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Didactics of Mathematics as a Scientific Discipline, 15-26.
© 1994 Dordrecht: Kluwer Academic Publishers. Printed in the Netherlands.

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building in an educational system without central control of such activity.
Listening to the voices in those forums also raises questions about the feasi-
bility of developing elementarization as a scientific activity in the didactics
of mathematics. The ferment of American debate about goals and methods
of school mathematics has led to production of imaginative curriculum ma-
terials and teaching ideas, but very modest and uneven implementation of
the possible innovations.
In this paper I will analyze, with examples from recent American experi-
ences, the influences of various factors in formation of school curricula. The
underlying goal is improving translation of mathematics as a discipline of
human knowledge and reasoning to a subject for school learning. But the
immediate question is how a broad range of interests and expertise can be
organized to perform that task effectively. What are the prospects for mak-
ing elementarization a rational activity in the science of mathematical didac-
tics?
2. INSIGHTS FROM MATHEMATICS
One of the most obvious places to look for guidance in construction of
school mathematics curricula is in the structure and methods of the root
discipline itself. It seems reasonable that the mathematical education of
young people should provide them, in some appropriate way, with the basic
understandings and skills that enable mathematicians to reason so effec-
tively about quantitative and spatial problems. Who could be better quali-
fied to identify the core concepts, principles, and techniques of mathematics
and the paths by which mastery of those ideas can be most naturally reached
than professional mathematicians?
As Kilpatrick (1992) notes, “mathematicians have a long, if sporadic,
history of interest in studying the teaching and learning of their subject.”
This concern for the content and organization of school mathematics curric-
ula was especially acute during the reform era of the 1950s and 1960s when
hundreds of research mathematicians engaged in curriculum development
and teacher education projects designed to update school programs. The in-
fluence of many of those mathematicians led to emphasis in the new pro-
grams on underlying abstract structures of mathematical domains, increased
attention to precision of language for expressing mathematical ideas, and
introduction to school mathematics of topics previously viewed as part of
collegiate study (NACOME, 1975).
In summarizing a conference of prominent research mathematicians and
scientists who gathered to think about directions for improvement of school
curricula and teaching, the psychologist Jerome Bruner (1960) recorded the
brave assertion that, “any subject can be taught to anybody at any age in
some form that is honest.” He, and many others, went on to suggest that
school mathematics should give students an understanding of the discipline
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ECLECTIC APPROACHES TO ELEMENTARIZATION

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17
and its methods that parallels (albeit in a weaker form) that of mathemati-
cians who are active at the frontiers of pure and applied research.
Unfortunately, proposals to use the structure and methods of advanced
mathematics as a guide to school curricula have proven problematic at best.
The concepts and principles of the major branches of mathematics can, in
some sense, be derived logically from a small set of primitive assumptions
and structures. However, the formal logical coherence of the subject masks
quite varied aspects of the way the subject is actually developed and used by
mathematicians. Almost as soon as the first new math reform projects got
underway in the United States, there were debates about the proper mathe-
matical direction of that reform. Differences of opinion on the balance of
pure and applied mathematics, the role of deduction and intuition in mathe-
matical work, and the importance of various mathematical topics reflected
the diversity of the discipline itself. There was little unanimity in the advice
about school mathematics coming from the professional mathematics com-
munity. Consequently, if school curricula are to convey images of mathe-
matics that faithfully represent the content and methods of the subject as
practiced in mathematical research and applications, it seems likely that
they will include a combination of topics chosen from many options, as a
result of competition among opinions that reflect the mathematical taste and
experience of concerned individuals, not scientific analysis.
In retrospect, promises that the content and organization of school math-
ematics curricula could be guided by following the deductive structure of
formal mathematics seem incredibly naive. While there is a certain plausi-
bility to the idea that all students can profit by acquiring something of the
mathematical power possessed by experts in the field, a little thought on the
subject reminds us that many people use mathematical ideas and techniques
in ways quite different than those taught in school and in settings quite dif-
ferent from formal scientific and technical work. Thus it seems quite rea-
sonable to ask whether school mathematics should be designed with an eye
on formal academic mathematics alone, or in consideration of the varied
ways that people actually use mathematics in daily life and work. This ten-
sion between images of formal and practical mathematics has always been a
factor in curricular decision-making. Research over the past 20 years has
added intriguing insights into the mathematical practices of people in vari-
ous situations (e.g., Rogoff & Lave, 1984), adding a new dimension to the
debate over what sort of mathematics is most worth learning and what
should be in school curricula.
