THE STRUCTURE OF KNOWLEDGE FOR MATHEMATICS
by
PAUL M. TUNSTALL, JR., B.S.Ed., M.Ed.
A DISSERTATION
IN
HIGHER EDUCATION
Submitted to the Graduate Faculty
of Texas Tech University in
Partial Fulfillment of
the Requirements for
the Degree of
DOCTOR OF EDUCATION
Approved
Accepted
Dean of the Graduate School
August, 1993
rs
NO- 101^
Copyright 1993, Paul M. Tunstall, Jr.
ACKNOWLEDGEMENTS
I will be forever grateful to Dr. Oliver Hensley, my
chairman and the great guru of knowledge structures.
He has
made it possible for me to express some ideas that I have
had and used for a long time.
I am also grateful for the assistance I received from
the other members of my committee:
Dr. Paul Randolph, Dr.
Cliff Fedler, Dr. Charles Kellogg, Dr. Diane Miller, and Dr.
Ron 0pp.
I thank them for their time and for the guidance
they have given me over the last five years.
I would like to thank my wife, Edie, and my two
daughters, Casey and Megan, who have surely wondered where I
have been this past year.
I love them all very much and
hope that my efforts will serve them in the future.
11
CONTENTS
ACKNOWLEDGEMENTS
ii
LIST OF TABLES
vi
LIST OF FIGURES
vii
CHAPTER
I.
INTRODUCTION TO THE PROBLEM
1
Purpose Statement
1
Problem Statement
1
Summary of the Problem
6
Thesis Statement
11
The Structure of Knowledge for Mathematics
. . 13
The Tunstall Model for the Structure of
Knowledge for Mathematics
Definition of Terms
Description of the Major Processes Side .
Description of the Knowledge Content
Areas Side
Knowledge Content Areas Explanations . . .
Description of the Dominant
Technologies Side
Definitions Related to Understanding,
Meaning, and Knowledge
Uses of the Tunstall Model
Incremental Learning in Mathematics . . .
Historical and Psychological Basis . . . .
Conclusion
II.
13
19
21
25
26
27
28
30
31
32
33
Assumptions
34
Hypotheses
34
Delimitation
35
Study Justification Statement
36
RELATED LITERATURE
37
Introduction
37
Early Mathematics
38
• • •
111
The Greek Contribution
The Mathematics Revolution (1500-1600)
38
....
41
Seventeenth, Eighteenth, and Nineteenth
Century Mathematics
42
Twentieth Century Mathematics
45
Mathematics Education through
the Late 1950s
Disciplinary Organizations and the New
Mathematics of the 1960s and 1970s
The Problem of Poor Mathematics Performance
from the 1970s to the Present
National Concerns
International Competitiveness
and Comparisons
Failure of Mathematics Recruitment
Programs
Perennial Solutions
Curriculum
Teaching Strategies
Professional Development
Recent Attempts to Structure
the Discipline
III.
IV.
47
. 74
99
99
100
104
106
107
108
110
Ill
RESEARCH METHODS AND PROCEDURES
116
The Setting
116
Sample Characteristics
117
Development of the Model
118
Development of the Survey Instrument
118
Procedures for Gathering Data
120
Procedures for Analyzing Data
121
PRESENTATION AND ANALYSIS OF DATA
123
Perceptions of Attributes
124
Overall Validity, Comprehensiveness,
and Usefulness
IV
124
Validity of the Major Processes
Component
Validity of the Knowledge Content Area
Component
Validity of the Dominant Technologies
Component
V.
126
128
131
SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS . . .133
Summary and Conclusions
133
The General Tunstall Model
134
The Major Processes Component
134
The Knowledge Content Areas Component . .135
The Dominant Technologies Component . . .135
Comments
136
Recommendations
REFERENCES
137
141
APPENDICES
A.
INSTITUTIONS INCLUDED IN THE SAMPLE
147
B.
SURVEY INSTRUMENT
159
LIST OF TABLES
4-1
4.2
4.3
4.4
Mean validity, comprehensiveness, and usefulness
ratings and hypothesis test results for
the General (overall) Tunstall Model
125
Mean validity ratings and hypothesis test results
for the Major Processes Component of
the Tunstall Model
127
Mean validity ratings and hypothesis test results
for the Knowledge Content Areas Component
of the Tunstall Model
129
Mean validity ratings and hypothesis test results
for the Dominant Technologies Component of
the Tunstall Model
131
VI
LIST OF FIGURES
1.1
1.2
The Tunstall Model for the Structure of Knowledge
for Mathematics: The General Model
14
The Tunstall Model for the Structure of Knowledge
for Mathematics: The Major Processes
16
1.3
The Tunstall Model for the Structure of Knowledge
for Mathematics: The Knowledge Content Areas . 17
1.4
The Tunstall Model for the Structure of Knowledge
for Mathematics: The Dominant Technologies . . 18
Vll
CHAPTER I
INTRODUCTION TO THE PROBLEM
Purpose Statement
The purpose of this investigation is to devise a model
of the structure of knowledge for mathematics.
The model
should graphically show the major content areas, the generic
processes, and the dominant technologies of mathematics.
This researcher intends to validate that model through the
expert opinions of mathematicians.
Problem Statement
There is, currently, no valid model of the structure of
the body of mathematical knowledge to assist mathematicians
in providing for the stewardship of knowledge in
mathematics.
In particular, there is not an accepted model
of the structure of knowledge for mathematics that can be
used by mathematicians as a heuristic for research in the
discipline, as an aid to scholarship in the organization of
knowledge, or as an impetus for further study in
mathematics.
There is not an accepted model that can guide
mathematics educators in curriculum construction, in group
or individualized instruction, in student career counseling
and advisement, in program articulation, or for assessment.
There is not an accepted model that can assist applied
mathematicians in their role as consultants in the practical
uses and service needs for the disciplinary organizations
associated with mathematics and mathematics education.
If
such a model were to be devised, it would have great
potential for assisting students in understanding the
generic processes of mathematics and in knowing the
relationship of the major areas of study.
The model should,
furthermore, facilitate the work of mathematicians and
educators in dealing with the organization of mathematical
knowledge in their role as stewards of the discipline.
Specifically, the model should advance mathematics through
scholarship, practice, administration, instruction,
grantsmanship, the disciplinary organizations, advising,
research, and, in general, disciplinary thinking.
The
validation of a model for the structure of knowledge for
mathematics would, therefore, serve as a basis for a second
generation of work involving specific content and
instructional pedagogy.
The social problems related directly and indirectly to
the structure of knowledge for mathematics are
multidimensional.
They extend deeply into the public
schools, reach throughout our businesses, and permeate every
part of our technological society.
The disciplinary
problems associated with developing a model for the
structure of mathematical knowledge are endemic in most of
our mathematics, science, and applied disciplines.
Structuring and articulating knowledge is a common problem
among all levels of education and all of the other areas
affected by disciplinary thinking.
The greatest problem associated with developing a valid
model of the structure of knowledge for mathematics is the
general abdication of responsibility for the problem.
Mathematicians are so knowledgeable about the entire field
and so dedicated to pure mathematics that they consider the
functional modeling of the discipline to be at such a
fundamental level as to have already been accomplished.
In
discussing the topic with mathematicians, some immediately
take the position that a model already exists or that it was
suggested by Pythagoras (Maziars, 1950) or Viete (Bell,
1945) prior to the discovery of the new world by the
Europeans.
When one assures the skeptics that none of these
mathematical geniuses ever created a model of functionally
structured knowledge for mathematics, many immediately
switch their argument to one which involves their own
motives for studying mathematics.
Some also declare it to
be a trivial problem for mathematicians or one that does not
pertain to their area of interest or specialization.
If it
has not been developed, they conclude, it is because
mathematicians perceive that the personal nature of
understanding mathematical structure would cause variation
from mathematician to mathematician thus making it
impossible to arrive at a consensus on any model.
The
problem of modeling the discipline of mathematics may be
acknowledged as legitimate by some mathematicians, but its
development is considered generally quixotic:
believed to be an impossible task.
it is
Unfortunately, most pure
mathematicians do not see its theoretical significance, and
they have no interest in its applied value.
What this study proposes to investigate is the
introduction of a new approach to the study of mathematics.
New paths are difficult to take as no one has seen fit to
traverse that direction before.
There is great hazard in
stepping into the unknown, and it is difficult to know how
to start.
The adage that "Those who can, do;
can't do, teach;
those who
and those who can't teach, teach teachers"
might serve as a common starting point. We have all heard
this bit of cynicism and recognize its basic truths.
Despite its pessimism, the adage lends great insight into
the problem of the division of the stewardship of
mathematical knowledge.
For centuries, many great
mathematicians have devoted much of their time to research;
thus expanding the knowledge of mathematics.
Teachers of
mathematics have given most of their time to instruction;
thus training the new generation in the use and the
understanding of mathematics.
Mathematics educators have
for years dealt with the preparation of teachers and the
problems of teaching mathematics in elementary and secondary
schools.
Little time, however, has been devoted by any of
these groups to the study of the stewardship of knowledge.
Now there is a new group of academicians, known as
epistecyberneticists (Fedler & Hensley, 1991), who are
advocating the study of mathematics using the structure of
knowledge validation design by Hensley and Tunstall (1993)
as their mode of inquiry.
These new pioneers do not fit
into any of the traditional mathematics associations.
Most
were trained in a discipline and practice scholarship,
teaching, pedagogy, and research; however, all at some time
changed their way of thinking from classical modes to the
radically different expert systems analytical method of
inquiry (Harmon, Maus, & Morrissey, 1988) which champions
the seven phases of expert systems development.
Epistecyberneticists, such as Hensley, Fedler, Tunstall, and
Sisler (1993), believe that the world is entering an Age of
Structured Knowledge and that the traditional ways of
thinking and the traditional classical roles must be changed
in order to provide for a more efficient and effective
stewardship of knowledge.
Epistecyberneticists (Fedler &
Hensley, 1991) maintain that academicians, teachers, and
educationists have neglected the study of the structure of
knowledge for their disciplines and that many of the
stewardship problems of mathematics stem from the lack of a
valid, conceptual model of the structure of knowledge for
the field of mathematics.
Efforts to improve scholarship,
instruction, advisement, practice, administration,
development, disciplinary service, disciplinary thinking,
and research in mathematics are, therefore, all impeded by
the absence of such a model.
They believe that a new way of
thinking about the totality of knowledge and its stewardship
is necessary if our society is to meet the knowledge needs
in an Age of Structured Knowledge.
Summary of the Problem
Throughout history, mathematicians have attempted to
organize mathematical knowledge.
The Pythagoreans failed in
their attempt to quantify the universe in terms of
mathematics because of their lack of understanding about
irrational numbers (Maziars, 1950).
Bell (1945) reports
that Archimedes also attempted to model the world through
mathematics.
Socrates and his fellow Greeks spent little
time wondering about the evolution of mathematics (Bochner,
1966) since they believed that basic mathematical knowledge
was innate (Jowett, 1968).
According to Bell (1945), the Europeans contributed
nothing to the development of mathematics until Boethius
introduced Greek mathematics in the eighth century, A.D.
Even as late as the sixteenth century, mathematicians were
more concerned with trying to implement symbolic notation in
order to reduce tedious computation as a better way to
organize the discipline.
Today, most mathematicians
continue this tradition rather than attempt to establish an
epistecybernetic (Fedler & Hensley, 1991) approach to the
fundamental organization and structure of disciplinary
knowledge.
Even the seven great achievements of the seventeenth
century (Bell, 1945) failed to impel mathematicians toward
the perception of a need for organized knowledge.
Leibniz
never completed his venture of using symbolic logic to
create an encyclopedia of mathematical information (Maziars,
1950).
Bell (1945) asserts that it was during the
eighteenth century that the four major branches of
mathematics began to develop, and, therefore, little time
was left for thought about the need for a system of
structured knowledge.
The Industrial Revolution and some prophetic thinking
by Rousseau was to give mathematics a push toward social
utility (Bloom, 1979). The philosophical views of Dewey
regarding schooling, in general, and mathematics, in
particular, was seen as a means of social change.
In the
nineteenth century, educators became the organizers of
mathematical knowledge (Dworkin, 1959).
The structural
theorists advocated meaningful learning (social utility) as
a vehicle for organizing mathematical subject matter (Van
Engen, 1949).
When the social utility movement (Brownell,
1954) failed to produce better mathematicians, the emphasis
moved toward the study of learning rather than the content
to be learned.
By the end of the 1950s, the Commission on
Mathematics of the College Entrance Examination Board (CEEB,
1959) was appointed to make changes in mathematics
instruction due to public outcry concerning the
effectiveness of the mathematics curriculum in the United
States.
The report of the Commission was based on a belief
that the United States could make a major change in the
curriculum by promoting strong preparation skills for
college mathematics, deductive reasoning, and an advocacy
for the teaching of mathematical patterns.
However, after
more than ten years of the new mathematics, the program
(Changing Times. 1970) was condemned when American students
placed eleventh out of the twelve countries involved in
international standardized tests.
Morris Kline (1973) criticized the new mathematics for
its corrosive effects on mathematics and mathematics
education by pointing out the fallacies and the weaknesses
of the system; however, even Kline failed to see how a model
of the structure of knowledge for mathematics might provide
a way to solve some of the problems which have been
encountered by the previous stewards of mathematical
knowledge.
Since the heyday of new math, mathematics achievement
scores have not increased.
Moreover, mathematicians and
mathematics educators have not analyzed mathematics as a
"disciplinary system" and have not addressed the problems of
the stewardship of knowledge in mathematics.
Today, student
achievement remains low when compared to the achievement of
8
students in other countries (Stevenson & Stigler, 1992).
International students (Kolata, 1985) make up the majority
of graduate populations in mathematics in many major
universities across the United States. As long ago as 1984,
Peterson (1984) reported that the supply of mathematics
teachers, researchers, grant writers, and practitioners was
threatened, while math anxiety (Hechinger, 1987) flourishes
in our schools.
Current research involving object oriented
approaches to knowledge (Hennessey, 1993) is only beginning
to touch on the possible advantages of having a mathematical
knowledge structure.
What can be done to improve the stewardship of
knowledge and, hence, mathematics achievement?
Many studies
attempt to find solutions to these problems by concentrating
on specific needs of the curriculum in the United States
(NCTM, 1989), on prevalent teaching strategies (Peterson,
1987), and on ideas for continued professional development
(Cooper, 1987).
Unless there are mathematics teachers who
are soundly trained to implement the proper strategies into
a valid curriculum, slow progress will be made in increasing
mathematics achievement on any level (Peterson, 1984).
While salaries and monies for local curriculum changes
and professional development are crucial to the development
of a high national standard in mathematics achievement
(Rotberg, 1985), the proper use of research and funding to
wisely direct these curriculum and professional development
programs should be the major goal of the leaders of the
discipline.
The persistence of these problems indicates that none
of these approaches has met with universal success. A
hastened substantive advancement toward solutions to these
knowledge organization problems should be possible with the
recognition and validation of a basic model of the structure
of knowledge for the field of mathematics.
Hensley et al. (1993) have described the Scholars' Ring
of Activities.
It interprets the stewardship of knowledge
to include the following disciplinary activities:
thinking,
administration, instruction, scholarship, grantsmanship,
practice, disciplinary organizational needs, and advisement.
The problems in each of these areas are quite evident to
mathematicians, but low student achievement scores in
mathematics is the most popular indicator of the major
difficulties within the discipline of mathematics and has
eclipsed other serious and practical knowledge organization
problems.
This researcher has identified in the related
literature several problems which lie within all of the
general stewardship activities of the discipline and which
might be alleviated with the validation and eventual use of
a model of the structure of knowledge for the field of
mathematics.
10
Thesis Statement
It is the author's thesis that although there are
several subdisciplines of mathematics containing a
tremendous amount of ataxic mathematical knowledge,
currently, there does not exist a valid model of the
structure of knowledge for mathematics that shows the
relationship among the major functional processes, the
knowledge content areas, and the dominant technological
knowledge.
Furthermore, there is no professional
association within or outside of mathematics that claims
responsibility for the problem of organizing the knowledge
of mathematics within some set of standard guidelines.
Neither the mathematics nor the mathematics education
associations have, as yet, developed an acceptable model for
the knowledge of the discipline.
Will mathematicians recognize the validity of a model
that shows the functional structure of knowledge, and will
they recognize the model as a solution to the problem of
knowledge organization in mathematics?
An expert systems
approach appears to be the best way to investigate this
critical national question.
Hensley and Tunstall (1993)
have developed the Structure of Knowledge Model Validation
Design based upon this approach. The first step of this
system involves the identification of a graphic or pictorial
representation, or model, of the structure of knowledge for
a particular discipline.
The second step entails the
11
identification and analysis of the knowledge in the
discipline through a scan of the literature.
Stage three
consists of the development of an infrastructure as evidence
of idea and use of the knowledge of the discipline, while
the fourth step demands the refinement of the overall
structure of the prototype by testing its validity on a
pilot population.
Step five requires that the system be
tested for validity by a group of experts in the field.
In
this study, the Structure of Knowledge Model Validation
Design will be used to test the validity of The Tunstall
Model of the Structure of Knowledge for Mathematics.
Two mathematicians from each of the 204 "national
universities" (Elfin, 1992) and a mathematician from a
university in thirteen other countries will be asked via a
survey to assess the validity, completeness, and usefulness
of the Tunstall Model of the Structure of Knowledge for
Mathematics.
These 421 experts in the discipline will be
invited to rate the general model as a whole in regard to
these three qualities and each of the 19 more specific areas
of its three main components as to their perceptions of its
validity.
If the Tunstall Model of the Structure of Knowledge for
Mathematics and its components are assessed as being valid,
the field of mathematics and its researchers will possess an
acceptable design for the organization of knowledge within
the discipline.
This new model and theory should provide an
12
alternative to new mathematics and its institutional dogma.
It should lead to an improvement in the efficiency and the
overall effectiveness of all of those who participate in and
have a role in the stewardship of mathematical knowledge.
The Structure of Knowledge for Mathematics
The Tunstall Model for the Structure
of Knowledge for Mathematics
The general model (Figure 1.1) for the structure of
knowledge for mathematics takes the shape of a cube.
Each
visible face of the cubic model represents an important
component of the way the work of mathematicians occurs and
is organized, e.g., how mathematics knowledge is structured
and used.
The front of the cube shows the generic processes
of mathematics.
The face on the right shows the major
subdisciplines of mathematics, and the top face shows the
dominant technologies used by mathematicians.
The partitions on the face on the front of the model
(Figure 1.1) represent the levels of the Major Processes of
mathematics.
Each level is meant to classify generic levels
for the processes of mathematics in such a way as to imply a
recurrence of systematic use.
The comprehension of the
intricacies and the generalities of such a system should aid
students in the construction of mathematical knowledge.
These generic process levels subsume the customary
operations of mathematics.
More specific information
regarding the Major Processes Component of the Tunstall
13
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found in Figure 1.2.
The partitions on the square face on the right side of
the cube in Figure 1.1 represent the content side of
mathematics.
Each partition of this side of the General
(overall) Model is meant to describe the Knowledge Content
Areas of mathematics into which all of the past, present,
and future knowledge of the discipline might be classified
and to provide a class into which material found in current
predominant courses and topics of study could be readily
subsumed.
Although some mathematical knowledge may be used
in more than one area, each piece of mathematical knowledge
should be able to fit into one of these areas so that
mathematics scholars may use the model as a guide toward
better comprehension of the organization of the discipline
and toward a better understanding for the uses of
mathematical knowledge.
More specific information
regarding the Knowledge Content Areas Componenent of the
structure of knowledge for mathematics can be found in
Figure 1.3.
The partitions on the top face of the cube (Figure 1.1)
represent some of the Dominant Technologies available to the
scholars of mathematics in the twentieth century.
More
specific information regarding parts of the Dominant
Technology Component of the Tunstall Model for the Structure
of Knowledge for Mathematics can be found in Figure 1.4.
15
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18
Definition of Terms
Algorithm:
A purely procedural process for solving a
problem in a finite number of steps.
Archetype:
The original pattern or model of all things
of the same type.
Ataxic:
Without classification or order;
disorganized.
Axiom:
A statement generally accepted as true.
Comprehensiveness:
Digit:
Being inclusive of the necessary.
Any one of the ten symbols which make up all of
the numerals in the decimal system.
Epistecybernetics:
The science of organized knowledge.
Essential Knowledge Elements;
Those specific generic
solutions which, by their placement as a necessary part of a
subdiscipline of mathematics, actually tend to define the
subdiscipline.
