M333L: Structure of Modern Geometry

M33L is designed to familiarize students with some topics primarily from two dimensional geometry. Topics include axiomatic systems, Euclidean geometry, transformations, and an introduction to non-Euclidean geometries. The course is designed to develop an understanding of the concepts related to plane geometry and the ability to use those concepts in proving theorems. Sections taught in the computer lab in the pharmacy building also include an introduction to the computer program, Geometer's Sketchpad. This program is used as an investigative tool to help students gain intuition and to arrive at conjectures and counterexamples. The ideas obtained from the investigations are then formalized into proofs.

 This course includes a substantial proof component, and therefore addresses the Logical Reasoning portion of SBEC Standard V: Mathematical Processes:

This course also addresses the first part of item 6.7s of SBEC Standard VI. Mathematical Perspectives:

• " The beginning teacher of mathematics is able to … analyze the structure of mathematical systems and use the structural properties of mathematical systems … "

 M333L also addresses SBEC Standard III. Geometry and Measurement:

These courses can help prepare the teacher to teach according to Basic Understanding (2) of the TEKS for Geometry:

• (2) Geometric thinking and spatial reasoning. "Spatial reasoning plays a critical role in geometry; shapes and figures provide powerful ways to represent mathematical situations and to express generalizations about space and spatial relationships. Students use geometric thinking to understand mathematical concepts and the relationships among them."
• (5) Tools for geometric thinking. Techniques for working with spatial figures and their properties are essential in understanding underlying relationships. Students use a variety of representations (concrete, pictorial, algebraic, and coordinate), tools, and technology, including, but not limited to, (powerful and accessible hand-held calculators and computers with graphing capabilities to solve meaningful problems by representing figures, transforming figures, analyzing relationships, and proving things about them.
• (6) Underlying mathematical processes. Many processes underlie all content areas in mathematics. As they do mathematics, students continually use problem-solving, computation in problem-solving contexts, language and communication, connections within and outside mathematics, and reasoning, as well as multiple representations, applications and modeling, and justification and proof.

This course can help prepare you to teach according to the 9-12 Geometry standards:

• analyze properties and determine attributes of two- and three-dimensional objects;
• explore relationships (including congruence and similarity) among classes of two- and three-dimensional geometric objects, make and test conjectures about them, and solve problems involving them;
• establish the validity of geometric conjectures using deduction, prove theorems, and critique arguments made by others;
• understand and represent translations, reflections, rotations, and dilations of objects in the plane by using sketches, coordinates, vectors, function notation, and matrices;
• use various representations to help understand the effects of simple transformations and their compositions.

Most importantly, this course can help you develop your skills in reasoning and proof so that you can teach according to the 9-12 Standard for Reasoning and Proof.

This course usually includes a project as a part of the course. The project requires the student to independently research an area of geometry and to relate geometry to other topics of interest to the student. The projects are presented in class, giving students the opportunity to communicate ideas of a mathematical nature to others. Click here to see a sample course syllabus, from which you can download a bibliography for possible project topics. The Miscellaneous Math Links page of this website also has several links to geometry websites that can give you ideas for projects in this course an in teaching geometry.