For centuries, mathematics has been taught in schools around the world. Although school mathematics has changed in content and emphasis over the years and across national borders, the school mathematics curriculum has until recently been rather constant and stable. From a distance, one sees school mathematics in the primary grades as essentially concerned with numbers, simple figures, and concrete operations with numbers, whereas in the secondary grades, it deals with more abstract material to prepare students for the liberal arts. The primary grades have tended to focus on problems involving practical arithmetic, measurement, and geometric figures. In contrast, the focus of school mathematics in the secondary grades has historically been more on theoretical problems from algebra, geometry, and analysis. In other words, the early focus has tended to be on applied mathematics, and the later focus on pure mathematics.

During the century from 1850 to 1950, secondary school enrolments expanded, with more and more students all around the world studying secondary mathematics. Before that time, the secondary curriculum had been mostly for those few students who were going on to universities. Therefore, it was rather pure and rather removed from real problems. The effect of the enrolment expansion was to increase the mathematical preparation of those students but also, because university mathematics was becoming more formal and abstract, to increase the differences between secondary and tertiary mathematics courses. In response, a number of projects launched in the mid-twentieth century attempted to change the school mathematics curriculum.

These efforts arose from various sources and took many forms, but they tended to have in common a desire to bring school mathematics closer to the academic mathematics of the twentieth century—to eliminate inane jargon and make it better preparation for the mathematics being taught in the university. (Kilpatrick, 2012, p. 563; see also Howson et al., 1981, and Kilpatrick, 1997/2009)

The New Math Era

The result of those efforts has been called “the new math” (Kilpatrick, 2012), a term used “to describe the multitude of mathematics education concerns and developments of the period 1955–1975” (National Advisory Committee on Mathematical Education, 1975, p. 137). The term does not characterise a single approach to, or style of, curriculum development. Instead, as the projects outlined by Howson et al. (1981) clearly demonstrate, the new math projects in the United States and Great Britain alone spanned a variety of approaches.

Nonetheless, the curriculum development projects of the new math era did have some common features. Many of the projects, for example, turned to university mathematicians for ideas about revising the school curriculum, and many of those mathematicians considered the twentieth-century concepts of sets, functions, groups, rings, and fields to be appropriate fodder for young learners. They thought more children would be attracted to the study of mathematics if it were organised around those abstract ideas. They soon discovered, however, that changing school mathematics is far from simple. It requires attention to local conditions and teachers’ preparation and attitudes (Kilpatrick, 1997/2009).

During the new math era, some mathematicians who played a strong and influential role in shaping the curriculum got a bit burnt. They thought they knew what primary school children should learn, and they wrote books about that content. Teachers and students alike, however, had trouble with some of the approaches taken in those books. Their response was not what the mathematicians expected. It is one thing for mathematicians to address the secondary curriculum, because the connections to what is happening in university classes are clear. What mathematicians have to say about the elementary or primary school curriculum, however, is a different story. Some mathematicians have stayed with that issue, but in general, not many feel comfortable working on the school curriculum. It is not a rewarding thing for them to spend time on school mathematics. They have their own area to work in, and they get their rewards from proving theorems and doing other things like that. In mathematics, there are not many rewards for mathematicians to spend time on the school curriculum.

In the past, individual mathematicians like Felix Klein and some others looked at the secondary school curriculum. They said that it needs to be made more like the university curriculum, and that was a part of what their contribution was. Felix Klein probably did the best job by introducing functions as a concept and making calculus the endpoint of secondary education. He really had a strong impact on the school curriculum. Throughout recent history, we have had mathematicians who helped us understand how the secondary curriculum could be made more like what the university curriculum was becoming. The question of what kind of help mathematicians could provide the primary curriculum, however, has proved to be much more difficult, and we have had fewer mathematicians working on that. On the topic of modelling, statistics, and that sort of thing, few mathematicians want to work. Many do not consider statistics to be mathematics. They do not really see the point of it. It is something, however, that students need to know; and most countries want to make it part of school mathematics. Therefore, we have to get more statisticians to help us understand what the mathematics of statistics should be in the curriculum.

