To differentiate this Practice Quiz, I distribute it to all students, and then I ask them to determine which sections they feel that they can master. I ask them to indicate this with a star. I tell them that I will circulate and check to see whether I agree with their assessment.
I tell them: “The goal is for you to choose sections that are neither too easy nor too hard—which sections will be worthwhile for you to focus on today? Which sections can you master to show me your full understanding of this unit?” Obviously this is the kind of flexibility that requires accountability, so I also tell them, “If you obviously underestimate yourself, then I will make the decision for you. Earn my trust and respect by making a thoughtful choice.” After this, I also circulate to check in with each student, asking them sometimes to add another section or two to challenge themselves, or telling them that they have chosen some sections that are obviously too easy for them.
When using this kind of flexible assessment approach, I think it is important to be transparent about my motives, by saying something like:
I am not sure if it is a good idea or not to let you choose your sections for the assessment. My goal is for you to be able to focus on what you really understand, so that you can feel successful and really master the content, and also for you to challenge yourself and not waste time doing problems that are obviously too easy. I want to try this but if it seems like you are choosing problems thoughtlessly or trying to be lazy, then I will take away this flexibility and assign you even more sections.
It doesn't always work perfectly, but I have found that over time, this approach actually gets students to hold themselves to a much higher standard of comprehension than a less flexible approach.
As with any unit, I find that by the end I have a good fraction of students who have basically fully mastered the skills (even at the most challenging level) of the unit. I distributed these Interesting Cubic Function Problems so that my students who felt that they had mastered the content on the practice quiz would still have a challenge to work on.
I allow all of my students to assess themselves and determine whether or not they want to tackle these problems. If I see students working on these challenge problems when I don’t think they have mastered the basic content of the practice quiz, I ask them to show me how to do a few problems so that we can both make sure they fully understand. On the other hand, when I see students who have probably already mastered the skills of the practice test still working on it, I ask them if they feel that they really want that extra practice, or if they want to push themselves. Students don’t always choose for themselves what I would choose for them, but I want to initiate these dialogues as much as possible so that they can be aware of the factors affecting their decision.
Each one of these problems (except perhaps #4) can take a bunch of time and warrant a hearty investigation. I generally have only a few students who choose to take these on, but as much as possible I try to check in with these students to help them think of strategies that they can use in their process.