My Fraction Rant

It is a common complaint among math instructors that entering freshman do not know how to add, subtract, multiply or divide fractions.  Judging by the look on their faces  I would say that they actually fear them.  The reasons, I speculate, are 1) Students are not developmentally ready to learn complex algorithms at the age fractions are commonly introduced, 2) The need for skill with fractions is not reinforced across the curriculum, and 3) We are making things too complicated and too abstract.  What follows is my proposal for helping students learn operations with fractions and have fun doing so.

Students need a complete familiarity with commonly used fractions like $\frac{1}{4},\frac{2}{5},\frac{5}{6},\frac{7}{8}$ etc.  In particular we are stuck with the English system of measurement.  So students should be completely  familiar with $\frac{1}{2}'s,\frac{1}{4}'s,\frac{1}{8}\text{'s and }\frac{1}{16}'s$.  I suggest that we give them a ruler and a tape and have them measure everything in sight, for days if necessary.  This part is easy. They just need to learn how to read a  ruler.  They should also  be drawing  lines to precise lengths with proper and improper fraction specificiations.  After they are proficient at using a ruler,  give them two rulers and show them how to use the two as an analog computer for adding  and subtracting lengths and even halving  and quartering.  Again require mastery.  Then have them construct rectangles and find the area by subdividing into square inches.  Then take away one ruler and have them add and subtract lengths.  This time they will counting along the ruler.  No algorithms should be mentioned much less taught.  Students  will discover short cuts on their own.  With cleverly designed exercises, they will discover the idea of least common denominators themselves.  Wait until much later to introduce lcd’s formally.

Another series of fractions that they need to know are $\frac{1}{3}'s,\frac{1}{6}'s,\frac{1}{12}'s$ and maybe even $\frac{1}{60}'s$.  This time use a clock-face.  Make a disk with the clock-face divisions on the  perimeter and also make another face the same size  with divisions on the outside.  Students can use these to add and subtract fractions.

The last series of fractions are $\frac{1}{5}'s,\frac{1}{10}'s,\frac{1}{20}\text{'s and } \frac{1}{100}'s$.   Here I envision a representation of a dollar bill with the appropriate divisions.  Again require mastery.

Now that students really know the common fractions, teach them algorithms.  The modern trend seems to be to teach for understanding.  As I interpret this, we somehow explain why the algorithms work as we teach them how they are done.  I certainly don’t remember any such justifications if I was taught any back in the 1950’s.  I think students are mostly empirical learners.  They will have already a good idea about what do to if they have spent all that time measuring and calculating with rulers.  In any event do not mention the dreaded “common denominator” phrase.  This diagram is all that is needed.

Quick Algorithms for Operating with Fractions

There are just three short phrases to memorize.  For adding or subtracting we could say “crisscross across” but if we say “crisscross applesauce” it is easy to remember that the algorithm is for adding and subtracting since the two words in applesauce begin with “a” and “s”.  I know that using a common denominator could make certain calculations more efficient but don’t teach this yet.   Right now get students in the habit of checking for lowest terms at the end of every calculation.  Also don’t let the students say for instance “7 times 5 is thirty-five”.  Take it from mental calculators( see one of my favorite books Calculator’s Cunning by Karl Menninger, page 56), this just slows them down.  They should just look at the seven and five and immediately say or write thirty-five.    Introduce these algorithms using common fractions so that the students  know the outcome before they do the operation.  The short cuts (cancelling) can be introduced, if the students haven’t already figured them out, after the algorithms are mastered – really mastered.  Use a timed test with all operations intermixed.  More than once.

So now we have students very familiar with common fractions and fast and accurate with operations using the common fractions.  The next step is mastery with a calculator.  Yes I said calculator.  I have heard something like this plaintive cry over the years: “My students actually use their calculator to multiply two by two.”  For some students this is true enough.  But my contention has always been that students don’t grab their calculators often enough or fast enough.  At twenty years old, the age that I meet them, students need to be able to operate with fractions quickly without fear.  If it takes a calculator for this, so be it.  At this point they need to be quick and accurate operating with all types of numbers. So have students practice with one of those scientific calculators that have a $a\frac{b}{c}$ key.  Then given them timed tests.  More than once.

Finally after all this is mastered, discuss why the algorithms work.  Here we can introduce common denominators as part of the explanation.  For the better students the common denominator will make sense and they will incorporate its use  automatically.

In summary, give students a large amount of empirical experience which can be made fun, teach them simple and quick algorithms, get them proficient with a calculator, and understanding will follow.  The next step is to be sure all other instructors in your institution are just as proficient and then ask those  instructors to require that their students to use these skills in their (non-math) classes.