Mathematical Errors

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.

Other ideas?

[*]
My friend Darren pointed out that this term means that these errors will cause you to die, as in a “vital injury.” Either that, or it was important to make these errors in order to live; I do not think this was the intent, though.

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4 Responses to “Mathematical Errors”

  1. ericakathryn Says:

    Maybe you could shame them: “If you make this mistake, you’re totally ordinary, just like everybody else.” Actually, this might work, if they think it’s a funny thing to say and thus remember it. But if it worked that way, it would only work once.
    I don’t know if it’s possible to erase this common mistake making just by talking about it. I’ve tried to do little “grammar fact of the day” things in my FYS, and they’re still making the errors I point out. (Sigh.)

    • bretbenesh Says:

      Yeah, I have pretty much decided that little interventions do not work for me. I think that big interventions are needed. Unfortunately, this would mean that “Calculus” class would turn into “algebra” class.
      Actually, that might be a good idea…
      Have you found anything that helps with the FYS grammer problems? Also, what sorts of errors are they making?

      • ericakathryn Says:

        Oh, the usual stuff like effect/affect, it’s/its, etc. But also, the one I’ve been harping on lately is pronoun-antecedent agreement, particularly in number. Like, “A person’s vocation is their calling.” Aaaagh! Oh, and “reason…is because…” (These last two are things we say in everyday speech, and I don’t object to them. But academic writing is different.)

  2. Anonymous Says:

    Student’s perspective
    Hey Bret,
    Scott Sorheim here…since I am not an educator, I will give my perspective, both from my days as a college student and now as a student of life (terribly cliche!).
    It seems to me that these “lists” should not be taught or even handed out at the start (or even the end) of a class (and from what I can tell, you concur). They should be developed by each individual student through a process of discovery. They have no meaning unless the student understands why they don’t result in a correct answer. And seemingly, if they know how to arrive at the correct answer, the “vital error” would never be made in the first place.
    I am now a computer programmer. If, when I started programming, I was handed a list of “what not to do’s”, they would have been completely meaningless to me, having no idea of when “not to do” those things. For example, if I was told something like “don’t forget to increment your counter to avoid getting stuck in an infinite ‘Do’ loop”, it would make no sense until I understood the application of a ‘Do’ loop, and what the negative impact of being stuck in that loop even meant.
    However, through “trial by fire” I have found those items that are impediments to me achieving my desired outcomes and created my own “what not to do” list. This also lines up with my “unofficial” method of “lean” learning, which is that information is acquired at the point of “customer” demand (where the student is the “customer”). Any irrelevant information, like it or not, is discarded by the student, even if the student would like to retain it. It is retained when it has relevance to some application for their current needs (in my case, solving some customer programming requirement).
    Not sure if any of that makes sense, but it does in my mind! I have additional thoughts, but seems like I could talk it out better than type it out. Good luck!

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