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.]

But I do know that students will be adding various types of rational expressions throughout their math careers and they will be doing the multiply and add operation (criss-cross) in various contexts (see Vedic Mathematics to get a flavor).  Also being able to do arithmetic operations efficiently gives a certain ease to problem solving.   Take a typical word problem that requires the adding of fractions.  Why do we even ask a student to work the problem?  I think to see if the student can pull the need for addition from a context.  We could, and why not, ask the student to write why he/she needs to add fractions but for efficiency’s sake, ours not theirs, we just look at the answer which tells us most of the time that  the correct fractions were added together. If adding fractions is associated with tedium and/or takes a long time, the student will get bogged down with the details of the computation or avoid even thinking of the possibility of using his/her arithmetic skills.  I want a student to be able to “chunk” it.  For the student to say to themselves that the situation requires addition and that is easily done.

Fast mental calculators do, and many children can if encouraged multiply pairs of one digit numbers and sum the results all in their head. Enforcing a least common denominator algorithm is like hobbling a race horse and many students are mathematical race horses looking for patterns and likely enjoy the discovery process.  Facility at the crisscross and add process makes two digit multiplication easier (the pyramid method) for example. Binomial multiplication goes faster – no FOIL and combine like terms but First, CrissCross, Combine and Last.  The formula for the cosine of the sum of two angles uses the pattern so does evaluating a two by two determinate.  These cases show that sometimes the quest for efficiency can lead to the discovery of a universal pattern – again see Vedic Mathematics.

There is no contradiction in wanting efficiency and understanding.  Teach both and test for both separately and repeatedly.

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About jrh794

I am a sixty-five year old math instructor at Southern Oregon University. I taught at the College of the Siskiyous in Weed California for twenty-six years. Prior to that I worked as a computer programmer, carpenter and in various other jobs. I graduated from Rice University in 1967 and have a MS in Operations Research from Stanford. In the past I have hand-built a stone house and taken long solo bicycle tours. Now I ride my mountain bike and play golf for recreation.
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