Assessments are a tricky business. Writing an exam that successfully tests a person’s knowledge or abilities, without inadvertently giving preference or advantage to certain demographics, is very difficult. The examinations I’ve written so far for my job fall into the category of testing whether a student has mastered a certain curriculum. After a couple of months of class, we give them an exam to check if they learned all that they were supposed to. Everybody is used to such tests, and everybody has experienced them.
I’ve now been given the task of writing a very different type of examination. Instead of testing whether a student has learned certain information, I instead need to write an exam focused on whether a student is prepared to learn new information. Of course, these ideas are tied together, but not quite as closely as may be naively thought. For example, a student’s ability to learn and understand exponents relies on their understanding of multiplication. Yet, just because a student knows multiplication, that does not mean they are prepared to handle exponents. In mathematics, we would say that understanding multiplication is a necessary, but not sufficient condition for learning exponents.
Why would this be? Why is the content knowledge an insufficient proxy for a student’s ability to move forward? The most clear consideration is mathematical maturity. This ties into the ability to extend ideas via analogy, work generally beyond specific examples, and reason abstractly about new concepts. The jump from simple multiplication to exponentiation is an excellent example of this. You can give students multi-digit multiplication problems, and if they are computationally fearless or sufficiently clever with the multiplication algorithm, they can get all of them. Yet, when you start introducing new notation, and explaining why n things multiplied have any meaning, they may falter. So, you also need to somehow address a student’s ability to grasp new contents, and their ability to grow in their mathematical maturity.
This poses a serious challenge. There has to be some give in the process, there must be a proxy, a compromise, at some point. Of course this is true, because the only surefire way to determine whether a student is ready to learn exponents is by teaching them exponents and seeing how they do. But that is rather inefficient for a school that must limit enrollment to students who will benefit from the pace, and from being around students who are incredibly prepared to learn this new information.
So, how do we address it? Part of it relies on a human element. No matter what problems I choose to include in these assessments, I have to trust that the folks on the other end giving the assessment to the student are prepared to engage in a conversation afterward. They have to go through the student’s answers, their explanations of how they did the problem, and attempt to glean how a student will handle new ideas. Furthermore, including a problem that only tangentially relies on content knowledge, but is really just a puzzle or a tricky problem, can be very useful. While we expect students to get the "content knowledge" questions correct, even if they did not take the fastest route, seeing how a student faces down a difficult challenge that they are incapable of overcoming in the moment is very important. How does the student deal with failure, with hitting a wall? This is important, because we want our students to hit a wall. That’s our job.
The problem surrounding assessments, particularly ones that ask Are you ready? continues. I know that I and the people around me will keep working hard to make sure our assessments are in the best interest of our students. Hopefully other larger players, who affect orders of magnitude more students, can shift in a similar way. Although that’s unlikely to happen, I can hope, and keep working on the small bits in my own sector of the education world.