How do we teach children to express and communicate ideas in a formal and informal mode? What type of language do they need in a concrete context? How should they determine a proper level of formalization of their descriptions? In an attempt to explore these issues we carried out a pilot experiment in the frames of the DALEST European project whose goal was to create environment for stimulating the 3D geometry understanding of young students and to assist them in developing some fundamental mathematical skills including spatial visualization and articulating ideas. The pilot experiment was carried out with 5th graders from five Bulgarian schools by means of specially designed educational scenarios and the Cubix Editor (a Logo based application for manipulating unit-sized cubes). The children were given tasks to describe compositions of unit-sized cubes and to build such compositions by means of the Cubix Editor when given their descriptions by peers. The students experienced the whole process of generating a good description - becoming aware of the ambiguity, producing counterexamples, reducing the ambiguity, eliminating the redundancy.
The pilot experiment aimed at specifying the structure, scope and methods behind the stereometry activities envisaged for 5th graders in the frames of the DALEST project.
The first impressions confirm our belief that the language is playing significant role in the learning experiences of the students, that the relationship between thoughts and words involves back and forth reshaping process. While constructing and describing cubical structures they articulated their own ideas, developed concepts collaboratively with others, moved between everyday and mathematical terms, between procedural and declarative style, exploring the boundaries of understanding. Such interplay with the step-wise refinement of their descriptions of cubical structures would hopefully enhance students' skills for working with mathematical definitions, on one hand, and prepare them for writing, debugging and explaining programs, on the other.
A reversible sequence of steps from the specification of the algorithm and the mathematical definition of the recurrent solution through the recursive procedure, the tail recursive procedure and finally to the iteration procedure, is shown. The notation for analysing recursive function execution as well as modified flow charts of an algorithm to identify the differences between the iteration and the tail recursion are proposed. All the procedures are written in Logo, so the lists are used as the data structure. Transformation from the recursive procedure to the iterative procedure and vice versa can be shown in such a way in every language in which the recursion is allowed. All examples are one-recursion-call examples and all except one are the functions of discrete mathematics.
Two years ago the Faculty of Mathematics and Informatics at Sofia University makes a decision to design a new series of Logo-based courses which make use of the modern technology. The pedagogical component of the challenge is to design a multidisciplinary course suitable for students with different skills and interests. From a development perspective the challenge is to build an entirely new one. And finally the course must be attractive regardless of the seriousness and complexity of the topics included in it.
The paper discusses the structure of the course including the final weeks when topics emerging from students' course projects are taught. Each lesson from the course is based on sets of sample programs representing the general lifecycle of software development. This includes designing, coding and debugging. Samples are created on-the-fly, thus different instances of the course results in different final projects. Lessons are interactive and students may interfere with the direction of demonstrated software development.