What follows and is attached, by Lynn Arthur Steen, is the last
chapter of NCTM's 1999 yearbook.
Twenty Questions about Mathematical
Lynn Arthur Steen, St. Olaf College
-- http://www.stolaf.edu/people/steen/index.html - other papers
The concluding chapter in NCTM's 1999
Yearbook which is devoted to mathematical reasoning: Developing
Mathematical Reasoning in Grades K-12. Lee Stiff, Editor. Reston,
VA: National Council of Teachers of Mathematics, 1999, pp.
We begin with two warm-up questions.
First, why is mathematics an integral part of the K-12
curriculum? The answers are self-evident and commonplace: to teach
basic skills; to help children learn to think logically; to prepare
students for productive life and work; and to develop quantitatively
Second, and more problematic: How does mathematical reasoning
advance these goals? This is not at all self-evident, since it
depends greatly on the interpretation of "mathematical
reasoning." Sometimes this phrase denotes the distinctively
mathematical methodology of axiomatic reasoning, logical deduction,
and formal inference. Other times it signals a much broader
quantitative and geometric craft that blends analysis and intuition
with reasoning and inference, both rigorous and suggestive. This
ambiguity confounds any analysis and leaves room for many
1. Is mathematical reasoning mathematical?
Epistemologically, reasoning is the foundation of mathematics. As
science verifies through observation, mathematics relies on logic. The
description of mathematics as the "science of drawing necessary
conclusions" given over a century ago by the philosopher C. S.
Peirce still resonates among mathematicians of today. For example, a
contemporary report by mathematicians on school mathematics asserts
that "the essence of mathematics lies in proofs" [Ross,
Yet mathematics today encompasses a vast landscape of methods,
procedures, and practices in which reasoning is only one among many
tools [e.g., Mandelbrot, 1994; Thurston, 1994; Denning, 1997].
Computation and computer graphics have opened new frontiers of both
theory and application that could not have been explored by previous
generations of mathematicians. This frontier has revealed surprising
mathematical insights, for example, that deterministic phenomena can
exhibit random behavior; that repetition can be the source of chaos as
well as accuracy; and that uncertainty is not entirely haphazard,
since regularity always emerges [Steen, 1990].
It took innovative mathematical methods to achieve these
insights-methods that were not tied exclusively to formal inference.
Does this mean that mathematical reasoning now includes the kind of
instinct exhibited by a good engineer who finds solutions that work
without worrying about formal proof? Does it include the kinds of
inferences from "noisy" data that define the modern practice
of statistics? Must mathematical reasoning be symbolic or deductive?
Must it employ numbers and algebra? What about visual, inductive, and
heuristic inferences? What about the new arenas of experimental
mathematics and computer-assisted problem solving? What, indeed, is
distinctively mathematical about mathematical reasoning?
2. Is mathematical reasoning useful?
For most problems found in mathematics textbooks, mathematical
reasoning is quite useful. But how often do people find textbook
problems in real life? At work or in daily life, factors other than
strict reasoning are often more important. Sometimes intuition and
instinct provide better guides; sometimes computer simulations are
more convenient or more reliable; sometimes rules of thumb or
back-of-the-envelope estimates are all that is needed.
In ordinary circumstances, people employ mathematics in two rather
different ways: by applying known formulas or procedures to solve
standard problems, or by confronting perplexing problems through
typically mathematical strategies (e.g., translating to another
setting; looking for patterns; reasoning by analogy; generalizing and
simplifying; exploring specific cases; abstracting to remove
irrelevant detail). Rarely do they engage in rigorous deduction
characteristic of formal mathematics. At work and in the home,
sophisticated multi-step calculations based on concrete
measurement-based mathematics is far more common than are chains of
logical reasoning leading to mathematical proof [Forman & Steen,
1995]. It is not the methodology of formal deduction that makes
mathematics useful for ordinary work so much as the mathematical
habits of problem solving and the mathematical skills of calculation
Can people do mathematics without
reasoning? Many certainly do-using routine methods ingrained as
habit. Can people reason without using mathematics? Obviously so, even
about situations (e.g, gambling, investing) that mathematicians would
see as intrinsically mathematical. Those few people who employ
advanced mathematics necessarily engage in some forms of mathematical
reasoning, although even for them the role played by reasoning may be
unconscious or subordinate to other means of investigation and
analysis. But how much mathematical reasoning is really needed for the
kinds of mathematics that people do in their life and work? Does
ordinary mathematical practice really require much mathematical
3. Is mathematical reasoning an appropriate goal of school
Mathematics teachers often claim that all types of critical
thinking and problem solving are really examples of mathematical
reasoning. But employers have a different view, rooted in a paradox:
graduates with degrees in mathematics or computer science are often
less successful than other graduates in solving the kinds of problems
that arise in real work settings. Often students trained in
mathematics tend to seek precise or rigorous solutions regardless of
whether the context warrants such an approach. For employers, this
distinctively "mathematical" approach is frequently
not the preferred means of solving most problems arising in
authentic contexts. Critical thinking and problem solving about the
kinds of problems arising in real work situations is often better
learned in other subjects or in integrative contexts [Brown,
The goals of school mathematics seem to shift every decade, from
"conceptual understanding" in the new math 60s to
"basic skills" in the back-to-basics 70s, from "problem
solving" in the pragmatic 80s to "mathematical power"
in the standards-inspired 90s. Will "mathematical reasoning"
