# The Beliefs Underlying UCSMP

#### UCSMP Director Zalman Usiskin presented this talk at the opening session of the project's second annual Users' Conference, held November 7 and 8, 1987. This transcript has been edited slightly for publication.

**MY TASK IS TO GIVE YOU INFORMATION** about
the principles which drive all of us, the beliefs underlying
UCSMP.
While we have held these beliefs since the
inception of the project, two of our important tasks are to
confirm them and, if confirmed, to disseminate them. People
must share our beliefs to make or be comfortable with the
kinds of changes we recommend.

UCSMP would not exist were it not for the funding we receive from the Amoco Foundation. From the start, Amoco guaranteed six years of funding at a high level-over $1 million a year. It is testimony to the strength of the components that each of them now has funding separate from Amoco. The Carnegie Corporation of New York has given us $1 million to support the development and evaluation of some of the secondary component texts. The National Science Foundation supports elementary teacher development and some of the translations of the resources component. The General Electric Foundation supports the final year of the secondary curriculum. And the General Telephone and Electronics Foundation has given money for the development of materials in grades K–2.

Why do we get so much support? For the most part, because each of these funding sources shares our basic beliefs. So let me turn to them.

## Belief 1: Mathematics is valuable to the average citizen.

**Most of you are the believers,** the converted or the
never doubtful. You can be misled by the Gallup Poll on
Education, which indicates that people say mathematics is the
most important subject in high school, to think that everyone
shares this belief. The plain fact is that for many people only
arithmetic matters, and the only reason they think you should
learn algebra, geometry, and all that other stuff is either that
it's on college entrance exams or it's good for students to
study something so hard.

We know mathematics is valuable to scientists and to virtually all researchers. We also know mathematical ideas are ubiquitous in business and economics. But we believe that mathematical knowledge and thinking are important beyond these technical areas valuable to the average citizen, the typical laborer, the voter.

Our evaluators visit both classes using UCSMP materials
and comparison classes. I think it is fair to say that when they
go into schools, our evaluation staff members are often
surprised to see how sterile, how banal, how repetitive, how
*boring* traditional curricula are. Yet mathematics is so
wonderful that, despite this miseducation, many are still
turned on by it. But so many are turned off.

Our second belief connects the situation in schools today to the first belief.

## Belief 2: Huge numbers of students leave high school mathematically ill-prepared for the activities they will undertake.

**Specifically, the average college-bound student**
leaves high school with a knowledge of mathematics
insufficient for college. The average non-college bound
student leaves with a mathematics background insufficient
for advancement in most jobs. But mathematics is too
valuable for the average citizen to be content with a situation
which denies mathematics to so many, or which turns off so
many from it.

We base this belief on some alarming statistics. According to the National Center for Education Statistics, in 1982 about 46 percentof 18-year-olds went on to two-year or four-year colleges. Three-quarters of college majors require mathematics: about a quarter in business, about a quarter in engineering or the physical sciences, and about a quarter in various social sciences or education. But, according to the Second International Mathematics Study (SIMS) conducted in 1981–82, only 13 percent of 17-year-olds were enrolled in what we would call senior-level mathematics.

Simply put, the number of students who take four years of mathematics is about the same as the number of those who go into engineering or the physical sciences. Thirty years ago, before the information explosion in business and the social sciences, this might have been enough. Now it is not enough.

**Furthermore**, the mathematical knowledge – even of
this elite – is not high enough. Even our advanced placement
students do not perform well when compared to their peers in
other countries. SIMS found:

In the U.S., the achievement of the Calculus classes, the nation's best mathematics students, was at or near the average achievement of the advanced secondary school mathematics students in other countries. (In most countries, all advanced mathematics students take calculus. In the U.S., only about one-fifth do.) The achievement of the U.S. Precalculus students (the majority of twelfth grade college-preparatory students) was substantially below the international average. In some cases the U.S. ranked with the lower one-fourth of all countries in the Study, and was the lowest of the advanced industrialized countries. (The Underachieving Curriculum, Stipes Publ. Co., Champaign, IL, 1987, p. viii.)

