by teachers in greater depth and detail. The problem-solving process, however, proved not to be bookable. Books are not good at handling tentative hypotheses, erroneous formulations, blind alleys, or partial solutions. Teachers misunderstood what they were to do and called for more problems instead.

Max Beberman, in the new math materials he designed for the University of Illinois Committee on School Mathematics, attempted to incorporate into many lessons what he called guided discovery. Students would be led to see patterns in mathematical expressions and thus to arrive at generalizations that would not need to be made explicit in the materials (until a later lesson). The materials apparently worked well when restricted to teachers who had been trained in their use. When they were later published in the form of commercial textbooks, however, the guided discovery feature was greatly attenuated in order to capture a larger market.

Several recent curriculum projects have run into the same phenomenon. When efforts are made to encourage students to think about mathematical ideas rather than having them enshrined in a text, teachers who are not familiar with how those ideas might be handled criticize the books as incomplete and unsatisfactory. Textbooks are expected to contain authoritative rules, definitions, theorems, and solutions. Consequently, asking students to think about and formulate their own versions of these things rather than providing them ready-made can make a textbook unusable for many teachers.

Lesson Three: Teachers' knowledge is more easily changed than their teaching.

The new math era is often criticized as a time when teachers were neglected and only the production of curriculum materials counted. Forgotten are the many institutes and courses that were provided to help teachers acquire the mathematical knowledge that the new materials demanded. For the most part, however, the courses were essentially college mathematics courses retooled for teachers. They provided new content knowledge but did not address the pedagogical problems of teaching that new content. Moreover, almost no attempt was made to deal with the conditions under which teachers work that inhibit their ability to change their teaching.

A wise colleague, a mathematics teacher in Athens, once said, "I know how to be a better teacher than I am." That profound statement is true, I think, for many of us—teachers who have learned the mathematics and have thought about the techniques we might use in teaching it but who have been unable to put that new teaching into practice day in and day out. A recent study my colleagues and I at Georgia conducted of an innovative precalculus course showed how risky and troublesome it can be for teachers to make substantial changes in their practice. They do not do it without having strong incentives and substantial support from their colleagues. The new math projects, for the most part, failed to address the conditions under which teachers work.

Curriculum reform efforts in the new math era tended to concentrate on out-of-class product rather than in-class process. It is interesting, for example, that most efforts supported by the National Science Foundation, essential as they have been in promoting reform in school mathematics, have dealt with what are termed materials development and teacher enhancement. The focus has been on materials and on teachers rather than on what is taught and how it is taught. That focus, I am afraid, characterizes many of today's reform proposals as well.

Lesson Four: You can compare apples and oranges, but not on what counts.


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