“If you love something set it free; if it comes back, it’s yours. If it doesn’t it never was.”
In Elizabeth Green’s book, Building a Better Teacher: How Teaching Works (and How to Teach It to Everyone) the story of A & W’s attempt in the 1980’s to compete with McDonald’s Quarter Pounder is recounted. A & W built what they thought was a better burger. The A & W burger consisted of a one-third pound beef patty, focus groups preferred it over the Quarter Pounder, and it was priced about the same as the Quarter Pounder. Unfortunately, the burger campaign flopped—It didn’t sell! The general public was not swayed to move from the Quarter Pounder at McDonald’s to A & W’s one-third pound burger. Asked why they did not prefer the A & W burger, the consensus was “Why would I want to pay the same amount for less beef?” (1/3 vs. 1/4) .
A few weeks ago I was in an elementary classroom in which a student was asked to compare the fractions 1/6 and 1/8 . She walked to the board and quickly wrote the following stating that “8, 8 times 1, was greater than 6, 6 times 1 so 1/6 is greater than 1/8.”
Wanting to better understand her reasoning I asked, “Can you tell me what you did?” She said, “It’s the butterfly method” I asked her to explain to me what she was doing and why it works and her comment was, “I don’t really know why it works, it just works!”
Is this good enough for us as a teaching community, “I don’t really know why it works, it just works”? I want to take a moment to share first, why I believe we are doing a disservice to our children if this is the understanding and reasoning that we expect from our students.
Bottomline, is we want our children to make better decisions about burgers than those people in the 80’s! (I think we want more than burger knowledge!) In a nutshell we want students to have greater power when it comes to reasoning about fractions. What does this power look like? Let me start by giving an example from my everyday life of the type of reasoning I would like our students to have. Each morning I wake up I make oatmeal. My oatmeal recipe requires 1/2 a cup of oatmeal and 1 cup of water. As mental exercise I grab whichever measure cup that’s clean (okay, I am a guy sometimes it does not have to be clean! And sometimes I grab it because it’s the only one remotely clean!), this morning it was a 2/3 cup. I try to figure out how I need to fill it to get what I need—a little over half full for the a cup of oatmeal and one and a half times for the water. I want our students to have the power to reach in the drawer of life, pull out any type of measuring cup, and have the mental capacity to complete the recipe. Why does everything come back to food! That’s a different blog!
At this point it is okay to ask, “If you don’t like the butterfly method, what type of a reasoning would you want from students?” The Mathematics Teaching Practices (NCTM, 2014) suggests that “effective teaching of mathematics builds fluency with procedures on a foundation of conceptual understanding so that students, over time, become skillful in using procedures flexibly as they solve contextual and mathematical problems.” I want students to be able to reason conceptually and know why the procedures they are using work. In this specific context, reasoning conceptual would mean that she can mentally reason that 1/6 is greater than 1/8 providing two aspects to her thinking: 1) sixths are bigger pieces than eighths because there are 6 sixths in one whole and 8 eighths in the same sized whole. Since there are less pieces to make a whole (i.e sixths) the lesser number of pieces (6 vs. 8) in the whole is the greater piece (i.e. sixths) ; 2) Comparing 1 sixth and 1 eighth involves comparing the same number of each type of object since there is 1 of each type (i.e. sixths and eighths) we know the bigger piece is the greater fraction.
“Do we really have to do away with the butterflies in our classroom? They are really pretty!”
It is certainly a shortcut and, agreeing with the student, it does work, however it is important that students understand why it works. The butterfly method provides the numerators, 8 and 6 in our example, if the denominators were the same.
So, though hidden by the procedure, the comparison is really 8 forty-eighths and 6 forty-eighths. Since the size of the pieces is the same, forty-eighths, the comparison involves only considering the number of each of those pieces, 8 vs. 6. [All of this is assuming students understand the meaning of equivalent fractions.]
As I said earlier I want students that have power with fractions. It would be great to have a generation that does not fear and loathe fractions and has the mental power to perform mental computations with fractions and knows the meaning of the procedures that they perform. It is my belief that teaching the “Butterfly Method” early in a child’s development of fraction understandings impedes their capacity to reason mentally about the meaning of fractions. The butterfly method has been around for a long time, but as the A & W story suggests, if it didn’t come back when it is needed to reason about contextual real-life situations (e.g. size of a hamburger patty and oatmeal) then why not set those butterflies free!