The “butterfly method” – it sounds fun and looks like a breeze, especially when you’re trying to make comparing fractions seem less daunting. You might have seen it as a colorful trick to quickly tell which fraction is bigger. But as educators and parents, we need to ask ourselves: is this visual shortcut truly helping students understand the essence of fractions, or is it just a temporary fix that might flutter away when they face more complex math problems?
While the butterfly method might offer a seemingly easy route to compare fractions, it’s crucial to examine its long-term impact on a student’s mathematical journey. In my experience teaching 6th-grade math, I’ve observed some significant drawbacks when students rely heavily on this technique, particularly if it’s introduced before they grasp fundamental fraction concepts.
In this article, we’ll explore the butterfly method for comparing fractions, delve into why it might not be the most beneficial approach, and discuss why focusing on conceptual understanding and common denominators builds a stronger foundation for future math success.
Unpacking the Butterfly Method for Comparing Fractions
For those unfamiliar, the butterfly method for comparing fractions typically involves cross-multiplying the numerator of one fraction by the denominator of the other, and vice versa. These products are then compared to determine which fraction is larger. Visually, lines are often drawn in a butterfly shape connecting the numerators and denominators being multiplied, hence the name.
Here’s what I’ve noticed about students who come to 6th grade already equipped with the butterfly method for comparing fractions:
- Mystery Behind the Magic: Often, students can perform the butterfly method steps without having a clue why it actually works. It’s perceived as a set of instructions to follow, rather than a method rooted in mathematical principles.
- Trick Over Technique: They often master this shortcut more readily than they learn how to find a common denominator, a foundational skill for fraction operations. This reliance on a trick can overshadow the development of deeper mathematical understanding.
- Numerator Misconnection: Students sometimes fail to connect the products from cross-multiplication to numerators of equivalent fractions. They may not realize that these products would be the numerators if they had converted the original fractions to have a common denominator (as illustrated in Figure 1).
The core issue isn’t that the butterfly method doesn’t work for comparing fractions in simple cases; it’s that students often use it successfully without understanding the underlying mathematical reasoning. And in mathematics education, understanding “why” is just as important as knowing “how.”
Beyond Comparison: The Butterfly Method and Fraction Operations
This year, I encountered something new: students telling me they were taught the butterfly method not just for comparing, but also for adding and subtracting fractions! This extension of the butterfly method involves cross-multiplying to get ‘numerators’, then adding or subtracting these ‘numerators’, and finally multiplying the original denominators to get a new denominator (as shown in Figures 2).
Figure 2: Demonstration of the butterfly method extended to adding fractions. Note the cross-multiplication and multiplication of denominators.
While this process can lead to a numerically correct answer if followed precisely, it completely bypasses the conceptual understanding of what adding or subtracting fractions actually means. Students become focused on executing steps rather than grasping the fundamental principle of combining parts of a whole using common units.
It appears this “trick” is being promoted to make fraction operations seem “easier” and more “engaging.” However, are we truly making math easier, or are we simply sidestepping the crucial process of developing conceptual understanding?
Consider this:
- Why avoid the Lowest Common Denominator (LCD)? In the example in Figure 2, 6 is clearly the LCD. Why not guide students to recognize and utilize this, which is a more efficient and conceptually sound approach?
- Why create larger denominators and more work? The butterfly method often results in larger denominators, necessitating simplification to reach the lowest terms. This adds extra steps and potential for errors, when using the LCD would have avoided this in the first place.
- Most importantly, why not foster a genuine understanding of fraction addition and subtraction? Mathematics is about making sense of numbers and operations, not just getting to an answer through a memorized procedure.
The Butterfly Method in Action: When Shortcuts Become Roadblocks
Let’s examine a real-world scenario of how over-reliance on the butterfly method can backfire, especially when dealing with more complex problems.
Recently, my students tackled problem-solving tasks that required them to apply all fraction operations. It was during this activity that I discovered the extent to which some students were relying on the butterfly method for adding fractions.
