Mathematics Learning in Virtual Learning Environments Márcia Rodrigues Notare 1,2, Patricia Alejandra Behar 1 1 Information Technology in Education Post Graduate Program - UFRGS 2 Universidade do Vale do Rio dos Sinos - UNISINOS Key words: e-learning, educational virtual environments, collaborative learning Abstract: The present article aims to discuss the learning of Mathematics in virtual learning environments. It is known that the main focus of online learning is communication mediated by interaction tools available in virtual environments; however, mathematical communication is so far insufficient because virtual learning environments do not offer edition resources for scientific expressions. Thus, this article presents the scientific editor ROODA Exata and its use by a group in a Differential Calculus discipline. 1 Introduction Problems inherent to the process of learning and teaching Mathematics have been experienced by teachers and students throughout the years. On one hand, students have real difficulties to understand Mathematics in its true essence as they see it as a set of rules, ready-made formulas, algorithms and definitions to be memorized. On the other hand, teachers realize and acknowledge the difficulties presented by students, they are aware of the problems presented by traditional teaching methods and they feel challenged to face the issue. Math can be understood as a form of precise, clear and objective language, and its comprehension implies, among other things, dominating this language. So, to understand Math, it is necessary to be able to express oneself correctly through this language because it is through communication and expression exercises that thought becomes concrete and reaches higher levels of understanding. Thus, the process of math knowledge construction should look for means that enable students to face situations that demand communication and expression, valuing the practice of argumentation and justification. We are now witnessing a new way of interaction with the emergence of communication networks. Such networks offer a new way to communicate and socialize and more room is made to develop new ways to teach and learn, with greater flexibility regarding time and space. Communication processes, in this context, depend on computers. And computers offer the opportunity to share and build knowledge, to share information and abilities. So far, these tools of virtual communication are predominantly written in that messages and replies are so. Thus, the integration of TICs to process of teaching and learning Math may favor the construction of Mathematical knowledge. That is because, from its use, it is possible to value action, communication and expression, important factors in this process. ICL 2009 Proceedings - Page 464 1(8)
However, current virtual learning environments do not offer the necessary resources to a smooth scientific online communication that allows the utilization of symbols and markings that are typical of the area in an effective, intuitive and friendly way. In this context, ROODA Exata was developed as an editor for scientific communication online, as a functionality integrated with different resources for interaction and communication offered in the virtual learning environment ROODA (Cooperative Learning Network) [1]. 2 Building Mathematical knowledge According to Machado, mathematical language has come up to allow the exercise of reason whose grammar permits precise expression, not allowing for dubious interpretation [2]. It is known that through reading and writing we are able to communicate. However, reading and writing not only mean understanding our mother language but also all forms of interpretation, explanation and analysis of the world. Mathematics is one of these ways, with its codes and language, in other words, with its own system of representation and communication. Understanding Math cannot be reduced to manipulating operational techniques, in a mechanical way, neither to memorizing formulas, rules and properties. Understanding Math means understanding what one reads and writes, attempting to grasp all the meanings of it. To understand Math it is not enough to know how to read, write and count. It is necessary to know how to express oneself because the expression helps achieving concrete thought, forcing the subject to order mental images creating the need to master appropriate vocabulary consisting of Math symbols. When one is able to express oneself, arguing about a theme or concept, he or she is at a higher level of understanding if compared to one who can only solve a problem numerically, through the use of a formula, rule or equation. Therefore, in Mathematical learning, it is important to incentive students to think and express their thoughts, either speaking or writing, so as to justify their ideas and reflect upon their concepts. If a subject is able to express his/her ideas about a certain theme, this is a sign that he/she is reflecting, or in a process of thought coordination [4]. Mathematical communication has been done, basically, in written form. Natural spoken languages can describe mathematical objects and their properties but it is the symbology that allows the same property to be described in a fast, precise and straightforward manner. Thus, when structural properties become more complex, their description becomes harder to be spoken and understood without the use of symbols that simplify and facilitate Math, allowing for clarity and quickness at the expression of ideas. For students, however, such symbolism may be difficult to understand. It is important that the student should argue about his solution and reflect upon ideas and concepts he has already learned, in a way to promote a re-organization of his knowledge. If the student manages to solve a problem analytically, he should also manage to justify his choices and procedures and analyze the results obtained, reflecting upon and establishing relationships among concepts. Very often students are able to solve problems and equations correctly but we realize they are not able to justify their procedure nor argue about what was done, how it was done and why it was done. We perceive, in such cases, how knowing how to do doesn t mean understanding what is done. These students can solve a problem, they find a solution for it but don t ICL 2009 Proceedings - Page 465 2(8)
