Do questions like these intrigue you? Do you think you would enjoy some hands-on experiences with mathematics --- building models, constructing visual representations, drawing pictures, engaging in some computer-based explorations? Are you interested in connections of mathematics to other disciplines --- art, history, literature, theater, ...?
You are invited on a journey of exploration into worlds you may have thought beyond imagining. This course will give you an opportunity to explore visual and spacial aspects of higher-dimensional geometries. The our text for this course, Beyond the Third Dimension by Thomas F. Banchoff, will serve as a jumping-off place for explorations into higher-dimensional spaces.
Each week a new chapter of this text will open up new vistas for our explorations. A standing weekly assignment for this course is to read a chapter of the text and write a short reflection on ideas that are sparked by your reading, and to come to class prepared to engage in class activities which will build on the ideas presented in the chapter. Specific topics to be studied include:
Mathematics is ubiquitous. The development of mathematical ideas has connections to the development of many other disciplines. Paging through our text, you will notice many connections to literature, history, the natural and social sciences, art, and the development of ideas about teaching. The development of perspective drawing helped to bring a realistic depth -- a third dimension -- to the artist's canvas; paintings looked more realistic. The study of perspective drawing led to the development of projective geometry. Thus, developments in the visual arts have served as a springboard for develoments in mathematics. This has happened frequently throughout history. Even today we enjoy the fruits of mathematical understanding in many areas of our lives. For example, an understanding of coordinate geometry has made possible some of the state-of-the-art lighting systems used in stage lighting for theater productions.
In this course, you are being invited to learn something about the mathematical way of thinking. You will be challenged to think about and visualize objects and ideas in new ways. You will learn how to use a variety of mathematical tools: computers and calculators, 2- and 3-dimensional models, visual representations of data. You will learn some coordinate geometry, some algebra, a little combinatorics, some applications of statistics, some non-Euclidean geometry.
This course will help you to grow in your ability to think critically. Again and again, you will be asked to think about how you know what you know, and how you learned it. You will be challenged to communicate ideas both orally and in writing. You will have the opportunity to explore many ideas with others in small groups as you work together on a variety of problems.
Back to TOCA student who chooses this course should have a vivid imagination, an intense curiosity about shapes and spaces, and a willingness to work with others using the mind and resources available in the library and on the World Wide Web to explore ideas that seem beyond all reason.
Back to TOCYou will do much of the work of this course in cooperative learning groups. It seems to work best if there are about three students in each group.
An unusual aspect of this course is that you will often be asked to experiment with an idea before we have discussed it formally in class. That is, you will be asked to experiment with an idea and make some of your own conjectures. Later, you will have an opportunity to try to prove or disprove these conjectures. This will give you opportunities to confront and think critically about problems; that is, you will be learning problem solving by solving problems. Research into how people learn mathematics has shown that students who learn mathematics using these methods have both deeper understanding and longer retention of what they have studied, and are better able to think about new problems than those taught using traditional methods.
The idea is to learn something about how to learn using a wide variety of resources.
Back to TOCComputers are legitimate tools for doing mathematics. In this course, the computer is used primarily as a learning tool, especially for visualization and for acquiring additional information.
There are many resources available on the World Wide Web. You will be expected to use a graphical web browser (such as Netscape or Internet Explorer), email, a word processor and simple spread sheets, and geometric drawing software tools.
Back to TOCAn essential part of a liberal education is the developement of writing and speaking skills for all students across all disciplines. You will have opportunities in every class session to speak with your peers informally in small-group and full-class discussions. You will also have some opportunities to give short presentations to your colleagues on some topic that you have been investigating. Any written work that you turn in -- graded or ungraded homework assignments, tests, papers, and the course project -- is to be written in standard English using complete sentences. Throughout the semester, you will receive feedback from the instructor (and from your peers) which is intended to help you to develop good speaking and writing skills.
Back to TOCRegular attendance is expected. Mathematics is not a "spectator sport". I expect you to be active, constructive participants in the learning process. During most class periods there will be time for group discussion, lab activities, questions, and/or presentation of homework exercises at the board. At the end of each class period, starting in the second week, I will ask you to fill out a class participation form. I will respond to your self-evaluation, and give you a score for your participation in each class.
If you miss a class, you are expected to find out what happened. If you must miss a class for any reason (excused or unexcused absence), your participation score for that day will be recorded as 0. However, if you wish to make up for these absences, you may turn in written evidence that you have done some work to make up missed lab/class activities. This make up work must be turned in within two class periods of the missed classes.
Some homework exercises will be collected, but in general these are not graded. I simply keep a record of whether you have seriously attempted the problems, and often write comments on homework that I have reviewed. There may be occassional quizzes for which there will be no "make-ups."
The content of this course opens wide vistas for course projects. We will discuss specific requirements for the project early in the semester. You will have an opportunity to work individually or with a project team to explore some aspect of higher-dimensional space of your on choosing, subject to my approval..
Your course project will be graded on format (professional presentation), writing style (spelling, grammar, sentence structure), mathematical correctness, and evidence of synthesis and integration of the material of the course.
There will be two or three equally-weighted tests. Some of the tests will have a section for you to work on in your groups. The material to be covered on each test will be announced one or two class periods prior to the scheduled test date. Ordinarily, I do not give make-up tests; exceptions to this policy will be considered on a case-by-case basis.
The schedule of dates for tests will be announced by the end of the second week of classes.
The written component of the cumulative final exam is scheduled by the University according to our 11 MWF time. There will also be a final individual or group presentation at a mutually convenient time.
The University of North Carolina has a strong tradition of academic integrity, codified int he Honor Code. Students who cheat violate their own integrity and the integrity of the University by claiming credit for work they have not done and knowledge they do not possess. All students are expected to recognize and to abide by the Honor Code. Group work is encouraged, except where explicitly restricted; it is up to each group to see that the work is fairly divided. Work submitted by a group should acknowledge specific contributions of each memeber.
If you have any special needs for alternative instruction and/or evaluation procedures, please feel free to discuss these needs with me so that appropriate arrangements can be made.
Back to TOCThe Mathematics Department is located in Phillips Hall. My office is 303 Phillips, close to the front end of the bridge. If you are looking for me outside of class, I have another class at 9 MWF.
The best way to reach me between classes is to send me an email message; you may leave me a hard copy message on my door or with the secretary in Philips 331.
I will be regularly on campus Monday through Friday and often Sunday afternoon. You are encouraged to make an appointment with me during the first week of classes; if you suddenly need to see me, try calling my office: 962-9607. The best way to contact me is via email, which I check regularly (several times a day) both while I am on campus and from home.
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