|
|
Peer Review, Summer 2004
Everything I Needed to Know about Averages
. . . I Learned in College
By Lynn Arthur Steen, professor of mathematics,
St. Olaf College |
Several months ago, the conservative-leaning American Council
of Trustees and Alumni (ACTA) excoriated America's leading
colleges and universities with a report documenting the "failure
of general education" (ACTA 2004). Among many cited
shortcomings, one--emphasized in bold face in the opening
paragraph--is that "mathematics is no longer required
at 62% of the examined institutions."
Much could be said about the educational merits of traditional
core curricula or the political agendas served by debates
about the core. But that is not what I found most interesting
about this report. Rather, it was the messages hidden in the
fine print. There, in the endnotes, lie intriguing clues about
collegiate mathematics--both about its place in general
education and its role in the ACTA study.
First, many colleges and universities call this core requirement
not "mathematics" but "quantitative reasoning,"
although variations abound: "quantitative or formal
reasoning," "mathematical thinking," "mathematical
and logical analysis," "quantitative and deductive
sciences," "formal reasoning and analysis,"
or "quantitative and deductive reasoning." All
of these stress the processes of mathematics (reasoning, deduction,
analysis) rather than its components (algebra, geometry, statistics,
calculus).
Second, these requirements are often fulfilled with courses
that help students build connections between mathematics and
other subjects, courses that reveal how quantitative reasoning
is used across the entire spectrum of collegiate studies:
- Counting People;
- Economics and the Environment;
- Health Economics;
- Introduction to Energy Sources;
- Introduction to Population Studies;
- Language and Formal Reasoning;
- Limnology: Freshwater Ecology;
- Maps, Visualization, and Geographical Reasoning;
- Practical Physics: How Things Work;
- Quantifying Judgments of Human Behavior.
Here's what caught my attention: in every case where colleges
allowed students to fulfill a quantitative reasoning requirement
with courses such as these, the ACTA study judged the institutions
as not including "mathematics" in its core curriculum.
These colleges wound up on the 62 percent blacklist. But colleges
that required a course in college algebra--whose pièce
de résistance is the manipulation of negative
fractional exponents--were checked off for having a suitable
"mathematics" core requirement.
Quantitative Literacy
This ACTA analysis demonstrates the presence of "two
mathematics" (see Bernard Madison's article in
this issue). One is an abstract, deductive discipline created
by the Greeks, refined through the centuries, and employed
in every corner of science, technology, and engineering. The
other is a practical, robust habit of mind anchored in data,
nourished by computers, and employed in every aspect of an
alert, informed life. This is what these many colleges call
"quantitative reasoning," what many other countries
call "numeracy," or what I'll call "quantitative
literacy" (or QL for short).
Although clearly related, quantitative literacy and mathematics
are not the same. Whereas mathematics rises above context,
QL is anchored in context. Whereas the objects of mathematical
study are ideals (in the Platonic sense), the objects of QL
are data, generally measurements retrieved from some computer's
data warehouse. Because quantitative reasoning relies on concepts
first introduced in middle school--averages, percentages,
graphs--many believe that QL is just watered down mathematics
(and thus should not satisfy a "mathematics" requirement).
Some academics, typically mathematicians, argue that students
should complete QL by the end of high school; in this view,
it is not a central (or even proper) responsibility of higher
education. Others, typically not mathematicians, argue that
QL is too important to be left to mathematicians, whose training
inclines them more toward Platonism than earthly practicality.
The issue of the core curriculum raised in the ACTA study
is exactly the central issue for quantitative literacy. Whereas,
typically, college-level mathematics serves primarily preprofessional
purposes (as prerequisites for particular courses), quantitative
literacy is essential for all graduates' personal and
civic responsibilities. College-level quantitative literacy
is inextricably connected to virtually all areas of undergraduate
study.
Understanding compound interest is a trite staple of QL expectations,
but it is nonetheless a good example whose significance is
not truly manifest until students are of college age. Only
when students become responsible for their own loans do the
formal calculations they may have learned in eighth grade
become personally meaningful. (Few adults realize the extraordinary
difference even a quarter-percent change in interest rates
can make on payoff time for a fixed payment loan.) More generally,
it is in college where many students study historical events
and first become personally engaged in social and political
causes whose roots often lie just beneath the surface in the
financial conditions of individuals or states. The habit of
thinking quantitatively--even more, of seeking quantitative
evidence--requires repeated practice in many different
contexts. For that reason, many colleges have replaced course
requirements (whether in mathematics or QL) with programs
of "QL across the curriculum."
