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.

Thanks! I’ve been watching from the sidelines, and keeping my opinion of IBL to myself. My prior experience with a thing called IBL was less than fulfilling. I was handed a book suggested by a professor, told to read and do problems and that that was IBL. The book had neither solutions, nor completely worked examples. Needless to say, I loathed the idea of IBL and basically considered it LPS, (lazy professor syndrome).

What you’re doing though, you know, real IBL, is fascinating! I would like to think this is the way lectures should happen if the students have the guts to ask questions and participate on their own. Using IBL to help them model constructive behavior is awesome!

Any chance you could tweet, or G+ microblog all the misconceptions you find? For those of us who got through to higher level calculus physics, and whatnot without taking a linear algebra course, it’s very useful information.

Just like everything else, IBL can be done well or it can be done poorly. What you describe may or may not be intentionally lazy, but it certainly lacked some qualities that I identify with good teaching:: like getting to know your students, and helping them navigate their difficulties. (I try to do all that stuff on the sly, so they can own the victories, but I still do it.)

I don’t know if I can report ALL of the misconceptions I find, but I will think about sharing others as they pop up.

I hit a local minimum on IBL recently, due to my lack of success with Moore Method-type courses (and my relative success with Peer Instruction classes). However, the function is increasing now thanks to this post (and one other thing that I am going to blog about, which you were also mostly responsible for).

Peer Instruction is great at rooting out misconceptions, since I can guess what the misconceptions are and force the students to directly confront them. This is seriously great, and I do not want to lose this.

But the problem is that I need to anticipate misconceptions in Peer Instruction (well, mostly. But it is easier to write in absolutes). In IBL, it sounds like the students often bring their misconceptions to you. This is the thing that is helping me get back on the IBL bandwagon.

I look forward to reading what you have!