Literasi Mathematis
Literasi Mathematis
Literasi Mathematis
The goal of the National Council on Education and the Disciplines (NCED) is to advance a vision that will unify and guide
efforts to strengthen K-16 education in the United States. In pursuing this aim, NCED especially focuses on the continuity
and quality of learning in the later years of high school and the early years of college. From its home at The Woodrow
Wilson National Fellowship Foundation, NCED draws on the energy and expertise of scholars and educators in the
disciplines to address the school-college continuum. At the heart of its work is a national reexamination of the core
literacies quantitative, scientific, historical, and communicative that are essential to the coherent, forward-looking
education all students deserve.
Foreword
ROBERT ORRILL
Quantitative literacy, in my view, means knowing how to reason and how to think,
and it is all but absent from our curricula today.
Gina Kolata (1997)
Increasingly, numbers do our thinking for us. They tell us which medication to take, what policy to
support, and why one course of action is better than another. These days any proposal put forward
without numbers is a nonstarter. Theodore Porter does not exaggerate when he writes: By now
numbers surround us. No important aspect of life is beyond their reach (Porter, 1997).
Numbers, of course, have long been important in the management of life, but they have never been
so ubiquitous as they are now. The new circumstances arrived suddenly with the coming of computers and their application to gathering, processing, and disseminating quantitative information.
This powerful tool has brought unprecedented access to quantitative data, but in so doing it also has
filled the life of everyone with a bewildering array of numbers that often produce confusion rather
than clarity. The possible consequences for our ability to direct our affairs are worrisome to say the
least. For some observers, the flow of numbers amounts to an inundation that calls forth images of
a destructive flood of biblical proportions. Looking toward the future, James Bailey warns that
today we are drowning in data, and there is unimaginably more on the way (Bailey, 1996). Even
if we manage to keep our heads above water, Lynn Steen writes, we can be sure that the world of the
twenty-first century will be a world awash in numbers (Steen, 2001).
Gina Kolata looks at this data-drenched environment from a special vantage point. She reports on
science and health issues for the New York Times and often hears from readers who complain that
numbers presented by experts seem to mean one thing one day and another thing the next. What
are they to believe, readers ask, when a regimen first said to promote well-being is later said to
undermine it? Kolatas response is that they must learn to interpret the numbers for themselves. The
only remedy, she says, is that they have to learn how to think for themselves, and that is what an
education in quantitative reasoning can teach them. Such an education, she writes, makes all the
difference in the world in peoples ability to understand issues of national and personal importance
and helps them evaluate in a rational way arguments made by the press, the government, and their
fellow citizens (Kolata, 1997).
But, as a practical matter, Kolata counts on no such well-prepared readership in her own reporting.
The attention to quantitative reasoning that she thinks so essential to sound judgment simply does
not exist in the academic programs of most of our schools and colleges. Thus, even the collegeeducated often lack an understanding of how to make sense of numerical information. For a
democracy, this is no low-stakes concern. If numbers are present everywhere in our public discourse,
and many are more confused than enlightened by them, what happens to decision making in our
society? If we permit this kind of innumeracy to persist, do we not thereby undermine the very
ground and being of government of, by, and for the people?
Robert Orrill is the Executive Director, National Council on Education and the Disciplines (NCED), and Senior Advisor
at The Woodrow Wilson National Fellowship Foundation, Princeton, New Jersey. NCED brings together university
faculty and secondary school teachers to address issues of educational continuity in the later years of high school and the early
years of college.
vii
viii
Foreword
To help launch this conversation, the National Council on Education and the Disciplines (NCED) sponsored a National Forum
on December 12, 2001, aimed at promoting discussion and
debate about Why Numeracy Matters for Schools and Colleges.
Held at the National Academy of Sciences, the Forum was designed to bring together many different points of view education, business, government, and philanthropy were all represented
in the deliberations. International perspectives on quantitative
literacy also were presented, making it clear that numeracy is a
growing global concern. The most immediate outcome is the rich
and abundantly informative proceedings presented in this volume, which we believe in giving voice to a wide range of
opinion provide a benchmark discussion from which the
needed national conversation can go forward.
References
NEED
FOR
WORK
AND
LEARNING
Four of these papers focus on the need for quantitative literacy, particularly in the context of
citizenship and work, while eight address components of QL education: curriculum, pedagogy,
articulation, and assessment. Of the eight, four deal directly with curriculum and four consider
policy issues involving curricular relationships and assessment. Although thoroughly grounded in
the realities of U.S. education, these papers explore a variety of paths to the goal of imbuing students
with quantitative habits of mind in addition to conveying facts and procedures. As with all such
Bernard L. Madison is Professor of Mathematics at the University of Arkansas where he previously served as Chair of
Mathematics and Dean of the J.W. Fulbright College of Arts and Sciences. During 1985 89, Madison directed the
MS2000 project at the National Research Council, including the 1987 Calculus for a New Century symposium. Madison has
worked in various roles for the Advanced Placement program, including serving as Chief Faculty Consultant for AP
Calculus and as a member of the Commission on the Future of AP.
CURRICULUM ISSUES
Four thought papers look specifically at the mathematics curriculum and its role in general education. Deborah Hughes Hallett,
Jan de Lange, and Lynn Arthur Steen address various aspects of
that curriculum, including some that often are classified under the
rubric of statistics. Hughes Hallett writes of the college experience
while de Lange offers an international perspective. Steen argues
for a mathematics curriculum in grades 6 to 12 that would expand
the current narrow focus on algebraic symbol manipulation. Finally, Randall Richardson and William McCallum discuss how to
extend college QL education beyond mathematics courses to develop authentic contexts for mathematical concepts in other disciplines.
POLICY CHALLENGES
Policy challenges in various QL areasarticulation, assessment,
relation to mathematics, and core curriculumare the subjects of
the final four thought papers. Michael Kirst addresses the complex
political and policy issues surrounding articulation, that is, how
QL education is affected by the decision making and transitions
from secondary to higher education. Bernard Madison looks at
articulation from within mathematics and analyzes features of the
NEED
FOR
WORK
AND
LEARNING
POLICY PERSPECTIVES
Although most participants at the Forum felt that education for
QL should extend beyond mathematics and statistics, Jan Somerville cites policy issues surrounding college and university mathematics that impede progress toward a more useful mathematics
education. She challenges mathematicians to take QL as a responsibility and to address more forcefully problems in mathematics
education. Margaret Cozzens reinforces Somervilles views by
identifying higher education policies and practices that hinder QL
INTERNATIONAL PERSPECTIVES
In addition to the background paper by Jan de Lange, this volume
contains a group of essays by authors from Brazil, Denmark,
France, Great Britain, and the Netherlands, offering views of QL
in those countries. Lynn Steen writes in his introduction to this
section that these glimpses of how mathematics educators in
other nations are coming to terms with the new demands of numeracy, mathematics, and citizenship open a window on approaches that move well beyond those normally considered in
U.S. curriculum discussions.
FORUM REFLECTIONS
AND
OBSERVATIONS
Rosen and Packer address the complex needs of a major component of U.S. culture, the workplace, and show how those needs are
or are not being met by mathematics and quantitative education.
Scheaffer confronts the doubly complex task of explaining how
the relatively new and poorly understood discipline of statistics
and statistics education fit into the haze of QL education.
Aside from the discussionalbeit mostly academicabout the
need for a better definition of QL, the Forum papers and essays
clearly point to two other needs:
1.
2.
References
Madison, Bernard L. 2001. Quantitative Literacy: Everybodys Orphan. Focus (6):10 11.
Steen, Lynn Arthur, ed. 2001. Mathematics and Democracy: The Case for
Quantitative Literacy. Princeton, NJ: National Council on Education and the Disciplines.
Patricia Cline Cohen is Chair of the History Department at the University of California, Santa Barbara, where she has also
served as Chair of the Womens Studies Program and as Acting Dean of Humanities and Fine Arts. Cohen specializes in U.S.
womens history and early American history. She is coauthor of The American Promise and author of A Calculating People:
The Spread of Numeracy in Early America, a cultural history of the diffusion of arithmetic skills and the propensity to use
quantification in American culture in the 18th and early 19th centuries.
in 1789 had quite limited numeracy skills, skills that got exercised
the most (if at all) in the world of commerce and trade, not in the
world of politics. Although at the outset, the U.S. Constitution
provided for a decennial census for apportioning the House of
Representatives, the Congress took two decades just to begin to
realize that the labor-intensive enumeration process could be augmented almost without cost to capture additional data that might
be useful for a government to have. Even the apportionment
function was carried out with considerable imprecision, showing
us that a general faith in representative institutions in 1789 did
not yet translate to anything so arithmetically concrete as one
man, one vote.
We have moved, over two centuries, from a country where numeracy skills were in short supply and low demand, to one in
which the demand is now very high indeedand in which the
supply, while greatly augmented, has not kept up with the need.
This Forum is primarily focused on the present and the future,
asking how much quantitative literacy is necessary to function in
these beginning years of the twenty-first century. But it helps to
look backward as well, to understand the development of the ideas
linking quantitative literacy to citizenship in a representative democracy. By mapping out the spread of numeracy in specific
arenas, we can begin to see what factors spurred the growth of
quantitative literacy in American history. More than just arithmetic, the shopkeepers skill, was highly valued in the early years
of our democracy; leaders also championed the study of geometry
as a pathway to the superior reasoning skills required by representative government. Our exploration of the practices of arithmetic
and geometry instruction will quickly lead us to another important problem: the differential distribution of mathematical skills
in past populations, which provides a further important clue to
the question of how quantitative literacy bears on citizenship. And
finally, we need to move beyond the nave enthusiasm for political arithmetick characteristic of the early nineteenth century,
which valued numbers for their seemingly objective, neutral, and
therefore authoritative status, to see the symbolic and constructed
uses of political numbers that can both convey and hide important
information. As we all learned so dramatically in the election of
2000, simple counting in politics is never really simple.
This essay looks at three historical eras: the founding generation,
from the 1780s to about 1810; the antebellum period, from the
1820s to the 1850s, when direct democracy came into full bloom
and the country underwent a market revolution; and the late
nineteenth century, when empirical social science became wedded
to government and when new citizensthe newly freed slaves and
the many thousands of immigrantsposed new challenges and
choices for the developing education system. For each period I will
sketch out features of the spreading domain of number in service
to the state and then assess the levels of numeracy prevalent that
10
11
merchant would grow very familiar with the particular and limited kinds of calculations his type of business required, and the
copybook would have served its purpose.11 But such a form of
training did little to enhance the generalized facility with numbers
that we now call quantitative literacy.
By 1796, the mint was producing the new money, triggering the
publication of dozens of new arithmetic textbooks with nationalistic titles, for example, The Federal Calculator, The Scholars Arithmetic: or, Federal Accountant, The Columbian Arithmetician, and
The American Arithmetic: Adapted to the Currency of the United
States.17 One book of 1796 spelled out explicitly the interconnections between common arithmetic, decimal money, and republican government:
Jefferson sketched out an ambitious system of education for Virginia that would have provided three years of publicly supported
schooling for all free boys and girls, covering reading, writing, and
common arithmetic (probably to the rule of three). From there,
the worthy boys (ones who could pay tuition plus a tiny fraction
invited on scholarships) could progress to a Latin grammar school
where the higher branches of numerical arithmetic would be
taught. At the pinnacle, a college would educate the most deserving; here was where algebra and geometry would be encountered.14 Notable in this plan was the provision for girls to be
taught basic arithmetic. But the two lower levels of Jeffersons
system were not built in his lifetime. Similar schemes for common
school systems in other states were equally hard to implement
because of the expense of public education. This meant that collegesthe handful that there were around 1800 generally
needed to offer first-year courses in basic arithmetic to compensate
for persistent deficiencies at the lower levels of instruction.15
Jefferson took another route, however, that had a much more
immediate impact on numeracy in the 1790s. As secretary of state
under Washington, he proposed a major reform in the monetary
system of the nation in 1793, abolishing pounds and shillings in
favor of decimal dollars, dimes, and cents. (Jefferson was equally
inspired by French Enlightenment plans for the metric system,
but that did not fly in the 1790s.) Ease of calculation was Jeffersons goal: The facility which this would introduce into the vul-
12
Although more children were learning basic arithmetic and learning it better than ever before, coverage was uneven, of course, only
reaching children who attended school; the days of mandatory
school attendance lay far in the future. What is perhaps most
striking about this early period of the flowering of numeracy is
that it was, in theory, as available to young girls as it was to young
boys. This was an unprecedented development. Before 1820, girls
had only limited chances to become proficient in arithmetic. The
spread of common schooling, the drop in the age at which formal
arithmetic instruction began (from 11 to 12 down to 5 to 6), the
disconnect between narrow vocational training and arithmetic,
andperhaps most significantthe large-scale entry of women
into the teaching profession: all these factors combined to bring
arithmetic instruction into the orbit of young women.
It was not an unproblematic development, however. It is ironic
that, when at long last basic arithmetic education was routinely
available to young girls in school, critics of that development
began to assert that girls had a distinctly lesser talent for mathematics than boys. It is a gender stereotype that was actually rather
new in the nineteenth century or, if not entirely new, appearing in
a new and more precise form. In the eighteenth century, when
proficiency with the rules of figuring was the province of boys
bound for commercial vocations, any gender differential in mathematical skill could easily be understood as the product of sex
differences in education. Women were less numerate than men,
and they were also less often literate, but no one needed to conclude that women had an innately inferior capacity for reading the
printed page just because fewer women could read. So too with
numbers: the divide between the numerate and innumerate was
traced to specific training and needs, not to sex-based mental
capacity. And, to be sure, many female activities of the eighteenth
century required, if not actual arithmetic performed via rules, then
some degree of what we now see as part of a mathematical intelligence counting, spatial relations, measuring, halving and doublingas women went about cooking, weaving, knitting, and
turning flat cloth into three-dimensional clothing without benefit
of patterns.25
But in the early nineteenth century, when young girls finally had
a chance to be included in formal arithmetic instruction, the perceived differences between the sexes were increasingly naturalized.
Critics of arithmetic instruction for girls questioned whether girls
needed it. Who is to make the puddings and pies if girls become
scholars, one critic wondered. A state legislature objected to masculine studies in mathematics at one school, studies with no
discernable bearing on the making of puddings and stockings.
What need is there of learning how far off the sun is, when it is
near enough to warm us? said a third.26 Of course, there were
champions of arithmetic instruction for girls. Most argued that a
knowledge of household accounts was highly valuable for thrifty
wives to have, but a few moved beyond the purely practical and
13
14
Congress slowly shed its earlier reluctance to maximize the information derived from the census. The 1810 enumeration,
launched during the failed embargo policies of Jefferson and Madison in the prelude to the War of 1812, was pressed into service as
a way to learn about the actual state of manufactures and industry
in the country. For the first time, data were collected that went
beyond population, but the actual results were riddled with errors
and many omissions. The 1820 census finally noted occupation,
but again the effort was rudimentary, sorting all working adults
into only three expansive categories. The 1830 census broadened
the scope to further fine-tune age categories for the white population (but not the black) and to count the numbers of deaf,
dumb, and blind in the population; here we see the start of federal
interest in social statistics. But it was in 1840 that Congress completely succumbed to the siren song of statistics. The census population schedule expanded to 74 columns, adding new inquiries
about the number of insane and idiot Americans, the number of
scholars and schools, a tally of literacy, and a headcount of revolutionary war pensioners, a category associated with direct government expense. A second schedule also filled in by all enumerators
contained 214 headings and answered Congresss blanket call for
statistical tables containing all such information in relation to
mines, agriculture, commerce, manufacturers, and schools, as will
exhibit a full view of the pursuits, industry, education, and resources of the country. From this massive aggregation, a person
could learn the number of swine, of retail stores, of newspapers, of
the bushels of potatoes and 200 more economic statistics (i.e.,
descriptive numbers) for every census district in the United States.
This deluge of statistics was eagerly awaited by the reading public.
A variety of statistical almanacs first appeared and gained popularity in the 1830s, and they were eager to carry news of Americas
progress to their readers. The American Almanac and Repository of
Useful Knowledge was an annual Boston publication dating from
1830, which was devoted to statistics, defined as an account of
whatever influences the condition of the inhabitants, or the operations of government on the welfare of men in promoting the ends
of social being, and the best interest of communities.30 This
almanac filled its pages with miscellaneous figures on banks,
canals and railroads, pupils and schools. Other annual publications had a strictly political focus, such as the Politicians Register,
begun in 1840, and the Whig Almanac and United States Register,
begun in 1842; both recorded elections back to 1788 for many
localities and provided county-level data for recent elections, giving readers information to strategize future campaigns.31 We take
this kind of data for granted now, but it was newly publicized
information in the years around 1840 not coincidentally, the
year when electoral participation was at an unprecedented high, a
high that was sustained for another five decades.
Another rough but very innovative act of political quantification
arrived on the scene in the 1850s, the straw poll of voters. Jour-
nalists roamed the public thoroughfares, targeting mixed assemblages of people, often passengers on a steamboat or a passenger
railroad, to ask about voter preference in an upcoming election.
Interestingly, women passengers usually were not excluded from
such polls even though they were not voters, but their votes were
tallied separately from mens (which is how we can know that
women were asked). These 1850s straw polls were the first American efforts to quantify public opinion.32
Antebellum newspapers, the everyday reading of many thousands
of Americans, studded their columns with facts and figures. A very
typical small item, from the New York Herald of 1839, titled
Railway and Stagecoach Travelling, drew on a return of the
mileage and composition duties on railway and stage carriages
respectively to show that over the previous two years, 4,800,000
fewer persons had traveled by stage while 14,400,000 more persons had traveled by railway.33 No meaning or analysis was attached to these data; they simply stood alone, in manifest testament to the railroad revolution that all the Heralds readers knew
was underway. Mileage of railroad tracks was another favorite and
frequent boast. But newspaper readers of the 1830s would not
have been able to learn the total number of lives lost in steamboat
explosions and accidents over that decade (unless they added up
the losses reported for each accident, a rather shocking sum that
historians have been able to reconstruct).
Not all statistical reports were cheery and boastful. The antebellum era has been tagged the era of reform by some historians for
the rich variety of civic movements dedicated to eradicating social
problems. Although the federal and state governments were not
yet counting and publicizing the numbers of inebriates, prostitutes, or runaway slaves, other associations werethe temperance, moral reform, and abolitionist movements. A faith in the
unimpeachable truth of numbers was part of the landscape now,
and the most powerful way to draw attention to and gain legitimacy for a political or social goal was to measure and analyze it
with the aid of arithmetic, giving the analysis the aura of scientific
result.
It was in the 1850s that statistics were finally harnessed to opposing sides of the most pressing political division in the history of the
United States, the conflict over slavery that led to the Civil War.
In this decade-long debate, we can most clearly see the political
constructedness of numbers and their mobilization to serve both
symbolic and instrumental functions. Although it is very unlikely
that anyonea voter, a member of Congress changed opinions
about the sectional crisis based on quantitative data, it is instructive to see how both sides tried hard to harness the numbers to
endorse their own predilections.
The quantitative dueling started in the congressional debate over
the 1850 census. This was the first census designed to gather
15
statistics collection, with a new focus on urban problems, immigration, labor conditions, and standards of living. Unlike midcentury censuses, which had been run by men with no particular
training in mathematics, the later census officials, such as Carroll
Wright and Francis Amasa Walker, came from the new ranks of
professionally trained economists and statisticians. Statistics was
no longer limited to descriptive number facts; work by European
thinkers such as Adolphe Quetelet, Francis Galton, and Karl Pearson had pushed the field into an increasingly sophisticated mathematical methodology. Federal censuses still were used to apportion Congress, but that was a minor sideline to a much larger
enterprise engaged in measuring social indicators that would be
helpful not only to legislators but to external commercial agencies
and businesses as well, including universities, private research organizations, and trade associations.36
This growing sophistication of government statistical surveillance
was not matched by a corresponding improvement in quantitative
literacy on the part of the public. Unlike the early nineteenth
century, when a public enthusiasm for numbers and arithmetic
developed along with a statistical approach to civic life, in the early
twentieth century the producers of statistics quickly outstripped
most consumers abilities to comprehend. The number crunchers
developed more complex formulations while the arithmetic curriculum stagnatedthis despite two further major attempts to
reform the mathematics curriculum, first in the 1910s to 1920s
and again in the 1950s to 1960s.
In the earlier phase of reform, a new breed of specialistthe
professional mathematics educators in the universityaddressed
the problem of a rapidly growing student population assumed to
have limited abilities. Foreign immigration and African-American
migration combined with new compulsory schooling laws shifted
the demographics of American schools. The percentage of youth
ages 14 to 17 who attended school went from 10 percent in 1890
to 70 percent by 1940; the decades of maximum change were the
1910s and 1920s. When primarily middle- and upper-middleclass students had attended high school or academy, higher mathematics was typically served up in two or three standard courses,
algebra, geometry, and trigonometry. But when the children of
immigrants, emancipated slaves, and industrial workers arrived on
the high schools steps, the wisdom of teaching the higher
branches for the intellectual development they promised was increasingly called into question. A leading educational theorist,
Edward Thorndike, reversed the truism of the early nineteenth
century and argued that mathematics did not encourage mental
discipline. Vocational education and the manual arts became
prominent themes in educational circles, promoting the line that
instruction should be geared to likely job placement. Several states
removed all mathematics requirements for graduation and, predictably, enrollments declined. One study of Baltimores schools
in the 1920s explicitly recommended that algebra and geometry
16
Conclusion
This brief survey of quantitative literacy and citizenship in the
nineteenth century has tried to demonstrate that although there is
a natural affinity between numerical thinking and democratic
institutions, that affinity was not necessarily predicated on quantitative sophistication on the part of citizens, at least not at first.
Representative democracy originated in a numerical conception
of the social order, under the U.S. Constitution. That same document ordained that government should promote the general
welfare and secure the blessings of liberty, a mandate that around
1820 was increasingly answered with a turn toward authentic
facts and statistics. Statistics soon became compressed into quantitative facts, an efficient and authoritative form of information
that everyone assumed would help public-spirited legislators govern more wisely. Schools, both public and private, correspondingly stepped up arithmetic instruction for youth, bringing a
greatly simplified subject to all school-attending children and
making it possible for them to participate with competency both
in the new market economy and in the civic pride that resulted
from the early focus on quantitative boasting.
student test scores, and the gyrations of the stock market as summarized in a few one-number indexes reported hourly on the
radio. The danger is that we may not realize we are in over our
heads. The attractiveness of numbers and statistics in the earlyand mid-nineteenth century arose from their status as apparently
authoritative, unambiguous, objective bits of knowledge that
could form a sure foundation for political decisions. That may
have been nave, but gains in numeracy enabled some, at least, to
learn to question numbers, to refine them, and to improve on their
accuracy. Now, however, numbers are so ubiquitous and often
contradictory that some fraction of the public readily dismisses
them as damned lies.41
The recent bandying about of the term fuzzy math furthers
suspicions about numbers; when used in the political context, it
seems to condemn arithmetic and political arithmetic alike.
Wrenched from its origins as a legitimate if esoteric mathematical
term dating from the 1960s, fuzzy math was first appropriated and
rendered perjorative by the critics of curriculum reform in the mid
1990s, most famously and nationally by then-National Endowment for the Humanities (NEH) chair Lynne Cheney in a 1997
Wall Street Journal essay. It was lifted to national attention by
George W. Bush in the first presidential debate in the fall of 2000,
when Bush used it to characterize Al Gore as a number-benumbed
pedant who was, in Bushs charge, eliding the truth with numbers.
In its most recent turnabout, the term has been slapped back on
Bush by the New York Times columnist and economist Paul Krugman, whose book Fuzzy Math: The Essential Guide to the Bush Tax
Plan excoriates the Bush administrations arithmetic on tax relief.42
So what is to be done? Statistics are not the perfect distillation of
truth that early nineteenth-century statesmen thought they were,
but neither are they the products of fuzzy math that can be safely
disregarded or disparaged. Statistical reasoning and the numbers it
produces are powerful tools of political and civic functioning, and
at our peril we neglect to teach the skills to understand them in our
education system. Some of this teaching needs to happen in arithmetic and mathematics classes, but some of it must be taken up by
other parts of the curriculum, in any and every place in which
critical thinking, skepticism, and careful analysis of assumptions
and conclusions come into play.
On my campus (the University of California at Santa Barbara)
and no doubt many others, two programs developed in the last
decade or two aimed to generalize basic skills. The first, Writing
Across the Curriculum, devised ways to implant intensive writing experiences in courses well beyond the expected domains of
the English department or writing programsay, in engineering
and the sciences. Additionally, composition teachers taught writing courses keyed to the science and social science curricula. And
in a related fashion, language instruction and practice branched
17
2.
18
4.
5.
6.
Consider national land policy during the nineteenth century with respect to the selling of the national domain. How
should government handle such a valuable resource? How
did the government, at various times, set up land sales?
What were the origins of the rectilinear survey idea? How
were the survey lines run? What were the procedures on size
of parcels? Who gained benefits and who did not?
7.
8.
What has been the average life expectancy over our countrys history? How is that number arrived at? How has it
changed over time? How does it vary by race, by sex, by
region, and why? Who first tried to frame this question and
answer it, and why? Why was/is it worth answering?
9.
10.
Ditto for the history of wealth. How has wealth been mea-
In the late nineteenth century and later, where did quantitative knowledge come from? Who generated it? Who processed it? Who abstracted it? Who defines the standard measuresof weight and quantity, of economic indicators
and what difference might that make?
12.
Notes
1. A Statistical view of the Commerce of the United States of America;
its connection with agriculture and manufactures, and an account of
the publick debt, revenues, and expenditures of the United States. . .
By Timothy Pitkin. . . , The North American Review and Miscellaneous Journal 3(9): (1816): 34554. The published review was unsigned, but the copy digitized on Cornell Universitys Making of
America Web site includes attributions for many articles in this
volume, added in an early nineteenth-century cursive hand. J.
Quincy was written at the head of this piece, on p. 345; Josiah
Quincy also was on a similar but very long review in the same
volume, a review of Moses Greenleafs Statistical View of the District
of Maine (pp. 362 426). The federalist Josiah Quincy (17721864)
had a distinguished career in politics and higher education. He
served in the U.S. Congress for Massachusetts from 18051814, in
the state senate from 18151821, as a judge in 18211823, as mayor
of Boston from 18231829, and finally as president of Harvard
University from 1829 1845.
2. The phrase Political Arithmetick was first used by the English
economist Sir William Petty in the late seventeenth century to describe what seemed to others to be an unorthodox combination of
high-level statecraft with arithmetic, which was then seen as a vulgar
art beneath the notice of leaders because of its associations with the
world of commerce. Petty promoted the expression of all political
and economic facts in terms of Number, Weight, and Measure.
William Petty, Political Arithmetick (London, 1690), reprinted in
Charles Henry Hull, ed., The Economic Writings of Sir William Petty,
2 vols. (Cambridge, UK: At the University Press, 1899), I: 244.
3. Margo J. Anderson, The American Census: A Social History (New
Haven, CT: Yale University Press, 1988), 1516. In the end, remainders were ignored, in accordance with Jeffersons plan; Hamiltons plan had provided for extra seats distributed to the states with
the largest remainders.
4. Quoted in Patricia Cline Cohen, A Calculating People: The Spread of
Numeracy in Early America (New York, NY: Routledge, 1999), 159
60.
5. Cohen, A Calculating People, 161 62.
6. John Sinclair, The Statistical Account of Scotland (Edinburgh, 1798),
vol. 20, xiii.
7. S.v. statistick, John Walker, A Critical Pronouncing Dictionary and
Expositor of the English Language (Philadelphia, 1803); statistics,
Noah Webster, A Compendious Dictionary of the English Language
(New Haven, 1806).
8. A list of 31 of these books appears in Cohen, A Calculating People,
254, n.3.
9. See Patricia Cline Cohen, Numeracy in Nineteenth-Century
America, forthcoming in George M. A. Stanic and Jeremy Kilpatrick, eds., A History of School Mathematics (Reston, VA: National
Council of Teachers of Mathematics, 2003).
10. Nicholas Pike, Pikes Arithmetick (Boston, 1809), 101. Small print
helpfully elaborated that more requiring more, is when the third
term is greater than the first, and requires the fourth term to be
greater than the second.
11. I had not considered this possibility until I read Zalman Usiskins
essay, Quantitative Literacy for the Next Generation, in Mathematics and Democracy: The Case for Quantitative Literacy, Lynn
Arthur Steen, ed. (Princeton, NJ: National Council on Education
and the Disciplines, 2001), 79 86.
19
29. Western Academician and Journal of Education and Science (Cincinnati) I (1837): 438.
14. Bill for the More General Diffusion of Knowledge, 1779, Virginia
State Legislature; and Thomas Jefferson, Life and Selected Writings,
Adrienne Koch and William Peden, eds. (New York, NY: Modern
Library, 1944), 262-63. Plan discussed in Richard D. Brown, The
Strength of a People: The Idea of an Informed Citizenry in America,
31. The Politicians Register was published in Baltimore by G. H. Hickman; The Whig Almanac was the work of Horace Greeley, editor of
the New York Tribune.
20
32. Susan Herbst, Numbered Voices: How Opinion Polling Has Shaped
American Politics (Chicago: University of Chicago Press, 1993), 74 79.
33. New York Herald, April 17, 1839.
34. Margo J. Anderson, The American Census, 40 41.
35. Margo J. Anderson, The American Census, 5355; Cohen, A Calculating People, 22224. Quote is from Samuel M. Wolfe, Helpers
Impending Crisis Dissected, quoted in Anderson, 55.
36. The story of the industrial eras censuses is well told in Anderson, The
American Census, ch. 4.
37. George M. A. Stanic and Jeremy Kilpatrick, Mathematics Curriculum Reform in the United States: A Historical Perspective, International Journal of Educational Research 17 (1992), 409 11; Herbert
M. Kliebard, Schooled to Work: Vocationalism and the American Curriculum, 1876 1946 (New York, NY: Teachers College Press,
1999), 93, 156.
38. No research that I am aware of yet addresses this question for the late
nineteenth century. See Danny Bernard Martin, Mathematics Success
and Failure Among African-American Youth: The Roles of Sociohistorical Context, Community Forces, School Influence, and Individual
Agency (Mahwah, NJ: Lawrence Erlbaum Associates, 2000); James
D. Anderson, The Education of Blacks in the South, 1860 1935
(Chapel Hill, NC: University of North Carolina Press, 1988); and
Donald Spivey, Schooling for the New Slavery: Black Industrial Education, 1868-1915 (Westport, CT: Greenwood Press, 1978).
39. James T. Fey and Anna O. Graeber, From the New Math to the
Agenda for Action, forthcoming in Stanic and Kilpatrick, eds., A
History of School Mathematics.
40. Diana Jean Schemo, Test Shows Students Gains in Math Falter by
Grade 12, New York Times, August 3, 2001. Perhaps the most
striking finding in this report is that only a quarter of all eighth
graders scored at or above the proficient level on the mathematics
portion of the National Assessment of Educational Progress test last
spring. (The scores fell into four groups, below basic, basic, proficient, and advanced.) For a long-range view of these issues, see David
L. Angus and Jeffery E. Mirel, The Failed Promise of the American
High School, 1890 1995 (New York, NY: Teachers College Press,
1999).
41. For the most recent articulation of the famous aphorism (attributed
variously to Mark Twain or Benjamin Disraeli), see Joel Best,
Damned Lies and Statistics: Untangling Numbers from the Media,
Politicians, and Activists (Berkeley, CA: University of California
Press, 2001), 5.
42. The Oxford English Dictionary, 2nd edition (1989), credits mathematician Lotfi A. Zadeh for the term fuzzy in mathematics, dating
from 1964. It was in use in the California debate over curriculum
reform in 1995, appearing in a Los Angeles Times article of December
19, 1995, by Richard Lee Colvin: Parents Skilled at Math Protest
New Curriculum, Schools: Vocal minority, many in technical fields,
deride fuzzy teaching. But reformers call them elitist. Lynne
Cheneys Wall Street Journal article deriding fuzzy math appeared
June 11, 1997, and it sparked a flurry of other newspaper usages in
the following months, according to a LEXIS-NEXIS database
search. Bushs debate use occurred on October 3, 2000; Paul Krugmans book was published in May 2001 by W. W. Norton (NY).
Anthony P. Carnevale is Vice President for Education and Careers at Educational Testing Service. Previously, he served as
Vice President and Director of Human Resource Studies at the Committee for Economic Development and as Chair of the
National Commission for Employment Policy. Carnevale has written numerous books and articles on competitiveness and
human resources, most recently The American Mosaic: An In-depth Report on the Future of Diversity at Work and Tools and
Activities for a Diverse Work Force.
Donna M. Desrochers is a Senior Economist at Educational Testing Service. Previously, she served as an economist at the
Bureau of Economic Analysis and at the Center for Labor Market Studies. Desrochers research examines how changes in
the economy impact the education and training needs of students and workers. She has co-authored several articles and
reports, most recently Help Wanted. . .Credentials Required: Community Colleges in the Knowledge Economy (2001).
21
22
humanities need to be lowered. Moreover, to fully exploit mathematics as a practical tool for daily work and living, mathematics
needs to be taught in a more applied fashion and integrated into
other disciplines, especially the applied curricula that now dominate postsecondary education.
(Murnane, Willet, and Levy 1995). Improvements in mathematical skills account for at least half of the growing wage premium
among college-educated women and is the most powerful source
of the wage advantages of people with postsecondary education
over people with high school or less. Moreover, although the wage
premium for college-educated workers has increased across all
disciplines, it has increased primarily among those who participated in curricula with stronger mathematical content, irrespective of their occupation after graduation (Grogger and Eide
1995).
Those with stronger quantitative skills thus earn more than other
workers. Data from the National Adult Literacy Survey (NALS)
show that workers with advanced/superior mathematical literacy similar to that of the average college graduate earn more than
twice as much as workers with minimal quantitative skills similar to average high school dropouts. Those with advanced/superior mathematical literacy earn almost twice as much as workers
with the basic quantitative skills typical of below-average high
school graduates. Moreover, the importance of quantitative skills
in labor markets will grow in the future. Almost two-thirds of new
jobs will require quantitative skills typical of those who currently
have some college or a bachelors degree (see Figure 1).
Success in the new information economy also appears to require a
new set of problem-solving and behavioral skills. These skills,
especially problem-solving skills, emphasize the flexible application of both mathematical and verbal reasoning abilities in multifaceted work contexts across the full array of occupations and
industries. Such skills most often require the versatile use of relatively basic mathematical procedures more akin to numeracy
and quantitative literacy than to higher knowledge of advanced
mathematical procedures.
Unlike many of the continental European systems, there are minimal earnings and benefits guarantees for the unemployed or the
underemployed in the United States. Even among those who are
fully employed, wages and benefits depend on skill. We know that
those who cannot get or keep good jobs are trapped in working
poverty, underemployed, or unemployed. Eventually many of
them drop out of the political system and withdraw from community life. In some cases, they may create alternative economies,
cultures, or political structures that are a threat to the mainstream.
If educators cannot fulfill their economic mission to help our
youth and adults achieve quantitative literacy levels that will allow
them to become successful workers, they also will fail in their
cultural and political missions to create good neighbors and good
citizens.
Higher levels of quantitative literacy increase both individual and
national income. Sweden is one of the most quantitatively literate
countries in the world. If the levels and distribution of quantitative skills in the United States mirrored those of Sweden, a backof-the-envelope calculation suggests that we could increase GDP
23
24
Although more is not always better, in this case it often is. For
instance, we have four times as many workers as France, Italy, or
the United Kingdom. Four pretty good engineers tackling a business problem often outperform one very good engineer working
alone. Similarly, four companies in the software business competing directly against each other in the highly competitive U.S.
product market are likely to produce better software than a single
company elsewhere.
A second advantage that allows the United States to get away with
relatively low levels of mathematical and scientific literacy is the
flexibility that allows us to make better use of what talent there is.
In the United States, minimally regulated labor markets allow
If we are to retain the lead in the global economic race and the
good jobs that go with it, we will at some point have to rely on
homegrown human capital for our competitive edge. Eventually,
we will have to close the education gap; however, because of demographic shifts we face at home, that may be surprisingly difficult.
for men, and that retirement ages have been declining steadily. By
2020, about 46 million baby boomers with at least some college
education will be over 55 years of age. Over the same period, if we
maintain current attainment rates in postsecondary education, we
will produce about 49 million new adults with at least some college educationa net gain of about three million (Carnevale and
Fry 2001).
Historical and projected increases in the share of jobs that will
require at least some college-level mathematical literacy far exceed
this small increase in the college-educated population, however.
Official projections on the share of jobs that will require at least
some college education through 2020 are unavailable, but the
U.S. Bureau of Labor Statistics projects a 22 percent increase by
2010 in such jobs. If the trend continues, we will experience a net
deficit in workers with mathematical skills at or above the some
college level of more than 10 million workers by 2020 (Carnevale
and Fry 2001).
25
26
Table 1.
Mathematical Literacy Paradigm from the National Adult Literacy Survey
Skill
Level
Minimal
Basic
Competent
Advanced
Superior
Approximate
Educational Equivalence
Dropout
Some postsecondary
education
Bachelors or advanced
degree
NALS
Level
Source: Carnevale, Anthony P., and Donna M. Desrochers. 1999. Getting Down to Business: Matching Welfare Recipients to Jobs that Train. Princeton, NJ: Educational
Testing Service; Barton, Paul E., and Archie LaPointe. 1995. Learning by Degrees: Indicators of Performance in Higher Education. Princeton, NJ: Educational Testing
Service.
Those who get the best jobs have taken the most mathematics. We
estimate that three-fourths of those in the top-paying 25 percent
of jobs have at least one year-long high school credit in algebra II.
