Student Experiences with Visualization of Abstract Entities in the Learning of Multivariable Calculus
ABSTRACTThis paper focuses on undergraduate students taking Calculus course in a BEd degree programme offered at Great Zimbabwe University. The paper analyses student conceptions and misconceptions on the graph of surfaces. The results indicated that students conceptualise better through use of visuals and also that technology plays a key role in enhancing better understanding of calculus concepts.Keywords: Visualisation, technology and surfacesINTRODUCTION:Students have interacted with drawing graphs of functions as early as primary level. Graphs sometimes are very easy to conceptualise the characteristics of a model in problem solving than explanations. Practices in mathematics problem solving are often based on verbal representations that make use of logical connectives in sequential reasoning. Recent research in mathematics teaching (Diezmann, 1997) however has advocated the use diagrammatic explanation to assist comprehension. In this study the researcher explores students' conceptions and misconceptions on the graphs of surfaces by undergraduate students taking the course of Calculus of several variables. The objective of the research is to reinforce students 'conceptions of graphs on single variable Calculus transferring such concepts to the graphs of surfaces. For example in single variable Calculus students were able to draw the sketch of y=sinx being able evaluate its domain and range. Are students going to use the same analytic methods for the graph of z=sinxy What properties are preserved by such functions in space geometry? However the main question is: what conceptions and misconceptions of the graphs of surfaces do students develop in a course which emphasises visualization uses of technology, and deemphasises symbolic manipulation?THEORETICAL FRAMEWORK:Stewart J.(1999) asserted that the primary aim of Calculus instruction is the emphasis on understanding concepts. In fact the impetus for the current Calculus reform movement came from the Tulane Conference in 1986, which formulated as their first recommendation focus on conceptual understanding. He also went on to emphasise that the most important way to foster conceptual understanding is through the problems we assign. In this study the researcher also used the Cognitive Dissonance theory (L.Festinger,1957) and Cognitive Flexibility theory (R.Spiro et al,1990) Dissonance theory applies to all situations involving attitude formation and change. It is especially relevant to decision making and problem solving. When there is an inconsistency between attitudes (dissonance) something must change to eliminate the dissonance. This theory shall be useful to students when constructing graphs of functions in space Some properties of functions of single variable are going to be inconsistent with functions of several variables, for example, the criteria for evaluating critical points. Flexibility theory is especially formulated to support the use of interactive technology. Its main principle is that learning activities must provide multiple representations of content. This will be applied when Computer Algebra Systems (Mathematica) is going to be used to draw graphs and level curves of functions of several variables. Students should then associate functions with their graphs and level curves creating a chain of symbolic representation, plane representation and space representation.VISUALIZATION: Is a picture worth a thousand words? It seems so ,as historical accounts of scientific discovery and invention have shown that visualisation is a powerful cognitive tool (Rieber,1995) The term visualisation is familiar to us from common usage and fundamentally means "to form and manipulate a mental image" In everyday life visualisation is essential to problem solving and spatial reasoning as it enables people to use concrete means to grapple with abstract images. The process of visualisation involves the formation of images, with paper and pencil or even mentally to investigate discover and understand concepts ,facts and ideas.