Paper for the course "Learning theories and the relationships between them, with special emphasis on the writings of Paul Cobb".

1. Connections between learning theory and e-learning
theory
Øistein Gjøvik, NTNU
Introduction
In this paper, I will focus on the use of ICT for learning mathematics. The paper can
be seen as a preliminary clarification of the theoretical framework I will be using in
my study. I will briefly define e-learning, and recapitulate the technology-inspired
terminologies of instrumental genesis and webbing. I will also summarise some main
points of the learning theories that I have considered using in my project. Finally I
will attempt to look at the learning theories, together with the use of ICT, and see how
these can feed into my preliminary research questions and research project.
Throughout, I will have in mind that learning in this situation is e-learning and that
there is some kind of digital tool involved in the educational setting.
What is e-learning?
The increasing use of ICT in schools has led to the belief that there is something
special about the digital learning tools, and it has been compared to other specially
influential technologies:
Perhaps there is something in the technology related to the computer that makes it a
very peculiar artifact, which makes it similar to a basic technology like writing.
(Mariotti 2002, p.696)
I want to use an alternative to the term ICT because, as it turns out, this is a rather
elusive concept, and I want to stress that tools are used for learning. Some people
have reacted to the technology and technical terms being used somewhat arbitrarily:
More fundamentally, the literature displays no uniformity regarding how ‘the
computer’ is conceived, and what is taken as the role of software. (Noss and Hoyles
1996, p.142)
Landing on an e-learning definition everybody can agree upon is not an easy issue,
and several definitions exist. I turn to (Powell, Knight et al. 2003) to find the
following figure (by Markos Tiris) and definitions, these being the definitions used by
BECTA1 and being fairly accepted terms:
Fig. 1
1
British Educational Communication and Technology Agency: http://www.becta.org.uk/
1

2. Model of e-learning (Powell, Knight et al. 2003)
Here, IT is the concrete equipment; this could be computers or calculators together
with skills for using them, for example, typing a letter in a word processor is IT. ICT
is what you get when you connect pieces of IT in some manner. Examples are
searching for online documents and e-mailing. If, in addition, ICT is used in education
in some sense, for example, managing an educational institution, it is called ILT
(Information and Learning Technology). Finally, e-learning is the part of ILT that is
occupied with teaching and learning (and not with the organization of education).
There is a distinction to be made between e-learning and online education. E-learning
is more content-oriented, and may or may not include online activities, while online
education focuses more on communication between learners and tutors.
E-learning can still, in its everyday-use, involve as diverse activities as having a video
conversation with a peer on another continent, as well as pondering a geometrical
theorem with Cabri Geometry in solitude. This indicates that in addition to the term elearning we will also have to clarify more precisely what kind of tools are being used
all along.
E-learning equipment stands between the students and the knowledge that students are
supposed to learn in one way or the other. Accordingly, we are led to focus on the
mediation of knowledge with these tools.2 How does one get to know something with
technology? How do students e-learn something? The first obstacle is getting to know
the tools. This has lead to the concept of an instrument, and one way of looking at the
process of tools and usage together becoming an instrument, is through the concept of
instrumental genesis.
Instrumental genesis
As mentioned when defining e-learning, technology consists of equipment together
with skills for using them. A way of connecting utilization schemes and tool is by way
of instrumental genesis. The concept of instrumental genesis is central to discussing
use of computers or calculators in education. This concept is discussed in (Artigue
2002) and it is elaborated on in for instance (Hoyles, Noss et al. 2004). According to
Artigue, there is a dialectic in which the learner and the artifact, that is, the physical
tool being considered are mutually constituted in action. This is called instrumental
genesis and includes the instrumentalisation where the subject shapes the artifact for
specific uses, and the instrumentation, where the subject is shaped by interacting with
the artifact.
Note that only part of the tool/artifact is incorporated in the overall instrument, i.e. we
can not be expected to know all the possibilities of one particular artifact.
2
This can be referred to as Computationally Mediated Mathematical Knowledge (CMMK).
