The Creation and Destruction of a Myth: Either You Have It In Math Or You Don't
Presented at the Annual Meeting of the National Council of Teachers of Mathematics in New Orleans.
by Zalman Usiskin
Since 1983, 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, with models for curriculum and teacher training, with materials for both students and teachers. The complexity of our enterprise mirrors the complexity of the task of school mathematics, and in attempting to improve school mathematics we have had to deal with virtually all of the problems with it. The subject of this paper is as difficult a problem as any in school mathematics. Perhaps for that reason, until recently there has been very little discussion of it. The problem is the fact that only a small percentage of our students survives their school mathematics experience.
It is impossible to determine exactly what the percentage of survivors is, because the percentage depends on one's definition of "survival" and because mathematics courses through the United States and Canada are not uniform. For instance, NCTM has since its Agenda for Action (1980) recommended that three years of high school mathematics be required of all students. Suppose we take "survival" to mean that a student completes the equivalent of two years of algebra and a year of geometry, the most common program. Then, a self-reported statistic by the 17-year-olds in the 1986 U. S. National Assessment would indicate that nearly half our students survive, since about 47% of 17-year-olds reported a half-year of Algebra II or precalculus or calculus as their "highest level taken". (Dossey et al., pp. 116-117) However, Algebra II 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. So NAEP data is likely to be optimistic. With all this, I would estimate that the percent of survivors under this first definition, those who take three years of secondary school mathematics, in 1986 was 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 four National Assessments, and there has been about a 5% increase in the number of 17-year-olds reporting having taken either second-year algebra or precalculus or calculus. In the 1990 Texas state adoption, the number of second-year algebra texts purchased (about 120,000 in a population of about 17,000,000) would indicate a survival rate in that state (perhaps typical of the entire U.S.) from 40-50%. Furthermore, according to ACT data the survival rate is as high for females as for males. The message that more mathematics is needed is getting around. Under this definition of survival, the cup is almost half full.
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 of graduating seniors now goes to college. It seems that we can equate survival in school mathematics with college entrance. This agrees with remarks by Donald Stewart, president of the College Board, "The link between math and college is 'almost magical'; math is the gatekeeper for success in college."1
If the cup is almost half full, then it is a little more than half empty. Among those who do not take three years of high school mathematics are huge numbers of students that have had no algebra or just a year of algebra, no geometry, no statistics, no probability. A third of the population enters the job market either before graduating high school or immediately upon graduation. A sixth of the population become unemployed. These youths, far more likely to be from our cities, are simply unprepared for the mathematics they will encounter on almost any job. It is not that they have not learned what they were taught; these students never had the content.
When we compare the situation to that in other countries, we realize how empty the cup is. In many countries of the world, the equivalent of two years of algebra and a year of geometry are completed by the end of 10th grade, when almost all students are still in school, and when almost all students are taking the same, required, mathematics.
The National Assessment data are the most optimistic data available regarding survival. Other statistics show that there is quite a dropoff after second-year algebra. Only about 12% of the age cohort in the U.S. in 1981-82 was 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. However, the Second International Mathematics Study (commonly known as SIMS), reported: "In most countries, all advanced mathematics students take calculus. In the U.S. only about one-fifth do." Kifer, one of those who worked on SIMS, has commented that we sort so early that by grade 8 the proportion of students taking algebra is about the same as those senior level mathematics in grade 12 in other countries (Kifer, 1986, 1989). An extensive look at this problem has been done by the sociologist Elizabeth Useem (1990).
If we make a standard of performance part of the definition of survival, then the situation is even more bleak. Even our best survivors do not perform so well. Here is what SIMS had to say about the performance of our seniors: "...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." (McKnight et al., p. vii)
The highest survival rate among the 23 countries that were involved in SIMS was in Hungary, where 50% of the age cohort took the standard mathematics curriculum as seniors. British Columbia (a "country" for the purposes of SIMS) was second, with 30%, 2.5 times the U.S. survival rate. Even with the higher survival rate, students in both of these geographic entities outperformed students in the United States. (It is not clear how the rest of Canada would have performed; only Ontario was represented in the study.) 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.
