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Socratic Seminars for Mathematics
Stephanie. I think it is graph abecause a Ferris
wheel goes round.
Tara. Graph agoes backward in time.
Laura. Time goes forward this way, along the
x-axis, so it cannot be a.
Quavis. It goes down like this. That would
make it go back in time.
Stephanie. Yes, but that is how a Ferris wheel
goes.
tephanie and Quavis gestured animatedly as they
stood in front of the graphs, shown in figure 1, that
had been drawn on the board. Tara and Laura were
speaking from their desks, and the rest of their
second-year algebra class watched intently. We will
subsequently share more of their discussion as we
discuss an example of a Socratic seminar, but we
first notice how these students struggled to “orga-
nize and consolidate their mathematical thinking
through communication” and to “communicate their
mathematical thinking coherently and clearly to
peers, teachers, and others” (NCTM 2000, p. 348).
The students were arguing vigorously and seemed
disappointed when the bell rang to end the seminar.
Mathematical discussions with this high level of
interest and involvement are a goal of the Standards
and are stimulating for both students and teachers.
At Forest Park High School in Forest Park, Georgia,
the entire mathematics department uses Socratic
seminars to create classroom settings that are con-
ducive to this type of discussion. Each mathematics
teacher conducts several Socratic seminars a year
in each class—with the whole class. For the sake of
this action research, each teacher who taught second-
year algebra did the seminar described in this arti-
cle with at least one class, but each teacher had a
control class, as well. As teachers compared stu-
dents’ achievement on tests that focused on the con-
cept of function, they found that students who had
participated in a seminar did better on the chapter
test and on a posttest instrument used to measure
students’ understanding of the concept of function.
In this article, we share the basics of using
Socratic seminars in a mathematics classroom.
BACKGROUND
For an example that indicates how Socrates taught
mathematics, we turn to the dialogue between
Socrates and Meno (Rouse 1956). Socrates used
what we have come to call Socratic questioning to
Karen Koellner-Clark, kkoellner@gsu.edu, teaches mathe-
matics education at Georgia State University in Atlanta,
GA 30338. She is interested in teacher development and
students’ mathematical thinking. L. Lynn Stallings,
lstalling@kennesaw.edu, teaches mathematics education
at Kennesaw State University, Kennesaw, GA 30144. Her
interests include teacher education and the use of comput-
ing technologies to teach mathematics. Sue Hoover,
shoover@fphs.ccps.ga.net, teaches algebra and trigonome-
try at Forest Park High School in Forest Park, GA 30297.
She is interested in using calculator-based labs and other
lab activities that allow students to be active learners.
S
MATHEMATICS TEACHER
Mathematics
discussions
with this
high level of
interest and
involvement
are a
goal of the
Standards
Karen Koellner-Clark, L. Lynn Stallings, Sue A. Hoover
Time elapsed
Distance from ground
a)
Time elapsed
Distance from ground
b)
Time elapsed
Distance from ground
c)
Time elapsed
Distance from ground
d)
A man takes a ride on a Ferris wheel.
Indicate which graph matches the statement.
Fig. 1
Ferris wheel problem for Socratic seminar discussion
Source: Van Dyke (1994)
This material may not be copied or distributed electronically or in any other format without written permission from NCTM.
Copyright © 2002 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.
Vol. 95, No. 9
•
December 2002 683
teach the Pythagorean theorem to Meno’s Greek
servant boy. Socrates later said that “no one taught
[the boy], only asked questions, yet he will know,
having got the knowledge out of himself ” (Rouse
1956, p. 50). Plato, a student of Socrates and
recorder of his dialogues, called this one-on-one
questioning the “art of midwifery—the art of assist-
ing at the birth of thoughts.”
John Dewey, Leonard Nelson, and Mortimer
Adler have elaborated on using Socratic methods
with more than one student. Dewey believed that
students who were actively involved in their educa-
tion learned more than those who were passive. He
called his version of the seminar the recitation.
According to Dewey (1933, p. 262), the goals of the
discussion should be to “stimulate intellectual
eagerness, awaken an intensified desire for intellec-
tual activity and knowledge and love of study.”
