1998NCTM03Dialogues

Dialogues
Volume1,Issue 1
A publication of the National Council of Teachers of Mathematics
March 1998
Math ematics
Ed ucation
Simultaneous with these develop-
ments, and perhaps because of them, the
learning of mathematics has emerged as
a high-priority educational issue. Teach-
ers are being expected to bring all stu-
dents to levels of performance if not
higher than those expected in previous
generations, different from those expec-
tations. Tests that originated in many of
our states and provinces to monitor the
performance of students and schools are
increasingly being adapted or replaced
by tests whose passage is required for
graduation.
Many questions naturally emerge from
these changes. Here are just a few:
• What mathematics should all students
be expected to learn? For how long in
their schooling should all students be
expected to learn the same mathematics
in the same classes? When and how
should the differentiation take place?
• Should four-function calculators re-
place any of the paper-and-pencil skills
of arithmetic? Do they require that new
ideas be taught? How, if at all, should
symbol-manipulating calculators affect
the paper-and-pencil skills traditionally
taught in algebra?
• Is it wise for high-stakes tests to be used
to drive the mathematics curriculum? If
so, what is the best way for this outcome
to be accomplished? If not, what are
the alternatives? When are we using too
many tests?
• To what extent do different students
learn in different ways? And if they do,
All-New Forum Makes Debut!
Perhaps the major development in mathematics education
in the twentieth century has been the transformation from
mathematics beyond arithmetic as a subject expected to be
learned only by some to a subject to be learned by all. At the same
time, within the past generation, the learning of traditional arithmetic
itself has been placed into question by the existence of hand-held
calculators, and calculators and computers have changed the ways in which
many adults work with mathematics.Inside this issue …
• Is Mathematics Necessary? 2
Underwood Dudley
• Executive Summary of
“Mathematics Equals
Opportunity” 3
Secretary Richard W. Riley
• Reactions
Clay Burkett 8
Roger Howe 8
Mercedes McGowen 9
Penny Noyce 9
John C. Souders Jr. 9
• Is Long Division Obsolete? 15
• Responses
Susan Addington 16
Stephen Willoughby 16 The purpose of Mathematics Education Dialogues is to provide a forum through which NCTM
members can be well informed about compelling, complex, timely issues that transcend grade
levels in mathematics education.
See All-New Forum Makes Debut, page 18

Underwood Dudley
DePauw University
Is mathematics necessary? Necessary, that is, for citizens of the
United States to function in the world of work? You would get that
impression from reading various recent documents, some coming
from high and official places. For example, Moving Beyond Myths,
published by the National Academy of Sciences, says so [5, p. 11]:
2 Mathematics Education Dialogues
March 1998
Myth: Most jobs require little mathematics.
Reality: The truth is just the opposite: more
and more jobs—especially those involving
the use of computers—require the ability to
use quantitative skills. Although a working
knowledge of arithmetic may have sufficed
for jobs of the past, it is clearly not enough
for today, for the next decade, or the next
century.
The anonymous author of that item
presumably had mathematical training
and thus should know that theorems are
not proved by assertion. But if we look in
the document for evidence for that sup-
posed reality, we look in vain, so an asser-
tion is all that it is.
Here is an excerpt from Everybody Counts
[6, p. 4], written anonymously for the Na-
tional Research Council. (Do documents
issuing from important national organiza-
tions gain more weight when their authors
are not identified? Some members of 
important national organizations evi-
dently think so.) This report too says that
mathematics is a vocational necessity:
Just because students do not use algebra any-
where except in algebra class does not mean
that they will not need mathematics in the fu-
ture. Over 75 percent of all jobs require pro-
ficiency in simple algebra and geometry, ei-
ther as a prerequisite to a training program
or as part of a licensure examination.
A quick reading of that passage might
leave the impression that algebra and
geometry are used in 75% of all jobs, but
that is silly. Just look at the next eight
workers that you see and ask yourself if at
least six of them require proficiency in al-
gebra to do their jobs. (If you are a
teacher of mathematics, it is not fair to
look at eight colleagues.)
The anonymous author was careful to
(This article is reprinted with the permission of the author and the Mathematical Association of America.
The article appears in the November 1997 issue of the College Mathematics Journal.)
See Is Mathematics Necessary?, page 4
Underwood Dudley
([EMAIL])
Underwood Dudley taught
his first calculus course in
1957 and is amazed that after
all the years he has been
teaching them, students are
still making the same mis-
takes. He hopes to see the
new millennium in at
DePauw University, here he
has taught for 19% of the
institution’s existence. His
last book,
Numerology, was
published by the MAA in 1997 .
Woody is the editor-elect of
the 
CMJ (the College
Mathematics Journal).
???IsMathematics
Necessary

This report highlights the following
findings:
• Students who take rigorous mathe-
matics and science courses are much
more likely to go to college than those
who do not. Data from the National Ed-
ucational Longitudinal Study (NELS) re-
veal that 83 percent of students who
took algebra I and geometry went on to
college within two years of their sched-
uled high school graduation. Only 36
percent of students who did not take al-
gebra I and geometry courses went to
college. While nearly 89 percent of stu-
dents who took chemistry in high school
went to college, only 43 percent of
students who did not take chemistry went
to college.
• Algebra is the “gateway” to advanced
mathematics and science in high school,
yet most students do not take it in middle
school. Students who study algebra in
middle school and who plan to take ad-
vanced mathematics and science courses
in high school have an advantage: approx-
imately 60 percent of the students who
took calculus in high school had taken al-
gebra in the 8th grade. However, 1996
NAEP data reveal that only 25 percent of
U.S. 8th graders enrolled in algebra, and
that low-income and minority students
Mathematics Education Dialogues
March 1998 3
EXECUTIVESUMMARYOF
MATHEMATICS
EQUALS
OPPORTUNITY
Richard W. Riley
U.S. SECRETARYOF EDUCATION
In the United States today, mastering mathematics has become
more important than ever. Students with a strong grasp of mathe-
matics have an advantage in academics and in the job market.
The 8th grade is a critical point in mathematics education.
Achievement at that stage clears the way for students to take rig-
orous high school mathematics and science courses—keys to col-
lege entrance and success in the labor force. However, most 8th
and 9th graders lag so far behind in their course taking that get-
ting on the road to college is a long way off.
(The essay above is the Executive Summary of a white paper, Mathematics Equals Opportunity, prepared for
U. S. Secretary of Education Richard W. Riley, 20 October, 1997.)
See Mathematics Equals Opportunity, page 7
RICHARDW. RILEY
Dick Riley was
governor of South
Carolina from 1978 to
1986 and served in
the state legislature
prior to that. His
goals for education
include voluntary
national tests to
ensure that all
students master the
basics of reading and
mathematics, and
organizing one million
volunteer reading
tutors across the
nation.

