For this investigation, I give students the document “Write Piecewise Function Graphs” and I assign them a partner. Depending upon what students have figured out already, I make the investigation more or less structured. I start by saying, “Figure out as much as you can in the next 5 minutes.” Then I walk around quickly to get a sense of how well students understand this with no direct instruction:
If most students are at Level 0 or Level 1, then I interrupt them after 5 minutes and ask them as a class if they want a hint. Even if I know they are going to say yes, I take the time to give them the choice. This is a small way of reiterating the fact that it would be possible for them to figure this out themselves, and it also includes them in the decision making process about their learning. Done consistently, this kind of thing creates a very cooperative classroom culture in which students feel that they are agents in the learning process.
If students do want a hint, I write down one part of the function (if we think of this as a 6-part function). Then I keep going back and forth with them like this, give them some time to work; if they are struggling I ask if they want a hint; give them a small hint and have them get back to work. This process feels chaotic for teachers who are used to delivering direct instruction in which all the students listen and write stuff down quietly, and it is a sort of chaotic process, because it leaves space for each student to figure stuff out at their own pace. I say this just so that if you feel like it is not as “tight” as your usual classes, you can know that this is okay. As long as all students are thinking and talking about the problem, the chaos is fine. I only interrupt them if any of them get off topic, and I then I ask them if they have lost focus because they don’t understand or just because they aren’t trying. A dialogue ensues and this helps me figure out if the whole class (or specific small groups) need more direction or if they are ready to get started on the assessment, which is the Problem Set. Once they have showed that they understand the problems in the investigation, I let them get started on the assessment. This is another way that they are the ones setting the pace of their learning.