UNIVERZITA KOMENSKÉHO
(dizertačná práca)
Referee:
1. Prof. RNDr. Ondrej edivý, CSc
3. Prof. RNDr. Helena Bereková, PhD
4. Prof. Filippo Spagnolo (Supervisor)
AIM OF THE THESIS
Thought and language in contextualising experience
The context or environment has an essential function in negotiating meaning as means of communication in the teaching system. According to Chevallard and Johsua (1982), the context comprises three components: the teacher, learner and the knowledge to be taught. Subsequently, Cornu and Vergnioux (1992) introduced the concept of the noosphere, thereby considering the social context which involves and interacts with the teaching system. The use of various linguistic registers and personal, cultural experience are important elements in encouraging communication. Vygotskij has drawn on Piagets assertion: Being aware of an operation indeed means passing from an action to a linguistic plan; it means, therefore, inventing it in your imagination, to be able to express it in words (Vygotskij 1990, p.227). Clearly, in order to pass from one to the other, the action must be contextualised and then recognised and accepted buy the learner.
In order for understand and encourage the process of passing from an antion to a linguistic plan in the teaching process, the connection between the word and its meaning must be taken into consideration; this is not stable but subject to a process of evolution (Vygotskij, Chapter VIII). Observing various experiments regarding the evolution of the brain, it has been seen that in general this evolution is strictly personal since it is tied to a process of cultural, spatial-temporal contextualisation of the experience, which, in certain cases, draws on an individual, emotional state (Changeux 2003). The basis of this contextualisation recalls the concept of a medium or environment (French, as used in English milieu), like that of a subsystem which interacts directly with the learner (materials, games, etc). This milieu can be initially defined as the totality of that which acts on the learner or that which the learner acts on (Brousseau 1977). One can think about the interaction between learner and milieu, devoid of the substantial involvement of the teacher (with the stated meaning of a teaching contract but the teacher can take on the role of tutor or supervisor), as that which defines an a-didactic situation. Whilst, if one also considers an explicit educational system (for example, the figure of the teacher), then one can speak of a teaching situation.
In a teaching situation, which has been prepared and created by a teacher, the learner generally has the task of interpreting the questions asked, information given and obligations imposed, which are constant in the teachers teaching method. These (specific) habits of the teacher, awaited by the learner and the learners behaviour awaited by the teacher, make up the teaching contract. (Brousseau, 1980a p.127)
The a-didactic situation has been described by Brousseau (1986, p.50) as:
The definitive and referential a-didactic situation, which typifies knowledge, can be studied in a theoretical way, but regarding the teaching situation, as much for the teacher as the learner, there exists a type of convergent idea: the teacher must ceaselessly help the learner to remove all their teaching strategies from the situation so that personal and objective knowledge is left.
Sometimes milieu are defined on the basis of real concrete arguments, sometimes the intention for choosing these arguments is added, and sometimes as something stable, on other occasions as something which is developed and modified in the learner. In general, the function of the milieu is:
in the teaching system, to define that part connected to specific a-didactic uses, planned by the teacher and teaching aims but without the necessary and constant presence of such aims (for example, without the direct participation of the teacher).
Brousseau (2000) has emphasised that
the learner learns by adapting to a milieu which is constitutes a factor of contradictions, difficulty, disequilibria, a little like that which takes place in human society. As the result of the learners adaptation, this knowledge reveals itself with new answers, which are the proof of learning.
My main research problem is the study of the conditions in which knowledge is constructed, the aim of which is its optimisation, checking and reproduction in schools. The teaching situations which I considered are specific to the knowledge, which I wanted to be inculcated. By teaching situations, I mean:
a totality of relationships established in an explicit or implicit way between the teacher, learner (or group of learners) and surrounding elements (instrumental or material), with the aim of students learning, that is, they construct a certain consciousness, which has been previously established.
In order that the learner construct their own knowledge, they must personally concern themselves with solving the problem, which has been set in the teaching situation, that is, they must involve themselves in the activity. It can, therefore, be said that the learner has now reached the devolution of the problem. Originally (Brousseau, 1986), devolution was defined as:
the action through which the teacher has the learner accept responsibility for a learning situation (a-didactic) or a problem and personally accept the consequences of this devolution.
