Mathematic standards today expect students to go beyond the basics since children will need to think critically, solve problems, and make decisions using mathematical reasoning and strategies in a technological world. This widening of math instruction involves the use of a relatively new branch of mathematics known as discrete math. The National Council of Teachers of Mathematics in 1989 elevated discrete mathematics to the level of algebra, geometry, and calculus by including it in their standards.
What is discrete math? While there is no agreed-upon definition, there is a general agreement that discrete math is a collection of interesting mathematical topics not covered in traditional math and often grounded in real-world problems. Problems in discrete mathematics are engaging because they are often visual and more geometric than algebra, and often have a puzzle-like quality. Sudoku, mazes, and dot-to-dot are widely known examples of discrete math. Why is discrete math important? According to a vision statement of the Discrete Mathematics in the Schools Conference, "Discrete mathematics is the mathematics that is used by decision-makers in business and government; by workers in such fields 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. Increasingly, 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."
Many discrete math problems have few mathematical prerequisites. They can be introduced at all grade levels, even to children who are not yet fluent readers. Here are a few examples of discrete mathematical problems:
Combinatorics
This area of discrete math involves combinations. Problems of this type encourage students to think about relationships and their various permutations. There are several ways to arrive at an answer which encourages logical thinking.
Problem: There are four different colors of fall leaves (red, yellow, orange, and green) and you want to place them in any order. How many possible ways are there to do that? Answer: 24 ways
Here is one way to get the answer. There are four colors of leaves, and you can make four lines to represent the choices.
4 3 2 1
For the first slot gives 4 choices; the second slot, has 3 choices (I used 1); for the third slot gives 2 choices (I used 2); for the fourth slot, there is just 1 choice left. The answer equals 24 (four factorial). Four factorial means 4 x 3 x 2 x 1.
Two other ways this problem can be solved is by listing all the possible combinations (systematic listing) or by making a tree diagram.
Iteration
Iteration means repeating an action over and over. Iteration is a type of pattern. Patterns are found everywhere in life and are certainly common in math. The ability to recognize patterns helps children to recognize the relationship of numbers and to make math predictions. Problem: King Frog decides to hop across a field that is 13 feet. He hops in a straight line. From the starting point, he takes a giant hop forward that is 5 feet, then a giant hop backward that is 3 feet. He then takes another giant hop forward that is 5 feet, and then a giant hop backward that is 3 feet. He continues this pattern until he has gotten across the field. How many hops must he take until he reaches this goal?
Answer: 9 hops
This problem can be solved by drawing a picture. It also can be solved by constructing a number sentence with a pattern that continues until the answer is 13.
This would be (5-3)+(5-3)+(5-3)+(5-3)+5=13
Graphs
Graphs consist of a finite set of vertices and edges that connect them. Graphs are usually depicted pictorially as a set of points (vertices) with lines (usually straight, but not necessarily so) connecting them to represent the edges. Math problems of this type strengthen a student's visual and spatial comprehension. Graphs help a student to organize and arrange data not just in math but in all subjects studied in school.
Problem: Six blocks (representing points on a graph) are spread out on the floor in a random manner.
Example:
One is labeled Start and one is labeled Finish. Children use string to connect the blocks. Then children measure the length from one block to another. They MUST touch each block once.
Here are some questions that can be asked:
Question 1: What is the length to the nearest inch of the shortest route from start to finish when you touch each block once?
Question 2: What is the length to the nearest centimeter of the shortest route from start to finish when you touch each block once?
Question3: How many routes are possible from start to finish when you touch each block once? Answers: This of course depends on the number of blocks used and the placement of the blocks. Discrete math enables teachers to engage students fully in a topic and explain in a different way, concepts that students may not have understood before. It helps students organize their thoughts, and to truly master math—which they are not able to do until they can look at a problem and solve it in more than one way. Finally the "thinking outside of the box" encouraged by discrete math may helps students tackle other problems that will arise in their lives.
For more information about discrete math, you can go to the homepage of "Making Math Engaging: Discrete Mathematics for K–8 Teachers" by Valerie A. DeBellis and Joseph G. Rosenstein:
(http://www.shodor.org/discretemath/index.php?content)
or to the Web site from Tufts University:
(http://www.cs.tufts.edu/research/dmw/).
Linda Wiegenfeld is an educator in the Boston area.
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