VIII.4 Numeracy
Eleanor Robson
1 Introduction
Most of this Companion is rightly concerned with the theories and practices of professional mathematicians. But all human beings have ideas about numbers, space, and shape, and ways of putting these ideas to use. It could be said that numeracy is to mathematics what literacy is to literature: everyday, routine application versus expert, elite innovation. But while literacy is now a wildly fashionable subject of academic study, the word “numeracy” is not even recognized by my mass-market word processor. Yet an array of interesting work has been done on nonprofessional mathematical concepts, practices, and attitudes. They range from historical studies and ethnographies to cognitive analyses and developmental psychologies, and cover such diverse periods and places as ancient Iraq, the pre-Columbian Andes, and the European Middle Ages, as well as many parts of the contemporary world. By surveying selected studies on five broadly construed topics in numeracy and artisanal mathematics, I hope to make the case in this essay that numeracy is as valuable a topic of academic research as professional mathematics on the one hand and literacy on the other.
Mathematics has rarely been considered part of the sociology or anthropology of knowledge, as it has often been assumed to stand outside culture. That is to say, many people have held the view that one can only think mathematics, not think about it. Furthermore, such work as has been done on the place of mathematics in culture is fragmented: mathematical thinking in the developed world has tended to be studied by sociologists, but in the developing world by anthropologists; historians of mathematics have mostly taken as their subject the literate mathematics of the professional elite, while psychologists have generally focused on the acquisition of numeracy, by adults and children.
But, as we shall see, the way that societies and individuals regard mathematics is strongly contingent on many environmental factors. Educational, linguistic, visual, and intellectual cultures all shape mathematical thinking in different ways. That is not to say that there are no constraints, however. Humans all share basic anatomical similarities that influence our ways of thinking: we are approximately symmetrical about one vertical axis, for instance, which gives rise to arguably innate concepts of left and right, front and back. And we all have fingers and opposable thumbs and the ability to subitize (that is, to recognize the size of a small set without counting its individual members). This, Reviel Netz has argued, makes human beings uniquely good at manipulating small groups of small objects, which has given rise to sophisticated systems of accounting and coinage. We shall return to Netz’s work later.
The examples in this essay have been selected from studies of three very different clusters of world cultures. The ancient Middle East and Mediterranean (Egypt and Mesopotamia, classical Greece and Rome) have strongly influenced modern global culture in a variety of ways. Most obviously, the Euclidean tradition has been central to Western educational ideals for centuries, along with the teaching of Latin. And while the languages and writings of ancient Egypt and Mesopotamia are essentially nineteenth-century rediscoveries, their cultural influence runs in deep undercurrents throughout Western thinking, having percolated through classical and biblical learning. We should not be surprised, then, to discover the familiar as well as the alien in the world’s oldest evidence for numeracy and artisanal mathematics. By contrast, the cultures of the pre-Columbian Americas are important for their very lack of contact with the premodern old world and thus their isolation from modernity. Virtually extinguished by the European conquests of the sixteenth and seventeenth centuries, and yet structurally similar to many old-world societies, they give a useful sense both of the constraints on numerate practice and thinking and of their diversity. Finally, this article also draws material from studies of the contemporary Americas, both South and North, in an attempt to break down the traditional disciplinary boundaries between past and present and between the developed world and the developing world. Numeracy is a feature of all human culture, wherever and whenever we have lived, and this should be reflected in how it is investigated.
2 Number Words and Social Values
Number words are usually studied for their mathematical content. French, for instance, shows traces of a vigesimal system in words such as quatre-vingts, meaning “four twenties,” while the English word eighty is clearly derived from “eight tens.” But in all languages number words also have social values attached, especially the counting numbers and words for sets. This is a rather different phenomenon from mystical numerology such as that of Late Antique Neo-Pythagoreanism. For instance, Nichomachus’s book The Theology of Arithmetic (written in the second century B.C.E. but now known only from later summaries) assigned esoteric meanings to the first ten integers, understanding those numbers to represent fundamental attributes of the cosmos. But the social values of number words are often much more prosaic than that. English, for instance, has a variety of words for “group of three,” each of which is applicable to a particular range of objects and has particular social connotations. “Threesome” is not a synonym of “trinity” in everyday language, just as in musical terminology “trio” does not have the same referents as “triad” or “triplet.” There is nothing mystical or esoteric in the use of these words; it is simply that, in addition to their semantic content, these words also carry implicit qualitative information about the sort of objects that are being grouped (sexually active adults, divine beings, musicians, musical notes, criminals, babies), about which society and individuals tend to form value judgments.
