The Role of Metaphors and Images in Learning and Teaching Mathematicshttp://ermeweb.free.fr/CERME3/tableofcontents_cerme3.htmlInternational audienceOur communication fits into a wider study dealing with the activity of thelearner and of the teacher during the solving of numeric problems at primary school.This article focuses on the first stage of our research, during which we examined theplace given explicitly by the teacher to the conversion of representation registers,within the framework of solving numeric problems at primary school. It describesand analyses a practice of education which allows us to examine on the one handhow a teacher delivers, this conversion of registers to the learner in an explicit wayand then how learners take into account this devolution in order to practice usefulconversions during the problem-solving activity

Priolet, Maryvonne

Regnier, Jean-Claude

Archive Ouverte a LUniversite Lyon 2

CONVERSION OF REGISTERS AT PRIMARY
SCHOOL:THE LEARNER’S OR THE TEACHER’S
RESPONSIBILITY ?
Maryvonne Priolet, Jean-Claude Regnier
To cite this version:
Maryvonne Priolet, Jean-Claude Regnier. CONVERSION OF REGISTERS AT PRIMARY
SCHOOL:THE LEARNER’S OR THE TEACHER’S RESPONSIBILITY ?. European Re-
search in Mathematics Education III. Third Conference of the European society for Research
in Mathematics Education, Feb 2003, Bellaria, Italy. European Research in Mathematics Ed-
ucation III, pp.1-10, 2003. <halshs-00405184>
HAL Id: halshs-00405184
https://halshs.archives-ouvertes.fr/halshs-00405184
Submitted on 19 Jul 2009
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destine´e au de´poˆt et a` la diffusion de documents
scientifiques de niveau recherche, publie´s ou non,
e´manant des e´tablissements d’enseignement et de
recherche franc¸ais ou e´trangers, des laboratoires
publics ou prive´s.

Thematic Group 1 EUROPEAN RESEARCH IN MATHEMATICS EDUCATION III
M. Priolet, J.-C. Regnier 1
CONVERSION OF REGISTERS AT PRIMARY SCHOOL:
THE LEARNER’S OR THE TEACHER’S RESPONSIBILITY ?
Maryvonne Priolet, Jean-Claude Régnier
Université Lumière Lyon 2, France
maryvonne.priolet@wanadoo.fr; regnierj@univ-lyon2.fr
Abstract: Our communication fits into a wider study dealing with the activity of the
learner and of the teacher during the solving of numeric problems at primary school.
This article focuses on the first stage of our research, during which we examined the
place given explicitly by the teacher to the conversion of representation registers,
within the framework of solving numeric problems at primary school. It describes
and analyses a practice of education which allows us to examine on the one hand
how a teacher delivers,  this conversion of registers to the learner in an explicit way
and then how learners take into account this devolution in order to practice useful
conversions during the problem-solving activity.
1-  INTRODUCTION
Having noticed in the first part of our research [Priolet, 2000] that on the one hand
mathematical textbooks effectively contained problem solving wordings having a non
exclusively discursive presentation but that on the other hand certain teachers tried to
take on this role of knowledge transmission either by totally ignoring the textbook, or
by curtailing the wordings and by giving them a completely discursive shape, we
questioned the role which the teacher is effectively going to assign to the various
registers of representations, according to R. Duval [Duval, 1995]. Having situated the
theoretical frame to which we refer, we will examine and analyse a problem-solving
situation during which the teacher leads learners to use the conversion of registers to
solve a problem.
2- THEORETICAL FRAMEWORK AND  RESEARCH QUESTIONS
R. Duval regrets the weak consideration, in education, of the diversity of semiotic
registers that nevertheless characterise mathematical activity. He establishes a
classification of the various registers [Duval, 1999] susceptible to being used in
mathematical stages and clarifies that the originality of mathematical activity is in the
simultaneous use of at least two registers of representation at the same time, or in the
possibility of changing the register of representation at any time. How is the pupil
going to proceed if he is confronted from the beginning with a single register only?
