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 3 ^{rd}
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).