A tetrahedron to model e-learning Mathematics

A tetrahedron to model e-learning Mathematics Giovannina Albano 1, Eleonora Faggiano 2, Flavia Mammana 3 1 Dipartimento di Ingegneria dell'informazione, Ingegneria Elettrica e Matematica Applicata, Università degli Studi di Salerno. galbano@unisa.it 2 Dipartimento di Matematica, Università degli Studi di Bari. eleonora.faggiano@uniba.it 3 Dipartimento di Matematica e Informatica, Università degli Studi di Catania. fmammana@dmi.unict.it Abstract In this paper we introduce a model to 'read' the impact of e-learning technologies on the teaching/learning process, usually interpreted through the didactical triangle - knowledge, student, teacher (Chevallard, 1985). The idea we present here starts increasing the number of actors -that are author (A), student/s (S), tutor (T), mathematical knowledge (M) - and makes the model to become three-dimensional, passing from a triangle to a tetrahedron. The mediation of technology is represented by a sphere inscribed in the tetrahedron. The tangent point of the sphere with each face of the tetrahedron allows us to look at the relations among the three actors (the vertices of the face) in the e-learning environment and then how they can be transmitted /fostered/created by the new technological tools. The e-learning tetrahedron We argue that the didactical triangle in a e-learning environment can be extended as follows: the actors are the author (A), the student/s (S), the tutor (T) and the mathematical knowledge (M) and the model no longer consists in a triangle but it becomes a tetrahedron. In literature we can find other extensions of the same triangle which make visible technology adding it as new vertex (Ruthven, 2012, Rezat & Sträßer, 2012). Our approach starts from the conviction that full exploitation of e- learning environment and its integration with results from research in mathematics education require to well design didactical interventions, both 429

in terms of contents/activities (didactical transposition) to be insert in the platform and of the methodologies through which a certain content is introduced (didactic engineering). This is why we introduce as a new vertex, the author, which is in charge of that planning activity. We also consider the milieu in both the technological and social dimensions as a sphere including the tetrahedron. This means that the milieu impacts anyway on all the vertices and the faces, but it is explicitly taken into account by the author. In an e-learning environment the author is a collective subject with different professional skills, from knowledge domain and disciplinary educational expert to instructional designer/manager, from ICT to pedagogical/sociological expertise, and so on. This subject acts as scriptwriter of didactical experiences mediated by the technology. #" #" #" $" $" %" $" %"!" %"!"!" Looking at the ASM face, e-learning technologies support the selfinstruction model, without tutor, i.e. the student builds up her own knowledge by interacting with mathematical knowledge. How? The interaction is driven by the didactic transposition done by the author on the platform (represented by the ATM face). The transposition in the platform has to be as wide as possible, i.e. the same knowledge should be transposed in several teaching methods and several didactic and a-didactic situations: such a richness of proposal allows the creation of personalized learning. The personalized learning can be self-regulated or guided from the computer or both at the same time, providing the student with a path from which she may deviate surfing freely in the knowledge available on the platform. 430

Looking at the STM face, e-learning acts as supporter for both the model of individual learning with tutor and the cooperative/collaborative model when more students are contemporarily engaged. The first puts beside what already said in the previous subsection, activities with interactions among tutor and students, such as task, through which the tutor poses to the student a stimulus (question, problem, etc.). The student has to answer by submitting her own product/result and, starting from this, a one-to-one dialogue between tutor and student is set up addressing and personalizing the student s learning path. In the latter case, instead, activities are carried out engaging more students, with a more or less explicit tutor s help, such as role-play activities, or discussions or realization of shared documents. Looking at the ATM face, e-learning supports the didactical transposition process (Chevallard, 1985) and didactical engineering (Artigue, 1992). Here the author focuses her efforts on the platform tools and on the analysis, the design and the implementation of educational resources and activities (with or without tutor). It is worthwhile to note that many of the tools offered by the e-learning platforms are not born for educational purposes and then their meaningful use from the educational point of view is not for granted, but it has to be attained (Chevallard & Ladage, 2008). A successful implementation of educational practices in e-learning environment requires the mastery of the integration between technology and pedagogy in order to let the learning occurs effectively (Govindasamy, 2002 and Engelbrecht, 2003, in Kahiigi et al., 2008). The teaching methods used in face-to-face courses cannot be simply transferred in distance settings, but they need to be revised and modified, as not always they effectively act or they can be immediately implemented in an e-learning environment (Kahiigi et al., 2008). This helps to understand how this face of the tetrahedron represents a key for the success of the learning experiences on the platforms. Looking at the AST face, e-learning supports the tridimensional approach of the relation between the students and the learning objects and activities produced by the author: in fact the student does not restrict herself to receive and elaborate objects (such as in the case of the book), but she can produce new ones starting from those available in the platform (Maragliano, 2000). Similarly the student s and tutor s feedbacks about the activities allow the author to adjust/refine the proposals available in the platform. Moreover the tutor can validate the new products created by the students. 431

