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

seventeen 15-year-olds:

 

          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...."

 

 

+++++++++++++++++++++++++++++++++

 

RAR comment:

 

     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-

graphy.

 

     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

discussing fractions.

 

     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

matter.

 

     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

solution.

 

     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)