"Tiger gotta hunt. Bird gotta fly.
Man gotta sit and wonder why, why, why.
Tiger gotta sleep. Bird gotta land.
Man gotta tell himself he understand."

Kurt Vonnegut Jr.


Introduction to Higher Mathematics (14011) - Fall 2003

TR 13:30-14:50 in BUSN 323

  • Instructor. Dr. H. Knaust, Bell Hall 122, tel. 747-7002,
    e-mail: helmut@math.utep.edu,

  • Office Hours. TR 11:00-12:00, or by appointment (call 747-6754).

  • Textbook. Laboratories in Mathematical Exploration - A Bridge to Higher Mathematics, Mount Holyoke College. Springer-Verlag, 1997. The textbook is required at all class meetings.

  • Course Description. An introduction to mathematical problem solving, experimentation, and proof writing, and the relationship among all three. The course will be built around a series of in-depth problems from a variety of areas of higher mathematics, especially those not encountered in pre-calculus and calculus courses. Corequisite: Math 1411.

  • Course Philosophy and Objectives. This course is built on the proposition that you learn mathematics, and how to construct mathematical proofs, better when you formulate the questions and discover the answers yourself. Upon successful completion of the course, you will be able to investigate mathematical questions, big and small, both experimentally and theoretically.

    This is very different from courses like pre-calculus, calculus and differential equations, which are primarily focused on computations. Although there are computations in this course, they are a tool for discovering, and proving, more general mathematical truths.

  • Contents. We will go through the following labs, at a rate of two weeks per lab:
    • Chapter 1 - Iterations of Linear Functions
    • Chapter 2 - Cyclic Difference Sets
    • Chapter 3 - Euclidean Algorithms
    • Chapter 7 - Polyhedra
    • Chapter 8 - p-adic Numbers
    • Chapter 11 - Sequences and Series
    You will also go through one additional lab, or another topic of your choosing (with my approval), on your own, during the second half of the semester.

  • Laboratories. Class time will be devoted exclusively to labs. Each lab will start with a brief explanation of the question or problem to be explored. You will perform experiments (usually with a computer or programmable calculator) and gather data. The data will lead you to make your own conjectures, which you will then test and refine by further experimentation. Finally, when you are more certain of your conjectures, you will prove them carefully. (In practice, this process is rarely as straightforward and linear as outlined here. You will often revisit earlier steps as you carry out later steps.)

    You will work in small groups of your choosing in class (as well as out of class). There will also be whole-class discussions about your experimental and theoretical discoveries.

    After two weeks of work in class (and while you are starting the next lab), you will have a week to write up your discoveries, both experimental and theoretical, into a clearly-written report. (Grading criteria are below.) Although you may work with other students during the lab, you must write your report yourself. After each report is graded and returned to you, you will have approximately one more week to revise your report for a better grade, if you like. Revised reports must be complete; in other words, it should be possible to understand your revised report without reading your original report.

  • Grades. Each lab will be graded on the following criteria:
    • Experimental design
    • Organization and presentation of data
    • Analysis of data
    • Statement of conjectures
    • Mathematical analysis (including proofs) of conjectures (see p. xvii of the text).

    The final grade for each lab will be the average of the grades you receive on your initial report, and on your revision. If you do not turn in a revision, it will simply be the grade of your initial report. Your grade for the course will be the average of the final grades for each of the seven labs.

    Deadlines for the various assignments will be announced in class. A late submission of an assignment will result in a grade of zero.

  • Drop Policy. The class schedule lists Monday, October 20 as the last day to drop with an automatic "W". After the deadline, instructors can only drop you from a course with a grade of "F". Consult the current Class Schedule, page 3, for details.

  • Information on the Web. This syllabus and other relevant material for this class can be found on my homepage http://helmut.knaust.info/mediawiki.

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