WHAT IF SOCRATES USED MATHLETS?
Perihan Sen, Ozlem Çezikturk
Private Oguzkaan Lycee, Bogazici University
Present study aimed to integrate the Socratic Dialogue and the online
interactive java applets (mathlets) for teaching derivatives. A group of 11 Lycee
3 students were given a little Socratic Dialogue enriched course on derivatives
after normal instruction. The design was a pretest/ posttest quasi-experimental
research. Due to the number of subjects, a qualitative analysis was carried out.
The results suggested en effective relationship between these two methods of
instruction. Students excelled in understanding of the derivative of a function,
they even demonstrated some understanding of integral of a function. In
addition, students were able to cross the boundaries of mathlets. The teacher
and the students have been motivated for not only technology integration but
also for the Socratic dialogue and derivatives and functions in general.
INTRODUCTION
Any educational design would be beneficial as long as it could associate the
technology and the teaching way of an old man who still guides us in many
respects. On the one hand, we have Socrates (469-388 B.C.) the head of the
critical thought, brainstorming, creativity, in other words dialectic. On the other
hand, we have java applets so called mathlets, which enable the unseen to be
seen. It may seem as a conflict to search for linkages between the teaching
method of an ancient philosopher and a new way of technology integration into
mathematics education. However, there are some points where technology
would feel insufficient and surely as a man beyond the vision of his time would
appreciate and make use of the applets in the meantime. This study aims to use
Socratic questioning or in other words, the power of recollection for the students
to analyze mathlets with in-depth mathematical reasoning.
Socrates believed that human beings had all the answers to all of the questions if
only they knew how to ask the right question. In the dialogue Meno, Plato left us
an example of a Socratic dialogue while teaching geometry to him. One of his
few quotes that we got from his student Plato is like this: “I shall only ask him,
and not teach him, and he shall share the enquiry with me: and do you watch and
see if you find me telling or explaining anything to him, instead of eliciting his
opinion.” within his conservations with slave. In other words, a teacher couldn’t
teach anything new to the students that they didn’t know already. Hence, the
questioning of the teacher should have employed the idea of recollection.
One could say that a lot of things have changed from the time of Socrates, hence
using his ideas inside a classroom in conjunction with java applets would be
nothing but a meaningless idea. We may want to remember though how he also
mentioned about the children of those times; as tyrants since they contradict
withy their parents, gobble their food, and tyrannise their teachers. Does it seem
familiar enough?
THEORETICAL FRAMEWORK
Even though the main corners of this literature review are Socrates, his thoughts
and Java Applets, we also tried to develop our instruction in favor of the
constructivist view of knowledge and knowing (Cobb and Yackel, Wood 1993;
Confrey , 1995; Greeno, 1988; Simon, 1995; von Glasersfeld, 1987) and from a
Vygotskian view of teaching as creating successive zones of proximal
development (Vygotsky, 1978). (cited in Fraivillig, Murphy, & Fuson, 1999).
We thought this would be a necessary attempt regarding the interactions taking
place in the classroom.
Socrates brought into attention the term “dialectic”, or “question-answer”. He
referred to this method as the only admissible method of education, which is no
matter of mere conjecture. According to Socrates, while all opinions are equally
true, one opinion is better than the other and wise man is the one who by
arguments causes good opinions to take place of the bad ones, thus reforming
the soul of the individual or the laws of a state by a process similar to that of a
physician or the farmer.
In education, we have been using his ideas in so called Socratic method.
Teachers were guiding students to find the answers within. Socratic Method has
been used two folded both as a seminar to reach many students at once and as a
one to one mathematical discourse between teacher and the student.
Recently, mathlets gained importance due to their potential in their capability of
giving students a sense of first hand experience of mathematical inquiry. A
mathlet is a small scale interactive learning environment which is designed to
address key ideas in science and mathematics (Confrey et al, 1998; Castro-Filho
et al, 1999). They are compact, free to use, easy to find, and easy to use since
they do not require any programming language knowledge.
Mathlets have the capability of introducing multiple, mostly dynamic
representations of the same topic in mathematics. This ensures students to be
able to see and observe the effect of different parameter changes and dynamic
visualization. Socrates was a man who was surely ahead of his time and if he
lived we believe that he would conjecture the possibility of usage of his ideas
reflected on the image of the mathlets. The present study aims to use the socratic
questioning for the students to analyze mathlets with indepth mathematical
reasoning.
RESEARCH DESIGN AND PROCEDURE
Subjects and realities of Turkey
The first author of this paper is a teacher in a private school for 10 years. The
subjects of the study were drawn out of the original set of students of hers from
her school. The third year high school students in Turkey go through a very
anxiety-provoking exam called “University Entrance Examination”.
Due to the pressure this exam provides, students may be allowed to take the last
semester off in some schools. As a matter of fact, parents support this kind of
decision from the schools and even they lead the school into this. Some parents
also take the main aim of this exam as being a life-threatening goal for students
to achieve. There may be some realities behind this belief. As a matter of fact, to
be able to be a good citizen one day, and to have a descent job, you need to enter
to a university department unless you have a very rich parent. However, being a
university graduate still doe not guarantee your status and having a good job due
to many circumstances which are beyond the scope of this paper. This
explanation above is given to give a description of the student body in the third
year high school especially in private schools. Yet these students reached to a
motivation level even we, as researchers did not expect from the beginning.
Although the class size is much larger, at that day the intervention was carried
with almost 11 students. This was also due to the computer lab constraints.
Procedure
The teacher tought the derivative of a given function and tangent line to the third
year high school students.
Pretest which measured the students understanding about the identification of
the derivative of a function and the function whose derivative was given. This
pretest was developed by the help of the web site that the java applet was
positioned (see Fig. 1). This web site included a set of graphics, which was
given in a 3X3 grid and each below function is the derivative of the above
function. Students were required to identify with each column of the grid either
the first and second derivatives of the function, the preceding function whose
derivative is the below or the integral function of the preceding function. This
kind of activity enabled students to see even the relationship between integral of
a function and the derivative of a function that they have not been taught yet.
The graphics from this web page with the same idea of grids established the
pretest questions.
Figure 1. Pretest graphics
http://www.univie.ac.at/future.media/moe/galerie/diff1/diff1.html
Derivative puzzle 1
During the intervention , students were taken to the computer lab, the teacher
used two techniques; socratic dialogue and the “Definition of a Derivative” (see
Fig. 2) mathlet at the same time. The teacher prepared a lesson plan to
accompany this applet. She used questioning in terms of Socratic dialogue and
in the spirit of the Socratic seminars. The class hour was tape recorded. Socrates
believed that a teacher could only teach to the students what they already know
but what they need to remember through the idea of recollection. As Richard
(1993) mentioned once, the role of the skilled teacher/facilitator is to keep the
“inquiry train on track," but, also, to allow the students to "travel to a viable
destination" of their own design. This way of thinking clearly could be seen via
Socrates himself “I shall only ask him, and not teach him, and he shall share the
enquiry with me: and do you watch and see if you find me telling or explaining
anything to him, instead of eliciting his opinion.” within his conservations with
Meno. Meno was a slave to whom Socrates used the first examples of a Socratic
dialogue while teaching geometry to him. Our set of questions was prepared
regarding these example questions. This way, it was hoped that the students
would manipulate java applets (mathlets) in a much enriching manner and they
could get used to these mathlets in a much productive way possible.
The mathlet of the study has been selected from the Internet regarding the topic
at hand of this time of the year in lycee three mathematics curriculum. On the
top left of this mathlet, student can see the value of the function and its
derivative at each point in the x-axis. Also this value is given in reference to the
zero so that the student can see if both of these values are greater than or smaller
than zero. When the student manipulate the arrow underneath of the graph, the
tangent line moves accordingly and the x, f(x), and f ’(x) are all meaningfully
situated in conjunction with the graph in the applet. Hence, the applet gives an
in-depth graphical representation where as it gives only a surface level
understanding about the symbolic representation of the derivative of a function.
The mathlet is not composed of two pieces but also there are some buttons on
the top so that the students can see some related exercises, some solutions and
some explanation. For the purposes of the study, we did not use these buttons
but we used the teachers’ didactical guidance throughout the class hour. And
this was named as Socratic Dialogue in this study.
Figure 2: On the definition of the derivative
http://www.univie.ac.at/future.media/moe/galerie/diff1/diff1.html
Finally, as a posttest, students filled out the pretest again but this time they
recorded their results with a colored pen on the same answer sheet. They were
able to see their pretest results since they have used the same sheet.
FINDINGS AND ANALYSIS
Students’ answers to the pre and posttests were analyzed for recurring themes.
