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.
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