Digital artefacts as representations: forging connections between a
constructionist and a social semiotic perspective
Professor Candia Morgan (corresponding author)
Institute of Education, University of London
20 Bedford Way
London WC1H 0AL
United Kingdom
c.morgan@ioe.ac.uk
and
Professor Chronis Kynigos
Educational Technology Lab, National Kapodistrian University of Athens, Greece
Abstract: This paper uses the methodology of cross-case analysis to clarifying connections and
differences between two specific conceptual frameworks, multimodal social semiotics and
constructionism, in particular in the ways they each deal with the idea of representation. It builds
especially upon the idea of ‘distance’ of digital representations with respect to usual
representations, stressing the multidimensionality of this notion and possibilities for progress
towards a shared framework for research about representations. The paper focuses on the crosscase analysis study of a ‘middle distance’ dynamic digital artefact, MoPiX, by two teams sharing
a common reference framework (constructionism), but with distinct views with regard to
representations. The research yielded a distinction in the ways connections between
representations are valued by the two approaches and their respective interpretations of meaning
generation.
Keywords: constructionism; social semiotics; multimodality; representation; cross-case analysis;
theory networking
1
Digital artefacts as representations: forging connections
between a constructionist and a social semiotic
perspective
Candia Morgan, Institute of Education, University of London, United Kingdom
Chronis Kynigos, Educational Technology Lab, National Kapodistrian University of Athens,
Greece
Abstract: This paper uses the methodology of cross-case analysis to clarifying connections and
differences between two specific conceptual frameworks, multimodal social semiotics and
constructionism, in particular in the ways they each deal with the idea of representation. It builds
especially upon the idea of ‘distance’ of digital representations with respect to usual
representations, stressing the multidimensionality of this notion and possibilities for progress
towards a shared framework for research about representations. The paper focuses on the crosscase analysis study of a ‘middle distance’ dynamic digital artefact, MoPiX, by two teams sharing
a common reference framework (constructionism), but with distinct views with regard to
representations. The research yielded a distinction in the ways connections between
representations are valued by the two approaches and their respective interpretations of meaning
generation.
Introduction
This special issue addresses the problem of fragmentation of theoretical frameworks and
constructs in the context of the use of digital media in mathematics education. This
fragmentation has resulted in polysemous use of terms and notions, difficulty in using
constructs across educational contexts and a sense of noise in the growth of knowledge in
the field (Kynigos & Lagrange, this issue; Artigue, Cerulli, Haspekian & Maracci, 2009;
Prediger, Bikner-Ahsbahs & Arzarello, 2008). The papers aim to contribute to the
production of and experience with tools and methods for the networking amongst
frameworks and constructs. The notion of networking and the ensuing agenda in the
mathematics education community has been given serious attention (Prediger et al, 2008)
but has taken on a range of different meanings. In this issue it is being used in the sense of
the forging of connections (Mariotti & Artigue, this issue, Artigue et al, 2009) and the recontextualization that occurs when theories and constructs are put to use in designing and
studying educational practices (Lagrange & Kynigos, this issue). This paper addresses a key
2
feature of the affordances and the uses of digital media: mathematical representations. The
nature of representations, their affordances, their uses and the respective relation with
meaning generation has given rise to frameworks and constructs, themselves a part of or
connected with more general frameworks in mathematics education (e.g. Dreyfus, 1994;
Font, Godino & d’Amore, 2007; Kaput, 1991; Radford, 2009). In this paper we do not
attempt to review all approaches to representations but rather we discuss the use of crosscase analysis, a cross-experimentation methodology pervading this special issue, as a tool
to forge connections between two particular approaches to representation, the multimodal
social semiotic approach and the constructionist approach. We narrow down our lens
precisely in our attempt to better understand what light two compatible but different
approaches can shed on inquiry into didactical phenomena in the context of the use of
digital media and what precisely is the commonality and the distinctiveness between them.
The idea of representation has held an important place within the field of mathematics
education for several decades, as is evident in the extent of literature addressing it (e.g.,
Cobb, 2000; Duval, 1999; Goldin, 1998; Janvier, 1987; Vergnaud, 1998). With the
development of new technologies, a particular focus has been developed on new forms of
representation afforded by the technology (e.g., Ainsworth, Bibby, & Wood, 1997; Borba &
Villarreal, 2006; Edwards, 1998; Falcade, Laborde, & Mariotti, 2007; Morgan, Mariotti, &
Maffei, 2009; Noss & Hoyles, 1996; Yerushalmy, 2005). Two characteristics of
representations with digital technologies are considered especially significant for
mathematics education. The first characteristic is the possibility of linking different
semiotic systems, for example algebraic expressions and Cartesian graphs, in such a way
that manipulation within one system effects a corresponding change in the other. This
linking of multiple representations of the ‘same’ mathematical object is seen to have
potential to enrich students’ conceptualisations. The second significant characteristic is the
possibility for representations to have a dynamic element. As such representations are
manipulated by students, the manipulation and the ensuing changes become part of the
representation itself, again enabling new forms of mathematical conceptualisation to
develop. New modes of communication and new means of interaction have thus emerged,
both between students and the technological artefacts themselves and between students
3
and mathematical ideas. There is, however, a wide range of different ways in which
researchers conceptualise ‘representation’, not only drawing on different theoretical
frameworks but also focusing on different types of phenomena.
As is evident in its title, the ReMath project (Representing Mathematics with Digital
Technologies)i took as its starting point a recognition of the significance of representation
in the development and use of new technologies in mathematics education. However, there
was no initial shared theoretical conceptualisation of the idea among the participating
teams. As discussed in Morgan et al. (2009), the maximal common agreed definition of
representation adopted at the start of the project was simply “something which stands for
something else from someone’s point of view”. This sparse definition proved sufficient to
form some useful distinctions. In particular, the agreed recognition that any discussion of
representations must take account of the context within which they are situated (that is,
considering the point(s) of view of the “someone(s)” for whom they are representations of
something), allowed us to develop a refined notion of “distance” as a characteristic of the
external representations offered by digital technologies, identifying different dimensions
(epistemological and social) along which teachers, students and researchers might
experience such representations as “near” or “far” from the traditional school mathematics
with which they are familiar (Morgan, et al., 2009).
Recognition of the extent of differences between our various theoretical orientations
towards representation was important in that it allowed us to develop and use a shared
language coherent with all our perspectives. This shared language did not imply that we
also developed shared theoretical perspectives but that we became more able to
communicate about our perspectives in a way that made similarities and differences
explicit. Our agenda with respect to representations was part of an important general aim
of the project: to attempt to build an integrated theoretical framework that would allow a
deeper and more productive conversation between researchers with different theoretical
starting points. In this article, we seek to illustrate the process of integration and some of
its outcomes through looking in detail at a cross-case analysis of a single example, bringing
into conversation the perspectives of teams of researchers from two institutions, the
Institute of Education, University of London (IOE) and the Educational Technology
4
Laboratory, University of Athens (ETL). The “cross-experimentation” methodology, initially
conceived during the TELMA project (see Bottino & Kynigos, 2009) and used as a central
method of ReMath, was further developed to use cross-case analysis as a means of
producing this conversation. In the next section, we present the initial theoretical
orientation towards representation of each team. We then discuss the crossexperimentation carried out by these two teams using a digital technology, MoPiX,
developed by the IOE team. This experimentation produced data from two distinct contexts
that were analysed by each team from its own theoretical perspective. The consequent
analyses then served as a focus for reflection between the two teams. We offer an example
of this cross-case analysis and discuss how it provided a basis for clarifying the similarities
and differences between the ways in which the two teams conceptualised representation
and for moving towards an integrated framework that would enable us to share our
research methods and outcomes more effectively.
Representations
IOE perspective:
The IOE team adopts a multimodal social semiotic perspective on representation (Halliday,
1978; Hodge & Kress, 1988; Kress & van Leeuwen, 2001). From this perspective, the
elements of spoken, written, diagrammatic or other forms of communication are not taken
to have a fixed relationship to specific objects or concepts or to indicate any internal
intention or understanding on the part of the author. Rather, the resources offered by
language, diagrams, gestures and other modes are considered to provide a potential for
meaning making. The term representation cannot therefore be taken to have an internal
reference to some individual mental image or structure. Nor can it be taken to refer to a
determined relationship between signifier (word, picture, symbol, etc.) and signified
(represented object or concept). As the elements of communication acquire meaning in
interactions within social practices, the notion of representation must also be understood
relative to specific social interactions and practices.
