Recently I’ve become aware of the reality that as math teachers, we often talk about what problems we will be using and even how we will implement them but frequently we miss opportunities to discuss why we’re using them. Consider Graham Fletcher’s amazing sugar cube task on volume of a rectangular prism. I can think of three different reasons why you would do this task, and all three of them greatly impact the decisions you make during the problem and consequently how the students learn.
Consider though how this could potentially look to two observers: one who understands the reason why you are using this problem and one who does not.
- An observer who understands that you are using this problem to introduce a new concept expects that students will be curious and possibly baffled by the problem. They’ll see how students will be eager to learn more about how to solve the problem. Whether or not you finish the problem is inconsequential because you can come back to it at the end of the unit. Either way, you invested time in developing a marvelous context that will pay dividends throughout the unit.
- An observer who doesn’t understand that you are using this problem to introduce a new concept may wonder:
- What was the purpose of this problem?
- Why didn’t you finish it?
- Why didn’t you let students struggle through it?
- Did the teacher end the problem because he or she was confused and gave up?
- An observer who understands that you are using this problem to let students productively struggle is looking to see that kids are working hard to make connections on their own. The observer realizes that your choice to give minor hints and not tell students what to do is not laziness or being unprepared, but rather trying to give students the least amount of help needed for them to keep going. This is similar to a bench presser and a spotter. A spotter’s job is to give the bench presser the least amount of help needed so that he or she can lift the weight. Too much help and it’s the spotter who gets stronger. Too little help and the bench presser dies. As part of this process, students will inevitably demonstrate many of the Common Core Standards for Mathematical Practice including 1, 3, and 6 as they try to make sense of the problem and convince their partners why they are right or wrong.
- An observer who doesn’t understand that you are using this problem to let students productively struggle may wonder:
- Why did the teacher let the students sit there confused instead of telling them what to do?
- Did the students even learn anything because they never figured out the answer?
- Why didn’t the teacher finish the problem? Did she lose track of time?
- An observer who understands that you are using this task to show a completed problem realizes that you had to make some sacrifices to complete it in a single class period. The observer is better prepared to imagine how different parts could be expanded, given more time, and is appreciative of having experienced the problem in Cliff Notes form.
- An observer who doesn’t understand that you are trying to complete the problem in one period may wonder:
- Who really did the work today: the students or the teacher?
- Why did the teacher not see all those great opportunities for students to make their own connections and take advantage of them?
- Why did the teacher give such obvious hints and tell them what to do?
My recommendation is to have conversations with your colleagues about why you are using specific problems. You may find that your “why” is different from theirs. Not having that conversation before the problem usually sounds like this after implementing the problem:
- Teacher 1: “Man, that problem was challenging for students. They spent the whole period working on it and didn’t even finish it.” (Goal is productive struggle)
- Teacher 2: “Yeah, it was challenging but I gave them some hints and we worked our way through it because we are beginning a project tomorrow.” (Goal is problem completion)
- Teacher 3: “We didn’t actually finish the problem. We started it but we’ll come back to it later once they’ve learned more about the concept.” (Goal is introducing a new concept)
What do you think about this? What other reasons for using a problem are you seeing?
It made me wonder:
– is using a problem for assessment its own category?
– does this categorization only apply to non-procedural problems?