That numbers have a “social life” was first recognized by Gary Urton in his ethnographic study of the Quechua-speaking inhabitants of the Bolivian Andes. Structurally, Quechua numeration is straightforwardly decimal, much like modern European number systems, and is written with Arabic numerals. This has ensured its survival side by side with Spanish, but the fact that it is not particularly exotic relative to Western norms has caused it to be somewhat neglected academically. However, as Urton shows, there are two predominant social aspects to Quechua numeration: family relations on the one hand, and the idea of completeness or “rectification” on the other. There are also clear boundaries around what may be counted and who may count them.
All Quechua number words are composed of a dozen basic lexemes—one to ten, hundred, and thousand—which may be combined additively or multiplicatively, just as in English the word thirteen means “three and ten” and thirty means “three tens.” Also as in English, Quechua number words tend to be a distinct lexical set; for instance, kinsa means “three” and nothing else. But where synonyms for cardinal numbers are fairly rare in English (one example is dozen for “twelve”), they are a normal part of Quechua speech:
- iskaypaq chaupin, “the middle of (sets of) twos,” used of the third item in a group of five;
- iskay aysana, “double puller” (because the symbol 3 looks like two handles);
- uquti, “anus” (because the symbol 3 also looks like a human bottom);
- uj yunta ch’ullayuq, “one pair, possessor of one standing alone” (2 + 1 = 3).
Family relations are most clearly visible in ordinal sequences, especially the names of the fingers, which are themselves important everyday counting tools. Urton lists six very similar sets of names, attested over the past 500 years. The most recent, collected by the Bolivian anthropologist Primitivo Nina Llanos in 1994, goes as follows:
- thumb, mama riru, “mother finger;
- index finger, juch’uy riru,”small[er] finger”;
- middle finger, chciwpi riru, “middle finger;
- ring finger, sullk’ci riru, “younger finger”;
- little finger, sullk’ciq sullk’crn riru, “younger sibling of the younger finger.”
Thus the thumb is considered both the oldest and the antecedent of the others and the little finger the youngest; this is true of all six attested variants of the finger names. The hands themselves are considered as two symmetrical halves of a unified whole—as are paired items in general. In Quechua, one hand alone (or indeed an odd number) is not in its natural state. As Urton explains:
[T]he motivation for two is the “loneliness” (ch’ulla) of one. “One” is an incomplete, alienated entity: it needs a “partner” (ch’ullantin). The principle and motivational force obtain . . . regardless of whether the unit that composes the “one” is indivisible (e.g., a single digit) or divisible (e.g., a hand with five digits).
And more generally, Urton shows that in Quechua, odd numbers (ch’ulla) are incomplete while even numbers (ch’ullantin, “the part together with its pair”) represent the normal state of being.
But in Quechua society not everything is permissibly countable, even when there is no obvious difficulty in doing so. For example, they inventorize their herds, on whom they are often heavily economically dependent, not by counting but by naming. It is thought that counting individualizes the constituent members of the inseparable group, and thereby threatens its unity and fertility. If a herd must be counted then only a woman may do so; it is an unacceptably effeminate action for a man to carry out.
While restrictions on counting are not a notable feature of contemporary English-speaking culture, taboos on particular numbers are still common Why is thirteen considered so unlucky, for instance, particularly in North American hotels or on Fridays, while seven is regarded as lucky? In ancient Babylonia (modern- day southern Iraq) in the second and first millennia B.C.E., seven was thought to be particularly uncanny and unworldly. There were seven heavenly bodies (the Sun, the Moon, five visible planets), seven books of the Epic of Creation, and seven nights in each phase of the Moon. Demons, both beneficent and malevolent, were said to operate in groups of seven.
The Babylonians’ primary numerical base for counting and recording groups of discrete objects was 60, factored into six groups of ten. The number 7 is, of course, the smallest one that is coprime to 60 and thus became a favorite subject of mathematics problems designed to be solved by trainee scribes. Further sexagesimal coprimes—11, 13, 17, 19—also featured prominently in ancient Babylonian mathematical problems and riddles. More often than not, however, the parameters were chosen in such a way that the tricky coprimes factored out or were otherwise disposed of, leaving an arithmetically innocuous answer:
I found a stone; I did not weigh it. I added a seventh. I added an eleventh. I weighed it: 1 mina. What was the original (weight of the) stone? The original stone was 
mina 8 shekels, 22 
grains. (180 grains = 1 shekel; 60 shekel = 1 mina, ca. 0.5 kg.)
