Assessment in a High-Density Classroom
“How do you assess this?”
This is the question I eventually field during every opportunity I get to share pieces of my classroom with other stakeholders in education–be it teachers, administrators, or pre-service teachers. I don’t mind fielding it; it is a good question, one teeming with complexities and littered with implicit values.
I was not the one presenting during my most recent encounter with the familiar script. Instead, I was eagerly awaiting its appearance as I thoroughly enjoyed a talk from an educator I hold in the highest regard. When it came, I tried to cling to his words so I might add his eloquent reply to my repertoire, but my mind wandered. I wondered why people are so obsessed with assessing students’ problem solving. I wondered why people seem to think that any opportunity for students to make-sense of problems with any degree of freedom must be void of assessment. I wondered how, in these moments of generative thought, people could become preoccupied with judgements of its worth. I became cognizant of my surroundings just in time to catch the look of surprise on participants’ faces when the speaker claimed that he used familiar forms of assessment (like quizzes and tests) alongside other, more collaborative assessments (like group quizzes and informal student-teacher conferencing).
But how? He just finished describing a classroom built on the premise of inter-activity. He just finished saying that students bring forth all sorts of different notations and strategies. He just finished labelling strategies as beautiful and surprising. How could such an unbridled classroom experience be held responsible to such a steady instrument such as the unit exam?
All of this came after the group was asked to think together about one of the speaker’s favourite problems. A problem with the potential to sponsor elegant reasoning, vivid imagery, and counter-intuitive twists. It was not the first time I’d encountered it; in fact, it was taken from one of my all-time favourite books, The Art of Problem Posing. Here, I’d like to introduce the problem, describe the solution our table group agreed upon, and then make a metaphorical tie back to how I view the role of assessment in a classroom built around a high density of student action.
Imagine a Cartesian forest with a tree planted at each location with integer coordinates. Each tree is perfectly straight, infinitely thin, and extends up well past your line of vision. This forest extends forever in each direction, and now imagine you become one of the trees. For convenience, you are standing at the origin of this infinite forest. The question that emerges from this suffocating landscape is simple: Is there a line of sight where you can see out of the forest?
We began by pursuing the opposite problem. In order to decide if we could stare past trees, we decided to ask what was required to stare directly at a tree. It was not long before the idea of slope emerged as relevant to our action, and we began to pursue the solution to the problem: What slopes will allow us to stare at specific trees? With the familiar rise-over-run mantra of linear functions dancing repeatedly through our minds (and finding its way into our conversational interchanges), we began to find paths of sight that would extend ever-further. Our problem drifted forms as we asked ourselves: What trees will not be blocked by other trees? We focused our efforts in quadrant one. If we are standing at (0,0), then every tree at (a,b) can be found by staring at a slope of b/a, providing that a tree has not already blocked this vision by being proportional to (a,b), like the two points (1,1) and (2,2). It seemed that we would need to solve the problem: What coordinate is guaranteed to not be proportional to any previous coordinate? From here we investigated primes, hypothesizing that if we could somehow find the largest prime, we could find an infinite gap in the trees and resolve the original problem.
Unfortunately, we knew this would not be possible due to the elegant reasoning of Euclid. We were then forced to conclude that we could stare very far, but not infinitely so. Then someone suggested that our slope model could stare at any rise-over-run, but we hadn’t thought about what it couldn’t stare at: Numbers that could not be represented by a rise-over run. The irrationals. The problem shifted again as we turned our attention to solving: Will an irrational slope ever hit a tree? Quickly, we were satisfied that it wouldn’t because the required slope could not, by definition, be written as b/a.
The problem had become one about number systems. After some discussion, the group settled on a startling (and beautiful) set of conjectures:
- There are infinitely many ways to stare at trees.
- There are infinitely many more ways to see out of the forest.
- If you were to turn at random and choose a line of sight, you are guaranteed to be looking out.
- Therefore, to see a tree, you need to be intentionally looking for trees.
Place yourself in a classroom where students are encouraged to work together. There are what seems like an infinite number of students each making decisions in real time. Add a curriculum document, a time deadline, the limitations of your senses, and a very real social dynamic not present when referring to Cartesian trees. This is a high-density classroom. Through all of this, you have anticipated a goal, a grade-level understanding that you intend on accentuating and applying in other contexts. As the teacher, you are infinitely surrounded with the necessity to make decisions. These decisions are made based on a complex history between you, the students, and the task.
There are times when a teaching decision is motivated by the need to assess specific student understanding. These moments are a necessary component to any classroom, and can take any number of an infinite forms (1). However, most teaching decisions are designed to keep action going, not to strictly evaluate it’s usefulness. They are made with an eye for the forest, the maintenance of class activity (2). Amidst all of this, there is one thing that all assessment activities have in common: They are intentional. In fact, although there are an infinite number of ways that assessment can take place, if you were to simply assume that assessment will take care of itself, you will miss the trees every time (3). Therefore, in any classroom there are moments when we purposefully stare at trees to orient ourselves as a group (4). These moments are not at odds with the moments when we act to maintain inter-activity, yet somehow a unit exam comes as a shock to people whose impression is that the pathway to high-density interaction is through the disregard for assessment. Assessment informs instruction in any classroom, including those focused on collaborative problem solving. It is not a mismatched burden to the knowing activity of learners, but a critical part of maintaining a healthy dynamic. It is unhealthy to ignore the forest of activity in favour of constant tree-gazing, but just as unhealthy to stare past every opportunity for assessment. The two are mutually beneficial.
After all, as we experienced through the problem, staring at trees can inform how to look past them.