be next? Not likely. In its strict (deductive) meaning, mathematical
reasoning is hardly sufficient to support the public purposes of
school mathematics. Everyone needs the practice of mathematics. But
who really needs to understand mathematics? Who really need
mathematical reasoning? Can one make the case that every high school
graduate needs to be able to think mathematically rather than just
4. Can teachers teach mathematical reasoning?
The Third International Mathematics and Science Study (TIMSS)
documented that U.S. mathematics teachers focus on teaching students
how to do mathematics and not on understanding what they do [NCES,
1996]. There are many reasons for this, including teachers' self-image
of mathematics as a set of skills, parents' demand that children
master the basics before advancing to higher order tasks; and the
constraining environment of state-mandated tests that emphasize
Many believe that curricular reform based on mathematical reasoning
will never succeed since there are far too few teachers prepared to do
justice to such a goal. Even if enough willing and able teachers can
be found (or educated), will the public allow them to teach
mathematical reasoning in school? Might the fear of "fuzzy
mathematics" [Cheney, 1997] constrain even those teachers who
might want to stress understanding?
5. Can mathematical reasoning be taught?
Just as we don't really know what mathematical reasoning is, so we
don't know very much about how it develops. Research does support a
few general conclusions. First, successful learners are mathematically
active [Anderson, Reder, & Simon, 1996]. Passive strategies
(memorization, drill, templates) are much less likely than active
tasks (discussion, projects, teamwork) to produce either lasting
skills or deep understanding. Second, successful mathematics learners
are more likely to engage in reflective (or "metacognitive")
activity [Resnick, 1987]. Students who think about what they are doing
and why they are doing it are more successful than those who just
follow rules they have been taught.
We also know that students differ: no single strategy works for all
students, nor even for the same student in all circumstances. Howard
Gardner's theory of multiple intelligences [Gardner, 1983, 1995]
supports the practice of experienced teachers who create multiple
means for students to approach different topics. Diverse experiences
provide implicit contexts in which mathematical reasoning may emerge.
But can we be sure that it will eventually emerge? Might some
students, or some types of reasoning, require explicit instruction?
Are there some types of mathematical reasoning that can only develop
through student construction and reflection? If some types of
mathematical reasoning cannot be taught explicitly, is it appropriate
to require it of all high school graduates?
6. Do skills lead to understanding?
Although mathematical performance generally involves a blend of
skills, knowledge, procedures, understanding, reasoning, and
application, the public mantra for improving mathematics education
focuses on skills, knowledge, and performance-what students
"know and are able to do." To this public agenda mathematics
educators consistently add reasoning and understanding-why
and how mathematics works as it does.
Experienced teachers know that knowledge and performance are not
reliable indicators of either reasoning or understanding. Deep
understanding must be well-connected. In contrast, superficial
understanding is inert, useful primarily in carefully prescribed
contexts such as those found in typical mathematics classrooms
[Glaser, 1992]. Persons with well-connected understanding attach
importance to different patterns and are better able to engage in
mathematical reasoning. Moreover, students with different levels of
skills may be equally able to address tasks requiring more
sophisticated mathematical reasoning [Cai, 1996].
Nonetheless, the public values (and hence demands) mathematics
education not so much for its power to enhance reasoning as for the
quantitative skills that are so necessary in today's world. It is not
that adults devalue understanding, but that they expect basic skills
first [Wadsworth, 1997]. They believe in a natural order of
learning-first skills, then higher order reasoning. But do skills
naturally led to understanding? Or is it the reverse-that
understanding helps secure skills? Does proficiency with mathematical
facts and procedures necessarily enhance mathematical reasoning?
Conversely, can mathematical reasoning develop in some students even
if they lack firm grasp of facts and basic skills? Might the relation
of skills to reasoning be like that of spelling to writing-where
proficiency in one is unrelated to proficiency in the other?