It is probably otiose to speculate whether the difficult mathematical properties of seven led directly to its cosmological demonization; the link is never made explicitly in any surviving cuneiform sources. But just as Babylonian demons failed to adhere to the norms of human behavior, so certain integers did not conform to the numerical patterns of the sexagesimally regular majority and the conceptual tools were not yet in place to explain that phenomenon in mathematical terms.
3 Counting and Calculating
While anyone can have views on whether particular numbers are lucky or unlucky, lonely or partnered, the ability to manipulate numbers arithmetically, and to take pleasure in doing so, is not universally shared. Both personal cognitive skills and social constraints are at work here. Patricia Cline Cohen argues that there were two key factors in the rapid rise in numerical competence in the early nineteenth-century United States. It was not that people suddenly became smarter. On the one hand, the decimalization of money in the late eighteenth century meant that at last accountants, shopkeepers, and business owners were working with a single number base. At the same time, a new educational movement forsook the rote learning of arithmetical rules, applied mechanically to particular situations, for inductive instruction that encouraged pupils to calculate with fingers and counters, and in their heads, before they progressed to pen and paper. In this way some basic structural impediments were removed, both to the learning of number relationships, and to their application in commercial life.
Because modern decimal notation is a calculating system as well as a recording device, it is easily forgotten that other methods are just as effective. Indeed, for most communities, most of the time, numerals were simply a means to record the outcome of operations performed on the body or with other calculating tools. Finger counting and abacus use remained ubiquitous in the medieval Islamic world and Christian Europe long after knowledge of decimal numerals, together with AL-KHW
RIZM
’s [VI.5] treatise on how to use them, and cheap paper on which to write them, began to spread outward from Baghdad in the ninth century C.E. Their retention was not a knee-jerk reaction in the face of an overwhelmingly superior technology; rather, it took into account such factors as portability, speed of use, and a long-established trust in and institutional sanction of the old methods.
Indeed, it is difficult to overestimate just how old abacus calculation is. Reviel Netz identifies two evolutionary prerequisites for what he calls “counter culture,” by which he means the uniquely and ubiquitously human use of small objects to represent other objects that are being counted, in one-one or one-many relationships. One is physiological: one needs to be able to pick up and manipulate small objects such as pebbles or shells. All primates share this ability thanks to prehensile fingers and opposable thumbs. The other is cognitive: one must be able to subitize, or recognize the size of a small set of up to about seven objects, without counting them individually. Stringed-bead abacuses exploit this most obviously, whether in the Russian-style ten- bead variety, whose fifth and sixth beads are always a different color from the others, or in the Japanese version, whose strings contain just four unit-beads and one five-bead each.
But, as Netz so powerfully puts it, “The abacus is not an artefact: it is a state of mind” All one needs is a flat surface and a pile of small objects to act as counters. This extreme ephemerality makes the use of abacuses almost impossible to detect in the archaeological record, except in the rare cases where abacus counters can be recognized as such. Denise Schmandt-Besserat has argued that a sophisticated accounting system was developed in the Neolithic Middle East from the ninth millennium B.C.E. She proposes that the tiny, unbaked pieces of clay, crudely shaped into various simple geometrical figures and found in preliterate archaeological contexts from eastern Turkey to Iran, are ancient accounting tokens. It is certainly true that the earliest written numerals in the area, from southern Iraq in the late fourth millennium B.C.E., are marks on clay tablets that look remarkably like stylized impressions of such objects, and are visually distinct from the signs for the objects that were being counted, which were scratched onto the clay rather than impressed. It is also true that these earliest written records are almost exclusively accounting records, drawn up by temple administrators in the management of assets such as land, labor, and agricultural products. And from the fifth millennium B.C.E. onward, those tiny clay tokens are found in archaeological contexts—sealed into jars, for instance, or wrapped in little clay bundles, or carefully piled in the corners of storerooms—that are entirely compatible with their use as abacus counters. But Schmandt-Besserat’s claim for a universally standardized system across the Middle East from several millennia before then is not provable: there is no way of establishing that they were not sometimes gaming pieces, for instance, or sling shot, or any number of other possibilities, and certainly no way of determining what specific shapes signified and to whom.
