Chapter 8. Three Skills For Algebra

Volume 2, Three Skills for Algebra

Talking about three skills and illustrating them with examples may be enough to go from a mastery of arithmetic to a mastery of algebra. In learning to talk, write, argue and possibly do arithmetic, we have mastered harder skills. In elementary school, we mastered the first two skills: the ability to talk about numbers and quantities and the ability to describe calculations. The third skill depends on the first two. The three skills are as follows.

• First, we can talk about numbers and quantities without doing any arithmetic. For instance, numbers and quantities may be big, small, known, measured, never known, changing or unchanging, private, top-secret, confidential, embarrassing, or simply forgotten. A number, measurement or quantity may be known to you but not to me. We can speak about numbers and quantities in many ways. Talking about numbers and quantities is an ability we all have. It is a part of mathematics that does not require us to do arithmetic. There is more to mathematics than just doing arithmetic or being given a formula and numbers to use in it.

• Second, we can describe calculations which we want to do or avoid or have someone else do, without doing any arithmetic. The description gives a recipe or a formula for doing a calculation. The description can be done with words alone or with shorthand notation. This shorthand notation is worth a thousand words. The first service of mathematics to other subjects lies in the description of calculations that can be done or repeated as needed. There is more to mathematics than just doing arithmetic or being given a formula and numbers to use in it.

• Third, we can change the way numbers and quantities are computed (or measured). Rules or properties of arithmetic tell us when different calculations or measurements give the same result. These rules are described using shorthand notation. That gives a second role to the shorthand notation. In the computation of numbers and quantities, we may replace a calculation by another, when both give the same result. And in the description of calculations, we may replace a calculation by a shorthand symbol that represents its result, and vice-versa. These replacement ideas, illustrated below with examples, allows us to compute or describe different ways to calculate a single number or quantity.
  Algebra or the manipulation of formulas is concerned with the shorthand description of different computations and with when one description can replace another. Description of one calculation can replace the description of another in any circumstance where the two calculations give the same result. Such replacements can be made one at a time, or one after another. There is more to mathematics than just doing arithmetic or being given a formula and numbers to use in it.

The description of calculations that might be done is a first service of mathematics to other subjects. The creation of new calculations by changing old ones is a second service to all subjects using arithmetic. Mathematics after arithmetic is based on the above three skills and the ability to read exactly rules, patterns and definitions. For the latter, see the previous chapters on logic.

Notes

1. The first skill, our ability to talk about numbers and quantities, is widely known. We can say whether or not a number is known, forgotten, unknown, small, large, changing or varying, constant or unchanging, confidential and so on. Thus we can talk about and describe numbers and quantities. This can be done before the very visible, but sometimes misunderstood, symbols, letters and written shorthand of algebra, is introduced. Talking about numbers and quantities represents a easily-spoken element of algebraic thought apart from the algebraic way of writing and recording such thoughts.
   
2. A number or quantity which may change in the circumstances of interest to us is called a variable. The common idea that all variables have to be given by letters has mislead many. As just suggested, talking about variables, that is numbers or quantities which may change or vary, can be done without from any reference to letters and symbols. That is the notion of a variable can be clarified or explained before any linkage to algebraic shorthand or symbols used to write and record calculations and further parts of algebraic thought.
   
3. How to compute the area of a rectangle can be described with words alone or with a formula A = WL. In contrast, the compound interest formula A = P(1+i)ⁿ and even more so, the quadratic formula

   \[x = \frac{-b+\pm\sqrt{b^2-4ac}}{ 2a } \]

   describe calculations in an algebraic and symbolic way. It would be a horrible exercise to describe what these formulas mean, do and represent with words alone and no symbols.

Learn More - Online Postscript for this chapter

See the essay What is a Variable. It will show you how to talk about variables and constants before and besides symbols. That represents the first skill in action.

Chapter 14 here on the compound interest formula introduces the forward and backward use of formulas. All the rules and patterns met in mathematics, science and logic will be used forwards and backwards. You may think of that as unifying theme. Talking about it recognizes it and points to its appearance, again and again in mathematics, science and logic. Using formula backwards deliberately is a sign of intelligence.

The following chapters will talk about the above three skills, including the forward and backward use of formulas. I am not sure whether to classify the latter as part of the third skill, or as a fourth skill. You decide.

Talking about the three or four skills above, and talking about finding counts and sums by forming and adding subcounts and subsums, and talking about finding products by forming and multiplying subproducts expands the role of words in both arithmetic and algebra. Earlier is better than later in mathematics skill development.

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