Reexamining the Beliefs Underlying UCSMP
Presented November 9, 2002
This talk was presented by UCSMP Director Zalman Usiskin at the Eighteenth Annual UCSMP Secondary Conference on November 9, 2002. It was published in UCSMP Newsletter No. 31: Spring 2003.
The 3rd UCSMP Secondary Conference, held in 1987, was the first UCSMP conference at which I wore the hat of director of UCSMP. I decided to write a talk for that conference whose purpose was to describe the beliefs that underlay the entire UCSMP enterprise at that time. I drafted some remarks and distributed the draft to the directors of all the components of UCSMP for ideas and comments. The ultimate product was a talk entitled "The Beliefs Underlying UCSMP". Now, 15 years later and in the 20th year of UCSMP, I think it is an appropriate time to look back at those beliefs and see where we have come, both in our project and nationally.
The 1987 talk identified 12 beliefs. They are here in the same order they were given then.
Belief 1: Mathematics is valuable to the average citizen.
By this belief, we meant that more than arithmetic is valuable to more than just scientists, researchers, and people in business. This belief has now been embraced by the broader community, by traditionalists and reformers. In the past 20 years, requirements for high school graduation and for college have increased so that high school enrollments in mathematics are at their highest level ever. Expectations at the elementary and middle school levels are also higher than ever before. People do differ strongly regarding how to reach everyone, but the goal of having "No Child Left Behind", the name of the President's plan, is widely embraced.
As requirements increase, two questions loom large. How much mathematics is valuable to the average citizen? And, even if a great deal of mathematics is valuable, should every child be required to learn a certain amount of mathematics before his education can continue?
We believe that a great deal of mathematics is valuable—the equivalent of at least UCSMP Advanced Algebra, and the statistics in our Functions, Statistics, and Trigonometry course. But we should not let poor performance in mathematics keep any student from advancing a grade or moving ahead in other parts of their education. And we should not require such knowledge by a particular age. One of the great strengths of education in the U.S. has been that we give students opportunities to catch up when they are older. We should not remove this strength of our system even though it allows students to dawdle.
Belief 2: Huge numbers of students leave high school mathematically ill-prepared for the activities they will undertake.
Judging from the statements of some critics of the NCTM Standards, you would think that we have made no headway in this regard and perhaps moved backwards. But in fact we have made extraordinary gains.
But do today's young adults know arithmetic? The evidence is that they do, despite a Brookings report that looked very selectively at NAEP items at the 12th grade and suggests that they do not know as much as students 30 years ago.
But there is a sense in which the problem of ill-preparedness is growing. The gap is widening between those who do not know mathematics beyond arithmetic and those who do, and the knowledge gap is not just in mathematics. Fifteen years ago, only 46% of 18-year-olds attended some sort of college. Today, around 68% of 18-year-olds attend some sort of college upon graduation. The percent of 12th graders who said they "definitely will" complete a bachelor's degree increased from 36% to 55% between 1983 and 1998. But do these 12th graders know the mathematics they need for college? The evidence from colleges is that most do not.
Our school culture is still holding the mistaken belief that a typical student can move from algebra to calculus in 4 years. We have 5 years in the UCSMP secondary curriculum—two years after advanced algebra—because we felt then and still feel now that only quite well-motivated and hard-working students can get from algebra to calculus in 4 years. But another reason that many students do not know the mathematics they need for college is that colleges have upped their mathematics requirements in the last 15 years. Also, some colleges think, mistakenly, that some students do not know the mathematics they need because the placement test is designed for mathematics departments and not for all the other departments that use mathematics.
The mismatch between high school preparation and college placement tests is a major problem in American education today. We should realize that this is a problem due to success—more young people than ever before wanting to go to college, and more mathematics needed for college majors than ever before. It requires that colleges change what they do at least as much as high schools change. But we also must understand that the people who are the most mathematically ill-prepared are the 25% who do not graduate with their age cohort, a quarter of the population that tends to be poor and contains a disproportionate number of Blacks and an even more disproportionate number of Hispanics, and more males than females. No system of high school tests is going to increase the mathematical knowledge of these students, because they are not in school. We need programs that stimulate students to remain in school, not testing programs that are perceived as threatening and biased.
Belief 3: We can learn from other countries.
Few people are so parochial as to assert that there is nothing we can learn from other countries. We have, in UCSMP, directly or indirectly learned much from thinking and practices in Japan, the former Soviet Union, and the Netherlands and other European countries.
