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Visualzing algebra using part/whole thinking

I recently presented at NCTM‘s regional conference in Seattle. The theme of the conference was Everyone Belongs Here. Here’s more on that from their website:

The conference will explore instructional practices that create safe learning spaces where students’ voices, thoughts, and ideas are heard; their math identities thrive; and they are secure in knowing they are valued, they matter, and they belong.

For many, the math classroom does not feel like a welcoming place. When we teach with a one-size-fits-all approach and  focus solely on memorizing procedures, students inevitably fall through the cracks. Methods that make sense for many, maybe even the majority, leave others in the dark. 

In keeping with the theme of the conference, my session was on using invented strategies for addition and subtraction. To kick off my session, I wanted the participants to grapple a little with a task. I gave each pair a baggie with 20 counters and the challenge pictured below. I also let them know that the little x in 3x meant equal groups, so 3x meant three equal groups

It was fascinating to watch as the participants enthusiastically went to work. Remember, this was a K-2 session! There was a LOT of great mathematical dialogue taking place. It didn’t take long for the participants to determine that x = 2, and the process used by most of the pairs was some version of what’s pictured below. Applying part whole thinking, we know the whole is 13 (fig. 1). Seven is one of the parts and the other three parts are equal (fig. 2), so each of the three equal parts must be 2 (fig. 3).

When I got home, I posted a picture from the session on my Facebook page. The comments rolled in and they truly highlighted why making math approachable for all learners is so important. Here’s a sampling:

Where was this when I was in high school 38 years ago. It would have made sense to me!! Love this!

Get rid of the 7 , 3x=6. X=2. Right. My way is much easier.

Whoa 🤯

Oh what a cool visual for this!

x=2. Not sure how the other box method works. 🤷🏻‍♀️


In other words, a method that brought new understanding to a very abstract procedure for many, made no sense to others who preferred the abstract method. One size does not fit all. It’s our job to make math doable for all learners!

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