When considering representation within mathematics and mathematics education
practices, there are formal systems of words, diagrams, algebraic notation etc. that are used
5
in conventional ways in relation to mathematical constructs. For example, within such
practices an algebraic expression ax 2 + bx + c and a parabola may both be used as forms of
representation of the construct quadratic function (of course the words quadratic function
may themselves be considered a further form of representation). These different forms of
representation, using different modes of communication, have different potentials for
meaning making. For example, an algebraic expression may orient participants in the
practice of mathematics education towards possible symbolic manipulations, e.g.
substituting numeric values or factorising, whereas a graphical representation may orient
participants towards specific visual features, e.g. minimum or maximum point; intercepts
with the axes; gradient. (See O’Halloran (2005) for a detailed social semiotic analysis of the
meaning potentials of these different modes.)
However, as a representation can only be interpreted within the context in which it is being
used, this context, the resources it provides and the resources that participants bring to the
interaction from wider contexts must be taken into account in analysing how
representations function in a given interaction. This conceptualisation results in an
analytical focus on episodes of interaction, tracking the functioning of particular forms of
representation in relation to other forms and to the multimodal interaction as a whole.
ETL perspective
ETL adopts a constructionist perspective where epistemology is at the centre, accompanied
by a theory and a style of pedagogical design and learning process (Kynigos, 2012). The
role of representations is important in the sense that they are perceived as integral
components of artefacts-under-change and as a means for expressing, generating and
communicating meaning. The nature of representations and the kinds of use to which they
are put are at the centre of attention to the extent to which they play a significant part in a
constructionist context, i.e. where verbal, written communication goes together with
communication through tinkering with digital representations which are also perceived as
artefacts. Constructionist artefacts can embody a wide range of complexity and have been
perceived and analysed as representations themselves (Edwards, 1998). Both the structure
and the functionality of the artefacts are important for the learning process. Some
connections can be made by Edwards' distinction between structural and functional
6
perspectives and the artefact-instrument distinction made by Vérillon and Rabardel
(1995). When a representation is put to use, precisely because it is seen as malleable, the
meanings conveyed change along with changes made to the representation itself. Unlike
the social semiotic perspective, digital artefacts are seen as representations designed by
pedagogues to embed one or more powerful ideas. At the same time, in a constructionist
setting, representations are not seen simply as objects to which some kind of meaning may
be attached but as artefacts for tinkering with. In a typical situation, you would have
humans dismantling an artefact, or improving it, or using it as a base or a building block to
create more complex structures, or simply considering it as a base from which to build
something different. So, the mathematical construct of quadratic function, to take an
example from the previous section, can be represented textually, by means of a
mathematical formalism, by means of a formalism within a programming language or by
means of a model of something created by such a functional relationship (such as a
trajectory of a projectile in a Newtonian space with gravity). In all cases, the attention is on
what is being tinkered with. Meaning is generated through the use of the artefact but is also
shaped by the representation itself. The representation plays the role of an important
element of a learning environment which is dense in opportunities to generate meanings
around a concept originally designed by pedagogues. It does allow for surprise, i.e. students
generating unexpected meanings and uses of the representations and in fact it often
welcomes creative and original meanings. However, there is didactical intentionality as to
the kinds of meanings that may emerge: the design of the learning environment and of the
representations within it are intended to support specific learning objectives.
As representations are seen as expressions of meaning, the ways in which representations
are manipulated also represent meaning. Writing an equation and then continually
changing its parameter by means of a slider is a particular type of representation of
continual change or rate of change between two variable values. Clicking on an object in
order to perform an action on that object is also an expression of meaning. Finally, also in
the core of the ETL perspective, was the idea of malleable artefacts as representations. In
line with the constructionist perspective, artefacts are considered not as entities with fixed
properties but as objects under change and in use. The activities of discussing behaviours,
7
planning and implementing changes and engaging in continual tinkering of the artefacts are
integral parts of learning. In the same sense, representations are seen as directly linked or
integral parts of the artefact.
Comparing the two perspectives
An important distinction between the perspectives of the two teams lies in their approach
to the cognition of individuals. While both teams conceive of learning as something that
happens in social interaction, the IOE’s perspective only allows one to speak of learning in
terms of changes in patterns of interaction, without any move to take such changes as
evidence of changes in individual cognition. On the other hand, ETL perceives cognition as
the generation of meaning within a complex context of communication and construction.
The attention is on studying what meanings are generated, how they are connected to and
emerge from the situation at hand (see Noss & Hoyles, 1996) and how they evolve through
these interactions.
Another distinction is the role attributed to representations in the learning process. The
ETL use of the notion of ‘tinkering’ with specially designed representational artefacts
identifies a specific form of student activity that is theorised to be conducive to learning
about the structure of the represented construct. The IOE on the other hand does not
privilege particular forms of representation, seeing them all as resources that are available
to students to make meanings. The focus of the IOE team’s research is on exploring the
kinds of meanings facilitated by the multiple resources available to students.
The IOE approach conceives of learning as changes in patterns of interaction rather than as
intra-personal changes. From this perspective, the adoption and use of new
representations in communications within mathematical contexts is a form of learning. The
ETL perspective considers the extent to which a specific representation, and the ways it is
connected to others, is conducive to the generation of meanings. A given representation
such as formalism, which in the pre-digital era may have been an obstacle to meaning
generation for children, may in some cases have been intentionally included in a digital
expressive medium so as to be used in a meaningful way. Formal and traditional
mathematical representations thus become texts that may be interrogated and new forms
8
of representations are sought and considered. ETL thus focuses on ways in which
traditional representations can be put to use as for example when mathematical formalism
becomes a programming language connected with graphical representations to construct
parametric models.
Cross-experimentation and cross-case analysis study
The ReMath project used the methodology of “cross-experimentation” (Bottino & Kynigos,
2009) as a means to investigate the ways in which the representations offered by digital
artefacts designed for learning and teaching mathematics function in mathematics
classrooms and to support the communication between different teams of researchers. Use
of this methodology recognises and takes account of the fact that the context within which
classroom experimentation takes place and the theoretical frameworks used by the
researchers affect the conduct and the outcomes of the research. In brief, two research
teams, working in different contexts, each devise and conduct a programme of classroom
research using the same digital artefact designed by one of the teams. A common research
question is agreed, but must then be made specific by each team, according to their
particular interests and theoretical frameworks. A set of instruments is used to gather
information from each of the teams before and after the classroom implementation,
enabling a careful elaboration of: the principles underpinning the design of the digital
artefact; the didactic functionalities (Artigue et al., 2009) of the artefact attended to by each
team; the local and global context in which each classroom experimentation takes place;
the pedagogical plan used in each classroom; the data collected and methods of analysis. In
elaborating these details of the methods used, each team makes explicit the ways in which
the context of their research and their theoretical orientations affect the design and
conduct of the research.
Having conducted such cross-experimentation, the analyses of data conducted by each
team of researchers are recognised to be strongly located within their separate contexts.
An important part of the context is the theoretical framework employed by each research
team. Understanding the role of context in the analysis is partly enabled by the rigorous
specification of context provided by the cross-experimentation instruments. Each team’s
9
specification of the role of theories in their analysis, however, while clarifying and
justifying the production of the research outcomes, is not sufficient to allow us to consider
fully the question of the impact of the theoretical perspective. In order to explore this issue
more deeply and to develop a fuller understanding of the similarities and differences
between our various perspectives a cross-analysis study was conducted. In this crossanalysis study, each team chose a selection of data and its analysis from their own
experiment. The data were re-analysed by the other team, using their own theoretical
perspective, drawing on all the contextual information provided by the crossexperimentation. The two analyses then provided an opportunity for reflection on the
similarities and differences between the perspectives and, importantly, on the ways in
which apparently similar constructs (such as in this case representation) are used. The
cross-analysis study methodology thus enables not only richer insights into the data but
also a means of articulating the theories more precisely. We use the word articulating here
with deliberate awareness of its dual meaning: both defining the connections between the
perspectives and speaking each of the perspectives more clearly by developing our
understanding of the ways its constructs are used.
