Matthew Leingang tweeted about a webpage on common errors in college mathematics. This list does a reasonable job cataloguing common mistakes, but one missing error is “inadvertently dividing by zero.” For instance, a student might “simplify” the equation “x2=2x” by dividing both sides by x to yield “x=2.” This division is allowable as long as x is not zero, so the student is implicitly assuming that x cannot be zero. Unfortunately for the student, he assumes away the solution x=0.
A second problem—I do not know if it should go on this list— is that
students everyone seems to be under the assumption that “if I wish it were true, then I’ll just assume that it is true without thinking about the evidence.” Perhaps students do this when they distribute square roots. This is also common in political discussions, as when Joe Wilson yelled “you lie” at President Obama (the proposal clearly states that illegal immigrants cannot receive health benefits. I am not taking a side on this issue, but am only pointing out a recent example of “if I wish it to be true, it must be true”).
It is interesting to see all of the errors enumerated and explained, although I am not sure how useful it is. On the page, the author writes of a professor who warned students that an exam would contain many extraneous solutions, and that he would penalize them heavily if the students included extraneous solutions in their answers. It sounds like this did little to prevent the students making these errors.
When I was a graduate student at the University of Wisconsin, our College Algebra course had a theme called “vital errors.”[*] These were a list of misconceptions that algebra students often make. For example, one vital error is the distribution of powers, such as (x+y)2=x2+y2. Our students were given a couple tests on problems that contained these “vital errors.” In spite of knowing that every question on these tests would contain one of the 7-8 vital errors they had been warned about, students still did very poorly.
It seems like telling the students “these are the errors that you might make, and so you will be punished heavily if you make them” is not effective; at least, it was not effective at Wisconsin and not for the professor referenced on the page. This begs the question: what is an effective way to get students to avoid making these errors?
I welcome proposals in the comments. My only contribution is that students need to understand arithmetic before they can abstract it to algebra. If I were teaching college algebra now, I would begin with a unit on arithmetic. Only after students felt comfortable with, say, the distributive law with numbers would we move to variables.