7. Can drill help develop mathematical reasoning?
Critics of current educational practice
indict "drill and kill" methods for two crimes against
mathematics: disinterest and anxiety. Both cause many students to
avoid the subject as soon as they are given a choice. Yet despite the
earnest efforts to focus mathematics on reasoning, one out of every
two students thinks that learning mathematics is mostly memorization
[Kenney & Silver, 1997].
And they may have a point. Research shows rather convincingly that
real competence comes only with extensive practice [Bjork &
Druckman, 1994]. Yet practice is certainly not sufficient to ensure
understanding. Both the evidence of research and the wisdom of
experience suggest that students who can draw on both recalled and
deduced mathematical facts make more progress than those who rely on
one without the other [Askew & Dylan, 1995].
Yet children who can recite multiplication facts may still not
understand why the answers are as they are or recognize when
multiplication is an appropriate operation, much less understand how
ratios relate to multiplication. High school students who memorize
proofs in a traditional geometry course may show good recall of key
theorems, but be totally unable able to see how the ideas of these
proofs can be used in other contexts. Is there, indeed, any real
evidence that practiced recall leads to reasoning and
8. Is proof essential to mathematics?
Despite the dominance of proof as the methodology of advanced
mathematics courses, contemporary advances in applied, computer-aided,
and so-called "experimental" mathematics have restored to
mathematical practice much of the free-wheeling spirit of earlier
eras. Indeed, these recent innovations have led some to proclaim the
"death" of proof-that although proof is still useful in
some contexts, it may no longer be the sine qua non of
mathematical truth [Horgan, 1993]. Although this claim is hotly
disputed by many leading mathematicians, it resonates with diverse
pedagogical concerns about the appropriateness (or effectiveness) of
proof as a tool for learning mathematics. Uncertainty about the role
of proof in school mathematics caused NCTM in its Standards
[NCTM, 1989] to resort to euphemisms-"justify,"
"validate," "test conjectures," "follow
logical arguments." Rarely do the Standards use the
crystalline term "proof."
In fact, most people understand
"proof" in a pragmatic rather than a philosophical way:
provide just enough evidence to be convincing. For many people, proof
is tantamount to the civil legal test of "preponderance of
evidence"; others require the stricter standard of "beyond
reasonable doubt." In routine uses of mathematics, what works
takes precedence over what's provable. So how much understanding of
formal proof is necessary for the routine practice of mathematics?
Probably not very much. But how much is needed for advanced study of
mathematics? Undoubtedly a great deal.
9. Does learning proofs enhance mathematical reasoning?
Nothing divides research mathematicians and mathematics educators
from each other as do debates about the role of proof in school
mathematics. Proof is central to mathematical reasoning, yet there is
precious little agreement on how, when, why, or to whom to teach it.
Its suitability for school mathematics has always been open to
question, both on the grounds of pedagogy and relevance.
The vocabulary of mathematical truth, rigor, and certainty is not a
natural habitat for most students; their world is more empirical,
relying on modeling, interpretation, applications. Only a very few
students in high school comprehend proof as mathematicians do-as a
logically rigorous deduction of conclusions from hypotheses [Dreyfus,
1990]. Students generally have very little comprehension of what
"proof" means in mathematics, nor much appreciation of its
importance [Schoenfeld, 1994]. Might early introduction of proof
actually do more to hinder than enhance the development of
Although mathematicians often advocate including proof in school
curricula so students can learn the logical nature of mathematics
[Ross, 1997], the most significant potential contribution of proof in
mathematics education may be its role in communicating mathematical
understanding [Hanna & Jahnke, 1996]. The important question about
proof may not be whether it is crucial to understanding the nature of
mathematics as a deductive, logical science, but whether it helps
students and teachers communicate mathematically. Is, perhaps, proof
in the school classroom more appropriate as a means than as an
10. Does "math anxiety" prevent mathematical
Mathematics is perhaps unique among school subjects in being a
major cause of anxiety. Many students believe deeply that they cannot
do mathematics and so learn to avoid it; a few are so paralyzed by the
prospect that they exhibit physiological evidence of acute anxiety
[Buxton, 1991; Tobias, 1993]. It may seem obvious that anyone
suffering even mildly from "math anxiety" would not engage
in much mathematical reasoning. But this is not at all the case. Many
students (and adults) who fear mathematics are in fact quite capable
of thinking mathematically, and do so quite often-particularly in
their attempts to avoid mathematics! What they really fear is not
mathematics itself, but school mathematics [Cockcroft, 1982].
Both research and common sense say that anxiety is reduced when
individuals can control uncertainties [Bjork & Druckman, 1994].
When percentages and ratios appear as impossible riddles, panic
ensues. But when self-constructed reasoning-under the control of the
individual-takes over, much valid mathematical reasoning may emerge,
often in a form not taught in school. How can schools respect each
student's unique approach to mathematical reasoning while still
teaching what society expects (and examines)? Would reduced panic
result in improved reasoning? Is this a case where less may be
more-where reduced instruction might yield deeper understanding?