MoPiX software: description and design principles
The example presented in this article involved the use by teams based in London (IOE) and
Athens (ETL) of the digital artefact MoPiXii. MoPiX was designed by the IOE team as an
exploratory environment for constructing animated models using the principles of
Newtonian motion. The objects of the MoPiX microworld were designed to behave in
mathematically coherent ways, providing an environment that, by exploring and building
models within the microworld, was intended to allow students to construct orientations to
concepts such as velocity and acceleration consistent with conventional mathematical and
physical principles.
The MoPiX microworld consists of objects whose properties are determined by sets of
equations. The basic properties include shape, colour and position (defined by values of
Cartesian coordinates). The equations determining these properties can include a time
parameter, thus enabling change in e.g. position over time. A library of equations is
provided as a starting point for users, including sets of equations that will cause an object
10
to move with a given velocity and acceleration. Other equations in the library include ones
which detect whether an object is in a given position (e.g. “hitting” the side of the screen or
another object). Conditional operators can be used within equations, thus allowing, for
example, a change of direction if a moving object “hits” another. Users can edit these
equations or create new ones in an editor. Objects and models (objects and their
associated equations) may be saved to a central server where they are immediately
available to all other users.
An object in its initial state appears as a small black square. It can be “created” by dragging
it onto the “stage” - the part of the screen where the animated model is constructed and set
in motion. Equations are then assigned to the object by dragging and dropping and these
equations are implemented by “running” the model. Although the motion of animated
objects on the screen appears continuous, the equations model velocity and acceleration by
discrete incrementation over time of position and velocity respectively (see Figure 1).
---- insert Figure 1 about here ---The IOE team’s multimodal social semiotic framework discussed above informed the
design of MoPiX (see Bezemer & Kress, 2008 for a discussion from this perspective of
issues involved in design of pedagogic texts). This theoretical framework highlights the
different potentials for meaning offered by different modes of representation. Each mode
involves its own distinctive system of elements, grammar and meaning potential.
Moreover, interaction between such different modes creates further opportunities for
meaning making. MoPiX was thus designed to provide a multi-semiotic environment with
rich potential for making meanings drawing on multiple resources, including those
occurring in the wider social environment of its useiii. In particular, MoPiX links a formal
notation of equations with visual animated models. The trace of motion of an animated
object can also be perceived as a Cartesian graph.
However, social semiotics is not a theory of learning. While it can suggest how students
may make sense of the multimodal texts they experience in the classroom and hence
suggests some characteristics of good learning environments, it is not sufficient by itself to
inform the design of activities for learning. The design of MoPiX was also influenced by a
11
broadly constructionist theoretical frame. The constructionist approach to learning (Harel
& Papert, 1991; Kafai & Resnick, 1996) promotes investigation through the design of
microworld environments. MoPiX was conceived as a constructionist toolkit (Strohecker &
Slaughter, 2000), a dynamic visual environment that supports construction activities in
social contexts, based on these constructionist principles. Learners use the fundamental
elements of the microworld (equations and objects whose properties and behaviours are
defined by the equations assigned to them) to build objects and models with new sets of
properties and behaviours. They may then activate their constructions to investigate them,
forming and testing hypotheses about their behaviours. While the ideas of both multimodal
social semiotics and constructionism provided principles for the design of MoPiX, the IOE
team involved in the cross-experimentation did not draw on constructionism, rather
conceiving of their research only from the multimodal social semiotic perspective. On the
other hand, the constructionist design principles fitted the constructionist framework of
the ETL team. Besides the general constructionist design principles already embedded in
MoPiX, the ETL team focussed on specific aspects of the representations such as the
concern to find a meaningful use of mathematical formalism and the different ways in
which representations can be manipulated.
‘Distance’ of MoPiX representations
Morgan et al. (2009) adopt the term distance to describe the extent to which teachers and
students may experience the representations provided by a piece of software as different
from or similar to the familiar representations used in their classrooms. Representations
embedded in digital media are characterised by their inter-connectivity and the ways in
which they can be manipulated dynamically. Both these characteristics have potential to
increase distance. The representational elements of MoPiX are compatible in some ways
with traditional mathematics semantically and syntactically but also differ in significant
ways from the traditional repertoire. For example, there are semantics for handling time
and position and for handling specific properties of objects such as colour. The formalism
of MoPiX has some similarities to traditional mathematics, but it also has characteristics
which transform it into a programming command. A further important aspect of the
distance of MoPiX representations from familiar classroom representations is connectivity.
12
MoPiX mathematical formalism is inextricably connected to graphical representations of
objects, including both features of their appearance and the nature of their movement,
which can be perceived as physical objects or as strictly mathematical objects according to
the properties given to them and the field in which they exist.
Lastly we consider what Morgan et al. (2009) call curricular distance: the relationship
between the ways MoPiX may be used and the content and methods of the usual
curriculum. In particular, we note the role of formalism in MoPiX to describe and control
the properties and behaviours of animated objects. A curriculum in which traditional
formalism is used only to solve abstract algebraic exercises may be considered as more
distant from MoPiX formalism than when traditional formalism is used to express real
world relations and behaviours. So, in contexts where the curriculum and pedagogy
construct mathematics as applied and mathematical learning as experiential, MoPiX is less
distant than in contexts where mathematics is considered as abstract and as an end in
itself.
The IOE experimentation
In designing the classroom experimentation in London, the IOE team drew on the
theoretical concerns discussed above as well as taking account of the educational context
within which they were intervening. In accordance with the original conceptualisation of
MoPiX as a means of addressing ideas about motion, we chose to work with students in
their final year of pre-university studies (17 years old), for whom the study of elementary
Newtonian mechanics formed a part of the mathematics curriculum. The high stakes nature
of the examination system in England meant that the research team’s access to students
necessitated making close connections to the existing curriculum and making these
connections explicit to students and to teachers. The pedagogical plan was thus devised
around constructs that could be identified within the standard curriculum, for example:
straight-line motion, constant acceleration and acceleration as a force.
The pedagogical plan and the organisation of the IOE teaching experiment were designed
to enable students to communicate using pencil-and-paper-based representations
involving conventional or informal notations or diagrams, using 'natural' language in face-
13
to-face speech and by sharing MoPiX objects, equations and models electronically. Some
parts of the plan explicitly directed students’ attention to connections between different
forms of representation, for example, making changes to values in given equations in order
to observe and describe the effects of such symbolic changes on the motion of an animated
object. At other times, students were given more open tasks in which they could choose
which type of representation to use, for example, making an animated model of their own
design, a task that in practice involved use of paper and pencil, a computer drawing
package, gestures and talk as well as equations drawn from the MoPiX library, new
equations formed by editing these and the moving objects produced on the MoPiX stage. In
Morgan and Alshwaikh (2009) we present an example of the work of a pair of students on
such an open task
The multimodal social semiotic approach proposes an analytic focus on the ways in which
different modes of representation function within teaching and learning interactions. The
analytic method adopted by the IOE team thus identified: the ways in which each mode of
representation was used; how students’ uses of the various modes of representation
related to each other; and how the use of each mode of representation, both separately and
in conjunction with other modes, contributed to the development of shared orientations to
the task and to mathematical concepts (also see Morgan & Alshwaikh, 2009).
The ETL experimentation
The experimentation by the ETL team took place in a Secondary Vocational Education
school in Athens. Eight 12th grade students (17 years old), studying mechanical
engineering, worked in groups of two or three for 25 school hours. These students had not
associated mathematics with engineering before and their experience reflected the lack of
experiential learning activity characteristic of Greek schooling. They were enthusiastic to
connect engineering with programming digital models. The researchers were not tightly
restricted by specific curriculum or examination demands, so they designed activities
which related to the engineering and mathematics curriculum while not necessarily tightly
matching its expositional-theoretical nature.
