I received an email last week from Bill Goffe who teaches economics at SUNY-Oswego (and contributed to this great guide to teaching economics with clickers) with a neat tip for practicing agile teaching. He noted that he’s heard Harvard physics professor Eric Mazur talk about bringing to class a folder full of transparencies, each with a different clicker question. Mazur asks his students the clicker question at the top of the stack and, depending on how well the students do, either moves on to the next question or skips a few to move to a question on a different topic. This is a great way to practice agile teaching by basing the selection of a clicker question on the results of a previous one.

Bill wrote that it can be challenging to take a similar approach if one’s clicker questions are embedded in PowerPoint slides. Breaking out of presentation mode, wandering through one’s slides in “Normal” or “Slide Sorter” mode to find one’s next question, then switching back to presentation mode to display the question–that seems like an awkward process, particularly if it’s visible to the students. However, Bill found a better way:

Yesterday I came across [the Inside Higher Ed article] “The Advantages of a 2,500 Slide PowerPoint Deck,” and it had the solution: number your slides and bring a printed version of the slides in outline view. Look at the latter, find the desired slide, type that number, and then PP takes you to that slide, still in display mode. It might seem like a small point, but it would make class much smoother.

I had no idea that you could just type a slide number while in presentation mode in PowerPoint and instantly go to that slide. (Try it, it really works!) That’s a handy trick for going nonlinear in PowerPoint. It might require you to have a printout of your slides handy so you can determine the number of your next slide, but as the comments on that article point out, if you use the same slide deck over and over, you’ll probably start to memorize some of those numbers. (The presenter described in the article uses the same 2,500 slide PowerPoint deck for all of his presentations. He just skips around nonlinearly in response to questions from the audience!)

This trick is probably better than the one that came to mind when I first read Bill’s email, which is to switch from the “slides” view in PowerPoint to the “outline” view. The outline view gives a more compact view of your slide deck, and, depending on how you’ve formatted your clicker questions, can show you your clicker questions particularly well. I think the type-a-slide-number trick is even slicker.

Another way to go nonlinear is to use Prezi. In Prezi, you can organize all your content (text, images, clicker questions, whatever) visually on a great big canvas. This means that finding a particular bit of content is pretty easy, assuming you’ve placed it on the canvas in a sensible location. And while you can set up a “path” in Prezi to follow somewhat linearly, you can always go “off path” and zoom around to other content at will. I’ve used Prezi for the visuals in a few presentations that also included clicker questions, such as this talk at the University of Louisville and this one at Central Michigan University. In both cases, I simply embedded my clicker questions in the Prezi (in a particularly clever way in the Louisville talk) and ran my clicker software on top of Prezi. Worked like a charm.

(See how I turned the entire Prezi into one big clicker question? The letters A, B, C, and D weren’t visible until near the end of the presentation when I zoomed out and posed my final clicker question.)

Of course, in those talks, I was mostly moving linearly through the Prezi. I can see, however, setting up a Prezi where your clicker questions are organized visually in groups and subgroups and using that to go nonlinear during a class session. It’s not quite as slick as typing a number and instantly moving to a different slide, but I would guess that some of us would be faster at navigating visually to a new question than remembering or looking up a question number.

Finally, one of the comments on the Inside Higher Ed article that Bill sent me links to a blog post describing a “Choose Your Own Adventure” session on information literacy designed and facilitated by librarians at the University of Dubuque. PowerPoint hyperlinks (yet another way to move nonlinearly through a slide deck) were used with clicker questions to have the students determine the progression through the slide deck as they grappled with information literacy tasks like finding and evaluating the quality of sources.

I’ve been eager to find more examples of this kind of classroom response system use since I first read about it in the David Banks book on response systems. That edited volume includes a chapter (Hinde & Hunt, 2006) on the use of a “Choose Your Own Adventure” style question tree in an economics course. For more on the Dubuque library use of CYOA / question trees, see this follow-up blog post and the PowerPoint deck itself.

Thanks for sharing, Bill!

Image: “Choose Your Own Adventure 1” by Flickr user Jason Permenter, Creative Commons licensed

Earlier this week, I gave a virtual presentation at the Muskegon Community College Math and Technology Workshop organized by Maria Andersen. The participants were all math instructors spending the week at MCC learning from Maria and others about various uses for educational technology in math instruction.