Expression:
A group of numerals and/or symbols which
stand for a number.
The Four Basic Operations of Mathematics: Addition,
subtraction, multiplication, and division.
Heuristic:
An example or model that leads to further
study.
Model:
A graphic or pictorial representation of a
generic solution to a problem.
Postulate:
A proposition taken for granted as true and
made the starting point in a chain of reasoning.
19
Procedure:
A non-unique ordering of steps that may
achieve an intended result (not necessarily an algorithm.)
Real World Form:
An orderly arrangement which suggests
an entity, e.g., the point-slope form of a linear equation.
Real World Model:
A graphic or pictorial
representation of an object, e.g., the graph of a supply
curve.
Real World Object:
Something that may be seen or felt,
e.g., a sphere, a counting stick, any numeral, etc.
Semantic Symbols:
The ten digits, the symbols for the
four basic operations, the basic figures of plane and solid
geometry, and grouping symbols.
Steward:
One actively caring for the affairs of an
organization or discipline.
Stewardship:
Active care for the affairs of an
organization or discipline.
Syntactic Symbols: All symbols of mathematics which
are not defined as semantic.
Truism:
Truth:
A truth, axiom, or postulate.
An accepted statement, declaration, or
proposition.
Usefulness:
Being serviceable for a beneficial or
productive end.
Validity:
Logical correctness;
sound or well founded.
20
the state of being
Description of the Major Processes Side
Regardless of the topic, there exists a hierarchical
structure of processes that permeate the whole of
mathematics.
The General (overall) Model (Figure 1.1) is
more specifically represented by one of its three major
components, the Major Processes, as shown in Figure 1.2.
Mathematical interrogatives, mathematical processes, and
mathematical demonstratives subdivisions are used to clarify
the Major Processes Component of the Tunstall Model.
The mathematical interrogatives in the structure
proposed in this thesis list some of the questions that
students may need to answer in order to perform the
processes of mathematics.
The mathematical demonstratives,
consequently, are the products that result from the use of
each of the process.
The structure, therefore, consists of
eleven interrogatives that correspond to eleven processes
that lead to eleven related demonstratives.
All eleven processes in the Tunstall Model of the
Structure of Knowledge for Mathematics are based on
principles of logic as well as on given axioms and
postulates that are to be taken as truth without proof.
While they are not a learning taxonomy (Bloom, 1956) or a
learning hierarchy (Gagne', 1977) the levels of the
processes are hierarchical with regard to mathematical
thinking with level one having the lowest rank and level
eleven having the highest rank.
21
Level one allows a student to ask how one can represent
mathematically.
The process by which we represent is called
ASSOCIATING THROUGH SYMBOLS and involves the use of a symbol
or symbols to represent real world objects, forms, and/or
models.
The use of this process is demonstrated by the
recognition of semantic symbols, common and decimal
fractions, units of measurement, function notation,
exponential form, set notation, the standard form of a
quadratic equation, etc.
Level two allows a student to ask how one can define
mathematically.
The process by which we define is called
DEFINING THROUGH SYMBOLS and involves the generalizations
that explain what symbols will produce.
The use of this
process results in the definitions and conventions for usage
of all semantic symbols and forms, i.e., "+" commands
addition, "8!" indicates the multiplication of 8 by all of
the positive integers that precede it, etc.
Level three allows a student to clarify semantically.
The process by which we clarify is called MANIPULATING
SEMANTIC SYMBOLS and involves the maneuvering of semantic
symbols in order to clarify a mathematical concept or model.
The use of this process may be demonstrated by the acts of
rounding, counting, ordering, drawing a transversal through
two parallel lines, borrowing, finding the union of two
sets, substituting numbers for variables, augmenting a
matrix, etc.
Level four allows a student to ask how to simplify
numerically.
The process by which we numerically simplify
is called PERFORMING SEMANTIC ALGORITHMS and involves the
procedures for achieving the four basic operations (e.g.,
addition, subtraction, multiplication, and division) using
semantic symbols.
The use of this process results in the
algorithms for performing the four basic operations with
semantic symbols.
Level five allows a student to ask how to combine
semantic algorithms in order to simplify numerically.
The
process by which we combine semantic algorithms is called
LINKING SEMANTIC ALGORITHMS and involves the chaining of two
or more semantic algorithms using the four basic operations
with semantic symbols.
The use of this process is
demonstrated by any multiple operations with semantic
expressions involving real numbers, exponent forms, sets,
angles, numbers of bases other than ten, etc.
Level six allows a student to simplify algebraically.
The process by which we algebraically simplify is called
PERFORMING SYNTACTIC ALGORITHMS and involves the procedures
for achieving the four basic operations using syntactic
(non-semantic) symbols.
The use of this process is
demonstrated by the algorithms for performing the four basic
operations with simple or complex polynomials, elementary
matrices, systems of equations, or any other syntactic (nonsemantic ) expression.
23
Level seven allows a student to ask how to combine
syntactic algorithms in order to simplify algebraically.
The process by which we combine syntactic algorithms is
called LINKING SYNTACTIC ALGORITHMS and involves the
chaining of two or more syntactic algorithms using the four
basic operations. The use of this process is demonstrated by
any multiple operations involving syntactic symbols.
Level eight allows a student to ask how to relate
mathematically.
The process by which we relate is called
DEFINING HIGHER RELATIONS and involves the generalizations
that explain what relation rules can produce.
The use of
this process results in any defined relation and function
rules including polynomial, trigonometric, exponential,
logarithmic, etc.
Level nine allows a student to orchestrate processes in
order to solve mathematically.
The process by which we
orchestrate is called COMBINING and involves the linking of
higher relations with lower processes that result in the
procedures for problem solving.
The use of this process is
demonstrated by solving equations and inequalities, for
finding a relative maximum for a polynomial function, for
performing integrations and finding derivatives, for
computing with formulas, for graphing equations, and for
solving non-routine word problems, etc.
Level ten allows a student to prove.
The process by
which we prove is called VERIFYING and involves the
24
justification or proof of a theorem, statement, or lower
process.
The use of this process results in proofs in
algebra, modern algebra, number theory, trigonometry, and
calculus, for example.
Level eleven allows a student to adapt and apply.
The
process by which we adapt and apply is called DERIVING and
involves the invention of new processes or the use of
previous processes in new ways.
The use of this process is
demonstrated by any addition to the content or process
knowledge of mathematics or in the applications of the
processes of mathematics to other fields.
Description of the Knowledge
Content Areas Side
The content of the knowledge of mathematics as a
discipline might be rather simply divided between discrete
and infinite processes (Bell, 1978) or, as has long been
thought, into "the four major divisions of modern
mathematics" (Bell, 1951, p. 7) that can be further divided
into subdisciplines as arranged on the square face of the
cube that serves as the General Tunstall Model of the
Structure of Knowledge for Mathematics (Figures 1.1 and
1.3).
At a later time, other epistecyberneticists (Hensley
et al., 1993) will specifically define the content of these
subdisciplines according to their essential knowledge
elements.
The representation of these major areas should be
thought of as hierarchical only in the sense that each
25
content subdiscipline is dependent upon parts of those that
might precede it as an elementary foundation without
pretending to wholly contain them.
Knowledge Content Area Explanations
LOGIC serves as the basis for all of mathematics
including the axioms and postulates, and/or truths, that
serve to anchor the logic of thought to the reasonableness
of mathematics.
NUMBER THEORY is thought to necessarily be a part of
all of the remaining areas of mathematics in that it serves
as a foundation for manipulating the mostly countable
objects of arithmetic as evidenced in probability,
statistics, and set theory, etc.
The NUMBER THEORY AREA is
composed of mostly finite processes with semantic symbols.
The ALGEBRAS and GEOMETRIES are made feasible because
of the advent of logic and number theory.
Analytic geometry
and trigonometry are generally considered subdisciplines of
mathematics that are blends of modern algebra and plane and
solid geometry.
Affine, projective, Lobachevskian, Reimann,
other geometries, traditional, vector, linear, other
algebras, and the subdiscipline of trigonometry are, also,
in varying degrees, mixtures of modern algebra and plane and
solid geometry.
The ALGEBRAS and GEOMETRIES AREAS are
composed of predominantly finite processes with syntactic
symbols and real world objects, respectively.
26
ANALYSIS with its principal tool, calculus, serves
topology, manifolds, and homology theory, etc., by employing
the occasional use of content from all of the previously
mentioned areas.
The ANALYSIS AREA is composed of
predominantly infinite processes with all symbols.
APPLIED MATHEMATICS is not considered a true
subdiscipline of mathematics although, in the reality of
common practice, there may be no clear demarcation as to
where pure mathematics ends and the actual application of
mathematics begins.
This study will not consider applied
mathematics as a subdiscipline of mathematics.
Description of the Dominant
Technologies Side
The top face of the cube (Figure 1.1) represents the
Dominant Technologies Component that mathematicians
currently access. Figure 1.4 lists specific technologies in
the three divisions:
computers and calculators, tables and
charts, and manipulatives and tools.
Technologies listed
include PC's, hand-held and graphing calculators, the slide
rule, the compass, the protractor, the abacus, the straight
edge, pencils, paper, trigonometric tables, tables of common
and natural logarithms, tables of differentiation and
integration formulas, software, counting rods, rulers, and
graph paper.
All instruments of mathematics technology should fit
into one of these three divisions.
27
Definitions Related to Understanding,
Meaning, and Knowledge
Mathematics must be made meaningful to students
through their understanding of the relationships inherent in
the structure of mathematics itself so that the knowledge
they acquire and/or construct will be useful to them.
Meaning, says Van Engen (1949), is simply the idea
behind a symbol in mathematics.
It is a substitution in
which one sees a symbol and is able to mentally replace it
with a "referent" (p. 323) (an object, event, or "overt act"
[p. 323]).
Meaning is "that which is read into a symbol by
an individual" (Van Engen, 1953, p. 76).
Knowledge is defined by Cooper and Hensley (1992) as
the acceptance of a generic solution into a person's problem
solving repertoire.
At that instant, there is a connection
made "in the mind of an individual" (Van Engen, 1949, p.325)
between a referent and a symbol.
Knowledge suggests an
external to internal acquisition that leads to an internal
construction.
Understanding is, then, according to Weaver (1972), "an
organizational process by which knowledge is fitted together
to form cause and effect relationships" (p. 20). It is,
therefore, an internal process of integrating concepts.
Meaningful learning (Van Engen, 1949) is learning as an
organizational process in which knowledge is constructed
when the learner is externally made aware of the
relationship between a "symbol for a referent" (p. 323).
28
Meaningful learning is a term that is broader than "meaning"
while similar to "understanding."
It is the learning
process that results in the construction of knowledge.
To facilitate the occurrence of meaningful learning,
there might be a need for some external stimulus that would
serve to create, develop, or promote this needed awareness
in an individual's mind between a referent and a symbol.
If
there was to exist a structure of knowledge of mathematics
that was based on mathematical content and processes, and,
yet, was consistent with the conventions and rules of the
discipline, such a structure could be used as the mental
scaffolding or schema that could serve to anchor the
processes related to the learning of the concepts of
mathematics.
Another important concept (Van Engen, 1949) in the
structure of knowledge of mathematics is the distinction
between "semantic" (p. 398) and "syntactical meanings" (p.
398) in mathematics.
Semantic mathematics would require
learners to master foundational or primary meanings whereby
an individual associates a referent with a symbol.
Semantic
symbols would denote only the ten digits, the four basic
operation signs corresponding to addition, subtraction,
multiplication, and division, and the basic figures of
geometry, and grouping symbols.
Syntactic mathematics, on
the other hand, would require a learner to associate symbols
with other symbols and/or rules with other rules.
29
Syntactic
symbols would include all symbols in mathematics that are
not defined to be semantic; hence, they may also be called
non-semantic.
An algorithm may be defined as a purely procedural
process for solving a problem in a finite number of steps.
Truths represent the accepted yet necessarily unproven
assumptions of mathematics:
axioms and postulates.
The student's familiarity with all of these kinds of
mathematical meanings are fundamental to the success that he
or she will have in becoming functionally literate in
mathematics.
Uses of The Tunstall Model
Each of the three divisions represented by information
on the faces of this model (Figure 1.1) should be utilized
daily in mathematics classes at all levels.
The model
should be represented graphically in textbooks, on posters,
and on overhead projector screens in order to show the
location of the current topic and its relationship with and
its connection to other content and/or processes and/or
technologies.
The model should also be represented verbally
so that students may begin to recognize the generic
processes and the knowledge content areas and subdisciplines
by name.
In this way, the students may begin to
discriminate or generalize between/among different types of
processes and/or content areas as an aid to the students'
30
ability to construct mathematical knowledge while using an
appropriate technology.
Incremental Learning in Mathematics
When a teacher instructs with the idea of demonstrating
progressive learning, the job of teaching as well as the
student's job of learning may well become one of being able
to master a closed set of workable mathematical ideas rather
than one of dealing with a seemingly infinite disarray of
meaningless tasks.
Student should be made to feel good
about mathematics and must have or should be encouraged to
gain some confidence in their ability to get the job done.
Students should also be encouraged, if necessary, to feel
comfortable about starting a course with a chance to do as
well as anyone else in their same readiness class regardless
of their perceived and possibly mistaken abilities.
The
teacher may then begin a progressive learning process by
attending to these needs by making every effort to ensure
that a majority of students are able to master their initial
lesson.
The progressive process of learning can then be
continued at a reasonable pace (Brownell, 1944).
Does there exist some chain that links together these
concepts?
Is there some set of operations that threads its
way through all subdisciplines of mathematics that might
serve to guide a student's mathematical thinking?
A student
who is made aware of the existence of a structure of
31
knowledge of the discipline of mathematics might possess
such a guide.
Historical and Psychological Basis
Our everchanging knowledge base of mathematics requires
us constantly to change our ways of teaching mathematics.
Although centuries ago, the average person could make it
through life by counting and by using self-taught or rotelearned mathematics, Plato writes that arithmetic is
"something which all arts and sciences and intelligences use
in common, and which every one must learn first among the
elements of education" (Jowett, 1968, p. 280). One hundred
years ago, algebra was exclusively a college course.
Recently, however, the usual length of time that public
school education gives to mathematics is between nine and
twelve years.
Mathematics educators have realized that
mathematical knowledge was continuing to grow too fast for
the amount of time previously allotted to teach it and
decided that what was needed was a good change in teaching
strategies.
They reasoned that, if the structure of the
material were to be emphasized over arithmetic facts, much
material that was being taught in high school could be moved
down into the elementary school level.
This change was so
abrupt, however, that the necessary shift to the new math
was somewhat jolting (NSSE, 1970) and disconcerting for many
veteran teachers.
32
The selection of an appropriate learning theory to
accompany the new mathematics was heavily debated (Kline,
1973).
While theories related to those of Ausubel, Bruner,
Skemp, and Gagne' were considered and parts of many were
incorporated (NSSE, 1970), no one theory was endorsed.
It
was concluded that there is, likely, no clear method of
teaching mathematics that is always appropriate and that the
selection of a theory of learning for classroom use depends
on the subject matter, the unique characteristics of the
learner, and the emphasis the teacher puts on relating one
topic to another.
The existence and use of a model for the structure of
knowledge for the field of mathematics could give
instructors another tool in the construction of mathematical
knowledge by helping to relate topic to topic and concept to
concept without dictating learning style.
Conclusion
One can begin to solve the problems of low mathematics
achievement in the United States by introducing this
structure of knowledge for mathematics to students as long
as its onset is combined with appropriate progressive and
meaningful learning instructional techniques.
The author
believes that this structure is valid, unique, and
consistent with the laws of the nature of mathematics, and
it should be utilized so that it might help students begin
33
to make the connections that lead to the understanding of
mathematics.
Moreover, if the model helps a student to
succeed in making the appropriate connections between each
successive topic, then knowledge of the subject matter as
well as knowledge about each topic will be constructed and
the jobs of the scholars, advisors, instructors, grant
writers, disciplinary organizations, thinkers,
administrators, practitioners, and researchers in the
discipline of mathematics might be facilitated.
Assumptions
The author believes that there exists a body of
knowledge in the discipline of mathematics that can be
partitioned into three parts. These parts, in theory, can
be modeled to reflect the structure of the content,
processes, and technologies of the discipline of mathematics
by providing a framework for the placement of the generic
knowledge elements of the discipline.
Hypotheses
The author of this study intends to test the following
hypotheses:
1.
the general Tunstall Model of the Structure of
Knowledge for Mathematics will be assessed as
being:
3^
a.
valid, (mean, standard deviation, hypothesis
test at the < .05 level),
b.
comprehensive, (mean, standard deviation,
hypothesis test at the < .05 level),
c.
useful, (mean, standard deviation, hypothesis
test at the < .05 level);
2.
each of the three main components of the Tunstall
Model of the Structure of Knowledge for
Mathematics will be assessed as being valid,
(mean, standard deviation, confidence interval and
hypothesis test at the <. .05 level).
Delimitation
The population of this study will be limited to the 204
"national universities" (U. S. News and World Report. 1992)
in the United States, to the largest institution of higher
education in the other countries represented in the First
International Educational Achievement in Mathematics
comparison study (Husen, 1985), and to Moscow State
University and the University of Beijing.
This mechanism
will tend to include domestic and international institutions
of higher education which have a faculty that may be
involved to some extent in mathematics instruction,
research, administration, academic advising, scholarship,
grant writing, disciplinary organizations, and public
service.
35
study Justification Statement
The fields of mathematics and mathematics education are
unorganized in that they do not, at this time, agree upon
the functional arrangement of the content, processes, and
technologies related to the knowledge of mathematics.
The
absence of such a model exacerbates a number of
disciplinary, interdisciplinary, and educational problems
that are endemic in the Scholars' Ring of Activities
(Hensley et al., 1993) and in our social, economic, and
political systems.
The following chapter identifies
precisely the problems of mathematical knowledge
organization as related to the respective activities of
knowledge stewardship.
36
CHAPTER II
RELATED LITERATURE
Introduction
E. T. Bell (1945) wrote that mathematics was born and
lives through a relationship among number, form, continuity,
discreteness, and inventiveness and could not exist without
deductive logic based on given and unproven assumptions.
The history of mathematics, he said, is usually divided into
seven natural periods that follow the general development of
Western culture.
Although there were short periods of rapid
mathematical expansion within the sixteenth, seventeenth,
and eighteenth centuries, this cultural progression resulted
in the reality that the bulk of mathematical knowledge as we
know it today had its beginnings in the nineteenth century.
Bell was, nevertheless, quick to emphasize that, although
the applications found in science and other applied fields
today are entrenched in the mathematics of the past, the
motivation for those ideas were not totally utilitarian or
economic.
They were, in the majority of cases, simply the
objects of "intellectual curiosity" (p. 21). Maziars
(1950), a mathematics philosopher, believes that the history
of mathematics is more readily divided into periods that are
based on three major crises of mathematics:
the struggle
for mathematical abstraction, the confusion of mathematics
with science, and a preoccupation with self-analysis.
37
Early Mathematics
Bell (1945), the dominant mathematics' historian of the
twentieth century, looked at the reasons for the development
of mathematics and determined that, during this time, there
was a need for arithmetic computation for commerce, for
crude engineering projects, for astrology in conjunction
with agriculture, navigation, mapping, and calendar
refinements.
Just after 2000 B.C., the Sumerians actually
invented a crude kind of algebra to help them with their
work.
Need for a place-value system and for the zero arose
due to the ambiguities of the written number.
Pre-Greek
mathematics was "mathematics for a purpose and enmeshed with
applications" (Bochner, 1966, p. 21).
The Greek Contribution
The Greeks, prior to Socrates in the 4th century B.C.,
began to work with numerical calculations called logistic,
and they also embraced the notion of number theory, called
arithmetica, that dealt with the properties of nximbers.
Euclid (365-275 B.C.) provided many algebraic proofs in
addition to his great work on deductive geometry, and,
around the 5th century B.C., the Pythagoreans contributed
the notions of figurative numbers and the arithmetic law of
musical intervals (that were reportedly gained from the
Orient.)
The Pythagoreans attempted to explain all worldly
objects mathematically but failed when confronted with the
38
notion of irrational numbers (Maziars, 1950).
Archimedes
(c. 287-212 B.C.), also attempted to advance the idea that
the world could be modeled through mathematics (Bell, 1945).
Socrates in the 4th century B.C. would not have
expected everyone to be proficient in mathematics (Jowett,
1968).
