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The Stages of Change

author: Dr. Zalman Usiskin

These remarks are abstracted from and update two previously-published talks. "Stages of Change", National Council of Supervisors of Mathematics Newsletter, July 1995, and "The Fundamental Problems in Implementing Curricular Change and How To Overcome Them", UCSMP Newsletter No. 4, Winter 1989. The earlier full talks contain somewhat more detail than given here. They are available from Zalman Usiskin, University of Chicago, 5835 S. Kimbark, Chicago, IL 60637.

published: 01/28/1999
posted to site: 01/28/1999

1999 LSC PI Meeting
Keynote Address

The Stages of Change

for NSF Local Systemic Change Projects

PI Meeting

January 22, 1999


These remarks are abstracted from and update two previously-published talks. "Stages of Change", National Council of Supervisors of Mathematics Newsletter, July 1995, and "The Fundamental Problems in Implementing Curricular Change and How To Overcome Them", UCSMP Newsletter No. 4, Winter 1989. The earlier full talks contain somewhat more detail than given here. They are available from Zalman Usiskin, University of Chicago, 5835 S. Kimbark, Chicago, IL 60637.


It is an honor to have been invited to speak here today. My remarks are in two parts. The first part is a look at the stages of change. It is adapted from a talk I gave in 1995 when the first signs of discontent were appearing in the current mathematics reform movement. This should take about 40 minutes. Then I will move into the second part of the talk, which is about the creation of significant change that lasts. This will take about 20 minutes. This should leave about 30 minutes for questions and discussion.

Before proceeding, however, I must use my role as a mathematics educator to correct the characterization of mathematics given last night by Dan Rubenstein. He said, "Math is the language of science. Why do we use it — because it’s precise." This description may have been true 100 years ago, but it is only a small part of today’s mathematics. Mathematics is the study of numerical and spatial information, patterns, and relationships. It is useful because it is flexible. It deals with estimates as well as exact values. It is a language for describing many aspects of the world, and it has as at least as many applications to business and social science as to science. It is so ubiquitous as a part of everyday language that it is a part of literacy. But that is another talk.

I know a little about science education, but not the subtleties of teaching science. So my examples are taken from mathematics, not science. Perhaps the subjects are different when it comes to change. Dan’s suggestions to get lasting change included getting rid of textbooks and basing the curriculum on problem situations that would change every couple of years to keep them fresh. This to me is a blueprint against lasting change. Schools are not colleges with experts teaching each course. They need materials like textbooks as an anchor. If you can get good books and other materials into your system, it is easier to have the change last. But there are parallels in the history of the subjects, and I am certain that some of these parallels will be obvious to you as I speak.

Stages of Change

Every so often we are reminded that certain aspects of education go in cycles. For instance, in the 1960s there was a major push for educating our top students. This was followed by a push for educating the students at the bottom in the 1970s, that was followed by another push for dealing with gifted students in the 1980s, and now in the 1990s we again have a push for raising the standards of the lowest-performing students.

In the 1930s, the era in which the progressive education movement was at its peak, there was a concerted effort to create a child-centered educational atmosphere. After World War II, this movement was replaced by a movement that was far more teacher-centered and based on objectives. A decade later, the new math began with a wave of teaching by discovery and giving broad-based objectives rather than specific ones. This was followed by an era of behavioral objectives in the 1970s. Today's Standards movement is more student-centered with broad-based goals rather than specific objectives, but the backlash is asking for specific objectives once again.

The emphases on skills and problems also cycle. The 1930s was a time in which people endorsed the use of practical problems and reasoning; skills were the dominant theme in the late 1940s and 1950s; problems came back into vogue in the 1960s; basic skills returned in the 1970s; the 1980s was an era of problem-solving, and now we are beginning to see a desire to return to skills. There may be a corresponding cycle in science between process and product.

The cycles are given mathematical models. Some have spoken of the sine wave of change; others invoke the image of the pendulum swinging back and forth. About five years ago, a panel about the tendency for the same educational movements to reappear in different guises was part of the program of the annual meeting of the Association for Supervision and Curriculum Development, ASCD. The panel was described with the title "Reforming the Schools: Is It Deja-vú All Over Again?"

Though not all aspects of education cycle, the examples that I have given cover practices relating to curriculum, instruction, and educational opportunity, and suggest that there are some general patterns underlying change that prompt these cycles. It may be easy to decry the cycles as indicating that we do not learn from history — and I have often felt this way — it also may be the case that we have little recourse but to expect that the cycles will occur. If that is the case, then if you want lasting change, do not pin your project to these ideas. Instead, combine the best from both sides of the issue.

This morning, in speaking of the stages of change, I will use two sets of changes more than any others for my case studies. The first case study is the set of changes known as the "new math", from the very beginning of the first ideas in 1951 to the end of its impact in 1975. The second case study is from the current era, and has to do with the set of changes we call the "standards". Both of these examples are of movements that were considered successes in their times, so what I am talking about are the stages of change that accompany success, not failure.