I’ve blogged often about teaching math with clickers here, but I don’t think I’ve shared slides from any of my presentations on this topic. Since Maria asked me to put my slides on Slideshare for the workshop participants, I thought I would share them here.

Continuing my reports from the contributed paper session on teaching with clickers I helped coordinate at the Joint Mathematics Meetings back in January…

“Using Prediction and Classroom Voting via Clickers to Address Students’ Overreliance on the Representativeness Heuristic,” Tami Dashley, University of Texas-El Paso [Slides]

Tami Dashley is a graduate student in math education and a student of Kien Lim, one of the organizers of the contributed paper session. She shared some of her thesis research, an investigation into the connection between classroom voting with clickers and certain misconceptions students have about probability. Her work focuses on the representativeness heuristic, which she defines as “determining the likelihood for events based on how well an outcome represents some aspect of its parent population.”

Tami gave the following example: Suppose you toss a coin six times, getting a sequence of heads (H) and tails (T). Which of the following is more likely to occur: TTHHTH or HTTHHH? Someone using the representativeness heuristic would say that TTHHTH is more likely to occur since it includes an equal amount of heads and tails, just like the coin does. The other option includes more heads than tails, so it would not seem as likely to someone using the representativeness heuristic. Actually, both of those outcomes are equally likely (each occurring with probability 1/64), so the representative heuristic is a misleading one in this example.

The issue is that the representativeness heuristic is useful in some cases, but not useful in all cases. The misconception that many students have is that it’s always useful.

How to help students stop over-relying on the representativeness heuristic? Tami has been investigating the use of prediction questions, ones that ask students to predict an outcome or probability without actually computing anything. For example, students might be asked to determine which of several outcomes is most likely to occur. Since students need not be as precise when responding to prediction questions, they have some cognitive processing power freed up to focus on concepts. Clicker questions are a natural match here, since they allow students to commit to their predictions and compare their predictions to those of their peers. Then discussion of the incorrect answer choices provides an opportunity to deal with misconceptions.

Tami conducted her research in a high school setting, using three groups of students. Her “control” group received a lesson exploring the representativeness heuristic that didn’t ask the students to predict any probabilities. A second group was asked several prediction questions but didn’t use clickers to respond to the questions. The third group used clickers to respond to prediction questions during the lesson. Tami used pre- and post-tests to determined the efficacy of these three different lessons.

Tami found that her “control” group did pretty well on the post-test compared to the two experimental groups. However, most of their success came from what she called a “learned response.” In this case, many of the students picked up on the fact that “all of the above outcomes are equally likely” is often the correct answer to questions exploring the representativeness heuristic. (These are what students might call trick questions!) When Tami looked at performance on questions where “all of the above outcomes are equally likely” was, in fact, not the correct answer, the prediction-with-voting group performed better than the control and prediction-only groups.

I was very impressed with Tami’s research design and the subtlety with which she explored student misconceptions in this teaching context. I don’t believe that Tami has published this work yet, but I look forward to reading it when she does.

Image: “Heads and Tails” by Flickr user canonsnapper, Creative Commons licensed

The summer meeting of the American Association of Physics Teachers (AAPT) wraps up in Portland, Oregon, today. There were several talks on teaching physics with clickers at the meeting, including one by Ian Beatty of the University of North Carolina at Greensboro physics education research group. Ian was the subject of my first podcast interview, and he’s been doing great work helping science instructors at the K12 and post-secondary levels teach effectively with clickers.

In Ian’s presentation, he identified and addressed several common concerns instructors express about teaching with clickers. For each concern, Ian identifies a belief about teaching and/or learning that likely underlies the concern, as well as an alternate belief that can be adopted to address the concern productively. Ian also includes some practical strategies and example clicker questions for each of these alternate beliefs.

For example, when many instructors hear about teaching with clickers, they’re concerned with having sufficient class time to cover what they need to cover in their courses given the time required by having students discuss and respond to clicker questions. Ian notes that this concern is likely a result of the following belief: “I must explicitly address in class everything students will be held accountable for.” Ian then presents an alternate perspective on this idea: “I can use class time to focus on core ideas and big-picture understanding, and charge students with filling in the details outside class.” This alternate perspective is, perhaps, non-intuitive to many instructors, but it’s a reasonable and useful perspective to have. Adopting this perspective leads to a shift from what Ian calls an understanding of class as a place to present content to an understanding of class as a place to help students digest content. Ian then shares five tips and techniques for implementing this shift in the classroom.

