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Digital Mathematical Performance:
Creating digital narratives in mathematics education.
Ricardo Scucuglia
Grupo de Pesquisa em Informática, outras Mídias e Educação Matemática
Universidade Estadual Paulista – UNESP, Campus Rio Claro, SP
Marcelo de Carvalho Borba
Universidade Estadual Paulista – UNESP, Campus Rio Claro, SP
Departamento de Matemática
Abstract
In this text, we present the foundations and procedures for a mini-course to be
administered during the IX National Mathematics Education Meeting (Encontro
Nacional de Educação Matemática – ENEM – 2007). Basically, participants will have
the opportunity to investigate and create videos using images (figures, photos) that
contextualize situations (narratives, stories, fables) involving mathematical problems.
The program we will use is Photo Story 3 for Windows1, and the target audience will be
elementary and high school teachers as well as university professors. The videos
produced are considered digital mathematics performances, as they are elaborated by
bringing together mathematics and arts, illustrating how different on-line media can
condition modes of production of knowledge. When one seeks convergence between
mathematics education, information technology, and the arts, different collaborative
possibilities emerge for acting in virtual educational contexts.
Key words: mathematics education, computers, videos, digital performance (arts)
1 It will also be necessary to use the program Windowns Media Player 10 to visualize the videos. Participants will have access to the Internet to select the images that will compose their videos. Speakers and microphones will be connected to the computers as well, since the videos have audio effects.
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Introduction
In this text we present some guidelines and perspectives on a mini-course that
will be administered during the IX National Mathematics Education Meeting (Encontro
Nacional de Educação Matemática – ENEM – 2007). Participants will have the
opportunity to investigate and create videos using images (figures, photos) that
contextualize situations (narratives, stories, fables) involving mathematical problems.
We will begin by discussing the theoretical perspective of Humans-with-Media and our
conceptualization of digital mathematics performance. This will be followed by a
presentation of two videos that will be investigated in the mini-course and the
procedures participants can use to create their own videos, i.e., their own digital
narratives of mathematical problems. Finally, we will develop some discussions
involving mathematical thought and the arts in virtual environments.
Humans-with-Media
Tikhomirov (1981), influenced by Vygotsky, proposes that computer and information
technology (CIT) neither substitutes nor merely complements humans in their
intellectual activities; rather, the processes mediated by the interfaces and tools of CIT
reorganize thinking. The process of digital mediation makes it possible to structure, in
diverse ways, the elements that constitute thought, (re) attribution of meanings,
learning, understandings, developments, knowledge.
Lévy (1993), in his discussion of Technologies of Intelligence, emphasizes that
orality, the written word, and CIT can be understood as technologies that condition the
“temporality” of humanity, calling to mind the following metaphors for memory:
circular, with orality; linear, with the written word; and hypertextual (web), with CIT.
Thus, as the written word reorganized orality as an intellectual technology, CIT
concentrates and potentiates the systems that preceded it: language, numerations,
alphabets, ideographs, etc. (LÉVY, 1998).
Is technology an autonomous actor, separated from society and culture which are only passive entities touched by an exterior agent? I defend, on the contrary, that the technique is an angle of analysis of global socio-technical systems, a point of view that emphasizes the material and artificial part of human phenomena, and not the real entity which exists independently of the rest, that has distinct effects and acts according to its own will. (LÉVY, 2000, p. 154 - Our translation)
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In harmony with these perspectives, the theoretical construct Humans-with-
media (BORBA; VILLARREAL, 2005; BORBA, 2004; BORBA; PENTEADO, 2002)
emphasizes the role of technologies in the process of (mathematical) knowledge
production, providing evidence that thinking is not conditioned by a human being, or
groups of human beings, alone, but by collectives composed of humans-with-media: Knowledge is not produced by humans alone, but also by non-human actors. Technologies are human products, and are impregnated with humanity, and reciprocally, humans are impregnated with technology. In this sense, the knowledge produced is conditioned by the technologies. (BORBA, 2004, p. 305).
Discussions emerge that focus on new characterizations of mathematics and the
ways the dimensions of teaching this discipline are engendered with the use of CIT.
Models emerge of the contexts where CIT conditions the production of mathematical
knowledge. Different CIT media, from calculators to on-line applications, condition,
characterize, and shape the way (mathematical) knowledge is produced.
In the educational context, CIT, its dimensions, and its plasticity (re)define the
roles of the human and technological actors involved in (mathematical) thought. The
teacher can create a collective context of questioning based on a variety of information
and potentialities contained in programs and hypertexts. (Graphing) calculators and/or
computers do not provide students with problems to be resolved; rather, students-with-
technologies (and other human-technological actors / collectives of humans-with-
technology) shape and resolve problems and think (mathematically) (SCUCUGLIA,
2006).