11. Do cooperative activities enhance individual understanding?
Arguments for cooperative learning and teamwork come from two
rather different sources: first, from those (primarily in the
education world) who view these activities as effective strategies for
learning mathematical reasoning and second, from those (primarily in
the business world) who view cooperative activities as essential for
productive employees [SCANS, 1991]. Advocates envision mathematics
classes as communities where students engage in collaborative
mathematical practice both with each other and with their teachers
[Silver, Kilpatrick, & Schlesinger, 1990]. In such classes
students would regularly engage in authentic forms of mathematical
practice by inventing strategies, arguing about approaches, and
justifying their work.
Parents often object to educators'
rationale for teamwork, since they view mathematics as an ideal
subject in which individual accomplishment can be objectively measured
and rewarded. They worry both that children who are above average will
be held back by slower students and that those who are behind will be
instructed not by teachers but by other children. Ironically, despite
their distrust of teamwork in subjects like mathematics, most parents
and students admire teamwork in sports and musical organizations. (Of
course, in sports and music-as in the workplace-success accrues
not to individuals but to the team as a whole.) Despite these
objections, there is considerable evidence that cooperative learning
is effective, especially for children [Bjork & Druckman, 1994].
For high school students and adults, however, the evidence is more
mixed. Older students bring to cooperative groups stronger individual
motivations, complex experiences in social interactions, and often
some defensiveness or embarrassment about learning.
Employers value teamwork because it produces results that no
individual could accomplish alone. But can teamwork in the classroom
also produce reasoning at a higher level than could be accomplished by
any single member of a team? Will individual members of a team learn
more mathematics as a result? Just how do group activities promote
mathematical reasoning in individuals? Even more difficult and
important: How can mathematics educators gain public support for
12. Can calculators and computers increase mathematical
At home and at work, calculators and computers are "power
tools" that remove human impediments to mathematical performance.
For example, spreadsheets and statistical packages are used by
professionals both to extend the power of mind as well as to
substitute for it-by performing countless calculations without error
or effort. Students certainly need to learn these empowering uses of
But in addition, calculators and computers are responsible for a
"rebirth of experimental mathematics" [Mandelbrot, 1994].
They provide educators with wonderful tools for generating and
validating patterns that can help students learn to reason
mathematically. Computer games can help children master basic skills;
intelligent tutors can help older students master algebraic
procedures. Many educators have argued that since programming enforces
logical rigor, computer languages such as Logo and ISETL can help
students learn to reason.
Calculators and computers hold tremendous potential for mathematics.
Depending on how they are used, they can either enhance mathematical
reasoning or substitute for it, either develop mathematical reasoning
or limit it. However, judging from public evidence, the actual effect
of calculators in school is as often negative as positive: for every
student who learns to use spreadsheets there seem to be several who
reach for a calculator to add single digit numbers or to divide by 10.
Why are the consequences of calculators in school mathematics so
mixed? Why is there such a big gap between aspirations and
13. Why do so many student feel that mathematics is a foreign
A substantial number of children find school mathematics opaque.
Part of children's difficulty in learning school mathematics lies in
their failure to reconcile the rules of school-math with their own
independently developed mathematical intuition [Freudenthal, 1983;
Resnick, 1987]. Too often, entrenched assumptions-like
"regular" grammar applied in contexts where irregularity
To what extent does the mathematical environment in a child's home
affect how the child responds to mathematics in school? Many people
believe that certain peoples or cultures are better suited to
mathematics than others. The thriving-and controversial-specialty
of ethnomathematics documents beyond reasonable doubt that all
societies have developed some form of mathematics, and that these
forms reflect the cultures in which they emerge. Historically alert
mathematicians can recognize similarities and differences in the
mathematics of different cultures and can trace the influence of
cultures on one another in the evolution of mathematics [Joseph,
1992]. Thus there are undeniable cultural differences in
But are there cultural differences in the
development of mathematical reasoning? Here the evidence is less
definitive. World-class mathematicians have emerged from societies all
around the globe, yet certain cultures put greater emphasis on the
kinds of rigor and reasoning that give mathematics its special power.
Students growing up in these cultures are more likely to recognize a
zone of comfort in school mathematics, whereas students growing up in
cultures that view the world through other lenses may feel as if
school mathematics is a foreign culture. Why do some students see
mathematics as the only welcoming subject in school, whereas others
see it as the most foreign of cultures? Why, indeed, do some children
find mathematics so unreasonably hard?
