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213 Pages·2008·2.8 MB·English

DOCUMENT RESUME
SE 020 896
ED 123 132
Cooney, Thomas J., Ed.; Bradbard, David A., Ed,
AUTHOR
Teaching Strategies: Papers from a Research
TITLE
Workshop.
ERIC information Analysis Center for Science,
INSTITUTION
Mathematics, and Environmental. Education, Columbus,
Ohio.; Georgia Univ., Athens. Georgia Center for the
Study of Learning and Teaching Mathematics.
National Inst. of Education (DREW), Washington, D,C.;
SPONS AGENCY
National Science Foundation, Washington, D,C.
PUB DATE
Hay 76
213p,
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DESCRIPTORS
Concept Teaching; Elementally Secondary Education;
*instruction; *Mathematics Education; * Research;
*Research Reviews (Publications); Teacher Education;
*Teaching Techniques
ABSTRACT
Nine papers presented at a research conference on
strategies for teaching rdathematics are presented in this volume. The
first paper provides an overview of research on teaching strategies,
defining a perspective on the subsequent papers. The second paper
reviews the major strategies frog a historical perspective. The third
paper discusses the role of a theory in the development of teaching
strategies. Your papers are concerned with research problems related
to teaching strategies. The first of these deals with studios of
efficacy of different strategies; the second concerns a comparison of
teaching strategies which differed in the amount of information being
taught and the amount of pupil-teacher interaction, More general
research papers concern problems of designing studies of teaching
strategies and a context for studying teaching strategies from a
delive-7-systems approach, The eighth paper discusses materials for
teacher training. The final paper provides an integrative summary of
research on teaching strategies.
(SD)
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Documents acquired by ERIC include many informal unpublished
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* to obtain the best copy available. Nevertheless, items of marginal
*
* reproducibility are often encountered and this affects the quality
*
* of the microfiche and hardcopy reproductions ERIC makes available
*
* via the ERIC Document Reproduction Service (EDRS), EDRS is not
* responsible for the quality of the original document, Reproductions *
* supplied by EDRS are the best that can be made from the original,
*
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Thom'as J. COoney, Editor
TEACHING
STRATEGIES
Papers from a Research Workshop
Sponsored by The Georgia Center
for the Study of Learning and
Teaching Mathematics
and the
Department of Mathematics Education
University of Georgia
Athens, Georgia
Thomas J. Cooney, Editor
David A. Bradbard, Technical Editor
ERIC Center for Science, Mathematics,
and EGironmental Education
College of Education
The Ohio State University
1200 Chambers Road, Third Floor
Columbus, Ohio 43212
1976
May,
These papers were prepared as part of the activities of the Georgia
Center for the Study of Learning and Teaching Mathematics, under
The opinions
Grant No. PES 7418491, National Science Foundation.
expressed herein do not necessarily reflect the position or policy
of the National Science Foundation.
This publication was prepared pursuant to a contract with the National
Institute of Education, U.S. Department of Health, Education and Welfare.'
Contractors undertaking such projects under Government sponsorship
are encouraged to express freely. their judgment in professional
Points of view or opinions do not, therefore,
and technical matters.
necessarily represent official National Institute of Education position
or policy.
ii
MATHEMATICS EDUCATION REPORTS
The Mathematics Education Reports series makes available recent
analyses and syntheses of research and development efforts in mathematics
We are pleased to make available as part of this series the
education.
papers from the Workshop on Teaching Strategies sponsored by the Georgia
Center for the Study of Learning and Teaching Mathematics.
Other Mathematics Education Reports make available information
concerning mathematics education documents analyzed at the ERIC
Information Analysis Center for Science, Mathematics, and Environmental
These reports fall into three broad categories.
Research
Education.
reviews summarize and analyze 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
literature 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 future Mathematics Education Reports
are established 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, and other professional
Individual comments on past Reports and
groups in mathematics education.
suggestions for future Reports are always welcomed by the ERIC/SMEAC Center.
Jon L. Higgins
Associate Director
iii
5
0111111.1.11.Wc4151MMEMMEMEMM.
Contents
Acknowledgements and Overview
vii
Leslie P. Steffe, Thomas J. Cooney, & Larry L. Hatfield
Research on Teaching Strategies:
An- Introduction
Thomas 3. Cooney
1
Teaching Strategies: Historical and Contemporary Perspectives
B. Othanel Smith
23
Toward the Development of Pedagogical Theory in Mathematics
Kenneth 8. Henderson .
