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Excellence and Equality in Mathematics Education

For most countries, mathematical literacy is an expected outcome of schooling. This has been true for a long time. Mathematical literacy initially encompassed basic arithmetic skills such as adding, subtracting, multiplying, and dividing whole numbers, decimals, and fractions; computing percentages; and computing the area and volume of simple geometric shapes. More recently, the digitization of many aspects of life, the ubiquity of data for making personal decisions involving health and investments—as well as major social, economic, environmental, and policy decisions—have all reshaped what it means to be mathematically literate and to be prepared to be a thoughtful, engaged, and reflective citizen.

Manufacturing is particularly illustrative of this dramatic change. The skills required of people in the new high-tech manufacturing sector are not what they were twenty years ago. Countries with advanced economies still require a manufacturing workforce, but those individuals are not doing the assembling. They are maintaining complex websites, tracking inventory using sophisticated software, coding to fix problems, and operating computerized assembly machines and robots—all of which require a level of proficiency in mathematics that was not needed in yesterday’s manufacturing plants. The old adage that all that is needed to work in manufacturing is a strong back is no longer true. A strong background in technology and mathematics is now necessary.

This has been called the “paradox of American manufacturing jobs in 2017.” Many jobs have disappeared but, according to one report, “American manufacturers have actually added nearly a million jobs in the past seven years. Labor statistics show nearly 390,000 such jobs open.”1 Even though this example comes from the U.S. economy, such issues occur in most developed countries around the world. Manufacturing workers who have labored for fifteen years or more and who have been displaced typically do not possess the skills required of those who work in the new type of manufacturing plants. To address this issue, many companies, especially in Europe, offer apprenticeship programs to train employees in “mechatronics,” a combination of electrical and mechanical engineering together with computer programming. These apprenticeships are often open to individuals with no more than a secondary school leaving certificate.

In other words, there is a new “basic” for basic skills, demanding a reconsideration of what it means to be “literate” in mathematics, including the role of twenty-first-century competencies as they relate specifically to mathematics.

It is this discontinuity between the last century and the future that drives the need for education reform and the challenge of achieving it. Educators, policy makers, and other stakeholders need to revisit public education standards and policies. In the course of these deliberations new or revised policy responses to two general questions should be generated: (1) what do schools need to teach students to know and to do, and (2) which students need to be taught which things?2

Opportunity to Learn: What Is It? Why Is It Important?

Responses to these two questions lead to policies that can impact both excellence and equality. Unfortunately, U.S. public policy discussions around these questions often lead the country into a proverbial rabbit hole, and soon become so complicated that little effective policy reform ever results. In part public policy around education is often limited by structure and politics. In the United States, education is left to the states and to the local education authorities or school districts within states.

When student test performance on the national assessment (NAEP) is poor, there is often a cacophony of possible policy solutions offered. These typically revolve around such issues as teacher quality, the way the school is organized, the need for increased parental involvement, the lack of student motivation, the means by which the schools are financed, and many other responses.

Some of the policy options that arise in such a context inevitably reflect political interests and are not based on research related to factors that influence student performance. Other issues such as teacher knowledge and school financing reflect important concerns but, to a large extent, they only have an indirect impact on student learning through the curriculum that students study.

Considerable research, both domestically and internationally, shows that there is a more direct and fundamental relationship to student learning: the curriculum that structures the educational experiences of students in their schooling. Those structured experiences are what have been called students’ opportunities to learn (OTL). Curriculum policy, which defines those opportunities, is directly related to student performance. Put simply, students learn what they are exposed to; and they do not, most typically, learn that to which they are not exposed. This is especially true for mathematics. Consider calculus, quadratic equations, or similarity in geometric figures; these are not topics most students would come to know if they were not exposed to such content in their school instruction (aside from an exceptional few who possess exceedingly unusual talents).

Schooling, in a simple sense and most fundamentally, has been characterized as a black box having three components: a student, a teacher, and the content, skill, or reasoning ability that the student is to learn—that which is defined by the curriculum. The student brings his or her background to the situation, including a particular propensity or ability to do mathematics and a degree of interest in or motivation to study mathematics. He or she also has a family background which includes cultural and economic capital. The teacher brings the mathematics content knowledge, the knowledge of how to teach mathematics to students at a particular grade level, as well as classroom management skills. Finally, the curriculum defines the mathematics content (skills and reasoning) to be learned, in what sequence, and for how long in terms of time allocation.

