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.

(not) Explaining “Half”

“If you can’t explain it to a 6-year-old, you don’t understand it yourself.” – Albert Einstein


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“I’ve drank half my water”


My daughter isn’t a 6-year-old… yet. I get a couple more months before Dr. Einstein’s quote applies. Regardless, today I found myself at the breakfast table trying to make sense of one of those topics that exists in the intersection of math and common language.

It’s good to make mathematics common place, but the problem with making technically-specific terms commonplace is that it often leads to the usage of the word becoming a bit less technical. ELA teachers will tell you that this has happened with the word “literally.” (For more on “literally”, this post will literally blow your mind.)

In that same spirit, today’s breakfast was abuzz with the word “half”. My daughter is comfortable with the word “half.” She will often take a couple of gulps from her cup leaving ~5 ounces in a 6-ounce cup. Then she’ll say, “I drank half my water.” To her that means that she has consumed a some portion of her water, more than “just a little”, and less than “all” or even “a lot”. It seems the developing spectrum looks something like this:

Water Drinking Spectrum

Water Drinking Spectrum

It’s clear that spectra like this make clear the need for fractions, but that’s a discussion for another time.

Well, today, she asked about what “half” means. Which was a question that I should be qualified to answer, but you wouldn’t know it from how the conversation went.

Her: “Dad, what does ‘half’ mean?”

Me: “Well, it’s like broken into two pieces that are the same size.”

Her: “No, like, I drank half my water”

[See? My explanation didn’t work because I was thinking the two pieces were the water in the cup and the empty space. She only sees water. That’s only one piece.]

Me: “That means that the amount of water that you have already drank is the same as the amount of water you have left.”

Sage: “I don’t get it.”

Me: [holding up clear cylindrical glass that is “about half-full”] “See? This glass has water up to here [I point at the water line], and then the rest of the glass is empty. It’s like there’s the same amount of water and empty.”

Sage: “Um… Maybe we should talk about this when I’m older.” [Sage continues eating oatmeal and the topic changes]


Good thing she’s only five. I have a few months to prepare for Dr. Einstein’s test of whether or not I really understand the idea of “half.”