These notes provide a collage, primarily in the author's own words, of issues and evidence in the history of quantitative literacy adapted from "A Calculating People." The notes do not represent a complete or coherent summary of the book, but merely a selection of ideas relevant to quantitative literacy.(QL Home Page)
In the middle ages, numbers were used to record transactions, not for
calculating. (Counting boards were used for that.) Arabic numerals
prevailed as a means of calculating, even though many feared that they
could be modified too easily to serve as a reliable means of keeping
records. Commercial calculation was complicated because different units
were used for different commodities. Up through the eighteenth century
most Englishmen though it perfectly reasonable to let the material being
measured dictate the unit of measurement. No one suggested a uniform
system of weights and measurements...not least because it would have
forced an unnatural dissociation between the product and its measure.
Before the seventeenth century the order of the cosmos was dictated by
Aristotelian classification. Quantification emerged as an alternative way
of making sense of the world, a way that could account for activities
newly perceived in the interstices of classical categories. In the 16th
and 17th centuries, numbers appeared to capture the vastness of the world
and the universe, the importance of money, and the unruliness of people.
They also introduced into Western thought the idea that it is possible to
remove values from anything that can be quantified. Nevertheless, by the
end of the seventeenth century numeracy had not made rapid progress in
England. Probably fewer than four hundred Englishmen could be said to be
mathematically minded, including teachers of navigation as well as members
of the Royal Society. These men shared a delight in numbers and a
keenness to measure.
In the seventeenth century, applications of numbers were still limited.
Measuring things was a kind of sport, and few people engaged in it. Yet
commerce thrived since all that was required to enter into a market
exchange was trust: merchants made change and kept records for their less
numerate clients. (Today we do the same with mortgage rates and FICA
deductions.) The common feature of the diverse instances of
quantification (e.g., counting parish membership, measuring height of
mountains) was their origin amongst a relatively few men. Those who took
up the craft were inspired by the ability of numbers to create certainty.
In 1690 Sir William Petty published Political Arithmetickin which
he argued that the specificity of numbers conferred objectivity. True
political knowledge, Petty believed, would arise out of full quantitative
accounting of social and economic facts.
Numbers are preeminently descriptive. On the most basic level they
enumerate. They create uniformity, make comparison possible. Moreover,
they can register the combined effect of several variables.
Quantification then creates a sense of uniformity and finitude; it
counts, and in the process, accounts. It can be a powerful explanatory
tool, an alluring way to impose order on the flux of the seventeenth
century world.
The shift in the last three centuries in how we view censuses illustrates
well the shift from qualitative to quantitative thinking. In 1694
Parliament passed a census law. However, it was not well received by the
populace, both because it was seen as foreshadowing more aggressive tax
collection but also because it violated a biblical warning which took
precedence over any advantage that numerical precision may have conveyed.
The Old Testament records a "sin of David" who brought a plague on Israel
for "numbering" the people. (Note: "census" and "censor" derive from the
same Roman practice of censuring immoral behavior.) In the seventeenth
century, most people thought of population in terms of qualities and
characteristics, but number was not among them. Political arithmeticians
talked about the importance of knowing the whole number of people, but
dictionaries still did not show the word "population." Not until
mid-eighteenth century was their sufficient interest in population as a
quantity to make a census sensible.
In 1735 Ben Franklin wrote about Mathematicks in the Pennsylvania Gazette
and argued for benefits beyond those of trade, including a logic which
stretched the mind and improved the faculty for exact reasoning. Indeed,
what progress there was in numeracy in pre-revolutionary America came not
as a tangent to commercial growth but as the result of a slow but
fundamental shift in the way men thought about human affairs and divine
intervention. Puritans did not keep totals of souls saved because of
their fatalistic belief that the number of the elect was predetermined.
The decline of fatalism and the discovery of peculiar regularities in
events once thought to be under inscrutable divine control encouraged the
evolution of a mathematical sense in the eighteenth century.
Death and mortality--God's ultimate province--provide a good example.
