William Zlot, a math teacher in New York State, once described student misconceptions in mathematical thinking.
These misconceptions seem to all revolve around an understanding of addition and its symbolic representation. But, it may be more.
- The sum of the whole numbers 3 and 4 is seen to be 7, but not 3+4
- The rational numbers 1/5 and 2/5 can be added, whereas the rational numbers 2/3 and 3/5 cannot be added
- The terms x and 2x can be added, whereas the terms x and y cannot be added
It may be a language problem. That is, the teacher is using a broad version of addition, while the student interprets "addition" as meaning "I must turn these two things into a single result." And, to this student, the notions of 2/3 + 4/5 and x + y are not "single" results.
Relative to 2/3 + 4/5, the student might reply that it is possible to add them if you first get a common denominator. This same student might wrongly feel the same-but-different change might be needed relative to the expression x + y.
Are these misconceptions mis-misconceptions on our part as teachers? That is, our internalized language may not match the students, and the latter bears the brunt of the blame.
Then, we also must face the old rule: "To multiply a number by 10, just add a 0 to it." My college students still recite this rule...yet, they are perplexed when I ask: What is pi x 10, or 1/3 x 10, or y x 10? As you might imagine, the rule produces some interesting results (and discussions).