TEXTS: Mathematical Thinking: Problem-Solving and Proofs
by
D'Angelo and West (Prentice Hall,1997)
Set Theory and Related Topics by Lipschutz (McGraw-Hill
SOS,1964?)
How to Solve It by G. Polya (Princeton, 19??)
SCOPE: This course will provide a foundation for further work
in mathematics.This will be accomplished primarily by informal but careful
and rigorous exploration of key topics related to mathematical reasoning.
This will include a discussion of many of the essential tools for any mathematical
discourse and problem solving: sets, functions, and relations; problems
and conjectures; evidence, proofs and refutations; and direct and indirect
arguments.
Several topics from discrete mathematics will provide additional opportunities
for using these tools.
Lectures will organize the topics to present materials not covered in
the texts as well as those treated in the texts. We will cover material
from D'Angelo and West contained in chapters 1 to 8 and parts of chapters
9 to 12; from Lipschutz chapters 1 to 9, 14, 15, and 17, and perhaps
others as time permits. By the end of the first two weeks students will
be expected to have read the textual part of the Polya, and references
to relevant words in his "dictionary" section should be read regularly.
Supplementary readings and materials will be supplied as appropriate.
Summaries
of lectures may be available through the course webpage.
TECHNOLOGY: We will use the computer at various stages of this course to illustrate and investigate some of the topics. No particular software will be required though at times we may use X(PLORE) or Geometer's Sketchpad.
TESTS AND ASSIGNMENTS:
Reading Assignment: Each student will be expected to read at least one proof from a mathematics periodical or web page per week. These need not be lengthy. A brief written summary, including an appropriate citation and analysis is to be passed on Wednesdays beginning September 9th along with an explanation of the weekly proof without words.
Besides a review of the proof, the analysis should discuss briefly the techniques of argument (direct, indirect, induction, etc.) and exposition (forward-backward organization, reference to prior work, definitions, etc.) used by the author in presenting the result. [See the Weekly assignments. The reading reports will be graded Honors(4)/Cr(3)/NCr(0). (Accepted one day tardy at most!) ]
Regular Homework: Shorter problem assignments (about 5-10 problems) will be made on a regular basis for each class. These will not be accepted after 5 p.m. of the due date and will be graded Well-done (++=4), Acceptable (+=3), Unacceptable (-=2), No Credit (--=0)
Problem of the Week: Every other week I will designate one or two Problems of the Week. Individuals or groups may submit solutions to these problems for extra credit supplementing the points available from quizzes and examinations only.
Reality Check Quizzes: During the term I will give several reality check quizzes. These will usually be distributed on Fridays and collected on Mondays, covering work from the previous week's assignments and class discussions.
Midterm Examinations: There will be two self-scheduled mid-term examinations.These will be announced a week in advance and will be worth 100 points each.
FINAL ASSESSMENT: The final assessment will be in two parts. Part I will be a cooperative team take home examination that will be due on the last day of the final examination period which will be distributed on the Friday before the last week of classes. Part II will be an individual self-scheduled 90 minute examination given during the final examination period. Part I will be worth 100 points. Part II will be worth 150 points.
GRADES: Final grades will be based on the accumulation of points in the various categories of assignments as indicated in the following chart:
| Reading Reports | 50 |
| Proof w/o Words | 50 |
| Homework | 50 |
| Reality Check Quizzes | 100 |
| Midterm Examinations | 200 |
| Final Assessment Part I | 100 |
| Final Assessmant Part II | 150 |
| Total Points........... | 700 |