White and Blue Elephants

First, a riddle…

Q. How do you shoot a white elephant?

A. With a white elephant gun.

Q. How do you shoot a blue elephant?

A. Paint him white, then shoot him with a white elephant gun.

Back to Units

In this TEDx Talk, Randy Palisoc argues that mathematics should be taught as a language.

While I think he oversimplifies the case (mathematics, natural language, and computer programming languages have definite overlap but also significant differences), I really like his primary example. As I’ve been discussing recently, he talks about how mathematical understanding can be improved and even generalized by reinforcing the units.

Basically, he’s saying that there’s no significant difference in mathematical processing between adding three apples to five apples, three million to five million, three sevenths to five sevenths, or 3x to 5x.

Combining Like Terms

In algebra, we call this “combining like terms”. What algebra is introducing (or at least ought to be) is formal terminology for a process students have already been doing for years. Each term of mathematical significance can be separated into a numeric portion and a unit. Addition involves combining the numeric portion, but only works when the unit is the same.

When the unit is something concrete, like apples or books or ladybugs, children don’t have any particular problem understanding the process. I have five crayons. Becky has ten crayons. Stevie has four pencils. Janet has a package of eight pens. We can add various ways with this information: How many crayons? How many discrete items? How many writing implements? As long as we specify what it is we’re counting, there’s no particular difficulty in counting them.

As students of mathematics develop, they need to start seeing units for the purposes of addition as possibly having a portion which looks numeric, or which is abstract. If a student can add three apples to five apples, there’s no reason that same student can’t successfully add three thirteenths to five thirteenths or \(3\sqrt{2}\) to \(5\sqrt{2}\) or \(3\pi\) to \(5\pi\). In the first case, the unit is “apples”; in the other two, the units are “thirteenths” and \(\sqrt{2}\) and \(\pi\), but the process is the same.

Naturally, though, apples are discretely countable. They’re salient. \(\sqrt{2}\) isn’t even an integer; \(\pi\) isn’t even algebraic. So that’s one part of the struggle: Students want to assess those numeric-unit-kernels before they’re willing to work with them. They struggle with understanding that those units are just to be set aside until the operation is done.*

Another part of the struggle is that the break between “numeric portion” and “unit portion” of a term is flexible. Consider: If Joe ate 2/3 of a pizza and Steve ate 1/3 of a pizza, how much did they eat together?

Fractions and Elephants

For the first step, we use “1/3 pizza” as a unit, and add 2 to 1. That gives us 3/3 pizza, but we’re expected to simplify that. To do this, we use “3/3” as a numeric value and “pizza” as a unit, giving us one whole pizza.

Fractions are an early exposure to this concept, and represent a persistent problem for students. Prior to fractions, while it might have been useful to think of terms in this way, it’s not crucial: 50 + 60 can be solved by either by lining up places and adding (without thinking about units) or by realizing that we’re adding 5 tens to 6 tens to get 11 tens.

But with fractions, addition only works if the “units” (denominators) are the same.

It’s like the blue elephant of the riddle. We might ask a student how to add 1/3 to 1/2. We might even give them an algorithm for doing so. The reality, though, is that we can’t add 1/3 to 1/2. That’s like shooting a blue elephant with a white elephant gun: We have to make the elephant white first. We have to get common units, then add the numerators.

I’ve also been surprised with the number of teachers (not just students) who insist that adding fractions involve finding the lowest common denominator. That’s not needed, either. We could convert 1/3 and 1/2 to 10/30 and 15/30 (respectively) if we wanted to. It doesn’t matter. All that matters is that we need to be able to add numerators. Granted, using a unit that’s too big does mean that we’ll need to simplify the result, so it’s usually more efficient to find the lowest common denominator, but there are also rare case where simplification still needs to take place.

As I’ve mentioned before, in fact, the very name “numerator” refers to this counting concept. The “denominator” is the name of a unit, as the name suggests. Adding fraction involves adding the numeric parts of fractions with shared denominated units. We just need to make sure are elephants are white before we try to shoot them.

* I’m reminded of an article in this month’s GAMES Magazine, on Scrabble strategies. In it, the author discusses how many amateur Scrabble players get caught up on the meaning of the obscure words. While that’s certainly interesting in its own right, the only thing that matters to the game of Scrabble is that hypoxic is worth more than zeuxite.

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