Versatile Learning & the Computer

IntroductionThe activities that are prized in mathematics are usually symbolic and logical. Farless often do we emphasize the visual and holistic. In this paper we discuss the useof the computer to encourage a more versatile approach to learning involvingboth types of mental activity. Empirical evidence is drawn from three studies: therelationship between equations and the graphs of straight lines [Blackett 1987],the introduction of the gradient of a more general function in the initial stages ofthe calculus [Tall 1986a] and the introduction of algebraic symbolism [Thomas1988]. In each case we find that traditional approaches lead to a narrow symbolicinterpretation, yet the use of the computer gives a visual framework for themental manipulation of higher order concepts.Many researchers have identified two distinct learning strategies (see, forexample, Brumby 1982). The first style has been labelled analytic or serialist,whilst the second has been called global, holist or intuitive. The essentialcharacteristics of the two learning styles have been listed as:(i) Immediately breaking a problem or task into its component parts, andstudying them step by step, as discrete entities, in isolation from eachother and their surroundings.(ii) An overall view, or seeing the topic/task as a whole, integrating andrelating its various subcomponents, and seeing them in the context oftheir surroundings. [Brumby 1982, p.244]Brumby’s observations suggested three distinct groups of students: those whoconsistently used only serialist/analytic strategies, those who used onlyglobal/holistic strategies, and those who used a combination of both, whom shedescribed as versatile learners. Overall 42% of her sample maintained aserialist/analytic style, 8% were global/holistic and 50% were versatile.Much mathematical instruction tends to be serialistic in style, so that, inisolated learning activities, serialistic thinkers may not seem to be disadvantaged.However, versatile learners are more likely to be successful in mathematics at thehigher levels where the ability to switch one’s viewpoint of a problem from alocal analytical one to a global one, in order to be able to place the details as partof a structured whole, is of vital importance: