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#### This volume contains a series of papers on algorithmic learning. Included are six reviews of research pertaining to various aspects of algorithmic learning, six reports of pilot experiments in this area, a theoretical discussion of "The Conditions for Algorithmic Imagination," and an annotated bibliography. All the papers assume a common definition of algorithmic learning as "the process of developing and/or applying methods or procedures, i.e., algorithms, with the goal of learning-how-to-learn." A common definition of algorithm is also used. Topics covered by literature reviews include algorithmic processes for cognition, algorithms and hierarchies, conceptual bases for the learning of algorithms, interference with the learning of algorithms, algorithmic problem solving, and algorithms and mental computations. Research papers report on studies related to algebra (3), arithmetic (2) and the use of desk calculators (1). The authors conclude that there are many open researchable questTélécharger gratuit ERIC ED113152: Algorithmic Learning. pdf

DOCaMEMT BESOHE

BD 113 152

AOIHOIl

.TITLE

INSTITOTION

SPONS AGENCY

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NOTE

AVAILABLE FBOH

SE 019 635

Ed.

Suydao, Harilyn H. Ed. ; Osborne, Alan B.,

Algorithnic- Learning.

EfilC Information Analysis Center for Science,

Mathematics, and Enirironmental Education, Colaobtis,

Ohio.; Ohio State Oni?. , Columbus. Center for Science

and Mathematics Education.

National Inst, of Education (DHEW) , Washington, D.C.

Career Education Program.

[75]

194p.

Ohio State University, Center for Science and

Mathematics Education, 24a,Arps Hall, Columbus, Ohio

43210 ($3.75)

EDBS PBICE

DESCEIPTOBS

IDENTIFIEBS

ABSIBACT

MF-$0.76 HC-$9.51 Plu

♦Algorithms; Cognitiv

Secondary Education;

Theories; ♦Literature

Education; Memory; ♦B

♦Algorithmic Learning

s Postage

e Processes; Elementary

Instruction; ♦Learning; Learning

Ueviews; ♦Mathematics

esearch; Teaching \Hethods

This volume contains

algorithBic learning. Included are si

to various aspects of algorithmic lea

experiments in this area, a theoretic

for ' Algorithmic Imagit^ation," and an

papers assume a common definition of

process^ of developing and/or applying

algorithms, with the goal of learning

definition of algorithm is also used,

reviews include algorithmic processes

hierarchies, conceptual bases for the

interference with the learning of alg

solving, and algorithms and mental co

report on studies related to algebra

of desk calculators (I). The authors

open xesearchable guestions in the* ar

(SD)

a series of |)apers on

X reviews of research pertaining

rning^ six reports of pilot

al discussion of "The Conditions

annotated bibliography. All the

algorithmic learning as "the -

methods or procedures, i.e.,

*how-to-learn. " 1 common

Topics covered by literature

fo^: rr.gnition, algorithms and

learning of algorithms,

orlthms, algorithmic problem

mputa^ions. Research papers

(3) , Arithmetic (2) and the use

concluiaie that there are many

ea of algorithmic learning.

• / - '

/

♦ Documents acguired by ERIC include many informal unpublished ♦

♦ materials not available from oth^ir soiirces. ERIC makes every, effort ♦

♦ to obtain the best copy available. Nevertheless, items of marginal ♦

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♦ of the microfiche and hardcopy reproductions ERIC makes available ♦

♦ via the EEI-C Document Reproduction Service (EDRS) . EDRS is not ♦

♦ responsible for the quality of the original document. Reproductions ♦

♦ supplied by EDES are the best that can be made from the original. *

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ATION

M ATI ON

THE ERIC SCIENCE, MATHEMATICS AND

ENVIRONMENTAL EDUCATION CLEARINGHOUSE

in cooperation with ^

Center for Science and Mathematics Education

The Ohio State University

Algorithmic Learning

edited by

> Marilyn N. Suydain

V and

Alan R. Osborne

ERIC ^

/

/

Mathematics Education Reports

Mathematics Education Reports are being developed to disseminate

information concerning mathematics -education documerits analyzed at the

^ERIC Information Analysis Center for Science, Mathematics and Environ-

laental Education. These reports fall into three broad categories.

Research reviews summarize and ajialyze recent research in specific

areas of mathematics education. Resource guides identify and analyze

materials and references for use by mathematics teachers at all levels.

Special bibliographies announce the availability of documents and review

the literatTore in selected interest areas of mathematics education.