2

3. In stru m e n t
I n s t r u m e n t a l is a t io n
U t iliz a t io n s c h e m e s
I n s t r u m e n t a l g e n e s is
A r t if a c t
I n s t r u m e n t a t io n
Fig.2
Instrumental genesis
Based on figure in (Strässer 2003, p.33)
One example of an instrumented technique is the windowing scheme, where students
were to draw a graph from a certain given function, and then goes hunting for a
suitable window to view the graph on screen (Artigue 2002). This is one example of
instrumented knowledge. One reason that using digital technologies in education has
not evolved to the extent that one earlier expected is the underestimation of the
complexity of these instrumentation processes (Artigue 2000, p.9).
Instrumental genesis is not restricted to merely digital devices; it is also one way of
looking at, say, pupils’ instrumentations of textbooks. Take, for instance, a student
noting that he or she can find answers to exercises in the back of the textbook. This is
instrumentalisation. The book is being readied for becoming an instrument. Using the
answers at the back before doing en exercise is now shaping the way the student uses
the textbook. This is then one particular utilization scheme for this student.
I now consider e-learning tools as mediators of mathematical knowledge.
Mediation
It seems that focusing on mediation of knowledge will be important for considering elearning or studying pupils in computationally rich environments.
A
B
Fig. 3
Computer as mediator of knowledge
(based on Figure 1.2 in (Noss and Hoyles 1996, p.6)
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4. When interpreting how students use autoexpressive3 artifacts like computers, we not
only get access to see students’ expression of mathematical ideas and knowledge, but
we may also gain insight into more general aspects of the learning process, like
cultural influences, gender issues, identity, and how they affect students constructions
of mathematical concepts.
Focusing on mediation of knowledge, means we consider the ‘something’ standing
between the individual learner or the learner-in-social action and the knowledge
intended to be learned. An important concept within this is the metaphor of webbing.
Webbing
Starting with Vygotsky’s appreciated zone of proximal development, ZPD, we find
this stated as:
The distance between the actual developmental level as determined by independent
problem solving and the level of potential development as determined through
problem solving under adult guidance or in collaboration with more capable peers
((Vygotsky 1978, p.86) as quoted in (Chaiklin 2003, p.40))
Imagining we exchange the human tutor in Vygotsky’s quote, with a digital tool like a
computer. This will sort of be ‘translating’ Vygotsky’s ZPD into the world of elearning. Where the teacher or more capable peer up to now has been seen as a person
providing scaffolding for the student, we are now shifting focus from scaffolding for a
child to voyage safely into the zone, to webbing, indicating shifting perspective from
teaching to learning and from restricting what can be learned to opening it wide up.
The fact that this term is inspired the world of the Internet is not coincidal, as the
structure found in webbing resembles that of the Internet.
We find in (Noss and Hoyles 1996, p.108) an introduction to this concept of webbing.
In the hunt for a metaphor of a support system, Hoyles and Noss seek a system that
can extend the concept of scaffolding and also be as applicable to mathematics
teaching as basket weaving. They want the metaphor to capture the following:
•
•
•
•
it is under the learner’s control;
it is available to signal possible user paths rather than point towards a unique directed
goal;
the structure of local support available at any time is a product of the learner’s current
understandings as well as the understandings built by others into it;
the global support structure understood by the user at any time emerges from
connections which are forged in use by the user. (ibid)
They further elaborate this with
“The idea of webbing is meant to convey the presence of a structure that learners can
draw upon and reconstruct for support – in ways that they choose as appropriate for
their struggle to construct mathematical meanings.” (ibid)
We can turn to CabriMathematics for an example of webbing. Given two mirrored
objects, students in a teaching experiment were to find where the mirror line was. The
students constructed the mirror line in an unusual way, dragging4 objects on screen
until they coincided. The observation the participants make is only meaningful, and
3
Meaning that the medium includes a language as well as elements to talk of the language itself.
This is a technical computer-term, involving moving objects on screen with peripheral devices like a
mouse.
