Earlier this year, I wanted to share student work on Graham Fletcher’s Krispy Kreme three-act task with a group of intermediate teachers. When I last facilitated this task, many students thought of multiplication as repeated addition (only). Others used the standard algorithm — few successfully. At that time, analyzing student work revealed what students really understood (or didn’t). Further, the teacher and I discussed implications on practice going forward. (This prompted my last post.) But with my group of teachers I wanted to talk partial product strategies and models and these samples weren’t helpful. So Marc and I faked it and created some possible approaches:

I’m using *approaches* to include and differentiate *strategies* and *models*. Pam Harris defines strategies as “how you mess with the numbers” and models as how you represent your strategy. For example, I might use an open number line to *model* my adding up *strategy* for 2018 − 1984. The same adding up strategy can be represented with a different model (e.g., equation). The same open number line can represent a different strategy (e.g., keeping a constant difference).

We shared the approaches with the group and after some noticing and wondering invited them to find as many connections as they could. Some intended connections:

- Students 1 & 5 thought of multiplication as repeated addition
- Students 2 & 4 & 7 think place value to decompose 32 into two (or more) addends
- Student 2 “splits” 32 symbolically; Student 7 partitions an open array
- The partial products in Student 3’s algorithm can be seen in Student 4’s open array
- Students 1 & 8 make use of the fact that four 25s make 100
- Students 4 & 8 make use of halves and doubles

Teachers then discussed the placement of these approaches within a learning progression and how they might “nudge” each student.

Analyzing student work has become my favourite professional development activity. Here, what is lost in terms of authenticity is gained in terms of diversity of thinking. Still, I was excited to see this from @misskwiatkaski5‘s *real* students: