In (Partial) Defense of Butterflies

Teaching students how to add fractions can be a real struggle. A big part of this is that we tend to get conceptually complicated about what fractions are. And a big part of this is because fractions can be conceptually complicated.

I teach Algebra II. (For an elementary teachers reading this, though, hold on: I’m not going to tell you you’re doing your job wrong!) As far as my own class is concerned, “fractions” are almost always just division. One book I keep handy is Milne’s 1893 primer, “Elements of Arithmetic”, precisely for this quote: “A fraction may be regarded as expressing unexecuted division” (§144).

From that standpoint, I could explain the challenge of adding fractions entirely from a “clash of tiers” standpoint: Division is on the multiplication tier, addition is on the addition tier, and our order of operations says to deal with the multiplication tier before the addition tier.

Consider: $\frac{2}{7}+\frac{3}{5}$

We can rewrite this is $$2 ÷ 7 + 3 ÷ 5$$, and this presents the same challenge as $$2 · 7 + 3 · 5$$: We can’t rewrite this to get rid of the addition sign because what do we add?

Indeed, what students often want to do is: $\frac{2}{7} + \frac{3}{5} = \frac{2+3}{7+5} = \frac{5}{12}$

which is as wrong as \9(2 · 7) + (3 · 5) = (2 + 3) · (7 + 5)\) is.

In elementary school, it is common to teach fractions from multiple perspectives: They’re division, but they’re also ratios, proportions, and subdivisions of a whole.

The last one is a major Common Core approach: 2/7 is 2 copies of a “seventh” unit. This is arguably the first historic use of fractions: The Egyptians apparently did their fractional math using unit fractions. 3/4 wasn’t a thing; Egyptian mathematicians instead wrote 1/2 + 1/4.

So it’s no surprise that, when it comes to adding fractions with unlike denominators, many teachers suggest the “butterfly” or “bowtie” method:

When it comes to conversations on “nix the tricks” or “rules that expire”, there is a lot of hate for the Butterfly Method.

But here’s the thing: It works. 100%, when applied to adding or subtracting fractions. It never expires. It works with this: $\frac{x-1}{x^2-3x+5}-\frac{x^2+3}{x^3-2x+4}$

The reason why it works is because it takes a basic algebraic identity and makes it cute: $\frac{a}{c} + \frac{b}{d} = \frac{ad + bc}{cd}$

This identity, written in this fashion, looks boring. The Butterfly Method, with its circles and antennae (I didn’t bother drawing those), is cute.

Okay, so? If we rail against cute math, we shouldn’t wonder that students get turned off.

The main problem I have with the Butterfly Method isn’t specific to it: I have a problem with teaching algorithms without any sort of understanding. Why does the method work?

When teachers reject the Butterfly Method, what do they often teach instead?

Here’s an example: $\frac{2}{7}+\frac{3}{5}=\frac{2}{7}\cdot\frac{5}{5}+\frac{3}{5}\cdot\frac{7}{7}=\frac{10}{35}+\frac{21}{35} = \frac{10+21}{35}$

… which gets us to the same spot, but with more steps. What’s going on here is that we’re multiplying each fraction by a fraction equaling 1. We can prove our identity using this: $\frac{a}{c}+\frac{b}{d}=\frac{a}{c}\cdot\frac{d}{d}+\frac{b}{d}\cdot\frac{c}{c}=\frac{ad}{cd}+\frac{bc}{cd} = \frac{ad+bc}{cd}$

And here’s a major key in defending the Butterfly: An adept mathematician will skip those middle steps. They’ll use their own version of the Butterfly Method, they just (a) will understand why it works and (b) won’t call it that.

My position:

a. If a student comes to you using the algorithm, help them understand why it works, rather than just throwing it out.

b. If a student understands why it works (and hence when and how to apply it), let them call it that.

The most common theme behind most of the “nix the tricks” complaints I’ve seen have been: Teach for understanding, not for simple algorithms.

I agree. 100%. What I don’t agree with is throwing out a valid algorithm, as the Butterfly Method is, just because it’s cute.