If 46 percent of 18-year-olds attend some sort of college, then 54 percent do not Half of these non-college bound students have never taken algebra, and virtually 90 percent of them have taken no formal geometry. Evidence from studies we had done here in the middle 1970s and early 1980s lead us to believe that students who enter non-honors courses in algebra and geometry, as these students customarily do, know very little if any of the subject matter in these courses.

**In other words,** *after 10 or more years of schooling,
half of our population knows nothing more than arithmetic.*
And when we examine the arithmetic those students do know,
we become even more dismayed. The worse the students
perform, the more time they are likely to have spent occupied
with arithmetical computation that can more easily, more
accurately, and more quickly be done with a calculator.

I detail this problem for an important reason. If you cannot get your colleagues to agree there is a major problem, you will not get them to join with you in work ing towards a solution.

## Belief 3: We can learn from other countries.

**Many people are bothered by the very idea** of
discussing the situations in other countries when trying to
look for solutions to American problems. The rugged individualism
of the United States, our history of isolationism
before we enter world wars, our success in new inventions and
Nobel prizes, our high standard of living – all of these lead
some to question the motives of anyone who sees too much
wrong in our system and who looks elsewhere for ideas.

Let me clear the air. We are still an exceedingly rich country. But we now import more than we export, keeping our unemployment rate higher than it should be. Foreigners own a growing percentage of companies and real estate in the United States. Eighteen years ago we put men on the moon; today, as I speak, we have not sent anyone into space for 18 months. We are a great nation but we have become complacent.

At one time the United States had far more students graduating high school than any other country in the world. In 1964, 77 percent of the age group ultimately graduated high school; in Japan it was 57 percent. But, by 1982, while our percentage had risen to 82 percent, Japan's was 92 percent. Other countries are catching up to us and some have surpassed us.

However, and this is a point seldom made in the media, we have by far the greatest college and university system in the world. It is not by chance that so many foreign students come here to study, both at undergraduate and graduate levels. The opportunities for study after age 18 in other countries do not match ours.

**Furthermore**, in many countries, including Japan,
Great Britain, the eastern European countries and Taiwan, the
educational systems are pressure cookers. Large numbers of
students take national exams in order to qualify for a small
number of spots in the universities. We do not wish to emulate
that kind of situation. This tells us that if we can improve
American pre-college education, then we will have the best
educational system *at all levels*.

UCSMP began because Prof. Izaak Wirszup was sounding the alarm. Our pre-college educational shortcomings were real and could lead to disastrous results for our nation, he knew.

Izaak had been translating materials from other countries
for over 25 years when the project began, but he had never
translated the texts actually used by students. Since 1983,
UCSMP's Resource Development Component has translated
a large number of texts from the Soviet Union, Japan, Hungary,
and Bulgaria. In a few months, we will have them all on
display in Lillie House as part of our International Mathematics
Education Resource Center. [See the article about the
opening of the resource center, p. 2-*Ed*.]

**We have learned from the translations** that other
countries do not even follow certain practices we consider
universal or immutable. For example, no country seems to
spend as much time on triangle congruence as we do. We see
other things done more poorl y and some things done better.
Just as American football is distinctly American, but some
time ago the teams recognized that recruiting soccer-style
kickers might have benefits, so it is with UCSMP. We are not
looking at other countries to copy their practices, but to
enhance what we do.

In the past, UCSMP has also provided some funds for the
research of Jim Stigler, who has worked with Harold Stevenson
at the University of Michigan in comparing the performance
of elementary school students in Taiwan, the United
States, and Japan. Jim also compared Soviet and United
States elementary school textbooks. The comparisons, either
of performance or of textbooks, are devastating. The Soviets
introduce a large number and variety of multi-step problems
in their grades 1 and 2, equivalent to our grades 2 and 3, while
we are preoccupied with single-step problems. It seems that
we never recover from this delayed start. Taiwanese and
Japanese 5th graders are so good that their *worst* classes
perform better than our *best* classes.

## Belief 4: A major cause of this problem lies in the curriculum.

**I have already mentioned the SIMS report.** No
one on our project was involved in this report, but it so
substantiates what we are trying to do that you must read it.
The SIMS study examines 8th and 12th grades. The report
notes five explanations commonly heard for American
mathematical deficiencies: classes are too large, teachers
spend too little time on mathematics, teachers have too little
preparation, the United States has a greater percentage of
students taking mathematics at these levels, and the quality of
mathematics instruction is low.

