Today, I introduced my junior students (Grade 5/6) to the idea of using models to expose their thinking. This was their, (and my), first a lesson that focused specifically on how to model their thinking in mathematics.
Meeting Students Where They Are
Students had some experience modelling their thinking when it came up in their math workbooks or textbooks, but never before had a lesson specifically on how to model their thinking and that highlighted to them all the different ways they can think about a problem and solve for it.
Unfortunately, I have been rather prescriptive about what I was looking for when students were asked to “model their thinking” up to this point: When students were asked to show their work I looked to see if they showed all the steps when they applied an algorithm, or that they have drawn representations of manipulatives, for example, base 10 materials.
I know. Yikes.
Understanding What It Means to Model Thinking
Over the last several months I have been gaining a better understanding of what it means to model thinking, why it’s important, and how to do it. I teach my students that “Show your work”, means that they are to communicate what they did to get the answer and that this may have nothing to do with showing all the steps in an algorithm or drawing out representations of base 10 materials.
Asking students to solve a math question mentally can nudge students into thinking about a problem more creatively because, often, using the algorithm in their head is impractical. If I had let students solve the problem however they wished I doubt I would have had any variety in how the questions was answered. Using the algorithm will typically be the most cumbersome and inefficient method. Mental math methods, on the other hand, opens up discussion since there are so many ways they could solve the problem, and requires a much deeper understanding of the underlying concepts.
I owe my inspiration for this lesson to Duane Habecker. I saw him run a sample Number Talk, at the NCTM Annual Meeting 2016, designed to show early-primary students how to model their thinking when solving an addition question.
Before the Lesson
I had done a dot card lesson with the students to give students some experience seeing multiple ways of getting a single answer. This lesson took on the same structure: They were asked to solve a problem mentally, students shared how they got it, and then helped to construct a model that accurate represented their thinking.
I told students that were were going to do a lesson similar to the dot card lesson, but this time they were going to be shown a math problem instead of a series of dots. I told them that I will be showing them a math problem that they were to solve mentally, or, “in their heads”. When students arrive at an answer they were to discreetly give me a thumbs up. I did this so that students wouldn’t see other students get answers quickly and give up. I reassured them that I was going to give them lots of time to get an answer, and that it was not a race.
The question I showed them was:
37 + 41 =
I started with a question that was below grade level to ensure all students would be able to participate, and to ensure that students didn’t get too muddled in their thinking, and me as well since this was my first time modelling their thinking in a lesson like this.
Once most of them had a thumbs up, I asked a student what they got for an answer and wrote it on the board. I asked if anyone got anything different. No one did.
I then asked students how they got their answer, and constructed a model that I thought represented their thinking and asked them if it was accurate. I put their name under the model, and asked if anyone else had solved it the same way and put their name under it as well.
This was a fairly straight forward math question, and had a small class of 8 students that day, but here is the what the share board looked like:
This lesson, as simple as it is, took some preparation because I wanted to have the models that might represent my students’ thinking in my back pocket ready to use. You have no idea what your students might come up with! Here was my cheat sheet for the lesson:
We did one more mental math question as part of this math talk, (76 + 25 = ), and a new strategy emerged. A couple of students noticed that 75 and 25 “go together” to make a nice round number. Then they adjusted their answer to 101 knowing that the compatible numbers would add up to a number that is too small by one.
Here is the picture of the share board:
I think I played it pretty safe using such straight-forward questions for my students to solve, but I’ve got quite the range in my classroom. Also, it can be a tricky thing: Not just solving a problem, but also being aware of the thinking they used to solve it, and then trying to articulate it.
Now that the students are a little more comfortable with the structure of this number talk, I am considering splitting them up into two groups, unless I manage to use a question with quite a low floor and high ceiling.