Notes from the NCTM volume, "Research Issues in the Learning and Teaching of Algebra" 1989, a symposium volume edited by Wagner and Kieran.
I begin with a lengthy quotation from Carolyn Kieran's paper, The Early Learning of Algebra: A Structural Perspective. On page 40 begins the section, Variables.
"High school algebra usually begins with instruction in the concept
of variable. In elementary school, children have already seen placeholders
in open sentences and used letters in formulas such as the area of a
rectangle. However, their past experiences cannot easily be related to
the many uses of variable to which they are exposed in high school
algebra. Usiskin (1988) has described some of these many uses of variable
and has related them to the different purposes of algebra.
"According to Usiskin, if we consider algebra as generalized
arithmetic, then variables can be viewed as pattern generalizers (e.g., in
generalizing 3+5=5+3 to the pattern a+b=b+a) and algebraic skills are
centered on translating and generalizing known relationships among
numbers. If we consider algebra as the study of procedures for solving
certain kinds of problems, then variables can be viewed as unknowns (e.g.,
in translating a word problem into an equation) and algebraic skills
involve simplifying and solving. If we consider algebra as the study of
relationships between or among quantities, then variables are either
arguments or parameters (e.g., in finding an equation for the line through
(5,2) with slope 9) and coordinate graphs are often used to represent
these relationships. In this conception of algebra, variables truly vary.
Finally, if we consider algebra as the study of structures such as groups,
rings, integral domains, fields, and vector spaces, then variables are
arbitrary objects in a structure related by certain properties.
"Thorndyke (1923) suggested in the 1920s that different letters be
reserved for different interpretations of the variable. However, this
suggestion was never seriously considered. Thirty years later, it was
still being remarked (Van Engen, 1953) that the symbol x was being used in
many different ways, ways which were causing confusion for students. But
the new math movement with its emphasis on unifying concepts altered this
situation. Ironically, it promoted just the opposite of what Thorndike
had earlier suggested. It encouraged the teaching of the concept of
variable in its most general form, right from the start. Variables were
considered as one of the unifying ideas of the high school algebra
curriculum, with all algebraic letters being referred to as variables.
Thus, it is not surprising that the same student confusion that was
pointed out by Thorndike and by Van Engen should be noted again in the
late 1970s. Matz (1979) remarked that lumping together symbolic
constants, parameters, unknowns, and unconstrained variables as simply
"variables" draws attention only to their common abstract nature. Such an
overly general concept of a variable, according to Matz, blurs dis-
tinctions that affect how the symbolic value is manipulated, by obscuring
restrictions about exactly how and where the variable varies.
"The resulting confusion that students experience over the different
ways that a single letter variable can be used in algebra often leads to
erroneous interpretations. Firth (1975) gave the following task to
If x is any number
(a) Write the number which is 3 more than x;
(b) Write the number which is 5 less than x;
(c) Write the number which is twice as big as x;
(d) Write the number which is 50% bigger than x.
"He found that only 10, 11, 9, and 7 students, respectively, answered
the parts correctly. Firth noted that 5 of the 17 students solved the
entire task incorrectly by first choosing a value for x and using that
value throughout the exercise. It has been suggested that this error may
be linked to the student's difficulty in considering x+3 as a final
answer... The operation of adding 3 directly to the letter x can only be
expressed in terms of the process: The process is also the product. ...
"A large-scale study of some of the various ways in which students
use algebraic letters was carried out by Kuchemann (1978, 1981) in 1976. As
part of the Concepts in Secondary Mathematics and Science (CSMS) project,
Kuchemann administered a 53-item paper-and-pencil test to 3000 British
high school students, aged 13, 14, and 15 years old. He classified each
item into one of six levels of interpretation of letters according to the
minimum level required for successful performance. Unsuccessful answers
were classified into a lower level of interpretation. The six levels that
Kuchemann used in the analysis of his data, based on levels originally
developed by Collis (1975), are the following:
(a) Letter evaluated: The letter is assigned a numerical value
from the outset;
(b) Letter not used: The letter is ignored or its existence is
acknowledged without giving it a meaning;
(c) Letter used as an object: The letter is regarded as a
shorthand for an object or as an object in its own right;
(d) Letter used as a specific unknown: The letter is regarded as
a specific but unknown number and can be operated on directly;
(e) Letter used as a generalized number: The letter is seen as
representing, or at least being able to take on, several values rather
than just one;
(f) Letter used as a variable: The letter is seen as representing
a range of unspecified values, and a systematic relationship is seen to
exist between two such sets of values.
"Kuchemann found that, even though the interpretation that students
chose to use depended in part on the nature and complexity of the
question, most students could not cope consistently with items that
required the use of a letter as a specific unknown. They erroneously used
one of the three lower-level interpretations instead.
"The result of the CSMS algebra research led to a follow-up study...."
Ms. Kieran distinguished four "concepts" of algebra, in one of which
"variables truly vary." I can see them wriggling on the page right now,
unlike the masked ones that are merely "unknowns" and don't much vary --
though they can have dopplegangers when a quadratic equation is involved.
How is it known in advance, I wonder, when (say) trying “to solve” an
algebraic equation, whether x represents (c) a "specific unknown" or (d) a
"generalized number" able to 'take on' more than one value? That's a
poser that Emmy Noether might have had some trouble with. Try x in
x2+y2=a2, for example; what sort of variable is x? Type (f)? But how
about when a=0? Does it change? Sure does; it lurches into type (d), I
think. Or is zero then a double root? Two zeros placed one atop the
other, as anyone with enough x-ray vision can plainly see. Damn it all,
where is the question?
In the earlier part, where "x is any number" forms the mysterious
hypothesis, students are asked to write down a number "which is twice as
big as x". Really, is x any number? If it can be any number, then surely
it can be six, so the answer is 12. Well, that's one interpretation,
though quite wrong, according to Kieren, and representative of some
serious misunderstanding of algebra.