Critique

Not all the mathematicians who worked on the new math curricula approved of the direction the reforms were taking. In an article signed by 64 mathematicians that was published in both the Mathematics Teacher and the American Mathematical Monthly in 1962, a strong critique was made of that direction. The article gave some guidelines for judging school mathematics curricula:

It warned against focusing the curriculum too exclusively on future mathematicians, urged that abstract concepts be built on concrete examples, and recommended greater attention to connecting mathematics with science. (Roberts, 2004, p. 1063)

Despite differences of opinion about the new math, it had the effect of awakening mathematicians and mathematics educators to the curriculum as a phenomenon to be studied, understood, and changed – at least potentially.

Before the new math era, no one thought of school mathematics as something to be reformed or updated; it simply was what it was. The new math reformers knew almost nothing about the school mathematics curriculum in other countries or, in some cases, in their own country. By the time the new math era ended, in contrast, everyone concerned with school mathematics had a much better sense of what was going on around the world. (Kilpatrick, 2012, p. 569)

Bipolarity

The school mathematics curriculum has two foci – two poles. The elementary curriculum did not originally have many pure aspects to it. It was mostly applied mathematics: arithmetic with some simple geometry. Over time that changed, and during the new math era, some abstractions and other ideas from pure mathematics were introduced into the earlier grades. The pure mathematics pole moved into those grades. The other pole, the other part of the bipolarity, is that pure mathematics had always dominated the secondary curriculum, and that curriculum was not intended for every child. As enrolments grew in secondary mathematics over the last century, however, the applied mathematics pole moved into the secondary level.

Subsequent Changes

By the 1980s, many were convinced that the new math had been a failed effort (Kilpatrick, 1997/2009, 2012, 2017). That verdict, however, depends upon one’s criteria for success. Certainly, the curriculum looked different from before even if it did not change in all the ways reformers wanted.

Since the 1980s in response to some of the changes fostered by the new math, there have been a number of projects to build school mathematics around the more applied parts of the subject matter and to take social and cultural aspects of the subject into account. Reformers have wanted to include such topics as statistics and other ways of looking at representations of practical problems. They have also wanted to connect the mathematics being taught more closely to the social-cultural context in which the students are learning.

An Applications Turn

One of the big arguments against the new math was that the pure mathematics being taught did not have applications – or at least that students were not being introduced to applications. In response, a number of projects in various countries sought to build the curriculum around the more applied parts of the subject matter, including statistics and other ways of looking at representations of mathematical problems. A special effort was made to look at how children might approach practical problems. Today we have many applications for the earlier grades that we did not have during the new math era.

When I was teaching junior high school mathematics in Berkeley, California, in the late 1950s, I took a summer school course at Stanford whose instructor was Morris Kline. Kline, the author of Why Johnny Can’t Add (Kline, 1973) and one of the authors of the critique cited above (for details, see Roberts, 2004), was probably the strongest U.S. critic of the new math. He was a professor of applied mathematics at New York University and wanted to build, if he could, a curriculum of applications of mathematics. He considered applications a better way than using pure mathematics to get into the subject matter.

Kline attempted to have us students collect applications that he might use, but we were not very successful. For the elementary algebra course that some of us taught, for example, we were able to develop a few problems involving projectile motion – to illustrate uses of parabolas – but because our students had no access to calculators or computers at that time, the calculations needed would have been too complicated for them to perform. Few of the applications we came up with used real world data that did not involve such calculations. Even when we had some good applications, we could not handle them very well in our classrooms because the students would get bogged down in the work. Today we can let the computer do the calculations. Then the students can go much farther, and I think that we are moving in that direction. I would guess that school mathematics is going to become a much more applied subject over the next few years.

School mathematics is likely to become more applied as teachers learn more about how to handle the applications of mathematics. I expect that programs that include modeling, statistics, and other applications of mathematics will grow once teachers learn what they want to do with applications. The focus of the push will be in that direction because technology is allowing us to deal in the classroom with applications that were never possible before.

