Discrete Mathematics and Theoretical Computer Science

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

A PostScript version of this document

Chapter 14 of the New Jersey Mathematics Curriculum Framework

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:

- Grades K-12 Overview
- Grades K--2 Overview
- Grades K--2 Indicators and Activities
- Grades 3--4 Overview
- Grades 3--4 Indicators and Activities
- Grades 5--6 Overview
- Grades 5--6 Indicators and Activities
- Grades 7--8 Overview
- Grades 7--8 Indicators and Activities
- Grades 9--12 Overview
- 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:*

All students will apply the concepts and methods of discrete mathematics to model and explore a variety of practical situations. |

**By the end of Grade 4, students:**

- Explore a variety of puzzles, games, and counting problems.
- Use networks and tree diagrams to represent everyday situations.
- Identify and investigate sequences and patterns found in nature, art, and music.
- 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.
- 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:**

- Use systematic listing, counting, and reasoning in a variety of contexts.
- Recognize common discrete mathematical models, explore their properties, and design them for specific situations.
- Experiment with iterative and recursive processes, with the aid of calculators and computers.
- Explore methods for storing, processing, and communicating information.
- 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:**

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

- 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.
*