**NYC HOLD Honest Open
Logical Debate on math reform**

**Are
our school’s math programs adequate?**

** Experimental mathematics programs and their
consequences**

**New York**** ****University**** ****Law**** ****School**

**New York City**

**June 6, 2001**

**opening
remarks**

**Research
Into K-12 Mathematics Education**

Bas Braams

Courant Institute of Mathematical Sciences

Studies
such as the Third International Mathematics and Science Study

(TIMSS)
and the National Assessment of Educational Progress (NAEP) paint an unfavorable
picture of United States K-12 mathematics education: TIMSS shows that the
relative performance of U.S. students decreases as they progress through the
educational system, and the NAEP long term trend shows only very modest
progress since the early 1970's and shows no clear relation between this small
progress and any educational policy -- for all we know it is just a weak
correlate of general economic prosperity.

There
is no shortage of talk about mathematics education reform and plenty of
government support for reform efforts, with preference for programs that are
based on the Standards of the National Council of Teachers of Mathematics and
on constructivist pedagogy and that emphasize process and
"understanding" over ability.
Unfortunately, it is hard for a professional mathematician that takes an
interest in the matter to avoid the conclusion that on the whole these reform
efforts are only pushing mathematics education in the wrong direction.

A
practicing scientist might think that reform efforts could, should, and
probably would be guided by a respected body of research into what works and
what does not, although within such a body of research there might still be
significant differences in research focus, methodology and results. With that in mind I started looking for
appropriate research, and this letter is a little report on my search. I'll say right away that the outcome has been
entirely negative.

I have
looked for K-12 mathematics education research that has the following
characteristics.

- Large scale longitudinal study. Many pupils from many schools

followed over multiple years. The primary unit of analysis is the

pupil, not the classroom or the school. Performance data are

obtained at least once a year using broad
tests -- sufficiently

broad to avoid teaching to the test. Of course one has some

personal data on the subjects (date of birth
and gender to begin

with, and one may appreciate to have things
like ethnicity and

social economic status), and then one has a
classification of each

pupil's educational history -- the effect of
aspects of this

history is what one wants to assess. Information on the pupils'

educational history would include at least:
what curricular

material, what teaching style, what
class-level environment, and

what homework policy; there will be many
other items.

- The study looks at pupils' progress over the
years. One first

studies the data in order to obtain best
predictions for pupils'

performance at later times based on their
performance at earlier

times (and, maybe, on certain
extracurricular factors), and then

one studies how the addition of data on
pedagogical practices

modifies these predictions. Basically one asks, does the

information that the XYZ pedagogical
practice was used change

materially the pupils' predicted
performance.

- The study must be managed by a group that is
independent of any

particular educational strategy and any
particular set of

curricular materials, or the study must be
jointly owned by

different groups.

- The study must be not more than 20 years
old, not less than a

few months old, and must be relevant to
current educational

practices in the

I
don't think this is asking for anything unreasonable, given the large total
scale of efforts funded by the Federal Government and by various
Foundations. Deliberately left out of
the above list, because of the many impediments that this would place in the
way of the study, is the feature of randomized assignment of pupils to
programs. I've looked for (references to
or reviews of) studies of the above kind in places such as the following:

- Douglas A. Grouws
(Ed.): Handbook of Research on Mathematics

Teaching and Learning. A Project of the NCTM. Macmillan

Publishing, 1992.

- Alan J. Bishop, Ken Clements, Christine Keitel, Jeremy Kilpatrick

and Colette Laborde
(Eds.): International Handbook of Mathematics

Education.
2 Vols. Kluwer
Academic,

- Lorna J. Morrow and Margaret J. Kenney
(Eds.): The Teaching and

Learning of Algorithms in School Mathematics
-- 1998 Yearbook.

National Council of Teachers of Mathematics
(NCTM), 1998.

- A. Sierpinska and
J. Kilpatrick (Eds.): Mathematics Education as a

Research Domain. 2 Vols.
Kluwer Academic,

- Anthony E. Kelly and Richard E. Lesh (Eds.): Handbook of Research

Design in Mathematics and Science
Education. Lawrence Erlbaum

Publ., 2000.

- National Research Council: How People
Learn--Brain, Mind,

Experience and School. Under the responsibility of the Committee

on Learning Research and Educational
Practice. National Academies

Press, revised edition, 2000.

- National Research Council: Adding it Up:
Helping Children Learn

Mathematics.
Edited by Jeremy Kilpatrick, Jane Swafford and

Bradford Findell. National Academies Press, 2001.

I've
also looked through the Journal for Research in Mathematics Education, the
journal Educational Studies in Mathematics, and other such journals, I've asked
people of various backgrounds, and I've generally paid attention to writings on
K-12 mathematics education reform.

Finally
I'm persuaded that there is nothing there.

To be
sure, there are plenty of efforts in mathematics education research. Many of them provide results that are of
anecdotal and perhaps of inspirational value.
Many appear to be tightly linked to a particular implementation of some
reform, limiting their scientific standing.
It really looks as if all the recent

Fortunately
we still have our common sense to guide mathematics education. Unfortunately (but it would take us too far afield to discuss it further here) present trends towards
discovery-based learning and constructivist pedagogy seem as little rooted in
mathematicians' common sense as they are rooted in education research.

In
conclusion I will just mention some other relevant references. The reader looking for a more positive view
of mathematics education research can start with the earlier mentioned thick
volumes, which do try to put the best face on things. On the other hand, my perspective is in line
with "Theories That Gyre and Gimble in the Wabe" by Lynn Arthur Steen, Journal for Research in
Mathematics Education, March 1999, pp. 235--241 (a review of one of the
mentioned volumes); in "Improving the <<Awful Reputation>> of
Education Research" by Gerald E. Sroufe,
Educational Researcher, October 1997, pp. 26--28; and in written testimony by
R. James Milgram presented before the U.S. House of Representatives Committee
on Education and the Workforce, February 2, 2000. As a general reference I mention also
"What's At Stake in the K-12 Standards Wars: A Primer for Educational
Policy Makers" edited by Sandra Stotsky, Peter Lang Publishing, 2000.