In fact, ad hoc means of counting and measuring are still everyday occurrences in all our lives, even among those with a high level of formal mathematics education. A team of anthropologists and psychologists, headed by Jean Lave, observed newcomers to a Californian Weight Watchers scheme in the 1980s as they adjusted to careful quantification of the food they were allowed to consume on the diet. One participant, who had taken a calculus course at college, was asked to modify a recipe calling for two-thirds of a cup of cottage cheese so that it contained three-quarters of that amount. Lave recalls: “He filled a measuring cup two-thirds full of cottage cheese, dumped it out on a cutting board, patted it into a circle, marked a cross on it, scooped away one quadrant, and served the rest.” She comments:
Thus, “take three-quarters of two-thirds of a cup of cottage cheese” was not just the problem statement but also the solution to the problem and the procedure for solving it. The setting was part of the calculating process and the solution was simply the problem statement enacted within the setting. At no time did the Weight Watcher check his procedure against a paper and pencil algorithm, which would have produced 
× 
= cup. Instead, the coincidence of problem, setting, and enactment was the means by which checking took place.
In other words, there are many situations in many people’s lives in which potentially applicable literate, school-taught mathematical procedures are ignored in favor of equally effective nonliterate ones that produce the correct result with the tools at hand. Numeracy takes many forms, not all of which entail writing.
4 Measurement and Control
The Weight Watcher invented a system of cottage-cheese measurement that satisfied him in its accuracy and fulfilled his immediate culinary needs. But as individuals and social groups we also accept the accuracy and consistency of standardized measurement systems, and the institutional necessity of counting and measuring particular things but not others. Theodore Porter has written eloquently of the twentieth century’s growing “trust in numbers,” whether of census statistics or environmental data. But institutionally sanctioned quantification is often contested, and it frequently alters the very phenomenon that is being pinned down. Cohen’s description of nineteenth- century North America is more generally apposite:
What people chose to count and measure reveals not only what was important to them but what they wanted to understand and, often, what they wanted to control. Further, how people counted and measured reveals underlying assumptions about the subject under study, assumptions ranging from plain old bias . . . to ideas about the structure of society and of knowledge. In some cases, the activity of counting and measuring itself altered the way people thought about what they were quantifying: numeracy could be an agent of change.
Cohen and Porter both explore problems raised by early nineteenth-century census taking. Porter describes the obstacles that the under-resourced Bureau de Statistique faced in obtaining accurate population data in post-revolutionary France. Without resorting to the old class categorizations of the ancien régime, it needed to acknowledge the huge diversity of occupations and social structures across the country. To do so it relied on local officials to return a mass of quantitative data that was simply not readily available—and so the prefectures commissioned qualitative descriptions of their regions instead. As Porter puts it, in 1800 “France was not yet capable of being reduced to statistics.” Cohen analyzes the U.S. Census of 1840, which appeared to demonstrate a much higher rate of insanity among the black population in the abolitionist northern states than in the south. Pro-slavery factions took this as irrefutable evidence that slavery suited the black population much better than freedom did; abolitionists queried the trustworthiness of the census itself. Whether or not one chose to believe the data was more or less a matter of what one’s preexisting political convictions were. As Cohen shows, the source of the error lay in clumsily designed recording sheets, in which the “idiot white” and “idiot black” columns were easily confused, resulting in the misrecording of many elderly senile inhabitants of all-white households. In the 1840s, however, the public debate was not about methodology, but whether fraud had been committed: the numbers themselves could not lie.
Two thousand years earlier, as Serafina Cuomo has shown, the Roman land surveyor Frontinus opined that the world was essentially unknowable without quantitative intervention, and that the trustworthiness of that measure was dependent on professional expertise:
The basis of the art of measuring lies in the experience of the agent. It is in fact impossible to express the truth of the places or of the size without calculable lines, because the wavy and uneven edge of any piece of land is enclosed by a boundary which, because of the great quantity of unequal angles, can be contracted or expanded, even when their number [that is, the number of the angles] remains the same. Indeed pieces of land which are not finally demarcated have a fluctuating space and an uncertain determination of iugera.
The natural world is problematically irregular, Frontinus believed, and must be disciplined into quantified straight lines—and, ideally, marked out into grids of 2400 foot squares (iugera)—in order to be brought under control. The Roman reshaping of the landscape through its quantification is still visible throughout Europe, the Middle East, and North Africa today, both on land and from the air.
