The role of visualization in the teaching and learning of multivariate calculus and systems of ordinary differential equations
The purpose of this study was to investigate the role of visualization in the conceptualisation and solution of problems in multivariate calculus and dynamical systems. The theoretical basis, and the visual and analytical aspects of evaluating multiple integrals, and the stability analysis of dynamical systems, were established. To address the research questions, a teaching experiment with activities to facilitate visualization of 3D objects and phase portraits of non-linear dynamic systems was conducted with an experimental class (n = 24) which received six activity sessions in the computer Laboratory in addition to traditional lectures. The control class (n = 26) received traditional lectures and tutorial instruction. Both groups were lectured by the researcher using the same set of class notes, assignments, worksheets and tutorials. Additional support materials were posted on the Blackboard on Web-City. The activities included tasks in the computer laboratory that reinforced visualization and spatial ability factors such as surface features, nets, projections, cross-sections and rotation of 3D objects as well as phase portraits of systems of differential equations. The students were tested at several time points, and over both the short and long term to assess the impact on their visual and analytical solutions to problems in the two study domains. The pre-test on prior knowledge indicated no significant differences between the means of the experimental and control groups. Results indicate that there were no significant differences between the achievement of the two groups in Test 1 and Test 2 while the activities were ongoing, but towards the end of the semester significant differences in favour of the experimental group were recorded. A multiple linear regression analysis confirmed that in addition to prior knowledge as measured by the pretest, two of the spatial factors were significant predictors of achievement for the domains under investigation. Students had difficulties in visualising 3D regions of integration and in switching the order of triple integrals. Very few (18%) recognised the need for split integrals to span the required area or volume. While students could find analytical solutions to systems of differential equations and describe the stability of individual equilibrium points using eigenvalues, they struggled with translating rates of change into slopes on the phase portraits, with the interpretation of the solutions and in describing the global behaviour of the system. Students had difficulties in visualizing the region of integration in R³, the stability of equilibrium points in the phase portraits, and in coordinating the treatments and conversions between the geometric, numerical, symbolic and algebraic registers. The tendency to work in the algebraic register to determine the limits of the integral was noted, and students opted to use analytic methods in conducting a stability analysis of the given dynamic system rather than the geometric method. This study adds to research on visualization in mathematics by examining how exposure to technologically enhanced representations complement and promote the conceptualisation of solutions to problems involving multiple integrals and systems of differential equations.