14
A pivotal feature of the pedagogical plan was to give students what the ETL group call a
'half-baked microworld' and ask them to make changes to improve it (Kynigos, 2007). This
is a digital artefact specially designed to be faulty or incomplete. The artefact incorporates
an interesting idea but the point is for students to see sense in deconstructing it, in building
on its parts, making changes and eventually constructing a new artefact, which may be
distinctly different than the original one. In this case the artefact was called “The Juggler”
(Kynigos, Psycharis, & Moustaki, 2010) and consisted of three interrelated objects: a red
ball and two rackets with which the ball interacted. The ball was animated using equations
to define its motion. Equations affecting the ball’s behaviour in relation to the rackets were
especially created by the researchers with the use of the MoPiX formalism. The rackets
were not animated, but it was possible to move them around using the mouse and thus
make the ball ‘bounce’ on them, forcing it to move away in specific ways.
The students were asked to execute the Juggler model, observe the animation generated
and identify the conditions under which each object interacted with the others as well as
the objects’ possible changes of behaviour because of these interactions. They were
encouraged to discuss with their teammates how they would change the Juggler
microworld and embed in it their own ideas regarding its behaviour. In the process of
making changes the students were expected to deconstruct the existing model, linking the
behaviours generated on the screen to the corresponding equations in order to reconstruct
the microworld, employing strategies that would depict their ideas about the new model’s
animated behaviours.
During the experimentation process one researcher acted as a teacher since she had been a
teacher in this school for several years and the other one as a co–researcher. During the
activity itself, the researchers circulated among the teams, posing questions, encouraging
students to explain clearly their ideas and strategies, asking for refinement and revision
when appropriate and challenging students to express openly their thoughts and put into
effect their ideas. At specific time points during the sessions the researchers orchestrated
whole class discussions.
For the ETL team, the representations are seen as artefacts which play the role of
expressions of meanings. There are different possible ways to group these representations.
15
One is similar to the IOE approach i.e. with respect to the mode: formalism, graphical
model, written text, spoken language. The other is with respect to the kind of artefact, for
example a conditional equation linked with the respective model behaviour. A third is to
group representations of a specific concept such as variable. In all cases, the constructionist
framework focuses on reification, i.e. on how an artefact plays the role of an expressive
object, how it is addressed, changed, discussed over and used.
In analyzing the data, the ETL group first looked for instances where meanings stemming
from the students’ interaction with the available formalism were expressed. The unit of
analysis was the episode, defined as an extract of actions and interactions performed in a
continuous period of time around a particular issue. The episodes which are the main
means of presenting and discussing the data were selected (a) to have a particular and
characteristic bearing on the students’ interaction with the available tools during which
MoPiX formalism was used to construct mathematical meaning and (b) to represent clearly
aspects of the reification processes emerging from this use (e.g. articulating variables and
invariants within an equation and conceptualising the structure of an equation as a system
of connections and relationships between its component parts).
Examples of cross-analysis
In this section we present two episodes, one from each of the two classroom
experimentations and the analyses of these episodes provided by each of the two research
teams. We then reflect on how these analyses differ and on the sources of these differences.
As space is restricted, we provide a narrative account of each episode, including brief
extracts of data that are particularly significant to our analyses.
Episode 1 from the IOE experiment
In the seventh session, students were introduced to the idea of acceleration applied to an
object at an instant. They experimented with applying acceleration equations of the form
Ax(ME,20)=3, observing the effect as a sudden change in direction. (Together with the
equation Vx(ME,t+1)=Vx(ME,t)+Ax(ME,t), this increments the current velocity with a “kick”,
applying an acceleration of 3 units in the horizontal direction when time is 20). Students
were then posed the task of using such acceleration in order to draw a square. In an earlier
16
session students had worked on the outwardly similar task of drawing shapes (not
including a square) by making changes in velocity. It was here that Roniv decided to start.
Rather than using acceleration, he first used velocity equations to turn the corners of his
square then started the task of drawing a square using acceleration equations.
After some initial hesitation he created his object, assigned it a basic set of motion
equations and, after a short period of trial and improvement using strategies such as
changing the signs or swapping the values of Vx and/or Vy, found the necessary equations
to turn the first corner of the square. He then completed the other corners of this square
efficiently and accurately. Ron’s initial systematic trial and improvement strategy of
changing the sign or swapping the values of the new velocity worked well in this case
because of the nature of the relationship between horizontal and vertical components of
velocities of perpendicular motions.
On completing the task, his growing confidence was apparent as he explained
spontaneously to his partner how to make an object turn right-angled corners.
“ When you want it to turn you got to say at 20, or whatever you want, Vy equals
zero, Vx equals three, whatever what happened, at 40, Vx equals zero, Vy equals
minus three, at … “
He then started the task of drawing a square using acceleration equations. As he did this, he
kept his model of a square formed by using changes in velocity on the screen and
constructed his second model next to it, running both simultaneously and comparing the
results at each stage. After creating the basic motion of the new object without hesitation,
Ron then seemed to run into difficulties.
As he tried to turn his first corner using trial and improvement as before, the change
sign/swap values strategy no longer worked. He turned to alternative strategies such as
doubling and trying extreme large and small values of acceleration. These strategies
focused only on the values of the acceleration and his exploratory attempts appeared to
take no account of the desired values of the velocity. After 11 minutes and 14 trials he
succeeded in finding the values of acceleration needed to turn the first corner. Having
17
achieved this, he proceeded to turn the other corners successfully and relatively efficiently,
having to make only minor corrections.
When he came to the final corner, wishing to make the object stop, he encountered new
difficulty as the pattern of changes of sign and values that was successful in turning corners
was not useful for coming to a stop. At this point he flipped over the two models and
examined the equations used in each case, apparently comparing the values of velocity and
of acceleration at each of the corners. With significant pauses for thought, he succeeded in
adding correct acceleration equations without further trials. Finally, having completed a
correct model, he spent time inspecting the equations of the original model built using
changes in velocity, pointing to the various values of velocity as if calculating what
acceleration would be needed to achieve the same effect. Table 1 presents a comparison of
Ron’s processes as he attempted the two tasks.
---- insert Table 1 about here ---Initial analysis from the IOE perspective
Ron’s activity on this task is characterised by alternation between working with MoPiX
formalism and running the animated model. The interaction between these two modes of
representation is fundamental to his developing strategy. Ron’s earlier experience with
MoPiX enabled efficient association of change of direction of motion with change in values
of horizontal and vertical components of velocity. Two representational features of MoPiX
seem to play an important role in this initial activity. The connectivity between MoPiX
formalism and the movement of a graphical object enables a trial and improvement
approach. Secondly, this approach works well because the symbolic language of MoPiX
embodies the separation of velocities into horizontal and vertical components.
By keeping his model of a square drawn using changes in velocity on the screen while
constructing his second square using changes in acceleration, Ron’s behaviour seems to
construe the two tasks as parallel, running both simultaneously and comparing the results
at each stage. In the initial stages of the second task, however, he only compared the visual
output of the animation, not the symbolic structure. Instead, he re-used his trial and
improvement strategy, making use of the connectivity between formalism and graphics
18
within the second model but not between the two models. Unfortunately, in the case of
acceleration, the value needed to produce a given velocity is dependent on current velocity
rather than on the previous acceleration. After an extended period of trialling, his eventual
success in finding the value of acceleration needed to turn the first corner enabled him to
turn subsequent corners, presumably by following a numerical pattern, as he still did not
appear to refer back to the desired values of velocity. This numerical pattern was not,
however, useful when faced with the task of making the object stop at the final corner of
the square.
At this stage he turned to a comparison between the two models within the symbolic mode,
flipping over both models and examining the equations used in each case. Engagement
with the symbolic mode in MoPiX and interaction between this and the animation mode
enabled him to complete the task successfully. His final period of inspection of the sets of
equations for both objects, pointing in turn to the velocity equations used at each corner of
the original model, suggests a move towards a focus on acceleration as change in velocity.
Considering the relationships between the available representations and Ron’s activity, it
appears that the congruence between simple numerical patterns in the values for velocity
used in the formalism and perpendicular motions in the animated graphics enabled
successful solution of the problem of constructing a square through using the number
patterns. The lack of congruence between similar patterns in the values for acceleration
and the desired motions, especially when bringing the model to a halt, demanded that Ron
engaged in a different way with the formalism. Rather than focusing simply on the values of
the variable to be changed, he had to make connections within the formalism between the
values of velocity and of acceleration.