14. Is context essential for mathematical reasoning?
For at least a decade, both educational researchers and reformers
have been preaching the message of "situated cognition" or
"contextualized learning." For much longer scientists and
engineers have fussed at mathematicians for persisting with
context-free instruction [Rutherford, 1997]. Recently vocational
educators have joined the chorus, citing persistent lack of context in
mathematics courses as one of the chief impediments to student
learning [Bailey, 1997; Hoachlander, 1997]. Yet according to a
National Research Council report, there is no consistent evidence that
performance is enhanced when learning takes place in the setting in
which skills will be performed [Bjork & Druckman, 1994].
Context can affect learning in two opposing ways: generally, it
enhances motivation and long-term learning, but it can also can limit
the utility of what is learned. Knowledge often becomes context-bound
when it is taught in just one context. Anyone who has ever taught
mathematics has heard complaints from teachers in other subjects that
students don't appear to know any of the mathematics they were
supposed to have learned in mathematics class. The pervasive problem
of compartmentalized knowledge has led many educators to assume that
transfer of knowledge from one subject to another is atypical. In
fact, transfer does occur, but not nearly as systematically or as
predictably as we would like.
Just how situated is mathematical cognition? Does instruction in
context facilitate learning mathematics? Does it limit or enhance the
likelihood of transfer to other domains? When, if ever, does
mathematical reasoning transfer to other domains?
15. Must students really construct their own knowledge?
One of the most widely accepted goals of the mathematics community
is that students should understand the mathematics they perform. For
centuries educators have known that understanding grows only with
active learning. This has led, in the argot of mathematics educators,
to a widespread belief that students "construct" their own
understanding [Davis, Maher, & Noddings, 1990; Hiebert &
Carpenter, 1992]. In this view, understanding cannot be delivered by
instructors, no matter how skillful, but must be created by learners
in their own minds.
The constructivist posits that children learn as they attempt to solve
meaningful problems. In this view, understanding emerges from
reflection catalyzed by questions [Campbell & Johnson, 1995]. The
teachers primary role is not to instruct but to pose problems and ask
questions that provoke students to reflect on their work and justify
their reasoning. In this way, activities such as explaining,
justifying, and exemplifying not only demonstrate understanding but
also help create it.
According to supporters, constructivism focuses education on the
learner (what happens in students' minds); on inquiry (seeking the
right questions, not just the right answers); on relevance (questions
of natural interest to children); and on activity (learning with both
hand and mind) [Brooks & Brooks, 1993]. Yet critics [e.g.,
Anderson et al., 1996; Wu, 1996] contend that constructivist methods
too easily slight the importance both of didactics (systematic
instruction) and drill (systematic practice). What is the appropriate
balance between teacher-directed and student-inspired learning? Do
students need to construct everything for themselves? What should be
memorized and what constructed?
16. How many mathematics are there?
Mathematics lives in many environments-home math, school math,
street math, business math, work math-and many students who succeed
in one mathematical world fail in another. Even though these are all
mathematics, these environments offer fundamentally different contexts
in which students learn and utilize mathematics. One might well
imagine that, like multiple intelligences [Gardner, 1983] there may be
multiple mathematics [Grubb, 1997].
Evidence of multiple mathematics abounds. Research documents what
parents and teachers know from rueful experience-that many children
see school mathematics as disconnected from sense-making and the world
of everyday experience [Silver, Kilpatrick, & Schlesinger, 1990;
Schoenfeld, 1991]. The widespread separation of symbols from meaning
and of calculation from reasoning is an inheritance of an educational
system whose historic purpose was to separate the practical from the
abstract and workers from scholars [Resnick 1987]. Only for an elite
was abstract or higher order reasoning a goal (much less an
accomplishment) of education. School has helped foster the public's
view of different mathematics for different purposes.
This history encourages a pervasive myth about mathematics
learning-that mathematical reasoning is appropriate only for the ten
percent of students who are destined for mathematically rich careers
in science and engineering. Yet in today's workplace, mathematical
thinking is needed by more students than every before. Nonetheless,
some students learn mathematics better in mathematics classes, some in
science or shop courses, and some on the job or at home. Do these
settings offer different mathematics? In what circumstances is
abstract mathematics appropriate? When is concrete mathematics better?
Can we trust students to know which type of mathematics is best for
them in particular contexts? Do teachers know enough to decide? Does
17. How does our brain do mathematics?
Recent research in neuroscience has begun to open a window into
what has heretofore been largely beyond the reach of science: the
neural mechanism of cognition. Intriguingly, this research suggests a
Darwinian mechanism of diversity and selection that operates within
the brain just as it does among species in an ecosystem [Edelman,
1992; Abbott, 1994; Changeux & Connes, 1995]. Such a mechanism may
help explain the stages of mathematical creativity noted in the
classic work of Jacques Hadamard  of preparation (trial and
error), incubation (often subconscious), illumination (frequently
sudden), and verification (requiring reasoning). According to this
theory, mathematical reasoning depends on the same two forces as the
evolution of species: a mechanism for generating diversity
(alternatives) and a strategy for selection that stabilizes optimal
choices among this diversity.
