Teaching & Testing in the 1990s

UCSMP Director Zalman Usiskin presented this talk at the Seventh Annual UCSMP Secondary Conference, held November2-3,1991. This talk has been abbreviated and edited slightly for publication.

FOR THE 1990s, THE TEN YEARS IN THE INTERV AL BETWEEN 1989 AND 2000, the agenda for mathematics education has al- ready been established. It has been defined, at one endpoint of the interval, by the 1989 NCTM Curriculum and Evaluation Standards for School Mathematics and by MSEB's Everybody Counts of the same year; and at the other endpoint by President Bush's call in America 2000 for United States students to be number one in the world in mathematics and science by the end of the century.

If these three documents are understood to define our goals in mathematics education for the decade, this year two new events occurred designed to provide the means to reach these goals. One of these is the appearance of NCTM's second "Standards" document, Professional Standards for Teaching Mathematics. The other is the scheduling in 1994 of the first official national test for students in mathematics, a project that was the catalyst for the National Summit on Mathematics Assessment held last April in Washington.

In my talk today I will discuss this agenda for the 1990s and how it is related to what we have been doing in UCSMP , to what you do in the classroom, and, most importantly, to your students.

Let me begin with the NCTM Curriculum Standards. These standards were motivated primarily by the poor performance of our students on mathematics questions requiring a modicum of thought, and by the changing world of mathematics itself, a revolutionary new world in which the drudgework is done by calculators and computers, statistics is taken by as many students as calculus, and virtually every student, worker, citizen, and consumer needs to be able to deal with mathematics.

In terms of content, the Curriculum Standards present wonderful goals in accord with those that we in UCSMP have had since the inception of our project in 1983. Al- though there are no new content recommendations in the Curriculum Standards, there are new ideas or at least ideas brought into prominence like never before. Of the three views of mathematics given, mathematics as problem-solving, mathematics as reasoning, and mathematics as communication, the last is new. Mathematics as communication, as described in the Curriculum Standards, includes expressing mathematical ideas orally and in writing, reading mathematics, reflecting upon mathematical ideas and relationships, asking questions related to mathematics, and more. It is a statement better than we have ever had in the literature of one of the major changes needed in mathematics classrooms.

We are also in agreement with the Curriculum Standards regarding the notion of mathematical connections. In fact, clarifying the connections between mathematical topics, and the connections between mathematics and the real world, are two of the paramount strengths of UCSMP materials.

The other major report of1989, Everybody Counts, was also motivated by the poor performance of students, and emphasizes many of the same points that are in the Curriculum Standards. Its title reflects a second aspect of the poor performance: the loss of national productivity due to the fact that certain minorities are underrepresented in mathematics and women are underrepresented in many areas requiring mathematics.

But there is something new in this report. Everybody Counts calls for "national goals and local implementation" to solve the problems in mathematics education.

School mathematics programs across the nation need to share a common philosophy and framework-a universal set of interrelated concepts and methods held together by a simple work able philosophy, yet flexible enough to allow for local and regional variations. In a highly mobile society, the basic framework should be transportable and adaptable (p. 91 ).

But is this not what we have today? Can't a student enter your schools from anywhere in the country and be placed in an appropriate course?

Evidently the authors of Everybody Counts believe that this is not the case and that one of the problems in the United States is that we do not have a uniform curriculum. And they want one, though the document speaks of "harnessing the power of a centralized system with the flexibility and initiative of the decentralized U.S. tradition (p. 90)." This statement, like the notion of national goals and local implementation, is Washington rhetoric to disguise what is a move towards a national curriculum. A national test will accelerate that move.

A National Curriculum?

THERE ARE A NUMBER OF ARGUMENTS FOR a national curriculum and a national test. One is that the problem of poor performance in the United States is so massive that something needs to be done at the national level. If people complain about the textbooks and tests, to change them, since they are all marketed nation- ally, we need to think in national terms. I f schools resist change by claiming that they dare not be too different from other schools due to the mobility of the student population, we must change all the schools at once.

Another argument is that the countries of the Far East, whose students perform so well on international tests, all have centralized curricula and national tests. If it works so well for them, why not copy what they do?

Still another argument is based on equity. Currently not all students receive the same curriculum; in particular, some minorities and females do not take as much mathematics as white males. To remove inequities in our country, why not have everybody take the same curriculum? The Curriculum Standards are viewed in Washington as containing widely accepted content goals for mathematics education. If we have the goals, why not put pressure on everyone to achieve them?

