Today my class structure let me learn and address some unexpected things about my students.
If you have read this blog before, you know that I try to structure lots of class time around student presentations. A single student will discuss his or her own solution to a problem that I set them for homework, and the rest of the class will engage in constructive critique of the argument for that proposed solution. Sometimes, this means that things slow down and I don’t discuss exactly what I thought we might. Usually, it means that we end up discussing exactly what we need to.
So, in linear algebra, one of my students was up at the board to present on this task:
Consider the matrix equation
Draw a diagram of the “row picture” of this equation.
Now, the student did a reasonable job. She translated the matrix equation into the system of linear equations
x + 2y + 4z = -1 and 2x + z = 3, and then she graphed (to the best of her ability) the first equation as a plane on the chalkboard. Next, she claimed that 2x+z=3 is a line.
At this point, I learned something that I would not have found so easily when I used to lecture. This student had some trouble contextualizing that equation and identifying it with the correct object. This is an easy mistake to make, from one point of view. The y has dropped out of the equation, so it looks like a relationship between only two variables. Students have seen that over and over again. And the presenter was not alone. Many students in the room had this misconception. “It looks like a slope-intercept form,” said a different student. (It isn’t but it can be changed to that rather quickly.) So, I asked questions, we had a discussion, we had a vote, we had arguments for each position (line versus plane). Ten or fifteen minutes later, everyone was on board, and we had sorted out that we preferred the language:
The set of points (x,y,z) which satisfy the equation 2x + z = 3 .
This is an opportunity to point out that mathematical language is very careful for a reason. I didn’t take that opportunity in class, which was a minor failing on my part, but no one is perfect.
This same phenomenon happened again later in class over the idea of what it means to say that a vector v is a solution to a matrix equation Av = b. I am not sure how I would find out that so many students are missing some important thing without letting them do all the talking. And in each instance, I could adjust in the moment and get class focused on fixing the misconception. We might not have gotten through a discussion of all of the items I had planned for today, but I am certain we spent all of our time well.
So, get the students to the board. Make them explain themselves, you might be surprised what you learn. And if you are, I bet you’ll be glad to learn it when you can still address it.
My first title for this post was “Why IBL? Unexpected Formative Assessment Moments.” That seemed a little too full of teacher jargon to catch new readers. But now you are part of the club. All that business above is formative assessment.