Conceptual Knowledge in Introductory Calculus

ION AND CONCEPTUAL KNOWLEDGE A feature of all advanced mathematics is the need for abstract concepts, that is, concepts formed by the process of abstracting. "Abstracting" is used here in the sense of Skemp (1986), who describes it as the process of identifying certain invariant properties in a set of varying inputs. The act of abstracting is based on generalizing these properties to other inputs, but it is seen as qualitatively different from simply identifying patterns in a set of examples. It is a many-to-one function where generalizations are synthesized from many inputs to form a new abstraction. Dreyfus (1991) summarizes the process as a sequence of generalizing -synthesizing -abstracting. Sfard (1991, 1992) and Dubinsky (1991) have further highlighted the qualitative difference between generalizations and abstractions, pointing out that abstracting is in effect a move to a higher cognitive plane. Sfard (1991, 1992) describes a threephase model similar to the sequence of Dreyfus: interiorization --condensation --> reification. The interiorization phase occurs when some process is performed on familiar mathematical objects. The condensation phase occurs when the process is condensed into more manageable units. Both of these phases are said to be operational because they are process-oriented. Reification is the leap from an operational mode to a structural mode where a process becomes an object in its own right. Dubinsky (1991) also describes this leap from dynamic process to static object as a form of reflective abstraction; he calls the leap "encapsulation." Kieran (1992) has applied Sfard's model to the learning of algebra. Initially, students operate with numbers; patterns in arithmetic are then interiorized or generalized and eventually give way to algebraic expressions that become objects in their own right. Tall (199 1b) notes that 2 + 3x can be considered as the process of adding 2 to the product of 3 and x and also as the object that is the result of the process. He also gives the example (cited in Kieran, 1992) that an operational (process) orientation results in 2(a + b) being seen as quite different from the resultant object 2a + 2b. Tall (1991b) argues that until algebraic expressions can be conceived as mathematical objects as well as processes on objects, algebraic manipulation can be a source of conflict. Gray and Tall (1994) cite examples in a wide range of mathematics where the same symbols are used to represent both a mathematical process and the resultant mathematical object. The amalgam of process, resultant object, and common symbol used to represent both is defined as an elementary "procept." They hypothesize that successful mathematical thinkers can think proceptually; that is, they can comfortably deal with symbols as either process or object. Once an abstraction has occurred, the generalizing -synthesizing -abstracting sequence can be repeated. The sequence certainly occurs in concept formation at all levels in mathematics, but it is a feature of advanced concepts that they are Paul White and Michael Mitchelmore 81 based on several repetitions. Each repetition has led to a higher order of abstraction and a further removal from what Skemp (1986) calls "primary concepts," that is, those that are formed by direct experience. Conceptual knowledge in mathematics has been characterized by Hiebert and Lefevre (1986) as relationships between mathematical objects and hence appears to be similar to what Skemp (1976) calls "relational understanding." Hiebert and Lefevre also make a key distinction between relationships that are constructed at the same level of abstraction as the constituent concepts and therefore do not involve an increase in abstraction, and those "reflective relationships" that are constructed at a higher level. They use the term "abstract" to refer to the degree to which a relationship is freed from specific contexts. Because advanced mathematical concepts are the result of several abstraction sequences, the network of relationships among concepts can be extremely complex. Hiebert and Lefevre describe procedural knowledge as knowledge of standard learned procedures that can be applied when some type of recognizable cue is presented. A key word for such procedures is "after" in the sense of "after this step comes the next step" (Hiebert & Lefevre, 1986, p. 8). Procedural knowledge may or may not be supported by conceptual knowledge. Unsupported procedural knowledge is similar to Skemp's (1976) "instrumental understanding," which he describes as knowing rules without knowing why they work. Steen (1988) and Cipra (1988) argue that skills-based calculus courses result in rote, manipulative learning. The result is instrumental understanding or unsupported procedural knowledge. As Skemp points out, skills-based courses are very efficient if the only criterion is the ability to perform routine manipulations. It is the application problems that appear to call on conceptual knowledge, and it is clearly the hope of those who design concept-based calculus instruction that students' ability to solve such problems will be improved as a result. APPLICATIONS, MODELING, AND VARIABLES In calculus, the context of an application problem may be a realistic or artificial "real-world" situation, or it may be an abstract, mathematical context at a lower level of abstraction than the calculus concept that is to be applied. We shall restrict ourselves here to problems that can be solved using algebra and symbolic calculus. In solving such problems, the given situation is first translated from the context to the abstract level of the calculus, the abstract problem is then solved, and the solution is finally translated back to the context (Tall, 1991 la). It is the first step that most obviously calls on conceptual knowledge because it depends on the identification not only of the appropriate concepts in the given context but also of the relationships among them. (The abstract problem might well be solvable using procedural knowledge alone, and the back-translation would probably require only the same understandings developed in the first step.) The identification of appropriate concepts might involve the selection of one or more symbolized variables from among several presented, or it might require the solver to define one or more new variables. 82 Introductory Calculus The identification of relationships requires the establishment of some algebraic relations among the variables or the selection of some calculus concept involving the variables (such as a derivative) and its expression in symbolic form. Only then can known manipulations be carried out. For the purpose of this article, we shall refer to the definition of new variables and the symbolic expression of relations between variables as (algebraic) modeling. The selection of a calculus concept and its expression in symbolic form we shall call symbolization. Modeling and symbolization together constitute translation. Translation as defined above encompasses both algebraic and calculus concepts. In particular, the use of symbols to represent changing quantities is crucial. The fact that students find this difficult in calculus contexts is supported by the work of Frid (1992). She showed that although students are able to manipulate symbols and perform operations with symbols when doing calculus problems, they do not generally use symbols to represent concepts. No other research on students' use of symbols in calculus has been found, so we have to rely on algebra research to guide our expectations. Kiichemann (1981) suggests that very few students understand variables at his highest conceptual level. Eisenberg (1991) agrees and in support quotes the research of Wagner (1981), who showed that 15% of 16-year-old students treated two equations as totally different when the only difference was the letter used to represent the variable. Such a superficial understanding of variables is in line with Kieran' s (1989) view that one of the main difficulties in learning algebra centers on accommodating the meaning of the letters involved. Booth (1989) suggests that the required meaning is often neglected in the teaching and learning of algebra, so that many students only learn manipulation rules without reference to the meaning of the expressions being manipulated. It is a matter of some interest to find out whether students who aspire to the advanced mathematical thinking involved in calculus have an adequate concept of a variable.