It is the coverage of the content to be learned that is central to student learning: the curriculum is what provides the opportunity to learn. The role of the teacher is to shape and deliver that opportunity, with his or her background influencing the quality of that opportunity. In that way, the teacher has an indirect effect on student learning through the curriculum. In addition, the teacher may well also have a direct effect, on student motivation for example.

All of this is to say that content coverage, as defined by the curriculum, plays a very direct, central, and fundamental role on the opportunities to learn that a student experiences and what the student learns about mathematics through those experiences.

Curriculum Policies: Why Are They Important?

It is frustrating that U.S. educational policy often addresses issues concerning mathematics performance by considering factors that are related to learning only indirectly, to the exclusion of curriculum-related policy. In most countries, curriculum policy—especially the specification of topics that are to be included in the curriculum at each grade—is the key policy lever for addressing issues of educational excellence and equality. Typically, this is done at the country level by ministries of education, resulting in a country-wide definition of the intended curriculum and curricular expectations for all students in the specified grades (the focus in this paper being on grades K–8). In the United States, however, there are no federal standards that identify what content should be taught at what grade because such policies are determined at the state level. Even there that authority often devolves informally to the local school district and sometimes even to the individual school.

In 2010, through the cooperation of the National Governors Association (NGA) and the Council of Chief State School Officers (CCSSO), a set of mathematics standards was created. These standards have been adopted in some form by over forty-five states, giving the United States, for the very first time, a set of de facto national standards.3 Those standards were based on the research that was done as a part of the Third International Mathematics and Science Study (TIMSS) and, as such, are internationally benchmarked.4 They are focused, coherent, and rigorous, demanding more of students than was previously the case.

The original TIMSS study found that these characteristics of the top-achieving countries were lacking in American schools. The U.S. curriculum prior to 2010 lacked focus and covered more topics at each grade level than was the case worldwide. This led to the U.S. curriculum being characterized as a mile wide and an inch deep. Too often, topics were not covered in sufficient depth and as such needed to be repeated year after year.

The second characteristic of coherence had to do with the nature of mathematics itself as a very logical discipline, which implies a particular ordering of content that should progress to more demanding topics over the years. The U.S. curriculum did not represent that logical structure; more advanced topics were often introduced before the necessary prerequisites had been covered.

Finally, the rigor of U.S. mathematics education was not at a level commensurate with that of the top-achieving countries—or in fact with most of the countries in the world. This was most evident at eighth grade, where the U.S. curriculum was one to two years behind the curriculum of the top-achieving (A+) countries. The eighth grade for the A+ countries encompassed algebra and geometry, while for most U.S. children the curriculum was still focused on arithmetic including fractions, proportions, and ratios.

Achieving Excellence

The Organisation for Economic Co-operation and Development (OECD) sponsors a mathematical literacy test for fifteen-year-olds (Program for International Student Assessment, or PISA) that is administered every three years with the focus on mathematical literacy every third time within the sequence—in other words, every nine years. PISA 2012 was the most recent assessment that focused on mathematical literacy; the next one with this focus is scheduled for 2021. The PISA 2012 results revealed that U.S. performance was below the OECD international average and ranked 27th out of the 34 participating OECD countries. Clearly, the United States needs to continue the quest for excellence and perform at a higher level on the next PISA mathematical literacy test. In 2012, the more demanding, new U.S. mathematics standards referred to previously were only beginning to be implemented across the states (and were not yet in place in some states). But to the point of this paper, what can we as a nation learn from the 2012 PISA results relevant to the goal of achieving excellence in student performance on the next mathematical literacy assessment?

PISA not only administered the literacy assessment to each country’s fifteen-year-olds (in the United States these are mostly tenth graders) but also collected data concerning the opportunities to learn mathematics that each student experienced. Students were asked questions about their cumulative—up to the grade they were in at that point—exposure to various mathematical topics such as linear equations, trigonometry, and geometric similarity. From these data an index was developed reflecting their formal mathematics content exposure. Across all of the participating OECD countries, this formal mathematics OTL measure was related to performance on the PISA literacy test.5 Over the thirty-three participating OECD countries (Norway did not provide OTL data), the average effect on performance was equivalent to about one year’s higher performance. In other words, a one-unit change in the formal mathematics OTL index predicted a performance on the test that would occur given about one year of additional coverage of mathematics content—a strong relationship. The effect for the U.S. students was larger than that of the OECD average—more than a year. For some countries, like Korea, the results predicted almost a two-year difference in performance.

These results bring us back to the issue of how the United States can improve the level of mathematical literacy for its fifteen-year-olds. The OTL indexes suggest that the answer is rather straightforward: U.S. students must be exposed to a more rigorous and demanding mathematics curriculum, one that is similar in content coverage to that of the countries that perform exceptionally well in PISA. Twenty years ago, that was what was determined from the 1995 TIMSS international benchmark. It was those results that led to the current set of U.S. standards which have been adopted by most states, at least in terms of content if not in name.

The adoption of those standards for mathematics was a policy decision that put the United States on the right path. Yet, from a policy perspective, the main issue now is providing the necessary support for the implementation of those standards. Again, it is the implemented curriculum—what is delivered by the teacher—that has the potential direct effect on student learning. One important policy change affecting this area would be to adopt statewide requirements regarding teacher professional development and providing the necessary financial support. Another would be to support state efforts to develop better instructional support materials.

In addition, PISA collected another OTL measure especially relevant to mathematical literacy, yet less clear in its implementation. Students were asked to indicate the extent to which they were exposed to real-life problems as a part of their mathematics instruction and testing. This OTL measure was called applied mathematics and it was also related to PISA performance on the literacy test even when controlling for the relationship of formal mathematics and socioeconomic status to performance. The effect of applied mathematics was similar in size to that of formal mathematics on average for the OECD countries—about one year. The effect, however, was not statistically significant for all countries, but was for 79 percent of the more than sixty participating countries in PISA 2012.

The nature of the relationship of applied mathematics was different than it was for formal mathematics. For formal mathematics, the relationship was linear, leading to the conclusion that more formal content coverage predicted higher test performance. For applied mathematics, this relationship was also true, but only up to a point, after which more exposure was negatively related. This means that more exposure to applied problems beyond a certain amount actually predicted a lower score on the test. The specific amount at which this was true could not be clearly established from this research.6

For the United States, the striking result was that the amount of real-world applied mathematics that students indicated they had been exposed to was not related to the performance, at least not enough to be statistically significant. The implication of this is clear—what students viewed as applied mathematics did not add any value to what was provided by formal mathematics in their schooling.

What exactly that means is not determinable from the PISA data. It may be that what students perceive as applied mathematics is not of high quality and, therefore, has little impact on their PISA performance. This interpretation will have to remain a hypothesis, but it does seem consistent with the relatively low level of U.S. average PISA performance.

Given these findings, rethinking the place of real-world mathematics problems in the U.S. curriculum may be necessary, especially focusing on the quality and improvement of such exercises.

Equality

Turning to equality, we move our focus from average performance to the variation in performance, both across U.S. schools and across students within schools. Equality of opportunity addresses the question posed in the introduction as to which students receive what opportunities to learn mathematics. If, as reported in the previous section, opportunities related to formal mathematics and applied real-world problems are both related to mathematical literacy in most countries, then who gets what mathematics content opportunities, and for what amount of time, become important questions of resource allocation and equitable distribution.

Recent discussion in the United States and around the world has focused on the question of income inequality and its impact on society in general and on democracy in particular.7 Clearly, concern over the impacts of economic inequality on U.S. society is far-reaching and has implications for public schooling. Inequality in schooling is both a consequence of and a contributor to economic inequality. As neighborhoods become more segregated by social class, and given that public schools are typically oriented around serving the children who live in the neighborhood, the socioeconomic status of a neighborhood could affect its school’s curriculum. In this way, economic inequality could influence what students learn.

Conversely, if the schools themselves are purveyors of social class inequality, in terms of student opportunity to learn, then it is likely that economic inequality will continue as these students enter the workforce with lower mathematical literacy. This outcome undermines the fundamental belief we as a nation have in the American dream: if you work hard and take advantage of the opportunities provided in school, you will have a better life and can climb the economic ladder.

In the end, to break the cycle of economic inequality leading to inequality in schooling, which in turn exacerbates economic inequality, students’ opportunities to learn must be defined and structured by the intended and implemented curriculum, and must be equitably distributed across all schools and for all students within schools. A strong K–8 curriculum must become a reality for all. But is this the reality in American schools?