My objective is to study the acceptance of this devolution, connecting it to affective learning in order to emphasise the tie between the word and image and their meaning to the learner.
With the aim of reconciling the rigidity of mathematical language with the evolutionary nature of the meaning of terms, Vygotskij has stated that:
a complete elimination of discordances in favour of general and unconditionally correct expression can be reached beyond language and its mathematical skill. We can only say one thing: our spoken language, by virtue of its own fluctuations and discordances between grammatical and psychological features, is habitually found in a state of equilibrium between the ideal of mathematical harmony and fantasy in a never-ending movement, which we call evolution (Vygotskij, 1990 p.339). In this direction an important perspective for teaching mathematics has been opened up which also encompasses my work: theoretical research into neuro-physiology regarding learning and the experimental analysis of interference, which moulds the learner.
This takes place between the meaning of the term in its internal language, common language (that is, everyday extra-curricular experience) and the specific meaning of mathematical terms.
Research Questions and Hypotheses
The choice of carrying out experimental research which includes games or cartoons does not arise only from the necessity of investigating concepts about learners, rather from the need to suggest a new way of doing mathematics, which appeals to a motivational state as regards personal needs. Various objectives have guided me in the selection of teaching tools which are to be used in structuring my experimentation. Some of these are:
· studying multi-sensorial aspects in teaching and learning activities for mathematics
· analysing the game-like characteristics of mathematics in relation to the motivation and interest of doing this type of mathematical activity (appetitus noscendi, J Changuex, 2003)
· developing a real sensitivity in the learners in interpreting and comprehending symbolic images
· organising a grammar which is the most characteristic possible in creating and interpreting a mathematical cartoon
· analysing, from a neuro-physiological point of view, the use of parallel and serial thought by means of diagrams
· analysing the role and meaning of the graphic tools, used in creating cartoons, for students which are recognised by the cartoons iconic code or those which have been introduced ad hoc by the teacher (the teachers implicit tools)
· analysing the problem of mathematical communication in multi-cultural environments.
Regarding the Guess the number game, my objectives are
· thoroughly analysing the relationship between natural and symbolic language
· analysing how the constructing of patterns intervenes in the process of anticipation
We can, therefore, outline the following research hypotheses:
H1 constructing teaching situations, involving a conscious use of cartoons to facilitate devolution
H2 constructing teaching situations, involving a conscious use of arithmetic games to facilitate devolution
H3 constructing learning/teaching milieu which encourage an instrumental use of functional emotions as regards mathematical knowledge (from the learners point of view)
H4 constructing learning/teaching milieu which encourage an instrumental use of functional emotions as regards mathematical knowledge (from the teachers point of view).
INDEX
Presentation Body, thought and language:
emotion as the reason for changes in information .. . pag.6
Aim of the thesis
- Thought and language in contextualising experience .pag.10
- Research questions and hypothesis . pag.12
Chapter 1 Theoretical frame of reference: the relationship between the bodys emotional state and the sensory experience of contextualising experience
Abstract pag.14
1.1 Introduction pag.17
1.1.1 Theoretical frame of reference (Clamats cartoon).
Context in a cartoon:
space-time relationships and the phenomenon of clousure .... pag.18
1.1.2 Theoretical frame of reference (Guess the number).
The use of games for negotiating meaning in passing from
a natural to a pre-algebraic language .. pag.21
1.2 The role of affect in learning mathematics and historical enquiry: the relevance of the history of mathematics in teaching .pag.22
1.3 Methodology: tools and the teachers role pag.24
1.3.1 The structure of the a-didactic situation in cartoons and games ..... pag.25
1.3.2 Validation of the two experimentations . ..... pag.27
Chapter 2 Affective learning: the emotional state of the body and sensory experience in a two-fold alternating of perceptive and recalled image.
