VOLUME Thrity Six
TITLE: "Discrete Mathematics in the Schools"
EDITORS: Joseph G. Rosenstein, Deborah S. Franzblau and Fred S. Roberts. Published by the American Mathematical Society and the National Council of Teachers of Mathematics

Introduction

This article contains the chapter of the New Jersey Mathematics Curriculum Framework ¹ which deals with discrete mathematics. The first three pages of the article describes what this document is and why it was written.

On May 1, 1996, the New Jersey Board of Education adopted core curriculum content standards in seven content areas, including mathematics.

These standards describe what all New Jersey students need to know and be able to do at the end of grades 4, 8, and 12. Statewide assessments reflecting these standards are being developed at these grade levels, and students will be expected to demonstrate that they meet these standards in order to graduate from high school.

The standards for mathematics includes a discrete mathematics standard; thus all New Jersey students will be expected to demonstrate understanding and proficiency in discrete mathematics.

The development and adoption of standards extended over a period of three years, and, as Director of the New Jersey Mathematics Coalition, I was very much involved at every step along the way. The mathematics standards represent what New Jersey mathematics educators believe are high achievable standards for all students in the state.

How will New Jersey teachers ensure that their students can meet these standards? During the past four years, the New Jersey Mathematics Coalition, working in collaboration with the New Jersey Department of Education and with an Eisenhower grant from the United States Department of Education, has developed a resource book, the New Jersey Mathematics Curriculum Framework; this 688-page document was developed to assist teachers and administrators in implementing the mathematics standards at both the classroom and the district level. The preliminary version was published in Spring 1995, and a revised version in December 1996. The preliminary version included the contributions of many New Jersey educators; the revised version incorporated the suggestions of many reviewers and reflected the standards adopted by the Board.

I am pleased to have spearheaded and directed this effort; we have produced a valuable guide for New Jersey teachers and, through its availability on the World Wide Web, for those of other states. The New Jersey Mathematics Curriculum Framework is not intended to be a curriculum; rather it is intended to be a structure (i.e., a ``framework'') around which a district can build its own curriculum (or curricula). This particular framework, however, provides much more detail about the content of K-12 mathematics than any other state framework of which I am aware; for that reason, it should be a valuable resource to all teachers of mathematics.

What should students be expected to know and be able to do? The discrete mathematics standard, like the other mathematics standards (and those in other content areas), consists of a general statement about discrete mathematics followed by five or six statements, called ``cumulative progress indicators'', which describe what students should be able to do at each of the three grade levels. The discrete mathematics standard and cumulative progress indicators appear at the end of this Introduction.

How will teachers be able to reflect these indicators in their curricula? The discrete mathematics chapter of the Framework (like each of the other chapters) is intended to respond to this question. The chapter consists of a K-12 overview of discrete mathematics, followed by sections addressing five different grade levels; for each grade level there is a (self-contained) overview of discrete mathematics for that grade level, followed by a number of classroom activities that illustrate how each indicator could be addressed at that grade level. These materials are arranged in this article in the following sections:

1. Grades K-12 Overview
2. Grades K--2 Overview
3. Grades K--2 Indicators and Activities
4. Grades 3--4 Overview
5. Grades 3--4 Indicators and Activities
6. Grades 5--6 Overview
7. Grades 5--6 Indicators and Activities
8. Grades 7--8 Overview
9. Grades 7--8 Indicators and Activities
10. Grades 9--12 Overview
11. Grades 9--12 Indicators and Activities

Note that because the materials for each grade level are self-contained, there is considerable overlap between the overviews (even numbered sections). Note also that all references for each grade level are provided at the end of the odd numbered sections.

The activities in this chapter are based on activities used by teachers in the DIMACS-sponsored and NSF-funded Leadership Program in Discrete Mathematics, which I have directed since its inception in 1989 (see article in this volume). The organization of discrete mathematics into five areas and the list of indicators, one for each area at each grade level, emerged from a series of discussions in 1993 by Rutgers University faculty associated with DIMACS. Although I have been responsible for the selection and writing of the activities, as well as the overall organization of the material, I would like to acknowledge the assistance I received from a number of people, including many participants in the Leadership Program, who reviewed and commented on drafts of this chapter. The expectation is that, through the wonders of the Web, the entire Framework and this chapter on discrete mathematics in particular will continue to evolve.

And now, Chapter 14 of the New Jersey Mathematics Curriculum Framework, which addresses the following standard and cumulative progress indicators of the New Jersey Core Curriculum Content Standards:

Cumulative Progress Indicators

By the end of Grade 4, students:

1. Explore a variety of puzzles, games, and counting problems.
2. Use networks and tree diagrams to represent everyday situations.
3. Identify and investigate sequences and patterns found in nature, art, and music.
4. Investigate ways to represent and classify data according to attributes, such as shape or color, and relationships, and discuss the purpose and usefulness of such classification.
5. Follow, devise, and describe practical lists of instructions.

Building upon knowledge and skills gained in the preceding grades, by the end of Grade 8, students:

6. Use systematic listing, counting, and reasoning in a variety of contexts.
7. Recognize common discrete mathematical models, explore their properties, and design them for specific situations.
8. Experiment with iterative and recursive processes, with the aid of calculators and computers.
9. Explore methods for storing, processing, and communicating information.
10. Devise, describe, and test algorithms for solving optimization and search problems.

Building upon knowledge and skills gained in the preceding grades, by the end of Grade 12, students:

11. Understand the basic principlies of iteration, recursion, and mathematical induction.
12. Use basic principles to solve combinatorial and algorithmic problems.
13. Use discrete models to represent and solve problems.
14. Analyze iterative processes with the aid of calculators and computers.
15. Apply discrete methods to storing, processing, and communicating information.
16. Apply discrete methods to problems of voting, apportionment, and allocations, and use fundamental strategies of optimizaion to solve problems.

1. Rosenstein, Joseph G., Janet H. Caldwell, and Warren D. Crown, New Jersey Mathematics Curriculum Framework, New Jersey Mathematics Coalition, 1996.

The remainder of this article is Chapter 14 of the New Jersey Mathematics Curiculum Framework.