Initial analysis from the ETL perspective
Initially Ron attributed to his object ready-made equations that he found in the Equations
Library classified under the “Horizontal” and “Vertical Motion Equations” categories. As he
had already gained familiarity with the equations of those two categories and the meaning
their symbols conveyed, Ron carefully selected only the equations that would assist him in
drawing a shape and disregarded others. Having worked before with shape drawing using
motion equations, Ron chose to bring to this task a strategy that he had previously followed
19
and that had proved successful. Thus, although he initially attributed to his object an
acceleration equation, he preferred to investigate the role of the velocity equations -instead
of acceleration equations- in drawing a square, which seemed to be consistent with what he
had achieved up to that point.
Having identified the meaning the symbols in the velocity equations conveyed and having
particularly articulated an understanding regarding the variable of time and its specific role
in the equations, Ron performed a series of changes not only on the right part of the
equation, substituting one numerical value for another, but also on the left part of the
equation, substituting the variable of time to specific arithmetic values. The continuous
changes in the values as part of his experimentations with the velocity equations were not
confined to substituting one arithmetic value for another but also involved sign changes to
signify changes in the object’s direction.
Ron’s initial systematic trial and improvement strategy of changing the sign or swapping
the values of the new velocity worked well also because he had deep structure access to the
symbolic facet of the model animated on the screen. In the process of debugging his model,
Ron pressed the “Play” button to observe the animation generated and flipped his object to
identify the equation responsible for the buggy behaviour several times, developing in the
way meaningful connections between the mathematical formalism and the graphical/visual
representation of the model. Specifying each time which equations needed to be fixed, Ron
performed a series of changes editing the symbols of the velocity equations.
Mentioning to his partner that he could use “20 or whatever you want” as the time point at
which changes to the values of velocities should be made so as to change the object’s
direction, Ron seems to have reached a higher level of abstraction as he appears to have
identified “20” not as a fixed arithmetic value necessary in drawing any square but as a
value that could be of the user’s choice.
Ron used the model he had developed before as a starting point to go further with his
experimentations with the acceleration equations. Thus, he initially attributed to his new
object the equations he used to make the first object move upwards and draw one of the
square’s sides bringing in once again strategies that he had successfully employed before.
20
As he had already created a model in which he used the velocity as the varying quantity
inducing changes to the object’s direction, Ron focused on producing a new model having
the exact same effect to the object’s direction, using this time the acceleration as the
varying quantity instead of the velocity. The strategies he selected to use in this case also
seemed to differentiate. In order to produce the first turn, Ron made several changes to the
arithmetic value on the right side of the x and y acceleration equations. Nevertheless, these
changes seemed to be coherent as he moved from doubling the value of the acceleration he
had previously attributed to his object to giving extremely large and small values,
observing in each case the animation generated. At that point Ron didn’t seem to have
developed concrete links between the changes in acceleration and the changes in direction.
The fact however that any actions he performed to the model’s symbolic facet (e.g.
editing/modifying or at several times inserting/removing acceleration equations)
produced a direct change to the visual result generated on the Stage, gave Ron the
opportunity to gradually move to a more solid conceptualization of the mapping between
direction and acceleration and to continue his construction having identified a pattern of
changes to be made so as to make the object turn.
Coming to the final corner, Ron seemed to have realised that the pattern he had previously
identified and successfully used wouldn’t make his object come to a stop. Thus, instead of
making any attempt to attribute the values 3 or -3 to the x and y accelerations as he had
before, he decided to explore the potential of attributing to both accelerations the value -6.
At this point, Ron seemed not to have identified exactly the way in which changes in the
acceleration equations affected the velocity of the object (so as to make the necessary
changes in the acceleration values and cause the object stop at a specific time point). In this
case the deep structure access the user has in the MoPiX environment was again proven to
be useful as Ron flipped both his objects to inspect the symbolic facet of the two models.
Recognizing an equivalence between these two models, Ron seemed to be calculating at his
second model the values of the X and Y velocities for each time instance (through the
equations Vx(ME,t)=Vx(ME,t–1)+Ax(ME,t) and Vy(ME,t)=Vy(ME,t–1)+Ay(ME,t)) and
compare them to the ones that appeared on his first model in the form of “V = an arithmetic
value”.
21
Initially Ron seemed to be attempting to make connections between a varying quantity (the
velocity in the first case and the acceleration in the second) and the changes in direction to
be produced so as to make his object turn. He started inserting and changing arithmetic
values in the velocity and acceleration equations and each time started the animation to
observe the graphical effect of his actions. The deep structure access and the linked
representations gave Ron the opportunity to develop an understanding between the
changes in direction (the visual effect) and the modifications made to the acceleration and
velocity equations (the manipulations performed using the mathematical formalism).
However, during the last part of his experimentations with the acceleration equations, Ron
seemed to develop an understanding regarding the relationship between the two varying
quantities (i.e. the velocity and the acceleration) in drawing the squares. Up to that point,
he didn’t seem to have made any connections between velocity and acceleration as in order
to construct the first model he merely manipulated and modified velocity equations, while
in order to construct the second one, he solely used acceleration equations. Any
modifications made to each one of them were regarded in isolation. Flipping the two
objects, the symbolic facets of the two models were put next to one other. Needing to
calculate the velocity at each time instance for the second model and match the values
calculated to the ones that appeared in the first one, Ron came to use the equations
Vx(ME,t)=Vx(ME,t–1)+Ax(ME,t)” and Vy(ME,t)=Vy(ME,t–1)+Ay(ME,t) which describe the
relation between the acceleration and the velocity at each time instance.
Reflecting on the two analyses
Before turning to consider theoretical differences reflected in the analyses, it is worth
noting that the relationship of each team to the research context affected their analytic
focus. The IOE team had been involved in designing and teaching lessons with a specific
curricular focus. Their interest in analysing the episode, while primarily concerned with
the ways the student made use of the multiple forms of representation, was also influenced
towards considering the ways in which the student’s activity within the set task may have
contributed towards the curricular goals envisaged in the task design (i.e. operating with
acceleration as change in velocity). ETL chose not to address this curricular focus in their
22
analysis, rather making links to the key issue addressed in their own teaching experiment:
the way symbolic formalism functioned as a resource in the problem solving process.
In particular, the IOE analysis noted that the separation of horizontal and vertical
components of motion in the MoPiX symbolic system appeared significant to the solution
process. This separation was interpreted as enabling a match between the student’s
pattern of positive and negative trials and the values of velocity required for perpendicular
motion - a match which is no longer valid as the student proceeds to deal with acceleration,
hence necessitating other strategies in order to achieve a successful solution. With its close
connection to the specific curricular aims of the task, this representational aspect of MoPiX
did not form part of the ETL analysis.
Both analyses focus on the ways in which particular characteristics of the MoPiX
environment appeared to affect the student’s behaviour and his pathway to successful
completion of the task. For both teams, the connectivity between symbolic and animated
graphic modes is identified as significant to the solution process and of interest
theoretically.
From the IOE perspective, this connectivity is of interest because of the additional meaning
potential it affords, enabling the various systems of representation to be used both singly
and in combination to construe mathematical meanings. The initial approach to the task
was identified as “trial and improvement”. Although this phase of activity suggests that the
student has a focus on pattern making rather than on relationships between the
components of motion, repeated movement between symbolic and graphic modes
combined with manipulation of the symbolic formalism also construes a causal relationship
between the formalism and resultant graphic behaviour.
ETL’s analysis also noted incidents in which the student, after starting and observing the
animation, flipped the object to look for the equations that needed to be fixed so as to
produce the desired visual effect. From their perspective, this behaviour is interpreted as
evidence that the student was developing meaningful connections between the
mathematical formalism and the graphical/visual representation of the model.
