Butterflies are Better Left Outside!

“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.”

Butterflies1

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.

Butterflies1

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!

– Patrick

“When Have You Ever…”

I just have a question for you (and I know some of us “in” the basement…pun intended…are as old as dirt so I will understand if it takes you longer to “remember” or “recall” if this has ever happened to you).  “When have you ever been asked to get out of a check-out line by the cashier at the grocery store because you couldn’t figure out as fast as you possibly could how much your total grocery bill would be ?”  Ok, let me explain why I wonder about this.  When I go to the grocery store, I sometimes just need a few items and sometimes I actually have some cash to pay the grocery bill.  So, as I place my groceries on the conveyor belt and the cashier proceeds to scan the items for the price, I calculate the approximate total grocery bill to confirm whether I can pay with cash or not.  For example, I might have 1 item @ ~ 2.00 and 3 items @ $3.00 and 7 items @ 80₵, etc.  I don’t “remember” a single instance where I have needed to do this mental arithmetic as fast as humanly possible or if I couldn’t do it fast “enough” that I was asked to go the back of the grocery check-out line so that I could get a “second attempt” at doing this arithmetic faster and customers who could do this arithmetic faster could go to the front of the check-out line for all other customers in line to see.

So, I noticed a couple of years ago when I was visiting a third grade classroom, the paper below was being sent home to parents and I wondered why being fast at math facts is emphasized instead of being fluent with math facts.

And, I noticed with my own child how being fast (100% on timed test) with math facts in third grade didn’t always pay off in remembering math facts in later grade levels (77 % in 5th & 95% in 6th) and I wonder how fast my child might be today with these same math facts (I think I know what my child is going to do tonight when he gets home from school…I’ll let you know the results).

I recall reading a very interesting article titled “Fluency Without Fear: Research Evidence on the Best Ways to Learn Math Facts” written by Jo Boaler and Cathy Williams (found at https://www.youcubed.org/fluency-without-fear/).  Boaler and Williams noticed, “in order to learn to be a good English student, to read and understand novels, or poetry, students need to have memorized the meanings of many words.  But no English student would say or think that learning about English is about the fast memorization and fast recall of words” and the authors wondered, “why is mathematics treated differently?”

I encourage you to read their article (please do not feel as though you must read it as fast as humanly possible because as we might agree, English is not about fast memorization or recall of words) and let us know what you notice and wonder.

– Gay