Reproduced
with permission from the April 2000 issue of
American
School Board Journal
Copyright
© 2000, National School Boards Association
MATH PROBLEMS
Why the
recommended math programs don't add up
By David Klein
What
constitutes a good K-12 mathematics program? Opinions differ. In October 1999,
the U.S. Department of Education released a report designating 10 math programs
as "exemplary" or "promising." The following month, I sent
an open letter to Education Secretary Richard W. Riley urging him to withdraw
the department's recommendations. The letter was coauthored by Richard Askey of
the
Although
a clear majority of cosigners are mathematicians and scientists, it is
sometimes overlooked that experienced education administrators at the state and
national level, as well as educational psychologists and education researchers,
also endorsed the letter. (A complete list is posted at
http://www.mathematicallycorrect.com.)
University
professors and public education leaders are not the only ones who have
reservations about these programs.
Thousands of parents and teachers across the nation seek alternatives to
them, often in opposition to local school boards and superintendents.
Mathematically Correct, an influential Internet-based parents' organization,
came into existence several years ago because the children of the
organization's founders had no alternative to the now "exemplary"
program, College Preparatory Mathematics, or CPM. In Plano, Texas, 600 parents
are suing the school district because of its exclusive use of the Connected
Mathematics Project, or CMP, another "exemplary" program. I have
received hundreds of requests for help by parents and teachers because of these
and other programs now promoted by the Education Department (ED). In fact, it
was such pleas for help that motivated me and my three coauthors to write the
open letter.
The
mathematics programs criticized by the open letter have common features. For
example, they tend to overemphasize data analysis and statistics, which
typically appear year after year, with redundant presentations. The far more important areas of arithmetic
and algebra are radically de-emphasized. Many of the so-called higher-order
thinking projects are just aimless activities, and genuine illumination of
important mathematical ideas is rare. There is a near obsession with
calculators, and basic skills are given short shrift and sometimes even
disparaged. Overall, these curricula are watered-down math programs. The same
educational philosophy that gave rise to the whole-language approach to reading
is part of ED's agenda for mathematics. Systematic development of skills and
concepts is replaced by an unstructured "holism." In fact, during the
mid-'90s, supporters of programs like these referred to their approach as "whole
math."
Disagreements
over math curricula are often portrayed as "basic skills versus conceptual
understanding." Scientists and
mathematicians, including many who signed the open letter to Secretary Riley,
are described as advocates of basic skills, while professional educators are
counted as proponents of conceptual understanding. Ironically, such a portrayal ignores the deep
conceptual understanding of mathematics held by so many mathematicians. But
more important, the notion that conceptual understanding in mathematics can be
separated from precision and fluency in the execution of basic skills is just
plain wrong.
In
other domains of human activity, such as athletics or music, the dependence of
high levels of performance on requisite skills goes unchallenged. A novice
cannot hope to achieve mastery in the martial arts without first learning basic
katas or exercises in movement. A violinist who has not mastered elementary
bowing techniques and vibrato has no hope of evoking the emotions of an
audience through sonorous tones and elegant phrasing. Arguably the most hierarchical of human
endeavors, mathematics also depends on sequential mastery of basic skills.
The
standard algorithms for arithmetic (that is, the standard procedures for
addition, subtraction, multiplication, and division of numbers) are missing or
abridged in ED's recommended elementary school curricula. These omissions are
inconsistent with the mainstream views of mathematicians.
In
our open letter to Secretary Riley, we included an excerpt from a committee
report published in the February 1998 Notices of the American Mathematical
Society. The committee was appointed by the American Mathematical Society to
advise the National Council of Teachers of Mathematics (NCTM). Part of its
report discusses the standard algorithms of arithmetic. "We would like to
emphasize that the standard algorithms of arithmetic are more than just 'ways
to get the answer'--that is, they have theoretical as well as practical
significance," the report states. "For one thing, all the algorithms
of arithmetic are preparatory for algebra, since there are (again, not by
accident, but by virtue of the construction of the decimal system) strong
analogies between arithmetic of ordinary numbers and arithmetic of
polynomials."