Although we had briefly reviewed adding and subtracting fractions, the butterfly method wasn’t part of our classroom instruction. However, during independent work time, some students who had previously learned the butterfly method reverted to it, perceiving it as the “easier” route compared to finding common denominators.
In one problem, students needed to add 5/6, 2/3, 7/12, and 7/10.
The Butterfly Method’s Breakdown
The limitations of the butterfly method became strikingly clear. Not only did it fail to promote conceptual understanding, but it also proved incredibly cumbersome and inefficient for adding multiple fractions.
Since the butterfly method is inherently designed for adding only two fractions at a time, students attempted to extend it by:
- Adding the first two fractions (5/6 and 2/3) using the butterfly method.
- Taking the result and adding the third fraction (7/12), again using the butterfly method.
- Repeating this process to add the fourth fraction (7/10).
The actual process looked something like this:
- Adding 5/6 + 2/3 resulted in a denominator of 18 and a sum of 27/18 (as seen in Figure 2).
- Then, adding 27/18 + 7/12 led to a denominator of 216 and a sum of 450/216 (Figure 3).
- Finally, adding 450/216 + 7/10 (Figure 4) resulted in an astronomically large denominator and a fraction that required significant simplification.
It might seem astonishing that students persisted with this approach despite the exploding numbers, but they did, because:
- It was the method they had been taught and had come to rely on.
- They were led to believe it was the “easy” way to handle fractions.
- They lacked a fundamental understanding of fraction addition and how to effectively find common denominators.
Witnessing this firsthand was quite revealing. It highlighted the potential pitfalls of shortcuts and the importance of prioritizing conceptual understanding over procedural tricks. Since then, we’ve dedicated considerable time to revisiting the core concepts of fraction operations and “unlearning” the reliance on the butterfly method.
3 Key Reasons to Rethink the Butterfly Method Shortcut
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Shortcuts Can Lead to Long-Term Struggles: While the butterfly method might appear helpful for students who initially struggle with fractions, dependence on such shortcuts can create greater challenges as they progress in mathematics. As educators and parents, we need to consider the long-term implications of our teaching methods. Shortcuts for simple problems should be evaluated for their effectiveness with more complex problems and ideally introduced only after conceptual mastery.
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Bypassing Conceptual Understanding: The butterfly method fundamentally skips over the conceptual understanding of fraction operations. Students don’t grasp what’s actually happening when adding or subtracting fractions using this method. We must prioritize teaching math concepts, number sense, and number connections, rather than just procedural shortcuts. If we, as educators, don’t fully understand the “why” behind mathematical methods, we need to deepen our own understanding to effectively guide our students.
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Avoiding Essential Skills: The butterfly method circumvents the crucial practice of finding common denominators. Even if students understand the concept of fraction operations, relying on this shortcut means they miss out on practicing a fundamental skill necessary for success in middle school, high school, and beyond. Finding the least common denominator is not just about getting the “right answer” efficiently; it’s about number sense and manipulating fractions effectively.
Students are capable of understanding fraction concepts. We need to have confidence in their ability to “get it” without relying on tricks. While some students may require more reinforcement and repeated exposure to concepts, they can achieve genuine understanding with proper instruction and support.
What are your experiences and thoughts on the “butterfly method”? Let’s prioritize methods that build lasting mathematical understanding, not just fleeting shortcuts.
Resources for Teaching Fraction Concepts and Operations
To support a deeper understanding of fractions, consider using resources that emphasize conceptual learning and skill development:
Math Wheels for Equivalent Fractions, Comparing Fractions, Adding and Subtracting Fractions: These resources guide students through the process of finding equivalent fractions, comparing fractions, and adding/subtracting fractions with unlike denominators, using models to make concepts more concrete.
Want to Explore Fractions Further?
Dive deeper into the discussion about the butterfly method and effective fraction instruction:
Listen to the podcast episode: Why the Butterfly Method Is Not the Best Method
Explore the comprehensive program: Fractions: From Foundations to Operations for a structured approach to teaching fractions from foundational concepts to operational fluency.