Why do we lose so many students? Of course some of the reasons are societal. Drugs are a scourge, some parents do not value education and discourage their children from giving their all, and many students do not have guidance from 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 1989 was aptly titled, we must ask why so few survive.
A Dangerous Myth
There are a number of myths that one must dispel if one is to make the kinds of changes we need to make to update and upgrade our school curriculum. 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.
Everyone knows some teachers who believe the first part of the statement, and the reader probably knows some teachers who act in accordance with the second part of the statement. This essay does not focus on those teachers, because I believe teachers are not the major cause. I believe the problem is more subtle than this. At the same time that we have people saying that everyone can learn mathematics, we have a system in which certain common practices serve to undermine that belief.
The difficulty of the issue is that a statement very similar to the myth is true: Some people can learn more mathematics and do know more mathematics than others. Do we treat people differently but offer the same opportunity, or do we give everyone the same treatment? We desire equality or equity in a democracy, but there are two common interpretations of equality: at times equality means equal opportunity; at other times equality means the same treatment.
A Story from Elementary School
Generally in education our rhetoric is that we give all students the same treatment, but our practices show that we prefer equal opportunity. The problem is that the opportunities are cut off at very early grades. I would like to give you a personal example of this. In telling this to other groups, the response has been that my story is not unusual and, in fact, is nowhere near a worst case scenario.
My son is now in 5th grade. Three years ago, at the end of first grade, he took his first college entrance exam. (I used to report that this occurred at the beginning of second grade, but I found out recently that it was earlier.) By this, I mean that he took an exam where the results will affect what colleges he might attend. The exam was not something that his mother and I pushed him to take; it was given by the school. I have been told that it included reading and the math covered addition and subtraction facts up to 18 and some names of geometric figures. That was it.
He knew everything on the exam, because my wife and I are educators and we had taught all that stuff to him before he even started first grade, and as a result of his performance he was placed in a special 2nd grade math class. (There was no special class for any other subject in the school.) That math class reviewed first grade quite a bit even though the students knew first 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 second grade math classes.
Our school district does not track. So, in third grade, the procedure was repeated. Students were tested and those that scored highest were put in a special 3rd grade math class. Now who was most likely to score highest? Those who were in the 2nd grade special class, of course, because they had done more the previous year. Each year the district does the same thing, and each year the students that have been in these special classes have a greater and greater chance to remain in the top class. You see that there is no tracking per se, only equal opportunity each year. 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 calculus in 11th grade. In 12th grade he will be able to take other such advanced courses and this will help his chances of getting into one of the better colleges. In this way, that test at the end of 1st grade will have influenced the colleges he will get into, and it seems appropriate to call it his first college entrance test.
What about those 2nd grade students who did not know their facts to 18 at the end of 1st grade? Like me, their parents were not told of the significance of the test; they were not even told 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 was nothing the special math class did that could not have been done in the standard classes. The special math class reviewed previous years less, which is wise policy for virtually all students. They did open-ended problems and "word problems", which is wise for all. The students had discussions about mathematics in their class, which is wise. In short, these 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 discussed at that grade. Something more should be done with them. It is boring and self-destructive to have them learn the same mathematics as other students who have not progressed as far. But, as I have tried to argue, it is self-destructive to the others to withhold reasonable opportunities from them.
There is a happy ending. The story you have heard is true, and I first told it in November 1989 at a UCSMP conference. There were people from my son's school district in attendance, and my remarks concerned them. The part of the story that got them thinking most dealt with the quality of the curriculum, and the math committee got together and they decided that in fact they were depriving all their students. So last summer they decided to upgrade the curriculum for the other students so that it would resemble the curriculum they have been giving to the best. This meant that they acceleratedevery student in those classes an entire year, and they did this with no advance warning. They simply gave students in grades 3-6 books that were meant for grades 4-7. The verdict so far is that other than a problem in grade 6 at the beginning (the grade 7 books in this basal series had never been used by the district and the jump was more than they expected), the entire venture has been an outstanding success.
It is possible to do this in almost any school or school district. As Fuson et al. (1988) has reported, we teach even the simplest of mathematics content later than many other countries. When SIMS entitled one of their reports "The Underachieving Curriculum", it was partially in recognition of the fact that the United States seems to have the weakest K-6 curriculum of any industrialized country.