add the qualification that the algebra and
geometry may be necessary only for train-
ing or licensing. But I find even this
claim, another bald assertion with noth-
ing to back it up, unbelievable. Over 75%
of all jobs? Incredible! I cannot imagine
how that wildly inflated percentage was
arrived at, unless the author was includ-
ing having a high school diploma under
“licensure examination.” Authors who
want to indulge in unsubstantiated per-
centages should be careful to have them
consonant with common sense. For ex-
ample, “99% of the mathematics done by
the average person relates to money”
[2]—now that I can believe.
Almost all jobs, I counter-assert, require
no knowledge of algebra and geometry at
all. You need none to be President of the
United States, none to be a clerk at Wal-
Mart, none to be a professor of philosophy,
… the list extends indefinitely. Few jobs re-
quire knowing any mathematics beyond al-
gebra. You might think that engineers, of
all people, would need and use calculus,
but this seems not to be so [7]:
Why do 50% (probably closer to 70%) of
engineers and science practitioners seldom,
if ever, use mathematics above the elemen-
tary algebra/trigonometry level in their
daily practice?
My work has brought me into contact with
thousands of engineers, but at this moment
I cannot recall, on average, more than three
out of ten who were well versed enough in
calculus and ordinary differential equations
to use either in their daily work.
If 70% of engineers don’t need calculus to
do their jobs, then how many of the
500,000 or so students that we put through
calculus every year will? Minutely few, so
we should not tell them how tremendously
useful calculus is going to be to them when
they go to work. If most engineers can do
quite well with only algebra and trigonom-
etry (or perhaps even less), is it not rea-
sonable that nonengineers can survive and
flourish with arithmetic, or even less? Yes,
it is.
Were algebra necessary for 75 percent
of all jobs, our algebra textbooks would
be filled with on-the-job problems, since
examples would be so plentiful. But they
are not. Open any textbook at random—
I will open a new one, just published—
and what you find are problems like this:
Through experience and analysis, the man-
ager of a storage facility has determined
that the function s(t) = –3t
2 + 12t + 10 mod-
els the approximate amount of product left
in the inventory after t days from the last re-
supply. We want to find when the supply of
the product will be exhausted and a new re-
supply needed.
Real inventories do not behave this way.
For one thing, they do not increase after
the resupply, from 10 at t = 0 to 22 at t = 2.
For another, they usually decrease linearly,
not quadratically. Besides, I doubt that
warehouse managers, even 75% of them,
use formulas to decide when to reorder.
Making fun of the “applications” that ap-
pear in textbooks is as easy as swatting
mosquitoes in a swamp in midsummer,
and as useful. What such problems actu-
ally illustrate is that the mathematics in
the textbooks has no application to the
world of warehouses and work. Does this
mean that we should teach less mathe-
matics? No; we should teach more. Every-
one should learn algebra, but not because
it is necessary for managing warehouses.
Does it mean that we should stop assign-
ing “applied” problems? Certainly not; we
should assign more. Problems expressed
in words are the best kind, but they should
all start with “Suppose that….” If we can’t
be realistic, we can at least be honest.
Those who know not history …
Let us look at history. Those ignorant of
history too often assume, knowing no bet-
ter, that the world has always been much
as it is now, which is seldom so. Today, with
near-universal instruction in arithmetic
and algebra, it is easy to suppose the cur-
riculum has always been like that. But it
has not. Algebra was not always taught to
everyone. Not only that, even arithmetic it-
self is a relative newcomer.
Here is a report from Massachusetts in
the early 1800s. Not the 1700s, nor the
1600s, the 1800s [3, p. 13]:
Until within a few years no studies have
been permitted in the day school but
spelling, reading and writing. Arithmetic
was taught by a few instructors one or two
evenings a week. But in spite of the most de-
termined opposition, arithmetic is now
being permitted in the day school.
Opposition to arithmetic! How could
anyone possibly be opposed to arith-
metic? It is difficult for us to imagine.
4 Mathematics Education Dialogues
March 1998
Is Mathematics Necessary?
continued from page 2
Making fun of
“applications”
…  is as easy as
swatting
mosquitoes
in a swamp in
midsummer,
and as useful.
If we can’t be
realistic, we
can at least be
honest.

The explanation is that arithmetic was a
vulgar subject. As Patricia Cline Cohen
tells us in A Calculating People: The Spread of
Numeracy in Early America, a book that de-
serves to be more widely known [1, p. 26]:
Those of high social rank, theoretically
above the world of getting and spending,
did not deign to study the subject. The most
respectable English public schools, like
Eton and Harrow, did not offer any instruc-
tion in arithmetic until well into the nine-
teenth century.
The English attitude was exported to
the colonies [1, p. 49]:
The founding generation arrived in Massa-
chusetts in the 1630s with the highest num-
ber of university degrees and the highest
rate of literacy of any migratory group.
Within a decade they instructed towns to es-
tablish local grammar schools and had set
up Harvard College to provide high-level
training for homegrown ministers. But
arithmetic was not among the subjects con-
sidered basic for Puritan children to learn.
Nevertheless, the colonies, and Eng-
land, not only survived but thrived, eco-
nomically as well as culturally. Some peo-
ple believe that the eighteenth century
represented a peak of civilization from
which we have declined. I would not go
that far, and I much prefer living in our
times, with its plumbing and penicillin,
computers and compact disks, anesthesia
and even its automobiles, yet history
clearly shows that arithmetic in the
schools is not needed for a high civiliza-
tion. How can that be? Easily enough:
workers learn what they need on the job.
What happens in the schools simply does
not matter.
Here is a report on the situation in
Boston in 1789 [ 1, p. 131]. See if it does
not sound familiar today:
[There was a requirement] that boys aged
eleven to fourteen were to learn a standard-
ized course of arithmetic through fractions.
Prior to this act, arithmetic had not been re-
quired in the Boston schools at all. Within a
few years a group of Boston businessmen
protested to the School Committee that the
pupils taught by the method of arithmetic
instruction then in use were totally unpre-
pared for business. Unfortunately, the edu-
cators in this case insisted that they were
doing an adequate job and refused to make
changes in their programs.
Of course the students were unpre-
pared for business, one reason being that
it is neither wise nor practicable to try to
prepare all students for all possible jobs.
Another is that the “applications” in
school books were just as phony as ours
[1, p. 122]:
Here is a typical word problem, typical in its
complexity and in its use of current events
to suggest the utility of arithmetic:
Suppose General Washington had 800
men and was supplied with provision for
but two months. How many of his men
must leave him, that his provision may
serve the remaining five months?
In this particular case the student mechan-
ically applied the Rule of Three, writing
2 : 800 :: 5 and then dividing 5 into 2 × 800
to get a final answer of 320. Now, 320 is the
number of men who can be fed for five
months, not the number who must leave.
So Washington’s troops would have gone
hungry if the schoolboy or his master had
been in charge of provisioning.
As Professor Cohen pointed out, if
Washington ran short of provisions, he
would try to get more instead of telling
part of his army to go away.
The conclusion cannot be avoided that
school mathematics is not now, and never
has been, necessary for jobs. There are a
few exceptions, of course, most being for
the jobs of teaching the subject. And of
course science—both physical and social—
cannot advance without a supply of scien-
tists able to use mathematics. But most of
these people did not need to be bullied or
cajoled into learning the subject.
Even more advanced mathematics
turns out to be all too often not needed
for work [8]:
Presumably, with degrees in mathematics
and statistics [students with mathematical
majors] could pursue careers in their disci-
plines. But, for mathematicians and statisti-
cians who would seek employment in com-
merce, i.e. in business, industry, or
government, this presumption is not
presently valid. In fact most, if not virtually
all, such mathematical scientists currently
employed in commerce do not work in
their fields of expertise.
This holds even for those with higher
degrees: the National Research Council
“reports that at least 90 percent of
nonacademically employed mathematical
scientists who received master’s degrees
in 1986 do not work as mathematical sci-
entists” [8].
A few years ago I heard an interesting
talk at an MAA section meeting on the
use of mathematics by employees of the
5Mathematics Education Dialogues
March 1998
Some people
believe that the
eighteenth
century
represented a
peak of
civilization from
which we have
declined.
How could
anyone
possibly be
opposed to
arithmetic?
See Is Mathematics Necessary?, page 6

