Constructivist Views on the Teaching and Learning of Mathematics || Chapter 3: Epistemology, Constructivism, and Discovery Learning in Mathematics

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  • Chapter 3: Epistemology, Constructivism, and Discovery Learning in MathematicsAuthor(s): Gerald A. GoldinSource: Journal for Research in Mathematics Education. Monograph, Vol. 4, ConstructivistViews on the Teaching and Learning of Mathematics (1990), pp. 31-47+195-210Published by: National Council of Teachers of MathematicsStable URL: .Accessed: 03/10/2013 01:47

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  • Chapter 3: Epistemology, Constructivism, and Discovery Learning in Mathematics

    Gerald A. Goldin Rutgers University

    What is the best way to characterize the body of knowledge that we call mathematics? How do children and adults learn mathematics most effectively? How can we best study their learning processes, and assess the outcomes of learning? Can meaningful learning be consistently distinguished from nonmeaningful or rote learning? What constitutes effective mathematics teaching, and how can elementary and secondary school teachers be enabled to provide it?

    In the accompanying papers, a number of my colleagues propose partial answers to these questions. Recurrent themes in these papers include the following: (1) a view of mathematics as invented or constructed by human beings, rather than as an independent body of "truths" or an abstract and necessary set of rules; (2) an interpretation of mathematical meaning as constructed by the learner rather than imparted by the teacher; (3) a view of mathematical learning as occurring most effectively through guided discovery, meaningful application, and problem solving, as opposed to imitation and reliance on the rote use of algorithms for manipulating formal symbols; (4) the study and assessment of learning through individual interviews and small-group case studies which go far beyond traditional paper-and-pencil tests of skills; (5) approaches to effective teaching through the creation of classroom learning environments, encouraging the development of diverse and creative problem-solving processes in students, and reducing the exclusive emphasis on mathematically correct responses; and (6) goals for teacher preparation and development which include reflections on epistemology consideration of the origins of mathematical knowledge, investigating and understanding mathematical knowledge as constructed and mathematical learning as a constructive process, and abstraction from teachers' and students' own mathematical problem-solving experiences.

    This set of ideas, with which I strongly concur, has a long and distinguished history of development in mathematics education. It is to be hoped that the monograph will contribute

    The ideas in this chapter evolved from discussions at the Conference on Models for Teacher Development in Mathematics, June 4-5, 1986, sponsored by the New Jersey Department of Higher Education and the Rutgers University Center for Mathematics, Science, and Computer Education. A brief version was presented at the 13th International Conference of PME in Paris (Goldin, 1989).


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  • 32 to their exposition and advancement, particularly at the present critical juncture-when, despite increasing recognition of the urgency of improving mathematics and science education, public policy in the United States and in many states remains uncertain of the most effective methods to adopt, and too often encourages movement in directions opposite to those described.

    Recently, considerable attention has been devoted by the mathematics education research community to questions of epistemology as they pertain to the psychology of mathematics leaming-particularly to the philosophical perspective known as radical constructivism. Many of those who have adopted a constructivist approach to learning and

    teaching base their theories on radical constructivist epistemology, which has emerged as

    offering a powerful justification for views such as those above (Cobb, 1981; Confrey, 1986; Steffe, von Glasersfeld, Richards, & Cobb, 1983; von Glasersfeld, 1984; 1987a). This essay is intended to raise some issues in criticism of our adopting radical constructivism as the foundation for our approach to mathematics education, even as I argue in favor of what I would prefer to call a "moderate constructivist" view. I hope to indicate how the six recurrent themes I have just listed can emerge from an empiricist epistemology that is consistent with scientific methods of inquiry as they are usually understood and applied, and that avoids some of the potentially damaging consequences of radical constructivism

    (Kilpatrick, 1987). In offering the arguments which will follow, I nevertheless would like to stress at the

    outset my respect and admiration for the thinking of many individual researchers and mathematics educators of the radical constructivist school, including those who are authors of the accompanying papers. Their contribution has, in my opinion, been especially valuable as a challenge to the premature conclusions which are sometimes drawn from empirical, quantitative, and apparently "scientific" research in mathematics education. Such a challenge is particularly important when, as is often the case, the principal variables that have been selected for empirical study are surface variables selected because they are relatively easy to make quantitative, while more difficult cognitive variables are disregarded. Radical constructivists have also sought needed alternatives to the overly mechanical and deterministic models sometimes offered by the artificial intelligence/cognitive science school.

    Epistemological Schools of Thought Epistemology is the branch of philosophy that deals with the underpinnings of how we know what we know, and in particular the logical (and sometimes the psychological) bases for ascribing validity or "truth" to what we know. To place radical constructivism in context, let

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    us consider on an elementary level some contrasting epistemological perspectives. I shall mention a few of the philosophical questions that these perspectives have addressed, and some of the implications for psychological research and educational practice that have been drawn from them-sometimes quite erroneously.

    Most epistemological reasoning begins with an analysis of the sources of what "I" (the reasoner) know. One major source of knowledge is via my senses-the world of sensory experience or "sense-data," directly accessible to me. Such direct, "inner" experiences can be taken to include "feelings" as well as "sensations" (e.g., my experience of pleasure, as well as of warmth). Another possible source of knowledge is logical reasoning and introspection-hypotheses and/or conclusions which I can reach through mental processes. Some of the many questions with which epistemologists then grapple are the following:

    Can I validly infer the existence of an external reality, apart from my own experience? If so, how? What can I know about it, if it exists, and how can I arrive at such knowledge?