By statistically combining TIMMS and NAEP, we have learned that there are places in the U.S. that perform as well as the highest-performing places in the world, and other places that perform as poorly as the lowest-performing places in the world. And, although there are exceptions both in the U.S. and elsewhere, the impact of socio-economic status (SES) on the performance of students, schools, and school districts cannot be ignored.
Why is SES so significant? One reason is the expected probability of success for students from higher-income families. If you are a student and you believe that by studying you can get a high-paying job and live in comfort as an adult, then you will be much more likely to study than another student without that belief. Furthermore, if you can do that studying without having to take a job to support yourself or your family, you will have the time to perform better. No educational policy that aims to leave no child behind can ignore the economic realities that face students in our poorest areas.
Consequently it seems that the countries we might try to learn from are successful countries in which there is a wide range of economic class.
Belief 4: A major cause of this problem (mathematical ill-preparedness) lies in the curriculum.
The political left and right tend to agree on this point. The disagreements are on what should be the curriculum, and whether the curriculum is the only cause. I shall discuss curriculum later so for now let's look at other possible causes.
One possible cause of mathematical ill-preparedness is quality of instruction. But research does not support quality of instruction as a major source of difference in performance of students, except as that quality impedes opportunity to learn. That exception is critical, because if teachers are not well-enough prepared to teach certain mathematics, then that mathematics will not be taught, and students will not learn. For instance, if a teacher is not comfortable with functions, then that teacher will not spend as much time on functions.
This is one of the reasons why over the past four years we have been working on a mathematics text for high school teachers in the High School Mathematics from an Advanced Standpoint (HSMFAS) project. Three sessions at this conference are about that work. I am pleased to report that just three days ago the manuscript was put to bed, that is, sent by the publisher Prentice Hall College Division to the printer. It can be ordered on line by going to the Prentice Hall College website (see also the article on p. 13 of the Spring 2003 UCSMP Newsletter).
Another benefit of good instruction is that students who feel they understand mathematics and who have enjoyed their mathematics classes are more likely to take advanced mathematics. Today, with so many students taking mathematics every year they are in school, this benefit does not occur until college. And a major problem is that college instruction is nowhere near the quality of middle school and high school instruction. Surely this is one of the reasons why there are fewer mathematics majors than we would like, and fewer students going into mathematics teaching than we would like. We need better mathematics instruction at the college level.
Belief 5: The existing (traditional) K–8 mathematics curriculum wastes time. It underestimates what students know when they enter the classroom and needlessly reviews what students have already learned.
This belief, which I think we at UCSMP were about the first to communicate widely, has been adopted by the mathematics and mathematics education communities. Virtually everyone now agrees that the traditional curriculum has wasted time. Acting on that belief, within the past 15 years most middle schools have changed the nature of 8th grade mathematics. Based on NAEP data, the percent of 8th graders who are taking algebra or have taken the course has doubled from 13% to over 28% in the past 20 years, and the percent of 8th graders in algebra or pre-algebra has increased in that time from 24% to 59%. This has caused a ripple effect down into 7th grade and, to a lesser extent, into 6th grade. At all earlier grade levels, the amount of mathematics learning expected of students has increased.
The desire not to waste time has also caused a ripple effect up. A full 25% of seniors taking this year's SAT exams reported having studied some calculus. Since 46% of seniors in the nation took the SATs, and good students in many states do not take the SATs, this means at least 12% of seniors and probably as many as 15-18% of seniors were studying some calculus last year.
So the UCSMP solution to the problem has been widely accepted at the elementary and middle school levels but not at the high school levels. Schools are getting more students in 8th grade algebra. But their reason for eighth grade algebra is not to even out the pace of instruction in grades K–12 but to accelerate it, to provide entry into calculus in high school. In contrast, our view is that most students should take eighth grade algebra in order to allow them to learn all the mathematics they need for college.
We are fighting an uphill battle. At the same time that some college level mathematics departments are bothered by the enrollment of too many students into high school calculus classes, the admissions departments of most colleges and universities look kindly on high school calculus enrollment.
I am not ready to give up this battle, and I am happy to report that some UCSMP-using school systems have adopted the UCSMP solution to this problem. They have strengthened their K–6 programs through the adoption of UCSMP Everyday Mathematics, and they have a significant percentage of students, sometimes 30% or more, in UCSMP Transition Mathematics or Everyday Mathematics in 6th grade and UCSMP Algebra in 7th grade. These students can take all six years of the UCSMP secondary curriculum and take BC calculus in high school.