He gave several guidelines for conducting a
recitation. The first was that students should be
given a new problem on which to apply previously
learned concepts. Then the teacher should use
questioning to guide the discussion to the subject
matter, with the process as important as getting
the correct answer. Throughout the seminar, the
line of questioning should be used to clarify and
extend students’ thinking. Finally, the discussion
should end with a sense of closure and anticipation
of what is to come next, in the form of a new prob-
lem or topic to tackle at the next seminar.
In the twentieth century, a German professor of
philosophy, Leonard Nelson, examined the Socratic
method and elaborated on the role of the teacher.
Nelson, a contemporary of the mathematician David
Hilbert, believed that the “teacher was forbidden to
utter judgment in the subject matter, including the
right-wrong evaluation of the students’ statements”
(Loska 1998, p. 238). The main role of the teacher
was to ensure “a genuine mutual understanding
among the students, the concentration on the
respective question to prevent digression, and the
preservation of the good ideas that had come up in
the course of the discussion” (p. 238). Nelson
believed that such a discussion, gently guided by
teacher questioning, would converge on the main
ideas and that any individual’s errors would be
challenged by other discussion participants.
Mortimer Adler’s notion of seminars was firmly
rooted in the progressivist tenets of Dewey. Adler
(1982, p. 22) believed that the three goals in the
teaching and learning process were as follows: the
“acquisition of knowledge,” the “development of
intellectual skills” (the skills of learning), and the
“enlarged understanding of ideas and knowledge.”
In Adler’s view, the first goal was delivered through
didactic teaching; the second through “coaching,
exercises, and supervised practice;” and the third
through the Socratic seminar.
For teaching older students by Socratic seminar,
Adler proposed that classes be longer than fifty
minutes. Ideally, participants in the discussion sit
in a circle instead of in rows. In The Paidea Propos-
al, he explains further:
The teacher’s role in discussion is to keep it going
along fruitful lines—by moderating, guiding, correct-
ing, leading, and arguing like one more student! The
teacher is first among equals. All must have the sense
that they are participating as equals, as is the case in
a genuine conversation. (Adler 1982, p. 54)
GUIDELINES FOR USING
A SOCRATIC SEMINAR
A few guidelines can help establish a respectful
atmosphere for the Socratic seminar and should be
discussed before the seminar begins. The guidelines
that mathematics teachers at Forest Park High
School found effective are as follows:
∑
Participants must respect one another’s opinions.
∑
Participants do not have to raise their hands to
speak, but they must not interrupt.
∑
Participants address their fellow classmates by
name (name cards can be placed on the desks, if
necessary) and should take notes.
∑
Participants’ comments address the topic and do
not digress.
∑
Participants settle points of disagreement among
themselves. The teacher is not used as a
resource.
The desks are rearranged so that students and the
teacher are all seated in a circle and they can easily
see one another.
The role of the teacher is to choose the topic and
guide the discussion by using skillful questioning,
including questions that focus the discussion on the
topic, prevent any improper comments or behavior,
and clarify important concepts brought out in the
discussion. The teacher does not state his or her
position on the correctness of the viewpoint of the
students during the seminar.
QUESTIONS AND PROMPTS
Thought-provoking mathematical problems or
questions are critical to the success of the seminar.
We found some problems from a Mathematics
Teacher article (Van Dyke 1994) to be very effective
in provoking a rich and lively discussion about
important mathematics. NCTM Addenda documents
and the Illuminations Web site are also useful
resources for good problems. The problems cannot
be too easy; they need to involve interesting mathe-
matics ideas that students struggle to understand.
Each seminar used three of Van Dyke’s problems:
the swing problem first, next the train problem,
and then the Ferris wheel problem.
Thought-
provoking
mathematical
problems are
critical to the
success of
the seminar
684 MATHEMATICS TEACHER
AN EXAMPLE SEMINAR
The goal of this seminar was to clarify students’
understanding of the definition of function. Eight
second-year algebra classes at Forest Park High
School used a Socratic seminar to address this
objective. The teacher initially asked students to
define function on notebook paper, and they then
discussed their definitions as a group. The teacher
asked students to share their definitions, when
needed, during the seminar so that any misconcep-
tions that may have been discovered through the
discussions could be clarified. Then the teacher dis-
tributed several choices of graphs to represent each
scenario. Each student decided on his or her own
answer before the discussion began. For example,
the first question asked the students to decide
which graph in the group shown in figure 2 best
illustrated the scenario of a child swinging on a
swing.
