| All students will develop their understanding of the concepts and applications of discrete mathematics through experiences which enable them to use a variety of tools of contemporary mathematics to explore and model a variety of real-world situations. |
However, during the past 30 years, discrete mathematics has grown rapidly and has become a significant area of mathematics. Increasingly, discrete mathematics is the mathematics that is being used by decision-makers in business and government; by workers in fields such as telecommunications and computing that depend upon information transmission; and by those in many rapidly changing professions involving health care, biology, chemistry, automated manufacturing, transportation, etc. Discrete mathematics is the language of a large body of science and underlies decisions that individuals will have to make in their own lives, in their professions, and as citizens.
Discrete mathematics has many practical applications that are useful for solving some of the problems of our society and that are meaningful to our students. Its problems make mathematics come alive for students, and helps them see the relevance of mathematics to the real world. Discrete mathematics does not have extensive prerequisites, yet poses challenges to all students. It is fun to do, is often geometrically based, and stimulates an interest in mathematics on the part of students at all levels and of all abilities.
Students should use a variety of strategies to systematically list and count the number of ways there are to complete a particular task. For example, elementary school students should be able to make a list of all possible outcomes of a simple situation such as the number of outfits that can be worn using two coats and three hats. Middle school students should be able to list and count the number of towers that can be built using four blue and red blocks, or the number of possible routes from one location on a map to another, or the number of different "words" that can be made using five letters. High school students should be able to determine the expected payback of a $1 investment in the state lottery.
Students should use discrete mathematical models such as graphs and trees to represent and solve a variety of problems based on real-world situations. For example, elementary school students should recognize that a street map can be represented by a graph and that routes can be represented by paths in graphs; middle school students should be able to find cost-effective ways of linking sites into a network using spanning trees; and high school students should be able to use determine efficient methods of ordering the tasks in a larger project using directed graphs.
Students should recognize and apply iterative patterns and processes in a variety of settings in nature and art, as well as in mathematics. For example, elementary school students should recognize and investigate sequences on pine cones and patterns on floor tilings; middle school students should generate fractal curves and construct tessellations; and high school students should understand the long-term behavior of infinite sequences.
Students should explore different methods of arranging, organizing, analyzing, transforming, and communicating information, and understand how these methods are used in a variety of settings. Elementary school students should investigate ways to represent and classify data according to attributes like color or shape, using relationships like family trees, and into structures like tables. Middle school students should be able to read, construct, and analyze tables, matrices, maps and other data structures. High school students should understand the application of discrete methods to problems of information processing and computing such as sorting, codes, and error correction.
Students should explore different methods of solving real-world problems, and determine what is the best solution using a variety of algorithms -- where best may be defined, for example, as most cost-effective or as most equitable. For instance, elementary school students should discuss different ways of dividing a pile of snacks, and should determine the shortest path from one site to another on a map laid out on the classroom floor; middle school students should be able to plan an optimal route for a class trip (see the vignette in Chapter 1 entitled "Short-circuiting Trenton") and, pretending to be the manager of a fast-food restaurant, devise work schedules for employees which meet specified conditions yet minimize the cost; high school students should be conversant with fundamental strategies of optimization and recognize both the power and limitations of computers in solving algorithmic problems.
In summary, discrete mathematics is an exciting and appropriate vehicle for working toward and achieving the goal of educating informed citizens who are better able to function in our increasingly technological society; have better reasoning power and problem-solving skills; are aware of the importance of mathematics in our society; and are prepared for future careers which will require new and more sophisticated analytical and technical tools. It is an excellent tool for improving reasoning and problem-solving skills.