The Incas, by contrast, brought time, space, society, and the gods under control through radial lines in the landscape, tied to the ceremonial year. Before Spanish-led Christianization in the sixteenth-century, the heart of the Inca cosmos was the sacred city of Cuzco in the Peruvian Andes. The Incas divided the world into unequal quarters or tawantinsuyu “the four parts together,” radiating out from the Temple of the Sun. Through each suyu ran nine to fourteen ceque paths through the mountains, forty-one in total, with an average of eight huaca shrines stationed on each. The local inhabitants performed a ritual at one of the 328 huacas every day of the sacred year (composed of twelve months of 27
days). Thus the religious focus of the Inca state moved systematically around its territory, day by day and from community to community, binding every social group into the same calendar, cult, and cosmos.
Numeracy, then, is a powerful institutional tool: measuring, quantifying, and classifying can transform an unknowable mass of individual people, places, or things into manageable categories of known entities; in turn, this institutionally imposed structure shapes the self-identities of those being managed. Institutional numeracy, while imposed from above, is always dependent to some degree on community-wide support and cooperation, if not necessarily for the objects of account then always for the counters. Attempts at censuses in the eighteenth century did not fail because people refused to be reduced to numbers in boxes, but because those charged with collecting the data had neither the infrastructural means to do so nor an intellectual outlook that valued quantification. Inca and Roman societies, by contrast, were able to produce whole classes of the professionally numerate who did.
5 Numeracy and Gender
In modern anglophone culture, academic mathematics is popularly considered a male pursuit—and women supposedly have to subordinate or compromise their femininity if they are to succeed in it. But such perceptions are far from universal: studies collected by Barbro Grevholm and Gila Hanna, for instance, show that in the early 1990s some 80% of Kuwaiti and over half of Portuguese undergraduate mathematics majors were women. However, as the following examples demonstrate, this has more to do with how particular societies construct ideals of femininity and masculinity and with what they count as mathematical activity than with any intrinsically gendered properties of mathematics itself.
For most of the second millennium B.C.E., Babylonian scribes understood professional numeracy to be a divine gift—not from the gods in general but from a handful of powerful goddesses. In the literary works that scribal students memorized as part of their professional training, creator gods bestowed land-measuring equipment and numeracy on those goddesses to enable them to manage household estates equitably. In a myth now known as Enki and the World Order the great god Enki announces:
My illustrious sister, holy Nisaba,
Is to receive the 1-rod measuring reed.
The lapis lazuli rope is to hang from her arm.
She is to proclaim all the great divine powers.
She is to fix boundaries and mark borders.
She is to be the scribe of the Land.
The gods’ eating and drinking are to be in her hands.
The scribes’ literary works also portrayed Nisaba as the patron of institutional numeracy in the real world: she in turn provided mensuration tools to scribes and kings to enable them to uphold justice in society.
Another scholastic literary genre was the scribal dialogue, in which the protagonists argue over the ideals of scribal professionalism. In one such debate the young scribe Enki-manshum explicitly relates metrological competence to social justice:
When I go to divide a plot, I can divide it;
when I go to apportion a field, I can
apportion the pieces,
So that when wronged men have a quarrel
I soothe their hearts and. . . .
Brother wilt be at peace with brother, their hearts . . . .
This was not merely a literary trope: law codes promulgated by real-life Babylonian kings often began with prologues claiming that they would uphold fairness in commercial measuring, weighing, and counting, and included provisions for punishing metrological fraud. Many hundreds of legal records survive, attesting to the settlement of land disputes through accurate professional measurement and calculation. In the nineteenth-century B.C.E. city of Sippar, the judges who held court in the temple of Shamash, god of justice, employed female scribes and surveyors as well as male (often from the same families) Further, the personal seals of fourteenth-century B.C.E. royal land surveyors were often dedicated to Nin-sumun, the divine mother of the legendary hero Gilgamesh: for them the numerate goddess who bestows numerate justice was no school story but at the very heart of their professional self-identity.