More than 80 percent have taken geometry. Twenty-seven percent of those in the top-paying jobs have at least a semester of
pre-calculus and roughly 20 percent have taken calculus. Among
the rest of those in the top half of the pay distribution, more than
half have taken algebra II and more than two-thirds have taken
geometry in high school. In the bottom half of the distribution of
earnings in American jobs, roughly three-quarters have at least a
27
28
29
30
Notes
1. The debate over whether mathematics should be taught as an abstract
deductive system or in a more applied fashion sets up a false choice
between purists and advocates of quantitative literacy and numeracy.
The validity of mathematics is founded on deduction but it develops,
and is most easily understood, in applied contexts. Similarly, the distinction between mathematical and verbal reasoning is also artificial.
In the real world, reasoning is a cognitive soup of words and numbers
that assumes the shape of social contexts. (Cole 1996; National Research Council 2000; Scribner 1997; Scribner and Cole 1997).
2. The most powerful call for a general curriculum comes from a study
released in 1945, sponsored by James Conant at Harvard, officially
entitled General Education in a Free Society and unofficially known as
the Red Book. The report argued for interdisciplinary learning to
foster an appreciation of the pluralism of ideasthe rational, subjective, and spiritualat the heart of western culture. The general curriculum was viewed as an antidote to single-minded ideologies and
fanaticism. In 1945 that meant communism and fascism. In the
twenty-first century, it applies to the global clash of cultures. The
development of a general curriculum remains difficult in the context
of specialization in the academic disciplines and the rise of vocationalism.
3. Teaching and learning that takes advantage of the synergy between
applied and abstract knowledge can be deeper and more accessible, if
done properly (see Barton 1990; Berryman and Bailey 1992; National
Research Council 1998; Resnick and Wirt 1995; Schoen and Zubarth
1998; Steen 1997; Steen 2001; Wood and Sellers 1996).
4. Data from the International Study of Adult Literacy shows that workers in Sweden have the following distribution of quantitative literacy:
Level 1 (lowest): 5 percent; Level 2: 17 percent; Level 3: 40 percent;
and Level 4/5 (highest): 38 percent. In contrast, the distribution of
workers quantitative literacy in the United States is much lower:
Level 1 (lowest): 16 percent; Level 2: 24 percent; Level 3: 33 percent;
and Level 4/5 (highest): 27 percent (OECD 1995). To estimate the
increases in GDP and taxes that would occur if we had a quantitative
literacy distribution similar to Swedens, we first calculated the number of workers in the United States at each literacy level and, second,
applied the distribution of literacy in Sweden to the total number of
workers in the United States to estimate how many workers would fall
at each skill level if the United States quantitative literacy levels resembled Swedens. Taking both of the distributions, we multiplied
the average earnings of U.S. workers at each skill level by the number
of workers at each level and summed to get aggregate earnings. The
difference in aggregate earnings using the U.S. and Swedish distribu-
References
Barton, Paul E. 1990. From School to Work. Princeton, NJ: Policy Information Center, Educational Testing Service.
Barton, Paul E., and Archie LaPointe. 1995. Learning by Degrees: Indicators of Performance in Higher Education. Princeton, NJ: Policy Information Center, Educational Testing Service.
Barzun, Jacques. 2000. From Dawn to Decadence: 1500 to the Present: 500
Years of Western Cultural Life. New York: HarperCollins.
Berryman, Sue E., and Thomas R. Bailey. 1992. The Double Helix of
Education and the Economy. Institute on Education and the Economy, Teachers College. NY: Columbia University Press
Carnevale, Anthony P., and Donna M. Desrochers. 1999. Getting Down
to Business: Matching Welfare Recipients to Jobs that Train. Princeton,
NJ: Educational Testing Service.
Carnevale, Anthony P., and Donna M. Desrochers. 2001. Help Wanted
. . . Credentials Required: Community Colleges in the Knowledge Economy. Princeton, NJ: Educational Testing Service.
Carnevale, Anthony P., and Richard A. Fry. 2000. Crossing the Great
Divide: Can We Achieve Equity When Generation Y Goes to College?
Princeton, NJ: Educational Testing Service.
Carnevale, Anthony P., and Richard A. Fry. 2001. The Economic and
Demographic Roots of Education and Training. Washington,
DC: Manufacturing Institute, National Association of Manufacturers.
Carnevale, Anthony P., and Stephen J. Rose. 1998. Education for What?
The New Office Economy. Princeton, NJ: Educational Testing Service.
Castells, Manuel. 1997. The Information Age: Economy, Society and Culture: Volume II. The Power of Identity. Oxford, UK: Blackwell.
Cole, Michael. 1996. Cultural Psychology: A Once and Future Discipline.
Cambridge, MA: Harvard University Press.
Ewell, Peter T. 2001. Numeracy, Mathematics and General Education. In Mathematics and Democracy: The Case for Quantitative
Literacy, edited by Lynn Arthur Steen, 37 48. Princeton, NJ: National Council on Education and the Disciplines.
Flynn, James R. 1998. IQ Gains Over Time: Toward Finding the
Causes. In The Rising Curve: Long-Term Gains in IQ and Related
Measures, edited by Ulric Neisser, 25 66. Washington, DC: American Psychological Association.
Gellner, Ernest. 1992. Postmodernism, Reason and Religion. London:
Routledge.
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Roey, Stephen, Nancy Caldwell, Keith Rust, Eyal Blumstein, Tom Krenzke, Stan Legum, Judy Kuhn, and Mark Waksberg. 2001. The High
School Transcript Study Tabulations: Comparative Data on Credits
Earned and Demographics for 1998, 1994, 1990, 1987, and 1982
High School Graduates. National Center for Education Statistics.
NCES 2001 498. Washington, DC: U.S. Government Printing
Office.
Romer, Paul M. 2000. Should the Government Subsidize Supply or
Demand in the Market for Scientists and Engineers? NBER Working Paper 7723. Cambridge, MA: National Bureau of Economic
Research.
Schoen, H. L., and S. Zubarth. 1998. Assessment of Students Mathematical Performance. Core Plus Mathematics Evaluation. Iowa
City, IA: University of Iowa.
Schooler, Carmi. 1998. Environmental Complexity and the Flynn Effect. In The Rising Curve: Long-Term Gains in IQ and Related
Measures, edited by Ulric Neisser, 6779. Washington, DC: American Psychological Association.
Scribner, Sylvia. 1997. Mind and Social Practice: Selected Writings of
Sylvia Scribner. Cambridge, UK: Cambridge University Press.
Scribner, Sylvia, and Michael Cole. 1997. The Psychology of Literacy.
Cambridge, MA: Harvard University Press.
Steen, Lynn Arthur, ed. 1997. Why Numbers Count: Quantitative Literacy
for Tomorrows America. New York: College Entrance Examination
Board.
Steen, Lynn Arthur, ed. 2001. Mathematics and Democracy: The Case for
Quantitative Literacy. Princeton, NJ: National Council on Education and the Disciplines.
U.S. Department of Education. National Center for Education Statistics.
2000. Digest of Education Statistics. Washington, DC: U.S. Government Printing Office.
U.S. Department of Education. The National Commission on Excellence in Education. 1983. A Nation at Risk: The Imperative for
Educational Reform. Washington, DC: U.S. Government Printing
Office.
Wood, T., and P. Sellers. 1996. Assessment of a Problem-Centered
Mathematics Program: Third Grade. Journal for Research in Mathematics Education 27(3): 33753.
The Challenge
The first order of business is to demonstrate that mathematics education is inadequate to todays
challenge. The challenge exists because of mathematics growing importance for both economic and
citizenship reasons. It is no accident that, along with reading, mathematics is one of the two subjects
that always are required on standardized tests. This implies that it is important for everyone, not only
for those few who love the subject and grow to see the beauty in it.
Two hundred years ago, only merchants, engineers, surveyors, and a few scientists were mathematically literate. Merchants had to calculate the price of cloth 2 yards 1 foot 4 inches square at 3 pence
2 farthings the square foot.3 Military engineers had to determine the angle needed for a cannon to
project a missile over a moat. Surveyors had to lay out site lines. Isaac Newton needed to invent
calculus to solve his physics problems. How many Americans are now mathematically literate is an
arguable question. By some estimates, it is less than one-fourth: that is how many adults achieve
levels of 4 and 5 in the National Adult Literacy Survey (NALS) and International Adult Literacy
Arnold Packer is Chair of the SCANS 2000 Center at the Institute for Policy Studies, Johns Hopkins University. An
economist and engineer by training, Packer has served as Assistant Secretary for Policy, Evaluation, and Research at the U.S.
Department of Labor, as co-director of the Workforce 2000 study, and as executive director of the Secretarys Commission
on Achieving Necessary Skills (SCANS). Currently, his work is focused on teaching, assessing and recording the SCANS
competencies.
33
34
ber, via rote recall and long enough to pass the final, how to plug
in numbers and chug through the formula.14 Do not, however,
ask either the passing or failing students to apply the technique to
a real-life problem. Weeks after the final, they cannot even remember that the formula exists. They have no way to recall the
formula from long-term memory. Imagine asking most adults the
formula for solutions to the quadratic equation or, worse yet, what
real-life process is described by the equation. Nor does the problem disappear as students take higher-level courses: . . . undergraduates never learn how calculus relates to other disciplines,
much less the real world.15 The situation is untenable for two
kinds of students: those who do not like mathematics and those
who do.
Thomas Berger of Colby College, former chair of the Committee
on the Undergraduate Program in Mathematics of the Mathematical Association of America, speaks of the mathematization of
society as symbolic models become the basic tools of engineers,
medical researchers, and business executives.16 These models,
however, simulate real-world processes and systems. They are used
to allocate resources, design technology, and improve system performance. Mathematization increases the significance of higherlevel skills for most citizens. Successful workers, citizens, and consumers need to know how to solve problems, analyze data, and
make written and oral presentations of quantitative results.
How does the traditional pre-calculus curriculum serve these purposes? How valuable, actually, is calculus? Even mathematicians
often replace calculus with finite mathematics to take advantage of
computer technology.17 The same technology can handle both
the traditional tasks of manipulating formulas and performing
long computations. Spending too much time teaching humans to
solve problems better handled by machines is not a wise strategy
when there is so much to know.
Conceptualizing how a problem can be stated mathematically has
become (and, indeed, always was) more valuable than factoring a
polynomial or taking a derivative. The tasks most college graduates face demand quantitative literacy (a.k.a. empirical mathematics)a way with numbers and comfort replacing concrete realities
with symbolswithout forgetting the reality beyond the symbols.
In other words, banish x and y from mathematics class at least until
the completion of college algebra. Use, instead, letters (even Greek
letters) that stand for something students can understand or picture: v for velocity, d for distance, P for price, n for number, p for
profit, and so on.
This idea is not so radical. In his introduction to Why Numbers
Count, Lynn Arthur Steen referred to scientific mathematics in
which mathematical variables always stand for physical quantitiesa measurement with a unit and implicit degree of
accuracy.18 In the same volume, F. James Rutherford said,
35
. . . citizens need to possess certain basic mathematical capabilities understood in association with relevant scientific and technological knowledge.19
Mathematicians, of course, want their students to understand
mathematics power to solve general problems, ones that are not
rooted in a specific problem. That point can be made and demonstrated near the end of the mathematics course. Teach that the
equation for velocity can be used in many contexts relating to
change. True generality can be saved for those mathematics students who still will be taking mathematics courses in their junior
year in college. (Indeed, there might be more such students if
mathematics were less abstract in the earlier years of school.) Make
each year of mathematics instruction worthwhile in itself, not just
preparation for the next mathematics course.
Many critics of this point of view believe it does not credit the
power of abstract mathematics to be generally applicable. Butas
guns dont kill but people with guns domathematics does not
apply itself: mathematically competent individuals do. Individuals require enough competence and creativity to structure a problem mathematically and to know when and how to use the tools.
On that basis, the current approach to achieving widespread
mathematical competence is failing. The real issue is whether
mathematics should be taught inductivelyfrom the concrete to
the abstract or the other way around, as it often is today. The
answer can be found empirically, not theoretically. Which approach will meet educations goals of productive workers, engaged
citizens, and well-rounded individuals who continue to learn after
graduation?
Quantitative Literacy:
Goals and Objectives
What mathematics should everyone know and be able to do?
The National Council of Teachers of Mathematics (NCTM) has
been attempting to answer this question for more than a decade as
council members developed mathematics standards. Wisely,
NCTM included the ideas of real-world problem solving and
being able to communicate in the language of mathematics. Its
report, however, is built around standard mathematical topics,
algebra, geometry, calculus, statistics, and so on. The problems
were what we might expect: Solve x2 2x 7 0, or derive the
equation describing the motion of a Ferris wheel car. Few individuals work on either of these problems outside of a school situation. One helpful criterion is to restrict problems to those that
American workers get paid to solve, those that American citizens
should have informed opinions about, or those that American
consumers actually need to solve.
36
Consider the challenge posed by Bergers mathematization of society. What if Bergers problem were stated in the language of
mathematical optimization? (Surely, mathematicians should applaud using their discipline to analyze this problem.) In the workplace, industrial psychologists analyze the frequency and criticality (sometimes called importance) of tasks and the various skills
needed to carry them out. The human resources department then
either hires folk with the more important skills or trains staff in
those skills. Transfer this thinking to the school situation in which
the challenge is how best to prepare students for their life beyond
the schoolhouse walls. Think first of the students professional life.
The optimizing school would seek to maximize the benefit that is
a function of:
1.
2.
3.
4.
A Scheme
E. D. Hirsch is widely known for his educational canon. He lists
things students should know at each grade level. I will try to avoid
such laundry lists and suggest, instead, a structure in which types of
problems can be placed or developed. This structure can accommodate a range of difficulties for each of the problem types. Some examples in the range are suitable for different school grades. A second
grader may, for example, be taught something about schedules while
mathematics post-docs may struggle with variations of the traveling
37
Information problems. Gathering and organizing data, evaluating data, and communicating, both in written and oral
form.
1.
2.
It is one of the only structures that have widespread recognition across the academic and occupational standards that
have been developed in the United States and other countries.
3.
4.
Four roles are used to fill out the structure. For most students
and policymakers, the role of work and careers is of foremost
concern. If not for this role, mathematics departments would be
competing for students with literature and art departments, not
with computer sciences. Unlike most school-based problems, real
work-based problems usually cannot be solved in a few minutes
but take hours of sustained effort. Preparation for this role requires that students engage in long-term projects. The ability to
carry out such tasks has been noted by a recent National Academy
of Sciences effort to define computer literacy and fluency.27 Academy member Phillip Griffiths, director of the Institute for Advanced Studies in Princeton, speaks to the need for the mathematically competent to function in various SCANS domains. We
asked . . . [about] . . . science and engineering PhDs. The employers told us that . . . they find shortcomings: . . . communications skills . . . appreciation for applied problems . . . and teamwork. . . . Skills like project management, leadership . . .
interpersonal skills . . . computer knowledge. Students will have
to work in teams, use computers to solve problems, and make oral
and written presentations if Griffiths requirements are to be met.
Many mathematics teachers may decry my emphasis on work and
careers, so I want to acknowledge, once again, the importance of
other domains (without relinquishing the idea that the primary
force behind the nations emphasis on mathematics is economic).
Individuals also require mathematics to succeed in their second
important role as consumer. Some buying problems can and must
be solved quickly and on the spot. Comparing the price per ounce
for similar products sold in differently sized bottles, understanding discounts, approximating a large restaurant or hotel bill are
examples. Other problems, such as comparing retirement or
health plans or comparing mortgage rates may take more time
(although the Internet can speed things up).
38
39
Computers take derivatives and integrals, invert matrices for linear programming, and perform other algorithms much better
than humans. I have written elsewhere about the end of algorithmic work as a means of making a livelihood in the United States.34
American workers cannot accept a wage low enough to compete
with a computer. Workers do, however, require enough mathematics
and creativity to structure a problem so that mathematics can be used.
They also need sufficient quantitative capacity to know when tools
are not working right or have been improperly used.
This appendix, whose structure is shown in the accompanying
table, offers a first stab at a canon of empirical mathematical
problems. The table lists mathematical tools for each of the five
SCANS competencies and each of the four adult roles. Some of
the entries in the table do not include all the tools needed because
they have been noted frequently in other boxes. For example, the
four arithmetic functions are needed to solve many problems that
an American will face, so they are not listed repeatedly. When the
phrase concept of . . . is used in the table, it means just that
knowing the concept of rate of change and change in the rate of
change (acceleration) does not require knowing how to take the
second derivative. The same thought applies to linear programming and to the idea that minimums and maximums occur when
the derivative is zero or inflection when the second derivative is
zero or that an integral takes the sum over an interval.
The main part of the appendix consists of examples of important
tasks arranged according to the SCANS taxonomy. The mathematical models described are expected to be developed and expressed in spreadsheets, graphics packages, etc. Students would be
asked to estimate rough numbers to ensure that the models have
been properly specified and the numbers properly entered.
Planning
SCHEDULE:
Worker: Using a spreadsheet (or other software) with algebraic
formulas, develop a schedule for a construction project, advertising campaign, conference, medical regime, or software
project. Require conversions from hours to workweeks. Understand the difference between activities done in sequence
and simultaneously. Understand PERT and Gantt charts.
Consumer: Using pencil and paper without a calculator, plan a
party or a meal. Convert hours to minutes.
Citizen: Understand why it takes so long to build a road or school.
Personal: Appreciate why Napoleon was beaten by the weather
and Russia.
SPACE:
Worker: Using a computer graphics package, lay out a storeroom or
office space in three dimensions. Develop a graphic for a brochure. Lay out material for a garment or a steel product. Lay out
a restaurant or hotel space. Place paintings in a gallery.
Consumer: Look at a builders plans and modify them. Understand your own living space.
Citizen: Understand plans for a public building.
Personal: Appreciate good design in products and buildings.
Hang paintings in your house.
BUDGET:
STAFF:
40
TEACH
AND
LEARN:
Worker: For teachers, help students do quantitative work in nonmathematical subjects.35 For workers, teach co-workers or customers the mathematics needed to carry out a task or use a
product. Should know enough mathematics to absorb training.
Consumer: Should know enough mathematics to learn how to use
a product when taught by a salesperson. Should be able to
teach a spouse how to use a product.
Citizen: Should be able to explain and debate policy issues when
quantitative issues are involved.
Personal: Should be able to discuss topics when quantitative issues are involved.
Information
MONITOR:
Worker: Use techniques of statistical process control to monitor a
manufacturing process or patient or customer complaints.
Consumer: Understand statements about the quality of the products or services purchased.
Citizen: Understand environmental safeguards.
Personal: Monitor changes in a local garden.
GATHER
AND
ORGANIZE:
DESIGN:
EVALUATE:
Worker: Use a statistical package to evaluate data. Read relevant
statistical studies and come to a judgment.
Interpersonal
NEGOTIATE:
Worker: Negotiate the price of a product or project and be able to
think on your feet, including manipulating numbers mentally. Participate in a labor-management negotiation.
Consumer: Be able to understand a construction contractors or
mechanics proposal and negotiate a fair agreement.
Citizen: Understand a government negotiation.
Personal: Understand a historically important negotiation.
COMMUNICATE:
Worker: Write a report about a quantitative issue, including tables and charts. Make a presentation on the material to more
senior colleagues.
Consumer: Read and listen to such reports critically and be able to
ask intelligent questions.
Technology
USE:
Worker: Use equipment, such as a numerically controlled machine tool, to produce a part.
Consumer: Use a computer.
Citizen: Use a countys Internet address to find tax data.
Personal: Use a chat room to engage in discussion.
41
Consumer: Analyze alternatives for video on demand, home security systems, or computers.
Citizen: Analyze a countys or school boards decision to purchase
technology, from fire engines to computer systems. Be able to
judge whether the antimissile system makes sense.
Personal: Analyze a historic technology decision, from the longbow to atomic energy.
MAINTAIN:
Worker: Follow maintenance instructions for a piece of industrial
equipment.
Consumer: Follow maintenance instructions for a consumer
product.
CHOOSE:
Worker: Analyze alternative medical, construction, manufacturing, or computer equipment and recommend a purchase.
Personal: Maintain rare books or valuable paintings when temperature and humidity must be controlled.
Mathematics Required to Solve Frequently Occurring Problems in Four Roles and Five SCANS Competencies36
Problem
Domains
Planning
Budget
Schedule
Space
Staff
Interpersonal
Negotiate
Teach and
learn
Information
Gather and organize
Evaluate
Communicate
Technology
Use
Choose
Maintain
Worker
Role
Model-building. Concept of
first and second
derivative and of
integral, average, and
standard deviation.
Mental arithmetic,
fractions,
percentages.
Consumer
Role
Mental arithmetic,
fractions,
percentages.
Citizen
Role
Mental arithmetic,
fractions,
percentages.
Personal
Role
Geometry, concept of
trade-offs.
Geometry.
42
Notes
formula (in a case where it worked easily)! This is the crux of the
problem I see from day to day, from freshmen on up.
15. Robin Williams, The Remaking of Math, Chronicle of Higher Education, 7 January 2000, A14.
16. Ibid.
17. http://www.stolaf.edu/other/ql/intv.html.
18. Lynn Arthur Steen, Preface: The New Literacy. In Why Numbers
Count, Lynn Arthur Steen, ed. (New York, NY: College Entrance
Examination Board, 1997), xvxxviii.
19. F. James Rutherford, Thinking Quantitatively about Science. In
Steen, Why Numbers Count, 60 74. Italics in the original.
20. Arnold Packer, Mathematical Competencies that Employers Expect, In Steen, Why Numbers Count, 137-54.
21. Ibid.
10. Ibid.
25. http://www.stolaf.edu/other/ql/intv.html.
12. Lynn Arthur Steen pointed out that adding odd fractions is preparation for adding mixed algebraic fractions. The preparation can, in
my judgment, wait until students reach such algebra problems (if
ever). The cost, in students who become convinced that mathematics is not for them is too high to justify the benefit.
13. Robert S. Bernstein and Michele Root-Bernstein, Learning to
Think With Emotion, Chronicle of Higher Education, 14 January
2000, A64.
14. An anecdote from Steve Childress of New York University: I just
finished grading an exam I gave to graduate students seeking admission to our Ph.D. program. One of the questions I asked (the subject
was complex variables) was of a standard kind requiring the calculation of a residue. Now there are various ways of doing this, certain
formulas that are useful in individual cases, but the heart of the
matter is that you are seeking a certain coefficient in a series and this
can usually be obtained directly by expanding the series for a few
terms. My problem could be solved in several lines by this direct
approach. I was astounded to see that almost everyone applied a
certain formula that, in this problem, led to impossibly complicated
mathematics. I asked around about this and learned that we had just
instituted a kind of prep course for the exams, and that the instructor
had given them a problem of this type and solved it with that special
27. National Academy of Sciences, Being Fluent with Information Technology (Washington, DC: National Academy Press, 1999).
28. I recall my quantitatively literate mother using her skills to figure out
when important eventssuch as births, marriages, and deaths
occurred.
29. Arthur Mattuck, The Remaking of Math, Chronicle of Higher Education, 7 January 2000, A15.
30. I heard one engineering dean wonder if his course in electronics was
not the best recruiting tool the School of Business had for transfers to
its program.
31. That is, we do not shine even on the disaggregated topics. As to real
problem solving, it is not even tested.
32. I, and many others, have been involved in creating CD-ROMs to
relieve teachers of the task of project construction.
33. Something advocated by NCTM and emphasized to me by Ivar
Stakgold in a private telephone conversation.
34. Hudson Institute, The End of Routine Work and the Need for a
Career Transcript, Hudson Institute Workforce Conference (Indianapolis, IN: September 2324, 1998).
35. Mathematics teachers would presumably have taken mathematics
courses beyond this level.
36. See Appendix for examples of the entries in this table.
Linda P. Rosen is an educational policy consultant. Previously, she was Senior Vice President for programs at the National
Alliance of Business. Prior to that she served as mathematics and science advisor to Education Secretary Richard Riley and
as executive director of the Glenn Commission on mathematics and science teaching, whose report, Before Its Too Late,
was issued in September 2000. Earlier, Rosen served as executive director of the National Council of Teachers of Mathematics (NCTM) and associate executive director of the Mathematical Sciences Education Board (MSEB).
Claus von Zastrow is Director of Institutional Advancement at the Council for Basic Education. Previously, he was Director
of Post-secondary Learning at the National Alliance of Business.
Lindsay Weil is Education and Marketing Manager at the Character Education Partnership. Previously, she was Program
Manager at the National Alliance of Business.
43
44
STATE REFORM
Advocacy: In collaboration with the nations governors, over the
past five years several CEOs6 have served as members of Achieve,
Inc., including participation in four national education summits.
These corporate leaders have committed themselves and their
companies to improving student achievement, increasing investments in and accountability for teachers, and promoting regular
assessments that are comparable across schools and districts.
Implementation: The business communitythrough state and local business coalitionsplans to work with state education officials to implement No Child Left Behind, the reauthorized Elementary and Secondary Education Act (ESEA). These efforts may
include dissemination of information about the legislation, mobilization of business leaders to participate in strategic planning,
identification of effective practices for business involvement, and
providing public officials with the business perspective on roadblocks and implementation successes.
ENHANCING
THE
TEACHING PROFESSION
The business community is now partnering with educational leaders, policymakers, and other stakeholders to bring these models to
fruition.
Micron Technology devotes considerable staff time and energy to K12 programs that demonstrate the importance of
mathematics to twenty-first-century careers.
SETTING BENCHMARKS
The American Diploma Project (ADP), recently launched by
Achieve, the National Alliance of Business, the Fordham Foundation, and the Education Trust, has three goals:
1.
45
2.
3.
IMPLICATIONS
These business initiatives all address education, yet only the last
one overtly addresses quantitative literacy, and then only in the
context of school mathematics. Although the business community
has demonstrated its sincere and long-standing commitment to
education reform, the issue of quantitative literacy is almost absent from its education agenda. Furthermore, business leaders are
not looking for new issues to champion, especially when substantial progress on existing issues remains elusive. Advocates for
greater quantitative literacy, therefore, cannot expect business to
take any position on the issuemuch less to promote it in its
principles for education reform unless they themselves raise
business awareness of the issues importance. To do so, they first
have to formulate a useful definition of quantitative literacy, one
that clearly addresses the business demand for necessary knowledge and skills and one that is widely understood.
46
Students and workers can document and advance their employability skills; and
Yet the WorkKeys Applied Mathematics scale is primarily arithmetic, with virtually no reference to the reasoning skills sought by
the GE quality manager. Although this scale might be appropriate
for some entry-level jobs in some industries, its usefulness across
the economy has not yet been demonstrated. Thus, its utility as
the basis for a workplace definition of quantitative literacy is questionable at best.
Skills Standards: The National Skills Standards Board (NSSB) was
created in 1994 to build a voluntary national system of skill
standards, assessment and certification systems to enhance the
ability of the U.S. workforce to compete effectively in a global
economy.10 These standards were intended to define the work to
be performed, how well the work must be done, and the level of
knowledge and skill required. Although a description of quantitative literacy in the workplace might emerge from this initiative,
this potential is far from being realized, because:
The rate of change in the workplace has outpaced the development of the standards, rendering them almost obsolete by
the time of release.
Formulating problems, seeking patterns, and drawing conclusions; recognizing interactions in complex systems; understanding linear, exponential, multivariate, and simulation models; understanding the impact of different rates of growth;
IMPLICATIONS
Attainment of quantitative literacy requires the ability to reason,
to make sense of real-world situations, and to make judgments
grounded in data. The description of lifelong literacylearning
to read and reading to learn could provide a good model for
the development of a similarly effective characterization of quantitative literacy. This model must capture the notion that people
who acquire quantitative literacy gain a foundation for future
learning, one that enables them to adapt to the demands of an
increasingly technological world.
Still, the business and education communities have yet to close
ranks around a single, well-known conception of quantitative literacy that could motivate a reform agenda advocated by both
parties. What is emerging, however, is a consensus that there is
something new needed by an educated adult, something more than
arithmetic proficiency. Business leaders are seeing the problem;
they have not yet seen the solution.
47
technological explosion. Innovations stemming from more advanced technology, remote satellite communication systems, fiber
optic cables, encryption, biotechnology and genomic discoveries,
laser scanners, and the Internet have launched the marketplace in
multiple, often uncharted, directions. As a result, a companys
competitive advantage rests with its workers ability to interpret
data, make decisions, and use available technology. This is true, to
some degree, of almost all jobs along the skill continuum, even
though each calls for different levels of quantitative literacy.
Although some jobs are becoming more complex, computers and
related technologies are simultaneously eliminating many traditional jobs. With the assistance of technology, one person, in
dramatically less time, now can accomplish tasks that once were
carried out by a team of people. ATMs have replaced bank tellers,
on-line databases have replaced travel agents, and computer-operated machines have replaced factory laborers.
JOB GROWTH
AND
DECLINE
48
IMPLICATIONS
Although gainful employment is not the sole purpose of education, it is a necessary and expected outcome. Education therefore
must be influenced by changes in the workplace. That some types
of jobs disappear and new ones emerge is certainly not a new
phenomenon; neither is the fact that education evolves to reflect
the changing employment market.
The accelerating rate of change in the workplace, however, heightens the challenge. Mathematics education reform in the late 1980s
criticized the lingering vestiges in school mathematics of a shopkeeper curriculum left over from the previous century for a nation no longer dominated by shopkeepers. We cannot afford the
luxury of such a slow response, if we ever could.
The pervasiveness of quantitative literacy among jobs showing the
greatest growthand the reasonable assumption that the trend
will continuerequires the education system to respond accordingly by incorporating quantitative literacy into schooling. In its
frequent calls for critical thinking abilities or real-world skills,
the business community has long been moving toward something
resembling a conception of quantitative literacy. Still, a vast gulf
separates this intuitive sense of new skill requirements from the
advocacy of education reforms that can actually result in a quantitatively literate citizenry.
IN EDUCATION
Business involvement in education, as described earlier, is focused
on policy. Calls for higher student achievement are often accompanied by calls for rigorous course work. What should comprise
that rigorous course workin mathematics or any subjectis not
discussed in detail. Instead of addressing specific pedagogical or
curricular questions in which it has little expertise, the business
community focuses on broader issues in terms of outcomes:
IN BUSINESS
Even in their own employee training programs, businesses do little
to encourage quantitative literacy.
Effectiveness: Reports from the American Society of Training and
Development, The Work in Northeast Ohio Council, and the
National Association of Manufacturers indicate that training programs are effective up to a point.14 These studies provide evidence that corporate training programs can improve employee
performance, firm productivity, product quality, and even company profitability. Indeed, such evaluations help business justify
the expenditure. Over the long term, however, such gains in productivity and profitability will inevitably remain limited as long as
the training is restricted to narrowly defined skill areas.
Course Content: Two types of corporate training programsremedial and computer-based could, but apparently do not, include quantitative literacy. Remedial programs tend to teach basic
arithmetic and fail-safe formulas with little emphasis on problem
solving. Employees are rarely taught to identify quantitative relationships in a range of contexts and settings, to consider a variety
of approaches to manipulate those quantitative relationships, or to
make data-based decisions on the job. As a result, few employees
acquire even rudimentary quantitative literacy on successful completion of such a program.
With businesses incorporating more technology into their daily
operations, the majority of workplace training both formal and
informal seminarsis computer related. Indeed, according to
Training Magazine, nearly 40 percent of all workers receive formal
training from their current employers.15 These classes run the
gamut from the use of spreadsheets to the use of advanced statistical analysis software such as SPSS.
Despite this universal access to computer training, such classes
apparently have little impact on quantitative literacy. Existing
computer training programs may fail to build strong quantitative
literacy because they devote scant attention to the connection
between computer applications and real-world scenarios. Because
accessing technology does not necessarily depend on a persons
ability to reason with the inputs or results, very few computerrelated training courses are contextualized. Consequently, train-
IMPLICATIONS
The candid answer to the questionWhat is business doing to
address quantitative literacy?remains: apparently very little, at
least little that consciously addresses the challenge.
We could attribute this inaction to uncertainty about effective
ways to broaden training or to participate meaningfully in educational discussions about curriculum. There could be a reluctance
to invest the time and money without a clear means of measuring
results. Business might be dismayed by the lack of a clear course of
study that leads to quantitative literacy, or by many other training
issues competing for attention and support in the business world.
Moreover, there is no clear leverage point to rally around. When
students are not yet achieving at acceptable levels in traditional
course work, the prospect of fighting for a new, somewhat amorphous concept with far-reaching curricular implications is daunting.
Most important, business leaders routinely measure investments
of time, resources, and commitments against the potential return
on the investment. Without a clear understanding of the means
and ends of quantitative literacythe ways in which young and
adult learners acquire the knowledge and skills, and the payoff for
such acquisition business will not likely make any serious investment.
This is not cause for discouragement but rather a window of
opportunity.
49
increase in quantitative literacy, we must adopt an aggressive strategy designed to improve the knowledge and skills of the current
and future workforce.
The following six action steps provide corporate America with a
blueprint for meeting this challenge:
1. Participate with education and workplace researchers to
better document the existing level of and anticipated need for
quantitative literacy in the workplace.
The vibrancy of the U.S. economy despite the recent downturnand the high level of innovation throughout history suggests that there has always been a cadre of people with the necessary skills and verve. Yet, a greater proportion of the workforce
needs quantitative literacy to sustain and grow business in the
twenty-first century.
There seems to be little readily available data that could inform
new policies to support broader acquisition of quantitative literacy. This is ironic, because in such a data-driven field, experts who
promote quantitative literacy apparently have not gathered the
ammunition to support the need expressed in their rhetoric.
Without data, questions pivotal to policy decision making end up
unanswered. For example: What proportion of the population
lacks quantitative literacy skills? What proportion of jobs requires
quantitative literacy? What can business expect as a return on
investment for implementing quantitative programs? Is it more
cost effective to achieve quantitative literacy through the educational pipeline or through workplace training or through both?
What are the educational characteristics of programs that would
yield quantitative literacy?
The inability to answer such questions has impeded, and will
continue to impede, progress in developing realistic options and
programs that demonstrate results.
2. Work with schools and colleges and among companies to
raise general awareness about the importance of quantitative
literacy in todays workplace.
Most businesses do not recognize quantitative literacy in the
workplace, making it difficult to design and support efforts to
increase it. Because many use computational capabilities as a
proxy for quantitative skills, they often develop and support educational programs that may rest on faulty assumptions. Quantitative literacy may even manifest itself differently from industry to
industry, from occupation to occupation, from task to task, further complicating the situation.
As more data about quantitative literacy are gathered, and businesses analyze the demand for quantitative literacy in the workplace, they will be better equipped to formulate a cogent message
50
to students, employees, educators, policymakers, and peer companies about the true implications of quantitative literacy in todays workplace. Such a public information campaign is needed to
institutionalize quantitative literacy as a fundamental goal of the
educational pipeline.
Because some information about workplace skills is proprietary
and may bear on a companys competitive advantage in the marketplace, impartial business organizations and researchers may be
best positioned to aggregate data on quantitative literacy and share
it among interested stakeholders. A broad group of stakeholders,
however, should work together to raise the level of awareness and
understanding of quantitative literacy.
3. Provide leadership and support to achieve quantitative literacy among elementary and high school students.
A specific and workable conception of quantitative literacy should
provide a foundation for long-term initiatives to improve U.S.
secondary school education.
Ultimately, the typical business manager has a right to be perturbed when he or she pays twice to educate an employeefirst
through taxes for public education that did not fully succeed, and
again through direct expenditures for that employee. While recognizing the need to shore up the skills of the current workforce,
business must promote improvement in public schools as a means
of increasing the skills of future generations.
Indeed, this need to promote improvement must extend beyond
mere advocacy for higher standards to include a call for fundamental reforms to the way we teach mathematics. Business leaders
regularly argue that even students who perform well in mathematics courses are often not prepared to function effectively in todays
workplace because they lack versatility and flexibility in dealing
with real-world obstacles. They become stymied by challenges for
which there are no prescribed textbook solutions. Young people
must learn this versatility and flexibility in school, long before they
enter the workforce.
Business is particularly well equipped to make a powerful case for
quantitative literacy in elementary and secondary schools, but first
it must acknowledge that widespread quantitative literacy will not
necessarily result from requirements that students take more
mathematics courses. To effect more meaningful changes over the
long term, business must become more fully engaged with the
content and delivery of those courses.
4. Engage education and training partners to help upgrade the
quantitative literacy of the workforce based on identified
quantitative needs.
Although efforts to create a new generation of quantitatively literate Americans will promote a stronger economic future, the
ily add up to a measurable national improvement every localized initiative must support a much larger vision.
As Intel CEO and President Craig Barrett argues, the processes of
continuous improvement common in the business world also can
promote successful education reforms. To advance the cause of
quantitative literacy, such processes must ultimately incorporate
all the action steps described above into what Barrett, quoting W.
Edwards Deming, calls a plan-do-check-act cycle.16
Advocates for quantitative literacy must first plan: define and measure the problem and then formulate a plan for addressing it.
Then they must do: implement the plan in both schools and
workforce training programs. Next, they must check: monitor the
plans results according to preestablished criteria for success. Finally, they must act: on the basis of these results they must enact
targeted changes to the original plan. Then the process begins
again, at a higher level. By learning from their mistakes while
capitalizing on their achievements, reformers can make incremental but significant progress toward the goal of quantitative literacy
for all Americans.
