Today we began our investigation of integers arithmetic by presenting the following problems pulled from Project Z.

The most important part for us as teachers today was the right hand column of this “worksheet.” Kristin Gray recently considered how a student’s representation of their thinking might differ based on the semantic difference of asking them to “show your work” or “show your thinking.” I decided to stay out of it and asked just to tell me how they figured it out. I’d love feedback on this prompt, as it is imperative that for students with cognitive processing challenges to represent their thinking with clarity, the directive must be as clear and concise as possible.

Anyway, back to integers. Kent Haines and Michael Pershan have pushed my thinking about integer instruction over the past couple months. I have always been an admirer of Cognitively Guided Instruction and be able to apply this instructional thinking to integers was too good to pass up.

Which leads to today’s class. Students got time to work individually before sharing with the group. If a student finished early, I asked them to make some part of their thinking more clear with questions like “What does that mean?” or “Can you tell me more about that?” or “Why did ___ go there?” and so on.

The student sharing was the class rainbow’s pot of gold. I made SMARTboard slides for each problem with space at the bottom for students to represent their thinking on each slide. I asked, “Who would like to share first?” A student would put on the board what they had written or drawn on their worksheet and then I asked, “Did someone do it a different way?” Inevitably, someone had done it differently, even if only slightly. The discussion of the large or slight differences in thinking allowed students who were “wrong” to reconsider their process and for students who were “right” to attend to the precision of their thinking.

Finally, here is the student work and my questions:

• Where would you go next?
• Which students seem to “get it” and which “don’t”?
• What aspect of their thinking strikes a chord with you, and why?