BRINGING THE VOICE OF PLATO IN THE CLASSROOM
TO DETECT AND OVERCOME CONCEPTUAL MISTAKES
Rossella Garuti
Paolo Boero
Giampaolo Chiappini
Scuola Media "Focherini", Carpi
I. M. A. - C. N. R., Genova
Dipartimento Matematica
Università di Genova
I. M. A. - C. N. R.
Genova
Abstract: The capacity of detecting conceptual mistakes and overcoming them by general
explanation is important in the approach to theoretical knowledge, and its development in
students calls for the teacher's intervention. Our working hypothesis is that the "voices and
echoes game" can function as an appropriate methodology to this end. In order to explore this
perspective in depth, a teaching experiment was performed in six classes (grades V and VII).
This report provides a partial account of this complex experiment, presents some results and
highlights some open research questions.
1. Introduction
Since 1995, the problem of approaching theoretical knowledge in compulsory
school has been of major concern for our research group. We have produced an
innovative methodology, the "voices and echoes game" (VEG), which is mainly
based on Vygotskian elaboration concerning common and scientific concepts and
Baktin's idea of "voice". This has been used as a guideline to plan and analyse
teaching experiments intended to mediate crucial aspects of theoretical knowledge
(see Boero et al., 1997, 1998; Garuti, 1997; for a brief account, see Subsection 2.1).
The research reported in this paper focuses on one aspect of the mastery of
theoretical knowledge not yet considered in preceding papers and that is especially
relevant to mathematics education: the capacity to detect conceptual mistakes and
overcome them by general explanation. The development of this capacity in students
calls for the teacher's intervention; our working hypothesis is that the VEG is an
appropriate methodology for achieving such development (see Subsection 2.2). In
order to explore this perspective in depth, a teaching experiment was performed in
six classes (grades V and VII). The object of the experiment was a well known piece
of the Plato's "Meno", that concerning the problem of doubling the area of a given
square by constructing a suitable square (this means overcoming the mistake which
consists of doubling the side length). This report provides a partial account of this
complex experiment (see Section 3), presents some results (see Section 4) and puts
some open research questions into evidence (see.Section 5).
2. Theoretical background
The purpose of this section is to provide essential background information, as
well as (in Subsection 2.2.) some development of the theoretical framework related to
the issue dealt with in this study.
2.1. About the VEG
What is the VEG? Some verbal and non-verbal expressions (especially those
produced by scientists of the past) represent in a rich and communicative way
important steps in the evolution of mathematics and science. Referring to Bachtin
(1968) and Wertsch (1991), we called these expressions 'voices'. Performing suitable
tasks proposed by the teacher, the student may try to make connections between the
voice and his/her own interpretations, conceptions, experiences and personal senses
(Leont'ev, 1978) and produce an 'echo', i.e. a link with the voice made explicit
through a discourse. The 'echo' was an original idea through which we intended to
develop our new educational methodology. What we have called the VEG is an
educational situation aimed at activating students to produce echoes through specific
tasks, for instance: "How might.... have interpreted the fact that...?"
Students' echoes: students may produce echoes of different types (depending
on the tasks and personal adaptation to them). In Boero et al. (1997), individual and
colletive echoes were classified. In this report we will focus particularly on individual
resonance echoes. In this case the student appropriates the voice as a way of
reconsidering and representing his/her experience; the distinctive sign is the ability to
change linguistic register or level by seeking to select and explore pertinent elements
('deepening'), and finding examples, situations, etc. which actualize and multiply the
voice appropriately ('multiplication').
What are the aims of the VEG? Our general, initial hypothesis on this issue
was that the VEG might broaden the students' cultural horizon, embracing some
elements of theoretical knowledge that are difficult to construct in a constructivist
approach and difficult to mediate through a traditional approach (see Boero & al,
1997). The need to exploit the potentialities that emerged in the first series of
teaching experiments led us to try to characterize better the elements of theoretical
knowledge to be mediated through the VEG (cognitive strategies, methodological
requirements, speech genre, etc.), in order to organize and analyse better their
interiorization by students (see Boero & al, 1998)
2.2. Conceptual mistakes and the VEG
The research reported in this paper concerns another important potential of the
VEG, namely the possibility of intervening in aspects of the student's mastery of
theoretical knowledge - those related to detecting conceptual mistakes and
overcoming them by general explanation.