51
Appendix
Business Coalition for Excellence
in Education:
Principles for K12 Education Legislation
In a world of global competition and rapid technological advances,
U.S. schools must prepare all students for the challenges and opportunities of the twenty-first century. To achieve this goal, our school
systems must adopt higher standards, use high-quality assessments aligned
to these standards, and hold schools accountable for results, so that all
students have the opportunity to succeed. Federal investments must help
each state implement a standards-based, performance-driven education system that is carefully aligned to the goal of higher student
achievement. The Business Coalition for Excellence in Education*
urges Congress to enact bipartisan legislation that embodies the following principles:
Standards: All states should have high-quality, rigorous academic standards that reflect the levels of student achievement
necessary to succeed in society, higher education, and the workplace. The federal government should provide all states with the
information and resources to develop, continuously improve,
and benchmark rigorous academic standards that can be used to
raise individual student performance to world-class levels.
Student Achievement: Assessments should be used as diagnostic tools to ensure that all students, particularly those identified as under-performing, receive the assistance they need to
succeed in reaching high academic standards. Similarly, federal leadership should ensure that preschool aid focuses on
helping prepare children to enter school ready to learn.
Accountability: States, districts, and principals should ensure that all students, including disadvantaged and underperforming students, meet high academic standards. States
should have policies of rewards and sanctions to hold systems
accountable for improving the performance of students,
teachers, and principals. Such policies should be based on
performance, including student achievement.
Conclusion
The jobs of the twenty-first century are more complex than ever
before. Technologies such as computers, e-mail, faxes, and the
Internet have created a world awash in data. To succeed in this
data-drenched society, employees need to have tools to make sense
of information in faster and cheaper ways than heretofore. The
notion of quantitative literacy promises to offer a mechanism for
making sense of this world.
The principles advanced in this essay cannot provide any immediate or easily implemented solutions to the shortage of quantitatively literate citizens. Rather, they call for a committed and sustained effort to specify emerging needs for new quantitative skills,
and then to rally stakeholders around carefully targeted programs
addressing these needs. Although such an effort presents great
challenges, it promises even greater rewards. If they work together
toward clearly articulated goals, the business and education communities will have an unprecedented opportunity to prepare every
U.S. citizen for success in a constantly changing world.
* An ad hoc coalition of leading U.S. corporations and business organizations that support these principles in the reauthorization of the Elementary and Secondary Education Act
52
Notes
1. Philip R. Day and Robert H. McCabe, Remedial Education: A
Social and Economic Imperative, American Association of Community Colleges (AACC) Issue Paper, Washington, DC: American
Association of Community Colleges, 1997.
2. American Management Association. Retrieved at http://www.
amanet.org/research/pdfs/bjp_2001.pdf.
3. Training Magazine, Annual Report, Minneapolis: Bil Communications, Vol. 36, No. 10 (1999). Retrieved at http://www.trainingsupersite.com.
4. Milton Goldberg and Susan L. Traiman, Why Business Backs Education Standards, in Brookings Papers on Education Policy, Diane
Ravitch, ed. (Washington, DC: Brookings Institution, 2001).
5. U.S. Department of Labor, 20 Million Jobs: January 1993November
1999, A Report by the Council of Economic Advisors and the Chief
Economist, U.S. Department of Labor, December 3, 1999. Retrieved
from http://clinton4.nova.gov/media/pdf/20miljobs.pdf
6. To date, the CEOs from Boeing, Eastman Kodak, IBM, Intel, Proctor
& Gamble, Prudential, State Farm, and Williams have served on
Achieve. Other executivesfrom Bristol-Myers Squibb, Pfizer, Washington Mutual, to name a few have attended the meetings as guests.
7. National Alliance of Business, Investing in Teaching (Washington,
DC: National Alliance of Business, 2001).
8. Skills and Competencies Needed by Arizonas Workforce: The Environmental Technologies Industry (Prepared for the Arizona Department
of Commerce by Advancing Employee Systems, Inc. 2001), 16.
9. Visit http://www.act.org/workkeys/.
10. Visit http://www.nssb.org/.
11. See http://www.pisa.oecd/pisa/math.htm. Although the United
States scored near the international average in the mathematical
literacy part of the 2000 PISA examination, students in eight countriesJapan, Korea, New Zealand, Finland, Australia, Canada,
Switzerland, and the United Kingdomsignificantly outperformed
American students. Results such as these underscore the urgency of
developing a more quantitatively literate citizenry as a means of
preserving our long-term economic competitiveness.
12. Lynn Arthur Steen, ed., Mathematics and Democracy: The Case for
Quantitative Literacy (Princeton, NJ: National Council on Education and the Disciplines, 2001), 16 17.
13. U.S. Department of Labor Bureau of Labor Statistics, Occupational
Outlook Handbook 2000 01 Edition (January 2000), 4 5.
14. See The Work in Northeast Ohio Council, The Impact of Basic
Skills Training on Employee and Organizational Effectiveness (October 2000); American Society for Training and Development,
Training Investments Improve Financial Success (September
2000); U.S. Business Views on Workforce Training, Price Waterhouse, prepared for American Society for Training Development,
National Retail Federation, National Association of Manufacturing,
and Student Loan Marketing Association (April 1994), 710.
Also visit www.nam.org/tertiary_print.asp?TrackID&CategoryID
678&DocumentID1419.
15. Training Magazine, 2000 Industry Report: A Comprehensive Analysis of Employer-Sponsored Training in the U.S.
16. Craig R. Barrett, speech delivered at the 33rd Annual Meeting of the
National Alliance of Business in Arlington, VA (November 6, 2001).
External Forces
Beginning with A Nation at Risk (National Commission on Excellence in Education 1983) and
continuing through Before Its Too Late, the report of the Glenn Commission (National Commission
on Mathematics and Science Teaching for the 21st Century 2000), countless hand-wringing reports
have documented deficiencies in mathematics education. Professional societies (American Mathe-
Lynn Arthur Steen is Professor of Mathematics at St. Olaf College, and an advisor to the Mathematics Achievement
Partnership (MAP) of Achieve, Inc. Earlier, Steen served as executive director of the Mathematical Sciences Education
Board and as president of the Mathematical Association of America. Steen is the editor or author of many books and articles,
including Mathematics and Democracy (2001), Why Numbers Count (1997), On the Shoulders of Giants (1991), and
Everybody Counts (1989).
53
54
matical Association of Two-Year Colleges 1995; National Council of Teachers of Mathematics 1989, 2000) have responded with
reform-oriented recommendations while states (e.g., California,
Virginia, Minnesota, Texas, and dozens of others) have created
standards and frameworks suited to their local traditions. Analysis
of these proposals, much of it critical, has come from a wide
variety of sources (e.g., Cheney 1997; Kilpatrick 1997; Wu 1997;
Raimi and Braden 1998; Gavosto, et al. 1999; Stotsky 2000). In
some regions of the country, these debates have escalated into
what the press calls math wars (Jackson 1997).
Nearly one-hundred years ago, Eliakim Hastings Moore, president of the young American Mathematical Society, argued that
the momentum generated by a more practical education in school
would better prepare students to proceed rapidly and deeply
with theoretical studies in higher education (Moore 1903). In the
century that followed, mathematics flowered in both its practical
and theoretical aspects, but school mathematics bifurcated: one
stream emphasized mental exercises with little obvious practical
value; the other stream stressed manual skills with no theoretical
value. Few schools ever seriously followed Moores advice of using
practical education as a stepping-stone to theoretical studies.
Now, following a century of steady growth based on rising demand and a relatively stable curricular foundation, a new president of the American Mathematical Society has warned his colleagues that the mathematical sciences are undergoing a phase
transition from which some parts might emerge smaller and others dispersed (Bass 1997). The forces creating this transition are
varied and powerful, rarely under much control from educators or
academics. I have selected only a few to discuss here, but I believe
these few will suffice to illustrate the nuances that too often are
overlooked in simplistic analyses of editorials, op-ed columns, and
school board debate. I begin with the contentious issue of tracking.
TRACKING
Until quite recently, mathematics was never seen as a subject to be
studied by all students. For most of our nations history, and in
most other nations, the majority of students completed their
school study of mathematics with advanced arithmeticprices,
interest, percentages, areas, and other topics needed for simple
commerce. Only students exhibiting special academic interest
studied elementary algebra and high school geometry; even fewer
students, those exhibiting particular mathematical talent, took
advanced algebra and trigonometry. For many generations, the
majority of students studied only commercial or vocational mathematics, which contained little if any of what we now think of as
high school mathematics.
In recent decades, as higher education became both more important and more available, the percentage of students electing the
academic track increased substantially. In the 1970s, only about
40 percent of U.S. students took two years of mathematics (algebra and geometry) in secondary school; 25 years later that percentage has nearly doubled. The percentage of high school students
taking three years of mathematics has climbed similarly, from
approximately 30 percent to nearly 60 percent (National Science
Board 1996; Dossey and Usiskin 2000).
This shift in the presumption of mathematics as a subject for an
academic elite to mathematics as a core subject for all students
represents the most radical transformation in the philosophy of
mathematics education in the last century. In 1800, Harvard University expected of entering students only what was then called
vulgar arithmetic. One century later, Harvard expected a year of
Euclid; two centuries laterin 2000 Harvard expects that most
entering students have studied calculus. In no other subject has
the expected level of accomplishment of college-bound students
increased so substantially. These changes signal a profound shift in
public expectations for the mathematical performance of high
school graduates, a change that is sweeping the globe as nations
race to keep up with rapidly advancing information technology.
Secondary school mathematics is no longer a subject for the few,
but for everyone.
In response to the increasing need for mathematical competence
in both higher education and the high-performance workplace,
the National Council of Teachers of Mathematics (NCTM) initiated the 1990s movement for national standards by recommending that all students learn a common core of high-quality
mathematics including algebra, geometry, and data analysis
(NCTM 1989). Dividing students into academic and nonacademic tracks, NCTM argued, no longer makes the sense it once
did when the United States was primarily an agrarian and assembly line economy. In this old systemremnants of which have not
yet entirely disappeared college-bound students were introduced to algebra and geometry while those in vocational tracks
were expected only to master arithmetic. Because algebra was not
needed in yesterdays world of work, it was not taught to students
in the lower tracks. This vocational tradition of low expectations
(and low prestige) is precisely what NCTM intended to remedy
with its call for a single core curriculum for all students.
Yet even as mathematics for all has become the mantra of reform, schools still operate, especially in mathematics, with separate tracks as the primary strategy for delivery of curriculum. They
are reinforced in this habit by teachers who find it easier to teach
students with similar mathematical backgrounds and by parents
who worry not that all children learn but that their own children
learn. Indeed, parents anxiety about ensuring their own childrens success has rapidly transformed an academic debate about
EMPLOYMENT
During the last decade of the twentieth century, just as the movement for academic standards began, business and industry
launched a parallel effort to articulate entry-level skill standards
for a broad range of industries (NSSB 1998) as well as to suggest
better means of linking academic preparation with the needs of
employers (Bailey 1997; Forman and Steen 1998).
Although preparing students for work has always been one purpose of education, teachers generally adopt broader goals and
more specifically academic purposes. Mathematics educators are
55
56
Systems: Understand, monitor, and improve social, organizational, and technological systems
Mathematical thinking is embedded throughout these competencies, not just in the set of basic skills but as an essential component
of virtually every competency. Reasoning, making decisions, solving problems, managing resources, interpreting information, understanding systems, applying technology all these and more
build on quantitative and mathematical acumen. But they do not
necessarily require fluency in factoring polynomials, deriving trigonometric identities, or other arcana of school mathematics
(Packer, see pp. 39 41).
TECHNOLOGY
The extraordinary ability of computers to generate and organize
data has opened up an entire new world to mathematical analysis.
Mathematics is the science of patterns (Steen 1988; Devlin 1994)
and technology enables mathematicians (and students) to study
patterns as they never could before. In so doing, technology offers
mathematics what laboratories offer science: an endless source of
evidence, ideas, and conjectures. Technology also offers both the
arts and sciences a new entree into the power of mathematics:
fields as diverse as cinema, finance, and genetics now deploy com-
57
depriving themselves of any possibility of recognizing or appreciating the unique certainty of mathematical deduction.
At the same time, computers and calculators are increasing dramatically the number of people who use mathematics, many of
whom are not well educated in mathematics. Previously, only
those who learned mathematics used it. Today many people use
mathematical tools for routine work with spreadsheets, calculators, and financial systems, tools that are built on mathematics
they have never studied and do not understand. This is a new
experience in human history, with problematic consequences that
we are only gradually discovering.
Finally, as the technology-driven uses of mathematics multiply,
pressure will mount on schools to teach both information technology and more and different mathematics (ITEA 2000; NRC
1999). At the same time, and for the same reasons, increasing
pressure will be applied on teachers and schools to ensure that no
child is left behind. Alarms about the digital divide already have
sounded and will continue to ring loudly in the body politic
(Compaigne 2001; Norris 2001; Pearlman 2002). The pressure
on mathematics to form a bipartisan alliance with technology in
the school curriculum will be enormous. This easily could lead to
a new type of tracking one track offering the minimal skills
needed to operate the new technology with little if any understanding, the other offering mathematical understanding as the
surest route to control of technology. Evidence of the emergence
of these two new cultures is not hard to find.
TESTING
Largely because of its strong tradition of dispersed authority and
local control, the United States has no system to ensure smooth
articulation between high school and college mathematics programs. Instead, students encounter a chaotic mixture of traditional and standards-based high school curricula; Advanced Placement (AP) examinations in Calculus, Statistics, and Computer
Science; very different SAT and ACT college entrance examinations; diverse university admissions policies; skills-based mathematics placement examinations; and widely diverse first-year curricula in college, including several levels of high school algebra
58
ALGEBRA
In the Middle Ages, algebra meant calculating by rules (algorithms). During the Renaissance, it came to mean calculation with
signs and symbols using xs and ys instead of numbers. (Even
today, laypersons tend to judge algebra books by the symbols they
contain: they believe that more symbols mean more algebra, more
words, less.) In subsequent centuries, algebra came to be primarily
about solving equations and determining unknowns. School algebra still focuses on these three aspects: following procedures, employing letters, and solving equations.
In the twentieth century, algebra moved rapidly and powerfully
beyond its historical roots. First it became what we might call the
science of arithmeticthe abstract study of the operations of
arithmetic. As the power of this abstract algebra became evident
in such diverse fields as economics and quantum mechanics, algebra evolved into the study of all operations, not just the four found
in arithmetic. Thus did it become truly the language of mathematics and, for that reason, the key to access in our technological
society (Usiskin 1995).
Indeed, algebra is now, in Robert Moses apt phrase, the new civil
right (Moses 1995). In todays society, algebra means access. It
unlocks doors to productive careers and democratizes access to big
ideas. As an alternative to dead-end courses in general and commercial mathematics, algebra serves as an invaluable engine of
equity. The notion that by identifying relationships we can discover things that are unknownthat we can find out what we
want to knowis a very powerful and liberating idea (Malcolm
1997).
Not so long ago, high school algebra served as the primary filter to
separate college-bound students from their work-bound classmates. Advocates for educational standards then began demanding algebra for all, a significant challenge for a nation accustomed to the notion that only some could learn algebra (Steen
1992; Chambers 1994; Lacampagne et al. 1995; Silver 1997;
NCTM and MSEB 1998). More recently, this clamor has escalated to a demand that every student complete algebra by the end
of eighth grade (Steen 1999; Achieve 2001).
The recent emphasis on eighth-grade algebra for all has had the
unfortunate side effect of intensifying distortions that algebra already imposes on school mathematics. One key distortion is an
overemphasis on algebraic formulas and manipulations. Students
quickly get the impression from algebra class that mathematics is
manipulating formulas. Few students make much progress toward
the broad goals of mathematics in the face of a curriculum dominated by the need to become fluent in algebraic manipulation.
Indeed, overemphasis on algebra drives many students away from
mathematics: most students who leave mathematics do so because
Others may cite, as grounds for emphasizing algebra, the widespread use of formulas in many different fields of work; however,
this use is only a tiny part of what makes up the school subject of
algebra. Moreover, most business people give much higher priority to statistics than to algebra. Some mathematicians and scientists assert that algebra is the gateway to higher mathematics, but
this is so only because our curriculum makes it so. Much of mathematics can be learned and understood via geometry, or data, or
spreadsheets, or software packages. Which subjects we emphasize
early and which later is a choice, not an inevitability.
Lurking behind the resurgent emphasis on algebra is a two-edged
argument concerning students who are most likely to be poorly
educated in mathematicspoor, urban, first generation, and minority. Many believe that such students, whose only route to
upward mobility is through school, are disproportionately disadvantaged if they are denied the benefits that in our current system
only early mastery of algebra can confer. Others worry that emphasis on mastering a subject that is difficult to learn and not well
taught in many schools will only exacerbate existing class differences by establishing algebra as a filter that will block anyone who
does not have access to a very strong educational environment.
Paradoxically, and unfortunately, both sides in this argument appear to be correct.
DATA
Although algebra and calculus may be the dominant goals of
school mathematics, in the real world mathematical activity usually begins not with formulas but with data. Measurements taken
at regular intervals be they monthly sales records, hourly atmospheric pressure readings, or millisecond samples of musical
tonesform the source data for mathematical practice. Rarely if
ever does nature present us with an algebraic formula to be factored or differentiated. Although the continuous model of reality
encapsulated by algebra and calculus is a powerful tool for developing theoretical models, real work yielding real results must begin and end in real data.
In past eras, mathematics relied on continuous models because
working with real data was too cumbersome. An algebraic or
differential equation with three or four parameters could describe
reasonably well the behavior of phenomena with millions of potential data points, but now computers have brought digital data
into the heart of mathematics. They enable practitioners of mathematics to work directly with data rather than with the simplified
continuous approximations that functions provide. Moreover,
they have stimulated whole new fields of mathematics going under names such as combinatorics, discrete mathematics, and exploratory data analysis.
59
ACHIEVEMENT
Strained by a growing number of forces and pressures (only some
of which are discussed here), U.S. mathematics educators have
found it very difficult to improve student achievement educations bottom line. For at least the last half-century, graduates of
U.S. secondary schools have lagged behind their peers in other
nations, especially those of the industrial world and the former
Communist bloc. Documentation of this deficiency has been
most consistent in mathematics and science, subjects that are relatively common in the curricula of other nations and that are
examined internationally at regular intervals. Some U.S. analysts
seek to explain (or excuse) poor U.S. performance by hypothesizing a negative impact of our relatively heterogeneous population,
or conjecturing that a larger percentage of U.S. students complete
secondary school, or arguing that other nations (or the United
States) did not test a truly random sample. But despite these
exculpatory claims, a central stubborn fact remains: on international tests administered over several decades to similarly educated
students, the mathematics performance of U.S. eighth- and
twelfth-grade students has always been well below international
norms.
The most recent headlines came from TIMSS, the Third International Mathematics and Science Study, and its repeat, TIMSS-R.
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Quantitative Practices
The forces created by differential tracking, needs of employment,
impacts of technology, misaligned testing, overemphasis on alge-
MATHEMATICS
During the last half-century, as mathematics in school grew from
an elite to a mass subject, mathematics expanded into a portfolio
of mathematical sciences that now includes, in addition to traditional pure and applied mathematics, subjects such as statistics,
financial mathematics, theoretical computer science, operations
research (the science of optimization) and, more recently, financial mathematics and bioinformatics. (It is a little-appreciated fact
that most of the advancesand fortunes being made in investments, genetics, and technology all derive from clever applications
of sophisticated mathematics.) Although each of these specialties
has its own distinctive character, methodologies, standards, and
accomplishments, they all build on the same foundation of school
and college mathematics.
61
If we look at these common uses of mathematics from the perspective of the school curriculum, we see that mathematics at
work is very different from mathematics in school:
Numbers are not just about place value and digits but about
notation and coding, index numbers and stock market averages, and employment indexes and SAT scores.
Statistics is not just about means, medians, and standard deviations but about visual displays of quantitative ideas (for
example, scatter plots and quality control charts) as well as
random trials and confidence intervals.
Proof is not just about logical deduction but about conjectures and counterexamples, scientific reasoning and statistical
inference, and legal standards such as preponderance of evidence or beyond reasonable doubt.
Mathematics is far more than just a tool for research. In fact, its
most common usesand the reason for its prominent place in
school curriculaare routine applications that are now part of all
kinds of jobs. Examples include:
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STATISTICS
The age of information is an age of numbers. We are surrounded
by data that both enrich and confuse our lives. Numbers provide
descriptions of daily events, from medical reports to political
trends and social policy. News reports are filled with charts and
graphs, while politicians debate quantitatively based proposals
that shape public policy in education, health, and government.
The study of numbers is usually associated with statistics. In
schools, the term quantitative literacy is often employed as an
informal synonym for elementary statistics. Although statistics
is today a science of numbers and data, historically (and etymologically) it is the science of the state that developed in the Napoleonic era when central governments used data about population,
trade, and taxes to assert control over distant territory. The value
of systematic interpretation of data quickly spread to agriculture,
medicine, economics, and politics. Statistics now underlies not
only every economic report and census but also every clinical trial
and opinion survey in modern society.
knownthat the subtle reasoning involved in data-based statistical inference is harder for students to grasp and explain than the
comparable symbol-based problems and proofs in a typical calculus course. Properly taught, statistics is probably a better vehicle
than algebra and calculus for developing students capacity to
reason logically and express complex arguments clearly.
Statistics is also very practical; far more so than any part of the
algebra-trigonometry-calculus sequence that dominates school
mathematics. Every issue in the daily newspaper, every debate that
citizens encounter in their local communities, every exhortation
from advertisers invites analysis from a statistical perspective. Statistical reasoning is subtle and strewn with counterintuitive paradoxes. It takes a lot of experience to make statistical reasoning a
natural habit of mind (Nisbett et al. 1987; Hoffrage et al. 2000).
That is why it is important to start early and to reinforce at every
opportunity.
NUMERACY
The special skills required to interpret numberswhat we call
numeracy or quantitative literacyare rarely mentioned in national education standards or state frameworks. Nonetheless,
these skills nourish the entire school curriculum, including not
only the natural, social, and applied sciences but also language,
history, and fine arts (Steen 1990). They parallel and enhance the
skills of literacy of reading and writing by adding to words
the power of numbers.
in data derived from and attached to the empirical world. Surprisingly to some, this inextricable link to reality makes quantitative
reasoning every bit as challenging and rigorous as mathematical
reasoning.
Mathematics teachers often resist emphasizing data because the
subject they are trying to teach is about Platonic idealsnumbers
and functions, circles and triangles, sets and relationships. Employers and parents, however, often are frustrated by this stance
because school graduates so frequently seem inexperienced in
dealing with data, and the real world presents itself more often in
terms of data than in the Platonic idealizations of mathematics.
Although numeracy depends on familiar mathematical topics
from arithmetic, algebra, and geometry, its natural framework is
commonly described in broader terms (Steen 2001). Some are
foundational, focused on learned skills and procedures:
Confidence with Mathematics: Being comfortable with numbers and at ease in applying quantitative methods
Number Sense: Estimating with confidence; employing common sense about numbers; exhibiting accurate intuition
about measurements
Prerequisite Knowledge: Using a wide range of algebraic, geometric, and statistical tools that are required for many fields of
postsecondary education
Interpreting Data: Reasoning with data, reading graphs, drawing inferences, and recognizing sources of error
Symbol Sense: Employing, reading, and interpreting mathematical symbols with ease; exhibiting good sense about their
syntax and grammar
63
School Mathematics
For various reasons having to do with a mixture of classical tradition and colonial influence, the school curriculum in mathematics
is virtually the same all over the world. Fifteen years ago, the
secretary of the International Commission on Mathematics Instruction reported that apart from local examples, there were few
significant differences to be found in the mathematics textbooks
used by different nations around the world (Howson and Wilson
1986). Even a country as culturally separate as Japan follows a
canonical western curriculum with only minor variations
(Nohda et al. 2000). Detailed review of U.S. practice in the mid1980s showed little significant change from the practice of previous decades (Hirsch and Zweng 1985). At the end of the twentieth century, therefore, a birds-eye view of school mathematics
reveals little substantive variation in either time or space.
Not surprisingly, however, a more refined analysis prepared in
advance of the TIMSS study reveals subtle differences in scope,
sequence, and depth (Howson 1991). The TIMSS study itself
included an extensive analysis of curricula (and of teaching practices) in participating nations. This analysis showed significant
variation in the number of topics covered at different grade levels,
a variation that appears to be inversely correlated with student
performance (Schmidt et al. 1997). In the case of mathematics
education, it seems, more really is less: too many topics covered
superficially lead to less student learning. The consensus of experts
who have studied both domestic and international assessments is
that neither the mathematics curriculum nor the classroom instruction is as challenging in the United States as it is in many
other countries (e.g., Stevenson 1998).
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CHALLENGES
Fixing school mathematics requires attention to many significant
(and overwhelming) issues such as teacher competence, recruitment, salaries, and performance; class size and classroom conditions; alignment of standards with textbooks and tests; and consistent support by parents, professionals, and politicians. Here I
merely acknowledge these issues but do not deal with any of them.
Instead, my primary purpose in this essay is to think through the
goals of mathematics in grades 6 to 12 in light of the significant
forces that are shaping the environment of school mathematics.
These include:
These environmental forces are not hidden. Everyone who is concerned about the quality of mathematics education is aware of
them. Nonetheless, school mathematics continues to serve primarily as a conveyor belt to calculus that educates well only a
minority of students. Many individuals and organizations have
developed proposals for change (e.g., California Academic Standards Commission 1997; MSEB 1998; NCTM 2000; Achieve
2001; Steen 2001), but these proposals represent contrasting
rather than consensus visions of school mathematics.
The traditional curriculum in grades 6 to 12 is organized like a
nine-layer cake: advanced arithmetic, percentages and ratios, elementary algebra, geometry, intermediate algebra, trigonometry,
advanced algebra, pre-calculus, and (finally) calculus. Each subject builds on topics that precede it, and each topic serves as a
foundation for something that follows. Although this sequence
has the benefit of ensuring (at least on paper) that students are
prepared for each topic by virtue of what has come before, the
sequence does this at the expense of conveying a biased view of
mathematics (because topics are stressed or ignored primarily on
the basis of their utility as a tool in calculus) and creating a fragile
educational environment (because each topic depends on mastery
of most preceding material). The inevitable result can be seen all
around us: most students drop out of mathematics after they
encounter a first or second roadblock, while many of those who
survive emerge with a distorted (and often negative) view of the
subject.
The intense verticality of the current mathematics curriculum not
only encourages marginal students to drop out but also creates
significant dissonance as states begin to introduce high-stakes
graduation tests. Inevitably, student performance spreads out as
students move through a vertical curriculum because any weakness generates a cascading series of problems in subsequent
courses. The result is an enormous gap between curricular goals
and a politically acceptable minimum requirement for high school
graduation. Consequently, in most states, the only enforced
mathematics performance level for high school graduation is an
eighth- or ninth-grade standard. This large discrepancy between
goals and achievement discredits mathematics education in the
eyes of both parents and students.
BREADTH
AND
CONNECTEDNESS
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66
MIDDLE GRADES
For several reasons, it is helpful to think of the seven years of
grades 6 to 12 in three parts: the middle grades 6 to 8; the core
high school grades 9 to 11; and the transition grade 12. To oversimplify (but not by much), the goal for grades 6 to 8 would be
numeracy, for grades 9 to 11, mathematical sciences, and for grade
12, options. Data analysis, geometry, and algebra would constitute three equal content components in grades 6 to 8 and in grades
9 to 11. (In this simplified synopsis, measurement and probability
can be viewed as part of data analysis, while number and operations can be viewed as part of algebra; discrete mathematics and
combinatorics are embedded in every topic.) The five NCTM
process standards cut across all topics and grade levels, but rather
than being left to chance, they do need to be covered intentionally
and systematically.
Careful planning can ensure that the foundational parts of school
mathematics are covered in grades 6 to 8, without tracking but
with multiple points of entry and many opportunities for mutual
reinforcement. There are many different ways to do this, one of
which is being developed by a dozen or so states belonging to the
Mathematics Achievement Partnership (Achieve 2001). In a curriculum designed for breadth and connections, anything not
learned the first time will appear again in a different context in
which it may be easier to learn. For example, graphing data gathered through measurement activities provides review of, or introduction to, algebra and geometry; finding lengths and angles via
indirect measurements involves solving equations; and virtually
every task in data analysis as well as many in algebra and geometry
reinforces and extends skills involving number and calculation.
Used this way, with intention and planning, linkages among parts
of mathematics can be reinforcing rather than life-threatening.
Instead of leading to frustration and withdrawal, a missing link
can lead to exploration of alternative routes through different
parts of mathematics. If middle school teachers give priority to
topics and applications that form the core of quantitative literacy,
students will encounter early in their school careers those parts of
mathematics that are most widely used, most important for most
people, and most likely to be of interest. More specialized topics
can and should be postponed to grades 9 to 11.
SECONDARY SCHOOL
In high school, all students should take three additional years of
mathematics in grades 9 to 11, equally divided between data analysis, geometry, and algebra but not sequentially organized. Parallel
development is essential to build interconnections both within the
mathematical sciences and with the many other subjects that students are studying at the same time. Parallel does not necessarily
mean integrated, although there certainly could be integration in
particular curricula. It does mean that in each grade, students
advance significantly in their understanding of each component of
the triad of data analysis, geometry, and algebra. Parallel development reduces the many disadvantages of the intense and unnecessary verticality.
The content of this curriculum would not differ very much from
the recommendations in the NCTM standards. The core of mathematics data analysis, geometry, and algebrais what it is and
can be neither significantly changed nor totally avoided. There is,
however, considerable room for variation in the implementation
of specific curricula, notably in the examples that are used to
motivate and illuminate the core. Appendix II, adapted from a
report of the National Center for Research in Vocational Education (Forman and Steen 1999), offers some examples of important
but neglected topics that can simultaneously reinforce mathematical concepts in the core and connect mathematics to ideas and
topics in the world in which students live. Some recent textbooks
(e.g., Pierce et al. 1997) build on similar ideas.
But perhaps even more important than an enriched variety of
examples and topics would be a powerful emphasis on aspects of
what NCTM calls process standards. As the practice of medicine
involves far more than just diagnosing and prescribing, so the
practice of mathematics involves far more than just deducing theorems or solving problems. It involves wide-ranging expertise that
brings number and inference to bear on problems of everyday life.
Part of learning mathematics is to experience the wide scope of its
practice, which is what the process standards are all about.
Some aspects of mathematical practice are entirely pragmatic,
dealing with real systems and situations of considerable complexity. A mathematics education should prepare students to deal with
the kinds of common situations in which a mathematical perspective is most helpful. Common examples include scheduling, modeling, allocating resources, and preparing budgets. In this computer age, students also need to learn to use the tools of modern
technology (e.g., spreadsheets, statistical packages, Internet resources) to collect and organize data, to represent data visually,
and to convert data from one form and system to another. Performance standards for mathematics in the age of computers means
performance with computer tools.
Other aspects of mathematics are anchored more in logic than in
practice, in drawing inferences rather than working with data.
Ever since Euclid, mathematics has been defined by its reliance on
deductive reasoning, but there are many other kinds of reasoning
in which mathematical thinking plays an important role. Students
finishing high school should have enough experience with differ-
Scientific Inference: Gathering data; detecting patterns, making conjectures; testing conjectures; drawing inferences; verifying versus falsifying theories
Mathematical Inference: Logical reasoning and deduction; assumptions and conclusions; axiomatic systems; theorems and
proofs; proof by direct deduction, by indirect argument, and
by mathematical induction; classical proofs (e.g., isosceles
triangle, infinitude of primes, Pythagorean theorem)
Statistical Inference: Rationale for random samples; doubleblind experiments; surveys and polls; confidence intervals;
causality versus correlation; multiple and hidden factors, interaction effects; judging validity of statistical claims in media
reports.
OPTIONS
Ideally, every student should study the same mathematics through
grade 8, with only minor variation in examples to support different student interests and abilities. Accommodation to student
differences in middle school should reflect student needs, not
67
68
SYNOPSIS
To summarize, all middle school students (grades 6 to 8) would
study a three-year, non-tracked curriculum that provides equal
and tightly linked introductions to data analysis, geometry, and
algebra. When students enter high school, they would move into
a second three-year mathematics curriculum that may provide
some options based on student interests. No matter the emphasis,
however, each high school program would advance equally the
three main themes (data analysis, geometry, algebra) without letting any lag behind. Different programs may emphasize different
contexts, different tools, and different depths, but each would
leave students prepared both for the world of work and for postsecondary education.
In this plan, minimum high school graduation requirements (certified, for example, by high-stakes state tests that are typically set at
tenth-grade competence) would represent citizen-level quantitative literacy, one year behind mathematics preparation for college
admission (eleventh grade), which would be one year behind qualification for mathematically intensive college programs. Aiming
for this three-step outcome of high school mathematics is more
logical and more achievable than the imagined (but never
achieved) ideal of having every student leave high school equally
educated in mathematics and equally prepared for college admission.
By studying a balanced curriculum, students would leave school
better prepared for employment, more competitive with their
international peers, and well positioned for a variety of postsecondary programs. By experiencing breadth and connectedness
rather than depth and verticality, students would have repeated
opportunities to engage mathematics afresh as their own interests
and attitudes evolve. By focusing on the symbiosis of computers
and mathematics, students would experience how mathematics is
practiced. And by studying a blend of mathematics, statistics, and
numeracy, students would be flexibly prepared for life and work in
the twenty-first century.
Content Standards:
69
formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them;
Process Standards:
understand numbers, ways of representing numbers, relationships among numbers, and number systems;
in problem solving, to
in algebra, to
in geometry, to
analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships;
specify locations and describe spatial relationships using coordinate geometry and other representational systems;
in measurement, to
in communication, to
in connections, to
70
in representation, to
create and use representations to organize, record, and communicate mathematical ideas;
Mental Estimation: Quick, routine mental estimates of costs, distances, times. Estimating orders of magnitude. Reasoning with
ratios and proportions. Mental checking of calculator and computer results. Estimating unknown quantities (e.g., number of
high school students in a state or number of gas stations in a city).
Numbers: Whole numbers (integers), fractions (rational numbers), and irrational numbers (, 2). Number line; mixed
numbers; decimals; percentages. Prime numbers, factors;
simple number theory; fundamental theorem of arithmetic.
Binary numbers and simple binary arithmetic. Scientific notation; units and conversions. Number sense, including intuition about extreme numbers (lottery chances, national debt,
astronomical distances).
Risk Analysis: Estimates of common risks (e.g., accidents, diseases, causes of death, lotteries). Confounding factors. Communicating and interpreting risk.
71
Bailey, Thomas R. 1997. Integrating Academic and Industry Skill Standards. Berkeley, CA: National Center for Research in Vocational
Education, University of California.
Growth and Variation: Linear, exponential, quadratic, harmonic, and normal curve patterns. Examples of situations
that fit these patterns (bacterial growth, length of day) and of
those that do not (e.g., height versus weight; income distribution).
Financial Mathematics: Personal finance; loans, annuities, insurance. Investment instruments (stocks, mortgages, bonds).
Exponential Growth: Examples (population growth, radioactivity, compound interest) in which rate of change is proportional to size; doubling time and half-life as characteristics of
exponential phenomena; ordinary and log-scaled graphs.
Normal Curve: Examples (e.g., distribution of heights, repeated measurements, production tolerances) of phenomena
that distribute in a bell-shaped curve and examples that do
not (e.g., income, grades, typographical errors, life spans).
Area as measure of probability. Meaning of 1-, 2-, and 3.
Parabolic Patterns: Examples (falling bodies, optimization, acceleration) that generate quadratic phenomena; relation to
parabolic curves.
Cyclic functions: Examples (time of sunrise, sound waves, biological rhythms) that exhibit cyclic behavior. Graphs of sin
and cos; consequences of sin2 cos2 1.
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(In)numeracy
In 1990, a newspaper reported:
Yesterday, Monday October 9, AVRO Television paid attention to analphabetism in The
Netherlands. From data collected for the transmission, it appeared that no fewer than 1 out of
25 people cannot read or write, that is, cannot read or write a shopping list, cannot follow
subtitles on TV, cannot read newspapers, cannot write a letter.
Just imagine, 1 out of 25 people, in a country that sends helpers to developing countries in order
to teach their folks reading and writing! 1 out of 25, which means 25% of our citizens.
How many citizens does The Netherlands have? 14 million? That means that in our highly
developed country no less than three and a half million cannot read or write.
Arent you speechless?
Speechless, indeed. Errors such as the one above often are not noticed by our literate, educated
citizens. Innumeracy, or the inability to handle numbers and data correctly and to evaluate statements regarding problems and situations that invite mental processing and estimating, is a greater
problem than our society generally recognizes. According to Treffers (1991), this level of innumeracy might not be the result of content taught (or not taught) but rather the result, at least in part,
of the structural design of teaching practices. Fixing this problem, however, requires dealing with
several issues: From a mathematical perspective, how do we define literacy? Does literacy relate to
mathematics (and what kind of mathematics)? What kind of competencies are we looking for? Are
these competencies teachable?
Introduction
Before trying to answer the question What knowledge of mathematics is important?, it seems wise
first to look at a comfortable definition of quantitative literacy (QL). Lynn Arthur Steen (2001)
pointed out that there are small but important differences in the several existing definitions and,
although he did not suggest the phrase as a definition, referred to QL as the capacity to deal
effectively with the quantitative aspects of life. Indeed, most existing definitions Steen mentioned
give explicit attention to number, arithmetic, and quantitative situations, either in a rather narrow
way as in the National Adult Literacy Survey (NCES 1993):
The knowledge and skills required in applying arithmetic operations, either alone or sequentially, using numbers embedded in printed material (e.g., balancing a checkbook, completing
an order form).
Jan de Lange is Director of the Freudenthal Institute at Utrecht University in The Netherlands. A member of the
Mathematical Sciences Education Board (MSEB), de Langes work focuses on modeling and applications in mathematics
education, implementation of mathematics curriculum reform, and assessing student learning in mathematics. De Lange is
chair of the Expert Group for Mathematics of OECDs new Program for International Student Assessment (PISA).