Outline of possible learning activities for each face In this section we want to give a more detailed look at how to implement the teaching/learning process assuming as point of view each face of the tetrahedron. The ASM Face From this viewpoint, we look at the student who learn directly interacting with the mathematical knowledge. The adjective directly actually does not mean that no mediators are involved, but that the student has many degrees of freedom with respect to the access to knowledge (e.g. she can choose what to learn). The main mediator is the author: in fact, differently from the Internet, the knowledge to be accessed is not unlimited, but it consists in what the author makes available on the platform. It is worthwhile to say that, differently from the Internet, the presence of the author is also warranty of validity of what is offered in the platform. Thus, the interaction among author, student and mathematics of this face of the model allow us to take into account the modality of self-learning. The ability of self-regulation is a fundamental competency required to the today citizen, that is to be able to control her own process of acquisition, elaboration and exploitation of her knowledge in relation to the learning goals posed by herself, to the awareness of her knowings, to the attitude with respect to the failure in the past and to the expectations for the future (Boscolo, 1997, p. 320). In particular, concerning mathematics, Zan has shown that repeated unsuccessful behaviors can be interpreted not much as dependent on not sufficient knowings but as dependent on a bad management of last ones and on negative attitudes (Zan, 2006, p. 238). How e-learning can support self-learning and self-regulation? The author has to exploit the tools offered by the platform to make available to the student a large range of learning resources and activities. This can give the student the chance of movement, of choice, of designing and management of her learning path. In this view, the author is not just the one that edits some contents, but she becomes an arranger of contexts where the contents can serve to a precise purpose. The exploitation of the potential of personalisation needs the availability of various resources, according to different learning styles, degree of difficulty, didactical approach, media, etc. This has a key role as it allows to support diversity in student s methods, which in the constructivist perspective is viewed being the driver of mathematical learning (Balacheff & Sutherland, 1999). This requires the author to prepare many didactical proposals as well as many resources such that each student can construct 432

her own learning path. For instance various learning objects (hypermedia, structured videos, lessons, animated slides, dynamic problems) and templates of activities (individual tasks or quizzes, cooperative writing of a document, collective discussion on a specific topic) can be foreseen according to the above cited parameters and to the specificity of the discipline (Albano, 2011). Sometimes it can be useful for the student to have a sketch to move on, then the intelligent potentials offered by some platforms (such as automatic generation of a learning path) can be exploited, allowing to create for each student a learning path as near as possible to her needs and her learning style (Albano, 2011). It is important to point out that, also in this last case, the student is always free to move away from what proposed, surfing freely the resources of the platform. It is worthwhile to note that the products of the author s work change in time according to the students outcomes and feedback (for instance, according to degree of successful in assessment after certain kind of contents or activities). The STM Face This face of the model allows to consider learning as the output of interactions among students, tutor and mathematics. Thus we can look at mathematics learning as the initiation to mathematical discourse, that is the initiation to a special kind of communication known as mathematics (Sfard, 2001). The cornerstone of the activities, hence, will be the discourse, seen as any specific communication instance, both diachronic and synchronic, both between student and tutor and between student and student, both predominantly verbal and with the help of other semiotic representations. This approach is well combined with the Laurillard s Conversational Framework (CFL) (1993), according to which the teaching/learning process is modeled through a continuous and interactive exchange between student and tutor and between student and herself. The tutor supports an environment where the student interacts through a cycle goal-action-feedback-modified action and the dialogue between student and tutor fosters the discursive thinking on the made experience in order to let her go on a new learning cycle. The CFL can be put in action also favoring the dialogue among students, other than between student and tutor. The e-learning platforms offer various opportunities to implement such approach. Modules such as workshop, forum, wiki, blog, task are, for their own nature, suitable to a discursive approach. From the STM face we can think at the learning process through individual or 433