4
4

5. the mirror line, is only constructible in this manner when webbed from the Cabri
MicroWorld (see (Papert 1980) for the introduction to MicroWorlds), and the
teachers’ challenge is connecting the CabriProof to a mathematical proof.
E-learning and individual perspectives
So far, I have defined what kind of artifacts we are considering in my study and what
kind of processes subjects goes by to get to grip with these artifacts and knowledge
mediated by them. I now consider one part of the learning theoretical framework I
will take on, the individual perspective. Speaking of an individual perspective on
learning, the theory (or philosophy) of constructivism needs to be clarified.
Sorting out constructivism
Constructivism is a theory of knowing and learning, and not one of teaching. There
are two main strands of constructivism, radical constructivism and social
constructivism. These are usually presented as a collection of tenets (depending on
what literature one consults, slightly different wordings may appear):
1. Knowledge is not passively accumulated, but rather, is the result of active
cognizing by the individual (Glaserfeld 1989, p.162)
Accepting the first tenet only, results in what is known as weak constructivism or
trivial constructivism. The second tenet is usually split in two:
2. The function of cognition is an adaptive process that functions to make an
individual’s behavior more viable given a particular environment.
3. Cognition serves the subject’s organization of the experiential world, not the
discovery of an objective ontological reality. (von Glaserfeld, as quoted in
(Jaworski 1994, p.16)
Accepting only the second tenet results into what is known as cognitive
constructivism. The radical constructivism, as founded by Ernst von Glaserfeld
embraces all of these tree tenets. A fourth tenet has been added due to recent research:
4. Knowing has its roots in both biological/neurological construction, and social,
cultural, and language based interactions. (Dewey, 1916/1980; Garrison,
1997,1998; Gergen, 1995; Maturana & Varela, 1992 as quoted in (Doolittle
1999)).
Also embracing this fourth tenet leads to social constructivism. The differences
between the strands of constructivism are not sharp, as can be seen in a quote from
von Glaserfeld:
The present interest among educational researchers and philosophically inclined
psychologists in social interaction and its role in the process of learning need not pit
them against radical constructivism. This topic certainly requires investigation and its
investigation should not be hampered by the unwarranted fabrication that there is a
conceptual contradiction between the principle of subjective cognitive construction
and the experiential reality of the phenomena that are called social. (Glaserfeld 2000,
p.6)
Central to any timbre of constructivism are the concepts of assimilation and
accommodation as outlined in for instance (Glaserfeld 1995a, pp.62-66). According to
von Glaserfeld, assimilation is the way we make the world fit into what expectations
and mental concepts we might have, while accommodation is the operation of
adapting ourselves, encountering conflicts between expectations and experiences.
Influences on e-learning and teaching
5

6. Embracement of constructivism has lead to changes in teachers’ pedagogical beliefs.
The instructionism earlier known to be the dominant educational praxis, has evolved
towards constructionism, the pedagogical outcome of having a constructivist
rationale, introduced in (Papert 1993, chapter 7).
Doolittle has listed eight pedagogical recommendations as the result of having a
constructivist theoretical background, and these are pedagogical principles
independent of strand of constructivism:
1.
2.
3.
4.
5.
6.
7.
8.
Learning should take place in authentic and real-world environments
Learning should involve social negotiation and mediation
Content and skills should be made relevant to the learner
Content and skills should be understood within the framework of the learner’s
prior knowledge
Students should be assessed formatively, serving to inform future learning
experiences
Students should be encouraged to become self-regulatory, self-mediated, and selfaware.
Teachers serve primarily as guides and facilitators of learning, not instructors.
Teachers should provide for and encourage multiple perspectives and
representations of content (Doolittle 1999).
Doolittle continues, explaining the effectiveness and usefulness of online education in
these recommendations and arguing for the benefit of the technology. For example, he
claims that online education (he does not distinguish online education and e-learning
and in my opinion, the latter would here be a more suited practical context) gives a
multitude of relevant activities, environments and settings for students to experiment
within or work on. A subtle point here is that computers really could not give realworld environments, but offer simulations of some sort. Also, there are fruitful
analogies to be drawn between interacting in MicroWorlds and practical, everyday
activities (Noss and Hoyles 1996, p.105)
Taking, as an example, the batteries episode, from (Cobb 2002, Appendix), where
students use a statistics minitool for reasoning about the lifespan of batteries, we see
that an educational situation has a context relevant to the learners, students become
engaged in social interaction and the teacher in this episode serve as a guide and
discussion facilitator.
E-learning changes education, and changes already reported are teachers’ raised
expectations of pupils, a more student-centered pedagogy with small-group and
independent student work and greater willingness for teachers to experiment (as
summarized in (Jarrett 1998)). This could be taken to mean that the incorporation of
e-learning in schools leads to a constructivist pedagogy. Indeed, this was also one of
Papert’s hopes for future education.
E-learning comes equipped with new experiential realities to ponder, and therefore,
new ways for students to construct knowledge. Different computer software,
computer activities and graphic calculators give students new ways of perceiving
mathematics. Software like Cabri Geometry5 has often been called cognitive tools, as
they give students direct access to mathematics. However, the relation to what many
may think of as real mathematics may be changing because of the introduction of new
technology:
5
http://education.ti.com/us/product/software/cabri/features/features.html
6

7. Focusing on technology draws attention to epistemology: for new technologies – all
technologies – inevitably alter how knowledge is constructed and what it means to
any individual. This is as true for the computer as it is for the pencil, but the newness
of the computer forces our recognition of the fact. There is no such thing as
unmediated description: knowledge acquired through new tools is new knowledge,
MicroworldMathematics is new mathematics. (Noss and Hoyles 1996, p.106)
For example, it is not always clear what are the connections between
LOGOmathematics and traditional school mathematics. Students in a LOGO
MicroWorld, given a series of tasks may produce and construct a lot of mathematics,
but it can take on different shapes and formulations than traditional textbooks provide.
E-learning from a social perspective
It is not easy to give a coherent summary of a social theory for learning. Consider
Wenger’s four principles for a social theory of learning:
1. We are all social creatures, and this is a central aspect to learning
2. Knowing means competence in a variety of appreciated domains
3. Knowing is connected to participating and active engagement
4. Learning should produce meanings, the possibility to perceive the world as
meaningful (Wenger 1998)
Influences on e-learning and teaching
When interpreting students’ mathematical meaning-making, adopting a socio-cultural
perspective will force us into considering the role of the computer as a cultural
artifact. Computers, technology, e-learning, ICT, online education, M-learning
(mobile learning), they are all an increasing part of society and culture, and students
may be motivated for e-learning because of the way knowing is connected to
participating and belonging to a community of practice or microculture. Also having
competence in appreciated domains is important, according to Wenger’s second tenet,
and the way technology has grown into the world gives us good reason to consider
matters like these.
Meanings taken-as-shared has a different impact within the scope of e-learning,
considering meanings can be shared among a wider audience geographically and
developmental-wise, compared to what an institutional setting would do.
Considering that participating (and belonging) is an aspect of learning from a social
perspective we also should consider gender issues, and building of an identity. Also,
the (increasing) presence of computers in schools has transformed what is
mathematical knowledge, and perhaps will cultural changes due to e.learning
transform traditional teaching and learning as we know it even further:
I believe that the computer presence will enable us to so modify the learning
environment outside the classroom that much if not all the knowledge schools
presently try to teach with such pain and expense and such limited success will be
learnt as the child learns to talk painlessly, successfully and without organized
instruction. (Papert 1980, p.8)
Regarding enculturation into a community of practice, the ways e-learning enters the
arena may influence students’ degree of belonging to such a community. For instance,
instrumented knowledge, say, the windowing scheme example given earlier, can
7

8. perhaps be compared to and e-learning version of street mathematics, and may not
gain the same status as what is considered school mathematics.