Regarding types of alterations of the semiotic representations, R. Duval distinguishes
procedures (alterations which are made by means of the same register) and
conversions (alterations which consist of changing register whilst keeping the same
denoted objects) [Duval, 2000]. When we calculate in the same system of writing
representation of numbers or when we resolve an equation or a system of equations,
we are speaking about procedures. When we pass from algebraic writing to an
Thematic Group 1 EUROPEAN RESEARCH IN MATHEMATICS EDUCATION III
M. Priolet, J.-C. Regnier 2
equation in its graphic representation or even from a natural language instruction to a
relation in its literal writing, we are speaking about conversions.
R. Duval underlined the importance of conversion in cognitive activity and regrets
that numerous studies in mathematics didactics are based around the search for
isomorphism, a strategy nevertheless already denounced in the 30s by L.S. Vygotski
[Vygotski, 1977]. However, is this activity of conversion, to which R. Duval assigns
an essential role in problem solving, present in the teaching of the resolution of
problems? And if it is, what are the effects of these practices on the pupils’ problem-
solving results? These are our main questions.
R. Douady based a part of her studies around "changement de cadres" [Douady,
1986] which she defined as being a means to obtain various formulations of a
problem which without being inevitably completely equivalent, allow a new access to
the difficulties met and the application of tools and techniques which were not
themselves imperative in the first formulation. She described and analysed situations
focusing at the same time on both the tool-object dialectic and framework games, the
first being creative of sense and the second being a source of imbalance, to analyse
the connections between the teaching and the learning of mathematical knowledge.
The framework changes [R. Douady speaks about algebraic, arithmetical and
geometrical frameworks, but also about qualitative or algorithmic frameworks] allow
the pupil to break with the method "I learn, I apply" to go towards an active practice
and a different form of education. These framework changes can be the initiative of
the pupil; they will be called "spontaneous", or provoked by another learner or by the
teacher. R. Douady noted the gains, for the majority of the pupils concerned by the
experiment, of acquisitions by the tool-object dialectic and by framework games, the
latter allowing the pupils to progress in problem solving, by freeing a situation or by
developing the concepts.
It seems to us that the studies led by R. Douady and those that we wish to engage in
are based on the same problematic: Can the passage (which she names change and
which R. Duval names conversion) from a state [a] to a state [b] of a data set in the
same group favour the mathematisation of the situation in order to resolve it more
easily?
But we also think that it is necessary to include a social perspective in our research.
Therefore, our research begins with observations in classrooms. Just as T. Wood and
Turner-Vorbeck have written [Wood, T and Turner-Vorbeck, 2001], “it is essential to
look into the classroom situation to see how teaching is contributing to the
mathematical meanings that are evolving and to examine how teachers’actions
contribute to children’s social learning.” One aim of our research consists in
analysing the different roles that the teacher has to play in the classroom, therefore
we refer to researchers such as Paolo Boero [Boero, 1995] and to M. Bartolini Bussi
and M. A. Mariotti when they analyse the introduction and the place of tools
Thematic Group 1 EUROPEAN RESEARCH IN MATHEMATICS EDUCATION III
M. Priolet, J.-C. Regnier 3
–technical and psychological tools- when they are introduced intentionally into
activity from the outside [Bartolini Bussi, M.G. and Mariotti, M.A., 1999]
3- OUR RESEARCH
Selection: Five primary school teachers participated in this first part of our study.
Drawing lots on a random choice basis. One of them was inexperienced (six months’
experience), two were very experienced (36 years’ experience) and two were
experienced (between 8 and 25 years experience).
Procedure and method:
We asked the five teachers – teaching in classes of pupils aged 10 and 11 years – if
we could observe a lesson concerning problem solving. They choose the subject. We
conducted lesson observations and teacher interviews. We made tapes and we took
video films of the five classes. Four teachers didn’t ask their pupils to use
representations in order to solve the problems and the fifth one (Mr C) asked his
pupils to use representations: Mr C introduced the conversion of representations.
Therefore, in this paper, we only use the results of the fifth teacher’s pupils in order
to examine how conversion belongs both to the learners and to the teacher. For more
details about the other observations  [Priolet, 2001].
Observation and analysis of a situation:
Teacher: Mr C; 21 ten-eleven year old pupils; four classes in the school; situation: a
small village in the centre of France. Length of the lesson : 55 min.
Mr C. having read the instructions of the problem below asked his 10 or 11-year-old
pupils to solve the problem in two stages:
- First to use drawings, graphs, diagrams, tables, or any other representation of their
choice,
- Then, to answer the question.