There are at least two potential pitfalls specific to the method. The first is that students could get confused about a subtraction sign. That’s a common problem with that pesky subtraction sign: If it were up to me, I’d get rid of the subtraction sign entirely. We don’t need it.

However, while the subtraction sign can trip students up a lot of places, it’s particularly easy to either forget it or overapply it here:

Why should the subtraction only apply to the red oval? Why not the blue or even the green?

I believe this is part of the impetus of “Keep-Change-Flip” when dividing fractions. One common strategy for dividing fractions is one form of “cross-multiplying”:

But students will forget which product goes on top. Plus, um, this looks an awful lot like the Butterfly Method, but it isn’t.

Before you get nervous that I’m about to defend cross-multiplication as a useful term in class, fear not: I won’t. There are three specific places that cross-multiplication can occur (properly), and they have three distinctly different effects. I’ve already mentioned two; the third is in solving equations like this:

So I’m totally on board with discouraging teachers from saying “cross-multiply”. Not because it doesn’t work, but because it does three different things in three different places and it really is confusing.

And I think some of the special hate that’s leveled against our poor butterfly is that it includes cross-multiplication. But I’m not saying, “Don’t cross-multiply, ever.” I’m saying, “Don’t call it that. Be clear about what you’re doing, and why.”

Another common pitfall with the Butterfly algorithm is that it works best with denominators that are coprime. In other words, it actively discourages students from trying to find a “lowest common denominator” (LCD).

Consider 1/2 + 1/8. The thing most mathematicians would be inclined to do is convert this to 4/8 + 1/8 = 5/8. Applying the Butterfly algorithm without consideration of why we’re doing it leads to 8/16 + 2/16 = 10/16, which then needs to be “reduced”.

“Simplified”?

While we storm on about the evils of the Butterfly Method, I don’t see us getting as upset about what I see as an even bigger problem: “Reduce” means to “make smaller”, but 5/8 = 10/16. “Simplify” means to “make simpler”, and is 5/8 really that much simpler than 10/16? And even if it is, why isn’t 0.6125 even simpler? And isn’t 10/16 simpler than 20/32?

And the biggest question of all: Why are we using two words that have distinctly different meanings in non-math English interchangeably?

I digress. The point here is that the Butterfly algorithm, intended to make less work, can make more work in many situations.

My rebuttal has two parts:

1. There are some cases where finding the LCD first still requires “simplification” later. For instance, 1/4 + 1/12 = 4/12 = 1/3.
2. The methods for finding the LCD can themselves be taught without understanding. A common one is to list a bunch of multiples for each number until you see one that’s the same.

The most reliable way to find the LCD is to prime factorize each number, then form a new number based on those. For instance, the LCD of 6 and 10 is 30. This can be found by noting that 6 = 3 · 2 and 10 = 5 · 2, so we need 3 · 5 · 2 = 30. But that requires an understanding of prime numbers, something that may be absent at the time that adding fractions is happening.

Even so, if I am faced with 5/7 + 2/9, I may do a version of the Butterfly algorithm, but if I am faced with 3/10 + 5/6, I might instead go with 9/30 + 25/30… depending on where my brain is at the time.

To sum up: The problem isn’t the algorithm. The problem is in teaching mnemonic algorithms without any understanding.

FOIL isn’t the problem. Teaching FOIL so students think they must multiply in that order, and have no way of generalizing to trinomials, is the problem.

KCF isn’t the problem. Teaching KCF so that students think that division “becomes” multiplication is the problem. (I was going to add that students might try to use KCF with subtracting a negative, then I realized… WAIT, THAT WORKS! And I know why, too.)

And we do a disservice to our students and colleagues by referring to valid algorithmic shortcuts as “tricks” undermine what understanding students may have developed. Yes, I have experienced the student who wants to just cross-multiply every time they see two fractions and an operator (even when they’re trying to multiply two fractions!). But casually throwing that out and starting fresh communicates a dangerous message about anything else the student may have learned in a prior class.

Most of the “tricks” are taught because they work. The problem isn’t the “tricks” themselves, but with teaching them without understanding.

Clio Corvid

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