The community does not have to be that of a classroom microculture either. Managing
to use mathematics by way of ICT in practice is a much sought skill in many different
job situations. Examples are spread sheets, architectural drawing, managing staff, etc.
Coordinating individual and social theories
We don’t have to choose between social or individual theories of learning. Adopting a
social point of view in research, does not mean denial of individuals’ constructions. In
the same vein, adopting an individualist point of view does not mean the denial of
social interaction and cultural influence. Social interaction is one of the ways of
gaining experiential material for individual construction, and we also see there the
blurred distinction between radical constructivism and social constructivism.
The dispute between the two perspectives considered here, can probably best be seen
in the Steffe/Lermann debate, with the statement:
I will suggest that it does not do justice to the implications of cultural psychology,
indeed that it cannot do so; that the assumption of complementarity leads to
incoherence; and as a consequence, that mathematics education would benefit from
abandoning constructivism as a view of how people learn. (Lerman 1996, p.133)
According to (Cobb 1994), what is meant by bringing theories together, is that in
interpreting educational situations, sometimes it is fruitful to let the cultural-historical
setting play the background of the situation and the individual constructions be the
focus of the situation. In other situations, the opposite might be the most illuminating
way of considering the situation:
The discussion of Rogoff’s, von Glaserfeld’s, Saxe’s and Steffe’s work indicates that
sociocultural analyses involve implicit cognitive commitments, and vice versa. It is as
if one perspective constitutes the background against which the other comes to the
fore. (Cobb 1994, p.18)
As an example, consider (Cobb 1994) where Cobb interprets the way the Oksapmin
learn their special way of counting with their body.
The constructivist analysis circumvents this difficulty by stressing that rather than
internalizing a cultural form that appears to be pregiven, the novice reorganizes his or
her own activity. (…) By the same token, the sociocultural perspective complements
the constructivist perspective by emphasizing that the novice trader reorganizes his or
her counting activities while attempting to achieve goals that emerge in the course of
his or her participation in the practice of economic exchange. (ibid)
Looking at e-learning in a similar manner; take the instrumental genesis, this is most
naturally viewed from an individual perspective. An individual constructs his or her
own instrument by interaction with the artifact. Then the focus can be seen as that of a
constructivist, the process of learning mathematics is one that takes place alongside
and after the student’s instrumentation processes. Seen from a socio-cultural
perspective, the instrumental genesis itself, and the learning of mathematics with this
artifact, is a way of becoming a participant in a community of practice.
8

9. My research project
My research project concerns the learning of mathematics through the aid of elearning tools. In particular, the parAbel website6 will be a key data source, and
perhaps other artifacts will be considered. The website will consist of mathematical
activities and games, learning material and possibilities for online collaboration. The
intention is to make the entire curriculum of mathematics (and later on physics and
chemistry) available through the website.
My preliminary main research question is
How can digital tools influence on students’ mathematical learning?
This very wide question immediately spawns a series of sub questions:
•
•
•
How does gender influence CMMK?
How does the timing of incorporation of digital tools into the learning process
influence students’ understanding of mathematical concepts?
How does the instrumental genesis of the digital tools influence students’
understanding of mathematical concepts?
Seeing the websites as a collection of MicroWorlds, this Internet resource could give
a diverse insight into students’ variant webbing processes. On example of a
MicroWorld is one where students alter an angle, drawn on a unit circle, and
simultaneously see changes occurring on a sine or cosine graph. This is in one way a
constructivist approach, seeing the computer mediating mathematical knowledge and
giving the user an experience from which to construct the relationship between angles
and trigonometry. Other ways of interpreting a situation in which a user interacts with
a computer like this, is taking into account the way the users perceives technology.
Are there gender differences in the way the computer is being used for such a
MicroWorld activity? Can one speak of fear of technology in any way? Is the real
mathematics “hidden” within the artifact somewhere? Taking such issues into account
will hopefully give richer tools with which to interpret mathematical learning.
References
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6
See http://www.parabel.no, for information and samples. The resource is provided by the Agder
University College, Department of Technology.
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