Stephanie. You don’t always start with a push.
Tara. Graph bcould represent one push starting
lower from the ground.
Deidre. Yeah, but it says swings, meaning you go
back and forth.
Sam. Yes, I agree.
This type of conversation was typical of the ones
in each of the second-year algebra classes. Deidre
confidently explained her position that graph a,
which was the correct answer, fit the given context.
The discussion that followed focused on the starting
time and the distance from the ground. Some stu-
dents initially seemed to attend only to particular
points on the graph, such as the starting time x = 0
and the initial distance from the ground. However,
Deidre continued to justify her thinking by using
the whole graph and the functional relationship
f(time) = distance.
Quavis. The arrow represents that it goes on and
on, that the graph continues.
Deidre. Graph ashows that it goes up and down
like the back and forth.
Tara. Look at b. Doesn’t it look like the first
hump in ais a ray?
Quavis. We need to look at elapsed time. It looks
like at that time, it stops at the humps so that can-
not be right. I think graph c shows it best because
someone pulls the kid back and lets him go, and the
arrow says that “it keeps going.”
Laura. But the swing always starts straight
down; how does the swing get up like in c?
Quavis. Somebody had to pull you back, and that
is where c starts.
This excerpt was typical of the seminars in that
some students, for example, Quavis, interpreted the
graph literally, as a visual representation of the
motion of the swing. When he explained that the
arrow indicated that it went on and on, he illustrat-
ed the movement with his hand, repeating the line
of the graph in c. Deidre continued to argue for her
initial interpretation, but she was not able to con-
vince the other students. Tara suggested another
approach but was ignored in the passion of the dis-
cussion, and the discussion returned to graphs a
and c.
Jarell. I don’t like b and c. I like either a or d
because b and c are rays and a and d are functions.
Quavis. You are missing my point. I said if you
think of it as a ray, then the arrow signifies that it
doesn’t stop—it keeps going.
Tony. A ray goes indefinitely in one direction. It
doesn’t mean that it comes back down.
Jarell. Yeah, so it means that it can keep on
swinging.
Time elapsed
Distance from ground
a)
Time elapsed
Distance from ground
b)
Time elapsed
Distance from ground
c)
Time elapsed
Distance from ground
d)
A child swings on a swing.
Indicate which graph matches the statement.
Fig. 2
Swing problem for Socratic seminar discussion
Source: Van Dyke (1994)
Deidre. I chose a because when you swing back
and forth, the longer you swing, the higher you go.
John. You do not start in the air though. You
have to get on the swing.
Quavis. It says the child swings—the kid is
already on the swing. Anyhow, you know that you
start from a higher point because someone holds
you up. That’s where you start. Graph a doesn’t
show you starting high enough above the ground.
Some
students
literally
interpreted
the graph as
a visual rep-
resentation
of the motion
of the swing
Vol. 95, No. 9
•
December 2002 685
This dialogue illustrates another typical scenario,
in that several people argued for their own mis-
conceptions in their mathematical understanding.
In retrospect, it might have been an opportunity for
someone to ask Jarrell to clarify what he meant by
ray and why it mattered whether or not the graph
was a function. In this excerpt, Quavis continued to
try to persuade others that graph c represents a
child swinging on a swing and that the arrow on the
graph indicates that the child will keep swinging.
Students who understood which graph represented
the situation were sometimes frustrated with their
peers who did not understand. These students often
glanced to the teacher for help but then realized
that convincing their peers was their own responsi-
bility. They reentered the discussion with renewed
determination to explain their correct mathematical
reasoning more precisely and persuasively. Their
explanations helped other students revise and
refine their mathematical thinking.
In this seminar, the teacher redirected the dis-
cussion on two occasions. In her first redirection, she
encouraged the students to discuss the quantities
associated with the x- and y- axes. Student discus-
sion then brought out that the distance was a func-
tion of time and suggested that function notation be
used. In her second redirection, the teacher asked
which graphs represented functions. These probes
aided in the teacher’s assessments of the students’
functional representations and extended their dis-
cussion to a higher level of thinking. On reflection,
the teacher wondered whether students truly under-
stood the connections between their mathematical
knowledge of functions and their application to
phenomena in the physical world.