23
Another aspect of MoPiX considered important by the ETL team is what they term the
“deep structure access” that the symbolic mode provides to the way objects behave. In the
final part of this episode, flipping objects and putting side by side the symbolic facets of two
equivalent models allowed the student to inspect and compare the equations comprising
these models, using in this way the mathematical formalism as a means to develop an
understanding of the relationship between the specific varying quantities appearing in
both models. The IOE team also found the final part of the episode of interest. However,
rather than interpreting this in terms of the student’s developing understanding, the
inspection of the equations of the two models, accompanied by gestures indicating focus of
attention, is of interest because it construes a connection between velocity and acceleration
that was not apparent at earlier stages of the episode.
Episode 2 from the ETL experimentation
In the episode we consider here, a pair of students had inserted a new object, an ellipse,
into the “Juggler” microworld and wanted to modify the behaviour of the moving ball so
that it would change its colour according to whether it was higher or lower than the ellipse:
S1
What I want to happen is that: when the ball is above the ellipse to become red and when
it is below the ellipse to become green. I don’t care about when it hits [the ground]. Can
we do this?
S1’s partner related this conceptualisation of the desired effect to other effects familiar
from their previous experience with the microworld:
S2
You have to define something. How did you define the plane which is the ground? How
did you define that on the right side there is a wall and that you can’t go beyond this wall?
[The “ground” and the” wall” are elements of already existing equations that the students
had used].
This need for definition was then related to the XY coordinate system:
S1
This. The: “I am below now”. How will we write this?
S2
Using the Y. Using the Υ. The Y. That is: when its Υ is 401, it is red. When the Y is
something less than 400, it’s green!
The students started to develop a new equation to express this behaviour in the Editor. As
there was no in-built MoPiX symbol to express the idea of an object becoming green under
24
certain conditions, they created a new symbol gineprasino (i.e. become green in Greek),
giving it the standard format of a MoPiX variable, varying over time t. The first version of
their equation was developed as:
gineprasino(ME,t)=y(ME,t)≤274
(1)
As this equation did not achieve the desired effect, the students decided to construct
another equation in which they attempted to integrate the gineprasino variable. They used
the structure of an equation (provided in the MoPiX library) that they had already used:
Vx(ΜΕ,t)=
(not(amIHittingASide(ΜΕ,t–1))×(Vx(ΜΕ,t–1)+Ax(ΜΕ,t–1))+(amIHittingASide(ΜΕ,t–1))×(Vx(ΜΕ,t–1)×–1)
(2)
which defines how the velocity of an object changes, including a change of direction if it hits
an edge of the MoPiX screen. This equation incorporates use of the variable
amIHittingASide(ΜΕ,t), taken from the MoPiX library and defined as:
amIHittingASide(ME,t)=(x(ME,t)≤ 0 or x(ME,t)799) and Vx(ME,t)≠0
(3)
The students appeared to recognise a similarity between changing velocity under a given
condition, as in (2), and changing colour under a given condition. They also attributed a
similar role to the variables amIHittingASide and gineprasino, each of which identifies the
condition under which the desired change is to take place. They thus duplicated the
structure of (2), eliminating its specific content and using it as a template to define what
would happen to the ball’s colour when below the ellipse:
greenColour(ME,t)=not(gineprasino(ME,t))×0 + gineprasino(ME,t)×100
(4)
greenColour is an in-built MoPiX variable, already familiar to the students, that may take
values from 0 to 100, controlling the saturation of green. Like the variable amIHittingASide,
the variable gineprasino acts as a truth function, taking values of 0 or 1 according to the
current value of the y-coordinate of the object. Equation (4), applied to the ball object, thus
assigns 100% green saturated colour to the ball when its y-coordinate is 274 or less (i.e.
below the fixed position of the ellipse) and 0% green when it is above the ellipse (see
Figure 2).
25
---- insert Figure 2 about here ---Initial analysis from the ETL perspective
In line with the analytical framework discussed before, the ETL group looked in this
episode for events that indicate qualitative changes in the ways the students used the
formal equations register. In their attempts to insert another object whose behaviour was
related conditionally to the first object and to a fixed parameter in the geometrical plane
(the y value), there were several events in which the students reified the equations as
expressions through their interaction with the available tools. These events suggest that
the students were able to develop insights into the consequences of the equations’
formalism on the behaviour of the two objects as well as to cope effectively with structural
aspects of the equations.
While building equation (1) the students thought to create a variable and to give it a name
corresponding semantically to the variable's function (i.e. the gineprasino - become green variable). They also related the symbols with a mathematical system (i.e. the x-y coordinate
system) and manipulated variables, numerical values, equals and inequality symbols.
Surprisingly, in attempting to link the behaviour of the ball to its position relative to the
ellipse, the way in which they used the y-coordinate concept for each object was distinct.
The ball’s y-coordinate was expressed in terms of a quantity varying over time (y(ME,t)),
while the ellipse’s y-coordinate was expressed in terms of the constant arithmetic value
corresponding to the object’s position at that time on the Stage (i.e. 274).
In building the second equation, the students’ meanings seemed to evolve by including a
view of equations as objects, i.e. as higher order representational units. First, the students
extracted mathematical meaning from an equation which as a whole seemed to describe a
behaviour similar to the one they wished to attribute to their own object. Conceptualizing a
mapping between the idea “the ball should change its velocity when it hits one of the
Stage’s sides” and the idea “the ball should change its colour when it is situated below the
ellipse”, the students used the structure of equation (2) but inserted new terms in order to
define a completely novel behaviour for their object. This constitutes a clear indication that
the students were able to relate the MoPiX equation they were constructing with an
existing equation and to recognise mutual connections between those two structures. Thus,
26
students appeared to recognise the existence of structures external to the symbols
themselves and use them as landmarks to navigate the construction process to produce
equation (4).
The manipulation of the terms of the equation reveals further their developing structural
approach to equations. Inserting the gineprasino variable and providing it with new forms
(i.e. not(gineprasino)), the students seem to have conceptualised equation (2) as an object
and used it as a means to encode meaning and structure in equation (4). From the ETL
perspective, this reflects a kind of mathematical thinking that relates to the development of
a good algebraic structural sense accompanied with the acquisition of a functional outlook
to equations as objects, which is considered to be crucial to relational understanding.
Initial analysis from the IOE perspective
As before, the IOE analysis focuses on relationships between the various modes of
representation and the students’ activity. In this case, the episode involves movement
between everyday language and the formalism of mathematical symbolic systems and
MoPiX equations. S1 first uses everyday language to describe the goal of the activity. By the
end of the initial discussion, this goal has been reformulated in terms of the XY coordinate
system. We suggest that some representational characteristics of the MoPiX environment
play an important role in this reformulation. In particular, there are a number of terms
within the MoPiX symbolic language that may be considered “boundary” terms, reducing
the distance of MoPiX from students’ previous experience: there is a convergence between
MoPiX terms such as ground, side, and amIHittingASide and their use in everyday discourse.
The interdiscursivity of such boundary terms allows the meaning potential of one system
to inform how the other system is understood. The creation of the new symbol gineprasino
(become green) relates MoPiX language to the original everyday expression of the goal of
the activity. This can be considered a new boundary term.
The construction of equation (1) may be seen as a direct movement from the everyday
formulation of the goal: “gineprasino(ME,t)=y(ME,t)≤274” is a very close translation of
“When the Y is something less than 400, it’s green”. However, this similarity also highlights
fundamental differences between the two semiotic systems, indicating a high degree of
epistemological distance. Whereas in everyday language “become green” is associated with
27
change in colour, in MoPiX formalism it has no such association. As defined in equation (1),
gineprasino is simply a variable that may take values of 0 or 1 depending on the ycoordinate of the position of the object.
By using the structures of everyday language, the students’ first attempt is not successful.
The surprise provided by the visual feedback from this first attempt seems to have
prompted a reappraisal of their approach. It is also likely that it prompted an association
with earlier experience (evident in data drawn from previous episodes) attempting to use
the terms amIHittingASide or amIHittingGround, which have similar properties to
gineprasino, that is: an ‘everyday language’ meaning; MoPiX variable taking values 1 or 0;
no visible effect on MoPiX animation unless integrated into other equations. The initial
exchange as the pair negotiated the task also seems key to making this association and
deciding to use equation (2) as a template: S2 had immediately recognised the relevance of
previous work with “the plane which is the ground”.
