Intro to Higher Math
Please feel free to come by my office any time during scheduled
You are welcome to
come at other times, but in that case you might want to make
an appointment, just to make sure that I will be there then. You can
make an appointment simply by talking to me before or after class, by
calling me at my
or at home, or by sending e-mail.
You may also ask any questions directly via phone or e-mail. If I'm
not in when you call, please leave a message on the voice-mail or
answering machine with your name, number, and a good time for me to
call you back. I will try to respond to your phone or
as soon as possible.
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
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
Laboratories in mathematical experimentation: A Bridge to
higher mathematics, by Mount Holyoke College.
This book consists of 16 mathematical "labs". We will go through the following labs, at
the rate of two weeks per lab:
- Ch. 1: Iterations of Linear Functions
- Ch. 2: Cyclic Difference Sets
- Ch. 3: Euclidean Algorithms
- Ch. 7: Polyhedra
- Ch. 8: p-adic numbers
- Ch. 11: Sequences and series
You will also go through one additional lab, or another topic of your
choosing (with my approval), on your own, at the end of the semester.
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 may 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.
Each lab will be graded on the following criteria:
Organization and presentation of data;
Analysis of data;
Statement of conjectures; and
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.
The deadline for student-initiated drops with a
W is Friday, March 7. After this date, you can only drop with the
Dean's approval, which is granted only under extenuating
I hope everyone will complete the course successfully, but if you are
having doubts about your progress, I will be happy to discuss your
standing in the course to help you decide whether or not to drop.
You are only allowed three enrollments in this
course, so please exercise the drop option judiciously.