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
Introduction
Lesson One
Lesson Two
Lesson Three
Lesson Four
Lesson Five


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 fiveyear 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 metalesson
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.