Florida Department of Transportation.
The department needs to calculate many
things, including areas, and its method of
finding the area of irregular shapes was
surprising to me. When I asked the
speaker how the department copes with
new workers with varying degrees of
mathematical training, the answer was
that it doesn’t: it had found that the only
safe assumption is that new workers know
nothing about mathematics, so they are
taught what they need as it is needed.
This is satisfactory to everyone. It does
not imply that the time that new employ-
ees had spent in school trying to do prob-
lems in arithmetic, algebra, and geometry
was wasted, but it had nothing to do with
their jobs. Boston, 1789, Florida, 1993:
some things do not change.
A way of thought.
Despite the initial opposition and con-
tinued irrelevance to jobs, mathematics
instruction spread in the United States in
the nineteenth and twentieth centuries.
As the History of Mathematics Education [3]
tells us, Harvard in 1816 required “the
whole of arithmetic” for entrance. Until
then addition, subtraction, multiplica-
tion, division, and the Rule of Three had
been enough. After 1865, geometry was
required as well. As the country was set-
tled, secondary education expanded, and
arithmetic moved from the academies
and high schools to become an elemen-
tary school subject by the end of the nine-
teenth century [ 3, p. 27]. Algebra was an
optional subject in some high schools,
and it became possible to study calculus
in the upper reaches of some colleges.
Today years and years of mathematics is
compulsory for all and calculus has be-
come a high school subject.
How come? Because parents, school
boards, society as a whole think that
mathematics instruction is worth doing.
On account of applications and jobs? Cer-
tainly not. The reason, I think, is that one
of the tasks of schools is to do their best
to teach students to think, and of all sub-
jects none is better suited to this than
mathematics. In no other subject is it so
clear that reasoning can get results that
are right, verifiably right. When you solve
x
2 + x = 132 and get x = 11, you can then
calculate 112 + 11 and know that you are
correct. No other subject has this capacity
at the elementary levels. Mathematics in-
creases the ability to reason and shows its
power, all at the same time.
It is not fashionable these days to assert
that mathematical training strengthens
the mind, perhaps because that proposi-
tion is as impossible to prove as the
proposition that music and art broaden
and enrich the soul. But it is still believed
by many people, including me. Some of
our forebears had more confidence, as
did John Arbuthnot (1667–1735) whose
On the Usefulness of Mathematical Learning
(c. 1700) proclaimed: “The mathematics
are the friends of religion, inasmuch as
they charm the passions, restrain the im-
petuosity of the imagination, and purge
the mind of error and prejudice” [ 4, p.
70]. Even better, “[M]athematical knowl-
edge adds vigour to the mind, frees it
from prejudice, credulity, and supersti-
tion” [4, p. 67]. Though we no longer say
such things out loud, the belief that they
hold quite a bit of truth goes a long way
toward explaining why people have sup-
ported and continue to support the mass
teaching of mathematics, though many of
them did not enjoy the experience when
they underwent it.
Once a graduate of my school, a mathe-
matics major, came back to campus to
visit. I said to him, after finding out that
his job was running a television station in
Knoxville, Tennessee, “Well, I guess all
that mathematics you learned hasn’t
been very useful.” “Oh no,” he replied, “I
use it every day.” I found this claim in-
credible (soap operas have no partial de-
rivatives), so I pressed him. It turned out
that he meant that he believed he used
the mathematical way of thinking every day.
That is impossible to quantify and im-
possible to prove, but we cannot tell him
that he is wrong. Nor should we.
It is time to stop claiming that mathe-
matics is necessary for jobs. It is time to
stop asserting that students must master
algebra to be able to solve problems
that arise every day, at home or at work.
It is time to stop telling students that
the main reason they should learn
mathematics is that it has applications.
6 Mathematics Education Dialogues
March 1998
… the only safe
assumption is
that new
workers know
nothing about
mathematics,
so they are
taught what
they need as it
is needed.
Can you
recall why
you fell
in love with
mathematics?
Is Mathematics Necessary?
continued from page 5
See Is Mathematics Necessary?, page 14

Students whose
parents are
involved in their
school work are
more likely to
take challenging
mathematics
courses early.
Many mathematics
programs are now
incorporating
fundamentals of
algebra and geometry
into the upper
elementary grade
curriculum
Mathematics Education Dialogues
March 1998 7Mathematics Education Dialogues
March 1998
were even less likely to take algebra in the
8th grade.
• Taking rigorous mathematics and sci-
ence courses in high school appears to be
especially important for low-income stu-
dents. Low-income students who took al-
gebra I and geometry were almost three
times as likely to attend college as those
who did not. While 71 percent of those
who took algebra I and geometry went to
college, only 27 percent who did not take
those courses went on to college. By way
of comparison, 94 percent of students
from high-income families, and 84 per-
cent of students from middle-income
families who took algebra I and geometry
in high school went on to college. Sixty
percent of students from high-income
families and 44 percent of students from
middle-income families who did not take
algebra I and geometry went to college.
• Despite the importance of low-in-
come students taking rigorous mathe-
matics and science courses, these stu-
dents are less likely to take them.
Students from higher-income families
are almost twice as likely as lower-in-
come students to take algebra in middle
school and geometry in high school.
They are more than twice as likely to
take chemistry.
Other important findings include:
• Mathematics achievement depends on
the courses a student takes, not the type
of school the student attends. Students in
public and private schools who took the
same rigorous mathematics courses were
equally likely to score at the highest level
on the NELS 12th grade mathematics
achievement test.
• Students whose parents are involved
in their school work are more likely to
take challenging mathematics courses
early. Students whose parents were in-
volved in their education were more likely
to take courses like algebra and geometry
in the 8th and 9th grade than students
whose parents were not involved.
• The results of the Third Interna-
tional Mathematics and Science Study
(TIMSS) reveal that the middle school
mathematics curriculum may be a weak
link in the U.S. education svstem. While
U.S. 4th graders scored above the inter-
national average in mathematics and sci-
ence, U.S. 8th graders scored below av-
erage in mathematics, and only slightly
above the international average in sci-
ence. Initial analysis of TIMSS data also
shows that the middle school mathe-
matics curriculum in the U.S. is less
challenging than in other countries. The
curriculum of average 8th-grade mathe-
matics classrooms in the U.S. resembles
7th grade curriculum elsewhere. Al-
though algebra and geometry are inte-
gral elements of the middle school cur-
riculum in other countries, only a small
fraction of U.S. middle schools offer
their students these topics. 
Mathematics Equals Opportunity
continued from page 3
Algebra in the Curriculum
Making a successful transition from arithmetic to more advanced mathematics, in-
cluding algebra and geometry, has often been difficult for students. As a result, many
mathematics programs in the U. S. are now systematically incorporating some funda-
mentals of algebra and geometry into the upper elementary grade curriculum. In these
programs, 5th, 6th and 7th grade students are representing and solving equations, char-
acterizing patterns and rates of change among variables, and using other fundamental
algebraic concepts.
In addition, some middle and high schools are taking a new approach to advanced
topics. While many schools offer the traditional model of separate courses for pre-
Algebra, Algebra I, Geometry, Algebra II, Trigonometry, pre-Calculus and Calculus,
these schools are integrating them. This approach is consistent with practices in other
industrialized nations, which integrate algebra, geometry, and other topics throughout
the elementary, middle, and high school years and offer a significant component of al-
gebra in the 8th grade. Building a firm foundation in algebra during the elementary
and middle school years eases the shift from arithmetic to advanced topics, whatever the
format of students’ new curriculum. NELS and NAEP, the two sources of national
mathematics course-taking data analyzed in this brief, employ traditional courses titles,
such as “algebra I” and “geometry.” Thus, these titles are used throughout the brief.  
G

8 Mathematics Education Dialogues
March 1998
Reactions to
“Is Mathematics Necessary?”and
“Mathematics Equals Opportunity”
A Classroom Teacher Speaks Out
by Clay Burkett
U
nderwood Dudley has hit the nail on the head in his ar-
ticle “Is Mathematics Necessary?” The mathematics that
is really necessary for the average citizen is arithmetic—the
mathematics used by the masses to get by in everyday life. I
love the point made that “were algebra necessary …, our al-
gebra textbooks would be filled with on-the-job problems,
since examples would be so plentiful.” Most of the applica-
tion problems I find in traditional textbooks are contrived at
best and do nothing to convince the students of the rele-
vance of mathematics.
However, Mr. Dudley’s conclusion that mathematics tran-
scends jobs because of the elegance and beauty that lies
within does little to assist the classroom teacher. The ele-
gance and beauty of mathematics do not intrigue the vast
Mathematics as Externality: Implications
for Education
by Roger Howe
U
nderwood Dudley sees mathematics as a sharpener of
minds, and underscores its beauty. Secretary Riley tells
students that mathematics is a key credential. Two justifications
for mathematics could hardly be more different, but strictly
speaking, they are not in conflict. Secretary Riley does not
parry, but neatly sidesteps, Professor Dudley’s thrust. He never
claims, as do the NRC documents that Dudley so deftly skew-
ers, that mathematics is necessary for careers. He only claims
that it is necessary, or almost so, for college attendance. His fig-
ures are not round, and they are differentiated by ethnicity.
They have an a priori plausibility that the NRC’s claims lack.
What Secretary Riley is giving is the bleakest reason imag-
inable to study mathematics: no beauty and no utility—just a
See Clay Burkett, page 10
See Roger Howe, page 10
Clay Burkett
Clay Burkett has been in-
volved in mathematics edu-
cation for nine years, teach-
ing everything from Saxon
to SIMMS (Systemic Initia-
tive for Montana Mathe-
matics and Science). One
year of his experience was spent writing curricu-
lum for the SIMMS and STEM (Sixth through
Eighth Mathematics) projects. He enjoys spending
time with his wife and three children in the great
outdoors of Montana.
Roger Howe
Roger Howe did graduate
study under Calvin Moore at
the University of California
at Berkeley. He received his
Ph.D. in 1969 and has been
at Yale since 1974. He
studies the implications of symmetry, especially
the theory of group representations. In the 1990s,
he has become more and more involved in
mathematics education at all levels.