Problem:
Alan, Maxime and Julie want to make a game for their younger brother Oliver who
goes to play school. They decide to cut out coloured stickers so that Oliver can make
boats by assembling them. For this purpose, the following rules are set:
- Boats can be white, green, red or black;
- We can choose between sails, an engine or oars;
- There are two sizes of boats: small and large.
The three children think that Oliver will be able to make lots of boats…
Thematic Group 1 EUROPEAN RESEARCH IN MATHEMATICS EDUCATION III
M. Priolet, J.-C. Regnier 4
They start to think about the number of different boats that their brother will be able
to make with the stickers.
Help them solve the problem.
For this purpose:
1°) Use a representation of your choice.
2°) Write down the number of different boats.
The work and the results of the pupils
The pupils worked individually. We were interested in the types of procedure or
conversion implemented by the learners. Starting from an exclusively textual
statement, did they stay in a discursive form or did they organize their data in the
form of diagrams, tables or other representations? Looking at the work and results of
the pupils, we could divide the conversion into five groups. I will list them according
to the type of procedure or conversion used.
From the textual statement to a textual form
[Pupil x01]  This pupil [pupil x01] suggests the following representation:
This pupil [x01] stays in a discursive register. On the other hand, he already operates
a sort of classification and we can identify signs that border on an organisation in the
form of a tree. This pupil first considered the size criterion then varied the other two
criteria: colour and propulsion mode. We observe the procedure and, in spite of a
visible textual form we are already in a process of conversion.
[Pupil x02]
Thematic Group 1 EUROPEAN RESEARCH IN MATHEMATICS EDUCATION III
M. Priolet, J.-C. Regnier 5
We find an aid by textual representation in the work above where, after an attempt to
use a sagittal diagram, the pupil returned to a textual register, there again very
structured and we notice that having first considered the colour criterion by
associating it to the propulsion mode criterion, he then added the size criterion:
Attempt 1: Attempt 2:
white with sails white with sails and small
white with engine white with engine and large
white with oars white with oars and small
In the second attempt only, by taking the results obtained by the two first criteria, he
added the unused criterion, the size criterion. He therefore advanced further:  
Attempt 3: 
white with sails and large
white with engine and small
white with oars and large
The first attempt of representation in the form of a sagittal diagram certainly
influenced the continuation of this pupil's work.
From the textual statement to the diagram
[Pupil x03]
In the following representation, we are very close to the representation in the shape of
a tree, where it would be sufficient not to repeat the same colour adjective three
Thematic Group 1 EUROPEAN RESEARCH IN MATHEMATICS EDUCATION III
M. Priolet, J.-C. Regnier 6
times. We are indeed in the presence of a conversion, because the pupil x03 went
from a register (textual) to another register (non-textual).
[Pupil x04]
We find the same organisational structure in the following example. Furthermore, this
pupil [Pupil x04] implemented the same step twice: his first attempt started with the
size criterion, then he started again by taking this time the colour criterion as a
starting point.
Thematic Group 1 EUROPEAN RESEARCH IN MATHEMATICS EDUCATION III
M. Priolet, J.-C. Regnier 7
From the textual statement to an iconic representation
[Pupil x05]
This fifth example [Pupil x05] shows the conversion of the textual representation
towards an iconic representation, which characterises a drawing. We find ourselves in
the presence of a pupil who, starting with textual data, did not look for the number of
boats, but the name of boats corresponding to the various categories. And so:
- For a power-driven green boat, he drew a fishermen's boat,
- for a red boat with oars, he drew a boat,
- for a power-driven black boat, he drew the "Titanic".
He systematically avoided the size criterion.
The work of the teacher [Mr C]
In the first part of this sequence (see above), the conversion of registers of
representations is the learner's responsibility. Then, they are the exchanges of work
during successive demonstrations in front of the group/class to present the means
used to come to a result ; this helped find the weak points and the key points of every
work. Exchanges created by the teacher will allow the intervention of an 11-year-old
pupil who will present the conversion in tabular form.
Tree
Thematic Group 1 EUROPEAN RESEARCH IN MATHEMATICS EDUCATION III
M. Priolet, J.-C. Regnier 8
                
                 Table 1 Table 2
These are the exchanges amongst learners that allowed the movement from
representations such as the tree (quoted by the pupils, drawn on the blackboard and
validated by both the learners and the teacher) to the representation in tabular form.
