It all started with a question…where should I put 3/4 on the number line? I was starting a unit with my 4th-grade RTI kiddos on fractions, and the question was meant to activate prior knowledge and give me a baseline of their understanding.
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I had passed out fraction tile kits, because I planned to use them in the lesson, so the students had the tiles in front of them when I posed the question. I had also drawn a number line showing 0 and 1 on my whiteboard easel. When I asked the first student, she told me that it wouldn’t be on my number line. I asked what she meant, and she said it would be further past the 1. Oh. I erased my number line and drew a new one showing 0 to 5. The student proceeded to tell me that 3/4 would be somewhere between 3 and 4 on the number line. I asked the four other students in the group and the five students in the next group, and I got responses of between 2 and 3, between 4 and 5, between 1 and 2, and more who thought it was between 3 and 4. Not one student told me it would be between 0 and 1. Time to punt.
I asked the students to show me 3/4 using their fraction tiles and, while they did that, I asked if I could borrow their ‘whole’ pieces. I used the wholes to draw a number line from 0 to 5 on my table and laid the whole pieces under my number line (see picture).
I would like to say that once they saw the concrete pieces together with the number line that the connection was immediate, but it was not. I still had students who thought that 3/4 would be between 3 and 4. We spent the next twenty minutes discussing the meaning of the numerator and denominator and placing different fractions on the number line. I’m confident that my kiddos are on their way to developing strong fraction sense.
What is Fraction Sense?
Fraction sense implies a deep and flexible understanding of fractions that is not dependent on any one context or type of problem. Fraction sense is tied to common sense: Students with fraction sense can reason about fractions and don’t apply rules and procedures blindly; nor do they give nonsensical answers to problems involving fractions.
If that sounds a lot like how you might describe number sense, you’d be right, because fractions are numbers. That’s the concept that so many of our students lack–the idea that fractions are numbers. Think back to the first student’s response about where 3/4 should be placed on the number line. She said between 3 and 4. She saw the 3 and the 4 as two distinct whole numbers, rather than understanding that the 3 and 4, when written as a fraction, imply a relationship between the numerator and denominator (McNamara/Shaughnessy, p xviii).
What Do We Do?
- KNOW YOUR STANDARDS Whether your standards are the CCSSM, the Texas TEKS, or some other set of standards, it’s essential that you understand the standards for your grade level. The standards for each grade level are perfectly aligned, like interlocking puzzle pieces, to develop deep understanding. If you don’t teach your part, it’s awfully hard to complete the puzzle. Check out this blog post for an explanation of the 2nd Grade fraction TEKS.
- DON’T SKIP THE MANIPULATIVES Fractions are such an abstract concept that without concrete support it is nearly impossible for students to develop conceptual understanding. Manipulatives and concrete learning are for every grade level, not just the primary grades. In Beyond Invert and Multiply, a companion resource to Beyond Pizzas & Pies, McNamara emphasizes the importance of providing students with visual models for fraction computation. Grab a freebie for hands-on learning for equivalent fractions here.
- QUIT TEACHING TRICKS We do our students a huge disservice when we teach them to cross multiply to compare fractions (also known as the “butterfly method”) or to “invert and multiply” to divide fractions. This is a short-term solution that does absolutely nothing to develop their fraction sense.
- IMPROVE YOUR OWN FRACTION SENSE If you are looking to revamp how you think about and how you teach fractions, the two books I referenced earlier are excellent resources. Not only are both bursting with activities you can try in your classroom tomorrow, but they also come with DVDs that show you what effective instruction looks like in the classroom. The format of the books make them perfect for a book study, a professional development session, or a team planning resource.
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