Geometry and algebra in multidimensional analysis: representations of curves and surfaces
Matija Bašić and Željka Milin Šipuš, University of Zagreb, Croatia
Different representations of curves and surfaces and their flexible conversions is often assumed to be a prerequisite for various mathematical courses in university mathematics education that involve multivariable calculus. They are essential, for instance, in determining tangent lines or planes, calculating curve integrals, applying Green-Stokes theorem, or in introductory lessons in differential geometry when manipulating with basic objects. In university mathematics education, requests for efficiency and economy in teaching, with high speed of introduction of new objects, frequently result in disconnected teaching modules or gaps between them. In our teaching practice we have noticed students’ difficulties regarding representations of curves and surfaces that may be associated with such requests. Regarding straight objects in space (straight lines and planes), students’ difficulties in using their algebraic representation and in recognizing the objects from their equations, in particular, converting between parametric and implicit points of view, are well evidenced. We have addressed the non-linear case, where the conversions are supported by the Implicit and Inverse function theorems. However, these theorems provide conditions under which conversions are locally possible and do not give any explicit procedure for obtaining them. In addition, the local character of non-linear problems requires students’ subtle reconstruction of possible procedures of “eliminating a parameter” or “substituting a variable by a parameter” used in the linear problems. The theoretical framework for our study comes from TDS, Theory of Didactic Situations. We have identified the impact of the didactic contract on students’ productions, and we aim to utilize the framework further to design tasks with linking and deepening potentials that could support students’ autonomous building of more comprehensive and coherent knowledge on the topic.
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This seminar is held online, on Zoom: https://ucph-ku.zoom.us/j/66967901059