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/Shaugnessy, 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.

Join in this important discussion by leaving a comment below.

Quit teaching tricks should be sung from the rooftops! I find teaching tricks leads to random number grabbing and manipulation and no idea if the answers make sense. There is a better way! I will definitely be getting this book!

Tara

The Math Maniac

Exactly, Tara! I think another danger of tricks is that it gives the kiddos a false sense of success. They are so proud and think they know how to ‘do’ fractions, but really have no understanding. You’ll love the book!

I would also recommend this book for teachers, “Extending Children’s Mathematics Fractions and Decimals” by Linda Levi and Susan B Empson. Teachers in grades 3 -6 who have been following the research and advice of these ladies in our school have seen a world of difference in their students’ understanding of fractions.

As always, your post is amazing and exactly right. Actually, your message is true to all subject areas. Thank you for always sharing your wisdom!

Kara

http://purposefulteachingandlearning.blogspot.com/

This was a fantastic article! I really appreciate that you explain what “fraction sense” looks like AND that teachers should stop teaching “tricks!” As a middle and high school math teacher, I had to re-teach so many concepts because students didn’t correctly remember some “trick” (and thus were solving problems incorrectly) or because a “trick” had given them a wrong understanding of a particular concept.

Thanks for sharing! 🙂

Bethany

p.s. Love you new site!! 🙂

I can’t agree enough with “stop teaching tricks”. I would add that fraction sense NEEDS to be developed and developed again for different fraction concepts before any algorithms are taught. I also wonder about manipulatives–I have had trouble seeing how fraction bars are transferable but do like how you connect them to a drawn number line. I’ve been working with teachers in grades 3 and 4 around using real world contexts such as pies, sub sandwiches, and brownies to develop fraction sense and math rival models such as the area model, numberline and clock. We are starting to see some great results in terms of students being able to “see” and reason with fractions. Love your posts and all of the big ideas and questions you raise! Thank you!

It says “math rival” and it should read “mathematical”. Sorry about the typo!

Donna,

This post is fantastic! I couldn’t agree more….fraction manipulatives are SO important for ALL grade levels. I’m currently working on a Comparing Fractions App Smash product and I would love to link in this post if that is okay with you!!

Julie

The Techie Teacher

Sure, Julie! I’m always on the look-out for great apps!

Awesome…thank you! This is the post I did about the mini app smash if you are interested: Click HERE . I’ll be posting a freebie guide for teachers (that links out to you!) so that they can easily implement this in their classrooms 🙂

Once again, thank you!! I LOVE learning from you. Reading your posts really helps me help teachers provide quality math instruction in their classrooms.

Julie

The Techie Teacher

Our scope and sequence and the testing calendar pushed our kids through the curriculum too quickly. They need the time to work with it. I have always made a fraction book that they cut the pieces and see the unit fraction pieces and how this makes a whole. This year we moved so fast that kids are still struggling and that is our focus the rest of the year. Making sure they leave with fractional understating.

Such a shame when testing and time constraints drive instruction, Terri. So glad you have time to work on it now!

We made a list of things (after talking to our 5th grade math teachers) on what the kids needed for shoring up before moving on. They had multiplication and division as on area and then talked about fractions and how they needed to truly understand the equivalence! Our conversations have been so deep ! I am impressed with their knowledge! I think we have cleared up some misconceptions! I hope it “comes back” next year!! I love this time of year, I am not stressed about testing and they are more relaxed and open to taking risks because they know it is ok to fail and we will keep going to fix it!

Coming from 17 years in kindergarten where building conceptual understanding and number sense is the core of our math instruction, I am frustrated that beyond 1 or 2 nd grade the use of manips is left by the side. Teachers go straight for teaching tricks on how to solve a problem on a test.

I understand your frustration, Lynda, but I do think it’s getting better! We just need to keep spreading the word. 🙂

The book Beyond Pizza and Pies totally changes how I teach fractions to my third graders. Another really great activity is a fraction clothesline.

http://illuminations.nctm.org/uploadedfiles/content/lessons/resources/6-8/fractionalclothesline-as.pdf

Just starting to work on fractions here…and never thought of the fact that my kids may not understand 3/4 is less than 1. Manipulative are definitely a must with fractions! Going to create some tonight.

Do you have any professional development books on fraction sense for younger grades? I will be second this year and feel I need to develop this a bit more.

I don’t know of any, maybe since the fraction standards are really limited prior to 3rd grade. Of course, Van de Walle’s Teaching Student Centered Mathematics is a terrific resource for all learning progressions.