Schoenfeld divided his talk into three parts.

**1. Students are currently not engaging in mathematical sense-making.**

He gave the example of the following problem posed to 97 first- and second-grade students.

There are 26 sheep and 10 goats on a boat. How old is the captain.

76 students (78%) tried to solve the problem by combining those numbers using various operations.

He explained that what you test is what you get (WYTIWYG). Current tests are skill-oriented but Common Core demands more, which it tries to explain with the mathematical process standards. He says that of course content matters (obviously students have to learn the skills of math), but that the real action is in the practices. You can’t just put these on a list and check them off.

**2. There are tools we can and should use more often in the classroom.**

He advocated more formative assessment more often. Scoring formative assessments rather than just giving feedback ruins their utility. Students see the grade and never look further. I can’t even tell you how many times a student has thrown a test away after checking the number on the top without looking once at the notes I’ve made.

Schoenfeld also recommended letting students experience material and using questions to elicit productive mistakes. These mistakes then lead to formal mathematical conversations that can build a student’s understanding.

He recommended the Shell Centre’s formative assessment lessons as “diagnostic situations” to help introduce students to these concepts.

**3. There are tools we can use to help us reflect on our teaching.**

Schoenfeld showed us two videos:

- A TIMMS video from a 1995 study.
- A video showing students explaining their work in different ways, including students who had gotten the problem wrong in the first place. (Link unavailable.)

It was obvious which video we should emulate and not emulate.

Schoenfeld claimed that often teachers deny their students the right to engage their mathematical muscles. I agree with this. It’s often a struggle, at least at first, to get students to explain their reasoning. However, I do think that once you push past that struggle, it will ultimately benefit the students. He posed these questions, which are a useful assessment for our math instruction:

- Was there honest-to-goodness math in what students and teachers did?
- Did students engage in “productive struggle,” or was the math dumbed down to the point where they didn’t?
- Who had the opportunity to engage? A select few, or everyone?
- Who had a voice? Did students get to say things, develop ownership?
- Did instruction find out what students know and build on it?

*LeeAnn Allen blogs at DIY: Math PD.*