This
statement deserves elaboration. How could the standard algorithms of arithmetic
be related to algebra? For concreteness,
consider the meaning in terms of place value of 572:
572 = 5 (102) +
7(10) + 2
Now
compare the right side of this equation to the polynomial,
5x2 + 7x + 2.
The
two are identical when x = 10. This connection between whole numbers and
polynomials is general and extends to arithmetic operations. Addition,
subtraction, multiplication, and division of polynomials is fundamentally the
same as for whole numbers. In arithmetic, extra steps such as
"regrouping" are needed since x = 10 allows for simplifications. The
standard algorithms incorporate both the polynomial operations and the extra
steps to account for the specific value, x = 10. Facility with the standard
operations of arithmetic, together with an understanding of why these
algorithms work, is important preparation for algebra.
The
standard long division algorithm is particularly shortchanged by the
"promising" curricula. It is preparatory for division of polynomials
and, at the college level, division of "power series," a useful
technique in calculus and differential equations. The standard long division
algorithm is also needed for a middle school topic. It is fundamental to an
understanding of the difference between rational and irrational numbers, an
indisputable example of conceptual understanding. It is essential to understand
that rational numbers (that is, ratios of whole numbers like ¾) and their
negatives have decimal representations that exhibit recurring patterns. For
example: 1/3 = .333..., where the ellipses indicate that the numeral 3 repeats
forever. Likewise, 1/2 = .500... and 611/4950 = .12343434....
In
the last equation, the digits 34 are repeated without end, and the repeating
block in the decimal for ½ consists only of the digit for zero. It is a general
fact that all rational numbers have repeating blocks of numerals in their
decimal representations, and this can be understood and deduced by students who
have mastered the standard long division algorithm. However, this important
result does not follow easily from other "nonstandard" division
algorithms featured by some of ED's model curricula.
A
different but still elementary argument is required to show the converse--that
any decimal with a repeating block is equal to a fraction. Once this is
understood, students are prepared to understand the meaning of the term
"irrational number." Irrational numbers are the numbers represented
by infinite decimals without repeating blocks. In California, seventh-grade
students are expected to understand this.
It
is worth emphasizing that calculators are utterly useless in this context, not
only in establishing the general principles, but even in logically verifying
the equations. This is partly because calculator screens cannot display
infinite decimals, but more important, calculators cannot reason. The
"exemplary" middle school curriculum CMP nevertheless ignores the
conceptual issues, bypassing the long division algorithm and substituting
calculators and faulty inductive reasoning instead.
Steven
Leinwand of the Connecticut Department of Education was a member of the expert
panel that made final decisions on ED's "exemplary" and "promising"
math curricula. He was also a member of the advisory boards for two programs
found to be "exemplary" by the panel: CMP and the Interactive
Mathematics Program. In a Feb. 9, 1994, article in Education Week, he wrote:
"It's time to recognize that, for many students, real mathematical power,
on the one hand, and facility with multidigit, pencil-and-paper computational
algorithms, on the other, are mutually exclusive. In fact, it's time to
acknowledge that continuing to teach these skills to our students is not only
unnecessary, but counterproductive and downright dangerous."
Mr.
Leinwand's influential opinions are diametrically opposed to the mainstream
views of practicing scientists and mathematicians, as well as the general
public, but they have found fertile soil in the government's
"promising" and "exemplary" curricula.
According
to the Third International Mathematics and Science Study, or TIMSS, the use of
calculators in U.S. fourth-grade mathematics classes is about twice the international
average. Teachers of 39 percent of U.S. students report that students use
calculators at least once or twice a week. In six of the seven top-scoring
nations, on the other hand, teachers of 85 percent or more of the students
report that students never use calculators in class.