Mathematics Education Dialogues
March 1998 9
The Splintered Vision:
Wayfarers in Search of Different Roads.
by Merecedes McGowen
The wayfarer,
Perceiving the pathway to truth,
Was struck with astonishment.
It was thickly grown with weeds….
Later he saw that each weed
Was a singular knife.
“Well,” he mumbled at last,
“Doubtless there are other roads.”
— Stephen Crane
The Wayfarer
M
ore than one-third of the 3.27 million undergraduate
students enrolled in mathematics courses at two- and-
four year colleges and universities in 1995 were enrolled in
Reaction of a Curriculum Developer
by John C. Souders Jr.
C
ould two more diametrically opposed pieces of writing
be found? In one, mathematics is presented as unneces-
sary for success in the world of work. In the other, mathe-
matics is seen as the “gatekeeper” for future academic and
career success. I’m not surprised that such a wide spectrum
of opinion exists. After all, the teaching and learning of
mathematics have been controversial for decades. I compli-
ment the authors. Both present substantive and thought-
provoking arguments.
The rallying call of Dudley’s article could well have been the
often-asked student question “Why do I have to learn this?” Ac-
cording to the article, a mastery of arithmetic is sufficient for
A Concerned Citizen’s Perspective
by Penny Noyce
U
nderwood Dudley asks, “Is Mathematics Necessary?”
and assures us that for the vast majority of jobs, calcu-
lus, geometry, and even algebra are not. He’s probably right.
For entry into medical school, I was required to take calcu-
lus; but I never needed calculus for learning anatomy or
pharmacology, and I have certainly never used it in patient
care. Calculus was a rite of passage, perhaps a weeding-out
mechanism. What I do use, both in reading the medical lit-
erature and in thinking about patients, is probability.
Similarly, several years ago, the UNUM insurance com-
pany surveyed what mathematics the members of its work
force actually required for their work. The findings? No al-
gebra was necessary, but probability and statistics were vital. 
See John C. Souders Jr., page 14
See Mercedes McGowen, page 11
See Penny Noyce, page 13
Merecedes McGowen
Mercedes McGowen, [EMAIL],
teaches mathematics at William Rainey Harper
College, Palatine, Illinois, a two-year college in the
northwest suburbs of Chicago. Previously, she
taught mathematics at Canton Junior High School
and Elgin High School in Illinois. Her interests
include the development of curricular materials
based on research about how students learn
mathematics.
Penny Noyce
Trained as a physician with a specialty in internal
medicine, Penny Noyce now spends most of her
time working on issues of K–12 public education 
reform as a trustee of the Noyce Foundation. The
Foundation focuses on the core academic areas of
literacy, mathematics, and science. For the past
three years, Noyce has served as coprincipal 
investigator of PALMS, the Massachusetts State 
Systemic Initiative.
John C.
Souders Jr.
John C. Souders Jr. is vice-
president for curriculum
materials with CORD
(Center for Occupational
Research and Development).
At CORD he developed the new textbook CORD
Algebra and four new units in the Applied 
Mathematics Series that deal with higher order
geometry topics. Previously Souders taught at the
United States Air Force Academy and worked in
nuclear engineering.

10 Mathematics Education Dialogues
March 1998
majority of my students. They are more interested in the el-
egance and beauty of the opposite sex, or music, or movies,
or sports …. I recently came across a profound statement
that reflects our culture and hence the attitudes of our stu-
dents. My paraphrase of this statement is, We in America
like to think that we value art and education, but what we
really value are sports and entertainment. Students want to
know how and where mathematics is applicable to their
lives. Relevance will motivate them to learn. We can get
them to pass our classes through other means, but let’s not
confuse a grade with learning and valuing mathematics.
What about college-bound students? Are they not moti-
vated to learn mathematics? Or, as the paper “Mathematics
Equals Opportunity” suggests, does taking rigorous mathe-
matics courses spur students on to attend college? It does
not surprise me that students who take rigorous mathe-
matics courses in high school tend to attend college in
greater numbers than students who do not. However, any
half-decent mathematician knows the difference between
correlation and causation. After all, don’t many colleges re-
quire these types of mathematics classes as prerequisites to
admission? The thinking behind this paper appears to be
that of mathematics as a pump—propelling students on to
achievement in college and beyond. I like this idea and
find it somewhat valid, yet to some degree mathematics will
also act as a filter—separating the less motivated, less disci-
plined, and yes, even the less able.
As for the idea that college-bound students are motivated
to learn mathematics, I submit the following informal survey.
As a quick background to the survey, I currently teach Hon-
ors Geometry and Integrated Math IV, both of which meet
college-entrance requirements. I had my students read the
two articles of interest here and then held a discussion of the
ideas contained therein. Many of my students believed that
their high school mathematics courses were of no relevance
to their lives or future careers—but merely hoops that they
must jump through. When asked for a show of hands, thirty-
one of forty-two Honors Geometry and seventeen of forty-
one Integrated Math IV students indicated agreement.
One thing that caught my attention was that a greater
proportion of students in the Integrated Math IV class
thought that the mathematics they were learning was rele-
vant to their lives in preparing them for college or a career.
This fact did not come as a surprise to me, as the curricu-
lum used in the Integrated Math IV class is one that is
based on problem solving and applications. These students
tend to see the relevance of mathematics more clearly be-
cause it is taught in a context of how and where the mathe-
matics is used.
In conclusion I find some merit and some fault in both
articles. Mathematics is not necessary on the one hand;
and yet on the other, mathematics equals opportunity (has
anyone been denied a job or college entrance for knowing
too much mathematics?) for college and beyond. To me
these articles point out the need for mathematics reform.
Will we continue on with a mathematics curriculum that
the TIMSS report describes as “a mile wide and an inch
deep,” or will we work to develop a curriculum that is
meaningful and relevant to our students? G
credential, a hoop to jump through. However, in a sense its
bleakness is a strength. It gets us an audience with no
promises made except for the carrot of college attendance.
If we then can help some member of this audience to
sharpen their thinking habits, and to see the beauty we
know in mathematics, then they have gained a precious
treasure, and so have we.
Why do we have this problem? Why is it necessary to “jus-
tify’ ’ mathematics to students. And what, exactly, is wrong
with the usefulness justification? Is mathematics not useful?
On the contrary, mathematics is more than useful; it is nec-
essary, but not for (most) individuals. This is a great paradox
of mathematics education: although our society is utterly de-
pendent on mathematics for many of our daily needs, and
even for the very shape of our civilization, for the most part
we do not need personally to be able to master very much of
the mathematics that serves us. It is built into our products,
encoded in our practices, or available at a fee. It is the 
enduring luck of the species, and the sometimes bane of the
profession, that mathematics is indestructible, infinitely recy-
clable, and totally fungible. A little goes a long, long way.
In Quantum magazine, an excellent publication, I read a
wonderful detective story starring Johannes Kepler, the man
who also discovered the laws of planetary motion. Kepler was
in Linz, Austria, to get married. He needed wine for the wed-
dding party. He was intrigued to see that the wine merchants
of Linz computed the capacity of their barrels by making a
single measurement, diagonally from the bunghole in the
middle of the side, to the top of the far wall. Barrels are not
uniform in shape or size, so how could a single measurement
provide an adequate estimate of the volume? This simple
method was a trade secret of the Linz Coopers (Barrelmak-
ers) Guild. Kepler wanted to understand their secret. He
modeled the barrels as two truncated cones joined end to
end, and reasoned that the barrels must be made to have ap-
proximately the shape that would make the volume as large as
possible for the given measurement. Maximizing the volume
would provide two benefits at once. First, the vintners could
charge the maximum amount for the given measurement.
Second, because a maximum is also a stationary point, the vol-
ume will vary very little if the barrel differs slightly from the
optimal shape, so customers will accept this method of calcu-
lation. After doing a little calculus (actually, precalculus, since
calculus had not yet been invented!), Kepler concluded that
the barrels should be cylindrical, with a ratio of height to di-
ameter of . Allowing for the realities of manufacture
Reactions—Clay Burkett
continued from page 8
Reactions—Roger Howe
continued from page 8
2

