Category Archives: Lesson Planning, Delivery, and Assessment

Lesson Plans and Curriculum Connections

Introducing My Grade 5/6 Students to How to Model Their Thinking in Mathematics

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.

The Lesson

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:


The Preparation

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.

Happy Learning.

Promoting Fluency in Math: How to Convince Your Students it’s not Just About the Answer


As I learn more and more about effective math instruction, one thing is clear:  Good math instruction has less to with your students getting the right answer, and a whole lot more to do with flexibility in approach, and strategy.

Try explaining that to your students though.

My students tend to focus on whether or not their answer is right, so when I ask students to explain to me what they did to get an answer students will often ask, “So, is it wrong then?”, or, “Just tell me what I was supposed to do.”

I remind them that them that I am far more interested in how they got the answer than I am in the answer itself, and that it’s their thinking that is important to me.  But their eyes glaze over, and I know I’ve lost them.  They just want to get the answer and move on.  Does this sound familiar?

Taking the Focus Away from Getting the Right Answer

This year, I started off with a math talk that Jo Boaler did with a class of 6th graders using a dot card.  I chose to start off the year with this one in particular because this one does a really good job of highlighting to my students that it’s their thinking that is important.

It manages to do this because the math itself is very easy, and the actual answer to the question is so mundane and uninteresting that their thinking gets to take centre stage.

When you see the question, you will see there there really is nothing to compete with the many ways students solve the problem.  They realize there are a variety of ways to get the answer, and that is what is interesting and important.  Students are engaged, and even fascinated by it.

I did this lesson as my first number talk of the year with my grade 5/6/7 split class.  For many of the students it was their first experience with a number talk.

The Lesson

Credit for this math talk goes to Jo Boaler.  A video of her teaching this lesson can be found here.

I ran my lesson very similarly to the way Boaler did.  I told my students that I was going to show them a collection of dots for just a couple of seconds and I wanted them to tell me how many dots they saw.  I explained that it was going to be shown for only a couple of seconds because I wanted them to do it without counting.

Instead of displaying a graphic with a pattern of dotcardpicdots on an overhead, I presented the dots on a piece of paper drawn by hand.  This ended up working well because some students struggled to explain how they got their answer and needed to point to the array to help them.  Walking over to each student with the dot card proved easier and less disruptive than having students get up and down out of their seats.

Best Practices

After showing the dot card for about 2 seconds I asked students to give me a thumbs up if they had an idea about how many dots they saw.  Once everyone had given a thumbs up, I asked a student to share how many dots they saw (they saw 7), and I checked to see if anyone got a different answer (no one did).

I then asked the student to tell me how they knew there were 7 without counting.  They initially struggled with how to explain, so I re-phrased the question:  I asked her what she saw to know there were 7.  She was able to tell me she saw a line with 2, then a line of 3, and another line of 2, which was 7 altogether.

I represented her thinking on the board, asked her if it accurately reflected how she got her answer, and put her name under it.  I asked if anyone else saw it the same way, and added their names.  Finally, I asked if anyone else saw it differently.  I repeated this until I had exhausted all of their ideas.  This is what the share board looked like by the end of the lesson:


Thoughts, Reflections, and Next Steps

The conversation was rich, and students were engaged for a long period of time.  It laid the foundation for placing an importance on student thinking rather than the answer.

I think the engagement was high enough to refer to this lesson when students just want to get the answer and move on.  I may even create a visual and put the dot card with the question “How many dots do you see?” on a sticky note, put the answer “7” on another sticky note, and all of their answers on a large piece of chart paper and put it in the hallway to share with the rest of the school.