Neither Plato (427-348 B.C.) nor the Pythagoreans
considered everyone necessarily capable or in need of using
mathematics even though they thought that everyone was born
with basic mathematical knowledge.
"The Greeks introduced
names, conceptions, and self-reflections into mathematics,
and at an early age they began to speculate on how
mathematics had come into being" (Bochner, 1966, p. 22).
In reality, the content and process knowledge of
mathematics is learned, and, although people have different
innate abilities that allow them to develop mathematical
knowledge and processes to various degrees, a person's
mathematical abilities are determined through the
individual's success in bringing together the actions of the
real world with the models and processes of mathematics.
Even though the content of mathematics certainly has changed
as knowledge in mathematics has increased over the centuries
since the time of Socrates and Plato, the basic structure of
the processes of mathematics has remained the same.
Bell
provided no hint to this structure nor to the generic
solutions all mathematicians use in performing even the most
trivial processes that may recur within many routine
39
mathematical procedures.
Centuries ago, it may have been
thought necessary or even beneficial for a person to wait
until they reached their twenties to study past arithmetic
calculations and on toward plane and solid geometries and
motion, but, now, nearly all students are able to master
more mathematical knowledge in ten years of study and
evaluation (Epstein, 1968) than even existed then.
All
students need not become mathematicians, but all should be
shown every means by which they might reach their fullest
mathematical potential.
Their introduction to a structure
of knowledge for the field of mathematics could be an
important step in this learning venture.
According to Bell (1945) the Europeans, including the
Romans, contributed virtually nothing to the development of
mathematics until the fifth century when Boethius, through
Latin translations in elementary schoolbooks, introduced
Greek arithmetic, geometry, and deductive logic to Europe in
Latin during the Middle Ages.
Between 400 and 1300 A.D.,
algebra with symbolism and trigonometry came to be
considered separate from arithmetic and geometry as branches
of mathematics, and although several great universities were
founded in Europe at the beginning of the thirteenth
century, very little mathematics found a place in liberal
education; however, with the use of the printing press in
the fifteenth century, a few important mathematics books
were circulated, and the mathematics used in navigation was
40
being refined.
During the period before the seventeenth
century, Galileo (1564-1642) and others used mathematics to
initiate future branches of what were to be the modern
sciences, and Fibonacci (1175-1250) introduced Hindu algebra
(with single-letter symbols) to Europe. Vieta (1540-1603)
and Pascal (1623-1662) innovated algebraic notation through
the use of uniform terminology to condense the laborious
manipulation prevalent in purely numerical procedures. No
effort was made during this period to model the structure of
the subdisciplines in the rapidly expanding field of
mathematics in order to begin to quantify the current set of
essential knowledge elements (Hensley et al., 1993).
Mathematics Revolution
(1500-1800)
Bochner (1966) believed that mathematics was a great
part of the "Great Scientific Revolution" (p. 37) of the
sixteenth, seventeenth, and eighteenth centuries.
The
period from 1637 to 1687, according to Bell (1945), marked
the beginning of modern mathematics during which the
development of mathematics is noted by seven great
achievements.
geometry.
(1) Fermat and Decartes invented analytic
(2) Newton and Leibniz pioneered differential and
integral calculus.
(3) The mathematical theory of
probability was devised by Fermat and Pascal.
(4) Fermat,
along with Gauss, also originated higher arithmetic and,
therefore, the basis for number theory.
41
(5) Newton designed
theories of universal gravitation by himself and dynamics
with the help of Galileo.
(6) Desargus and Pascal devised
synthetic projective geometry, and (7) Leibniz conceived the
beginnings of symbolic logic through which he hoped "to
unify all knowledge by a universal" (Maziars, 1950, p. 59)
system of notation by amassing "an encyclopedia as an
inventory of all human knowledge and a universal
characteristic or system of signs representing the concepts
and the legitimate manners of combination" (p. 60). Leibniz
never followed through on the project.
Seventeenth, Eighteenth, and Nineteenth
Century Mathematics
According to Bell (1945), the development of four major
branches of mathematics began around 1725. Arithmetic
change was based upon a new way of looking at the natural
numbers.
The operations of addition and multiplication, the
inverse operations of subtraction and division, and the
inverse of raising a number to a power necessitated the
formulation of common fractions, negative and irrational
numbers, and the invention of complex number forms by Carl
Friedrich Gauss. Modern arithmetic was expanded by the
introduction of concepts involving fields, finite groups,
and cardinal and ordinal numbers.
Algebra was partly an attempt by mathematicians to make
arithmetic increasingly general and abstract.
Bell (1945)
reminds us that the systems of mathematics are not natural.
42
They are manmade and for the utilization of future
mathematicians.
The notions of congruence, vector,
polynomial, and modular numbers, among others, were
originated during this period.
Geometry was forced to
become less rigorous and less traditional at this time in
order to expand.
The result was the realization that other
geometries could be constructed that were non-Euclidean in
nature; that is, those that were just as valid as Euclidean
geometry while based on a different set of postulates.
Mathematicians such as Bolyai, Lowbachewsky, and Riemann
synthesized geometries of their own.
the brainchild of Descartes.
Analytic geometry was
He found that there was a
connection between a set of ordered pairs as defined by an
equation in two variables and curves drawn on a plane.
Although he first used only one axis in one quadrant, his
discovery soon led to further innovations in
multidimensional calculus, functions, normals, and
transformations.
All of these innovations were most affected, according
to Bell (1945), by four critical milestones in the years
from about 1795 through 1910: the use of equivalence
relations for mappings, the recognition of the abstract
qualities of algebra, the beginning use of numerical
analysis by Gauss, and the inception of mathematical logic
through paradox.
In each case, mathematics is shown to
become more and more abstract and general with no thought as
43
to how the scholars of the discipline might begin to develop
methods of classification for the knowledge elements and the
processes that comprise the essence of mathematical
erudition.
Jean-Jacques Rousseau (Bloom, 1979) believed that a
person's education, in general, should come from nature or
from the natural ways of things of the world and,
subsequently, Rousseau would have been pleased to have had a
structure of knowledge of mathematics to use as a tool for
the teaching of mathematics based on the true meanings of
the processes of mathematical knowledge.
teaching through essay.
Rousseau disliked
He wanted his student, Emile, to
have some deeper understanding of subject matter than that
which is achieved solely through lecture.
To Rousseau, the
best thing for society was what was best for Emile, and
being able to find function in the performance of
mathematics as it suited his student was judged as best for
Emile.
In this respect, a model that delineated the content
and processes of mathematics would have served Emile well.
Today, our society is quick to judge the potential of a
course of study on how secure it will make the student as a
future wage earner in our capitalistic system.
This idea
was not lost on Rousseau; however, he would be more apt to
embrace such a disciplinary model if he were convinced that
it was the only truly informed way to approach the content
and, eventually, to apply the processes of mathematics.
44
Bell (1945) was convinced that, since science was
related so closely to mathematics, the evolution of analysis
has had the greatest influence on inventiveness.
He wrote:
Of subsequent additions to mathematics originating
at least partly in science, the most highly
developed are the vast domain of differential
equations, the analysis of many special functions
arising in potential theory and elsewhere,
potential theory itself, the calculus of
variations, integral equations and functional
analysis, and differential geometry. By 1800, the
calculus of variations and differential equations
had advanced sufficiently to be recognized as
autonomous but interdependent departments of
mathematics; the statistical method was still an
embryonic possibility in the theory of
probability; while the theory of functions of a
complex variable had yet to wait a quarter of a
century for systematic development by Cauchy,
although some of the basic results were implicit
in the applied mathematics of Lagrange and others
in the eighteenth century, (p. 361)
Bell's comments seem to suggest the need for an
epistecybernetic (Fedler & Hensley, 1991) approach to the
quantification of mathematics as a way to allow for the
expansion of mathematics and of its promise for application
but failed to realize the importance of such a disciplinary
imperative and of its possible effect on the future stewards
of mathematics.
Twentieth Century Mathematics
Bell (1945) asserted that mathematical logic was the
basis for most of the deductive reasoning found in the
twentieth century, and that, without it, scientific and
technological analysis would, simply, not exist.
45
He
defended charges by critics who contended that logical
mathematics had contributed little to philosophy in recent
years and predicted an explosion of innovations that, in the
near future, would come about due to advances in the field
of mathematics.
Bell (1951) claimed that the task of
showing mathematics to be largely an endeavor in symbolic
logic "before 1930 was instigated by the patent need for
putting a sound foundation under the enormous mass of
nineteenth-century mathematics—both pure and applied" (p.
407).
Although many mathematicians at this time were
divided between those who favored pure mathematics and those
who favored applied mathematics. Bell (1951) regarded the
two as "inseparable" (p. v.) by noting that the "pure serves
the applied, the applied pays for the service with an
abundance of new problems that may occupy the pure for
generations" (p. 2). Other uncertainties about the
discipline (Bell, 1945) were emerging as several
philosophers took it upon themselves to challenge the
Platonic concept of number and the Pythagorean notion of
observable numerology as a way to question the logical
foundations of mathematical truth.
Smith (1959) confidently
divided the whole of mathematics into four fields in order
to provide teachers and students with a source book for
mathematics in which he presented "excerpts from the works
of the makers of the subject considered" (p. v.).
But Bell
(1945) insisted that the changing nature and flexibility of
46
mathematics held the greatest promise for the future of
civilization and that "there has yet to be devised a method
more efficacious than the mathematical for enabling human
beings to reason about the results of scientific
observations and experiments" (pp. 593-594).
Although Bell certainly was an expert on mathematics
and its history, he failed to recognize or address the
organization of knowledge within the discipline.
While he
mixed historical data with mathematical example and
technical jargon to the point of confusion, his use of
detail was very interesting, and his insights into the
motivations of other mathematicians continue to make his
work an invaluable tool in the search for ways to assess and
execute the disciplinary imperatives of mathematics in
higher education.
Bell did not, however, provide a model to
mathematics scholars that might have helped to represent the
knowledge base of the field to those who may desire to
practice it, teach it, or learn it.
Mathematics Education through
the Late 1950s
Many educational learning theorists impacted
mathematics education in the twentieth century.
Their
efforts were concentrated on either how the students learn,
how teachers teach, or how both could better reach their
respective goals.
None of them, however, sought to advance
the teaching or learning of mathematics by concentrating on
47
the content and processes of knowledge in the discipline
itself or by recognizing or advancing a model of either in
order to promote learning, to facilitate instruction, or to
better represent the field of mathematics.
Dworkin (1959) saw John Dewey as a turn-of-the-century
innovator in American education through his views on
education as the proper vehicle for social change.
Dewey
may have been since proven correct to the extent that the
school has become the dominant social entity in our culture
but not in the way that he would have desired.
Dewey saw
the school and the teacher as facilitators of society and of
the work place.
But in order for a student to realize the
democracy of being able to control his own education, a
great amount of time and other resources must be placed at
the individual student's command.
Dewey failed to recognize
the stewardship of knowledge as the primary goal of any
educational system.
Education with respect only for the
good of each student without the proper consideration for
the knowledge elements of a discipline will result in a
deviation from true scholarship.
Instruction may become too
individualized for the good of the person in relation to its
cost on the resources of society.
While education must
never forget the individual needs of a student, it will
succeed by helping each person become the best individual
that they can without expecting them necessarily to learn
basic knowledge elements in different ways.
48
In mathematics,
for example, a student must be able to use the most obvious
tools available in order to create opportunities for himself
or for herself (and for society) by striving for excellence.
One way for a student to make the most of his or her
mathematics ability is to be exposed to a model of the
structure of knowledge in the field of mathematics.
Since
such a model is the most fundamental tool that recognizes
and relates the source of knowledge in the discipline to
real world problems, only then can any potential toward
vocational or occupational goals that may be harvested as
societal, cultural, or economic opportunities be fully
realized.
During the first quarter of the century, the Drill
Theory of Arithmetic, the Incidental Learning Theory, and
the Meaning Theory of Arithmetic Instruction had become
three notable ways of thinking about elementary mathematics,
and each are to be found to some extent in modern
mathematics classrooms.
While each seeks to explain how
arithmetic might be taught and learned and, although not one
of them can be totally isolated from the others, dictates a
special circumstance of the learning situation, neither has
its foundations in the knowledge base of the discipline
itself (Brownell, 1935.)
Drill Theory is based on the belief that the learning
of facts occurs only in association with a student's
involvement in the repetition of unitary facts.
49
In this
theory, no explanations are given as to why these subdivided
facts are to be learned or even as to how they relate to
each other.
Brownell (1935) seemed to feel that drilling
every math fact into a child by repeated practice is timewise utterly impractical and that, furthermore, subjecting a
student to learning by drill alone is inevitably boring.
Studies were cited that indicate that children actually
learn by many methods even when it was thought that they
were being subjected entirely to intense repetition and that
drill produces a kind of learning acceptable only when
likened to the teaching of certain trainable animals.
The Incidental Learning Theory expects children, when
left on their own, to obtain a knowledge of mathematical
facts as a byproduct of an involvement with some sort of
self-inclined activity.
Brownell (1935) began to argue that
such experiments not only develop no systematic arithmetical
logic but may even pique the child's interest in the method
rather than the method's mathematical significance.
He felt
that the theory provides no assurance of learning—only the
possible occurrence of it.
Brownell (1935) introduced the Meaning Theory of
Arithmetic Instruction that developed the notion that
students should actually make sense of what they learn.
It
attempts to bring together many learning concepts and
teaching strategies by envisioning a goal of purpose over
method, and, in doing so, actually raises the standards of
50
mathematical instruction to a newly acquirable level.
Although, at times, drill and experiment are accepted within
Meaning Theory, it excludes drill and incidental learning as
valid alternatives to teaching with reason and meaning.
Brownell proposed that Meaning Theory of Arithmetic
Instruction, in the long run, could simply reduce the
complexity of mathematics by stressing how being able to
capitalize on certain relationships that exist between many
basic concepts might tend to improve quantitative thinking.
Without question, many disagreements (Brownell, 1935)
can be found with a theory that narrowly preaches a one way
and only one way approach to learning (Drill Theory) and
there are just as many ways to attack a kind of learning
based on hit-or-miss fun and games (Incidental Learning
Theory); but what modern day theories can possibly argue
against teaching toward meaning in mathematics?
The answer
here is "probably the uninformed, the stubborn, and the lazy
who find Meaning Theory a threat to their simple, onevariable worlds" (p. 23).
When elementary students are taught from the beginning
to understand (Brownell, 1935) what they are doing without
regard to getting the correct answer they get a feel for
what mathematics means rather than the idea that arithmetic
is strictly imitation.
So often, young students are,
because of time limitations or a lack of educational wisdom
in their teachers, bribed into the happy idea that they can
51
always do well in math if they continue accurately to
reproduce problems and situations given them and are,
therefore, lulled into a kind of false security about
numbers.
Later, when high levels of thinking are needed for
problem solving and complex computation, these same
students, still trying to rely on memory and patterns alone,
get confused and do not understand concepts simply because
their priorities have been systematically misplaced and
erroneously reinforced by their trainers. Many textbooks
along with countless departments, schools, and county
curriculum lend to this problem by giving teachers only
enough time to hint at the fact that mathematics even has
meaning.
Meaning Theory of Arithmetic Instruction (Brownell,
1935), in practice, amounts to as little as giving a thirtysecond background to a new topic or simply showing how
percents relate to a just-completed unit on decimals or as
much as introducing the possibility of negative numbers
where before there were only positive ones.
Saying that one
teaches meaningfully is not the cure all. A teacher must
practice meaningful teaching techniques while using drill,
motivation, lecture, experiment, and example everyday in
order to give students a varied program where, at a very
early age, they can adopt the idea that it is really easy to
fully understand what is actually occurring in an arithmetic
class.
52
The Meaning Theory of Arithmetic Instruction (Brownell,
1935) is a theory of instruction as opposed to the other two
notions mentioned here that are theories of learning.
The
Meaning Theory of Arithmetic Instruction by its own titular
description projects mathematics from a level of "how to do"
(p. 26) to a level of "sensibly why" (p. 26).
In a 1938 summary of studies on the inventories of
skills of beginning first graders, William Brownell showed
that children possess a potential number sense that
surpasses simply being able to count to twenty, performing
addition or limited subtraction, or having a vague idea
about what fractions represent; however, these ideas are
limited and these abilities are usually performed without a
good mathematical foundation and upon mostly concrete
objects.
These studies seem to indicate either that
teachers need to postpone arithmetic instruction in order to
allow children more time to develop mentally or that any
immediate instruction should begin to capitalize on this
diverse bank of number knowledge.
The latter being a direct
endorsement of the idea that readiness is a product of
experience and not the slow unfolding of the mind.
Through tests (Brownell, 1938) in 1928, the Committee
of Seven found that by the sixth grade students who had been
introduced to structural arithmetic earlier than others
carried a distinct advantage with them.
In 1934, a similar
test showed that Scottish children were at least fifteen
53
months ahead of their American counterparts in computation
and problem solving skills due to the fact that the first
grade, in Scotland, was comprised of five-year-old students
as compared with the six-year-old students in the American
first grade.
By reviewing controlled experiments like those that
attempted to determine whether early or postponed education
was better or which attempted to fix the ideal temporal
placement of topics, Brownell (1938) ruled that the findings
in favor of postponement were invalid due to testing
procedures and to the emphasis on unimportant variables. He
proposed that, when arithmetic is taught in the right way,
it can be taught at an early age which, in turn, can give
more meaning to a topic at a later age for even
traditionally difficult subjects.
The advent of a model of
the content and process knowledge in mathematics might help
a student to clarify where he or she is in being able to
master the meanings related to the work of the discipline.
Brownell's readiness argument (1938) was that students
are able to be taught many concepts in the first grade that
can help them to understand arithmetic then and that will
stay with them forever no matter what skills they start with
as long as the teaching is meaningful.
He seemed to believe
that we should not waste our time bickering about when to
teach a certain topic if we are not going to teach it
meaningfully anyway.
Why wait until a student has an 8.9
54
mental age, for example, before teaching him/her subtraction
when the same student could learn even one meaningful
concept that would aid in the student's eventual, real
understanding of subtraction at age 6?
the important issue.
Again, the goal is
Only when one teaches meaningfully and
with a knowledge of the basic functional structure of the
discipline, what a student is able to learn, regardless of
age, the environment, the topic placement, or the teacher's
instructional method will be of potential benefit to the
student in the future.
Brownell (1944) began to realize that there were many
weaknesses in a system of learning that emphasized teaching
through connectionism.
He saw connectionism as a theory
that advocated product over process.
Connectionism
necessarily accepts the idea that a child has the ability to
go from a complete lack of knowledge of a particular topic
to a full understanding of that topic in one reinforced
leap.
This view, he said, does not consider the ability
that a student has to step from a basic concept to a more
difficult one by advancing through several stages of
understanding.
Brownell also felt that the lack of
"progressive learning" (p. 157) in this aspect of
connectionism makes getting a correct answer the most
important part of mathematics, provides no way of proving
answers, and does nothing to promote the idea of relational
mathematics.
Using a discipline-centered approach to
55
accessing the content knowledge of mathematics as a way to
promote knowledge and understanding was never considered.
Brownell (1944) also recognized that a practice makes
perfect attitude can only hold true when practice is allowed
to vary in a way that increases a student's capacity to see
how ideas relate to one another.
Repetitive practice, on
the other hand, only succeeds in establishing proficiency at
a single and usually low level of understanding.
A model
for the structure of knowledge for mathematics would
increase a student's ability to construct such relational
knowledge without falling victim to repetitive practice.
At a time when many were beginning to agree with
meaningful learning in practice and/or theory, Brownell
(1945) raised thoughts about two more problems encountered
when instructors try to make arithmetic meaningful:
that of
identifying the meanings to be taught and that of coming up
with the activities which will guide the meaningful
learning.
Brownell (1945) was more concerned with the
former of these two questions, and by viewing frequent
student errors, he noted three areas which pertain to
essential meanings in arithmetic.
The concepts of whole
numbers, place value, fractions, decimals, and percentages
are, in themselves, fundamental and essential in working at
any level within the field of mathematics.
The four basic
operations are also essential and meaningful, if not taught
by memorization and drill, in promoting future learning.
56
The basic operation algorithms are necessary for speed and
for the convenience of computation and are effective when
they are taught with understanding rather than merely as a
series of isolated rules.
Brownell (1945) acknowledged four major objections to
teaching meaningfully.
One objection asserted that basic
mathematics computation is all a student would ever need in
the way of mathematics instruction.