To answer this question, it is helpful to turn to the same PISA 2012 data used in the previous section on excellence. One additional variable included in the following analyses is a measure of the socio-economic status (SES) of the student and his or her family. This index is based on variables including the mother’s and father’s educational attainment and occupations, together with the presence of various family possessions, particularly those related to education, such as the number of books in the household. The model fitted to the data specified both a direct and an indirect relationship of SES to performance on the PISA literacy test.

The indirect effect occurs if the SES of the student is related to the students’ school-related opportunities to learn formal mathematics. As reported in the previous section, opportunity to learn was found to be related to mathematical literacy in all OECD countries. In fact, that relationship was quite strong.8

Consequently, if wealthier students are given greater exposure to the type of mathematics needed to become mathematically literate, while poorer students are given exposure to less demanding, more remedial mathematics content, then the indirect effect of SES on student performance is added to the direct effect of social class—mostly reflecting family background and home environment—to produce the total effect of social class on mathematical literacy. The size of this total SES effect–related inequality across the OECD countries is found in the figure below.

Equality in schooling only occurs when SES is not a determinant of a student’s opportunities to learn: when all children in the elementary and middle school years are exposed to the same mathematics content coverage. To do otherwise makes the American dream a myth and, in the long run, likely reinforces economic inequality. Put simply, the rich get richer and the poor get poorer—a proverbial saying that unequal opportunities in education can make literally true.

Public schooling is viewed worldwide as the essential way to a better future for students from lower classes; it is supposed to be the equalizer by reducing if not eliminating social class inequality, thus creating a level playing field. From these data we can see that the opposite is the sad reality. Education systems are actually exacerbating inequalities. Across thirty-three OECD nations, the percent of the total SES effect that was attributable to school related inequalities (the indirect effect) was one-third. In all OECD nations except Sweden, there was a significant indirect effect related to inequalities in mathematics content coverage related to social class. However, there was also considerable variation as the indirect effect of SES inequality accounted for 9–58 percent of the total SES inequality. The somewhat more optimistic interpretation of the variation across countries is the fact that there are some countries, such as Sweden, Estonia and Poland, in which the size of the additional school related inequality is substantially lower—about 15 percent or less.

For the United States, the estimated percentage of the total SES effect related to inequalities in schooling was 37 percent, about average. A good illustration of the impact of public policies around equality of opportunity is provided by a comparison of Sweden and the United States. If the total SES inequality of Sweden and the United States is compared, the United States, as many would expect, has greater inequality. On closer examination, however, the disparity in Swedish students’ performance directly related to home/family background is actually greater in Sweden, but Sweden adds virtually no additional SES inequality through schooling. The coverage of the country curriculum is distributed equitably, with virtually no relationship to SES. The United States, on the other hand, adds almost 40 percent to its total SES inequality because the coverage of the mathematics content in schools is strongly influenced by the SES of the student.

Although perhaps not as severe as many would expect, the PISA data clearly suggest that there is significant inequality in the distribution of OTL in the United States. Such school-related inequalities are probably a consequence of the large economic inequalities present in U.S. society, particularly as they create housing silos of SES homogeneity. These silos contribute to the perpetuation of economic inequalities into the next generation. There are inequalities between schools within states, within districts, and even within cities, often reflecting housing patterns. From an educational policy perspective, implementing rigorous, national curriculum standards may offer the best remedy in this case.

Less intuitively, perhaps, additional analyses reveal that the within-school inequalities in the United States are among the five worst in the world, accounting for almost half of the total U.S. SES effect (47 percent). Middle schools in the United States typically track students in mathematics based on decisions concerning what the school staff feel the student is capable of accomplishing. Sometimes those decisions are based on tests and other times teacher judgements or both, but the clear intent, no matter the method, is to sort children into different groups covering different content. Although not intentional, such sorting tends to be related to SES, thus creating inequalities in opportunity to learn.

Of course, the standard rationale for tracking is to avoid holding back high-aptitude students. But high-income American students, while scoring better than low-income Americans, still significantly underperform wealthy student cohorts from other countries. One explanation may be that tracking within schools is currently driven more by SES sorting, rather than focusing on delivering a more rigorous, internationally comparable curriculum for all children. In any event, the current methods of within-school tracking deserve more careful examination, as they contribute significantly to indirect SES inequality while any corresponding gains in performance seem insufficient to justify that effect.

Excellence and Equality: Can Both Be Achieved?