Eager empiricists showed death rates to be different in different places,
thus suggesting that they are somewhat under man's control. The gradual
realization that man could, in theory, control mortality accelerated the
decline of the notion of an omnipotent God and hastened the arrival of
autonomous man, directing his own destiny. Fatalism and uncertainty began
to give way to control and predictability, exercised through the medium of
numbers.
Bills of mortality led religious leaders (e.g., Cotton Mather) to argue
for preordained mortality percentages, applying data from European cities
to Boston without a second thought (because they were believed to be
divine, thus universal). A 1721 smallpox epidemic in Boston occasioned
the first suspicion that man could alter God's plan. Mather proposed
inoculations to fight spread of the disease, and marshalled medical
statistics to prove the efficacy of this approach. The inoculation
controversy gripped Boston almost as tightly as the disease itself.
However, the public debate was not so much over whether (much less how)
inoculation worked, but how inoculation affected God's power to ordain
deaths and send punishments in the form of epidemics. Aiding the sick did
not interfere with God's power, but preventing illness removed smallpox
from God's arsenal of warnings. "It was a problem that engaged the best
minds of Boston in 1721."
Samuel Grainger, a mathematical practitioner in Boston and one of the most
numerate of the colonists, considered argument by number to be irrelevant
in the face of religious obstacles. "Do not be seduced by its [smallpox
vaccination] supposed success. For to urge the lawfulness of this
practice from its success is a very weak argument to prove it so. For
should success become a sufficient plea for the lawfulness of any action,
every wicked action successfully acted would become lawful at that
rate."
The smallpox record illustrates not only colonists' skepticism about the
proper role of numbers in thought, but also the relative innumeracy of
even the educated New Englanders. Numerical arguments in newspapers about
smallpox were filled with errors and sloppy reasoning. Numbers were used
to impress, even calculations that could impress only the inattentive.
But in 1730 no one gave numbers close scrutiny.
In the eighteenth century "mathematicians" were primarily surveyors and
measurers who taught short courses in evenings and on weekends. Since
(applied) mathematics was learned this way, it was slow to penetrate other
parts of the educational system. It was viewed more as a trade and craft
than as part of basic education. However, a few individuals, Jefferson
among them, were inveterate calculators--people who quantified everything,
who exercised an "esoteric proclivity for numbers" in a climate not
supportive of that style of thought. Quantification, for these men, held
a psychological attraction every bit as strong as whatever practical value
they may have perceived.
The drive for numeracy in the colonies coincided with the great debates
about how to implement a democratic society. Jefferson's plan to
decimalize money, weights, and measures discarded centuries of tradition
in order to simplify arithmetic so that "men of ordinary capacities could
become as facile as he in the calculations of daily life." His plan to
impose a grid survey on the Western wilderness was meant to eliminate
ambiguous property boundaries. Things that were counted and measured
perfectly were fixed for certain; precision defied ambiguity. That was
the attraction of mathematics for Jefferson, and for the new nation.
Decimal money, begun in 1792, democratized commerce by putting computation
within reach of all. At the same time, the self-consciously utilitarian
spirit of the new nation invaded education and elevated arithmetic to the
status of a basic skill along with reading and writing. Decimal money and
arithmetic education were justified as fruits of republican ideology:
numeracy was hailed as the cornerstone of free markets and free society.
"Republican money ought to be simple and adapted to the meanest
capacity."
Before arithmetic became a necessity, it was taught and learned as an
abstract system of knowledge. Between 1660 and 1750, various practical
arithmetics gradually replaced Robert Recorde's standard theoretical text
which taught arithmetic as a system for dealing with abstract quantities.
The practical texts omitted explanations as being unnecessary and beyond
the grasp of their audience of tradesmen. For the most part, authors
stuck to rules and examples, making no attempt to weave the whole into an
integrated system of thought. They omitted reasons because they believed
them to be "tedious and inconvenient."
In 1800 arithmetic was still laborious, depending on rules, catechism, and
memory. In the process of making arithmetic a required subject for all,
no one had stopped to wonder whether there might be a better pedagogic
approach. Educators finally had to conclude that the traditional method
of instruction simply did not work very well. Elementary arithmetic had
for so long been associated with commerce that it had been overlooked as a
purely intellectual exercise for the mind. "We do boys a great injustice
by supposing that they cannot reason," said one educator. But with the
desire for a populace well attuned to reason as a solid foundation for
republican government, there emerged the idea that even basic arithmetic
could help train citizens to think well.