If there were a national curriculum, we can guess what might happen in the country by examining the state of New York, which has its own curriculum in grades 9-11, known as the sequential curriculum, and standardized tests over each of the three courses in that curriculum. Success on these tests is measured in what is called Regents credits, and a student must have earned a certain number of Regents credits to enter the state university system in New York. This has its advantages. During the 1970s, when standards were falling in most of the rest of the country, New York held its own. But there are disadvantages now, because while many people in the country are moving forward quickly, the curriculum in the state of New York remains unchanged. There are no applications in the Regents curriculum; there is no use of technology. In fact, the most recent Regents exam for the third course still had log tables. Furthermore, in many Regents courses for average students, the entire course is designed around the test. Many teachers finish the book months early and spend the rest of the year administering practice test after practice test.

The New York system is not conducive to freedom or to change. As far as I know, at a time when there area multitude of curriculum improvement projects in the country, there are no major curriculum improvement projects centered either in universities or in school systems in the state of New York. No school there can pilot or adopt anything significantly different than what is already on the test without a great deal of work, for they must apply to the state to be exempted from taking the standard Regents exam and to have an alternate test approved. It is true that Regents tests are being developed for UCSMP materials, but it has not been an easy process for the schools involved.

For these reasons, I feel that a national test would signal the death of independent projects like UCSMP. No one could afford to experiment at the grade levels slated for testing. I may seem overly sensitive in this regard, but the world-wide experience of national curricula backs me up. The only curriculum projects one sees in Japan and the Soviet Union and other countries that have had centralized curricula are those managed by the government. Individual initiative is virtually absent.

As it happens, some of these national curricula have been of high quality, so we should not assume that the centralization of authority necessarily means a lessening of standards. However, it is because we have no national curriculum that at this time the U.S. is ahead of the rest of the world in the use of technology in classrooms and in developing curricula involving statistics and other applications. There is more work going on in the United States than in any other country to incorporate new ideas into the curriculum.

The Year 2000 ...

NOW LET ME TURN TO THE OTHER ENDPOINT of the 1990s and the six goals for education that the nation's governors and the president have endorsed in America 2000.

Goal I: All children in America will start school ready to learn.
This goal is exceedingly important, and I certainly hope the federal government puts its wallet where its mouth is on this one. A concomitant responsibility of kindergarten and first- grade teachers must be to spend more time teaching and less babysitting. In UCSMP we have materials to help.

Goal 2: The high-school graduation rate will increase to at least 90 percent.
This is not as hard to reach as it might seem. Although the percent of students who graduate with their class is only about 75% in our country, by the age of 22 about 85% of students have the equivalent of a high-school diploma, having either graduated late, or gone to night school, or picked up a GED. So if we pick the more relaxed definition of graduation rate", we only have to increase by 5%. This is a full third of those who today do not finish high school, so it is not a trivial task, but it is not unreachable, particularly as there become fewer and fewer jobs for high-school dropouts.

Goal 3: American students will leave grades four, eight, and twelve having demonstrated competency in challenging subject matter including English, mathematics, science, history, and geography; and every school in America will ensure that all students learn to use their minds well, so they may be prepared for responsible citizenship, further learning, and productive employment in our modern economy.
The first part of this goal is clearly the means by which it is hoped we attain the second part This requires the national test of which I have spoken. The first grade to be tested will be grade 4, because that is thought to be the easiest of the three to develop. The aim is to test first in 1994, and there are already plans to pilot items later this school year.

Goal 4: U.S. students will be first in the world in science and mathematics achievement.

Goal 5: Every adult American will be literate and will possess the knowledge and skills necessary to compete in a global economy and exercise the rights and responsibilities of citizenship.
Goal 4 should actually be considered a subset of Goal5. That is, literacy should include mathematics and science. Sup- pose literacy is defined as tire ability to read a daily news- paper with comprehension. Over the years at UCSMP we have done surveys o f newspapers and found that the median number of numbers on a newspaper page is well over 100. The mean is over 500, being driven up by the sports section, the want ads, and the stock-market listings. Thus, in order to understand a newspaper, you need to be able to understand numbers. And these numbers are not always small numbers or simple counts. They include Richter scales, Dow-Jones averages, and consumer price indexes; percents as taxes, discounts, growth rates, and more; statistics of all sorts both as numbers and graphs; large numbers for populations and distances in space; and even the various infinities of our national debt.

Goal 6: Every school in America will be free of drugs and violence and will offer a disciplined environment conducive to learning.
Usually the drug-free aspect of this goal is emphasized, but I would like to focus on the other part, the disciplined environment, which is actually more important.