understand what they have really done and, many times, they cannot even produce an interpretation for the solution they found. Situations like this show us that calculus are made in a mechanical way, meaninglessly; therefore, conceptualization is missing. In Math, situations like that are frequently observed when problems with different representations are presented and there is no way to solve them following a ready-made model, when understanding of the concepts involved is called for. For Piaget (1978, p. 179), to understand means to isolate the reason for things while to do means to use them sucessfully [5]. 3 Math Education and Technologies of Information & Communication Technological advances have been modifying world inter-relation dimensions. Computer networks have modified the concept of time and space as individuals who live far away and isolated can be connected to big research centers, libraries, other professionals of their area as well as to several services. A new way of communication comes up in the virtual world where people, upon their interaction, explore and update a common memory, configurating what we call a space of collective creation and intelligence. In the educational context, it is possible to integrate classroom moments with extra-class virtual learning moments through the Internet, allowing students to expand their learning moments otherwise restricted to presential moments. Thus, the teaching learning process is not limited to work done in classrooms as the Internet opens a wide spectrum of possibilities that enables flexibility and evolution from the presential class mode. The virtual environment is a great ally that makes communication and long distance contact easier. Virtual interaction is important for the learning process: it is possible to debate topics that are being studied or to use the space to solve doubts and consolidate concepts. One of the main contributions of semi presential or virtual courses is the active learning which implies social and cognitive commitment. To be part of these courses, one needs to give opinions, reply to colleagues and share ideas because a student is only socially online when he/she makes a comment. Active participation leads to learning because the act of writing ideas and information demands intellectual effort and helps both understanding as well as retention. To formulate and articulate statements is a cognitive action and constitutes a valuable process of human development. To make comments, students need to organize ideas and thoughts coherently and that is intellectual work. Besides, when information and ideas are published in forums or discussion lists, they can trigger new answers, as upon clarification requests, deeper development of the idea or even disagreements. These exchanges make authors of messages more accountable for their ideas as they have to improve their concepts or re-evaluate them in a process of cognitive reconstruction. Thus, ideas are developed interactively having motivated reflection, interaction and, consequently, knowledge construction. Learning to teach and to learn in this context, integrating presential and virtual, is one of the greatest challenges education is facing nowadays. Regarding the role of the teacher, it changes the relation of space, time and communication with students. Exchanges and interactions go beyond classroom to virtual as well as the time for these exchanges and interactions that expand to any time or hour. ICL 2009 Proceedings - Page 466 3(8)
However, to ensure that a student is successful in a virtual course, he/she must be self motivated and be self disciplined because the online environment is free and parallel to this freedom, there should be responsibility, commitment and discipline. Moreover, a student of an e-learning course must know how to work together with peers to reach learning objectives and course objectives. Knowing that the teacher is a mere facilitator, the student takes charge of his learning process. However, Math learning and communication processes have been impaired. One of the reasons for such situation may be related to the limitations of the e-learning environments that have not yet brought about enough resources to promote quality interaction in the scientific area. What we know is that natural language alone is not sufficient to promote a mathematical conversation once it is formed by its own symbols that are necessary to express ideas and concepts in a precise way. Smith, Ferguson and Izubuchi highlight that virtual learning environments have emphasized written communication, through natural language, to promote debates and discussions but that these environments do not supply tools that enable mathematical communication, something vital for the process of learning of this discipline [7]. Mathematical communication takes place through scientific markings; due to the lack of such tools in virtual environments, communication becomes difficult, requiring attached files that break the flow and naturalness of communication. 4 Scientific Editor ROODA Exata In this context, ROODA Exata [3] was projected and developed as an editor for scientific communication. One of the main requirements considered for the development of ROODA Exata was to conceive it so as not to need language for formatting and marking so its use could be transparent and intuitive to the user, following usability criteria. Thus, interaction in the editor is made through icons and buttons that allow the insertion of symbols and formulas through a simple mouse click. ROODA Exata has three tabs: symbols, formulas and Greek alphabet, as illustrated in Figure 1. The tab with symbols has the most frequently used symbols of exact science communication, such as relational symbols, operators, arrows, logic symbols, number groups symbols, number groups, subscript and superscript, sum, product and integral, among others. The tab containing formulas has the main Math, Physics and Chemistry formulas and was elaborated to reduce the effort of the user in communicating, enabling faster communication as the most commonly used formulas can be inserted with just a click. Finally, the Greek alphabet tab, which contains capitals and lower-case Greek alphabet letters, can be widely used in communication and scientific expression. Figure 1. ROODA Exata editor tabs ICL 2009 Proceedings - Page 467 4(8)