Whereas these first five beliefs are shared by virtually everyone in the mathematics and mathematics education communities, the next belief is the first of three beliefs that are not shared by all in the community.
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.
The first sentence in Belief 6 is a variation of the law of trichotomy. No other option is possible except to ignore this technology, to say that nothing changes even though these incredible tools exist. Some years ago at this conference, I discussed these issues in a talk entitled "Paper and Pencil Algorithms in a Calculator and Computer Age". That talk appeared as an article in the 1998 NCTM Yearbook, so I will not repeat here what I said then. But I do wish to discuss this belief.
The technology world has changed dramatically since the beginning of UCSMP. In 1983, a student could pick up an appropriate calculator (a scientific calculator—there were no graphing calculators) and learn how to operate it within minutes. Today it can take many lessons to learn all that a graphing calculator can do. In 1983, I was able to buy a Lisa computer, the forerunner of today's Macintosh, during the day and use it and the accompanying software that very evening to type the next Transition Mathematics lesson. Today's word processing software is so complex that even with the experience of having written tens of thousands of pages, I still do not know all that my Microsoft Word software can do.
The effect of technology on mathematics problem-solving is profound. In 1985, the year these conferences began, I coincidentally began to write a math contest for the Metropolitan Mathematics Club of Chicago. The contest problem is designed to be accessible to a wide range of students in age and background and a month is given for people (teachers and students) to solve it. Because it is impossible to control what a student or teacher might use to help solve the problem, we have to allow calculators, computers, and books. So the problem has to be one that is not found in books and one that cannot be answered easily even when powerful technology is available. I thought the following problem might be appropriate.
Approximate the 25 primes less than 100 by a linear combination of the four irrational numbers √2, π, e, and ∅. Each approximation must be of the a√2 + bπ + ce + d∅. where a, b, c, and d are positive or negative integers from -100 to 100.
(For example, if you were asked to approximate 34, then you might use 1⋅√2 + -6π + 10e +15∅ = 34.0178... In this case a = 1, b = -6 , c = 10, and d = 15, so the approximation is off by 0.018 (rounded to three decimal places).)
The amount by which the approximation is off, always a positive number, is the error for the approximation to 34. The sum of the errors for the 25 primes is your total error. The lowest total error wins.
This seemed like a reasonable problem until David Witonsky of our staff said it would take him about 10 minutes to write a program that could find the best answer. It would take less than a second for the program to run because it would have to run through only about 16 billion possibilities. It is getting far more difficult to find problems for this contest that cannot be answered using computers.
Allow me to repeat a message from a talk at an earlier one of these conferences. We all know that some students get answers with calculator or computer technology without knowing what they are doing. Getting answers without understanding is not new. Paper and pencil also constitute a technology. Generations of students—some would claim even a majority in every generation—have learned how to do get answers to arithmetic and algebra problems with paper and pencil and have no idea whether their answers are correct or not. The difference is that with calculators and computers, one can get answers to far more questions, deeper questions, more complicated questions, and obtain understandings difficult to obtain with paper and pencil.
Surely one of the reasons that fewer students are majoring in pure mathematics is that there have been few changes in the pure mathematics major despite the changes in available technology. A bright student naturally does not want to major in a subject that does not incorporate the latest in technology. I would love for some of the mathematicians who are spending their time criticizing the school mathematics curriculum in which they are amateurs to turn and spend their time examining and updating the university mathematics curriculum in which they are professionals.
Belief 7: The scope of mathematics should expand at all levels.
Here again there is a fundamental difference between reformers and conservatives. Reformers hold the belief that mathematics should expand while conservatives have tended to think that the scope of mathematics should contract. The phrase "mile wide, inch deep", appropriately applied to get rid of unnecessary review, is also used inappropriately by some as a rationale to cut back on topics such as applications of mathematics.
We maintain, as do virtually all teachers, that a responsibility to teach all students means a responsibility to look at mathematics and the mathematical sciences as a whole and to teach from that entire body of knowledge. There is much important mathematics that is not needed for success in a college mathematics department, including many consumer applications, geometry, statistics, and probability, among others.