The Vygotskian elaboration about consciousness as a condition for accessing
scientific concepts, clearly pointed out by Vygotskij in his seminal work about
"common concepts" and "scientific concepts", seems to be useful to frame this
complex operation. According to Vygotskij (1990, chap. VI), consciousness is related
to mastery of scientific concepts for different reasons: "scientific" concepts are not
isolated (and consciousness is needed to control connections and inner coherence of
the system); "scientific" concepts are explicit (and consciouness is needed to manage
explicitation and especially the relationship between mediating signs and meaning);
"scientific" concepts are in dialectic relationship with common ones (and
consciousness is needed to be aware of the borders between them). During an activity
in which students participate effectively in examining their conceptual mistakes, all
these aspects where consciousness intervenes can come into play: contradictions with
known properties are frequently a motive the teacher advocates for helping students
recognize a conceptual mistake; explicitation of some concepts is needed in order to
point out ambiguities that may be the root of mistakes; in many cases the teacher
must point out that common intuition is a possible source of mistakes.
But how can productive classroom activities concerning students' conceptual
mistakes be organised? According to Bachelard (1977) many conceptual mistakes
come from ancient knowledge that is appropriate in earlier situations but which is no
longer suitable. The teacher must take the responsibility for selecting and proposing
appropriate tasks (those which lead to crisis of the ancient knowledge) and for
helping the student to overcome his/her mistakes. The teacher's role is central for
other conceptual mistakes as well: for instance, those related to misunderstandings or
ambiguities. What's more, the student must be aware of the role played by the teacher
and his own role as a condition for being able to reproduce by himself, in the future,
the sequence of actions needed to detect and overcome conceptual mistakes (cf.
Brousseau, 1997).
Our working hypothesis was that the VEG could intervene as an appropriate
educational methodology for attaining both the aims pointed out in preceding
analyses: to develop students' consciousness about the functioning of theoretical
knowledge when conceptual mistakes come into play; and to promote awareness of
the teacher's and student's roles during classroom activities concening conceptual
mistakes. Indeed, detecting and overcoming conceptual mistakes plays a crucial role
in the evolution of mathematics and science. It is therefore natural that the history of
mathematics and, more generally, the history of culture should offer "dialogical
voices" that speak about this issue (exchange of letters, imaginery debates, etc). The
production of "echoes" of well chosen "dialogical voices" during suitable tasks could
lead students to participate consciously in the process of detecting and overcoming
conceptual mistakes as a preliminary step towards interiorization.
The teaching experiment reported in this paper was planned and performed in
order to test and develop our working hypothesis.
3. Method
3.1. The choice of voice
Plato's "Meno" presents some crucial aspects of Plato's theory about learning
(how the "learner" can reach truth) and teaching (how the "teacher" can help the
"learner" to reach truth). Plato's general, underlying hypothesis is that forgotten truth
can be restored ("recollection") through an effort by the "learner" led by the "teacher"
and motivated by the fact that "now not only is he ignorant[...] but he will be quite
glad to look for it". The crucial mediational tool is the "socratic dialogue", i. e. a
dialogue intended to provoke crisis and then allow it to be overcome. In this
framework, the excerpt concerning doubling the area of a given square is crucial as a
practical demonstration:
Phase A) Socrates asks Meno's slave to solve the problem of doubling the area
of the square by constructing a suitable square; the slave's answer (side of double
length) is opposed by Socrates through direct, visual evidence (based on the drawing
of the situation).
Phase B) Then the slave is encouraged to find a solution by himself - but he
only manages to understand that the correct side length must be smaller than three
halves of the original length. Socrates' comment is that from this moment on, the
slave can learn: "Nor indeed does he know it [the solution] now, but then he thought he knew it
[...] Now however he does feel perplexed. Not only he does not know the answer; he doesn't even
think he knows".
Phase C) Socrates interactively guides the slave towards the right solution
(achieved through a construction based on the diagonal of the original square).
