Designing hypermedia tools for solving problems in mathematics
Introduction
Historically, the processes of solving problems and the analysis of transfer mechanisms have been the two mainstays of the cognitive paradigm as it is applied to the study and analysis of Mathematics learning (Polya, 1973). Hence, in developing our theory of the design of computer-based mathematical instruction, thought was given as to how to represent such processes in computerisable models. The Global Problem Solving (GPS) computer application is the basic historical reference for research into the implementation of the executive-heuristic system in computer application designs.
From a theoretical point of view, these processes should give an answer to the following basic question: to what extent can heuristic processes be represented algorithmically? Newel and Simon (1972) gave the first solution with their method of “analysis of means and ends”. However, the poverty of technology at that time meant that this methodology was essentially reductionist. Only a limited group of problems, such as the classic Towers of Hanoi, could be solved in this way.
The teaching of Mathematics has been used as one of the more widely extended conceptual laboratories for proving the efficacy and efficiency of computer science experiments in teaching and learning. These experiments were used as a basis for many of the Computer Aided Learning (CAL) programs which focused on such topics as arithmetical calculation, geometrical simulations, and formal expansions in algebra. The 1950s and 1960s saw the design of instructive material in which the student had to reply to a set of questions programmed by means of a linear flow diagram, with no possibility of interaction between the computer and the user; i.e. the behaviourist model (for a more extensive analysis see López Fernández, 1999). Subsequently, an attempt was made to adapt the software to the student's level of learning, depending on his/her answers. Hence, branched programmes allowed some interactivity (for example the programmes developed in the context of TICCIT experiences).
In the 1970s and 1980s the influence of theories of discovery learning came to the forefront, and simulations and games were used, not least because of their perceived ability to motivate the learner. Parallel to this, a number of programming languages were developed (such as BASIC, PASCAL and LOGO) which allowed the student to take control of the computer. At this time Mathematics was used as a reference framework for developing these languages, because of the conceptual proximity between them and the logical/deductive thought proper to Mathematics.
Currently, the appearance of hypertext and, latterly, that of multimedia and hypermedia systems, have given rise to new possibilities for the technological implementation of models that carry out and develop capacities characteristic of the executive-heuristic system.
In this article, we describe an interactive CD-ROM, developed to exploit the new possibilities offered by hypermedia systems to improve the processes of heuristic learning of secondary level students. The theoretical underpinning of the design draws out the understanding of metacognitive mechanisms, Polya's gestalt theory (1973), Schoenfeld's (1994) heuristic models, the theory of incremental analogy (Keane, 1987) as well as being cognisant of hypermedia structure.
The prototype interactive CD-ROM, which is the basis of this study, is designed to facilitate students' ability to use heuristics to solve problems, following Polya's (1973) stages. Section 2 provides a fuller description of theory and of the implications for the design of the CD-ROM, while in Section 3 the curriculum content and modes of use are outlined. There are two modes of use. The first is designed to support students' learning (or as an information source for teachers) in relation to their heuristic capacities (the Theoretical Foundations). The second mode is a practice mode designed to promote the aforementioned capacities (the Practical Development). The hypermedia tools used for the development and implementation of this product are given in Section 4. Finally, Section 5 presents the conclusions deriving from this study, together with the future research lines and action transpiring from it.
Section snippets
Hypermedia systems in relation to mathematics learning
This section lays the foundations on which the design of this CD-ROM rests.
An interactive CD-ROM
The CD-ROM prototype presented here exemplifies the application of the executive-heuristic system in a hypermedia environment.
Briefly two options are presented to the student at the start of the program: Theoretical Foundations and Practical Development. The first takes the student to the main principles and the underlying theory for solving problems, whereas the second one lets him/her practise this knowledge (below we comment each of these options). Fig. 1 presents the main screen for access
Hypermedia tools
The hypermedia tools used for implementing this CD-ROM, with a view to achieving the greatest possible interactivity with the student, were the following:
Conclusions and future works
As has already been mentioned, hypermedia systems in the educational sphere are still in the phase of trial and experimentation. Currently, many of the multimedia applications designed and implemented are just resources conceived and developed as texts, but adapted to computer media. This makes them, in many cases, excessively textual materials or mere galleries of images, thus giving few possibilities for interaction. Moreover, considering the teaching methodology of many countries, their use
Acknowledgments
This study was partially subsidised by the Ministry of Education and Culture of the Regional Government of Castile and Leon (Spain). Our most sincere gratitude to William B. Rawson and Dennis Almeida, both teachers of School Education at Exeter University (England), for the careful correction of the original work. The authors wish to thank to the anonymous referees for their valuable suggestions for improving the original manuscript of this paper.
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