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76
Diagrams
Lists
Maps
77
in the work of many others representing many countries (as indicated by Neubrand et al. 2001):
1.
Mathematical thinking and reasoning. Posing questions characteristic of mathematics; knowing the kind of answers that
mathematics offers, distinguishing among different kinds of
statements; understanding and handling the extent and limits
of mathematical concepts.
2.
3.
4.
5.
6.
Representation. Decoding, encoding, translating, distinguishing between, and interpreting different forms of representations of mathematical objects and situations as well as understanding the relationship among different representations.
7.
8.
Tools and technology. Using aids and tools, including technology when appropriate.
To be mathematically literate, individuals need all these competencies to varying degrees, but they also need confidence in their
own ability to use mathematics and comfort with quantitative
ideas. An appreciation of mathematics from historical, philosophical, and societal points of view is also desirable.
It should be clear from this description why we have included
functionality within the mathematicians practice. We also note
that to function well as a mathematician, a person needs to be
literate. It is not uncommon that someone familiar with a mathematical tool fails to recognize its usefulness in a real-life situation
(Steen 2001, 17). Neither is it uncommon for a mathematician to
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following the suggestions made by Schoenfeld and Hughes Hallett and extrapolating from experiences in the Netherlands and
other countries), the gap between mathematics and mathematical
literacy would be much smaller than some people suggest it is at
present (Steen 2001). It must be noted, however, that in most
countries this gap is quite large and the need to start thinking and
working toward an understanding of what makes up ML is barely
recognized. As Neubrand et al. (2001) noted in talking about the
situation in Germany: In actual practice of German mathematics
education, there is no correspondence between the teaching of
mathematics as a discipline and practical applications within a
context (free translation by author).
What Is Mathematics?
To provide a clearer picture of literacy in mathematics, it seems
wise to reflect for a moment on what constitutes mathematics.
Not that we intend to offer a deep philosophical treatmentthere
are many good publications around but it is not unlikely that
many readers might think of school mathematics as representing
mathematics as a science. Several authors in Mathematics and Democracy (Steen 2001) clearly pointed this out, quite often based on
their own experiences (Schoenfeld, Schneider, Kennedy, and Ellis, among others). Steen (1990) observed in On the Shoulders of
Giants: New Approaches to Numeracy that traditional school mathematics picks a very few strands (e.g., arithmetic, algebra, and
geometry) and arranges them horizontally to form the curriculum:
first arithmetic, then simple algebra, then geometry, then more
algebra and, finally, as if it were the epitome of mathematical
knowledge, calculus. Each course seems designed primarily to
prepare for the next. These courses give a distorted view of mathematics as a science, do not seem to be related to the educational
experience of children, and bear no relevance for society. A result
of this is that the informal development of intuition along the
multiple roots of mathematics, a key characteristic in the development of ML, is effectively prevented. To overcome this misimpression about the nature of mathematics left by such courses, we
will try to sketch how we see mathematics and, subsequently, what
the consequences can be for mathematics education.
Mathematical concepts, structures, and ideas have been invented
as tools to organize phenomena in the natural, social, and mental
worlds. In the real world, the phenomena that lend themselves to
mathematical treatment do not come organized as they are in
school curriculum structures. Rarely do real-life problems arise in
ways and contexts that allow their understanding and solutions to
be achieved through an application of knowledge from a single
content strand. If we look at mathematics as a science that helps us
solve real problems, it makes sense to use a phenomenological
approach to describe mathematical concepts, structures, and
ideas. This approach has been followed by Freudenthal (1973)
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the components of form and recognize shapes in different representations and different dimensions. The study of shapes is closely
connected to the concept of grasping space (Freudenthal
1973)learning to know, explore, and conquer, in order to live,
breathe, and move with more understanding in the space in which
we live. To achieve this, we must be able to understand the properties of objects and the relative positions of objects; we must be
aware of how we see things and why we see them as we do; and we
must learn to navigate through space and through constructions
and shapes. This requires understanding the relationship between
shapes and images (or visual representations) such as that between
a real city and photographs and maps of the same city. It also
includes understanding how three-dimensional objects can be
represented in two dimensions, how shadows are formed and
interpreted, and what perspective is and how it functions.
Change and Relationships. Every natural phenomenon is a manifestation of change, and in the world around us a multitude of
temporary and permanent relationships among phenomena are
observed: organisms changing as they grow, the cycle of seasons,
the ebb and flow of tides, cycles of unemployment, weather
changes, stock exchange fluctuations. Some of these change processes can be modeled by straightforward mathematical functions:
linear, exponential, periodic or logistic, discrete or continuous.
But many relationships fall into different categories, and data
analysis is often essential to determine the kind of relationship
present. Mathematical relationships often take the shape of equations or inequalities, but relations of a more general nature (e.g.,
equivalence, divisibility) may appear as well. Functional thinkingthat is, thinking in terms of and about relationshipsis one
of the fundamental disciplinary aims of the teaching of mathematics. Relationships can take a variety of different representations,
including symbolic, algebraic, graphic, tabular, and geometric. As
a result, translation between representations is often of key importance in dealing with mathematical situations.
Uncertainty. Our information-driven society offers an abundance
of data, often presented as accurate and scientific and with a degree of certainty. But in daily life we are confronted with uncertain
election results, collapsing bridges, stock market crashes, unreliable weather forecasts, poor predictions of population growth,
economic models that do not align, and many other demonstrations of the uncertainty of our world. Uncertainty is intended to
suggest two related topics: data and chance, phenomena that are
the subject of mathematical study in statistics and probability,
respectively. Recent recommendations concerning school curricula are unanimous in suggesting that statistics and probability
should occupy a much more prominent place than they have in
the past (Cockroft 1982; LOGSE 1990; MSEB 1993; NCTM
1989, 2000). Specific mathematical concepts and activities that
are important in this area include collecting data, data analysis,
data display and visualization, probability, and inference.
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A Matter of Denitions
Having set the context, it seems appropriate now to make clear
distinctions among types of literacies so that, at least in this essay,
we do not declare things equal that are not equal. For instance,
some equate numeracy with quantitative literacy; others equate
quantitative and mathematical literacy. To make our definitions
functional, we connect them to our phenomenological categories.
Spatial Literacy (SL). We start with the simplest and most neglected, spatial literacy. SL supports our understanding of the
(three-dimensional) world in which we live and move. To deal
with what surrounds us, we must understand properties of objects,
the relative positions of objects and the effect thereof on our visual
perception, the creation of all kinds of two- and three-dimensional paths and routes, navigational practices, shadows even
the art of Escher.
Numeracy (N). The next obvious literacy is numeracy (N), fitting
as it does directly into quantity. We can follow, for instance,
Treffers (1991) definition, which stresses the ability to handle
numbers and data and to evaluate statements regarding problems
and situations that invite mental processing and estimating in
real-world contexts.
Quantitative Literacy (QL). When we look at quantitative literacy,
we are actually looking at literacy dealing with a cluster of phenomenological categories: quantity, change and relationships, and
uncertainty. These categories stress understanding of, and mathematical abilities concerned with, certainties (quantity), uncertainties (quantity as well as uncertainty), and relations (types of,
recognition of, changes in, and reasons for those changes).
Mathematical Literacy (ML). We think of mathematical literacy as
the overarching literacy comprising all others. Thus we can make
a visual representation as follows:
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(b)
This problem has been thoroughly researched with 16-year-old students. It illustrates very well the third cluster on reflection and insight.
Students recognized the literacy aspect immediately and quite often
were able to make some kind of generalization; the heart of the solution lies in recognizing that the key mathematical concepts here are
absolute and relative growth. Inflation can of course be left out to
make the problem accessible to somewhat younger students without
losing the key conceptual ideas behind the problem, but doing so
reduces the complexity and thus the required mathematization. Another way to make the item simpler is to present the data in a table or
schema. In this case, students have no preliminary work to carry out
before they get to the heart of the matter.
SPACE
AND
SHAPE
QUANTITY
The Defense Budget. In a certain country, the defense budget was
$30 million for 1980. The total budget for that year was $500
million. The following year, the defense budget was $35 million,
whereas the total budget was $605 million. Inflation during the
period between the two budgets was 10 percent.
(a)
They then are asked to draw the shadows created by the lamp
(Top View A) and also the shadows cast by the sun (Top View B):
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The fastest sprinter in the world is the cheetah. Its legs are shorter
than those of a horse, but it can reach a speed of more than 110
km/h in 17 seconds and maintain that speed for more than 450
meters. The cheetah tires easily, however, whereas a horse, whose
top speed is 70 km/h, can maintain a speed of 50 km/h for more
than 6 km.
A cheetah is awakened from its afternoon nap by a horses hooves.
At the moment the cheetah decides to give chase, the horse has a
lead of 200 meters. The horse, traveling at its top speed, still has
plenty of energy. Taking into consideration the above data on the
running powers of the cheetah and the horse, can the cheetah
catch the horse? Assume that the cheetah will need around 300
meters to reach its top speed. Solve this problem by using graphs.
Let the vertical axis represent distance and the horizontal axis time
(Kindt 1979, in de Lange 1987).
CHANGE
AND
RELATIONSHIPS
Cheetahs and Horses. Some animals that dwell on grassy plains are
safeguarded against attacks by their large size; others are so small
that they can protect themselves by burrowing into the ground.
Still others must count on speed to escape their enemies.
An animals speed depends on its size and the frequency of its
strides. The tarsal (foot) bone of animals of the horse family is
lengthened, with each foot having been reduced to only one toe.
One thick bone is stronger than a number of thin ones. This single
toe is surrounded by a solid hoof, which protects the bone against
jolts when the animal is galloping over hard ground. The powerful
leg muscles are joined together at the top of the leg so that just a
slight muscle movement at that point can freely move the slim
lower leg.
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Teacher: Explain.
Student C: Well, thats simple: the maximum is 2 meters, the
low is 2 meters, and the period is around 12 hours
33 minutes or so. Thats pretty close, isnt it?
Teacher: So?
Student C: 2 sin (x/2) will do.
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The discussion continued with What happens if we go to a different city that has smaller amplitudes and where high tides come
two hours later? How does this affect the formula? Why is the
rate of change so important?
Why do we consider this a good example of mathematics for ML?
Given the community in which this problem is part of the curriculum, the relevance for society is immediately clearand the
relevance is rising with global temperatures. The relevance also
becomes clear at a different level, however. The mathematical
method of trial and error illustrated here not only is interesting by
itself, but the combination of the method with the most relevant
variables also is interesting: in one problem setting we are interested in the exact time of high water, in another we are interested
in the exact height of the water at high tide. Intelligent citizens
need insight into the possibilities and limitations of models. This
problem worked very well for these students, age 16, and the fact
that the real model used 40 different sine functions did not
really make that much difference with respect to students perceptions.
UNCERTAINTY
Challenger. If we fail to pose problems properly or fail to seek
essential data and represent them in a meaningful way, we can very
easily drown in data. One dramatic example concerns the advice
of the producer of solid rocket motors (SRM) to NASA concerning the launch of the space shuttle Challenger in 1986. The recommendation issued the day before the launch was not to launch
if the temperature was less than 53 degrees Fahrenheit; the low
temperature (29 degrees) that was predicted for the day of the
launch might produce risks. As beautifully laid out by Tufte
(1997), the fax supporting the recommendation was an excellent
example of failed mathematical and common-sense reasoning.
Instead of looking at the data on all 24 previous launches, the fax
related to only two actual launches (giving temperatures, with
ensuing damage to rubber O-rings). NASA, of course, refused to
cancel the launch based on the arguments found in the fax. Simple
mathematics could have saved the lives of the seven astronauts.
The scientists at Morton Thiokol, producer of the O-rings, were
right in their conclusion but were unable to find a correlation
between O-ring damage and temperature. Let us look at the problem systematically. The first thing to do if we suspect a correlation
is to look at all the data available, in this case, the temperatures at
the time of launch for all 24 launches and the ensuing damage to
the O-rings. At that point, we then order the entries by possible
cause: temperature at launch, from coolest to warmest. Next, for
each launch, we calculate the damage to the O-rings and then
draw a scatter plot showing the findings from all 24 launches prior
to the Challenger.
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Reections
I have not answered the question I was asked to address, namely,
what mathematics is important for ML? But I have attempted to
offer some directions: the desired competencies, not the mathematical content, are the main criteria, and these are different at
different ages and for different populations. From a competencies
perspective, mathematics for ML can coexist with calculus or
even better, should coexist with a calculus track but with opportunities to develop intuition, to explore real-world settings, to
learn reasoning, and so on. It goes without saying that the line of
reasoning I have tried to follow holds for all ages, including university students. We also need mathematicians to become mathematically literateas such, they are much better prepared to
participate in society at large and, even more important, can contribute in a constructive and critical way to the discussion about
mathematics education. We all need to understand how important, how essential, ML is for every student, and mathematicians
in particular need to understand that ML will contribute to a
better perception about what constitutes mathematics and how
important that field is to our lives.
We have not addressed several other questions. One of the most
important is: How do we teach mathematics for ML? What are the
pedagogical arguments and didactics of mathematics for ML? But
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Cappo, M., and de Lange, J. 1999. AssessMath!, Santa Cruz, CA:
Learning in Motion.
Cockroft, W. H. 1982. Mathematics Counts. Report of the Committee of
Inquiry into the Teaching of Mathematics in Schools. London: Her
Majestys Stationery Office.
De Lange, J. 1987. Mathematics, Insight, and Meaning. Teaching, Learning, and Testing of Mathematics for the Life and Social Sciences.
Utrecht: Vakgroep Onderzoek Wiskundeonderwijs en Onderwijscomputercentrum (OW&OC).
De Lange, J. 1992. Higher Order (Un-)Teaching. In Developments in
School Mathematics Education around the World, edited by I. Wirszup and R. Streit, vol. 3, 49 72. Reston, VA: National Council of
Teachers of Mathematics (NCTM).
De Lange, J., ed. 1994. Rapport Studiecommissie Wiskunde B-VWO.
Utrecht: Onderzoek Wiskundeonderwijs en Onderwijscomputercentrum (OW&OC).
De Lange, J. 1995. Assessment: No Change Without Problems. In
Reform in School Mathematics and Authentic Assessment, edited by
T. A. Romberg, 87172. New York, NY: State University of New
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De Lange, J. 1999. Framework for Assessment in Mathematics. Madison,
WI: National Center for Improving Student Learning and Achievement in Mathematics and Science (NCISLA).
De Lange, J. 2000. The Tides They are A-Changing. UMAP-Journal
21(1): 1536.
Ellis, Wade Jr. 2001. Numerical Common Sense for All. In Mathematics and Democracy: The Case for Quantitative Literacy, edited by
Lynn Arthur Steen, 61 66. Princeton, NJ: National Council on
Education and the Disciplines.
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National Council of Teachers of Mathematics. 2000. Principles and Standards for School Mathematics. Reston, VA: National Council of
Teachers of Mathematics (NCTM).
Neubrand, N. et al. 2001. Grundlager der Erganzung des Internationalen PISA-Mathematik-Tests in der Deutschen Zusatzerhebung.
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Deborah Hughes Hallett is Professor of Mathematics at the University of Arizona and Adjunct Professor at the Kennedy
School of Government, Harvard. Hughes Hallett is an author of several college mathematics textbooks, including those
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Quantitative Literacy:
Who Is Responsible?
High school and college faculty may be tempted to think that
because the underpinnings of quantitative literacy are middle
school mathematics, they are not responsible for teaching it.
Nothing could be further from the truth. Although the mathematical foundation of quantitative literacy is laid in middle
school, literacy can be developed only by a continued, coordinated
effort throughout high school and college.
The skill needed to apply mathematical ideas in a wide variety of
contexts is not always acquired at the same time as the mathematics. Instructors in middle school, high school, and college need to
join forces to deepen students understanding of basic mathematics and to provide opportunities for students to become comfortable analyzing quantitative arguments in context.
Also key to improving quantitative literacy is the participation of
many disciplines. Quantitative reasoning must be seen as playing
a useful role in a wide variety of fields. The development of quan-
93
Calculus provides other examples of how easy it is to learn procedures without being able to recognize their meaning in context.
Formulas, although a small part of quantitative literacy, are central to calculus. We expect literacy in calculus to include fluency
with formulas for basic concepts. Problems such as If f (t) represents the population of the United States in millions at time t in
years, what is the meaning of the statements f (2000) 281 and
f (2000) 2.5? look as though calculus students should find
them easythere are no computations to be done, only symbols
to read.3 Yet such problems cause great difficulty to some students
who are adept at calculations.
As another example, in 1996, a problem on the Advanced Placement (AP) Calculus Exam4 gave students the rate of consumption
of cola over some time interval and asked them to calculate and
interpret the definite integral of the rate. All the students had
learned the fundamental theorem of calculus, but many who
could compute the integral did not know that it represented the
total quantity of cola consumed.
These examples suggest how teaching practices in mathematics
may differ from those required to develop quantitative literacy.
Mathematics courses that concentrate on teaching algorithms, but
not on varied applications in context, are unlikely to develop
quantitative literacy. To improve quantitative literacy, we have to
wrestle with the difficult task of getting students to analyze novel
situations. This is seldom done in high school or in large introductory college mathematics courses. It is much, much harder
than teaching a new algorithm. It is the difference between teaching a procedure and teaching insight.
Because learning to apply mathematics in unfamiliar situations is
hard, both students and teachers are prone to take shortcuts. Students clamor to be shown the method, and teachers often comply, sometimes because it is easier and sometimes out of a desire to
be helpful. Learning the method may be effective in the short
runit may bring higher results on the next examination but it
is disastrous in the long run. Most students do not develop skills
that are not required of them on examinations.5 Thus if a course
simply requires memorization, that is what the students do. Unfortunately, such students are not quantitatively literate.
Another obstacle to the development of quantitative literacy is the
fact that U.S. mathematics texts often have worked examples of
each type of problem. Most U.S. students expect to be shown how
to do every type of problem that could be on an examination.
They would agree with the Harvard undergraduate who praised a
calculus instructor for teaching in a cookbook fashion.6 Both
college and school teachers are rewarded for teaching practices
that purposefully avoid the use of new contexts.
94
College mathematics faculty frequently fail to realize how carefully a course must be structured if students are to deepen their
understanding. Many, many students make their way through
introductory college courses without progressing beyond the
memorization of problem types. Faculty and teaching assistants
are not trying to encourage this, but are often blissfully unaware of
the extent to which it is happening. This, of course, reinforces the
students sense that this is the way things are supposed to be,
thereby making it harder for the next faculty member to challenge
that belief.
K12 teachers are more likely than college faculty to be aware of
the way in which their students think. They are, however, under
more pressure from students, parents, and administrators to ensure high scores on the next examination by illustrating one of
every problem type. So K-12 teachers also often reinforce students tendency to memorize.
There are strong pressures on college and K12 mathematics instructors to use teaching practices that are diametrically opposed
to those that promote quantitative literacy, and indeed much
effective learning. Efforts to improve quantitative literacy must
take these pressures into account.
95
Another elaborated:11
When I began learning mathematics everything was so simple. As I got older there were many more rules taught to me.
The more rules I learned, the easier it became to forget some
of the older rules.
Unfortunately, the attitudes toward mathematics displayed in
these responses are diametrically opposed to the attitudes required
for quantitative literacy. In attempting to improve quantitative
literacy, we ignore these attitudes at our peril.
ESTIMATION
The ability to estimate is of great importance for many applications of mathematics. This is especially true of any application to
the real world and, therefore, of quantitative literacy. Unfortunately, however, estimation is a skill that falls between the cracks.
Mathematics often does not see estimation as its responsibility;
teachers in other fields do not teach it because they think it is part
of mathematics. Many students therefore find estimation difficult.
The solution is for all of us to teach it.
Worse still, because of mathematics emphasis on precision, students often think that estimation is dangerous, even improper. In
their minds, an estimate is a wrong answer much like any other
wrong answer. The skill and the willingness to estimate should be
included explicitly throughout the curriculum.
Given the current concern about calculator dependence, some
people claim that students would be better at estimation if they
were not allowed to use calculators. It is certainly true that proficiency with a slide rule required estimation; however, even in
pre-calculator days, many students could not estimate. Instead of
grabbing a calculator to do their arithmetic, past students
launched into a memorized algorithm. For example, some years
ago I watched a student use long division to divide 0.6 by 1, then
0.06 by 1, and then 0.006 by 1, before he observed the pattern.12
Even then, he did not recognize the general principle. He never
thought to make an estimate or to see if the answer was reasonable.
Because this was a graduate student, we might reasonably conclude that his education had failed to develop his quantitative
literacy skills.
96
PROBABILITY
AND
STATISTICS
Positive HIV
Test Positive
Test Negative
Totals
Negative HIV
Totals
495
995
1,490
98,505
98,510
500
99,500
100,000
97
Conclusion
Achieving a substantial improvement in quantitative literacy will
require a broad-based coalition dedicated to this purpose. Higher
education should lead, involving faculty in mathematics and a
wide variety of fields as well as people from industry and government. Classroom teachers from across grade levels and across institutionsmiddle schools, high schools, and collegesmust
play a significant role. The cooperation of educational administrators and policymakers is essential. To make cooperation on this
scale a realistic possibility, public understanding of the need for
quantitative literacy must be vastly improved. Because the media
informs the publics views, success will require a new relationship
with the media.
Notes
1. Paper on Pedagogy and the Disciplines, 1990. Written for the
University of Pennsylvania.
2. For example, Harvard students who had already had some calculus in
high school.
3. These statements tell us that in the year 2000, the U.S. population
was 281 million and growing at a rate of 2.5 million per year.
4. AP Calculus Examination, Questions AB3 and BC3, 1996.
5. A few students will independently develop the skill to apply mathematics beyond what is asked of them in courses and on examinations.
98
These students are rare, however; they are the students who can learn
without a teacher. If we want to increase the number of people who
are quantitatively literate, we should not base our decisions about
teaching practices on such students.
6. From a Harvard course evaluation questionnaire.
7. DeFranco, T. C., & Curcio, F. R. (1997). A division problem with
a remainder embedded across two contexts: Childrens solutions in
restrictive versus real-world settings. Focus on Learning Problems in
Mathematics, 19(2), 58 72. Reported by Erik De Corte in a presentation entitled Connecting Mathematics Problem Solving to the
Real World at a conference on Mathematics for Living, Amman,
Jordan, November 18 23, 2000.
8. From David Matthews, University of Central Michigan; reported at
a conference at the University of Arizona, fall 1993.
9. A student made this request in linear algebra when she wanted a
picture showing why a result was true, but was not yet ready to hear
the proof.
10. Published in Chick, Stacey, Vincent, and Vincent, eds., Proceedings
of the 12th ICMI Study Conference: The Future of Teaching and Learning Algebra (University of Melbourne, Australia, 2001), vol. 2: 438
46.
11. Challenges, Issues, and Expectations of Pre-Service Teachers at
CBMS/Exxon National Summit on Teacher Preparation, Washington, DC, 2000. Available at http://www.maa.org/cbms/
NationalSummit/Speakers/McGowan.pdf.
12. This example may seem improbable; unfortunately, it is real.
13. http://europe.cnn.com/2001/WORLD/europe/UK/08/10/coniston.
dna/index.html. The article described the identification of a body in
a lake in the United Kingdom as speed record breaker Donald
Campbell. (CNN: August 10, 2001.)
14. http://www.unfoundation.org/campaign/aids/index.asp; accessed
October 31, 2001.
15. In South Africa alone, it is estimated that there will be 2.5 million
orphans by the year 2010. From Impending Catastrophe Revisited,
prepared by ABT Associates (South Africa) and distributed as a supplement to South Africas Sunday Times, 24 June 2001.
16. http://www.cdc.gov/hiv/pubs/rt/rapidct.htm; accessed July 22,
2001.
17. Reported in Mass Testing: A Disaster in the Making, by Christine
Johnson at http://www.healtoronto.com/masstest.html, accessed
February 24, 2002.
18. High Risk of False-Positive HIV Tests at http://www.healtoronto.
com/pospre.html, quoting from To Screen or Not to Screen for
HIV in Pregnant Women, by Jeffrey G. Wong, M.D., in Postgraduate Medicine 102, 1 (July 1997); accessed February 24, 2002.
19. The word certain is not correct here; however, that is not what
confuses many students. The central issue is that the large number of
false positives is a consequence primarily of the low prevalence of the
HIV virus in the U.S. population, not of inaccuracies in the test.
20. Some states apparently give all pregnant women AIDS tests whether
or not they consent; others suggest an AIDS test but require consent.
21. Some medical schools explicitly require calculus for admission and
many students take calculus as part of their premed program. Statistics is less frequently required for admission.
22. Lewis Carroll, Through the Looking Glass, http://www.literature.
org/authors/carroll-lewis/through-the-looking-glass/chapter05.html; accessed September 9, 2001.
23. Arizona and Massachusetts both have included probability and data
interpretation on the state-mandated high school graduation tests;
however, the contents of these tests are subject to sudden changes, so
there are no guarantees for the future.
24. This was quite reasonable because it was the prevailing view for most
of the last century.
William G. McCallum is Professor of Mathematics at the University of Arizona. A founding member of the Calculus
Consortium based at Harvard University, McCallum has done mathematical research at the Institut des Hautes Etudes
Scientifiques in Paris and at the Institute for Advanced Study in Princeton. McCallums professional interests include both
arithmetic algebraic geometry and mathematics education; he has written extensively in both areas.
Randall M. Richardson is Professor of Geosciences and Vice President for Undergraduate Education at the University of
Arizona. Richardson is responsible for the universitys general education program and recently chaired a university-wide
taskforce on mathematics across the curriculum. He is also involved in NSF-sponsored projects for reforming K-12 teacher
preparation and for mentoring early-career geosciences faculty.
99
100
ports engineers, physicists, biologists, chemists, computer scientists, statisticians, and mathematicians explicitly mentioned
understanding as a key goal of the mathematics curriculum, and
they made it clear that they thought that it was a proper role of
mathematics classes to teach it (Curriculum Foundations Project
2001):
Physics: Students need conceptual understanding first, and some
comfort in using basic skills; then a deeper approach and
more sophisticated skills become meaningful.
Life Sciences: Throughout these recommendations, the definition
of mastery of a mathematical concept recognizes the importance of both conceptual understanding at the level of definition and understanding in terms of use/implementation/
computation.
Chemical Engineering: . . . the solution to a math problem is
often in the understanding of the behavior of the process
described by the mathematics, rather than the specific closed
form (or numerical) result.
Civil Engineering: Introductory math content should focus on
developing a sound understanding of key fundamental concepts and their relevance to applied problems.
Business: Mathematics departments can help prepare business students by stressing conceptual understanding of quantitative
reasoning and enhancing critical thinking skills.
Statistics: Focus on conceptual understanding of key ideas of calculus and linear algebra, including function, derivative, integral, approximation, and transformation.
What is striking about these reports is that so many science, mathematics, engineering, and technology (SMET) disciplines feel the
need to explicitly request conceptual understanding from mathematics courses preparing their students. All the more must we
worry about the state of conceptual understanding in students
who are not preparing for SMET disciplines but simply need
quantitative literacy as a basic life skill. Thus, our first criterion:
A curriculum for quantitative literacy must go beyond the basic
ability to read and write mathematics and develop conceptual
understanding.
101
Another example is a business mathematics course recently developed at the University of Arizona by a collaboration between the
Department of Mathematics and the College of Business and
Public Administration. In this course, students use mathematical
and technological tools to make business decisions based on realistic (in some cases, real) data sets. In one project, for example,
students decide whether to foreclose on a business loan or work
out a new payment schedule. They have available some information about the value of the business, the amount of the loan, and
the likely future value if the business is allowed to continue but
still fails. They also have some demographic information about
the person running the business. Using historical records about
the success and failure of previous arrangements to work out a
payment schedule, they make successively more sophisticated calculations of expected value to arrive at their decision. Students are
expected to understand both the mathematics and the business
context, and to make professional oral presentations of their conclusions in which they are expected to express themselves mathematically, with clarity, completeness, and accuracy.
102
algebra, geometry, logic, probability, and statistics) is foundational to quantitative literacy for everyday life? Looking at the
curriculum as a list of topics, however, misses an important point:
quantitative literacy is not something that a person either knows
or does not know. It is hard to argue that precollege education in
writing fails to cover the basics of grammar, composition, and
voice, for example. Yet it is widely accepted that writing is a skill
that improves with practice in a wide variety of settings at the
college level. We argue here that quantitative literacy at the college
level also requires an across-the-curriculum approach, providing a
wide variety of opportunities for practice.
The challenges to incorporating quantitative literacy across the
curriculum are many, including math anxiety on the part of both
faculty and students, lack of administrative understanding and
support, and competing pressures for various other literacy requirements. We discuss below a variety of approaches that have
demonstrated success at the college level in moving quantitative
literacy across the curriculum. A more comprehensive discussion
would address how these approaches should be coordinated with
efforts to improve K12 education, an issue we do not feel qualified to address. It is worth pointing out, however, that improving
quantitative literacy at the college level would have an important
effect on K12 education for the simple reason that it would
influence the mathematics education of K12 teachers.
103
104
GATEWAY TESTING
IN
WORKSHOPS
FOR
105
106
Acknowledgments
The authors wish to thank H. Len Vacher and Stephen Maurer for
very constructive reviews of a preliminary version of this manuscript.
References
Bauman, Steven F., and William O. Martin. 1995. Assessing the Quantitative Skills of College Juniors. College Mathematics Journal 26:
214.
Mathematical Association of America. 2001. Curriculum Foundations
Project. Washington, DC.: Mathematical Association of America.
Retrieved at: http://academic.bowdoin.edu/faculty/B/barker/
dissemination/Curriculum_Foundations/.
Center for Mathematics and Quantitative Education at Dartmouth College. 2002. Hanover, NH: Dartmouth College. Retrieved at: http://
hilbert.dartmouth.edu/mged.
Fox, J., and J. Powell. 1991. A Literate World. Paris, France: UNESCOInternational Bureau of Education.
Hockney, David, and Charles M. Falco. 2000. Optical Insights into
Renaissance Art. Optics and Photonics News 11(7): 5259.
Mathematical Literacy (ML): The basic skills of arithmetic, algebra, and geometry that historically have formed the core of school mathematics;
Quantitative Literacy (QL): Reasoning with data in their natural contexts, especially in situations that citizens encounter in judging public issues (e.g., pollution, taxes) or private decisions
(e.g., cell phone plans); and
Symbol Literacy (SL): Fluent use of algebraic notation as a second language, typical of students
in science and engineering, but at a level beyond what states or districts would consider as a
requirement for all students.
Although these same terms often are used by others with different interpretations, these are the
definitions that apply throughout this essay.
Michael W. Kirst is Professor of Education at Stanford University. A member of the National Academy of Education, Kirst
is co-director of Policy Analysis for California Education (PACE), a policy research consortium including Stanford and UC
Berkeley. Previously, he served as staff director of the U.S. Senate Subcommittee on Manpower, Employment, and Poverty
and as President of the California State Board of Education. His current research centers on the relationship between state
education reform efforts and educational outcomes.
107
108
AND
HIGHER
cross paths even less often. The only large-scale, nationally aligned
K16 standards effort that involves K16 faculty is the Advanced
Placement Program (AP)a stalactite extending from universities
to high schools that influences the course syllabus and examination. An examination grade of 3, 4, or 5 on an AP examination is
one indicator of college preparation. But roughly one-third of all
AP students do not take the AP Examination, which means that
many AP students may not be benefiting much from APs close
link to postsecondary standards (Lichten 2000).
With the exception of the AP Program, there are no major efforts to
provide curricular coherence and sequencing for grades 10 to 14. Nor
has anyone proposed a conception of liberal education that relates the
academic content of mathematics in secondary schools to the first two
years of college. Instead, students face an eclectic academic muddle
in grades 10 to 14 (Orrill 2000) until they select a college major. In
Ernest Boyers metaphor, postsecondary general education is the
spare room of the university, the domain of no one in particular
whose many functions make it useless for any one purpose (Boyer and
Levine 1981). The functional rooms, those inhabited by faculty,
are the departmental majors.
There are no recent assessments of the status of general education.
C. Adelman (1992) analyzed college students transcripts from the
National Longitudinal Study, containing data from the early to
mid-1970s, which proved to be a low point in general education
requirements. He reported that students took very few courses in
the fields comprised by general education. Less than one-third of
college credits were from courses that focused on cultural knowledge, including Western and non-Western culture, ethnic, or gender studies. Among bachelors degree recipients, 26 percent did
not earn a single college credit in history, 40 percent did not study
any English or American literature, and 58 percent had no course
work in foreign languages.
When attention is paid to general education, two contending theories
predominate. One holds that courses used for general education
should be the same courses as those used to prepare prospective majors for upper-division specialization. Another view contends that the
purpose of general education is as an antidote to specialization, vocationalism, and majors. Clark (1993) hoped that somehow the specialized interests of the faculty could be arranged in interdisciplinary
forms that would provide a framework for mathematical literacy, but
there is little evidence that this is happening.
In sum, the high school curriculum is unmoored from any continuous vision of quantitative literacy. Policymakers for secondary
and postsecondary schools work in separate orbits that rarely interact, and the policy focus for community colleges has been more
concerned with access to postsecondary education than with academic preparation. Access, rather than preparation, is also the
theme of many of the professionals who mediate between high
109
Some state K12 assessments permit students to use calculators, but many college placement examinations do not.
AND THE
K16
Californias newly augmented K12 assessment, the Standardized Testing And Reporting (STAR) program, includes
mathematics that is considerably more advanced and difficult
than the SAT or ACT, but Texas high school assessment,
Texas Assessment of Academic Skills (TAAS) includes less
algebra and geometry than the SAT.
BABEL
OF
ASSESSMENTS
High school students receive confusing messages about the mathematical knowledge and skills that they need to acquire in high
school to succeed in college. In deciding how many years of math-
110
Table 1.
Distribution of Topics on Standardized Mathematics Tests
Geometry
Data,
Probability,
Statistics
Number Theory,
Arithmetic, Logic,
Combinatorics
Algebra
II
Trigonometry,
Pre-Calculus
14
29
23
21
Stanford 9 m/c
29
25
25
21
33
17
18
20
23
28
13
18
13
New York
29
26
26
Texas (TAAS)
12
23
53
47
23
23
ACT
25
27
18
12
14
23
19
25
15
Accuplacer (algebra)
25
75
Accuplacer (calculus)
16
63
21
admitted to college based on one set of skills but then are given
placement tests that cover different topics.
Higher education must be an integral part of any attempt to
improve mathematics articulation. Higher education policymakers need to be involved in the design and implementation of K12
standards and assessments to ensure K16 mathematics articulation. Some states, such as Illinois, California, and New York, are
moving ahead on this. Illinois is giving ACT mathematics to all
eleventh graders, but augmenting it with test items based on the
Illinois mathematics K12 standards. The 19 campuses in the
California State University (CSU) system are eliminating the
placement test designed by CSU faculty and using the K12 California standards test for placement. The City University of New
York (CUNY) system allows students with high scores on the
K12 Regents examination in mathematics to be exempt from
taking CUNY mathematics placement tests.
Some K12 state assessments are rigorous, with content that more
closely resembles college placement tests than the SAT-I. The
Massachusetts and Kentucky K12 assessments include intermediate algebra and trigonometry. Then again, many state K12
REACHING CONSENSUS
ON
MATHEMATICAL LITERACY
2.
3.
4.
5.
6.
If you do not hear appeals from the public for specific content
changes (e.g., inclusion of calculators), you will be criticized
for not having public participation at the highest level and
leaving crucial decisions to a technical panel of nonelected
officials. If the commission hears all these protests, it will
111
Insiders
- Higher education policymakers
- Professors in mathematics, mathematics education, and related disciplines
- State curriculum framework policymakers
- Textbook publishers/testing agencies (private industry)
- National Science Foundation (NSF) collaboratives, partnerships, and curriculum development projects
- Legislative leaders in educational policy
Near Circle
- Teacher preparation institutions
- Teacher certification organizations (e.g., National Council
on Accreditation of Teacher Education (NCATE), National Board for Professional Teaching Standards (NBPTS),
American Association of Colleges for Teacher Education
(AACTE))
- Ideological interest groups
- Federal agencies (Office of Educational Research and Improvement (OERI), National Science Foundation (NSF),
U.S. Department of Education (DOE))
Far Circle
- National Governors Association
- Education Commission of the States
- Council of Chief State School Officers
- National Academy of Sciences
Sometimes Players
- School accrediting agencies (e.g., North Central)
- Business organizations, minority organizations
- National Research Council
112
MATHEMATICS ARTICULATION
CONFLICTS
AND
POLITICAL
AND
VALUE
As the national debate about curriculum content standards demonstrates, policymaking concerning mathematics content and
standards is a political as well as a technical process (Ravitch
1995). Disputes over such issues as the inclusion of AIDS education or creation science in a curriculum highlight the existence of
value conflicts embedded in the development and maintenance of
curriculum standards (Wirt and Kirst 1992). The math wars in
the California standards debate of 1997 included intense debate
concerning the relative emphasis on mathematical literacy versus
quantitative literacy. Because of these conflicts, curriculum policymaking often requires complex trade-offs between groups of
competing interests. Articulation of the mathematics curriculum
in grades 11 to 14 not only involves these conflicts but also others
surrounding the relative priority of symbol literacy in the total
mathematics curriculum linking lower and higher education in
such fields as science and engineering.
