Problem Solving Skills

This is the story of what a student did when confronted with a procedural problem for which she did not ‘remember’ the standard procedure.

One of the in-class assessments I used is a ‘worksheet’; it’s like an open-note quiz over a set of material.  During our last class (intermediate algebra), the worksheet is longer than usual because it ‘covers’ the entire course.  Item 6 on this worksheet is:

Rationalize the denominator 

As I said, she could not remember ‘what to do’.  However, she did a great thing … she recognized that both numbers could be written as a power of 2:




I was very pleased that she did this, but the student was frustrated … she then could not see what to do.  This is pretty typical when novices dive in to the world of ‘non-standard problems’ — problems for which we lack a remembered process.

Of course, it was pretty easy to guide her through the remainder of the work:




Obviously, the expectation (this is our traditional intermediate algebra course) was that students would apply the standard procedure (multiplying top & bottom by the cube root of 4).  Students do not like that procedure, and I tell them that the procedure itself is seldom needed.

The alternate method worked only because there was a common base between numerator and denominator, and I doubt if the student will gain any long-term benefit from this experience.  This was more of a positive thing for me, as a teacher and problem-solver: Noticing a special pattern within a problem is a critical problem solving skill.

I’m sharing this story just because I had fun with it!

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1 Comment

  • By schremmer, December 30, 2016 @ 12:21 pm

    There is an interesting shift in emphasis here: on the first line, we have “did not ‘remember’ the standard procedure” while, thereafter, we now have remember ‘what to do’ (my boldface).

    I believe that this shift is the core of the problem: We do realize that the problem is with remember (something totally anti-learning) but we immediately shift to what to do (something unfortunately just as anti-learning).

    What to do ought to proceed from the question. (As John Hold wrote once upon a time, in How Children Fail, we force the students to be “answer oriented” instead of letting them remain “question oriented”.)

    The issue above was purely a linguistic one. The student did not know the meaning of the “code”. It is as if I asked an English speaker to do something in, say, Norwegian. Sure, even languages can be superficially forgotten but most of the time they come back very quickly.

    To use a related example: After I have coerced students into reading 3x^{+4} as “3 multiplied by 4 copies of x” and 3x^{-4} as “3 divided by 4 copies of x”, I immediately ask them to “do” 2x^{+4}•5x^{+3}, 2x^{+4}•5x^{-3}, 2x^{-4}•5x^{+3}, 2x^{-4}•5x^{-3}, … and … they “do it”. (Without me showing them anything of course. Just insisting that they go back to reading the code.) Then, after they have made their peace with “division of fractions”, I can ask them without further ado to “do” 2x^{+4}÷5x^{+3}, 2x^{+4}÷5x^{-3}, 2x^{-4}÷5x^{+3}, 2x^{-4}÷5x^{-3}, … and … they “do it”.

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