Mathematics is independent of culture

The counterpoint to the “mathematics is independent of culture” perspective expressed above is that knowledge of any type, but specifically mathematical knowledge, is a powerful vehicle for social access and social mobility. Hence lack of access to mathematics is a barrier – a barrier that leaves people socially and economically disenfranchised. For these reasons, noted civil rights worker Robert Moses declared that “the most urgent social issue affecting poor people and people of color is economic access…. I believe that the absence of math literacy in urban and rural communities throughout this country is an issue as urgent as the lack of registered Black voters in Mississippi was in 1961.” (Moses, 2001, p. 5). Who gets to learn mathematics, and the nature of the mathematics that is learned, are matters of consequence. This fact is one of the underpinnings of the math wars. It has been true for more than a century.

Anthropologists and historians may differ with regard to details, depending on their focus. Rosen (2000), an anthropologist, argues that over the past century or more there have been three “master narratives” (or myths) regarding Education , “Each of which celebrates a particular set of cultural ideals: education for democratic equality (the story that schools should serve the needs of democracy by promoting equality and providing training for citizenship); education for social efficiency (the story that schools should serve the needs of the social and economic order by training students to occupy different positions in society and the economy); and education for social mobility (the story that schools should serve the needs of individuals by

providing the means of gaining advantage in competitions for social mobility).” (Rosen, 2000, p. 4)

Stanic (1987), an historian of mathematics education, describes four perspectives on mathematics that battled for dominance in the early 1900s, and then throughout the century. Humanists believed in “mental discipline,” the ability to reason, and the cultural value of mathematics. That is: learning mathematics is (by virtue of transfer) learning to think logically in general; mathematics is also one of civilization’s greatest cultural achievements, and merits study on that basis. Developmentalists focused on the alignment of school curricula with the growing mental capacities of children. (During the heyday of Piagetian stage theory, some developmentalists argued that topics such as algebra should not be taught until students became “formal thinkers.”) Social efficiency educators, identified above by Rosen, thought of schools as the place to prepare students for their predetermined social roles. In opposition, social meliorists (similar to those who believed in education for social mobility) focused on schools as potential sources of social justice, calling for “equality of opportunity through the fair distribution of extant knowledge” (p. 152).

To these social forces shaping mathematics curricula I would add one more. Mathematics has been seen as a foundation for the nation’s military and economic pre-eminence, and in times of perceived national crisis mathematics curricula have received significant attention. This was the case before and during both world wars, the cold war (especially the post-Sputnik era, which gave rise to the “New Math”), and the U.S. economic crises of the 1980s (see A Nation at Risk: National Commission on Excellence in Education, 1983).

With this as background, let us trace both numbers and curricular trends. The 20th century can be viewed as the century of democratization of schooling in the United States. In 1890 fewer than 7% of the 14 year-olds in the United States were enrolled in high school, with roughly half of those going on to graduate (Stanic, 1987, p. 150). High school and beyond were reserved for the elite, with fewer students graduating from high school back then than earn Masters and Ph.D. degrees today. In short, “education for the masses” meant elementary school. In line with the ideas of the social efficiency educators, an elementary school education often meant instruction in the very very basics. For example, one set of instructions from a school district in the 1890s instructed teachers that their students were to learn no more mathematics than would enable them to serve as clerks in local shops (Resnick, 1987). In contrast, the high school curriculum was quite rigorous. High school students studied algebra, geometry and physics, and were held to high standards. In the 1909-1910 school year, roughly 57% of the nation’s high school students studied algebra and more than 31% studied geometry. (A negligible 1.9% studied trigonometry, which was often studied at the college level;

calculus was an upper division college course.) (Jones & Coxford, 1970, p. 54).

By the beginning of World War II, almost three-fourths of the children aged 14 to 17 attended high school, and 49% of the 17 year-olds graduated (Stanic, 1987, p. 150). This expanding population put pressure on the system. Broadly speaking, the curriculum remained unchanged while the student body facing it was much more diverse and illprepared than heretofore. (The percentage of students enrolled in high school mathematics dropped steadily from 1909 to 1949, from 57% to 27% in the case of algebra and from 31% to 13% in the case of geometry. It must be remembered, however, that there was a ten-fold increase in the proportion of students enrolled in high school, as well as general population growth. In purely numerical terms, then, far more students were enrolled in algebra and geometry than previously.) As always, in times of national crisis the spotlight tends to focus on mathematical and scientific preparation for the military and for the economy. “In the 1940s it became something of a public scandal that army recruits knew so little math that the army itself had to provide training in the arithmetic needed for basic bookkeeping and gunnery. Admiral Nimitz complained of mathematical deficiencies of would-be officer candidates and navy volunteers. The basic skills of these military personnel should have been learned in the public schools but were not.” (Klein, 2003) The truth be told, however, there was not a huge amount of change in the actual curriculum, before or after these complaints.