What, indeed, is the neural mechanism of mathematical thought? This is
now a researchable question, and the implications of such research are
profound. For the first time, we may be able to connect mathematical
thinking to the biology of the brain. We now know, for instance, that
memory involves several anatomically different structures. As improved
understanding of physiology has moved athletes' performances to the
edge of human potential, might we soon be able to scientifically
improve individuals' mathematical performance? Can we identify the
biochemistry of mathematical reasoning? Might neuroscience help
educators understand the vexing problem of transfer? Or of the
relation of skills to reasoning?
18. Is our brain like a computer?
We tend naively to think of the brain as a computer-especially
when it is engaged in mathematical activity. Store basic facts in
memory; provide key algorithms for calculation; then push a button.
Much of the drill-oriented pedagogy of traditional mathematics
education is rooted in this metaphor. In fact, as contemporary
neuroscience reveals, the brain is less like a computer to be
programmed or a disk to be filled than like an ecosystem to be
nourished [Abbott, 1996; ECS, 1996, 1997].
Although the evidence against the
brain-as-computer metaphor is overwhelming [e.g., recovery patterns of
stroke victims], the paradigm persists in large measure for lack of a
compelling alternative. But that may be about to change. Research in
the intersection of evolutionary genetics and neuroscience suggests
potentially important neurological differences between those cognitive
capacities that are evolutionarily primitive (e.g., counting) and
those such as arithmetic (not to mention algebra!) that are more
recent social constructs [Geary, 1995]. Capacity for reasoning is
created by a continually changing process of natural selection of
neuronal groups responding to an individual's goals (called
"values" by Edelman ). Thus both the processes of
cognition and the elements on which these processes act-if you will,
procedures and facts-are subject to the evolutionary pressures of
diversity and selection within the living brain.
19. Is the capacity for mathematics innate?
For years linguists and neuroscientists have studied the way
babies learn language in an effort to understand the relation of human
language to the genetic endowment of our species. As children
naturally develop their own rules of grammar-regularizing irregular
verbs, for example-so they also invent rules to explain patterns
they see around them. To the extent that making patterns is a
mathematical activity [Steen, 1988; Devlin, 1994], young children
learning language are doing mathematics!
There is abundant evidence that young children, on their own, develop
simple mathematical rules that they use to solve problems in their
environment [Resnick, 1987]. Yet these patterns often lead to
mathematical misconceptions-e.g., that multiplication makes things
bigger-that persist despite subsequent contrary evidence and
instruction [Askew & Dylan, 1995]. Does this mean that young
children have the same innate capacity to learn mathematics as they
have to learn language? How might mathematical reasoning be enhanced
if babies were bathed in an environment as rich in mathematical
patterns as it is in natural language?
20. Is school too late?
Although certain aspects of the brain are determined by genetics
and by the environment in the womb, both neurons and synapses grow and
change rapidly during the early years of life. How they grow is
determined by the environment of the infant. What they become-after
five or six years-determines to a considerable degree the cognitive
capacity of the child and adult. Although much of the brain is formed
at birth, much remains plastic, amenable to being shaped by
experience. The capacity for abstract thinking is particularly
plastic. Synapse growth occurs at a phenomenal rate until age two or
three, and then gradually diminishes for the rest of life [ECS, 1997].
"Use it or lose it" is a fitting description of the early
Everyone knows the importance of aural stimulation for the learning of
language in the first years of life. Recent research has provided
rather firm evidence that musical stimulation in these early years
enhances capacity for spatial and mathematical abstraction later in
life [Rauscher & Shaw, 1997]. (Whether early musical stimulation
enhances musicality is less clear.) Apparently the acoustical bath of
aural structure provided by classical music does for the abstract
centers of the brain what hearing phonemes does for language
This research leads to many questions that are hardly touched on in
mathematics education. Are there "windows" for learning
arithmetic or algebra, or for mathematical reasoning, as there surely
are for learning language? What, besides music, can enhance the young
brain's capacity for mathematical thinking? How sensitive is
mathematical ability to the sensory environment of a baby? Just how
does learning change the brain's physiology? Might we someday be able
to sculpt children's capacity for mathematical reasoning?
Abbott, John. "The Search for Next-Century Learning."
AAHE Bulletin, 48:7 (March 1996) 3-6.
