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 apparent when, in fact, they are not. An example is simple to illustrate. Most students are conditioned to make statements such as two plus two equals four, but the basis behind those statements is never completely understood. If one is asked, "What is two plus two?" the answer will invariably be four, and the student will be congratulated for that statement. Later, when we begin abstract algebra, when asked to add a plus b, we usually get a stutter and an incorrect response, usually ab. Why? A more profound question can be why one can add 1 + 2, 3 + 4, and so forth, with the concurrent explanation that we cannot add a + b? Another question should be how do the symbols 1, 2, 3 differ from the symbols a, b, c — and why do they act differently, if in fact they do?
If these questions are carefully examined, the discrepancies become a product of the lack of clarification of basic terms.
a. What is a number?
b. What is a numeral?
c. How are they different — and how can you distinguish one from the other?
The answer to similar simple questions will begin the process of removing some of the mystery that math imposes on many students. That will also help to answer several questions.
a. What are addition and subtraction?
b. What are numerals and what are numbers?
c. Why can one add 1 + 2 — but not a + b?
d. How is counting different from adding?
e. What are multiplication and division?
f. The importance and uses of zero and one in mathematics.
g. The understanding and use of operations in mathematics.
h. What is closure and why is it important?
i. Understanding the correct mathematical symbols and their uses.
Those are just a few of the problems with understanding basic math that will demystify some of the tenets of basic mathematical concepts. Many of the consistent problems encountered by math students can be mitigated by a thorough grounding in the basic, concise, workable definition of a number and how it is used to create the system of numbers in use today.
This book will mute many of the problems encountered by new math persons.
It is suggested that you create a dictionary of terms to be used for easy reference as needed.
Comparison of Math and English
Before we continue, a very important but little stressed point of mathematics is a concept called a truth table, which basically states that:
• you cannot deviate from the topic;
• only verified truths can be considered; opinions do not count as verified truths and cannot be considered;
* Example 1: If the topic is about girls, then boys are not to be considered because they are not girls.
* Example 2: If the topic is about students at school, both boys and girls can be considered because they are students.
• in general, if P is a statement and Q is another statement and ;
• if P is true and Q is true, then both P and Q are true (true conclusion);
* Example: If P is a girl is true, and Q is a girl is also true, both P and Q are girls are also true.
• if P is true and Q is false, then both P and Q are not true (false conclusion);
* Example: If P is a girl is true and Q is a girl is false, then both P and Q are girls are not true and the statement that both P and Q are not girls (boys) is also not true.
• if P is false and Q is true, then both P and Q are not true (false conclusion); or
* Example: If P is a girl is false and Q is a girl is true, then both P and Q are girls is not true and the statement that both P and Q are not girls is also not true
• if P is false and Q is false, then both P and Q are not true (false conclusion).
* Example: If P is a girl is false and Q is a girl is false, then both P and Q are girls is false.
The statements in are about girls, the subject is about girls, the conclusion must be about girls, the true conclusion must be about girls and they are a basis for the concept of addition. A very important concept of mathematics, illustrated by the above truth table, is that "one must stick to the subject."
An attempt will be made to flush out the verifiable truths in the following presentation of the structure and properties of the basic system of mathematics. Notice that this attempt is only a beginning and will be used to construct the system of mathematics.
Mathematics is a language — just as there are many different languages in use throughout the world today. The differences in languages are attributed to the differences in the original sounds (sounds of the alphabet) of which the language is composed. The different original sounds determine the sound of the combinations (words) that can be uttered using the basic sounds of the language. This can easily be corroborated by comparison of the sounds by use of the basic sounds of any language. An example can be illustrated by the use of the sounds in one language that cannot be produced using the basic sounds of another language, which partly determines the different words used to describe similar objects.
You may wish to experiment by using the alphabet of another language to try to produce certain sounds that are easily produced using your native language. If the sound of the alphabet in English B (bee) E (eee), T (tee) can be used to create the sound bet, and using the sound of those same alphabets in another language were B (bae), E (eae), T (tae), could the sound bet be produced? Notice that the words that can be produced are determined by the sounds of the alphabet. The languages of the world are different in part because of the differences in the original sounds of the native language that determines the words that can be produced using those sounds and the objects to which the sounds refer.
Notice also that there is a difference between the alphabet and the symbol that represents that alphabet. Can you distinguish the alphabet from the symbol that represents the alphabet? If asked to give me an A, and you yell out the sound of A and are given an A written on a piece of paper, one is the actual object and one is the symbol that represents that object. Notice also that both can be interchanged and used in communication of ideas, but the circumstances of their use determine which form will be used to convey the desired information. One can write information in a book or broadcast on the radio or other device. Notice that the medium determines the method.
If mathematics is treated as a language, it will also conform to the basic communication modes as shown above with at least one significant difference. In languages, words are assigned to objects, but in math, the object or symbol determines the word. In English, the word dog is assigned, and in German the word Hund is assigned to [??]. In mathematics, the symbol 1 determines the name assigned to it in any language. In English, it is one, and in German, it is ein. It will be called by different names in different languages but the symbol will remain constant, hence the universal language.
There are basic differences between the language of mathematics and any language.
• Basic symbols and combinations can produce only one outcome in math, but they can produce more than one outcome in a spoken language.
• The combination bat can be assigned to more than one object in English. In math, only one symbol can represent any object (1, 2, 25, 1001, etc.).
• In languages, only a finite number of combinations can be produced without repetitions. In math, an unlimited number of combinations can be produced with no repetitions.
• In languages, combinations can have more than one meaning. In math, all combinations have only one meaning. (5 + 3) can have only one meaning, but bat can have more than one meaning or object to which it is assigned.
Mathematics is a language similar to any other language, and it must have some basic symbols that represent the alphabets of the system. Just as determined in spoken language, you will discern that a difference exists between the alphabet and the symbol that represents the alphabet. Can you determine the alphabets of math? Can you determine the symbol that represents each alphabet? If a numeral is the name of a number, and 2 is a numeral, then what represents the number 2? Conversely, if 2 represents a number, then what represents the numeral and are they one and the same thing? That is a relationship similar to a chair and a picture of a chair where the chair is the actual object and the picture is a representative of the chair but is not an actual chair.
To begin any system or project, one must have a basic plan or idea. Our basic plan will begin with symbols that are representative of groups (or members of a sequence) of items. The basic symbols (the alphabet of mathematics) are named 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The symbols are called numerals. Each numeral represents a group (or unit of distance), and each group (or unit of distance) represents items in a sequence. Each numeral represents one group (one unit of distance) of items, which are numbers. Therefore, the numbers represented by the symbols (numerals) 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 are 1(0), 1(1), 1(2), 1(3), 1(4), 1(5), 1(6)1(7), 1(8), and 1(9). You will also notice that numbers and symbols (numerals) that represent them are different from each other. Numbers (actual item) consist of two parts, and numerals (name) consist of only one part. The names of the numbers are 0, 1, 2, and so forth, but the numbers are 1(0), 1(1), 1(2), and so forth.