Middle and Junior High School Practices
When we move into middle and junior high school, a different phenomenon begins: the beginning of remedial classes. We take those who are behind and slow everything down so they get even further behind. We implicitly tell them they are dumb by not giving them anything of interest, often not giving them anything new, but spending the entire year reviewing things they have had before. 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 problem is complicated: it is not the grouping that causes the problem, but what we do in the groups. The mathematics of this is simple. If student A is behind student B, and student A is being taught at a rate x that is slower than the rate y at which student B is being taught, then student A will not only remain behind student B, but become further and further behind.
We must change our rhetoric to students who are in remedial programs. We must tell them: Mathematics is important. It is on every test you might take to get a job in business or industry. You need it as a consumer. It is part of literacy. It is 50% of the SATs. But you are behind. You must work harder because you are behind, and we will help you do so.
Senior High School Practices
In high schools, the tracking issue is overt. There are levels of algebra in almost every school of sufficient size. If you are at low level of algebra, you can get into the low level of the next course. If you are at the middle level, you can get into the middle level, unless you do not perform well. Then 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, you can almost never go up a level, because the courses are designed to widen the gap in performance. If you have just one bad year, you are dropped from the highest level and cannot return.
We insure that some of the best students go down levels because, in the name of high standards, we give them wipe-out courses. In 1981-82 about 13% of the students in the country took algebra in 8th grade (McKnight et al., 1987). Since about 7% took precalculus or calculus as juniors in 1986 (Dossey et al., 1988), 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 districts that of the two or three classes of 8th grade algebra offered, there will only be one class of survivors that reach calculus.
What seems to cause the wipe-out are two practices which we must work to remove. The first is to expect 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 math contests. 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 even more important than the content of the course.
We have direct evidence of this belief. We know of people who look at the UCSMP courses at the secondary level and believe "They have nice stuff, but they are too easy for our best students." My response is a question: 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.
The second practice causing the wipeout is the tendency to have moving standards. This phenomenon is utilized with far more than classes for the best students. Regardless of how well your students perform, if a teacher would give them all As or Bs, in many schools that teacher's job would be in jeopardy. The teacher grades too easy. Success is not expected by all, particularly in mathematics. Again it traces 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?
It is useful to realize that five hundred years ago the arithmetic we teach in fourth 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. We have better algorithms, among them the punching of keys on a calculator for computation and for graphing.
The Influence of Tests
Curves are related to another of the pernicious practices, 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. And this tells you nothing about how well they are performing. You would think that mathematics teachers would understand these relative standards better than others, but I have heard many teachers identify children as being in a particular stanine as if this is a tattoo that cannot be removed. "We put 2nd-4th stanine children in that class, 5th-8th in the other class,..." It is that myth again: not only do some people think they can determine who has it and who does not, but they act as if they can determine precisely the extent to which every student has it.
Normed tests give rise to some interesting rhetoric: the rhetoric of the overachiever. The child performed well but surprised us; according to the test the child was not supposed to be able to do that. To me there are no overachievers, there are just tests that incorrectly were employed to predict that someone could not do something.
How Many Are Really Disabled?
Allow me another true story that is too familiar to many parents. In the early days of UCSMP, I was writing and teaching the first draft of one of our courses. Every day for the school year, I taught a class of 9th graders who did not get into algebra, the 10th to the 25th local percentile in this school. Since I was only teaching this one class, and even that class was legally assigned to a faculty member of the school, I was not aware of all the rituals of the school, and in November I received a computer list of my class on which there were asterisks (*) by four student names, and I was puzzled. I went to the department chair and asked what the asterisks meant, and he responded that those students were LD (learning disabled). I responded that I had been grading the papers of these four students every day since the beginning of the year and that I had no evidence that they were learning disabled. I asked rhetorically: Were they cured?
There are students who are truly learning disabled, but it also happens that we in the United States seem to have the greatest percentage of learning disabled students in the world. These students that I was teaching were not identified as LD in 9th grade; in this school district they are identified as early as 1st grade! Can you imagine going through school with an asterisk by your name that announces to the world of teachers that you have a learning problem? How many students are learning disabled because they are doing what was expected of them? One of the reasons that so few survive is that we write off students, often very early.