The next major crisis did affect curricula, at least temporarily. In October 1957, the USSR caught the United States off guard with its successful launch of the satellite Sputnik. That event came amidst the Cold War and Soviet threats of world domination. (It was more than 40 years ago, but I still remember Nikita Khruchschev banging his shoe on a table at the United Nations, and his famous words “We will bury you.”) Sputnik spurred the American scientific community into action. With support from the National Science Foundation, a range of curricula with “modern” content were developed in mathematics and the sciences. Collectively, the mathematics curricula became known as the New Math. For the first time, some of the content really was new: aspects of set theory, modular arithmetic, and symbolic logic were embedded in the curriculum. The full story of the New Math should be told (though not here); it shows clearly how curricular issues can become social issues. Specifically, it provides a cautionary tale for reform. One of the morals of the experience with the New Math is that for a curriculum to succeed it needs to be made accessible to various constituencies and stakeholders. If teachers feel uncomfortable with a curriculum they have not been prepared to implement, they will either shy away from it or bastardize it. If parents feel disenfranchised because they do not feel competent to help their children, and they do not recognize what is in the curriculum as being of significant value (and what value is someone trained in standard arithmetic likely to see in studying “clock arithmetic” or set theory?) they will ultimately demand change.

The one-liner, which is an over-simplification but represents accepted wisdom: “By the early 1970s New Math was dead.” (Klein, 2003). In a reaction to what were seen as the excesses of the New Math, the nation’s mathematics classrooms went “back to basics” – the theme of the 1970s. In broad-brush terms, the curriculum returned to what it had been before: arithmetic in grades 1-8, algebra in 9th grade, geometry in 10th, a second year of algebra and sometimes trigonometry in 11th, and “pre-calculus” in 12th.There are various opinions about the level of standards and rigor demanded of students – some will argue that less was being asked of students than before, and some will disagree – but in broad outline, the curricula of the 1970 resembled those of the pre-Sputnik years. In compensation for the “excesses” of the 1960s, however, the back-to-basics curricula focused largely on skills and procedures.

By 1980, the results of a decade of such instruction were in. Not surprisingly, students showed little ability at problem solving – after all, curricula had not emphasized aspects of mathematics beyond mastery of core mathematical procedures. But, performance on the “basics” had not improved either. Whether this was due to back to basics curricula being watered-down versions of their pre-Sputnik counterparts, to a different social climate after the 1960s where schooling (and discipline) were de-emphasized, or because it is difficult for students to remember and implement abstract symbolic manipulations in the absence of conceptual understanding, was (and is) hotly debated. What was not debated, however, is that the mathematical performance of U.S. students was not what it should have been.There was one lasting change due to the post-Sputnik reforms. Calculus entered the high school curriculum for those students on a more accelerated track.

In response, the National Council of Teachers of Mathematics published An Agenda for Action” in 1980. NCTM proposed that an exclusive focus on basics was wrong-headed, and that a primary goal of mathematics curricula should be to have students develop problem-solving skills. “Back to basics” was to be replaced by “problem solving.” From the jaundiced perspective of a researcher in mathematical thinking and problem solving, what passed for “problem solving” in the 1980s was a travesty. Although research on problem solving had begun to flower, the deeper findings about the nature of thinking and problem solving were not generally known or understood. As a result, the problem solving “movement” was superficial. In the early 1980s, “problem solving” was typically taken to mean having students solve simple word problems instead of (or in addition to) performing computations. Thus a sheet of exercises that looked like 7 – 4 = ?

might be replaced by a sheet of exercises that looked like John had 7 apples. He gave 4 apples to Mary.

How many apples does John have left? But otherwise, things remained much the same. Part of the difficulty lay in the mechanisms for producing textbooks, a topic discussed in greater detail below. Soon after the publication of An Agenda for Action, major publishers produced “problem solving editions” of their textbooks.