These themes were searched from the individual pre and posttest results. Then
the data was triangulated from the transcription of the tape of the class hour,
from pre and posttests and from the observations of the teacher herself.
Posttest results were almost same for all of the students. Actually, students were
pretty successful for the posttest and this was thought to be a result of the
specific intervention and the specific case that the class was in, in spite of the
fact that due to the number of subject constraints, it would be very hard to
generalize. Pretest results indicated some recurrent themes as the following:
-Some students took derivative and the function becoming zero at the same point
(as Student 1)
-Some students chose the similar graphs for representing the derivative and
function relationship (as Student 3)
-Some students seemed to throw an answer arbitrarily (as Student 4)
-We have found an interesting group of students who directed their attention to
complex graphs and one line graph in between. This we named as “Sandwich
method” (as Student 7).
-Another group of students (as Student 8) were able to get the relationship of the
function and its derivative. But within the answers of this group, we have
realized of task interference since the order of this relationship was not correctly
established.
These groupings became very important regarding the transcription analysis, and
the triangulation of the data with respect to both. We wanted to see what
changed and how it changed.
We have divided the transcription of the discourse between the teacher and the
students into the themes that emerged as a part of the study. The first part (see
Appendix 1) is where the teacher starts with asking where the derivative
becomes zero. This theme goes around with three students (students 8, 4 and 3).
We see that students use their own wording to explain what they think they see
on the mathlet. Here “hills and depths” is very interesting since, they already
learn about maxima and minima of a function but they prefer to state it like that.
Teacher does not interfere with what the students bring to the situation, instead
builds upon where they are and directs their attention to the characteristics of the
tangent line. This attempt is a way to incorporate the concept of tangent line and
how it is related to the derivative of a function. Students catch the teacher’s
point and they identify the tangent line becoming parallel to the x-axis. They
once more use their own wording as “horizontal”. Teacher does not leave it here
unattached but she continues with asking the relation with a very open ended
“how” question. As a matter of fact, “how, why” questions are thought as a very
good reasoning promoting questions from the Socratic Dialogue. Last but not
least, Student 4 give s a clear example of deductive thinking in saying “there are
zeros as many as hills and depths”.
Appendix 1.
Teacher
: Yes we defined derivative.Now by moving arrows I want you to
investigate where derivative becomes zero?
8
: At hills and depths.
Teacher
: At hills and depths ,does that tangent line show a common
characteristic?
4
: Yes, it does get horizontal.
Teacher
: How can we relate the derivative being equal to zero and the
tangent line being
horizontal at the hills and depths ?
4 ( see pre posttest of student number 4): There are zeros as many as the hills and
depths.
Teacher
: what was zero?
3
: The derivative .
Appendix 2.
Teacher
7
2
Ogretmen
3
Ogretmen
2
Ogretmen
2
:Lets play with the arrow and see if there is a point where the
derivative and the function coincide to zero.
: I wondered about it , too. But it was not there.
: I looked for too, but could not find.
: So, it took your attention. You are rigth, it is impossible for this
given function. Or can it be?
: Of course, it can.
: Okey then, at that special point how does the graph of the function
look like?
: It looks like ( She draws a parabola with her hand) . But its
vertex is at the origin.
:Is it necessary to be at the origin?
: Hımmm, no. I got it this time, hill or depth will be tangent to the
x axis.
In the second part (see Appendix 2), the teacher researcher points to the question
of if there is any point where both function and its derivative becomes zero. We
understand from the transcription that, some students (students 7 and 2) already
thought about this possibility and checked for that. Yet some students still
believe that could be (2 and 3). Teacher guides them with new questions to see
how this could be possible. Students 2 and 3 give us a special case since, they
somehow does not restrict themselves with the boundaries of the function that is
given with this mathlet. They see beyond and they point to that. Teacher flows
with them through that imaginary layout and organizes their thinking to make
them realize what an important thing they are referring to. We close this theme
with identifying students’ catch and how they see a parabola coming from the
midsts of their imagination added to the boundaries of the mathlet. We do
believe that this is one of the most important points of this study since, we did
not expect going this far. We had an intuitive feeling that the mathlets with the
help of the Socratic Dialogue would help students to unsolved some of the
mysteries of derivative of a function etc but we did not see that they could go
beyond the boundaries of this mathlet. This prejudice of ours actually took its
roots from the previously done research with mathlets and how they were seen
as a very restrictive learning environment since their compactness could
somehow shape their utmost characteristics. This could be due to the great
example of the Socratic Dialogue that the teacher researcher was able to carry or
it could be due to the specific characteristics of this mathlet. We hope that this
research should be a light on these types of research on mathlets and on different
instructional strategies accompanied with mathlets so that we could answer these
questions in the future.
