Mathematics is our "invisible culture"(Hammond, 1978). Few
people have any idea how much mathematics lies behind the artifacts and
accoutrements of modern life. Nothing we use on a daily basis–houses,
automobiles, bicycles, furniture, not to mention cell phones, computers, and Palm
Pilots–would be possible without mathematics. Neither would our economy nor
our democracy: national defense, Social Security, disaster relief, as well as
political campaigns and voting all depend on mathematical models and quantitative
habits of mind.

Mathematics is certainly not invisible in education, however. Ten years of
mathematics is required in every school and is part of every state graduation
test. In the late 1980s mathematics teachers led the national campaign for high,
publicly visible standards in K-12 education. Nonetheless, mathematics is the
subject that parents most often recall with anxiety and frustration from their
own school experiences. Indeed, mathematics is the subject most often
responsible for students' failure to attain their educational goals. Recently,
mathematics curricula have become the subject of ferocious debates in school
districts across the country.

My intention in writing this paper is to make visible to curious and
uncommitted outsiders some of the forces that are currently shaping (and
distorting) mathematics education. My focus is on the second half of the school
curriculum, grades 6-12, where the major part of most students' mathematics
education takes place. Although mathematics is an abstract science, mathematics
education is very much a social endeavor. Improving mathematics education
requires, among many other things, thorough understanding of pressures that shape
current educational practice. Thus I begin by unpacking some of the arguments
and relevant literature on several issues–tracking, employment, technology,
testing, algebra, data, and achievement–that are responsible for much of the
discord in current public discussion about mathematics education.

Following discussion of these external forces, I examine the changing world of
mathematics itself and its role in society. This leads to questions of context
and setting, of purposes and goals, and quickly points in the direction of
broader mathematical sciences such as statistics and numeracy. By blending the
goals of mathematics, statistics, and numeracy, I suggest–in the final
section of the paper–a structure for mathematics education in grades 6-12
that can help more students leave school equipped with the mathematical tools
they will need for life and career.

### Algebra

In the Middle Ages, algebra meant calculating by rules (algorithms). During
the Renaissance, it came to mean calculation with signs and symbols–using
*x*'s and *y*'s instead of numbers. (Even today, lay persons 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
arithmetic–the 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 today's 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 unknown–"that we can find out what *we* want
to know"–is 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. Then advocates for
educational standards 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, 1995; Silver 1997; NCTM, 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 a distortion 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 on the broad goals of
mathematics in the face of a curriculum dominated by the need to become fluent in
algebraic manipulation. Moreover, overemphasis on algebra drives many students
away from mathematics: most students who leave mathematics do so because they
cannot see any value in manipulating strings of meaningless symbols.

What's worse, the focus on formulas as the preferred methodology of school
mathematics distorts the treatment of other important parts of mathematics. For
example, despite the complexity of its algebraic formula, the bell-shaped normal
distribution is as ubiquitous in daily life as are linear and exponential
functions, and far more common than quadratic equations. As citizens, it is very
helpful to understand that repeated measurements of the same thing (length of a
table) as well as multiple measurements of different although similar things
(heights of students) tend to follow the normal distribution. Knowing why some
distributions (e.g., salaries, sizes of cities) do not follow this pattern is
equally important, as is understanding something about the tails of the normal
distribution–which can be very helpful in thinking about risks (or SAT
scores).

Yet despite its obvious value to society, the normal distribution is all but
ignored in high school mathematics, whereas quadratic and periodic functions are
studied extensively. Many reasons can be advanced to explain this imbalance,
e.g., that mathematicians favor models of the physical over the behavioral
sciences. But surely one of the most important is that the algebraic formula for
the normal distribution is quite complex and cannot be fully understood without
techniques of calculus. The bias in favor of algebraic formulas as the preferred
style of understanding mathematics–instead of graphs, tables, computers, or
verbal descriptions–causes mathematics teachers to omit from the high school
curriculum what is surely one of the most important and most widely used tools of
modern mathematics.

That a subject that for many amounts to little more than rote fluency in
manipulating meaningless symbols came to occupy such a privileged place in the
school curriculum is something of a mystery, especially since so many parents, as
students, found it unbearable. Perhaps more surprising is algebra's strong
support among those many successful professional who, having mastered algebra in
school, found no use for it in their adult lives. Why is it that we insist on
visiting on eighth graders a subject that, more than any other, has created
generations of math-anxious and math-avoiding adults?

Many argue on the simple pragmatic "civil right" ground that algebra is,
wisely or unwisely, of central importance to the current system of tests that
govern the school-college transition (not to mention providing essential
preparation for calculus which itself has taken on exaggerated significance in
this same transition). But this, of course, is the ultimate circular argument.
We need to study algebra to pass tests that focus on algebra. And why do the
tests focus on algebra? Because it is a part of mathematics that virtually all
students study.

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
mathematics–poor, urban, first generation, and minorities. 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.

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