There are many problems associated with bringing applications into the curriculum. Parents may say, “Why is this in here? I didn’t study this when I was in school. Why are you having students do this? This is not mathematics.” And some mathematicians may agree: “This is not mathematics. These are applications. They are not part of mathematics.” For those mathematicians, it ruins the subject to bring in applications. Even if it makes students happy, it is not staying true to what mathematics really is. If we stick with pure mathematics with no applications, however, students are likely to say, “When will I ever use this?” And it is not surprising, therefore, that those students do not pursue more mathematics. I think that for self-preservation, mathematicians and mathematics educators should work on the question of: how we orchestrate the curriculum so that applications play a major role.

One of the things that will slow any change is that teachers do not necessarily know about applications, and they are not always sure how to handle them in class, especially if they have not seen that done. When an application works, however, it can work very well. After having seen a mathematical topic put into practice, the students can say, “Oh! Now I understand where I would use this mathematics.”

There is even a problem with the word applications because it implies that first you do the mathematics and then you apply it. In class, however, it can actually go the other way. You can start with a good application, with a situation where mathematics can be applied, and then students can learn how mathematics can be brought into the situation. “I’m learning quadratic functions, and now I see what good that might do me.” If you have a good application, then you can convince students that the mathematics does work, and they need to know that.

The whole idea of trying to organise the applications into a coherent curriculum is a special problem of its own. In a sense, pure mathematics is easy to organise into a curriculum because everything is sort of logical, connected, and so on. In a more applied curriculum, there are some big questions: Where do we start with applications? In what order do we take them? Which ones do we use where? Nonetheless, in a project we did with an upper secondary precalculus course that we studied in several places in the United States (Kilpatrick et al., 1996), the teachers told us that their students loved the examples of applications of mathematics and that it really helped them understand why they were doing this mathematics. The students understood much more about functions, for example, than they would have from just a pure mathematics approach.

I think there are pedagogical values in working with applications even though it is difficult to put together a sensible curriculum made up largely of applications. How do we weave together the pure mathematics and the applied mathematics? Whatever we construct needs to be some kind of coalescence of pure and applied. We can downgrade the applied part, and we have done that in the past. There are good pedagogical reasons, however, for raising the level of the applications and the number of applications. It is just that we have to be careful about how we choose and arrange them.

Today, teachers can look online for some problems, but they may not be comfortable with that. Any change will likely be a slow process. Teachers, however, are the ones who know the students in front of them. They know what these students can do or cannot do, and we need to trust the teachers to bring in the applications that these students will be able to learn from.

A Social and Cultural Turn

A second movement that one finds in curriculum projects today has to do with what has been called the ‘social turn’ in mathematics education (Lerman, 2000), or more precisely, the ‘sociocultural turn’ (Lerman, 2004). Rather than just looking at how individual children are learning, curriculum developers are looking at how classes of students learn and how we can incorporate the socially and culturally relevant aspects of mathematics learning into our work. The sociocultural turn has been a focus of many recent projects because people recognise that the situation in which you learn mathematics affects the mathematics that you learn. That idea was not well understood or even thought about much before the 1980s. (For a recent critique of the sociocultural movement, see Jorgensen, 2014.)

One of the most difficult lessons learned during the new math era (see the last chapter of Howson et al., 1981) was to recognise that the teacher was the critical person in curriculum reform. That is, if the teacher did not understand why the change was being made or what the change was, it did not matter what materials you gave to the teacher. The teacher had to be part of the process of understanding what is going on and fitting it into the culture of the classroom.

Many new math reformers began their efforts with the view that the curriculum would be brought up to date mathematically if they could simply get their new syllabuses and textbooks into the hands of students and teachers. By the end of the era, they had come to see that much more was required. At the crux of any curriculum change is the teacher. The teacher needs to understand the proposed change, agree with it, and be able to enact it with his or her pupils—all situated in a specific educational and cultural context. (Kilpatrick, 2012, p. 569)

Another lesson was that every country has a unique classroom culture when it comes to the teaching of mathematics. Some countries have connections to each other’s classroom cultures. Around the world, however, there are many differences between cultures. In some cases, for example, the teacher is expected to pose all of the problems, and in other cases, the book is supposed to have the problems; all the teacher does is help the students work. Countries differ quite a bit on that question.

