If you are teaching or planning to teach discrete mathematics, or if you are a mathematics supervisor assisting teachers in using discrete mathematics, you may be wondering what sorts of classroom materials are available. What books should you own or ask your school to buy for the library? Where can you get ideas for interesting student activities? Are there good videos or software? What have other teachers used successfully? This article addresses these questions, and provides recommendations for a selection of the best resources known to us. This is not a comprehensive listing, but rather a description of a core library recommended for anyone teaching discrete mathematics in grades K-12. Several of the resources mentioned here (especially videos or stand-alone activities) are also useful for convincing parents, school boards, or administrators of the value of teaching discrete mathematics
We have collected these recommendations from a number of sources. The most important of these are teachers and instructors from the Leadership Program in Discrete Mathematics (LP). Both of us have been involved in the LP for over two years, Janice Kowalczyk as an alumna, lead teacher, workshop leader, and program evaluator; Deborah Franzblau as instructor and summer institute organizer. Kowalczyk maintains a resource list for the program, and edits a resource column, ``The Discrete Reviewer,'' in the LP newsletter, In Discrete Mathematics: Using Discrete Mathematics in the Classroom (see section 3), which Franzblau has edited. Several of the resources described here have been reviewed in the newsletter. Kowalczyk has taught mathematics in middle school. Franzblau has taught mathematics at Vassar College and is currently teaching at the College of Staten Island; she also edits the ``Education Forum'' of the SIAM Activity Group on Discrete Mathematics Newsletter.
Both of us believe strongly in activity-centered teaching methods which focus on student exploration and discovery first, and abstraction and precision later. Although we agree that hard work is essential for understanding, and drill and practice are useful at the right time, we believe that student motivation and problem-solving must always come first. Thus, our focus is on materials and activities which either provide motivation for learning or lead to interesting mathematical discoveries. For this type of teaching, it is essential that the teacher have sufficient background and confidence to recognize and help students articulate their discoveries, so we have included resources providing background and breadth for teachers.
We believe that teachers are the best judges of what is appropriate for the ages or grade-level of their students. In our experience, the best activities and ideas work at many levels, and skillful teachers can adapt them to the level of their own classroom. We do however give suggested grade-level ranges for each resource, where appropriate, based on comments from teachers who have used the materials.
An important resource to teachers at all grade levels is , which presents the discrete mathematics chapter from the New Jersey Mathematics Curriculum Framework. This chapter is the first comprehensive attempt to describe what activities and topics from discrete mathematics are appropriate at each grade level cluster from K-2 to 9-12. The materials in the Framework are also based on the experiences of LP teachers.
Many of the resources listed here were developed for grades 7-12 (or the college level). Until recently, there has not been much available material that is labeled ``discrete mathematics'' at the elementary level. Nevertheless discrete mathematics often appears in publications such as Wonderful Ideas or The Elementary Mathematician (see Section 3). Moreover, there are many activities and topics that can be adapted from materials written for a higher level, as well as some excellent children's literature (see Section 4) that can be used to to introduce these activities.
In addition, we recommend the following catalogs for browsing; a number of the resources mentioned here can be found in them. The first three are good sources for physical models (``manipulatives'') and software. Addresses of the publishers are given in the Appendix.
The resource descriptions are organized by category, as follows.
Each title is followed by a suggested grade-level range and the resource topic is given (in bold italics) if it is not clear from the title or the context. Within each subsection, unless stated otherwise, resources are arranged by approximate grade-level.
For each resource, we list the publisher and/or a distributor (as of 1995/96). Print resources with no distributor listed can usually be found at large bookstores. The prices given are approximate retail cost, based on 1995 or 1996 catalogs or bookstore quotes (prices may vary a few dollars between different sources).
The article is accompanied by several appendices to assist the reader in locating resources: