A word about fractions

This story begins with a tweet that I read.

This tweet poses a nice engaging situation where addition of fractions would be a very useful tools. But, addition of fractions involves common denominators. And, then I began to remember my students attitudes toward fractions, which can be summed up by the following…


… clearly fractions are so difficult that it requires someone with the reputation of Chuck Norris to be able to deal with them effectively.

Except, they aren’t. Or maybe they are, but they certainly don’t need to be. The logic that says that 2 min + 31 sec doesn’t equal 33 of anything is perfectly understandable to most. It’s the exact same premise as requiring common denominators to complete a fraction addition problem. And THAT is confounding to many. It seems like an arbitrary rule that math teachers invented to trick students.

And the teaching of it carries with it some strong opinions, too. I remember during my undergrad, one of my professors asked this:


One of my classmates changed his major that day. He got so angry that it we would be discussing the possibility that a student could write that equation and could be thinking something mathematically accurate. Dude literally stormed out of class and I never saw him again.

It is possible, by the way:

About the same time I was reading the IES Practice Guide for teaching Fractions. Are you familiar with the IES Practice Guides for mathematics? The Institute for Educational Studies gathers high quality research studies on educational and catalogs them in the What Works Clearinghouse.

The Practice Guides are documents that synthesize the multiple research studies that exists on a certain subject and operationalize the findings. Recently, I explored the IES Practice Guide for Fraction Instruction K-8.

I’d encourage you to check it out. To summarize, making fractions and conversations about portioning and sharing things a common part of math conversation from the beginning can help take the natural understanding that kids have and build fractions into that context. That will give us a chance to use math talk as a tool for students to need more exact language. My preschool son right now uses “half” extremely loosely at the moment. (I’ve drank “half” my water could really mean anything quantitatively.) In order for him to effectively communicate, he’s going to need to develop a more precise definition of “half”. That will require him adding additional fraction vocab to his toolbox.

As teachers, this gives us a chance to build in some more effective language, clearly defining the fractions as numbers. As such, encouraging a lot of conceptual sense-making about the different operational quirks that are required to effectively compute when fractions are involved. (If fractions aren’t numbers, but instead are just made-up, goofy ways of writing numbers, then the rules for computing them are goofy and made-up, too.)

The practice guide provides some tangible steps to achieve this. I’d encourage you to check it out. Lots of steps forward to take in the area of student comfort and effectiveness with fractions.

Excellent Classroom Action – Art and Geometry in the Elementary Classroom

I’ve written about the connections in math and art before. The visual nature of Geometry lends itself quite nicely to this. I found that the right pairing could bring in the engagement of the visual arts while maintaining fidelity to the content.

Exhibit A: Sarah Laurens, 5th grade teacher at North Elementary in Lansing. Mrs. Laurens reached out to me excitedly while I was in her building to see a math activity that she was leading her students through. They involved quilts hand sewn by Sarah’s grandma.

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The activity went like this: Students were in groups of threes and fours gathered around one of grandma’s quilts. Each quilt was made of a series of geometric shapes. Sketch the primary “unit” shape of each quilt and identify each of the polygons that are contained within it. On the surface, it is a fairly simple activity, but listening to the students talk to each other.

[Students looking at the black and blue quilt above left]

Student 1: “Those are just a bunch of hexagons.”

Student 2: “Hexagon’s, no… no… those are octagons”

Student 1: “Yeah, yeah… same thing.”

Student 2: “They’re not the same, one’s got six sides and one’s got eight.”

Student 1: “Well… wait… one… two… three… four… five… six… six sides! See I told you!”

[Students looking at the purple and while quilt on the above left]

Student 1: “That’s an octagon with kites around the outside.”

Me: “Are they kites?”

Student 2: “They look like kites.”

Me: “They sure do. How many sides do they have?”

Student 1: “One… t, th, f… oh! five…. They’re pentagons!

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Ms. Laurens and her students were comfortably saying and hearing words like “regular”, “tessellation”, and using definitions to make sense of what these shapes are, and using the definitions to settle disagreements (the foundations of proof…)

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Interesting images like these were leading to some interesting conversations as well. I’m thinking of  a conversation I heard between two students who were trying to make sense of the shapes they were in the picture directly above to the right. They were trying to to determine if the purple section in the middle was one big shape or two smaller shapes put back-to-back.

Then after some discussion, they realized that their answer would be the same either way. (Quadrilateral was their choice for the shape name. The word “trapezoid” was getting thrown around, but the students were having to be prompted for it).

I very much enjoyed getting to see these fifth graders exploring. Ms. Laurens was excited, the students were engaged (and this was clearly not the first time they were expected to be a self-directed and collaborative).

I’m just bummed that my schedule forced me out the door before I got to see Ms. Laurens’ closure of the activity. The students were wrapping up their discussions as I had to head out the door.