Abstract pag.30
2.1 History pag.36
2.2 Let us see what is inside the box pag.37
2.3 Memory and learning: the role of the hippocampus and the amygdala pag.43
2.4 Perception and perceptive images pag.48
2.5 Typical elements of affective learning from the neuro-scientific point of view pag.49
2.6 . but can learning become appetitus noscendi (the wish to learn) pag.52
2.7 Motivation: what are its origins?...........................................................................pag.53
2.7.1 Motivation and reward .pag.56
2.8 Cultural and environmental experience: neuronal Darwinism and the plasticity
of the brain pag.59
2.9 Knowledge and social life: language and inferred communication pag.61
2.9.1 The triadic model of sign pag.61
2.9.2 Syntax and understanding pag.62
2.9.3 Sharing knowledge:
- inferred communication . ... pag.62
- gestures in contextualisation pag.64
- Mirror neurons and reciprocity in inferred communication .. pag.64
2.10 The Theory of Embodiment. . ... pag.64
Chapter 3: Cartoons as environments of mediation
Abstract pag.76
3.1 Introduction: explaining the choice of cartoons as tools of mediation . pag.80
3.2 Cartoons: general and specific objectives . pag.81
3.2.1 Explaining the choice of cartoons as a substitute for a written text . ... pag.82
3.3 The five senses of the cartoon . pag.83
3.3.1 Perceiving images: seeing ≠ understanding .pag.84
3.3.2 Seeing sounds and hearing silence .. pag.87
3.3.3 The sensations of taste, smell and touching images .... pag.88
3.4 The visual code of cartoons: structure and educational characteristics pag.89
3.5 History: the origins of comics pag.93
3.6 The iconic role of images: Invisible Art and Scott McClouds point of view pag.96
3.7 The language of cartoons as an environment for understanding written text pag.100
3.8 First Experimentation: Clamats cartoon
3.8.1 The experimental context: a sample .. pag.102
3.8.2 Methodology: instructions and organisation of cartoons .. pag.102
3.8.3 Structural analysis of each single cartoon and
the role of the tools of mediation ... pag.104
3.8.4 Conclusion of the experimentation:
- qualitative analysis of the experimental work, the teachers role and the tools used in describing the problem ...pag.110
- results of the experimentation .. pag.111
Chapter 4: Second Experimentation: introduction to pre-algebraic language in primary and lower secondary schools. Experimental analysis of a a-didactic situation:
Guess the number
Abstract pag.112
4.1 Introduction .. pag.115
4.2 What is meant by the term language .. pag.116
4.3 References to various works on algebraic language:
- the use of a symbol to indicate a number pag.117
- discussions in class pag.117
- the aim of the teaching contract: teaching aim pag.117
- learning by discovery pag.118
4.4 Explanation of the activity and experimentations results . pag.118
4.5 The experimental context: the sample . .. pag.120
4.6 Phases in the game and a qualitative description of the experience .. pag.120
4.7 Instructions and teachers strategies .. pag.122
4.8 Qualitative analysis: results of teaching phase .. pag.123
4.9 Conclusions pag.128
Chapter 5: Conclusion
Abstract .pag.129
5.1 The role of visual images and symbols in establishing a teaching context, which is easily recognisable by the learner pag.129
5.2 Considerations on the importance of the social-cultural context in a teaching-learning context, which permit the use of cartoons or arithmetical games pag.132
Chapter 6: Information for constructing the milieu
Abstract pag.136
6.1 Mathematical descriptors for interpreting a mathematical cartoon: a possible grammar for a mathematical cartoon pag.137
6.2 Textual analysis of a mathematical cartoon:
Donald Duck in the land of mathemagics pag.143
6.3 Components of affective learning from a didactic-neurophysiological
point of view pag.149
6.3.1 The teaching aim: the importance of contextualisation in teaching ...... pag.152
6.4 A teaching activity for introducing the use of cartoons in geometry: Math Maps pag.154
Appendix
1 The discrete and the continuous:
aspects and methods for a philosophy of mathematics . pag.160
2 How the brain constructs the visual image pag.165
3 Perceiving shapes and movement ..pag.174
4 An interview with Claudio Stassi (designer-advertising cartoonist) ..... pag.181
5 Teaching units for Clamats cartoons pag.188
6 An a-priori analysis of Clamats cartoons .pag.196
7 Learner-secretarys protocol for the Guess the number game .... pag.200
Bibliography ..pag.203