Reports in each of these categories may also be targeted for specific

sub -populations of the mathematics education community. Priorities

for the development of furture Mathematics Education Reports are estab-

lished by the advisory board of the Center, in^cooperation with the

National Council of Teachers of Mathematics, the Special Interest Group

for Research in Mathematics Education, the Conference Board of the

Mathematical Sciences, and other professional groups in mathematics

education. Individual comments on past Reports axid suggestions for

future Reports sure always welcomed.

This publication was prepared pursuajit to a contract with the

National Institute of Education, U.S. Department of Health, Educa-

tion and Welfare. Contractors undertaking such projects under

Government sponsorship are encouraged to express freely their

judgment in professional and technical matters. Points of view or

opinions do not, therefore, necessarily represent official National

Institute of Education position or policy

ii

d

/

\

CONTENTS

. Algorithmic Learning

Section

Page ....

V

*-

I

3

II

Conditions for Algorithmic Imagination, by

13

III

Revievs of Previous, Research on Specific Aspects

A.

Algorithmic Processes for Cognition, by

25

B.

Algorithmic Learning and Hierarchies, by-

ko

c.

Investigations of Conceptual Bases Under-

lying the Learning of Algorithms, by

56

D.

Algorithms: Interference, Facilitation, and

73

E.

Algorithmic Problem Solving, by Richard W.

85

F.

Algorithms and Mental Computation, by

97

IV

Reports of Some Exploratory Research

A.

Algorithmic Processes in Arithmetic ajid

107

B.

A Comparison of Different Conceptual Bases

for Teaching Subtraction of Integers, by

119

. c.

Solving Quadratic Inequalities: .More Than

137

D.

A Comparison of Two Strategies for Teaching

Algorithms for Finding Linear Equations, by

ll|-7

E.

Seme Computational Strategies of Students

Using Desk Calculators, by Raymond Zepp...

iii

15^^

ERLC

5

I

/

CONTENTS (continued)

Section Page

V Annotated List of Selected Research References

Related to Computational Algorithms, by

Marilyn N. Suydam... I65

VI . Summary,, 197-*

IV

8

•1 \

I. Algorithmic Leaxning: Introduction

ii

ERIC

Algorithmic Learning ; Introduction

Marilyn Suydam

To many people, "algorithmic learning" means "the learning of

algorithms". They think of algorithms for addition, subtraction,

multiplication, and division vri. th whole numbers, such as:

5^ i2 86 l8j23ir

±37 ^ ^ 18 .

11 27 775" ^ ^

80 ' 602 5i

^ ^79^

They think of algorithms for operations with fractions and decimals, of

a square root algorithm, of procedures in the content of algebra and

calculus and other mathematical areas.

But algorithmic learning involves more than just the learning of

specific algorithms. - It connotes having learners generalize from

specific skills to broader process applications. It is related to

learning~how-to-learn. As Simon (1975) pointed out, teaching the algo-

rithm and teaching the characteristics of an algorithmic solution are /

two different things.

The importance of algorithmic learning is being increasingly-

recognized, across other content areas as well as within mathanatics.

In the past few years, it has been developed as the approach in at

least one textbook. The research interest in artificial intelligence

is built on a foundation of algorithmic learning. Several Russian

psychologists, among others, have been Very much concerned with the

implications of algorithmic learning (Gerlach and Brecke, 1974; Landa,

The focus of much current writing is still on algorithms, hut the

need to provide for algorithmic learning is becoming increasingly more

evident. The use of hand-held calculators at all levels fVom the ele-

mentary years through life has raised new questions about algorithins —

3

8

and emphasizes the need to explore ways in which algorithmic learning

can be promoted, as calculators decrease the need to focus' so much of

our attention on the algorithms for calculation.

Explanation

This document is not intended to be all-inclusive (although we had

dreams of being comprehensive' at one early pointi). It. is basically the

report of a year of emphasis on algorithms and algorithmic learning in a

seminar for mathematics education doctoral students at The Ohio State

University. It do^.sn't include all that the seminar encompassed. But it

does present some results, both in the form of research reviews and mini-

research studies. It is hoped that it will serve to have others do more

thinking .about what is known about algorithmic learning, and, even more

iiaportant, to think about what still needs to be explored and learned^

In proposing the seminar, it was noted that there is a tradition of

concern for algorithms in the computational orientation of elementary

school mathematics, but new information-processing models of learning

seem to be stimulating a new body of research problems exid studies. A

more general interest is suggested, in broadly conceived algorithmic

learning non-specific to the computational needs of young children. The

relation of algorithmic learning to problem solving, logical ability,

creativity, and the like have not been explored. And they should be.

