If Everybody Counts, Why Do So Few Survive?
UCSMP Director Zalman Usiskin presented this talk at the opening session of the project's Fifth Annual UCSMP Conference, held November 11 and 12, 1989. This transcript has been edited slightly for publication.
FOR SIX YEARS NOW the scope of the University of Chicago School Mathematics Project has been such that we have dealt with all the grades, from kindergarten to grade 12. The complexity of our enterprise mirrors the complexity of the task of school mathematics. The subject of my remarks today is as difficult a problem as any we face and, perhaps for that reason, there is very little discussion of it.
Only a small percentage of our students survives its school mathematics experience. What percentage depends on one's definition of survival. For instance, NCTM has, for the entire decade, said that three years of high school mathematics should be required of all students. If we view as survival that these three years should include the equivalent of two years of algebra and a year of geometry, then according to National Assessment data about 47% of students survive. But National Assessment calls the second course in algebra by the title Algebra II, which in many schools is the title of the second semester of first-year algebra. Also, some students take second-year algebra before geometry, and so the "highest level taken" is somewhat misleading. Furthermore, already at this age some students have dropped out of school. With all this, I would estimate that the percent of survivors, those who take three years of secondary school mathematics, is no more than 40%.
Regardless of the accuracy of the figure, the number of survivors is increasing. The same item has been on the last three National Assessments, and there has been about a 5% increase in thenumberof17-year-oldsreportinghavingtakeneithersecond- year algebra or precalculus or calculus. The message that more mathematics is needed is getting around.
But in many countries of the world, the equivalent of two years of algebra and a year of geometry are completed by the end of l0th grade, when almost all students are still in school, and when almost all students are taking the same, required, mathematics.
From recent Bureau of Labor statistics and unemployment data, using the age cohort of the class that graduated high school in 1988, best estimates are that 48% of the population now goes to college. Since at most 40% take a second year of algebra, at least one-sixth of today's entering college students, 8% out of that 48%, have not had three years of high-school mathematics. Among the other 52% are huge numbers of students who have had no algebra or just a year of algebra, no geometry, no statistics, no probability. They are simply unprepared for the mathematics they will encounter on almost any job.
The National Assessment data are the most optimistic data available regarding survival. Other statistics show either that there is quite a drop-off after second-year algebra, or that the National Assessment data are way off. The Second International Mathematics Study, SIMS, found that only about 12% of the age cohort in the U.S. is enrolled in precalculus or calculus classes as seniors in high school. This is another estimate of our survival rate. That 12% is about the same survival rate as in Japan and many other industrialized countries.
The highest survival rate in the world seems to be in the province of British Columbia, where 50% of the age cohort takes the standard mathematics curriculum as seniors. The population of British Columbia is about the same as the population of Chicago, a little less than 3 million, but in Chicago less than 50% of the age cohort graduates high school and the percentage who have taken senior mathematics is probably less than 10%. The survival rate in our inner cities is comparable to the survival rate in third-world countries.
And even our best survivors do not perform so well. Recall what SIMS had to say about the performance of our seniors:
In most countries, all advanced mathematics students take calculus. In the U.S., only about one-fifth do....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. The achievement of the U.S. Precalculus students (the majority of twelfth grade college-preparatory students) was substantially below the international average.
Why do we lose so many students? Of course some of the reasons are societal. Drugs are a scourge, and many students do not get adequate support for their schoolwork in the home. But this does not account for most of the math dropouts. Even in our best schools, even from the best homes, there are children who need to make it who don't. If "everybody counts", as the National Research Council report of earlier this year was aptly titled, we must ask why so few survive.
A Dangerous Myth
LAST YEAR I SPOKE ABOUT THE MYTHS that one must dispel if one is to make the kinds of changes we need to make. One of those myths is fundamental to the problem of survival. The myth is: Either you have it in math or you don't, and the job of the teacher (or school) is to find out who has it.
You probably know some teachers who believe the first part of the statement, and you probably know some teachers who teach in accordance with the second part of the statement. My remarks will not focus on those teachers, because I do not believe they are the real problem. The real problem is more pervasive than this; it is systemic. Even as we assert that everyone can learn mathematics, certain common practices in our system serve to undermine that belief.