Abbott, John. Learning Makes Sense: Recreating Education for a
Changing Future. Letchworth, UK: Education 2000,
Anderson, John R.; Reder, Lynne M.; and
Herbert A. Simon. "Applications and Misapplications of Cognitive
Psychology to Mathematics Education." Carnegie Mellon
University Web Site URL:
Askew, Michael and Dylan William. Recent Research in Mathematics
Education 5-16. London, UK: Her Majesty's Stationery Office,
Bailey, Thomas. "Integrating Vocational and Academic Education."
In High School Mathematics at Work. Washington, DC: National
Academy Press, 1997.
Bjork, Robert A. and Daniel Druckman, Editors. Learning,
Remembering, Believing: Enhancing Human Performance. Washington,
DC: National Research Council, 1994.
Brooks, Jacqueline G. and Martin G. Brooks. In Search of
Understanding: The Case for Constructivist Classrooms. Alexandria
VA: American Society for Curriculum Development, 1993.
Brown, Patricia (editor). Promoting Dialogue on School-to-Work
Transition. Washington, DC: National Governors' Association,
Buxton, Laurie. Math Panic. Portsmouth, NH: Heinemann
Educational Books, 1991.
Cai, Jinfa. A Cognitive Analysis of U.S. and Chinese Students'
Mathematical Performance on Tasks Involving Computation, Simple
Problem Solving, and Complex Problem Solving." Journal for
Research in Mathematics Education, Monograph No. 7, 1995.
Campbell, Patricia F. and Martin L. Johnson. "How Primary
Students Think and Learn." In Prospects for School
Mathematics, edited by Iris M. Carl, pp. 21-42. Reston, VA:
National Council of Teachers of Mathematics, 1995.
Changeux, Jean-Pierre and Alain Connes. Conversations on Mind,
Matter, and Mathematics. Princeton, NJ: Princeton University
Cheney, Lynne. "Creative Math or Just 'Fuzzy Math'? Once Again,
Basic Skills Fall Prey to a Fad." The New York Times,
August 11, 1997, p. A13.
Cockroft, Sir Wilfred. Mathematics Counts. London: Her
Majesty's Stationery Office, 1982.
Davis, Robert B., Carolyn A. Maher, and Nel Noddings.
"Constructivist Views on the Teaching and Learning of
Mathematics." Journal for Research in Mathematics
Education, Monograph No. 4, 1990.
Denning, Peter J. Quantitative Practices. In Why Numbers Count:
Quantitative Literacy for Tomorrow's America, edited by Lynn
Arthur Steen, pp. 106-117. New York, NY: The College Board, 1997.
Devlin, Keith. Mathematics: The Science of Patterns. New York,
NY: W. H. Freeman, 1994.
Dreyfus, Tommy. "Advanced Mathematical Thinking." In
Mathematics and Cognition, edited by Pearla Nesher and Jeremy
Kilpatrick. Cambridge UK: Cambridge University Press, 1990. pp.
Edelman, Gerald. Bright Air, Brilliant Fire. New York, NY:
Basic Books, 1992.
Education Commission of the States. Bridging the Gap Between
Neuroscience and Education. Denver, CO: The Commission, 1996
Education Commission of the States. Brain Research Implications for
Education. 15:1 (winter 1997)
Forman, Susan L. and Lynn Arthur Steen. "Mathematics for Life and
Work." In Prospects for School Mathematics, edited by Iris
M. Carl, pp. 219-241. Reston, VA: National Council of Teachers of
Freudenthal, Hans. "Major Problems of Mathematics Education."
In Proceedings of the Fourth International Congress on Mathematical
Education, edited by Marilyn Sweng, et al., p. 1-7. Boston, MA:
Gardner, Howard. Frames of Mind: The Theory of Multiple
Intelligences. New York, NY: Basic Books, 1983.
Gardner, Howard: "Reflections on Multiple Intelligences: Myths
and Messages." Phi Delta Kappan. Nov. 1995 pp.
Geary, David C. "Reflections on Evolution and Culture in
Children's Cognition: Implications for Mathematical Development and
Instruction." American Psychologist 50:1 (1995).
Glaser, Robert. "Expert Knowledge and the Processes of Thinking."
In Enhancing Thinking Skills in the Sciences and Mathematics,
edited by Diane. F. Halpern, pp. 63-75. Hillsdale, NJ: Lawrence
Earlbaum Assoc., 1992.
Grubb, Norton. "Exploring Multiple Mathematics." Project
EXTEND Web Site (URL:
Hadamard, Jacques. The Psychology of Invention in the Mathematical
Field. Princeton, NJ: Princeton University Press,
Hanna, Gila and H. Niels Jahnke.