Mathematics Education Dialogues
March 1998 11
(nonzero thickness of the barrel top, etc.), one finds that 1.5
is a sufficiently accurate, numerically manageable proxy. Once
the shape is known, translating the length measurement into
volume is a simple calculation, and the volumes for given
lengths can be inscribed on the ruler. Having deduced this
fact, Kepler sought an interview with the Elder of the Coopers
Guild, to whom he revealed that he had discovered their se-
crets of barrel construction and measurement. The
amazed Elder acknowledged that Kepler was correct, and
said that these were trade practices handed down through
the generations since the days of the Venerable Cooper.
Here was a sophisticated and very useful piece of mathe-
matics, embedded as a simple rule of thumb in the
cooper’s trade. It took a genius to reverse engineer it, and
undoubtedly also, a genius to invent it, but it could be
learned by rote by any apprentice cooper.
A very large part of the mathematics that we use is like the
cooper’s ratio—embedded in rules of thumb of various
trades and professions. On the one hand, we don’t have to
understand it to use it. Also now, mathematics gets built into
products. Especially, very sophisticated mathematics can be
incorporated in silicon, which will reliably perform compu-
tations that would boggle us but about which we, like the six-
teenth-century coopers of Linz or the English nobility be-
fore the nineteenth century, need not bother ourselves.
On the other hand, some of us have to know this mathe-
matics, or the rest of us cannot benefit from it. And we
aren’t sure beforehand who those some of us are. This
huge gap, between the mathematics that most of us will ob-
viously have to cope with on a daily basis and the mathe-
matics that some of us have to know so that the rest of us
can enjoy it in ignorance, is a major cause of the conun-
drum of prescribing mathematics for education.
So how much mathematics, and of what type, should
every person learn today? Our democratic ideology is an
important factor in these calculations. It demands that all
students be educated to maximize their personal capacities.
This ideal means that every student should be given a shot
at qualifying for a job that really does require mathematics
(since these are obviously the desirable jobs!). Advisors do
not tell students that they should not take mathematics—on
the contrary, as Secretary Riley properly does, they tell them
that they should. For most students, any reason given, other
than Secreatry Riley’s, will be false (even, alas, the beauty
reason, since beauty is in the eye of the beholder—if mathe-
matical beauty were abundantely evident to the average per-
son, Professor Dudley would have to use his sword to fend
off students rather than to make shishkebab of anonymous
NRC authors). It follows that many if not most students in
the more advanced mathematics courses should not be
there. They will not find it useful in further life, or beauti-
ful, or otherwise worthwhile. They will decide not to take
the next one. I think this situation is essentially unavoid-
able. Better teaching can ameliorate it but not eliminate it.
The attrition rate throughout high school and college is
widely quoted as 50 percent per year. If we double the effec-
tiveness of our teaching system, a tremendous accomplish-
ment of which we could justly be extremely proud, we
would delay losses by only one year. Attrition in mathe-
matics courses will continue to be a challenge to the profes-
sion. We should strive to improve mathematics instruction
and mathematics curriculum. However, mathematics educa-
tion will continue to serve as a sorting process.
Clearly, some waste occurs in this system. But we are talk-
ing about reproduction here—the reproduction of the
mathematical expertise vital to the functioning of modern
society—and reproduction is so important that vast re-
sources are lavished on it. Think of how many maple seeds
a maple tree produces each year to create a few viable off-
spring over its lifetime. How wasteful is our system of
mathematics education? Consider calculus. Suppose that 1
percent of us will actually use calculus in our jobs (Which
may be a wild overestimate, but it’s a lot harder to debunk
than 75%!). That 1 percent is actually a lot of people—
over 30 000 a year. So if about 600 000 people per year are
taking calculus, as many as 5 percent of those studying cal-
culus may actually use it. (Probably most of these people
are learning it in an AP high school course, which under-
lines the importance of getting this course right and of
having good teachers at this level.) Perhaps two to three
times as many will understand and enjoy it, and remember
some of it. Even larger numbers will take away a few nice
examples or lessons, or find it somewhat interesting at the
time, or at least like their teacher. Also, a lot of these peo-
ple are premedical students. Given the importance of tech-
nical expertise to society as a whole, and the doctrines of
liberal education and of letting people discover their own
limitations, I find these numbers broadly acceptable. 
With all this said, I think that Professor Dudley is right.
We should try to sell mathematics on the basis of its beauty,
its power, its rigor. However, for me, the usefulness of
mathematics, even in fairly mundane situations, is a part of
its beauty. And since I know that students may not always
on the first crack find the inspiring teacher who communi-
cates the wonder of mathematics, I am glad that Secretary
Riley is telling them how necessary it is as a credential.
Reference
Bak, M. B. “The Venerable Cooper.” Quantum, May 1990,
36–39. G
Reactions—Roger Howe
continued from page 10
remedial mathematics courses, that is, arithmetic, algebra,
and geometry (MAA 1997). Dropout rates as high as 50
percent in the traditional remedial courses have been cited
(Hillel at al. 1992). According to the National Center for
Education Statistics (1997), only 27 percent of students
who enrolled in college completed four years despite the
fact that 68 percent of incoming freshman at four-year 
See Mercedes McGowen, page 12
Reactions—Mercedes McGowen
continued from page 9

