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November 2007 | Volume **65** | Number **3**

**Making Math Count**
Pages 38-42

Zalman Usiskin

Having one set of national standards for mathematics is no guarantee of improved student performance.

At its 2007 Annual Meeting in March, the National Council of Teachers of Mathematics (NCTM) organized a panel on the topic, “Would U.S. Students and Teachers Benefit from Consensus on National Curriculum Standards for Mathematics?” I was one of the panelists,^{1}
and the question struck me as strange, particularly because it originated from the NCTM itself. I thought we had consensus on national standards in 1989, when the NCTM came out with its *Curriculum and Evaluation Standards* (1989). The entire mathematical sciences community, including the Mathematical Association of America and the American Mathematical Society, endorsed the standards.

What happened? In 1996, a small group of mathematicians, worried that the 1989 standards did not convey their view of mathematics, rallied other mathematicians to rail against NCTM's standards. The mathematics education community had thought there was consensus, but there wasn't.

The fact is that if we have any standards with backbone in them, there will be those who do not agree, and we will not have consensus. However, lack of consensus is not an argument against the existence of standards. I believe it is helpful to have guidelines against which we can examine what we are doing in schools and what we would like to do.

So the question before the panel was not just about national curriculum standards in mathematics, because we already have them. We have seen examples of standards in all the subject areas, and it seems that most people like having some framework for discussion, even when they disagree with its details. The question was, more accurately, whether the United States should have national standards *with teeth in them*, that is, a single set of standards agreed to by the states with assessments attached, creating alignment of curriculum and testing.

As I see it, five basic arguments have been given for having a single set of national standards in mathematics. Each argument is tied to a problem that national standards would purportedly solve. I believe that not only are these links between problem and solution questionable, but given recent and past history, analysis of each argument suggests that national standards with teeth might exacerbate rather than solve the problem.

**Argument 1:** U.S. performance in mathematics is weak on international assessments. This endangers our economy. The highest-performing countries have national curriculums and national tests. It follows that having national standards would be more likely to improve U.S. performance in mathematics and help the economy.

*Rebuttal*: There are two steps to this counterargument. The first deals with the connection between national curriculums and international performance; the second deals with the connection between international performance and the economy. Regarding the first, most countries have national curriculums. However, some of these countries are the *lowest-performing*
countries of the world. The link between having a national curriculum and a country's performance in mathematics is tenuous at best (Wolf, 1998).

Moreover, we can draw a parallel between many U.S. states and countries with national curriculums. State curriculums have been in place for decades. No state has tried harder than California to have a strong central curriculum and textbooks aligned to that curriculum, yet mathematics performance in California remains near the bottom end of the distribution. During the past 15 years, even as California has flexed its muscle, its 8th grade scores on the National Assessment of Educational Progress (NAEP) have not kept up with the rest of the states. More generally, a study by Amrein and Berliner (2002) showed that states that instituted high-stakes tests have generally not outperformed states that rely on national measures, such as NAEP, ACT, SAT, or advanced placement scores.

This brings us to the second step of the counterargument—the connection between international performance and the economy. On the International Study of Achievement in Mathematics (later sometimes called the First International Mathematics Study) in 1963–64, the United States was the lowest-performing country for students in the last year of secondary school^{2}
and second-lowest at the 13-year-old level^{3}
(Husén, 1967). In 1980–81, in the Second International Mathematics Study, the United States was among the lower-performing countries at both age levels (McKnight et al., 1987; Robitaille & Garden, 1989).^{4}
The U.S. economy did not suffer as a result, either in the short or long term. In fact, U.S. performance in the Third International Mathematics and Science Study (TIMSS, later called the Trends in International Mathematics and Science Study) was better than at any other time (Mullis, Martin, & Foy, 2005). How we fare on the statistic used for international comparisons—mean national performance—seems to have little or no effect on the economy.

**Argument 2:** State standards show great variety in the expectations and in the grade levels of common expectations. This creates large textbooks and unnecessary redundancies and inefficiencies.

*Rebuttal*: The fact that state standards show variety is evidence that wise people do not agree on what should be in the curriculum or when it should be taught. But even our national standards show disagreement. Currently, three documents are viewed as attempts at national standards: NCTM's *Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics* (2006), which are ostensibly based on NCTM's *Principles and Standards for School Mathematics*
(2000), but which, in reality, are quite different; the College Board's *Standards for College Success* (2006); and the American Statistical Association's *Guidelines for Assessment and Instruction in Statistics Education, PreK–12* (2005). A draft document is also floating around from Project Achieve, titled *Secondary Mathematics Expectations* (Forgione & Slover, 2007), which is based on the mathematics benchmarks in an earlier document,
*Ready or Not: Creating a High School Diploma That Counts* (American Diploma Project, 2004).

