Abstraction in mathematics learning

Abstract

ion has been a frequent discussion topic since the days of Aristotle and Plato. Constructivist theories often espouse the notion of levels of abstraction. Hiebert and Lefevre (1986, pp. 4-5) put it this way: It is useful to distinguish between two levels at which relationships between pieces of mathematical knowledge can be established. One level we will call primary. At this level the relationship connecting the information is constructed at the same level of abstractness (or at a less abstract level) than that at which the information itself is represented. That is, the relationship is no more abstract than the information it is connecting. Some relationships are constructed at a higher, more abstract level than the pieces of information they connect. We call this the reflective level. Relationships at this level are less tied to specific contexts. They often are created by recognising similar core features in pieces of information that are superficially different. The relationships transcend the level at which the knowledge currently is represented, pull out the common features of different- looking pieces of knowledge, and tie them together. Of particular interest are the assertions that relationships at the "higher, more abstract level" are "created by recognising similar core features" and "are less tied to specific contexts". These two features are often referred to as the hierarchical and decontextualisation views of abstraction. Why, then, did Hiebert and Carpenter (1992) write a whole chapter on learning mathematics with understanding in a major treatise on mathematics education without even mentioning abstraction? They defined understanding as "making connections between ideas, facts, or procedures" (p. 67) and discussed two methods for doing this: investigating similarities and differences, and establishing inclusion relationships — both of which are crucial to learning by abstraction. Clearly they had the notion of abstraction in mind, but did not dare speak its name. Carpenter (personal communication, May 1999) informed us that the term "brought too much baggage with it." Abstraction had gained a bad reputation (in some circles) because of the criticisms expressed by the situated cognition movement. According to this movement, "the primary concern of schools often seems to be the transfer of ... abstract, decontextualized concepts" (Brown, Collins, & Duguid, 1989, p. 32), based on the theory that "knowledge acquired in 'context-free' circumstances is supposed to be available for general application in all contexts" (Lave, 1988, p. 9). However, "much of what is taught turns out to be almost useless in practice"