What We Have Learned From Seven Years of UCSMP
UCSMP Director Zalman Usiskin presented this talk at the Sixth Annual UCSMP Secondary Conference, held November 2-3, 1990. This transcript has been edited slightly for publication.
EIGHT YEARS AGO, late in 1982, the Amoco Foundation gave $25,000 to the university of Chicago to plan a large project to improve mathematics and possibly other subjects in our schools. In the first half of 1983, professors from the departments of mathematics and education met regularly and out of these meetings developed the overall mission and the structure of UCSMP . Official funding for the project began in the fall of 1983. In September, 1983, even before funds were in hand, I was writing the first draft of Transition Mathematics.
Before UCSMP began, I had worked on mathematics curricula for eighteen years, written books for each of the four years of college preparatory mathematics in high school and conducted three large-scale studies testing these books, and taught in nine high schools; I had learned quite a bit. But in seven years of the project, I have learned quite a bit more. And it is this I wish to share with you today.
LET ME BEGIN WITH what we have learned about technology. From 1973 to 1976, under a grant from the National Science Foundation, I wrote and tested an algebra course that assumed fourfunction, hand-held calculators. In 1983, when I wrote and taught the first draft of Transition Mathematics, I went no further. But when we convened the secondary component advisory board for the first time, they were concerned. Why aren't you using scientific calculators? I had no ready answer; I suppose I had thought it would be enough just to get calculators in.
The 1984-85 revision introduced scientific calculators, and we learned our first lesson. The sequence had to be changed. It is very obvious now but it was not so obvious then. In order to work in scientific notation with the extremely large and small numbers made accessible by the scientific calculator, exponents had to be done early. Even negative exponents were needed early. The general principle new technology requires changes in the sequence was established.
That school year, during the formative evaluation of Tramsition Mathematics, we learned that calculators are easy to implement. Just make sure that every student has a calculator; distribute them to or require them of all students just as you do textbooks. Like many others, we found that paper-and-pencil skills did not suffer at this level when calculators were used, and we found that test scores on concepts and applications in arithmetic, algebra, and geometry were substantially improved.
A few years later, though, we found that the same thing cannot be said for computers; computers are not so easy to implement. While many have focused on the lack of good software as a reason for the underutilization of computers in schools, we found the logistics to be a formidable barrier. Here is what happened:
In 1986 we had a small conference of teachers and administrators representing a wide variety of schools to talk about the implementation of computers in math courses. In preparation for the first draft of what was then called Functions and Statistics with Computers, we asked the teachers whether they had a computer lab in their schools (they all did) and whether they could get this lab one day a week for regular use by a math class. Even those in schools with only a few computers and not particularly supportive administrations felt that this could be done.
But when we evaluated the pilot year of the course, we found that even this one-day-a-week had not been implemented in many places. Despite the good intentions of those who had agreed to be in the study, all sorts of things went wrong. The computer lab was not always available; or the day set aside for it was a holiday; or the manuals for the computers were locked up; or the software was written for an earlier version of the equipment. Many things can go wrong.
We had thought the one-day-a-week idea was a good compromise between not requiring computers and requiring them at all times. We found out that this compromise did not work; it was necessary for the class to have continual access to the computers. Not that they would need the mall the time, but for homework, for demonstrations, and for maximum teacher flexibility, computer time had to be an omnipresent option. A principle emerged: In curriculum, going halfway is not always a good compromise.
Yet the UCSMP curriculum is in many ways a middle-of-the-road curriculum, positioned midway between those who want to base the entire curriculum on problem-solving and those who do not want to change; between those who want no applications and those who want only applications; between those who want students to use technology always and those who eschew technology; between those who want no paper-and-pencil skills and those who want even more paper-and-pencil work than we have had; between those who want to keep the distinctions between the courses algebra, geometry, and trigonometry and those who want to break them down completely. But we have not chosen our middle road because we decided in advance on some sort of compromise. W e just tried to create the best curriculum we could, and it so happens that this curriculum does keep much of what we have traditionally done even as it introduces a number of newer ideas.
