I share a classroom with the math coach at my school, and lately we have been making a lot of cross-content connections. In the four years that I’ve known her, she has talked about partial models hundreds of times. It was only recently, however, that I made a connection between this math term and my reading work. I’m happy to report that while it’s not brand new information, it has really helped me deepen my understanding of how to support student thinking.
I learned the term “partial model” from my math coach, Cindy, who learned it from Kathy Richardson, the author of the Developing Number Concepts series. Kathy Richardson says there are four stages of using models, each stage more abstract than the one before it. I will briefly describe them in this post, and you can click here if you’d like more information.
Stage One: Moving the Model
In this stage, students actually manipulate the model, whether it consists of color tiles, beans, or cubes. In the photo, the student is touching a model to solve the problem 8+7.
In stage two, students look at a complete model without touching it. This is more abstract, but all numbers are still represented visually. You can see that 8+7 is fully modeled, but the student is not manipulating the model.
In this stage, only part of the model is represented visually. In the photo, you can see that only one ten frame (the eight) is present. Students must visualize the other part of the model (in this example, the seven) for themselves.
Stage Four: Solving the Problem Mentally
In this stage, students solve a problem (like 8+7) mentally, without the need for a model. This is the most abstract stage.
I will be the first to admit that I often forget to provide students with partial models. Many other teachers have also said that this is easily forgotten. But Rachel Scott, a 2nd grade teacher in my building, had an epiphany while watching her students solve the problem 8+6 with Cindy. With the complete model, every student could say the answer quickly. They easily saw how to make a ten with four leftovers and knew the answer was fourteen. However, when doing the same problem with a partial model moments later, the students ran into some trouble. When they could only see the eight, they forgot what they knew about making a ten, and they reverted back to counting by ones. This was not an efficient strategy, and some of them didn’t land on the correct answer. Rachel and Cindy were shocked to uncover so much by simply removing part of the model. Rachel told me later that she felt confident her students could have put the right answer on paper, but this work with a partial model told her much more about their thinking than a correct answer. “The partial model is where you really get to see where kids are struggling and where you get to see the meat of their thinking. It’s also where the bulk of their learning happens,” she said.
This made me really stop and think about this whole idea of partial models and how they might be present in other classroom situations.
It’s no secret that I’ve been diving into reader’s notebook work, so when I tried to connect partial models to my world, I immediately thought of the work we’ve been doing in notebooks. The kindergarten, 1st, and 2nd grade teachers are really pushing their students to write about their reading. They have modeled this heavily, and they are pushing for independence. They’ve seen success with the literacy prompt cards I wrote about, so they are using those as springboards for writing. They came up with sentence stems for each of the prompt cards, and they saw an instant improvement in notebook work. These sentence stems are, as you might have guessed, a partial model! The pictures below are from Christine Kuipers’ 1st grade classroom. She charted all of the stems, and her students refer to them often. These prompts get students started, but they must fill in the rest with their own thinking. This is just like Rachel holding up a model of eight and asking her students to imagine adding four more. They must fill in the blank with their own thinking.
I’m so excited to learn more about partial models, and we have enjoyed making connections to the real world. (Training wheels are the partial model for learning to ride a bike, for example.) I’d love to hear how you’re using partial models. Feel free to leave a comment below!