About Mathematics Commentary
Evaluation
Because the approach of About Mathematics is not on the kind of problem-solving tasks that usually lead to examination questions,
it is necessary to rethink the means of evaluating students in classes using this text.
The completion of assigned tasks like chosen text exercises can form one basis, but I have found that two others serve far better.
Reaction PapersFirst, I require that a brief reaction paper (restricted to one side of a sheet of paper) be turned in at each class session after
the first. This paper is a direct private communication between each student and me to which I respond marginally when I return it.
I make a number of features of this assignment clear at the outset:
1. “Reaction” is the key word here. This assignment requires students to react to the text contents, to the text and
classroom presentations, to other classroom activities and to me personally.
2. The paper should include comments on mathematical ideas and approaches that are new and unfamiliar to them.
3. The paper can raise questions about content that the students do not understand, thus helping me to direct class
attention to particular points.
4. Critical comments are as important as commendations.
5. The limitation to one page is significant. This assignment calls for a precis of their thoughts. It does not
provide an opportunity to write without limits.
My own responsive reactions, especially to the initial papers, provide additional guidance in how to address this regular
assignment. In particular I make a point of welcoming critical comments.
ProjectsEarly in the course I offer a list of topics from which students choose their personal course project. They are required
to provide a written progress report on their project work at mid-semester and to complete the final written project report
a week before the end of the semester. A listing of sources is required on the written report. The project also includes a
class report strictly limited to ten minutes. The final class sessions and examination period are devoted to those reports.
I take care to point out that the idea of these projects is not to solve mathematical problems but to extend the course focus
on information about additional topics. This may, however, include the solution of reasonable problems related to the
assigned topic.
These projects serve not only to involve individual students with additional mathematical concepts but, through their reports,
they also extend the scope of the entire class.
Completion of this course project can count for half or more of the evaluation of individual students. I divide this credit
among the mid-term progress report, the final written report and the class presentation. Although this is in a real sense a
test of student initiative and research technique, I respond to student requests for assistance throughout the course.
Here is a list of project topics I have offered. You can easily supplement this list with topics of interest to you, but
care is suggested that the topics you choose not involve prerequisite mathematical understanding that the students do not have.
The Platonic Solids and Regular Polyhedra
Continued Fractions
Polygonal Dissections
Geometric Transformations
Fibonacci Numbers and the Golden Section
Benford’s Law
Mobius Strips and the Klein Bottle
Infinities
Teteraflexagons
Wallpaper Patterns
Fuzzy Logic
The Konigsberg Bridges and Mathematical Maps
Leonard Euler
Isaac Newton
Archimedes
The Bernoulli Family
The Pigeonhole Principle
The Soma Cube
The Game of Life
Divisibility Rules
The Harmonic Series
The Collatz Conjecture
Proofs without Words
Voting Alternatives
Mathematical Paradoxes
Latin and Graeco-Latin Squares
Magic Squares
The Mathematics of Map Coloring
Duplication of the Cube
Angle Trisection
Prime Numbers
Famous Problems Recently Solved
The Mathematics of Knots
Packing Circles and Spheres
The Games of Sprouts and Brussels Sprouts