BEFORE UCSMP BEGAN, I knew transformations to be gorgeous and powerful and motivating in both geometry and algebra, and I knew that applications could have those same qualities. But still there was more to learn.
To my knowledge, we are the first secondary curriculum in the world to attempt to sequence pure mathematics and applied mathematics simultaneously. As you might expect, we have found many ways in which doing pure mathematics and applied mathematics are alike. Consider, for in- stance, the sequence used to teach pure mathematics. We give students the properties of numbers and geometric figures years before we teach them proofs.
Modeling, the process of utilizing mathematics to solve real-world problems, is to applied mathematics what proof is to pure mathematics. Confronted with a real-world problem, one attempts to create a mathematical model of that problem to work with and then translate back.
In the UCSMP secondary curriculum, we sequence modeling. It begins in Transition Mathematics with the identification of models for operations, which are the applied mathematics analogue to the properties of those same operations in pure mathematics. In Algebra students work with linear and quadratic models; in the first chapter of Geometry, students learn how points and lines can model things that do not look like points and lines, like land masses, and they learn how algebra models geometry. These connections are the postulates that enable one to use mathematics. As early as the second chapter of Advanced Algebra, students learn how functions model various situations. In FST, students use computers to find and test lines of best fit and other models that have been statistically generated, and of course there is continued work with function models.
We have learned that pure and applied mathematics can be taught simultaneously. Each enriches the other. Applications need not be done last and compartmentalization is unnecessary. The SPUR approach works.
Before we began, we knew that there would be hard applications and easy applications just as there are hard skills and easy skills. We knew that there would be messy applications and elegant applications just as there are messy manipulations and easy ones. What we have learned in the past seven years, though, is that contrary to what many expected, applications and other rich problems enable more students to get involved in the action rather than fewer. We thought pure and applied mathematics would be the same in this regard, but we have found that the richness of a good application offers many handles for students to grab onto, handles often inaccessible even in rich pure mathematical contexts.
In algebra, for instance, if you ask for the line with slope 20 and y-intercept 5, and a student does not know how to do it, there is little you can do except tell him or her the answer. But if you remind students that to mail a first-class letter costs 25¢ for the first ounce and 20¢ for each additional ounce, and then you ask for a formula which gives the cost C of mailing a first-class letter in terms of the number of ounces w, there are things a student can do even if he or she is unable to get the answer. The student can make a table. Or graph the points. Or find the costs for mailing third-class parcels. And so on.
As most of you know, we strongly urge that students read a lesson before it is explained by the teacher. We did not appreciate the full impact of reading, combined with applications, until the 1987-88 school year, when we had our first study of UCSMP geometry students who had been through the two previous UCSMP courses. Here is what happened:
We had 9th graders in a senior high school who had taken Transition Mathematics in 7th grade and UCSMP Algebra in 8th grade in a feeder school. These were average students who normally would not have taken algebra in 8th grade. They were viewed by the senior high with suspicion; there was little confidence in the algebra they might have had and they were reluctantly placed in geometry. The geometry teacher, who shared those suspicions, subsequently reported the following to the evaluators: The first surprise was that the students expected to do a lesson a day and to be assigned all the problems. Freshmen do not usually know what is expected of them. The second surprise was that their knowledge of geometry was very good, perhaps as good as the school's honors students. The third surprise was that their knowledge of algebra was as good as the school's typical geometry students. But the biggest surprise was the desire of these students to discuss mathematics, something the teacher had hitherto seen only among the very best students. The UCSMP students expected to talk about the mathematics in class. Let me emphasize here: Without rich problems, there is very little to talk about.