Brownell responded
that, when one looks at the high school curriculum, mere
computation falls drastically short of secondary goals.
Some critics of meaningful learning theory also said that
meanings were being taught as a byproduct of computation.
Brownell said that these false meanings were nothing more
than unconnected rules. Another objection was that meanings
are ideas which are too sophisticated for young people to
understand.
Brownell emphasized that the key to proper
instruction is to organize the material on a meaningful
level.
Finally, some wanted proof that teaching
meaningfully was really better, so Brownell returned with
four reasons why teaching meaningfully is important.
He
offered that knowledge, when learned meaningfully as opposed
to other methods, is useful only when it is understood, when
it helps promote future learning, when it increases the
chances for transfer, and when it makes knowledge more
easily retained.
He accentuated the need for a total
reorganization of the way arithmetic is learned and taught
57
by citing a twenty-five-year period in which arithmetic
knowledge had been taught to children in a way that was
neither personally practical nor mathematically sound.
Brownell failed in each case, however, to suggest where the
knowledge to be taught is to be found in reference to other
knowledge in the discipline.
Two of Brownell's ideas (1945) are of great concern
today:
that of the practicality of mathematics and that of
using skills only as a result of understanding.
When
students are led into a real understanding of arithmetic,
they are not limited by the mere act of doing.
Students are
able to think mathematically for themselves by applying what
they have truly learned to new situations that confront
them.
Sometimes those new situations are not new at all,
but are simply new ways of expressing a previously learned
relationship.
Students who have been meaningfully taught
have an advantage when functioning in everyday life over
those who have experienced only rote learning in
mathematics.
Such students are able to use knowledge in
ways more varied than could have possibly been presented to
them, and are, also, able to bring their own kind of meaning
to the knowledge and the situation through relational
understanding.
Arithmetic may be meaningful, says Brownell (1945),
only when the students, in their own lives and separate from
a worksheet, can use what they have learned in a rational
58
way.
Real progress results from teaching in a way which
allows methods of computation to arise naturally as a child
ponders on a method to solve a problem rather than in a way
which forces a child into trying to simply start guessing
the answer.
Accuracy in computation is useless to those not
in a position to call upon the proper problem solving
techniques or to those not even in a position to begin to be
able to define the problem.
Understanding is not
necessarily a product of acquiring skills.
Useful skills
are a result of understanding, and understanding is what
makes the knowledge of mathematics meaningful and useful to
students.
Brownell again neither mentioned the importance
of the scholarly development of mathematics as a discipline
nor of the structure of the content and processes of
mathematics.
Both must be addressed before an instructor
can know what subject matter is supposed to be taught in the
first place—meaningfully or otherwise.
A model for the
structure of knowledge for mathematics might help to further
discern and quantify the knowledge elements of the
discipline.
In the late 1940s (Brownell, 1947) attempted to define
meaning in mathematics further by making several comparisons
and clarifications.
He suggested three ways that educators
use the term meaningful and then described how the "meaning
of" (p. 256) something is different from the "meaning for"
(p. 256) something.
He gave definitions of "'meaningful'
59
arithmetic, in contrast to 'meaningless' arithmetic" (p.
257).
Brownell (1947) divided the meanings in arithmetic into
the four categories of understanding:
basic facts,
fundamental operations, important principles and
relationships, and the decimal number system.
He then
compared the accepted definition of meaningful arithmetic
with how meaningful arithmetic might have been defined in
the past.
Brownell failed to arrange the processes and
content of the discipline of mathematics in any kind of
heuristic model in order to facilitate a student's
understanding of the structure or of the relationships
between the subdisciplines as a way to promote understanding
of mathematics.
Another article by Brownell (1947) showed a need for
meaningful arithmetic by providing evidences of the failures
of other programs.
Brownell helped to clarify meaningful
arithmetic by answering objections to declarations that
teaching meaningfully was unnecessary, too difficult, overly
time consuming, and/or ultimately useless.
He summarized
twelve advantages of teaching meaningfully which he
supported in the article with logic, common sense,
educational innovation, and psychological theory.
Brownell (1947) did more than just define meaningful
arithmetic.
He showed that, whether there was proof or not,
there was plenty of evidence that meaningful learning, while
60
not perfect, had several advantages over traditional
theories.
His two teacher-related and ten student-related
evidences tended to substantiate the theory more than if he
were to have merely conjured up several technical
descriptions.
Brownell's arguments (1947) seemed always to
concentrate on the shortcomings of traditional programs.
And why not?
One of the best strategies in any revolution
is a needed and well-organized attack on the status quo.
In chapter four of part one of the Learning and
Instruction NSSE Yearbook. William Brownell and Gordon
Hendrickson (1950) provided a thorough discussion about the
way arbitrary associations, concepts, and generalizations
are taught and learned.
The ideational learning of each of
these three learning products was discussed in the major
portion of the text with an emphasis on the verbal
characteristics they employ.
Each of the three learning
products were also discussed in terms of their prevalence in
school and/or life, their conventional applications for
learning, their modern applications toward learning, and
their implications with regard to teaching procedures.
Brownell and Hendrickson (1950) talked about arbitrary
associations as existing at all educational levels.
These
associations were judged as being learned when responses are
quick and correct.
They said that such knowledge need not
be rotely memorized and, in fact, is learned in a meaningful
61
way even when methods of rote learning were employed.
Motivation must usually come from an external source.
In
regard to teaching arbitrary associations, they noted that
it should be remembered that memorization is sometimes good,
that repetitive practice makes associations more useful when
speed is required, that wrong answers should be corrected
without explanations as not to fix a wrong answer in the
mind of the student, and that without maintenance one will
easily forget the association.
Brownell and Hendrickson
(1950) noted that process should not be arrived at by
viewing product alone.
Concepts were defined (Brownell & Hendrickson, 1950) as
abstractions that may be complex in relation to meanings
that were not to be viewed as just new words.
Such concepts
have many dimensions and involve a step-like hierarchy of
complexity and sophistication that ranges from a total lack
of understanding to meaningful habituation.
In reference to
the teaching of concepts, the authors felt that
consideration should be given to many factors.
Mastery is hard to achieve since time is a factor. A
student's needs, background, and abilities must be
considered.
Intrinsic motivation, purposeful activities,
and many varied experiences relating to the concept being
taught should be reorganized and put to use.
Errors should
not be avoided as in the case of arbitrary associations.
They should be included in an overall program designed to
62
help a student show the use of the specific concept being
developed.
By definition, a generalization (Brownell &
Hendrickson, 1950) states some abstract relationship between
concepts.
Generalizations are thought to be solely the
products of problem solving through which the student must
be guided.
A student must be given a problem which can be
solved with the help of his/her background and a procedure
suggested by a teacher or deduced by the student.
There are
several points to consider about the teaching of
generalizations.
Students should not be taught in reference
to concepts that are hard to define.
given generalizations.
Students cannot be
The teacher must guide the student
toward the understanding of a generalization a deductive or
inductive method in relation of varied experiments.
If the
generalization is learned properly, as in the case of
concepts, maintenance will not be necessary.
In closing, the authors (Brownell & Hendrickson, 1950)
stressed that nonverbal concepts and generalizations are
common but that language makes it possible for us to go
beyond reality in our thoughts.
They cautioned teachers to
be careful that an over emphasis on verbalizations does not
cause items that need to be meaningfully learned to be
viewed incorrectly as products of rote memorization.
If one assumes correctly that this paper (Brownell &
Hendrickson, 1950) was not written exclusively for
63
mathematics educators, it is interesting, in brief, to note
how Brownell and Hendrickson feel when meaningful concepts
are applied to topics other than arithmetic.
Much, or even
most, of what was written is a general educator's version of
what was becoming so familiar to mathematics educators in
terms of understanding, but some statements do stand out as
being different from the Meaning Theory of Arithmetic
Instruction.
In the case of arbitrary associations, rote
memorization is condoned at times without any kind of
substitution process at hand.
Errors are handled without
explanation in a very stimulus-response sort of way.
Repetitive practice is somewhat over emphasized.
In regard
to concepts and generalizations, very little is said about
referents, meaning, symbols, concrete materials, etc.
The
student is to be guided toward understanding, and in some
cases, it is actually recommended that the student be given
a problem and left entirely on his or her own to figure out
how to arrive at a solution.
This methodology, they claim,
is reminiscent of product teaching.
When maintenance is
viewed (Brownell & Hendrickson, 1950) as a byproduct of
meaningful learning, it is rendered as useless when the
subject matter in question is taught properly.
Varied
practice should be stressed as a maintenance tool and not
only as a means of initial learning.
Henry Van Engen (1949) asserted that Meaning Theory
needed to be more precise in order to provide a way to test
64
for meanings.
He explained that the purpose of this paper
was to clarify meaningful learning by writing about overt
acts instead of writing more about what has already been
written.
In order to clarify the Meaning Theory of Arithmetic
Instruction, in general. Van Engen (1949) defined the three
elements that he said were to be present in every meaningful
situation:
(1) an event, object, or action, called the
referent, (2) the symbol for the referent, and (3) an
individual to interpret the symbol.
He used, by analogy, a
cat to demonstrate a meaningful activity in which the mind
reaches out for an object even when the object is not there
as long as the person has been presented a symbol for the
object.
In arithmetic, the mind takes on a mind set when
presented a symbol for an already learned concept which is
related to other groups or classes of objects.
Van Engen (1949) then gave the insight into the
definitions of meaning and knowledge in regard to Meaning
Theory.
He said that the symbols are not knowledge and that
the awareness of an action or operation is not knowledge,
but that when the two become associated in the mind there is
knowledge.
Van Engen is careful, here, to make clear that
the operations in question are not the same as those basic
fundamental operations.
Here they describe referents:
actions on concrete objects.
He also detailed several
arithmetic examples using proper meaningful techniques.
65
Subtraction, addition, division are illustrated meaningfully
and operationally in contrast.
The meaning of zero is also
debated operationally and meaningfully.
The second half of
this paper concerned itself with "the social-meaning, the
structural, and the nihilistic theories of meaning" (p.
395).
The proponents of social-meaning theory prevalent in
this period believed that a student will learn best when he
can observe and use numbers in social situations that are as
real and true to life as possible.
They were careful to
promote the use during instruction of real life situations
which develop meanings instead of merely endorsing lifelike
situations which may never lead to any kind of learning.
Structural theorists at this time (Van Engen, 1949)
endorsed the learning of meanings through the internal
structure or organization of the subject matter.
This
emphasis on interrelationships led to a discussion
contrasting semantic and syntactic meanings.
Structural
theorists proposed that an understanding through
interrelationships in structure is a syntactic meaning which
may or may not be made up of more basic semantic truths.
Those who espoused (Van Engen, 1949) the nihilistic
theory of meaning believed that, when given the proper
sequence of experiences, a student can be made to understand
how symbols may represent some operation or overt action
without actually manipulating a concrete object.
using the symbols gives meaning to them.
66
The act of
An article by Fred Weaver (1950) began by pointing out
that, although many groups were beginning to accept Meaning
Theory of Arithmetic Instruction, there were still those
supporters who had frequent misunderstandings about meanings
in arithmetic.
One group consisted of teachers who were
still in the process of learning about Meaning Theory and
another of those practicing teachers who seem to be
misrepresenting it.
Weaver (1950) noted three reasons for these
misconceptions but devoted the paper to only two of them.
He claimed that a minor reason for a lack of clarity in
meaningful arithmetic is the different terminology and
emphasis each individual writer seemed to use and place on
many meaningful learning concepts.
Each author's
articulations specifically depended upon mathematical or
social needs.
The first major reason that Weaver (1950) saw as
causing misconceptions was the relationship between a
student's ability to rationalize a computational process and
his ability to use those processes. Weaver strongly
emphasized the fact that neither is a prerequisite for the
other nor does either guarantee the other.
He assured
instructors that rationalization helps students to overcome
the initial hesitation related to learning a new concept and
that they should be prepared to regroup when remediation is
necessary.
67
The second major reason for misconceptions. Weaver
(1950) felt, may have been due to the misuse of concrete
materials within a meaningful arithmetic program.
The
article listed three common inaccuracies regarding those
concrete materials and attempted to underline the importance
of the three levels of representation as related to
meaningful learning.
If Van Engen (1949) and Weaver (1950) seemed to be
refreshingly exact about meanings, it may have been because
they were not plagued with being one of its originators.
They were able to pick and choose all of the things that are
related in other's theories and to weave them into a single,
plausible theory.
In fact, they were fortunate to have been
able to have enough people experimenting with meaningful
arithmetic to see what some of the problems were.
It should not be forgotten, however, that their
clarifications were rather informative to be so simple.
Their proposed system is a logical way of explaining exactly
what meanings are.
Referents, overt actions, and semantic
and syntactic meanings describe very nicely all the things
Brownell never came out and said.
Brownell (1954) further attempted to build a case for
the Meaning Theory of Arithmetic Instruction by showing
chronologically how arithmetic had changed since the 1900s
and, therefore, how a further change toward teaching
meaningfully was to be expected.
68
Brownell (1954) reminded educators that, before 1910,
the theory behind the teaching of arithmetic depended upon
memorization and the training of one's mental faculties
(Skinner, 1968).
But, between 1910 and 1935, research,
along with the social utility movement, disproved the Formal
Discipline Theory and gave arithmetic a new basis for its
content:
functional use.
Brownell noted that the
reductionist movement, which carried social utility to an
extreme, failed because there was too much concentration on
the study of content and not enough on the study of
learning.
Many learning theories which emphasized "process"
(p. 3) over "product" (p. 3) had already been advocated by
several educational psychologists and teacher experience had
particularly shown Meaning Theory to be a classroom success.
Teachers and psychologists were beginning to see that a
child was not merely a little adult.
Children have
attitudes and interests of their own that must be taken into
consideration during instruction.
The student, according to
Brownell (1954), must see sense in arithmetic while being
taught meaningfully and enthusiastically.
These ideas seem
to reflect Brownell's continued attempt toward the
establishment of Meaning Theory as the logical alternative
to traditional teaching theories.
Brownell (1954) developed the history of the arithmetic
curriculum in such a way as logically to show Meaning Theory
as an obvious culmination of the wisdom of the ages by
69
showing how instructional methods have changed.
In the
beginning mathematics educators concentrated on teaching
written facts.
Then they tried to teach only written facts
that were determined to be socially useful. When the social
utility movement failed to produce better mathematicians,
the emphasis moved toward the study of learning rather than
that of the content to be learned.
Brownell seemed to say
that the study of learning along with knowledge about the
individual learner leaves a responsible educator no choice
but to teach in a meaningful way.
Only then does
instruction emphasize "transfer of training" (Rosskemp,
1953, p. 205) through process over product in a way that can
readily make sense to every student.
Brownell apparently
wanted educators to believe that meaningful learning is
simply the next ordered step in the evolution of educational
mathematics.
According to William Brownell (1956), students who were
being taught meaningfully had been shown to do poorly on
standardized tests.
Since standardized tests of mathematics
were mainly computational, some problems may have been
attributed to a lack of the proper use of computation within
a meaningful arithmetic program.
Brownell (1956) described four major sources of this
problem.
One source may have been in the fact that
meaningful arithmetic had been misrepresented as being
ng>ressarilv noncomputational.
Another source may have been
70
in the misunderstanding that, while other theories had their
objective, they relied heavily upon drill and computational
skills.
A third source could have been attributed to
educators who adopted only portions of Meaning Theory and,
thereby, left gaps in a student's learning.
The final
source mentioned was in the observation that teachers were
not teaching the Meaning Theory of Arithmetic Instruction as
it should be taught.
In moving toward a solution, Brownell (1956) attempted
to describe levels of learning and types of practice.
He
said that students demonstrate various levels of performance
depending upon what level of understanding they have
achieved.
He pointed out that each student should go
through whatever stages he finds necessary to learn the
subject matter meaningfully but that meaningful learning is
not complete until a student reaches the final stage:
"meaningful habituation" (p.143).
Meaningful habitation
describes the level of understanding/performance where a
student knows both how to get an answer as a matter of habit
and why.
Brownell (1956) noted that a child necessarily reaches
the level of meaningful habituation through practice.
are, however, two kinds of practice:
There
"repetitive and
varied" (p. 135). Repetitive practice holds a student at a
particular level of understanding by making the student an
expert in dealing with one aspect of the subject matter in
71
question.
Brownell endorsed varied practice which, after
meanings have been properly taught, allows a student to use
his full knowledge of a subject matter to refine technique
and progress to higher levels of dealing with related
materials.
In noting the dramatic changes concerning arithmetic
instruction since the beginning of the century, Brownell
(1959) predicted four kinds of innovations that he says will
take place in the near future.
First, he predicted that the
subject matter familiar to elementary courses will be made
available to students at a much earlier age—the opposite of
postponement will occur, he says, as many ideas are pushed
down and interwoven.
Second, Brownell (1959) saw a greater emphasis being
given to the mathematical principles behind the arithmetic
concepts then being taught.
Teaching meaningful mathematics
may become teaching meaningfully related mathematics.
Brownell (1959) also predicted the development of
separate textbooks and workbooks for high- and low-level
students.
The curriculum would also be different for
students with different abilities.
Finally, in terms of the secondary curriculum, Brownell
(1959) saw a "stepped-up curriculum" (p- 42) which would,
thereby, allow more time to be spent on secondary
statistics, intuitive geometry, and calculus.
Social and
scientific changes would cause these changes to occur
72
(Brownell, 1959).
But what purpose might Brownell have had
for publishing a paper on the future of arithmetic
instruction at this very time?
Surely he had investigated
the CEEB (1959) report which had just been released:
a
report which would launch the new mathematics toward
changing mathematics instruction in exactly the four ways
Brownell predicted.
There was even a clever reference to
the CEEB in his paper.
Why might Brownell have been
outlining the expectations of the new mathematics program?
One clue (Brownell, 1959) that might lead to an answer
to this query is in his statement about what meaningful
arithmetic necessarily emphasizes.
This statement along
with the continuing paragraphs seem to contradict what he
and other proponents had said before about Meaning Theory:
in "meaningful arithmetic" (p. 43) instruction, meanings
were supposed to be based upon objects from a child's
environment and not based on the mathematical system of
related ideas in totality.
The ideas a student learned were
not to be explained using another mathematical concept a
rule to give the correction.
Brownell may have been trying to rally some forces
against new mathematics without really saying so, and in
what may have turned out to be his only truly prophetic
statement is his comment (1959) about into what he hoped the
movement would not turn arithmetic.
Surely new math did
become, in a way, the "sterile" (p. 44) scapegoat of the
73
1970s for those who wished to see arithmetic move backwards
to basics.
Disciplinary Organizations and the New
Mathematics of the 1960s and 1970s
"What kind of mathematics should be studied by today's
American youth capable of going on to college work" (CEEB,
1959, p. xi)?
The desire in 1955 for an answer to this
question, along with the realization that some colleges were
already making some curriculum changes that might affect
secondary schools, prompted the College Entrance Examination
Board (CEEB) to appoint the Commission on Mathematics.
The Commission "would be asked to review the existing
secondary school mathematics curriculum, and to make
recommendations for its modernization, modification, and
improvement" (p. xi). "To the credit of the Board, it
should be pointed out, it chose a committee of professionals
who represented the three groups most directly concerned
with the secondary school program in mathematics—college
mathematicians, high school teachers, and college teachers
of mathematics education" (NCTM, 1970, p. 260).
The final Commission roster (CEEB, 1959) was composed
of Albert W. Tucker, Princeton University, who served as
chairman; Carl B. Allendoerfer, University of Washington;
Edwin C. Douglas, The Taft School, Watertown, Connecticut;
Howard F. Fehr, Teachers College, Columbia University;
Martha Hildebrandt, Proviso Township High School Maywood,
74
Illinois; Albert C. Meder, Jr., Rutgers, the State
University of New Jersey; Morris Meister, Bronx High School
of Science, New York, New York; Frederick Mosteller, Harvard
University; Eugene P. Northrop, University of Chicago;
Ernest R. Ranucci, Weequahic High School, Newark, New
Jersey; Robert E.K. Rourke, Kent School, Kent, Connecticut;
George B. Thomas, Jr., Massachusetts Institute of
Technology; Henry Van Engen, Iowa State Teachers College;
and Samuel S. Wilks, Princeton University.
"It should be noted that the members of the Commission
came from many parts of the country and . . . was, in this
period, the first truly national group to concern itself
with the high school curriculum" (NCTM, 1970, p. 260). The
board (CEEB, 1959) hoped that the body would use their
influence to bring action to any recommendations that might
come out of the report.