Belief 7 is widely supported in Europe, if the PISA study of the Organization for Economic Cooperation and Development, OECD, is any indicator. PISA stands for the Programme for International Student Assessment. Its first survey of knowledge and skills for life took place in the year 2000 and involved 265,000 15-year-olds in 28 countries, including the U.S. Thirteen more countries took the survey this year. The U.S. finished towards the middle of all countries in all three major areas: reading literacy, mathematical literacy, and scientific literacy.
|PISA||TIMMS||NAEP and NCTM|
|Numbers & Equations||Number & Operation|
|Algebra||Algebra &. Functions|
|Probability & Statistics||Probability & Statistics|
|Discrete Mathematics||Validation & Structure Calculus|
The curricular strands of PISA in mathematical literacy are quite close to our own NAEP and NCTM divisions of content, as the table shows. They are a little more distant from the TIMSS 12th grade Advanced Mathematics, as you might expect.
Besides identifying strands, PISA includes a categorization of situations and contexts. Situations are classified in order by distance from the student's life:
- personal (personal life)
- educational (school life)
- occupational (work and sports or leisure in general)
- public (local community and society)
- scientific (scientific contexts, proofs of abstract conjectures, etc.).
This is a view of mathematics literacy that we endorse, a realization that a good mathematics curriculum must recognize the links between mathematics and all aspects of life. It is the next belief from the talk of 1987.
Belief 8: The classroom should not be divorced from the real world.
There are many aspects to being wedded to the idea that the real world is important in the mathematics classroom. One aspect is the use of applications with real data and questions that might actually occur in the real world. Another aspect of the use of applications is the discussion of mathematical ideas such as estimation that are so important when mathematics is applied. Notice that the PISA study has a separate content category entitled "Estimation". A third aspect is the realization that multiple methods exist for solving most problems, and students should be allowed the flexibility in the classroom that exists outside it. Fourth is that the technology that is widely available outside the classroom should be a part of instruction. We are astonished that there are people who think that any of these is unwise, but there still are. Time is on our side, and so is most of the industrial world.
Belief 9: To make any significant change at the elementary level, we need specialized mathematics teachers.
Some of you may have forgotten, and others may not have known, that one of the first programs of UCSMP was to prepare mathematics specialists in grades 4–6. We still carry this belief, that to teach mathematics well at the elementary level requires both knowledge well beyond that level but also significant preparation time. We who are subject-matter specialists tend to forget that a typical elementary school teacher has five preps a day: reading, language arts, mathematics, science, and social studies, and many other things to do. Unless a teacher can teach the same lesson more than once, there simply is not enough time to prepare for all these subjects day after day. And unless a teacher is a specialist, it is almost impossible to keep up with the latest developments in a field.
Liping Ma's study of teachers from Shanghai and Michigan shows us how much elementary school teachers can learn if they teach only mathematics, for the majority of her Shanghai sample are those kinds of teachers. It would be interesting to find elementary schools in the U.S. where the teachers teach only mathematics and see how they would fare on the items given by Ma.
With the increasing sophistication of mathematics at all levels, we have come to a related belief for the middle school.
Belief 9: To make any significant change at the middle school level, teachers of mathematics in grades 7 and 8 should have at least a college minor in mathematics or its equivalent.
This belief supports and extends the recommendation in the 2001 report from the Conference Board of the Mathematical Sciences, The Mathematical Education of Teachers, which recommends specialized mathematics teachers at grades 5–8.
K–8 certification should not be considered sufficient for teaching at these grades. The years when students move from a mathematics experience devoted to arithmetic to one containing a substantial amount of algebra and geometry require significant changes in mathematical skills and understandings, including the following general transitions:
- from whole number to real number;
- from number to variable;
- from arithmetic to algebra;
- from properties of individual figures to theorems about classes of figures;
- from inductive arguments to deductive ones;
- from operations on two numbers to statistics with sets of numbers;
- from informal descriptions to formal definitions of mathematical ideas; and
- from a view of mathematics as a set of facts accessible through memory alone to seeing mathematics as interrelated ideas accessible through a variety of means.
In most areas of the United States, these transitions, which are critical to the understanding and doing of high-school mathematics, used to take place in grades 9 and 10 if they ever took place at all. Over the past two decades, research in learning theory and student performance and developments in curriculum have created a consensus that these transitions should begin earlier and constitute major ideas in grades 6–8.
Recognizing the importance of good mathematics at this level, we have three sessions at this conference about a new effort for teachers of mathematics in grades 5–9. We call this project MEMS, "Mathematically Empowering Middle Schools." Our plan is to write a mathematics text for middle school teachers along the same lines as the text for high school teachers that has just been finished.