Less obvious, perhaps, than compound interest are the many
examples of public policy issues requiring voters' attention
that depend significantly on subtle quantitative reasoning.
I'm not referring to obvious, although nonetheless complex,
issues such as projecting future deficits or counting votes
accurately, but to situations where quantitative traps lie
hidden beneath routine calculations of percentages and averages.
I offer a few examples from issues in public education; similar
examples abound in every area of public policy.
Percentages
Major problems beset public education, leading to significant
gaps in performance and to high dropout rates. Measuring the
gap between expectations and accomplishment is a complex,
multidimensional challenge that every parent recognizes as
a task requiring judgment and interpretation. But measuring
dropout rates seems simple: just apply the formula for percentages
everyone learned in the seventh grade. Here's one result,
as reported by the New York Times on August 13, 2003,
under the headline "The ‘Zero Dropout' Miracle" (Winerip
2003):
Robert Kimball, an assistant principal at Sharpstown High
School [in Houston], sat smack in the middle of the "Texas
miracle." His poor, mostly minority high school of
1,650 students had a freshman class of 1,000 that dwindled
to fewer than 300 students by senior year. And yet--and
this is the miracle--not one dropout to report!
Nor was zero an unusual dropout rate in this school district
that both President Bush and Secretary of Education Rod
Paige have held up as the national showcase for accountability
and the model for the federal No Child Left Behind law.
Westside High here had 2,308 students and no reported dropouts;
Wheatley High 731 students, no dropouts. A dozen of the
city's poorest schools reported dropout rates under
1 percent.
Now, Dr. Kimball has witnessed many amazing things in his
58 years. Before he was an educator, he spent 24 years in
the Army, fighting in Vietnam, rising to the rank of lieutenant
colonel and touring the world. But never had he seen an
urban high school with no dropouts. "Impossible,"
he said. "Someone will get pregnant, go to jail, get
killed." Elsewhere in the nation, urban high schools
report dropout rates of 20 percent to 40 percent.
A miracle? "A fantasy land," said Dr. Kimball.
"They want the data to look wonderful and exciting.
They don't tell you how to do it; they just say, ‘Do
it.'"
As it turns out, there are a number of different ways to
"do it," each with its own justification. Finding
one that produces the desired answer of zero may take some
effort, but it is not beyond the realm of plausible quantitative
argument. One simple way is to divide the number of high school
graduates by the number of entering freshmen four years earlier,
and then to subtract from 100 percent. If a high school is
growing, it is not unreasonable that this dropout calculation
yields a number close to zero. Another approach--the
one used in higher education--tracks a specific entering
cohort of students through their four years of high school,
ignoring all other students in the school (e.g., transfers).
A third common method is to classify the reasons students
leave school each year (transfer, work, jail, death, dropout,
etc.) and then report only the "dropout" classifications.
Each method has distinct characteristics that may make it
more or less useful for a particular purpose. The first and
simplest calculation is highly sensitive to irrelevant circumstances
such as growth and transfers. The second, being limited to
a subset of students, may not represent the quality of education
received by all students. The third attempts to account for
why students leave a school, thereby limiting the meaning
of "dropout" to students for whom no other reason
may apply. (Setting aside the possibility of deliberate misrepresentation,
this may explain the Texas miracle: "Do it" can
be taken by teachers as a challenge to find any reason other
than "drop out" to explain why students left school.)
In the seventh grade students might be asked, "if 500
students enter Abraham Lincoln High School as freshmen and
400 graduate, how many dropped out?" The next time they
may be asked to consider what "drop out" means
is when they vote for school board candidates or on a school
levy referendum. Unless their quantitative literacy has been
significantly enhanced, citizens are likely to enter the voting
booth with a seventh-grade concept of dropout rate. That's
why courses such as those discounted by the ACTA study are
so important: students who took a course on, say, "Economics
of Education" would be far better equipped to fulfill
their responsibilities as educated citizens than those who
met their mathematics requirement by simplifying rational
functions in a college algebra course.
Averages
Students in my hypothetical "Economics of Education"
course are likely to learn that averages, like percentages,
are also a source of mysteries. A recent study shows that
the average verbal SAT score did not improve during the two
decades between 1981 and 2002 (Bracey 2004). But during that
same period, the average scores of each of the six major ethnic
categories used in reporting SAT data (white, black, Asian,
Puerto Rican, Mexican, and American Indian) increased by amounts
ranging from eight to twenty-seven points. Yet the overall
average did not budge--enabling skeptics to claim that
all the money invested in education during the last two decades
has produced no noticeable improvement.