Thus, it is our belief that collectives of humans-with-media, for example, can
transform mathematics and reorganize the form of (on-line) collaboration between
teachers (BORBA; ZULATTO, 2006). More recently, we have been investigating how
the Internet modifies and brings together mathematics, education, and the arts
(performing arts, music, poetry, and others).
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Digital Mathematical Performance2
The idea of digital mathematical performance (GADANIDIS, 2006) is being
developed in an international project coordinated by George Gadanidis3 and Marcelo C.
Borba. A theoretical framework for this approach has been developed as well as
practical examples of what can be considered a digital mathematical performance.
In the symposium Digital Mathematical Performance (A Fields Institute
Symposium, 2006 - Faculty of Education – University of Western Ontario – Canada),
specific discussions were elaborated and illustrated among mathematics educators from
different countries seeking to bring arts and mathematics education closer together4.
Currently, with the financial support of the Social Sciences and Humanities Research
Council of Canada (SSHRCC), the Digital Mathematical Performance project has permitted
theoretical and practical convergences between researchers and two countries in particular:
Canada and Brazil. Information about the project, the participating researchers, digital
performances that have been developed, interviews with teacher-artists, and scientific papers
produced can be found at http://www.edu.uwo.ca/dmp/.
2 The term performance transcends the context of its genesis in the arts, where the human body is the main instrument of design, painting, or artistic expression. Performance Art can be conceived of as a border region constituted of convergences among expressions that involve and emphasize the aesthetics of communication-interaction between performers in the performance space-time of creations and productions (theatrical, musical, poetic, etc.) (http://en.wikipedia.org/wiki/Performance_art). It has also been characterized in fields such as sociology, anthropology, and others (GEROFSKY, 2006). 3 University of Western Ontario, Canada. 4 • Performance happens in the theatre, at poetry readings and on the screen. What would happen if mathematicians and math educators moved outside of the domain of assessment (where performance takes on a different meaning), and used an artistic lens to ‘perform’ mathematics? If we view mathematics (doing, teaching, learning) as embodied performance, what will it look like and how can we talk about, and better understand, it? How might we express and further develop mathematical concepts physically through drama or virtually through multimodal digital tools? • Thinking of mathematics and mathematics teaching and learning as performance may help to destabilize and reorganize our thinking about what it means to do and teach mathematics with technology. (http://www.edu.uwo.ca/mathstory/index.htm).
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Initiating the Mini-Course
Based on these considerations about the Digital Mathematical Performance
project, the possibility arose to propose a mini-course to be offered during the IX
National Mathematics Education Meeting (Encontro Nacional de Educação Matemática
– ENEM – 2007), in which participants would have the opportunity to investigate and
create videos using images (figures, photos) that contextualize situations, narratives,
stories, or fables involving mathematical problems. The target audience will be
elementary and high school teachers as well as university professors, who will then be
able to direct their activities to the educational contexts they consider most appropriate.
The first activity proposed will be exploration of the Digital Mathematical
Performance site where participants can investigate, collectively in groups, various
digital mathematical performances. Two videos in particular, created with the program
Photo Story 3 for Windows5, will be emphasized: The Problem of the Waiter and The
Problem of the Camels.
The Problem of the Waiter
A specific author for The Problem of the Waiter is not currently known. Like
Valladares (2003), we consider this problem to be a “Mathematical Fable”, which can
be told as follows:
The Problem of the Waiter
Once upon a time, a person sitting in a bar … observed an interesting occurrence! Three friends were having a conversation at one of the tables. They were talking about parties, work, and other things. After a while, one of them asked for the check. Right away, the waiter came and gave them the check. The friend looked and said, “The bill is $30.00!” Each friend gave $10.00. The waiter went to the cash register, and the owner of the bar decided to give a $5.00 discount to the group of friends . . . However, before giving back the $5.00 . . . the waiter had an idea. He decided to keep $2.00 for himself.
5 Photo Story is a free application developed by Microsoft to create personalized “slideshows” with digital images (mainly photos). In a dynamic manner, it is possible to drag, rotate, and organize the order of the images, and add special effects, legends, soundtracks, and even its own narration. The program even corrects imperfections in color, light, and contrast, removes red eyes, and adds special effects and, finally, creates a Windows Media Video (WMV) extension file to exhibit the presentation.
