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Tina reflects:

“Kids seem to forget their skills when the level is increased. They revert back to intuition like “subtract numerators, subtract denominators” when faced with trig functions, but as soon as I ask “how do you subtract fractions?” they immediately recall “common denominators.” How do we get them to stop and think when the cognitive load is higher?”

That’s a great question.

  • My first instinct is that when we can break problems down to smaller goals, then the pressure – the cognitive load (LOVE that phrase) – is not as obvious in the process of problem-solving. I am torn on this as I feel that it is an important goal to have our students know how/when to break things down on their own. Exhausted now – that’s all I’ve got.

  • kcota

    Maybe what you are asking is not even possible. Perhaps the misguided thinking is actually part of the learning process. If a student in my class made this mistake I would write a more simple subtraction problem such as 3^2/4^2 – 1/4^2 = 3^2 and ask them what they think about my steps.

    • But how many times do I have to do that before they can do it themselves? When we go back to basics (simplifying square roots, fraction operations etc.) they’re okay, but layer trig on top and they collapse. Is it a sign of weak understanding? Weak skills? Inattentiveness?

    • I think it’s possible, but what needs to occur is enough training in the basics of manipulating fractions such that correctness for dealing with fractions becomes routine, and then you can layer in additional math without added “cognitive load”. It’s generally true one cannot consciously pay attention to many different things simultaneously, so my strategy to prevent errors is to develop good habits so that I don’t *need* to pay attention to too many things at once.

      I think it also helps to go back and review basic topics like fractions using complicated expressions, just to reinforce the idea that the procedures learned using real numbers (probably integers, in fact) generalize to any numerator and denominator.

  • Owen Thomas

    my best guess: slow down.

    coursework math is typically
    in much too much of a hurry.
    a lot of what *i’ve* (slowly)
    come to understand about
    maths i’ve understood by
    considering *very simple
    examples*… the *same*
    ones… *over and over*.

    instead one is made to remark,
    for example, that such-and-such
    needed fact or process was
    presented last week at the board
    or can be found in the text or
    what have you and stick to
    the schedule. whereupon
    such simple fact… which might
    even have been clearly understood
    by much of one’s audience earlier…
    *adds to the confusion* and, whatever,
    the “cognitive load” causes the “wow,
    i’m actually *getting* this” response
    to shut down for the rest of some
    long-suffering students hour or lifetime.

    certain trig classes oughta probably have
    a 45-90-45 and a 60-60-60…
    and *i* say (why not?) a 72-36-72…
    (plato’s metaphysics seems to ignore
    the “golden” triangle but he sure got
    the first two in there right at the found-
    ation)… right in plain sight indelibly on
    the whiteboard or something for frequent
    reference. and some unit circles…
    cut in six and in eight slices, say
    (the golden case is an exercise).
    then, *every time* i’m the *least bit unsure*
    of where the signs or the radicals go…
    i’ll go back to the drawings and wave my
    hands around or scribble (in *erasable*
    whiteout marker) some symbols.

    a “side calculation”.
    *many* students *vigorously* resist
    my efforts at developing this
    (cognitive-load lessening)
    technique.

    specifically, the *very idea* that
    there is some *main line of discussion*
    (where we will [ideally] present, not only
    our calculations, but our *reasoning*)…
    a “place”, as it were, that we can, say,
    “move away” from (to another part of
    the page or board, e.g.) and temporarily
    “forget about”.

    then think as much as you need about
    this *little* problem. if it’s being done
    right, you’ll *know for sure* you can
    work it out if you just keep looking
    at the diagram and checking everything.

    and then you “plug it in”
    back in your main line of discussion.
    (and, very likely, you work through
    the whole thing so far all over again
    just to be sure you remember how
    this new piece *fits* in.)

    but.
    students merely needing to get through
    a certain number of exercises in a certain
    amount of time famously don’t *want* to
    talk about *why* certain pieces of code
    are caused in our presentations to appear
    somewhere near certain others. (“just
    show me how to *do* it.)

    many of them won’t slow down enough
    to find out what the heck the very
    *sign of equality*
    even (@#$!!) means.

    and it’s more or less hopeless coaching
    such students on trickier cognitive skills
    until they’ve learned some pretty good
    habits-of-the-equal-sign. or so i’ve found.

    your milage may vary.