Here the teacher has a real intermediary role: he sets up a situation aimed at
improving the exchanges amongst the pupils and he obtains the approval of the group
which then works together. “ Implicit in any discussion on teaching is that it is
intended to create learning” [Pearson, 1989]. The "table" representation evolved
exclusively from the exchanges between learners. The above photos show the
conversion of registers: we move from "the tree" to a first table, which did not receive
the support of the group/class, to a second more functional table. More functional
because, without losing sight of the direction of the activity which consisted of
counting all the boats, the pupils managed, by using the table, to find the exact
number of boats. Indeed, these different tools were introduced by the group of
learners, but we underline the role of the teacher who stimulated the interaction
between the pupils and “created learning”.
4- CONCLUSION
The above study is aimed at outlining the place effectively reserved for this phase of
conversion, which seems to us to be at the core of the problematisation process,
within a sequence of problem solving in the elementary school. Once we consider
that the conversion of representation is a compulsory and essential passage of
problem solving, it is necessary to examine various types of practices. Is it a question
of leaving the conversion in its entirety to the learner who should pass from a
problem instruction either in natural language, or in tabular or graph form, to another
type of representation, without the support of "learning vectors" (teacher, school
textbook, software)? Or is it a question of favouring the conversion by giving the
pupil outside tools (on paper, or via software)? Or is it a question for the teacher to be
responsible for this phase of conversion within the framework of a problem solving
educational method?
Thematic Group 1 EUROPEAN RESEARCH IN MATHEMATICS EDUCATION III
M. Priolet, J.-C. Regnier 9
The observation of various practices will lead us later to devise a typology of
situations in comparison to the place given by the teachers to the conversion of
registers. The above study placed us in the presence of a teacher who allowed the
learners to regulate this conversion:
- on the one hand by giving them clues in the instructions to intervene in the process
of problem construction or reconstruction, without doing the conversion for them,
- on the other hand by favouring the exchanges between pupils and between pupils
and teacher during the phase of correction.
Can we consider this use of conversion just as a tool of “semiotic mediation” that is
introduced intentionally into the activity from the outside [Bartolini Bussi, M.G. and
Mariotti, M.A., 1999] offered by somebody else, such as the teacher and the other
pupils or can we consider that this construction is more than an external guide and
that it will help the learners to  activate a process to find the different solutions of the
problem?
Our later works will lead us to examine the contributions but also the limits of the
various practices met in educational-learning situations which put the conversion of
registers in the foreground or in the background.
REFERENCES
BARTOLINI BUSSI, M.G. AND MARIOTTI, M.A., (1999), Semiotic mediation: from
history to mathematics classroom, For the Learning of Mathematics
BOERO, P., DAPUETO, C., FERRARI, P., FERRERO, E., GARUTI, R., LEMUT, E.,
PARENTI, L., SCALI, E. (1995), Aspects of the mathematics-culture relationship
in mathematics teaching-learning in compulsory school, Proc. of PME-XIX,
Recife.
DOUADY R., (1986), Jeux de cadres et dialectique outil-objet, Recherche en
Didactique des Mathématiques, vol.7, n°2, Grenoble, La Pensée Sauvage, p.5-31
DUVAL R., (1995), Sémiosis et pensée humaine, Berne, Peter Lang
DUVAL R., (1999), Conversion et articulation des représentations analogiques,
IUFM Nord-Pas de Calais
DUVAL R., (2000), L’analyse cognitive des problèmes de compréhension dans
l’apprentissage des mathématiques, Conférence faite au Tercero en didactica de
la Matematica, Universidad Catolica de Valparaiso.
PEARSON, T.A. (1989)  The Teacher:  Theory and Practice in Teacher Education.
London:  Routledge
PRIOLET My, (2000), Résolution de problèmes arithmétiques et registres
sémiotiques, Mémoire de Maîtrise en Sciences de l’Éducation, sous la direction
de J.Cl. REGNIER, Université Lumière Lyon 2, 363 p.
Thematic Group 1 EUROPEAN RESEARCH IN MATHEMATICS EDUCATION III
M. Priolet, J.-C. Regnier 10
PRIOLET My, (2001), Résolution de problèmes numériques au cycle des
approfondissements de l’Ecole Primaire, recours à la production de
représentations écrites et vecteurs d'apprentissage, Mémoire de D.E.A. de
Sciences de l’Éducation, sous la direction de F. CLERC, Université Lumière
Lyon 2, 358 p.