In the past decade, the task of selecting content goals for school curricula
has been further complicated by a dramatic revolution in the structure and
methods of mathematics itself. Electronic calculators and computers have
become standard working tools for mathematicians. In the process, they
have fundamentally altered the discipline. For centuries, if not millennia,
one of the driving forces in development of new mathematics has been the
JAMES T. FEY

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ECLECTIC APPROACHES TO ELEMENTARIZATION
search for algorithmic procedures to process quantitative and geometric in-
formation. But execution of those procedures was always a human activity,
so school mathematics had to devote a substantial portion of its program to
training students in rapid and accurate execution of algorithms. With calcu-
lators now universally available at low cost, few people do any substantial
amount of arithmetic computation by traditional methods; with powerful
personal computers also widely available to anyone engaged in scientific or
technical work, few people do algebraic symbolic computation by tradi-
tional methods. Furthermore, the visual representations provided by modern
computers provide powerful new kinds of tools for mathematical experi-
mentation and problem-solving. The effect of these changes in the techno-
logical environment for mathematics is to change, in fundamental ways, the
structure of the subject and its methods. For those who look to the structure
and methods of mathematics as guides to school curricula, it is time for re-
consideration of every assumption that underlies traditional curriculum
structures (Fey, 1989; NRC, 1990). Of course, this fundamental change in
mathematics wrought by emergence of electronic information-processing
technology underscores another factor in the curriculum design process –
we plan curricula to prepare students for lives in a future world that will un-
doubtedly evolve through continual and rapid change. Our experience of the
recent past suggests that we can hardly imagine what that future will hold,
and this uncertainty itself must be a factor in the curriculum decision-mak-
ing process.
What then are the insights from mathematics that play a role in the task of
elementarization for school curriculum design? The structure of mathemat-
ics obviously provides some guidance to selection and organization of top-
ics in school curricula. However, it now seems clear that, in making content
choices, we must consider a very complex web of insights into the ways that
the subject can and will be used by our students. Those judgments can be in-
formed by analyses of alternative conceptual approaches to the content, by
assessments of how the subject is used, and by implications of new tech-
nologies. However, such analyses will ultimately be blended into personal
judgments by people who must make choices based on incomplete evi-
dence, not by following an algorithm for curriculum design.
3. INSIGHTS FROM PSYCHOLOGY
When mathematicians become concerned about school curricula, their first
instinct is usually to focus on the content of textbooks and instruction at var-
ious grade levels. Quite reasonably, they feel most expert at judging the rel-
ative importance and correctness of the topics and their presentation. How-
ever, anyone who remains engaged with the reform process long enough to
work on the production and testing of alternative curricula for schools will
soon realize that selection of content goals is only the easy part of the task.
The naive faith expressed in Bruner's assertion that any child can learn any
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mathematics in some honest form led many curriculum innovators to try
some daring experiments. However, those who watched the classroom ex-
periments carefully and listened to voices of teachers and students soon
found that the search for accessible honest representations of mathematical
ideas is a deep problem that gets entangled quickly in questions of how
young people learn.
It is natural to turn to psychology for insight into the mechanisms by
which humans learn facts, concepts, principles, skills, reasoning processes,
and problem-solving strategies. There is a long tradition of research by
American and European psychologists on questions related to mathematics
learning and teaching (Kilpatrick, 1992; Schoenfeld, 1992). Sometimes that
research has focused on mathematics, because the subject appears to offer a
domain of well-defined content in which knowledge can be objectively
measured, but psychological investigations have also addressed questions
that are fundamental in mathematics education.
In the heyday of connectionist and behaviorist psychology, studies of
arithmetic learning examined questions in the procedural aspects of arith-
metic and algebra. Psychologists in the Gestalt tradition were more inter-
ested in problem-solving and concept formation, with mathematical subject
matter useful in both types of investigation. Developmental psychologists
have used mathematical tasks in their studies aimed at understanding stages
and rates of cognitive development. The work of Piaget and his descendants
in the constructivist school of learning and teaching has been enormously
influential in thinking about school mathematics teaching and learning.