35
The Role of Relative Efficacy Studies in the Development
o,f Mathematical Concept Teaching Strategies:
Some Findings
and Some Directions
John A. Dossey
51
-An Empirical Comparison of Teaching Strategies Where the Amount
of Content Information and TeacherPupil Interaction is Varied
Earl W. Swank
87
..
Design Problems in Research on Teaching Strategies in Mathematics
Richard L. Turner
115
A Research Context for Delivery Systems Research on Strategies
for Teaching Mathematics
Kenneth A. Retzer
141
The Materials of Teacher Training
David Chessman
171
Focusing on Teaching Strategies from
Reflections from Research:
Various Directions
Marilyn Suydam
185
Participants
207
v
6
Acknowledgements and Overview
The Georgia Center'for the Study of Learning and Teaching Mathematics
(GCSLTM) was started July 1, 1975, through a founding grant from the
Various activities preceded the founding
National Science Foundation.
The most significant was a conference held at Columbia
of the GCSLTM.
University in October of 1970 on Piagetian Cognitive-Development and
This conference was directed by the late Myron
Mathematical Education.
F. Rosskopf and jointly sponsored by the National Council of Teachers of
Mathematics and the Department of Mathematical Education, Teachers
College, Columbia University with a grant from the National Science
Following the October 1970 Conference, Professor Rosskopf
Foundation.
spent the winter and spring quarters of 1971 as a visiting professor of
Mathematics Education at the University of Georgia. During these two
quarters, the editorial work was completed on the proceedings of the
October conference and a Letter of intent was filed in February of 1971
with the National Science Foundation to create a Center for Mathematical
Professor Rosskopf's illness and
Education Research and Innovation.
untimely death made it impossible for him to develop the ideas contained
in that Letter.
After much discussion among faculty in the Department of Mathematics
Education at the University of Georgia, it was clear that a center devoted
to the study of mathematics education ought to attack a broader range of
As a result of these
problems than was stated in the Letter of Intent.
discussions, three areas of study were, identified as being of primary
interest in the initial year of the Georgie Center for the Study of
Learning and Teaching MathematicsTeaching Strategies, Concept Develop-
Thomas J. Cooney assumed directorship of the
ment, and Problem Solving.
Teaching Strategies Project, Leslie P. Steffe the Concept Development
Project, and Larry L. Hatfield the Problem Solving Project.
The GCSLTM is intended to be a long-term operation with the broad
goal of improving mathematics education in elementary and secondary schools.
To be effective, it was felt that the Center would have to include
mathematics educators with interests commensurate with those of the
Alternative organizational patterns were available- -
project areas.
resident scholars, institutional consortia, or individual consortia.
The latter organizational pattern was chosen because it was felt maximum
In order to operationalize a
participation would be then possible.
concept of a consortia of individuals, five research workshops were held
These workshops
during the spring of 1975 at the University of Georgia.
were (ordered by dates held) Teaching Strategies, Number and Measurement
Concepts, Space and Geometry Concepts, Models for Learning Mathematics,
vii
7
Papers were commissioned for each workshop.
and Problem Solving.
It
was necessary to commission papers for two reasons.
First, current
analyses and syntheses of the knowledge in the particular areas chosen
for investigation were needed.
Second, a catalyst for further research
and development activities was needed - -major problems had to be
identified in the project areas on which work was needed.
Twelve working groups emerged from these workshops: three in
Teaching Strategies, five in Concept Development, and four in Problem
The three working groups in Teaching Strategies are:
Differential
Solving.
Effects of Varying Teaching Strategies, John Dossey, Coordinator;
Development of Protocol Materials to Depict Moves and Strategies, Kenneth
Retzer, Coordinator; and Investigation of Certain Teacher Behavior That
May Be Associated with Effective Teaching, Thomas J. Cooney, Coordinator.
The five working groups in Concept Development are Measurement Concepti,
Thomas Romberg, Coordinator; Rational Number Concepts, Thomas Kieren,
Coordinator; Cerdinal and Ordinal Number Concepts, Leslie P. Steffe,
Coordinator; SI .ce and Geometry Concepts, Richard Leah, Coordinator; and
Models for Learning Mathematics, William Geeslin, Coordinator.