cooperative/collaborative activities according to the engagement of one or more students. In the first case, the activities can consist in assigning the students some homeworks, to be completed and returned to the teacher by a fixed deadline, by means of the task tool, which allows the teacher to open a one-to-one private and multiple-steps communication with each student, giving all of them the same homework. In the second case we can think for instance to cooperative activities such as reciprocal peer questioning in a role-play setting where students assume both the teacher and the learner role (Albano & Ferrari, 2008). The digital platform traces of all the messages exchanged during the activities can be accessed anytime for seeing students advancements. The ATM Face From this viewpoint, we look at the potentialities that the e-learning tools offer to create learning resources and to design individual or group learning activities. Here the author and the tutor consider the problem of setting new educational practices which maximize the tools potential. For instance, the platforms allow the multisemioticity, then this is the time to put in practice and to design activities fostering the coordination of various semiotic representations, which is a hub of learning (Duval, 1995). Similarly, as the platforms allow a discursive approach to learning, then the problem to be faced here is to design activities which let the use of various registers (meant as linguistic variety based on use, Ferrari, 2004), and promote the passage from the colloquial ones to the literate ones, the latter having much in common with the language of mathematics. The interactions between the author and the tutor is very important since the tutor can report students feedback to the didactical proposals already delivered in the platform. Taking advantages of them, the author can make adjustments and provide new and more effective proposals. This means that the educational resources and activities are not static objects in the platform but they evolve according to the practices feedback. Finally it is important to point out that the didactical proposals, related to certain goals, are not unique, but there are a large range of objects and activities with the same purpose, eachone implemented according to various approaches and methods, so to match various students needs and profiles. 434

The AST Face From this viewpoint, we consider the student with respect to her involvement as author of new contents. For instance, in some platforms the student can make changes to the resources created by the author, annotating her own remarks or deepening, and can share the so created new objects (preferably validated by the tutor). Sometimes, the author designes suitable activities to involve a group of students in creating, under the tutor s help/supervision, collaboratively written documents (experimentations report, design of lectures in teachers training, etc), that can become new resources available in the platform for other students. Then the interaction among the three vertices A, S and T should lead to enlarge the opportunities offered by the platform, through the active engagement of the students in becoming herself author. Conclusions Herein, we have introduced a model to help the educational community to frame e-learning in the teaching/learning process: the tetrahedron with vertices author, student, knowledge and tutor. In this ongoing work we have tried to look at each face in order to see how e-learning supports the relations between the involved vertices. Then, we have investigated how e- learning tools can be exploited for setting up educational activities which can take advantages from research in mathematics education. It can be seen that, looking at the various faces of the tetrahedron, several educational approaches are supported by e-learning, which impacts on all the learning levels (cognitive, meta-cognitive and non-cognitive). It is worthwhile to note that each face allow to stress one or one other learning approach or activity, but it is also evident that there is no sharp separation among the faces. Each of them simply acts as magnifying lens to focus on some aspect. References Albano, G., Ferrari, P.L. (2008). Integrating technology and research in mathematics education: the case of e-learning. In Garcia Peñalvo (ed.): Advances in E-Learning: Experiences and Methodologies. DOI: 10.4018/978-1-59904-756-0, ISBN13: 9781599047560, ISBN10: 159904756X, EISBN13: 9781599047584 - pp. 132-148. Albano, G. (2011). Mathematics education: teaching and learning opportunities in blended learning. In A. Juan, A. Huertas, S. Trenholm and C. Steegmann (eds): Teaching Mathematics Online: Emergent Technologies and Methodologies pp. 60-89. 435

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