12 Mathematics Education Dialogues
March 1998
colleges and universities had taken four years of mathe-
matics in high school (NCES 1997). These students paid
college tuition for courses that do not count for credit to-
ward graduation at most colleges and universities.
What do we tell these students that makes sense to them
about mathematics and why they should take mathematics
courses? Dudley’s conclusion that “mathematics is more
important than jobs” and that “it transcends them” reflects
the beliefs and experiences of a successful practicing math-
ematician. However, his experiences are not those of a ma-
jority of students studying mathematics today. The students
enrolled in undergraduate remedial mathematics courses
are commonly left with feelings of failure and a belief that
mathematics is irrelevant, feelings very different from
those described by Dudley. For these students, mathe-
matics inspires fear not awe, discouragement not jubila-
tion, a sense of hopelessness not amazement. Why do so
many students who attempt rigorous mathematics courses
not succeed? Even many of those who complete three or
four years of “rigorous” high school mathematics are un-
successful in subsequent college-level mathematics courses. 
We need to have a clearer understanding of the differ-
ences and needs of the individual students in our classes,
and we must take these differences into account in our cur-
ricular design and instructional practices. We have not yet
figured out how to deal with those differences in ways that
are appropriate to achieve the goal of mathematical power
for all our students. The beliefs about what constitutes
mathematics, what skills should be taught, when they
should be taught, and to whom vary from individual to in-
dividual and community to community. The recent United
States report on the Third International Mathematics and
Science Study curriculum analysis (Beaton et al. 1996) cites
these conflicting beliefs and practices, describing the cur-
rent United States mathematics curriculum as unfocused,
“a splintered vision,” which is reflected in our mathematics
curricular intentions, textbooks, and teacher practices. In
comparison to other countries, the U.S. “adds many topics
to its mathematics and science curriculum at early grades
and tends to keep them in the curriculum longer than
other countries do. The result is a curriculum that superfi-
cially covers the same topics year after year—a breadth
rather than a depth approach.”
Terms whose meanings were once commonly understood
by those engaged in the practices of mathematics now have
different meanings and serve as flashpoints for increasingly
vehement discourse. Dialogue based on a common lan-
guage and definitions has become extremely difficult, as
Humpty Dumpty pointed out to Alice in Through the Look-
ing Glass: “You see, it’s like a portmanteau—there are two
meanings packed up into one word.”
In the absence of mutually agreed-on definitions and ac-
cepted meanings, the debate continues among those who
favor a “return to basics” and those who are attempting to
implement reforms in the teaching and learning of school
mathematics, with increasingly high costs. Our vision has
become not only fragmented but clouded by emotion. Wit-
ness the ongoing saga in California, where efforts to estab-
lish a set of statewide mathematics standards have gener-
ated contentious debate and vehemence on both sides.
Competing visions of what mathematics students should
learn have polarized mathematics practitioners and educa-
tors, students, their parents, and the community at large.
Robert Davis, in an electronic mail communication (1996),
described the position in which we trap students: “There is
at present a tug of war going on in education between a
‘drill and practice and back to basics’ orientation that fo-
cuses primarily on memorizing mathematics as meaning-
less rote algorithms vs. an approach based upon under-
standing and making creative use of mathematics.” Does
the current splintered vision of mathematics really serve
the best interests of mathematicians, teachers, students,
and the public? What do we really mean by “Algebra for
All,” and what mathematics should we be teaching? In our
efforts to make mathematics accessible and attractive to a
large number of students, are we, as Cuoco (1995) worries,
“changing the very definition of mathematics?” Will it be a
fundamentally different discipline in the future? Should it
be a different discipline for some, for all, for none?
I agree that we should not tell students lies about why they
should study mathematics. I also agree with the authors of
“Mathematics Equals Opportunity” that students should
study more mathematics earlier so as not to close off their
options. However, I am no longer certain what it is that we
should tell students. Even those who are successful in their
mathematical endeavors in school often fail to recognize
how what they learned in school is used in their work envi-
ronment. The extent to which problem-solving skills and the
use of symbols to mathematize situations are recognized in
the workplace frequently goes unnoticed by employers as
well. The assumption that algebra is the key to well-paying
jobs and a competitive workforce requires the efforts of
mathematically knowledgeable observers to support this as-
sertion with data and to question the beliefs about what
should be taught and how it should be taught in schools. 
I disagree with Dudley’s conclusion that mathematics is
sufficient, not necessary. The failure to take rigorous
mathematics courses has significant economic conse-
quences in terms of future earnings, productivity, and sta-
ble employment. The United States Bureau of Labor Statis-
tics (1997) predicts that, in the years between 1994 and
2005, occupations that require a bachelor’s degree or
above will average 23 percent growth, almost double the 12
percent growth projected for occupations that require less
education and training, and that jobs requiring the most
education and training will grow faster than jobs with lower
education and training requirements. Typically, state col-
leges and universities recommend, and often require, that
students take at least three years of mathematics in high
school for entrance. Graduation requirements often in-
clude several more rigorous courses in mathematics or 
Reactions—Mercedes McGowen
continued from page 11

Mathematics Education Dialogues
March 1998 13
At the same time, “Mathematics Equals Opportunity”
argues for the prime importance of algebra, and urges
us to introduce algebra and geometry into the middle
school or even earlier. What are we to make of the no-
tion that first-year algebra is a “gateway” course? Is it just
that the kinds of students who are likely to take first-year
algebra early—higher-income students with involved
parents—are the same group who are likely to attend
college? Is it that success in algebra selects students ca-
pable of abstract reasoning? Or does algebra actually in-
crease the ability to think, as Dudley asserts for all of
mathematics? 
I suspect that mathematics study is valuable because it
accustoms us to a rigorous, quantitative approach to prob-
lems. But whatever the reason, algebra success is associated
with college attendance; and major experimental interven-
tions, chiefly the Equity 2000 project, are under way to de-
termine whether extending algebra and geometry to poor
and minority students does in fact increase their rate of
college attendance. Still, we are confused about how best
to move algebra and geometry down the grade levels. The
temptation is to transfer the traditional high school course
without adjustment, but such a strategy may crowd out the
possibility of allowing middle school students to explore
rich content in discrete mathematics, probability, or num-
ber theory. We may succeed only in creating more students
who do mathematics but dislike it.
One barrier to progress in the mathematics curriculum
is the conflict between what we want to do and how we
count progress. For example, the most effective strategy for
teaching advanced mathematics earlier may be to weave
concepts of algebra and geometry through several years of
the elementary and middle school curriculum, but what we
count is the number of students taking a course called al-
gebra in eighth grade.
Undoubtedly, much of the current upper elementary
and middle school mathematics curriculum is repetitive
and unchallenging. Students can do more. Diverse and
challenging curriculum materials exist. Graphing calcula-
tors allow students, through experimentation, to reach an
intuitive understanding of how functions behave with
much less tedium and more delight than was true for those
of us who once graphed everything by hand.
But if teachers are to use these new materials and tools
successfully, they need to make significant changes in their
teaching. Nor is change at one or two grade levels enough.
High school, middle school, and even elementary teachers
need to work together to plan, prepare, and evaluate a pro-
gression of courses, including integrated courses, that in-
troduce not only algebra and geometry but probability and
statistics, discrete mathematics, and number theory. Suc-
cessful acceleration of all students, not just those most pre-
cocious at abstract thinking, will require long-term invest-
ment in professional development, examination of student
work, and continual adjustment of teaching strategies to
make sure that students actually understand. It won’t hap-
pen overnight.  G
science, and employers generally require applicants to pass
standardized mathematics and reading tests. 
The question is not “Is mathematics necessary?” but
“What mathematics do we want students to learn?” and
“How do we stop building Alban houses with windows shut
down so close some students’ spirits cannot see?” (Dickin-
son 1950). To answer those questions, we must find ways to
reconcile the different understandings of mathematics
held by parents (“What I learned in school”); by students
(“A hoop to jump through” requirement for high school
graduation, entry into college, college graduation); by
teachers (“What’s in the textbook”); and by such mathe-
maticians as Dudley, for whom mathematics is the subject
that is “the human race’s supreme intellectual achieve-
ment”—that increases the ability to reason, inspiring awe,
jubilation, and a sense of power and amazement.”
References
Beaton, Albert E., et al. Mathematics Achievement in the 
Middle School Years: IEA’s Third International Mathematics
and  Science Study (TIMSS) . Chestnut Hill, Mass.: TIMSS
International Study Center, 1996.
Cuoco, Al. “Soundoff: Some Worries about Mathematics
Education.” Mathematics Teacher88 (March 1995): 186–87.
Davis, Robert B. Electronic mail discussion. 1996.
Dickinson, Emily. “Bring Me the Sunset in a Cup.” In Com-
bined Edition of Modern American Poetry and Modern British
Poetry, edited by Louis Untermeyer, 99. New York: Har-
court, Brace & Co., 1950.
Mathematical Association of America (MAA). Statistical Ab-
stract of Undergraduate Programs in the Mathematical Sciences
in the United States: Fall 1995 CBMS Survey. MAA Reports
No. 2. Washington, D.C.: MAA, 1997.
Hillel, Joel, Lee, Lesley, Laborde, Colette, and Linchevski,
Liora. “Basic Functions through the Lens of Computer
Algebra Systems.” Journal of Mathematical Behavior 11
(1992): 119–58.
National Center for Education Statistics (NCES). Findings
from Education and the Economy: An Indicators Report. Wash-
ington, D.C.: U. S. Government Printing Office, 1997.
http://nces.ed.gov/pubs97/97939.html
U. S. Department of Labor, Bureau of Labor Statistics. Oc-
cupational Outlook Handbook. Washington, D.C.: U. S. Gov-
ernment Printing Office, 1997.
http://stats.bls.gov:80/oco2003.htm
G
Reactions—Penny Noyce
continued from page 9
Reactions—Mercedes McGowen
continued from page 12