A quantitatively literate college graduate would recognize
this mystery as a classic example of Simpson's Paradox:
changes in composition can cause the whole to show trends
opposite to each of its parts when considered separately.
Demagogues rely on the public's simplistic seventh-grade
understanding of how numbers work to ply their trade. But
in today's data-drenched society, sometimes no one really
understands what is going on.
Society's reliance on data as a justification for decisions
increased gradually throughout the nineteenth and twentieth
centuries (Desrosières 1998; Porter 1995), but it has
taken a significant leap during the last two decades--the
chief reason being the vast quantity of data that computers
disgorge. More recently, the importance of QL (and the consequences
of quantitative illiteracy) has been greatly magnified, if
not totally transformed, by the behavior of computer networks.
Recent financial scandals, for example, were enabled by clever
bookkeeping that displayed apparent corporate gains while
every part of the business was actually losing money. Yet
even professionals well aware of Simpson's Paradox did
not detect these machinations.
Something deeper than just clever or illegal accounting seems
to be at work. Two years ago, an analysis of this new "culture
of finance" was presented at the International Congress
of Mathematicians and subsequently published by the American
Mathematical Society (Poovey 2003). It suggests that the invisible
impact of "mathematical abstractions" on modern
society has generated a "new form of value" that
is unhinged from work or experience. "In the new culture
of finance, the numbers one writes and the computations a
computer performs upon them generate the only value that matters."
In this purely quantified culture, value is created "without
labor," decisions rely on "an unstable mixture
of mathematical equations and beliefs," and "responsibility
is simply dispersed." In short, the mechanisms of quantification
that began with averages and percentages have become just
as abstract--and hence as powerful--as mathematics
itself. We just haven't realized it yet.
QL on Campus
My main point in these examples is not to argue that QL is
important; I've rarely met anyone who doubts that. Rather,
my point is that QL is sufficiently sophisticated to warrant
inclusion in college study and, more important, that without
it students cannot intelligently achieve major goals of college
education. Quantitative literacy is not just a set of precollege
skills. It is as important, as complex, and as fundamental
as the more traditional branches of mathematics. Indeed, QL
interacts with the core substance of liberal education every
bit as much as the other two R's, reading and writing.
Quantitative literacy differs from mathematics primarily
by being anchored in real contexts. While this anchor is generally
a source of strength--notably for improved student motivation
and learning--it is also a source of structural weakness.
Since QL is not a discipline in the traditional sense, it
lacks the academic infrastructure of departments, journals,
and professional associations. By its nature, QL is dispersed
and, thus, almost invisible. Many efforts are now underway
to make QL visible and to establish a strong presence in the
ecology of liberal education. Some are described in the box
on pages 6-7, others later in this issue.
From all these sources one clear priority has emerged: the
need to develop benchmarks for quantitative literacy that
can guide both curriculum and assessment in grades 10-16.
Since QL is relatively new and since it lives in the matrix
of other disciplines, neither higher education professionals
nor public leaders have a clear understanding of suitable
performance expectations. Consensus on expectations is a desirable
(but not inevitable) outcome of various approaches to mathematical
and quantitative literacy in core curricula and, more broadly,
general education. This issue of Peer Review is an
important step in the process of building consensus.
References
American Council of Trustees and Alumni. 2004. The hollow
core: Failure of the general education curriculum. Washington,
DC: American Council of Trustees and Alumni.
Bracey, Gerald W. 2004. Simpson's paradox and other
statistical mysteries. American School Board Journal,
February.
Committee on the Undergraduate Program in Mathematics. 2004.
CUPM Curriculum Guide 2004. Washington, DC: Mathematical
Association of America.
Desrosières, Alain. 1998. The politics of large
numbers: A history of statistical reasoning. Cambridge,
MA: Harvard University Press.
Ganter, Susan L. and William Barker, eds. 2004. Curriculum
foundations project: Voices of the partner disciplines. Washington,
DC: Mathematical Association of America.
Poovey, Mary. 2003. Can numbers ensure honesty? Unrealistic
expectations and the U.S. accounting scandal. Notices
of the American Mathematical Society 50(1): 28-35.
Porter, Theodore. 1995. Trust in numbers: The pursuit
of objectivity in science and public life. Princeton,
NJ: Princeton University Press.
Winerip, Michael. 2003. The "zero dropout" miracle:
Alas! alak! a Texas tall tale. The New York Times.
August 13.
|
 |
|