At the time of its appointment, the Commission (CEEB,
1959) believed that the United States was on the brink of an
explosive era in technological development and, more
crucially, understood what an important role the secondary
school mathematics curriculum would have to play in the
country's ability to stand prepared for this explosion.
The Committee saw mathematics as "a dynamic subject,
characterized in recent years by such impressive growth and
such extensive new applications that these have far outrun
the curriculum" (p. 9), which meant that, as new mathematics
75
were being born, our school's programs would not just be
forced to catch up, but would constantly have to expand to
meet the future growth.
The Committee reasoned, therefore,
that the older mathematics would have to be reorganized to
stress understanding in doing rather than simply to stress
doing and that the ability to understand the modern uses of
the old and the new mathematics would have to be handled in
our schools as well.
Since, however, it would be impossible
simply to keep adding more and more new information to the
course plans daily, a process was needed that would drop
outdated material, add relevant new material, and "use new
methods of approach" (p. 6) to teach all the mathematics
that was necessary.
The Commission also felt compelled to improve
mathematics instruction due to pressures from unspecified
public arguments concerning the competency of the then
current mathematics program.
The recommendations for making
these changes would take four years to complete.
In recommending a complete program, the Committee
(CEEB, 1959) did not try to hide the fact that the proposals
were intended to be college preparatory in nature, and they
specifically wanted mathematics to be taught to these
college worthy students "for at least three years" (p. 11).
The members also endorsed ability grouping within the
program so as to increase "the challenge to the student and
the likelihood of his developing his talents and ability to
76
the full" (p. 13). According to the committee, the student
in this program:
will not be exposed to certain so-called
"practical" courses such as consumer mathematics,
installment buying, principles of insurance, and
so forth. The Commission does not regret this
fact. For it believes that the student's will
ordinarily develop sufficient mathematical power
to acquire such information independently if they
need it. (p. 13)
In terms of learning objectives, the major proposals,
called the nine-point program, read as follows:
1. Strong preparation, both in concepts and in
skills, for college mathematics at the level of
calculus and analytic geometry
2. Understanding the nature and the role of
deductive reasoning—in algebra, as well as in
geometry
3. Appreciation of mathematical structure
("patterns")—for example the properties of
natural, rational, real, and complex number
4. Judicious use of unifying ideas—sets,
variables, functions, and relations
5.
Treatment of inequalities along with equations
6. Incorporation with plane geometry of some
coordinate geometry, and essentials of solid
geometry and space perception
7. Introduction in grade 11 of fundamental
trigonometry—centered on coordinates, vectors,
and complex numbers
8. Emphasis in grade 12 on elementary functions
(polynomial, exponential, circular)
9. recommendation of additional alternative units
for grade 12; either introductory probability with
statistical applications or an introduction to
modern algebra, (pp. 33-34)
77
The position was also taken that calculus should be
made a college-level course since "all too few schools are
now adequately prepared" (p. 15) to teach the subject.
The proposal was not as much an attempt to completely throw
out the traditional curriculum as it was an attempt to
change the emphasis from the concrete to the abstract.
The following with the twelfth year as optional would
be typical as a minimum sequence for a student in the
program:
Grade
Grade
Grade
Grade
nine - Elementary Mathematics I,
ten - Elementary Mathematics II,
eleven - Intermediate Mathematics,
twelve - Advanced Mathematics, (pp. 36-47)
A priority was placed on the requirement that three
years in the study of mathematics in high school was a
minimum for these students.
"To this end, schools must have
teachers trained to teach the subject matter in the spirit
of twentieth-century mathematics" (p. 50). "The Commission
suggests that an irreducible minimum of mathematics study as
essential background for teachers of arithmetic ought to be
the completion in full of the first three years of a
secondary school curriculum comparable to that outlined in
this report" (p. 49) on the sound principle that one cannot
teach any subject well unless the knowledge the instructor
has about the subject exceeds his or her knowledge of the
material to be taught.
To aid implementation, new courses
"must be designed for the needs of secondary school
78
teachers" (p. 51). These courses might have included full
credit summer classes, short summer conferences, regular
study group meetings, professional society meetings, or
short programs of continued study.
Also in reference to
implementation, several appendices that went with the report
were designed to help teachers to understand exactly how the
program related to the subject matter.
Response by the professional community was more than
immediate since "representatives of the Commission presented
the developing program to innumerable local, regional,
state, and national conferences of teachers of mathematics,
mathematicians, and educators" (Halavarty, 1961, p. 27).
There was every indication that the work of the Commission
would have "profound effects on the secondary school
curriculum" (p. 28). The interest in the report was
"country wide and world wide" (p. 28) as acknowledged by the
number of questions that some mathematics teachers and
textbook writers had of the Board.
Some proponents noted
that "in doing this, of course, they carry out the very
intentions of the Commission" (pp. 28-29).
"The School
Mathematics Study Group (Yale Report) accepted in essence
the recommendations of the Commission.
In its tremendously
productive writing project in the summer of 1959, the SMSG
produced sample textbooks for grades 7 through 12" (p. 29).
At a symposium for high school and college educators on
engineering mathematics just after the report was released,
79
a general acceptance survey was distributed.
The one
hundred eighty-six responses indicated "clearly that there
is overwhelming support for the major part of the
Commission's report" (Fisher & Leissa, 1960, p. 114), but
one specific question seemed to indicate "that neither
teacher group seems to agree that the logical defects in
Euclid's treatment of geometry can be regarded as a valid
reason for modifying high school geometry" (p. 117);
however, "in the DBE Bulletin for December 1964 . . . Irving
Allen Dodes, chairman of the department of mathematics,
Bronx High School of Science, ...refers to the movement away
from the convincing but incorrect Euclidian geometry to a
more convincing and correct geometry based on properties of
numbers" (Bacon, 1965, p. 35) as a positive approach.
Later in the 1960s, "criticism of the program flared
up...when American 13-year-old students placed next to last
on standardized tests administered to about 133,000
teenagers in 12 countries" (Changing Times. 1970, p. 22).
In five independent words, the report was ambitious,
positive, complete, challenging, and historic.
But it might
have surpassed its goal to better mathematics instruction if
it had established a functional model outlining the content,
process, and technological knowledge of the discipline that
might have constructed a cornerstone on which to erect the
essential knowledge elements of the field of mathematics.
By failing to provide such a model, the Commission
80
relinquished the job of deciding what should be taught in
the discipline into the waiting hands of the educators and
the textbook writers.
Morris Kline (1973) fashioned a scathing criticism of
the new or modern mathematics curriculum for secondary
schools.
Since the initiation of the movement in the early
1960s, its effect on students, teachers, other curricula,
teaching methods, politics, finances, and postsecondary
education has been colossal.
He attempted to justify why
reform in mathematics should take another direction by
explaining the traditional mathematics curriculum, by
exposing some of the fallacies held by educators about the
development of traditional mathematics, and by pointing out
the weaknesses of some basic assumptions in the new
mathematics system.
While he was correct about the nature
of the problem with mathematics education, Kline failed to
see the need for a model of the structure of knowledge for
mathematics as a solution to the dilemma brought about by
the failure of educators and/or mathematicians to deal with
the assumptions made by the proponents of the new
mathematics.
Kline (1973) offered several examples of what might
occur in a mathematics class if students were used to
working with traditional notions of arithmetic and if
teachers were trying to explain concepts using new math
terminology.
His point seemed to be that, even if the
81
teacher is sincere and accurate in the use of nomenclature,
students and others may be justified in their right to be
confused.
A model for the structure of knowledge for
mathematics would help to alleviate some of this confusion
by providing students with a way to begin to organize the
great amount of terminology essential to a discipline as
complex as mathematics.
Kline (1973) presented several problems that confronted
the traditional curriculum in mathematics.
First, he found
that it relied too heavily on certain pitfalls that
necessarily must occur in such an algebra-burdened plan
which broadly included memorization, the discontinuity of
processes, confusing nomenclature and notation, and a lack
of deductive reasoning.
He also found that the traditional
curriculum tended to be inflexible (even when topics were
outdated) and to be lacking in motivational qualities (due
the abstractness of the subject matter and the sheer
dullness of the multitude of contrived textbooks).
Certainly, there did exist problems with the traditional
mathematics curriculum; however, each of these problems
might be eased by the onset of a model which presents the
structure of the content and processes of mathematics as a
way to give students disciplinary confidence, cohesion, and
direction.
Kline (1977) also indicted the practices of liberal
arts mathematics education for choosing to emphasize
82
publishing over teaching.
He believed that a growing
shortage of education funds and the enticement of research
monies and prestige has caused college professors to neglect
pedagogy at the expense of their students.
Universities
maintain, he said, that research professors "advance
knowledge and improve all levels of education directly and
indirectly" (p. 70) and, therefore, believe that good
teachers are good researchers and vice versa so that
"appointment, promotion, tenure, and salary are based
entirely on status in research" (p. 70).
Even before the Russians launched their first Sputnik
satellite in 1957, educators were already touting a
mathematics curriculum (as opposed to teaching reform or the
better utilization of new technology) which would do a
superior job of educating students in secondary schools in
order to prepare them better for college, for the military,
or for professional life.
The term modern or new in
reference to mathematics was actually used to describe
curriculum reforms which were being advised from many
legitimate sources.
Since the new curriculum would take a
fresh approach to mathematics as well as change its content,
Kline (1973) said that the approach should reflect a concern
for the realistic goals of high school students since not
every elementary student will become a mathematician.
Unfortunately, the designers failed to represent the new
content in a model based on knowledge elements and the
83
processes which define mathematics that could be utilized by
the primary or secondary teachers in order to enhance
instruction.
One way that the new curriculum would change
traditional mathematics was by encompassing the deductive
approach in courses other than high school geometry.
By
using several examples, Kline (1973) showed how, although
the use of deductive methods in arithmetic and algebra is
proper, the processes taken to even moderately larger
numbers or expressions could be tedious, at best.
He also
submitted that at some point the use of deductive axioms
become rules to be learned anyway.
Why not teach the rules,
he argued, in the first place?
Kline (1973) also investigated whether using the
logical or deductive approach on subject matter provides for
better understanding in mathematics.
He argued that the
earliest developments in mathematics did not come about by
deductive thinking.
Even Euclid's ideas came about as a
result of intuitive thinking.
The deductive approach of
teaching geometry certainly does not make it an easier
subject than algebra.
Intuitive concepts were naturally
"accepted and utilized first" (p. 38).
Neither approach is going to be successful if the
content of the discipline has never been codified; their
goal should be the stewardship of the knowledge of the
discipline.
84
Another complaint that the backers of the new math
movement had about the traditional curriculum concerned a
lack of rigor in the development of topics.
Kline (1973)
claimed that to make geometry and algebra more rigorous,
that is to make it rely less on unproven axioms and
postulates, requires extra axioms and postulates to be
introduced and then proven by deductive reasoning.
He
believed these to be confusing substitutions for intuitive
models and that time is better spent on more relevant study.
He was correct but failed to see the need for a content and
a process model which would give students an intuitive edge
on the content and process structure of the discipline.
A third problem that Kline (1973) had with the
traditional mathematics was that of imprecise language.
He
found fault with how the new mathematics proposed to correct
it:
overdefinition.
Since the new math sought to be
precise, every concept must be strictly defined even when
the terminology is mostly abstract.
Kline believed that the
use of excessive terminology created a great burden for high
school mathematics students in that it "labels what need not
be labeled" (p. 67) and that "the purpose of their symbolism
is to make the obvious inscrutable and so repel the
understanding" (p- 71). What Kline failed to see was that
when students are made aware of the overall structure of a
system through a physical model, the symbols, labels, forms,
and definitions may be more easily remembered.
85
Kline also addressed the notion that "all of
mathematics can be arrived at by raising questions about
ideas already studied" (p. 75). He asserted that the
development of mathematics came from a need to deal with
physical situations in the real world.
Mathematics is not
an isolated body of knowledge that exists solely for its own
sake and should not be taught in that way.
Application was
neglected in the new mathematics in favor of the development
of structure, again, through deductive reasoning.
Kline
believed that the teaching of structure without making it
meaningful produces "the usual hodgepodge of disconnected
topics that it is in the traditional curriculum" (p. 82).
Again he saw the need for a way to represent mathematics as
a dynamic structure but failed to provide the model of the
structure of knowledge for mathematics as a way to begin to
codify its content and processes.
Maziars (1950) believes that there are two classes of
mathematical thought.
Mathematicians are at work only when
they are considering meanings of symbols or are performing
mathematical operations.
As soon as they begin to reflect
on the existence or meaning of mathematics itself, they have
turned from the study of mathematics to the study of
metaphysics.
Kline (1973) also criticized the role of
deductive reasoning, rigor, terminology, and structure over
application in the modern mathematics curriculum.
addition, he was concerned with its content.
86
In
Set theory,
for example, was said to hold a premier place as a new part
of the modern curriculum by unifying several branches of
mathematics as a basic concept.
Kline called set theory "a
hollow formalism which encumbers ideas that are far more
easily understood intuitively" (pp. 94-95).
The
introduction of bases, the congruencies of modular numbers,
inequalities, matrices, symbolic logic. Boolean algebra, and
the idea of groups and fields, likewise, only leads to
confusion on the part of the student through further
reliance on more and more abstraction.
Kline contended that
the new content was composed of topics that had been pushed
down from college courses or ones that were frivolous and
should have been preserved for exploration by future
mathematicians.
He felt that it was only reasonable to
assume that the new mathematics curriculum had been
thoroughly tested in experimental class situations that pit
traditional mathematics against its new opposition.
Actually very little testing had been carried out in the
more than two decades since the new mathematics movement
began, and such experiments that had been carried out were,
at best, suspect.
Many were abandoned in order to meet
competitive publishing deadlines.
Some studies were faulty
in that they were localized and/or were not standardized.
All attempts to prove the superiority of the new mathematics
curriculum have failed, so far, simply because of the
failure to isolate the new mathematics from the traditional.
87
Even international test results showed that students in
other countries do better without the benefit of having the
knowledge of new mathematics.
Kline cited instances where
some major proponents and architects of the original modern
mathematics curriculum have indicated that mistakes had been
made and some already see the need for revising the program.
No one has provided the discipline with a prototype in which
the essential elements of each subdiscipline in the field
can be organized.
Kline (1973) preceded to blast pure mathematicians for
their role in the development of the new mathematics.
He
proclaimed that mathematicians were basically out of touch
with the rest of the scientific community and with the world
of education.
Kline also believed that, because of their
affinity toward specialization, abstraction, research, and
pious prestige and away from pedagogy, the fate of the
curriculum was left in the wrong hands.
The job belongs to
those who have a knowledge base in the discipline coupled
with the organizational skills and the educational
experience necessary to model and use the knowledge for
mathematics as an innovative teaching tool.
Without a model
to guide them, many students may never be able fully to
understand or appreciate the true source of mathematics
knowledge.
Kline (1973) also proposed several factors which caused
the ready acceptance and adoption of the modern mathematics.
88
Among all of these factors, the propaganda of well organized
lobbies for the curriculum groups and their textbooks,
threats about what kind of questions might soon be included
on college entrance tests, the possibility of higher
prestige in the profession by teachers at all levels, and
the hope of an easier workload brought about by the welldefined deductive approach were the most influential.
Kline (1977) challenged the notions that the
mathematics' researcher—like researchers in other
disciplines—possessed superior knowledge, that
specialization broadens ones ability to teach, that pure
research has relevance to teaching, that the researcher's
knowledge is widely accessible, and that the researcher will
automatically relay their enthusiasm and love for
mathematics to their students. Kline (1973) believed the
answer to curriculum reform to be simple.
Teach the uses
and values of mathematics in a broad way at the elementary
and secondary level that touches on other activities,
interests, and disciplines so that students might be
motivated to pursue it further.
Mathematics, he asserted,
is not an isolated body of knowledge, and it, therefore,
should be taught by showing what mathematics accomplishes in
order intuitively to cultivate critical thinking. Again,
Kline made an excellent argument against curricular reform
by purists in the mathematics profession.
Kline (1977)
complained about the inadequacy of mathematics textbooks on
89
the elementary, secondary, and postsecondary levels. He
claimed that they are written by professors who "are either
indifferent to pedagogy" (p. 208) or "are totally ignorant
of it" (p. 208). Kline declared that professors receive
little training in writing and that explanations are
insufficient since "mathematicians do not take the trouble
to find out what students should know at any particular
level, they do not know how much explanation is called for"
(p. 209).
For there to exist a successful revolution in
mathematics education in primary and secondary schools in
this country, there will have to be strong input on behalf
of educators who know the client and the pedagogy as well as
they know the subject matter.
The task of reform will have
to come from those scholars in higher education who see the
value of putting control of the content and the processes of
the field of mathematics in the hands of mathematicians and
educators through the stewardship of disciplinary knowledge
instead of cowering to the misguided doctrine of the
purists' calls for mathematics for mathematics' sake or the
educational theorists' cry for appeasement through another
learning theory.
Under the direction of Dr. Edward G. Begle, the Board
of Directors of the National Society for the Study of
Education published its Sixty-ninth Yearbook of the NSSE
(1970) for the purpose of addressing some of the issues
90
brought about by the then recent changes in the elementary
and secondary school mathematics program.
The changes
involved the emphasis being put on the structure of
arithmetic and algebraic material over the acquisition of
arithmetic facts.
The abruptness of these changes was hastened by the
wide adoption of new mathematics textbooks and by a new
psychology had contributed a new way of applying old
teaching techniques.
Lloyd Scott (1966) condensed these ideas in the
following:
1. The structure of mathematics should be
stressed at all levels. Topics and relationships
of endurance should be given concentrated
attention.
2. Children are capable of learning more abstract
and more complex concepts when the relationship
between concepts is stressed.
3. Existing elementary arithmetic programs may be
severely condensed because children are capable of
learning concepts at much earlier ages than
formerly thought.
4. Any concept may be taught a child of any age
in some intellectually honest manner, if one is
able to find the proper language for expressing
the concept.
5. The inductive approach or the discovery method
is logically productive and should enhance
learning and retention.
6. The major objective of a program is the
development of independent and creative thinking
processes.
7. Human learning seems to pass through the
stages of pre-operations, concrete operations, and
formal operations.
91
8. Growth of understanding is dependent upon
concept exploration through challenging apparatus
and concrete materials and cannot be restricted to
mere symbolic manipulations.
9. Teaching mathematical skills is regarded as a
tidying-up of concepts developed through discovery
rather than as a step-by-step process for
memorization.
10. Practical application of isolated concepts or
systems of concepts, particularly those
applications drawn from the natural sciences, are
valuable to reinforcement and retention, (pp. 1516)
The report (NSSE, 1970) attempted to deal with relevant
questions which were being asked of mathematics educators at
that time.
Major aspects of the theories of several
contemporary educational psychologists were easily adaptable
to the new psychology.
Lee Shulman (NSSE, 1970), another
contributing author to the NSSE, compared some of their
views in relation to what he calls the three basic
components of instruction:
"the objectives of instruction,
the readiness of learners, and tactics and sequence of
teaching" (p.34).
The major objective of instruction, according to
Jerome Bruner (NSSE, 1970), detailed a matter of process
over product with emphasis on discovery (Wertheimer, 1959).
Robert Gagne' (NSSE, 1970) also believed in process over
product but with emphasis on levels of learning.
David
Ausubel (1963) was not a believer in process as much as he
was a supporter of product through the transmission of
knowledge as related to cognitive structure.
92
He reasoned
that something that may be discovered was not necessarily
meaningful.
Readiness was, to Jean Piaget (NSSE, 1970), simply a
matter of determining whether or not a child was at the
right stage of growth to be able to incorporate the
knowledge being offered.
Inhelda and Piaget (1958) surmised
that intelligence grew as one passed through several
biologically and environmentally related stages.
Bruner
(1960) theorized that a child would be ready to learn a
topic only when the material was presented relative to his
three modes of representation.
Gagne' (NSSE, 1970) asserted
that readiness is a "function of the presence or absence of
prerequisite learning" (p.46).
In regard to the tactics and sequence of instruction,
Bruner (1966) felt that learning should be encouraged
through the presentation of general topics by which the
student could, in a rather unsystematic way, begin to
concentrate on specifics.
Gagne' (1977), in contrast, felt
that the teaching of subtopics would result in the eventual
learning of broader and more general concepts. Ausubel
(NSSE, 1970) accepted Gagne's theory here except that he
also endorsed the use of the advanced organizers to replace
the smaller subtopics.