The man seeking to test students' understanding of something he calls
algebra is asking questions as rigid as all those conventional drill
exercises he has been damning all his career. He thinks his wording has
obvious meaning, and that students who haven't reached it, i.e. haven't
learned the expected response, have failed to learn something about
algebra, maybe at level two. But it is not until the notion he has in his
own mind, about x being 'any number', is converted into a question with
meaning involving x as something that matters, that he can decide what the
student does understand about mathematics, and not about his lexico-
For this is what is interesting about this excerpt, which goes on in
the same manner for pages and pages and for years and years: its total
abstraction from mathematical questions written in English sentences. The
author has the notion that "variable" is a "concept" of some difficulty,
that has to be learned in a vacuum somewhere. In some other vacuum she
doubtless learned, and maybe teaches, about truth tables and quantifiers,
but never thought to relate all that to English prose, just as she here
never thinks to relate English prose to "algebra." This business of
"concepts" is heavy stuff in math education, as in education circles
generally. But mathematics is not about concepts, whether in four layers
or six. The mischief here is obvious to anyone who understands mathe-
matics, however ignorant he may be of the total literature of education,
and what it is doing to his children in the schools. In short, I would
hate to have Carolyn Kieran teaching my children algebra in the public
schools, but I'm afraid I did.
This conference was financed by the National Science Foundation,
and the participants were professors of education, who took time out from
a busy schedule (as the saying goes) to talk and listen to such nonsense
at the expense of the public. It wouldn't be so bad if they didn't all go
back and teach variables to future teachers of mathematics.
From the same conference volume, a paper by a certain John A. Thorpe, of
Stony Brook, contains the following (p.23):
"For example, when numbers are expressed in decimal form, the order
relations between them are transparent. Children would internalize number
facts like '2/7 is less than 1/3, which is less than 3/8' (.286 < .333 < .375)
much sooner than they do now."
This is part of his argument against learning much about fractions in
school; and indeed it is true that decimal fractions are easily compared
as to order. (2/7 is not .286, by the way, but is close enough for
Thorpe's purpose here, a purpose that in fact has now created a generation
of students who do use "2/7" and ".286" interchangeably.) But does
Thorpe really imagine that anyone but an idiot savant has internalized
"2/7 < 1/3 < 3/8"? It doesn't take me long to determine the truth of
those inequalities, but I can't say I know them the way I know 5X9=45, nor
would I expect children to internalize them either. It is not this that
we are teaching when we teach children about fractions; and converting
fractions to decimals has virtually nothing to do with our purpose in
In another place Thorpe condemns the following problem as not
"real-life": Find two consecutive integers whose product is 110. He
calls it a puzzle-problem, and would rid all school mathematics of such
problems, just as Tolstoy would have abolished Aida as not true-to-life.
People don't really sing their conversations, do they? Or wear such
costumes in the streets of Petrograd?
Professor Thorpe's acquaintance with mathematics is probably
best illustrated in the following insight, Footnote 2 on page 23:
"The use of zero in this context seems to have caught on in
school algebra. I, for one, cannot get used to it, and I do not think its
use here is pedagogically sound. The concept of zero as a real number
is sufficiently mysterious without compounding the problem by using the
same word to describe an entirely different concept. I marvel when
anyone seems to be comfortable making a statement like 'Two is a zero
of x2 - 4.' "
Comfort is of course a matter of familiarity. It is no curiouser to
call two a zero than to call it a root, as if it were a carrot or radish.
All that Professor Thorpe is confessing here is that he has spent somewhat
less time with contour integrals than with pondering the factorization of
the difference of two squares. (Thorpe later became Executive Director of
the National Council of Teachers of Mathematics, and in this capacity he
defended Secretary of Education Riley’s 1999 announcement of the notorious
“exemplary” and “promising” NSF-sponsored math textbook series, against
a notorious polemic from some 200 mathematicians.)
I wonder: Is the difference of two squares formula still true when
one of them is the square of a number and the other is the square of an
as-yet-unclassified "variable"? I would pose this proposition (that the
formula remains true) as a theorem for Professor Thorpe to prove to me, if
I could understand what it said. Fortunately I do not need to understand
this, nor any of Professor Usiskin's four interpretations of the word
"variable", for seventh grade purposes. Or for any other purpose, for that
We don't solve equations, after all; we solve problems. Problems are
written in English, and when a word like "variable" turns up it turns up
as the name of something easily identified as a feature of the problem.
When an equation turns up, it is as a clause in a sentence, not
necessarily as a thing to be solved. It is the sentence (generally in
some conventional abbreviation) that expresses the problem wanting
This distinction is at the root of all the palaver about the difference
between 'identities' and 'equations'. There is no difference, and the
digging around in children's subconscious for reasons for their
misunderstanding is futile; the digger should look in the mirror.
What we call an identity is one sort of proposition involving an
equation and the replacement values that render it true; and what we call
an equation -- I'm speaking of school algebra problems here -- is the very
same sort of question involving an equation and the replacement values
that render it true. It is the custom of schoolbooks to call the thing an
equation if it has one or two roots, and an identity if it has a lot more.
It happens that the analytic method ("Suppose x satisfies the equation;
then...") is convenient in practice for smoking out those values in the
case of some equations, but not for others, but this is another story.
The logical structure of what is being asked must be expressed and
understood in English before there is any value in teaching techniques.
Conventionally abbreviated nomenclature should not be confused with deep
questions of philosophy, and occasions for NSF grants.
I recommend the entire volume to anyone who is new to theory
and research in math education, as I have been until the past year.
Ralph A. Raimi
10 February 1997
(modified 31 March 2001)