After a consensus was reached, the teacher posed
the second problem, which asks students to deter-
mine the graph that represents a train pulling into
a station and letting off its passengers. Students
were to choose the graphical representation in fig-
ure 3 that best matched the scenario. (The correct
answer is b.) In the eight seminars that used this
train problem, students focused on the end behav-
ior of the graph and laughingly reenacted how it
would feel to be on train d, which stopped suddenly;
on trains a or c, which did not stop for passengers
to disembark; or on train c, which alternately accel-
erated and decelerated. This problem and students’
reenactments helped them grapple with the depen-
dence of speed on time. Typically, the discussion of
the train graph did not take as long as either of the
other two discussed here.
The last problem, shown in figure 1, represented
a man taking a ride on a Ferris wheel. (The correct
answer is b.) This problem is designed to help stu-
dents deal with a common conceptual error, that is,
mistaking the motion of the event for a graph of its
distance and time (Dugdale 1993). When the stu-
dents initially shared their ideas regarding the
graph that matched the Ferris-wheel problem, their
answers were split evenly between graph a and
graph d, with a very few bor canswers.
Later in the discussion, the students tried to con-
vince Stephanie and eight or nine other students
that graph a is not a function. They tried to explain
their reasoning using the line of the graph and
what it represents. However, Stephanie and other
students argued adamantly that graph a represent-
ed a ride on a Ferris wheel for that very reason,
that the graph represents how a Ferris wheel goes
around. They did not seem aware of the functional
relationship between distance and time.
Tony. I think it’s d.
Donald. I am with Tony. A Ferris wheel doesn’t
go up and down. It goes around.
Laura. It’s a circle. . . . The height of the Ferris
wheel stays the same. It never changes, and b
shows that it goes the same height.
Tara. That is not what the graph is saying.
Graph b is saying that you go up. Then it is going
forward. It has a definite shape—it peaks and rep-
resents the highest point. A Ferris wheel goes
around, and graph a shows the movement you feel.
Patrick. There is no answer to this one.
Tara. You start at ground level and go up. I’m
talking from firsthand experience. You first get up
on a pedestal—that’s ground level. See how that
happens in a?
Time elapsed
Speed
a)
Time elapsed
Speed
b)
Time elapsed
Speed
c)
Time elapsed
Speed
d)
A train pulls into a station and lets off its passengers.
Indicate which graph matches the statement.
Fig. 3
Train problem for Socratic seminar discussion
Source: Van Dyke (1994)
Students
focused on
the end
behavior of
the graph
686 MATHEMATICS TEACHER
Kia. I agree with Tara. You are not touching the
ground.
Quavis. No, I agree with how you start, but I look
at d and it shows the whole story. You walk up, get
on the pedestal, get in a seat and go up, and then it
goes around.
Stephanie. So it is d.I can see that. You still go
around, not up and down.
At this point (late in the seminar), a lot of the
students were still trying to make sense of the
graph and the Ferris-wheel context. They wanted
the graph to illustrate literally the movement
they felt on a Ferris wheel. Although some stu-
dents were confident and vocal in the beginning
about a and d not being functions because “they
are going back in time,” this discussion briefly fell
by the wayside while students struggled with
their misconceptions.
Quavis. No! That would be backward in time
again.
Stephanie. See; this is how Ferris wheels go.
[Traces the path of d in the air.]
Laura: You are trying to make this the track—it
is not the track. Okay, look at this. [She goes up to
the board and draws a copy of d.] Look at this point
where the lines cross. See, you cannot be in two
places at once! [She draws a straight line from the
cross to the loop.] So it cannot be dor a.
Quavis. Yeah, and d and a are not functions
because they do not pass the vertical-line test.