14 Mathematics Education Dialogues
March 1998
most people to survive in today’s high-technology world and
the learning of advanced mathematics topics is unnecessary.
For a while, I thought Dudley was basing this premise strictly
on the operational mathematics required for most jobs and
was ignoring the thought processes that mathematics acts as
a catalyst to build. Then in the section “As a Way of
Thought,” Dudley makes a strong case that mathematics
plays an important role in developing reasoning skills. How-
ever, since this assertion can be proved only anecdotally, he
never definitively uses it as a counterbalance to his “math is
not necessary” argument.
I would like to provide the counterbalance. The residual ef-
fect of studying mathematics is not a mastery of how to ma-
nipulate variables or perform operations on them. For most
people, mathematics provides a platform for developing prob-
lem-solving skills that are rooted in logic and based on avail-
able facts. As people study more rigorous mathematics, their
mastery of these skills continues to grow and their ability to
discern cause-and-effect relationships sharpens. As businesses
continue to flatten their organizational structures, employees
will assume positions of greater responsibility and will be ex-
pected to solve problems more quickly and accurately. There-
fore, any process that enhances problem-solving is valuable,
indeed necessary, because it empowers people to reach their
fullest potential as citizens and employees. If Dudley can as-
sert without proof that mathematics transcends jobs, I feel
comfortable stating that mathematics complements jobs.
Next, let me address the viewpoint expressed by Secretary
Riley. Many hold the opinion that mathematics in today’s
world is essential and has meaning for the vast majority of
our citizenry. Therefore, in principle, it is not difficult to sup-
port the Executive Summary and its proclamation that all
high school students should take a rigorous mathematics cur-
riculum. Even though this proclamation is timely and appro-
priate, is it practical and implementable in today’s educa-
tional environment? Although many barriers stand in the way
of successful implementation, one deserves special attention.
Most of the rigorous mathematics taught in our schools is
presented in an abstract manner. As research (e.g., Perkins
[1995]) has shown, many of our students are not adept at ab-
stract thinking. This mismatch must be addressed if all stu-
dents are to take a rigorous mathematics curriculum. 
The good news is that such progressive organizations as
NCTM and, more recently, the American Mathematical As-
sociation of Two-Year Colleges have published standards
that provide a framework for presenting mathematics from
a more concrete perspective. Both sets of standards advo-
cate a contextual, applied, and hands-on approach to pre-
senting mathematics. This stance does not mean that these
standards abandon abstract presentations; rather they seek
a balance between abstraction and the concrete insertion of
mathematical relevance and real-world experiences. Such
balance is supported by research (e.g., Caine and Caine
[1991]; Kolb [1984]) on learning styles and will address the
needs of a much broader cross section of students. When
Reactions—John C. Souders Jr.
continued from page 9
this balance is achieved at the classroom level through new
teaching and learning strategies, the proclamation can
meet its goal and be truly viable for all students.
References
Caine, Renate Nummela, and Geoffrey Caine. Making Connec-
tions: Teaching and the Human Brain.Alexandria, Va.: Associa-
tion for Supervisors and Curriculum Development, 1991.
Kolb, David A. Experiential Learning: Experience as the Source
of Learning and Development. Upper Saddle River, N.J.:
Prentice-Hall, 1984. 
Perkins, David. Outsmarting IQ, the Emerging Science of Learn-
able Intelligence. New York: The Free Press, 1995.  G
We should not tell our students lies. They will find us
out, sooner or later.
Besides, it demeans mathematics to justify it by appeals
to work, to getting and spending. Mathematics is above
that—far, far above. Can you recall why you fell in love with
mathematics? It was not, I think, because of its usefulness
in controlling inventories. Was it not instead because of the
delight, the feelings of power and satisfaction it gave; the
theorems that inspired awe, or jubilation, or amazement;
the wonder and glory of what I think is the human race’s
supreme intellectual achievement? Mathematics is more
important than jobs. It transcends them, it does not need
them.
Is mathematics necessary? No. But it is sufficient.
References
1. Patricia Cline Cohen, A Calculating People, University of
Chicago Press, Chicago, 1982.
2. Philip J. Davis, review of Math Curse, SIAM News 29:7
(1996) 7.
3. Philip S. Jones, ed., A History of Mathematics Education in
the United States and Canada, National Council of Teach-
ers of Mathematics, Washington, DC, 1970.
4. Robert Edouard Moritz, Memorabilia Mathematica, reprint
of the 1914 edition, Mathematical Association of Amer-
ica, Washington, DC, no date.
5. National Academy of Sciences, Moving Beyond Myths, Na-
tional Academy Press, Washington, DC, 1991.
6. National Research Council, Everybody Counts, National
Academy Press, Washington, DC, 1989.
7. Robert W. Pearson, Why don’t most engineers use un-
dergraduate mathematics in their professional work?,
UME Trends 3:3 (1991) 8.
8.Michael Sturgeon, The occupational displacement of
mathematical scientists in commerce, UME Trends 3:4
(1991) 8. G
Is Mathematics Necessary?
continued from page 6

Mathematics Education Dialogues
March 1998 15
I
n arithmetic, “long division” is an algorithm for finding
the quotient of two numbers. The display of 4320 ÷ 75
shown below is typical and leads either to the quotient
57 with remainder 45 or to 57 45/75 or 57.6 if carried out to
another decimal place. Long division is characteristically
taught in fourth grade with whole numbers, and then in
fifth or sixth grade with decimals. All books that teach long
division start with one-digit divisors and then extend it to
more digits.
The NCTM’s (Reston, Va.: NCTM 1989) Curriculum and
Evaluation Standards for School Mathematics recommends that
long division, and long division without remainders, be given
decreased attention in K–4 mathematics (p. 21). Long division
is not specifically mentioned in the 5–8 Standards, but they
make the following statements: “Performing two-digit compu-
tations with whole numbers or decimals aids students in un-
derstanding connections between computation and numera-
tion. Even though students can explore paper-and-pencil
computations with numbers of any size and with various sys-
tems, they should not be expected to become proficient with
paper-and-pencil computations with several digits. A curricu-
lum that incorporated this standard would not include paper-
and-pencil practice for proficiency with tedious computations,
such as those with three-digit multipliers or divisors…” (p. 96).
A second and similar long-division algorithm is used with
polynomials. The display of ( x
2 + 5 x + 9) ÷ ( x + 1) shown
above at the left leads either to the quotient x + 4 with
remainder 5 or to the quotient
The Standards do not mention this long division explicitly
but say, “For college-intending students who can expect to use
their algebraic skills more often [than other students], an ap-
propriate level of proficiency remains a goal. Even for these
students, however, available and projected technology forces a
rethinking of the level of skill expectations” (p. 150). The
technology that was projected in 1989 has appeared: two com-
panies manufacture calculators that can symbolically divide
polynomials and obtain the answers found by long division.
Is Long Division Obsolete?
What are your personal opinions about teaching these
long-division algorithms, and why do you hold these opin-
ions? Should the recommendations in the next version of
the Standards with regard to long division be different from
the recommendations given in 1989?
E-mail your comments to [EMAIL]. You can
also mail your response to “Dialogues, 1906 Association
Drive, Reston, VA 20191-1593, or fax your response to
NCTM at [PHONE]— Attention: Dialogues. Selected re-
sponses will appear in a future issue of Dialogues.  G
Is Long Division
Obsolete?
“Long division—something else I don’t understand … I’ve just about
resigned myself to leading a life of quiet desperation.”
)75 4320
375
570
525
4
57
  
  
    5
)xx x
xx
x
++ +
+
+
15 9
4
4
5
4
2
2
       x + 9
       x + 4
               
x
x
++
+
4 5
1
.

