LOS Mistake

My new favorite game is trying to classify math mistakes. (See: Classifying Math Mistakes)

Right now, I see three big categories of mistakes:

  1. Mistakes Due To Limited Applicability of Models
  2. Mistakes Due To Applying Properties of a Familiar Model in an Less Familiar Situation
  3. Mistakes Due to Quickly Associating Something In Place Of Another

I think this is pretty clearly an example of the third category. The student’s brain was working hard, and they swapped the 10 and the x.

These sorts of mistakes are interesting to me because I think a lot of teachers see these and say, “Oy, this student thinks that you can just swap out the x with the angle.” Or others would say, “Oy, this student has no conceptual understanding of trigonometry.”

Nah. This kid needs more practice with the Law of Sines so that you’ve got enough brain power available to pay attention to all the moving parts while you’re trying to solve the problem.

There’s something else that’s interesting about these associational errors, and it’s about the associations that students make. Isn’t it interesting that the x*sin(10) is more familiar to this student than 10*sin(x)? Maybe this also points to the need for more practice that mixes up missing angle and missing sides Law of Sines problems?

  • Owen Thomas

    i just found this blog. this looks like a great project so far.
    i might have more to say if the robots let this much through.

  • Owen Thomas

    is that x = 46 *degrees* i see here? because *that*’s
    kind of interesting. one of the most persistent and
    harmful errors: failure to “sanity-check” at every
    opportunity. the student appears to have noticed
    that since we began with “sin x” we should expect
    an “angle” answer. so far so good. but the “angles”
    have “cancelled” out of the last calculation (in some
    sense) and we would maybe hope for this intuition
    to’ve been more helpful (anyhow, long story shorter:
    the “real mistake” here in my eyes is *failing to check
    the work*). probably “check your work” can only be
    instilled as a matter of routine by routinely requiring
    that such work be *defended* in some serious way.

    • The_L1985

      Your phrase “sanity-check” makes me think of Dungeons and Dragons. “You enter the next room of the dungeon. On the wall is a difficult math problem. Roll for sanity.” 🙂

      And yes, failure to notice that “angle measure in degrees” isn’t the sort of answer we’re looking for means that somebody’s rushing through and not checking the answer!

  • Wonder what this kid would do with proportions that didn’t involve trig functions.

    Also wonder if the student thinks that sin(42)/sin(10) = sin(21/5)

    • The_L1985

      That’s another nasty little sine error that I’ve seen in the past. To the point that the Laws of Sines and Cosines, and the laws for double- and half-angles, became my trig class’s “Trig Bible.” (i.e., these were as important to understanding trig as the Bible is to Christians.)

      • Owen Thomas

        as to half- and double- angles, all i’ve got time to memorize is e^{it} = cos(t) + i*sin(t). the rest is left as an exercise (and there’s sure as heck no harm in *doing* it over and over a few hundred times until, whattaya know, you’ve learned it after all). so you could say that’s *my* “trig bible” (the rest is commentary). [this model would have sin^2(x) + cos^2(x) = 1 as its “ten commandments”, i suppose…]

    • Owen Thomas

      i’m guessing that it’s some other bug since sin(21/5) < 0. but suchlike "cancel"-as-if-at-random pathologies are indeed common. a lot of the time the real tragedy is that they could've just copied the code into their calculators before they munged the code.

      • Owen Thomas

        “in degree mode”, one has sin(21/5)>0 but 12sin(21/5) not= 46. i very nearly forgot about this “mode” business which shows how long it’s been since i worked one of these things with actual students.

      • Owen Thomas

        oh. egad. degrees-and-radians. anyhow. i’d *still* like to convince some beginners to let the calculator do the hard bits.

  • D. Ebert has posted an interesting question about classifying “math mistakes” on the Stack Exchange network: http://matheducators.stackexchange.com/questions/1609/

    D. Wees leaves an interesting comment:
    I’m wondering if there is a way of framing
    this question differently. Instead of thinking about issues students
    have as errors, what if we thought of (most of them) as flawed or
    incomplete models for understanding? In other words, instead of “what
    are the typical ways students do not understand mathematics” we could
    ask “what are the typical ways students understand mathematics”?
    Related: How do these models of understanding manifest as errors or
    mistakes?

    I do not know if this is of any use to you…

  • Well, this is actually a good thing to be shared to those people who wanted to become more aware in using sine particularly on some of their equations. Also, this might going to affect on the usual sine laws that was being implemented and being used by many.