Before the University of Chicago School Mathematics Project began in 1983, many national reports had issued recommendations for improving the state of mathematics education. Most of these recommendations reappeared in the recent Standards documents published by the National Council of Teachers of Mathematics. In general, our project's purpose has been to ascertain whether key recommended changes could actually be implemented. What we have found should be considered by anyone attempting to change curriculum in the direction of the NCTM Standards.
- use in schools the technology that is widely available outside schools;
- incorporate real-world data and applications into mainstream mathematics;
- upgrade student performance on the traditional content that is still important;
- lower the percentage of students who receive no mathematics instruction beyond arithmetic; and
- support the implementation of such changes by developing materials and training models for teachers.
So far, the project has developed a complete mathematics curriculum for grades K–3, teacher development materials for grades K–and a complete secondary mathematics curriculum for grades 7–12. (Large-scale evaluations of all these materials have been conducted.) The project has also translated 40 mathematics schoolbooks (from Japan, the former Soviet Union, and Hungary) and has held three international conferences on mathematics education.
Findings About Technology In School Mathematics
All recent national reports on school mathematics have recommended the incorporation of calculator and computer technology into the study of mathematics.
This possibility is relatively recent. A 1973 article by E. G. Begle summarizing lessons learned by the School Mathematics Study Group, the largest of all the New Math projects, does not even mention the words “technology” and “calculator.” Why? Because the first hand-held calculators didn't even appear until 1971.
In our project, however, the use of technology has been a major thrust at all levels, K–12. Our very first materials, developed in 1983–84 for 7th grade students, required calculators. It was a bold move then. Since that time, our discoveries about technology in mathematics have included the following principles:
Calculators are easy to implement. Calculators can be distributed to (or required of) all students just as is done with textbooks. This practice has worked in public and private schools with good students and poor students in rich areas, poor areas, suburbs, and cities.
Computers are not nearly as easy to integrate into secondary school mathematics as calculators are. In 1986, before we developed an 11th grade course called Functions and Statistics with Computers, we wanted to find out about teachers' access to computers. We invited in two dozen teachers and administrators from a wide variety of schools. We asked whether teachers had a computer lab in their schools (all did) and whether they could reserve this lab one day a week (all could, even those in schools with few computers and not particularly supportive administrations).
Thus, we included one computer activity a week in our new course. Ultimately, we were surprised to find that, in many places, the computers were not used one day a week, despite good intentions.
All sorts of things prevented access to the computers. The lab day was a holiday. Or the computer manuals were locked up. Or the software was written for an earlier version of the hardware. Or a school assembly required shortened periods. Or the day set aside for computer work did not occur at a logical time in the study of the material. And so on.
If computers are to be integrated into students' experiences with mathematics, it is easier to have them always available than sometimes available. In short, the one-day-per-week compromise between not requiring computers and requiring them all the time did not work. Students needed continuous access. Not that they would use the computers all the time, but for homework, demonstrations, and maximum teacher flexibility, computer time had to be an option that was always available. (It surprised us to discover that teachers found it easier to schedule a class to meet every day in the computer room than to go to the lab one day a week.)
An implication that can be drawn from this experience is that in many school districts, computers may be easier to incorporate into instruction in elementary schools than in secondary schools because more elementary classrooms have computers in them.
Technology requires changes in sequence and content. For instance, if you want to know the national debt per person in the United States, you need to enter a number in the trillions for the debt and the hundred millions for the population. On most calculators, such large numbers must be entered in scientific notation, which means you will need to know about exponents, a topic normally placed late in books. Thus, if students are to make the best use of calculators, exponents have to be introduced early.
At a more advanced level, the impact of technology on content is evident in the way graphing calculators have transformed our approach to functions. With calculators, questions can now be asked of students that once could only be raised in calculus courses. This observation leads to something we learned about teachers.
Teachers do not ordinarily embrace technology until the technology has capabilities teachers want students to have that they cannot learn easily or at all without the technology. When the curriculum lacks these additional requirements, many teachers maintain the view that “if I can do something without technology, you should learn to do it, too.” Since graphing calculators empower teachers and students to deal with functions earlier and more thoroughly, many mathematics teachers have completely reversed their negative views toward calculators.
Teachers are strongly affected by ease of use of technology. Most teachers did not begin to graph functions when graphing calculators first appeared in 1985. But, with the advent of a more user-friendly model in 1990, mathematics teachers enthusiastically embraced the new technology.
If calculators can do the same thing computers can do, even if not as well or easily, it is more effective to use calculators than computers. The reason is access. Students can have the calculators at all times, including at home. Computers still need to be available, however, to do explorations in geometry, crunch statistics, perform simulations easily, do spreadsheets, and get good resolution with graphs.
