ERIC Identifier: ED265050 Publication Date: 1985-00-00
Author: Weaver, J. F. Source: ERIC Clearinghouse for
Science Mathematics and Environmental Education Columbus OH.
The Learning of Mathematics. ERIC/SMEAC Mathematics Education
Digest Number 1.
Twenty years ago Macdonald (1965) pointed to the unfortunate fact that
"There is no consistently clear distinction in the use of much educational
terminology. One definition of curriculum may well turn out to be the same as
the next definition of instruction, and this definition of instruction could
quite likely be synonymous with another's definition of teaching" (p.2).
Macdonald then characterized teaching "... as the behavior of the teacher,
learning as the change in learner behavior, (and) instruction as the
pupil-teacher interaction situation" (p.6).
The distinction between teaching and learning, for instance, has been phrased
in somewhat similar ways by others at later times: "By teaching I mean any
activity on the part of one person intended to influence learning on the part of
another" (Gage 1978, p.14), and "Learning is a change in human disposition or
capability, which persists over a period of time, and which is not simply
ascribable to processes of growth" (Gagne 1977, p.3).
In keeping with such distinctions, the present report focuses on findings and
observations associated with relatively recent research on the learning of
mathematics and is not intended to embrace research on mathematics teaching and
Closely related to these distinctions is another one: a theory of learning is
descriptive rather than prescriptive, as is the case for a theory of teaching or
a theory of instruction.
PIAGET AND THE LEARNING OF MATHEMATICS
Piaget's "theory" is a theory of intellectual development. It is not a theory
of learning, teaching, instruction or curriculum -- either in general or with
respect to mathematics in particular. But much writing and many investigations
have been directed toward hypothesized implications of Piaget's theory for
mathematics learning, teaching and instruction, for the nature and sequence of
curricular content, and for classroom structure, organization and management.
And concerns have been expressed regarding ways in which Piaget's theory has
been applied to different aspects of education and mathematics (Weaver l972).
Frequently it has been suggested that certain Piagetian tasks (conservation
tasks, for instance) provide a good basis for determining students' readiness
for learning one aspect or another of school mathematics. But Hiebert and
Carpenter (1982) have indicated that "The available research evidence suggests
that ... Piagetian tasks are not useful readiness measures. (With respect to the
question of whether) Piagetian tasks can be used to identify children who will
be unable to benefit from instruction in mathematics, all the available evidence
clearly suggests that the answer is no; many children who fail Piagetian tasks
are able to learn mathematical concepts and skills" (pp.339-340).
Such evidence is not to be construed as refuting Piaget's theory of
intellectual development, nor as reflecting unfavorably upon the nature of
school mathematics. Instead, it indicates that a certain kind of "match" simply
does not seem to exist. This is part of the reason why many persons now are
turning away from, rather than looking toward, Piaget for help in dealing with
issues associated with mathematics learning (and teaching and instruction).
A DIFFERENT DIRECTION
Kirby and Biggs (1980) indicated that "Cognition returned to psychology in
the 1960s and flourished in the 1970s, permeating most areas of psychology. Its
metaphor, information processing, became dominant in that discipline" (p. xi).
More directly to the point, Groen and Kieran (1983) pointed out that "A few
years ago, research on children's mathematics was dominated by Piaget. To many
in the field, the task was to extend Piaget's theory or reinterpret it.... In
contrast, the research described in this volume relegates Piaget to the
background. Information-processing theory, broadly conceived, has replaced the
Piagetian framework as a broad explanatory model.... The significance of
information-processing theory in cognitive development has grown concomitantly
with a retreat from the Piagetian framework" (pp.351-352).
Within the contemporary context of "cognitive science" emphasis is placed
upon "understanding" and "comprehension" -- recalling Brownell and his
exposition of the "meaning theory" 50 years ago! The learning of mathematics,
and research associated with such learning, is more and more commonly being
described or explained in relation to a system that includes provision for the
intake of information, for its "chunking" within a working memory, and for
interaction there with other information stored within and retrieved from a
It is not at all surprising that an information-processing system often is
associated closely with, and described in terms related to, a computer or
SOME ADDITIONAL NOTEWORTHY OBSERVATIONS
1. Contemporary research associated with the learning of mathematics is much
more likely to focus upon individuals than upon groups. That is to say, learning
is an individual phenomenon and, although much learning takes place within the
context of groups such as classes, the focus of attention is not upon the
group/class but upon individuals.
2. Contemporary research associated with the learning of mathematics looks
beyond observable performance related to so-called "behavioral objectives" for
its data base. Romberg and Carpenter (in press) have indicated that "internal
cognitive processes are acknowledged. Rational task analysis, which is based on
a logical analysis by experts, has evolved to empirical task analysis, which
focuses on what children actually do when they solve mathematics problems"
(p.852). Some of this "doing" may be observable in a student's overt actions,
but much of the doing may be terms of a student's observable but self-reportable
3. There is a growing body of evidence that young children invent or
construct much of their own mathematical knowledge, and that they come to school
with some well-developed, although understandably immature, ways of dealing with
various mathematical situations. We do not seem to take full advantage of
children's conceptions in our programs of mathematics teaching and instruction.
4. The term metacognition refers to a person's awareness of and sensitivity
to her or his own thought processes, and includes the ability to monitor and
control (to some degree) such processes. There is growing evidence that learning
associated with problem solving is facilitated or enhanced by a student's
increased awareness of metacognitive aspects of the problem-solving process.
Also, there is evidence that some of the differences between expert and novice
problem solvers may be attributed to differences in metacognitive skills.
Romberg and Carpenter (in press) believe that "We currently know a great deal
more about how children learn mathematics than we know about how to apply this
knowledge to mathematics instruction. Research is clearly needed to explore how
knowledge of children's learning of mathematics can be applied to the design of
instruction" (p.859). Furthermore, "Although the emphasis in research on
learning has changed dramatically in the last 15 years, the connection between
theories of instruction and theories of learning remains an issue" (p.851).
FOR MORE INFORMATION
Gage, N. L. THE SCIENTIFIC BASIS OF THE ART OF TEACHING. New York: Teachers
College Press, 1978.
Gagne, R. M. THE CONDITIONS OF LEARNING (3rd ed.). New York: Holt, Rinehart
and Winston, 1977.
Groen, G., and C. Kieran. "The Many Facets of Piaget." In H.P. Ginsburg
(Ed.)., THE DEVELOPMENT OF MATHEMATICAL THINKING. New York: Academic Press,
Hiebert, J., and T. P. Carpenter. "Piagetian Tasks as Readiness Measures in
Mathematics Instruction: A Critical Review." EDUCATIONAL STUDIES IN MATHEMATICS
13: 329-345, 1982.
Kirby, J. R., and J. B. Biggs, (Eds.). COGNITION, DEVELOPMENT, AND
INSTRUCTION. New York: Academic Press, 1980.
Macdonald, J. B. "Educational Models for Instruction--Introduction." In J. B.
Macdonald and R. R. Leeper (Eds.). THEORIES OF INSTRUCTION. Washington, DC:
Association for Supervision and Curriculum Development, 1965.
Romberg, T. A., and T. P. Carpenter. "Research on Teaching and Learning
Mathematics: Two Disciplines of Scientific Inquiry." In M. C. Wittrock (Ed.).,
THIRD HANDBOOK OF RESEARCH ON TEACHING n press.
Weaver, J. F. "Some Concerns About the Application of Piaget's Theory and
Research to Mathematical Learning and Instruction." ARITHMETIC TEACHER l9
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