The most common way to determine curriculum standards is to
endow an individual or group (e.g., a state school board or a
national subject-matter association) with the authority to make
decisions about curricular content using professional and, presumably expert, judgment (Massell and Kirst 1994). But what
procedures do the developers of curriculum standards follow? Past
efforts best can be described by what Lindblom and Braybrooke
(1963) call disjointed incrementalism, a strategy in which decision makers use pragmatic methods that result in minimal changes
at the margin. Conflict is avoided by adopting vague language
concerning standards and covering so many topics that no major
interest group feels left out. Content priority is sacrificed to the
political necessity of coverage. Disjointed incremental strategies,
however, will not solve the grades 11 to 14 mathematical literacy
TRENDS
IN THE
POLITICS
OF
MATHEMATICS REFORM
Until the 1950s, mathematics curricula were selected by individual school systems in response to the perceived desires of local
communities. The successful launch of Sputnik I in 1957 created
demands for stronger federal and state roles in the education system using two broad strategies: more mathematics content at all
levels and different content and instructional foci (Yee and Kirst
1994). Nevertheless, there were strong demands for preserving
local control over some traditional curriculum matters, and the
political conflict surrounding curricula escalated in the 1970s
(Dow 1991).
An attempt by a federal agency to influence the development of all
subjects in the local curriculum was rebuffed in the 1970s when
Congress cut back the role of the federal government in social
studies curriculum development (Dow 1991). An example of resistance to the federal governments efforts was the charge by
Congressman John Conlan (R.-Ariz.) that this curriculum was a
federal attempt to use classrooms for conditioning, to mold a
new generation of Americans toward a repudiation of traditional
values, behavior, and patriotic beliefs (cited in Wirt and Kirst
1992, 102). Yet 20 years earlier, the federal government had entered the curriculum and text development field because of concern that mathematics and science curricula were outdated, inaccurate, dull, and lacking in diversity (Dow 1991). Scholars and
experts in education who advise federal and state governments
have been criticized for trying to impose their own cosmopolitan
and secular values on diverse local communities. Curricular reform itself has become professionalized through government and
foundation grants. No longer are perceived crises such as the economic recession of 1980 to 1983 required to generate curricular
change because curricular change now has a self-starting capacity.
Curricular conflict has many roots. Military threats or changes in
public sentiment about issues such as the womens movement
generate value conflicts about curriculum. Other forces, such as
court decisions favoring bilingual education or the pronouncements of influential individuals, can result in changes to the curriculum without the direct development of new materials. To
incorporate all these influences, the process of new mathematics
textbook creation is managed, whereby a writing team prepares
a series of texts. The actual author is frequently the publishers
internal editor, not the authors listed on the title page. States that
adopt textbooks (mostly in the Southeast) have disproportionate
influence on what is offered in the national market. Thus any
attempt to change mathematics curriculum must involve rethinking textbook creation and adoption policies.
113
THE COMPLEXITIES
OF
SYSTEMIC CHANGE
114
AND
POLITICAL
Previous education reform efforts, especially large-scale curriculum reforms, often have been criticized for ignoring the social,
political, and technical realities of implementation in schools and
classrooms (Dow 1991; McLaughlin 1991; Yee and Kirst 1994).
The new math projects that were sponsored by the National Science Foundation from the 1950s to the 1970s are good examples
of programs that were criticized because parents, teachers, community leaders, administrators, and others had only limited, if
any, involvement in the development of the new curriculum, were
uninformed about the changes they were expected to make, and
were ill-prepared to defend the reforms when challenges arose at
the local levels (Massell 1994a, 186-87). Because of the failure of
these past reform efforts, todays educators are well aware of the
types of problems that will arise if notions of change are not widely
shared at the community level (Carlson 1995). Most of todays
K12 standards projects thus try to gather diverse input by engaging in a broad review and feedback process with professional educators, business, community members, and others who have an
interest in the standards. Gathering diverse input alone, however,
will not achieve the development and implementation of leadingedge content standards because the broad range of ideas frequently
blocks consensus. Might a variety of passionate individuals pursuing very different goals in very different ways produce much
greater student learning and commitment than the same individuals constrained by a forced consensus?
NCTM achieved a degree of initial consensus around the content
standards that they designed in 1989 (Massell 1994a). The impact
of those standards was enhanced by a long period of preparation
prior to convening the writing committees. This preparation laid
some of the intellectual groundwork for mathematics reform and
ensured broad involvement in the developmental process. In contrast to past large-scale curriculum reform efforts, NCTM engaged more educators as well as subject-matter specialists on its
drafting committees. Its efforts also were enhanced by far-reach-
115
AND
SPECIFIC STANDARDS
116
SUMMARY
OF
POLICY
AND
POLITICAL ISSUES
117
PROMISING POLICIES
Despite these complex issues, some progress has been made. For
example, NCTM content revisions in 2000 appear to have reduced objections by opponents. CUNY has agreed to use the New
York Regents mathematics tests for initial freshman placement,
and Oregon has developed a system for K16 educators to rate
high school student work samples in their college preparatory
courses as one criterion for university admissions. Representatives
of the University of California, California State University, California community colleges, and K12 are devising a college admissions test that will be an integral part of the California state
secondary school assessment. Georgia has regional P16 Councils
that help improve student preparation for postsecondary education. These diverse K16 efforts have the potential to send clearer
signals to students, but articulation requires that these signals be
received: they must culminate in specific and clear student understanding of mathematical literacy.
Signaling theory suggests that streamlined messages have a positive
impact on students learning and achievement but that mixed signalsthe current state of affairs have the opposite effect. Crucial
aspects of signals and incentives are clarity and consistency. Consistency occurs when signals and institutional policies are alignedfor
example, when state and local K12 assessments are coordinated with
the ACT and SAT. In the emerging climate, a simple rule of thumb
will likely apply: the more that incoherent and vague signals are sent
by universities to students, the less adequate student preparation for
higher education will become. Better mathematics articulation will
require basic changes in curriculum policymaking and K16 integration. It can be done but will require persistent leadership and institutional structures that provide effective deliberative forums. The
United States has made some initial promising steps but there is still
a long way to go. The incentives for the higher education community
to work with K12 to develop quantitative literacy are weak, so this
must become a higher priority for college and university presidents,
provosts, deans, and faculty.
100
Math Analysis
Stanford 9
58
12
18
25
33
10
15
24
26
22
21
12
12
42
22
26
18
18
23
75
10
16
12
13
Diagrams
S
RO P
16
10
12
18
15
25
10
18
15
Formulas
M
G
MC multiple-choice items
GR fill-in-the-grid items
22
10
31
IA
19
12
12
23
19
16
15
19
33
14
28
19
29
23
86
14
14
12
25
Formulas
M formula needs to be memorized
G formula is provided
C contextualized items
TR trigonometry
PG plane geometry
CG coordinate geometry
IA intermediate algebra
EA elementary algebra
PA prealgebra
Content
13
60
14
30
37
31
15
52
32
20
22
EA
Content
CG
PG
Context
OE open-ended items
Graphs/Diagrams
13
23
42
17
PA
Format
GSE A California state exam based on the New York Regents end-of-course structure.
Graphs
S RO P
17
OE
Context
C
Source: Vi-Nhuan Le, Alignment Among Secondary and Post-secondary Assessments (Santa Monica: Rand Corporation, 2002).
Legend:
100
100
SATII-Level IIC
100
25
Format
QC GR
SATII-Level IC
58
92
SATI
95
GSE (Geometry)
100
CSU
95
100
Algebra Readiness
GSE (Algebra)
100
MC
ACT
Test
Appendix I
12
11
63
16
26
34
32
13
62
52
19
28
16
40
31
82
54
58
53
82
23
38
76
70
84
53
20
15
15
10
PS problem-solving
PK procedural knowledge
CU conceptual understanding
Cognitive Requirements
4 40
18
0 13
0 15
10
0 10
2 22
Cognitive
Requirements
TR SP MISC CU PK PS
118
Quantitative Literacy: Why Numeracy Matters for Schools and Colleges
119
Educational Standards and Testing: A Response to the Recommendations of the National Council on Educational Standards and
Testing. Santa Monica, CA: RAND.
Lichten, William, 2000. Wither Advanced Placement? Education Policy
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Lindblom, C., and D. Braybrooke. 1963. A Strategy of Decision. New
York, NY: Free Press.
Marsh, D. D., and A. R. Odden. 1991. Implementation of the California Mathematics and Science Curriculum Frameworks. In Education Policy Implementation, edited by A. R. Odden, 219 40. Albany, NY: State University of New York Press.
Marshall, C., D. Mitchell, and F. Wirt. 1989. Culture and Educational
Policy in the American States. New York, NY: Falmer.
Marzana, R. J. 199394. When Two World Views Collide. Educational Leadership 15(4): 18 19.
Clark, B. 1985. The School and the University. Berkeley, CA: University of
California Press.
Massell, D., and M. Kirst, (eds.). 1994. Setting National Content Standards [Special issue]. Education and Urban Society 26(2).
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David, J. L. and P. D. Goren. 1993. Transforming Education: Overcoming
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Frederickson, N. 1984. The Real Test Bias: Influences of Testing on
Teaching and Learning. American Psychologist 39: 193202.
Fuhrman, S. H. 1993. Politics of Coherence. In Designing Coherent
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Francisco: Jossey-Bass.
Kirst, M. W. 1984. Choosing Textbook. American Educator (Fall).
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American Journal of Education 102: 38393.
Koretz, D. M., G. Madaus, E. Haertel, and A. Beaton. 1992. National
120
Purkey, S. C., and M. Smith. 1983. School Reform: The District Policy
Implications of the Effective Schools Literature. Elementary School
Journal 85(4): 358 89.
Ravitch, D. 1995. National Standards in American Education. Washington, DC: Brookings Institution.
Smith, M., and J. ODay. 1991. Systemic School Reform. In The
Politics of Curriculum and Testing, edited by S. Fuhrman and B.
Malen, 223 67. New York, NY: Taylor & Francis.
Steen, Lynn Arthur, ed. 2001. Mathematics and Democracy: The Case for
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Tyson-Bernstein, H. 1988. The Academys Contribution to the Impoverishment of American Textbooks. Phi Delta Kappan 70: 19398.
Venezia, A. 2000. Texas Case Study: Bridge Project Report. Palo Alto, CA.:
Bridge Project, Stanford University School of Education. www.
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Vi-Nhuan, Le, Alignment Among Secondary and Post-secondary Assessments
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Whitty, G. 1984. The Privatization of Education. Educational Leadership 41(7): 5154.
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Stocking, C. 1985. The United States. In The School and the University.
Berkeley, CA; University of California Press.
Wirt, F., and M. Kirst. 1992. Schools in Conflict. Berkeley, CA: McCutchan.
Sykes, G., and P. Plastrik. 1992. Standard-Setting as Educational Reform. Paper prepared for the National Council on Accreditation of
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Yee, G., and M. Kirst. 1994. Lessons from the New Science Curriculum
of the 1950s and 1960s. Education and Urban Society 26(2): 158
71.
What is the meaning of the phrase statistical tie in the sentence The result of the 2000
election in Florida was a statistical tie, even though George Bush was declared the winner?
Extra credit: Sketch out a mathematically sound and politically palatable solution to the problem of close elections.
2.
Respond to the following claim, made by a student to his geometry teacher: Well, you may
have proven the theorem today, but we may discover something tomorrow that proves the
theorem wrong.
3.
Guesstimate quickly, please: If you want the most money for your retirement, should you (a)
invest $500 per year in an index-based mutual fund from the time you are 16 years old to the
time you are 30, or (b) invest $1,000 per year in a bank savings account from the time you are
25 until you are 65?
4.
Is mathematics more like geography (a science of what is really out there) or more like chess
(whose rules and logical implications we just made up)? Did we discover the truth that 1
1 2, or did we invent it? Based on our work this semester, give two plausible reasons for each
perspective. Then give your own view, with reasons.
5.
Study the data on the last 10 years of AIDS cases in the United States from the newspaper
clipping in front of you. What are two trends for charting future policy?
6.
At current rates of revenue and payout the Social Security fund will be bankrupt by the time
you retire. Explain how this statement could be both true and false, mathematically speaking,
depending on the definitions and assumptions used.
7.
Grant Wiggins is the President and Director of Programs for Relearning by Design, a not-for-profit educational organization that consults with schools, districts, and state education departments on a variety of issues, notably assessment and
curricular change. Wiggins is the author of Educative Assessment (1998), Assessing Student Performance (1999), and (with Jay
McTighe) Understanding by Design (2000). Wiggins many articles have appeared in such journals as Educational Leadership
and Phi Delta Kappan.
121
122
8.
9.
10.
123
1.
Statistical Tie
C, E, F, H
2.
Fragile Proof
A, D, I
3.
Investment Estimate
E, F, G, H
4.
Discover or Invent
A, B, D, I
5.
AIDS Data
C, F, G, I
6.
Social Security
A, B, D, E, G, H
***
7.
Silly Proof
D, I
In an essay designed to stimulate thought and discussion on assessing quantitative literacy (QL), why not start with a little concrete provocation: an attempt to suggest the content of questions
such an assessment should contain? (Later I will suggest why the
typical form of mathematics assessmenta secure quiz/test/
examination can produce invalid inferences about students QL
ability, an argument that undercuts the overall value of my quiz,
too.)
8.
Solar System
C, E, F, G, H
9.
D, I, J
10.
Testing Memo
C, D, E, F, H
11.
Cataclysmic
11.
B.
Cultural Appreciation
C.
Interpreting Data
D.
Logical Thinking
E.
Making Decisions
F.
Mathematics in Context
G.
Number Sense
H.
Practical Skills
I.
Prerequisite Knowledge
J.
Symbol Sense
124
The result of students endless exposure to typical tests is a profound lack of understanding about what mathematics is: Perhaps
the greatest difficulty in the whole area of mathematics concerns
students misapprehension of what is actually at stake when they
are posed a problem. . . . [S]tudents are nearly always searching for
[how] to follow the algorithm. . . . Seeing mathematics as a way of
understanding the world . . . is a rare occurrence.9 Surely this has
more to do with enculturation via the demands of school, than
with some innate limitation.10
Putting it this way at the outset properly alerts readers to a grim
truth: this reform is not going to be easy. QL is a Trojan horse,
promising great gifts to educators but in fact threatening all mainstream testing and grading practices in all the disciplines, but
especially mathematics. The implications of contextualized and
meaningful assessment in QL challenge the very conception of
test as we understand and employ that term. Test items posed
under standardized conditions are decontextualized by design.
These issues create a big caveat for those cheery reformers who
may be thinking that the solution to quantitative illiteracy is simply to add more performance-based assessments to our repertoire
of test items. The need is not for performance tests (also out of
context)most teacher, state, and commercial tests have added
some but for an altogether different approach to assessment.
Specifically, assessment must be designed to cause questioning
(not just plug and chug responses to arid prompts); to teach
(and not just test) which ideas and performances really matter; and
to demonstrate what it means to do mathematics. The case statement challenges us to finally solve the problem highlighted by
John Dewey and the progressives (as Cuban notes11), namely, to
make school no longer isolated from the world. Rather, as the case
statement makes clear, we want to regularly assess student work
with numbers and numerical ideas in the field (or in virtual realities with great verisimilitude).
What does such a goal imply? On the surface, the answer is obvious: we need to see evidence of learners abilities to use mathematics in a distinctive and complicated situation. In other words, the
challenge is to assess students abilities to bring to bear a repertoire
of ideas and skills to a specific situation, applied with good judgment and high standards. In QL, we are after something akin to
the test faced by youthful soccer players in fluid games after they
have learned some discrete moves via drills, or the test of the
architect trying to make a design idea fit the constraints of property, location, budget, client style, and zoning laws.
Few of us can imagine such a system fully blown, never mind
construct one. Our habits and our isolationfrom one another,
from peer review, from review by the wider world keep mathematics assessment stuck in its ways. As with any habit, the results
of design mimic the tests we experienced as students. The solu-
125
126
In mathematics, the facts are arguably far worse than this dreary
general picture suggests. Few tests given today in mathematics
classrooms (be they teacher, state, or test-company designed) provide students with performance goals that might provide the incentive to learn or meaning for the discrete facts and skills learned.
Typical tests finesse the whole issue of purpose by relying on items
that ask for discrete facts or technical skill out of context. What
QL requires (and any truly defensible mathematics program
127
The aims in the case statement are not new ones. Consider this
enthusiastic report about a modest attempt to change college admissions testing at Harvard a few years back. Students were asked
to perform a set of key physics experiments by themselves and
have their high school physics teacher certify the results, while also
doing some laboratory work in front of the colleges professors:
allow appropriate opportunities to rehearse, practice, consult resources, solicit feedback, refine performances, and revise products.
Secrecy, enforced quiet, solitary work, and other artificial
constraints imposed by large-scale testing are minimized.
Nothing new here. Benjamin Bloom and his colleagues made the
same point almost 50 years ago, in their account of application
and synthesis:
[S]ituations new to the student or situations containing new
elements as compared to the situation in which the abstrac-
128
CONSTRUCTION
OF
KNOWLEDGE
1.
2.
DISCIPLINED INQUIRY
3.
4.
5.
7.
129
130
131
132
Is the setting realistically noisy and messysufficiently illstructured and ill-defined that the learner must constantly
consider what the question really is and what the key variables
are?
Are there apt opportunities to self-assess, to get lots of feedback, and to self-adjust en route as needed?
133
consider both the intellectual content and the interpersonal wisdom needed here, now, in this case.
THE
How to calculate surface area and volume for various threedimensional figures
Consult to the United Nations on the least controversial twodimensional map of the world, after having undertaken Exploration 22.
What we are really seeking evidence of in context-bound assessment is a combination of technical skill and good judgment in its
use. A good judge, said Dewey, has a sense of the relative values
of the various features of a perplexing situation, has horse
sense, has the capacity to estimate, appraise, and evaluate, and
has tact and discernment. Those who judge well, whether it be
in matters of numeracy or human interaction, bring expertise to
bear intelligently and concretely on unique and always incompletely understood events. Thus, merely acquiring information
can never develop the power of judgment. Development of judgment is in spite of, not because of, methods of instruction that
emphasize simple learning. . . . [The student] cannot get power of
judgment excepting as he is continually exercised in forming and
testing judgments.31
134
135
2.
3.
Is the mathematically sound solution always the most objectively fair solution?
Predictable misunderstandings:
1.
136
2.
Interesting investigations:
1.
2.
3.
New grading systems for students (median/standard deviation/throw out high and low, etc.)
2.
3.
4.
5.
We now see that the scoop of ice cream has a volume that is well
over 50 cm more than the cones volume. Therefore it is unlikely
that the melted ice cream could fit completely inside the cone.
However, as all ice cream lovers like myself know, there is a certain
amount of air within ice cream [therefore experiments would have
to be done].
137
What if mathematics teachers routinely had to use multiple criteria with related rubrics in the assessment of performance? Here are
five possible criteria, with the top-level descriptor from each rubric
(used in the pilot statewide performance assessments in North
Carolina, mentioned above) 40:
Contextual Effectiveness of Solution. The solution to the problem is effective and often inventive. All essential details of the
problem and audience, purpose, and other contextual matters
are fully addressed in a graceful and effective way. The solution may be creative in many possible ways: an unorthodox
approach, unusually clever juggling of conflicting variables,
the bringing in of unobvious mathematics, imaginative evidence, etc.
Accuracy of Work. The work is accurate throughout. All calculations are correct, provided to the proper degree of precision/measurement error, and properly labeled.
Quality of Communication. The students performance is persuasive and unusually well presented. The essence of the research and the problems to be solved are summed up in a
highly engaging and efficient manner, mindful of the audience and the purpose of the presentation. There is obvious
craftsmanship in the final product(s): effective use is made of
supporting material (visuals, models, overheads, videos, etc.)
and of team members (when appropriate). The audience
(4/3) r 3 (1/3) r 2h
4 r 3 r 2h
4r h
4r h
From this final comparison, we can see that if the height of the
cone is exactly 4 times the radius, the volumes will be equal . . .
[The student goes on to explain why there are numerous questions
about ice cream in real life that will affect the answer, e.g., Will the
ice creams volume change as it melts? Is it possible to compress ice
cream?, etc. He concludes by reminding us that we can only find
out via experiment.]
The second explanation is surely more sophisticated, displaying a
number of qualities we seek in QL (e.g., attention to context,
comfort with numbers and data). The students analysis is mature
in part because it subsumes the particular mathematics problem
under a broader one: under what conditions are the volumes of
different shapes equal? In the first case, all the student has done is
calculate the volumes based on the formulas and the given numbers. The second explanation is mature and indicative of understanding by showing perspective: the student has written a narrative as if he were explaining himself to his teachermindful, in a
humorous way, of audience and purpose.
Nonetheless, the teacher in question gave these two papers the
same grade. Both papers gave the correct mathematical answer,
after all. Even more alarming, the second paper was given lower
grades than the first paper by a slight majority of middle school
mathematics teachers (who seemed to take offense at the students
flippancy) in a national mathematics workshop a few years ago.
Of course, when scoring criteria are unclear, arbitrariness sets
in usually in the form of scoring what is easiest to see or scoring
based on the teachers unexamined and semiconscious habits.
That is why learning to score for inter-rater reliability (as is done
138
shows enthusiasm and/or confidence that the presenter understands what he/she is talking about and understands the
listeners interests.
Note that these criteria and rubrics provide more than a framework for reliable and valid scoring of student work. They also
provide a blueprint for what the assessment tasks should be. Any
assessment must be designed mindful of the rubrics so that the
criteria are salient for the specifics of the proposed task. That
compels teachers and examination designers to ground their designs in the kinds of complex and nonroutine challenges at the
heart of QL. Rather than requiring a new array of secure tests with
simplistic items, we should be requiring the use of such rubrics in
all assessment, local and statewide.
139
The tacit assumptions at work that have not been made explicit;
Level 2. Pupils select the mathematics they use in some classroom activities. They discuss their work using mathematical
language and are beginning to represent it using symbols and
simple diagrams. They explain why an answer is correct.
Level 3. Pupils try different approaches and find ways of overcoming difficulties that arise when they are solving problems.
They are beginning to organise their work and check results.
Pupils discuss their mathematical work and are beginning to
explain their thinking. They use and interpret mathematical
symbols and diagrams. Pupils show that they understand a
general statement by finding particular examples that match
it.
The basic blueprint for tasks that can help us assess insight was
provided by a grumpy workshop participant many years ago: You
know the trouble with kids today? They dont know what to do
when they dont know what to do! But that is because our assessments are almost never designed to make them not know what
to do.
Longitudinal Rubrics
Level 5. In order to carry through tasks and solve mathematical problems, pupils identify and obtain necessary information. They check their results, considering whether these are
sensible. Pupils show understanding of situations by describing them mathematically using symbols, words and diagrams.
They draw simple conclusions of their own and give an explanation of their reasoning.
AND
APPLYING
Level 1. Pupils use mathematics as an integral part of classroom activities. They represent their work with objects or
pictures and discuss it. They recognise and use a simple pattern or relationship.
140
of symbols that is sustained throughout the work. They examine generalisations or solutions reached in an activity,
commenting constructively on the reasoning and logic or the
process employed, or the results obtained, and make further
progress in the activity as a result.
2.
2.
3.
4.
5.
6.
With respect to the next to the next-to-last step, the authors note:
Evaluating and Reflecting, Again. This time, it is common for
all members of the school faculty to participate in a long
meeting. Sometimes an outside expert will be invited to attend as well. . . . Not only is the lesson discussed with respect
to what these students learned, but also with respect to more
general issues raised . . . about teaching and learning.51
Is it any wonder, then, if this process is customary, that the typical
Japanese teacher would develop more sophisticated curricular and
141
142
the making public of all design work, and peer review against
design standards.
Finally, let us never forget that although the issues here seem
technical and political, at bottom they are moral. The aim in any
realistic assessment process is to gather appropriate evidence and
render a considered judgment, much like what the judge has to do
in a civil trial. The analogy is useful because such a judgment is
always fraught with uncertainty; it is never neat and clean; it is
always in context. The evidence is weighed, considered, argued
about. The standard for conviction is that there be a preponderance of evidence of the right kind. To convict a student of
understanding similarly requires compelling and appropriate evidence and argument: the student should be considered innocent
of understanding unless proven otherwise by a preponderance of
evidence. That is a high standard, and appropriately so even
though impatient teachers, testers, and policymakers may wish it
were otherwise. Alas, for too long we have gotten away with verdicts in mathematics education using highly circumstantial, indirect evidence. It is high time we sought justice.
Notes
1. From N. Movshovitz-Hadar and J. Webb, One Equals Zero, and
Other Mathematical Surprises (Emeryville, CA: Key Curriculum
Press, 1998).
2. New York Times, 13 August 2001, A35.
3. Lynn Arthur Steen, ed., Mathematics and Democracy: The Case for
Quantitative Literacy (Princeton, NJ: National Council on Education and the Disciplines, 2001), 122.
4. Steen, Mathematics and Democracy.
5. Steen, Mathematics and Democracy.
6. Deborah Hughes-Hallett, Achieving Numeracy: The Challenge of
Implementation, in Mathematics and Democracy, Steen, ed., 9398.
7. Hughes-Hallett, Achieving Numeracy, in Mathematics and Democracy, Steen, ed.
8. Peter T. Ewell, Numeracy, Mathematics, and General Education,
in Mathematics and Democracy, Steen, ed., 37 48.
9. Howard Gardner, The Unschooled Mind: How Children Think and
How Schools Should Teach (New York, NY: Basic Books, 1991), 165.
10. See, for example, the theme issue The Mathematical Miseducation
of Americas Youth in Phi Delta Kappan 80:6 (February 1999), and
Schoenfeld, A. 1992. Learning to Think Mathematically: Problem
Solving, Metacognition, and Sense Making in Mathematics. In
D. A. Grouws, ed., Handbook of Research on Mathematics Teaching
and Learning. New York: Macmillan, 334 371.
11. Larry Cuban, Encouraging Progressive Pedagogy, in Mathematics
and Democracy, Steen, ed., 8792.
12. J. Stigler and J. Hiebert, The Teaching Gap: Best Ideas from the
13. Cf. Dan Kennedy, The Emperors Vanishing Clothes, in Mathematics and Democracy, Steen, ed., 55 60.
14. Grant Wiggins, Assessing Student Performance (San Francisco: JosseyBass, 1996).
15. See Grant Wiggins, Educative Assessment: Assessment to Improve Performance (San Francisco: Jossey-Bass, 1998).
16. P. T. Ewell, Numeracy, Mathematics, and General Education, in
Mathematics and Democracy, Steen, ed., 37:
. . . the key area of distinction [between QL and mathematics] is
signaled by the term literacy itself, which implies an integrated ability
to function seamlessly within a given community of practice. Literacy as generally understood in the verbal world thus means something quite different from the kinds of skills acquired in formal
English courses.
17. Daniel Resnick and Lauren Resnick, Standards, Curriculum, and
Performance: A Historical and Comparative Perspective, Educational Researcher 14:4 (1985): 521.
18. A cautionary note, then, to professional development providers: information, evidence, and reason will not change this habit, any more
than large numbers of people quit abusing cigarettes, alcohol, and
drugs because they read sound position papers or hear professionals
explain why they should quit. Habits are changed by models, incentives, practice, feedbackif at all.
19. The Present Requirements for Admission to Harvard College,
Atlantic Monthly 69:415 (May 1982): 67177.
20. See, for example, G. Cizek, Confusion Effusion: A Rejoinder to
Wiggins, Phi Delta Kappan 73 (1991): 150 53.
21. Earlier versions of these standards appeared in Wiggins, Assessing
Student Performance, 228 30.
22. Benjamin Bloom, ed., Taxonomy of Educational Objectives: Book 1:
Cognitive Domain (White Plains, NY: Longman, 1954), 125.
23. B. S. Bloom, G. F. Madaus, and J. T. Hastings, Evaluation to Improve Learning (New York, NY: McGraw-Hill, 1981), 265.
24. Bloom, et al., Evaluation to Improve Learning, 268.
25. Newmann, W. Secada, and G. Wehlage, A Guide to Authentic Instruction and Assessment: Vision, Standards and Scoring (Madison,
WI: Wisconsin Center for Education Research, 1995).
26. Wiggins, G. & Kline, E. (1997) 3rd Report to the North Carolina
Commission on Standards and Accountability, Relearning by Design,
Ewing, NJ.
27. Federal Aviation Report, as quoted in A Question of Safety: A
Special Report, New York Times, Sunday 13 November 1994, section 1, page 1.
28. Jay McTighe and Grant Wiggins, The Understanding by Design
Handbook (Alexandria, VA: Association for Supervision and Curriculum Development, 1999).
29. Grant Wiggins and Jay McTighe, Understanding by Design (Alexandria, VA: Association for Supervision and Curriculum Development, 1998).
30. William James, Talks to Teachers (New York, NY: W. W. Norton,
1899/1958).
31. John Dewey, The Middle Works of John Dewey: 1899 1924 (vol. 15)
(Carbondale, IL: Southern Illinois University Press, 1909), 290.
32. A historical irony. For it was Descartes (in Rules for the Direction of
the Mind) who decried the learning of geometry through the organized results in theorems presented in logical order as needlessly
complicating the learning of geometry and hiding the methods by
which the theorems were derived. See Wiggins and McTighe, Understanding by Design, 149 ff.
143
40. Wiggins, G. & Kline, E. (1997) 3rd Report to the North Carolina
Commission on Standards and Accountability, Relearning by Design,
Ewing, NJ.
41. Stallings, Virginia and Tascione, Carol Student Self-Assessment
and Self-Evaluation. The Mathematics Teacher. NCTM. Reston,
VA. 89:7, October 1996. pp. 548 554.
42. Cf. Hughes-Hallett, Achieving Numeracy, in Mathematics and
Democracy, Steen, ed., 96 97.
43. Edward Rothstein, Its Not Just Numbers or Advanced Science, Its
Also Knowing How to Think, New York Times, 9 March 1998, D3.
44. From the Oxford English Dictionary CD-ROM.
45. From the National Curriculum Handbook for Secondary Teachers in
England, Department of Education and Employment, 1999. Also
available on the Internet at www.nc.uk.net. Note that these rubrics
are called Attainment Targets.
33. Norm Frederiksen, The Real Test Bias, American Psychologist 39:3
(March 1984): 193202.
46. Paulos, John Allen. 1990. Innumeracy: Mathematical Illiteracy and Its
Consequences (New York: Vintage Books).
Because much of the early work in quantitative literacy was led by statisticiansindeed, many K12 programs in
probability and statistics are named quantitative literacystatistics bears a very special relation to quantitative
literacy, with respect to both substance and education. This essay provides a perspective by leaders of statistics education
on issues raised in the other background essays prepared for the Forum on quantitative literacy.
Richard L. Scheaffer is Professor Emeritus of Statistics at the University of Florida and in 2001 was President of the
American Statistical Association (ASA). Scheaffer served as the first Chief Faculty Consultant for the College Boards new
AP Statistics course. He has written extensively in statistical education and on sampling theory and practice. For many years
Scheaffer has been active on ASA education committees where he has fostered joint work with NCTM in statistics
education.
145
146
skills which enables an individual to cope with the practical demands of everyday life(Cockcroft 1982, 11). More recently, the
International Life Skills Survey, as quoted in Mathematics and
Democracy: The Case for Quantitative Literacy (Steen 2001), offers
a slightly broader definition of quantitative literacy as an aggregate of skills, knowledge, beliefs, dispositions, habits of mind,
communication capabilities, and problem-solving skills that people need in order to engage effectively in quantitative situations
arising in life and work (Steen 2001, 7).
There are strong ties between statistical thinking, data analysis,
and quantitative literacy in terms of historical developments, current emphases, and prospects for the future. As pointed out in
Mathematics and Democracy (Steen 2001), the American Statistical Association (ASA) conducted a National Science Foundationfunded project called Quantitative Literacy in the mid-1980s that
produced materials and workshops to introduce mathematics
teachers at the middle and high school levels to basic concepts of
data analysis and probability. The project was built around a
hands-on, active learning format that involved student projects
and appropriate use of technology.
The ASA QL program was motivated by the Schools Project in
England that had introduced statistics into the national curriculum, using the report Mathematics Counts (Cockcroft 1982) as
one of the supporting documents. This report noted that statistics
is essentially a practical subject and its study should be based on
the collection of data . . . by pupils themselves. To this end it
urged in-service training courses on the teaching of statistics not
only for mathematics teachers but also for teachers of other subjects as well as teaching materials which will emphasize a practical approach (Cockcroft 1982, 234). Even then, 20 years ago,
the Cockcroft commission recognized that micro-computers . . .
offer opportunities to illuminate statistical ideas and techniques
(Cockcroft 1982, 235). All these points were taken to heart by the
ASA QL team, and all are still valid concerns.
The emphasis on statistical thinking and data analysis that was
introduced in Britain migrated to Canada and was picked up as a
main theme for U.S. K12 education by a Joint Committee of the
ASA and the National Council of Teachers of Mathematics
(NCTM). The ASA-NCTM QL project served as a model for the
data analysis and probability strand in Curriculum and Evaluation
Standards for School Mathematics published by NCTM (1989), a
strand that is even stronger in the updated edition (NCTM 2000).
The movement to include data analysis and probability in the
school mathematics curriculum thus has some of the same historical roots as the current QL movement, and has similar emphases.
Properly taught, statistical thinking and data analysis emphasize
mathematical knowledge and skills that enable an individual to
cope with the practical demands of everyday life. They also de-
velop knowledge, beliefs, dispositions, habits of mind, communication capabilities, and problem-solving skills that people need to
engage effectively in quantitative situations arising in life and
work. It is no accident that almost all of the examples given in the
opening paragraphs of Mathematics and Democracy (Steen 2001)
are statistical in nature.
Simultaneous with the K12 effort, many statisticians began emphasizing statistical thinking at the college level. As mentioned
above in the discussion of statistical literacy, excellent textbooks
and other materials as well as numerous college courses have been
developed around this theme. These deal with issues of quantitative literacy in much more authentic ways than almost any mathematics text seems to.
Because statistics and quantitative literacy share so much in common, we hope that statisticians and mathematics educators will
work together to build a strong emphasis on QL in the school and
college curriculum. Many statisticians would probably disagree
with the statement in Mathematics and Democracy (Steen 2001)
that QL is not the same as statistics. Indeed, many think that a
very large part of QL is statistics (statistical thinking or data analysis), just as the Cockroft commission thought that statistics was a
large part of numeracy. In what follows, we take a more detailed
look at the common ground between statistics and QL and suggest ways of building on that commonality for the good of all.
QL and Citizenship
Patricia Cline Cohen, quoting Josiah Quincy, notes in her essay
that one of the duties of responsible government is to provide
statistical knowledge about the general welfare of its citizens. Hard
data are to be sought and ought to be studied by all who aspire to
regulate, or improve the state of the nation. . . (Cohen, see p. 7).
In fact, the very word statistics derives from its use to collect
information on and about the state. A good example of the growth
of statistics in government can be seen in the development and
expansion of the U.S. Census Bureau over the years and the widespread uses to which its data are put. Developing an informed
citizenry is one of the tasks of public education and, in light of the
emphasis on data within the government, a large part of that task
involves improving the quantitative literacy of all citizens. That
statistics can be misused by politicians (and others) is one of the
reasons citizens need some skill in statistical thinking and reasoning with data.
According to Cohen, statistics are a powerful tool of political and
civic functioning, and at our peril we neglect to teach the skills
required to understand them. In large measure, Cohen equates
quantitative literacy with statistics and makes a strong case for
including statistics in everyones education. With this, statisticians
147
certainly can agree. They would not agree, however, with Cohens
statement that statistics has become a branch of mathematics.
Statistics has many roots, including business, engineering, agriculture, and the physical, social, and biological sciences; it deals with
many issues that would not be considered mathematical. Emphasis on context is one such issue; emphasis on the design of studies
is another. Although statistics uses mathematics, the key to statistical thinking is the context of a real problem and how data might
be collected and analyzed to help solve that problem. Some would
say that the greatest contributions of statistics to modern science
lie in the area of design of surveys and experiments, such as the
demographic and economic surveys of the Census Bureau and the
Bureau of Labor Statistics and the experiments used in many
health-related studies.
In fact, statistics has much broader uses than its mathematical
roots might suggest, and many, including the federal government
itself, are attempting to enlighten citizens about the proper collection, analysis, and interpretation of data. One example of this is
the effort of the FedStats Interagency Task Force to develop a
statistical literacy program for users of the Federal Statistical System. A related effort is embodied in a recent report from the
National Research Council entitled Information Technology Research for Federal Statistics, which talks about the importance of
literacy, visualization, and perception of data:
Given the relatively low level of numerical and statistical
literacy in the population at large, it becomes especially important to provide users with interfaces that give them useful,
meaningful information. Providing data with a bad interface
that does not allow users to interpret data sensibly may be
worse than not providing the data at all, . . . . The goal is to
provide not merely a data set but also tools that allow making
sense of the data. (NRC 2000, 20)
These and other efforts by the federal government to improve statistical literacy are supported by Katherine Wallman, chief statistician of
the US government, who said in a 1999 speech (Wallman 1999):
Electronic dissemination is truly a boon to national statistical
offices anxious to make their data more accessible and usefuland to user communities equipped to handle the wealth
of available information. But this technology remains to a
degree a bane, for while we have taken monumental strides in
making our nations statistics electronically available, attention to documentation in electronic media has lagged. And I
continue to argue, as I have for almost a decade, that the gap
between our citizens computer literacy and their statistical
literacy remains significant.
Citizens encounter statistics at every turn in their daily lives. Often, however, they are ill-equipped to evaluate the information
148
These recommendations fit well with current efforts in the statistics community to build bridges between the academic community and business, industry, and government to ensure an effective
statistics education for the workforce of the future. Somewhat
surprisingly, however, the level of skills attached to quantitative
literacy varies greatly among those quoted by Rosen, ranging from
merely knowing basic arithmetic to making judgments grounded
in data. If such judgments are thought of in the sense of statistical
thinking and data analysis, they are much deeper than basic mathematical skills and require an educational component that is not
found in traditional mathematics courses. Statistical thinking has
a stochastic component (could this variation be caused by chance
alone?) that is essential to intelligent study of business, industry,
and government processes.
