Course Description: The main focus of this class is on learning to do, and actually doing mathematics, specifically geometry. Here I distinguish doing mathematics from doing mathematics problems (getting answers to pre-specified problems is something that a professional mathematician, academic or industrial, rarely does). I also distinguish doing mathematics from learning mathematics content (in the sense of having a certain list of concepts that you should master by the end of the course). So what do I mean by “doing mathematics?” We will explore geometric structures, identify the most interesting ones and give them names, develop our intuitions and abilities to argue about their properties and relationships, then look for what implications they have and ways in which they can be generalized. These are all things that a “mathematician” would recognize as being part of their daily work. Consequently, I will spend very little time lecturing or telling you about content. Instead, you will work in groups, argue, present ideas to each other, and hold each other accountable to rigorous standards of reasoning. My role will be to help you become better at all of these things by listening to what you say and helping to steer your investigation. Roughly, we will organize this exploration around material and questions presented in the class text. On a regular basis, we will also take classroom time and use homework to explore additional topics related to the content from the text. Notably, the text makes several references to differential geometry, and we will regularly explore the meanings behind these statements using mathematics covered in multi-variable calculus, such as MAT 272. Depending on interest, additional topics may include definitions of volume, area and length, the Banach-Tarski paradox, Euler’s formula (V-E+F=2) and it’s connection to curvature and vector fields, mappings of the earth and Thale's measurement of the earth's circumference, Foucault pendulums, astronomical coordinate systems, special and general relativity, the shape of SO₃ and other interesting 3-dimensional spaces, and 4-dimensional spaces.

Class Participation: I expect everyone in the class to contribute constructively to the class and group discussions. This can take several forms from clearly articulating points of confusion to uncovering problems with previous lines of reasoning to providing key ideas and breakthroughs. If your entire group is stuck and/or confused (there is something wrong if this doesn’t happen on a regular basis), you should not give up and wait for me to come around, but you should be resourceful in trying to find new ways to attack the problem. Also, you should look for ways to draw other students into the conversation – since much of what you need to learn is how to listen and evaluate, you are not participating fully if you are doing all of the talking.

Homework:  Written homework assignments will be based directly on classroom discussion questions covered each week and collected the following week at the beginning of class on Wednesday. In general only a few questions will be assigned, but they will require significant depth of discussion. In order to earn an A or B for the course, I anticipate that most people will need to spend approximately six to eight hours per week outside of class on homework, studying, and writing papers (see “Proof and Argumentation” below). For each homework assignment, you should place a star or asterisk next to the solution which you feel represents your best work, and while I may grade any of your responses, over the semester I will generally focus on the ones you mark. Getting a good grade on an assignment is mostly independent of solving the problems. A correct solution that is unclear or provides little understanding will not receive credit. A good discussion of the issues with strong reasoning, without finding a solution may often receive full credit. Remember FLT! I will drop your lowest three homework grades, and late homework will not be accepted for any reason.

Proof and Argumentation:  As described above, this course is aimed at developing your skills of mathematical inquiry. Throughout the semester, you must pick episodes from four different topics during which the class or your group was trying to generate a proof. You will write up a description of how the thinking progressed regarding that proof, describing fully any difficulties or misunderstandings encountered, how they were overcome, definitions created or revised, and key insights. Also describe how you personally contributed to the progress of the class or group discussion during this episode. Finally, include a revised exposition of the proof improving on its rigor, depth, and clarity. A first draft of each of the four papers must be submitted to me within two weeks of the featured episode. After you get my comments back, you will rewrite the paper. (Note that since my turnaround time will increase during the semester, if you wait too long or submit several at once, you will have little to no opportunity to produce a revision.)

Midterm and Final:  The midterm exam will be taken on a non-class day at the testing center (location and instructions will be provided) and the final will be given at the time and location appointed by the university. Both will assess your understandings of the mathematical content covered in the course. Before each exam, I will provide an overview of what will be covered.

Graduate Credit: Students enrolled in MTE 585 will complete an additional project. This may consist of an in-depth paper on some area of geometry not covered in the class or a small teaching experiment with some classroom materials that you develop with a subsequent write-up. Project proposals must be approved by me no later than Wednesday, February 8.