By Peter Rosnick
Rosnick, P. (1981). Some misconceptions concerning the concept of variable. Are you careful about defining your variables? Mathematics Teacher, 74(6), 418-420,450.
I have since returned to mathematics, tackling a's, b's and g's left and right. But, the memory of the difficulties I had has increased my interest in the problems associated with learning about mathematics and its symbols.
Much has been written in the last decade concerning the reasons why so many students have difficulty with, and an aversion to, mathematics. Compelling arguments can be presented about cultural, political, and psychological factors that have caused the mathematical downfall of many a would-be mathematician, scientist, business person, and computer programmer. As valid as these arguments may be, we must not neglect the inherent difficulty of the subject of mathematics itself. The curriculum of mathematics, from elementary school to graduate school, follows a path of increasing abstraction. As the curriculum becomes more abstract, the symbols used become more obscure. For many students, as was true for me, unfamiliarity with mathematical symbols and the abstract concepts to which they refer breeds contempt for mathematics.
In this article, I will discuss a study that underscores the extend to which our students do not understand the use of letters in equations. This study is an extension of a body of research done by the Cognitive Development Project at the University of Massachusetts that has focused on students' ability to translate English sentences into algebraic expressions, and vice versa. The results of Previous research--based on diagnostic tests, videotaped interviews, and the microanalysis of those interviews--indicate the students (including, for the most part, first-year engineering students and/or students who have taken one or two semesters of calculus) have a very difficult time with these translations (see entries 1, 2, 3, and 6 in the Bibliography). In addition, we have found that the misconceptions that students have surrounding the use of letters in equations contribute significantly to this difficulty.
One of the problems on which much of our research has been based is the Students and Professors problem. It reads as follows:
Write an equation, using the variables S and P to represent the following statement: "At this university there are six times as many students as professors." Use S for the number of students and P for the number of professors.
Fully 37 percent of a group of 150 entering engineering students at the University of Massachusetts were unable to write the correct equation, S = 6P, in any form. The most common error was what we refer to as the reversed equation, 6S = P (Clement, Lochhead, and Monk 1981).
The error rate increases to over 73 percent when the problem deals with a 4 : 5 ratio rather than a 6 : 1 ratio. Again, a reversed equation is the most common mistake. Furthermore, students in those mathematics classes that are oriented more toward business and the social sciences did appreciably worse on these problems.
We have several taped interviews that support the belief that many students who write 6S = P believe that S is a label standing for students rather than a variable standing for number of students. They will read the equation 6S = P as "there are six students for every one professor," pointing to S as they say students and P as they say professors. Conversely, they will read S = 6P as "one student for each six professors" instead of the appropriate "the number of students is equal to six times the number of professors." That letters in equations can stand abstractly for number may sound obvious to the initiated, but it is apparently not at all obvious to the students.
The following question, a version of the
Students and Professors problem, was given to 33 sophomore and junior
business majors in my statistics course. Most of these students had
had two semesters of calculus. It was also given to 119 students in a
second-semester calculus course designed for the social sciences.
At this university, there are six times as many students as professors. This fact is represented by the equation S = 6P.
A) In this equation, what does the letter P stand for?
- Number of professors
- None of the above
- More than one of the above (if so, indicate which ones)
- Don't know
B) What does the letter S stand for?
- Number of students
- None of the above
- More than one of the above (if so, indicate which ones)
- Don't know
The results were quite startling. Over 40 percent of the 152 students were incapable of picking "number of professors" as the only appropriate answer in part A. Similarly, over 43 percent did not answer part B correctly. We believe that these results alone are significant and important. They support the hypothesis that students tend to view the use of letters in equations as labels that refer to concrete entities. P stands for "professor" or "professors," not the more abstract "number of professors." Furthermore, these results underscore the fact that students do have a great deal of difficulty with translations.
