Discrete Math Should Be Taught in High School

Discrete Mathematics is not "discreet" - nor should it be! Here's what Webster's has to say:

discreet capable of preserving prudent silence

discrete  constituting a separate entity; consisting of distinct or unconnected elements

Discrete Math includes a number of distinct areas of study such as mathematical logic, set theory, number theory, combinatorics, graph theory, and mathematical relations.What all of these disciplines have in common is that they use distinct, separate mathematical values. The best way to explain the mathematical meaning of the word "discrete" is in contrast to the mathematical concept of "continuous".

Things that are measured have continuous values. The length of a board can be measured at, let's say 8 or 9 inches, or anything in between - perhaps 8.1 inches, or 8.01 inches, or 8.001 inches or 8.0001 inches or.......... You get the picture. These possible values for the length of the board are continuous -  no matter how close two numbers are there is always another number between them.

Calculus is an example of an area of math that is continuous. Most of the time, algebra is treated as continuous mathematics. For example, here is what a continuous graph looks like:

On the other hand, things that are counted have discrete values. You can count 8 people or you can count 9 people, but you cannot count any number of people between 8 and 9. There is a distinct separation of the values .

This area of math is becoming critically relevant to many of the issues that face our society today, and to the technology that is used to address these issues. Not the least of this technology is the hardware and software aspects of computer science.

The body of knowledge and applications that comprise Discrete Mathematics has been growing by leaps and bounds in the last fifty or so years and, unlike traditional mathematics, is a wide open field of opportunities for mathematicians to make original and creative contributions.

It is said that the discipline dates back to 1735, with Leonhard Euler's "Konigsberg Bridge Problem".

It was asked whether the citizens of this German city could traverse each of the seven bridges over the River Pregel exactly once, and return to where they started. Euler determined that if there are certain properties related to the number of bridges and the number of land masses that they connect, then the answer to this question is "yes". If these properties are not met then the answer is "no". This is considered the "birth" of Graph Theory but it is really only since the 1970's that the body of discrete mathematics knowledge has grown with new questions, new answers, new conjectures, new theorems, and new applications.

At the college level, the topics that comprise Discrete Mathematics have traditionally been treated as distinct and separate areas of study; That is, in depth study of only logic, or only number theory, etc. This is fine for those who are focusing on pure mathematics and will be pursuing a career in mathematical research in one of these areas. It is only fairly recently that it is the integration of all of these topics that is necessary for those moving into careers in the application of mathematics.

It is also growing in popularity at the high school level but this is not as widespread as I believe it should be. The National Council of Teachers of Mathematics (NCTM) recognizes Discrete Mathematics as one of the core bodies of knowledge that should be integrated into mathematics education at all levels, i.e.K-12. But I think that this should be taken a step further; Advanced high school curriculum should be expanded to include not only the traditional areas of calculus and statistics, but also the continually evolving theory and skills that have become necessary factors in our computer driven society.

Here's a case in point. Sometime last year I was reading an article in Time Magazine entitled How to Raise the Standard in America's Schools. The author of the article commented that although he had learned a lot of calculus, he hadn't found it very useful in his career. He went on to say that it was dismaying when, in a staff meeting about digital strategies, the entire committee was stumped when trying to determine how many direct two-way links there needed to be in a fully connected network of 50 nodes. He brought out the point that "It was a long time before any of us could figure out even how to begin figuring this out". Yet any student of beginning graph theory could easily answer that question in less than a minute - even my high school students!

Such varied tasks as calculating UPC codes, scheduling parts of a project to minimize total time, finding shortest or most efficient routes among a network of locations, (or telephones, or computers, or....), and efficiently storing and retrieving data from computer memory are just a few examples of where discrete mathematics is applied today.

In recent years disturbing statistics have emerged showing that the US lags behind other countries in the level of math education. Our students rarely win international mathematics competitions. The U.S. ranks 25th in math scores on international tests! Of course this is a complex problem that needs to be addressed in many varied aspects, but I believe that offering Discrete Mathematics as an area of study for high school students to pursue is one of the ways that would help make our country more globally competitive in mathematics.

There is no better time than now to bring all of this to the advanced high school level so as to encourage growth in mathematical thinking among our high school students, and to better prepare them to pursue an area of critical need in college and in their future professions.

If you are interested in math, you should read my hubs about algebra and pi.

You can also learn more about how I feel about math philosophy and math education, or about overcoming math anxiety.