I have been thinking a lot about placement and assessment. So many questions fly through my mind:
Who belongs in my class? What can I assume about the students who ARE in my class? What effect do my assumptions have on my students? What effect do my assumptions have on my teaching? What should I do if I discover that I’ve made an assumption that isn’t true? These are just a smattering of the questions I’ve been asking myself this last year. My opinions and answers are still in development. I’ll share where I’m at right now.
- Trained in logic and proof, I understand that conclusions are unreliable if the corresponding assumptions are not met. So while I have to make some assumptions in my teaching, I know that I absolutely should not base my work on any assumptions that I know to be false. For example: There is so much that we learn and forget! If I know that many of my students will be ill prepared to recall the background knowledge needed for a new topic, then I had better not assume that they will be ready with that background knowledge. I don’t have to teach them it, but if that bit of knowledge is on my wish list of what they’re good at, I think they’ll be more successful if I give them an exercise that reminds them of that stuff that they learned once and need to recall now.
- I used to think that some students did not belong in my class. I used to think that something went awry in some cases. “How could that person POSSIBLY have made it through all the other classes and yet still be so ill-prepared to succeed in my class?!” I even used to try to figure out what had gone wrong. But now, I know that it really doesn’t matter. And I believe that every student in my class belongs there. Some will find it easier to succeed and some will find it harder to succeed. But if they’ve placed into my class or if they’ve taken classes that serve as prerequisites for my class, then they do have the mathematical background to tackle the ideas in my class. For example: Many calculus students entering my Math 1C class still struggle with the difference between power functions and exponential functions, topics from algebra and precalculus. I used to think that one solution was to send them back to an earlier class. Now I think that that might be the worst thing in the world for them. They are intellectually ready for Math 1C, and their understanding of power and exponential functions will deepen with the kind of work we do together. But the moment I assume that they understand those functions as well as I do, I am bound to create activities that are not helpful, except to the lucky few who remember more than their peers.
- No one would ever look at a group of students and tell them that they should remember everything they’ve ever learned. But I have been repeatedly guilty of believing just that. Whenever it suited my needs, I would think “they should remember how to do this; they learned it in algebra!” I’m speaking in the past tense because I hope that I never do that again. Now I tell my students, “you can’t possibly remember everything you’ve ever learned. But when you find that you need a skill that you’ve forgotten, well, that’s the time to relearn it or remind yourself of what you used to know. And if there’s something that you need and you never learned it before, that’s OK too. There are lots of resources; and your teachers will always be happy to help you out if you run into trouble along the way.”
- There are two experiments that I am dying to play with.
- I’d love to take a group of students who pass a class and give them a final exam a second and third time, maybe one month after they pass and again one year after they pass. I’m really curious to know how they’d do.
- What would happen if I tried to complete an exam from a class that I haven’t taught in 1 year, 3 years, 5 years, ever? I think it’s safe to predict that I would not do well in some cases, depending on the class, the style of the exam, and the length of time since I thought of anything related to it. And what conclusions would be reasonable? I am thinking of this because of the conclusions we draw from placement exams. Are they reasonable? A student who just completed a course in differential equations could expect to outperform me on a differential equations exam (unless I prepared myself in advance). Does that make them more qualified than me to teach differential equations next fall? Of course not! I think we need to think of these scenarios when we’re working on placement mechanisms for students.
I don’t know how to end this, because the ideas are still bouncing around in my head. But thank you for listening to a first attempt to get some of them on paper. You know, they say that the act of writing helps you to form the ideas…