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“I will give back only $3.00!” Each friend received $1.00. They were happy with the discount! Each ended up paying just $9.00. But the person observing all this was puzzled; If each friend paid $9.00 … … then they paid a total of $27.00. $27.00 plus the $2.00 that the waiter took is equal to $29.00. But the friends had given $30.00! One dollar was missing! Where is the one dollar that is missing?
As will be emphasized during the mini-course, to present the above fable,
researchers with the Digital Mathematical Performance project created the images
(photos) used. With the participation of various people (invited colleagues-actors) a
script was created and the scene set up in a bar. In other words, in this presentation with
Photo Story, the theatrical (dramatic) dimension was emphasized, which intensified the
artistic dimension of the digital mathematical performance created.
Convergences between theater and education inevitably emerge: From an epistemological point of view, some time ago, the foundations of theater in education were conceived of based on directed questions or formulated by psychology and education, indicating the path to follow. Today the history and aesthetics of the theater provide contents and methods to guide educational theory and practice. We can say that (…) specialists in various fields and at various levels of teaching – from pre-school to the university – seek the contribution that the field of theater can bring to education (KOUDELA; SANTANA, 2006 – our translation).
Koudela and Santana (2006) comment that, in recent years, the expression
Pedagogy of the Theater has been used to describe an approach that investigates the artistic
language of the theater and its insertion into various levels and modalities of teaching.
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Simultaneously, this movement between theater and education embraces conceptions and
tensions related to epistemological, historical, and cultural nuances.
Some studies along these lines conceive of theater as a cultural action.
Contemporary social problems, such as the environment and violence, have emerged as
themes in work carried out with children and teenagers. These projects have been
developed mainly as part of university research and extension activities and have enabled a
broadening of the discussion of cultural policies in the search for and continuation of
support to deal with contemporary social problems through theater and art, while at the
same time showing concern for the criteria established to guide the projects, including those
related to the continuing education of the teachers themselves. This perspective may enable
the Digital Mathematical Performance project to transcend the status of traditional
education, which is less interactive and more uni-directional. In fact, we understand
digital mathematical performance as one path to using the Internet in the classroom and
virtual learning environments.
The Problem of the Camels6
The Problem of the Camels was originally proposed by Malba Tahan7 and was
adapted by researchers of the Digital Mathematical Performance project:
Once upon a time, two friends decided to cross a desert. But the friends, who traveled with only one camel, came upon three brothers. The brothers did not know how to fulfill final wishes left by their father who requested that his 35 camels be divided in the following manner: One half (1/2) for the oldest brother; One third (1/3) for the second brother; One ninth (1/9) for the youngest brother. It was in the division of the inheritance that the problem emerged! The divisions did not result in whole numbers! But the two friends had an idea “to solve the problem”! First, they donated their camel to the brothers. The brothers then had a total of 36 camels. Now, in addition to the divisions resulting in whole numbers, the brothers now had more camels than before! Still, there were two camels left over, since 18 + 12 + 4 = 34. Finally, the brothers, to show their appreciation, donated the two camels to the friends.
6 There is an excellent documentary entitled The Story of the Weeping Camel. Nominated for the Oscar in 2004, this film was created by Byambasuren Davaa and Luigi Falorni and won awards for best documentary in festivals in Buenos Aires, Bavaria, Indianapolis, Karlov Vary and others (For more information, acess http://www.nationalgeographic.com/weepingcamel/index.html.) 7 http://www.geocities.com/g10ap/matematicos/mat27.htm; http://www.champ.pucrs.br/matema/malba_tahan.htm; http://www.matematicahoje.com.br/telas/cultura/historia/educadores.asp?aux=A; http://www.mat.ufrgs.br/~portosil/malba.html
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The friends thanked them and continued on their way. Everyone benefited in this story: (a) the friends ended up with one more camel; (b) the brothers ended up with more camels than before. Why did the brothers and the friends benefit from the solution proposed?
Although the digital mathematical performance of The Problem of the Camel
was elaborated using existing photos and images, the humorous dimension of the
resulting video stands out, with the two friends in Malba Tahan’s story represented by
the acclaimed duo Laural and Hardy, and the three brothers by the Three Stooges.
Some images from the story are presented below:
The participants in the mini-course will be asked to explore, in groups, both
digital narratives (The Problem of the Camel and The Problem of the Waiter), seeking to
discuss the following questions: (a) What are some possible solutions to the problems
presented? (b) In what teaching-learning situations could these videos be used? (c)
What would you change in the videos? What is your opinion of them? (d) Do you
consider the videos to be “good” examples of digital mathematical performance? (e)
What criteria would you use to make this judgement? Additional questions are expected
to emerge from the participants’ investigations and discussions during the mini-course,
as well.