We may remark that in Phase A) the slave's answer is similar to those usually
produced by young students when they tackle the same problem - this fact offers an
opportunity to involve students strongly in the VEG! And we may recognize in
Phases A), B), C) a sequence of activities not dissimilar from some present-day views
about how to guide students towards taking into charge and overcoming their
conceptual mistakes. This is true especially from a Vygotskian perspective, where the
teacher takes a strong mediating role in the evolution of students' culture.
We chose to ignore "recollection theory" in classroom activities. This entails a
violation of the authenticity of the historical source. But our aim was to lead students
to grasp that a general explanation must be reached in order to definitively overcome
a mistake. And, in general, compromises of this kind appear unavoidable if we want
to exploit historical sources in the classroom (cf. Fauvel, 1991).
3.2. The choice of classes
Six classes (114 students) took part in this teaching experiment: five fifthgrade classes and two seventh-grade classes.These classes belonged to different
school settings (four primary school classes and two junior high school classes) and
to different educational contexts (in particular, three fifth-grade classes were
following the Modena Group Project on "mathematical discussion", one was
following the Genoa Group Project). Their socio-cultural backgrounds were
extremely different. This set of classes was chosen in order to reveal "invariant"
elements and significant conditions for the productivity of the methodology.
As concerns the mathematical background, the students were able to measure
lengths and construct squares; they had met the concept of area of a plane surface in
the preceding months and knew how to calculate the area of a given square.
3.3. Teaching sequence planning and observations
The teaching sequence can briefly be described as follows:
i) students are briefly informed about the whole activity to be performed; then
they individually try to solve the same problem posed by Socrates to the slave. The
aim of this activity is to involve students in the problem dealt with in the voice;
ii) students approach the voice under the teacher's guidance: firstly, they read
and try to understand (with the help of the teacher) the three phases of the dialogue;
then they read the whole dialogue aloud (some students playing the different
characters); finally, they discuss (Disc. I) the content and the aim of the whole
dialogue, trying to understand (under the teacher's guidance) the function of the three
phases. After negotiation with students, a wall poster is put up summarising the three
phases in concise terms. This suggests the structure of the following echo;
iii) the teacher presents the students with some, possible mistakes that could
become the object of a dialogue similar to Plato's, and they are invited to propose
other mistakes. The aim is to negotiate and agree on a mistake that is appropriate for
the echo (i.e. a relatively frequent student mistake that is recognized as a mistake by
students and can be exhaustively explained through a discussion guided by the
teacher). Here is a sample of the 5 mistakes that were chosen in the 6 classes:
"By dividing an integer number by another number, one always gets a number smaller than the
dividend" (the two seventh-grade classes: see Annexe for two examples of echoes).
"By multiplying an integer number by another number, one always gets a number bigger than the
first number" (one V-grade class).
"By multiplying tenths by tenths, one gets tenths" (another V-grade class).
iv) students discuss (Disc. II) about the chosen mistake, trying to detect (under
the teacher's guidance) good reasons explaining why it is a mistake, then trying to
find partial solutions, and finally arriving to a general explanation. The aim of this
discussion is to create the common base of mathematical knowledge needed to
construct the echo, and prepare its three phases;
v) students individually try to produce an echo, i.e. a "socratic dialogue" about
the chosen mistake;
vi) students compare and discuss (under the teachers' guidance) some
individual productions.
All the individual productions and recordings of the Disc. I and II are
available; for the other discussions the teachers took notes.
3.4. Analysis of students' behaviours
In line with the educational aims of the teaching experiment we drew up the
following guidelines for the analysis of students' protocols:
I. How the student keeps to the roles of Socrates and the slave in each phase of
the dialogue. This point is related to the aim of developing awareness about the
teacher's and the student's roles in the activities concerning conceptual mistakes;
II. How the student appropriates the roles of the phases of the dialogue in
detecting and overcoming the mistake. This point is related to the aim of promoting
consciousness about the mechanisms of detecting and overcoming mistakes;
III. How the targeted mathematical content (the knowledge allowing the
students to overcome the mistake) is appropriated by the student: are the choice and
presentation of counter-examples appropriate? Is the guiding of the slave performed
through general, theoretical considerations about the knowledge in play?