In the third part (Appendix 3), The teacher lets the students took the initiative.
With the intrinsic motivation they have developed, students do not stop there,
and they follow through the mathlet. This time, the students (7,3, and 9) direct
the class to where the derivative of the function changes sign. Teacher leads
with the question of “is it random?” to make them identify a pattern if there is
one. Then, one student (student 7) catches it and ties it to the tangent line
pointing to the left. The teacher takes from here and points to the angle, which
the tangent line makes with the x-axis. Here, it is very important to note that the
very distinct role of the teacher is to be active 100% and be only one step ahead
of the students i.e. in the zone of proximal development of the students. It is
also important to note that students go with the flow and do not feel threatened
but motivated and challenged instead. Students make the deductive realization
that the sign of derivative of a function changes in different ways while between
maxima /minima and minima / maxima. Yet, they are not eager to generalize
that easily.
Does this application make students skeptical? Yes. Does this application let
students to deduce some of the facts through a continuous exercise with the
restricted face of the mathlets. It seems so. Do they see the unseen? Amazingly,
yes. Can we make generalizations out of this? We should be very careful not to
fall into a dead trap but there are some points this study proves important.
Mathlets invites us to the very secret world of mathematical reasoning of
students. They let the students to be researchers, detectives, players, and
anything that they want to be. But they also let them and us to realize that we
can promote at least higher cognitive level of functioning with students if we
make effective use of Socratic Dialogue. Socratic Dialogue gives the power of
finding right questions to the students to understand how these mathlets work?
And how the mathematical topic at hand is interrelated and inter connected to
the other concepts of mathematics.
It is conjectured not only the students will take the responsibility for their own
understanding but also they will master how to ask the right questions to get to
the thoughtful answers. Their metacognitive abilities could be heightened and
they could use this ability while learning other topics in mathematics.
Appendix 3.
Teacher
: Now ,I want you to play with that aplet whether or not you may
discover anything else?
7
: Yes , sometimes +, sometimes -.
Teacher
: Is this random?
3
: Is it - when x is a -?
7
: No, Look at ! it becomes minus when the tangent line is
inclined to the left.
Teacher
: While it is inclined to the left what can I say about the angle that it
makes with the x axis? Can it be an obtuse angle?
3
: Yes, the angle that the tangent line makes with the x axis is an
obtuse angle. When it is obtuse , derivative is negative.
Teacher
: Okey, is this always obtuse?
9: Between a hill and depth it is always minus then plus. But it is not like that on
the other side?
Teacher
: Can we generalize it?
7
: It can not. When it is inclined to the left it is minus , otherwise it is
plus.
References
Chang, K., Lin, M, & Chen, S. (1998). Application of the Socratic Dialogue on
corrective learning of subtraction., Computers and Education, 31, 1, 98, 55-68.
Fraivillig, J. L., Murphy, L., Fuson, K. C. (1999). Advancing children’s mathematical
thinking in everyday mathematics classrooms, Journal for research in mathematics
education, 30, 2, 148-171.
Koellner- Clark, K., Stallings, L.L. & Hoover, S. A. (2002). Socratic seminars for
mathematics. The Mathematics Teacher, 95, 9, 682.
Plato (380 B.C.E.). Meno. Available at [classics.mit.edu/Plato/meno.html]
Richard, P. (1993). Critical Thinking: How to Prepare Students for a Rapidly
Changing World, The Foundation For Critical Thinking, Santa Rosa, CATanner, M.
L. & Casados, L. (1998). Promoting and studying discussions in math classes,
Journal of Adolescent & Adult Literacy, 41, 5, 342-351.
Richard, P. (1990).Introduction to Socratic
[http://okra.deltastate.edu/~bhayes/socratic.html]
Questioning.
Available
at
[www.memphis-schools.k12.tn.us/schools/magnolia.es/socraticquest.htm
Tanner, M. L. & Casados, L. (1998). Promoting and studying discussions in math
classes, Journal of Adolescent & Adult Literacy, 41, 5, 342-351.