Common Visual Representations as a Source for Misconceptions of Preservice Teachers in a Geometry Connection Course

In this paper, we demonstrate how atypical visual representations of a triangle, square or a parallelogram may hinder students’ understanding of a median and altitude. We analyze responses and reasoning given by 16 preservice middle school teachers in a Geometry Connection class. Particularly, the data were garnered from three specific questions posed on a cumulative final exam, which focused on computing and comparing areas of parallelograms, and triangles represented by atypical images. We use the notions of concept image and concept definition as our theoretical framework for an analysis of the students’ responses. Our findings have implication on how typical images can impact students’ cognitive process and their concept image. We provide a number of suggestions that can foster conceptualization of the notions of median and altitude in a triangle that can be realized in an enacted lesson

Drawing Area-Proportional Euler Diagrams Representing Up To Three Sets

Area-proportional Euler diagrams representing three sets are commonly used to visualize the results of medical experiments, business data, and information from other applications where statistical results are best shown using interlinking curves. Currently, there is no tool that will reliably visualize exact area-proportional diagrams for up to three sets. Limited success, in terms of diagram accuracy, has been achieved for a small number of cases, such as Venn-2 and Venn-3 where all intersections between the sets must be represented. Euler diagrams do not have to include all intersections and so permit the visualization of cases where some intersections have a zero value. This paper describes a general, implemented, method for visualizing all 40 Euler-3 diagrams in an area-proportional manner. We provide techniques for generating the curves with circles and convex polygons, analyze the drawability of data with these shapes, and give a mechanism for deciding whether such data can be drawn with circles. For the cases where non-convex curves are necessary, our method draws an appropriate diagram using non-convex polygons. Thus, we are now always able to automatically visualize data for up to three sets

Capstone project : Zometool : from 0 to the 4th dimension

Abstract: My capstone project was the development of a geometric construction workshop at Mckinnon Elementary School in Salinas using Zometool, a manipulative whose components are based on the Fibonacci Sequence and the Golden Ratio . By building the Platonic Solids like the cube, icosahedron, and dodecahedron as well as the Archimedean solids such as the truncated cube and truncated octahedron. the students were able to begin to think componentially and see that an individual solid shape may be broken down into several smaller components with attributes of their own.Besides learning to think in two and three dimensions, this workshop also introduced my students to 4 dimensional thinking by building such structures as the hypercube (8 cubes) and the hyperdodecahedron (120 dodecahedrons) which were three dimensional projections of four dimensional objects. I have generated enough interest in this workshop that I will be continuing what I have started for the rest of this year and hopefully for many years to come