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DESIGNING MATHEMATICS TOOLS FOR BLIND CHILDREN Emanuela Ughi Dipartimento di Matematica e Informatica, Università di Perugia, Italy Mathematical manipulatives for blind children are planned, focusing the interest in offering them sensorial experiences they usually miss.

Manipulatives are commonly recommended and used in the field of education for blind children, for several teaching purposes. In particular, several authors stress the importance of this kind of tools for giving blind children the possibility of acquiring mathematical concepts though concrete activities (e.g, Del Campo 1986, Russo 2001, Tindell 2006). There is indeed a large offer of concrete materials for blind students, but just a few are specifically planned for teaching mathematics. On the basis of direct experiences, and of studies on the problems of blind students, in this paper I want to suggest that a deeper reflection can help in designing better concrete tools, specifically planned to offer them a concrete ground of experience surrogating the visual experiences they miss. Here I report some stories and facts, to later introduce my proposals. A) An experience in a 4th grade I planned a manipulative experience in a classroom, in which there is A. , severely visually impaired child, 9 years old. She is smart, and well adjusted in the classroom; her mates are very protective with her. I organized an activity offered to the whole classroom, with materials sligthly modified for A.'s needs, starting with an activity about cubes. While her mates were making a cardboard cube, A. was asked to play with a cardboard cubic box, containing perfectly 27 small wood cubes. First of all, after an external exploration, she reversed it, and she was very surprised that in that cubic box “there were so many things!”. Then she played to fulfill again the box, then to build the 3x3 cube without the box (this task was difficult, because of the mobility of her constructions! A magnetic structure would be better, but too difficult for our handcraft skills). There are two things that I want to report here: first of all, A. was extremely pleased. She was asking for intellectual challenges, and that was new and interesting to her. The poorness of her sensorial experience was in contrast with her natural intellectual curiosity! Secondly, after a while, she said to her teacher: “Oh, is this the same thing as the cube we used in first class?”. She was talking about a well known didactic material, a cube representing 1000 small cubes glued together, where the small cubes are just represented by lines engraved on the surface of the six faces of the big cube.; so, while her classmates could easily “imagine” that the cube, realized that way, was suggesting that it was obtained by glueing 1000 small cubes, she had no idea of this fact. In other words, the tool (planned for sighted children) was not sufficient to convey all the informations to her; so, A. deeply understood it, through the use of our rough and minimal manipulative, only 3 years later.

B) A true anecdote, as reported by Prondzinsky, 1998 A girl, 24 years old, blind from the birth, brilliantly graduated in philosophy, was following a course for sense of direction and mobility. Her tutor asked her to count the angles of a room (completely new to her). The room was rectangular, but the girl didn't stop at the fourth angle, and was continuing to count. At the 32nd angle, the tutor stopped her, and asked “what is the definition of a rectangle?”. The girl answered perfectly.

C) Several studies point out some specific difficulties of blind students, and in particular of those blind from the birth; they have problems in translating and transferring three-dimensional obiects into two-dimensional iconic form, so that they cannot understand perspective drawings; also, they have problems in enlarging an minimizing even two-dimensional forms, so that they are not able to recognize different-sized shapes as being the same (e.g, Kohanová 2007).