The new math is now viewed as a failure, and many of the ideas that were its hallmarks — ideas like the use of sets, and the teaching of understanding by going into mathematical theory — are now viewed as silly ideas. But at the time they were viewed as wonderful. What happened? Are there any ideas that we are promulgating today that will be viewed as silly 30 years from now?

It is obvious that in any successful movement there is an upcycle, followed by a time in which the idea is in vogue, followed by a downcycle. Now let me describe the aspects of each of these stages in more detail.

Stage 1: Work by the pioneers

Change begins with the pioneers, the person or persons or school of thought that works on the change. Pioneers often work alone, and their work may be known to only a few people. In the case of new math, the original pioneers were in a group called the University of Illinois Committee on School Mathematics, UICSM. There were three principals in that group: Gertrude Hendrix, a professor of education who wrote about what was called unverbalized awareness; Herbert Vaughan, a professor of mathematics who felt that if mathematics were made rigorous by the precise use of language and notation, then children would better be able to learn it; and the person who put it all together, Max Beberman, who believed fervently that one learned better if one was led to discover the mathematics rather than being told it.

Beberman began his work in 1951 or 1952, five years before the beginning of any other new math curriculum group, and well before the term "new math" was in vogue. This meant that the first UICSM materials were tested, revised, tested again, revised again, and so on for quite a while. As a result of all this testing, the UICSM people created extensive teacher materials which showed how they wanted each lesson to be taught, and they felt that the only people who should use the materials were those who came to the University of Illinois for a summer of teacher training. I mention all of these characteristics of UICSM because in the later implementation of the new math materials they were missing.

Stage 2. Proselytizing of and by the apostles

The first people who used the UICSM materials were people I call the apostles. Many teachers in high schools and professors in colleges spoke and wrote about its successes and told others about it.

Stage 3. Use by those disenchanted with the old

These first users taught in schools where most of the students were above average for the nation. They picked up the UICSM materials not because they were part of a national movement to change, but because they felt their students were not being reached by existing materials. They and most of the other first users were the disenchanted. Because of their disenchantment with the status quo, as often as not they sought out the new materials; the creators of the new materials did not seek them.

The Advanced Placement program began in 1955. There were NSF institutes in the summer of 1957. The Boston College Mathematics Program began in 1957. And, in the summer of 1957, a few months before Sputnik, the School Mathematics Study Group, SMSG, was organized at Yale under the direction of Ed Begle. So the disenchantment began before the Soviet Union launched Sputnik in October 1957. The disenchantment was present as much in science and in mathematics and led to the famous science curricula: PSSC physics, BSSC biology, at least two chemistry projects, and a number of elementary science projects.

Mathematics, fortunately, was never split into its constituent areas — the big projects dealt with all of mathematics. After Sputnik, UICSM became eclipsed by SMSG in importance. Unlike UICSM, SMSG had representation from across the nation, and when moneys from NSF became available due to Sputnik, SMSG became funded to a level we have never seen since.

Stage 4. Acceptance by the establishment

With the appearance of the SMSG texts, the new math became in vogue. Government reports and reports of all the national teacher organizations mentioned the various new programs that were being developed. New math had reached the establishment.

In this stage, with the endorsement of national organizations, comes all sorts of money. There was money for teacher training, for meetings, for curriculum work, and — for the first time — for research in mathematics education. The NSF institutes that began in 1957 became greatly expanded and through the late 1950s and most of the 1960s there were summer institutes and academic year institutes to retrain high school teachers. It is a myth that new math failed because there was not enough teacher training. I will return to the question of failure later — that new math failed is a doubtful point — but there certainly was a lot of teacher training. It was available to almost anyone who wanted it. But it was almost all at the high school level.

5. Joining by the piggybackers

At this point in the stage of change, there are many piggybackers who, for a variety of reasons — some good and some not so good, latch on to the movement of the time. If you disagree with the establishment, you are less likely to get money, so it is useful to agree. Besides, no one wishes to be behind the times, and once the establishment picks up an idea, you are subject to scorn if you do not accept it. NSF encouraged piggyback theorists and supported many other curriculum projects. Although the National Council of Teachers of Mathematics, NCTM, did not adopt any official policy towards the new math curricular materials, its support was obvious. Both of its journals, The Mathematics Teacher and The Arithmetic Teacher had article after article about various aspects of curriculum and instruction that were to be different, and the yearbooks of NCTM reflected the need to change in these directions.

6. Forcing of the enchanted

Only so many people at a given time are disenchanted, so every movement for change has to deal with those who were quite satisfied with what was going on. These people, for good reason, need to be cajoled, coerced, and otherwise made to change. Often this is done through mockery — the view that if you are against the particular changes being adopted, then you are against all change and against everything that is wholesome and good. This aspect of the change process I call the forcing of the enchanted. The change is not likely to fare well when implemented by those who are forced, often against their will, to adopt it. Some of the people who were against the change do wind up being new apostles and adopting the spirit and substance of the change, but most will give up the change at the first chance.

