Classroom Voting in Calculus

Kelly S. Cline

 

 


How do you get your students to take an active role during a calculus class?  How do you get them to form opinions and to participate in discussions about the difficult concepts?  There is a large body of education research demonstrating how teaching methods that actively engage the students can be very effective at stimulating learning, especially in comparison to traditional lectures.  Even when supplemented with dramatic and flashy PowerPoint, lectures allow students to be passive observers, and passive students are rarely learning.  Classroom voting is a powerful technique, which can easily be incorporated into an otherwise traditional class, and requires every student to take an active role.  This technique breaks students out of their inert lecture-mode and requires them to participate, creating a dramatically more effective learning environment.

The basic idea is to break up the class period with a series of voting events, in which the teacher poses a multiple-choice or true/false question to the class, from PowerPoint or on an overhead projector.  After a brief period for consideration and informal discussion (usually about two minutes) the students then vote on the correct answer, either by holding up a colored index card (A = blue, B = green, etc.) or using an electronic "clicker" device much like a television remote control.  The results of card voting can be assessed with a quick glance around the room.  Electronic votes are received and tabulated on a computer at the front of the class, and after the vote a bar graph of the results is projected for the entire class to see.  The teacher can then guide the class through a discussion of the concepts involved.

Advantages of Classroom Voting

This technique is useful for a variety of reasons:

·        It requires every single student in the class to consider a question, to form an opinion, and to actively cement their opinion by voting.  This prevents students from just sitting through class and instead compels them to participate and discuss calculus with the people around them.

·        It provides immediate feedback to the teacher:  Is this a concept the students understand, or does it need to be covered further?  If a large majority gets the question right, then the teacher can move on to the next topic, but if there is controversy, then the teacher knows right then and there that something must be done.

·        It provides immediate feedback to the students as well:  In a lecture situation, if a student learns a technique incorrectly, they often don't realize this until they do a problem wrong on a homework assignment, and get their paper back several days later.  Classroom voting allows students to discover and to correct their mistake before they’ve even left class.

·        It provides an efficient springboard for creating classroom discussions:  Students are generally reluctant to voice opinions about topics, especially in a calculus class.  However because they've all had time to think and decide on a vote, and because they have seen that others voted the same way, students are much more willing to get involved.

·        Perhaps most importantly:  It's fun!  The students love classroom voting.  They really like the act of participating, and enjoy the “game show” atmosphere of the process, competing to register their votes first.  Several students have commented that with voting, the class “goes faster” than regular classes, and if we choose to skip the voting for a day, they complain endlessly.

Drill or Conceptual Questions?

What sorts of questions work best?  This technique can be used with any question that can be asked in a multiple-choice or true/false format.  One strategy is to begin with fundamental drill questions and then move to more advanced questions that probe the difficult conceptual issues. 

Some of the most effective questions are designed to elicit common errors and misconceptions.  Knowing where many students will go wrong, the teacher can use voting to confront and deal with their mistakes right from the beginning.  For example, knowing the sorts of problems that students often have taking the derivative of power functions with negative exponents, we ask:

What is the derivative of

a) 

b)

c)

d)

e)

            One of the most emotionally intimidating moments in a math class comes when students are asked to go off by themselves and do their homework in isolation.  Classroom voting can help to break through this fear by allowing them to first practice each new type of problem in a safe environment, where they can get help from their peers and from the teacher.

On the other hand, the goal of this technique is to get students to engage with the material, to provoke a discussion that forces them grapple with the complexities of calculus.  It can be challenging to create multiple-choice questions of this nature, but it is far from impossible.  For example:

Suppose that when we integrate some function  we get .  What does this tell us about ?

a) 

b) 

c) 

d)  Not enough information is given.

            A question like this can really challenge the students’ understanding, requiring them to discuss it carefully and hopefully to draw some diagrams, so that they can see that the integral of f(2x) will simply compress the function f, causing you to get half as much area as you did before.  Classroom voting allows the teacher to pose deeper questions like this in a non-threatening environment, so that the students aren’t worried about grades, and can focus on the deeper issues.

Pre-Vote Discussions?