Virtually anyone could do a little better than he or she are doing, so in that regard almost any student is an underachiever. When a student comes up to me and says, "You know, Mr. U, 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 thought." When a student comes up to me and says, "My parents couldn't do that homework either," I would respond, "Five hundred years ago [in 1489] the symbols for + and – 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. Every generation needs to learn more than its parents know; otherwise there would be no progress."
Curriculum as a Cause
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 did some shifting of areas, combining plane and solid geometry, moving trigonometry in with algebra, integrating analytic geometry in with other high school mathematics. But new math assumed that all students were motivated like potential math majors, that is, it assumed students did not need motivation, and the texts of that era omitted 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 any mathematics, and incapable of 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 keep going on studying a subject beyond arithmetic when the only reason given for studying it "you need it for the next course"?
Although the problem-solving movement of 1980s 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 had some impact on elementary school books but almost no impact at the high school level. It 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.
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 fortunately for all of us many of the solutions are being promoted by NCTM and by the Mathematical Sciences Education Board.
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 (Flanders, 1987). Our primary curriculum takes advantage of what students know to go farther than any published curriculum at kindergarten and first 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 as meaning that you should never review. We believe strongly in reviewing, but in reviewing only what needs reviewing and, whenever 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. We see states like Mississippi and Louisiana and cities like Chicago requiring all students to take algebra without changing the curriculum sufficiently before algebra to enable them to succeed. The first year of the required algebra course in Louisiana, fewer than half the students completed the course. 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 top advanced placement calculus students. But the contract these students sign 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) Allow students of different ages to do the same mathematics. There will always be students who are quite 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 because "if I teach this, what will next year's teacher do?" We would not tolerate keeping a piano student on a particular piece even though the student has learned it. We would not tolerate having a child read the same story over and over. We keep them interested by moving on. Part of the reason that there is so much dropout rate is that school mathematics curriculum is so boring for so many students, both the poor and the good. The boredom starts in elementary school but it continues through high school and even college.
Our evaluation component has completed a number of studies at grades 7-10 with students at different grade levels who come in with the same knowledge. Equating for entering knowledge, 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.
Remediate immediately and powerfully.
Students do differ. Unless a teacher takes pains to keep them at the same level, there will be students who are unable to proceed at the same pace as most of their peers. Remediation must be immediate. Do not allow students to fall behind. For instance, in October, in most classes there are students who are not learning what is being taught. Teachers often think, "If that student does not perk up, next year that student will be in real trouble." That thinking is too lackadaisical. If a teacher is content to wait until next year, the student is content to wait also, and the importance of learning the idea this year is diminished. Students need to shape up as soon as the deficiency is noted. 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. The Algebra Transition Project in Philadelphia is an example that this approach can work. 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 such programs in the early grades, when the differences among students are the narrowest.
Even with these suggestions, powerful remediation for the students who are behind and moving students up who are ahead, classes will necessarily have a wide range of students in them. There are two aspects of current reform which help reach such a wide range. The first is technology.
Use technology. Technology is the great equalizer. For instance, there is research to indicate that calculators seem to diminish differences in performance between girls and boys. We have had evidence from teachers for many years that calculators help to diminish differences between better and poorer students, we believe because even poor students can think and use strategies. Often they became 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 must be treated in a different way than computers, because they are so cheap and so widespread. On the 1986 National Assessment, 82% of 9-year olds, 94% of 13-year-olds, and 97% of 17-year-olds reported that there is a calculator at home. (Dossey et al, 1988) 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 smoke screen to hide the fact that they are against calculators.
I am convinced that one of the reasons that calculators are not on some standardized tests is that they would make many of the traditional questions unusable, because they would make them too easy. It is that myth in another form: mathematics has to be hard to be good. Another way of putting the myth: if mathematics does not sort, then it is not real mathematics.
With respect to equity, the computer issue is different, because computers require significant expenditures, and some districts can afford far more than others. Fortunately, however, computers are not as controversial as calculators. Perhaps 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 them. 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 that the most important use for the computer is as a tool, not as a teacher. There is no mention of CAI 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 to use it there as it is used 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.
Incorporate applications and real (not contrived) problem solving into the mainstream of the curriculum.