Our focus was indicated by this flow diagram for our initial discussions:

What is an algorithm?

Definitions frm texts

Computational

algorithms

(examples, research,

selected studies)

Other elementary

Secondajry school

algorithms

Computers

J

Broad meaning/purpose

of algorithms

Learning how to learn

ALGORITHMIC LEARNING:

the evolving focus

k

9

Thus, this document attempts to:

(1) review the status of some aspects of research related to

algorithmic learning across the mathematics curriculum,

and

(2) indicate a few of the directions which reseai'ch on algo-

rithmic learning and on computational algorithms has

..taken and might take.

It is not intended to be a state-of-the-art paper, but only another con

tribution to the increasing documentation on algorithms and algorithmic

learning.

Definitions

We worked for many hours trying to find good definitions for '^algo

rithm" and "algorithmic learning". In the course of this serarch, we

found that algorithms have been defined in two ways :\

..(l) By examgle, especially at the elementary school level

and in elementary school mathematics content and method

textbooks for teachers.

i

(2) By simple definitions , such as:

\

(a) "A computational procedutre, especially one that

involves several steps,

rithm." (Bouwsma, Corle

ko)

is often called aji algo-

and Clemson, 1967, p.

(b) "Each arrangement of nxiijibers for purposes of com-

putation was called an algorism. . . .Many algorisms,

or ways of .setting down and arranging the figures,

were tried for each of the four processes before

those we now use finally prevailed." (Buckingham,

19^7, p. 15)

(c) "The most natxiral algorism, or written record of

the children's thinking, ..." (Clark and Eads,

195i+, p. 75)

(d) "An algorism is both the procedure for carrying out

an operation and the arrangement of the numerals

and operational symbols for computation." (Hollister

and Gunderson, 196!+, p. 29)

(e) "An algorithrfj is a set of procedures for perform-

ing a computation ..." (Kelley and Ri chert,

1970, p. k7)

(f) '\ general procedure, called an algorithm,"

(Muel3:er, I96I+, p. 71)

(g) . . the usual term algorittuii will be used to

refer to any computational device [where 'device'

i3 a written procedure]." (Ohmer and Aucoin,

1966, p. 89) . ' .

(h) "... the advocates of the' Hindu -Arabic system

with its algorithms, or procedures, for computa-

tion." (Peterson and Hashisaki; 1963, P- 18)

(i) "^ ; arithmetic based on the Hindu-Arabic ninner-

als, more especially those that made use of the

zero, came to be called algorism as distinct from

the theoretical work with nmbers which was still

called arithmetic ... we have the word loofiely

0 used to represent any work related to computation

by modern numerals and also as synonymous with

the fundamental operations themselves and even

with that form of arithmetic which makes use of

the \abacus.'-* (Smith, 1925, PP. % lO-ll)/

(j) "From a mathematical standpoint we may c^iaracter-

ize an algorithm in terms of a finite alphabet t'the

digits 0. to 9 plus a few additional symbols in the

case of arithmetic), an\ infinity of wo^^ds made up

of a sequence of elementary steps or rules that

are required to handle any initial woi*k in a unique

way. The algorithm for column addition is a good ■

example of such a scheme . . -."(Supp'es, Jerman,

and Brian, I968, pp. 289-290) /

We attempted to evolve a more inclusive definition, one not

specific to mathematics: , i

algorithm : a method (e.g., for computatio^) consisting of

a finite number of st^ps, the /steps being taken

in a preassigned ordei* and reproducible, that

is specifically adapted to th^ solution of prob-

lems of a particular category^

And for

\

algorithmic learning : the process of developing ajid/or

applying methods or procedures, i.e., algo-

rithms, with the goal of lear)aing-how-to-learn.

6

1i

Beilin (l97^) summarizes the problem in discussions and explora-

tions of work on algorithms and algorithmic learning:

The difficulty over the use of algorithmic methods stems

in part from the lack of differentiation between con-

ceptual algorithms and instructional algorithms. Instruc-

tional algorithms are devices, usually symb'olic, that

provide standardized ways of apprbaching the analysis or

solution of problems and are essentially pedagogical

instruments. ... . -

Although practical considerations are important ^ih con-

sidering the value of algorithms, even more impor;tant

is the need to detennihe what is essential for thought

and problem solving to occur. ...

