## Fractions are Hard!

### 1.5 I give up!

We spend the first formative years of our fraction story focused on parts-of-a-whole thinking. Here fractions are “calls to action” – circle a third of the kittens, color half the stars, give a person a quarter of the pie – and are not numbers per se.

Most people are comfortable with this thinking.

We might be pushed to start thinking of some arithmetic with fractions. Although it makes no sense to add portions of different objects

it does seem to make some sense to add portions of pie. (However, multiplying pieces of pie is meaningless!)

Next the the number line model is introduced and we look at parts of line segments. That is, we take that parts-per-whole thinking and apply it specifically to line segments. By placing line segments on the number line we start to associate locations on the line with numbers. We can see how to add and subtract numbers on the number line simply by stacking together section lengths.

Most people are actually comfortable with this thinking too. Fractions aren’t really numbers – they are part of a whole – but if you apply this parts-of-a-whole thinking specifically to line segments, then you can start to associate fractions with locations on the number line.

Then matters get crazy.

We are usually just told that fractions are numbers. That $$7 \div 3$$ and seven copies of a third, $$7 \times \frac{1}{3}$$, are “clearly” the same. We come to learn that each fraction can be expressed in an infinitude of ways: $$\frac{2}{5} = \frac{4}{10} =\frac{6}{15} =\frac{34}{85}$$, for example. (Whoa!)

Because fractions are now allegedly full-fledged numbers in their own right, we can not only add and subtract fractions with like denominators, we can also add and subtract fractions with unlike denominators ($$\frac{4}{5} + \frac{3}{7}$$, for example). And further, we can also multiply and divide fractions.

But explanations of how multiply fractions are typically hazy! We are told that “of means multiply” (it just does), and we are asked to draw pictures of square pies again to see that $$\frac{2}{3}$$ of $$\frac{4}{5}$$, for example, is a picture of 8 pieces of a pie sliced into 15 parts. (Hang on! I thought we weren’t thinking about pies anymore?)

Next an edict is usually passed down: To divide fractions multiply by the reciprocal.

The message is that, really, fractions are too hard to actually understand (Why this edict? Why can we sometimes go back to parts-per-whole thinking and sometimes not?). Yet the typically curriculum forges on nonetheless. The mechanics of fractions – performing fraction arithmetic worksheets – often becomes the norm and the best survival technique for a human being in this environment is to memorize the fraction rules and just do.

UPSHOT: We shut down. Most people, it seems, do hold on to their lovely intuitive picture of fractions – the parts of a whole thinking we were first taught. But as soon as the “math” of fractions enters the scene – being asked to add, subtract, multiply, and divide fractions –  there is usually nothing but memorized rules to hold on to. (And if you forget those rules, as we all do, then, well, there is nothing to hold on to.)

By the time students finish middle school in North America, they are typically expected to have mastered fractions.

How is that possible?

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