The SIMS researchers dismiss all of these explanations as not valid, at least at 8th and 12th grades. American class sizes, teacher preparation, time spent on mathematics, the percentage of students taking mathematics, and the quality of instruction all lie in about the middle of the range found in other countries. But the average performance of U.S. students does not. It is lower. The SIMS researchers conclude that the cause is the curriculum. This supports a conclusion we at UCSMP had before we began the project.

## Belief 5: The existing mathematics curriculum wastes time. It underestimates what students know when they enter the classroom and needlessly reviews what students have already learned.

**In the early 1980s, Max and Jean Bell** conducted
a study of what kindergarten and first grade children knew
upon coming to school. They conducted their study in city
schools, not in the suburbs. One thing they found was that
most students could count past 100. The children certainly
didn't learn this in school; in fact, 1st grade books stop at 99.
Just three weeks ago I received a phone call from a parent of
a 2nd grade child in a private school; she had been dismayed
because the school told parents not to go past 99 with their
children until 2nd grade. A school official had said, "Otherwise
what would the students learn there?" The scene is too
often repeated: "We can't teach that because then what would
next year's teacher do?" Max has been working on a primary
curriculum which takes advantage of what children know
rather than neglects it.

For the past four years, we have supported some of Karen Fuson's work at Northwestern University in Evanston, IL. Karen's work substantiates the beliefs of Max Bell, Izaak Wirszup, and Jim Stigler. Her first and second graders can perform the multi-step problems of the type found in foreign curricula. They can even do far more symbolic manipulation than most people think.

**American expectations are low at higher grades**
as well. To solve the problem, one school district in the
Chicago area just this year decided to adopt books for grades
one year above grade level. That is, the district adopted 2nd
grade books for grade 1, 3rd grade books for grade 2, and so
on. While there has been the need to supplement curricula in
the early grades, there has been almost no such need in grades
5 and 6. Those years start with so much review of earlier
material that the teacher can use the book for the higher grade
and feel that it is just as appropriate.

That district is very affluent, but if one can move one grade up there with no preparation, in other places one can certainly move up at least by portions of grades. Ruth Hoffman of our elementary component advisory board tells of walking into a school in inner-city Denver and asking the 4th grade teachers to start at page 180. The teachers, forced to do what she requested due to an agreement they had made, were nevertheless very reluctant to do so. By the end of the year, they all thought they had never had classes learn so much.

**The needless review has been documented** by
Jim Flanders, one of the many doctoral students who work for
UCSMP. I'm sure that many of you have read his article
entitled "How Much of the Content in Mathematics Textbooks
is New?" in the September, 1987, issue of *Arithmetic
Teacher*. It shows that there are places where improvement
can occur in the mathematics curriculum with relative ease,
and without changing content at all. Grades 2, 6, 7, and 8 are
particularly vulnerable. Do not be misled by the rhetoric that
these years "consolidate" or "review" or "bring together"
skills and concepts of previous years. For most students, that
is equivalent to doing nothing.

Some publishers have special books for grades 7 and 8.
We too have special books, beginning with *Transition Mathematics*
at grade 7, but the big difference is that we believe any
student who is at or above grade level at 7th grade and
successfully completes this course is ready for algebra in 8th
grade.

There are those who would call what we recommend "acceleration." It is not; we are trying to avoid forced deceleration. Aside from Canada, no other industrialized country waits as long as the United States does to give students concentrated work with algebra.

If you postpone learning, most people feel, nothing is
lost. They do not realize that *time* is lost, and that time is all
we have to work with. But we lose more than time. When a
student does not learn anything new, boredom and disinter-
est-attitudes which lead students to believe there is little else
to learn – set in. Would we have students read the same story
two and three years running? Yet American curricula do this
with mathematics, sometimes as many as *four or five* times.

## Belief 6: Calculators and computers render some content obsolete, make other content more important, and change the ways we should view still other content. New technologies also present new possibilities for instruction.

**Technology has already changed the
workplace.** I wrote this talk on my Macintosh. I have had
a computer on my desk for only four years this month, since
the beginning of this project. Still, even in that short time, I
have become wedded to it. I cannot believe I used to write
talks while sitting at a typewriter.

