Date: Apr 14, 2013 9:14 PM Author: Jerry P. Becker Subject: Twenty Questions about Mathematical Reasoning ****************************************

What follows and is attached, by Lynn Arthur

Steen, is the last chapter of NCTM's 1999

yearbook.

****************************************

Twenty Questions about Mathematical Reasoning

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,

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 [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

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, 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

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

understanding?

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 mathematical

reasoning?

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 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

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 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

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 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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Copyright © 1999.

Contact: Lynn A. Steen

URL: http://www.stolaf.edu/people/steen/Papers/reasoning.html

Disclaimer

--

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]

Fax: (618) 453-4244

E-mail: jbecker@siu.edu