            There are several ways you can conduct your class during the vote itself.  First, it’s good to put a specific amount of time on the clock (e.g. two minutes), so students know that they have time to work, but not to dawdle.  Some teachers prefer to have complete silence after the question is projected up for the class.  This requires the students to work individually, so that each person comes up with their own answer, and no one can simply copy their neighbor’s vote.  However I think it is much more powerful when I encourage students to informally discuss each question with their classmates:  I often explicitly tell them to please compare their answer with at least one other person and discuss any differences before voting.  The real goal is to get the students to engage, to think through things from different perspectives and to explore various viewpoints on each problem, and small group discussions can accomplish this.  The discussion is effectively motivated because each student must choose their own vote.  It gives them a specific and immediate reason to ask opinions of their peers, to listen carefully, and to consider contrasting views.  Because it’s such an effective way to get students to talk to each other about calculus and to explain their thinking, my perspective is that this more than makes up for the problem of occasional freeloading.

After the Vote

            If the vast majority of the class gets the question correct, then the teacher can briefly comment on why this was the right answer and confidently move on to the next topic.  However if there is a diversity of answers, there are more options.  Usually I will go around the class, Socratically asking different students to explain their votes, and I am often impressed with how clear it is when we get to the right answer.  I regularly hear students gasp “ah!” from around the room, as they recognize the logic of the solution.  Another option after a controversial vote is to simply ask the students to discuss the problem with their neighbors before holding a re-vote.  Most of the time, the numbers will move in the right direction, but be prepared to get burned once in a while too!

Let Go of the Classroom

            Effectively using classroom voting can be difficult, because it involves giving up a measure of control, and embracing a new and more boisterous classroom atmosphere.  All their lives our students have been taught that in school they need to be quiet and listen attentively, then all of a sudden you announce that from now on during a substantial portion of class time they will be required to discuss mathematical questions with the people around them, to vote their opinions, and often to describe their thinking to the rest of the class.  It’s important to explain to your class what you’re doing, and why.  You must tell them that this isn’t just some strange idea, but that they will learn more if they spend a good part of the time talking, discussing, voting, and debating, rather than just listening attentively for the entire period.  Some students may have trouble coping with the new atmosphere.  I had one very successful student tell me “I don’t know how you put up with it!”  However most students take to the new structure with relish, so much so that you may have to speak up to get control back when you want it.  You may find that once your students get used to the structure, they speak out in class more readily, and occasionally break up into small groups discussions even when you haven’t asked them to.  Never forget that this is a good thing.  Now that you’ve created a new environment, it may take a little more effort to herd the class along, to keep them together, and to regain their attention when you want it.  But that effort is a fair price to pay for students who are engaged, who are involved, and who are having fun learning calculus.

Although this technique has been used successfully in very large lecture classes with hundreds of students at large universities, we have found classroom voting to be even more useful in our small classrooms of twenty or thirty.  Smaller classes make it much easier to shift gears between lecture segments and voting questions, and they allow more people to participate in a Socratic post-vote discussion.

Electronic versus Non-Electronic Voting

            As described above, classroom voting can be done with no technology at all:  All you need are a few packages of colored index cards and a few transparencies and you are ready to go.  However we have found that using electronic clickers and a computer with a receiver can make this technique more effective.  The technology certainly makes it easy to know exactly which student voted for which answer.  Sometimes I get no volunteers when I ask, “Could someone who voted for B explain their choice?”  However with the electronic system, I can simply glance down at the screen and find someone.  The technology also makes it easier to be sure that everyone participates.  With a quick headcount I know how many votes I should receive, and I can keep the voting open until everyone has voted.  Another difference is that the act of voting is more private if clickers are used.  With colored card voting, it’s not too difficult for a student to quickly glance around and see what color is winning, but with the clickers, no one knows until the vote is closed and the bar graph appears.  Some of our teachers have even experimented with formally counting the votes towards student grades, either simply for participation, or grading them on right/wrong votes, which would be quite impossible using colored cards.

The main drawbacks of the technology are cost and ease of use.  In addition to a laptop and a projector, our department paid about $1,250.00 for each package containing a receiver, the software, and 30 clickers.  The electronic system means that I have to get to my classroom at a few minutes early to boot up the laptop, warm up the projector, and mount the receiver up on the wall.  (We’ve found that a Velcro patch is an effective way of securing a receiver up where it can see all the clickers.)  The effort isn’t huge, but time is precious, and the more involved classroom voting becomes, the more likely I am to just skip it when I’m running late.

On balance, I think the technology is a real plus, but there are definitely situations where simpler is better.