Belief 10: To make significant changes in any school, teachers and administrators (and parents) must work together.
The changes that are being recommended for students require them to devote more time to mathematics, if not in the school, then outside it as homework. They require new materials and new technology. Implementing such changes cannot be done by a mathematics teacher alone, even if that teacher is as enthusiastic as one of the great teachers at this conference, Cindy Boyd, or as good as one as the other of our great teachers, Dave Aggen. Support from the entire education community—teachers, administrators, and parents as well as the students themselves—is needed.
Belief 11: Reality does not always coincide with our impressions of reality, so impartial examinations of reality are necessary.
This belief is exceedingly important to us and to policymakers in Washington. We need impartial examinations of what we do. But some people seem to ignore the first half of this belief and view impartial examinations merely as a vehicle for confirming what we already believe rather than as a vehicle for testing whether what we believe is actually true.
In this regard, we have over the past month been amassing the research about UCSMP at the request of a committee of the Mathematical Sciences Education Board of the National Research Council in Washington. Notice I did not say the research "supporting UCSMP" but the research "about UCSMP': It happens that virtually all research we have supports us, but when we do our testing we do not know for sure what will happen. That is why we test!
Our collection of studies is, in my opinion, quite impressive. It begins with studies predating UCSMP that established some of the main tenets of our program: (1) that by teaching applications along with skills, students develop the same skills with a little less effort and learn the applications; (2) that students who know more geometry entering a full-year geometry course will be more successful in the course; (3) that delaying complicated manipulative skills in algebra need not deprive students from learning a great deal about functions and related concepts; (4) that students need more than one exposure to proof; (5) that ideas from statistics can be integrated into a standard curriculum and need not be segregated to separate courses.
The collection continues with the formative and summative studies of UCSMP's first edition, and then the market research and studies that went into the second edition materials. The typical lesson that you see in a second-edition book is the 6th or 7th iteration of the lesson originally tested with students.
Our collection includes a couple of studies like our summative studies, done as master's papers or dissertations. Some we have only recently found by searching the web.
The collection ends with studies done in school districts. Some of these are comparison studies of UCSMP vs. other texts in use in the same schools. Others are studies comparing school-wide scores before and after adoption of UCSMP. And others are studies comparing the school's place in state-wide rankings before and after adoption of our texts. We know we do not receive a random or even a representative sample of such studies. We know, in fact, that schools that see large positive increases in their scores are more likely to let us know how they did. Still, they indicate that there are schools who attribute the improvement in their mathematics program to us, and such endorsements should not be ignored.
The National Research Council, though not an official government agency, does independent studies for the government. In the field of education, both NRC and the government are particularly interested in scientific studies, i.e., studies that pass muster statistically. The idea is simple: the results of the research should be replicable so that the results can be applied to make wise policy.
Belief 12 : We cannot improve education alone. We need help from the entire education community.
The education community is now being dichotomized by an administration that seems to disagree with the long-held view that education policy as well as many other policies in the U.S. should be the province of local rather than national government. We will not improve education if this dichotomization continues, and the history of our nation indicates that people will not allow themselves to be forced into doing things that they believe to be unwise, particularly if they do not have a voice in the policy.
If there were substantial agreement on what and how we should teach mathematics, then one might countenance a national policy. But this is an entirely inappropriate time to have any person or agency come down asserting there's just one way, whether that be the NSF way or the California way or the Department of Education way or the UCSMP way. The wisest schools know that even within a school one has to be flexible. The wisest teachers know that even within a class one has to be flexible.
We say it year after year because each year so many people who attend our conference are first-timers. One thing that you will see here is a variety of styles and suggestions for how to teach from some of the best teachers that we know, and no two of these teachers have the same style. And none of these great teachers will tell you to use only one kind of teaching. Teachers differ as much as students differ, and one size will not fit all. You will learn that different schools use UCSMP texts in different ways, and select students for courses in different ways. We offer guidance to schools and teachers in the professional sourcebook in each of our texts, but we do not wish to prescribe one way you must follow in using our texts. There are alternate ways to approach almost every lesson, and ancillary materials to help you in a variety of ways. We understand that local traditions, strengths of local teachers, and the make-up of your student bodies are critical factors in what you do in the classroom. Having a choice from multiple ways is the democratic way, and we are proud to give you that choice.
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