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Jelena Beban Brkic1, Marija Simic Horvath2
1Department for Mathematics and Physics, Faculty of Geodesy, University of Zagreb, Croatia
2Faculty of Architecture, University of Zagreb, Croatia
Here we deal with different approaches in solving some math/geometry tasks. Two simple examples are presented: distance from a point to a line and distance from a point to a plane. We list some of the approaches:
One approach supposes the ability to find the derivative of a function of one and several variables, and a necessary and sufficient condition for extrema; another approach is based on the Lemma: the extreme distance of a given point from a curve/plane is realized on a normal of a curve/plane which runs through the given point. Besides, one needs to be acquainted with the equation of a line in parametric form; a third way supposes the knowledge of a scalar product and a definition of a projection of a vector to the axis determined by the vector; tasks without using formulas can be solved constructively, using methods of descriptive geometry, etc.
Students of the fourth grade of high school should be able to use each of the mentioned approaches to solve the distance from a point to a line. This applies also to students in the first year of any technical faculty in solving the distance from a point to a plane. However, we know from experience that it is not so. It is partly due to the reason some approaches to solving problems are tightly connected to the areas of mathematics which are dealt with in certain classes in high school, or some courses at the faculties. If there was more time perhaps we could use more of a “horizontal” approach to problems and thus overcome this situation. Some ideas in this regard appeared earlier in [1] and [2].
But, in view of the present curriculum we see a goal could be reached involving project based learning. Demand for different approaches to solving some math/geometry task would shift students from mere use of prepared recipes to understanding the principles of solving the problem by applying a broader idea. As project tasks, various topics could be processed in such a way. In order to help students consider such solutions, we prepared posters where different approaches are presented. The first such poster was presented at the 16th International Conference on Geometry and Graphics held in Innsbruck, Austria, in 2014. As we then announced we have continued our work the result of which we would like to shown at the 18th SEFI MWG Seminar.
[1] Beban-Brkić, J. and Jovičić, D.: Distance From Point to Line and From Point to Plane. MIŠ 42, 2007, 61–68.
[2] Beban-Brkić, J. and Šimić Horvath, M.: Interaction Among Courses, Proceedings of the 16th International Conference on Geometry and Graphics, 2014, 690-696.
[3] Divjak, B. et al.: Learning Outcomes in Higher Education. TIVA Varaždin, Faculty of Organization and Informatics, Varaždin, Croatia, 2008.
Jelena Beban Brkic, Ph.D. math, http://www.geof.unizg.hr/
Marija Simic Horvath, Ph.D. math, http://www.arhitekt.unizg.hr/