Ian addresses other concerns in a similar manner, including concerns about having enough time to write good clicker questions, concerns about poor student participation during class, and concerns about changing one’s teaching style. His visuals, which use the online presentation tool Prezi, are included below and are well worth checking out.

Over on the Vanderbilt University Center for Teaching blog, my Vanderbilt colleague Isabel Gauthier, professor of psychology, has shared her experiences asking her students to write their own clicker questions. I met with Isabel a few years ago and briefly discussed ways to use clickers in her courses, and she’s really taken the technology (and pedagogy) and run with it. She’s got a great handle on how to have students write their own clicker questions, and I’ve been wanting to share her experiences here on the blog for a while. Here’s her article, in her own words:

It is difficult to write meaningful and discriminative multiple-choice questions that students find clear and fair. Years ago, I met with CFT assistant director Derek Bruff, who gave me useful pointers to perfect this skill. But a side effect of this interaction transformed entirely the way I teach: I learned so much by working on writing better questions, surely my students could learn too! Derek said something like, “You know, some teachers ask their students to generate questions…” This idea took me on a path to use this strategy, cautiously at first, and then more boldly, as the central pedagogical and evaluative strategy in some of my courses, including Brain Damage and Cognition and Principles of Experimental Design.

I teach these courses three days a week. On two of these days each week, I lecture on course material. These lectures are informed by questions about the readings posted online by students and issues that emerge from a hands-on, semester long project I assign my students. On the third day, we use clickers to go through student-generated multiple choice questions.

Each week each student is responsible for turning in a single question on the weeks’ readings. Students use a PowerPoint template to submit their questions which facilitates use of the question in my clicker software, TurningPoint. In the notes area of the slide, each student includes their name, the correct answer, the page(s) that inspired the question, and, optionally, a justification for the correct answer. Before class, I concatenate all the questions in a single file and read them, grading each on a scale of 1 to 5. The grade goes in the notes area, and, in a textbox on the slide, I write comments about the question. This allows me to print the slides as a PDF with student names removed so that all questions and comments can be distributed to students. I then reorder the slides to choose the right mix of questions I want to use in class with the clickers.

This provides me in a single step with my preparation for the next class, an idea of what I need to focus on during my lecture days, an evaluation of each student, and a mechanism for providing students with feedback on their learning. This weekly feedback allows students to realize how difficult it is to write a good question, one that raises an important issue clearly and is appropriately challenging for their peers. Students eventually learn to key in on critical concepts and relationships in the readings and sometimes even go beyond the readings in interesting ways. They take a more active part in their own and their peers’ learning, and their questions keep me focused on what is most challenging for these students at each point in the course.

Each week students answer the best of these questions in class using clickers, accumulating points for their answers using a generous but motivating grading scheme. If there’s controversy over the correct answer to a question, the class can decide to eliminate a question or to accept multiple answers as correct, provoking interesting discussions. As needed, I can lecture for a few minutes, but issues are generally clarified in class discussion. Questions are used anonymously in class, but students want their question to be picked and use wit and humor to this effect, making the experience more enjoyable for everyone.

This method completely replaces any exams I used to give: They are no longer needed since my students now share the responsibility to evaluate their own learning throughout the semester.

Isabel and her use of clickers were featured on Nashville’s NewsChannel 5 last year. Here’s the video clip:

Reference: Webking, R., & Valenzuela, F. (2006). Using audience response systems to develop critical thinking. In Banks, David A. (Ed.), Audience Response Systems in Higher Education: Applications and Cases. Hershey, PA: Information Science Publishing.

Summary: Webking and Valenzuela describe ways they use classroom response systems in their political sciences courses at the University of Texas-El Paso to foster critical thinking through active participation and class discussions. After noting some commonly cited advantages of teaching with clickers—easier attendance and participation record-keeping, greater participation through anonymity and accountability, and the collection of data to inform agile teaching decisions—the authors provide several concrete examples of clicker questions they have found valuable for developing their students’ critical thinking skills.

