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

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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


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


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

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

In my work for the SMSG, I assisted with the National Longitudinal Study of Mathematical Abilities, a five-year project to assess the effects of new math curriculum programs. I learned many lessons from that experience, but one of the most important was that when curricula have different goals, they can be compared either on the goals they share in common, in which case important things are not measured, or on the entire set of goals, in which case each curriculum is at a disadvantage on the goals it did not attempt. If a new curriculum teaches different mathematics from that of the old curriculum, students who were not taught that mathematics cannot be expected to know it. It is discouraging and unfair to such students to attempt to go deeply into their understanding of something they have not been taught. Legitimate comparisons can only be made on common goals, which necessarily fail to capture much of what makes each curriculum unique.

That lesson seems simple enough, but it has not been learned by many who would use the results of a single mathematics test to compare the performance of students in different mathematics programs across different states or countries. The tests used for such comparisons are never optimally attuned to the goals of each program; they cannot be. Nor do they represent consensus judgments of what all students need to know and be able to do. Instead, they are essentially political compromises that reflect what administrative bodies are willing to fund as minimal indicators of student knowledge. If we want to know what mathematics our students are learning from the programs they are in, we need to use instruments that are sensitive to all facets of those programs.

Lesson Five: “Mathematics education is much more complicated than you expected, even though you expected it to be more complicated than you expected.”

This quotation comes from Ed Begle, who formulated it in relation to what he had learned about research and evaluation in mathematics education during the 1960s. I have adopted it as a kind of meta-lesson that not only covers the others I have offered but also doubles back on itself. It reminds us not to be too easily tempted by nice words about lessons to be learned from our experience.
The new math reforms, like all educational reforms I know about, accomplished neither what their supporters wanted nor their detractors feared. Instead, their major contribution, apart from preparing the ground in which current reform efforts have been planted, was to develop the field of mathematics education in schools and universities. The strength of mathematics education in the U.S. today is due in no small part to the recruitment into the community during the new math era of some of the most talented people of that generation.

Educational reform is often construed as a technical problem. The curriculum project, which came into being during the new math era and which remains one of its enduring contributions to the curriculum field, was modeled consciously or unconsciously on the scientific and military projects of World War II and its aftermath that set out to harness the atom, cure disease, or put a satellite into space. A nation that can put a man on the moon ought to be able to reform its educational system—so goes the standard claim. But changing how and what mathematics is taught to our children is not a technical problem. It is a human problem that demands an understanding and appreciation of how people work together in classrooms to learn and teach and do mathematics. What brings them there? What is important to them? How can they be helped to do their work better? What does society want them to do with mathematics, what do they want to do, and how can these be reconciled? Why should they change what they are doing? These are the sorts of tough questions that were overlooked during the new math era but that anyone who would reform education must somehow address.

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