Here are two situations.
A visual patterns warmup routine, focused on writing expressions for the number of squares in the nth step of this pattern:
Groups come up with these two expressions and explain how they relate to the pattern:
Another way to write an expression that no student found looks like this:
The final expression is a different way of conceptualizing the pattern and offers a potentially useful perspective for future problems. Do I present it to students?
A number talk warmup routine, trying to find a strategy to mentally multiply or approximate 0.48*650. Students offer several different ways to break it down, either by breaking up the 0.48 or by breaking up the 650. Another approach that I’ve found useful for problems like this is to solve it in terms of percents — finding 10%, then 1%, then 50%, 2%, and finally 48%. Do I present it to students?
When to Interject?
There are multiple possible next steps.
- I could explain the additional perspective and why it’s useful
- I could present the expression or the outline of the number talk strategy and have students discuss in small groups to figure out how to assign meaning to it
- I could send students back to groups with an open-ended question of trying to find another method
- I could move on and keep my ideas to myself
I’m not sure what the best answer is. My fundamental struggle here is that these routines work with a great deal of student ownership. I’m doing plenty of work selecting and sequencing students to share their ideas and building norms for students to be attentive to each other and make connections. But I speak very little, and I typically let students present all of the math. Which is more important: norms of student ownership, or a pedagogically useful piece of more explicit instruction?