MaLT was conceived as a constructionist microworld environment (Papert, 1980, Harel & Papert, 1991, Kafai & Resnick, 1996) for programmable constructions in 3d geometrical space. It is designed to provide opportunities for learners to engage in expression of mathematical ideas through *meaningful formalism* by means of programming and *dynamic manipulation* of graphically represented mathematical objects.

It constitutes a new digital medium within the context of more than a decade of ETL R&D work on designing constructionist exploratory media based on the principle of integrated and interdependent mathematical representations. A principal construct emerging from this work is that of *half-baked microworlds* (Kynigos, 1992 and in press). These microworlds are designed for *instrumentalization* (Guin &Trouche) through constructionist activity, i.e. they incorporate an interesting idea but at the same time invite changes to their functionalities and are mediated to the targeted users as unfinished artefacts which need their input.

ETL work is based on the idea of instrumentation *though*
instrumentalization (see Verillon and Rabardel, 1995 for these terms) which gives rise to a dialectic by which learner and artefact are
mutually shaped in action. We have focused particularly on the process by which implicit mathematical knowledge is constructed during shared student activity (*theorems in action*, Vergnaud, 1990 and *socio-mathematical norms*, Cobb & Yackel, 1996). We are strongly sensitive to the ways an instrument is *internalised *collaboratively
by the students (Mariotti, 2002) while it is being changed often quite
distinctly to what was designed by the researchers. Relatively, the implication of this perspective is that students' expressions can gain mathematical legitimacy, even if they differ from and/or they are shaped and structured by the artefact in ways that lead them to diverge from curriculum mathematics.

*Constructionism* is a theory of learning and designing that
builds upon the connotation of learning as "building knowledge structures" in a context where learners are consciously engaged in constructing (or de/re constructing) something on the computer (such as animations simulations, geometrical figures etc.), implying that the notion of construction refers both to the ‘external’ product of this activity as well as to the theories constructed in pupil’s minds. As developed by Papert (1980) "better learning will not come from finding better ways for the teacher to instruct, but from giving the learner better opportunities to construct."

The constructionist environments designed at ETL so far provide dynamic visual means that support immediate visualization of *multiple
linked representations* (i.e. any action carried on a specific representation provides immediate change and feedback in all representations, Kaput, 1992). A specific subset of these environments enables direct and sensual manipulations of mathematical objects with *body syntonic metaphors* (e.g. the turtle in the 2d space of
E-slate ‘Turtleworlds’ and in the 3d space of MaLT). In such settings learners are engaged in constructing public entities (constructions) implying an explicit apreciation of the relationships between mathematical objects within any situation (i.e. a mathematical model of the situation).

The key point here is that students can build their models into the medium that can act as a support for developing new meanings by investigating their hypothesis and argumentating in social contexts. Manipulating objects and articulating relationships between them can thus provide an action/notation context which can be a new resource for activity and construction of meanings, not so dependent on the medium for its expression. Noss and Hoyles (1996) introduced the notion of *situated abstraction* to describe how learners construct mathematical ideas by drawing on the linguistic and conceptual resources available for expressing them in a particular computational setting which, in turn, shapes the ways the ideas are expressed. Yet, from a social constructivist perspective, psychological and social aspects of learning can never be considered separately and the term situated abstraction captures also the synergy between them: student’s activity within a community (Lave & Wenger, 1991) both shapes and is shaped by their interaction with the available tools and those around them.

A key feature of the approach of ETL is to design artifacts afforded with integrated graphical and symbolic representations and functionalities including dragging modalities which provide a kinesthetic sense of the dynamic manipulation of mathematical objects (e.g. geometrical figures). As an example, in the last decade ETL has been involved in the design of E-slate, an educational authoring system with which many different microworlds have been developed for mathematics and science. These microworlds can be characterized as hybrids between symbolic programming (such as Logo-based Turtle Geometry) and dynamic manipulation (such as Dynamic Geometry Environments) systems.

In designing MaLT, we intend to create learning environments extending the above design representations and functionalities to the 3d space. This decision is also related to the following facts (a) most of the powerful ICT tools for the learning of geometry (Logo, DGS) have been designed to operate within 2d environments, and (b) relatively little research has been carried out on spatio-visual mathematical concepts concerning geometry in 3d spaces.

New representations enabled by digital media can place spatial visualization concepts in a central role for both controlling and measuring the behaviours of objects and entities in virtual 3d environments. We have chosen the notion of vector as a means to represent the link between 2d and 3d representations, since vectors can be considered as basic components underpinning the study of geometry and motion in space facilitating the study of 3d spatial thinking. In MaLT the mathematical nature and identity of vectors is considered as embedded in turtle’s movement which takes place in a specially designed 3d Turtle Geometry Environment where students will be able to capitalise upon the power of Logo as a programming language. It is a dynamic visual environment in which learners will be able to create programmable 3d geometrical constructions and then using the provided tools as a means of manipulating them and investigating their properties. Our perspective is centered on the utilization of the different representations and the feedback that they can provide so as to facilitate multiple didactical decisions within open-ended exploratory tasks.

Our approach to learning promotes also investigation through the design of activities that offer a research framework to investigate purposeful ways that allow children to appreciate the utilityof mathematical ideas (Ainley & Pratt, 2002). In this context, our approach is to design tasks for either exclusively mathematical activities or multi-domain projects containing a mathematical element within the theme which can be considered as marginalized or obscure within the official mathematics curriculum (Kynigos & Psycharis, 2003, Keisoglou & Kynigos, 2006, Kynigos & Gavrilis, 2006). This educational plan is designed to bring in the foreground issues concerning the mathematical nature of 3d geometrical objects and how interactivity, control and experimentation in virtual reality microworlds can be a versatile vehicle for enhancing mathematically driven navigation, orientation and spatial visualization. One such idea is the dynamic manipulation of mathematical variation as it is realised in 3d geometrical objects.