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Encountering abstraction with clickers

February 27, 2012, 8:00 am

I had the great pleasure this weekend of leading a session at Math In Action, which is Grand Valley’s annual K-12 educators’ conference. My session was called “Classroom Response Systems in Mathematics: Learning math better through voting” and was all about the kinds of learning that can take place in a class where active student choice is central and clickers are mediating the voting. (Here are the slides.)

It always seems like a bait-and-switch when I do a “clicker” workshop, because although people come to learn about clickers, I don’t really have much to say about the technology itself. As devices go, clickers are about as complex as a garage door opener, and in fact they work on the same principle. There’s not a lot to discuss. So instead, we spend our time focusing on the kinds of pedagogy that clickers enable — which tends to excite teachers more than technology does.

The audience were all high school math teachers, and the centerpiece of the workshop was having them make up their own clicker questions, in groups, for a class they teach. They came up with some really good questions, one of which I wanted to put up here for you to see and comment on.

The question was created by three guys who teach beginning algebra. It says, simply:

\(x\) is a number.
(A) True
(B) False

This is a good clicker question because it’s simple, and because it gets to the heart of a lot of beginning algebra students’ difficulties with algebra and the abstraction layer of variables. After looking at it as a large group, we modified this question to say:

\(x\) is a number.
(A) True, and I am very sure
(B) True, but I am not sure
(C) Not sure whether it’s true or false
(D) False, but I am not sure
(E) False, and I am very sure

Giving students an “I don’t know” option, makes sense at this level. I think it’s important for students in a class like this to realize that it’s OK not to be sure about some things when you are learning them. And there are some interesting things you can learn by allowing students to rate their confidence levels along with stating true/false. In fact, I decided on the fly to let the session audience vote on this question. A little over 50% voted (A), while 30% voted (E)!

The question I posed to the participants was, What do you do with this information if you’re the teacher? You have two groups of students, each a sizable portion of the class and each totally opposite in their answers. The first instinct of the teachers was to say, we step in and explain the right answer to the students. But since the teachers themselves had opposing views on what the right answer was, I had the teachers try to explain it to each other first.

The teachers realized there was a semantic issue involved — \(x\) is a placeholder that represents a number, but it is not a number itself; unless \(x\) is specifically set equal to a number in which case it is no longer a placeholder. It’s this very layer of abstraction that trips up students in algebra and later. Coming to grips with abstraction is crucial for understanding algebra; and yet, I have never heard of an algebra class actually discussing this. It’s always an edict from a lecturer telling you what to think, and students tend to accept it on authority rather than on the basis of whether it makes sense. Even if the teacher’s explanation is correct, it does the learner no good if the student doesn’t personally make sense out of it.

What’s great about using clickers in this way is that they enable a situation where students can think about this issue and start to form their own conception of the meaning of a variable, using not only the teacher’s ideas but also the ideas of their peers. Hopefully they can take teaching ideas like this back to their schools and put it to good use, which they can do even if they don’t have clickers.

Image: http://www.flickr.com/photos/reway2007/

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