I do not do long division or long multiplication any more.
This is not because I do not do arithmetic; I have more need
for multiplication and division now than I ever have. It is not
because I *cannot* do them or because I am poor at paper-and-pencil
math; quite the contrary, I am very good at it. Rather,
it is because I have the spirit of a mathematician – I am lazy
and I always look for better algorithms. I have found one. It
involves pushing a few buttons on my calculator.

Hand-held calculators, which first appeared in 1971, are here to stay. Long division, which in its present form fust appeared in the late 1400s, is dying. It is no longer in the tested curricula of several industrialized countries, including Sweden and England. Contemporary textbooks are devoting less and less attention to it, and for good reason. If you want accuracy and speed, use your calculator. If it is not available, go to the next room and find it. If you do not need accuracy, use estimation schemes.

**At the high school level,** Iogarithms for computation
have died, though on occasion I hear of some math teachers
who conduct seances, trying to revive the dead. Trig tables
are dying. In their place are scientific calculators, which in
effect give you a book containing more tables than you ever
dreamed you would have access to – tables of cube roots, nth
roots, factorials, the inverse trigonometric functions, all
powers, and so on.

Recently we also have begun to see function-graphing calculators. The hand graphing of any but the simplest functions is in its dotage, on its way to the grave. And manipulative algebra will be the next big thing to go. The time to get rid of it completely is not here, despite earlier optimistic claims of many – including me – about muMath or the HP-28C. But it seems close at hand, within a decade or so in the marketplace.

**We believe these developments force us** to
rethink the educational experiences of the current student
population. Today's kindergarteners will graduate from high
school in the year 2000. If we wish to prepare our students for
the year 2000, we can't wait until 1992 or 1995; we must
change now. Today's high school freshmen, 14 years old in
1987, will graduate from college in 1995. If they go to work
immediately following college, they will spend six years of
their working lives in the 20th century and 38 years – over 85
percent of their careers – in the 21 st century. We cannot be
content with a curriculum which does not prepare them for
that.

In UCSMP's Secondary Component, we have a course,
*Functions and Statistics with Computers*, that true to its name
requires computers. As far as we know, it is the first course
in a full curriculum for average students that does so. We did
not develop this course because we wished to satisfy the
computer zealots. We devloped it because to study functions
without computers, or to do statistics without computers, is as
dumb as doing arithmetic without calculators. We believe the
future will prove us correct.

## Belief 7: The scope of mathematics should expand at all levels.

**Again and again people ask me** about projects like
this dying. What ever happened to the new math? they ask.
Even my colleagues, well-versed and well-read, will say, "I
guess new math had an effect in high school, but not much in
elementary school."

They are wrong. There was at least as big an effect at the elementary level as at the high school level. In the 1950s, arithmetic was the ideal elementary curriculum. In the 1960s, mathematics became the ideal. The titles of the books changed and so did their content. Of course it wasn't all taught, but it was there. Now, one generation later, we are seeing the fruits of those labors: some algebra and some geometry on state tests, in guidelines, lots more measurement in the primary grades, and much more variety. Without the road paved by new math, the current movement would be much more difficult, if not impossible.

In Illinois, we have recently instituted state exams at grades 3, 6, 8, and 10. These exams required objectives. The objectives fall into seven categories which can be described quickly: arithmetic, ratio and proportion, measurement, algebra, geometry, statistics, and estimation and problem solving. Of all these, statistics causes the most fuss. High school teachers in particular do not see where the time to teach the statistics will come from. Of course, UCSMP has a solution, but that is not the point I wish to make. The fact is that the expansion of mathematics at the high school level is not something everyone believes in.

**However**, people outside the mathematics education
community support this broadening. In Illinois, the state
objectives came from the legislature, and the oversight panel
included people from all walks of life. At various times, the
oversight panel wondered why the Pythagorean Theorem
should be an objective for all students, and why all students
should know the quadratic formula. But this lay panel never
questioned the need for knowledge of statistics. They understood
the reality of today's world: statistics are everywhere
and statistics should be a part of everyone's education. They
understood the need for technology and, frankly, did not
question calculators as long as other things were not neglected.

