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