As we are looking at a process of development of a solution over time, the IOE analytic
approach tracks the connections between the semiotic resources used as the episode
progresses, reconstructing a semiotic chain as a means of examining how the everyday
description of the visual characteristics of the imagined solution is transformed into a set of
MoPiX equations that will realise that solution. The chain is reconstructed by identifying at
each step of the episode the source of the semiotic resources used by the students (cf.
Carreira, Evans, Lerman, & Morgan, 2002). In this episode the sources identified include
‘everyday’ language (e.g. “the ball is above the ellipse”), mathematical language (e.g. “x, y
coordinates”), MoPiX formalism (e.g. “gineprasino(ME,t)=y(ME,t)≤274”). We additionally
identify what we have called ‘boundary terms’ such as ground which are both part of
everyday language and components of pre-defined MoPiX terms. Such boundary terms
seem important in the formation of the semiotic chain, as they are both meaningful in the
everyday discourse and functional within the MoPiX environment. Indeed, we see the
newly formed term gineprasino to serve as a boundary term that plays a linking role in the
transformation of the students’ original vision into a functioning MoPiX model.
28
Reflecting on the two analyses
Comparing the IOE analysis of this episode with that provided by the ETL team, we note
that both teams highlight the students’ identification and exploitation of similarities
between their current goal and previous MoPiX experience with equations including terms
such as amIHittingGround. The IOE focus on links in a semiotic chain, however, leads to
some different interpretations. In particular, the construction of the equation
gineprasino(ME,t)=y(ME,t)≤274, seen as surprising in the ETL analysis because of the
different forms of expression used for the y-coordinates of the ball and of the ellipse, is
explained in the IOE analysis as a close translation of the students’ earlier everyday/
mathematical language expression “When the y is something less than 400, it’s green”. The
IOE analysis sees structural similarities both between expressions in different semiotic
systems and between expressions within the MoPiX system as playing crucial roles in
enabling the students to move towards the solution of their problem. While the ETL
analysis also lays importance on links between expressions within MoPiX, in particular the
use of existing equations as templates for the construction of new ones, it additionally
posits the use of conceptual similarities between existing and new equations. While not
denying the possibility of such conceptual connections, the IOE approach does not address
them.
Reflections on the cross-analysis
The analytic approaches taken by the two teams are not on the whole incompatible and
yield interpretations of the data that have some similarities. For example, in analysing
episode 1, both analyses consider interaction between the symbolic formalism of MoPiX
and its graphical effects as critical to the problem solving process and identify the student’s
final inspection of the two sets of equations of the two models as key to his making
connections between velocity and acceleration. However, the two teams emphasise
different aspects of student use of the representations and a significant difference may be
identified in the two analyses in the ways in which the two teams treat the representations
offered by MoPiX and the relationships between different systems of representation. For
ETL, the symbolic formalism of MoPiX plays a particularly important role because of the
way that it provides access to the ‘deep structure’ of the environment. Furthermore, the
29
analysis is connected with a pedagogical intentionality, focusing on the ways in which
formalism, through its connections with the other representations and affordances for
manipulation, may become meaningful to the students. While the IOE perspective does not
deny the role symbolism may play in meaning making, it does not have a privileged role in
the IOE analysis. Rather, the various systems of representation are considered of equal
significance, the interest being in analysing what each brings, both individually and in
combination, to the possibilities for meaning making and problem solving. This difference
is highlighted in the interpretations of the equation gineprasino(ME,t)=y(ME,t)≤274 in
episode 2. Whereas the ETL analysis focuses within the symbolism on the different ways in
which the value of the y-coordinate is expressed, the IOE analysis attends to the
relationship between the structure of the equation and that of the students’ everyday
expression of their goal.
Discussion of issues of representation raised by crossanalysis
In this final section, we reflect on how the cross-analysis has highlighted and sharpened
our awareness of similarities and differences in the ways the two theoretical perspectives
deal with representation. We also consider how this awareness moves us forward in the
search for integration of theoretical perspectives.
The issue of connections between different modes of representation has been significant in
the analyses by both teams. Not only is this issue widely recognised as one of the important
characteristics of representations enabled by digital technologies but the use of and
movement between multiple modes was also one of the principles underpinning the design
of MoPiX. While the importance of connections between representations was a common
assumption at the start, the cross-analysis has enabled us to identify differences in the
ways that “connection” may be conceptualised. For both teams the significance lies not
simply in the formal presence of connections embodied in the design of the software but in
the ways in which students interact with the various modes of representation. From the
constructionist perspective, the connections made by students between graphical/dynamic
mode and symbolic mode closely parallel the connections embodied in the design of MoPiX.
30
By interacting (“tinkering”) with the digital environment, these connections become
meaningful for students, enabling them to access the “deep structure” of the system. The
connection between two modes of representation is thus at a systemic level: the system of
animated graphics and the system of symbols can be mapped onto one another. This
mapping may be considered an example of conversion between registers, argued by Duval
(2006) to play an important cognitive role in development of mathematical concepts.
In contrast, connections are considered more pragmatically from the multimodal social
semiotic perspective, analysing the ways in which the potentialities of each mode are
realised as they are used separately and together and the ways in which combinations of
modalities are utilised at each moment. While congruencies between modes of
representation may provide an explanation for some of the ways they are used, their
differences are also significant and serve a function in enabling meanings to emerge during
student activity. Rather than seeking for structural connections between systems of
representation, the following of chains of signification seeks for associations between
specific signs (especially boundary terms) and the contexts and domains of activity within
which they may have been encountered. This entails a wider consideration of the various
modes of representation available to students, not just those made available by the
technological environment. For example, in analysing episode 2, the IOE analysis focuses on
connections with representations in natural language as well as the graphical and formal
symbolic representations in MoPiX. Analysis using this perspective also entails drawing on
knowledge of the domains of social activity in which particular representations play a part
in order to interpret students’ use of these representations and the connections they may
be making.
Whereas a theoretical perspective on representation can adopt a neutral position on the
value of a particular conjunction of representations, from a point of view situated within
mathematics education, there clearly must be value ascribed to students’ developing
relationships with mathematics. As anticipated, the context in which the experimentations
took place, including the curricular context and the relationship between researchers and
the site of research, may be seen to play a role in shaping the focus of each team’s analysis.
While the IOE’s theoretically informed focus on the roles and relationships of different
31
systems of representation is seen in the analysis of both episodes, it is also apparent that in
analysing their own classroom experimentation (episode 1), the IOE analysis attends to the
development of connections between velocity and acceleration as this was an important
part of the didactic aim of the experimentation. For ETL, the relationship to the official
curriculum was less critical and their main focus on the students’ use of formalism can be
seen equally in both analyses.
This highlights a difference at a fundamental level between the two theoretical
perspectives: while multimodal social semiotics is concerned with understanding how each
mode of representation functions in producing meanings, constructionism has a major
concern with educational relevance. Thus the constructionist perspective addresses the
question of whether a given form of representation serves as a facilitator or an obstacle to
the generation of meanings that are consistent with those of mathematics. Researchers
working with this perspective also seek to design representations for facilitation of
particular types of meanings and may give primary importance to one representation over
others. For example, in using MoPiX, mathematical formalism is placed in a driving role of
creating graphical representations with the didactical intention to find ways in which
formalism may become meaningful to students. Recently, educational researchers using
multimodal social semiotics have also sought to develop a related theory of learning that
addresses the issue of designing representations for meeting educational goals (Bezemer &
Kress, 2008; Kress & Selander, 2012). It will be interesting to see how outcomes of this
development of the scope of social semiotics into didactic design relate to those arising
from a constructionist perspective.
A further issue that has become apparent to both authors as we have constructed this
article but which has been largely deferred until this point has been the different ways in
which we use the term meaning. For both of us, students “make meaning” with or from the
use and manipulation of representations in interaction with others in the social
environment. Moreover, in many instances we find ourselves largely in agreement about
the relationships between the ‘meanings’ thus made and the mathematical objects and
structures involved. However, while the constructionist perspective is concerned with
meaning as (an ultimately individual) cognitive phenomenon, for social semiotics meaning
32
is conceptualised as the establishment of shared orientations through communication in
interaction between individuals – meaning is located in the interaction rather than
acquired by the individual. This difference reflects a fundamental difference in the object of
the two theoretical perspectives: constructionism may be characterised as a theory about
cognition and its development while social semiotics is a theory about signs and the ways
they function. Awareness of this difference in our theoretical objects helps us to
understand differences in the focus of our analyses and alternative explanations of
phenomena. However, it also suggests that, where our interpretations converge, for
example in identifying the significance of Ron’s shifting attention between graphic and
symbolic modes, there is potential for building a richer, multi-faceted understanding of the
phenomenon.