In ancient Babylonia, then, numeracy and metrology gained institutional authority and power as much through association with divine femininity as with royal masculinity. Many modern societies, by contrast, defeminize numerate thought and activity by denying its mathematical status when it is carried out by women. Gary Urton’s study of Quechua numeration started out as an ethnography of Bolivian weaving, which, he discovered, was based on highly intricate symmetrical patterns that the (female) weavers know by heart. They count off threads effortlessly, unerringly picking up where they have left off after interruptions to nurse babies, prepare food, or attend to other domestic matters. And yet the men of the area categorically told Urton that the weavers “can’t count”—because when a woman sells her finished weavings at market she will invariably ask another woman of the group to check her takings to ensure that she has not been cheated.
Urton was taught to weave by Irene Flores Condori, a twelve-year-old girl. He recalls:
On one occasion, a stern old woman. . . asked me point blank if, by weaving, I was trying to be like a woman. I answered by telling her that in some villages I know of, it is the men rather than the women who do the weaving. . . . The old woman gave us both a wry look and asked, if that was the case, then is it the women in those villages who have the penises!
Weaving was such a strongly gendered activity that this and other incidents led Urton to feel that “my behavior was being tolerated to the degree that it was only because, as an outsider, I was not subject to the same rules and expectations as local men.” Weaving is exclusively women’s work and therefore its intrinsically numerate character is socially invisible; women are more reluctant than men to trust strangers to handle money fairly and are therefore considered innumerate.
Mary Harris shows how a similarly powerful gender divide developed in Victorian Britain as primary education became available to an ever-widening section of the populace. Mathematics was regarded as the quintessentially male school subject, while needlework was the epitome of femininity. Yet:
Every garment knitted to fit a particular body depends on the principle of ratio. Every pinafore pattern copied from a blackboard requires visual interpretation of scaling and the ability to draw a smooth curve. All the fine stitching that the early Inspectors were unable to tell from machine stitching depended on the ability to judge equal distances by eye and maintain them in a straight row.
In other words, wherever girls and women weave, knit, or sew they are unwittingly engaging numerate aptitudes and skills, often highly creatively, just as Molière’s Monsieur Jourdain had been speaking prose all his life “without knowing anything about it.”
6 Numeracy and Literacy, School and Supermarket
Perhaps one reason that women’s work is not often thought to belong to the realm of professional numeracy is that numeracy is so often considered (when it is considered at all) as a subset of literacy. As Reviel Netz puts it,
With Arabic numerals, numbers appear as secondary to writing, benefiting from tools that were largely invented to record verbal systems and not numerical symbols. In broad historical perspective, this is the exception and not the rule. The rule is that, across cultures, and especially in early cultures, the record and manipulation of visual symbols precede and predominate over the record and manipulation of verbal symbols.
Netz is thinking here of counters and abacuses, but the Bolivian weavers remind us that numeracy does not have to entail symbolic manipulation at all. One may count threads, llamas, ideas, anything, and perform calculations without the intervention of external tools. The use of fingers and other body parts has cropped up repeatedly in the examples presented in this essay. Much of the weavers’ mental work is so naturalized within the rhythms and movements of their bodies that they can no longer verbalize the mental or physical processes involved. (That is why Urton chose a young girl as his teacher, who was still learning the craft, rather than a fully competent adult woman.) Nonliterate numerate practices and ideas, especially in the developing world, are often labeled by academic observers as “ethnomathematics.” But this raises difficult questions about the appropriate use of the “ethno” prefix and about the border between numeracy and mathematics. How do we distinguish numeracy from mathematics, and where does ethnomathematics fit in?
When Ubiratan D’Ambrosio coined the term “ethnomathematics” in the mid 1970s it was to describe the study of mathematics “in direct relation to [its] social, economic, and cultural backgrounds,” a subject lying “on the borderline between the history of mathematics and cultural anthropology.” However, for many, particularly within mathematics education, it has come to mean the study of culturally “other” mathematics, as if only the academically marginalized have ethnicity (just as, according to some lazy academic views, only women have gender). This semantic shrinkage is doubly damaging, for it implies that “ethnic” cultures are not fully numerate, while rendering the mainstream of academic mathematics, both past and present, invisible to sociological, anthropological, or ethnographic research. Nor does it distinguish between the intellectual creativity that is mathematics and the routine application of numeracy.