The authors’ first example is a sequence of clicker questions that serve to guide students through a close reading of a few passages in the play Antigone. At one point in the play, Antigone makes a statement that seems to very clearly express her belief that obedience to the gods trumps obedience to the king. At another point, however, she makes a somewhat cryptic statement that calls this previous assertion into question. Webking and Valenzuela start with an understand-level question that asks students to clarify this second statement. They follow this with an application-level question asking students to identify a logical consequence of her cryptic statement, one which seems to run counter to her earlier statement about serving the gods. Their third question is an analysis-level one, and it asks students to reconcile the two seemingly contradictory statements by Antigone by identifying a hidden motivation of hers that makes her statements consistent.

Webking and Valenzuela also describe how they use a particularly challenging, analysis-level question about Plato’s Euthyphro. The question asks students to identify the central argument of a particular passage, one that deals with the relationship between justice and piousness. The question is one that Jean McGivney-Burelle would call a “horizontal question” since students answering the question are typically split evenly among three answer choices. Webking and Valenzuela note that one of the three popular responses can’t be supported by the text. Students who argue for this answer choice quickly realize that they were projecting their own perspectives on the text, not arguing from the text. This is a useful metacognitive moment for these students. The class discussion then focuses on the remaining two popular answer choices. Making sense of these two choices requires the students to grapple with categorical logic, the kind that is well-represented by Venn diagrams. Once the students have discussed their way to the correct answer, they realize the value of categorical logic in making sense of arguments like the ones Plato makes—another metacognitive moment.

The Plato example comes from one of the authors’ smaller, upper-level courses, and they assert that “it is in a smaller class that the [classroom response] system is at its best in encouraging discussion and precise argument.” They reach this conclusion, in part, because of the ability of their classroom response system to report to the instructor individual student responses to clicker questions as those responses are submitted. The authors use these individual, real-time results to guide their post-vote discussions, focusing on “groups which had difficulties in reaching consensus, students or groups which answered particularly quickly or particularly slowly, students who disagreed with their groups, students who changed their minds, and so on.” They argue that the ability to see individual, real-time results is important in leading effective post-vote discussions since it allows instructors to analyze “each student’s rational odyssey with each question.”

Also in the article are two examples of student perspective questions the authors use to motivate particular topics in their courses. In one example, they ask students to identify questions they aren’t likely to ask someone they’ve just met. Invariably, students identify the questions about religion and politics. The authors point out to students that one reasonable conclusion from this is that religion and politics are the least important things to know about when getting to know someone. This motivates students to want to learn why this social phenomenon exists.

Comments: This would be a great article to give a faculty member in political science or philosophy who’s interested in getting started teaching with clickers. Webking and Valenzuela provide a concrete, interesting example of a guided close reading of a text (Antigone) using clicker questions of increasing difficulty. This is a great model for instructors in the humanities and social sciences interested in helping their students develop critical thinking and close reading skills. I wish, however, that they had included some voting data in this example and had discussed how they use the results of these questions to guide discussions, as they did with their Plato example.

The Plato example is a great model of clicker use in text-based courses, too. One reason is that the approach Webking and Valenzuela use leads students to appreciate the nature of argument in their discipline. They write, “In time, and actually not very much time, students learn to care more about the strength of the argument than about having their initial position defended as right.” The authors present a useful list of options for leading these kinds of class discussions—focusing on groups that were conflicted, students who answered quickly or slowly, students who changed their minds, etc.

The authors assert that the quality of discussions they can foster depends on the availability to the instructor of real-time, individual voting data. Not all classroom response systems have this feature and, in my experience, instructors who have the option of looking at individual results as they come in don’t frequently take advantage of this option. I think that perhaps the availability of real-time, individual results isn’t as critical as Webking and Valenzuela assert. I’ll often have my students vote on a question individually, then discuss it in groups, then vote again. I’ll sometimes ask for a student who changed his or her mind from the first vote to the second vote to explain his or her reasoning. I can also see asking for a student who disagreed with his or her group to contribute to the post-vote discussion.  (That’s a nice idea, one that I’ll have to try soon!)

My approach, using the aggregate and not individual voting data, relies on students who fit certain profiles volunteering to share their perspectives with the class. Webking and Valenzuela’s approach doesn’t rely on volunteers, but it isn’t quite cold-calling, either, since they select students only after the students have had a chance to consider and respond to the clicker question. I’d like to call this “warm-calling” since the students have had a chance to warm up to the question and since the instructors aren’t calling on students without any knowledge of what those students might contribute to the discussion. I’m not familiar with many instructors who practice warm-calling.  If you do, I’d love to hear from you in the comments about your experiences doing so.

