Designed for middle and high school teachers
This is an intensive, interactive course.
This course takes an algorithmic approach to the study of abstract mathematical concepts. Mathematical study requires a high level of understanding of the fundamental abstract ideas, such as those of sets, tuples, the propositional calculus, relations, functions, mathematical induction, matrices, linear spaces. Using the high-level mathematical programming language ISETL, we hope to demonstrate how students can construct a mature and highly developed understanding. And, we hope that each student will experience the process of construction for himself or herself.
For example, when a mathematician thinks of a function, he or she thinks of it in at least two ways: as a process and as an object.
But how can students understand these concepts more deeply? Using the ISETL computing language has been shown to be effective in aiding students to construct these concepts by writing short programs which become the concrete applications of the abstractions.
The ISETL language is interactive, easy to learn, and has a syntax very similar to the standard notations of mathematics. In addition, its richness allows for representing functions in more than one way, which can help make the constructions that are necessary to view functions the way mathematicians see them. Furthermore, complex abstract structures, such as groups, linear spaces, and graphs can be modeled in ISETL.
In addition to the study of abstract mathematical concepts, we may also look at algorithms from computer science (finite state machines), number theory (e.g., number bases, greatest common divisor, continued fractions, cryptography), graph theory (e.g., finding shortest paths), and linear algebra.
Meeting times:
Wednesdays, 6:00-9:00 pm
Room: Skalny Science 225
Textbook: "Learning Discrete Mathematics with ISETL" by
Baxter-Hastings, Dubinsky and Levin (Springer-Verlag NY, ISBN 0387968989).
Another interesting book is "Discrete Algorithmic Mathematics" by Maurer and
Ralson, published originally by Addison-Wesley, but
now published by A. K. Peters, Ltd. (ISBN 1-56881-091-1).
Another one that I might consult is "Discrete Mathematics with Algorithms", by
Albertson and Hutchinson (Wiley, ISBN 0-471-84902-2). This may be out of
print, too. I may take a few problems and topics out of this, too.
Grading:
Grading will be based on classroom presentations and written solutions to problems. There may be other methods of assessment, such as requiring a paper on how some of this material may be used in your classroom.