The second feature of our materials and current recommendations which makes possible work with a wider range of students is the discussion of problems which relate to a lot of different ideas, problems rich with context. Then the students who have 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 more than just dealing with the wide variety of students. Imagine a reading curriculum in which day after day students would learn new words without knowing what they mean and reading sentences but not in stories. Furthermore, imagine a story in which there was no plot, in which there is Goldilocks and the Three Bears on one page and four bears appear on the next and two on the third. That is the way that mathematics is taught to and consequently assimilated by many 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 division. The rules come without reason: "In multiplying fractions, multiply numerators and denominators but don't do that when adding fractions." "Add exponents when the bases are alike; subtract when dividing; multiply when its a power of a power." No context; no way to check 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 gibberish, 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 one hour before a plane flying overhead in the opposite direction going three times the velocity that the train had when it was twice as old as the plane which is three years younger than the station itself. The number of the train is a three-digit number; the tens digit is bigger than the units digit and the sum of the digits is 26. And on the train there is an engineer who himself is a third as old. 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 her 15 minutes. The question is: How long would it have taken them if they had gone back together?
As bizarre as that example is, it only caricatures what is found in most books. If there is anything that we have come to believe more and more strongly at all levels as we work on UCSMP materials, it is the need for context. That 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 rich problem-solving context, for instance, the search for a function that has certain properties. But the real world supplies wonderful contexts 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. We receive continual reports from (obviously surprised) teachers, "Not once this year has a student asked 'Why are we learning this?'"
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 utility, if not directly for them, at least for someone in whose place they could imagine being.
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 these people 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 a level no higher than an average first-year algebra class. One reason we believe in UCSMP that you must have specialist mathematics teachers in the upper elementary school (a belief also held by NCTM) is that we do not see how all teachers can be retrained to be able to give students the mathematics they need.
Summary of Problems and Solutions
The systemic problems causing the lack of survival can be summarized as follows: In many school districts, the better students are put into classes that enrich them with a curriculum that would be appropriate for all students, and the poorer students are put into remedial courses that cause them to lag further behind. There are many levels, and though few school systems have rigid tracks, it is far easier to go down a level and almost impossible to go up. Yet wipe-out courses are created with the expectation that not all will survive, purposely made difficult.
These beliefs are supported by the treatment of normed tests as if they are indicators of absolute intelligence or ability to perform, and by a guidance structure that identifies students as learning disabled but never seems to cure them.
We know that virtually all students can learn a significant amount of mathematics, because they do so in other countries and because programs in the United States and Canada have demonstrated that it is possible. Yet, if the solutions were easy, these systemic problems would not be with us. Here are what seem to be the common threads of successful programs: They start from where students are at. They do not needlessly review. They have high expectations, but they do not destroy students by placing them in classes for which they are unprepared. They allow students of different ages to do the same mathematics. As soon as deficiencies are found, powerful remediation begins. They use technology. Their curricula incorporate applications and problem solving. Perhaps most important, however, is a fundamental underlying belief quite different from the myth mentioned earlier: Virtually everyone has it in math, and the job of the teacher (or school) is to help the student realize that potential.
I have tried here to deal with what is perhaps the major problem for the health of our country, the growing disparity between the haves and the have-nots, between those who are successful survivors of schooling and those who are not. Although my remarks discussed teachers and students, many of them could be extended to administrators and school districts.
One of the gnawing problems with trying to improve the mathematics education of average students, of the vast majority is that we have greater difficulty reaching the school districts that need us most. One can only conclude that, judging from performance on tests, we have remedial school districts in our nation, not just remedial students. For example, in 1989 over half of the schools in the city of Chicago, 25 of 47 schools in which over 40 students took the ACTs, scored in the bottom 1% of schools in the nation on those tests. The level of performance in Chicago and many other school districts is low on more variables than student test scores; such school districts find it more difficult to send people to conferences like this one and their teachers, who often are not so well-prepared in the first place, fall further and further behind.
Just as remedial students need to work harder, those school districts need to work harder, and they need opportunities for teachers. Projects like ours cannot do it alone. I implore those readers who are in higher-performing school districts to work with your neighboring 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 as well as our suburbs, small towns, and rural areas 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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