UNIVERZITA KOMENSKÉHO
(dizertačná práca)
Referee:
1. PaedDr. Soňa Čeretková, Ph.D.
2. Doc. RNDr. Daniel Palumbiny, CSc.
3. Prof. RNDr. Ján Čimár, PhD
5. Prof. Filippo Spagnolo, PhD. (Supervisor)
Abstract.
My research was born from the idea that difficulties and problems students of different grades have mainly come from language and, in general, from the formal aspect of mathematical concepts.
It appeared extremely important to me to consider two sides apparently divergent
1) The specific quality of mathematics and its own language;
2) The role of the context (space, time, people) in communicating mathematics.
Do not forget that language is not only a source of trouble; it is also a necessary player in every learning process, this point is really discussed in recent works about mathematics education (Maier, Radford, Duval).
My personal idea is that, in order to make easier the mathematical communication, it is necessary to create a proper context.
The theoretical idea I go with is the one from Guy Brousseau who defines the milieu: the environment where the student and his knowledge building process happen.
The choice of this theoretical framework is due to this statement:
Learning-teaching process can follow two different ways, the first one is a well known one, based on frontal lessons that are the traditional way to make people learn contents, the second one refers to a way of learning based on the emotional side (that is going to be discussed in chapter 4 about neuro-sciences) that teachers difficultly control but, on the other side, allows the building of cognitive-conflict based situation (chapter I).
I built an a-didactical situation where the relation learning-teaching into the knowledge-pupil-teacher triangle is controlled and analysed in connection with the outside environment and the emotional sphere of the student.
But I did not use only the Theory of Didactical situations; this research aims to link two different theoretical frameworks:
1) The Theory of Didactical situations that structures the a-didactical situation and has a methodological control role.
2) The Embodiment theory regarding the body experience learning, that leads to the process of creating metaphors and learning into an emotional context.
3) A look at Neuroscience which can give suggestion about new hypothesis of work.
Once again I have to say that the theory of situation comes at a methodological level, the play context is built according to the theory, this play choice has been made with a look at this theory. But in this scenario another element is crucial, the play situations has also some corporal times, so that also the embodiment theory and its approach is very important as well as a theoretical reference and a way to analyse data.
It is possible to link these two theories?
We can probably say that the link between the two theoretical frameworks is that the student is the only player of his knowledge process; by the devolution act according to the theory of Didactical situations, by his own senses, brain and mind according to the embodiment theory.
What I tried to build recalls the book Cognitive space of action production and communication by F.Arzarello, whose basic elements are:
ü The body and the brain;
ü Physical world;
ü Cultural environment;
Here we can find culture, sense and motion related experiences (embodiment), languages, representations, signs and objects such as pens and computers. But Arzarello does not include in his work the theory of Didactical situations.
What does embodiment mean?
The concept of Embodiment is relatively new within the field of mathematical education, I would like to clarify, according to Lakoff and Nuñez, the use of the term in this thesis, and distinguish it from other notions concerning the role of the physical and concrete in mathematics learning. Embodiment is not simply about an individuals conscious experience of some bodily aspects of being or acting in the world. Embodiment does not necessarily involve conscious awareness of its influence. Nor does embodiment refers to the physical manipulation of tangible objects, or to the virtual manipulation of graphical images and objects instantiated through technology. Although there is a relation between such experiences and the technical concepts of embodiment, and an embodiment perspective does not constitute a prescription for teaching in a concrete way. Similarly, although embodiment may provide a theoretical grounding for understanding the teaching and learning of realistic or contextualized mathematics, it is not directly concerned with contextualization or situatedness in subject matter teaching.
Rather, embodiment provides a deep understanding of what human ideas are, and how they are organized in vast (most unconscious) conceptual systems grounded in physical, lived reality.
Some other important elements I consider are the conceptual metaphors. They are mapping that preserve the inferential structure of a source domain as it is projected onto a target domain. Thus the target domain is understood, often unconsciously, in terms of the relations that hold in the source domain. For instance, within mathematics, Boolean logic is an extension of the container scheme, realized through a conceptual metaphorical projection of the logic of containers. So a mathematical concept is build via physical experience, and later unconsciously mapped to a set of abstract mathematical concepts.
The projections or mappings are not arbitrary, and can be empirically studied and precisely stated. They are not arbitrary, because they are motivated by our everyday experience, especially bodily experience, which is biologically constrained. Unlike traditional studies of metaphors, contemporary embodied views dont see conceptual metaphors as residing in words, but in thoughts.
