MICHAEL: Absolutely! And I hope all the students begin using that strategy,
but most will have to construct the whole building before they’re ready to
think about a shortcut.This exchange illustrates a typical interaction between us. Together we have
discussed the benefits and drawbacks of just telling our students what we can see
so clearly ourselves. I strongly believe that students need to make sense of math-
ematics ideas—often incrementally, as they would when working floor by floor to
the top of a building even if these ways seem less “efficient.”
Pam’s conception of appropriate support has changed in many important
ways during the year. When she first began assisting her two students, she often
did most of the work for them. When they were working on story problems, she
picked the manipulatives they would use, counted out the appropriate number of
cubes, determined the operation for them, and then talked them through how to
count and find the total. Their papers had all the correct answers, but they were
not developing their own understanding of addition situations or strategies to
solve these types of problems. I decided to model the kind of support I thought
she could offer. This turned out to be exactly what she needed. She was able to
see that her students could do much of the work and that her role was to support
their learning through questions such as: “What’s happening in the problem? Will
your answer be more than 20 or less than 20? How do you know? What could you
use to help you solve it?” By asking questions, she helped the students become
learners.
Although she had developed many effective strategies to help her students,
she would sometimes revert back to a more traditional method so her students
would solve problems quickly. We continued to have conversations about how to
best support her students’ understanding.
PAM: I know Steven will be able to double, but Robert will have to build it.
MICHAEL: Steven should build it too, though. Otherwise he won’t really un-
derstand why he’s doubling. In fact, I’m probably going to require that each
student build to 10 floors a few times so they get a sense of what’s happen-
ing to the building.Steven was one of her students who had been diagnosed with developmental
delays and needed a great deal of support with work that emphasizes reasoning
and problem solving. However, his computation skills were fairly strong. I had no
doubt that he would be able to use the doubling strategy if someone pointed it out
to him, but he wouldn’t understand why it worked. To me, making sense of what
was happening as each floor was added and figuring out a strategy from that ex-
perience was the most significant aspect of the activity.
Collaborative Planning