The myth is reinforced by a fact of life: Some people do know and can learn more mathematics than others. Do we treat people differently in order to offer them the same opportunity, or do we give everyone the very same treatment? We desire equality or equity in a democracy, but there are two competing interpretations of equality: at times equality means the same opportunity; at other times equality means the same treatment.
Generally in education our rhetoric suggests that we give all students equal treatment, but our practices show that we prefer equal opportunity. The problem is that opportunity itself is circumscribed at very early grades.
Practices That Perpetuate the Myth
Classes that enrich the best students with a curriculum that would be appropriate for all students.
I would like to share with you a personal experience of this. I have told this story before, and judging from the response it receives, it is not at all unusual.
My son is now in 4th grade. Two years ago, at the beginning of 2nd grade, he took his first college entrance exam. By this I mean that he took an exam the results of which will affect what colleges he may attend. The exam was given by the school; it included reading and the math covered addition and subtraction facts up to 18.
My son knew everything on the exam, because my wife and I are educators and we had taught it all to him even before he started 1st grade, and as a result of his performance he was placed in a special 2nd grade math class. That math class reviewed 1st grade quite a bit even though the students knew 1st grade math, but it did not spend as much time reviewing as the other 2nd grade classes. This enabled my son's class to cover more material than the other 2nd grade math classes.
Our school district does not track. Instead, in 3rd grade, it repeated the procedure. Students were tested and those who scored highest were put in a special 3rd grade math class. Now who was most likely to score highest? Those who were in the special 2nd grade class, of course, because they had done more the previous year. Each year the district does the same thing, and each year the students who have been in these special classes have a greater and greater chance of being placed in the top class. When 7th grade comes, this school system has an algebra class, and my son will have a high probability of being in it, and if he continues he will take advanced placement and this will increase his chances of getting into one of the better colleges. You see that there is no tracking per se, only equal opportunity each year.
What about those 2nd grade students who did not know their facts to 18 at the beginning of 2nd grade? Like me, their parents were not told of the significance of the test; they were not even told that there would be a test. Each year that a student is not in the best class, that student falls farther and farther behind the best students. But there is something even more significant As far as I could tell, and I was monitoring the work all year, in both 3rd and 4th grade there has been nothing the special math class did that could not have been done in the standard classes. The special math class reviewed previous years' work less, which is wise policy for virtually all students. They did open-ended problems, which is wise for all. The students discussed mathematics, which is wise. In short, these special students got the kind of education all students should have had the opportunity to have.
Obviously there is a need to do something for students in a grade who know all or virtually all of the mathematics to be taught at that level. Something more should be done with them. It is boring and destructive to have them "learn" the mathematics they already know. But, as I have tried to argue, it is destructive to the others to withhold reasonable opportunities from them.
Remedial courses that put students further behind instead of catching them up.
When we move into middle and junior high school, a different phenomenon begins: the on set of remedial classes. We take those who are behind and slow everything down so that they get even further behind. Implicitly we tell them they are dumb by not giving them anything of interest, often giving them nothing new and spending the entire year reviewing. The goal of remediation is to save these students, to bring them back into the mainstream, but how many come back? Very few. As far as I know, we are the only country in the world that takes students as young as 12 years old and sorts them out of the mainstream.
The issue is complicated: it is not the grouping that causes the problem, but what we do in the groups. We must change our message to students who are in remedial programs. We must tell them: Mathematics is important You need it as a consumer. It is 50% of the SATs. It is one very test you might take to get a job in business or industry. But you are behind. You must work harder because you are behind, and we will help you do so.
Tracks that make it easy to go down a level but almost impossible to go up.
In high schools, tracking is overt. There are levels of algebra in almost every school of sufficient size. If you are in a low level of algebra, you take the low level of the next course. If you are at the middle level, you take the middle level, unless you do not perform well, in which case you go into the low level. If you are at the high level, you can stay at that level or, if you do not succeed, you can go down a level. From one year to the next, students almost never go up a level, because the way the courses are designed, they can only widen the gap in performance.
Wipe-out courses created with the expectation that not all will survive and taught as if all students were mathletes or future mathematicians.
We insure that some of the best students go down levels because, in the name of high standards, we give them wipe-out courses. In1981-82 about 13% of the students in the country took algebra in 8th grade. Since about 7% took precalculus or calculus as juniors in 1986, we wiped out about half of these students- our very best students—in three years. More will choose not to take calculus as seniors, and I estimate that only about a third of all students who are identified in 7th grade as advanced placement capable actually become advanced placement students. The practice is ingrained: it is the expectation in many school districtsthatofthetwoorthreeclassesof8thgradealgebraoffered, there will only be one class of survivors that reach calculus.
The belief that the difficulty o f a course is more important than its content.
What seem to cause the wipe-out are two practices which we must work hard to remove. The first is expecting all these students to be future mathematicians or mathletes. We teach them theory without any application; we give them content that is not useful even for future mathematicians but is found on con- tests. We look at books for such students and say: that book is OK but it's not hard enough. That is, we consider difficulty- mere difficulty-to be a virtue in itself, and even more important than the content of the course.
We at UCSMP have direct evidence of this belief. We know of people who look at our secondary books and say, "They have nice stuff, but they are too easy for our best students." Which is more important, that students be exposed to the right stuff or that the course be difficult? In many schools, the first objective in selecting the curriculum for the best students is that the course be hard. If ever there was an example of mixed-up priorities, it is this.
Relative standards that are used as indicators of absolute intelligence or ability to perform.
The second practice causing the wipe-out is the tendency to use moving standards. This phenomenon is utilized with far more than classes for the best students. Regardless of whether they have earned them, if a teacher gives students all As or Bs, in many schools that teacher's job would be in jeopardy. "The teacher grades too easy." Success is not for all, particularly in mathematics. And so the teacher makes questions hard enough so that some are sure to fail. Again this goes back to the myth that you either have it or you don't. And there are levels of having it-Becky is a B student, Charles a C student. What is wrong with giving all students As?
Five hundred years ago the arithmetic we teach in 4th grade, namely partial product multiplication and long division, was a college subject It was unfamiliar and new and therefore thought to be difficult. That is no longer the case. Mathematics is easier today. Numbers are everywhere. We have better algorithms, among them the punching of keys on a calculator. Our subject is getting easier, so more should succeed.
Another of the pernicious practices is the normed test which gives grade-level equivalents or percentiles. The problem here is that someone has to fall below the 50th percentile; in fact, half the population has to fall there. This tells you nothing absolute about how well they are performing.
Normed tests give rise to some interesting rhetoric: take the notion of the overachiever. I believe that there are no overachievers, there are just tests that were employed incorrectly to predict that somebody could not do something. Of course, there are underachievers because virtually anyone can do a little better than he or she is doing. We have evidence that almost all of our students are underachievers, and SIMS aptly entitled one of its reports The Underachieving Curriculum. When a student comes up to me and says, "You know, Mr. Usiskin, I could have gotten an A on that test if I had studied," my answer is simply: "I don't have any evidence of that, and neither do you. But I think you're right, except you may have to study more than you think." If a student came up to me and said, "My parents couldn't do that homework either," I would respond, " Five hundred years ago this year the symbols for plus and minus were first used. If every generation knew only what its parents knew, you wouldn't even know how to check whether you are being charged the right amount in a store. No one would know how to drive a car ..."
A curriculum that is taught without context or appropriate motivation.
The curriculum is not only underachieving, it is often unappealing, and this constitutes still another cause of the dropout rate. The curriculum we have today is, for the most part, the curriculum of 50 years ago with changes caused by a combination of two movements, new math and back-to-basics. On the good side, new math gave us the first impetus for a broad-based curriculum in the elementary school, a mathematics curriculum rather than an arithmetic curriculum. At the high-school level, new math also shifted some areas around, combining plane and solid geometry, moving trigonometry in with algebra, integrating analytic geometry with other high-school mathematics. But new math assumed that all students were motivated like potential math majors, that is, not in need of motivation, and took out virtually all the real-world connections that had been in the curriculum.
