Procedural and conceptual aspects of standard algorithms in calculus

This research studies the different methods students use to carry outalgorithms for differentiation and integration. Following Krutetskii, it mightbe conjectured that the higher attainers produce curtailed solutions giving theanswer in a smaller number of steps. However, in the population studied(Malaysian students in the 50th to 90th percentile), some higher attainingstudents wrote out solutions in great detail, so little correlation was foundbetween the attainment of students and the number of steps taken. On the otherhand, the higher attainers had less fragile knowledge structures and weresignificantly more likely to succeed. But with problems that can be simplifiedby a non-algorithmic manipulation before using a standard algorithm, thehigher attainers were more likely to use some form of conceptual preparation.IntroductionIn his renowned study of the different problem-solving styles of children, Krutetskii(1976) showed that, of four groups (gifted, capable, average, incapable), the gifted werelikely to curtail solutions to solve them in a small number of powerful steps, whilst thecapable and average were more likely to learn to curtail solutions only after considerablepractice, and the incapable were likely to fail. This may be related to the strength of theconceptual links formed by the more successful students in their cognitive structure(Hiebert and Lefevre, 1986) which helps the individual utilise knowledge in an efficientand powerful way.The brain is a huge simultaneous processing system that must filter out most of itsactivity to be able to focus attention on a small amount of data for decision making(Crick, 1994, p. 61). Therefore the ability to code information efficiently—to makeappropriate links between concepts and to develop methods that economise onprocesses—is likely to increase the brain’s capacity to perform mathematics.Davis (1983) suggested that at least two kinds of procedures exist: a visually moderatedsequence (VMS) and an integrated sequence. In a VMS, the whole sequence is not yetapparent and the student carries out a manipulation to produce new written informationwhich is then operated on in turn until the problem is solved. In an integrated sequence,the student is aware of the whole algorithm built up from smaller component sequences.Hiebert and Lefevre et al (1986) contrasted procedural and conceptual methods ofprocessing mathematical information. Following Dubinsky (1991) and Sfard (1991),who focused on the way in which process becomes encapsulated (or reified) as mentalobject, Gray & Tall (1991, 1994) introduced the notion of procept: the amalgam of