The design of the editor of formulas was structured with tabs so as to follow the pattern of the virtual learning environment ROODA where it is. The agreeable interface of ROODA allows intuitive and fast navigation. Its concept was based on the concept of interaction design which means computational systems are created to optimize or facilitate routine activities providing solutions to users. ROODA Exata is available in the discussion forum and chat tools and to access it, just a click on the access button is enough. Math expressions are built through the editor s buttons, illustrated above in Figure 1. For example, if the user wishes to insert a fraction in the text, a click on the button y x will do it. An editor box is open which allows the insertion of variables wanted. It is worth highlighting that new structures can be inserted in the fraction, allowing composition, for instance, of fractions with radicals, as it is illustrated in Figure 2. This shows the potential of ROODA Exata in the edition of mathematical expressions with complex structures. Figure 2. Edition of mathematical expression 5 Analysis of ROODA Exata: An Experience with Differential Calculus To analyse the scientific editor developed, the virtual learning environment ROODA was used as a support tool to a presential class of Differential Calculus whose students were encouraged to use ROODA as an extra class support area for interaction and communication and for the posting of doubts and for the solution of problems. From students participation in the environment, it was possible to analyse the feasibility of long distance mathematical communication, according to Piaget s theory of genetical epistemology [4,5,6]. There were initially thirty five students enrolled in the discipline. Twenty nine of them registered to take part in extra class work and to have follow ups in the ROODA environment. The activities were proposed through the discussion forum tool and the scientific editor ROODA Exata was used according to the need to use mathematical expressions. To follow, a debate sparked among some classmates around the issue of derivative calculus through differentiating rules is presented, as shown in Figure 3. Figure 3. Problem about differentiating technique Student ANT presented his solution to a problem, as illustrated in Figure 4, but failed to explain his reasoning and that reveals the first level of awareness, in other words, his understanding in action (supported by what can be observed) and not in thoughts [6]. His classmate REN presents his solution to the same problem, reaching a different result (Figure 5), as he did not use the rule of the product in the differentiating process. It is possible to ICL 2009 Proceedings - Page 468 5(8)
realize that REN is a little skeptical about his result when he states now, how come my result is different from that of our classmate ANT?. We can see that REN does not know yet the reasons for his actions, or it seems he just acts even if he does not know why he is acting in such way, without knowing if he has been successful in his result. Conversely, his participation shows he understands the learning environment as a space to overcome obstacles, face problems, elucidate doubts, all in all, a learning place. Classmate ADE compares the solutions presented by ANT and REN and identifies the mistake made by REN, as shown in Figure 6. It is clear from their interaction that this is a space for collective learning where students, upon reflecting about a problem, search for clarification and justification that lead them into a successful resolution of the problem. Figure 4. Solution lacking argumentation Figure 5. Wrong solution Figure 6. Classmate intervention The teacher tries to answer REN s questions in order to encourage him to reflect about and revise his solution (Figure 7). After analyzing the comments made, REN re-organizes his ideas, understands where he was wrong and shows he has understood the rule of the product, as shown in Figure 8. Figure 7. Teacher s comment Figure 8. REN s reflection The situation described above shows a sequence of contributions that led to the correct resolution of the problem and, above all, led to the awareness of the rule of the product by REN, who initially seemed not to understand the problem presented but, by following and participating in the debate that was started, overcame his difficulties. Thus, these contributions have shown that it is possible to have a long distance mathematical dialogue ICL 2009 Proceedings - Page 469 6(8)