When facing the problem of the comprehension of the concept of being perpendicular, through body experience, it is a duty to take in account the biological laws on human beings perception of the concepts of being vertical and being perprendicular.
To better understand this point the thesis provides an analysis of the Vestibular System (A.Berthoz)inside whom there are some receptors sensible to gravity direction.
Gravity can indeed be measured by specialized receptors: the otoliths; gravity is an external reference system, a plumb line the body refers to in a geocentric reference system.
The vestibular system is a very important egocentric reference tool that allows the perception of the plumb line model.
Mittelstaedt(1995/96) would say that we needed a new sense to add to the ones involved in gravity vertical perception, he discovered some neural receptors placed in the stomach that react to gravity (he recently ended up saying that these structures were rather placed on kidneys or blood system).
The experimental work has been led within the S.P.O.R.A. project, where I was teacher for the lessons of Curriculum with structure, organized by the file-leader school D.D.F. Ferrara of Palermo, in collaboration with G.R.I.M. (Research Group for Teaching of Mathematics) of the University of Palermo.
The schools involved were at a quite high degree of risk (based on social and economic indicators). The students involved were from 3 to 11 years old (first grade schools).
The field research has required some answers on pupils concepts about perpendicularity:
1. Do pupils have the inner model of the plumb line?
2. Does the misunderstanding of the notion of perpendicular come from a linguistic misunderstanding?
The quantity analysis has been done with the Chic software, the quality one according to the theory of the situations.
Final considerations have been done by the process of metaphors building through bodily activities that pupils have joined.
2.0 And now let go working it is time to communicate ...11
3.0 Some notes about the national and international research .13
4.0 Thesis stucture .... 19
1.1 The triangle teacher-Knowledge-pupils .... 20
1.2 The didactical contract ... 22
1.3 Theory of didactical situation . .... 23
1.4 Obstacles in the Theory of Situation . . 31
1.5 Mistakes and conceptions .. . 34
1.6 Cognitive conflict in obstacle theory .. .... 35
2.1. Introduction . .41
2.2. Mathematics as a language .... 41
2.3. The language of mathematics in a classroom . 44
2.4 Properties of mathematical language . 45
2.4.1 Mathematics as an ideal reference ....... 45
2.4.2. The not ambiguous mathematical language.. . ... 47
2.4.3. Common meaning and the one of that mathematical language. .. 47
2.4.4 Self-suffiency, consequentiality, density ...47
2.5 Mathematical language and communication in a classroom. ... 49
Chapter III A teaching proposal 50
3.1. Introduction ... .... .. 50
3.2 Research goals .. 51
3.3 Research tools .. . 53
3.3.1 Tom and Jerry test 53
3.3.2 Play-path .. 58
3.4 Students involved in the experience . . 59
3.5 Test analysis ... 60
3.6 A-priori analysis .... 61
3.7 Evaluation criteria for the quantity analysis .. 62
3.7.1 Qualitative and quantitative analysis................................................................... 62
3.8 Play-Path analysis .. 67
3.9 Conclusions of the experience........................................................................................ 69
Chapter IV. The Embodiment theory .. .. 75
4.1 Introduction 75
4.2 The philosophic point of view of the embodiment theory 75
4.3 Strong and weak aspects of the embodiment theory 77
4.4 World interaction defines math properties .78
4.5 Embodied mathematics properties 79
4.6 Relation between culture and Embodied mathematics 80
4.7 Cognitive science in studying math learning ..82
4.8 New discovers on mid nature 83
4.81 The conceptual metaphors ... 83
4.9 Brain areas dedicated to mathematical thinking 84
4.10 Daily conceptual procedures determining in maths learning .84
4.10.1 Spatial relation concepts and Image schemas .85
4.10.2 Motor control and mathematical ideas ..89
4.10.3The source-path-goal schema ... .89
4.11 The structure of the conceptual metaphor 90
4.12 The theory of Situation and the building of an environment to create metaphors.93
ChapterV Learning: a neuronal approach ................. .96
5.1. Learning and memory . ..96
5.2 From perception to memory ..97
5.3 The brain-body link in learning . 99
5.4 The subjective vertical ..103
5.4.1 The vestibular system .. .. . .103
5.4.2 Distance perception.. .. .105
5.4.3 Reference systems . .105
5.4.4 Allocentric and Egocentric reference 106
5.4.5 Natural reference systems: gravity .. 107
5.4.6 The subjective vertical as a multisensorial result .. .. 107
5.4.7 Vestibular receptors in the stomach . ... 108
5.4.8 The turn-spit experience . 109
5.4.9 Conclusions 110
Bibliography . 113