Each of these documents was written by a committee ostensibly selected to represent different constituencies, if not diverse opinions; each represents some sort of consensus that disallows extreme positions. Even so, these documents are quite different from one another in basic beliefs. For example, NCTM's Curriculum Focal Points do mention technology, the College Board recommendations embrace technology, the American Statistical Association assumes technology in doing statistics, and Project Achieve views calculator technology negatively. The existence of such different positions in consensus documents indicates that there is no consensus today about the use of technology in school mathematics, certainly one of the key issues of our time.

Why are there such differences? For one thing, the research we have does not support the definitive placement of topics by grade. This is significant from a policy perspective: If state and national leaders agreed on what we should teach, then it might be reasonable to codify that agreement in national standards. But there is no agreement, and any set of national standards will disenfranchise those who disagree. In fact, no subject matter is as prescribed in school curriculums as mathematics. Yes, states may differ in their choice of grades at which fractions should be taught, but they do not differ in the *idea*
that fractions should be taught (Reys, 2006). Contrast this with literature, U.S. history, or science. The differences in mathematics curriculums are minor compared with these other content areas.

Our arithmetic curriculum in the United States is remarkably uniform in this era of calculators. In fact, when there is even one change suggested in the arithmetic curriculum, such as teaching a different algorithm for long division, people get up in arms. One mathematician criticized the algebra text developed by the University of Chicago School Mathematics Project because it used the acronym FOIL—first, outside, inside, last—in teaching the multiplication of binomials. As an example of the uniformity of the arithmetic curriculum, an algebra teacher anywhere in the United States can generally assume what arithmetic students have had regardless of the state they live in. However, that same teacher cannot make assumptions about what geometry students have had. In fact, geometry and statistics have more inconsistencies and need more guidance than other areas in mathematics, but the people who are most concerned about student performance are least concerned about these areas.

As for large textbooks, they may actually help teachers adapt material to the diverse students in their classes. They help, rather than harm, students.

**Argument 3:** Opportunities for students in the United States are unequal. National standards would ensure that all students are on the same playing field.

*Rebuttal*: Some people push the equity argument as *the* argument for national standards. Experience shows that this argument, too, is faulty. Children come into 1st grade in some communities two years ahead of children in other communities. These differences among students are generally not due to schooling but to the richness of the early environment. Are we to teach all these children in the same way? Around 7th grade, the difference in students' willingness to do homework becomes a major factor in student performance. Can we ignore that?

In Chicago a number of years ago, it was decided that all students had to receive credit in algebra to graduate from high school. This gave teachers two choices: to teach a standard algebra course and fail three-quarters of the class or teach a course at the students' level—but that would not be algebra. Any national standards that arrange goals by grade level will result in large numbers of student failures and, ultimately, usher in the More Children Left Behind era.

It is sometimes stated that many high-performing countries do not track their students. This statement is based on the fact that virtually all students in some of those countries seem to be taking the same set of courses. However, students in these countries take these courses at different schools with vastly different expectations, and they may take these courses at different ages. In Singapore, the slowest students in grade 4 are held back a year. In Singapore, Japan, and China, students are tracked into different schools beginning, usually, at grade 7. The algebra course taught in one school can be as different from the algebra course taught in another as are honors and regular algebra courses in the United States. Further, getting into these schools is one's ticket to getting into the better colleges and universities. By having tests to get into better
*secondary* schools, these countries track more stringently than the United States does. That is why the national tests at the end of grade 6 that determine a student's track are the source of so much study and pressure.

**Argument 4:** The United States is in a time of change. Schools, however, are often slow in adopting the latest developments. We could change the curriculum more quickly with national standards.

*Rebuttal*: We are without question in a time of change. However, judging from what we see internationally, having a national curriculum slows down rather than promotes change. There is almost no curriculum development in Japan, Singapore, China, Korea, and other countries we have often looked to as high performers (Usiskin & Willmore, 2007). The development comes from countries that do not have strong central curriculums. As we look to these high-performing countries for reasons for their high performance, they look to the United States for new ideas to improve their practices. In the United States, we have a tradition of local control of education, which, among other things, fosters innovation and improvement in curriculum and pedagogy. There is also much curriculum development in Australia and the Netherlands, where there are no strong central curriculums. A strong tradition of mathematics curriculum development in England has waned with the establishment of a national curriculum there in 1988.

What seems like good policy does not always translate automatically into practice. Thoughtful change requires that schools of all kinds have an opportunity to test new materials. Those of us in curriculum development in the United States always have more difficulty getting a school to try out new materials in states with state adoption or strong state assessments tied to their curriculum than in states with more lax control. Teachers are naturally reluctant to teach something that is not in their state curriculum and on the state tests. Personally, I would not have had a 40-year career in mathematics curriculum development had we had a national curriculum because neither I nor anyone else *could* have had such a career.

**Argument 5:** Local and state curriculums are often weak. With the best people forming national standards, we would be better assured of having a good curriculum.