Even
at the eighth-grade level, the majority of students from three of the top five
scoring nations in the TIMSS study (Belgium, Korea, and Japan) never or rarely
use calculators in math classes. In Singapore, which is also among the top five
scoring countries, students do not use calculators until the seventh grade.
Among the lower achieving nations, however, the majority of students from 10 of
the 11 nations with scores below the international average--including the
United States--use calculators almost every day or several times a week.
Of
course, this negative correlation of calculator usage with achievement in
mathematics does not imply a causal relationship. There are many variables that
contribute to achievement in mathematics. On the other hand, it is foolhardy to
ignore the problems caused by calculators in schools. In a Sept. 17, 1999, Los
Angeles Times editorial titled "L.A.'s Math Program Just Doesn't Add
Up," Milgram and I recommended that calculators not be used at all in
grades K-5 and only sparingly in higher grades. Certainly there are isolated,
beneficial uses for calculators, such as calculating compound interest, a
seventh-grade topic in California. Science classes benefit from the use of
calculators because it is necessary to deal with whatever numbers nature gives
us, but conceptual understanding in mathematics is often best facilitated
through the use of simple numbers. Moreover, fraction arithmetic, an important
prerequisite for algebra, is easily undermined by the use of calculators.
A
number of the programs on ED's list have specific shortcomings--many involving
use of calculators. For example, a "promising" curriculum called
Everyday Mathematics says calculators are "an integral part of
Kindergarten Everyday Mathematics" and urges the use of calculators to
teach kindergarten students how to count. There are no textbooks in this K-6
curriculum, and even if the program were otherwise sound, this is a serious
shortcoming. The standard algorithm for multiplying two numbers has no more
status or prominence than an Ancient Egyptian algorithm presented in one of the
teacher's manuals. Students are never required to use the standard long
division algorithm in this curriculum, or even the standard algorithm for
multiplication.
Calculator
use is also ubiquitous in the "exemplary" middle school program CMP.
A unit devoted to discovering algorithms to add, subtract, and multiply
fractions ("Bits and Pieces II") gives the inappropriate instruction,
"Use your calculator whenever you need it." These topics are poorly
developed, and division of fractions is not covered at all. A quiz for
seventh-grade CMP students asks them to find the "slope" and
"y-intercept" of the equation 10 = x - 2.5, and the teacher's manual
explains that this equation is a special case of the linear equation y = x -
2.5, when y = 10, and concludes that the slope is therefore 1 and the
y-intercept is -2.5. This is not only false, but is so mathematically unsound
as to undermine the authority of classroom teachers who know better.
College
Preparatory Math (CPM), a high school program, also requires students to use
calculators almost daily. The principal technique in this series is the
so-called guess-and-check method, which encourages repeated guessing of answers
over the systematic development of standard mathematical techniques. Because of
the availability of calculators that can solve
equations, the introduction to the series explains that CPM puts low
emphasis on symbol manipulation and that CPM differs from traditional
mathematics courses both in the mathematics that is taught and how it is
taught. In one section, students watch a candle burn down for an hour while
measuring its length versus the time and then plotting the results. In a
related activity, students spend a whole class period on the athletic field
making human coordinate graphs. These activities are typical of the time
sacrificed to simple ideas that can be understood more efficiently through
direct explanation. But in CPM, direct instruction is systematically
discouraged in favor of group work. Teachers are told that as "rules of
thumb," they should "never carry or grab a writing implement"
and they should "usually respond with a question." Algebra tiles are
used frequently, and the important distributive property is poorly presented
and underemphasized.
Another
program, Number Power--a "promising" curriculum for grades K-6--was
submitted to the California State Board of Education for adoption in California.
Two Stanford University mathematics professors serving on the state's Content
Review Panel wrote a report on the program that is now a public document.
Number Power, they wrote, "is meant as a partial program to supplement a
regular basic program. There is a strong emphasis on group projects--almost the
entire program. Heavy use of calculators. Even as a supplementary program, it
provides such insufficient coverage of the [California] Standards that it is
unacceptable. This holds for all grade levels and all strands, including Number
Sense, which is the only strand that is even partially covered."