    Can I validly infer the existence of the internal experiences of other people? If so, how? What comparisons can be made between their internal experiences and my own?

    Can I consistently verify the validity of my own logical reasoning processes? Are logical reasoning and mathematical reasoning in some sense intrinsically valid, are they merely social conventions adopted by a subset of human society, or are they essentially private and non-comparable between individuals?

    What does it mean to say that a statement in mathematics is "true"? What does it mean to assert that an empirical statement in science is "true," one that seems to pertain to "external reality"? Is there any sense in which either of these "truths" is "objective?" Is the science we call psychology different in this respect from the physical and biological sciences because its domain includes "the mind?"

    Over the centuries, radically different perspectives have been proposed by the exponents of various epistemological schools, leading to very different answers to such questions. The following brief, highly simplified overview is intended to convey the flavor of a few main approaches (see, e.g., Turner, 1967, for more detail at an introductory level).

    Idealism is the view that all reality is, in fact, mental. All that I experience is mental, and no external, physical "real world" can be validly inferred from that experience. It is thus a rather extreme version of empiricism. But idealism broadly construed may allow for the existence of other minds, or even for a universal mind with which individual minds share experience. It is possible to be an idealist in one's metaphysics, but an epistemological realist. Solipsism is the still more radical view that the only reality is in my mind.

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    At the other end of the spectrum, causal realism is the view that the external world exists and in fact is what causes me to have the sense experiences that I have, although it is distinct from those experiences. This view falls within the more general framework of rationalism, as it asserts that one can acquire knowledge about the physical world through reason and logical inference. Sensory experiences in this view are not very trustworthy. They play a role, but they may be misleading or illusory. They are not the most fundamental, ultimate reality; and one must reason one's way through them to arrive at knowledge of the external world.

    Empiricism relies much more heavily on sense-data as the initial "givens" of

    epistemology. They are the elements of a world-as-experienced. Observation and measurement become fundamental processes for recording and organizing sense-data, and inferences drawn from patterns in sense-data account for the validity of knowledge. For

    example, the view that real-world statements about physical objects function as useful summaries of patterns in observed and predicted sensory experience is an empiricist perspective.

    Many epistemologists have distinguished between "analytic" and "synthetic" truth, though they have not always agreed on their definitions, and the distinction itself has been

    questioned. Roughly speaking, analytic statements, such as "All brothers are siblings," are those that are true by virtue of the meanings or definitions of the terms involved and are not

    subject to empirical contradiction, whereas synthetic statements, such as "George Polya was a mathematician," depend for their truth on empirical evidence. Analytic statements may nevertheless require reasoning to verify them. One point of view is that mathematics itself consists of a body of analytic knowledge, while another (20th-century) view is that mathematics is in essence a purely formal symbol system, with an internal logic but without intrinsic content.

    One of the more influential forms of radical empiricism, known as logicalpositivism, adopts the "verifiability criterion" of meaning, whereby the only meaningful content of a synthetic statement consists of the methods whereby it can in principle be confirmed or disconfirmed observationally.

    Radical constructivism is a school of epistemology which emphasizes that we can never have access to a world of reality, only to the world that we ourselves construct out of our own experience. All knowledge, whether mathematical or not, is necessarily constructed. Without evidence for some form of telepathic perception, no individual has direct knowledge of anyone else's world of experience; we can only construct personal models of the knowledge and experience of others. Thus, one can never conclude that one's own

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    knowledge is "the same as" that of another person. Likewise, we can only construct models of reality, and can never conclude that our own knowledge is in fact "knowledge of the real world."

    In the radical constructivist view, knowledge about mathematics, science, psychology, or the everyday world is never communicated, but of epistemological necessity constructed (and reconstructed) by unique individuals. It is constructed out of our in-context experience of each other's speech and actions. Thus, social conventions and social interactions in contexts, rather than any more "absolute" criterion of "truth" or "objectivity," often function as the most important determinants of whether an individual's knowledge is regarded as valid, or whether a mathematical or scientific concept to be taught has been "correctly" learned. Constructivism in this sense is to be distinguished from an earlier use of the term to describe the intuitionist view that an existential assertion in mathematics (e.g., the existence of a set with certain properties) acquires meaning only by means of an effective construction (Lerman, 1989).

    Two Sets of Epistemological Influences on Mathematics Education At times each, school of epistemology has influenced research on the psychology of mathematics education, as well as classroom practices. Here I mention two such sets of influences. First we consider the impact of logical positivist views, which were partially responsible for the ascendancy of radical behaviorist psychology, and lent support to the "behavioral objectives" approach to mathematics education (Skinner, 1953; Mager, 1962; Sund & Picard, 1972). Then we explore some aspects of the current radical constructivist influence.

    Logical Positivism The idea in psychology that there exist mental states, knowable through direct experience, is

    quite compatible with idealist epistemology: since in the idealist view all reality is mental, there is no fundamental basis for distinguishing between behavior (or, in a more precisely idealist characterization, those mental experiences that are classified as behavior), and mind (or, the full set of mental experiences th...


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