By Michael Flynn, posted February 13, 2017 —

When I was writing, my book, Beyond Answers: Exploring Mathematical Practices with Young Children, I had a number of teachers submit vignettes from their classrooms that capture one or more of the Common Core’s Standards for Mathematical Practice (SMP) in action. The power in these vignettes is that they derive from real moments from the classroom with student-teacher dialogue, teachers’ reflections, and examples of student work—allowing for a deep analysis of student thinking and instructional moves that are present. 

If you haven’t read the vignette, I would encourage you to do so before you read my analysis of it. Analyzing vignettes is somewhat subjective, allowing for engaging conversations about the work they present. Feel free to share your own thinking about the vignette, ask questions, and agree with or push back on things I say. If you have comments or questions, please use the space below or reach out on twitter. 

Analysis

The vignette captured the whole-group discussion that ensued as a result of students’ work from the previous day. It is a really nice example of students working on SMP 8: Look for and express regularity in repeated reasoning. Earlier, students had noticed a similarity between the two sets of problems they had worked on, and their teacher decided to guide them in a deeper exploration about what was happening by using representations and contexts. The use of these representational tools allows students to see the underlying structure (SMP 7) in the way addition behaves when you modify one element of an equation (Flynn 2017). 

One thing that stood out for me was how much the students owned the mathematical ideas. Rather than telling students that adding one to an addend will increase the sum by that amount, the teacher set up a scenario for them to discover and explore this idea. This deliberate move positioned students up engage in this work from their own understanding and perspective. 

Throughout the discussion, the teacher asked questions to encourage students to share their current thinking about the regularity they noticed and then encouraged others to respond to their peers’ thinking. Her role in the discussion was as a facilitator rather than instructor, and as a result, the students built their understanding on the basis of collective contributions from their classmates. You get the sense from reading this that she was very attentive to students’ thinking, and although she had an agenda in terms of where students should eventually go with this exploration, she did not force it. She guided the discussion along but did not interject her thinking into the discussion. 

During the first part of the discussion, the teacher asked students for representations and stories that would describe what was happening with the pairs of equations. This served a number of purposes. First, it provided a visual representation for the operation of addition that all students could see and work from as they discussed the regularity. Second, it helped illustrate what happened to the first equation when an addend was increased by one. The teacher also encouraged students to consider Kim’s story about marbles and how the story changed as the equation changed. This move worked to solidify the students’ understanding by connecting it to a tangible context that made sense to them. 

Spending time crafting stories that reflect the particular mathematics at hand can strengthen students’ conceptual understanding of the mathematical ideas in the situation. In this case, students could use the marble example to explain or justify why increasing an addend by one will also increase the sum by one. However, depending on how students structure their story context, it may or may not accurately reflect what is happening. 

Students began with Kim’s story about two children playing with marbles, each of them having six (6 + 6). To account for the increase in one addend, Cam altered the context by having one child go home and a different child come back with seven marbles. Although this does create a scenario that reflects 6 + 7, if we were to express Cam’s context in an equation, it would look more like this: 6 + 6 – 6 + 7 = 13. Compare Cam’s context to Luke’s: Luke adjusted the story to more accurately reflect the change in the added. He kept the same two kids, each with six marbles, and had one of them gain one more marble. Luke’s context could be expressed as 6 + (6 +1) = 12 + 1. Both scenarios are fine in this situation, but it is interesting to consider how subtleties in context affect the mathematics.

This work can be challenging for young students, which is why you’ll notice the teacher did not dwell too long on the nuances between the contexts. However, she did ask them to consider how the story changed as different ideas were presented. This gave them some experience connecting representations and contexts without delving too far into the minutiae. As students have more opportunities to explore contexts and representations as tools to make sense of a particular idea, their skills will grow alongside their developing mathematical sophistication.

After students explored the contexts and representations, their teacher asked them if they could describe what was happening using an “if-then” statement. This framing was designed to help them articulate their claim. Anthony started by saying, “If you add one to the addend, then the answer gets bigger.” Then Bridgette articulated further that it gets bigger by one. 

Next, the teacher asked students to try other examples. This repeated reasoning was designed for a few reasons. Those who hadn’t quite noticed the regularity were exposed to more examples. Those who were thinking about the general claim, it gave them an opportunity to see if the claim held with other examples. In either case, all students could approach the task from their own understanding.

At then end of this session, the teacher had students consider the regularities they had noticed and work toward articulating their general claim. This is the only place where she interjected an idea by suggesting they revise their claim to be more precise (SMP 6) on the basis of what they had noticed. Although the idea to revise the claim did not come from the students, the revision was based on ideas that students noticed. In other words, it still feels like the claim belongs to the students and not the teacher.

At this point, students were primed for the next step in the process (see Russell, Schifter, Bastable et al. 2017). Using the following criteria, based on work from Russell, Schifter, and Bastable (2011), students had to make representations that showed their claim works.

1. The representation must show addition.
   
2. It should show that your claim works for all whole numbers.
   
3. We should see why your claim works from your representation.

Although I do not have examples from her classroom for this post, I do have some samples from my second graders when my students were working on this idea. I would like to share them as we think about what students might do to show how this claim works.

This first example is very typical of second graders. Notice that it shows the action of addition by combining the red and the blue cubes. The blue represents one addend (part), and the red represents another addend (part). Combined, they represent the sum (whole). The yellow cube represents the addition of one to an addend. Therefore, when the yellow is added to the blue, it increases its value by one, at the same time increasing the total by one. The only drawback with this representation is that it shows only that it works for 3 + 5 because it has a finite number of cubes. 

When this issue arose with my students, a few suggested that we could just use rectangles instead of cubes.

This representation is almost identical to the cube model, but it has undefined quantities. We still have two addends of some amount being combined to form a total. The yellow shows an increase in one addend and the sum at the same time. The other difference is that the rectangles specify no amount. Therefore, it applies to all positive whole numbers. 

Overall, this was a nice example of a teacher setting up a scenario for which students could notice and express regularity in repeated reasoning (SMP 8). The numbers were small enough to allow students to focus on the structure of the problems rather than struggling with computation. This work about regularity and structure is very engaging for students and teachers, as it deepens number sense and conceptual understanding of the operations. I encourage you to try a similar kind of exploration with your students.

References

Flynn, Michael. 2017. Beyond Answers: Exploring Mathematical Practices with Young Children. Portland, ME: Stenhouse Publishers.

Russell, Susan Jo, Deborah Schifter, and Virginia Bastable. 2011. Connecting Arithmetic to Algebra. Portsmouth, NH: Heinemann.

Russell, Susan Jo, Deborah Schifter, Reva Kassman, Virginia Bastable, and Traci Higgins. 2017. But Why Does It Work? Mathematical Argument in the Elementary Classroom. Portsmouth, NH: Heinemann.

Michael Flynn is the Director of Mathematics Leadership Programs at Mount Holyoke College in South Hadley, Massachusetts. Previously, he taught second grade in Southampton, Massachusetts, for fourteen years. He is currently interested in how primary and elementary school students develop algebraic reasoning and how teachers can support that work. He tweets at @mikeflynn55.