The
report goes on to note, "It is explicitly stated that the standard
algorithms for addition, subtraction, and multiplication are not taught."
Like CMP and Everyday Math, Number Power was rejected for adoption by the state
of California.
Interactive
Mathematics Program, or IMP, an "exemplary" high school curriculum,
has such a weak treatment of algebra that the quadratic formula, normally an
eighth- or ninth-grade topic, is postponed until the 12th grade.
Even though probability and statistics receive greater emphasis in this
program, the development of these topics is poor. "Expected value," a
concept of fundamental importance in probability and statistics, is never even
correctly defined. The Teacher's Guide for "The Game of Pig," where
expected value is treated, informs teachers that "expected value is one of
the unit's primary concepts," yet teachers are instructed to tell their
students that "the concept of expected value is nothing new...[but] the
use of such complex terminology makes it easier to state complex ideas."
(For a correlation of lowered SAT scores with the use of IMP, see Milgram's
paper at ftp://math.stanford.edu/pub/papers/milgram.)
Core-Plus
Mathematics Project is another "exemplary" high school program that
radically de-emphasizes algebra, with unfortunate results. Even Hyman Bass--a
well-known supporter of NCTM-aligned programs and a harsh critic of the open
letter to Secretary Riley--has conceded the program has problems. "I have
some reservations about Core Plus, for what I consider too shallow a coverage
of traditional algebra, and a focus on highly contextualized work that goes
beyond my personal inclinations," he wrote in a nationally circulated
e-mail message. "These are only my personal views, and I do not know about
its success with students."
Milgram
analyzed the program's effect on students in a top-performing high school in
"Outcomes Analysis for Core Plus Students at Andover High School: One Year
Later," based on a statistical study by G. Bachelis of Wayne State
University. According to Milgram, "...there was no measure represented in
the survey, such as ACT scores, SAT Math scores, grades in college math courses,
level of college math courses attempted, where the Andover Core Plus students
even met, let alone surpassed the comparison group [which used a more
traditional program]."
And
then there is MathLand, a K-6 curriculum that ED calls "promising"
but that is perhaps the most heavily criticized elementary school program in
the nation. Like Everyday Math, it has no textbooks for students in any of the
grades. The teacher's manual urges teachers not to teach the standard
algorithms of arithmetic for addition, subtraction, multiplication, and
division. Rather, students are expected to invent their own algorithms.
Numerous and detailed criticisms, including data on lowered test scores, appear
at http://www.mathematicallycorrect.com.
Perhaps
Galileo wondered similarly how the church of Pope Urban VIII could be so wrong.
The U.S. Department of Education is not alone in endorsing watered-down, and
even defective, math programs. The NCTM has also formally endorsed each of the
U.S. Department of Education's model programs
(http://www.nctm.org/rileystatement.htm), and the National Science Foundation
(Education and Human Resources Division) funded several of them. How could such
powerful organizations be wrong?
These
organizations represent surprisingly narrow interests, and there is a revolving
door between them. Expert panel member Steven Leinwand, whose personal
connections with "exemplary" curricula have already been noted, is
also a member of the NCTM board of directors. Luther Williams, who as assistant
director of the NSF approved the funding of several of the recommended
curricula, also served on the expert panel that evaluated these same curricula.
Jack Price, a member of the expert panel is a former president of NCTM, and
Glenda Lappan, the association's current president, is a coauthor of the
"exemplary" program CMP.
