File this under “Better Late Than Never.” Back in January 2010, I coordinated (with Kelly Cline and Kien Lim) a contributed paper session on teaching with clickers at the Joint Mathematics Meetings in San Francisco. Shortly after the conference, I blogged about some clickers talks that didn’t fall in our session, but I never got around to blogging about the talks in our session! Five months later, I’m finally getting around to sharing my notes from those talks…
Jean teaches a “math for the liberal arts” course called Contemporary Mathematics taken by music and arts majors among others. She finds her students come to the course with relatively little interest or self-reported ability in mathematics, so it’s a tough crowd to teach. A few years ago, she started teaching with clickers in order to appeal to what she calls the “thumb generation”–students used to spending a lot of time sending text messages.
Jean interspersed some clicker questions throughout her lectures and encouraged students to discuss them in small groups before voting. She and the students liked this, but she found that most of her questions were answered correctly by most of her students and that the small group discussions didn’t involve much debate among students. The next year, Jean decided to ask tougher questions. She calls them QEDs–Questions to Encourage Discussion. She aimed for the analysis, synthesis, and evaluation levels of Bloom’s Taxonomy.
For example, in Jean’s first year using clickers she gave her students a preference schedule–a list of how each voter in an electorate (only four of them to keep things simple) ranked all of the candidates. She then asked her students to determine which candidate would win the election using the instant run-off voting scheme. This is a straight-forward application of a particular algorithm.
The next year, Jean asked another question in which she shared a preference schedule with her students and asked them to analyze it. However, this time, she asked the question at the beginning of the unit on voting schemes and asked her students to indicate which of the candidates had the best case for winning the election. There’s no single correct answer to this question since winner of an election (well, one involving at least three candidates) depends on what scheme you use to count the votes. This is a great example of a one-best-answer question (since students are asked to select the one answer they think is best among multiple reasonable answers) used to create a time for telling (since it’s used to make the point that which voting scheme you use matters).
Jean found that these more challenging and ambiguous questions generated longer and more engaged small group discussions as well as more “horizontal” bar graphs–ones indicating significant disagreement among the students. Looking ahead, she plans to build on this success by writing questions designed to develop mathematical habits of mind–an important goal of this course. For example, here’s a sample question she shared aimed at pointing students towards the notions of proof and counterexamples:
Suppose there is a majority winner in an election. Will all of the voting methods we have studied thus far always pick that winner? Yes or no? If you answer yes, prepare to defend your answer. If you answer no, have a counterexample ready.
I really like this question. It has a degree of ambiguity that students often find disconcerting, but it also reminds students of how they’ll need to defend their answers, which should help put their minds at ease. As Jean noted in her talk, in a course like this one, it’s more important students develop mathematical habits of minds (like the ones surfaced by this question) than learn particular math content areas. I hope this kind of question helps with this objective.
Stay tuned to the blog for more notes on these talks over the coming days…
Image: “Deep Down Inside, We All Love Math” by Flickr user Network Osaka / Creative Commons licensed