Algorithms, thus, are not pimply arbitrary devices for

solving school problems but enter into the very nature

of the processes by which cognition develops. They may

serve as ;instructional devices as well, but developments

in computer similation of thinking show that algorithms

serve 'a much more serious and necessary function in

reasoning and learning. . . . Th.e task for mathematics

education is to develop instructional algorithms \diose

structure and content will articulate most adequately^

with the structure and nature of conceptual algorithms,

(pp. 1^9-130)

References • ^" " .

^ i

Beilin, Harry. Future Research in Mathematics Education: The View from

Developmental Psychology. In Cognitive PsychojLogy and the Mathematics

Lal3 oratory (edited by Richard Lesh). ColTombus^ Ohio! ERIC/SMEAC

Science, Mathematics, and Environmental Education Information Analysis

Center, 197^. ERIC: SE OI9 075. '

Bouwsma, Waj^d D.; Corle^, Clyde* G. ; and Clemson, Davis F., Jr. Basic

Mathematics for Elementary Teachers . New York:' The Ronald Press

Company, I967.

Buckingham, Burdette R. Elementary Arithmetic , Its Meaning and Practice .

Bostjon: Ginn and Company, 19^7-

Clark, John R. and Eads, Laura K. Guiding Arithmetic Learning . Yonkers-

on-Hudson, New York: World Book' Company, 195^. .

Gerlacli,' Vernon S. and Brecke, Fritz H. Algorithms in Lear ning and

Instruction ; A Critical^ Review . Paper presented at the Aniiual

Convention of -the American Psychological Association, Viugust 1973«.

ERIC: ED O85 250. . ' \ '

Holliater, George E. and Gunderson, Agnes G. Teaching Arithmetic in

the Primary Grades. Boston: D. C. Heath and Company, 196^.

Kelley, John L. and Ri chert, Donald. Elementary M athematics for Teachers.

San Francisco: Holden-Day, Inc., 1970.

Lajida, L. N. Algor i thmi zati on in Learning and Instruction . Englewood

Cliffs, New Jersey: Educational Technology Publications,' 197^+ (first

published in Moscow, I966). ERIC: ED 097 032

Mueller, Francis J. Arithmetic : Its Structure and Concepts . Englewood

Cliffs, New Jersey: Prentice -Hall, Inc., 196^+.

Ohiuer, Merlin M. and Aucoin, Clayton V. Modern Mathematics for Elementary

School Teachers . Waltham, Massachusetts: Blaisdell Publishing Company,

1966. . . '

Peterson, John" A. and Hashisaki, Joseph. Theory of Arithmetic . New York:-

John Wiley & Sons, Inc., I963.

Simon., Herbert A. Learning with Understanding . Paper prepared for

Invited Address, Special Interest Group for Res.earch in Mathematics

Education, at the Annual Meeting of the American Educational

Research Association, March 1975.

/

Smith, D. E. History of Mathematics , Volume II, New York: Dover Pabli-

ca^tions, Inc., 1958: Girm^.-and Company, 1925.

Suppes, -Patrick; Jerm'aiC^ax; and Brian, Dow.' Computer-Assisted Instruc-

tion : Stanford-'^965-66 Arithmetic Program. Nev York: Academic Press

II. Conditions for Algorithmic Imagination

11

ERIC

15

There is nothing either good or bad but thinking makes it so.

(liamlet— Act 2, Scene 2, Shakespeare)

Vanditions for Algorithmic Imagination

Alan R. Osborne

Computers and small electronic calculators have recently become a

part of our culture. What was/a futuristic fantasy of science fiction

(Asimov 1957) is now a portioti of the reality requiring the thought and

attenti9n of educators. There is a reasonable expectation that calcu-

lators and computers will become more accessible and common in the

immediately foreseeable future.. Some would argue that this decreases

the importance of teaching computation in the schools: Others would

remark that the concern for "Why Johnny Can't Add" is misplaced and

inopportune. Although such argiHiients may have more credence than they

would have had even five short years ago, they are^strawmen diverting

the attention of .designers of curricula and theoreticians of the in-

structional process from more pressing and vital questions about the

experience of children and youth with arithmetic and mathematics.

The purpose of this paper is to raise some questions about the

focus of mathematical experiences in the school given the fact of ready

access to calculators and computers during the adult life of children

presently in today's schools. The questions and issues • raised by the

community of scholars in mathematics education within the context of

philosophizing about or considering needed research within the domains

of computational proficiency and instruction for algorithms indicate

some profound oversights in terms of the future needs of children.