Image: “Coffin Sculpture of Antigone” by Flickr user Xuan Rosamanios / Creative Commons licensed

Continuing my reports from the contributed paper session on teaching with clickers I helped coordinate at the Joint Mathematics Meetings back in January…

“Preservice Elementary Teachers’ Perceptions of Clicker Use in their College Mathematics Course,” Travis K. Miller, Millersville University of Pennsylvania [Slides]

In my last post, I mentioned that Janet White first used clickers in her courses for pre-service teachers at Millersville University of Pennsylvania. Another speaker in the contributed paper session back in January was her colleague, Travis Miller, who shared results of a student survey he conducted in the pre-service teacher course he taught. Travis used clickers for only six lessons during that course in each of the four sections he was teaching. His clicker questions weren’t graded, and he followed the “classic” peer instruction model each time, having students vote individually, then discuss the question in small groups, then vote again.

Travis’ students overwhelmingly (96%) liked using clickers in the course. Travis mentioned that there are very few things he does as a teacher that are as uniformly popular with his students! Almost as many students (89%) believed that the clicker activities helped them learn the material in those six lessons. Travis drilled down on this, asking students to say why the clickers were useful. The number one answer (59% of students) was that the clicker questions provided students with an opportunity to discuss and think about course content. The number two answer (23%) was that the clickers provided a sense of accountability and involvement.

Travis didn’t stop there, either. He asked his students which topics they understood better because of the clicker activities. Of the six topics that Travis addressed using clickers, sets and Venn diagrams was cited by 52% of the students as the one that most benefited from clickers. Numeration / base arithmetic was a distant second with 15%, and deductive reasoning came in third with 13%. When sharing these data, Travis floated a very interesting hypothesis. He wondered if the fact that the number one topic (sets and Venn diagrams) was a visual one led to the students selecting it as most benefited by clicker questions. I’m a big fan of visual thinking, so this comment caught my attention. Is there something special about peer instruction with clickers and visual thinking?  I’d appreciate your thoughts in the comments.

Travis’ other interesting hypothesis was that his more competitive students liked the competitive aspects of clickers (being the first to answer, answering correctly more frequently than other students, and so on), while the non-competitive students didn’t mind those aspects since they were essentially opt-in. That is, the students who didn’t want to compete could still participate fully with the peer instruction and voting process without feeling any pressure to treat it like a game. Graham, Tripp, Seawright, & Joeckel (2007) found that most students who are hesitant to participate in class liked clickers as well as those who were fine with participating, but I don’t think I’ve seen any research that compared competitive students with non-competitive students. That would make for an interesting research question.

Travis also taught some sections of his pre-service teacher course without using clickers, and he surveyed students in these sections about the potential advantages and drawbacks of using clickers. What concerns did they have about using clickers? They worried about the cost of the devices, that clickers weren’t necessary in small classes, that clicker activities take up too much class time, and that the technology might not be reliable. I found it interesting that these are among the common concerns of faculty members not already using clickers, too!

Image: “Happy Pi Day!” by Flickr user Mykl Roventine / Creative Commons licensed

Continuing my reports from the contributed paper session on teaching with clickers I helped coordinate at the Joint Mathematics Meetings back in January…

“Using Personal Response Systems (Clickers) in Liberal Arts Mathematics Courses to Support a Lecture Format,” Janet A. White, Millersville University of Pennsylvania [Slides]

Just like Jean McGivney-Burelle and Kimberly Burch, Janet White shared her experiences teaching with clickers in a “liberal arts” mathematics course taken by non-majors. Unlike Jean and Kimberly, who teach relatively small sections of this kind of course, Janet teaches in a large lecture hall with 75 students per section. Janet had used clickers in courses for pre-service math teachers in the past and found them useful, so when it was her turn to teach this larger course, she decided to use them again. A classroom response system was hardly the only technology Janet used in this course: She also had students complete online homework and quizzes and she annotated her PowerPoint lecture slides using an Interwrite Mobi.

Janet used clickers on a daily basis in her course, usually either to assess students’ prior knowledge or to assess their understanding of a topic taught during lecture.  Her questions came from a bank of multiple-choice questions provided by her textbook publisher.  She counted the clicker questions as part of her students’ participation grades, but in a low-stakes manner.  Given her use of the questions as well as the source of the questions, many were on the lower levels of Bloom’s taxonomy, aimed at recall and application of procedural knowledge.  She shared an example of a prior knowledge question that asked students to find the measure of an angle that complements a 36 degree angle.  A slightly harder question aimed at assessment of something taught during the course asked students to identify the cut edge in a given graph (or to assert that the graph had no cut edge).