A report of this kind occurred in three separate schools. These led us to wonder why we had not encountered them in our earlier tests. We think we know the answer. When our materials were still in loose-leaf or spiral format, we did not have many schools adopt them for large numbers of students. They were used only for one or two pilot classes. And when these students went on to senior high school, they were almost always put in classes with students who did not have any UCSMP background, even if the senior high used our materials. We have come to the conclusion that there is little benefit from previous years' work unless at least 80% of the class has had that work. Put another way, if more than 5 students in a class of 25 have not had something, most teachers feel obligated to review the content as if no one had had it; you cannot build on the background.
I TURN NOW TO INSTRUCTION, from the content to the process of teaching. In all of our studies, we have had a wide range of quality of students and quality of teachers. From the very first study of Transition Mathematics up until our most recent studies, one thing about instruction has come up again and again in the notes of evaluators: Math classes in the standard curriculum are dull. And if they are dull when there are observers present, what must they be like at other times?
We are often asked how to teach from our materials. That is why we have sessions at all our conferences in which UCSMP-experienced teachers report on what they do. Our advice is not so complicated: don't lecture before students have had a chance to read because they need a reason to read; go over questions with students the next day, in the order in which they are in the book - give your brief explanatory lectures in the context of discussing the questions; give students additional examples in class if there is time; assign the reading and the questions on the next lesson for homework.
It's the discussion that seems to be the hard part. Sharon Senk, co-director of the secondary component, put it in an interesting way: There are some teachers who have not actually taught for years. These teachers present the material, they show how to work the problems, and they hand out worksheets; but they think their job is done when the homework is assigned. They do not get into the problems; they do not discuss their significance; they do not provide alternative ways of looking at them; they do not respond to student difficulties except on a one-to-one basis. For such teachers, our books present a challenge because many of our questions are written to encourage class discussion.
Allow me to digress. I have two children, a boy 9 years old and a girl 7 years old. They are very bright: Each learned a foreign language before they were3 years old. The language was English. Well, it was foreign to them. But we spoke that language in our house. We talked to them in English. We used English to describe the world around them. We used English to explain things. We used English to converse. If we had not spoken English, they would have had a hard time learning it. Mathematics is in many ways a language. It is spoken in some homes, like mine and perhaps yours. But it is not spoken in all classrooms, not even in many mathematics classrooms. If we do not speak the language, it is much more difficult to learn. The classroom in which there is conversation about mathematics is a classroom in which more mathematics is likely to be learned. Not one-way conversation; that's like trying to learn a foreign language by turning on a radio or cable TV station broadcasting in that language. Conversation needs to be two-way, three-way: n-way, where n is greater than 1.
Last month at an NCTM regional meeting in Wichita, Kansas I was shown a letter written by a Kansas teacher who wrote that she was skeptical at first, but as the year wore on she began to see.
...remarkable changes occurring in the reasoning powers of my students, their attitudes towards algebra, and yes, in their achievement. The rich variety in the content, the constant application to the real world, the attention to the need for reading and trying on one's own- all combine to make students sense a purpose for mathematics that has been rather obscure before.
As she points out, it is the combination of these things- not applications alone, not variety alone, not just the reading or the expectation that students learn it for themselves – that works. For best results, the package has to come together.
I have mentioned in other talks that we were told at our very first conference with school personnel that we should not tamper with instruction, that is, with the behaviors used to teach. Aside from general suggestions, we have tried not to tell teacher show to teach. But even so, we have gotten into trouble. As soon as our books became more widely used, we heard of distressing stories about teachers who had the students read on their own and do the questions, and then the next day gave the answers perfunctorily and assigned more reading. No explanations at all. Never have we suggested that teachers should not explain the material! We only ask that students be given a chance to read the material first Obviously, there are times when explanation is required, in order to summarize, in order to emphasize key points, in order to caution about errors. And it is best to put these explanations into some context, such as the solution of a particular problem.
No two teachers teach our materials in the same way, and we know of no capable teacher who does not depart from our advice on occasion. But the best teachers do not depart from the spirit of our advice.