The era of new math lasted from about 1959 to 1972. The reaction to new math, back-to-basics, carried through the early 80s, assumed that all students were incapable of understanding or applying any mathematics and gave us a curriculum even more devoid of motivation. The lack of appeal in the curriculum-day after day of drill sheets and repetitive activities in the elementary and junior high school, algebra taught as a foreign language, geometry taught without connections to any objects in the physical world-is certainly a major cause of the high attrition rate in school mathematics. Why go on studying a subject beyond arithmetic when the only reason given for studying it is that you need it for the next course"?
Although the problem-solving movement of the past decade was clearly conceived to counter back-to-basics, the problem- solving movement can also be seen as an attempt to bring some appeal, some motivation back into the curriculum. This movement has had some impact on elementary school books but almost no impact at the high-school level. It has never picked up steam because of the vagueness of the idea of problem solving. What is done in one grade is almost never followed up in the next grade. Thus there are no long-range effects.
It is significant that NCTM got off its singular devotion to problem-solving in the Standards. The work of UCSMP, of COMAP, and of many other groups has already had its effects. The mathematics curricula being recommended by virtually every group in the country are far more appealing than any we have seen in the past 30 years. It is a good time for mathematics education.
Some UCSMP Solutions
HAVING IDENTIFIED PROBLEMS at all the grade levels, let me now speak of the solutions from the perspective of our project. Some of the practices I have been encouraging or discouraging are not in the domain of UCSMP work; you could do them or undo them with any curriculum. But some solutions are intimately related with our work. At this point allow me to reiterate that we are not the only people working for solutions, and that fortunately for all of us many of the solutions are being promoted by NCTM and by the Mathematical Sciences Education Board.
1. Start from where students are.
This axiom of good curriculum and instruction has three corollaries. (1) Do not needlessly review. UCSMP is known for our views on this subject. Our primary materials take advantage of what students know to go farther than any published curriculum at kindergarten and 1st grade. Our secondary curriculum avoids the counterproductive time spent on review that is inflicted on most students in grades 7 and 8.
Do not interpret this advice to mean that you should never review. W e believe strongly in reviewing, but only in reviewing what needs to be reviewed, and when possible in the context of learning new material.
(2) Do not destroy students by giving them courses for which they are unprepared. We are careful to indicate that the first course in our secondary curriculum requires that students be at the 7th grade level in mathematics, yet we see some school districts using the materials with students who are not that well prepared. W e see states like Mississippi and Louisiana and cities like Chicago requiring all students to take algebra without changing the pre-algebra curriculum sufficiently to enable them to succeed. Equal opportunity is hollow when the failure rate in a course is over 50%.
Many of you have seen the movie Stand and Deliver. The success at Garfield High in Los Angeles is real; it is possible to take students who are far behind and bring them up even to the level of advanced placement calculus students. But the contract these students sign with Jaime Escalante requires them to spend 30 hours a week on mathematics. Simply put, they need to make up for all the lost years. You are behind and so you must work harder to catch up. There is no quick fix to our problems in school mathematics; to change student performance significantly requires a significant change in what students encounter.
(3) Students of different ages should be allowed to do the same mathematics. You will always have students who are ahead of their peers, either because their parents have taught them, or because they are interested and read on the side, or because they are in some special programs. It is senseless to keep someone at a particular level in any subject, asking "If I teach this now, what will next year's teacher do?" We would not tolerate keeping a piano student on a particular piece even though he or she had learned it. W e would not tolerate making a child read the same story over and over. We keep them interested by moving on. Part of the reason for the high dropout rate is that the school mathematics curriculum is so boring for so many students, both the poor and the good. It starts in elementary school but it continues through high school and even college.
Our evaluation component has completed a number of studiesatgrades7-10 of students at different grade levels coming with the same knowledge. Entering knowledge being equal, the performance of these students is independent of age. We have yet to find any evidence of that thing some people call "mathematical maturity" which is supposed to be obtained at some particular age.
