Fractions are Hard!


The truth is that fractions are hard! And the reason why is that we are never told what a fraction actually is.

First, fractions are “calls to action” – identify a third of the kittens, take a fourth of the pie – and certainly are not quantities we can add or multiply. (How do you add kittens and pie?) But then we draw fractions on the number line to imply that they are numbers after all. Okay then. We can add and subtract numbers on the number line, but how do we multiply and divide them? For these latter two operations we might be told that “of means multiply” and to divide fractions just “multiply by the reciprocal.” (What’s the reciprocal of a portion of kittens?)

Our first eight years of learning fractions is a bit of a confusing mash-up of ideas that drift in and out of grade-levels in an apparently random way. (Grade 4/5: Don’t draw portions of pie any more – fractions are numbers in their own right. Grade 6/7: To multiply fractions, do think portions of pie. Grade 7/8: To divide fractions – just follow this mechanical rule.)

Here’s my approach to fractions to help young adults (and not so young adults) ready to make some personal sense of the fraction story. Part 1 of these notes goes through the various potentially confusing stories we typically see in grade and middle school and illustrates why we are are justified in being confused. Part 2 of these notes takes a more solid approach, providing one consistent model that seems to give good solid intuition for it all, one that bounces off grade-school thinking. But we’re being mathematically honest here, noting that we are really just playing the game of identifying the key beliefs that seem to make all the fraction models we talk about “click.” This leads to the university-level approach to explaining what fractions are: look at what we’re choosing to believe from our intuition and have a “meta” discussion to derive an explanation of fractions, necessarily abstract. (And this shows why we can’t give the true story to grade-schoolers and middle-schoolers: it is just not pedagogically appropriate.) I explain this approach in these notes too. Part 3 fills in all the extra bits you might want to do with fractions, and part 4 offers some cool mathematics extensions.

These notes are not classroom notes – just a detailed outline of the ideas of how a set of curriculum notes might go for an audience of students and adults grappling with the mathematical and philosophical underpinnings of the story.  Explore them for your own personal edification and enjoyment. I hope they help.

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