But, at the college level, we have a large problem. College entrance requirements and the entrance exams given by individual colleges and universities are controlled by people who are responsible for teaching calculus in mathematics departments-r, I should say, departments which used to be called "mathematics." Now they are likely to be known as departments of "pure and applied mathematics," "mathematics and computing," "mathematics, statistics, and computing," or the broadest phrase, "mathematical sciences." The people who control the exams forget that years ago their calculus course was considered inappropriate for the business majors, or the biologists, or the social scientists; that other departments had to start teaching statistics because the mathematics departments taught statistics without applications; and that computer science has caused high enrollments in new math department courses – in discrete mathematics, in recursion theory, and so on.

**Fortunately, colleges adopt books** every semester
or every year. The colleges can change quickly. I believe they
will change when they find that virtually all their students
have function graphers and symbol manipulators. Then, the
colleges will see that the world has changed. And, if they have
any belief like the next one, they too will change. Colleges
now house applied mathematics, statistics, discrete mathematics,
operations research, and so on – what we call the
mathematical sciences. The K–12 curriculum of the future
inevitably will cover the mathematical sciences. UCSMP is
offering steps in that direction. Like new math, it will
probably take more than a generatio n to actually be implemented,
but, also like new math, there are those who will
change immediately and be the better for it.
The biggest weakness of new math was the divorce of the
classroom from the mathematics found outside the classroom.

The biggest weakness of new math was the divorce of the classroom from the mathematics found outside the classroom.

## Belief 8: The classroom should not be divorced from the real world.

**As most of you know,** there is a national board for
mathematics education, under the auspices of the National
Academy of Sciences, which is attempting to gather information
and set directions for the nation . This board is not called
the Mathematics Education Board. Rather, it is called MSEB,
for the Mathematical *Sciences* Education Board. This name
recognizes that mathematics now permeates virtually every
endeavor. This is the real world.

The real world involves more than just different or expanded content horizons. UCSMP is committed to technology because we believe students should be taught to do problems as adults do them, and not be asked to go through torturous work simply because there is a long way to get an answer. In the real world, solutions arise from a variety of methods. Mental work is used. Estimation can be found at all stages of the solution process. Addition doesn't occur only in the addition chapter in a textbook. Algebra doesn't just occur in algebra.

However, many of us did learn that algebra occurred only in algebra, that even area and volume in geometry had no application to the sizes of containers or how much they could fill. Expanding conceptional horizons may be easy for some teachers, but others need quite a bit of education. This is particularly true in elementary schools.

## Belief 9: To accomplish any significant change at the elementary level, we need specialized elementary mathematics teachers.

**At the elementary school level,** teachers provide us
with three simultaneous problems. First, they are well prepared
to teach reading but are often untrained in mathematics.
Many have had only one course dealing with what
math they do teach, a course almost solely devoted to arithmetic.
It is no wonder that on international comparisons the
United States does *not* come up short in reading. But the kinds
of changes we and others feel is necessary in mathematics
cannot be accomplished with what these teachers currently
know.

Second, many who desire to become elementary school teachers want to work with kids but do not want to deal with complicated mathematics or science. Often the elementary school teacher's grade level preference is determined by a feeling that mathematics at any higher level would be too much to handle, or, worse, by a negative feeling towards all mathematics.

Third, even if the teacher knows and likes mathematics, that teacher cannot be expected to keep up with developments in mathematics as well as all the other areas she or he has to teach. The mathematics important for students to know is changing. A teacher without a strong allegiance to the subject cannot be expected to be aware of all the new developments in it.

**At UCSMP, we believe mathematics** should be
taught by mathematics teachers from at least as early as 4th
grade. In the city of Chicago, where still about half of the 7th
and 8th grade classes are taught in self-contained classrooms,
we hope to be one of many catalysts for change. For grades
4 to 6, Sheila Sconiers of our Teacher Development Component
has devised a way of converting teachers into subject
matter specialists without increasing a school's staff. These
subject specialists then can guide other teachers in their
school or district. We are creating the written guides for them
and for their K–3 colleagues.

This kind of change points out an important belief too often ignored by teacher education or curriculum projects.