A theme pervading Pirandello's plays is that reality is determined

by the thinking and feeling of an individual. In Six Characters in

Search of an Author (Pirandello, 1922), each character constructs his

own reaUtyT Historians of 'science hypothesize the same type of opera-

tional determination of reality for individuals contributing ideas to

the evolution of science. Boring (1929) defines and documents the

concept of Zeitgeist operating within the field of psychology in his

A History of Experimental Psychology . The prevailing philosophical :

orientation and spirit of the time^, the Zeitgeist, is a context thajfc

determines the categories of ideas , to be prized and the questions arid

research important for psychologists of a given era to advance the /

state of knowledge. This provides limits to the imagination in-jjiuch

•the same sense of T. S. Kuhn's concept of paradigm as explicated hio

13

study of the historiography of science ^ The Structure of Scientific

Revolutions (I962). Kuhn extends the concept of Zeitgeist with the

-concept^ of ^paradigm to encompass the model of the science held or

believed by practitioners in the field. • Ihis paradigm determines and

is determined by what are considered legitimate questions and problems

in that fields allowable research procedures ^ the philosophical orien-

tation of the fields the type of apparatus used^ and what is considered

to be known with a degree of certainty. For both Boring and Kuhn^ the

limitations apply to the individ.ual scholar and to the community of

scholars as a, whole. For the individual^ this provides the matrix of

beliefs^ understandings and' procedures from whence develops his sense

of appropriateness for his own activities and the delimitatio.ns of his

interests. Induced by his membership in the community of scholars in

his fields it is a function, of the nurture provided by that field which .

yields both the Vellsprings of creativity and the limits on the imagina--

tion for an indo^vidual scholar.

Have the s^me sorts of factors operated within the field of

mathematics? We would argue that t'lis must be the case. Many creators

of mathematics have -demonstrated keen awareness of the legitimatizing

charai.cter of tjhe paradi^ held by the community of scholars In

mathematics. Consider Cardan's apologies in reporting his work with

complex, non-:^eal numbers or the hesitancy evidenced by the inventors

of non-Euclidean geometry in publishing their studies . These two ex-

amples suggest a retarding effect on dissemination was operant if a

creator of mathematics was (or is) aware of the existence of a paradigm

within his discipline when hi3 creation doefi not fit the paradigm. Many

other examples can be found in the history of mathematics .

Of greater interest for our purposes is the set of ideas and^

approaches to mathematical problems and theories which were not created

because of the existence of a paradigm. That is to say, have paradigms

had a retarding effect (other than slowing dissemination and the spread

of ideas) on the advance of the field of mathematics? No historical

answer to this interesting question exists. One cannot provide histori-

cal evidence for the causes of a non-event; one must limit the arguments

to supposition. ' Some examples of such arguments do exist. For example,

Osborne (1968) argues that the Greeks' careful sense of closure con-

cerning operations with lengths , areas and volumes prohibited their

understanding and quantification of momentum even though the writings

of Aristotle indicate that momentum was an important concept to the

Greek scientists. Understanding of this rudimentary concept of science

would await Galileo in an era in which the paradigm of Greeks ' careful

reasoning was relaxed and freed by the impact of the Dark Ages and the

probable non-understanding of the niceties of Greek thought by the Arabs .

The history of modem algebra suggests the impelling force of

mathematical paradigms or traditions, ^i^emming from a Greek tradition

of geometrical algebra, it was the mid-i'ifteenth centur:,/^ before Bombelli

would formulate algebraic arguments free of the hampering restriction of

17

providing a magnitudinal base for numerical arguments. Vieta, approxi-

mately 25 years later^ moved algebra somewhat in the direction of its

own notation, yet it would be the turn of the eighteenth century before

Peacock would attempt to free algebra, completely from the. need to pro-

vide 'real* referents for algebraic symbols. The traditions of pro-

viding real referents for the symbols of algebra suggest a hampered

development of quaternion algebra by Hajnilton and the more generalized

description of a vector space by Grasgman. Indeed, both Hamilton and

Grassman were concerned with the question of whether a 'real' base for

their algebras existed.- One wonders what the retardation effect of the

mathematical paradigm of . needing real referents v/as on the field of

modem algebra. , , .