 VYGOTSKI, L. (1977), Pensée et langage, Paris, La Dispute, 3ème éd., 537 p.
WO O D , T AND TURNER-VORBECK, T.  (2001) ‘Extending the Conception of
Mathematics Teaching’ in  Wood T., Scott-Nelson B & Warfield, J. (Eds)
Beyond Classical Pedagogy in Elementary Mathematics


Conversion of registers at primary school: the learner's or the teacher's responsibility

HAL-ENS-LYON

CONVERSION OF REGISTERS AT PRIMARYSCHOOL:THE LEARNER’S OR THE TEACHER’SRESPONSIBILITY ?Maryvonne Priolet, Jean-Claude RegnierTo cite this version:Maryvonne Priolet, Jean-Claude Regnier. CONVERSION OF REGISTERS AT PRIMARYSCHOOL:THE LEARNER’S OR THE TEACHER’S RESPONSIBILITY ?. European Re-search in Mathematics Education III. Third Conference of the European society for Researchin Mathematics Education, Feb 2003, Bellaria, Italy. European Research in Mathematics Ed-ucation III, pp.1-10, 2003. <halshs-00405184>HAL Id: halshs-00405184https://halshs.archives-ouvertes.fr/halshs-00405184Submitted on 19 Jul 2009HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.Thematic Group 1 EUROPEAN RESEARCH IN MATHEMATICS EDUCATION IIIM. Priolet, J.-C. Regnier 1CONVERSION OF REGISTERS AT PRIMARY SCHOOL:THE LEARNER’S OR THE TEACHER’S RESPONSIBILITY ?Maryvonne Priolet, Jean-Claude RégnierUniversité Lumière Lyon 2, Francemaryvonne.priolet@wanadoo.fr; regnierj@univ-lyon2.frAbstract: Our communication fits into a wider study dealing with the activity of thelearner and of the teacher during the solving of numeric problems at primary school.This article focuses on the first stage of our research, during which we examined theplace given explicitly by the teacher to the conversion of representation registers,within the framework of solving numeric problems at primary school. It describesand analyses a practice of education which allows us to examine on the one handhow a teacher delivers,  this conversion of registers to the learner in an explicit wayand then how learners take into account this devolution in order to practice usefulconversions during the problem-solving activity.1-  INTRODUCTIONHaving noticed in the first part of our research [Priolet, 2000] that on the one handmathematical textbooks effectively contained problem solving wordings having a nonexclusively discursive presentation but that on the other hand certain teachers tried totake on this role of knowledge transmission either by totally ignoring the textbook, orby curtailing the wordings and by giving them a completely discursive shape, wequestioned the role which the teacher is effectively going to assign to the variousregisters of representations, according to R. Duval [Duval, 1995]. Having situated thetheoretical frame to which we refer, we will examine and analyse a problem-solvingsituation during which the teacher leads learners to use the conversion of registers tosolve a problem.2- THEORETICAL FRAMEWORK AND  RESEARCH QUESTIONSR. Duval regrets the weak consideration, in education, of the diversity of semioticregisters that nevertheless characterise mathematical activity. He establishes aclassification of the various registers [Duval, 1999] susceptible to being used inmathematical stages and clarifies that the originality of mathematical activity is in thesimultaneous use of at least two registers of representation at the same time, or in thepossibility of changing the register of representation at any time. How is the pupilgoing to proceed if he is confronted from the beginning with a single register only?Regarding types of alterations of the semiotic representations, R. Duval distinguishesprocedures (alterations which are made by means of the same register) andconversions (alterations which consist of changing register whilst keeping the samedenoted objects) [Duval, 2000]. When we calculate in the same system of writingrepresentation of numbers or when we resolve an equation or a system of equations,we are speaking about procedures. When we pass from algebraic writing to anThematic Group 1 EUROPEAN RESEARCH IN MATHEMATICS EDUCATION IIIM. Priolet, J.-C. Regnier 2equation in its graphic representation or even from a natural language instruction to arelation in its literal writing, we are speaking about conversions.R. Duval underlined the importance of conversion in cognitive activity and regretsthat numerous studies in mathematics didactics are based around the search forisomorphism, a strategy nevertheless already denounced in the 30s by L.S. Vygotski[Vygotski, 1977]. However, is this activity of conversion, to which R. Duval assignsan essential role in problem solving, present in the teaching of the resolution ofproblems? And if it is, what are the effects of these practices on the pupils’ problem-solving results? These are our main questions.R. Douady based a part of her studies