Chapter 3
Two Notions of a Variable
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Algebra
The concept of a variable is not simply described in most algebra texts. A
clarification follows. This clarification is not for the expert, but for the
novice. The specialized use of the term variable should not be the first
one given in an algebra text or dictionary, mathematical or not.
Words and Notions before and then Besides Symbols
A First Notion: Variables Without Symbols. We can talk about numbers
and quantities, and among them identify those which are changing or varying, and
those which are constant, known, unknown, given, confidential and so on. Here a
number and quantity which may vary, or take many values in the circumstances of
interest, is called a variable. We can talk about variables without using
the shorthand notation, that is, letters and symbols, employed in algebra.
Second
Notion: Variables with Symbols. Formulas use shorthand notation, symbols
or letters, to represent numbers and quantities. This suggests that when a
symbol or letter is the shorthand notation for a number or quantity which may
vary, we may also call that symbol or letter a variable.
Remark 1. The
association of symbols and letters with numbers and quantities which may vary is
so much a taken-for-granted part of the algebraic way of writing and thinking
(amongst the mathematical adept) that the observation that we can talk about
variables apart from symbols has been overlooked. But this symbol free notion
clarifies and refines the concept of a variable in mathematics.
Remark 2. The notion
that a variable may be given by a symbol, that is shorthand
notation (or a placeholder) for a number or quantity which may change,
relies on our ability or skill to talk about numbers and quantities and also on
our ability or skill to employ shorthand notation (symbols) for them in and
possibly outside calculations[4].
Two Notions of A Constant
The term constant is a reference to a number or quantity which will
remain unchanged, or is fixed in the situation of interest. A symbol or letter
is called a constant if it stands for a number or quantity not expected
to change in the situation of interest. There are two notions of a constant, one
with and one without symbols.
[4] Pages 44 and 46 in the 1965 book Secondary
School Mathematics by J. J. Kinsella, published by The Center for Applied
Research in Education, Inc., New York, indicate efforts in modern math
instruction to define a variable. The definitions described formally introduce a
variable as a symbol acting as a place holder for a number, specified or
not, and possibly restricted to some set of allowable values. The latter notion
associates the notion of a variable with a symbol, and does recognize the second
symbol free notion identified above. Moreover, understanding the latter requires
a previous mastery of IT, the algebraic way of writing and thinking, and
possibly some set-based concepts, and so cannot be an introduction to IT. (The
efforts essentially describe the role of algebraic thought with the formality
and precision demanded by modern mathematical thinking. I suspect the efforts
also represent the historical understanding of algebraic thought by osmosis. The
three skills represent an advance on this situation.)
[5] Kinsella's work also provides a perspective, still
pertinent today, on some of the problems and issues facing mathematics
instruction. It will be mentioned again.
Postscript (Sept 96): See also the article On the Meaning of Variable,
pp 420-427, Mathematics Teacher, September 1988, by A. Arcavi and A. H.
Schoenfeld. Included is a conclusion that the meaning of variable is
variable. Several meanings of the term have to be mastered.
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