"Proof and Proving." In International Handbook of
Mathematics Education, edited by Alan J. Bishop, et al., pp.
877-908. Dordrecht: Kluwer Academic Publishers, 1996.
Hiebert, James and Thomas P. Carpenter. "Learning and Teaching
with Understanding." In Handbook of Research on Mathematics
Teaching and Learning, edited by Douglas A. Grouws, pp. 65-97. New
York: Macmillan, 1992.
Hoachlander, Gary. "Organizing Mathematics Education Around
Work." In Why Numbers Count: Quantitative Literacy for
Tomorrow's America, edited by Lynn Arthur Steen, pp. 122-136. New
York, NY: The College Board, 1997.
Horgan, John. "The Death of Proof." Scientific
American 269 (April 1993) 93-103.
Joseph, George Ghevergheze. The Crest of the Peacock: Non-European
Roots of Mathematics. London, UK: Penguin Books, 1992.
Kenney, Patricia Ann and Edward A. Silver, editors. Results from
the Sixth Mathematics Assessment of the National Assessment of
Educational Progress. Reston, VA: National Council of Teachers of
Mandelbrot, Benoit. "Fractals, the Computer, and Mathematics
Education." In Proceedings of the 7th International Congress
on Mathematical Education, edited by Claude Gaulin, et al. pp.
77-98. Sainte-Foy, QB: Les Presses de l'Université Laval, 1994.
National Center for Educational Statistics. Pursuing
Excellence. Washington, DC: U. S. Department of Education,
National Council of Teachers of Mathematics. Curriculum and
Evaluation Standards for School Mathematics. Reston VA: The
Packer, Arnold. "Mathematical Competencies that Employers
Expect." In Why Numbers Count: Quantitative Literacy for
Tomorrow's America, edited by Lynn Arthur Steen, pp. 137-154. New
York, NY: The College Board, 1997.
Rauscher, Frances and Gordon Shaw. "Music Training Causes
Long-Term Enhancement of Preschool Children's Spatial-Temporal
Reasoning." Neurological Research 19 (Feb 1997) 2-8.
Resnick, Lauren B. Education and Learning to Think. Washington,
DC: National Research Council, 1987.
Ross, Kenneth A. "Second Report from the MAA Task Force on the
NCTM Standards." Mathematical Association of America Web
Site (URL: www.maa.org/past/nctmupdate.html) 1997.
Rutherford, F. James "Thinking Quantitatively about Science."
In Why Numbers Count: Quantitative Literacy for Tomorrow's
America, edited by Lynn Arthur Steen, pp. 60-74. New York, NY: The
College Board, 1997.
Schoenfeld, Alan H. "On Mathematics as Sense-Making." In
Informal Reasoning and Education, edited by J. F. Voss, D. N.
Perkins, and J. W. Segal, pp. 311-343. Hillsdale, NJ: Lawrence
Earlbaum Associates, 1991.
Schoenfeld, Alan H. "What Do We Know About Curricula?"
Journal of Mathematical Behavior, 13 (1994) 55-80.
Secretary's Commission on Achieving Necessary Skills (SCANS). What
Work Requires of Schools: A SCANS Report for America 2000.
Washington, DC: U.S. Dept. of Labor, 1991.
Silver, Edward A., Kilpatrick, Jeremy, and Beth Schlesinger.
Thinking Through Mathematics. New York, NY: College Entrance
Examination Board, 1990.
Steen. Lynn Arthur. "The Science of Patterns. Science 240
(29 April 1988) 611-616.
Steen, Lynn Arthur (editor). On the Shoulders of Giants: New
Approaches to Numeracy. Washington, DC: National Academy Press,
Thurston, William P. "Mathematics Education." Notices of
the American Mathematical Society 37:7 (September, 1990)
Thurston, William P. "On Proof and Progress in
Mathematics." Bulletin of the American Mathematical
Society, 30 (1994) 161-177.
Tobias, Sheila. Overcoming Mathematics Anxiety (Rev. Ed.). New
York, NY: W. W. Norton, 1993.
Wadsworth, Deborah. "Civic Numeracy: Does the Public Care?"
In Why Numbers Count: Quantitative Literacy for Tomorrow's
America, edited by Lynn Arthur Steen, pp. 11-22. New York, NY: The
College Board, 1997.
Wu, Hung-Hsi. "The Mathematician and
the Mathematics Education Reform." Notices of the American
Mathematical Society 43:12 (December, 1996) 1531-1537.
Copyright © 1999.
Lynn A. Steen
Jerry P. Becker
Dept. of Curriculum & Instruction
Southern Illinois University
625 Wham Drive
Mail Code 4610
Carbondale, IL 62901-4610
Phone: (618) 453-4241 [O]
(618) 457-8903 [H]
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