Computers and calculators enable a wider range of students to function in a single class. A lack of paper-and-pencil skills has prevented many students from having the opportunity to reason mathematically. When we freed students from the need to calculate like robots, many turned out to be much smarter than we thought they were.
Real-World Applications
Mathematics teachers' views toward the students best suited to be taught real-world applications of mathematics are noticeably bipolar. Some think applied mathematics is not as lofty as pure mathematics and would relegate applications to those students who are not adept enough to understand pure mathematics. Others think that applied mathematics is actually harder than pure mathematics, so the only students who should study applications are those who already understand pure mathematics.
All our project experiences and data analyses contradict both positions.
Virtually all students are able to learn to apply mathematics. Our testing supports a comment Begle made in 1973. After 10 years of work with the School Mathematics Study Group, he noted: a decade ago . . . the opinion was frequently expressed that modern mathematics was suitable only for students with high IQs. However, all our data analyses point in the opposite direction. No such cutoff point appears. There is no reason why any student should be deprived of the advantages of modern mathematics.
Begle also wrote that “the best predictors of performance at the higher cognitive levels of understanding, application, and analysis seldom include computational skills.” From what we have learned, we agree.
What's interesting is that people today are agreeing with Begle's conclusions, but resting their case on the availability of technology. Our data indicate, however, that Begle was correct whether or not a school uses calculators or computers.
Pure and applied mathematics should be approached simultaneously. Appropriately challenging applications need not be delayed until after paper-and-pencil skills or after theory. Each enriches the other. With good applications, however, there is one immediate advantage. The question “Why do we have to learn this?” almost completely disappears.
Applications let more students get involved in mathematics activity. The richness of a good application provides many handles for students to grab, so even students who do not know the mathematics may be able to contribute some data relevant to the situation.
Applications provide rich opportunities to teach both pure and applied mathematics. For instance, a close-to-linear relationship exists between the latitudes of the most populous cities in the Northern Hemisphere and their mean high temperatures in any given month. A graph of the latitudes and high temperatures shows that the points lie nearly on a line. Students can use this line to predict, for instance, the mean temperature in April for the place where they live. They learn not only some algebra, but geometry and geography as well.
This practical knowledge helps a student understand ideas like what a line of best fit is about and why we might want to have such a line. Contrast that image with the thought of a student who is able to graph and to find an equation of a line through two points, but has no idea why anyone would ever want to do it.
Improving Performance and Access
The traditional K–12 mathematics curriculum has been built around arithmetic, algebra, geometry, functions, and precalculus ideas, very much in this order. Over the past two decades, however, this lockstep sequence has been challenged.
Begle, for instance, wrote: We now know that it is not necessary to restrict the study of algebra to grades 9 and 11 and of geometry to grade 10. Through its elementary school texts, the School Mathematics Study Group demonstrated that informal geometry is an appropriate topic at any level.
Begle went on to assert that “the location of specific topics in the curriculum should be based not on the age of the student, but rather on the overall structure of mathematics.”
Mathematics educators have incorporated Begle's views into their thinking, but this consensus does not mean, however, that we have as yet acted on our thoughts. Judging from our comparisons of American high school textbooks with their Japanese and Soviet counterparts, a good deal of mathematics instruction in the United States is delayed and therefore denied to many students.
The traditional high school mathematics curriculum in the United States covers about the same content and has the same degree of difficulty as in other countries, but the coverage comes about two years later, and consequently reaches a lower percentage of students. Traditionally, for instance, more of our students have been taught algebra and geometry in 9th and 10th grades than in any other grade, whereas in other countries that same content is covered in 7th and 8th grades.
All of our studies indicate that this later placement for mathematics is unnecessary and unwise. In grades 7–10 we have not been able to detect any optimal age to begin learning algebra or geometry.
The earlier secondary students encounter algebra and geometry, the better these subjects are learned. We believe that the view of a developmentally appropriate middle school experience that downplays algebra and geometry is based on old algebra and geometry curriculums that were inappropriate for almost everyone of any age! They required the maturity necessary to tolerate and work diligently on tasks that have no meaning.
We are losing valuable years by thinking that social adjustment and intellectual development cannot grow side by side. The maturity needed for self-control has little relation to cognition. Anyone who has seen children work with video games knows that middle school students can concentrate for long periods of time on complex tasks requiring long logical trains of thought. In fact, as long as the tasks are interesting, the younger secondary students' ability and curiosity are often greater than that of older students.
The stark reality is that the United States has the weakest elementary school mathematics curriculum in the world. In our project, we have examined curriculums from countries where student performance has appeared to be superior to that in the United States or Canada. Our first translations, done within six months of the start of the project, were of Soviet elementary texts for their grades 1–6, which are equivalent to our grades 2–7. The contrast was astounding. The Soviet books were interesting, well written, and replete with a careful sequence of rich problems.
