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UMI: GRAPHICS OMITTED (...)
A few years ago, I noticed a student in an elementary algebra course having particular difficulty on an examination with such items as the following:
Find the equation of the line through the points (3, 2) and (-1, 4). (5 pts.)
Her difficulty surprised me, since I had thought that she was comfortable with the material before the examination. She continued to struggle with the test, turning in her paper at the last minute. When I asked her how she felt about the test, she replied that she had done what was asked on the problems but that she had no idea what to do with all those "pints." Her reply left me speechless. She had interpreted "5 pts."-that is, the value of each item, 5 points-as additional information-5 pints-to be used in the item!
Granted, the interpretation in this example was not essential to the mathematics involved and the miscommunication was ,easily corrected, but it opened my eyes to the possibility-perhaps the certainty-- that students do not see what I see, or intend, in my notations. The NCTM's Curriculum and Evaluation Standards for School Mathematics (1989) advocates a decreased focus on symbol manipulation in high school algebra, encouraging instead a view of algebra as a "means of representation." If algebra is to be a means of representation, teachers must be explicit about what is being represented. I thought that I was representing the value of each test item; my student thought that I was representing additional information to be used in the task. How often do such miscommunications as one this occur in algebra classes? What do students think is being represented by algebraic notation?
Investigating students' use of algebraic notation is not a new idea. Perhaps the most familiar example is the "student-professor" problem:
Write an equation using the variables S and P to represent the following statement: "There are six times as many students as professors at this university." Use S for the number of students and P for the number of professors. (Clement, Lochhead, and Monk 1981)
Although this question appears to be a straightforward one that is well-defined in the sense that the variables are identified and labeled, the "reversal error" of writing 6S =...