If “ethnomathematics” is an unhelpful term, there are useful alternatives. An influential Brazilian study of childhood numeracy, by Terezinha Nunes and colleagues, distinguishes formally learned “school mathematics” from “street mathematics” created informally by the same children. Jean Laves ethnography of adult numeracy in 1980s California likewise contrasts “school arithmetic” with “supermarket arithmetic.” The participants in her study often described themselves as arithmetically incompetent and “were unaware of the efficacy of their math practice in the supermarket, and some did not know, even that they used arithmetic practices there.” Yet often the supermarket setting required the solution of mathematical problems of much greater complexity than superficially similar scholastic “word problems”:
The shopper was standing in front of a produce display. As she spoke she put apples, one at a time, into a bag. She put the bag in the cart as she finished talking: “There’s only about three or four [apples] at home, and I have four kids, so you figure at least two apiece in the next three days. These are the kinds of things I have to resupply. I only have a certain amount of storage space in the refrigerator, so I can’t load it up totally. . . . Now that I’m home in the summertime, this is a good snack food. And I like an apple sometimes at lunchtime when I come home.”
While explicitly considering such variables as the number of apple-consumers in the household, their rate of consumption, fridge storage space, and perhaps implicitly the apples price and probably shelf life, the shopper selected nine apples to buy. She might also have compared the prices of different varieties of apple and/or considered whether loose or prepackaged apples were the better buy—all typical supermarket activities that Lave and her researchers observed and correlated with the same subjects’ performance in written tests of arithmetically similar skills. They found not a single significant correlation between frequency of calculation in a supermarket, and scores on math test, multiple choice test or number facts. . . . Success and frequency of calculation in supermarket and simulation experiment bear no statistical relationship with schooling, years since schooling was completed, or age.”
Rather depressingly for educators, perhaps, Laves work suggests that training in school mathematics has little or no impact on numerical competence in adult life. (Interestingly, this finding conflicts with Cohen’s historical argument discussed above, relating improvements in mathematics education to rising standards of numeracy in early nineteenth-century North America.) Rather, as she and Étienne Wenger argue, learning takes place most effectively when it is situated in the social and professional context to which it pertains, through interaction and collaboration with competent practitioners, rather than through abstract, decontextualized classroom learning. Learners become part of a “community of practice” that inculcates not only the necessary technical skills but also the beliefs, standards, and behaviors of the group. Through gains in competence, confidence, and social acceptance, the learner moves from the periphery toward the center of the practice community, in due course becoming accepted as a fully fledged expert. It is perhaps in this light, then, that we should understand the process of becoming professionally numerate. But if situated learning is so effective, the development of supra-utilitarian educational mathematics in the societies of the ancient Middle East and Mediterranean is a major historical conundrum that has hitherto gone unrecognized.
7 Conclusions
This essay began by suggesting that “numeracy is to mathematics what literacy is to literature.” But the case studies presented here show that numeracy has a far greater cognitive reach than that. Throughout time and across the world countless individuals and societies have managed perfectly well, and continue to thrive, without writing; none has yet been attested without counting, measuring, or pattern-making in some form or other. In this light a better formulation might be that “numeracy is to mathematics what language is to literature.” Indeed babies, toddlers, and young children learn many essential mathematical skills through engagement with their immediate environment well before formal school learning begins. Just as some children grow into more articulate adults than others, with or without highly developed skills in reading and writing, so they may become more or less numerate in their everyday practices, independently of their competence in school mathematics.
There are many deep and important questions about the relationships between numeracy and mathematics, language and literacy that have hardly yet been formulated, let alone explored: this is perhaps one of the most open fields of enquiry in academia today. This essay has only scratched the surface of a fascinating and complex subject that has paradoxically been overlooked because of its very ubiquity and centrality to human existence. In the next few decades, a wide range of interdisciplinary approaches will almost certainly yield important and surprising discoveries about numeracy that today we can only guess at.
Further Reading
Ascher, M. 2002. Mathematics Elsewhere: An Exploration of Ideas Across Cultures. Princeton, NJ: Princeton University Press.
Bloor, D. 1976. Knowledge and Social Imagery. London: Routledge & Kegan Paul.
Cohen, P. C. 1999. A Calculating People: The Spread of Numeracy in Early America, 2nd edn. New York and London: Routledge.
Crump, T. 1990. The Anthropology of Numbers. Cambridge: Cambridge University Press.
Cuomo, S. 2000. Divide and rule: Frontinus and Roman landsurveying. Studies in History and Philosophy of Science 31: 189-202.
D’Ambrosio, U. 1985. Ethnomathematics and its place in the history and pedagogy of mathematics. For the Learning of Mathematics 5:41-48.
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