In recent years, there has been a growth of using visuals and vocabulary in mathematics and mathematics classes. The purpose of this Master’s Report is to illuminate research done in the realm of mathematics education related to the increasing use of visuals and visual devices as models for mathematical concepts, as well as visuals for quick reference or “short cuts.” Also discussed is mathematics vocabulary, the words most likely seen on mathematics exams, standardized state tests, and overall, any vocabulary most likely to trigger problem solving strategies and solutions. Trends such as “word walls” and “graphic organizers,” as well as vocabulary strategies aimed at oral, visual, and kinesthetic learners have all emerged in the classroom. Other strategies implemented and researched include student mathematics journals, student created mathematics dictionaries, children’s literature, graphic organizers, and written explanations of open ended word problems. All proved to enhance students’ mathematical vocabulary, increase comprehension and increase ability in communication of mathematical ideas. Furthermore, the use of visual models has emerged in mathematics courses in order to promote more mathematical understanding. “Proofs without words” and patterns and pictures are growing in their use to explain mathematical concepts and ideas. Visual devices that help students arrive at probable answers also have grown in their implementation in the classroom and beyond. Overall, has the increased use of visuals and vocabulary in both mathematics education and mathematics in general improved the mathematical understanding of our society? What research, if any, has been done to document the effects of word walls, graphic organizers, and etcetera? The research will show that, yes, an overwhelming amount of data shows that the implementation of such visual and vocabulary strategies can improve the mathematical understanding of those exposed to the strategies and devices. / text

There are three main results in this dissertation. The first result is the construction of an abstract visual space for rational numbers mod1, based on the visual primitives, colour, and rational radial direction. Mathematics is performed in this visual notation by defining increasingly refined visual objects from these primitives. In particular, the existence of the Farey tree enumeration of rational numbers mod1 is identified in the texture of a two-dimensional animation. ¶ The second result is a new enumeration of the rational numbers mod1, obtained, and expressed, in abstract visual space, as the visual object coset waves of coset fans on the torus. Its geometry is shown to encode a countably infinite tree structure, whose branches are cosets, nZ+m, where n, m (and k) are integers. These cosets are in geometrical 1-1 correspondence with sequences kn+m, (of denominators) of rational numbers, and with visual subobjects of the torus called coset fans. ¶ The third result is an enumeration in time of the visual hierarchy of the discrete buds of the Mandelbrot boundary by coset waves of coset fans. It is constructed by embedding the circular Farey tree geometrically into the empty internal region of the Mandelbrot set. In particular, coset fans attached to points of the (internal) binary tree index countably infinite sequences of buds on the (external) Mandelbrot boundary.