## Belief 10: To make significant changes in any school, teachers and administrators must work together.

**At all Ieve Is of this project we ask for** substantial
change: more use of calculators and computers; a substantially
different curriculum; algebra one year earlier; specialist
teachers in the elementary grades; new textbooks; and so on.
We are not looking merely for a statistically-significant
improvement in performance at the .05 level on some standardized test. *We are asking for substantial changes in what
students experience as they learn mathematics in schools.*

Lasting change requires that the entire school agree to the goals of the change, and that all personnel work together to accomplish it. We have had administrators contact us to get us working in their districts before they contacted their teachers; that seldom works. We have had enthusiastic teachers tell us that certain things could be done in their schools, and who then had their best efforts thwarted by administrations busy with other tasks. Lasting change requires both top-down and bottom-up problem solving; it requires that all involved in the education process agree on basic principles and on a direction for movement.

I might have added "parents" to this belief. Parents are an important voice and, if they are oppose a change, they can effectively destroy almost any program in any school. The mood today is in our favor; most parents want schools to expect more from students. We need to harness that support because, if the pendulum changes as it seems to about every dozen years, when the complaints that schools are too tough come in there will be no one to defend the new status quo.

**To change**, all parties need to be convinced not only
that the principles are sound but that the execution is also
sound. Lousy materials can thwart good ideas. An idea is
only as good as the instructional technology (the materials,
the software, and so on) which supports it. We use evaluation
not only to keep us honest but also to provide evidence about
how we are doing. Another belief on my list motivates this.

## Belief 11: Reality does not always coincide with our impressions of reality, so impartial examinations of reality are necessary.

**We have, since the inception of UCSMP,** allocated
about 15 percent of our funds to evaluation. Evaluation
helps us improve and gives potential users and others some
ideas about what to expect from UCSMP programs. Also, we
are all biased by our beliefs, so we need evaluation to check
out those beliefs impartially.

There is still another reason for evaluation. Without evaluation, our beliefs would be tested by the experiences of only the few schools and districts with which we are personally involved. We need school districts we do not know. We need school districts far from the Chicago area. Many schools have been involved in one or more of our studies. We expect to continue these studies because without them we might as well pack up shop.

**These studies have confirmed the obvious.**
Teachers and schools differ. The same ideas do not work in
all places. Teachers change behavior only after much contemplation.
However, the studies also confirm what is not so
obvious. Teachers and administrators almost universally
view calculators as wonderful for the classroom, the studies
tell us. Also, the studies say students are capable of much
more than we expect from them, and will read mathematics,
but not automatically. And at the junior high level, educators
can produce achievement gains in a wide variety of areas
without giving away gains in other areas. But at the high
school level, where expectations are higher, you must give to
get.

I should warn you that, even within UCSMP,someofthe evaluation conclusions are controversial. Professors differ, and even they are swayed by their beliefs. However, if professors were not swayed by their beliefs, this project would not exist.

## Belief 12: We cannot improve education alone. We need help from the entire education community.

We wish to entice teachers and administrators into adopting or adapting the ideas that we espouse. To do so on a large scale requires far more than the resources of this project. We recognize that professional organizations like NCTM and its local affiliates, states and state supervisors, the various people who compose the standardized tests students take, and commercial publishers must also be allied if we are to make substantial improvement. We are pleased that again this year we have anum ber ofrepresentatives from publishers here.

Well over a year ago we told publishers that we were ready to engage in negotiations to publish the Secondary Component texts. Five publishers submitted proposals to publish these materials. Many rumors regarding what is happening have circulated; let me explain exactly where the negotiations stand.

It's easiest to proceed by analogy. When a house is about
to be sold, you may see at the front of the house the sign "under
contract." That means that a deal has been tentatively set, but
the closing has not occurred. So it is between UCSMP and
Scott, Foresman and Co. regarding the publication of the six
books of the Secondary Component. If all goes well, some of
our secondary level books will be commercially available for
the 1989–90 school year. *Transition Mathematics* will definitely
be ready, but we are not certain which of the other texts
will be. That is all I can report now. Until then, our books will
continue to be available directly from the project.

### Contact

UCSMP

1427 East 60th Street

Chicago, IL 60637

T: 773-702-1130

F: 773-834-4665

ucsmp@uchicago.edu