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Creating Digital Narratives in Mathematics Education
After investigating the Problem of the Waiter and of the Camels, the participants
will be invited to create a video using Photo Story that represents a mathematical
problem. Initially, each group of participants will be asked to define a problem to be
represented then develop a script of the presentation (for example, select from the
Internet or create images to be used, define in what person the presentation will be
narrated, whether the video will be dramatic, humorous, etc.)8. Following these initial
procedures, experimentation with Photo Story will be initiated, based on five steps:
(a) Inserting (in order) and editing images.
(b) Inserting and editing legends: images may be re-edited.
8 There may be one obstacle to the creation of images: it may not be feasible, as the participants will not be provided with digital cameras. In addition, the creation of a script that involves the creation of images may not be possible in the limited time of the course.
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(c) Editing transitions between images (effects, timing) and inserting narration
(voice).
(d) Selecting or creating music.
(e) Finalizing presentation: generating a.wmv file .
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At the end of the mini-course, each group of participants can present their video,
even if not entirely finished. Through the discussions that take place, various
theoretical references may be articulated and proposed9 that provide foundations for the
digital mathematical performances created as well as the scientific proposals of the
Digital Mathematical Performance project. In addition, with the permission of the
groups, some videos may be made available for viewing on the project’s site.
Cyberart and Experimentation with On-Line Technologies
The investigative process carried out by students in a course using CIT media
and fundamental mathematical concepts can be denoted experimentation with
technologies. This process includes: the use of trial-and-error and educational
processes that enable students to generate conjectures; the possibility of testing a
conjecture using a large number of examples; the availability of different types of
representations; the possibility of discovering mathematical results not known before
the experimentation. Consequently, visualization can be considered a fundamental
element in the thought process and the production of knowledge and mathematical
meanings. Its use conditions information and shapes the (re)elaboration of conjectures,
understandings, and justifications (BORBA; VILLARREAL, 2005).
With the various on-line applications now available, the possibilities for
experimentation are increasing in number and intensity, from the perspective of
communication, including collective experimentation and development. A distance
course in geometry based on a synchronous application, for example, as described by
Borba and Zullatto (2006), made possible intense interaction among the participants,
which led, among other things, to the collective development of a geometry test,
structured based on argumentations inferred (contributed) by various participants: a
collective intelligence context. This diversity of on-line applications and breadth of
communication offered by the Internet has also made possible, in a different way, the
exploration (and creation) of mathematical contents based on (artistic) performances.
9 Humans-with-Media (BORBA; VILLARREAL, 2005); Design (NORMAN, 2004); The Hollywood Eye (BOORSTIN, 1990); Literacy and Multimodality (KRESS, 2003); Philosophy, Mathematics and Poetry – Wisdom & Metaphor (ZWICKY, 2003).
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In addition to Photo Story, another application (involving flash language) that we
have been using in the Digital Mathematical Performance project is Digital Storytelling.
As Gadanidis (2006) explains, he first began to use this on-line application to create digital
mathematical performances inspired by research conducted by Janette Hughes, who was
using Digital Storytelling to create digital poetic performances.
The dynamic character of digital mathematical performances, the various links
available, and the videos and simulations inherent to the application provide a propitious
environment for intense experimentation with (on-line) technology. This intensity in the
experimentation conditions the way in which mathematical contents can be approached or
discussed, implying emergent possibilities related to the production of meanings and
knowledge. Inevitably, the status (process) of visualization, the way problems are posed
(open-ended), the possibility of repeating previous explorations, and the performing artistic
dimension characterize diversified educational demands. In addition, the mathematical-
artistic applications are constantly being updated.
Lévy (2000) uses the term CyberArt to discuss the artistic-aesthetic dimension of
Cyberculture, which could be understood as a set of techniques (material and
intellectual), practices, attitudes, ways of thinking, and values that develop in
cyberspace. Lévy (2000), emphatically analyzing the configurations of communication
and interaction that emerge in this context, clearly shows the possibilities for
participation/collaboration and the continuous-collective character of the process of
artistic creation as a fundamental characteristic of CyberArt.
Both collective creation and the participation of the performers walk side by side as a (...) special characteristic of cyberart: continuous creation. Virtual art is “open” for construction. Each update reveals a new aspect to us. In addition, some devices are not content to refuse but give rise to the emergence of absolutely unforeseeable forms. Thus the event of creation no longer finds itself limited to the moment of conception or the realization of the work: the virtual device proposes a machine for making the event emerge (LÉVY, 2000, p. 136 – our translation)
To explore diverse mathematical questions, the on-line applications used in the
Digital Mathematical Performance project compile simulations, videos,
argumentations, explanations, etc. In this sense, they enable and condition modes of
thinking; humans-with-Internet, upon delving into webs of meaning, can produce
mathematical knowledge involved in artistic performances, created in a context that
amalgamates thinking collectives and collective intelligences.
