More on Troublesome Fractions

Astute readers may have seen something fishy in the quote below from this recent post:

Although the operator and the state are mathematically equivalent, . . . they appear to be psychologically distinct for young children (Kieren, 1995) who treat their partitioning activities and their descriptions of the results of partitioning as separate entities (Empson, 1999).

The topic here is fractions, and the authors are observing that although fractions as expressions (i.e., operators) are not mathematically different from fractions as "answers" (i.e., states), students tend to see them that way.

Why do you suppose that is? Is it some hard-wiring in the brain that fragments these perspectives psychologically? Or could it be, at least in part, that students have difficulty seeing the equivalence between expressions and "answers" because we teach them to have this difficulty:

Look at the model below. Something like this appears in many primary mathematics textbooks to introduce the idea of subtraction. In this case, the model is meant to represent the expression 5 &minus 2:

We teach students from the very beginning to see expressions solely as processes that must be worked through in order to obtain "answers" (start with 5, cross out 2, that gives you 3), and we reinforce this idea so much in elementary school that the more accurate perspective becomes almost inaccessible when students start working with fractions.

Mathematical operations do NOT, in truth, represent actions. Mathematical expressions that employ one or more of the standard operations are descriptive statements—one can think of them as speaking about something that has already happened. The model below is a more accurate representation of 5 – 2:

This is not to say that a "temporal" perspective on mathematical ideas is to be completely discouraged. Indeed, simply showing three circles and calling the picture 5 &minus 2 would be debilitatingly accurate. But dramatically improving the presence of a "static" perspective on operations in the lower grades may help students transition more easily into working with fractions.

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