Paradigms provide limitations on the mathematical imagination and

creativity of both an individual and for the community of users and

doers of mathematics. On the one hand, it may be at' the attitudinal

level' for specific individuals, forcing them into a construction of

their own. form of reality in the sense of a Pirandello character. On

the other hand, it may be the more direct result of the traditions or

appeal^nce of traditions in tln^ sense of the mathematical paradigms

described above.. In schoal mathematics at the elementary and secondary

levels, the traditions and perceptions of what is legitimate mathematics

is communicated through the experience of each individual child. The

experiences of the child determine his Zeitgeist or paradigm from whence

his imagination and creativity will well. The modes of thought and

processe-^ that both limit and facilitate the child's productive use of

mathematics are imprinted in much the same sense as the imprinting of

intuition on the very young. The thesis of this paper is that if the

child's experiences within the context of his school mathematics environ-

ment extablish, and determine the paradigms of his thought, then mathe-

matics educators need address the problem of whether an appropriate

paradigm for our present and future ages in mathematics is being

established*

We would argue that present school mathematics programs, and the

' associated supportive research concerning their effectiveness, does not

address the problem of whether the goals and activities of the programs

build paradigms and/or a Zeitgeist fitting children's future adult

needs in mathematics. The school mathematics program at the elementary

and secondary school levels has been oriented by a need to produce

students who are computationally proficient. Throughout our history

this has been an important goal. Imagination and creativity, and the

setting up of these attributes of individual performance, has been

directed to the necessity of performing in the traditions of the existing

mathematical thought and uses. The goal of computation has been quite

appropriate. Individuals have needed to possess computational skills c

in order to participate fully in an adult life. Further, the very

nature of the scientific and mathematical world has required computational

skill. Note that by computational skill we mean much more than the

capability of working with numbers but also are including the ability

15

to work with higher-order mathematics even through the undergraduate

level.- Computational skills have been necessary to the individual in

gaining a modicum of control over his personal environment beyond his

application of mathematics to science or to mathematics per se. The

•housewife in coping with her budget, the golfer computing his score,

and the home -improvement nut constructing a new patio each need a leyel

of computational proficiency in order to fulfill expected roles in

their personal life. In order to maximize participation in life, chil-

dren needed to build^ computational competence.

Clearly computational jproficiency is still important. A student

of mathematics needs to know enough and be able to do enough computation

so that teachers and other individua.ls can communicate with him. > But

it is an open question yhether the operational proficiency of the^past

and present is sufficiei^t to provide the Zeitgeist or paradigm needed

for the future adult lifW of today's children. Does the present treat-

ment of school mathematics prepare a child for a world characterized by

rea(3y access to electronip calculators and to computers? Is the scope

and sequence and approach! of the school mathematics" program sufficient

to prepare individuals for intelligent application of devices capabl^^

of carrying out complex computations with the application of pressure'

on some buttons? Are we limiting the sort of problems which children x

can solve with the aid of machines? >;

■ • N

The world of the future will be characterized by extensive use of

the computer at many levels of our society. Individuals need to under-

stand algorithmic processes if they are to take maximal advantage of

computers. Although computer .programmers are presently oeing trained . .

on the base of present curricular orientation and content, is the

efficiency of this training impaired because of a failure to stress the i

development of algorithmic thinking? Inadvertently are curriculum

designers building limits on students' future creat^-vity in the use

and application of computers? Are habits of thinkfrJg or mind sets

acquired during the early childhood experiences with mathematics that

limit or retard algorithmic learning? Are students building appropriate

intuitions? ',

The advent of the machine is changing the basic nature of mathe-

matical endeavor. Algebra, number theory, and analysis are each evolving

around new processes and styles of thinking which are directly attribut-

able to the machine. Birkhoff 's article, '^Current Trends in Alget)fa"

(1973), argues persuasively that the machine orientation of matheniatical

research in algebra is here to stay. Not only are new processes being

used in modern algebra, but also a different style or type of problem,

is being considered as significant by the algebraist. The paradigm is

shifting. \ "

Finally, the student entering college today\often encounters the

use of the' computer as an instructional device. We do not refer to.

computer-monitored instruction or computer-assisted instruction that

16

uses the power of the machine as a means' of teaching the usual mathe-

matics by controlling individualization, administering drill and

practice, or the general administration of instruction. Ratherl^re are .

speaking, of the use of the computer to exhibit and do mathematics that

an individual with a pencil and paper could not accomplish. An example

of this might be the examination of the limit of a function in a

particular neighborhood. With rudimentary pro grapmlng skill the student

has access to mathematical examples unimagined in the -instructional

sense in the immediate past. An algorithmic sensibility would' facilitate

a student's perception of exactly what was happening in the example and

perha.ps make it real to him in a sense that is not available to many of

our is tudents today. We would argue that this entails more than the

experience of programming; to know and be able to use a language is not

sufficient. We would question whether the desired intuitions can be

established through experience gained as late as junior high school and

whether they 'can be acquired simply on the base of instruction in "

computers without attention to the mathematical orientation of algorithmic

thought. The modes of thought necessary to successful use of the com-

puter are isssentially mathematical. A basic component of this mode of

thought is algorithmic in character. :

