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682

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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