We in the U.S. are often told that the main reason for the high performance of children in the Far East is that society and the family are different there. I have only been to the Far East twice, but I am not convinced by this argument. I think the schools are different, and as a result a more ambitious curriculum is possible. In the U.S., learning before college is viewed as an unnatural behavior, something that people and particularly children do not want to do. Homework begins in earnest in many places only in the ninth grade. The pre-college classroom, from kindergarten through twelfth grade, is viewed as a place where talking off task and disruption are expected, and being on task and concentrating are unusual. The schoolteacher is often depicted on U.S. television as someone to be outwitted, not someone to be respected. It is the Welcome Back, Kotter view of education.

In the U.S., college is when we first get serious about education. In most of our college classrooms, a student would not think of talking out of turn or in any other way disrupting what might even be a lousy lecture. There are no laws governing this - it is simply part o f the school culture. Homework is taken seriously in college and viewed as natural. College students do homework without a policy, without written rules, because it is expected. In the Far East, the attitude towards elementary school and high school resembles our attitude towards college learning. I saw nothing in Chinese middle and high schools that was much different than our attitudes about learning in college trans- mitted to an earlier age.

If in fact we are to achieve Goal 4, being first in the world, then we will have to achieve Goal 6, a disciplined school environment, first. We do not have long to do this, either, because the twelfth-graders of the year 2000 are now in fourth grade.

On Testing

ON THE FIRST PAGE OF THE EVALUATION STANDARDS there appears the following statement: "Evaluation is a tool for implementing the Curriculum Standards and effecting change systematically." This use of tests to drive the curriculum constitutes a complete reversal of the prevailing view in past decades that tests should not be used to drive the curriculum. But it is in accord with the spirit of Goal 3 of America 2000 with its call for students to leave grades 4, 8, and 12 having demonstrated competence on "challenging" subject matter. So one reason for the national test is to strengthen what is taught, and the profession is on record as sanctioning this reason. But another reason has also been discovered for having these tests. How can we determine the relative status of the U.S. in the world without them? With two reasons for the tests now, it is unlikely that the idea will die.

Thus our leaders in Washington seem to have given up trying to change the curriculum first; now they will change the tests, and hope that others will change the curriculum to meet the tests. The problem with this is: How can you possibly test if you do not have the curriculum in place that is being tested? Fortunately for NCTM, the Curriculum and Evaluation Standards are widely believed in Washington to be the right model to be followed in creating any test Fortunately for those o f you using UCSMP materials, you have taken steps to modernize your curriculum and your students should be in good shape for the tests to come.

The Evaluation Standards emphasize four principles.
  1. That student assessment be integral to instruction.
I have said that we at UCSMP consider ourselves rather in line with the evaluation standards. A t the secondary level we have closely allied assessment with instruction, through the modified mastery learning strategy we have at the end of each chapter. I know of no materials that integrate testing more with instruction than ours do.
  2. That multiple means of assessment be used.
By multiple means of assessment is meant the use of written, oral, and demonstration formats; the use of calculators, computers, and manipulatives; and evaluation using a variety of scoring schemes-grades, written comments, portfolios of student work. In all UCSMP courses we want students to be able to use calculators on tests. In our last two secondary courses, Functions, Statistics, and Trigonometry and Precalculus and Discrete Mathematics, more advanced technology is required and we encourage students to do projects and present them orally in class. The response to the idea of projects has been very positive, and we will likely include them in other books in our second editions.
  3. That all aspects of mathematical knowledge and its connections be assessed.
This principle is one we adhere to religiously. All of our tests, virtually all of our assignments, and even all of the ancillaries encompass a wide range of questions.
  4. That instruction and curriculum be considered equally in judging the quality of a program.
    "The main purpose of evaluation, as described in these standards, is to help teachers better understand what students know and make meaningful instructional decisions (p. 189)." The fourth principle, that curriculum and instruction be considered equally in evaluating a program, stresses that point But I do not agree with it. In general, I don't think means should be as important as ends. For instance, how one solves a problem should not be as important as somehow solving it. Here the means in this case are the instruction and the ends the curriculum. W e offer curriculum materials, we provide suggestions that we think make it easier to implement these materials, but we do not test over the means you use to develop concepts.

On Teaching

THOUGH WE MAY NOT TEST OVER INSTRUCTION, we think teaching is exceedingly important But what is teaching? At the Third International UCSMP Conference on Mathematics Education that directly preceded this conference the first speaker, David C. Johnson of the University of London and Louisiana State University, reminded us of paragraph 243 from the British equivalent to the NCTM Curriculum Standards, the 1982 report Mathematics Counts. (Yes, the title Everybody Counts is a variant.) This paragraph identifies six aspects of teaching:

  • expositing, i.e., lecturing
  • management of consolidation and practice
  • facilitating communication (teacher-pupil and pupil-pupil)
  • guiding practical work and investigations
  • consulting on investigations
  • advising on problem-solving

We think all of these are important, and encourage you to use all of them with your class.

