Beware of the pies

As a teacher, you come to rely on certain Sherlock Holmes-like deductive shortcuts.

Is the student hesitating, even for a fleeting moment? You’ve found an area that needs more work.

Are we in eye-rolling, heavy-sighing territory? You’re in a minefield, but so is the student. Step lightly, flag this spot. There’s some discomfort here.

And when you see a student dutifully drawing pies to solve a math problem involving fractions, you know they are not yet a fraction expert (to put it kindly).

I really don’t know what it is with the pies. Maybe students (and teachers) cling to them when studying fractions because fractions are the only place that such pies (that is, circles divided into equal pieces) show up. It’s a shorthand for, “Hey, we’re doing fractions right now — look at these pies!”

And that’s exactly the problem. If you see 3/7 and think, “Let me draw a pie to represent that,” you’re going to run into trouble. It’s easy enough to divide a circle into halves, thirds, fourths, sixths, eighths, ninths, and so on — 360 (the number of degrees in a circle), conveniently, has a lot of factors — but 7 is going to be ugly. You’re going to have to get out your protractor and measure out approximately 51.4 degrees per part. (I recently got into an argument with a bartender about this very thing.) It’s messy.

To fully understand fractions, we need to be able to visualize them in lots of different ways: On a number line; as ratios (and then as not only part-to-whole relationships but also part-to-part); in relation to decimals; as part of a collection of objects or part of an array; as pieces of other shapes; as increments of time or distance; as part of a quantity (“half of this water”); as something that is already used up or otherwise abstract; and so on.

I want to make sure that a student has a lot of different ways of thinking about fractions. If the problem is about a person traveling 4 1/2 miles and another person traveling 3 1/4 miles and you have to find the difference in the distance traveled, a pie is going to be a laborious way of modeling this problem. A bunch of pies will do the trick, but there are better and easier ways. For instance, we could create a map, aligning both travelers at the same starting point and measuring their distance with tick marks.

This post is partly about fraction pies, but it’s also about questioning the conventions of school. There are other places that we find “school thinking” to be limited and unhelpful. For instance, the idea that a piece of writing is made better by listing the perceptions of each of the five senses; the notion that misspelled words should be looked up in the dictionary (it’s 2019 — we Google them); and the belief that a scientific experiment requires fizzy liquids or anything other than our powers of observation and deduction.

My job as an educator is to systematically identify the issues that potentially block good thinking along with the clues that such a block is present. Fraction pies are just one; it’s a game to find more. Join me in my nerdy quest.