Stage 7. Oversimplification and overapplication of the change

The SMSG or UICSM materials differed in virtually every lesson from what was in the typical materials of the time. To understand these approaches, a person needed to read years worth of student materials. This cannot be done in a short period of time. Yet only so many people can be retrained in institutes lasting for an entire summer or an academic year. For those who are unwilling or who cannot take the time to go through the training necessary to teach the new math materials, the whole approach — something that took years and years to perfect by the first of the new math projects — has to be simplified so that it can be explained in an hour or two. We often oversimplify in order to convert the enchanted — we say to them that it really isn't so difficult to implement change, you only have to do a few things — or we minimize the amount of change that is necessary. It is a fact of change that this oversimplification of the change will happen, but it is not useful for the change.

When the establishment enters the picture, so too do the people in publishing. Now the time for development of materials becomes shortened. As fast as materials can be produced, they are published. All the subtleties get removed from the materials, and oversimplifications of the available materials themselves are now the norm. Perhaps the most obvious manifestation of this in the case of new math was the appearance of books in which the first chapter was devoted to sets, but then the rest of the book never used them. Instructional subtleties are also ignored; whereas both UICSM and SMSG had extensive teachers notes with each lesson, commercially published materials in the 1960s usually had minimal notes.

A myth about new math is that the new math materials were not tested. Nothing could be further from the truth. The UICSM materials were tested for so long before they commercially appeared — something like 8 years — that by the time they appeared in 1962, they were actually too late to pick up the piggybackers. The SMSG materials were classroom-tested and then revised by the project; later, from 1964-1967 SMSG itself ran the largest comparison study of mathematics textbook performance ever conducted, a longitudinal study involving 125,000 students at various grade levels. The results of this state-of-the-art study, the National Longitudinal Study of Mathematical Abilities, were published in over 30 volumes.

The performance of students using SMSG materials was not lower even on paper-and-pencil tests of basic skills. But there were some new math materials that did result in lower performance. These tended to be the extreme types of materials, those that were very theoretical and rigorous.

One of the characteristics that comes with the establishment takeover of a change, and with its oversimplification, is that the change which started as a reaction to a particular problem with a particular population is now felt to be a solution to all sorts of problems with a much larger population. It's a natural human tendency to feel that if something works well for some people, it should work well for many more. We then get the overapplication of the change. In the case of new math, the ideas which had been developed with very good students, or with college-bound students in high-performing schools, now were being touted as being appropriate for all students. And publishers will market to anyone who will buy, so the overapplication of the change becomes either tacitly or actively accepted.

Stage 8. Failure of the oversimplified and overapplied theory

You may now be thinking ahead of me. We now have quickly put-together, oversimplified materials — sometimes even caricatures of the original materials — being taught by many people who did not believe there were problems in the first place to students who were not originally felt to be covered by the theory. The result is inevitable: the failure of the simplified theory in many places, particularly in those places in which the new ideas are only newly being used.

No reform is a panacea, but in order to institute reform those who are in favor make all sorts of claims for it, and they raise expectations. When those expectations are not met, the public — always skeptical of reform in mathematics — starts to perceive that the reform, just like the mathematics they took in schools, is not successful. So the public perception is of failure even if the overall situation is for the better.

Stage 9. Test scores that do not bear out people's desires

The public perception was mainly built from news reports of test scores. From 1955 through 1963, test scores rose, but then they began to decline and declined through the 1960s and early 1970s. In late 1974, the results of the National Assessment of 1972 were published. They showed what every National Assessment in every subject has shown — that students are not as good as we would like them to be.

Stage 10. Public perception of the failure of the change

Adults over age 26 were tested in this first assessment so that people who had old math could be compared with those who had new math, and even though the adults were given unlimited time while students were given timed tests — the adults performed worse than the students. Still The Wall Street Journal reported that new math had failed. A time of retrenchment was at hand — back to basics was here. In 1976, when the College Board issued a report that pointed out that the decline in scores from 1963 to 1970 had been due to an increase in the population of students taking the test, and could not be traced to any curriculum, either no one wanted to hear or the knowledge of statistics of the populace was so low that they did not understand the argument.

Stage 11. Fatigue of the establishment

An unfortunate side effect of the failure or the perception of the failure is that, because it occurs first with those who waited the longest to change, it inures those reluctant changers or those in school systems most difficult to change, against change itself. Thus a cycle of reluctance to change is perpetuated in these places that may need to change the most. On the other hand, when the change does not live out its panacean dream, even the pioneers start to move on. Even those for whom the reforms were successful are ready to move on. America is built on progress, and it does not seem like progress if you write the same report that you wrote ten years ago. This fatigue of the establishment seals the doom of the change, for the establishment is not prepared to defend the change that it once championed.

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