*Rebuttal*: On the surface, this seems a reasonable argument. However, our education system has become politicized. Both President Ronald Reagan and our current president came into office with the agenda of getting rid of the Department of Education. Instead, President George W. Bush dismantled the Eisenhower National Clearinghouses that collected research over many decades, and the Department of Education has filled its mathematics advisory committees with people who are not experts in school mathematics and who have been vocal against the use of technology, cooperative learning, and applications of mathematics in the curriculum.

By spreading the control of the mathematics curriculum over the states and into local districts, we may have disasters in some places, but we avoid a national disaster. We are able to adapt to the extraordinary differences in students who come into these districts as well as to the major differences in state and local economies. We are able to take advantage of the strength and imagination of our nation's teachers, a core of individuals who are getting battered rather than assisted by the federal government's intervention in schools.

In the opening session of the 2007 NCTM Annual Meeting, author and foreign affairs columnist Thomas Friedman spoke of the strength of the horizontal nature of communication among individuals today, rather than the vertical, top-down nature of communication in the past. He compared the
*Encyclopedia Britannica* with Wikipedia to point out the power that individuals now have to help one another in this information age.

Everyone knows that one gets the best performance from students when they are actively engaged in their own learning. Similarly, we know that we get the best performance from teachers when they have a voice in what they teach. National standards with teeth have a tacit assumption: Our teachers cannot be trusted to make decisions about which curriculum is best for their schools. That is a recipe for disaster, a recipe for pushing the best people out of our profession, a recipe that in the long run will result in a devastated teaching force and, as a consequence, poorer performance from our students.

American Diploma Project. (2004). *Ready or not: Creating a high school diploma that counts*. Washington, DC: Achieve.

American Statistical Association. (2005).
*Guidelines for assessment and instruction in statistics education (GAISE), preK–12*. Available:
www.amstat.org/education/gaise

Amrein, A. L., & Berliner, D. C. (2002, March 28). High-stakes testing, uncertainty, and student learning. *Education Policy Analysis Archives, 10*(18). Available:
http://epaa.asu.edu/epaa/v10n18

College Board. (2006). *Standards for college success: Mathematics and statistics*. New York: Author. Available:
www.collegeboard.com/prod_downloads/about/association/academic/mathematicsstatistics_cbscs.pdf

Forgione, C., & Slover, L. (2007, February).
*Secondary mathematics expectations (Achieve)*. Paper presented at the national meeting of the Center for the Study of Mathematics Curriculum, Arlington, VA.

Husén, T. (Ed.). (1967). *International study of achievement in mathematics* (Vols. 1–2). New York: Wiley.

McKnight, C. C., Crosswhite, F. J., Dossey, J. A., Kifer, E., Swafford, J. O., Travers, K. T., et al. (1987). *The underachieving curriculum: Assessing U.S. school mathematics from an international perspective*. Champaign, IL: Stipes.

Mullis, I. V. S., Martin, M. O., & Foy, P. (2005). *IEA's TIMSS 2003 international report on achievement in the mathematics cognitive domains: Findings from a developmental project*. Chestnut Hill, MA: TIMSS and PIRLS International Study Center, Boston College.

National Council of Teachers of Mathematics. (1989). *Curriculum and evaluation standards for school mathematics*. Reston, VA: Author.

National Council of Teachers of Mathematics. (2000). *Principles and standards for school mathematics*. Reston, VA: Author.

National Council of Teachers of Mathematics. (2006). *Curriculum focal points for prekindergarten through grade 8 mathematics: A quest for coherence*. Reston, VA: Author.

Reys, B. (Ed.). (2006). *The intended mathematics curriculum as represented in state-level curriculum standards: Consensus or confusion?* Charlotte, NC: Information Age.

Robitaille, D. F., & Garden, R. A. (Eds.). (1989). *The IEA study of mathematics II: Contexts and outcomes of school mathematics*. Oxford, UK: Pergamon Press.

Usiskin, Z., & Willmore, E. (Eds.). (2007).
*Mathematics curriculum in Pacific rim countries—China, Japan, Korea, and Singapore: Proceedings of a conference*. Charlotte, NC: Information Age.

Wolf, R. M. (1998). National standards: Do we need them? *Educational Researcher, 27*(4), 22–25.

^{1}This article is based on the remarks I made as a panelist at NCTM's 2007 Annual Meeting.

^{2}The U.S. mean was 8.3, Sweden's was the next lowest at 12.6, and all other countries had means of at least 20.7.

^{3}The U.S. mean was 17.8, Sweden's was 15.3, and 9 of the 12 participating countries had means of at least 21.0.

^{4}For 13-year-olds, of 20 participating countries, the U.S. mean was 8th highest on statistics items, 10th highest on arithmetic items, 12th on algebra items, 16th on geometry items, and 18th on measurement items (all units in measurement items were metric units). For 17-year-olds “still engaged in the serious study of mathematics,” of 15 participating countries, the U.S. mean was 12th on number items, 13th on algebra items, 12th on geometry items, and 12th on functions/calculus items.

**Zalman Usiskin** is Professor in Education and Director of the University of Chicago School Mathematics Project;
z-usiskin@uchicago.edu.

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