Depth of Understanding in Math

mathlessons commoncoreThe first few days of school in September are precious. You’re setting the tone of your community, establishing expectations and routines, and keeping your fingers crossed that you’ll be able to squeak some content in along the way.

I’ve noticed that a lot of beginning-of-the-year activities center around literacy. In a sense, that’s fantastic, because high-quality children’s literature has an incredible power to bring people together. But I can’t shake the niggling feeling that yet again, math receives the short end of the stick.

After years of gentle coaxing from colleague Siobhan Chan (you’ll hear plenty about her in the year to come, I assure you), last fall I committed to starting my year using The Art of Problem Solving from Teacher to Teacher. The Teacher to Teacher curriculum has its flaws, and it hasn’t been aligned to our district and state standards in a bajillion years, but I have yet to find a better way to kick off math than with The Art of Problem Solving (I refer to it as AoPS in my lesson plans, but I don’t know if that’s an official acronym).

Last year, I launched AoPS at the same time as Math Minutes (again, a practice not without its flaws, but my students ADORE it, so I’m willing to concede the three minutes of class time it takes, start to finish, including transitions). I struggled with a way to share with my students that although Math Minutes DID place a focus on speed, they couldn’t let it hamper the work they were doing to deeply understand problems. So I came up with this visual:


While explaining it, I used hand motions to indicate that understanding was the biggest, most vital piece of our work in math, then we assure accuracy, then we strive for fluency. It guided our practice throughout the year, and they emerged the most successful class of mathematicians I’ve had in six years.

So this year, based on the success I saw in taking the Gallon Man activity to the next level, I decided to give more ownership to my kids with these three levels of understanding in math. I redid my introduction lesson, providing this as a metaphor:


Understanding: This is the whole ocean. Nothing further can happen until we get here. If we’re not there yet, that’s fine, because at least we know what we’re striving for — our understanding is the most critical piece.
Accuracy: Kelp and other seaweed will die if it’s out of the water. Our accuracy is meaningless unless it is grounded in our understanding.
Fluency: If we work on our accuracy in a dedicated way, fish and other critters will come to live among the seaweed. It often happens naturally, but we can use strategies that improve conditions for fluency to flourish.

Then, students created their own representation of the three levels of understanding in their math notebooks. They provided remarkable metaphors, and also gave me insight into their thinking. I’m not really permitted to post student work on a non-district site, but you can view the work of everyone whose parents signed a release at our Artsonia site. So I’ve cropped a lot of these and removed student names, etc.

Many kids drew something very close to my example, which was totally fine.

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And some started with my basic example, but took it a step in a different direction. Behold, two representations that take place on the savannah and on a farm, respectively.

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And then, some of my students blew my mind. I didn’t have them write their explanations, as this was our first activity of the year and I wasn’t ready for that piece. I did record some of their comments, though.

“Understanding is the garden, the accuracy is a sturdy tree, and the flowers are fluency because it’s hard to get them to grow right.”
“The longer banners are the more important ones, and they get smaller.”
“You need sun or your won’t have plants and flowers, and the bees won’t come unless you have plants and flowers.”
“You have to stop and make sure you understand first before you go any further. And then when you get to fluency, you’re going fast but you’re still safe.”
“In the city is where all the action happens, so that’s understanding. Then you go to the neighborhoods in the suburbs for accuracy, and eventually you get to fluency that’s far away.”
“In Minecraft, your character is the most important, then you get a chest, then you get items to put in the chest. But you can’t do any of that without your character.”

Although I’m all for partner collaboration and I don’t mind if students’ work is similar, I was concerned that my CLD student (CLD, or Culturally & Linguistically Diverse, is the new, politically correct way to refer to ELL or ESL students) couldn’t explain his picture, while the person sitting next to him had an extremely similar representation and could explain his. This was a signal that I need to follow up with my CLD gentleman.


I’m excited to see what comes up next week in math! I always love comments and suggestions!


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