Although there seemed to exist no one method of
methodology that works in every instance in the teaching of
mathematics, the report cited that children seem to have
93
more success when nondirected learning techniques are used
along with manipulatives and other sensory aids (Avital &
Shettleworth, 1968).
In regard to how often one should vary
their method of teaching, change for change sake may
actually be appropriate.
Drill (Weaver, 1972) is advised
only when understanding has been achieved.
When used not
too often and with the right kind of motivation, drill can
be made much less laborious.
It is generally accepted,
however, that a higher degree of transfer comes with the
mastery of general principles rather than with repetition
from drill (Skemp, 1971).
A word of caution was given (NSSE, 1970) concerning
teaching for understanding as a cure all. A student may
find the presentation of several algorithms or explanations
of the same concept tedious when the student may have gotten
the message on the first try.
This situation can be avoided
when a problem is broken down into presentations that cover
smaller amounts of subject matter, is preceded by a general
overview of the topic (Joyce & Weil, 1972), and is laced
with pertinent examples and applications.
Teaching method (Brownell, 1951) is of no importance
when there is no idea as to the readiness of the student.
Different students are able to learn different things at
different times (Copeland, 1979).
Biological events (Beard,
1969) and environmental differences make it impossible for a
definite standard to be set for readiness in any curriculum.
94
There is plenty of arithmetic to be taught.
A teacher
should concentrate on teaching well the subject matter
relevant to that particular grade level at that particular
time for that particular child (Adler, 1966).
Learning
(Aichele & Reys, 1977) in which subject matter is presented
to the learner beginning with the topic's specific
characteristics and progressing toward its general ideas
should be avoided.
Advanced material should be left for
another grade.
Discovery learning (Shulman, 1968) may be considered as
a viable strategy when the teacher and the student are able
to justify the amount of time the inductive process takes.
The instructor must make the decision as to how much guiding
the individual student needs while remembering that all
students should be motivated toward achieving a reasonably
equal standard of performance.
The commission (NSSE, 1970) seemed to accept that there
were a sufficient number of theories and programs to meet
the needs of just about any student.
They believed that
their biggest problem was that of getting the word out to
teachers.
This yearbook effectively discussed the ideas of
several leading psychologists and put into perspective the
curriculum changes since the advent of the new mathematics.
The question and answer form of the material presented on
the issues in the teaching of mathematics seemed especially
95
effective in attempting to educate all educators without
insulting or alienating them.
Although the yearbook provided a broad and useful look
at the professional storehouse of the profession, it failed
to break new ground in realizing that the mathematics field
had no grasp of what it considered to be its disciplinary
imperatives in relation to content or process.
Just what
topics and processes were supposed to be taught?
The School Mathematics Study Group (SMSG, 1972) was
created in 1968 to review "the state of research in
mathematics education" (p.3) and to make recommendations and
plans for implementing suggestions for research that might
help solve "the basic problems facing us" (p. 4). In its
final report, the panel's recommendation to "construct one
or more theoretical models of mathematical learning and
teaching and to make them available to the mathematical
community as a source of suggestions for research in
mathematics education" (p. 3) was "incomplete since no
theoretical model of mathematics teaching and learning" (p.
3) was supplied.
Although the panel determined that facts,
principles, concepts, and skills to be the objects of
mathematics learning and considered the structure of
"mathematics to be a set of interrelated, abstract, symbolic
systems" (p. 5), they suggested it "unlikely that we will
ever accumulate the knowledge base we need to formulate
useful theoretical models of mathematics learning" (p. 4).
96
In trying to recover from the widespread failure of the
new mathematics of the 1960s and early 1970s, many educators
had endorsed going back to "basic arithmetic skills" (Bell,
1978, p. 83) in mathematics in order to return to the days
when standardized test scores were high and public
confidence in the methods of mathematics education was
higher.
The advocates of Meaning Theory, however, did not
accept any theory as being superior to theirs.
Richard Skemp (1978) was willing to take up the fight
for the Meaning Theory movement by writing a well-organized
article which emphasized the existence of different levels
of understanding in the teaching or learning of mathematics
and the possibly of two uses of the word mathematics itself.
The article showed how making false comparisons between
these sometimes misunderstood educational concepts could
bring about a lack of real understanding in mathematics.
By analogy and example, Skemp (1978) illustrated how
instrumental understanding (how to do) differs from
relational understanding (how to do and why) and made it
clear that a classroom is more productive when shared by
students and teachers who think relationally.
Skemp
believed that mathematics taught with relational
understanding in mind is the only real mathematics being
taught, but he played the Devil's advocate and weighed the
advantages of teaching instrumental mathematics (which is
easier to understand, offers more immediate rewards, and is
97
a faster method of instruction) against the advantages of
teaching relational mathematics (which is more easily
adaptable to new tasks, more easily remembered, more
effective as a long-range goal-setting device, and more
desirable in terms of the quality of the schemas it
produces.)
Skemp (1978) also conceded that certain esoteric,
teacher-related, and situational factors might force
instructors to teach toward instrumental understanding.
These factors might include the experience of the teacher,
the overloaded course requirements, and/or the teaching of a
topic for use in another subject.
Finally, in another analogy, Skemp (1978) showed that,
in the long run, students do themselves more good through
relational learning since in this kind of learning the means
become independent of the ends:
schemas are erected which
become goals in themselves, confidence is built up making
the student more independent, and the student is more aware
of learning as a never ending process.
By using one's own insight and knowledge, it may be
obvious after reading Skemp's article (1978), that one would
choose to teach/learn relationally rather than
instrumentally when given the choice.
The beauty of Skemp's
work might be in that his reasoning is so very effective in
bringing out the fact that there are two different kinds of
understanding which lead to two kinds of mathematics.
98
In regard to the problem of how to use Skemp's ideas
(1978), a large part of the difficulty with the way students
are educated, he said, may be in the employment of long term
versus short term goal setting by teachers. Although many
elementary teachers do a great deal of concept and
relational teaching (Skemp, 1978) for long-term reasons,
much of the teaching of mathematics is seemingly short term
and instrumental (by the end of the year students will be
able to recite multiplication tables through the fives).
Skemp (1978) believed that teachers must be educated on
long term goal planning (before students can possibly be
expected to practice it) even though they deal with short
term goals more often within an isolated year of a student's
schooling.
Some elementary and secondary teachers know
better than to teach as they do, but time, ability, and lack
of insight tend to limit their adoption of long term goals:
students will become better goal educated and will grasp
mathematics only after they see what the teachers do.
The Problem of Poor Mathematics Performance
from the 1970s to the Present
National Concerns
The general problems of knowledge stewardship as
related to mathematics that will be addressed in this
section involve instruction and the needs of the
disciplinary organizations as evidenced through student
achievement.
59
Fundamental questions have surfaced in regard to the
quality of instructional materials, teaching strategies,
teacher training, funding, and even our "national
curriculum"
(Freeman et al., 1984, p. 47). Mathematics
educator groups and mathematics associations maintain
separate agendas, and Peterson (1987) has determined that
solutions to these problems are as complex as the problems
themselves.
Good and Grouws (1987) found that many basic
misconceptions about mathematical knowledge stand in the way
of quick solutions.
Rotberg (1985) warns us to "define
precisely the problems we wish to solve and consider the
broad implications of our solutions.
Or else, in our rush
to meet the putative demands of the future, we may neglect
the students and issues in most need of attention" (p. 27).
The use of a proposed model for the structure of knowledge
for mathematics would allow mathematicians and mathematics
educators to begin to have a common frame of reference from
which to view their problems.
International Competitiveness
and Comparisons
The general problems of knowledge stewardship as
related to mathematics that will be addressed in this
section are associated with instruction and advisement as
evidenced through international competitiveness, national
welfare, and minority inclusion.
100
In a study of achievement in mathematics by Stevenson,
Lee, and Stigler (1986), Chinese, Japanese, and American
elementary school children were tested over a five-year
period using a battery of achievement tests in order to
compare mathematics performance.
The instruments were
designed to eliminate as much cultural bias as possible by a
team of bilingual researchers.
American kindergartners
scored about the same as Chinese preschoolers but much lower
than the Japanese students of that age.
By the first grade,
Chinese children had surpassed the Japanese, while the
American scores had actually declined.
At the end of the
fifth grade, the Japanese students had regained their lead
over the Chinese; however, the American scores had gained
only to a level which indicated that their lowest average
fifth grade score was hardly higher than those in the best
first grade classes in China.
In fact, at the end of the
study, only one American child scored in the top 100 out of
the 720 students in the sample.
The study also included information relevant to student
experiences and advisement.
In grade five, Americans spent
about 65 percent of their classroom time engaged
academically as compared to about 92 percent and 87 percent
for the Chinese and the Japanese students, respectively.
The American children spent more than twice as much time on
language arts subjects than on mathematics, while Chinese
and Japanese students spent about equal amounts of time on
101
each.
Children in China said that they liked homework, but
American children offered the opposite response even though
Chinese students spent 77 minutes doing mathematics homework
as compared with an average of 14 minutes for their American
counterparts.
The Chinese and Japanese children also
surpassed American students in the amount of homework
assigned, in the use of outside workbooks and tutors, and in
parental support.
Outside of the school setting, high parental
expectations coupled with a strong belief in the values to
be gained from hard work influenced the more positive
student attitudes of the Chinese and Japanese children with
respect to the American's attitudes.
In the classroom, the American teachers spent more time
performing nonacademic functions than was spent in the other
two countries.
Japanese and Chinese students come out of a
highly structured environment with great respect for
authority at home and in school, but Americans have
autonomy.
A model for the structure of the content of mathematics
should help to prepare students to make career choices in
mathematics by making them more aware of the branches and,
hence, the career paths of the discipline.
Multinational studies by the International Association
for the Evaluation of Educational Achievement (lAE) have for
the past 20 years sparked concern over the quality of
102
mathematics education in the United States.
In its 1985
study (Travers & McKnight, 1985), the lAE surveyed a group
of secondary students who specialized in mathematics from 22
countries, and summarized:
Overall, the U.S. sample performed at a level
markedly below the average level of performance
found in the final year of secondary school
mathematics in the other nations, (p. 412)
In an even more recent study, Stevenson and Stigler
(1992) noted that when American and Asian first and fifth
graders were tested on items which related to the
understanding of the structure and operations of
mathematics, the
Asian students' superiority was not restricted to
a narrow range of well-rehearsed, automatic
computational skills, but was manifest across all
the tasks. Our data do not support the stereotype
of Asian children as successful only in performing
what they have learned rather than in applying
what they know. (p. 41)
With the introduction of a model for the structure of
knowledge for mathematics, many of these seemingly
insurmountable problems might be solved more expediently.
Students who learn mathematicians using a structured
approach may better understand what they are studying, how
it relates to other topics, and where they are headed which,
in turn, can affect achievement and time spent on task.
Secondary educators and those responsible for teacher
education may find a model for the structure of mathematical
knowledge to be an invaluable tool for raising the level of
103
understanding of their students and of those who would teach
in elementary and secondary classrooms.
Failure of Existing Mathematics
Recruitment Programs
The general problems of knowledge stewardship as
related to mathematics that will be addressed in this
section involve scholarship, research, the needs of the
disciplinary organizations, grantsmanship, and practice.
Many roadblocks seem to stand in the way of finding
solutions to problems concerning mathematics achievement in
the United States. Differences among the many schools of
thought in the United States have traditionally caused a gap
in the time between program development, funding, and
implementation (Peterson, 1984) to the point that people
outside of the mathematics community do not understand even
the need for mathematics research and development.
Without
basic research, practical applications may be missed or
ignored, and fewer doctorates are earned.
In fact, out of
over 30,000 doctorates given in the 1986 academic year, less
than 400 Ph.D.s in mathematics were awarded to Americans
(Hechinger, 1987).
Kolata (1985) notes that, at some major
universities, sixty out of one hundred mathematics graduate
students are foreign.
Foreign students from Asia and
Central and South America dominate research and teaching
assistantships in nearly all geographic regions of the
United States.
Americans may feel that they are able to
104
make more money in other fields or that their elementary and
secondary schooling never properly exposed them to rigorous
mathematics.
His prediction of a shortage of mathematicians
and classroom mathematics teachers has become a reality
(Stevenson & Stigler, 1992).
With the help of a valid model of the structure of
knowledge for mathematics, American students may find a
renewed interest in performing needed basic mathematics
and/or related research by providing a heuristic that might
motivate further study.
Existing mathematics programs must also share the blame
for poor mathematics achievement.
Research results have
shown that many curricula in mathematics overly concentrate
on computation without regard to concept.
Unfortunately,
"too many teachers view mathematics merely as the production
of correct answers" (Good et al., 1987, p. 778). Students
have not learned to integrate mathematics with other
subjects.
them.
As a result, mathematics may seem foreign to
Mathematics textbooks do not read like other more
friendly literature.
Students might begin to avoid
mathematics completely, and the blame sometimes gets placed
on math anxiety.
Hechinger (1987) elaborates in the
following:
The myth persists that mathematics is an esoteric
subject only for scientific geniuses. Young
people think of it as dull, and teachers often
make it so. They give students little opportunity
to debate math in class as they do other subjects.
Instead, math teachers often press for the one
105
right answer, stress memorizing rules and force
students to keep their eyes on the clock for quick
answers, (p. 19)
According to Good and Grouws (1987), too little
attention is given to five fundamental mathematical skills
which might be used to promote student learning:
proper and
quality subject development, emphasis on procedural detail,
instruction in problem solving, the importance of
estimation, and the benefits of understanding mathematics.
This problem has particular relevance to the problem of
developing a model of the structure of knowledge of
mathematics.
How can any student understand a subject
without having a model that relates the functional
operations of the discipline?
The low achievement brought about by the lack of
importance given to these topics cannot be remedied by
merely changing class size, the school year, or even teacher
qualifications (Peterson, 1987).
The curriculum must be
functionally restructured to show the logical interrelations
among areas and processes.
Perennial Solutions
While each of the following solutions to low
mathematics achievement has its merits, they each might
become more effective with a common disciplinary model as a
guide.
106
Curriculum
The mathematics curriculum in the United States is, to
say the least, varied.
Whether or not the many variations
are good for student achievement is the subject of a report
on a study by Freeman et al. (1984) in which the existence
of a national curriculum based on the content of textbooks
and standardized tests is indicated.
In the study,
textbooks were found to be a major influence on "content
decisions" (p. 48) in all classrooms tested.
Unfortunately,
textbook changes are likely even in the same school.
It is
assumed that the variety of textbooks being used in
conjunction with pressure from principals, parents, district
policies, and standardized test reports makes "significant
differences in math instruction content . . . almost certain
to exist among elementary classrooms" (p. 48).
Although some excellent guidance toward curricular
reform has been furnished with the publication of what has
become to be known as the Standards (NCTM, 1989), no
physical model for the content or processes of mathematics
has been provided.
Such a model should serve as a guide for
curriculum developers who might see a need for a layout of
the structure of the knowledge in the discipline.
Other suggestions for the revision of the mathematics
curriculum include the total removal of any tracking which
might tend to remediate mathematics, the better utilization
of computers and calculators in the classroom ("U.S. Math,"
107
1987), the increase in the number of mathematics courses
required for high school graduation (Travers, 1985), and the
early intervention of positive role models in mathematics
for females and minority students (Lytle, 1987).
These solutions to the problem of low mathematics
achievement are valid only when a series of textbooks which
stress the content and processes of the field of mathematics
can be implemented in every classroom from kindergarten
through college.
Stevenson and Stigler (1992) agree that:
American textbooks tend to be excessively long,
repetitive, and distracting and underestimate what
children can understand. Rather than believing
that children can be active participants in the
construction of their own knowledge, many textbook
writers offer curricula that depend more on
memorization than on understanding, and on
offering step-by-step solutions to problems than
on enabling students to create their own solutions
to novel problems, (p. 213)
Teaching Strategies
All students need encouragement, good role models, a
little room for creativity and improvisation, and a
listener.
Often, students with fresh ideas and novel
approaches to old problems are not encouraged to verbalize
their ideas.
Teachers need to listen to students
(Hechinger, 1987).
Repetition of subject matter by teachers
from one topic to another is not effective unless the spiral
approach is used in which the instructor returns to the same
concept over and over at a higher level of sophistication.
For example, a mathematics teacher might reinforce the idea
108
of fractions with whole numbers, radicals, exponents, and
polynomials when employing a spiral strategy.
Peterson
(1987) found that students retain more when the subject
matter has some organized focus.
In comparing how our classrooms might be organized to
facilitate learning, McNergney and Haberman (1986) conducted
experiments with classes of mathematics students which had
been grouped by ability, nongrouped in whole-class
instruction, and given individualized instruction.
The
study concluded that teachers were significantly more
effective at teaching computational skills in the grouped
and individualized classes than they were in the nongrouped
classes.
Teacher and student involvement was considered the
key to higher achievement in those grouped and
individualized classes where the organization of the class
itself made the involvement possible.
Everyone involved in
the study "agreed that when children work together and
respect each others' needs and abilities, they're likely to
learn math" (p. 17).
Good and Grouws (1987) believed that proper skill
development was lacking and redefined development in
mathematics as a "process whereby teachers facilitate the
'meaningful acquisition' of an idea by learners" (p. 780) in
order to point out ways to enhance learning.
The study
emphasizes the roles of prerequisites, mathematical
relationships, concepts, abstract representations, and
109
terminology in the teaching of mathematics.
Such an
approach illustrates the need for a model of the structure
of knowledge for mathematics in order to facilitate basic
understanding and to provide a direction for learning that
can follow a validated model for the structure of knowledge
for mathematics.
Professional Development
Professional development is a way in which teachers can
maintain their subject skills and teaching competencies
(Walsh, 1984) while they remain in the classroom.
Through
local workshops and summer teaching internships, mathematics
instructors have the opportunity to learn about programs in
areas not directly related to mathematics which might still
benefit their mathematics students.
In some areas, local
support groups have been established to provide programs in
career guidance and counseling, to set up field trips to
work and research sites, to organize guest-lecture series
and student tutoring (Cooper, 1987).
Teachers of
mathematics, in many cases, indicate that a very rigorous
course was the catalyst that made them so interested in
mathematics (Kolata, 1985).
Such a course might be
attempted in an effort to spark waning student body
interest, and a model of the content and the processes of
the discipline could easily serve as the foundation for that
renewed interest.
110
Recent Attempts to Structure the Discipline
The Mathematical Sciences Advisory Committee of the
College Entrance Examination Board (1985) attempted to
outline how teachers should achieve the results of the goals
which the College Board (1983) had set forth in "what became
widely known as the Green Book" (CEEB, 1985, p. 4). The
board listed the "Basic Academic Competencies" (p. 15) in
mathematics as:
*
The ability to perform, with reasonable
accuracy, the computations of addition,
subtraction, multiplication, and
division using natural numbers,
fractions, decimals, and integers.
*
The ability to make and use measurements
in both traditional and metric units.
*
The ability to use effectively the
mathematics of:
- integers, fractions, and decimals
- ratios, proportions, and percentages
- algebra
- geometry
*
The ability to make estimates and
approximations, and to judge the
reasonableness of a result.
*
The ability to formulate and solve a
problem in mathematical terms.
*
The ability to select and use
appropriate approaches and tools in
solving problems (mental computation,
trial and error, paper-and-pencil
techniques, calculator, and computer).
*
The ability to use elementary concepts
of probability and statistics. (p. 19)
The Board (CEEB, 1985) also deemed that the college
students of the future would need to be proficient in all of
llx
the fundamental mathematical skills as noted in the
folowing:
*
The ability to apply to apply mathematical
techniques in the solving of real-life
problems and to recognize when to apply those
techniques.
*
Familiarity with the language, notation, and
deductive nature of mathematics and the
ability to express quantitative ideas with
precision.
*
The ability to use computers and calculators.
*
Familiarity with the basic concepts of
statistics and statistical reasoning.
*
Knowledge in considerable depth and detail of
algebra, geometry, and functions. (pp. 19-21)
Schools (CEEB, 1985) would need to identify and place
students into a college preparatory track so that they might
be given special mathematics courses meant to develop them
for college-level work.
Students deficient in the skills of
computational arithmetic would be allowed to take a course
which would help them build their computational problem
solving skills.
Teaching strategies were described as those
which make "preparatory mathematics a more vital,
stimulating subject" (p. 42) and were presented through
"vignettes" (p. 42) which were designed to illustrate a
particularly relevant pedagogical approach to a "specific
teaching situation" (p. 42).
