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.”


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!

– 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

A Mathematicians Lament – Lockhart

“In fact, if I had to design a mechanism for the express purpose of destroying a child’s natural curiosity and love of pattern-making, I couldn’t possibly do as good a job as is currently being done (in mathematics education)— I simply wouldn’t have the imagination to come up with the kind of senseless, soul-crushing ideas that constitute contemporary mathematics education.”

Originally written in 2002 (I think), and change in education is particularly slow, but what are we current educators doing to impact this?  I think we are making some progress with an emphasis on the mathematical practices from the Common core, but I also believe that this change is more evident at the elementary and middle levels.  What can be done to incorporate the mathematical practices at the secondary level  (if you want a real challenge, consider Algebra I and Algebra ll). Yes we do have more tools available than in the recent past (we have GeoGebra and applications from Desmos), but what is actually changing in the secondary classroom? Are we enhancing discourse, and looking for structure and making generalizations (just to mention a few)?

– Kurt

The James Brown Brain

Have you ever had the following mathematical experience? You persevered through a math problem, found a solution, saw the related mathematics in a completely new way, and your brain felt “So good! So good! I got you! Whoa!”  That type of experience is what I am referring to when I talk about the James Brown brain.

Unfortunately, the first time I can remember experiencing my brain feeling “so good” from solving a math problem occurred after I had already been teaching middle school math for eight years. Sadly, I now realize that if my brain was not experiencing any wonderful feel-good mathematical moments, I am confident my students’ brains were unaware of how great a math problem could be.

The math problem that changed my attitude towards the teaching of and learning about mathematics was so monumental that I simply refer to it as THE Square Tile problem.  I’ve seen this problem in a variety of forms, but below is a version that is similar to the one I encountered during a professional development opportunity in the summer of 1990. (Were you even born yet?)

The presenter posed the following task:

Create the shape below using square tiles. What’s the area of this shape? What’s the perimeter of this shape?

Square tiles

After counting square tiles and edges (several times), I was satisfied that the area was 8 in2 and the perimeter was 18 in. Not entirely interesting to solve, but it was fun trying to build the irregular shape.

The next task posed by the presenter was this: “If possible, add more square tiles to the area without changing the perimeter.”

The phrase “if possible” did not even motivate me to pick up another square tile. I knew that if I the area increased, the perimeter would obviously increase. After all, I had been teaching students for the last eight years about how to calculate area and perimeter. What else was there to know?

I sat there with my arms across my chest waiting for the other participants to join me on realizing the obvious conclusion. After listening to the sounds of square tiles moving around on tabletops, I finally heard one of the teachers say, “I found a way to add one more square tile without changing the perimeter.” Then someone else said, “I can increase the area by 2 and still have a perimeter of 18.”

What the heck? Now they had my attention. My arms were off my chest, and my hands were back on the square tiles. How could the area increase while the perimeter stayed the same? I placed a blue square on the irregular shape (see figure below) knowing it would change the area but wondering how it would change the perimeter.

square tiles with blue

While I was adding more blue square tiles to the shape, the presenter posed one more question by asking, “What’s the maximum number of square tiles you can add to the area of the original shape without changing the perimeter?”

It was a great moment for my brain when I figured out how and why you could add square tiles without changing the perimeter. But the real James Brown so-good mathematical moment came when I finally saw the maximum number of square tiles that could be added without changing the perimeter….I didn’t even see that solution coming.

Not wanting to rob any reader from experiencing their own James Brown brain moment, I am not revealing any of the answers to this problem. Get out your square tiles, and let your brain enjoy the pursuit.

How about you? Tell us about a math problem you have solved that made your brain feel “So good! So good! I got you! Whoa!”

– Joann

The Questions I Still Ask…and Shouldn’t!

Several weeks ago in my statistics class at different points in the lesson I asked the questions, “Does everybody understand?” and “Does this make sense?” After each question, multiple students responded with clear explanations articulating the important ideas from the lesson. And then I awoke from my dream because this NEVER happens after I ask those questions! Almost always I gain very little information about students’ reasoning from asking these questions. I recognize this is a problem and I am working to find solutions. I have told my students that if I ask the class either of those questions you are more than welcome to ask me, “What is it that you want me to understand?” or “What is it that you want me to make sense of?” The greater challenge for me is that I need to think about better, more specific questions, that help me get a sense of what I want my students to truly understand. The charge I have given myself is to think about the three to five most important understandings I want students to gain from the lesson and create questions that will help me gain information about their understanding of those ideas.

This charge has forced me to think more deeply about what I want students to understand and how I need to structure learning to make this happen. For example, in statistics, this past week we spent time examining the meaning of correlation, a measurement of the strength and direction of a linear relationship between two variables. At the end of the explanation instead of  asking, “Does everybody understand?” I asked “What would be the correlation between two variables whose scatterplot revealed a relationship represented by a horizontal line?” Instead of simply a head nod response to “Do you understand?” , the question forced students to reflect deeply on the meaning of the measurement of correlation and the meaning of strength, direction, and linear relationship in this context. The ensuing discussion required students to present an explanation that provided clear evidence of their understanding. I would consider this a success because I learned so much more about their understanding of correlation and I believe the ensuing discussion deepened their understanding of the idea. Also, it forced me to think more deeply about what I wanted them to understand about correlation.

– Patrick

Teaching Badly

“The depressing thing about arithmetic badly taught is that it destroys a child’s intellect and, to some extent, his integrity. Before they are taught arithmetic, children will not give their assent to utter nonsense; afterwards they will.”  —  W.W. Sawyer, Mathematician’s Delight

This quote makes me think of those milestones in a child’s mathematical education when she gets several unintended truths about math. One is that math is about following procedures and memorization. Another is that speed and efficiency are valued above all. Another is that math needn’t make sense. Maybe the greatest is that not everyone is good at math. This quote also reminds me of Christopher Danielson’s post on standard algorithms.

One of the most concerning problems is a collective agreement that, at least at the lower grades, mathematics is arithmetic. I would venture to claim that, based on my limited experiences, the general public conflates mathematics and arithmetic. My wife does or at least she enjoys pointing out how bad I regularly am at arithmetic, despite my mathematical education. I think many or most elementary teachers do, too. How do we change this for teachers for whom the bulk of their experiences as math students is/was in classrooms where arithmetic was the goal of mathematics when our opportunities to change the beliefs of these very important and influential teachers are limited to one or two semesters?

None of these ideas are revolutionary. We thought this might be a good place to start. Ease you in to all the revolutionary ideas to come.

– Adam