The public at large thinks of mathematics as a sequence of basic facts and algorithms, and considers the goal of the educational process, especially at the K--8 level, to be the training of children to memorize the facts and become proficient at the algorithms. Most college students entering the dreaded "math content" course for prospective elementary school teachers share these views. Thus part of the goal of such a course---in addition to handing down important knowledge and conveying insight about elementary arithmetic and geometry and their applications---is to disabuse them of this destructive notion. Over the past several years, I have added some components to a traditional "Math for Elementary Education Majors" course to do this, with gratifying results. This talk will discuss two aspects of these efforts.

The minor aspect, which can be easily implemented, is to insist that students buy and read the relevant sections of the NCTM's Curriculum and Evaluation Standards for School Mathematics ($25). This somewhat controversial document, published in 1989, deals with both content and pedagogical issues, offers many helpful suggestions for teachers, and certainly shows how mathematics is much broader, deeper, and more applicable than is popularly believed. I tell my students to sell back the textbook for the course at the end of the term if they must (we use Addison-Wesley's fine text by Billstein et al.), but to hang on to the Standards throughout their careers. Only about 5% of class and test time is taken up with this material directly, but the students are pleased to see this component in the course.

The major aspect is extended projects. I usually assign three or four of them per semester, and the students typically have at least two weeks to complete each one. They are encouraged to work in pairs ("cooperative learning" is a popular buzz phrase in educational circles), and the final write-ups are expected to be nicely presented. The 2- to 3-page project descriptions present the concepts involved and list several tasks that must be performed, which range from computing examples, to formulating conjectures, to explaining why something is true. There is often no one right answer. Little class time is spent discussing the assignments---they are meant to be done independently, partly in a discovery or investigative mode. A downside for the overworked faculty member is that the reports must be read and graded; they count for one sixth of the final grade.

Here are some of the topics I use. Keep in mind that each sentence here translates into two or three pages of exposition and directed problem solving. Discover and prove the inclusion--exclusion formula for three sets, apply it to a particular example, and for extra credit, generalize. Figure out some divisibility gimmicks for Base~6, determine which Base~6 fractions have terminating "heximals", and compute pi to eight heximal places. Find the twelve non-isomorphic intersection patterns that three sets can have. Learn how to square two-digit numbers mentally. Learn an efficient mental day-of-the-week algorithm. Investigate continued fractions. Look at partitions; discover the fact that the number of partitions of N into odd parts equals the number of partitions of N into distinct parts; find the partition of N with largest possible product. Learn about the codes used for drivers license numbers and ISBN numbers. Compute square root 2 using five different algorithms. Investigate two-person perfect-information games, like Nim and Chomp, and determine winning strategies in some cases.

In a second semester geometry course, the topics have a similar wide range. Investigate Conway's game of Sprouts. Solve some angle problems involving hands on a clock. List all essentially different arrangements of two lines and one plane in 3-space. Solve some pentomino problems (building problems, symmetries, enumeration). Investigate Pick's area theorem on a geoboard. Do some ruler and compass constructions, such as a regular decagon. Experimentally verify Heron's formula. Investigate tessellations of the plane. Look at Euler and Hamilton circuits in graphs. Find ways of generating Pythagorean triplets of various types (this is really number theory, of course).

The point is that these open-ended explorations deal with nontrivial mathematics, some very pure and beautiful, some quite applied, some quite related to the rest of the subject matter of the course, some far removed. Almost all of it is fun; it engenders excitement among the students; it teaches them things they haven't seen before (I'd wager that few university faculty members are acquainted with all of the topics mentioned here). Students learn to treat mathematics as something much deeper and more interesting than pushing symbols and solving artificial problems, and they get to try an experimental and hands-on approach to the subject. The feedback has been very positive all around.

Jerrold W. Grossman, Oakland University