establishing this way a collective construction where participants interact searching for a common goal: the understanding of the rules of differentiation. Such dialogues leave no doubt regarding the value of the virtual learning environment ROODA to the construction of mathematical knowledge. It is important to highlight, however, that this dialogue was only possible because of the use of the scientific editor ROODA Exata, which enabled the edition and publication of mathematical expressions in a clear and transparent way. The analyses of students participation in the virtual learning environment ROODA with the help of ROODA Exata were developed more deeply and, in this article, we presented a situation of collaborative learning to illustrate the potentialities of the editor ROODA Exata. 6 Students perception In the end of the school year, students had a questionnaire to answer to provide feedback on what their perception was about the activities made at ROODA and about the utilization of the scientific editor ROODA Exata. Their answers confirmed that ROODA Exata was necessary to their mathematical online communication as it is possible to observe in statements like ROODA Exata was an indispensable tool. ; The editor makes it possible to express radicals, rationalization, potency in a simplified way. The images it generates make understanding easier. ; and All the formulas we needed to do the activities were there. Regarding the difficulties they may have encountered to use the tool, students felt some in the beginning but, with time, these difficulties were overcome, as illustrated by the following comments: The first impact using the tool was difficult but with the construction of more formulas, we can see it is simple and useful to work with expressions. Yes, I had never seen anything like this before on the Internet. It is only a matter of habit. ; and To start using ROODA Exata is quite difficult but after some time, the tool becomes rather useful. In fact, it is only natural that the first contact with a new tool should offer some level of difficulty especially considering that students are not used to editing mathematical expressions routinely. However, in a short while, handling the tool becomes natural. Students were also asked about the methodology using ROODA as support to presential classes adopted during the course. Here are some of the answers: Yes, very much. I liked the idea of the teacher. The activities helped me to commit and to clarify doubts. ; I think it helps to solve doubts that come up when you are studying at home. ; Yes, because of the exchange of information. ; Yes, besides communicating with the teacher, we can communicate with our colleagues. Thus, some of the students saw in the environment a chance to solve doubts while others felt more committed with their studies and others, still, emphasized the exchange of information and communication. It is evident that their perception about this methodology making use of a virtual learning environment as support to presential classes was positive. Clearly, those who seriously committed themselves with the activities proposed and participated actively in the debates in the environment progressed in their construction of mathematical knowledge. 7 Final Considerations TICs are more and more used in the academic environment to promote extra class learning. The use of a virtual learning environment as support to presential disciplines has shown to be efficient as it favors the exercise of communication and expression in Mathematics. From the solutions presented by students, it was evident that they needed to justify the means by which they achieved their results. This exercise in argumentation leads to reflection about action, ICL 2009 Proceedings - Page 470 7(8)
leading into awareness of concepts involved in the resolution of problems. The interactions that occurred in the environment triggered dialogues that, many times, allowed students to advance in their mathematical knowledge, overcoming difficulties, identifying mistakes, reflecting about their conceptions and building new cognitive structure that made the understanding of mathematical concepts easier. However, for the interactions to occur, the scientific editor ROODA proved necessary to make them feasible and to potentiate them. It became evident that ROODA Exata can promote mathematical debate online, the use of symbols and formulas accounting for an easier communication and expression in Math. Mathematics still widely underuses these resources. To a certain extent, Math teachers resist virtual and semi presential modalities because current tools do not offer the necessary resources to good communication online. Thus, the scientific editor ROODA Exata was developed to allow online mathematical communication to happen efficiently. It is known that Mathematical learning cannot be reduced to algebraic communication.. Environments that enable the transition among multiple representations of the same mathematical object, such as tables, diagrams and graphs are necessary as well as the construction of these objects in a dynamic way. Nevertheless, the main focus of this work has been to make Mathematical communication feasible once long distance learning is based on exchanges that occur virtually and these exchanges occur through writing. References: [1] Behar, Patricia et al. Capacitando professores para o uso do ROODA: uma plataforma voltada para a construção de conhecimento. In: V Congresso Brasileiro de Educação Superior a Distância (ESUD) e 6º Seminário Nacional de Educação a Distância (SENAED), 2008, Gramado. Anais... V ESUD e 6 SENAEAD. São Paulo: ABED, 2008. [2] Machado, N. J. Matemática e língua materna: analise de uma impregnação mútua. São Paulo: Cortez: Autores Associados, 1990. [3] Notare, Márcia Rodrigues. Comunicação e Aprendizagem Matemática On-line: Um Estudo com o Editor Científico ROODA. Porto Alegre, Tese de Doutorado, UFRGS, 2009. [4] Piaget, Jean. Abstração Reflexionante Porto Alegre: Artes Médicas, 1995. [5] Piaget, Jean. Fazer e Compreender. São Paulo: Melhoramentos, Editora da Universidade de São Paulo, 1978. [6] Piaget, Jean. A Tomada de Consciência. São Paulo: Melhoramentos, Editora da Universidade de São Paulo, 1977. [7] Smith, G. Grackin, J. Ferguson, D. Izubuchi, R. Math and Distance Learning threaded discussions. Disponível em: http://www.linksystems.com/ext_pnqudt9ciqiaadkebh4/glennsmithedmedia42902.pdf (Acessado em 14 abril 2006). Author(s): Márcia Rodrigues Notare, Dra. UNISINOS Universidade do Vale do Rio dos Sinos Av. Unisinos, 950 - B. Cristo Rei / CEP 93.022-000 - São Leopoldo (RS) - Brasil marcia.notare@gmail.com Patricia Alejandra Behar, Dra. UFRGS - Information Technology in Education Post Graduate Program Av. Paulo Gama, 110 - prédio 12105 - sala 332-90040-060 - Porto Alegre (RS) - Brasil pbehar@terra.com.br ICL 2009 Proceedings - Page 471 8(8)