# Mathematical misconceptions

Alex Reinhart – Updated February 21, 2018

• Philipp, R. A. (1992). The many uses of algebraic variables. The Mathematics Teacher, 85(7), 557–561. http://www.jstor.org/stable/2796777

High school students struggle with the concept of a variable. Is an equation like y = ax^2 + bx + c a quadratic? What if a = x? Students struggle to separate the variables from the constants. There’s also a fascinating example problem, the “student-professor problem”: if “a university has six times as many students as professors”, students write this as 6S = P, since there are six students for each professor, rather than the correct S = 6P. They see the variables as labels for literal words rather than standing for the number of students and professors.

• Clement, J. (1982). Algebra word problem solutions: Thought processes underlying a common misconception. Journal for Research in Mathematics Education, 13(1), 16–30. http://www.jstor.org/stable/748434

Some of the research underlying the previous article, covering the student-professor problem. Revealing interviews with students as they work through it, switching between several mutually inconsistent interpretations of the algebraic variables on the way.

• Cohen, E., & Kanim, S. E. (2005). Factors influencing the algebra “reversal error”. American Journal of Physics, 73(11), 1072–1078. doi:10.1119/1.2063048

A systematic exploration of possible explanations for the student-professor problem, by using a variety of questions with different phrasings in a college physics class. Is it just that S and P suggest interpreting variables as labels, and x and y would be better? No. Do students simply follow the sentence structure in setting up their equation? Some (about 10%) do. Does extra context (so the student can check the numerical validity of their answer) help? No, it actually harms some students.

Suggests some students may have a “correspondence” interpretation of the equals sign, e.g. 6S = P meaning six students correspond to one professor.

• White, P., & Mitchelmore, M. (1996). Conceptual knowledge in introductory calculus. Journal for Research in Mathematics Education, 27(1), 79–95. http://www.jstor.org/stable/749199

Interesting approach: gives calculus questions to students in a series of forms, from a broad word-problem form that requires the student to figure out the appropriate variables and quantities to compute to a very specific form stating what to compute. Students found the more general form much more difficult, and had a “manipulation focus” leading to three main errors:

• “failure to distinguish a general relationship from a specific value” (in a problem giving the volume of a cube, students found it confusing to be told that V = x^3 at the same time as knowing that V = 64)
• “searching for symbols to which to apply known procedures regardless of what the symbols refer to” (e.g. substituting in the value of a variable before differentiating to get a rate of change, or trying to pick what to differentiate based on symbols, not the meaning of the problem)
• “remembering procedures solely in terms of the symbols used when they were first learned” (students wanted to calculate dy/dx simply because it had y and x, regardless of whether its meaning in the problem was appropriate)

The authors hypothesize that students “have learned to operate with symbols without any regard to their possible contextual meaning”, since this is easier and works well in narrowly chosen example problems.