In his summary of the subcommittee report (NCTM, 1989)
of the Curriculum and Evaluation Standards for School
Mathematics as published in 1989 by the National Council of
112
Teachers of Mathematics, John Dossey (1990) reported that
the Committee had been charged with the task of defining the
broad curricular goals that the nations schools should be
striving to reach and of suggesting methods by which
progress toward those goals should be assessed at the
student and the school levels.
The author wrote
specifically about the changes that would need to occur in
teaching methodology and about the changing nature of the
content of mathematics, in particular.
Dossey (1990) believed that mathematics must come to be
a helping discipline.
Teaching, he reported, will need to
become much more interactive by acknowledging the advent and
beginning the employment of small group discussions, the
increased use of mathematical manipulatives, and the writing
of learned mathematical experiences.
Time for these
investigations will need to be found by deleting unnecessary
and outdated computational practices.
Dossey (1990) also believed that the nature of the
content of mathematics itself must be rethought as to also
consider the overemphasis on procedural skills.
Recent
breakthroughs in graphics calculator technology have made
computations with numbers having many digits or large
fractions of less importance than simply being able to know
the proper computations to be used since those calculators
can instantly put at ones fingertips the graphs of equations
that cannot be solved by algebraic techniques by hand alone.
113
Dossey (1990) also reported the six recommendations of
the committee:
Recommendation 1: Schools should make a zerobased analysis of the content of their mathematics
curriculums. Time for mathematics is already too
crowded with content. Each topic allotted time in
the curriculum must be justified on the basis of
the role it plays in the students' overall
mathematical growth.
Recommendation 2: Calculators and computers must
be integrated into the curriculum as "fast
pencils." Their primary use must be to support
the overall growth of concepts and processes, not
just to quickly check answers or serve as objects
to teach programming skills.
Recommendation 3: Recent and relevant
applications of mathematics must be given
attention to illustrate the value of mathematics,
as well as to provide motivation for the study of
mathematics.
Recommendation 4: Teachers must participate in
continued professional development to keep abreast
of changes in the content of the curriculum and to
learn how to structure it for classroom settings.
Recommendation 5: States, schools, and teachers
must work with publishers and test developers to
develop materials that adequately present
mathematics to children and assess their progress
toward the goals outlined by the Standards.
Recommendation 6: Revisions of school mathematics
programs must involve teachers, administrators,
parents, mathematics educators, and others
interested in the mathematical development of our
nation's youth, (p. 24)
Dossey (1990) is a good reporter who is able to grasp
the basics of the Committee's report and to present it
accurately; however, he fails to analyze the report or the
recommendations and to recommend a single innovative means
by which the goals of the report might be achieved.
114
He is
content to be satisfied with a report which places its
finger squarely on the pulse of a discipline with major
problems without bothering to make any judgment or decision
as to how these recommendations might possibly be carried
out.
Mathematicians like Dossey need to stop simply
reporting the faults of the way the discipline is presented
and should begin, instead, to propose solutions to the
problems at hand.
Being able to define the structure of the
content and processes of the field of mathematics might help
educators and mathematicians to begin to implement the
recommendations which will send our nation into the twentyfirst century.
The Mathematical Sciences Education Board of the
National Research Council (1990) published a document whose
purpose was to propose reform in school mathematics based on
changes indicated in recent studies (NCTM, 1989).
The Board
(MSEB, 1990) recommended that programs be developed which
would help teachers implement mathematics.
These programs
would reflect expanded goals, a better description of
mathematics as a science and as a tool for other sciences, a
technological viewpoint, and a research perspective that
should stress higher-order thinking skills, constructed
knowledge, procedural knowledge, problem solving, and
"making sense of mathematics" (p. 32). Principles involving
applications, computers, and "symbol sense" (p. 45) would
need to be followed in order for change to take place.
115
Ch »TER III
RESEARCH METHODS AND PROCEDURES
The Setting
Since a model for the structure of mathematics, which
is recognized as a valid representation of the way knowledge
is organized does not exist, it was necessary to gather
basic information in reference to this concept.
The discipline of mathematics is disorganized.
No
model has been validated that can serve as a heuristic for
the advancement of mathematics through instruction,
research, scholarship, or practice.
There exists no
archetype to guide mathematics thinkers, teachers,
administrators, advisors, grant writers, and professional
organizations.
The purpose of this study was to develop a
model for the structure of knowledge in mathematics and to
have it validated by the experts:
professional
mathematicians.
Mathematics faculty at the 204 American universities
(Elfin, 1992) and at thirteen institutions abroad were
surveyed in order to characterize their opinions regarding
the proposed model for the structure of knowledge in
mathematics (see Appendix A for a complete listing of
institutions included in the study).
Specifically, two
mathematicians at each of these 204 American and one at each
chosen foreign university were surveyed to determine:
116
(a)
their perceptions of the overall validity of the
Tunstall Model of the Structure of Knowledge for
Mathematics;
(b)
their perceptions of the overall comprehensiveness
of the Tunstall Model of the Structure of Knowledge for
Mathematics;
(c)
their perceptions of the overall usefulness of the
Tunstall Model of the Structure of Knowledge for
Mathematics;
(d)
their perceptions of the specific validity of the
eleven major processes represented in the Tunstall Model of
the Structure of Knowledge for Mathematics;
(e)
their perceptions of the specific validity of the
five knowledge content areas represented in the Tunstall
Model of the Structure of Knowledge for Mathematics;
(f) their perceptions of the specific validity of the
three dominant technology areas represented in the Tunstall
Model of the Structure of Knowledge for Mathematics;
(g)
any comments and/or changes that they would
recommend with respect to the model.
The complete study included a pilot survey.
Sample Characteristics
Two faculty members in 204 domestic institutions and
one mathematician from 13 international institutions were
selected for the final survey.
117
A majority of these domestic
institutions were defined as "prestigious" (Elfin, p. 114)
because they reported that 80% of their faculty held a Ph.D.
Each of the non-American universities, along with Moscow
State University and Beijing University, were chosen because
they represent the largest institution in a country that
participated in the first international study on mathematics
achievement (Husen, 1967).
A random sample of 50 of the 204
domestic universities were chosen for a pilot survey.
Development of the Model
Early conceptualization of what is now known as the
Tunstall Model of the Structure of Knowledge for Mathematics
occurred during the spring and fall semesters of 1988 as a
part of graduate research classes conducted by Dr. Oliver
Hensley.
Since that time, the model has been assessed,
cultured, and refined into its present form through
suggestions offered by research, and through the expert
opinions of colleagues and local mathematicians and
mathematics educators.
Development of the Survey Instrument
The original conceptualization of the survey instrument
began in 1989.
The final instrument is the culmination of
many ideas and prototypes that might allow an unfamiliar
model and its components to be scrutinized and judged in an
expeditious manner.
In addition to a cover letter with a
118
brief explanation of the investigation, the survey packets
contained a descriptive set of four pages that diagramed the
general model and supplied more detailed explanations of the
three subdivisions of the model.
The actual survey
instrument was a duplicate of the described package with
space for questions and responses. It was sent to the head
of the department of mathematics at each university as
listed in The College Catalog Collection (1992) on
microfiche for the 204 domestic universities and in The
World of Learning (1992) for the thirteen international
institutions.
A second, identical survey instrument was included in
each American mailing, and the department head was asked to
pass the second survey on to one of his or her faculty
members.
This technique was suggested by colleagues as a
method that might be used in order to place a sense of
responsibility on the department head so that he or she
might be more likely to respond to the survey and/or to use
his or her influence to encourage a faculty member to do the
same.
Only one survey was sent to each of the foreign
institutions in the study.
All of the institutions are likely to have a
mathematics' faculty that is involved in one or more of the
scholarly functions associated with disciplinary knowledge
(Fedler et al., 1993).
119
Procedures for Gathering Data
The instrument entitled "Survey of the Validity of the
Tunstall Model of the Structure of Knowledge in Mathematics"
(see Appendix B) was mailed to the head of the department of
mathematics at the 204 institutions identified by Elfin
(1992, pp. 118-119).
These 204 institutions, including the
50 institutions from the pilot survey, along with the
thirteen international universities comprised the total
survey.
Each envelope directed to the 204 domestic
institutions contained two complete and identical surveys
except that one survey carried a mailing label that matched
the address on the outside of the envelope and the other
provided space for the second respondent to provide his or
her mailing information.
All surveys were equipped with a
self-addressed, stamped return envelope and appropriate
directions and definitions.
The mathematicians were asked
for their opinions regarding the validity, usefulness, and
comprehensiveness of the overall model and their opinions on
the validity of the 19 smaller divisions of the model. All
twenty-two questions solicited interval ratings from 0 to 5,
inclusive.
The respondents were also asked to make
"changes" directly on the model and to use the back of the
pages for additional comments. A requested date for the
return of responses was included in the letter that
accompanied the survey.
The pilot survey information has
been incorporated in the concluding findings with the data
120
from the major survey.
Further responses were not solicited
from the institutions involved in the pilot survey.
Procedures for Analyzing Data
The survey instrument contains questions that result in
narrative comments or in data which is interval in nature.
Summaries of the narrative comments are provided for the
study.
All twenty-two questions on the survey asked for
discrete responses of 0 (low) to 5 (high), inclusive,
whether rating validity, comprehensiveness, or usefulness.
In order to lessen the chances of a Type II error, a rating
midpoint of 3 was used as the division between validity and
non-validity instead of simply using the mean rating point
of 2.5.
One question sought a response concerning validity
for the overall model.
One question solicited a response
concerning the comprehensiveness of the overall model, and
one question solicited a response concerning the usefulness
of the overall model.
Nineteen questions sought responses
concerning validity for the specific areas within the model.
Means and standard deviations were calculated for each of
the three universal questions and were used in conjunction
with a hypothesis test at the .05 and .01 levels of
significance in order to determine the mathematicians'
perceived validity, usefulness, and comprehensiveness of the
overall model.
The sample responses have been assumed to
121
have a normal distribution.
Confidence intervals and
hypothesis tests (.05 and .01 levels) were applied to the
mean of the means of the 11 questions related to the
validity of the process area of the model, of the 5
questions related to the validity of the content area of the
model, and of the 3 questions related to the technology area
of the model.
These measures have been used to establish
the validity of each of these three major components as a
whole within the overall, general model.
One hundred fifty-
six out of the 421 surveys (37 percent) have been returned.
One survey, kept on file, contained missing data and has,
therefore, been deleted from the data analysis.
122
CHAPTER IV
PRESENTATION AND ANALYSIS OF DATA
The purpose of this study was to identify the
perceptions of mathematicians regarding the validity,
comprehensiveness, and usefulness of the Tunstall Model of
the Structure of Knowledge for Mathematics.
Mathematicians from 217 national and international
universities were surveyed in an attempt to collect this
data.
A total of 157 (37 percent) of the surveys were
returned.
Ninety-six (63 percent) of the 152 domestic
responses were from faculty identified as the department
head for mathematics in their respective colleges. Four
responses were identified as being from the 13 foreign
universities (31 percent).
Of the 152 domestic responses, 16.4 percent were from
the northeast region of the United States.
Fourteen and
one-half percent of the responses were from the Mideast,
21.1 percent were from the Southeast, and 11.8 percent were
from the Great Lakes regions.
Of the responses, 7.2
percent, 8.6 percent, 9.2 percent, and 11.2 percent of the
responses were from the Plains, Southwest, Rocky Mountain,
and Far West regions, respectively.
One response, kept on
file, was deleted due to missing data.
Confidence intervals were employed at alpha levels of
.05 and .01 for the group of eleven processes, five content
12
areas, and three technologies.
Hypothesis testing was used
to test the attributes of the model at the same levels of
significance.
SPSS was used to tabulate the statistical
data from the survey.
Chapter IV presents a series of tables that show the
results of these 156 surveys and reports the results of the
statistical tests used to determine the significance of the
associated hypotheses.
Perceptions of Attributes
Overall Validity. Comprehensiveness,
and Usefulness
Responding mathematics faculty did not in every case
give the validity, comprehensiveness, and usefulness of the
overall Tunstall Model the highest rating; however, on a
scale of 0 through 5, with 0 being the lowest value and 5
being the highest value, the mean validity (3.82), mean
comprehensiveness (3.39), and mean usefulness (3.33),
according to the 156 complete responses, were well above the
3.0 rating midpoint which was used in the analysis of data
(see Table 4.1).
124
Table 4.1
Mean validity, comprehensiveness, and
usefulness ratings and hypothesis test
results for the General (overall) Tunstall
Model. (Ho: /u< 3.0, Ha: ^ > 3 . 0 )
OVERALL
N
MEAN
ATTRIBUTE
STD
HYPOTHESIS ACCEPTED
DEV
AT ALPHA LEVELS OF
.05
.01
Validity
156
3.83
1.05
Ha
Ha
Comprehensiveness
156
3.39
1.42
Ha
Ha
Usefulness
156
3.33
1.39
Ha
Ha
In separate hypothesis tests at the .05 level of
significance, the null hypotheses that declared the Tunstall
Model not valid, comprehensive, or useful were rejected; and
therefore, the alternative hypotheses that proclaimed the
Tunstall Model to be valid, comprehensive, and useful were
accepted.
Ratings for validity, comprehensiveness, and
usefulness, in fact, were high enough at the .01 level of
significance to have provided significant acceptance (see
Table 4.1).
The responses indicated that the attribute regarding
validity received the highest rating for the overall model
and that the attribute regarding usefulness received the
lowest rating.
125
One might speculate from some of the comments that,
although the respondents may feel comfortable with the
overall validity of the model, many of them may need more
explanation or time to reflect upon the usefulness of the
model.
Narratives concerning the development and explanation
of the model and its components were not supplied to the
respondents.
Validity of the Major Processes Component
Responding mathematics faculty did not in every case
give the validity of the Major Processes Component of the
Tunstall Model the highest rating; however, on a scale of 0
through 5, with 0 being the lowest validity and 5 being the
highest validity, the mean validity for "Associating Through
Symbols" (3.72), for "Defining Through Symbols" (3.60), for
"Manipulating Semantic Symbols" (3.65), for "Performing
Semantic Algorithms" (3.51), for "Linking Semantic
Algorithms" (3.52), for "Performing Syntactic Algorithms"
(3.46), for "Linking Syntactic Algorithms" (3.54), for
"Defining Higher Relations" (3.47), for "Combining" (3.64),
for "Verifying" (3.73), and for "Deriving" (3.96), according
to the 156 responses, were all significantly above the 3.0
rating midpoint which was used for all of the data analysis
(see Table 4-2).
126
Table 4.2:
Mean validity ratings and hypothesis test
results for the Major Processes of the
Tunstall Model. (Ho: /u < 3.0, Ha: JJ > 3.0)
MAJOR
N
MEAN
PROCESSES
STD
HYPOTHESIS
DEV
ACCEPTED
AT ALPHA
LEVELS OF
Associating Through Symbols
156
3.72
1.03
Defining Through Symbols
156
3.60
1.03
Manipulating Semantic Symbols
156
3.65
1.08
Performing Semantic Algorithms
156
3.51
1.09
Linking Semantic Algorithms
156
3.52
1.17
Performing Syntactic Algorithms 156
3.46
1.07
Linking Syntactic Algorithms
156
3.54
1.09
Defining Higher Relations
156
3.47
1.12
Combining
156
3.64
1.14
Verifying
156
3.73
1.12
Deriving
156
3.96
1.05
SUMMATIVE (MEAN OF MEANS) DATA
11
3.62
0.15
.05
.01
Ha
Ha
Since a 99 percent confidence interval (alpha of .01)
of the summative (mean of the means) data indicated that the
population mean should lie between 3.50 and 3.73, one can be
127
confident that the true population mean is higher than the
rating midpoint of 3.0.
A 95 percent confidence interval
(alpha of .05) would have, obviously also excluded the
rating midpoint.
An hypothesis test was used to determine
if these statistics were significant (see Table 4.2).
In the hypothesis test at the .01 level of significance
with the summative (mean of the means) data from all eleven
processes, the null hypothesis that the Major Processes
Component of the Tunstall Model is not valid was rejected;
and therefore, the alternative hypothesis that the Major
Processes Component of the Tunstall Model is valid was
accepted (see Table 4.2). The responses indicated that the
process of "Deriving" received the highest validity rating
within the eleven processes and that the process of
"Performing Syntactic Algorithms" was given the lowest
validity ranking.
One might speculate from some of the
comments that, while great care was taken in choosing the
titles of the processes, some respondents may have had
difficulty with the meanings of or the differences between
semantic versus symantic terminology.
Narratives concerning the development and explanation
of the model were not supplied to the respondents.
Validity of the Knowledge Content
Areas Component
Responding mathematics faculty did not in every case
give the validity of the Knowledge Content Areas Component
128
of the Tunstall Model the highest rating; however, on a
scale of 0 through 5, with 0 being the lowest validity and 5
being the highest validity, the mean validity for the
subdiscipline "Geometries" (3.34), for the subdiscipline
"Algebras" (3.28), for the subdiscipline "Logic" (3.32), for
the subdiscipline "Number Theory" (3.34), and for the
subdiscipline "Analysis" (3.36), according to the 156
responses, were above the 3.0 midpoint (see Table 4.3).
Table 4.3:
Mean validity ratings and hypothesis test
results for the Knowledge Content Areas of the
Tunstall Model. (Ho:)Li<3.0, Ha: ^ > 3 . 0 )
KNOWLEDGE
N
MEAN
CONTENT
STD
HYPOTHESIS
DEV
ACCEPTED
AREAS
AT ALPHA
SUBDISCIPLINES
LEVELS OF
.05
Geometries
156
3.43
1.17
Algebras
156
3.28
1.16
Logic
156
3.32
1.14
Number Theory
156
3.34
1.18
Analysis
156
3.36
1.17
SUMMATIVE (MEAN OF MEANS) DATA
129
5
3.33
0.03
Ha
.01
Ha
since a 99 percent confidence interval (alpha of .01)
of the summative (mean of the means) data indicated that the
population mean should lie between 3.29 and 3.33, one can be
confident that the true population mean is higher than the
rating midpoint of 3.0.
A 95 percent confidence interval
(alpha of .05) would have, obviously, also excluded the
rating midpoint.
An hypothesis test was used to determine
if these statistics were significant (see Table 4.3).
In the hypothesis test at the .01 level of significance
with the summative (mean of the means) data from all five
processes, the null hypothesis that the Content Knowledge
Areas Component of the Tunstall Model is not valid was
rejected and, therefore, the alternative hypothesis that the
Knowledge Content Areas Component of the Tunstall Model is
valid was accepted (see Table 4.3).
The responses indicated that the subdiscipline of
"Analysis" received the highest validity rating within the
five subdisciplines and that the subdiscipline of "Algebras"
was given the lowest validity ranking.
"Number Theory" and
"Geometries" were rated almost as highly as "Analysis."
"Logic" was rated only slightly higher than "Algebras." One
might speculate from some of the comments that the
respondents feel that the subdiscipline of "Algebras" was
too broad.
Narratives concerning the development and
explanation of the model and its components were not
supplied to the respondents.
130
Validity of the Dominant Technologies
Component
Responding mathematics faculty did not in every case
give the validity of the Dominant Technologies Component of
the Tunstall Model the highest rating; however, on a scale
of 0 through 5, with 0 being the lowest validity and 5 being
the highest validity, the mean validity for the technology
division "Calculators and Computers" (3.42), for the
division "Tables and Charts" (3.39), and for the division
"Manipulatives and Tools" (3.48), according to the 38
responses, were all well above the 3.0 rating midpoint (see
Table 4.4).
Table 4.4:
Mean validity ratings and hypothesis test
results for the Dominant Technologies of the
Tunstall Model. (Ho: (j < 3.0, Ha: fj > 3.0)
DOMINANT
N
MEAN
TECHNOLOGIES
AREA
STD
HYPOTHESIS
DEV
ACCEPTED
AT
ALPHA
LEVELS OF
DIVISIONS
Calculators and Computers
156
3.42
1.15
Tables and Charts
156
3.39
1.13
Manipulatives and Tools
156
3.48
1.16
3.43
0.05
SUMMATIVE (MEAN OF MEANS) DATA
131
.05
.01
Ha
Ha
Since a 99 percent confidence interval (alpha of .01)
of the summative (mean of the means) data indicated that the
population mean should lie between 3.36 and 3.50, one can be
confident that the true population mean is higher than the
rating midpoint of 3.0.
A 95 percent confidence interval
(alpha of .05) would have, obviously, also excluded the
rating midpoint.