Another part of the sociocultural turn relates to whether teachers work together on mathematics instruction. In some countries, each teacher just closes the door and does what she or he wants to do. In other countries, teachers, at least in principle, work together and help each other change. In our study of a precalculus course in the United States (Kilpatrick et al., 1996), we found that only when groups of teachers worked together did one see good curriculum change. When the teachers tried to make the change individually and alone, there were so many barriers and problems that it was not successful. It was teachers working together that made the difference.

Another lesson that I hope has been learned is that people who want to research the curriculum cannot do it without engaging with the people in the classroom. The work of those educators who are going to be doing the reforms, creating the materials, and creating the teacher development plans cannot be separated from the research and has to be tied into it. I think some researchers have made the mistake of going to study the curriculum as if it was out there. But they need to be a part of the change in order to study it.

One theme of the book Mathematics Curriculum in School Education (Li & Lappan, 2014) is that mathematics educators have not done a good job of studying how the curriculum change process works or could work in schools around the world. We just do not know, and that is a sort of a first step. Despite an enormous amount of curriculum development work, we do not have an enormous amount of curriculum development research. That is a challenge for the future.

The Context of the United States of America

During the new math era, when we wrote Howson et al. (1981), U.S. politicians did not have any connection to the school mathematics curriculum. There were almost no cases of politicians anywhere saying, “Vote for me, and we will have this curriculum in the schools.” An exception was West Germany, where there were politicians who took different sides on the school mathematics curriculum. That was, however, the only case I ever heard of. In the United States today, however, there are politicians who say, “If you elect me, we will go back to that curriculum; we will not follow this curriculum.” And in particular, the proposed Common Core State Standards in Mathematics (Li & Lappan, 2014, pp. 38–40; see Chap. 33)) have been debated.

We have a movement to privatise school education, and that movement is caught up with some politicians on one side and other politicians on another side. Somehow the mathematics curriculum gets connected with that conflict. It started largely with the question: Should we teach mathematics to everyone, should we teach it just to the people who deserve it, or should we have different curricula for different pupils? Politicians have gotten into that conversation to say, “Well, these people are trying to teach the same mathematics to everybody; they are ruining mathematics.” There are some mathematicians who say that, too. Somehow U.S. politicians, mathematicians, and mathematics educators are involved in discussions today that they were never involved in during the new math era. It was not a political issue at the time.

The United States is almost unique in the fact that we do not have a ministry of education that establishes the school curriculum. One of the articles of faith for the U.S. public is that we do not want Washington telling us what our curriculum should be and what we should be teaching. All of our efforts in recent years have been to bring some structure into the school mathematics curriculum across the country, and having to face up to a public that says, “We don’t want this,” and “Who are you to tell us what to do?”

The fact that we have a National Council of Teachers of Mathematics (NCTM) setting up a curriculum standards program is very unusual. I do not know of any other country that has something like that happening. I have heard people saying, “Who chose the NTCM to do this work?” Well, they decided to do it on their own, and the government did not set it up.

The government has, however, in some cases embraced it. That is one of our problems. We have had political problems that can be attributed mostly to the fact that we do not have a national curriculum. Some people think we should have one, and other people say no. We have never had a national curriculum except informally. Therefore, there are a lot of divisions about that. If you start offering something as a core curriculum that everybody should work on, you will get some politicians saying, “Go ahead,” and parents and others saying, “Don’t do that.” We have a somewhat special situation.

