INTRODUCTION
Many students who have difficulty with math have a problem with the basic concepts, mainly because of conditioning. They spend much of their instructional time seeking answers to problems posed by the instructor or from the text. The answer usually consists of a number derived by use of a process, usually committed to memory, designed to find the answer without an understanding of the basic concepts involved. As a consequence, a habit of trying to find the answer becomes paramount — and all activity is directed toward that end. The conditioning is carried forth as an integral part of the math experience, and they immediately seek the answer before the proposed situation is evaluated. As a consequence, the focus is on the answer to the neglect of the evaluation necessary to arrive at a valid defensible conclusion.
This book is intended to diminish the immediate seeking of the answer and to magnify the idea of seeking to evaluate the situation to determine the information contained in the proposed situation (problem) before an attempt is made to find that elusive answer. To accomplish that aim, a thorough grounding in the basic fundamentals of the system is attempted by introducing the basic concepts without the stress of trying to immediately solve situations by applying memorized processes rather than gathering an understanding of the basic underlying concepts.
Basic Math in Plain English is an attempt to demystify math by comparing it to the basic structure of the spoken language, emphasizing the similarities, and stressing the differences. It is an attempt to show that mathematics is an integral part of our existence and how a basic understanding of the concepts that are paramount to the system of mathematics will create a greater appreciation of the role that math plays in our daily lives.
I implore the reader to carefully consider each salient idea and try to internalize its meaning for future reference. Algorithms (rules) are developed as shortcuts for arriving at defensible conclusions, but they do not necessarily enhance your understanding of the basis for the algorithms.
I have not taught mathematics for many years, but I was in administration and had many opportunities to interact with students. One of the basic questions was always if they were doing well in school. The answer was almost universal that math was the subject with which students were befuddled and could not understand or see any use for. The same questions posed to adults invariably elicit the same response. Why do intelligent persons have such difficulty with basic mathematics? Is math too difficult for the average person to comprehend — or is it presented in a manner that makes it incomprehensible to the average person?
I believe that the focus on using numbers without a clear definition of the nature of a number and how the numeration system is structured is one part of the problem. Another part of the problem is the focus on problem-solving techniques by getting answers rather than comprehending the structure and its use in constructing the algorithms commonly stressed in the teaching of math. If a teacher asks, "Why are two and two equal to four?" the answer is invariably, "It just is."
If the teacher asks, "What are the results of adding two and two?" the answer will be four. The four will be stressed since — from early in math instruction — students are conditioned to respond with the numeral four, but when asked if that is always true, the answer will always be yes. When asked to add two apples and two oranges, the answer will almost always be four fruits without the implied meaning of addition, which is often assumed and not explicitly expounded. Many other examples can be demonstrated, but we will not continue.
Mathematics is a tool like a hammer, telephone, or computer. Each tool has a function and can be used to attain certain objectives even without full comprehension of the structure and function. The structure and function of a simple tool such as a hammer is easy to decipher, but as tools become more complex, the function can be ascertained without a concurrent understanding of the structure. To attain a more complete understanding of the capabilities of the tool, both the structure and function need to be understood.
A telephone can be utilized without an understanding of how it functions or is constructed, but an understanding of the structure of a phone will create conditions for expanded use. The same is true for any constructed system. Mathematics is a constructed system, and a clear understanding of its structure and function will allow more utility in its use.
Imagine that you discovered some ancient coins and wished to decide the best use for them. Would they serve a better purpose if they were smelted into ingots or if they were kept intact? Now imagine that you spoke a different language. Would you come to the same conclusion? The same conclusion would be reached by the same person. The language would not change the reasoning process, hence the universality of the process. That process is devoid of numbers and is mathematical. The language of mathematics adds credence to the process.
This book is an attempt to mitigate some of the problems as discerned through interactions with students and other adults. It is hoped that by reading and rereading this book, some of the problems experienced with math will be tempered, the phobia associated with the word math will be proportionately diminished, and a rational discourse can be initiated by almost anyone who has a better understanding of the structure of the basic system of mathematical ideas and practices. This is an attempt to help readers understand the rationale for the current system of mathematics and how all the basic tenets of the system are explained by the basic model. Understanding the structure of the system will mitigate much of the confusion and apathy toward mathematics.
Exercises and problems to solve are not included in this book because the focus is directed toward the understanding of the basic structure and its uses in developing methods (algorithms) currently in use by most persons. A thorough scrutiny of the points of discussion will make it possible for the reader to develop algorithms (methods for solving problems) based on a complete understanding of the principles and not on rote memory without a clear understanding of mathematical sense.
From comments from students and others about math, most persons do not have a clear understanding of the function of mathematical reasoning and its implications to events in everyday life. There is little correlation between the relationships of mathematical ideas to situations in our daily lives. If "Do no act that will cause harm to others" is a deep and abiding principle of a society, it will help determine the actions of the citizens of that society. If the principle and its implied implications are not fully comprehended, then that will also impact the actions of the citizens. The same is true for the core principle (s) of mathematics, which if fully understood, would also affect the actions of the participants. One of the problems with math is that the basic core principles are not fully comprehended with the consequent result of math phobia, confusion, and apathy — any of which will cause a deficiency in comprehension.
Many problems in understanding basic math begin with the basic premise presented to beginning students. The fundamentals of math are assumed to be...