Resources

            Classroom voting, sometimes called “Peer Instruction” was pioneered in introductory physics classes by Eric Mazur at Harvard (Mazur 1997; Crouch and Mazur 2001; Fagen, Crouch, and Mazur 2002), and now has been applied in a wide variety of fields like astronomy (Green 2003) and chemistry (Landis et al 2000).  In the past few years the method has been applied to calculus at a variety of institutions with very positive results (Pilzer 2001; Schlatter 2002; Loman 2004).

There are currently several great sources for these multiple-choice and true/false questions that have been designed specifically for classroom voting in calculus, often called “ConcepTests.”  The Cornell GoodQuestions project (http://www.math.cornell.edu/~GoodQuestions/) has created an extensive library of questions and provides a great set of resources for anyone wanting to try out classroom voting for the first time.  Another set of questions comes with the instructor’s resource package for the Harvard Consortium Calculus Text (Hughes-Hallett, Gleason, McCallum, et al 2001).  The majority of their ConcepTests are multiple-choice questions ready for classroom voting, which they intersperse with other more free-response questions.  For multivariable calculus, Mark Schlatter of Centenary College has created his own set of voting questions (http://personal.centenary.edu/~mschlat/conceptests.pdf).

There are currently three major companies that are selling electronic classroom voting systems of this type that can be run from a computer with a projector.  The system we purchased is called the InterWrite PRS (Personal Response System) and is sold by GTO CalComp (formerly Educue; more information from http://www.gtcocalcomp.com/interwriteprs.htm).  This system has worked quite well for us, as long as we’re using a laptop with a parallel port.  The PRS receiver will not plug into a USB port without a special adapter, which posed a problem when we tried to attach it to a Compaq laptop.  Other electronic classroom voting systems are from Hyper-Interactive Teaching Technology (http://www.h-itt.com) and the CPS classroom performance system from eInstruction (http://www.einstruction.com).

Conclusions

            The math teachers at our college have been amazed at the power of classroom voting and have quickly incorporated it into our classrooms.  Our department has purchased three sets of clickers/laptop receivers that are currently being used by five teachers in all six sections of calculus, and both sections of multivariable calculus.  To build on this success, we’re currently working to create voting questions for our courses in linear algebra and differential equations.

            Give classroom voting a try.  It takes just a few minutes to hand out the packages of colored index cards, and to make up transparencies with a few interesting questions.  If our experience is any indication, you’ll be surprised at quickly your students become engaged and how much fun they have learning calculus together.


References

Crouch, Catherine H., and Eric Mazur.  “Peer Instruction:  Ten years of Experience and Results.”  Am. J. Phys. 69 (September 2001):  970-977.

 

Fagen, Adam P., Catherine H. Crouch, Eric Mazur.  “Peer Instruction:  Results from a Range of Classrooms.”  Phys. Teach. 40 (April 2002):  206-209.

 

Green, Paul J.   Peer Instruction for Astronomy.  Upper Saddle River, New Jersey:  Prentice Hall, 2003.

 

Hughes-Hallett, Deborah, Andrew M. Gleason, Daniel E. Flath, Sheldon P. Gordon, Patti F. Lock, David O. Lomen, David Lovelock, David Mumford, William G. McCallum, Brad G. Osgood, Andrew Pasquale, Douglas Quinney, Wayne Raskind, Karen Rhea, Jeff Tecosky-Feldman.  Calculus, Single and Multivariable 3rd Edition.  New York:  John Wiley and Sons Inc., 2002.

 

Landis, Clark R., Arthur B. Ellis, George C. Lisenky, Julie K. Lorenz, Kathleen Meeker, Carl C. Wamser.  Chemistry ConcepTests:  A Pathway to Interactive Classrooms.  Upper Saddle River, New Jersey:  Prentice Hall, 2000.

 

Loman, David O., Robinson, Maria K.  “Using ConcepTests in Single and Multivariable Calculus.”  In Proc. 16th Annual International Conference on Technology in Collegiate Mathematics.  New York:  Addison Wesley, 2004.

 

Mazur, Eric.  Peer Instruction:  A User’s Manual.  Upper Saddle River, New Jersey:  Prentice Hall, 2003. 

 

Pilzer, Scott.  “Peer Instruction in Physics and Mathematics.”  PRIMUS 11 (June 2001):  185-192.

 

Schlatter, Mark.  “Writing ConcepTests for a Multivariable Calculus Course.”  PRIMUS  12 (December 2002).