The NCTM Professional Teaching Standards discuss these means. I particularly like Standard 1, Worthwhile Mathematical Tasks. This standard not only lists the tasks, but also describes what it means for a teacher to facilitate. It uses the verbs engage, develop. stimulate, call for, promote, represent, display sensitivity, and draw on. It isn't easy to do all this, but these actions are certainly more interesting than the alternatives lecture, transmit, or show.

The newest idea in the Teaching Standards is that of dis- course, an ongoing discussion between students and be- tween students and teacher about specific problems and about mathematical ideas. It is a natural follow-through to Standard 1. The message is important and one we support at UCSMP .

On Students

A KEY INGREDIENT IN ALL THIS REFORM is the students. With all the rhetoric coming from Washington about improving the curriculum, I have seen no statement saying that student attitudes and behavior outside the teacher's control must change, and change rather significantly. There seems to be an assumption that if the curriculum changes, if teachers change, and if the tests change, then all students will automatically change. There is much evidence that this assumption is false.

In discussing this issue, I would like to divide students into four types: the elite college bound, the non-elite college bound, the work-force bound, and the disenchanted. This is an oversimplification, of course.

The elite college bound comprise 10-20% of all students in the nation. These students have valued education from the beginning of their school careers, they listen attentively in class, they tolerate even boring classes without disrupting them, and they do homework diligently, even going the extra mile. They are found mainly in our honors classes but there are many such students in our average classes, too. These students will learn whatever they have to in order to succeed; they will change if the curriculum changes, if the tests change, or if expectations change. They are already successful and they will stay successful.

The non-elite college bound comprise about 30-40% of students; along with the elite college bound they give the U.S. the largest percentage of college-attending students in the world, currently half our student population. This group also believes that education is important, and they do home- work, but not so diligently, because they have been taught by television and by society as a whole that serious schoolwork does not begin until college. Whereas the elite college bound have been doing homework whenever it was assigned even from grade 1, many in this second group start to do home- work daily only in 9th grade. For the most part, they go with the flow. They are usually docile in class, but they can become disruptive if that is the norm. Because of their docility, despite their numbers the non-elite college bound have been ignored in school for years. They can be challenged to perform virtually as well as the elite college bound.

The work-force bound includes another 25% of all students. This group graduates from high school. Members of this group go directly into the armed forces or get jobs, go to specialized schools for various trades, or are unemployed. Students in this group tend not to realize the value of education, or of mathematics. In upper grades, they are in classes for average or below-average students with lower expectations, and they barely meet these expectations. They, too, are usually docile and go with the flow, but their flow is different. They have learned from television that the natural behavior of students is to be inattentive or disruptive, and that teachers are expected to spend much of their time coping with this behavior. They do only the minimal amount of homework expected, and even this they feel is an intrusion on their lives, because they have not been successful in school and so have not often seen rewards for learning. This group is more difficult to reach than the college bound, because its members' attitude towards homework must be changed. These students need to be taught how to study, how to read for understanding, what to do when a question stumps them and the teacher is not around, how to use time outside of class effectively, where to go for help. Current national policy seems to ignore the existence of this group.

The fourth group, consisting of disenchanted students, has not been ignored, but their problems are generally greater than any curriculum project can overcome. This group, consisting of about 25% of the population, drops out before high school graduation, and over half of them are on the streets, unemployed. They have been unsuccessful in school and naturally do not like being unsuccessful, so they avoid or are disruptive in school. They seldom if ever do home- work because they have never received rewards for doing it National tests, a revised curriculum, and new ways of teaching will not change these students unless accompanied by a change in the climate in which they exist. They need schooling but they also need social programs and reasonable assurance of a job and decent living conditions. Current programs like the OED seem to work for about half of this group, but the other half remains disenchanted and outside the mainstream of school and society.

We cannot attain the significant improvement needed in student performance unless the students themselves change. Students who are behind should not be given a weak curriculum-they need to work harder to catch up-but they will never catch up if they do not do their home work, because those ahead of them got there by working and will continue to work. This must be made clear. Changes in student behavior are necessary for the disciplined environment conducive to learning that the president has called for. And these changes will take more than your individual efforts. They require the entire school community parents, teachers, administrators, business and government leaders working together. It is a difficult job, and in many places it will take years to achieve, but it is worth striving for.


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