16 Mathematics Education Dialogues
March 1998
Responses to
“Is Long Division Obsolete?”
From the Pen of a K–16 Educator
by Stephen S. Willoughby
The Standards ought to be briefer and less prescriptive
than they are, limited to describing the important things
people should be able to do and the attitudes they should
acquire toward mathematics. Among many other things,
people should be proficient with basic facts and most
multidigit algorithms; they should see mathematics as
something they can figure out rather than memorize, and
that is fun and useful rather than unpleasant and useless.
They should also understand division. Whether they be-
come proficient at long division is probably of very little
consequence as long as they don’t spend too much time
learning it. 
Today, long division is analogous to square roots when I
was teaching eighth grade in Massachusetts. The curricu-
lum called for teaching square roots. I asked how the stu-
dents knew that the square root of 25 was 5. They ex-
plained that 5 times 5 is 25. I asked them to find the square
root of 30. They knew that 6 was too large so tried 5.5. That
number was a bit too large. They tried 5.4. Too small. They
Our Methodology Is Obsolete
by Susan Addington
Long division is not obsolete, for pedagogical reasons:
the process of long division, if carefully taught, can rein-
force important elementary mathematical concepts and
lead the way to advanced mathematics. Although calcula-
tors can compute quotients, they give the result as a trun-
cated decimal, and, as a result, obscure concepts related to
remainders and infinite sums. Specifically, long division
has these advantages over “calculator division”:
• Long division gives a constructive (in the mathematical,
not the educational, sense) way of obtaining a quotient of
integers. A calculator is essentially a black box that gives the
answer with no explanation. In particular, the long-division
algorithm can reinforce the repeated-subtraction model of
division; see the subsequent cookies problems.
• Long division naturally gives a whole-number remain-
der. A calculator doesn’t. (In actuality, some calculators do,
but these remain hard to find—a busy parent shopping for
school supplies at the supermarket won’t find one.)
See Responses—Stephen S. Willoughby, page 18See Responses—Susan Addington, page 17
Susan
Addington
Susan Addington is 
an associate professor of
mathematics at California
State University, San
Bernardino. Her interests
include preservice and in-service education for
elementary and middle school teachers, writing
mathematics for the World Wide Web, and pre-
senting mathematics to the public.
Stephen S.
Willoughby
Steve Willoughby has
taught all grades from first
through graduate school.
He is interested in improv-
ing learning and teaching
of mathematics at all levels. He has written more
than 200 articles and books on mathematics and
mathematics education. He is professor of mathe-
matics at the Univeristy of Arizona and principal
author of the K–6 series SRA Math: Explorations
and Applications.

Mathematics Education Dialogues
March 1998 17
Not only do remainders come up naturally in simple ap-
plied problems, but remainder arithmetic is also of central
importance in such high-powered applications as cryptogra-
phy and coding theory.
Problem: How many 44-passenger school buses will
450 students need?
Problem: Figure out a method to find the whole-
number quotient and remainder using only
a standard calculator. Try it on 15263794
divided by 3 572.
• Long division gives the first natural context in which
you should regularly disbelieve your calculator. Different
calculators will give the quotient of 158 by 9 as
17.555 55556, 17.555 555, and 17.56. Which is right? Why
does the apparent pattern of 5s in the first answer change
to a 6? Why is the third answer so short? Students need to
be aware that calculators round off, and that different cal-
culators use different algorithms for the same calculation.
• Long division gives the first formal exposure to the
concept of infinity—the algorithm generates infinite
repeating-decimal expansions. This idea should be used as
a prologue to calculus, since it is a convincing demonstra-
tion that an infinite set of numbers can have a finite sum.
Pedagogy
Long division has traditionally been the most disliked
part of elementary arithmetic. Its many steps require care-
ful attention to the rules of the procedure, neatness in
aligning the digits, and sophisticated mental arithmetic. I
suspect that the efficiency of this algorithm is an artifact
of the days when students did their work on small slates—
too many steps wouldn’t fit. Here I offer some pedagogi-
cal suggestions for teachers of students who have lots of
paper.
Since multiplication is repeated addition, division is re-
peated subtraction. You can convince second graders of
this conclusion:
Problem: You are filling bags with 7 cookies each. You
have 35 cookies. How many bags can you fill?
35
– 7
(fill one bag)
28
– 7 (fill one bag)
21
– 7 (fill one bag)
14
– 7 (fill one bag)
7
– 7 (fill one bag)
0
Conclusion: 5 bags of 7, and no cookies left over
The method works just as well with large numbers, such
as 15 263 794 divided by 3 572, although it might take a
long time.
The “ladder” method, which I learned in the New Math
era, is almost as transparent. Instead of subtracting the divi-
sor, you subtract 10, or 100, or 1000 times the divisor, as
large as possible. Keep track of what multiples of the divi-
sor were subtracted in a column on the right. 
Example: 1619 divided by 7 has quotient 231 and
remainder 2.
Responses—Susan Addington
continued from page 16
)7 1619
1400 200
219
210 30
9
71
2 231
–
–
–
This method can be speeded up with exactly the kind
of estimating that is required by the standard long-
division algorithm. In fact, if you estimate optimally and
leave out some of the bookkeeping, you have the stan-
dard algorithm. 
)7 1619
14
21
21
9
7
2
231
–
–
–
)7 1619
700 100
919
700 100
219
70 10
149
70 10
79
70 10
9
71
2 231
–
–
–
–
–
–
I see no need to “teach” long division using divisors of
more than two digits. Such computations could be assigned
(once!) as a problem of the week, or the ladder method
could be used.
I conclude that it is not long division that is obsolete but
the traditional method of teaching it mechanically.  G

kept trying: 5.477 squared was 29.997529. That result was
close enough. 
The next day a student asked when we would learn the
“real way” to find square roots. I asked what he meant. He
started to show me the traditional algorithm but stopped
after two or three steps. I asked why. “That’s all my father
remembers,” he replied. I asked the class whether they
would ever forget how to find a square root by guessing
and multiplying. They agreed that they wouldn’t, but they
still wanted to learn the “real way” to do it. I agreed on the
condition that they would help me explain why it works.
Together we developed the standard geometric proof that
the traditional algorithm produces answers as precise as
needed. We agreed that the guess-and-check method pro-
duces equally precise answers but might take longer. We
also learned the divide-and-average technique, sometimes
known as Newton’s method, and, at their insistence, discov-
ered a cube-root algorithm based on analogous three-di-
mensional figures.
The student’s father had forgotten the “real way” to find a
square root because he didn’t use it and probably never un-
derstood why it works. The students from my class probably
no longer remember the traditional square root algorithm
and also don’t use it. But I hope that from that experience,
and many others like it, they learned that mathematics is
something to understand and think about rather than
something to be memorized and regurgitated. I also hope
that from many experiences in that class, they learned that
mathematics can be fun and useful.
It is not evil to teach students to divide either whole
numbers or polynomials. Nor is it evil to fail to teach
these algorithms. But more knowledge is generally better
than less.
Dividing by a single-digit number is easily taught by shar-
ing some amount of play money “fairly” among several chil-
dren I have in my “bank” only money with denominations
that are powers of ten when doing so. By keeping records
of the process, children arrive at the standard algorithm.
Converting to the short-division algorithm is easy. By mov-
ing decimal points so that the divisor’s decimal point is in
the place after the first digit ( not, generally, the rightmost
digit), students estimate quotients no matter how many
digits are in the divisor. If greater precision is needed, they
estimate and multiply.
When, and if, polynomial division is learned, it should be
related to the algorithm with whole numbers so that stu-
dents see that connections occur in mathematics and that
they can figure things out.  G
Responses—Stephen S. Willoughby
continued from page 16
18 Mathematics Education Dialogues
March 1998
to what extent do these learning styles imply that we
need to teach them differently?
• What additional training, if any, do today’s teachers of
mathematics need to deal with these developments?
These questions introduce complex issues that do not
lend themselves to simple answers. They have led to some-
times heated debates.
The Board of Directors of NCTM believes that rea-
soned discussion of questions like these is valuable for the
entire education community: teachers, parents, students,
administrators, and other concerned citizens. Recogniz-
ing that no vehicle has been specifically designed for this
purpose, the Board formed a task force of six people
(named below) to oversee the creation of two prototypes
of a new publication that could serve as a forum for the
identification and reasoned discussion of important is-
sues in mathematics education.
This publication is the first of those two prototypes. We
want your feedback. Please either copy and use the form (see
page 19) or e-mail your comments to [EMAIL].
Your reply can also be faxed to NCTM (attention: Dialogues)
to [PHONE] or sent by regular mail to Dialogues, 1906
Association Drive, Reston, VA 20191-1593.  G
Task Force Members:
Zalman Usiskin, Chair
University of Chicago
Cynthia Ballheim
Saint Mary’s High School, Calgary, Alberta
Peggy House
Northern Michigan University
Johnny Lott
University of Montana
Barbara Marshall
Philadelphia Public Schools, Pennsylvania
Hung-Hsi Wu
University of California at Berkeley
Mathematics Education Dialogues is published as a supple-
ment to the News Bulletin by the National Council of
Teachers of Mathematics, 1906 Association Drive, Reston,
VA 20191–1593. Pages may be reproduced for classroom
use without permission.
All New Forum Makes Debut
continued from page 1