Psychologists exploring the contemporary information-processing models of
learning have found it convenient to use mathematical procedural knowl-
edge in their studies.
There is now a very strong and active collaboration of research psychol-
ogists and mathematics educators that has resulted in focusing investiga-
tions of human learning on issues that are central to mathematics education
in school. Several examples illustrate that collaboration and its potential for
productive influence on design of mathematics curricula and teaching.
For instance, in modern cognitive theories, one of the central issues is the
representation of knowledge in memory. Representation of facts and rela-
tionships is a very important aspect of mathematical thinking and learning,
so mathematics educators have become vitally interested in psychological
research that contributes to understanding of representations. At the same
time, many mathematics educators, stimulated by the notion of representa-
tion, have launched independent work in curriculum development and re-
search on teaching that tests hypotheses about representation in practical
settings. The capability of computers for simultaneously displaying graphic,
numeric, symbolic, and verbal representations of mathematical information
and relationships has led to important work aimed at helping students ac-
quire better mathematical understanding and problem-solving power. Fur-
JAMES T. FEY
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thermore, the computer representations have made deep ideas and difficult
problems accessible to students in new ways – altering traditional curricu-
lum assumptions about scope and sequence. For example, with the use of
inexpensive graphing calculators, students in elementary algebra can solve
difficult equations, inequalities, and optimization problems with visual and
numerical successive approximation methods, long before they acquire the
symbol manipulation skills that have been the traditional prerequisites for
such work.
In contemporary psychological research, there is also considerable inter-
est in processes of metacognition and self-regulatory monitoring of mental
activity. Since mathematics education is especially interested in developing
student ability to work effectively in complex problem-solving situations,
there has been considerable interaction between psychological research and
mathematical education on that issue.
By any reasonable measure, the power of mathematics as a tool for de-
scribing and analyzing patterns and solving problems comes from the fact
that common structural concepts and procedures can be recognized and ex-
ploited in so many different specific contexts. The central problem of math-
ematical education is to help students acquire a repertoire of significant
conceptual and procedural knowledge and the ability to transfer that knowl-
edge from the specific contexts in which it is presented to new and appar-
ently different settings. The problem of transfer is a central issue in psycho-
logical research, and, in a 1989 review, Perkins and Salomon noted that
much research suggests, “To the extent that transfer does take place, it is
highly specific and must be cued, primed, and guided; it seldom occurs
spontaneously.” However, they go on to report recent work, much focused
in mathematics, which shows that, “When general principles of reasoning
are taught together with self-monitoring practices and potential applications
in varied contexts, transfer often is obtained.” On the other hand, recent re-
search on situated cognition (Brown, Collins, & Duguid, 1989) has coun-
tered this optimistic conclusion by suggesting that it is impossible to sepa-
rate what is learned from the activity and context in which learning takes
place, that “learning and cognition... are fundamentally situated.”
What then is the actual and potential contribution of psychological re-
search to the problem of curriculum design in school mathematics? The top-
ics that have been investigated by cognitive and developmental psycholo-
gists are relevant to central issues in teaching and learning of mathematics.
However, far from providing clear guidance to construction of optimal
teaching strategies and learning environments, the results are more sugges-
tive than prescriptive – incomplete and often contradictory. A curriculum
developer or teacher who turns to psychology for insight into the teaching of
key mathematical ideas and reasoning methods will find provocative theo-
ries, but also a substantial challenge to translate those theories into practical
classroom practices.
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4. INSIGHTS FROM CLASSROOM TEACHERS
Effective mathematics teaching certainly depends on knowledge of mathe-
matics and knowledge of ways that students learn mathematics. But there
remains an artistry about superb teaching that weaves mathematical and
psychological insights into workable curricula and engaging and effective
teaching activities. The findings of scientific research must still be informed
and enhanced by wisdom of practice. It is precisely this blending of theoret-
ical and practical knowledge that occurred in the recent National Council of
Teachers of Mathematics' efforts to establish and promote Standards for