|A&JS, June 1998|
This central idea is applies to learning generally and is not limited to any particular setting such as problem solving or listening to lectures. "Knowledge" refers to the contents of an individual mind, not to abstract or public knowledge. "Constructed" refers to the result of mental activity, such as reflecting on one's work or perceptions. It does not necessarily include the physical or abstract aspects of construction, such as assembling a jigsaw puzzle, constructing the rational numbers from the integers, or making a proof, although doing or discussing these might engender mental constructions.
New knowledge is seen as constructed out of older knowledge, where "knowledge" refers to more than just remembered "facts," e.g., the area of a triangle is one half its base times its height or the Pythagorean theorem. In addition, it refers to the largely tacit mental structures that allow one to interpret the meaning of words, bring ideas to mind, or effectively explore a new mathematical field. Below we give two simple illustrations of knowledge construction. (There are several views of knowledge and how it is constructed; however, the Piagetian and the Vygotskian theories are too complex to include here.)
Illustration 1. A beginning topology student's knowledge about continuous functions can be viewed as a vast web of ideas, images, and personal experiences, some based on mathematical relationships and some not. The word "continuous" may bring to mind the image of a sine wave or of a teacher saying "you can draw it without lifting your pencil from the paper," which, in turn, might bring to mind other associations, e.g., "but to prove it, you need to use an epsilon-delta argument." Consider now how the student might comprehend the proposition that the continuous image of a connected set is connected and develop it into lasting, usable knowledge. Suppose the proposition is presented by a teacher together with a couple of examples. If the student merely passively observes the presentation, he/she might be able to repeat some of it for awhile, but later the proposition might not come to mind when needed. However, as the student works with the proposition - perhaps examining its proof, applying it, even discussing its significance - he/she might come to associate it with many ideas and images, including aspects of his/her prior knowledge of "continuous" and "connected." Thus, the student "constructs" a mental structure that enhances the probability of the proposition coming to mind when needed. This structure is idiosyncratic and is neither directly observable nor fully communicable.
Illustration 2. Consider an abstract algebra student who is fairly familiar with groups, but has just met the definitions of field and vector space. Suppose the student is searching for finite fields, guided by his knowledge of how to find new algebraic structures -- a mainly tacit understanding based on his experiences with groups. The student's way of searching for algebraic structures is not yet likely to be very effective and may be altered, not necessarily consciously, based on new experiences, perhaps including that fields are also vector spaces which constrains somewhat their possible orders. This alteration can be seen as knowledge construction. Note that it might be more appropriate to evaluate such knowledge, which amounts to a way of searching, as effective or not, rather than true or false.
B. Constructivism in its radical form can be characterized by two additional ideas. (1) The function of cognition is adaptive in the biological sense of tending toward "fit" or viability. (Trees in a forest have more leaves where there is more light and runners have strong legs. Both have adapted to fit a situation.) (2) Cognition organizes the experiential world, but does not allow discovery of objective reality. [Cf. E. von Glasersfeld, "An Exposition of Constructivism: Why Some Like it Radical," in R. B. Davis, C. A. Maher & N. Noddings (Eds.), Constructivist Views on the Teaching and Learning of Mathematics (pp. 19-29). Reston, VA: NCTM.]
This position does not deny the existence of the external world, although it does deny the world can be mirrored objectively by one's own knowledge. It is consistent with one's knowledge appearing to approximately mirror much of the external world, perhaps because of the tendency toward "fit."
Currently much research in mathematics education is guided by, or consistent with, a constructivist perspective. Piaget was perhaps the best known constructivist. He did a great deal of empirical work with children and developed theoretical explanations including a description of the way knowledge construction occurs. Vygotsky developed a view that depends more on social interaction. Ernst von Glasersfeld, is known for his articulation and development of the ideas of radical constructivism.
In the mathematics education community, constructivism appears to have been very influential in legitimizing studies of students' mental processes, such as how they learn to add or develop the concept of function. It has facilitated moving away from behaviorism, with its logical positivist viewpoint, which tended to ignore the internal workings of the mind and thus effectively discouraged research on many interesting mental questions.
Illustration 1. Certain parents dislike a " . . . teaching philosophy called constructivism that says children learn better if they construct something on their own - use a string to measure a circle and 'discover' that the circumference is 3.14 times greater than the diameter . . ." [June Kronholz, "Numbers Racket," The Wall Street Journal, November 5, 1997.]
Illustration 2. Constructivism "apparently urges that students construct their own mathematics. It is not clear where this will lead. Perhaps every student will construct his own algorithm for long division." This doctrine "holds that students must somehow construct in their own minds all (?) the basic notions of mathematics." [Saunders Mac Lane, Letter to the Editor, Focus, February 1998.]
This popular meaning of constructivism is very different from the meaning in the mathematics education research community. According to it, what is constructed (discovered) is mathematics, e.g., that the ratio of the circumference of a circle to its diameter is independent of the circle's size. In contrast to this, according to the constructivism of mathematics education research, what is constructed is some kind of (personal) knowledge, i.e., a structure in an individual mind which may not even be fully describable in words and which might, or might not, arise from discovery of mathematics. Furthermore, the constructivism of mathematics education research does not refer to a particular teaching method, but rather provides a way of viewing learning which can be used when analyzing many kinds of teaching. "Constructivist teaching," if it is used at all in the mathematics education research community, is likely to be an umbrella term referring to teaching that is informed by constructivism, i.e., that takes into consideration the idea that a student uses his/her prior knowledge in mentally constructing new knowledge.
Most popular meanings of constructivism appear to be somewhat similar to that of "discovery learning," which refers to a kind of teaching and learning based very roughly on the idea that students learn well when they discover what is to be learned for themselves. Even in the 1960's, when it was espoused by Harvard psychologist J. S. Bruner and others, it was considered an untested hypothesis. [Cf. L. S. Schulman and E. R. Keislar (Eds.), Learning by Discovery: A Critical Appraisal, Rand McNally, 1966.] Currently, "discovery learning" is neither a major area of interest in mathematics education research nor a driving principle in research-based curriculum reform projects.
Here "knowledge" means public knowledge, not the contents of an individual mind and the ideas of truth and objectivity have been metamorphosed from their traditional meanings into a form of social acceptance. For example, in the traditional view, ones says it true that gold is heavier than aluminum because of the correspondence to the facts of the matter, as established by observation. Whereas in this view, gold is heavier than aluminum because it has been agreed upon by the physics community.
Constructivism is a philosophical position within the sociology of scientific knowledge (SSK) which is a subarea of the history and philosophy of science. However, SSK is often identified with constructivism which is its most conspicuous stance. Most research in mathematics education does not appear to be guided or inspired by this kind of constructivism although there is some interest in a fairly similar philosophy of mathematics. [Paul Ernest, Social Constructivism as a Philosophy of Mathematics, SUNY Press, 1998.] Many scientists and mathematicians hold views of the nature of science and mathematics which appear to be inconsistent with this kind of constructivism and some are quite concerned about its growing influence. [Paul R. Gross, Norman Levitt, and Martin W. Lewis, The Flight from Science and Reason, New York Academy of Sciences, 1996.]
Perhaps the best known constructivist was the Dutch mathematician, L. E. J. Brouwer. There appears to be little connection between the meaning of constructivism in mathematics and its meanings in mathematics education research, the various popular views, or the philosophy of science. Constructivism in mathematics is an absolutist philosophy, i.e., it is concerned with absolute truth, independent of individuals or communities, while the SSK form of constructivism is relativist, i.e., it is concerned, not with truth, but with acceptance, which is dependent on particular communities.
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