Excellence and equality can both be achieved. Indeed, Kate Pickett and Richard G. Wilkinson have observed that countries with greater equality have shown higher performances for students. From the PISA data, Canada in particular stands out.9 Comparable in many ways to the United States, Canada’s performance offers a note of encouragement that excellence and equality, although difficult to achieve simultaneously, can be achieved in an educational context similar to our own.

In the arena of equality of opportunity, the United States needs to re-examine previous policies that have led to where we are presently, most notably the sorting of students into different content-defined tracks—particularly in grades 7 and 8. The adoption of the new, internationally competitive standards by almost all of the states in one form or another over the last seven years is clearly the right step toward excellence. The next step is to ensure the effectiveness of this policy by encouraging all 15,000 local school districts to follow their state standards and not to create pockets where standards are not implemented.

As for the importance of achieving both excellence and equality in education, it is worth noting Elizabeth Anderson’s argument that social equality is the normative basis for democracy. We might, as I argued on an earlier occasion,

ask ourselves what the consequences would be for democracy and social order if we fail to support that equality. If a substantial number of citizens decided that their life opportunities, including those related to schooling, were not equal, that success in life had more to do with who one’s parents are than what one does, would they continue their obligations to the society that they would view (with some justice) as exploiting them?10

Perhaps we are beginning to see today the consequences of these developments. The sad irony is that if we achieve greater excellence in education without addressing inequality, the social class inequality only becomes more damaging for those left behind due to the indirect effect—and these are the very students for whom schooling is the best hope.

This article originally appeared in American Affairs Volume I, Number 4 (Winter 2017): 32–45.

Notes
1 Dan Sewell and Christopher S. Rugaber, “High-Tech US Plants Offer Jobs Even as the Laid-Off Struggle,” Associated Press, Aug. 15, 2017, http://www.philstar.com/world/2017/08/16/1729598/high-tech-us-plants-offer-jobs-even-laid-struggle.

2 The above paragraphs are drawn from a draft of an internal working report submitted to OECD.

3 The added emphasis on the word “national” reflects the widespread state acceptance of the standards, not that they were federally developed. In the final analysis, each state had the choice of whether or not to adopt them.

4 William H. Schmidt, et al., eds., Facing the Consequences: Using TIMSS for a Closer Look at U.S. Mathematics and Science Education (Dordrecht: Kluwer, 1999).

5 William H. Schmidt, et al., Why Schools Matter: A Cross-national Comparison of Curriculum and Learning (San Francisco: Jossey-Bass, 2001). See also William H. Schmidt, Curtis C. McKnight, and Senta A. Raizen, A Splintered Vision: An Investigation of U.S. Science and Mathematics Education (Dordrecht: Kluwer, 1997). See also William H. Schmidt, Hsing Chi Wang, and Curtis C. McKnight, “Curriculum Coherence: An Examination of US Mathematics and Science Content Standards from an International Perspective,Journal of Curriculum Studies 37, no. 5 (2005): 525–59. All these analyses were controlled for student socioeconomic status (SES).

6 William H. Schmidt, Leland Cogan, and S. Guo, “The Role That Mathematics Plays in College- and Career-Readiness: Evidence from PISA 2012” (unpublished manuscript, 2017).

7 See Thomas Piketty, Capital in the Twenty-First Century, trans. Arthur Goldhammer (Cambridge: Belknap, 2014); Ganesh Sitaraman, The Crisis of the Middle-Class Constitution: Why Economic Inequality Threatens Our Republic (New York: Knopf, 2017); Robert D. Putnam, Our Kids: The American Dream in Crisis (New York: Simon and Schuster, 2015); and Richard G. Wilkinson and Kate Pickett, The Spirit Level: Why Greater Equality Makes Societies Stronger (New York: Bloomsbury, 2010).

8 William H. Schmidt, et al., “The Role of Schooling in Perpetuating Educational Inequality: An International Perspective,” Educational Researcher 44, no. 7 (Oct. 2015): 371–86.

9 William H. Schmidt and Nathan A. Burroughs, The Trade-Off between Excellence and Equality: What International Assessments Tell Us,” Georgetown Journal of International Affairs 17, no. 1 (Winter/Spring 2016): 103–9.

10 William H. Schmidt and Curtis C. McKnight, Inequality for All: The Challenge of Unequal Opportunity in American Schools (New York: Teachers College Press, 2012).

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