# Rethinking How We Teach Operations with Fractions and Decimals

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Let me start by saying I have been meaning to write this blog post for a long time... a very long time! It was around November of this past school year that I had a revelation (thanks to a fantastic graduate course and professor, the amazing research-based book Extending Children's Mathematics: Fractions and Decimals, and students who were so willing and able to communicate their math understanding and struggles). The way that I have been teaching the 5th grade math standards was not allowing students to make critical connections between fractions and decimals.

Here me out...

There are certain concepts in math that are pretty obvious in which order they should be taught.  Clearly students should learn to add before they learn to multiply, or divide whole numbers before learning to divide decimals. What I found is not so clear among many teachers and those district leaders who write our pacing guides is what comes first? Operations with fractions or operations with decimals? Both of these concepts play a huge role in 5th grade math and provide a critical foundation for math learning through high school and beyond.

Don’t believe me? Hear what Sherry Parrish and Ann Dominick, authors of Number Talks: Fractions, Decimals, and Percentages have to say about the lack of strong fraction understanding in high school students.

“The National Mathematics Advisory Panel conclude that the most important foundational skill not presently developed appears to be proficiency with fractions... The panel’s findings were corroborated with a survey of 1,000 U.S. algebra teachers, who indicated that a lack of fraction knowledge was the second biggest problem students faced in being prepared to learn algebra.”

If this is one of the biggest problems students face in learning algebra in high school, it is important that we do all that we can to teach fractions (and decimals and percentages) in a way that allows our students to truly and deeply understand fractions. Now that you know how important mastery of these concepts is, which should you plan to teach first?

The very short answer: fractions.

Then why do many pacing guides do the exact opposite? Honestly, I believe it is because they expect by 5th grade students have built a strong enough fraction foundation. This is a big assumption to make for such an important mathematical concept. Even with the strongest foundation in fractions, there are still things students need to understand conceptually about operations with fractions before diving into operations with decimals. This is so very true for multiplication and division.

Truth moment…

For a long time I taught operations with decimals and then operations with fractions because that is what the pacing guides have always said to do and I thought surely this was the best way. It was not until I did my own learning and challenged what I had always done did I realize that decimal understanding is strongest when it builds on students’ understanding of fractions. See if any of this sounds familiar:

Why is 0.01 (one-hundredth) smaller than 0.1 (one-tenth)? Why is 0.2 one-tenth of 2? To explain this, you need to build on students' fraction understanding of what “one-tenth” truly is.

Where do I put the decimal when I multiply two decimal numbers together? No matter how many models we used, this question never seemed to disappear UNTIL we began working on multiplication with fractions. At that point, they could explain why 0.1 x 0.01 = 0.001 or 1/10 x 1/100 = 1/1000.

What does it even mean to divide 5 by 0.2 and why do I get a bigger answer? Why do I get a smaller answer when I multiply 5 by 0.2? I thought multiplication was supposed to make numbers bigger and division was supposed to make numbers smaller! Yes, technically you can explain this without necessarily getting into fractions, but clarifying this by building on students’ understanding of fractions is much less abstract than attempting to explain this by building on students’ struggling understanding of decimals. In addition, the models used to teach multiplying and dividing fractions answer these two questions in a visual way perfectly.

Check out the learning progression for decimals and notice how embedded many of the fraction standards are in the learning of operations with decimals in 5th grade! Believe it or not, decimals are much more abstract than fractions.

The authors of the book Extending Children's Mathematics: Fractions and Decimals state that "children's understanding of decimals simultaneously draws on their understanding of fractions and place value" (2011, pg. xxiv). I will say, sometimes students find "temporary success" more quickly with operations with decimals than they do operations with fractions if the instruction is focused on procedures rather than conceptual understanding. Procedures for operations with decimals are typically more familiar to students than the procedures for operations with fractions.

But, is this really true understanding? Does this type of learning (procedural rather than conceptual) last and allow them to make critical connections in their growing understanding of fractions, decimals, and percents?

Instead of teaching decimals and constantly jumping ahead into my fraction unit to explain why or how, I finally decided to just take a break from our decimal unit, move right into our fraction unit, and let students discover all the why's and how's for themselves! By the time we came back to our decimal unit, I could direct students to think about their own understanding of fractions to answer their many questions about decimals.

Moving forward, I intend to teach operations with fractions and decimals simultaneously because they have so many connections. I would be doing my students a disservice to not give them the freedom to see and make these connections!

At the end of my fraction and decimal units, I spent a week or two reviewing and taking time to intentionally highlight the connections many students had already made about fractions and decimals throughout the units. Luckily I have the flexibility to deviate from the pacing guide, so I am excited to use what I have learned from watching and listening to my students' thinking, paired with some great research, to come up with a new plan!

What are your thoughts? Which comes first on your pacing guide, fractions or decimals? How do you set up your fraction and decimal units?

Brittany Hege