Elsewhere around the world, there seems to be more acceptance of a national curriculum. There is a wonderful quotation in the book by Howson et al. (1981, p. 58) that I am fond of citing. Essentially, it goes back to the time when the United Kingdom did not have a national curriculum. At that time, a French school inspector was quoted as saying that in the UK, every teacher was supposed to be going his or her own way, but nobody was, whereas in France, everyone was supposed to be doing the same thing, but nobody was. That about sums up the difference between what politicians say and what teachers do.

The United States continues to grapple with the question of whether we should teach the same mathematics to everyone. Can everyone learn the same mathematics? One of the ideas during the new math was that we ought to have a standard curriculum. It might take some students longer than others to learn that mathematical material, but it ought to be the same for everybody. That was the general idea proposed in the new math era. That idea is, however, not widely accepted in the United States today. We have many cases in which students are given a test at the end of a certain grade. If they do not do well on the test, they are put into one set of classes; and if they do well, they are put into another set of classes. So, we have layers of school mathematics. If you do well on a test, you get a certain mathematics, and if you do not do well, you get something else.

This sorting happens in different ways in different parts of the country. There are schools that have different primary courses in mathematics for different students, but the differentiation is usually made in the middle grades. Typically, a line is drawn around Grade 8, and if you pass, you go into one program, and if you do not pass, you go into another. But in some cases, it happens earlier than that, in the primary grades. It almost never happens that students are kept together as a group all the way through to the twelfth grade. So, we have not figured out what we as a country want to do. Some reformers say, “We should keep kids together in the same class to learn mathematics regardless of what mathematics we are teaching.” But there are others who say, “No, we have to separate them. Some of them are going to do well, and others are not going to do well. We should not put those students into the same class.” It is a political issue in many places.

Each country has to deal with the question of when to start differentiating the curriculum. How do we give students choices, how do we give anybody a choice, and who chooses? The teacher? The parents? The students? What are the paths that students can take? When do they start taking mathematics, and do they have to take it every year? Those are all questions that each school system, or each nation, has to decide. Are we going to teach the same mathematics all the way through school? Most places say no, we are not.

International Comparisons in Mathematics Performance

The rise of international comparative studies of mathematics performance – such as the Trends in International Mathematics and Science Studies (TIMSS) and the Programme for International Student Assessment (PISA) studies (Li & Lappan, 2014, see theme D) – has had both positive and negative impacts on the school mathematics curriculum. One positive impact has been that it has made some countries more aware of what is happening in other countries, and what their curriculum looks like. For all of us, it has allowed us to see across the world what students can do and what they cannot do. I think it has, in general, been positive for people to see what students in their country can do, and then to compare that with the performance of students in other countries.

One negative impact stems from the problem that all these studies make use of artificial curricular frameworks that have been drawn up for a different purpose. I have criticised efforts by American educators to try to use a mixture of data to make points about U.S. schools (Kilpatrick, 2011), because TIMSS is one thing, and PISA is another. You cannot mix the two – that is one issue. Another is that these frameworks are pretty arbitrary. PISA is trying primarily to get a picture of how fifteen-year-olds can deal with applications of mathematics, whereas TIMSS is trying to get a picture of how well kids at different levels, say eighth grade, come out of the mathematics program. What can they do, and what can they not do? All of that is somewhat arbitrary.

I remember a recent conference in Malaysia where I heard a mathematics educator from Singapore say that they were going to look at how the Singapore students did on the different kinds of questions in PISA, and then they were going to change their curriculum to deal with the places where the students were not doing so well. That struck me as completely backwards. You do not want to use such a framework to say this is how our curriculum should be. You should decide what your curriculum is, and if it does not match what PISA has, okay, it does not match it. I do not accept the idea that the people from Singapore should be taking the PISA framework as the gold standard.

I have worked with measurement people in putting some of these framework documents together. Those documents reflect judgments as to what content questions should be included on the assessment instruments and what should not.

I remember an international content experts’ meeting for PISA in which at one point we discussed questions about conversion from Fahrenheit to Celsius units. For U.S. educators, such questions would be reasonable to include. We still use both the imperial system and the metric system, and U.S. students are expected to be able to convert measurements from one system to the other. Further, such conversion is a worthwhile mathematical exercise. For most of the rest of the world, however, items dealing with conversion do not make much sense. They are not part of the school curriculum. The experts threw those questions out of the PISA pool because they applied to only one country.

