This Small Business Innovation Research (SBIR) Phase II project will develop an interactive symbolic geometry system that integrates algebra and geometry and focuses on high school mathematics. The absence of such a system has led to a technology gap in mathematics education between the geometry year in high school and the college level calculus sequence. The result of this project will be a software system along with learning materials which fills that gap. The National Council of Teachers of Mathematics (NCTM) standards include the visualization of three-dimensional figures and the mapping between certain three-dimensional surfaces and their two-dimensional unfolding or projection onto the plane. To address this, the project will create a three dimensional symbolic geometry system and in the process will break new ground both from an algorithmic and a user interface perspective. The creation of geometric models dependent on discrete but possibly indeterminate parameters, for example, a general n-gon, is an important pedagogic device for the study of the limits of geometrical figures. Such a facility poses new design and user interface challenges ranging from the definition of the general form of the dependence to the display of a geometrical figure with an indeterminate number of primitives.
This Phase II project addresses the need for solid mathematics skills required for college-bound students and for those going directly into the workforce. Specifically, this project focuses on the learning of algebra, and its linkages with geometry. To date, no application exists that integrates algebra and geometry. The integration of technology itself within the learning of mathematics is one of the NCTM's six key principles of school mathematics. The project will incorporate geometrical constraints in addition to geometrical constructions and hence, unlike any other current educational system, directly address the workforce/professional requirements of a geometry system.
An outcome of this NSF project is software, for use in schools, which can take a geometric figure and output algebra. The use of such software lies in teaching mathematical modeling: the process whereby real life phenomena are represented as mathematics. Along with the software, we developed a range of books containing lessons and activities for use in the classroom. A variety of real-life phenomena can be represented geometrically: from the geometric optics of reflected light to the design of parabolic solar cookers to the placement of central pivot irrigation systems. The software developed in this project allows the student to model such situations, then creates mathematical expressions for measured quantities from their model. These mathematical expressions may be copied into a Computer Algebra System or an algebra enabled calculator for solution. This process is different from the way existing modeling tools work. In such tools (used by engineers), model formulation and solution is presented as a single operation. At no stage is an explicit mathematical model accessible. The value of the explicit model is that it allows students to apply and reinforce techniques which they are learning in the classroom to real-life problems. The range of phenomena which can be modeled with the software was extended in the latter stages of the grant to include basic mechanics, by the addition of mass, force, velocity and acceleration elements to the model. Modern algebra calculators can perform most of the symbolic manipulations we teach students in the algebra class through the calculus class. Were we to wholeheartedly adopt such technology, what would we teach instead? One answer to this question would be to de-emphasize problem solution and instead to emphasize problem formulation. Our software would help.
