What’s in that Fraction?

Sometimes students conceptualize math problems in ways that are mysterious to experts, but make sense to them.  On occasion, a bad conceptualization seems to be reinforced by features in the technology they are using.

I was helping a student work with rational expressions in our intermediate algebra course.  This particular student finds the material difficult, and often puts off dealing with the course.  Today, he was starting the first section which includes this problem:

If f(x) = 10/(x+1), find f(½)        [Presented in typical rational expression format.]

I think the student conceptualized fractions as two connected buckets (one for numerator, one for denominator) without seeing any particular meaning for the buckets together.

This student was doing most of their work on an older Casio graphing calculator, which shows fractions like this:

In other words, the calculator has a “a b/c” key used to enter fractions.  The student was trying to type in “10_½+1” so the calculator was showing ’21’ for an answer (which was a mystery answer for this student).  When I suggested using the division symbol instead of the fraction key, there was a resistance … until he discovered that it gave the correct answer for the online homework system.

I think it is pretty common to have students missing concepts in the meaning of fractions.  Frequently, they have trouble connecting a fraction with both one division AND with a combined product and quotient … where this last meaning allows for most of our algebraic work on rational expressions. Our instructional materials frequently emphasize the first concept (a single division), and never make explicit that a fraction also means multiplying and dividing … that “(3x)/(x²+2x)” means multiplying by 3x and dividing by (x²+2x).  Result: memorized rules for how we reduce a fraction.  It’s so much easier to focus on ‘multiplying and dividing by the same factor results in one’ as a concept … rather than ‘cancel common factors’ alone.

We might blame such misconceptions on an over-use of technology, or on a given calculator providing the ‘a b/c’ fraction key.  I think students have the misconception independent of the technology, and that the technology my student was using made it easier for me to identify the issue.

When a person looks for either research on learning fractions, or for suggested instructional sequences, there is agreement that a flexible and more complete set of concepts is critical for the diverse settings where fractions are used.  Our course materials (especially in developmental math, both in pre-requisite and co-requisite models) tend to focus so much on procedures that we never develop any further concepts about fractions.  That is really a shame, since students will forget the procedures; the concepts have a longer shelf life in the human brain.

We should always start with meanings and concepts … especially with fractions.

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