4. Some Results
This section reports a selection of results we consider to be of interest. Here we
will consider only the 102 students from grade V to VII who took part in the whole
activity. Only 6 students completely fail their echo (do not produce a dialogue, or
mixed up Plato's original dialogue with the new situation).
Roles in the echo: Among the other 96, 10 show serious difficulties in keeping
to the roles of Socrates and the slave in Phase I. Appropriate "deepenings" (including
original expressions intended to highlight the mistake and provoke the slave's crisis)
and "multiplications" (including choice of appropriate counterexamples not
presented in Disc. II) are found in almost all the other students' texts.
Detailed comparison between fifth- and seventh-graders is inappropriate, as
different mistakes were tackled in different classes. However, the percentages related
to success in keeping to the roles in Phase I do not differ much.
It is not easy to detect Phase 2 in the students' protocols: students were not
asked to separate the three phases. And for some of the mistakes chosen it is
objectively difficult to create a specific Phase-II dialogue!
At least 67 (out of the 96 students ) have serious difficulties keeping to the
roles of Socrates and the slave in the last phase of the echo. In many cases the
quality of the interaction between Socrates and the slave suddenly changes from
Socrates' questioning in order to make the slave understand, to Socrates' presenting
some formulas or procedures in order to avoid the mistake, (see end of TEXT 1) with
the slave reduced to a passive role of listener. In the same cases the quality of
deepenings and multiplications falls: the expressions become "assertive", not
"explanatory", and the examples (if any) stick closely to those discussed in Disc. II.
Consciousness about how to detect and overcome mistakes: We obtained
good overall results about the consciousness of the fact that appropriate counterexamples can reveal the mistake. This kind of consciousness was attained by
practically all the students (86) who kept to the roles in the first phase of the
dialogue, as shown by the "multiplications" and "deepenings" in their echoes.
At least 50 students out of these try to give a general "explanation" of the
mistake or find a general "rule" in the third phase of the dialogue, showing to be
aware of the necessity of doing it.
Mathematical content: We must distinguish between: I) consciousness about
the fact that a statement is false; II) consciousness about the reasons that may
provoke the mistake; and III) consciousness about the theoretical reasons "why it is
false" and how to overcome the mistake in general (i.e. the connection with
systematic mathematical knowledge that can frame the mistake and the correct
solution). As remarked above, the first level of consciousness seems to be reached by
all the students who keep to the roles during the first phase of the dialogue. One half
of these students reach the second level of consciousness (see TEXT 1 and TEXT
2).It is interesting to note that there is also an almost complete coincidence, across
tasks and classes, between the students who are able to attain the third level of
consciousness and those who are able to keep to the roles throughout the third phase
of the dialogue (see TEXT 2). The breaking point in reaching the third level is well
exemplified in TEXT 1: the student tries to explain why the result of the division is
larger than the dividend if the divisor is smaller than one. An appropriate geometric
example (similar to those considered during Disc. II) is provided. The interactive
structure of the presentation is kept. Then the student tries to move to a general
explanation. Some lines are written and then crossed out. At the end, a rather
confused rule is provided and Socrates takes the role of he who "gives the rule".
Compare with TEXT 2: here a real interaction is maintained in the last phase as well,
and seems to be perfectly functional to the development of a complex inner discourse
concerning the mathematical knowledge in play. Remarkably, "Socrates" considers
both the operational side (how to divide an integer number by a fraction) and the
explanatory side (why it is necessary to behave in such a way). The dialogue allows
these two sides to be represented in a clear way.
5. Discussion
The above description of students' behaviours raises an interesting research
question about the reasons why there is an almost complete coincidence between
students who keep to their roles during the different phases of the discussion, and
students who reach the different levels of consciousness about the knowledge in play.