In conclusion, the use of cross-analysis study on data derived from cross-experimentation
has helped us sharpen and highlight our awareness of aspects of representation and
meaning. Through examining the use of these terms within the two theoretical
perspectives, two aspects in particular, which had seemed very similar before the study,
came to be distinguished more clearly. One was the way in which connections between
representations were conceptualized by each approach, the kinds of value given to each
connection and the prioritisation of educational relevance or of explanatory relevance. The
other development was a more nuanced understanding of how our perspectives perceive
“meaning”, which in the case of constructionism seemed to address the individual in social
settings and give an emphasis to cognition, while multimodal social semiotics identified it
in the semiotic interactions. Our experience points to the potential usefulness of crossanalysis as a tool to elicit such distinctions and to forge connections in approaches to
representations and their uses in mathematics education. It further opens the floor for the
production of other such tools in an attempt to contribute to the networking of theoretical
frames, augmenting the ways in which they can be used to design, impact and understand
educational practices in a diversity of contexts.
33
Notes
i
ReMath was co-funded by the European Commission FP6-IST4-26751. Related resources
can be found at http://remath.cti.gr.
ii
MoPiX can be found at http://remath.cti.gr/ and will run in any browser with Flash.
iii
MoPix was designed to run on tablet computers in order to facilitate face-to-face
communication among students as well as the sharing of digital models on-line.
iv
Student names used here are all pseudonyms.
References
Ainsworth, S. E., Bibby, P. A., & Wood, D. J. (1997). Information technology and multiple
representations: new opportunities - new problems. Technology, Pedagogy and
Education, 6(1), 93-105.
Artigue, M., Cerulli, M., Haspekian, M., & Maracci, M. (2009). Connecting and integrating
theoretical frames: The TELMA contribution. International Journal of Computers for
Mathematical Learning, 14, 217-240.
Bezemer, J. & Kress, G. (2008). Writing in multimodal texts: a spcial semiotic account of
designs for learning. Written Communication, 25(2), 166-195.
Borba, M. C. & Villarreal, M. E. (2006). Humans-with-media and the reorganization of
mathematical thinking: Information and communication technologies, modelling,
experimentation and visualization. New York: Springer.
Bottino, R. M., & Kynigos, C. (2009). Mathematics education & digital technologies: Facing
the challenge of networking European research teams. International Journal of
Computers for Mathematical Learning, 14, 203-215.
Carreira, S., Evans, J., Lerman, S., & Morgan, C. (2002). Mathematical thinking: Studying the
notion of 'transfer'. In A. D. Cockburn & E. Nardi (Eds.), Proceedings of the 26th
Conference of the International Group for the Psychology of Mathematics Education
(Vol. 2, pp. 185-192). Norwich: School of Education and Professional Development,
University of East Anglia.
Cobb, P. (2000). From representations to symbolizing: introductory comments on
semiotics and mathematical learning. In P. Cobb, E. Yackel & K. McClain (Eds.),
Symbolizing and Communicating in Mathematics Classrooms: Perspectives on
discourse, tools and instructional design (pp. 17-36). Mahwah, NJ: Lawrence Erlbaum
Associates.
Dreyfus, T. (1994). The role of cognitive tools in mathematics education. In R. Biehler (Ed.),
Didactics of mathematics as a scientific discipline, pp.201-211, Dordrecht: Kluwer
Duval, R. (1999). Representation, vision and visualization: Cognitive functions in
mathematical thinking. Basic issues for learning Proceedings of the Annual Meeting
34
of the North American Chapter of the International Group for the Psychology of
Mathematics Education (pp. 3-26). Cuernavaca, Morelos, Mexico.
Duval, R. (2006). A cognitive analysis of problems of comprehension in the learning of
mathematics. Educational Studies in Mathematics, 61(1 - 2), 103-131.
Edwards, L. D. (1998). Embodying mathematics and science: Microworlds as
representations. The Journal of Mathematical Behavior, 17(1), 53-78.
Falcade, R., Laborde, C., & Mariotti, M. (2007). Approaching functions: Cabri tools as
instruments of semiotic mediation. Educational Studies in Mathematics, 66(3), 317333.
Font, V., Godino, J. D. & D’Amore, B. (2007). An onto-semiotic approach to representations
in mathematics education. For the Learning of Mathematics, 27(2), 2-7.
Goldin, G. A. (1998). Representational systems, learning, and problem solving in
mathematics. The Journal of Mathematical Behavior, 17(2), 137-165.
Halliday, M. A. K. (1978). Language as Social Semiotic: The Social Interpretation of Language
and Meaning. London: Edward Arnold.
Harel, G., & Papert, S. (Eds.). (1991). Constructionism. Norwood, NJ: Ablex.
Hodge, R., & Kress, G. (1988). Social Semiotics. Cambridge: Polity Press.
Janvier, C. (1987). Problems of Representation in the Teaching and Learning of Mathematics.
Hillsdale, NJ: Lawrence Erlbaum Associates.
Kafai, Y., & Resnick, M. (Eds.). (1996). Constructionism in Practice: Designing, thinking and
learnign ina digital world. Mahwah, NJ: Lawrence Erlbaum Associates.
Kaput, J. J. (1991). Notations and representations as mediators in constructive processes. In
E. van Glasersfeld (Ed.) Radical constructivism in mathematics education, pp.53-74.
Dordrecht; Kluwer
Kress, G., & Selander, S. (2012). Multimodal design, learning and cultures of recognition.
The Internet and Higher Education, 15(4), 265-268.
Kress, G., & van Leeuwen, T. (2001). Multimodal Discourse: The modes and media of
contemporary communication. London: Arnold.
Kynigos, C. (2007). Half-baked microworlds as boundary objects in integrated design.
Informatics in Education, 6(2), 335-359.
Kynigos, C. (2012). Constructionism: theory of learning or theory of design? Proceedings of
the 12th International Congress on Mathematics Education. Seoul, S. Korea.
Kynigos, C., Psycharis, G., & Moustaki, F. (2010). Meanings generated while using algebraiclike formalism to construct and control animated models. International Journal for
Technology in Mathematics Education, 17(1), 17-32.
Morgan, C., & Alshwaikh, J. (2009). Mathematical activity in a multi-semiotic environment.
Sixth Congress of the European Society for Research in Mathematics Education,
Working Group 6 Language and Mathematics, 993-1002. Retrieved from
http://www.inrp.fr/editions/cerme6
Morgan, C., Mariotti, M. A., & Maffei, L. (2009). Representation in computational
environments: epistemological and social distance. International Journal of
Computers for Mathematical Learning, 14(3), 241-263.
Noss, R., & Hoyles, C. (1996). Windows on Mathematical Meanings: Learning Cultures and
Computers. Dordrecht: Kluwer.
O'Halloran, K. L. (2005). Mathematical Discourse: Language, symbolism and visual images.
London: Continuum.
35
Radford, L. (2009). Why do gestures matter? Sensuous cognition and the palpability of
mathematical meanings. Educational Studies in Mathematics, 70(2), 111-126
Strohecker, C., & Slaughter, A. (2000). Kits for learning and a kit for kitmaking. CHI '00
Extended Abstracts on Human Factors in Computing Systems, 149-150.
Vergnaud, G. (1998). A comprehensive theory of representation for mathematics education.
Journal of Mathematical Behaviour, 17(2), 167-181.
Vérillon, P., & Rabardel, P. (1995). Cognition and artefacts: a contribution to the study of
thought in relation to instrumented activity. European Journal of Psychology of
Education, 10(1), 77-101.
Yerushalmy, M. (2005). Functions of interactive visual representations in interactive
mathematical textbooks. International Journal of Computers for Mathematics
Learning, 10, 217-249.
36