These frameworks and item pools are arbitrary constructions by experts. Who says, however, that they should be what the people in a given country are using as their gold standard – as their framework? That is a problem, I think, with these international comparative studies. They are being misused when the framework is taken as the thing that we want students to be able to do. A framework can be helpful. It can give some general idea of how your students are doing on this topic or that topic. To use it, however, as an overall evaluation of what your country is doing is, I think, a big mistake. Many of the concepts you are treating in school mathematics may not be on the test, but they might be important concepts that your students are learning. So why not keep those concepts there?

I understand that in order to make comparison you have to have a common measuring stick, but you do not have to take that measuring stick as the endpoint for your curriculum. That is where I think the problem is. If you using the measuring stick to say this is what we want, you have not solved the curriculum problem for your country. The frameworks are a kind of consensus documents. You and your country may be teaching something important and very good, and getting good outcomes. But it may not be measured on TIMSS or PISA. Does that mean you should throw it away? I do not think so.

I understand that TIMSS and PISA can a strong influence on educational policy, but that has its downside as well. In the United States, our students do not do well on some problems, but we very seldom look closely at the PISA results. TIMSS seems to dominate our attention when compared with PISA. That is kind of crazy, too, because I think both projects have something to tell us. It is just that the PISA message does not come through very clearly. People get into comparisons between states, for example, or between school systems on the basis of these tests. That is not a good idea.

We have not yet learned to put a distance between ourselves and these results. I think that as the results pile up, and as people get used to these situations, it may get better, because then they may stop being attracted by the disparities. The results tend to stay relatively constant, so there is not much to be gained from the way the results are being reported. I think there is a kind of lack of attention to what is happening, which is probably a good thing.

A potential contribution of PISA is to increase attention to mathematical literacy as an educational outcome related to effective citizenship. In some countries – including Japan, Korea, and Denmark – the curriculum is based on competencies: not contents but processes. By considering mathematical literacy rather than specific content knowledge, mathematics educators in these countries are looking for better outcomes from school mathematics, which is a good thing. To the extent that PISA gives us some ideas about students’ mathematical literacy, it can be quite helpful. Unfortunately, however, what usually happens when the results are reported, at least in the United States, is that all we get are numbers in the newspapers: Japan was here, and the U.S. was there. We do not get any discussion of the mathematical literacy of the U.S. students or the Japanese students.

Final Comments

The book Adding It Up (Kilpatrick et al., 2001) talks about mathematical proficiency and offers a framework for studying mathematical proficiency. It was an attempt to say that if you are aiming for mathematical proficiency, you need to think about more than just content and process, you need to think about other dimensions that are being dealt with in school mathematics. The metaphor of a braid for proficiency – strands that are being developed along the way – is a metaphor for how the curriculum might work that is different from the metaphors discussed by Howson et al. (1981).

So, I think the idea of curriculum as a process, and one that needs to be shaped by the situation in the school, the situation in the country, the situation in the classroom – all of that has changed from what it was in the 1980s. Today, I would say we are moving much more toward recognising that the goals for school mathematics may be different across different school systems, countries, and situations. Each country has to figure out what its goals are, and in what directions it wants to go.

The school mathematics curriculum is diverse and multi-dimensional, which makes it impossible to capture well in a single study, framework, or trend (Li & Lappan, 2014, pp. 6–9). The levels of school mathematics range from the intended curriculum (goals prescribed in policy documents) to the implemented curriculum (what is taught in classrooms) to the achieved curriculum (as seen in students’ mathematics performance). A single research study can address only some aspects of those curriculum levels.

The bipolar nature of school mathematics, in contrast, shines through regardless of the curricular context or level. We have learned since the new math era that school mathematics is complicated, contextualised, not easily changed, and not easily studied. The bipolarity of school mathematics offers a possible entryway into studying it in context and retaining much of its complexity.