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.
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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
- 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.
- 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.
-
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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