Since debate and argumentation are a regular part of my math classes, I also find it important to incorporate them into assessments. That is, I want students to create arguments on tests and quizzes. I’ve played with a few varieties over the years, but what I use most often (and what my students have come to expect) is a question like the following on every test.
This question is from a test early in the year in PreCalc. I don’t intend these questions to be too involved in the beginning of the year. I mostly want to see if they can give a decent argument, and I spend a lot effort giving written feedback on their arguments. My goal is to make clear what I expect on future tests (when I will be a little pickier in my grading).
Question B is one of my favorites to give feedback on and talk about with the whole class afterward. It is pretty clear to all students that the angle drawn is way more than 200 degrees. However, this is where I really get to talk about a quality and convincing argument.
Three typical pitfalls include:
- Assuming – every year I get students who say that the pictured angle is 300 degrees because that is roughly what matches a point on their unit circle diagram. These students are only considering the key angles that our class has memorized coordinates for (every 30 or 45 degrees). My comments include the question: what if this is 302 degrees? They are correct that the angle is not 200, but their argument is not necessarily a true statement.
- Vagueness – other students correctly say the angle is not 200 degrees, but then say vague ideas, presumably to have something written but possibly unsure how to be convincing. This includes statements like: This angle is not 200 and my warrant is that it is big and probably bigger than 200. A student who says this might have been able to give a convincing argument but was never pushed or had it made clear what was expected. This is where I like to talk with the whole class about precision.
- Missing Connections – This last group is the one I really want to talk about with the whole class. This includes students who might say something like: This angle is not 200 and my warrant is that it is in the 4th quadrant. This student is SO CLOSE to having a solid argument (and at the beginning of the year I may give them credit, but with lots of comments). They are just missing a connection. I ask what about the 4th quadrant makes that angle not 200?
There are definitely some students who give a good argument from the start, and I show those to the whole class as well. I really like to show when someone says something like
My claim is that angle is not 200 degrees and my warrant is that it is in the 4th quadrant and angles in that quadrant are between 270 and 360 degrees.
This gives me a great contrast to show the students who have a missing connection to see what exactly they were missing.
*One more note about these argument questions is that they really help emphasize that there is not just one correct way to answer a problem. You don’t have to talk about the 4th quadrant to be correct. Other students might say:
- That angle is not 200 degrees and my warrant is that 200 degrees is in the third quadrant and this angle is not.
- That angle is not 200 degrees and my warrant is that it is greater than 270 degrees.
And this is just the start. I love building from here!