A possible interpretation refers to the dialogical nature of the acquisition of
theoretical knowledge (cf. Brown, 1997, Chapter 2, for some hints in this direction)
and could be summarised as follows: Plato's voice presents a model of dialogical
treatment of mistakes; at the beginning, by echoing this voice, students are forced
(during both the discussions and the individual production of the echo) to make
explicit the knowledge which originated the mistake. Indeed the need to keep to the
roles can entail a shift to an inner questioning: "What idea about the knowledge in
play should the slave bear". This interpretation is justified by the fact that (as
happens in TEXT 1 and in TEXT 2: see (°)) the "idea about the knowledge in play" is
expressed in most cases by the slave and not by Socrates. In this way the transition to
the second level of consciousness about the knowledge in play is realised. The
passage to the third level is performed in two steps: first, by considering examples
(with the dialogical function of bringing the slave to see how the appropriate
knowledge could work, and the inner function of better understanding how it does
work); and second, by shifting to theoretical framing of the examples. Here, the
breaking point described at the end of Section 4 could be interpreted as follows:
under the necessity of posing appropriate questions to the slave, some students are
able to answer the inner question: "How is the correct rule related to the meaning?"i. e. to the examples tackled by the slave. Indeed in these cases (see TEXT 2)
Socrates keeps to his questioning role: the inner question is transformed into
appropriate questions posed to the slave. The other students are not able to keep to
the role of Socrates: perhaps because the shift to the inner question "How is the good
rule related to the meaning?" is too difficult. Or, more probably, because it no longer
concerns the slave in an immediate way and it is difficult to find appropriate
questions for him, and so a traditional model of teaching prevails (it is the teacher
who provides the rule!).
Further experiments (possibly with on-the-spot interviews with students who
fail at the "breaking point") are needed to test the validity of this interpretation,
whose research and didactical implications might be significant as concerns the
potential of exploiting dialogical voices for the VEG.
References
Bachtin, M.: 1968,Dostoevskij, poetica e stilistica, Einaudi, Torino
Bachelard, G.: 1977, La formation de l'esprit scientifique, Vrin, Paris
Boero,P.; Pedemonte, B. & Robotti, E.: 1997, 'Approaching Theoretical Knowledge Through
Voices and Echoes: a Vygotskian Perspective', Proc. of PME-XXI, Lahti, vol. 2, pp. 81-88
Boero,P.; Chiappini, G.P.; Pedemonte, B. & Robotti, E.: 1998, 'The "Voices and Echoes Game"...',
Proc. of PME-XXII, Stellenbosch, vol. 2, pp. 120-127
Brousseau, G.: 1997, Théorie des situations didactiques, La pensée sauvage, Grenoble
Brown, T.: 1997, Mathematics Education and Language, Kluwer Acad. Pub., Dordrecht
Fauvel, J.: 1991, 'Using History in Mathematics Education', For the Learning of Mathematics,3-6
Garuti, R.: 1997, 'A Classroom Discussion and a Historical Dialogue: a Case Study', Proc. of PMEXXI, Lahti, vol. 2, pp. 297-304
Leont'ev, A. N.: 1978, Activity, Consciousness and Personality, Prentice-Hall, Englewood Cliffs
Vygotskij, L. S.: 1990, Pensiero e linguaggio, edizione critica a cura di L. Mecacci, Laterza, Bari
Wertsch, J. V.: 1991, Voices of the Mind, Wheatsheaf, Harvester
Annexe
Seventh graders. Mistake chosen: "Dividing an integer number by another number, one
always gets a number smaller than the dividend." Two texts from the same classroom.
SO=Socrates, SL=Slave
TEXT 1 (mean level production)
SO:Tell me, my boy, do you know the result of this division: 15:5? SL: It is simply 3, Socrates. SO:
And now try to perform the following division: 15:3 What is the result? SL: Clearly 5, Socrates.
SO: Look at the results and try to tell me how they compare to the dividend. SL: They are smaller,
Socrates. SO: In your opinion, does this happen for all divisions? SL: Yes, surely SO: Could you
explain why? SL:(°)Naturally: this happens because if I divide one thing I do not have it, but only
one piece of it, so the part is smaller in relationship to the whole. SO: Fine. Can you tell me the
result of the following division: 15:1 ? SL: Fifteen, Socrates. SO: Look at this number and think.