Student survey results indicated that 85% of Janet’s students who used clickers regularly liked using them, and 71% said that using clickers helped them learn the material.  Students who used clickers regularly during the course ended up with higher grades in the course than students who didn’t, but, of course, that can’t necessarily be attributed to the use of the clickers.  (And since clicker questions were factored in the course grade, students who participated more frequently in clicker questions would almost certainly have higher grades in the course anyway.)

Student comments about the clickers were generally positive.  My favorite one was, “I liked getting the wrong answer anonymously.”  Other comments addressed the usual points that students like about clickers: They liked the interactivity, they liked discussing questions with classmates, they liked seeing where they stood relative to their peers, and they liked the feedback on their own learning the clicker questions provided.  The only significant negative aspect for the students was the cost, about $50 in Janet’s case.

Janet found that having students discuss clicker questions in small groups led to very engaged students, even in the large auditorium environment.  In the future, she plans to write more of her own questions, instead of relying on ones from the textbook’s question bank.  She hopes to write more difficult questions that will generate even more engaged discussion during class.  She’s also hoping to find ways to reduce the technology cost to the students, either by selecting a different vendor or facilitating the resale of clickers after each semester to students taking the course the next semester.

Also, Janet mentioned that the earth science faculty at Millersville are big users of clickers.  Earth science instructors looking for advice on using clickers might want to investigate!

Image: “Recursive Daisy” by Flickr user gadl / Creative Commons licensed

Continuing my reports from the contributed paper session on teaching with clickers I helped coordinate at the Joint Mathematics Meetings back in January…

“Clickers in the Classroom,” Kimberly J. Burch, Indiana University of Pennsylvania [Slides]

Kimberly teaches a “Math 101″ survey course called “Foundations of Mathematics.”  Topics covered include set theory, graph theory, and counting methods (among others), and Kimberly shared several interesting clicker questions on each of these topics.  For example, here’s one of her questions from the unit on graph theory:

How many vertices are there in a tree with 19 edges?

1. 19
2. 18
3. 20
4. Not enough information given

Kimberly practices the “classic” peer instruction technique of having students vote individually first, then discuss the questions in small groups, then vote again.  She finds that students often converge to the correct answer on the second vote.

In the example above, her students were split between 18 and 20 on the first vote, but after the peer discussion time, most students went with the correct answer, 20.  I found this interesting because the “Not enough information given” seemed to be the obvious wrong answer to this question.  A graph with 19 edges might have any number of vertices, but a tree with 19 edges can only have 20 vertices.  Students who don’t realize that trees are graphs with very specific properties might be tempted to go for the “Not enough information given” option.

I suspect that Kimberly used this question after the students learned the relationship between the number of edges and number of vertices in a tree and that this question was meant to assess whether students remembered that relationship.  Some students likely remembered that one of these numbers was one more than the other but weren’t sure which one was higher.  That would account for the split vote between 18 and 20.  Had this question been asked as an exploratory question and not a review question, I’m betting the split would have been between 20 and “Not enough information given.”

Kimberly also mentioned that she uses her clicker system’s priority ranking questions to have her students decide what topics should be emphasized during exam review sessions.  Kimberly gives her students a list of 8-10 exam topics, and the students indicate the top three or four toughest topics in order.  Kimberly said that this helps her make good use of limited exam review time by focusing on the topics the students find the most difficult.

Kimberly also shared some data from a quasi-control group experiment she conducted.  She taught two sections of this survey course and alternated which topics she covered with clickers in the two sections.  For example, section A might cover topic 1 with clickers while section B covered topic 1 without.  Then for topic 2, section B used clickers and section A didn’t.  She then compared test scores for the two sections by topic.  For some topics, students using clickers performed better on exams but for other topics, the students not using clickers performed better.  And for other topics, there was no difference.  The data was generally favorable to using clickers, but the “quasi” part of this quasi-control group experiment made it difficult to draw firm conclusions.

Image: “Point Marian Bridge” by Flickr user timmenzies / Creative Commons licensed

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…

“The Evolution of Classroom Voting in Contemporary Mathematics,” Jean McGivney-Burelle, University of Hartford [PowerPoint Slides]

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