In the summer of 2007 the minister of Education highlighted that mathematical teaching is an emergency in the Italian mainstream education system and “it risks to prevent Italian students to compete with their peers in Europe”. Moreover, the PISA international test has revealed that Italian students lack in mathematics skills:
among all OECD countries that participated in PISA, Italy comes in the bottom places. It is known that to teach and learn mathematics deep changes and new levels of teaching mediation are needed.
Within ReMath EC project the ITD research group designed and developed a digital learning environment - Alnuset, which can be effectively exploited to improve mathematics teaching and learning with special reference to algebra. Our idea is that technology can be used to make available new operative and representative possibilities to structure a new phenomenological space which allows teachers to configure algebraic knowledge as objects that can be investigated by students whio, at the same time improve their learning. Alnuset includes three closely integrated components
the Algebraic Line component,
the Symbolic Manipulator component and the Cartesian Plan component. The Algebraic Line ComponentThe Algebraic Line structures a new phenomenological space where processes, relationships and objects of algebraic nature can be investigated through a quantitative approach.
Digital technology characterizes the traditional numbers line with new operative and representative possibilities, namely:
- with three different editors to construct and represent algebraic expressions on the line, specifically with a geometrical editor, a linear editor and a bi-dimensional editor;
- with the drag of mobile points associated to letters on the line which determine the automatic movement of points of expressions containing the dragged letters on the same line;
- with a graphic and computational model to determine the roots of polynomials;
- with graphical models to define and validate the truth set of algebraic propositions.
The user can directly and dynamically control these new operative and representative possibilities exploiting their visual, spatial and motor experience.
These possibilities can be used to recognize properties and relationships between properties in the form of expressions and proposition respectively and to explore what expressions and propositions denote within the considered numerical domain.
The Symbolic Manipulator Component
The Symbolic Manipulator structures a new phenomenological space where norms, rules and conventions of algebra can be investigated in order to structure an idea of Algebra as the science of formal operations, i.e., as the theory and practice of formal operations that preserve equivalence in the performed transformations. This space is characterized by new operative and representative possibilities such as:
- exploring the hierarchical structure of the expression or proposition that has to be manipulated;
- exploring the rules of transformation of the interface that can be applied on each selected structure;
- exploring the effects that the applications of a rule produce;
- verifying that in the performed transformation the equivalence has been preserved;
- creating a new rule, once it has been proved using the rules available with the interface;
These new operative and representative possibilities can be exploited to pursue important teaching results.
On one hand, they can support the construction of general schemes for algebraic transformations by mediating the solution of cognitive problems involved in such construction, i.e., problems of perception, memory and language.
On the other hand, they can be exploited to develop a theoretical approach to algebraic transformation accessible to students.
Cartesian Plan Component
The Cartesian Plan structures a new phenomenological space where important algebraic processes and relationships can be investigated through a functional approach.
This environment is characterized bythe following possibilities:
- to define a letter contained in an expression as its independent variable and obtain the graph of the corresponding function in the Cartesian plan
- to drag the mobile point associated to a letter defined as independent variable on the Algebraic line obtaining automatically both the movement of that point on the line and the movement of the point corresponding to the pairs of values (variable; expression) on the graph of the Cartesian plan
- to drag the mobile point associated to a letter which is not defined as independent variable of the expression, obtaining automatically both the movement on the same line of the point corresponding to the expression, and the dynamic transformation of the graph of the expression according to the role of parameter of the dragged letter.
These new operative and representative possibilities can be exploited to manage important processes and relations of algebraic nature and to understand the various meanings that letters can assume in algebra, i.e., that of variable, parameter and unknown.