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Topic: Twenty Questions about Mathematical Reasoning
Replies: 0

 Jerry P. Becker Posts: 16,576 Registered: 12/3/04
Posted: Apr 14, 2013 9:14 PM
 Steen/ Twenty Questions about Mathematical Reasoning.webarchive (58.0 K) att1.html (46.7 K)

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What follows and is attached, by Lynn Arthur
Steen, is the last chapter of NCTM's 1999
yearbook.
****************************************

Lynn Arthur Steen, St. Olaf College --
http://www.stolaf.edu/people/steen/index.html -
other papers given at
http://www.stolaf.edu/people/steen/Publications/papers.html

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. 270-285.

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 literate citizens.
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 questions.
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, 1997].
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,
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 [Packer, 1997].
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
reasoning?
3. Is mathematical reasoning an appropriate goal of school mathematics?
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
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, 1995].
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 perform
mathematically?
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 routine
calculations.
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
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
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
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
understanding?
8. Is proof essential to mathematics?
Despite the dominance of proof as the methodology
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
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 mathematical
reasoning?
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 end?
10. Does "math anxiety" prevent mathematical reasoning?
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
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
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
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 cooperative activities?
12. Can calculators and computers increase mathematical reasoning?
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 technology.
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
accomplishment?
13. Why do so many student feel that mathematics is a foreign culture?
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 rules-impede learning.
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 mathematics.
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
anyone?
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 [1945] 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
[1992]). 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
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 brain.
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 learning.
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?

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Contact: Lynn A. Steen
URL: http://www.stolaf.edu/people/steen/Papers/reasoning.html
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Jerry P. Becker
Dept. of Curriculum & Instruction
Southern Illinois University
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