around "changement de cadres" [Douady,1986] which she defined as being a means to obtain various formulations of aproblem which without being inevitably completely equivalent, allow a new access tothe difficulties met and the application of tools and techniques which were notthemselves imperative in the first formulation. She described and analysed situationsfocusing at the same time on both the tool-object dialectic and framework games, thefirst being creative of sense and the second being a source of imbalance, to analysethe connections between the teaching and the learning of mathematical knowledge.The framework changes [R. Douady speaks about algebraic, arithmetical andgeometrical frameworks, but also about qualitative or algorithmic frameworks] allowthe pupil to break with the method "I learn, I apply" to go towards an active practiceand a different form of education. These framework changes can be the initiative ofthe pupil; they will be called "spontaneous", or provoked by another learner or by theteacher. R. Douady noted the gains, for the majority of the pupils concerned by theexperiment, of acquisitions by the tool-object dialectic and by framework games, thelatter allowing the pupils to progress in problem solving, by freeing a situation or bydeveloping the concepts.It seems to us that the studies led by R. Douady and those that we wish to engage inare based on the same problematic: Can the passage (which she names change andwhich R. Duval names conversion) from a state [a] to a state [b] of a data set in thesame group favour the mathematisation of the situation in order to resolve it moreeasily?But we also think that it is necessary to include a social perspective in our research.Therefore, our research begins with observations in classrooms. Just as T. Wood andTurner-Vorbeck have written [Wood, T and Turner-Vorbeck, 2001], “it is essential tolook into the classroom situation to see how teaching is contributing to themathematical meanings that are evolving and to examine how teachers’actionscontribute to children’s social learning.” One aim of our research consists inanalysing the different roles that the teacher has to play in the classroom, thereforewe refer to researchers such as Paolo Boero [Boero, 1995] and to M. Bartolini Bussiand M. A. Mariotti when they analyse the introduction and the place of toolsThematic Group 1 EUROPEAN RESEARCH IN MATHEMATICS EDUCATION IIIM. Priolet, J.-C. Regnier 3–technical and psychological tools- when they are introduced intentionally intoactivity from the outside [Bartolini Bussi, M.G. and Mariotti, M.A., 1999]3- OUR RESEARCHSelection: Five primary school teachers participated in this first part of our study.Drawing lots on a random choice basis. One of them was inexperienced (six months’experience), two were very experienced (36 years’ experience) and two wereexperienced (between 8 and 25 years experience).Procedure and method:We asked the five teachers – teaching in classes of pupils aged 10 and 11 years – ifwe could observe a lesson concerning problem solving. They choose the subject. Weconducted lesson observations and teacher interviews. We made tapes and we tookvideo films of the five classes. Four teachers didn’t ask their pupils to userepresentations in order to solve the problems and the fifth one (Mr C) asked hispupils to use representations: Mr C introduced the conversion of representations.Therefore, in this paper, we only use the results of the fifth teacher’s pupils in orderto examine how conversion belongs both to the learners and to the teacher. For moredetails about the other observations  [Priolet, 2001].Observation and analysis of a situation:Teacher: Mr C; 21 ten-eleven year old pupils; four classes in the school; situation: asmall village in the centre of France. Length of the lesson : 55 min.Mr C. having read the instructions of the problem below asked his 10 or 11-year-oldpupils to solve the problem in two stages:- First to use drawings, graphs, diagrams, tables, or any other representation of theirchoice,- Then, to answer the question.Problem:Alan, Maxime and Julie want to make a game for their younger brother Oliver whogoes to play school. They decide to cut out coloured stickers so that Oliver can makeboats by assembling them. For this purpose, the following rules are set:- Boats can be white, green, red or black;- We can choose between sails, an engine or oars;- There are two sizes of boats: small and large.The three children think that Oliver will be able to make lots of boats…Thematic Group 1 EUROPEAN RESEARCH IN MATHEMATICS EDUCATION IIIM. Priolet, J.