How does it compare with the dividend? SL: It is equal, Socrates. SO: So then a division does not
always generate a result smaller than the dividend! SL: My Zeus! It is true. SO: Now try to perform
the following division: 15:0.5. What is the result? SL: Thirty, Socrates. SO: But 30 is greater than
the dividend 1. SL: It is true!
II- SO: Tell me, slave: which divisions generate a result that is smaller, equal to or bigger than the
dividend? SL: Smaller: 15:3=5; equal: 15:1=15; bigger: 15:0.5=30
SO: How do the three divisors compare with one? SL: In the first division, greater; in the second,
equal; in the third, smaller. SO: Is there any link? SL: There may be. SO: Could you say what it is?
SL: For Zeus, no! SO: You see, Meno: before your slave was sure in answering, while now he finds
himself in difficulties. Before he was convinced he knew, while now he does not know. But he
knows his mistakes and will no longer fail.
III - SO: Slave, could you say how many times 0.5 is contained in 1? SL: Twice, Socrates.
SO: And 0.25? SL: Four times, Socrates. SO: Try to write the decimal number 0.5 as a fraction.
SL: The fraction is 1/2. SO: But how much is 1/2 compared to 1? SL: It is one half. SO: Then if 1 is
contained an integer number of times in another number, how many times one half will be
contained? SL: Double, Socrates. SO: But in your opinion is it correct to divide a division by
another division? SL: No. SO: Indeed you have now seen that one half of one is contained twice. It
will be sufficient to turn over the division, i.e. to perform the multiplication of the given integer
number by the denominator and we will got the result. What we have performed shows us how
many times a decimal number can be contained in an integer number, SO we have got a ratio
between them. SL: Now I understand my mistake. Bye, Meno is calling me.
TEXT 2 (High level production)
SO: Tell me, my boy, what is the result of 15:3? SL: Five. SO: Is it smaller than 15? SL: What a
question! That is clear! SO: And yet, how much is it 20:5? SL: Obviously 4, Socrates SO: Then is it
smaller than 20? SL: Exactly. SO: Then, how do you think that results of the divisions are? SL: I
think that they are always smaller than the dividend. SO: Are you sure? SL:(°) Yes, because "to
divide" means "to break in equal parts."SO: Now perform this division: 15:1. SL: Uhm, ...it makes
15. SO: But 15 is equal to the dividend. SL: It is true. SO: Why is it equal? SL: Because dividing
by one is how to give an amount to one person, it remains equal. SO: So does your theory still
work? SL: Not completely. Now I see that in some cases it does not work. SO: Are you still sure
you are right? SL: Yes... Pehaps... No... Perhaps there is one case in which the result is larger...or
perhaps not... My Zeus, I understand nothing! (five minutes elapse). SO: What is the result of
2:0.5? SL: These are difficult questions. I am no longer able to answer. SO: Take this square
(drawing)and divide it into small squares! SL: This way?(the drawing is divided into 16 pieces by
drawing 3 horizontal and 3 vertical lines, all equally spaced) SO: Yes, good. Now the unit is the
small square [drawing]. How much is 0,5 compared to 1? SL: One half. SO: Now make one half of
the small square. SL: Done. SO: Do the same for all the small squares. SL: Just a moment...Done.
SO: How many halves? SL: 1,2,3... 32, Socrates. SO: How many unit squares, at the beginning?
SL: Sixteen, Socrates. SO: Then you got a result greater than the starting number. SL: Uhm... Of
course. SO: And how is one half written as a fraction? SL: Uhm... perhaps 1/2. SO: Good! Are you
able now to divide a number by a fraction? SL: Yes, surely! SO: Then divide 2 by 1/2. How many
times is 1/2 contained in 2? SL: According to the preceding rule, I must invert the fraction and
then multiply. OK, it makes 4. SO: How can you represent this? SL: I'll try... Two
squares..[drawing] One half twice for each [drawing]. It works: 4. SO: Good! SL: I understand: the
division is not only "breaking into equal parts", but also seeing how many times a number is
contanined in another! SO: Make an example by yourself! SL: 1:1/4 [he performs and illustrates it]