At this point my question is: Is it possible to reach some conclusion summing up from the above points? I have looked at them through a “geometer” point of view. In this way, they suggest me that, despite the efforts, the manipulatives offered to blind children are planned by sighted people, who didn't recognize enough that it is not sufficient that a obiect “contains” an information. For that information to be received by a blind student, it has to be grounded on the experiences the student already knows, All those examples offer a striking evidence that they are strictly related to the kind of experience that blind children miss most: projections and sections. Indeed, sighted people have a huge experience of sections: thinking of “seeing” somehow as “drawing in perspective”, the sighted persons can easily imagine the result obtained by cutting the visual cone by a plane. This lack of experiences explains of course the problems for blind people in understanding perspective, but also the other problems arising from the previous examples. This relationship is clear in the first example: indeed, think of a sighted person imagining the cube obtained as 1000 glued small cubes: he/she is indeed “seeing” the cubes obtained by cutting the big cube along the engraved lines. Let's look now at the second example: it seems very difficult to imagine why the girl was not able to recognize a rectangular room; but, of course, no one suggested her to explore the room by touching the angles on the floor. So, the missing experience is that of cutting a dihedral angle by planes, and in particular by a plane ortogonal to the edge of that dihedral angle, while a sighted person can easily recognize the floor as that ortogonal section.. Missing this observation, I suppose that it would be difficult even for a sighted person to size a dihedral angle simply by touching the two planes. Moreover, having a clear idea of the behaviour of those sections is necessary even to understand how to assign a measure to that dihedral angle. Finally, even the problems to recognize similar (say two-dimensional) shapes can be related to lacking section experiences: indeed, a sighted person seeing two similar planar shapes can imagine them as obtained by a cone cut by two parallel planes, and the homothety between them obtained by cutting that cone by all the intermediate planes. To recognize similarity between two-dimensional shapes can be difficult even for a sighted person, if the portion of the shape he/she can explore is too small: for example, to compare crop circle shapes, it is necessary to fly over them. A similar situation is that of a blind person that can have the perception just of “local” properties of that shape, i.e. those contained in a surface corresponding to a fingertip, while the “global” properties can be gathered from the previous ones, but cannot be perceived (it would be interesting indeed to think about this observation from the point of view of local and global properties in the sense of differential geometry). So, the main conjecture of this paper is the following: Lacking of experiences in sections (and projections) is strictly related to problems of blind students (and persons) about the perception of geometric (mostly, but not only, 3-dimensional) facts. So, coming to the subject of this paper, the question becomes: assuming this conjecture as correct, how to design tools specifically planned to offer blind children experiences to replace those experience they miss? To the best of my knowledge there are no manipulatives focused on offering this kind of experiences. Sometimes blind children are asked to cut some soft material, but nothing else. So, I started a project to make a collection of manipulatives, tools, tactile exhibits to convey in an haptic form the section experiences I told before. The idea is to build three-dimensional puzzles, corresponding to a dihedral angle (represented as a prism over a triangle, with a raw irregular surface where it “continues”), a circular cylinder, a circular cone, a pyramid, but also some irregular shapes (for example, an egg, and cylinders and cones over irregular basis); each puzzle will be contained (whenever it will be possible) in a cardboard box, to be fulfilled by the pieces of the puzzle. Each puzzle will be in two copies. One of them will be cut by parallel planes, while the cuts of the other one will correspond to planes passing through a line; in both cases it will be possible to study the shape of planar sections. The cardboard boxes of several different dihedral angles will form a new puzzle themselves, in order to fulfill a whole 2π radians angle. Through this collection of puzzles, it will be possible to point out (in an haptic form) that the same section can belong to different shapes and that the sections of a given object can be rather different between themselves. Moreover, through the use of cones and pyramids, it will be possible to explain what “to see” means, and to offer an evidence explaining, for example, why circles are drawn usually as ellipses (so difficult to understand for blind students, as pointed out, for example, in Del Campo). More generally, perspective and drawing tecniques are quite an abstract subject to blind students, and it seems difficult to try to explain them in an “intellectually honest way”, in the sense of Bruner hypotesis. An attempt in this direction was made by Piochi and Baldeschi, 2004, by a rather complex manipulative tool . I am trying here a different attempt, through a small exhibit which is under construction, in two parallel tiny versions: one will be the usual thread model about the drawing of a cube, seen by a given point of view, over a plane representing “the plane of the picture”; the second will represent, in a “solid version” the same scene, with the same sizes, in order to offer also an haptic perception of the visual cone embracing the cube, and of its section. The objets described, despite the difficulties in realizing them, are alredy planned; a different question to think about, to which I have no proposal right now, is how to experience haptically the fact of a concave-convex puzzle fitting perfectly? In other words, what are the sighted experiences that allow to recognize, for example, that a given solid perfectly fits a given hole?

Is this work worth trying? I suppose yes, and like to remind the story of Mademoiselle Melanie de Salignac, blind from the birth, able to master geometric facts, as told by Diderot in his “Lettre sur les aveugles”:

“Je lui disais un jour: «Mademoiselle, figurez-vous un cube. - Je le vois. - Imaginez au centre du cube un point. - C'est fait. - De ce point tirez des lignes droites aux angles; eh bien, vous aurez divisé le cube. - En six pyramides égales, ajouta-t-elle d'elle-même, ayant chacune les mêmes faces, la base du cube et la moitié de sa hauteur. - Cela est vrai; mais où voyez-vous cela? - Dans ma tête, comme vous.” REFERENCES Fernández del Campo, J.E.: 1986, “Enseñanza de la matemática a los ciegos” , ONCE, Madrid. Kohanová, I.: 2007, “Teaching mathematics to non-sighted students: with specialization in solid geometry”, Doctoral Thesis (Comenius University Bratislava), Quaderni in Ricerca in Didattica del G.R.I.M. N. 17 Supplemento n.5, in http://math.unipa.it/~grim/quad_17_suppl_5.htm . Piochi, B., Baldeschi, M.: 2004, “Sussidi didattici per l'introduzione della prospettiva e della geometria proiettiva con alunni non vedenti”, in “Insegnare la matematica nella scuola di tutti e di ciascuno”, Ghisetti e Corvi editori. Russo, D.: 2001, “L'insegnamento della matematica ai ciechi”, Tiflologia per l'integrazione 2, in www.bibciechi.it/pubblicazioni/tiflologia/200302/Russo.rtf . Tindell, M.: 2006, “Technology and Life Skills - A Beginner’s Guide to Access Technology for Blind Students: Part Two”, Future Reflections 25, No. 2, in http://www.nfb.org/Images/nfb/Publications/fr/fr22/fr06sumtc.htm . Von Prondzinsky, S.: 1998, "Premesse psicopedagogiche e presupposti didattici per l’uso degli strumenti informatici nelle attività logico- matematiche", in http://www.handimatica.it/Handi1998/Convegni98/matematica/C98stefa.htm .