-C. Regnier 4They start to think about the number of different boats that their brother will be ableto make with the stickers.Help them solve the problem.For this purpose:1°) Use a representation of your choice.2°) Write down the number of different boats.The work and the results of the pupilsThe pupils worked individually. We were interested in the types of procedure orconversion implemented by the learners. Starting from an exclusively textualstatement, did they stay in a discursive form or did they organize their data in theform of diagrams, tables or other representations? Looking at the work and results ofthe pupils, we could divide the conversion into five groups. I will list them accordingto the type of procedure or conversion used.From the textual statement to a textual form[Pupil x01]  This pupil [pupil x01] suggests the following representation:This pupil [x01] stays in a discursive register. On the other hand, he already operatesa sort of classification and we can identify signs that border on an organisation in theform of a tree. This pupil first considered the size criterion then varied the other twocriteria: colour and propulsion mode. We observe the procedure and, in spite of avisible textual form we are already in a process of conversion.[Pupil x02]Thematic Group 1 EUROPEAN RESEARCH IN MATHEMATICS EDUCATION IIIM. Priolet, J.-C. Regnier 5We find an aid by textual representation in the work above where, after an attempt touse a sagittal diagram, the pupil returned to a textual register, there again verystructured and we notice that having first considered the colour criterion byassociating it to the propulsion mode criterion, he then added the size criterion:Attempt 1: Attempt 2:white with sails white with sails and smallwhite with engine white with engine and largewhite with oars white with oars and smallIn the second attempt only, by taking the results obtained by the two first criteria, headded the unused criterion, the size criterion. He therefore advanced further:  Attempt 3: white with sails and largewhite with engine and smallwhite with oars and largeThe first attempt of representation in the form of a sagittal diagram certainlyinfluenced the continuation of this pupil's work.From the textual statement to the diagram[Pupil x03]In the following representation, we are very close to the representation in the shape ofa tree, where it would be sufficient not to repeat the same colour adjective threeThematic Group 1 EUROPEAN RESEARCH IN MATHEMATICS EDUCATION IIIM. Priolet, J.-C. Regnier 6times. We are indeed in the presence of a conversion, because the pupil x03 wentfrom a register (textual) to another register (non-textual).[Pupil x04]We find the same organisational structure in the following example. Furthermore, thispupil [Pupil x04] implemented the same step twice: his first attempt started with thesize criterion, then he started again by taking this time the colour criterion as astarting point.Thematic Group 1 EUROPEAN RESEARCH IN MATHEMATICS EDUCATION IIIM. Priolet, J.-C. Regnier 7From the textual statement to an iconic representation[Pupil x05]This fifth example [Pupil x05] shows the conversion of the textual representationtowards an iconic representation, which characterises a drawing. We find ourselves inthe presence of a pupil who, starting with textual data, did not look for the number ofboats, but the name of boats corresponding to the various categories. And so:- For a power-driven green boat, he drew a fishermen's boat,- for a red boat with oars, he drew a boat,- for a power-driven black boat, he drew the "Titanic".He systematically avoided the size criterion.The work of the teacher [Mr C]In the first part of this sequence (see above), the conversion of registers ofrepresentations is the learner's responsibility. Then, they are the exchanges of workduring successive demonstrations in front of the group/class to present the meansused to come to a result ; this helped find the weak points and the key points of everywork. Exchanges created by the teacher will allow the intervention of an 11-year-oldpupil who will present the conversion in tabular form.TreeThematic Group 1 EUROPEAN RESEARCH IN MATHEMATICS EDUCATION IIIM. Priolet, J.-C. Regnier 8                                 Table 1 Table 2These are the exchanges amongst learners that allowed the movement fromrepresentations such as the tree (quoted by the pupils, drawn on the blackboard andvalidated by both the learners and the teacher) to the representation in tabular form.Here the teacher has a real intermediary role: he sets up a situation aimed atimproving the exchanges amongst the pupils and he obtains the approval of the groupwhich then works together. “ Implicit in any discussion on teaching is that it isintended to create learning” [Pearson, 1989]. The "table" representation evolvedexclusively from the exchanges between learners. The above photos show theconversion of registers: we move