The
four working groups in Problem Solving are: Instruction in the Use of
Key Organizers (Single Heuristics), Frank Lester, Coordinator; Instruction
Organized to use Heuristics in Combinations, Phillip Smith, Coordinator;
Instruction in Problem Solving Strategies, Douglas Grows, Coordinator;
and Task Variables for Problem Solving Research, Gerald Kulm, Coordinator.
The twelve working groups are working as units somewhat independently
of one another:
As research and development emerges from working groups,
it is envisioned that some working groups will merge naturally.
The publication program of the Center is of central importance to
Research and development monographs and school mono-
Center activities.
The
graphs will be issued, when appropriate, by each working group.
.
school monographs will be written in nontechnical language and are to be
Reports of single
aimed at teacher educators and school personnel.
studies may be also published as technical reports.
All of the above plans and aspirations would not be possible if it
were not for the existence of professional mathematics educators with
the expertise in and commitment to research and development in mathematics
The professional commitment of mathematics educators to the
education.
betterment of mathematics education in the schools has been vastly under-
estimated.
In fact, the basic premise on which the GCSLTM is predicated
is that there are a significant number of professional mathematics
educators with a great deal of individual commitment to creative scholar-
There is no attempt on the part of the Center to buy this scholar-
ship.
ship --only to stimulate it and provide a setting in which it can flourish.
viii
8
The Center administration wishes to thank the individuals who wrote
the excellent papers for the workshops, the, participants who made the
workshops possible, and the National Science Foundation for supporting
financially the first year of Center operation.
Various individuals have
provided valuable assistance in preparing the papers given at the work-
shops for publication. Mr. David Bradbard provided technical editorship;
Mrs. Julie Wetherbee, Mrs. Elizabeth Platt, Mrs. Kay Abney, and Mrs. Cheryl
Hirstein, proved to be able typists; and Mr. Robert Petty drafted the
Mrs. Julie Wetherbee also provided expertise in the daily
figures.
operation of the Center during its first year. One can only feel grateful
for the existence of such capable and hardworking people.
Thomas J. Cooney
Leslie P. Stefie Larry L. Hatfield
Director
Director
Director
Teaching Strategies Problem Solving
Concept Development
and
Director, GCSLTM
ix
-1-
Research on Teaching Strategies:
An Introduction
Thomas J. Cooney
University of Georgia
The Teaching Strategies Project has as its main objective the study
of teaching and teacher training. In particular, the project is concerned
with logical aspects of teacher behavior and the way in which these aspects
The work of the Teaching Strategies
may relate to effective teaching.
Project originated in the theory of teaching advanced by Smith and
The papers in this collection focus on theory and research
Henderson.
concerning the teaching of mathematics and possible directions for further
In this introduction, the contents of the papers are highlighted
research.
and promising research on teaching mathematics that was done outside the
project is identified and discussed.
Hyman (1971) described teaching as a triadic relationship, involving
a. teacher, at least one pupil, and the subject matter to be taught and
Henderson (1971) provided a more formalized conceptualization of
learned.
teaching by interpreting teachingas a ternary relation T(x, y, z). To
Henderson, the domain of x constitutes "sequences of 'actions' of an object
which, in terms of some criteria; is identified as a teacher" (p. 137).
The domain of z according
The domain of y is the set of'teachable objects.
to Henderson is "sequences of actions or behaviors of a person who, in
terms of some set criteria, is identified as a learner" (p. 138).
It is
clear that the domain of z is a factor influencing the conscious actions
of a teacher, as teachers behave differently with respect to the nature of
An artistic teacher
the learner and the learner's respective actions.
alters. his or her teaching strategies according to his or her perception
of the status of the learner as evidenced by the learner's behaviors.
Although not as obvious, the domain of y is also a factor in determining
the strategies a teacher utilizes.
Henderson (1972) made the following
observation with reference to the nature and influence on teaching of
the domain of y:
One can hypothesize that the kind of teachable object
(value of y) makes a difference in teaching (value of x)
just as the kind of behavior of the student (value of z)
Surely a teacher should go about teaching an "Item
does.
of analytic knowledge or belief differently than he would
an item of empirical knowledge or belief. And just as surely
a teacher should draw a distinction between a factual state-
ment and a value judgment and hence teach them differently.
(p. 4)
10