WE HAVE AN EXTRAORDINARY AMOUNT of evidence that people want to change. The fact that there are almost 400 educators from 37 states at this conference is just one indication of this. Publishers, unlike most industries in the U.S., zealously guard their sales figures, and since we do not know how many schools nation- wide adopted books last year, I cannot say what percent of schools across the country are using our books. I can tell you that the usage of our books is far more than even the most optimistic projections of people at UCSMP or at Scott, Foresman. The key point is that every school that adopts our books is deliberately opting for change. Educators may be concerned about whether their students will be able to handle the material, and they may be concerned about whether their teachers will be able to handle the material, but these concerns are not enough to keep them from using our curriculum. Despite what you may have heard to the contrary, in the current climate people are ready for change, and the decision announced this week by the Educational Testing Service to allow calculators on the SATs beginning in 1993-94 can only accelerate that change.
Evaluating the Scott, Foresman editions
AT THIS CONFERENCE, there are two sessions reporting on recent evaluations, one the formative evaluation of Precalculus and Discrete Mathematics (see page 3 of this newsletter) and the other a study of students who have been through the first four years of a UCSMP secondary curriculum. I say a UCSMP secondary curriculum because there have been various editions of our books. As you no doubt realize, all of the evaluation studies conducted by UCSMP were done with our own versions of the books.
The Scott, Foresman editions have only been available for teaching for one full year and we know of only one national study. It was commissioned by Scott, Foresman and done by a market research firm, I believe in New Jersey. The general results of this study, which included 40 users of Transition Mathematics, 40 users of Algebra, and 20 users of Advanced Algebra selected at random from schools having ordered 20 or more textbooks, are as follows:
- 90% of TM, 92.5% of Algebra, and 95% of AA users described themselves as either satisfied or very satisfied with the text.
- Rating the UCSMP text better than others were 92.5%ofTM, 90% of Algebra, and 95% of AA users.
If you read Consumer Reports, you know that these results are extremely high. If this is representative of all users, then we have incredible customer satisfaction.
I believe the reason for this, if indeed it is accurate, is that we have tried to listen to you, the teachers. Year after year people attend our conferences and are surprised that we want to know what isn't going well. Of course, we also want to know what is going well. But we need to hear from you what kinds of things you think will improve our materials. In these times of changing technology and the explosive proliferation of uses of mathematics, we need to be bold but we also need to be prudent. If you don't get the chance to speak your piece at any of the sessions, jot it down on paper and give it to any of the UCSMP staff members. I can guarantee that it will be read.
Magnitude of the Task
WHEN UCSMP BEGAN, we thought that there would be three or four more projects like ours. But there has been no other with our breadth, and at first we were surprised by this. There is not one other complete secondary curriculum trying to move the country forward. By now we have learned the reason: the immensity of the task.
When Functions, Statistics, and Trigonometry and Pre-calculus and Discrete Mathematics are published by Scott, Foresman, the total number of pages in the student editions of our six books will be about 5,100 pages. Because we have final say over their content, we see these pages many times while they are being readied for the Scott, Foresman edition. There are about 4,650 pages in the current Scott, Foresman K-8 mathematics series, the most in any math series Scott, Foresman has ever published. Not only are the 5,100 UCSMP pages much more complicated on the average than elementary school pages, but each of our books has gone through at least three versions. And there are the teacher's editions. And finally, when Scott, Foresman asked us to help with ancillaries, we agreed to do so.
As I mentioned earlier, I had already written four high- school texts before UCSMP began, and I thought I knew what it would take to do a six-year curriculum. But there is a difference between pouring it on for one year or two, and pouring it on for what is now the eighth year. It is the wonderful people we have working on the project, and the excitement you bring us, and the pride we feel in what we have done that keeps us all going. One thing we knew when we started this project was that there was a lot of talent out there in the schools across this country. We have been helped in the writing, the testing, and now the inservices of our materials by hundreds of teachers. Teachers know better than anyone that students cannot change unless classrooms change. All of those attending this conference are evidence of that desire to change. We would not exist without you, and we thank you again for coming.
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