In this last excerpt, Laura uses the intersection
of two parts of the graph in d and the fact that “you
cannot be in two places at once” to show that the
line of the graph is not the track. In doing so, she
reminds Quavis and others of the vertical-line test
for showing whether a graph is a function. Most of
the class verbally agreed that graph b fit the Ferris
wheel context. Further, they seemed to be able to
make sense of the functional relationship that the
graph illustrates. However, Stephanie and one
other student still were not convinced and left that
day’s seminar without resolution. The teacher took
notes and planned to follow up with them. She also
decided that the next day’s lesson would include a
discussion of why it was important that these
graphic interpretations of the relationship between
time and distance involved in different events
(swing, train, and Ferris wheel) should be functions.
In the discussion at the beginning of this article,
Tara and Quavis both understood that graphs a
and bshow “going back in time,” and Laura sug-
gested using the vertical lines to test whether the
graphs of the Ferris wheel were functions, but nei-
ther the teacher nor the teacher-educator observers
could determine whether students understood that
graphs in which time is the independent variable
must be functions.
REFLECTION
We found that the method of Socratic seminars was
very effective in encouraging students to assume
responsibility for reasoning and communicating
convincingly about mathematics. Student and
teacher feedback confirmed that all participants,
especially those who appeared to be more verbal
learners, found that the seminars were fun. Those
students showed understanding that they had not
shown on pencil-and-paper assessments. The sem-
inar helped the teacher assess students’ conceptual
understanding of functions and their graphical rep-
resentations, in addition to providing a forum for
rich discussion of an important mathematical topic.
The teachers and two teacher educators who
observed the eight seminars were surprised at how
literally the tenth- and eleventh-grade students
tried to make sense of the graphical situations pre-
sented, because most of these students were able to
solve typical textbook problems relating to func-
tions with relative ease. The Socratic method pro-
vided rare opportunities for the participating stu-
dents and teachers. Students had a forum for
articulating and organizing their mathematical
understanding; meanwhile, their teachers could
focus on listening to and reflecting on students’
understanding.
The six Forest Park High School teachers and the
two teacher-educator observers were all impressed
with the results of the Socratic seminar in the eight
classes that tried them. The assessments used after
the seminars showed that students who participat-
ed understood the concept of function better than
students who had not participated in a seminar.
Our observations of the seminars indicated that the
students were actively involved in reasoning and
communicating about mathematics and explaining
functions to their classmates who had misconcep-
tions. We saw repeatedly that when students dis-
cussed their ideas with others, they continued to
revise, refine, and improve them (Borasi 1992;
Moschkovich 1998). Consider adding the Socratic
seminar to your teaching repertoire as a productive
and entertaining way to promote your students’
mathematical reasoning and communication skills.
REFERENCES
Adler, Mortimer J. The Paideia Proposal. New York:
Macmillan Publishing Co., 1982.
Borasi, Rafaella. Learning through Inquiry.
Portsmouth, N.H.: Heinemann, 1992.
Dewey, John. How We Think: A Restatement of the
Relation of Reflective Thinking to the Educative
Process. Boston: D. C. Heath, 1933.
Dugdale, Sharon. “Functions and Graphs—Perspec-
tives on Student Thinking.” In Integrating Research
The Socratic
method
provided rare
opportunities
for the
students and
teachers
Vol. 95, No. 9
•
December 2002 687
on the Graphical Representation of Functions, edited
by Thomas A. Romberg, Elizabeth Fennema, and
Thomas Carpenter, pp. 101–3. Hillsdale, N.J.:
Lawrence Erlbaum Associates, 1993.
Loska, Rainer. “Teaching without Instruction: The
Neo-Socratic Method.” In Language and Communi-
cation in the Mathematics Classroom, edited by
Heinz Steinbring, Maria G. Bartolini Bussi, and
Anna Sierpinska, pp. 235–46. Reston, Va.: National
Council of Teachers of Mathematics, 1998.
Moschkovich, Judit N. “Resources for Refining Mathe-
matical Conceptions: Case Studies in Learning
about Linear Functions.” Journal of the Learning
Sciences 7 (2) (1998): 209–37.
National Council of Teachers of Mathematics (NCTM).
Principles and Standards for School Mathematics.
Reston, Va.: NCTM, 2000.
Rouse, William H., trans. Great Dialogues of Plato.
New York: New American Library, 1956.
Van Dyke, Frances. “Relating to Graphs in Introducto-
ry Algebra.” Mathematics Teacher 87 (September
1994): 427–32, 438–39.
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