QL and Curriculum
QL and Mathematics
Closely related to the issue of curriculum is the relationship between QL and mathematics. Deborah Hughes Hallett asserts in
her essay that QL is the ability to identify and use quantitative
arguments in everyday contexts, that it is more a habit of mind
than a set of topics or a list of skills. QL is more about how
mathematics is used than about how much mathematics a person
knows. For this and other reasons, a call to increase QL is a call for
a substantial increase in most students understanding of mathematics. It is, therefore, not a dumbing down of rigor but an increase in standards. According to Hughes Hallett, this increase is
essential because the general level of quantitative literacy is currently sufficiently limited that it threatens the ability of citizens to
make wise decisions at work and in public and private life
(Hughes Hallett, see p. 91).
Statisticians will find it interesting (and gratifying) that probability and statistics are the only subject areas that Hughes Hallett
mentions specifically. Indeed, she finds the absence of these subjects in the education of many students remarkable given that they
are so extensively used in public and private life. Simply requiring more students to study advanced mathematics is not the answer: they actually must be taught QL by solving problems in
context. Courses must demand deeper understanding, which
will require a coordinated effort to change both pedagogy and
assessment.
Although there is much to agree with in Hughes Halletts essay,
statistics educators would probably disagree with the claim that
. . . the teaching of probability and statistics suffers from the fact
that no one can agree on when or by whom these topics should be
149
150
The detailed statistical content may vary, and may be accompanied by varying levels of study in computing, mathematics,
and a field of application. (ASA 2001, 1)
Reports from the MAA (CUPM 1993) recommend that all undergraduate mathematical sciences majors should have a datacentered statistics course. Taken together, the standards, guidelines, and curriculum materials fashioned by the statistics community
(with support from the mathematics community) give solid evidence that many pieces of the coordinated effort needed to
improve quantitative literacy are in place. The QL reform that
may be coming should make good use of the projects and related
ideas already afloat within the statistics education community.
To be honest, however, many statistics courses still are taught in a
manner that misses the QL point. This is partly because tension
always exists between breadth of coverage and deep understandingthe latter of most importance to QL. Although the statistics
education community may have reached consensus on how to
deal with the tension, this consensus does not always play out
easily in the classroom. Courses serve many clients, some of whom
demand coverage of many specific topics in statistical inference.
Jan de Langes paper, also about QL and mathematics, introduces
two new and important ideas (de Lange, see pp. 75 89). First, it
extends the definition of quantitative literacy to the term mathematical literacy because of the indisputable fact that much more
in mathematics is useful besides numbers. Indeed, many aspects of
statistical thinking (which de Lange includes under the name
uncertainty as one of his core phenomenological categories) are
not about numbers as much as about concepts and habits of mind.
For example, the idea of a lurking variable upsetting an apparent
bivariate relationship with observational data is a conceptual idea,
part of statistical thinking but not particularly about numbers.
The notion that designed experiments are more reliable than observational studies is another very important nonquantitative idea.
De Langes second important idea is that if mathematics were
properly taught, the distinction between mathematical content
and mathematical literacy would be smaller than some people
suggest it is now. The issue is part of the aforementioned tension
between breadth of coverage and depth of understanding, but it
also suggests a resolution of the dilemma of QL courses. Separate
courses in QL create serious problems. First, students are pigeonholed into those capable of taking real mathematics and those
who will only need QL, thereby entrenching two classes of students in a structure that serves the nation poorly. Second, although all students need to be quantitatively literate, there is
growing evidence that those who take regular mathematics
courses (and who in a segregated system may not encounter much
QL) are not learning many of the critical thinking skills they need.
QL and Articulation
Articulation of the K16 mathematics curriculum is difficult to
attain because it involves inextricably linked political and policy
issues. Michael Kirsts essay (Kirst, see pp. 107120) outlines the
main areas of political tension: between professional leadership
and political consensus, between flexible and specific standards,
between dynamic standards and reasonable expectations for
change, between professional leadership and public understanding of standards, between expectations and requirements. Progress
toward improving articulation requires a clear signal up and down
the line as to what is required. Part of that signal should be a clear
message about QL.
As subject-matter standards and examinations have evolved in
recent years, one of the widespread changes has been increased
emphasis on data analysis and statistics; however, one of the main
limiting factors is the quality of materials for teachers. Any attempt to change mathematics curriculum, Kirst observes, must
involve rethinking textbook creation and adoption policies. Another limiting factor is the ever-present standardized examination.
Multiple-choice basic skills tests do not adequately emphasize
complex thinking skills such as statistical inference and multistep
mathematical problem solving.
The statistics community would argue that an emphasis on statistics and QL in the mathematics curriculum could help alleviate
some of these tensions. The movements to infuse school mathematics with data analysis and to enhance undergraduate statistics
offerings owe much of their success to the fact that leaders from
business and industry supported the efforts. It helped, of course,
that these efforts began when quality improvement was a high
national priority; that theme is still important for garnering support for statistics among business and political leaders. Another
theme that allows statisticians to enter doors that might be more
difficult for mathematicians to open is data: everyone is collecting
tons of it and few know what to do with it. The public understands something of these issues. Indeed, many see the need for
statistics education much more clearly than they see the need for
mathematics education (although they might view statistics as a
part of mathematics).
Will college faculty buy into an articulated program in mathematics education that includes a strong component of QL? Statistics
faculty are likely to do so, if the success of the AP Statistics course and
the support for the changes promoted by the NCTM standards and
the NAEP framework are any indication. A QL emphasis would not
look as radically new to a statistician as it might to a mathematician.
QL and Assessment
Many of the exhortations in the background essays about the
importance of assessment to a successful QL program are subsumed in the comprehensive and detailed paper by Grant Wiggins
(Wiggins, see pp. 121143). In Wiggins view, echoed by others,
we have often sacrificed the primary client (the learner) in the
name of accountability. Wiggins seeks to put the interests of the
learner back in the center of assessment.
Assessment plays a central role in QL reform. Wiggins argues for
a realignment of assessment with QL that puts more emphasis on
open-ended, messy, and authentic assessment tasks. Much of
this realignment will require challenging changes in the focus of
traditional instruction, including much more formative (diagnostic) assessment. To develop reliable examples of high-quality assessment strategies that are focused on a few big ideas will require
significant collaboration. In addition, instructors will need training to design, administer, and grade these new types of assessment.
Wiggins makes much of context but seems to use the term in at
least two different ways. One relates to determining the source of
a problem (who is asking the question, how was the information
gathered, who is the answer for, what are relevant issues in the
discipline that may affect the solution). Another suggests a more
philosophical, historical point of view (where do laws or theorems
come from, are they debatable, can you understand the history
and how it affects our present state of knowledge). Although historical perspective is important, Wiggins seems to overemphasize
the role of this type of context for beginning students. To statisticians, the first definition of context is absolutely essential for any
problem; the second, although helpful for some problems, is not
nearly as essential.
Data analysis problems usually have a built-in context that may
make them easier for teachers to attack (although not many such
examples are found in Wiggins essay). They have less of the
baggage of the years of formalism that has accompanied mathematics instruction and that can be difficult for new teachers to
break free from.
Wiggins differentiates between meaning making and statistical
reasoning, whereas statisticians would not see these as so different. His interpretation of meaning making as what is mathematics and why does it matter seems a bit narrow. Many levels of
reasoning and conceptual understanding are important in mathematics even when historical perspective is incomplete. The focus
should be on students abilities to reason with their own knowledge and understand how it works, even if their ability to question and debate is limited. Mathematics that is relevant to students direct experiences is more meaningful to many beginning
151
Conclusion
Statistics and quantitative literacy have much in common. Although few would disagree with this, statisticians would probably
argue that QL is mainly statistics while mathematicians and mathematics educators tend to argue that QL is only partly statistics.
Statistics emphasizes context, design of studies, and a stochastic
view of the world. Although statistics is clearly not the same as
mathematics, nor even a part of mathematics, it uses mathematics
as one of its main tools for practical problem solving. Being one of
the most widely used of the mathematical sciences, statistics is well
entrenched in many places across the curriculum. At the K14
level, statistics already has embarked on a program that emphasizes active learning, much in the spirit recommended by modern
cognitive science. All this suggests that students will reap dividends if the two disciplines work together.
Although statistics education has gained acceptance (even respect)
over the past 15 years as a key component of the K12 mathematics curriculum, this acceptance does not always translate into classroom practice. The taught curriculum is far from reconciled with
the recommended curriculum. In addressing this challenge, statistics and QL should be mutually reinforcing. Simply put, statistics has opened the door for quantitative literacy. In his background essay on curriculum in grades 6 12, Lynn Arthur Steen
argues that in a balanced curriculum, [D]ata analysis, geometry,
and algebra would constitute three equal content components in
grades 6 to 8 and in grades 9 to 11 (Steen, see p. 66). Real work
yielding real results, he emphasizes, must begin and end in real
data (see p. 59).
On the pedagogical side, statistics educators have learned to emphasize both engagement and relevance. There is ample evidence
that both teachers and students like a hands-on, activity-based
152
References
American Statistical Association. 2001. Curriculum Guidelines for Undergraduate Programs in Statistical Science. http://www.amstat.
org/education/Curriculum_Guidelines.html.
Chance, B. L. 1997. Experiences with Authentic Assessment Techniques in an Introductory Statistics Course Journal of Statistics
Education, 5(3). www.amstat.org/publications/jse/ v5n3/chance.html.
Cockcroft, W. H., ed. 1982). Mathematics Counts. London: Her Majestys Stationery Office.
Garfield, J. B. 1994. Beyond Testing and Grading: Using Assessment to
Improve Student Learning. Journal of Statistics Education 2(1).
www.amstat.org/publications/jse/v2n1/ garfield.html.
Mathematical Association of America. 1993. Guidelines for Programs
and Departments in Undergraduate Mathematical Sciences.
http://www.maa.org/guidelines.html.
Moore, D. 1998. Statistics Among the Liberal Arts. Journal of the
American Statistical Association 93:125359.
***
Moore, D. 2001. Statistics: Concepts and Controversies, 5th ed. New York,
NY: W. H. Freeman.
Moore, T., ed. 2000. Teaching Statistics: Resources for Undergraduate Instructors, MAA Notes No. 52. Washington, DC: Mathematics Association of America.
National Council of Teachers of Mathematics. 1989. Curriculum and
Evaluation Standards for School Mathematics. Reston, VA: National
Council of Teachers of Mathematics.
National Council of Teachers of Mathematics. 2000. Principles and Standards for School Mathematics. Reston, VA: National Council of
Teachers of Mathematics.
National Research Council. 2000. Information Technology Research for
Federal Statistics. www4.nationalacademies.org/cpsma/cstb.nsf/
web/pub_federalstatistics?OpenDocument.
Porter, T. 2001. Statistical Futures. Amstat News (291) (September):
61 64.
Steen, Lynn Arthur, ed. 2001. Mathematics and Democracy: The Case for
Quantitative Literacy. Princeton, NJ: National Council of Education and the Disciplines.
Utts, J. 1999. Seeing Through Statistics, 2nd ed. Belmont, CA: Duxbury Press.
153
154
is dominated by issues of vertical articulation. Nonetheless, horizontal and environmental articulation likely are more important
levers in improving U.S. education, especially in quantitative literacy.
THE CULTURE
OF
MATHEMATICS
Mathematics research is the principal activity of what Paul Halmos called the mathematics fraternity, which he described as a
self-perpetuating priesthood. Mistakes are forgiven and so is
obscure expositionthe indispensable requisite is mathematical
insight (Halmos 1968, 381). Prestige in mathematics is gained
through manifestations of mathematical insight developing
new mathematicsand those who have prestige wield the greater
power over academic mathematics.
Mathematics research is a demanding taskmaster requiring dedication, concentration, even obsession. Although most mathematics research does not aim at immediate applications, the history of
unanticipated uses of mathematics provides strong support for its
value to society. Consequently, educating mathematicians and
creating new mathematics often dominate educating people to use
mathematics.
Mathematicians see great value and power in abstract mathematical structures and seek students who can master advanced mathematics. This strongly influences views of the goals of mathematics courses and curricula, and those views are reflected in school
and college mathematics. Anthony Carnevale and Donna Desrochers argue that the implicit trajectory and purpose of all disciplines is to reproduce the college professoriate at the top of each
disciplinary hierarchy (Carnevale and Desrochers, see p. 28).
Mathematics, as they go on to analyze, is no exception. Lynn
Arthur Steen has compared mathematics teachers concentrated
attention on the best students to hypothetical physicians who
attend primarily to their healthiest patients (Steen 2002).
The efficiency of the path to calculus and advanced mathematics
has led to rigid linearity of the GATC sequence. No other disci-
pline, save perhaps foreign language, exhibits such linearity. Foreign language education is built on using the language, however,
whereas students use of mathematics is usually far in the future.
Most students in the GATC sequence never get to any authentic
uses for what they learn.
Fortunately, there are some signs that the mathematics fraternity
is turning its attention and vast talents to issues other than its own
reproduction and expansion. Among the most recent signs are
three publications: Towards Excellence: Leading a Mathematics Department in the 21st Century from the American Mathematical
Society5 (Ewing 1999); Adding It Up: Helping Children Learn
Mathematics from the National Research Council6 (Kilpatrick et
al. 2001); and Mathematical Education of Teachers from the Conference Board of the Mathematical Sciences7 (CBMS 2001).
OF
COLLEGE MATHEMATICS
155
bra a requirement for some majors e.g., for prospective elementary teachersis even more misguided. The traditional college
algebra course is filled with techniques, leaving little time for
contextual problems. Students, many of whom have seen this
material in prior algebra courses, struggle to master the techniques; three of four never use these skills and many of the rest find
that they have forgotten the techniques by the time they are
needed in later courses. No wonder the course is uninspiring and
ineffective. Success rates are very low often below 50 percent
and student dissatisfaction is high. Fortunately, many faculty and
administrators realize this and reform efforts are growing. The
task is nonetheless monumental.
COLLEGE STATEMENTS
ON
EXPECTATIONS
IN
MATHEMATICS
Comprehensive and useful statements from higher education institutions about mathematics expectations for entering students
are rare. In spring 2001, with the help of the Education Trust, I
requested from a number of states whatever statements concerning mathematics content were available from colleges and universities about expectations for the mathematics knowledge and skills
of entering students. I received responses from 11 states, seven of
which had such statements. The other four states had processes or
policies that addressed the transition from school to college mathematics, but these did not include statements on mathematics
learning, content, or skills.
The seven statements of college expectations range from comprehensive documents that look very much like a set of complete
standards for grades 9 12 mathematics to explanations of skills
(mostly algebraic) needed to survive in entry-level courses. Californias expectations are of the first type, Marylands and Nebras-
156
TO
SCHOOL
During various periods in the past, college and university mathematics faculty have played significant roles in supporting school
mathematics. During the 1960s, research mathematicians were
involved in developing new school curricula and in conducting
AND
COLLEGE
157
THE DILEMMA
OF
158
About 20 years ago when I was chair of the department of mathematical sciences at the University of Arkansas, I was struggling with
ways to reduce enrollments in intermediate algebra, the one remedial mathematics course we taught. The state was pressuring us to
reduce remedial enrollments, but my most pressing reason was to
reduce the range of courses we had to cover. We were the only
doctoral and research institution in the state and our resources were
stretched very thin, covering responsibilities from high school algebra to postgraduate seminars.
My local school system, which had one high school (from which
both my son and daughter later graduated), was revising its mathematics offerings and I was invited to meet with the superintendent
and associate superintendent to give them advice. I took the opportunity to talk about how they could help reduce our remedial enrollments.
Typically, they were offering two tracks of mathematics. One was a
college preparatory track with the usual courses geometry, Algebra I and II, trigonometry, and AP Calculusactually a very strong
offering. The second track was general or business mathematics, I
dont remember the exact terminology. I asked why they offered
this clearly weaker track and why they didnt keep all the students
on the track that would prepare them for college-level mathematics,
since at the time, any student who graduated from high school
could enroll at the University of Arkansas. Because we did not
require that they had followed a college preparatory track, students
from this weaker track would almost surely land in remedial algebra.
My superintendent and his associate were very frank: they were not
going to take the heat for students failing. I noted that they were
passing that heat on to us at the university and they did not disagree.
OF THE
SAME THING
In the mid-1970s, I was named director of the mathematics component of the Academic Skills Enhancement Program (ASEP) at
Louisiana State University. The goal of the program was to increase
the success rate of students in remedial mathematics. We instituted
a moderately complex system of four courses, each a half-semester
long, whereby students would progress to the next course or start
over based on the results of the previous course. I taught several of
these classes, including one section of the first course in which all the
students had failed to progress on their first try.
Never have I had a more challenging assignment. I was helping
college-age (and older) students to succeed in ninth-grade mathematics after they had all failed to do so in the previous eight weeks.
It was there that I learned the many different reasons why students
have trouble with elementary algebra. I also learned why remedial
algebra in universities faces almost insurmountable obstacles given
the levels of success expected in most academic enterprises. Perhaps
30 years later, with the use of technology, the obstacles can be
overcome.
159
160
COLLEGE MATHEMATICS
AS A
FILTER
Unfortunately, many mathematics faculty accept the long tradition of their discipline as a filter and expect a large number of
students to fail. This expectation casts a pall that hangs over many
mathematics classrooms, causes additional students to fail, and
increases resentment toward mathematics.
STATISTICS ARTICULATION
In most colleges, statistics courses are spread across several departments including statistics, mathematics, engineering, social sciences, agriculture, and business. By and large, college statistics is
taught to support majors in other disciplines, often by faculty
whose appointments are in the disciplines served. Statistics has
been viewed as a research method in agriculture and the social
sciences consistent with Richard Scheaffers characterization of
statistics as keeper of the scientific method (Scheaffer et al., see
p. 145). In many institutions, there is little interaction or synergy
among the statistics courses taught in various disciplines. Partly
because of this dispersion, college statistics departments have
never had sufficient enrollments to justify large departmental faculties. Measured by degree programs, statistics is largely a graduate
discipline.
But now statistics is also a high school discipline. The AP Statistics
course, first offered in 1997, has grown remarkably fast, with
about 50,000 examinations in 2002. Ten years ago, when the
College Boards AP Mathematics Development Committee was
first asked to make a recommendation about developing AP Statistics, they were stymied because there was no typical first college
course in statistics, which was necessary for the standard prototype
of an AP course. This lack of a standard first college course was
indicative of the dispersion of statistics teaching in higher educa-
tion. Therefore, in a reverse of the traditional pattern, AP Statistics, which was developed by college and school faculty outside the
muddled arena of college statistics, has become a model for a first
college course in statistics. This history illustrates the degree of
difficulty in changing college curricula without outside impetus.
The position of AP Statistics in school and college curricula differs
from AP Calculus in that the former does not sit in an established
sequence of prerequisite and succeeding courses. This freedom
promotes access to AP Statistics and does not affect students
course choices nearly as dramatically as does AP Calculus. AP
Statistics has been well served by the introduction of a strand on
data analysis and probability in the K12 mathematics standards,
which has increased the visibility of statistics to students and
teachers.
161
ARTICULATION
WITH THE
ENVIRONMENT
162
The content of [mathematics] curricula will have to be modernized at least every five to 10 years.
Similar to mathematics research, learning mathematics at the college level need not be linear. Students can learn mathematical
concepts and reasoning through combinatorial mathematics,
through data analysis, and through geometry, as well as through
calculus. Even fundamental concepts of calculusrate of change,
approximation, accumulation can be understood outside the
infrastructure of calculus methodology. A major impetus for the
calculus reform movement was a 1983 conference convened to
discuss discrete mathematics as an alternative gateway to college
mathematics (Douglas 1986). By developing multiple interconnecting pathways to the advanced study of mathematics, introductory college mathematics can become more appealing and
more useful to students. Further, a broader view of college mathematics can support a broader school mathematics curriculum
and remove much of the emphasis on a failed system of courses
dominated by algebraic methodology.
Because of their easy experience learning mathematics, most
mathematicians do not relate well to a student struggling with
factoring quadratics or mangling the addition of algebraic fractions. We mathematicians see the larger algebraic architecture and
the logic underlying the operations; however, some of us can
identify with that bewildered student by reflecting on how we first
use a new graphing calculator or software package. Here the architecture and underlying logic of the hardware or software are
obscure. So what do we do? We begin to use the calculator or the
software package and refer to the manual primarily when needed.
No one would first spend days pouring over the manual trying to
commit to memory procedures or keystrokes to accomplish thousands of unconnected operations. Many of our students see college algebra and trigonometry in this same illogical light. Every
operation is new and independent, making retention of skills until
the end of the semester unlikely and until the next year almost
impossible.
Just as computer software and calculators are useful to all of us, so
is algebra. For education to be effective, these uses of algebra must
be given priority over techniques, not only to accomplish tasks
that use algebra but also to master algebra. This approach may
help break the rigid GATC verticality and can increase access to
and success in both mathematics and its applications. And technology can surely help.
Much of the GATC sequence consists of learning skills that can be
performed by technology. Unfortunately, mathematicians do not
agree on what manual (paper-and-pencil) skills are essential or on
how technology helps; some even ban technology. Mathematicians know their own algebraic skills served them well, so when
they see students falter because of poor algebraic skills it reinforces
the beliefs that help maintain the GATC stranglehold.
Notes
1. The Bridge Project, housed at the Stanford Institute for Higher
Education Research, has as its aim to improve opportunities for all
students to enter and succeed in postsecondary education by
strengthening the compatibility between higher education admissions and placement requirements and K12 curriculum frameworks, standards, and assessments.
The Education Trust was created to promote high academic standards for all students at all levels, kindergarten through college. The
Education Trust publishes Thinking K16, an occasional newsletter
that contains discussions of issues in K16 education and how they
are being addressed by various coalitions. See www.EdTrust.org.
The American Diploma Project (ADP) is aimed at aligning high
school academic standards with higher education and the needs of
the new economy. ADP is sponsored by Achieve, Inc., the Education
Trust, the Thomas B. Fordham Foundation, and the National Alliance of Business.
2. Personal communication. Attributed to William Schmidt by Alfred
Manaster.
163
164
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1987. Report of the Panel on Calculus Articulation: Problems in
the Transition from High School Calculus to College Calculus.
American Mathematical Monthly 94: 776 85.
Committee on the Undergraduate Program in Mathematics (CUPM).
1991. The Undergraduate Major in the Mathematical Sciences.
Reprinted in Heeding the Call for Change, edited by Lynn Arthur
Steen, 225 47. Washington, DC: Mathematical Association of
America, 1992.
Conference Board of the Mathematical Sciences (CBMS). 2001. Mathematical Education of Teachers. Providence, RI and Washington,
DC: American Mathematical Society and Mathematical Association of America.
Council of Chief State School Officers (CCSSO). 2002. State Indicators
of Science and Mathematics Education 2000. Washington, DC:
Council of Chief State School Officers.
Douglas, Ronald G. 1986. Toward a Lean and Lively Calculus. Washington, DC: Mathematical Association of America.
Education Trust. 1999. Thinking K16: Ticket to Nowhere. Washington,
DC: Education Trust.
Ewing, John, ed. 1999. Towards Excellence: Leading a Mathematics Department in the 21st Century. Providence, RI: American Mathematical Society. http://www.ams.org/towardsexcellence/.
Gollub, Jerry P., Meryl W. Bertenthal, Jay B. Labov, and Philip C.
Curtis, eds. 2002. Learning and Understanding: Improving Advanced
Study of Mathematics and Science in U.S. High Schools. Washington,
DC: National Academy Press.
Halmos, Paul. 1968. Mathematics as a Creative Art. American Scientist
56(4): 375 89.
Kilpatrick, Jeremy, Jane Swafford, Bradford Findell, eds. 2001. Adding It
Up: Helping Children Learn Mathematics. Washington, DC: National Academy Press.
Lutzer, David, et al. 2002. Statistical Abstract of Undergraduate Programs
in the Mathematical Sciences in the United States, Fall 2000 CBMS
Survey. Washington, DC: Mathematical Association of America.
National Commission on Mathematics and Science Teaching for the
21st Century (National Commission). 2000. Before Its Too Late.
Washington, DC: U.S. Department of Education.
National Council of Teachers of Mathematics (NCTM). 1989. Curriculum and Evaluation Standards for School Mathematics. Reston, VA:
National Council of Teachers of Mathematics.
National Council of Teachers of Mathematics (NCTM). 2000. Principles
and Standards for School Mathematics. Reston, VA: National Council of Teachers of Mathematics.
Organization for Economic Cooperation and Development (OECD)
Center for Educational Research and Evaluation. 2001. Education
Policy Analysis 2001. Paris: Organization for Economic Cooperation and Development.
Steen, Lynn Arthur. 2002. Achieving Mathematical Proficiency for All.
College Board Review. (Spring)196: 4 11.
White, Robert M. 1988. Calculus of Reality. In Calculus for a New
Century, edited by Lynn Arthur Steen, 6 9. Washington, DC:
Mathematical Association of America.
David F. Brakke is Dean of Science and Mathematics at James Madison University. A limnologist, Brakke has studied
ecosystem assessment, lake management, and climate change in the U.S., Canada and northern Europe. He has been
actively involved with professional organizations concerning science and mathematics education, teacher preparation, and
undergraduate research. Brakke also writes a quarterly column on science and society for the Association of Women in
Science (AWIS) Magazine.
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Arnold Packer is Chair of the SCANS 2000 Center at the Institute for Policy Studies, Johns Hopkins University. An
economist and engineer by training, Packer has served as Assistant Secretary for Policy, Evaluation, and Research at the U.S.
Department of Labor, as co-director of the Workforce 2000 study, and as executive director of the Secretarys Commission
on Achieving Necessary Skills (SCANS). Currently, his work is focused on teaching, assessing and recording the SCANS
competencies.
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4.
How will students know whether they want to be mathematicians or research engineers unless they are exposed to abstract
mathematics early? some will say. Yeah, replies the quantitatively literate after looking at some data, as if students especially minorities and womenare beating down the doors to
graduate from mathematics departments now. The current system clearly needs to be radically improved.
Adding courses in quantitative literacy will not do; formal schooling already takes too long. Instead, we must change basic mathematics education, at least until grade 14. What does that mean in
practical terms? Replacing trigonometry with data analysis and
statistics as the first post-algebra course is one step. De-emphasizing calculus is another.
Even more radically, eliminate the use of xs and ys as variable
names until the junior year in college. Eliminate x and y in the
NAEP exams. Move understanding the transferability power of
mathematics to the end of the chapterwhere the applied problems now languishto be learned after students see how mathematics can solve relevant problems. See, now that you understand how rates of change apply to prices to produce measures of
inflation, you can use similar equations to determine the speed of
tennis serves or changes in the incidence of AIDS. The hallmark
of quantitative literacy (QL), in my judgment, is its emphasis on
learning in a meaningful context. The Humble Pi algebra examples
are not meaningful contexts, and neither are most consumer
math problems.
I do not mean to denigrate the importance of transferability and
the power of mathematics in this regard. The issue of transferability is quite complex but data clearly indicate that the majority of
students do not transfer what they learn in mathematics class to
problems in the outside world. A full conversation would bring us
into the field of learning theory, which I hardly understand. I do,
however, suggest How People Learn: Brain, Mind, Experience, and
School by the Committee on Developments in the Science of
Learning of the National Research Council, which reviews recent
developments (Bransford et al. 2001). The authors make the following pertinent points:
1.
2.
What learners already know may get in the way of new learning (for example, in ordering fractions).
3.
2.
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3.
4.
Mathematics needs to be taught in relevant contexts of reallife problems that productive workers and engaged citizens
need to be able to solve.
References
5.
One way to achieve this is to eliminate xs and ys from mathematics until the junior year in college and from the NAEP
and other high school exit examinations.
Bransford, John D., Ann L. Brown, and Rodney R. Cocking, eds. 1999.
How People Learn: Brain, Mind, Experience, and School. National
Research Council. Washington, DC: National Academy Press.
Smith, Michael K. 1994. Humble Pi: The Role Mathematics Should Play in
American Education. Amherst, NY: Prometheus Books.
Grounding Mathematics in
Quantitative Literacy
JOHNNY W. LOTT
We owe our children no less than a high degree of quantitative literacy and mathematical
knowledge that prepares them for citizenship, work, and further study. (NCTM 2000, 289)
Quantitative literacy (QL) is a major goal of the National Council of Teachers of Mathematics
(NCTM) for the teachers of mathematics in this country and in Canada. It is worth emphasizing
that our children means all children. Equity is a core principle for NCTM as well: All students,
regardless of their personal characteristics, backgrounds, or physical challenges, must have opportunities to studyand support to learnmathematics (NCTM 2000, 12).
Equity does not mean identical instruction for all, but it does mean that all students need access each
year to a coherent, challenging mathematics curriculum taught by competent and well-supported
mathematics teachers. Well-documented examples demonstrate that all children, including those
who have been traditionally underserved, can learn mathematics when they have access to highquality instructional programs that support their learning (NCTM 2000, 14).
As we think about quantitative literacy (or more broadly mathematical literacy), we must acknowledge that the mathematics community as a whole has provided neither access to nor a coherent
challenging mathematics curriculum for all students. In fact, underserved groups include not only
students from poor communities but also those from affluent communities that are college-bound.
If anything, college-bound students may have been the most ill-served. Locked into a mathematics
curriculum that has calculus as a single-minded focus, these students have been denied the most
elementary understanding of mathematical literacy. Only in selected schools with a selected curriculum might we find the rudiments of mathematics that lead to quantitative literacy.
Since the release of Curriculum and Evaluation Standards for School Mathematics (NCTM 1989),
NCTM has worked to make mathematics a foremost consideration in this country whenever
education is debated. The Curriculum and Evaluation document focused attention on the need to
improve the mathematical knowledge of all students. That document and its successor, Principles
and Standards for School Mathematics (NCTM 2000), have become magnets for criticism from
certain members of the higher education mathematics community. It is my hope, and that of
NCTM, that this Forum will set the stage for a common national push for mathematical literacy.
Mathematical literacy is a responsibility of precollege mathematics teachers, but it is not their
responsibility alone. A mathematical literacy curriculum must begin early in students school careers,
long before high school; otherwise it is doomed to failure. Many students do not take mathematics
beyond the tenth grade, and some have dropped out of school by that age. Thus, a major portion of
mathematical literacy must be achieved in grade school and early high school.
Johnny W. Lott is Professor of Mathematics at the University of Montana and President of the National Council of
Teachers of Mathematics (NCTM). Lott served as co-director of the SIMMS project that developed a new mathematics
curriculum for grades 9 12 called Integrated Mathematics: A Modeling Approach. An author or co-author of several books
and many articles, Lott has been chair of the editorial panels for three NCTM periodicals.
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References
Montana Council of Teachers of Mathematics. 1996. SIMMS/MCTM
Integrated Mathematics: A Modeling Approach Using Technology, Levels 1 6 Objectives/Content Outline. Bozeman, MT: Montana Council of Teachers of Mathematics.
National Council of Teachers of Mathematics. 1989. Curriculum and
Evaluation Standards for School Mathematics. Reston, VA: National
Council of Teachers of Mathematics.
National Council of Teachers of Mathematics. 1991. Professional Standards for Teaching Mathematics. Reston, VA: National Council of
Teachers of Mathematics.
National Council of Teachers of Mathematics. 2000. Principles and Standards for School Mathematics. Reston, VA: National Council of
Teachers of Mathematics
Quantitative Literacy:
A Science Literacy Perspective
GEORGE D. NELSON
I begin with a note of personal bias: I believe that the mathematics goals of the American Association
for the Advancement of Science (AAAS) are closer to the quantitative literacy (QL) goals discussed
in Mathematics and Democracy: The Case for Quantitative Literacy (Steen 2001) than either the goals
of the new National Council of Teachers of Mathematics (NCTM) standards or current school
science or mathematics curricula. Indeed, Benchmarks for Science Literacy, published by AAAS, has
many of the same QL goalsthey are just not called by that name (Project 2061 1993). In one
respect, therefore, QL is very much part of what we think good science teaching should be about.
On the other hand, Principles and Standards for School Mathematics (NCTM 2000) abandoned, for
many reasons, the vision of the original standards (NCTM 1989) that described both the mathematics important for all students to learn and the mathematics that goes beyond basic literacy
important for those students going on to higher education or technical careers. Principles and
Standards for School Mathematics is a notable and useful description of the goals of school mathematics, but it goes well beyond the goals of QL (QL may be an undefined subset) and may be an
unrealistic vision of the mathematics that all children can learn in 13 years. (I am willing to make the
same statement about the amount of science content in the AAAS Benchmarks and the National
Science Education Standards (NRC 1995).)
I also must point out that Mathematics and Democracy is very mathematics-centric, even as it makes
the case for the interdisciplinary nature of quantitative literacy. The references are almost all from the
field of mathematics and mathematics education, not from the places where QL really livesthe
natural and social sciences. QL is not something new, nor is it something that exists in isolation. It
exists in many places but always in specific contexts. Yet for lack of appropriate contexts, QL rarely
is seen in school classes.
For example, mathematics in science classes is typically independent of mathematics in mathematics
classes. In school science, there is almost no consideration of mathematics scope and sequence,
nor is much effort made to use consistent terminology and symbols. Typical science classes make
little effort to reinforce mathematical concepts or to demonstrate their application in scientific
inquiry. Mathematics classes, in turn, may employ a science setting (e.g., counting whales or planets)
but not science content appropriate to the local scope and sequence. Current mathematics classes
abound in inappropriate, inconsistent, or unrealistic situations and data. Units, when necessary, are
often absent or incorrect. QL-type applications are rare. On an optimistic note, some of the new
reform or standards-based K 8 curriculum materials in mathematics do a much better job of
offering realistic and appropriate examples and contexts.
The knowledge and skills that make up quantitative literacy can be defined through careful sets of
learning goals, specific concepts and skills that together paint a coherent and complete picture. There
are two types of goals: targets for adult knowledge and skills such as those in Science for All Americans
George D. Nelson is Director of the Science, Mathematics and Technology Education Center at Western Washington
University. Immediately prior to assuming this position, Nelson directed Project 2061, a national initiative of the American
Association for the Advancement of Science to reform K-12 science, mathematics and technology education. An astronomer
by training, Nelson earlier served as a NASA astronaut and flew as a mission specialist aboard three space shuttle flights.
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(AAAS 1989) (targets), and benchmarks to monitor progress toward the adult goals (NCTM 1989, AAAS 1993) (steps along the
way, or standards). We need both. Benchmarks are especially important as a strategy to reach our targets because they define the
content around which curricula can be designed and built. So far,
most of what we have in QL are targets without standards. And
those targets span the disciplines.
Where does QL live, or where might it thrive? School mathematics is typically formal and theoretical, thus not yet a welcoming
environment for QL. In comparison with the NCTM standards,
QL involves the sophisticated use of elementary mathematics
more often than elementary applications of advanced mathematics. Although science can be data-rich, natural science often is
taught more like what Arnold Packer and others call x, y math.
Because the contexts of QL are most commonly personal or social,
the social sciences may offer the most natural home. Of course,
this assumes that curriculum developers, teachers, and teacher
educators in the social science disciplines are willing to take on the
responsibility for helping students build on the prerequisite mathematics to learn QL skills and concepts and that the sum of any
students experience totals a coherent vision of QL.
Recommendations:
QL has a strong partner and advocate in the science community. Read and criticize the mathematics in Science for All
Americans (Project 2061 1989), Benchmarks for Science Literacy (Project 2061 1993), and the Atlas of Science Literacy
(Project 2061 2001).
Promote the pedagogical advances that the K12 mathematics community has made through its curriculum development
work.
Develop reliable and valid assessments of experiments in curriculum and instruction that target QL (i.e., do science). And
publish the results.
References
National Council of Teachers of Mathematics. 1989. Curriculum and
Evaluation Standards for School Mathematics. Reston, VA: National
Council of Teachers of Mathematics.
National Council of Teachers of Mathematics. 2000. Principles and Standards for School Mathematics. Reston, VA: National Council of
Teachers of Mathematics.
National Research Council (NRC). 1995. National Science Education
Standards. Washington, DC: National Academy Press.
American Association for the Advancement of Science. 1989. Project
2061. Science for All Americans. Washington, DC: American Association for the Advancement of Science.
American Association for the Advancement of Science. 1993. Project
2061. Benchmarks for Science Literacy. Washington, DC: American
Association for the Advancement of Science.
American Association for the Advancement of Science. 2001. Project
2061. Atlas of Science Literacy. Washington, DC: American Association for the Advancement of Science.
Steen, Lynn Arthur, ed. 2001. Mathematics and Democracy: The Case for
Quantitative Literacy. Princeton, NJ: National Council on Education and the Disciplines.
Arithmetic: Having facility with simple mental arithmetic; estimating arithmetic calculations;
reasoning with proportions; counting by indirection (combinatorics).
Data: Using information conveyed as data, graphs, and charts; drawing inferences from data;
recognizing disaggregation as a factor in interpreting data.
Computers: Using spreadsheets, recording data, performing calculations, creating graphic displays, extrapolating, fitting lines or curves to data.
Chance: Recognizing that seemingly improbable coincidences are not uncommon; evaluating
risks from available evidence; understanding the value of random samples.
William G. Steenken recently retired as a Consulting Engineer in Engine Operability from General Electric Aircraft Engines
in Cincinnati, Ohio. During his career, he published thirty-four papers and reports. An elected member of school boards in
Ohio for over 22 years, Steenken is president of the National Alliance of State Science and Mathematics Coalitions, past
chair of the Ohio Mathematics and Science Coalition, and a member of the Mathematical Sciences Education Board.
