Cruislet was conceived as a digital
medium for mathematically
driven navigations in
virtual 3-d geographical spaces. Users can view avatar positions and define
their displacements by employing either a Cartesian lat-long-height system or a
vector-differential (φ,θ,ρ) system where ρ is the length
of the vector of displacement. The users also have at
their disposal:

(a) a geographical
(non-mathematical) option of defining displacements by naming locations.

(b) a programming language
(Logo) which can be used for programming multiple or relative displacements.

Cruislet
is primarily a navigational medium but it is also constructionist (Papert,
1980, Harel & Papert, 1991, Kafai & Resnick, 1996) since avatar
trips can be constructed by using as well as displacement rules in 3d
geometrical space (Papert, 1980, Harel & Papert, 1991, Kafai &
Resnick, 1996). It is designed to provide opportunities for learners to engage
in expression of mathematical ideas
through *meaningful formalism* by means of programming and
interdependent representations of Cartesian and Vector-differential geometrical
systems.

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, 1999) 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.
With respect to mathematical content, the approach is to identify *conceptual fields* (Vergaud, 1990) which
with the use of this kind of media becomes rich in the potential to generate
mathematical meanings, irrespective of the ways in which they might be
structured (or fragmented) in the mathematics curricula.

ETL work is 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 artifact in ways that lead them to diverge from curriculum mathematics. Work and play with
cruislet is based on
the idea of instrumentation and instrumentalization since displacement rules
can be questioned and re-defined by learners resulting in a variety of
artefacts (see Verillon
and Rabardel, 1995 for these terms) which gives rise to a dialectic by which learner and artifact 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).

The constructionist environments designed at ETL 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). In such settings
learners are engaged in constructing public entities (constructions) implying
an explicit appreciation of the relationships between mathematical objects
within any situation (i.e. a mathematical model of the situation). In the case of Cruislet,
learner constructions are avatar trips as
well as the rules of displacement. The mathematics are those underlying the use
of analytic and/or vector-differential geometry, including functions,
co-variation and rate of change. However, these mathematics are integrated with
geo-spatial representations and information, providing opportunities for
processes of mathematisation of geographical space.

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 argumentation
in social contexts. Displacing avatars and articulating rules of and
relationships between the displacements 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
representations. 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), dynamic
manipulation (such as Dynamic Geometry Environments), simulations, information
handling and geographical systems.

In designing Cruislet we wanted to integrate
programming, mathematical and geographical concepts, relations and
representations.

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 Cruislet, a
vector-differential geometrical system co-exists with a Cartesian-geographical
one in an inter-dependent way. 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.

Moreover, navigations in virtual 3d
geographical spaces within Cruislet could be conceived as game play
simulations. There is a growing interest about the
ways in which game–based learning environments facilitate new ways of
learning (Gee, 2003). The key feature of this approach is that games can provide
a context for the development of valuable skills (Kirriemuir and McFarlane et
al., 2004) in the transitional stage between intuitions (informal) and formal
mathematics. Using games with an appropriate set of tasks and pedagogy,
students can be engaged in exploration, problem solving, rule-based thinking
and other forms of mathematical thinking (Goldstein et al, 2001; Mor et al,
2004). From this point of view, the process of building game play
activities involving navigation within the 3d representational space can be
seen as the design of the terrain within which
instrumentation/instrumentalisation processes may take place by student’s
interactions with the microworld and the mathematical concepts and rules
embedded in it.

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
& Yiannoutsou, 2002, Yiannoutsou & Kynigos, 2004). This educational
plan is designed to bring in the foreground issues concerning the mathematical nature of 3d navigation and how interactivity, control and experimentation
in virtual reality microworlds can be a versatile vehicle for enhancing mathematically driven navigation, orientation
and spatial visualization.