Aside
from institutional interconnections, there is a unifying ideology behind
"whole math." It is advertised as math for all students, as opposed
to only white males. But the word all is a code for minority students and women
(though presumably not Asians). In 1996, while he was president of NCTM, Jack
Price articulated this view in direct terms on a radio show in San Diego:
"What we have now is nostalgia math. It is the mathematics that we have
always had, that is good for the most part for the relatively high
socioeconomic anglo male, and that we have a great deal of research that has
been done showing that women, for example, and minority groups do not learn the
same way. They have the capability, certainly, of learning, but they don't. The
teaching strategies that you use with them are different from those that we
have been able to use in the past when ... we weren't expected to graduate a
lot of people, and most of those who did graduate and go on to college were the
anglo males."
Price went on to say: "All of the research that has been done with gender differences or ethnic differences has been--males for example learn better deductively in a competitive environment, when--the kind of thing that we have done in the past. Where we have found with gender differences, for example, that women have a tendency to learn better in a collaborative effort when they are doing inductive reasoning." (A transcript of the show is online at (http://mathematicallycorrect.com/roger.htm.)
I
reject the notion that skin color or gender determines whether students learn
inductively as opposed to deductively and whether they should be taught the
standard operations of arithmetic and essential components of algebra.
Arithmetic is not only essential for everyday life, it is the foundation for
study of higher level mathematics. Secretary Riley--and educators who select
mathematics curricula--would do well to heed the advice of the open letter.
David Klein is a professor of mathematics at
California State University at Northridge.
It
is impossible to specify all of the characteristics of a sound mathematics
program in only a few paragraphs, but a few highlights may be identified. The
most important criterion is strong mathematical content that conforms to a set
of explicit, high, grade-by-grade standards such as the California or Japanese
mathematics standards. A strong mathematics
program recognizes the hierarchical nature of mathematics and builds coherently
from one grade to the next. It is not merely a sequence of interesting but
unrelated student projects.
In
the earlier grades, arithmetic should be the primary focus. The standard
algorithms of arithmetic for integers, decimals, fractions, and percents are of
central importance. The curriculum should promote facility in calculation, an
understanding of what makes the algorithms work in terms of the base 10
structure of our number system, and an understanding of the associative,
commutative, and distributive properties of numbers. These properties can be illustrated by area
and volume models. Students need to develop an intuitive understanding for
fractions. Manipulatives or pictures can help in the beginning stages, but it
is essential that students eventually be able to compute easily using
mathematical notation. Word problems should be abundant. A sound program should move students toward
abstraction and the eventual use of symbols to represent unknown quantities.
In
the upper grades, algebra courses should emphasize powerful symbolic techniques
and not exploratory guessing and calculator-based graphical solutions.
There
should be a minimum of diversions in textbooks. Children have enough trouble
concentrating without distracting pictures and irrelevant stories and projects.
A mathematics program should explicitly teach skills and concepts with
appropriately designed practice sets. Such programs have the best chance of
success with the largest number of students. The high-performing Japanese
students spend 80 percent of class time in teacher-directed whole-class
instruction. Japanese math books contain clear explanations, examples with
practice problems, and summaries of key points. Singapore's elementary school
math books also provide good models. Among U.S. books for elementary school,
Sadlier-Oxford's Progress in Mathematics and the Saxon series through Math 87
(adopted for grade six in California), though not without defects, have many
positive features.--D.K.
__________________________________________________________________________________________
Askey,
Richard. "Knowing and Teaching Elementary Mathematics." American
Educator, Fall 1999, pp. 6-13; 49.
Ma,
Liping. Knowing and Teaching Elementary Mathematics. Mahwah, N.J.: Lawrence
Erlbaum, 1999.
Milgram,
R. James. "A Preliminary Analysis of SAT-I Mathematics Data for IMP
Schools in California."
ftp://math.stanford.edu/pub/papers/milgram
Milgram,
R. James. "Outcomes Analysis for Core Plus Students at Andover High
School: One Year Later."
ftp://math.stanford.edu/pub/papers/milgram/andover-report.htm
Wu,
Hung-Hsi. "Basic Skills Versus Conceptual Understanding: A Bogus Dichotomy
in Mathematics Education." American Educator, Fall 1999, pp. 14-19; 50-52.