New texts emerged offering various reforms--to eliminate fractions and use
only decimals; to eliminate units; to use counters and bead frames.
Warren Colburn in 1821 introduced "mental arithmetic" for children as
young as five years old based on "inductive reasoning." The goal was to
abandon slavish reliance on rules and memory, and let children develop
their own rules by manipulating tangible objects. The old system worked
from rules to examples, the new from examples to rules. Colburn's mental
arithmetic became quite popular. Children competed in arithmetic contests
as they did in spelling bees; they learned to think quickly in a changing
marketplace; and their growing mental skills reinforced the early
nineteenth century prestige of inductive reasoning.
When arithmetic became required in school, girls for the first time began
to learn arithmetic. Soon pamphlets arguing against this practice began
to appear. According to critics, women failed the utilitarian test of
advanced mathematics (they had no use for it) and the capacity test for
learning (women at that time were believed to be intuitive, not inductive
beings). Thus a stereotype formerly hidden became dogma because the issue
had to be confronted. In geometry, one branch of mathematics that had
never fallen victim to the deadening memory-based approach of the
commercial arithmetic texts, logic dominated. So it is not surprising
that particular objections were raised whenever girls took up the subject.
The irony of these arguments became evident in the 1840s when the
feminization of the teaching profession presented new and compelling
reasons for women to be more fully numerate.
Even in postsecondary education, numeracy was slow to take root. Harvard
College in the seventeenth century did not embrace arithmetic, which it
considered a vulgar subject. Although Harvard established a mathematics
professorship in 1726, it relegated the teaching of arithmetic, geometry,
and astronomy to a mere two hours a week in the senior year. In 1740 Yale
made arithmetic a first year subject, and Harvard followed two decades
later. Arithmetic became a requirement for admission at Yale in 1745, at
Princeton in 1760; and at Harvard in 1802.
Debates about the new census focused on whether it should measure people
or products. In 1790 Madison argued for a more detailed census that
categorized people and occupations. Congress threw out the census of
occupations as of no benefit. By 1800 Congress was asked by leading
scientists for an unprecedented social survey, but they rejected that too.
"Few shared Madison's enthusiasm for marking the progress of society."
The movement to comprehend society through quantitative facts accelerated
in precisely the same decade in which arithmetic instruction in the
schools was being thoroughly revamped. Arithmetic not only improved the
logical faculties and prepared young boys for commerce, but it also opened
the doors to useful political knowledge in the form of data on the state.
Example: a children's board game dating from 1806 included many facts and
figures from the 1800 census.
However, innumeracy persisted just beneath the surface. The temperance
movement used statistics widely in their efforts, but never thought of
distinguishing between correlation and causation. Random samples were
unknown. Moreover, this empiricism was freighted with unacknowledged
values. They counted acres of land and export tonnage, miles of roads
built and postage stamps sold, inebriate poor and hymns memorized. But
what they did not count in 1820--for instance, the number of slave owners,
black mortality, female illiteracy--tell as much about their society as
the things they chose to notice. "In the nineteenth century what was
counted was what counted."
Many years later analysts showed how a routine error of recording a few
senile whiles in an adjacent "idiot black" column, combined with the two
demographic gradients (larger numbers of elderly in the north and larger
number of blacks in the south) produced a gradient of idiot blacks rising
from south to north. That on the eve of the Civil War no one offered this
explanation for a matter of major public dispute "says something about the
degree of quantitative sophistication that they all shared."
Data were used in other ways in the pre-war political debates. Some
argued against slavery by displaying statistics showing that the South
trailed the North in every measurable aspect. But on what basis did they
expect these statistics to persuade? Anti-slavery advocates assumed the
form an argument should take to be convincing. But there are other ways
one could contrast two cultures as dissimilar as North and South. Suppose
one compared per capita wealth instead of total wealth. Would slaves then
be counted as population in the denominator, or as wealth in the
numerator?