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The technological conditioning of human productions, in the arts and in
education, and the intensity of interactions in the context of (digital) performance, make
explicit reciprocal and collective configurations between humans and technologies.
Actors immerse themselves in hypertext; different cinematographic resources condition
different feelings (BOORSTIN, 1990). Shaped by the design of CIT tools and interfaces
(NORMAN, 2004), actors can interact (intensely) with spectators and with other (digital)
actors. Reciprocally, on-line education CIT can be shaped based on understandings
regarding possible interactive optimizations - ways to optimize interaction between
teachers and students10.
References
Boorstin, J. (1990). The Hollywood eye: What makes movies work New York: Cornelia & Michael Bessie Books.
Borba, M. C., Penteado, M. G. (2003). Informática e Educação Matemática. 3. ed. Belo Horizonte: Autêntica (Tendências em Educação Matemática).
Borba, M. C. (2004). Dimensões da Educação Matemática à Distância. In: Bicudo, M. A. V. & Borba, M. C. (Org.) Educação Matemática: pesquisa em movimento. São Paulo: Cortez.
Borba, M. C. & Villarreal, M. (2005). Humans-with-Media and Reorganization of Mathematical Thinking: Information and Communication Technologies, Modeling, Experimentation and Visualization. USA: Springer. (Mathematics Education Library).
Borba, M., Zullato, R. (2006). Different media, different types of collective work in online continuing teacher education: Would you pass the pen, please? In: Proceedings of the PME 30, Jarmila, N., Moraová, H, Krátká, M, Stehlíková, N., eds., 2: 201-208. Charles University, Faculty of Education. Czech Republic, Prague.
Gadanidis, G. (2006). Exploring Digital Mathematical Performance in an Online Teacher Education Setting. The Society for Information Technology and Teacher Education 17th International Conference, Orlando, Florida.
Koudela I. D., Santana, A. P. (2006). Pedagogia do Teatro & Pedagogia e Educação. In: Carreira, A. et al. (Org.). Metodologias de Pesquisa em Artes Cênicas. Rio de Janeiro: Editora 7 Letras.
Kress, G. (2003). Literacy in the New Media Age. London: Routledge, 2003.
Lévy, P. (1993). As Tecnologias da Inteligência: O futuro do pensamento na era da informática. Rio de Janeiro: Editora 34.
10 O propósito do trabalho artístico se desloca para o acontecimento, ou seja, em direção à reorganização da paisagem de sentido que, fractalmente, em todas as escalas, habita o espaço de comunicação (...). Ocorre alguma coisa na rede dos significados, assim como no tecido humano (LÉVY, 2000, p. 154).
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Lévy, P. (1998). A Máquina Universo: Criação, Cognição e Cultura Informática. Porto Alegre: Artimed.
Lévy, P. (2000). Cibercultura. São Paulo: Editora 34.
Norman, D. (2004). Emotional Design: Why we love (or hate) everyday things. New York, Basic Books.
Scucuglia, R. (2006). A Investigação do Teorema Fundamental do Cálculo com Calculadoras Gráficas. 1v. Dissertação (Mestrado em Educação Matemática) – Instituto de Geociências e Ciências Exatas, Universidade Estadual Paulista, Rio Claro.
Tikhomirov, O (1981). The psychological consequences of the computerization. In: Werstch J. The concept of activity in soviet psychology. New York: Sharp.
Valladares, R. J. C. (2003). O jeito matemático de pensar. Rio de Janeiro: Editora Ciência Moderna.
Zwicky, J. (2003). Wisdom & Metaphor. Kentville, NS: Gaspereau Press.
http://en.wikipedia.org/wiki/Performance_art
http://www.edu.uwo.ca/mathstory/index.htm
http://www.edu.uwo.ca/dmp/.
http://cinema.uol.com.br/dvd/2006/03/07/camelos_tambem_choram.jhtm.
http://www.webcine.com.br/filmesso/gesckame.htm
http://www.geocities.com/g10ap/matematicos/mat27.htm;
http://www.champ.pucrs.br/matema/malba_tahan.htm;
http://www.matematicahoje.com.br/telas/cultura/historia/educadores.asp?aux=A;
http://www.mat.ufrgs.br/~portosil/malba.html