An hypothesis test was used to determine
if these statistics were significant (see Table 4.4).
In the hypothesis test at the .01 level of significance
with the summative (mean of the means) data from all three
technology divisions, the null hypothesis that the Dominant
Technologies Component of the Tunstall Model is not valid
was rejected; and therefore, the alternative hypothesis that
the Dominant Technologies Component of the Tunstall Model is
valid was accepted (see Table 4.4).
The responses indicated that the technology division of
"Manipulatives and Tools" received the highest validity
rating within the three divisions and that the technology
division of "Tables and Charts" was given the lowest
validity ranking.
One might speculate from some of the
comments that, while the respondents feel that "Tables and
Charts" is a valid technology for mathematics, they no
longer consider it a necessity.
Narratives concerning the development and explanation
of the model and its components were not supplied to the
respondents.
132
CHAPTER V
SUMMARY, CONCLUSIONS AND RECOMMENDATIONS
Summary and Conclusions
This survey has been the first stage in testing the
validity of a model of the structure of knowledge for
mathematics by presenting it to a group of mathematics
experts for validation.
Mathematicians have responded to
the survey, and data that rated the perceptions of these
experts regarding the validity of the Tunstall Model of The
Structure of Knowledge for Mathematics have been obtained.
Of the two faculty members at each of the 204 American
institutions surveyed and the one faculty member at each of
the 13 foreign institutions surveyed, one hundred fiftyseven (37 percent) of them returned the "Survey of the
Validity of the Tunstall Model of the Structure of Knowledge
for Mathematics" questionnaires.
One survey, kept on file,
contained incomplete data and was deleted from all
statistical analyses.
The Tunstall Model was rated overall by questions
regarding three attributes of the General (overall) Model
and were statistically tested with respect to each
attribute.
The three components that comprise the General
Tunstall Model were also tested individually for validity by
examining summative (mean of the means) statistics for each
of the items within each component.
133
The General Tunstall Model
The General (overall) Tunstall Model was rated by the
respondents in regard to its validity, comprehensiveness,
and usefulness.
Of these three measures, validity received the highest
mean rating (3.83), while usefulness garnered the lowest
(3.33).
Claims that the overall Tunstall Model is "valid,"
"comprehensive," and "useful" were able to be accepted at
the 99 percent level of significance (alpha of .01).
The General (overall) Tunstall Model of the Structure
of Knowledge for Mathematics was perceived by mathematicians
in the sample to be valid, comprehensive, and useful.
The Major Processes Component
The Major Processes component of the General (overall)
Tunstall Model was rated by the respondents in regard to its
validity.
Of the eleven processes, "Deriving" received the
highest mean rating (3.96), while "Performing Syntactic
Algorithms" was rated the lowest (3.46).
A 99 percent
confidence interval (alpha of .01) indicated that it is
highly probable that the population mean lies above the
rating midpoint (3.0) in an interval between 3.50 and 3.73.
The claim that the Major Processes component of the Tunstall
Model is "valid" was able to be accepted at the 99 percent
level of significance (alpha of .01).
134
The Major Processes component of the Tunstall Model was
perceived by mathematicians in the sample to be valid.
The Knowledge Content Areas Component
The Knowledge Content Areas component of the General
(overall) Tunstall Model was rated by the respondents in
regard to its validity.
Of the five areas, "Analysis" received the highest mean
rating (3.37), while "Algebras" was rated the lowest (3.28).
A 99 percent confidence interval (alpha of .01) indicated
that it is highly probable that the population mean lies
above the rating midpoint (3.0) in an interval between 3.29
and 3.33.
The claim that the Knowledge Content Areas
component of the Tunstall Model is "valid" was able to be
accepted at the 99 percent level of significance (alpha of
.01) .
The Knowledge Content Areas component of the Tunstall
Model was perceived by the mathematicians in the sample to
be valid.
The Dominant Technologies Component
The Dominant Technologies component of the General
(overall) Tunstall Model was rated by the respondents in
regard to its validity.
Of the three divisions,
"Manipulatives and Tools" received the highest mean rating
(3.48), while "Tables and Charts" was rated the lowest
135
(3.39).
A 99 percent confidence interval (alpha of .01)
indicated that it is highly probable that the population
mean lies above the rating midpoint (3.0) in an interval
between 3.36 and 3.50.
The claim that the Dominant
Technologies component of the Tunstall Model is "valid" was
able to be accepted at the 99 percent level of significance
(alpha of .01).
The Dominant Technologies component of the Tunstall
Model of the Structure of Knowledge for Mathematics was
perceived by the mathematicians in the sample to be valid.
Comments
Several responses included comments about the model or
about the survey.
Some of the comments were favorable, but
most of the comments were negative.
A negative comment,
however, did not necessarily translate into an extremely low
rating.
Although one rater called the model "ignorant
garbage" and gave the model zeros for all three overall
attributes, he gave higher marks on most of the other
nineteen questions.
On the other hand, one faculty member
replied to all 22 questions with a zero, but made the
comment of "really very interesting."
Many comments seemed
to angrily ask why a particular topic like calculus or
topology was excluded from the list of subdisciplines.
On the positive side, a few comments reflected a desire
to understand more about the model, especially in reference
136
to the Major Processes Area.
Several comments of "very
interesting" and "I agree" were noted.
Some raters seemed
to place checks by parts of the model that they gave a high
rating.
The rater that gave a five on every question
claimed to have "never seen anything like this before."
Recommendations
Since the results of the survey reflect a general
acceptance of the Tunstall Model, there may exist many
opportunities ahead in regard to its dissemination and use.
The possibility that the Tunstall Model may eventually be
accepted and used by mathematicians is a provocative and
engaging notion that could have many positive consequences
for the development of mathematics.
It may now be argued that mathematicians are ready to
give this new approach to the functional structuring of
knowledge in mathematics a chance to show how it might help
to improve the organization of the discipline.
Based on the
results of the survey, it is recommended as follows:
1.
The Tunstall Model will be presented to
mathematicians as a heuristic for research in the
discipline and as an aid to scholarship.
2.
The Tunstall Model will be offered to students as
an impetus for further study in mathematics.
3.
The Tunstall Model will be disseminated to
mathematics educators for use as a guide in
137
curriculum construction, in group or individualized
instruction, in student career counseling and
advisement, in program articulation, and for
assessment.
4.
The Tunstall Model will be distributed to applied
mathematicians so that they might strengthen their
role as consultants in the practical uses of
mathematics and for the service needs of the
disciplinary organizations associated with
mathematics and mathematics education.
5.
The Tunstall Model will be shared with students so
that they might be helped to understand the generic
processes of mathematics and to know the
relationship of the major areas of study.
6.
The Tunstall Model will be circulated to
mathematicians and mathematics educators so they
might better govern the organization of
mathematical knowledge in their role as stewards of
the discipline.
7.
The Tunstall Model will be used as a foundation for
the development of the essential knowledge elements
for each topic in the knowledge content areas of
mathematics.
8.
The Tunstall Model will be constantly refined,
upgraded, and improved to meet the needs of the
dynamic discipline of mathematics.
138
9.
The Tunstall Model will be used in the development
and structuring of computer software for
mathematics and the related educational pedagogies.
One might speculate that because of the impact of this
study, there might soon be a need for a change in the way
mathematical information is cataloged, stored, and accessed.
Computer aided information and instructional systems may
need to be revised in order to incorporate the structure of
the major processes and the content knowledge areas of
mathematics.
A panel of mathematicians may need to be
established by disciplinary organizations in order to
oversee the organization of mathematics knowledge as
reflected in the structure of the model.
One may begin to solve the problems of low mathematics
achievement in the United States by introducing this
structure of knowledge for mathematics to students as long
as its onset is combined with appropriate progressive and
meaningful learning instructional techniques.
The author
believes that this structure has been judged by experts in
the field of mathematics as valid, comprehensive, and
useful, and it should be utilized so that it might help
students begin to make the connections that lead to the
understanding of mathematics.
Moreover, if the model helps
a student to succeed in making the appropriate connections
between each successive topic, then knowledge of the subject
matter as well as knowledge about each topic may be
139
constructed and the jobs of the scholars, advisors,
instructors, grant writers, disciplinary organizations,
thinkers, administrators, practitioners, and researchers in
the discipline of mathematics might be facilitated.
140
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APPENDIX A
INSTITUTIONS INCLUDED IN SAMPLE
147
INSTITUTIONS INCLUDED IN SAMPLE
Included
Pilot
Major
Institution
Survey Survey
Harvard University
X
Princeton University
X
Yale University
X
Stanford University
X
California Institute of Technology
X
Massachusetts Institute of Technology
X
Dartmouth College
X
Duke University
X
University of Chicago
X
Columbia University
X
Cornell University
X
Rice University
X
Northwestern University
X
University of Pennsylvania
X
Johns Hopkins University
X
University of California at Berkley
X
Georgetown University
X
Brown University
^
Carnegie Mellon University
X
Washington University
X
148
INSTITUTIONS INCLUDED IN SAMPLE
Included
Pilot
Major
Institution
Survey Survey
Emory University
X
University of Virginia
X
University of California at Los Angeles
X
University of Michigan
X
Vanderbilt University
X
Boston College
X
Brandeis University
X
Case Western Reserve University
X
College of William and Mary
X
Georgia Institute of Technology
X
Lehigh University
^
New York University
^
Pepperdine University
^
Rensselaer Polytechnic Institute
X
Rutgers State University at New Brunswick
X
SUNY at Buffalo
^
Tufts University
^
Tulane University
^
University of California at Davis
X
University of California at Irvine
X
149
INSTITUTIONS INCLUDED IN SAMPLE
Included
Pilot
Institution
Survey Survey
University of California at Riverside
X
University of California at San Diego
X
University of Florida
X
University of Illinois at Urbana
X
University of North Carolina at Chapel Hill
X
University of Notre Dame
X
University of Rochester
X
University of Southern California
X
University of Texas at Austin
X
University of Washington
X
University of Wisconsin at Madison
X
Major
American University
X
Boston University
X
Clark University
X
Clarkson University
X
Clemson University
X
Colorado School of Mines
X
Florida State University
X
Fordham University
X
George Washington University
X
150
INSTITUTIONS INCLUDED IN SAMPLE
Included
Pilot
Major
Institution
Survey Survey
Howard University
X
Illinois Institute of Technology
X
Indiana University at Bloomington
X
Iowa State University
X
Loyola University Chicago
X
Miami University
X
Michigan State University
X
North Carolina State University at Raleigh
X
Ohio State University
X
Pennsylvania State Univ. at Univ. Park
X
Polytechnic University
X
Purdue University at West Lafayette
X
Rutgers State University at Newark
X
SUNY at Albany
X
SUNY at Binghamton
X
SUNY at Stony Brook
X
Southern Methodist University
X
Stevens Institute of Technology
X
Syracuse University
X
Texas A & M University at College Station
X
151
INSTITUTIONS INCLUDED IN SAMPLE
Included
Pilot
Institution
Survey Survey
University of Arizona
X
University of California at Santa Barbara
X
University of California at Santa Cruz
X
University of Cincinnati
X
University of Colorado at Boulder
X
University of Connecticut at Storrs
X
University of Delaware
X
University of Denver
X
University of Georgia
X
University of Iowa
X
University of Kansas
X
University of Maryland at College Park
X
University of Miami
X
University of Minnesota at Twin Cities
X
University of Missouri at Columbia
X
University of Missouri at Rolla
X
University of Oklahoma
X
University of Pittsburgh - Main Campus
X
University of Tulsa
X
University of Utah
X
152
Major
INSTITUTIONS INCLUDED IN SAMPLE
Included
Pilot
Major
Institution
Survey Survey
University of Vermont
Virginia Tech
X
Arizona State University
X
Auburn University - Main Campus
X
Baylor University
X
Brigham Young University at Provo
X
Catholic University of America
X
Colorado State University
X
Drake University
X
Drexel University
X
Duquesne University
X
Florida Atlantic University
X
Florida Institute of Technology
X
Hofstra University
X
Kansas State University
X
Louisiana State University at Baton Rouge
X
Marquette University
X
Mississippi State University
X
New School for Social Research
X
Ohio University
X
153
INSTITUTIONS INCLUDED IN SAMPLE
Included
Pilot
Institution
Survey Survey
Major
Oregon State University
X
St. John's University
X
St. Louis University
X
Temple University
X
Texas Christian University
X
Texas Tech University
X
University of Alabama at Birmingham
X
University of Alabama at Tuscaloosa
X
University of Hawaii at Manoa
X
University of Houston - Univ. Park
X
University of Idaho
X
University of Illinois at Chicago ^
X
University of Kentucky
X
University of Louisville
X
University of Maryland - Baltimore County
X
University of Massachusetts at Amherst
X
University of Mississippi
X
University of Missouri at Kansas City
X
University of Nebraska at Lincoln
X
University of New Hampshire
X
154
INSTITUTIONS INCLUDED IN SAMPLE
Included
Pilot
Institution
Survey Survey
University of New Mexico - Main Campus
X
University of Oregon
X
University of Rhode Island
X
Major
University of San Francisco
X
University of South Carolina at Columbia
X
University of South Florida
X
University of Tennessee at Knoxville
X
University of Wisconsin at Milwaukee
X
University of Wyoming
X
Utah State University
X
Virginia Commonwealth University
X
Washington State University
X
Wayne State University
X
Yeshiva University
X
Adelphi University
X
Andrews University
X
Ball State University
X
Biola University
X
Bowling Green State University
X
Cleveland State University
X
155
INSTITUTIONS INCLUDED IN SAMPLE
Included
Pilot
Major
Institution
Survey Survey
East Texas State University
X
Georgia State University
X
Hahnemann University
X
Idaho State University
X
Illinois State University
X
Indiana State University at Terre Haute
X
Kent State University
X
La Sierra University
X
Louisiana Tech University
X
Memphis State University
X
Middle Tennessee State University
X
Mississippi College
X
Montana State University
X
New Mexico State University
X
North Dakota State University
X
Northeastern University
X
Northern Arizona University
X
Northern Illinois University
X
Nova University
X
Oklahoma State University
X
156
INSTITUTIONS INCLUDED IN SAMPLE
Included
Pilot
Major
Institution
Survey Survey
Old Dominion University
X
Portland State University
X
Southern Illinois University at Carbondale
X
Tennessee Technological University
X
Texas Woman's University
X
Union Institute
X
United States International University
X
University of Akron
X
University of Arkansas at Fayetteville
X
University of Maine at Orono
X
University of Missouri at St. Louis
X
University of Montana
X
University of Nevada at Reno
X
University of New Orleans
X
University of North Carolina at Greensboro
X
University of North Dakota
X
University of North Texas
X
University of Northern Colorado
X
University of South Dakota
X
University of Southern Mississippi
X
157
INSTITUTIONS INCLUDED IN SAMPLE
Included
Pilot
Major
Institution
Survey Survey
University of Texas at Arlington
X
University of Toledo
X
West Virginia University
X
Western Michigan University
X
Beijing University, China
X
Helsingfors Universitet, Finland
X
Moscow State University, USSR
X
Rijksuniversiteit Te Utrecht, Netherlands
X
Tel-Aviv University, Isreal
X
Universitat Hamburg, Germany
X
Universite De Paris I, France
X
University of Cambridge, England
X
University of Edinburgh, Scotland
X
University of Sidney, Australia
X
University of Tokyo, Japan
X
Uppsala Universitet, Sweden
X
Vrije Universiteit Brussel, Belgium
X
158
APPENDIX B
SURVEY INSTRUMENT
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i TEXAS TECH UNIVERSIPl'
College of Educanon
Box4lO"l
Lubbock. TX'^09 10-1
FAX (80ot"^2 2r9
April 24, 1993
Dear Colleague:
The problems relating directly and indirectly to the organization of knowledge
in mathematics are multidimenaional. They extend deeply into the public achoola,
reach throughout our busineaaee, are endemic in moat of our mathematics, science,
and applied disciplines, and permeate every part of our technological society.
Structuring and articulating )inowledge is a common problem among all levels of
education.
It has been suggested that, rather than relying upon educators and textbook
writers to organize the content, processes, and technologies of the discipline
of mathematics, a model be devised which shows the organization of mathematical
Icnowledge. Such a model could have great potential for assisting students and,
therefore, future mathematics' stewards in promoting the understanding and
construction of mathematical knowledge. Specifically, the ideal model should aid
in the advancement of mathematical knowledge through SCHOLARSHIP, PRACTICE,
ADMINISTRATION, INSTRUCTION, ADVISING, GRANT WRITING, RESEARCH, the DISCIPLINARY
ORGANIZATIONS, and, in general, MATHEHATICAL THINKING.
There is, currently, no valid model to assist mathematicians in providing for the
stewardship of mathematics. The purpose of this study is to present a proposed
model of the structure of knowledge in mathematics to the experts in the
discipline for validation.
You and/or your immediate colleagues are being asked to assist in this
investigation by responding to the 'Survey of the Validity of the Tunstall Model
of the Structure of Knowledge in Mathematics' which asks for your response to
twenty-two questions regarding the (enclosed) Tunstall Model.
The leaders of the International Society for EpistEcybernetics (ISEE) have
already assessed the Tunstall Model to be a valuable heuristic for the future of
our discipline. You are one of the few that have been chosen to participate in
this ground-breaking approach to the problem of ataxic mathematical knowledge.
Please take a few minutes to answer and return the questionnaire. If you decide
to pass it on to a colleague, just have them change the name and/or address on
the label so that neither of you will receive a 'second wave' mailing.
A self-addressed, stamped envelope has been enclosed for your convenience. I
would be grateful if the BLUE SURVEY PAGES could be completed and returned by May
20, 1993. THERE IS NO NEED TO RETURN THE WHITE OR YELLOW PAGES. Thank you for
your participation in this study. If you have any questions, feel free to call
me at (806) 742-2377.
Sincerely,
f
Paul M. Tunstall, Jr.
An AjJtrmatiieAction histtlulion
170
MJ TEXAS TECH UNIVERSITY
CoUege of Education
Bcw 410-1
Lubbock TX 79409 ICri
(806)-•12 23"^
fiOC (806; 742 2 r 9
l u r c h 24, 1993
Dear Colleague:
The problems relating directly and indirectly to the organization of knowledge
in mathematics are multidimensional. They extend deeply into the public schools,
reach throughout our businesses, are endemic in most of our mathematics, aeicnce,
and applied disciplines, and permeate every part of our technological eociety.
Structuring and articulating knowledge is a common problem among all levels of
education.
It has been suggested that, rather than relying upon educators and textbook
writers to organize the content, processes, and technologies of the discipline
of mathematics, a model be devised which shows the organization of mathematical
knowledge. Such a model could have great potential for assisting students and,
therefore, future mathematics' stewards in promoting the understanding and
construction of mathematical knowledge. Specifically, the ideal model should aid
in the advancement of mathematical knowledge through SCHOLARSHIP, PRACTICE,
ADMINISTRATION, INSTRUCTION, ADVISING, GRANT WRITING, RESEARCH, the DISCIPLINARY
ORGANIZATIONS, and, in general, MATHEMATICAL THINKING.
There is, currently, no valid model to assist mathematicians in providing for the
stewardship of mathematics. The purpose of this study is to present a proposed
model of the structure of Icnowledge in mathematics to the experts in the
discipline for validation.
You and/or your immediate colleagues are being asked to assist in this
investigation by responding to the 'Survey of the Validity of the Tunstall Model
of the Structure of Knowledge in Mathematics' which asks for your response to
twenty-two questions regarding the (enclosed) Tunstall Model.
The leaders of the International Society for BpistEcybemetics (ISEE) have
already assessed the Tunstall Model to be a valuable heuristic for the future of
our discipline. You are one of the few that have been chosen to participate in
this ground-breaking approach to the problem of ataxic mathematical knowledge.
Please take a few minutes to answer and return the questionnaire. Zf you decide
to pass it on to a colleague, just have them change the name and/or address on
the label so that neither of you will receive a 'second wave' mailing.
A self-addressed, stamped envelope has been enclosed for your convenience. Z
would be grateful if the BLUE SURVEY PAGES could be completed and returned by
April 5, 1993. THERE IS NO NEED TO RETURN THE WHITE OR YELLOW PAGES. Thank you
for your participation in this study. If you have any questions, feel free to
call me at (806) 742-2377.
.Sincerely,
\'^SXJ>
L'
O^.-^\-L'V..--><C
Paul M. Tunstall, Jr.
An Ajjtrmanve Action Insntuiion
171