Date: Jan 20, 1997 8:41 PM
Author: Robert B. Davis
Subject: Definitions of "New Mathematics"

Continuing the discussion of "the new mathematics", Andrew Phelps wrote:

Dr. Davis does not have his facts straight. My father (who had been
on the faculty at Rutgers from 1946 to 1958!) was Head of the
Institutes Section of the N.S.F. in the late 60's. These textbooks
which Davis criticizes were developed in the earlier part of the 60s
as a kind of "phase I" of the *new math*. "Phase II* was to be
training the teachers of this country to teach it. My father was
in charge of funding this training. However this plan was aborted
and the result was those textbooks that the teachers did not
know how to teach (or criticize and upgrade). There was BTW
Piaget type talk all around our house!

Actually I do have the facts straight. We are only partly arguing over the
history - we are in part arguing about interpretations, and even definitions.

Where and when did the large effort to revise school mathematics arise? As
with the origins of democracy or of the classical common practice period in
European music, one can argue about where it all started, or even where it
all ended.

Among the earlier efforts at changing school mathematics, certainly the early
work by CEEB must be included, although that was mainly a matter of
manifestos and similar publications, and not a "down in the trenches" effort
to get into schools and to cause them to change. Having worked in math
departments at MIT, the University of New Hampshire, and elsewhere, when I
found myself getting interested in educational questions I realized that this
was new territory for me, and that I needed some guidance. I called people
who had worked on the CEEB reports, among them Al Meder from Rutgers. (I
myself was then in the Math Dept. at Syracuse University.) Meder was very
helpful to me and -- among other things -- asked me if I knew the work of Max
Beberman and the UICSM project in Urbana/Champaign. I didn't, but I began
exchanges with Beberman and Herb Vaughn (Math Dept at U of I), including
reciprocal visits. Beberman was then (and still) in my opinion headed in
the wrong direction with the most extreme formal notation I have ever seen at
a pre-college level (Vaughn was a logician), but I greatly admired his
determination to bring about truly serious changes in high school math. (He
later went on to work with younger children, and to try to make use of
alternative approaches, some of which he imported from England.)

Beberman's UICSM was started years before SMSG, or Sputnik, or all that. In
fact, it was partly started by Electrical Engineering faculty at the U of I
who were dissatisfied with incoming students at Illinois.

Now most people didn't know about UICSM in those days - but it definitely DID
exist, and along with some things I considered erroneous, it also embodied
some really good ideas (for example, the very high level of training and
support Beberman provided to teachers BEFORE he would ALLOW them to teach
from his books).

Through Beberman I met his colleague David Page, and through Page I met
Jerrold Zacharias, the main creator of PSSC (which also antedated SMSG).
Through them I also came to work with the ESS project, at EDC - one of the
very best of the programs in those days.

SMSG when it came along was simultaneously the new kid on the block and also
a three thousand pound gorilla. SMSG NEVER spent the kind of time in
classrooms trying our trial versions of material, as all of the better
projects did in those days. SMSG wrote textbooks DURING THE SUMMER - which
meant that necessarily they were based on guesses. When ESS, or Page, or my
own project (operaing out of the Math Dept at Syracuse University, AND KNOWN
AS (oops - I apologize for the caps - with this miserable line editor I can't
go back and fix it) ... to continue: my own project, operating out of the
Math Dept at Syracuse University, and known as the "Madison Project" [named
for Madison School in Syracuse, where we carried out trials]) ... when ay of
these projects had some ideas, the FIRST thing they did was to work with some
ACTUAL CHILDREN and see how the children understood - and perhaps
re-interpreted -- the key ideas. None of us wrote books early in the
process. Only after working with kids and teachers for some time (often
years) did we get formalized to the level of something akin to books.

But when SMSG came along, this degree of carefulness was left out, and people
who thought they had ideas went to the level of putting together books AFTER
which they ran so-called "trials". By then is was too late to reshape the
material in really basic ways.

Suppose we want to help younger children learn more about geometry. What
can we do?

In the case of ESS, or the Madison Project, or Bob Karplus's project, we
would think about many different geometry-related ideas, and see what
children did with them.

We could make graphs of children's heights, and consider how fast they were
growing (as Edith Biggs and ESS both did). We could have children make maps
of their school or of their neighborhood. We could have children design and
build furniture (as Earle Loman's project, at MIT, did - knwon as USMES, for
"Unified Science and Mathematics for Elementary Schools). We could have
children design pieces of clothing, and acutally make patterns and use these
patterns to make actual clothing (another activity from USMES, which was
funded by NSF). We could use computational differenf*tial geometry, as
Papert did, having children tell a computer how to tell a turtle how to move
around on the floor in a circle (early LOGO activities). We could play games
on Cartesian coordinates (like "battleship", or even a version of Go). We
could work with vector geometry, by using "forward one giant step" and "Take
one tiny step to the right" and so on. We could focus on inference or
implication, for example by asking children to make up examples where, if you
told someone certain statements, you would NOT NEED to tell them certain
other statements (one 10-year-old boy said: "My cousin plays in the Little
League. Only boys play in the Little League. I don't need to tell you "My
cousin is a boy" You can figure that out from the things I have told you.")
You can build on this to the point where 4th or 5th grade children INVENT the
idea of a "proof by cases" (as in proving that you have built every possible
tower 3 cubes tall, if you have blue and red cubes to work from: Every tower
must include either exactly no blue cubes, or exactly 1 blue cube, or exactly
2 blue cubes [in which case there is esactly one red cube; you can keep track
of this red cube, which must be either in the bottom slot, or the middle one,
or the top one], or exactly 3 blue cubes. And the children can - and do -
invent mathematical induction: make towers 3 cubes high by making all
possible towers TWO cubes high, and then put either a red cube or a boue cube
on top of that, etc. As one 9 year old girl figured out on her own
initiative, this meant that there must be 1,024 different towers exactly 10
cubes tall.

You can use finite geometries, etc. etc.

Now the Madison Proj