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structures, and it goes on. It did not happen in one course or one
place but slowly became a way of life that was continually honed
over a long career that continues to this day. I am always looking
for a better way to present complicated numerical results that can
be easily understood by a broad array of audiences.
Learning in Context
I probably gained most of my mathematical literacy from the
engineering courses that I took and the subsequent need for solutions to problems encountered in my daily work. I found what I
call xyz mathematics difficult throughout my educationprimary, secondary, undergraduate, and graduate. Now as I read
works written by many of you in this audience, I am beginning to
understand why. If I had had the privilege of studying under some
of you when my schooling was starting, I suspect my knowledge
and appreciation of xyz mathematics would be far greater. I see
great beauty in how concepts are now being developed for students and I smile to myself as I read about them, thinking Oh, if
only I could have started that way. To quote Hughes Hallett
again, One of the reasons that the level of quantitative literacy is
low in the U.S. is that it is difficult to teach students to identify
mathematics in context, and most mathematics teachers have no
experience with this (see p. 94).
Let me illustrate context with an example from my field. Consider
the following equation:
X C 1Y 2 C 2
In context, the symbols come alive for me because they are associated with usually measurable and understood physical properties
or quantities. They excite me intellectually, they hold my interest,
and they make me think about how they relate. They are not
simply xyz.
To summarize, I quote again from Mathematics and Democracy
. . . skills learned free of context are skills devoid of meaning and
utility (Steen 2001, 16). I could not agree more.
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of cases of fraud in industry and government, had those in positions of responsibility been quantitatively literate, fraud could not
have succeeded and careers would not have been ruined.
Turning from professionals to skilled craftsmen, the need for
quantitative literacy has increased enormously: operators of manufacturing cells need to know when and how many parts need to
be delivered to their position so they can maintain flow in their
part of the production chain; watch tolerances so their parts do
not fall outside the limits of variability; understand the trends in
variation curves and know which tool needs to be replaced; and be
responsible for self-inspecting their production and reporting the
results.
One last comment about the place of quantitative literacy in the
high-technology workplace. Most of you are probably aware of
the Six Sigma (less that 3.4 defects per million) quality initiative
originated by Motorola and instituted by many other companies.
General Electric was one of the latter. Knowledge of Six Sigma
technology was deemed so important that an extremely large
training effort was undertaken to give almost all employees an
understanding and appreciation of variability in our products and
processes. Jack Welch, the chairman of our board, drove this
change, for he recognized very early that to be the number one
supplier in a field and be profitable, one had to drive out defects as
never before and shorten the ordermanufacture delivery cycle.
There was no better way to accomplish these objectives than to
give every employee the tools needed to perform analyses in support of Six Sigma goals. Thus ordered bar graphs (Pareto charts) to
determine which parameters had the biggest impact (you tackle
them first), statistical analysis spreadsheet tools, tests for significance, flow charting to improve processes, etc., all became daily
tools in our corporate lives.
Instituting Six Sigma technologies has done more than anything
else to raise the overall level of quantitative literacy in corporate
America. Regrettably, nothing in the school curricula gives students this type of knowledge nor do I see it happening during the
next decade. This brings me to my final point.
Mathematics Standards
Ohio has been writing mathematics standards for the K12 grades
for the past two or three years. I had the opportunity to be a
member of a business team that reviewed the proposed standards.
None of this group of approximately 20 businessmenrepresenting small to very large businesses had any problems with the
content of the standards, but the discussion was dominated by
comments regarding the perceived lack of required quantitative
skills and demonstration of them. Businessmen wanted graduating students to be able to understand profit-loss sheets (the basis
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References
Blackwell, David and Leon Henkin. 1989. MathematicsReport of the
Project 2061 Phase I Mathematics Panel. Washington, DC: American Association for the Advancement of Science.
Steen, Lynn Arthur, ed. 2001. Mathematics and Democracy: The Case for
Quantitative Literacy. Princeton, NJ: National Council on Education and the Disciplines.
Roger Howe is Professor of Mathematics at Yale University; his research focuses on symmetry and its consequences. A
member of the National Academy of Sciences, Howe is chair of the Committee on Education of the American Mathematical
Society. Previously, Howe served on the Mathematical Sciences Education Board and as a member of steering committees
that produced two recent reports on mathematics education: Adding It Up (National Academy Press, 2001) and The
Mathematical Education of Teachers (CBMS, 2001).
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(b) 2
(c) 19
(d) 21
If you are comfortable with numbers, you notice that each of the
two fractions to be added is slightly less than 1, so the sum must be
close to 2, and the correct choice is answer (b). You may wonder
why such outlandish answers as 19 and 21 are even offered as
possibilities. Who would guess them? It turns out over half the
students chose one of these answers. The majority of 13-year-olds
apparently have no effective techniques for dealing with approximation of fractions, and perhaps little intuitive grasp of what a
fraction means. This level of understanding provides a weak foundation for using numbers to deal with the world.
The call for quantitative literacy is part of a broader movement of
mathematics education reform that has been growing since the
publication of the National Council of Teachers of Mathematics
(NCTM) Curriculum and Evaluation Standards (NCTM 1989).
Certainly there are good reasons for changing the mathematics
curriculum. What used to be key skills have became much less
important, and a host of new capacities is required to deal with the
diverse numerical data with which we are presented on a daily
basis; however, a major lesson of mathematics education research
during the 1990s is that, to enable significant change in mathematics instruction, we must attend closely to what teachers know
and can do (Ball 1991; Kilpatrick et al. 2001; Ma 1999; Conference Board of the Mathematical Sciences (CBMS) 2001). Failure
to emphasize this point was in my view a major failing of the 1989
Standards and a significant contributor to the math wars in
California and elsewhere. Such failure is the more regrettable because it is frequently cited as a reason for the earlier failure of the
New Math of the 1960s, and because the standards were produced by a combination of mathematics educators and teachers,
who might have been presumed to know better.
I believe that calls for quantitative literacy that ignore the contribution of the elementary years, and the need for attention to
capacity-building among teachers, are unlikely to be widely effective. The capacity of the teaching corps is not a peripheral issue, to
be resolved after formulation of the ideal curriculum. It is a central
issue.
2.
3.
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Population
California
Washington
1890
1,213,398
357,232
1940
6,907,387
1,736,191
1990
29,785,857
4,866,669
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References
Ball, Deborah L. 1991. Research on Teaching Mathematics: Making
Subject Matter Understanding Part of the Equation. In Brophy, J.,
ed. Advances in Research on Teaching, Vol. 2: Teachers Knowledge of
Subject Matter as it Relates to their Teaching Practice. Greenwich,
CT: JAI Press, 1 48.
Conference Board of the Mathematical Sciences. 2001. Issues in Mathematics Education, Vol. 11: The Mathematical Education of Teachers.
Providence, RI: American Mathematical Society.
Kilpatrick, Jeremy, Jane Swafford, and Bradford Findell, eds. 2001. Adding It Up: Helping Children Learn Mathematics. Washington, DC:
National Academy Press.
Ma, Liping. 1999. Knowing and Teaching Elementary Mathematics:
Teachers Understanding of Fundamental Mathematics in China and
the United States. Mahwah, NJ: Erlbaum Associates.
National Council of Teachers of Mathematics. 1989. Curriculum and
Evaluation Standards for School Mathematics. Reston, VA: National
Council of Teachers of Mathematics.
Post, T. R., G. Harel, M. J. Behr, and R. Lesh. 1991. Intermediate
Teachers Knowledge of Rational Number Concepts. In Integrating Research on Teaching and Learning Mathematics, edited by E.
Fennema, T. P. Carpenter, and S. J. Lamon. Albany, NY: State
University of New York Press, 194 217.
Steen, Lynn Arthur, ed. 2001. Mathematics and Democracy: The Case for
Quantitative Literacy. Princeton, NJ: National Council on Education and the Disciplines.
General mathematical competencies that will allow them to successfully learn more specific
competencies as, later, their career goals come more clearly into focus; and
An appreciation for the power of mathematics coupled with a confidence that they are capable
of learning and applying it.
Teachers who are not well trained in their content area (true of a significant percentage of middle and
high school teachers of mathematics) tend to teach what they feel most comfortable with (usually
skills) and what the end-of-course test assesses (usually skills and procedures, with some concepts).
They are least likely to use multiple representations or mathematics in context or to help students
gain the confidence with numbers and graphs and charts that seems to be a shared vision at this
Forum.
Many teachers of mathematics are currently doing a truly outstanding job, sometimes under extremely difficult circumstances, of preparing their students for the world of work or advanced studies
beyond school. These teachers are well-trained in mathematics and use a variety of pedagogical
approaches to make mathematics accessible to their students. They supplement the textbook and
course syllabus with rich explorations that make use of multiple approaches to mathematics. Students of teachers like these receive sufficient mathematical grounding to know what questions to ask
and to have the confidence to seek and find answers in new numerical settings. They become
mathematically competent, thus quantitatively literate. Although they may not have been taught
Bayes theorem or other specific applications they might require in the future, they are confident and
competent learners who will be able to pick up new concepts or skills as needed.
J. T. Sutcliffe holds the Founders Master Teaching Chair at St. Marks School of Texas in Dallas. A recipient of the
Presidential Award for mathematics teaching, as well as Siemens and Tandy Technology Scholars awards, Sutcliffe has
served as a member of the AP Calculus Test Development Committee and as an AP Calculus Exam Leader. Sutcliffe also
helped develop Pacesetter: Mathematics with Meaning, a teacher professional development project for the College Board.
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Most states have standards (to which end-of-course assessments are tied) that resemble the table of contents from a
typical textbook. State end-of-course assessments that determine whether students are able to clear a minimum bar are
often very high stakes for teachers and for students.
Because many teachers do not have strong content and pedagogical training in mathematics, they tend to rely heavily on
a page-by-page textbookdriven pedagogy. Stronger leadership from textbook publishers to incorporate QL as a standard and a goal therefore would help bring QL ideas and
problems to many more students.
2.
3.
Teachers teach according to what will be assessed on highstakes tests. If those in charge of designing the blueprint for
such tests increase the proportion of QL-like items, more QL
will be taught in the schools.
4.
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Janis I. Somerville is staff officer at the National Association of College and University System Heads (NASH) and directs
the State K16 Network that is jointly sponsored by NASH and Education Trust. Previously, Somerville led Marylands
Partnership for Teaching and Learning K16, helped found the Philadelphia Schools Collaborative, and served as the senior
academic officer for undergraduate education at Temple University and at the University of Pennsylvania.
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these assessments to determine whether students can begin creditbearing versus remedial courses or even whether they can enter a
four-year college at all. And as national transcript studies indicate,
the more remedial courses students have to take the less likely they
will even make it to the sophomore year, much less complete a
college degree. High stakes indeed.
In addition to tests, many states also specify Carnegie unit requirements that students must complete to meet college admissions
requirements. Once again, whenever specific courses are named,
these requirements are all about algebra and geometry.
The obvious conclusion is that higher educationmore specifically, the higher education mathematics communityis sending
very clear and consistent messages about the important mathematical knowledge and skills that students should have to succeed
in college, and this message is not at all about QL. For those of us
who are particularly concerned about closing achievement gaps
among poor and minority students who are especially vulnerable
to weak or mixed signals, the clarity of this message is especially
important. To be ready for college, you need Algebra II.
Thus as a matter of policy, school districts must focus curriculum
and instruction on courses that ensure that high school graduates
are prepared to meet these college-ready algebra requirements.
Needless to say, enabling all students to achieve levels of performance previously reserved only for a few requires intensive teacher
development as well as instructional support for teachers and students, support that is not focused in the direction of QL.
So my first recommendation is to think very hard about what
represents college-ready mathematics and where, or if, QL fits in.
If you really believe that QL is essential for all students, that it is
more than just an add-on elective for some, then you cannot duck
this issue. It is not enough to say by the way, here is something
new that you might like to add if there is time. QL advocates
must be very clear about what all students need to know and be
able to do, starting with where QL fits in the mathematics program.
Related to this is the issue of high school students who meet the
minimum college-ready standard early. Right now we rush students to Advanced Placement (AP) Calculus, a course that has
become the universal answer to rigorous school mathematics. For
many politicians, enrollment in AP Calculus is thought of as a
measure of the quality of school mathematics.
Of course, we all know better. Paradoxically, the fastest-growing
part of the high school curriculum is college-level study, while the
biggest part of our college mathematics program is remedial, that
is, high school course work. And it is not clear that either of us
does the others work particularly well. More important, in high
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Margaret B. Cozzens is Vice Chancellor for Academic and Student Affairs and Professor of Mathematics at the University
of Colorado at Denver. Previously Cozzens served as director of the Division of Elementary, Secondary and Informal
Science Education at the National Science Foundation and as Chair of the Department of Mathematics at Northeastern
University. She is a member of the American Council on Educations Task Force on Teacher Preparation, and served as
co-chair of the Technical Review Panel for TIMSS-R, the recent repeat of TIMSS.
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References
National Council of Teachers of Mathematics. 2000. Principles and Standards for School Mathematics. Reston, VA: National Council of
Teachers of Mathematics.
Usiskin, Zalman. 2001. Quantitative Literacy for the Next Generation.
In Mathematics and Democracy: The Case for Quantitative Literacy,
edited by Lynn Arthur Steen, 79 86. Princeton, NJ: National
Council on Education and the Disciplines.
Sadie Bragg is Senior Vice President of Academic Affairs and Professor of Mathematics at Borough of Manhattan Community College. A former president of the American Mathematical Association of Two-Year Colleges (AMATYC), Bragg
has served as a member of the Mathematical Sciences Education Board, the Advisory Board to the Education and Human
Resources Directorate of the National Science Foundation, and the Academic Assembly of the College Board. Bragg is
co-author of many mathematics textbooks for grades K12.
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Prociency Requirement
All senior college students must pass the CPE by the time they have
completed 60 credits. All community college students must pass the
CPE, which is a requirement for the associate degree, before they
graduate. The examination also affects the ability of community
college students to transfer to the senior colleges. This high-stakes
examination demands that faculty within a college and faculty
across the CUNY colleges throughout the system work together to
find ways to prepare the students for the demands of the CPE.
Like other community colleges, BMCC has the responsibility for
preparing students to enter the workforce and/or to transfer to a
four-year institution. With the advent of high-stakes testing, many
community college students may find it difficult to get jobs or transfer
to a four-year institution. Hence, the community colleges role in
preparing students to write well and think quantitatively is imperative.
References
Coley, Richard J. The American Community College Turns 100: A Look
at its Students, Programs, and Prospects. Princeton, NJ: Educational Testing Service, 2000.
Office of Academic Affairs, Test Development Program, The City University of New York. Spring 2001. A Description of the CUNY Proficiency Examination. New York: The City University of New York.
Background
The New York City public schools have published mathematics performance standards that clearly
define what students need to know and be able to do in mathematics. These standards were
developed through a careful process of review and alignment and within the context of major
instructional reform in all content areas.
In 1996, the New York City Board of Education adopted a plan to introduce and implement
learning standards in all major content areas. The board recognized that the process of becoming a
standards-driven system would be a multiyear effort, requiring us to reexamine policies and practices, with special emphasis on aligning assessments, redirecting resources, and improving professional development. Board members understood that we would need to work closely with the New
York State Education Department (SED) to ensure that our standards and assessments would align
with those at the state level.
Because we believed that performance standards in the New York City public schools should reflect
the best thinking of practitioners and researchers nationally in each of the content areas, we chose to
work with the New Standards project codirected by the National Council on Education and the
Economy and the Institute for Learning at the University of Pittsburgh. In the area of mathematics,
the New Standards reflect the recommendations of the National Council of Teachers of Mathematics (NCTM).
Although our belief in the need to align our city standards with state and national ones was
paramount, we also understood that for our schools and communities to accept the standards and
feel a sense of ownership, they needed to be directly involved in the process. To accomplish this, we
began the work of customizing the standards. The customization process brought together teachers,
principals, and other pedagogical staff to review the New Standards, the NCTM standards, and the
New York State learning standards to ensure alignment. The customizers also were engaged in a very
important effort that would directly connect the standards with the New York City teachers and
students. The customizers returned to their districts, schools, and classrooms to find evidence of
student work that reflected the standards. The discussions about which work samples met the
standards were very rich and provided an excellent model of professional development. This experience resulted in a standards document that included student work as a critical component and that
influenced the process of professional development in our districts and schools.
Progress to Date
The New York City public school system published the first set of standards in 1997: for English
language arts, Spanish language arts, and English as a second language. In 1998, we produced the
Judith A. Rizzo is Executive Director of the James B. Hunt Institute for Educational Leadership and Policy in Chapel Hill,
NC. Until 2002, Rizzo was Deputy Chancellor for Instruction for the New York City public schools. Previously, Rizzo
served as a K12 teacher, principal, and central administrator, and as adjunct professor at Columbia University. Rizzo was
a member of the blue-ribbon New York City Commission on Mathematics Education, chaired by Matthew Goldstein,
Chancellor of The City University of New York.
201
202
Mathematical Process
Geometry; Measurement
Algebra
Representation
Gather data;
Other suggested activities include asking students to design mathematical models of physical phenomena such as those used in
science studies; design and plan a physical structure, including a
presentation or report on how the project was carried out; or
manage and plan an event with cost estimates, supplies, and
scheduling.
Each section of the performance standards is followed by student
work samples demonstrating the activities and including teacher
commentary assessing the degree to which the student work meets
This section also includes suggested extension activities that provide students with real-world, practical applications of mathematics.
203
Challenges Ahead
A nationally validated mathematics curriculum does not teach
students, nor does a standards document. Common sense supported by research tells us that the single most important factor
affecting student learning is a well-qualified teacher.
204
The major challenge we face in New York City, along with many
urban school systems throughout the country, is the dearth of qualified mathematics teachers. We simply do not have enough welltrained teachers of mathematics to teach to the standards. One partial
response to this problem is quality professional development.
There are two broad categories of teachers with different professional development needs: those who are certified to teach mathematics but lack the skills and knowledge to teach to the new
standards and those, particularly at the elementary and middle
school levels, who do not have a solid base of content knowledge
in mathematics.
A one-size-fits-all approach to professional development will not
meet the needs of these two categories of teachers. Clearly, professional development as we generally understand it will not suffice for the second category. These teachers need to have more
in-depth training similar to course work rather than traditional
workshops in mathematics pedagogy. Even the best instructional
strategies are virtually useless in the hands of teachers lacking
content knowledge.
Even within the first category, there is a continuum of professional
development needs. Some of the teachers in this category are
novice teachers. Like other school systems, New York City has
begun to recruit teachers who are changing careers from mathematics-related fields. These novice teachers require very different
professional development opportunities from veteran teachers.
The challenge is to provide professional development opportunities
that are tailored to the needs of individual teachers but that also reflect
what we know about the learning needs of the students in their
particular schools and districts. The concept of differentiated approaches to teaching students is now commonplace; it should be
applied to adult learning as well. We do have some tools in New York
City to assist us in determining individual teacher needs and they are,
coincidentally, the same tools that we use to diagnose student needs:
the item analyses from our assessment results.
Conclusion
The New York City public schools have clearly defined content and
performance standards and a variety of instruments to support implementation at the district and school levels. We believe that our
standards and supporting materials begin to address the issue of quantitative literacy, but we need to continue to work on increasing and
deepening quantitative literacy experiences for students.
We have come a long way toward improving the alignment of our
assessments with our standards and collecting good data from
those assessments. We are putting into operation the recommendations of a mathematics commission formed last year to improve
our programs and support systems. We have begun to co-design
and co-teach mathematics courses for staff in collaboration with
some of our universities.
We began to generate item analyses a few years ago to assist teachers in identifying the learning needs of their students. The results
were intended to provide teachers with useful data that would
inform their instructional planning and help them address the
needs of individual and small groups of students. This information was intended to be used in completing report cards, progress
reports, and student portfolios and in parent conferences, and as
an integral ingredient in making promotion decisions.
1. http://www.wnycenet.edu/dis/standards/Math/index.html.
Notes
Creating Networks as a
Vehicle for Change
SUSAN L. GANTER
A critical feature of the national movement for quantitative literacy (QL) is the development of a
strong networking component supported by common vision and direction. To this end, much can
be learned from the role of networking in the national movement during the past two decades to
change the undergraduate curriculum in mathematics. In particular, aspects of calculus reform and
subsequent efforts to improve other areas of undergraduate mathematics can be helpful in assessing
possible strategies for improving quantitative literacy.
The use of alternative learning environments, such as computer laboratory experiences, discovery learning, and technical writing; and
An emphasis on a variety of student skills, such as computer use, the use of applications, and a
focus on conceptual understanding (Ganter 2001).
When reviewing data by institution type, the results are fairly consistent with those from the total
population. A few exceptions are worth noting, however:
Research and comprehensive universities were more likely to involve high schools in their efforts
than liberal arts and two-year colleges. In fact, only 36.3 percent of the two-year colleges with
NSF-funded calculus projects reported the involvement of high schools, while 57.8 percent of
the comprehensive universities and 49.3 percent of the research institutions reported such
involvement.
Minority student involvement was significantly more prevalent at two-year colleges than at any
other type of institution. This was perhaps because of the different composition of the student
population at two-year colleges, but is certainly a noteworthy observation.
Susan L. Ganter is Associate Professor of Mathematical Sciences at Clemson University. Earlier, Ganter served as Director
of the Program for Institutional Change at the American Association for Higher Education and as Senior Research Fellow
of the American Educational Research Association at the National Science Foundation. Ganter is Chair of the Committee
on Curriculum Renewal and the First Two Years of the Mathematical Association of America and editor of two recently
published books analyzing the impact of undergraduate mathematics reform.
205
206
Ongoing cooperation and collaboration with other disciplines to work on curriculum development;
Emphasis on developing analytical thinking and careful reasoning in all courses and for all students; and
An important aspect of the new recommendations is strong encouragement to mathematics departments to work with students
of a broad range of abilities. Increasing the number of students
with quantitative experiences holds promise for increasing the
overall quality of our scientific workforce and creating a general
appreciation for the importance of QL.
207
Increased recognition of the need for (and often actual) conversations with faculty in other disciplines;
208
aminations, activities, laboratories, projects, readings, publications, and examples of student work. Such a collection of materials
is very important to the work of NNN, serving as a resource for
outreach efforts and as a means of teaching interested individuals
and organizations about QL. In addition, the QL Resource Library soon will include a database of individuals, projects, and
institutions involved in the development of QL curricula.
By focusing on different aspects of policy, practice, professional
development, dissemination, and assessment, the National Numeracy Network will provide a catalyst for quantitative literacy,
especially in grades 10 to 14. Quantitative Literacy programs participating in the network already are working with organizations
that can directly influence a wider audience to create public pressure for QL. NNN institutions and organizations are developing
QL course materials and programs to share through professional
development opportunities, the QL Resource Library, and the QL
Web site (www.woodrow.org/nced/quantitative-literacy.html).
Through networking, QL education is becoming a reality at many
institutions.
References
Bookman, J., and C. P. Friedman. 1998. Student Attitudes and Calculus Reform. School Science and Mathematics (March): 11722.
Buccino, Al. 2000. Politics and Professional Beliefs in Evaluation: The
Case of Calculus Renewal. In Calculus Renewal: Issues for Undergraduate Mathematics Education in the Next Decade, edited by S. L.
Ganter, 121 45. New York, NY: Kluwer Academic/Plenum.
Dossey, John, ed. 1998. Confronting the Core Curriculum: Considering
Change in the Undergraduate Mathematics Major. MAA Notes No.
45. Washington, DC: Mathematical Association of America.
Eiseman, J. W., J. S. Fairweather, S. Rosenblum, and E. Britton. 1996.
Evaluation of the Division of Undergraduate Educations Course and
Curriculum Development Program: Case Study Summaries. (Prepared
under contract for the National Science Foundation.) Andover,
MA: The Network, Inc.
Ganter, Susan L. 2001. Changing Calculus: A Report on Evaluation Efforts
and National Impact from 1988 to 1998. MAA Notes No. 56. Washington, DC: Mathematical Association of America.
Ganter, Susan L., and William Barker, eds. 2002. A Collective Vision:
Voices of the Partner Disciplines. MAA Notes Series. Washington,
DC: Mathematical Association of America.
Ganter, Susan L., and M. R. Jiroutek. 2000. The Need for Evaluation in
the Calculus Reform Movement: A Comparison of Two Calculus
Teaching Methods. In Research in Collegiate Mathematics Education, IV, edited by E. Dubinsky, A. Schoenfeld, and J. Kaput, 42
62. Providence, RI: American Mathematical Society.
Keynes, Harvey B., Andrea M. Olson, D. J. OLoughlin, and D. Shaw.
2000. Redesigning the Calculus Sequence at a Research University:
209
211
212
syllabus dominated by four content areas: geometry (the education of vision); computation (both approximate and exact, including its relation to reason); statistics (stochastic literacy), and
computers (especially data structures and algorithms). (See pp.
223225.)
A. Geoffrey Howson, also a former secretary of ICMI, in What
Mathematics for All?, takes on what he and many others see as a
disastrous decline in the mathematical competence of British
school-leavers. He attributes this undisputed decline to many
causes, among which is the piece of tape curricular philosophy
in which all students study (snip off ) a certain length of a subject
(a piece of tape) whose courses and examinations are designed for
a university goal they never reach. Howson suggests that for most,
a (QL-like) curriculum deliberately designed to focus on the
mathematics of citizenship, culture, personal finance, health, . . .
would yield greater success. (See pp. 227230.)
Mieke van Groenestijn of the Netherlands writes about the ALL
literacy assessment project, focusing in particular on the ALL
characterization of numerate behavior in adults. Using ALLs
rather detailed description as a foundation, van Groenestijn examines the problem of educating adults for numerate behavior,
which is far different from passive (or worse, inert) knowledge.
She notes that because adults learn principally through action, a
predisposition to numerate behavior can best be learned in reallife situations. (See pp. 231236.)
Ubiratan DAmbrosio of Brazil, a former vice president of ICMI,
takes note of the political and cultural roles played by mathematics
in all countries and at all ages, especially the recent role of data
control and management as a tool for excluding cultures of the
periphery. DAmbrosio, who years ago introduced ethnomathematics as a way to restore cultural dignity in societies whose
mathematics was invisible in school, here advocates a three-part
endeavor he calls literacy, numeracy, and technocracy as a
means of providing access to full citizenship. (See pp. 237240.)
These glimpses of how mathematics educators in other nations are
coming to terms with the new demands of numeracy, mathematics, and citizenship open a window on approaches that move well
beyond those normally considered in U.S. curriculum discussions. In addition, by revealing great differences in fundamental
assumptions and objectives concerning mathematics education,
they suggest important limitations on the inferences that can
safely be drawn from comparative international assessments. To
the degree that numeracy and mathematics are important features
of our culture, differences in national traditions will necessarily
create significant differences in both the objectives and outcomes
of mathematics education.
References
Cockcroft, Wilfred H. 1982. Mathematics Counts. London: Her Majestys Stationery Office.
Gal, Iddo, Mieke van Groenestijn, Myrna Manly, Mary Jane Schmitt,
and Dave Tout. 1999. Numeracy Framework for the International
Adult Literacy and Lifeskills Survey (ALL). Ottawa, Canada: Statistics
Canada.
Kahane, Jean-Pierre, et al. 2002. Commission de reflexion sur
lenseignement des mathematiques: Presentation des rapports et recommandations. Retrieved January 25, 2002, at: http://smf.emath.fr/
Enseignements/CommissionKahane/RapportsCommissionKahane.
pdf.
213
Mogens Niss is Professor of Mathematics and Mathematics Education at the innovative Roskilde University in Denmark at
which studies are based on project work. From 1987 to 1999, Niss served as a member of the Executive Committee of the
International Commission on Mathematical Instruction (ICMI), the last eight years as Secretary. He is currently Chair of
the International Program Committee for ICME-10, a member of the Mathematics Expert Group for OECDs PISA
project, and director of a Danish national project on mathematics curriculum.
215
216
The main reason I prefer mathematical literacy is that the broadness of the term mathematical captures better than the somewhat narrower term quantitative what we actually seem to be
after, for instance, when providing examples. Of course, we could
argue on the basis of the history and epistemology of mathematics
that many aspects of those mathematical topics that are of particular importance to real life, such as geometry, functions, probability, and mathematical statistics, among others, were in fact
arithmetised in the nineteenth and twentieth centuries, so that
we are not restricting the animal greatly by referring to it as quantitative literacy rather than as mathematical literacy. To see that,
however, a person has to possess a fairly solid knowledge of modern mathematics and its genesis, and that is most certainly a prerequisite that we cannot and should not expect of all those with
whom we want to be in dialogue.
Now, how is mathematical literacy related to mathematical
knowledge and skills? Evidently, that depends on what we mean
by mathematics. If we define mathematics in a restrictive way, as a
pure, theoretical scientific disciplinewhether perceived as a unified, structurally defined discipline or as a compound consisting of
a number of subdisciplines such as algebra, geometry, analysis,
topology, probability, etc.it is quite clear that mathematical
literacy cannot be reduced to mathematical knowledge and skills.
Such knowledge and skills are necessary prerequisites to mathematical literacy but they are not sufficient.
This is not the only way to define mathematics, however. We may
adopt a broaderpartly sociological, partly epistemological
perspective and perceive mathematics as a field possessing a fivefold nature: as a pure, fundamental science; as an applied science;
as a system of tools for societal and technological practice (cultural techniques); as an educational subject; and as a field of
aesthetics (Niss 1994). Here, being a pure, fundamental science is
just one of five natures of mathematics. If this is how we see
mathematics, the mastery of mathematics goes far beyond the
ability to operate within the theoretical edifice of purely mathematical topics. And then, I submit, mathematical literacy is more
or less the same as the mastery of mathematics. By no means,
however, does this imply that mathematical literacy can or should
be cultivated only in classrooms with the label mathematics on
their doors. There are hosts of other important sources and platforms for the fostering of mathematical literacy, including other
subjects in schools and universities.
All this leaves us with a choice between two different strategies.
Either we accept a restrictive definition of mathematics as being a
pure, fundamental science and then establish mathematical literacy as something else, either a cross-curricular ether or a new
subject. Or we insist (as I do) on perceiving mathematics as a
multi-natured field of endeavor and activity. If we agree to use
such a perception to define the subject to be taught and learned,
Mathematical Literacy
and Democracy
Traditionally, we tend to see the role of mathematical literacy in
the shaping and maintenance of democracy as being to equip
citizens with the prerequisites needed to involve themselves in
issues of immediate societal significance. Such issues could be
political, economic, or environmental, or they could deal with
infra-structure, transportation, population forecasts, choosing locations for schools or sports facilities, and so forth. They also
could deal with matters closer to the individual, such as wages and
salaries, rents and mortgages, child care, insurance and pension
schemes, housing and building regulations, bank rates and
charges, etc.
Although all this is indeed essential to life in a democratic society,
I believe that we should not confine the notion of democracy, or
the role of mathematical literacy in democracy, to matters such as
the ones just outlined. For democracy to prosper and flourish, we
need citizens who not only are able to seek and judge information,
to take a stance, to make a decision, and to act in such contexts.
Democracy also needs citizens who can come to grips with how
mankind perceives and understands the carrying constructions of
the world, i.e., nature, society, culture, and technology, and who
have insight into the foundation and justification of those perceptions and that understanding. It is a problem for democracy if
large groups of people are unable to distinguish between astronomy and astrology, between scientific medicine and crystal healing, between psychology and spiritism, between descriptive and
normative statements, between facts and hypotheses, between exactness and approximation, or do not know the beginnings and
the ends of rationality, and so forth and so on. The ability to
navigate in such waters in a thoughtful, knowledgeable, and reflective way has sometimes been termed liberating literacy or
popular enlightenment. As mathematical literacy often is at the
center of the ways in which mankind perceives and understands
the world, mathematical literacy is also an essential component in
217
Traditionally, in Denmark and in many other countries, a mathematics curriculum is specified by means of three types of components:
Validly describe development and progression within and between mathematics curricula;
1.
2.
3.
The general idea is to deal with this task by identifying and making use of a number of overarching mathematical competencies.
This gave the stimulus (and the most important part of the brief)
for the Danish KOM project, directed by the author of this paper.
KOM stands for Kompetencer Og Matematiklring, Danish
for Competencies and Mathematical Learning. (More information is available at http://imfufa.ruc.dk/kom. By the end of August 2002 an English version of the full report of the project can be
found at this site.) The project was established jointly by the
Ministry of Education and the National Council for Science Education. It is not a research project but a development project to
pave the way for fundamental curriculum reform in Denmark,
from kindergarten to university. In fact it is a spearhead project in
that similar projects are now being undertaken in Danish, physics
and chemistry, and foreign languages; the natural sciences are
soon to be addressed.
More specifically, the project is intended to provide inspiration by
discussing and analyzing the possibility of dealing with the task
just presented by means of the notion of mathematical competencies, and accordingly to propose measures and guidelines for cur-
218
riculum reform. It is not the intention that the project itself shall
propose detailed new curricula at all the different educational
levels it addresses. Specific curriculum implementation is up to
the curriculum authorities responsible for each of these levels;
however, it is more than likely that the collaborators in the project
will be asked to take part in that implementation in the sectors in
which they work.
Mathematical Competencies
and Insights
2.
3.
4.
Decoding existing models, i.e., translating and interpreting model elements in terms of the reality modelled; and
5.
6.
Understanding and utilizing the relations between different representations of the same entity, including knowing
about their relative strengths and limitations; and
7.
8.
Decoding and interpreting symbolic and formal mathematical language and understanding its relations to natural language;
219
Understanding the nature and rules of formal mathematical systems (both syntax and semantics);
Making use of aids and tools (including information technology), such as:
The competencies and insights can be employed both for normative purposes, with respect to specification of a curriculum or of
desired outcomes of student learning, and for descriptive purposes
to describe and characterize actual teaching practice or actual student learning, or to compare curricula, and so forth.
The first four competencies are the ones involved in asking and
answering questions about, within, and by means of mathematics,
whereas the last four are the ones that pertain to understanding
and using mathematical language and tools. It should be kept in
mind, however, that these eight competencies are meant neither
to establish a partitioning of mathematical competence into dis-
In this paper there is room only to describe the core ideas of the
KOM project. It is also a key intention of the project to specify in
some detail how these competencies will actually be developed at
different educational levels in schools and universities, to specify
and characterize the relationships between competencies and
mathematics subject matter at different levels, and to devise ways
220
to validly and reliably assess students possession of the mathematical competencies in a manner that allows us to describe and
characterize development and progression in those competencies.
Skovsmose, Ole. 1994. Toward a Philosophy for critical Mathematics Education. Dordrecht: Kluwer Academic Publishers.
References
Niss, Mogens. 1994. Mathematics and Society. In Didactics of Mathematics as a Scientific Discipline, edited by R. Biehler, R. Scholz, R.
Straesser, and B. Winkelmann, 36778. Dordrecht: Kluwer Academic Publishers.
Steen, Lynn Arthur, et al. 2001. The Case for Quantitative Literacy. In
Mathematics and Democracy: The Case for Quantitative Literacy, edited by Lynn Arthur Steen, 122. Princeton, NJ: National Council
on Education and the Disciplines.
Geometry
Teaching geometry from elementary to high school levels is still necessary today. We first showed the
importance of geometry in order to grasp space, to develop the vision of space, a vision that plays
an essential role in our image-oriented society. Second, the report emphasized the fact that geometry
is a fundamental subject for the learning of reasoning. Finally, the report recalled how important
geometry is in the training of all scientists (technicians, engineers, researchers, and teachers).
We proposed to develop space geometry as an education in vision. For plane geometry, we suggested
reinforcing the use of elementary invariants (such as angle and area) and reintroducing criteria for
congruence and similarity of triangles. To avoid too dogmatic a teaching approach, the report also
proposed to favor open problems (research on geometric loci and construction problems) and to
make room for some rich geometries. For instance, at the end of high school, circular geometry
could be taught together with complex numbers.
As regards methods, the commission focused on two main goals: teaching pupils and students to see
and to think geometrically and teaching them to reason. Among the other important suggestions
included in the report was to establish strong links with other disciplines, in particular with the
sciences.
Michel Merle is Professor of Mathematics at the University of Nice in France. His mathematical research is mainly focused
on algebraic geometry, including applications of this theory to computer vision. Merle is currently a member of the national
Commission de reflexion sur lenseignement des Mathematiques and of the Conseil national des programmes, a
committee involved in the elaboration of the curriculum for French primary and secondary schools.
221
222
Computation
The commission tried to intertwine epistemological and didactical issues by questioning the nature and role of computation in the
mathematical sciences and their evolution, the cultural and educational representations of the theme, and the current curriculum
from elementary to university levels. In the epistemological dimension of the report, we were especially sensitive to:
The increasing diversity of the objects and practices that computations involve;
The relationships between exact and approximate computation, and between computation and reasoning; and
Statistics
Developing stochastic literacy for professional or individual purposes is widely understood to be part of obligatory education;
nonetheless, it is surrounded by controversies and many questions
set out in the commissions report are under investigation in many
European countries.
Computers
In 1992, UNESCO published a report on The Influence of
Computers and Informatics on Mathematics and Its Teaching
(Cornu and Ralston 1992). (Some of the contributors to this report
were participants in the Forum.) Reflecting ideas from this report,
our commission examined three issues created by computers:
223
References
Cornu, Bernard and Anthony Ralston, eds. 1992. The Influence of
Computers and Informatics on Mathematics and Its Teaching. 2nd
edition. Science and Technology Education No. 44. Paris:
UNESCO.
