## We Get What We Test For

My post on “fast fractions” elicits the occasional vehement objection to the mechanical nature of the algorithms.  I too would object, though not apoplexically, if the post claimed that those rules were all you need to know (plus reducing to lowest terms) about operations on fractions.  Understanding an algorithm is important.  So is efficiency and speed.  Our homework and testing often conflate the two.

Say I give a student a worksheet with twenty pairs of fractions to add and I demand that each problem be done using the “find the least common denominator” method and then I put five such problems on a test.  Are my expectations that they will understand the reasoning behind addition of rational numbers because they worked through a series of tedious steps that they have been drilled on?  I think not.  If I want them to understand the principle behind combining fractions, I would design a series of exercises/homework problems/class discussions/reading assignments that explain in various ways, geometrically algebraically,..,the principle  and then have them produce their own versions.  I would also be working on their understanding of how the “fast” algorithms work.  There would be two types of test questions.  One would be about the “why’s” and the other would measure speed and accuracy.  Shouldn’t I want both types of learning?  At least if I do, I should test for both.

One of the things I like about the Common Core without knowing much about it, is that it has focused on fewer topics, topics chosen presumably by deep thinking about what student really need to know.  I have a horror of wasting students’ time, of teaching something that is or will be useless to most of them.  The traditional algebra curriculum has many examples.  Teaching combining fractions is not in this category.

Let’s talk the real world for a moment.  I worked as a carpenter for five years.  Everyday, trapped in the archaic English measurement system, I added and subtracted fractions and never (almost) did I employ the crisscross applesauce method.  Adding eighths and sixteens or halves and quarters required fast conversion to the smaller fractional unit and quick reducing to lowest terms and proper fractions.  I therefore have a visceral understanding of adding fractions and have visual models – the tape and the lengths of wood I cut and nailed together.  Carpenter’s calculators apparently exist now, but I can’t imagine anyone working on a job that values speed grabbing one to do such simple operations. There are other occupations and situations where adding fractions with different denominators are best done with crisscross applesauce, say adding a third of a cup and three quarters of a cup when cooking.  For a given individual student these situations are hard to predict and if we lived in a decimal world would be less important. [I may have lied.  Every time I converted a proper fractions to an improper fraction I was using crisscross applesauce with one in a denominator.]