Since that time, others have looked at elementary curriculums elsewhere in the world, and there is now general agreement that we proceed more slowly through concepts. When we finally get to a concept, the tasks we give to students are, for the most part, simplistic. Furthermore, we dwell more on review than any other industrialized country does. The predictable result is that our 8th grade students score well below those of comparable countries.
Implementing Change
If we wish to change the curriculum, important considerations include what mathematics teachers think about it and how changing it might best be done. Our project has gathered a great deal of information—some anecdotal and some systematic—on these two matters.
The majority of teachers do not want to teach complex manipulations with rational algebraic expressions to most algebra students. In this respect, American teachers' opinions are in sync with the practices of many other countries. Nonetheless, we have traditionally expected first-year algebra students to do problems like this one from a standard 1990 U.S. first-year algebra text:
Figure
The only Russian students who study such complex manipulations are older students who are interested in mathematics contests or who are studying for entrance to the country's best universities. The omission of complex manipulations from algebra is certainly one reason that the Russians have been able to teach the subject to all students.
The only reasons now given for teaching this material are: (1) a mistaken view that these skills are needed before applying algebra, (2) a mistaken view that this material is on standardized tests, (3) a realistic belief that the placement tests of some colleges still regard these skills as important, and (4) a realistic belief that a mathematics major of today should be able to do these things.
As a result of rethinking the mathematics curriculum, complex manipulations are quickly becoming a topic for honors students only. The algebra courses developed in our project continue a century-long trend of diminishing attention to manipulative skills in general.
Among mathematics teachers, applications, transformations, and coordinates in geometry are gaining in popularity. This thinking, too, is compatible with a long trend to broaden the scope of geometry instruction and increase the connections with other areas of mathematics and to the real physical world. The NCTM Standards support this point of view, as do our project's geometry materials.
If we are to revise curriculum successfully, we must make changes systematically. It is shortsighted to think of curriculum change in terms of single years or courses. In our schools, the typical student takes mathematics for 10 or 11 years; changing the student's experiences in only one or two of those years is not likely to have significant long-term effects.
Thus, to have significant effects on students, projects that create materials and implement them in schools require more than a few years of funding. Our project began with a six-year grant from the Amoco Foundation, which later awarded us a five-year continuation grant. The project has also been supported by grants from the National Science Foundation, Ford Motor Company, Carnegie Corporation of New York, General Electric Foundation, GTE Corporation, Citicorp/Citibank, and the Exxon Education Foundation. In all, external support has totaled about $16 million, not including royalties from publications.
Long-term support is facilitating multiyear curriculum development. We have been told that the long duration of Amoco's original grant influenced the National Science Foundation such that it increased the maximum length of its grants from three to five years. Within the past three years, the National Science Foundation has funded 13 multiyear mathematics curriculum development projects at primary, middle school, and high school levels. (Our project was awarded one such grant to develop a mathematics curriculum for grades 4–6.) The prevalence of these multiyear curriculums is important when undertaking change.
Changing a multiyear experience is different from changing a single year or single course. A teacher, acting more or less alone, can change a single year's experience, but for multiyear curricular change, administrators, guidance counselors, parents, and school boards need to be involved. From our experience, we believe that such change cannot occur solely top-down or bottom-up. It must occur both ways at once.
A fundamental challenge in using a multiyear curriculum is building on the experiences that students have had in previous years. There is little sense to changing one year if the next ignores those changes. Teachers must be educated in how to take advantage of the new experiences that their students have encountered in previous years and in how the current year's activities will affect students in subsequent years.
As a general rule, an experience in a previous year can be built upon only if 80 percent or more of the students had that experience. That is, if more than 5 students in a class of 25 have not had a particular experience, most teachers think that too many students will be left behind unless the entire class repeats the experience, despite the fact that a large majority of the students already have had it.
Where We Stand Now
- We have broadened our idea of what mathematical knowledge is important for students to learn.
- Calculators—four-function, scientific, or graphing, depending on the grade level—are being used more and more in courses and on standardized tests. Beginning in March 1994, calculators will be allowed on the SATs. Computers are used in many classrooms as well.
- An increasing number of materials emphasize applications. In some places, the basic skills tests no longer deal with paper-and-pencil arithmetic but focus on the ability to deal with real-world applications.
- The number of students taking algebra in the 8th grade is increasing. According to the National Assessment, 16 percent of students did so in 1990, up from the 13 percent noted in 1981 by the Second International Mathematics Study.
Together with an increasing knowledge of what is going on in other countries, these changes have generated a great deal of new knowledge about the learning and teaching of mathematics. We now know that under appropriate conditions, virtually all students can learn significant amounts of arithmetic, algebra, geometry, statistics, functions, and trigonometry, and they can learn to use all the mathematics that they know in real situations.