Kahane, Jean-Pierre, et al. 2002. Lenseignement des sciences mathematiques (Rapport au ministre de lEducation nationale de la
Commission de reflexion sur lenseignement des mathematiques.) CNDPOdile Jacob: Paris. Retrieved January 25, 2002,
at:
http://smf.emath.fr/Enseignements/CommissionKahane/
RapportsCommissionKahane.pdf .
The teaching of mathematics in English secondary schools is far from satisfactory. Concerns arise
regarding how the needs of different types of students are being met, the standards attained, the
manner in which students attainments are assessed, and the provision of adequately qualified
teachers. In the longer paper from which this extract is adapted (Howson 2002), we look at each of
these aspects and propose possible ways ahead. Here we focus on one issuethe curricular implications of mathematics for all.
The coming of comprehensive education in many western countries raised important issues concerning curriculum design. Just what did mathematics for all mean? Nowadays the phrase often is
taken to mean that mathematics was not to be found in the old English secondary modern
schoolsthat only arithmetic was taught in them. Although this view is perhaps a caricature, there
were clear distinctions between the aims of mathematics teaching in the grammar schools and in the
bulk of technical and secondary modern schools. Put baldly, in the former, students were prepared
for further academic study, in other schools, for taking their place within society. This was perhaps
most clearly typified in the teaching of geometry. For the academic students there was Euclid-style
theorem and proof, for the others practical geometry, the classification of shapes and solids,
mensuration and, to varying extents (depending on the nature of the school), elementary scale and
technical drawing.
The introduction in 1965 of a Certificate of Secondary Education (CSE) intended for those within
the fortieth to eightieth percentiles of the ability range was not intended to threaten this dichotomy
of aims. The comprehensive schools introduced just after that time, however, contained all types of
students and their early differentiation would have defeated the aims of comprehensive schooling
and produced only multilateral schoolsthose in which students are separated into different curricular streams within the same school. Yet postponing differentiation meant postponing streaming
for CSE or the traditional GCE O-level (intended for the top 25 percent of students) and this, in
turn, resulted in the former becoming a watered-down version of the latter.
A. Geoffrey Howson is Professor Emeritus of Mathematical Curriculum Studies at the University of Southhampton in
England. A former Dean of Mathematical Studies at the university, Howson served from 1982 through 1990 as secretary
of the International Commission on Mathematical Instruction (ICMI). He is author of several books on mathematics and
mathematics education, most recently the TIMSS monograph Mathematics Textbooks: A Comparative Study of Grade 8
Texts.
225
226
The Cockcroft report (Cockcroft 1982), the outcome of a government committee established to counter criticisms of current
school mathematics standards, proposed to rectify this by a bottom-up approach to curriculum design that concentrated first on
the needs of the lower attainers. Accordingly, it drew up a foundation list: what the report saw as a basic mathematical kit for all
school leavers. That, in its turn, was supplanted in the late 1980s
by the National Curriculum, which appeared to be designed not
so much to meet the needs of students but rather those of an
untried and ambitious assessment scheme. All students now were
to follow the same curriculum, but at their own rates. A hierarchy
of levels was defined that was to be followed by students at varying
speeds, but the question of depth of treatment within a level was
ignored. No one, apart from the highest attainers, had any fixed
curriculum goal: students had simply to swallow as much of the
curriculum as they could before the age of 16. No heed was taken
of the wise words of the 1947 Hamilton Fyfe report:
Whatever be the values of the subject carried to its full term
in university study, they cannot be achieved for the child of
16 by simply snipping off a certain length of the subject like
a piece of tape. . . . Every course must have its own unity and
completeness and a proper realism requires that content and
methods alike be so regulated as to reach their objective
within the time available (Fyfe 1947).
The piece of tape mentality still persists and, for example, forces
weaker students to learn algebraic techniques that they will never
develop into usable knowledge. Of course, such students are no
longer being denied the opportunity to learn algebra, but instead are simply forced to learn techniques that might conceivably
(but with a fairly low probability) lead to something more useful
and valuable. It is difficult to see exactly what the aims of the
present curriculum are. There is an attempt to please everyone and
do everything, at the expense of a focus on clear aims and the
provision of sound and secure learning. That after 11 years of
compulsory mathematics it should be felt necessary to institute
post-16 courses and tests in key skills for sixth-formers (16- to
18-year-olds) illustrates the problems.
Such curricular considerations led recently to the publication of a
collection of essays entitled Why Learn Maths?, edited by two
philosophers of education, S. Bramall and J. White, which questioned the arguments put forward in the defense of teaching
mathematics and its status as a compulsory subject within the
national pre-16 curriculum (Bramall and White 2000). Although
reported in newspapers as a polemic, the monograph contained
contributions that genuinely merit consideration. The present
mathematics curriculum cannot be justified solely by the repetition of pious cliches or such foolishness as the National Curriculums claims of mathematics promoting spiritual development
through . . . helping pupils obtain an insight into the infinite, or
the nature of a pilot study, however, and did not test all aspects of
mathematical literacy.)
Another, significant offering is a report, Mathematics and Democracy: The Case for Quantitative Literacy, published by the National
Council on Education and the Disciplines (NCED) (Steen 2001),
that seeks a complete reorientation of the traditional U.S. school
mathematics syllabus. This report distinguishes between what it
terms quantitative literacy, which stresses the use of those mathematical and logical tools needed to solve common problems (e.g.,
percentages and mensuration), and mathematical literacy, which
emphasizes the traditional tools and vocabulary of mathematics
(e.g., formal algebra and, later, calculus).
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Paradoxically, students are being trained to perform those operations that can now be dealt with using suitably chosen software,
but all too often students share the computers inability to analyze
a problem and to reason. I wonder to what extent students have
been empowered to use the mathematics they have been taught in
new contexts, rather than merely to answer stock examination
questions on it. There would, then, appear to be a requirement in
many countries for a clearer definition of goals for school mathematics linked more closely to the differing needs and aspirations of
students.
References
Bramall, S., and J. White, eds. 2000. Why Learn Maths? London: Institute
of Education.
Cockcroft, Wilfred H. 1982. Mathematics Counts. London: Her Majestys Stationery Office.
Mieke van Groenestijn is a researcher at Utrecht University of Professional Education in the Netherlands. Her main research
interests concern the mathematical knowledge and problem- solving strategies of adults in Adult Basic Education (ABE)
programs. She is a member of the international numeracy team of the Adult Literacy and Lifeskills survey (ALL), a trustee
of Adults Learning Mathematics (ALM) and a board member of the Dutch National Organization for the Development of
Mathematics Education.
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2.
Problem transparency, varying from obvious/explicit to embedded/hidden. How difficult is it to identify the mathematical problem and decide what action to take? How much
literacy proficiency is required?
Plausibility of distractors, from no distractors to several distractors. How many other pieces of mathematical information are present? Is all the necessary information there?
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3.
4.
5.
232
Based on the five facets and five complexity factors, the ALL team
set up a grid and developed a bank of about 120 items on five levels
for the numeracy domain of the ALL survey. Each item received
an individual identity. The ALL team thought that this way of
working also could serve as a framework for the development of
mathematical content and actions for numeracy programs in educational settings (van Groenestijn 2002).
Implementation of Numeracy
in Education
For the implementation of numeracy in educational programs,
however, it is not sufficient only to determine mathematical content and actions. It also is necessary to look for components that
help develop numerate behavior, for now and in the future. In my
own study of numeracy in adult basic education (van Groenestijn
2002) I started from the ALL definition but added a second part to
include attention to the future:
Numeracy encompasses the knowledge and skills required to
effectively manage mathematical demands in personal, societal and work situations, in combination with the ability to
accommodate and adjust flexibly to new demands in a continuously rapidly changing society that is highly dominated
by quantitative information and technology (p.37).
To make this definition operational for the implementation of
numeracy in educational settings, four components were identified:
Problem-solving skills to identify, analyze, and structure problems, plan steps for action, select appropriate actions, actually
handle problems, and make decisions; and
Reflection skills to be able to control the situation, check computations, evaluate decisions, and come to contextual judgments.
Such management skills often are assumed to evolve spontaneously in the course of life. We argue that it is necessary to pay
explicit attention to teaching these skills in educational settings.
Training enables adults to develop appropriate skills for different
types of mathematical situations.
The same can be said concerning the third component, developing skills for processing new information in real-life situations.
The way students learn in school differs from the way in which
adults acquire and process new information in out-of-school situations, independently from teachers. Adults almost always process new information in the course of action (Greeno 1999). For
this, people need to learn to:
Conclusion
Becoming numerate is as essential as becoming literate for all
citizens in all nations. The case for quantitative literacy, or numeracy, or mathematical literacy, by whatever name, therefore
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Notes
1. The Young Adult Literacy Skills (YALS) study was conducted in
1986. The National Adult Literacy Survey (NALS) followed that
study in 1992. The International Adult Literacy Study (IALS) in
1996, was a follow-up of the NALS. In the first phase of the IALS
(1994, 1996), adults from 14 countries were tested based on methodology that combined household survey research and educational testing. A second cohort of 10 countries conducted surveys in 1998 and
1999 (the Second International Adult Literary Survey, or SIALS). In
1994 and 1996, participating countries were Canada, France, Germany, Ireland, the Netherlands, Sweden, Switzerland, and the United
States; in 1996, Australia, the Flemish community in Belgium, Great
Britain, New Zealand, and Northern Ireland participated. The second
full round of data collection in 1998 and 1999 (SIALS) included
Chile, the Czech Republic, Denmark, Finland, Hungary, Italy, Malaysia, Norway, Slovenia, and Switzerland.
2. The OECD Programme for International Student Assessment (PISA
2000) was an international assessment of 15-year-olds that looked at
how well they were prepared for life beyond school and was fielded in
32 countries. Four types of skills were assessed: skills and knowledge
that prepare students for life and lifelong learning, reading literacy,
mathematical literacy, and science literacy.
3. The international Adult Literacy and Lifeskills (ALL) survey is the
follow-up to the IALS and is planned for the years 2002 and 2003.
4. The ALL study is being organized by the National Center for Education Statistics (NCES) and Statistics Canada. Participating countries
in the ALL pilot study are Argentina, Belgium, Bermuda, Bolivia,
Brazil, Canada, Costa Rica, Italy, Luxembourg, Mexico, the Netherlands, Norway, Spain, Switzerland, the United States, and Venezuela.
5. The international ALL numeracy team is comprised of Yvan Clermont, Statistics Canada, Montreal, project manager; Iddo Gal, University of Haifa, Israel; Mieke van Groenestijn, Utrecht University of
Professional Education, Utrecht; Myrna Manly, enjoying her retire-
234
References
Cockcroft, Wilfred H. 1982. Mathematics Counts. London: Her Majestys Stationery Office.
Dossey, John A. 1997. National Indicators of Quantitative Literacy. In
Why Numbers Count: Quantitative Literacy for Tomorrows America,
edited by Lynn Arthur Steen, 4559. New York, NY: College Entrance Examination Board.
Gal, Iddo, Mieke van Groenestijn, Myrna Manly, Mary Jane Schmitt,
and Dave Tout. 1999. Numeracy Framework for the International
Adult Literacy and Lifeskills Survey (ALL). Ottawa, Canada: Statistics
Canada, 1999. Retrieved January 25, 2002, at http://nces.ed.gov./
surveys/all.
Manly, Myrna, Dave Tout, Mieke van Groenestijn, and Yvan Clermont.
2001. What Makes One Numeracy Task More Difficult Than
Another? In Adults Learning Mathematics: A Conversation between
Researchers and Practitioners, edited by Mary Jane Schmitt and
Kathy Safford-Ramus. Proceedings of the 7th International Conference of Adults Learning Mathematics (ALM-7), Medford, MA.
Cambridge, MA: NCSAL, Harvard University Graduate School of
Education.
Manly, Myrna, and Dave Tout. 2001. Numeracy in the Adult Literacy
and Lifeskills Project. In Adult and Lifelong Education in Mathematics, edited by Gail FitzSimons, John ODonoghue, and Diana
Coben. Melbourne, Australia: Language Australia.
Organization for Economic Cooperation and Development (OECD).
1997. Literacy Skills for the Knowledge Society: Further Results from
the International Adult Literacy Survey. Ottowa, Canada: Statistics
Canada.
Greeno, James G., Penelope Eckert, Susan U. Stucky, Patricia Sachs, and
Etienne Wenger. 1999. Learning in and for Participation in Society. In How Adults Learn. Washington, DC: Organization for Economic Cooperation and Development and U.S. Department of
Education.
Groenestijn, Mieke van. 2002. A Gateway to Numeracy. A Study of Numeracy in Adult Basic Education. Utrecht, Netherlands: CD-Press.
Steen, Lynn Arthur, ed. 2001. Mathematics and Democracy: The Case for
Quantitative Literacy. Princeton, NJ: National Council on Education and Disciplines.
Ubiratan DAmbrosio is Emeritus Professor of Mathematics at the State University of Campinas/UNICAMP in Sao Paulo,
Brazil, where he served as Pro-Rector for University Development from 1982 to 1990. DAmbrosio has served as President
of the Inter-American Committee of Mathematics Education (IACME), Vice-President of the International Commission
on Mathematics Instruction (ICMI), and as a Member of the Council of the Pugwash Conferences on Science and World
Affairs (the organization that was awarded the Nobel Peace Prize in 1995).
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To build a civilization that rejects inequity, arrogance, and bigotry, education must give special attention to the redemption of
peoples that have been for a long time subordinated and must give
priority to the empowerment of the excluded sectors of societies.
A consequence of this program for a new curriculum is synthesized in my proposal of three strands in curricular organization:
literacy, matheracy, and technoracy (DAmbrosio 1999b). The
three provide, in a critical way, the communicative, analytical, and
technological instruments necessary for life in the twenty-first
century. Let me discuss each one.
Literacy is the capability of processing information, such as the use
of written and spoken language, of signs and gestures, of codes and
numbers. Clearly, reading has a new meaning today. We have to
read a movie or a TV program. It is common to listen to a concert
with a new reading of Chopin. Also, socially, the concept of literacy has gone through many changes. Nowadays, reading includes
also the competency of numeracy, the interpretation of graphs
and tables, and other ways of informing the individual. Reading
even includes understanding the condensed language of codes.
These competencies have much more to do with screens and
buttons than with pencil and paper. There is no way to reverse this
trend, just as there has been no successful censorship to prevent
people from having access to books in the past 500 years. Getting
information through the new media supersedes the use of pencil
and paper and numeracy is achieved with calculators. But, if dealing with numbers is part of modern literacy, where has mathematics gone?
Matheracy is the capability of inferring, proposing hypotheses, and
drawing conclusions from data. It is a first step toward an intellectual posture, which is almost completely absent in our school
systems. Regrettably, even conceding that problem solving, modeling, and projects can be seen in some mathematics classrooms,
the main importance is usually given to numeracy, or the manipulation of numbers and operations. Matheracy is closer to the way
mathematics was present both in classical Greece and in indigenous cultures. The concern was not with counting and measuring
but with divination and philosophy. Matheracy, this deeper reflection about man and society, should not be restricted to the
elite, as it has been in the past.
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References
DAmbrosio, Ubiratan. 1998. Mathematics and Peace: Our Responsibilities. Zentralblatt fur Didaktik der Mathematik/ZDM, 30(3):
6773.
DAmbrosio, Ubiratan. 1999a. Ethnomathematics and its First International Congress. Zentralblatt fur Didaktik der Mathematik,
ZDM. 31(2): 50 53.
DAmbrosio, Ubiratan. 1999b. Literacy, Matheracy, and Technoracy: A
Trivium for Today. Mathematical Thinking and Learning, 1(2):
13153.
Identify and explore the right questions and take time doing this;
Have the right people at the table, those who bring a diversity of experiences and responsibilities
to the process of identifying, exploring, and implementing;
Take the kaleidoscopic perspective, recognizing that the work is to change the system, not tinker
at the edges;
Focus on getting something done, moving in a timely and expeditious fashion from discussing
to doing;
Jeanne L. Narum is Director of the Independent Colleges Office and the founding Director of Project Kaleidoscope
(PKAL), an informal national alliance working to strengthen undergraduate programs in mathematics, engineering, and
science. Educated as a musician, Narums prior experience includes administrative positions at Augsburg College (Vice
President for Advancement), Dickinson College (Director of Development), and St. Olaf College (Director of Government
and Foundation Relations).
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Talk about what works, not about what does not work. There
is not enough time to do both and opportunities will be lost if
attention is not given to solutions;
learning, students collaborate with one another and gain confidence that they can succeed, and institutions support such
communities of learners.
Learning is experiential, hands-on, and steeped in investigation from the very first courses for all students through capstone courses for science and mathematics majors.
How can we help geology majors at liberal arts colleges become more math literate without increasing the total number of mathematics courses we require them to take?
In a large, heterogeneous class, how do we include appropriate quantitative content that will not intimidate the mathphobic student but will challenge and not patronize the
math-able student?
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Are there gender-based differences in the effectiveness of different approaches to teaching quantitative reasoning?
One or two people who take responsibility to be the connectors, people with credibility in the communities of stakeholders and potential collaborators;
242
needed in the early nineteenth century but they are certainly very
different from those needed in the twenty-first century.
An affective component, so that people come to enjoy participating and have a sense of belonging.
We now come back full circle to the Conant text. I close with an
excerpt from an April 2000 report issued by the White House
Office of Scientific and Technology Policy (2000), Ensuring a
Strong U.S. Scientific, Technical, and Engineering Workforce in the
21st Century. It is a wonderful statistical analysis (lots of charts and
graphs) of present and future workforce needs based on demographics, school and college enrollments, and workplace opportunities. The report calls for greater nationwide attention to ensuring a strong workforce, but the most compelling line also best
describes why we are here, . . . . it is the fundamental responsibility
of a modern nation to develop the talent of all its citizens.
References
Greenspan, Alan. 2000. Testimony before Congressional Committee.
MacLeish, Archibald. 1960. Mr. Wilson and the Nations Need. In
Education in the Nations Service: A Series of Essays on American
Education Today. New York: Woodrow Wilson Foundation.
Office of Scientific and Technology Policy. 2000. Ensuring a Strong U.S.
Scientific, Technical, and Engineering Workforce in the 21st Century.
Washington, DC: U.S. Government Printing Office.
Project Kaleidoscope. 1991. What Works Building Natural Science Communities, Volume I. Washington, DC: Project Kaleidoscope.
Rita Colwell is Director of the National Science Foundation. Previously, Colwell was President of the University of
Maryland Biotechnology Institute. A member of the National Academy of Sciences, Colwell has served as President of the
American Association for the Advancement of Science (AAAS), of the American Society for Microbiology, of Sigma Xi, and
of the International Union of Microbiological Societies. Colwell is also a former member of the Mathematical Sciences
Education Board (MSEB).
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244
Quantitative literacy, just like English literacy or historical literacy, exists in degrees. If you asked historians what information
they would ideally want each American to know, they probably
would suggest topics critical to our nations future but not relevant
to our daily lives: the Whiskey Rebellion,1 Sewards Folly,2 the
Jethro Tull3 of circa 1701 Britain, as opposed to the Jethro Tull4
of circa 1971. And historians would want us to know more than
the necessary facts; they would say that we should know the historical context and the insight the events shed on the nature of
human experience.
When asked what information all Americans must know, however, historians would probably bring up subjects such as the
Constitutional Convention, or Standard Oil, or Brown v. Board of
Educationissues that are critical to understanding our present
society.
Ours would be a more effective, and perhaps more rational, society if all Americans felt the same fascination for the magic of
numbers and the elegance of graphic representations that we, as
scientists, do. The public, however, is most concerned with issues
affecting them daily, and it is the role of quantitative literacy in
our daily lives that must be understood. People are comfortable
using numbers in daily activities with which they are familiar
shopping, tracking sports statistics, even day-trading.
In schools, we likely can make daily quantitative activities a bridge
to higher levels of understanding. More may choose to elevate
their literacy, coming to appreciate what that master of quantitative representation, Edward Tufte, called the clear portrayal of
complexity. Not the complication of the simple; rather . . . the
revelation of the complex (Tufte 1983, epilogue).
So what are our standards for literacy in the United States? In
1988, Congress passed the Adult Education Amendments, mandating the U.S. Department of Education to define literacy and
measure the extent of literacy among Americans. The definition
eventually accepted by Congress characterizes literacy as an individuals ability to read, write, and speak in English and compute
and solve problems at levels of proficiency necessary to function
on the job and in society, to achieve ones goals, and to develop
ones knowledge and potential.
The Department of Educations first National Adult Literacy Survey was conducted in 1992. It questioned 26,000 Americans ages
16 and older and measured not just quantitative literacy but also
prose and document literacy. As we would expect, individuals
with less formal education dominated the lower levels. Of great
concern, minorities tended to have less formal education and were
overrepresented in the lower literacy levels.
Similar trends were observed in the Third International Mathematics and Science StudyRepeat (TIMSS-R) and the recent
National Assessment of Educational Progress reports on mathematics and science. In the TIMSS-R evaluation of the mathematics and science skills of eighth graders from around the world, the
United States ranked only about average in both mathematics and
science; however, students from disadvantaged minorities ranked
below average. Students from higher-income school districts
ranked on a par with their highest-ranking international counterparts.
Americans who are given access to excellent resources are, for the
most part, receiving an excellent education. In our country, literacy is most frequently linked to socioeconomic factors. Not all of
U.S. education is in crisis, but the unequal distribution of resources is a cause for great concern. For several years, the National
Science Foundation (NSF) has funded systemic reform initiatives
in both urban and rural school districts to improve overall science
and mathematics education. The results have been very encouraging.
Comprehensive and constructive assistance always works better
than berating education systems as a whole. Teachers are not the
root cause of all problems. We must recognize that there are great
educators out there for our young people. The problems that exist
are complex and the solutions are complex as well. Unequal distribution of resources and poor attitudes about mathematics
stretch across all age groups. Innovation in teaching should be
recognized and rewarded. Successful efforts to reach out to and
motivate students must be recognized and supported.
We all know that bringing quantitative literacy to our schools is
only one facet of a complex solution. We also must bring a recognition and, more important, an appreciation, of quantitative
knowledge to our daily lives. This is important particularly for
adults. People will seek out knowledge that directly affects them.
As proof, they are already gravitating to science topics on primetime TV. Shows produced by National Geographic, Discovery,
the Learning Channel, and others draw devoted audiences. NSF is
proud to support dynamic childrens shows such as The Magic
School Bus, Bill Nye the Science Guy, and Find Out Why, a
series coproduced with Walt Disney Television Animation for
broadcast between Saturday morning cartoons.
All these efforts recognize that everybody confronting a topic for
the first time has difficulty. As Ralph Waldo Emerson said, The
secret of education lies in respecting the pupil. Many audiences
come to the table with misconceptions and preconceptions, some
of which can be shocking but they need to be respected if we are
ever to reach them.
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Notes
1. Angered by a 1791 federal excise tax on whiskey, farmers in the western counties of Pennsylvania began attacking tax agents. On August 7,
1794, President George Washington issued a proclamation, calling
out the militias to respond. Thirteen thousand troops led by Washington and General Harry Lee, Robert E. Lees father, quelled the
uprising. This was the first use of the Militia Law of 1792, setting a
precedent for the use of the militia to execute the laws of the union,
246
References
Bernstein, Peter L. 1996. Against the Gods: The Remarkable Story of Risk.
New York, NY: John Wiley.
Tufte, Edward R. 1983. The Visual Display of Quantitative Information.
Cheshire, CT: Graphics Press.
Dening QL
Although we have no precise definition of QL, the case statement in Mathematics and Democracy:
The Case for Quantitative Literacy and the background essays contributed to this Forum give us a
rich, and not always consistent, set of characterizations and expressions of it. A common characterization seems to be this: QL is about knowledge and skills in use, so it is a kind of applied knowledge
that is typically illustrated in particular contexts. But these contexts are extremely diverse, and many
of them, if treated in more than a caricature fashion, are quite complex. This presents a challenge to
the design of curricula for QL. What is its focus? What is its disciplinary locus?
Voices at this Forum offered a very broad perspective. In our collective minds, QL appears to be
some sort of constellation of knowledge, skills, habits of mind, and dispositions that provide the
resources and capacity to deal with the quantitative aspects of understanding, making sense of,
participating in, and solving problems in the worlds that we inhabit, for example, the workplace, the
demands of responsible citizenship in a democracy, personal concerns, and cultural enrichment.
Urgency for QL arises primarily from the effects of technology, which exposes us to vastly more
quantitative information and data. Therefore, the tools of data analysis, statistics, and probabilistic
reasoning (in risk assessment, for example) are becoming increasingly important. Yet there is broad
agreement, with some evidence cited, that most adult Americans are substantially deficient in QL,
however it may be defined. This is viewed as a serious societal problem in several respects
economic (capacity of the workforce), political (functioning of a modern industrial democracy),
cultural (appreciation of the heritage and beauty of mathematics), and personal (capacity for a
responsible and productive life).
I agree with the views expressed that it is neither urgent, nor even necessarily productive, to attempt
to achieve a precise consensus definition of QL. At the same time, this is not an entirely benign
consideration. To illustrate, one speaker proposed that university mathematicians send a collective
letter to the College Board requesting more QL on the SAT and other examinations. Such a
recommendation, if implemented, is not immediately actionable by the College Board without an
operational interpretation of what QL should mean in that context, and that interpretation is open
Hyman Bass is the Roger Lyndon Collegiate Professor of Mathematics and Professor of Mathematics Education at the
University of Michigan. His mathematical research interests cover broad areas of algebra. A member of the National
Academy of Sciences, Bass is president of the American Mathematical Society and of the International Commission on
Mathematics Instruction (ICMI). A former chair of the Mathematical Sciences Education Board, Bass is currently helping
investigate the mathematical requirements involved in teaching mathematics at the elementary level.
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Educating for QL
Remedies to the problem of QL are generally assumed to be primarily the responsibility of the education system, principally in
grades 10 to 14. In fact, QL must be taught starting in the earliest
grades if we are to make any headway on this problem. Nonetheless, most of the discussion at this Forum centered on ideas about
QL in later grades. I note three recurrent themes:
1. The curriculum should include much more statistics and other
alternatives to the calculus trajectory that are focused more on data
analysis, modeling, etc.
This recommendation often has been accompanied by disparagement of the teaching of traditional mathematics topics. This reminds me of some of the debates about the teaching of computational algorithms. At root the objections were not to the skills and
concepts being taught but rather to the pedagogy, to the oppressive or obscure ways in which these topics have often been taught,
which the debaters could not see as distinct from the subject
matter. Although a full exposure to calculus may not be appropriate for a majority of students, algebra and geometry remain fundamental to all developed uses of mathematics.
2. Mathematics instruction should be contextualized and avoid the
abstraction associated with the traditional curriculum.
This common refrain of current reforms is more complex than
most of its advocates appreciate. One argument, which goes back
to John Dewey and others, is that learning best starts with experience, to provide both meaning and motivation for the more
general and structured ideas that will follow. Deweys notion differs in two respects from the above recommendation. First, it does
not eschew abstraction. Second, it speaks of the experience of the
learner, not of the eventual context of the application of the ideas,
which may be highly specialized and occur much later in adult
experience.
Another argument is that mathematics is best learned in the complex contexts in which it is most significantly used. This idea has a
certain appeal, provided that it is kept in balance. Authentic contexts are complex and idiosyncratic. Which contexts should we
choose for a curriculum? Their very complexity often buries the
mathematical ideas in other features so that, although the mathematical effects might be appreciated, there is limited opportunity
to learn the underlying mathematical principles.
The main danger here, therefore, is the impulse to convert a major
part of the curriculum to this form of instruction. The resulting
failure to learn general (abstract) principles then may, if neglected,
deprive the learner of the foundation necessary for recognizing
how the same mathematics witnessed in one context in fact applies
to many others.
Finally, contextualization is seen as providing early experience
with the very important process of mathematical modeling. This
is a laudable goal but it is often treated navely, in ways that violate
its own purpose. Serious modeling must treat both the context
and the mathematics with respect and integrity. Yet much contextualized curricular mathematics presents artificial caricatures of
contexts that beg credibility. Either many of their particular features, their ambiguities, and the need for interpretation are ignored in setting up the intended mathematics, which defeats the
point of the context, or else many of these features are attended to
and they obscure the mathematical objectives of the lesson. Good
contextualizing of mathematics is a high skill well beyond that of
many of its current practitioners.
3. Quantitative knowledge and skills for QL should have a much
more cross-disciplinary agenda, rather than one situated primarily in
mathematics curricula.
I am generally sympathetic to this recommendation. Because
mathematics is a foundational and enabling discipline for so many
others, it is natural that mathematics learning in general, not just
for QL, should evolve from an ongoing conversation and sometimes collaboration with client disciplines. At the same time, the
historical reasons for situating the learning of QL skills in mathematics study have not lost their relevance. And I am speaking of
more than the learning of basic arithmetic and measurement.
Take, for example, the learning of deductive reasoning, which
most of us would count as an important component of QL. Al-
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Moving Forward
Where do we go from here? Has this Forum accomplished its
goals? Many speakers have argued that, given the alarmingly low
rates of quantitative literacy among American adults and the already lengthy discussions of this problem, we should move
quickly to programs of dramatic action to improve the situation,
with a strongly articulated vision of what we want to accomplish.
Although I do not want to rain on your parade, I suggest that our
knowledge base about quantitative literacy is not yet adequate for
designing major interventions in the school curriculum. The comprehensive agenda of providing QL to all students is one measured
in decades, not years, but it is work that can productively begin in
incremental ways right now.
This Forum has taken an important step. The case statement in
Mathematics and Democracy and the collection of very interesting
and provocative background essays prepared for this Forum provide a rich articulation of questions and concerns regarding QL,
many analyses of the problems we face, and many stimulating but
somewhat divergent suggestions for what to do about them. Together, these provide a rich resource for an ongoing, disciplined,
and coordinated national (or even international) conversation
about these issues.
References
Steen, Lynn Arthur, ed. 2001. Mathematics and Democracy: The Case for
Quantitative Literacy. Princeton, NJ: National Council on Education and the Disciplines.
Reflections
A selection of brief observations by participants at the national Forum, Quantitative Literacy: Why Numeracy Matters for Schools and Colleges, offering different
perspectives on issues covered at the Forum.
Don Small
Robert Cole
Russell Edgerton
Charlotte Frank
Edward Tenner
William G. Steenken
Peter Ewell
Jo Ann Lutz
Gene Bottoms
William Haver
Andrea Leskes
Philip Mahler
Stephen B. Maurer
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252
Although several people I spoke with seemed interested in developing curricula around organizing questions and themes rather
than disciplinary content, they were, initially, uneasy with the
idea. Because their mind was still on a set of content to be covered,
it took a while for them to begin to see how this could be done.
The notion of designing curricula around important questions
was, at first, quite a stretch. One person said that it was a great idea
but that it would never fit into the departments he knew. I took
that to be a measure of how ingrown and isolated higher education
has become from the society that supports it.
One way of exposing how banal curriculum design questions have
become at most universities would be to do a QL analysis of
departmental and curricular structures. By analyzing the currencies we use to justify our academic enterprise (numbers of majors,
time to graduation, course sequencing, needs of majors, graduate
school preparation, hiring priorities, etc.), we might gain some
insight into what really drives curricular design. My hunch is that
such an analysis would not engage the important questions facing
homo sapiens on this planet at this time in history. Little wonder
that students often find our courses disconnected from the real
world.
The twenty-first century will be a crunch time for our species.
We currently are engaging in wholesale destruction of the ecosystem and far too many of us are chasing far too few natural resources. To the degree that the academic curriculum does not
organize itself to confront these important questions, it will continue a decline into irrelevance. Designing QL to address some of
Reflections
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From the papers and discussions, I would conclude that the most
promising approach at the college level might be to acknowledge
that only a minority of the mathematics, science, and social science faculty are potentially strong QL teachers. I would favor
identifying and working with this motivated group rather than
trying to bring everybody on board at once. For QL to succeed, it
must be perceived as an intellectual challenge by the faculty, not
just as a remedial activity. I can easily imagine that an economist
or sociologist might feel that teaching QL skills just delays developing the substance of their own courses.
An inspiring example for QL might be the late Edward Purcell,
whom I got to know when I was in the Society of Fellows at
Harvard. He had a Nobel Prize in physics, but he equally loved
simple and elegant explanationsfor example, how to tell if a set
of numbers might have been tampered with, or why quantum
theory is necessary for the world as we know it. For many years he
wrote a wonderful column for the American Journal of Physics that
consisted largely of Fermi problems back-of-the-envelope calculations mixing common sense with sensible estimatesproving
that QL can be a high art in its own right.
At least one Forum participant mentioned a paradox that also
occurred to me. Some people with very limited formal mathematics instruction, such as market traders in developing countries, are
proficient in handling numbers, while many westerners growing
up with advanced calculators are not. This suggests that we should
pay more attention not just to the pedagogical side of quantitative
literacy but also to the changing role of numbers in everyday and
professional life.
Two aspects of this changing role have especially interested me.
The first is the rhetorical side of numbers, the fact that people use
tables and graphs to prove points in which they have emotional or
financial stakes. There is surely a message in the failure of organizations such as Long Term Capital and Enron that were packed
with quantitatively sophisticated people yet succumbed to selfdeception. The second, and the main subject of my own investigations, is the tenuous nature of many vital measurements. Costof-living indexes measure shifting breadbaskets of goods,
including changing tastes and spending patterns. Television ratings measure a self-selected sample of the population, and the
presence of monitoring technology also may change viewer behavior. I think of these problems not so much as obstacles but as
opportunities to help students and adults achieve a deeper understanding of measurement and its uses.
Edward Tenner, Department of English,
Princeton University
Modeling: Formulating problems, seeking patterns, and drawing conclusions; recognizing interactions in complex systems;
understanding linear, exponential, multivariate, and simulation models; understanding the impact of different rates of
growth.
Statistics: Understanding the importance of variability; recognizing the differences between correlation and causation, between randomized experiments and observational studies, between finding no effect and finding no statistically significant
effect (especially with small samples), and between statistical
significance and practical importance (especially with large
samples).
Because these skills defining quantitative literacy have their foundations in mathematics, most participants believed that the primary responsibility for introducing concepts associated with the
tools of quantitative literacy lies with mathematics departments;
however, most also thought that developing special courses in
quantitative literacy would be the wrong approach.
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on what teachers are allowed to do. Until a quantitative literacybased curriculum is valued by these outside forces, high school
teachers will not be able to do what is best for all students.
Jo Ann Lutz, North Carolina School of Mathematics
and Science, and College Board Trustee
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literacy. Moreover, textbook publishers should provide more coordination between mathematics and science textbooks to align
mathematics concepts and the quantitative literacy potential of
science. In visiting hundreds of schools and high school classrooms over the past 15 years, it has been my observation that more
than any other teachers in high school, mathematics teachers depend on textbooks. Therefore, without quality text materials to
give students opportunities to use mathematics in a variety of
challenging contexts, QL simply will not happen.
Finally, one of the points made by many Forum participants was
that if quantitative literacy is not viewed as something for all
students, it will lead to further tracking in mathematics, which
these participants saw as undesirable. This is certainly a valid
point, but I fear that by stressing this distinction the quantitative
literacy movement runs the risk of being interpreted as saying that
what we are now doing is not working. We cannot simply overthrow one system and substitute another. We must develop mathematics course sequences that are appropriate for all students and
that offer a suitable balance between the more procedural emphases that now are taught to too many students and a QL-like emphasis that engages students in using mathematics to do real
things in contexts that have meaning for them.
Gene Bottoms, Director, High Schools That Work,
Southern Regional Education Board (SREB)
258
Perhaps most critically, we need support to include quantitative literacy on high-stakes tests, particularly on the mathematics portions of the tests many states are requiring for high
school graduation and the new grade-level tests now being
mandated at the national level.
All of this said, I think it is a big mistake to focus on the distinctions between mathematics and quantitative literacy, as was sometimes the case at the Forum and in Mathematics and Democracy.
Such artificial distinctions let mathematicians and the broader
mathematics community off the hook. In my opinion, the abilities and mind-set described as quantitative literacy are central to
mathematics (definitely including research mathematics) as well
as to effective teaching and learning of mathematics. Many in
positions of influence, however, would prefer to keep all students
focused on technical manipulation skills in the elementary algebra, formal geometry, intermediate algebra, college algebra, and
pre-calculus courses that are studied by masses of students today,
students who have no intention of entering fields that require
calculus.
If we make distinctions that can be translated as quantitative
literacy is not mathematics, we run the risk of giving ammunition
to those who oppose reforming the mathematics curriculum and
instruction in ways encouraged by quantitative literacy advocates.
The chance that leadership in implementing quantitative literacy
programs will come from anywhere but the mathematics community is slight to nonexistent. States require testing of mathematics,
language arts, and often social science and science. The chances of
adding a fifth test on quantitative literacy are nonexistent. The
federal government now requires states to test mathematics at
every grade level between three and eight. There is no possibility
of adding an additional test on quantitative literacy. Mathematics
is taught to all students from kindergarten through at least grade
10. An additional quantitative literacy subject will not be added
to the curriculum. Large numbers of colleges and universities
require mathematics course work of all their students. Some may
replace a mathematics requirement by a quantitative literacy requirement, but very few would add a quantitative literacy requirement on top of a mathematics requirement. Finally, colleges and
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School
Mathematics
High school
College
to
School
Mathematics
Middle school
Adult workforce
and citizenry
Other
Disciplines
At
Work
In the
Community
Pre-K5
Other
Disciplines
High school
College
This expansion requires many groups to shoulder the responsibility for helping to create a more quantitatively literate populace.
The usual suspects high school and four-year college mathematics educatorsare joined by faculty from other disciplines as
well as by elementary educators on one side and, on the other, by
community college and adult basic education teachers as well as a
host of informal education venues for adults such as the media, the
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