from "the tree" to a first table, which did not receivethe support of the group/class, to a second more functional table. More functionalbecause, without losing sight of the direction of the activity which consisted ofcounting all the boats, the pupils managed, by using the table, to find the exactnumber of boats. Indeed, these different tools were introduced by the group oflearners, but we underline the role of the teacher who stimulated the interactionbetween the pupils and “created learning”.4- CONCLUSIONThe above study is aimed at outlining the place effectively reserved for this phase ofconversion, which seems to us to be at the core of the problematisation process,within a sequence of problem solving in the elementary school. Once we considerthat the conversion of representation is a compulsory and essential passage ofproblem solving, it is necessary to examine various types of practices. Is it a questionof leaving the conversion in its entirety to the learner who should pass from aproblem instruction either in natural language, or in tabular or graph form, to anothertype of representation, without the support of "learning vectors" (teacher, schooltextbook, software)? Or is it a question of favouring the conversion by giving thepupil outside tools (on paper, or via software)? Or is it a question for the teacher to beresponsible for this phase of conversion within the framework of a problem solvingeducational method?Thematic Group 1 EUROPEAN RESEARCH IN MATHEMATICS EDUCATION IIIM. Priolet, J.-C. Regnier 9The observation of various practices will lead us later to devise a typology ofsituations in comparison to the place given by the teachers to the conversion ofregisters. The above study placed us in the presence of a teacher who allowed thelearners to regulate this conversion:- on the one hand by giving them clues in the instructions to intervene in the processof problem construction or reconstruction, without doing the conversion for them,- on the other hand by favouring the exchanges between pupils and between pupilsand teacher during the phase of correction.Can we consider this use of conversion just as a tool of “semiotic mediation” that isintroduced intentionally into the activity from the outside [Bartolini Bussi, M.G. andMariotti, M.A., 1999] offered by somebody else, such as the teacher and the otherpupils or can we consider that this construction is more than an external guide andthat it will help the learners to  activate a process to find the different solutions of theproblem?Our later works will lead us to examine the contributions but also the limits of thevarious practices met in educational-learning situations which put the conversion ofregisters in the foreground or in the background.REFERENCESBARTOLINI BUSSI, M.G. AND MARIOTTI, M.A., (1999), Semiotic mediation: fromhistory to mathematics classroom, For the Learning of MathematicsBOERO, P., DAPUETO, C., FERRARI, P., FERRERO, E., GARUTI, R., LEMUT, E.,PARENTI, L., SCALI, E. (1995), Aspects of the mathematics-culture relationshipin mathematics teaching-learning in compulsory school, Proc. of PME-XIX,Recife.DOUADY R., (1986), Jeux de cadres et dialectique outil-objet, Recherche enDidactique des Mathématiques, vol.7, n°2, Grenoble, La Pensée Sauvage, p.5-31DUVAL R., (1995), Sémiosis et pensée humaine, Berne, Peter LangDUVAL R., (1999), Conversion et articulation des représentations analogiques,IUFM Nord-Pas de CalaisDUVAL R., (2000), L’analyse cognitive des problèmes de compréhension dansl’apprentissage des mathématiques, Conférence faite au Tercero en didactica dela Matematica, Universidad Catolica de Valparaiso.PEARSON, T.A. (1989)  The Teacher:  Theory and Practice in Teacher Education.London:  RoutledgePRIOLET My, (2000), Résolution de problèmes arithmétiques et registressémiotiques, Mémoire de Maîtrise en Sciences de l’Éducation, sous la directionde J.Cl. REGNIER, Université Lumière Lyon 2, 363 p.Thematic Group 1 EUROPEAN RESEARCH IN MATHEMATICS EDUCATION IIIM. Priolet, J.-C. Regnier 10PRIOLET My, (2001), Résolution de problèmes numériques au cycle desapprofondissements de l’Ecole Primaire, recours à la production dereprésentations écrites et vecteurs d'apprentissage, Mémoire de D.E.A. deSciences de l’Éducation, sous la direction de F. CLERC, Université LumièreLyon 2, 358 p. VYGOTSKI, L. (1977), Pensée et langage, Paris, La Dispute, 3ème éd., 537 p.WO O D , T AND TURNER-VORBECK, T.  (2001) ‘Extending the Conception ofMathematics Teaching’ in  Wood T., Scott-Nelson B & Warfield, J. (Eds)Beyond Classical Pedagogy in Elementary Mathematics

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Conversion of registers at primary school: the learner's or the teacher's responsibility

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