Asking students to solve problems mentally, without paper and pencil, is always revealing and often surprising. I thought that asking students to solve 100 ÷ 3 would be sort of a slam dunk. My, was I wrong!

I’m often surprised by what I learn when I interview students. Watch this 46-second video clip of Jonah solving 100 ÷ 3. Then read how I used the clip in a lesson with a class of fifth graders, and also read the letters the students wrote to Jonah.

I believe strongly that mistakes are learning opportunities. At least that’s what I regularly tell students. But it sometimes feels different when the mistakes are mine . . . and especially when they are pedagogical mistakes that I make while teaching. That happened to me recently when teaching a lesson to fourth graders.

A friend and I were talking recently about how much work we put into planning lessons. Even after all these years of teaching, I have to think through lessons as carefully as possible, both about the logistics and about the mathematical thinking I want to keep in mind and support. Here’s an example.

The children's book 17 Kings and 42 Elephants by Margaret Mahy is one of my long-time favorites. In this post I describe a division lesson that I’ve taught to third graders but recently revisited with fourth- and fifth-grade classes. With the older students, we tried extensions that differentiated the experience and put students in charge of deciding on problems for themselves. It was exciting to me to expand a lesson I've taught many times into a multi-day investigation.