Bronowski "(1965) , whose, field is the foundations of mathematics,

argues persuasively that our mathematical imaginations are limited -by

what We know and do in mathematics. His supposition is that we cannot

conceive readily of scientific and mathematical ideas that do not have

a basis in the real number system. McLuhan (196^) hypothesizes that

number concepts operato^ng- within the context of printing has channeled

our imagination in direbtions accounting for the development of our

scientific-technological society. Computational machines are going to

have a comparable impact and influence on thinking. A new paradigm or

a different Zeitgeist will be established with both limits and facili-

tates our creativity in coping with our environment.

A significant question for curxicular specialists is suggested by

tfee shift to computational machines: Doe-o the present: Surricular,

experience of the child facilitate use of the machine? ^ ^at is.;*^o say,

do present materials help establish , a machine - zeitgeisf and crea-eivity

that will enchance the child's "future work with computational devices?^

Or do presently designed mathematical experiences inadvertently establish

inhibiting paradigms and modes of thought? We woxild argue that the

\ latter is the case.

At the heart of productive use of computational devices is a

Capability for algorithmic thought. Whether the device is a low-level,

hand-held calculator or the more sophisticated, programmable computer,

ek'fective power in using the machines depends .upon developing a paradigm

o^ Zeitgeist facilitating rather than limiting algorithmic understanding

irii and or mathematics. But our thinking and research about algorithms

hak been limited for the most part to purely computational algorithms in

terjns of the elementary school arithmetic program. Even when algorithms

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are implicit in the content of^the secondary mathematics sources, 'the

algorithms are seldom treated as such but examined as a means to another

content goal. At the elementary level, curricular development and

related research has been limited almost exclusively to the establish-

ment of computational competency rather than encompassing an understand-

ing of algorithmic processes.

The phrases algorithmic thinking and algorithmic learning have been

used. above. A word of explanation is in order. Textbooks at the school

level do not present algorithms as processes constructed by people which

entail evaluative decisions. ..Within texts algorithms are defined

explicitly as having a limited capacity of solving problems and are

seldom considered as providing mathematical problems in and of

themselves. Rather,. a mathematical context is defined to which a

specific, previously constructed algoritlim applies. Now it may be the

^ case that to this same contest more than one specific algorithm may

apply, but the texts, if they present an alternative algorithm, rein-

force the idea that no decisions are involved concerning the algorithm.

For excimple , givqn an addition problem 238 + 95? the child is taught to

use the regrouping or carrying algorithm: .

"^r 238

333

The child may encounter an alternative algorithm such as

238 ■

■ . — 1_95

200

120

13

300

30

3-'

. ^.^ ' 333-

But this second algorithm is used with the intent of stregthening the

. student ^s understanding of place value and of the ^Initial algorithm.

The first algorithm is the favored technique for the addition problem.

At no point, be it the context of addition at the early elementary

school level or other computational contexts, is the learner let in on

tlTe fact that he has a choice of algorithms to apply. He is not allowed

to make decisions concerning the efficacy and efficiency of algorithms.

We would argue that choice decisions between alternative algorithms

constitute an important component of algorithmic thinking.

TTie . example considered above does not argue that the presentation

of alternative algorithms is not an effective teaching device within

the context of current curricular practices. (It should be remarked

/

■ . ■ ' " ' * ' /

*that researchers have aniassed little firm evidence concerr^ing how and

when alternative algorithms should be presented or what outcomes may,

be predicted.) ^Rather it is to point out, through the us^ of an example

from elementary^. school arithmetic^ a characteristic of algorithmic

thinking.-: Algorithmic thinking invblves more than the application of a

decision-free algorithm with the litoited capability of only treating a

single mathematical context. We argue that algorithmic thinking entails

selection and decisions concerning alternate algorithms \ Lxch apply to

a single problem.

The most common strategy for instruction concerning an algorithm

is a progression through three distinct steps:

1. The necessary, prerequisite mathematics for the

conceptual base is developed carefully.

- 2. The algorithm is presented, typically with a

rationale in terms of the conceptual base.

3. Opportunity for practice is provided.

Each of these steps is developed with the learner to ^establish an

algorithm which has been constructed or borrowed for the learner by the

author of the instructional materials. Students are not expected to

construct or develop an algorithm themselves even though the necessary

conceptual base has been established as the first step of the instruc-

tional strategy.

Most ciorricular reform of the past twenty years assumed a founda-.

tional precept of the learner needing to behave like a sclentltst if

he were to understand the processes of science.- The exhortation leveled

at and by mathematics teachers was, ^Mathematics is not a spectator,

sport." Students were expected to behave like mathematicians. But,

curiously, this expectatio^ did not extend to algorithms . Students

were protected from behaving like mathemathicians with respect to

algorithms. A mathematician does construct algorithms; this is a

portion *of the task of being a mathematician. For the curricular

designers in mathematics of the late fifties and the sixties to pro-

claim that mathematics is not a spectator sport and then to design

materials not allowing students to create their own algorithms is at

the least ironic. ,

Mathematics educators have little or no experience in either '

allowing or expecting students to construct their own algorithms. The

effect of this type of constructivist orientation on student achivement

o/ computational proficiency is not. known. The impact on attitudes and

values may only be conjectured. It is not known if or how understanding

would be extended beyound the traditional, objectives which are con-

sidered important today. Woua.d students display the confidence and

sense of self -competence which contributes to being creative? Do

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maturity and experience factors contribute to the child ^s being able

to construct algorithms? If younger children have limited capability

for creating and evaluating algorithms ^ then what are the limiting

constraints of their problem-solving ability which provide the

interferencfe? These questions are important if we are to extend parti-

.eipation in doing mathematics to algorithmic subject matter • A study

in this vein is being conducted by Hatfield (197^) • Preliminary re-

sults indicate that children have a capability for constructing algorithms

as early as grade two^ given an appropriate problem solving context.

Clearly some knowledge of how students cope with algorithmic

learning exists in the literature of mathematics education. Some of

this may be suggestive of questions and problems of import. Some of

is may suggest hypotheses in need of testing. Perhaps the most com-

parable learning in mathematics which a child experiences is the idea

of mathematical structure. This important unifying concept of mathe-

matics is a set of ideas .which taken together possess significance far

beyond their significance taken separately. Research suggests learners ■

need to acquire cognitive maturity and to have some experience with the

separate ideas before they. acquire the concept of a mathematical

structure. If an algorithm is a fitting together of several processes

into a complex decision^ network designed to solve each .of a specific

category of problems ^ .then it is very similar to the concept of

structure. Perhaps the leaniihg of characteristics of algorithms and

the consideration of algorithmic, thinking as a process are subject to

the same order of maturity factors. We do not presently have a research,

base which suggests when and What first experiences in constructing

algorithms are most appropriate. We suspect that algorithmic learning

is very similar to children acquiring a feel for mathematical structure.

The child ^s preliminary experience with the important unifying concept

of algorithm should be infonnal^ intuitive and early* Formal expecta-

tions of students being able to construct algorithms probably should

follow considerable experience in construction on an informal^ explora-

tory basis. The task of the teacher in the early elementary grades may

best be considered as providing foreshadowing experiences.';. But the

precise nature of these early experiences has yet to be detern^lned. It

seems reasonable to expect the child's experiences to*mirror the mathe-

matical Judgments to be made concerning algorithms. That is^ students

should begin early to compare algorithms as to their efficiency ^ to

identify the types of j ^oblem contexts to which they apply^ to assess

their complexity ^ to note whether there are sub -algorithms within the

primary algorithm^ and the like. These are precisely the sorts of

evaluative judgments that are needed when one shifts from one sort of

electronic calculator to another or vfhen one encounters a new programming

language. .

A.nother aspect of algorithmic thinking is identified with the word

"awarene.ss". A student should expect and be aware of the pervasiveness

of algorithmic processes ^ particularly in mathematics but also in other

fields. Many topics in mathematics at the secondary-school level are

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appropriately considered algorithms but are seldom treated as such in

our curriculum. For example, a student typically encounters at least

six. different algorithms for solving simultaneous linear equations in

the college-bounci track of high school mathematic-s . But these approaches

are seldom treated as algorithms and the algorithmic character of the

approaches are not considered. The approaches are developed around a

limited set of mathematical principles, namely substi-^^ution and the

field properties. Students need to develop an awareness of the charac-

teristics which suggest the application of each of tne particular

algorithms in order to become proficient in. using each

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