The "butterfly method" for comparing, adding, and subtracting fractions  WHY teach this?
I don't understand why this method is being used.... I understand that it works and that it can make finding an answer easier for students (for simple fractions), but I am discovering definite drawbacks to this method. I don't want to seem negative, but I don't like to see students adding and subtracting fractions this way! Let me explain I used to have students come to me, having learned the butterfly method for comparing fractions.....they often had no idea why it worked....they knew the trick better than they knew how to find a common denominator...they didn't seem to understand that the products from the crossmultiplying were the actual numerators they would get if they made certain equivalent fractions (figure 1); they just knew it worked.This bothered me, because I believe they should understand why things work. So I always made sure to explain why the method worked.
But this year, I had students tell me they were taught to use the butterfly method to add and subtract fractions (cross multiply and add those products, then multiply the denominators together figure 2). But again, they didn't really have a conceptual understanding of WHY it works. It seems to me that many students are being taught "hacks" like this, to make learning fraction operations "easier and fun." In reality, they aren't learning what it means to add or subtract fractions. (And, really, why wouldn't we want them to see that 6 is the LCD in the problem in figure 2? Why would we want to them to use a larger denominator and then have to do more reducing...?)
I recently gave students problem solving that required them to use all fraction operations. Since adding and subtracting is in the 5th grade curriculum (and I teach 6th), I did just a brief review of adding and subtracting fractions before students worked on these problems  to see what they remembered. This is when I found out that many of them had been taught the butterfly method, among others.
In the problem solving, students had to add 5/6, 2/3, 7/12, and 7/10. And here's where the butterfly method totally fails the students who have learned to rely on it, not only because they don't understand why it works, but because it becomes so cumbersome! They couldn't use the butterfly method to add 3 or 4 fractions at a time, so they added two fractions at a time. Instead of finding a common denominator for all 4 fractions, they found a new common denominator each time they added on the next fraction. For example, they added 5/6 and 2/3, getting a denominator of 18, as in figure 2. Then they added 27/18 and 7/12, as in figure 3, getting a denominator of 216. From there, they added 450/216 and 7/10, shown in figure 4. They ended up with a HUGE denominator that they then had to work really hard to reduce! It might seem surprising that they continued this process to get such huge numbers, but some of them did, because this was the method they learned. I was so shocked to see this....and this is the first year I saw this method used in this way.
When we as teachers (or parents) find certain tricks that work for simple math problems, we need to look ahead to what our students will experience in future years. We need to try these methods with more complicated problems, to see if they will still be effective. We need to think about whether the "hacks" teach them math concepts, or number sense, or number connections....or do they just teach shortcuts? I have no problem with teaching shortcuts once the conceptual understanding is there. But shortcuts before understanding is detrimental to our students. They are capable of understanding the concepts and we need to have faith that they can "get it" without the tricky methods.
In the video linked below, Phil Daro stresses the value of teaching mathematics in greater depth and avoiding "clutter" in the curriculum  one of his examples includes the butterfly method. Don't Leave Out the Math Thanks for reading!
Grab this free fraction operations math wheel!
3 Comments
K Bailey
1/3/2019 02:04:19 pm
I agree with you 100%. The lack of understanding in creating common deniminators to add and subtract fractions is compounded when students are expected to manipulate rational expressions for addition or subtraction.
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Anne Santiago
4/10/2019 04:54:44 am
Hi Ellie, your knowledge on fractions is very profound. Your students must be very lucky to have you as their teacher. I like your butterfly method in adding and subtracting fractions. This is very suitable for complicated and bigger fractions because you no longer need to find the LCM. For me, finding the LCM is good only for smaller numbers. Let me also share my idea on dealing with fractions using your butterfly method.
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4/15/2019 03:11:26 am
It is hard to solve fraction without a thorough understanding of basic fraction. This is so true even in the higher grade levels. Student should understand first how to add, subtract, multiply, and divide fractions. They should have a deep understanding of LCM so that they can decide when to use it or use the butterfly method. They should also have the understanding of equivalent fractions, mixed fraction, and improper fractions. By having a depth understanding, student may find it easy and even fun to deal with fractions.
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