2. Remediate immediately and powerfully.
Students do differ. Unless you take pains to keep them at the same level, there will be those who are unable to proceed at the same pace as most of their peers. You must remediate immediately. Do not allow students to fall behind. It is now November. If you are like most teachers, you have students who are not learning what you are teaching. You may have thought to yourself, "If that student doesn't perk up, next year he's going to be in real trouble." This kind of thinking is too lackadaisical. If you are content to wait until next year, what you are doing is to convey the unimportance of learning it this year. Students need to shape up now. They need after-school or before-school help with their work. Many teachers use the quicker students to help the slower students.
If students don't shape up during the school year, they need summer programs to get them back with their peers. School boards and school administrators generally understand that mathematics is important, a key to success; acting on that understanding requires that programs be instituted to insure that as many students as possible succeed. It is best to start this in the primary grades, when the differences among students are the narrowest
Even with these suggestions, powerful remediation for the students who are behind and starting students where they are at, your classes will contain a wide range of students. Two aspects of current reform can help you reach such a wide range. The first is technology.
3. Use technology.
Technology is the great equalizer. Just last week, after I had written most of this talk, newspapers reported a study concluding that calculators seem to diminish differences in performance be- tween girls and boys. We have had evidence from teachers for many years that calculators help to diminish differences between better and poorer students-because, we believe, even poor students can think and use strategies. Often they have been identified as poor students only because they were not so good at paper-and-pencil arithmetic.
Ironically, technology is also considered by some people to be a problem in terms of equity. Here I believe strongly that calculators are different than computers, they are so cheap and so widespread. On the most recent National Assessment, 82% of9- year-olds, 94% of 13-year-olds, and 97% of 17-year-olds re- ported a calculator at home. The cost of a single personal computer and a modicum of software, which schools seem very willing to spend, equals the cost of three or four classes of scientific calculators and a school's worth of four-function calculators. You do not get plaudits from your professional organization if you are against calculators in classrooms, but we know there are many people who are opposed to their use. Those who state that equity is a problem when it comes to calculators are either misinformed or purposely putting up a smokescreen to hide the fact that they are against calculators.
I am convinced that one o f the reasons calculators are not on standardized tests is that they would make many of the traditional questions unusable, because they would make them too easy. This is another form of the myth that mathematics has to be hard to be good.
With respect to equity, the computer issue is different, because computers do cost money, and some districts can afford far more than others. Fortunately, however, computers are not as controversial as calculators. I suppose some people do not realize that computers can do all the arithmetic calculators can. Maybe I shouldn't have let the cat out of the bag. Anyway, computers are glamorous, and we have found that PTAs and bake sales can earn good money for buying computers for classrooms even in places where the district cannot afford the computers. One computer in a mathematics classroom is essential. Here is what the NCTM Standards have to say about this.
Because technology is changing mathematics and its uses, we believe that-
- appropriate calculators should be available to all students at all times;
- a computer should be available in every classroom for demonstration purposes;
- every student should have access to a computer for individual and group work;
- students should learn to use the computer as a tool for processing information and performing calculations to investigate and solve problems.
The fundamental difference between what is said in the Standards and what has been said in many previous reports is the recognition that the most important use for the computer is as a tool, not as a teacher. There is no mention of Cal in the old sense of the computer as a tutor. The role of the teacher is at least as important with computers as without them. Most leaders today have come to realize that the best use of the computer in classrooms is the same use it is put to in the world outside the classroom- as an extraordinary storage device and tool and picture generator with a multitude of uses. With its ability to capture, inform, and motivate students, the computer is something we cannot afford to ignore. In general, the power of calculators and computers to do mathematics is what makes them an essential ingredient of a good mathematics program.
4. Incorporate applications and real (not contrived) problem solving into the mainstream of the curriculum.
The second feature of UCSMP elementary and secondary materials which makes possible work with a wider range of students in the classroom is the discussion of problems which relate to many different ideas, problems rich with context. Then the students who have gotten the easier ideas can focus on the harder ideas. Everyone who has done this notes that students who cannot compute often can do everything else. So make certain there are calculators to help them.