There is a striking element of hubris in nineteenth-century statistical
thought, a hubris that is still at work in the twentieth: to measure is
to initiate a cure. In early nineteenth century America, numbers were
celebrated because they were genuinely useful, because they were thought
to discipline the mind, because they marked the progress of the era, and
because they were reputedly objective and precise, and hence tantamount to
truth. Yet despite the argument that numbers would provide a convincing
and objective basis for political decisions, social values always left
their imprint on supposedly objective empirical facts.
The essence of literacy is the manipulation and interpretation of
writtensymbols. Even the most exalted speech in a Shakespearean
play does not require literacy until it is written and read. But what of
numeracy? Reckoning over twenty is tantamount to being able to read and
write. (In 1701 an English mathematicians descried Americans as
"barbarous" because they could "hardly reckon above twenty.") But what of
the abacus? What of logical thinkers (Plato, Aquinas) or calculating
prodigies? Defining numeracy is not as easy as defining literacy.
Clearly it is a mistake to think of numeracy as either wholly present or
wholly absent.Numeracy in Early America
Settlers in the American colonies lacked interest in arithmetic only
partly because of their relatively low level of education. Puritans in
New England had the highest levels of university education and literacy,
but arithmetic was not among the subjects considered basic for Puritan
children to learn. Even when the church membership declined, Puritans
never thought about counting members as a means of documenting the slide
from faith.Numeracy and Democracy
By the end of the eighteenth century a new attitude emerged towards
numbers and the respective powers of God and man. This consisted of three
notions that were, at the time, novel: That man could alter the course of
nature; that quantification was essential both as a tool for altering
nature and as a means of assessing the extent of the alteration; and that
one was entitled to live out his "full" life.Learning Arithmetic
Numeracy made slow progress in the schools. The English assumption that
arithmetic was too difficult to explain persisted in America. Eighteenth
century methods of problem solving confounded all but the best students
because they deliberately relied on memory, not on understanding. Texts
contained a plethora of rules to match every conceivable situation,
sometimes in verse to aid in memorization. Most students' education ended
with the rule of three (given three parts, find the fourth)--what we now
call ratio and proportion. In New England many students did not get even
that far, since the ultimate aim of a Puritan education was theological,
not mercantile or scientific. Indeed, in 1648 the rules for a grammar
school in New Haven directed that children should be taught to "cipher for
numeracion and addicion and noe further."Statistics and the State
In 1800 "statistics" meant a statement about the civil condition of a
people. (Its root comes from "state" not "statics.") Leaders of the new
republic limited statistics to definite facts about populations--wealth,
trade, industry, occupations, civil and religious institutions--arguing
that these were the data most appropriate for assessing the American
experiment in republican government. Soon compiling civil facts and
figures became common. The many new and practical uses of numbers
contributed to the belief that whatever was quantifiable was objective.
(Observer bias in measurement never seems to have troubled anyone.)
Puritanism enhanced the scientific world-view by demanding direct,
personal experience of nature (as of God) and by creating civil chaos that
liberated creative individuals.
By the late eighteenth century, facts had come to be seen as indisputable
and objective in contrast to opinions which are idiosyncratic and
debatable. Statistical thought offered a way to mediate between
contending political ideas based on a homogeneous social order and
economic realities that were fast undermining homogeneity. An extension
of this respect for facts was the idea that if only enough facts were
known, disagreement on public issues would end. Inventories of facts were
touted as providing objective basis for determining the common good.
Complete possession of facts, it was hoped, would eliminate factionalism.
Wrote one influential editorialist, "If all men know alike, though
imperfectly, their opinions must be the same."Early Nineteenth Century
The census of 1840 generated intense political debate because it seemed to
support slave-owner's arguments that slaves could not survive freedom:
the census revealed a gradient of black insanity rising from south to
north. Experts and politicians fought over the data. Fortuitously, the
American Statistical Association was founded in 1839 and made itself
available to resolve errors in the 1840 census. Despite this "expertise,"
nobody was able to find an explanation.