But there is more! If you notice, my choice of S standing for "professor" for answer (1) in part B seems to border on the absurd. In fact, I must confess to having felt a little whimsical and capricious in including it as an option. Imagine my surprise when 34 people (over 22 percent!) chose as their answer, "S stands for professor."
It is important to note that every person who chose "professor" for the answer in part B chose "none of the above" for their answer in part A. The latter is a consistent response in that those who would view S as standing for "professor" would also view P as standing for "student." Since that option was not provided, they chose "none of the above."
These results support a conclusion that we have believed for some time: that the tendency on the part of many students to write the reversed equation, 6S = P, is not only a common one but is one that is deeply entrenched. Our hypothesis is that most students who believe S stands for "professor" also believe that 6 students = 1 professor (6S = P) is really the correct equation. And they believe it so strongly that when presented with S = 6P, they assume that the meanings of the letters have been switched.
The implications for secondary school are important and suggest the following objective: Students need to develop a better understanding of the basic concepts of variable and equation. More specifically, they should be able to distinguish between different ways in which letters can be used in equations. They should learn to distinguish when letters are used as labels referring to concrete entities or, alternately, as variables standing abstractly for some number or number of things. The ability to make this distinction would prevent students from reading the equation S = 6P as "one students for every six professors" or worse as one professor for every six student," indicating the S for professor.
Having stated the objective, I recognize that reaching it is another matter altogether. In a related study, we experimented with several different teaching strategies with several different students, usually to no avail. Our conclusion was that there doesn't seem to be a quick solution to the difficulties our students face concerning the concepts of equation and variable (Rosnick and Clement, in press). It might even be the case that many secondary school students and, for that matter, college students have not yet reached the necessary level of intellectual development to be able to make that distinction.
An important step, however, is that we, as mathematics educators, should be aware of the distinction ourselves. In teaching word problems in an algebra class, I have been, in the past, somewhat careless about writing "P = professors" rather than "P = number of professors." As a result of my experience, I make every effort to say and write the latter and to verbalize the distinction.
As students progress from year to year in mathematics, the letters they use, like the concepts they are learning, become increasingly abstract and, to them, ambiguous. It is important that we stay aware of the difficulties that our students are having in trying to understand labels, variables, constants, parameters, and all the rest of the uses of letters. It is equally important that we become aware of all the conceptual pitfalls to which our students can succumb. After all, if we can't help our students understand the x's and y's, they will never know the joy of understanding a's and b's. More important, they will drop out of, or be inept in, mathematics, a subject that has become a prerequisite for more and more careers in today's world.
Do you know how many of your students would say "S stands for professor"?
Clement, John, John Lochhead, and George Monk. "Translation difficulties in Learning Mathematics." American Mathematical Monthly 88 (April 1981):286-90.
Clement, John, and James Kaput. "The Roots of a Common Reversal Error." Journal of children's Mathematical Behavior 2 (1979):208.
Clement, John. "Solving Algebra Word Problems: Analysis of a Clinical Interview." Paper presented at the annual meeting of the American Educational Research Association, April 1980.
Doblin, Stephen. "Why Did It Work, and Will It Always?" Mathematics Teacher 74 (January 1981):35-36.
Herscovics, Nicolas, and Carolyn Kieran. "Constructing Meaning for the Concept of Equation." Mathematics Teacher 73 (November 1980):572-80.
Rosnick, P., and J. Clement. "Learning without Understanding: The Effect of Tutoring Strategies on Algebra Misconceptions." Journal of Mathematical Behavior, in press.
Research reported in this article was supported by NSF Award No. SED78-22043 in the Joint National Institute of Education-National Science Foundation Program of Research on Cognitive Processes and the Structure of Knowledge in Science and Mathematics. I would like to acknowledge the influence, suggestions, help in editing, and support of John Clement of the Cognitive Development Project, whose extensive work in this area provided the foundation for this article.
Reprinted with permission from The Mathematics Teacher, copyright September, 1981 by the National Council of Teachers of Mathematics. All rights reserved.