Having a context helps you to do more than just deal with a wide variety of students. Imagine a reading curriculum in which day after day students learned new words without knowing what they meant and read sentences but not in stories. Furthermore, imagine a story in which there was no plot, in which Goldilocks and three bears appear on the first page but there are two bears on the next page and four on the third. This is the way that mathematics is experienced by students. One week they get addition of fractions, the next week it is subtraction; or one week they get addition and subtraction of polynomials, the next week it is multiplication and the following week division. In multiplying fractions you can multiply numerators and denominators but you don't do that when adding fractions. You add exponents sometimes and multiply at other times and subtract at still other times. No story, no context, no way to check the problems, nothing to hang it on except memory. Mathematics is called abstract, but when taught this way it is not abstracting anything; it is jibberish, not much different than nonsense syllables.
Word problems are not automatic substitutes for applications. For instance, consider this word problem.
There is a train. The train leaves a station 1 hour before a plane flying overhead in the opposite direction going 3 times the velocity that the train had when it was twice as old as the plane which is 3 years younger than the station itself. The number of the train is a3-digitnumber; the tens digit is bigger than the unitsdigitandthesumofthedigitsis26. And on the train there is an engineer who himself is a third as old.
He is just a third as old. If you ask as what, you have missed the point of this problem.
There is a club car at the back of the train. In the club car they sell mixed nuts, some at $1.89 a pound. The engineer has a niece and a nephew on the train and sends the nephew back to the club car to get the mixed nuts. He gives the nephew 14 coins, some dimes and some quarters. It takes the nephew 20 minutes to get to the back of the train. The engineer is getting hungry and so sends his niece to the back of the train. It takes her15minutes. How long would it have taken them if they had gone back together?
As bizarre as the example is, it only typifies what is found in most books, from 1st grade through 12th grade. If there is anything that we have come to believe in more and more strongly at all levels as we work on UCSMP materials, it is the need for context. This context need not always be a real-world application; it can be a mathematical context, for instance. If multiplication is known, division can be related to it It can also be a problem- solving context, for instance, the search for a function that has certain properties. But the real world supplies wonderful con- texts for virtually all of pre-college mathematics, and one of the proudest things we can say about UCSMP materials at all levels is that we take advantage of the wonderful applicability of mathematics.
The movement towards applications, now one of the strongest movements in all of mathematics education, can be viewed as a refinement of the problem-solving movement. Do not give students only abstract problems; give them problems that have meaning, if not for them, at least for someone.
Implementing technology and rich mathematics experiences on a wide scale in the elementary school is difficult if not impossible with the teaching corps we have today, because so many of its members were sorted out of mathematics when they went through school. It is ominous that the one mathematics course required of prospective elementary school teachers is taught at a level no higher than that of an average first-year algebra class. One reason we at UCSMP believe that you must have specialist mathematics teachers in the upper elementary school is that we do not see how all teachers can be retrained to be able to give students the mathematics they need.
I HAVE TRIED IN THIS TALK to deal with what is perhaps the major problem for the health of our country, the growing disparity between the haves and the have- nets, between those who are successful survivors of schooling and those who are not. Although my remarks covered teachers and students, many of them could be extended to administrators and school districts.
One of the nagging problems for a project such as ours, funded to improve the mathematics education of average students, of the vast majority, is that we have the greatest difficulty reaching the school districts that need us most. You see, judging from performance on tests, we have remedial school districts in our nation, not just remedial students. For example, it was re- ported last week by the state of Illinois that over half of the schools in the city of Chicago, 25 ofthe47 schools in which over 40 students took the ACTs, score in the bottom 1% of schools in the nation on those tests. The level of performance in many of these districts is low on more counts than student test scores; such school districts find it difficult to send people to conferences like this one and their teachers, who often are not as well-prepared in the first place, fall further and further behind.
Just as remedial students need to work harder, these school districts need to work harder, and their teachers need opportunities. Projects like ours cannot do it alone. I implore those of you who are in higher performing school districts to work with neigh- boring districts. Show them what you do; show them what your students do; show them what technology you use and how you use it. Just as the health of a school depends upon how all its students perform, the health of your community depends upon how all its schools are performing. The good performance of the schools in our cities is necessary for our collective health as a society. Everybody counts, and let us work together to see that as many as possible measure up.
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