As part of homeschooling my son, I recently started teaching a mathematics class to a group of 8 to 10 year olds on Saturday. In this class, I decided to use the TERC investigations curriculum. After reading through the curriculum overview, I decided that the major focus of the first unit of each year is about helping students preview the mathematics to be learned for the year and investigating as a teacher what students currently understand.

The very first investigation for grade 3 is on counting out 100 snap blocks. In this task, students are prompted to explain how they are sure they have 100 snap blocks. The goals of the task are to investigate how students understand numbers represented in a visual way and to learn how they communicate mathematical ideas to each other orally as well as in writing.

Before I decided to use the task with students, I decided to try a few different strategies myself to try and anticipate what strategies students might use. Here are a few of them below.

As it turned out, only one of these strategies was actually used by students in the class. I uncovered this, while students were working on this task, by circulating between groups and looking to see what students were actually doing.

One group started with a 5 by 5 square they had already built during the free exploration time, and built out into a 10 by 10 square. As it turns out, I built my 10 by 10 square by snapping together 10 columns of 10 blocks so even this very similar strategy did not exactly match my thinking. This group finished quite quickly and when asked to find another way of proving they had 100 blocks, did not do so. Instead they worked on improving their explanation.

Another group decided to split the task of counting in half, and realized this meant that each of them would be responsible for counting out 50 blocks. However, when they started working, they did not communicate with each other very well on their plan for counting and for the first several minutes they just put blocks together. Upon observing this, I asked them what their plan was, which required them first to communicate about a plan. One of the students decided that 50 blocks could be counted as two rows of 25 blocks, and that 25 blocks could be represented as a 5 by 5 square, and so they each individually created a 5 by 5 by 2 block, and put it together on the height 2 side, forming a 5 by 10 by 2 block.

The final group just started by snapping blocks together as quickly as they could without keeping track of how many blocks they had, but they did attempt at least to keep their blocks in a rectangular prism form. I asked them what their strategy was, and one of them indicated that they were not counting because they planned on counting later once they were pretty sure they had 100 blocks. I then decided to keep asking them how many blocks they had every time I checked in with them. Finally I asked them to stop building and just count how many blocks they had.

At this stage the students had a 5 by 5 by 4 rectangular prism. As the students counted their blocks, I noticed that they had to count the lengths of each side each time, and had to count the number of blocks on each face each time, and did not seem to be subitizing this collection of objects. I also noticed that their strategy for counting all of the blocks was to count the surface area of four sides of the whole block ending up with 20 + 20 + 25 + 25 for 90 blocks total. I wondered why they had chosen only four sides with this strategy instead of a more consistent six sides, so I asked them how many sides the block had. They counted it out by rotating the shape, found the block had six sides, and decided that in fact they must have 20 + 20 + 20 + 20 + 25 + 25 blocks.

At this point, I thought about ending the counting activity and having groups come together, but I decided that because the students did not know each other very well, and because I had not done very much work on constructing community norms around how work is shared, it would not be very productive for this group to come and present their model at this time. Instead, I pointed at one of the corner blocks asked the students how many times this one block was counted with their scheme, and one of the students quickly noticed that it would be counted three times, at which point one of the other students said, “Oh” and decided to count the blocks by “counting the groups of blocks”. He basically counted out one row of 5 by 4 to find out how many blocks were in one row and then counted out the 5 rows of blocks to come up with 20 + 20 + 20 + 20 + 20 = 100 blocks. The first student looked clearly convinced that this meant they had 100 blocks so I decided to continue to be uncertain as to whether the third student in this group was convinced and end the activity.

As the students got together, I decided to sequence their explanations from the group that counted out half of the blocks, the group that counted out the big rectangular prism, and end with the group that counted out a 10 by 10 square. Each group shared their strategies while I prompted them to explain their work more completely.

After the class was over, I took time to write notes to myself on what I noticed during the class, to help me plan the class for the following week. I looked at what the students wrote down as their explanations on paper, and decided that these did not capture enough of the thinking students did to be very useful to me, except that I became acutely aware that all of my students need work constructing coherent explanations.

My goal for this group of students is to develop their capacity to use arguments and evidence to justify the mathematical ideas they uncover with each other to form a community of mathematicians. I want them to be curious about how each other understands the mathematics and to use their shared understanding to extend and build on their individual understanding of the mathematics.

The formative assessment process I used during this lesson could be summarized with the following steps:

- Do the mathematics myself and anticipate how students might do the task,
- Give the task to students to do,
- Observe what students actually do and what they say to each other while they work and ask questions to clarify my understanding of the strategy they are using,
- Intervene in the student thinking only when necessary and only when a useful intervention seems clear to me.
- Use my growing knowledge of how the students understand the mathematics as a basis for my decisions, both in-the-moment and to inform future work with these students.

The process I’ve described above does not require an exit ticket, it does not require different coloured cups on tables, it does not require daily quizzes. It requires me to plan what thinking I hope and expect to see and to build a model of how students understand the mathematics and to carefully select how I will support students in developing their understanding.

Formative assessment is more of a mindset on using student thinking as the basis for teaching and learning rather than a quick checklist or a list of strategies.