Figure 1. is schematic representation of the four factors that are necessary and sufficient to intellectual growth as well
as how these factors interact over time to develop higher forms of knowledge. The interaction between
maturation, physical
experience, social transmission, and equilibrium is through intellectual
functions and structures.
Structures refer to organized aspects of intelligence, while functions consist of various modes of acting on the world.
Intellectual structures are constructed of simpler organized aspects of knowledge. The simplest forms of these structural
components of knowledge are the innate reflexes with which we are born. These reflexes are initially isolated from one another
but soon they are integrated together to form more powerful organizations of knowledge. Piaget (1952) postulates the existance
of two functional invariants:
organization and adaptation. Organization refers to the tendency to integrate various
experiences by integrating parts into wholes and wholes into more comprehensive wholes. As we interact with the world around us
we tend to organize various parts of our experience into integrated wholes. This is true of perception, memory, language competence,
and other intellectual activity. As we organize we also adapt, that is, we tend to seek to adjust to our physical and intellectual
world in increasingly more flexible ways. Thus, the second functional invariant is
adaptation. Adaptation - as a mode
of coping with the world - takes two forms:
assimmilation and accomodation.
The Piaget's Theory versus Mathematics
The cognitive structures are fundamentally similar to mathematical structures. Transformation is the key idea in all cognitive structures.
What happens when an object is transformed to another state or when an idea is applied to a different situation, is for Piaget the basic
question used by the child to generate intellectual development. How similar this stucture is to mathematics, whose fundamental concept,
function, is illuminated by looking at mapping from one set to another. Piaget believes that the most powerful tool to understanding
transformations is the idea of reversibility of transformations. Testing transformations for reversibility allows the child to examine
variables for constancy and equivalence. But these elements are precisely the tools used for studying mathematical functions
and mathematical structures. The basic structures of mathematics are analogous to the basic cognitive structures used in
intellectual functioning by every human being. Mathematics is not just formalized expression of thought processes used by a small
group of unusual individuals employed in departments of mathematics. Mathematics is the formalization of basic structures which
underlie the thought processe used by every child and every adult. The difference in intelligence between children and adults
is that some of these structures are in the process of developing for children. This development places the classroom teacher
in a dilemma. On the one hand, mathematics is "natural" for children and should stand at the center of our school curriculum.
On the other, the cognitive structures necessary for learning much mathematics may not be present at all in early elementary school
years and will be present only in differing stages in the junior high school and early senior high school years.
In fact, Karplus (1970) has done work which suggets that many adults never fully develop the ability for abstract symbolic reasoning
which characterizes the ultimate development in the final period of formal operations.
The Vygotsky's Theory of Human Development
Vygotsky through his work tried to answer the questions: How do humans, in their short life trajectory, advance so far beyond
their initial biological endowment and in such diverse directions (Vygotsky, 1981) ?
He saw that, in order to arrive at an adequate answer, it would be necessary to look not only at individuals but also at the social and material
environment with which they interacted in the course of their development. The development of human being requires that this development be seen,
not as an isolated trajectory, but in relation to historical change on a number of other levels: of the individual level, of the institutions - family,
school, workplace - levels, and of wider culture in which those institutions are embedded, and finally that of the species as a whole.
However, in order to understand why Vygotsky placed so much emphasis on adopting a historical perspective - a "genetic approach" - it is necessary to
consider a second fundamental feature of his theory, that of the mediating role of artifacts in activity.
Human beings are not limited to their biological inheritance, as other species are, but are born into an environment that is shaped by the activities
of previous generations. The third key feature of Vygotsky's theory: mutually constitutive relationship between individuals
and the society of which they are members. In contemporary developed societies, these involve activity systems of education, health care, the arts, law, etc.
as well as the multifarious activity systems concerned with the exploitation of material resources for the production and
distribution of the products required to support the society's way of life. Seen from this perspective, therefore, a society is maintained and
developed by the particular individuals who contribute to their activity systems at any particular point in time.
Seen from the complementary perspective, the formation of individual persons, their identities, values and knowledgeable skills, occurs through their
participation in some subset of these activity systems, starting with activities in which they are involved with family
members, then in school and on into activity systems of work, leisure and so on. However, it is equally important to see these same situated activities as the
site of the potential change and renewal. Every situation is to some degree unique, posing challenges that in some respects require the participants jointly
to construct solutions that go beyond their past experiences. Each instance of joint activity is thus an occasion of transformation: transformation of the
individual participants and of their potential for future participation; of the tools and practices or of the ways in which they are deployed; and of the
situation itself, opening up possibilities for certain kinds of further action and closing down others.
This latter perspective allows us to gain a better understanding of learning. It is simply a way of referring to the transformation
that continuously takes place in an individual's identity and ways of participating through his or her engagement in particular instances
of social activities with others. Seen from this point of view, all participants continue to learn throughout their lives, as each
new situation makes new demands and provides opportunities for further development. In the context of the broader conceptualization of learning
Vygotsky construct the "zone of proximal development' - the zone in which an individual is able to achieve more
with assistance than he or she can manage alone.
Vygotsky in Thinking and Speech discussed how social speech comes to function as the medium for individual as well as inter-individual thinking.
Speech for self is "inner" directed; it serves as a means that "facilitates intellectual orientation, conscious awarness, the overcoming of difficulties
and impediments, and imagination and thinking" (Vygotsky, 1934/87). With the differentiation of speech for self from speech for others, there opens up
the possibility of dialogue, that is to say a form of collaborative meaning making in which both individual and collective
understandings are enhanced through the successive contributions of individuals that are both responsive to the contributions
of others and oriented to their further responses.
Social Constructivism as a Philosophy of Mathematics
The social constructivist is that mathematics is a social construction, a cultural product, falliable like any other branch of knowledge. This view entails two claims.
Firts of all, the origins of mathematics are social or cultural. Secondary, the justification of mathematical knowledge rests
on its quasi-empirical basis. A social constructivist epistemology could be developed from the two principles of radical constructivism, which are:
a) "knowledge is not passively received but actively built up by the cognizing subject;
b) the function of cognition is adaptive and serves the organization of the experiental world, not the discovery of ontological reality". Von Glasersfeld (1991)
Ernest (1991) extended these principles to elaborate the epistemological basis of social constructivism:
c) the personal theories which result from the organization of the experiential world must fit the constraints imposed by
physical and social reality;
d) they achieve this by a cycle of theory-prediction-test-failure-accommodation-new theory;
e) this gives rise to socially agreed theories of the world and social patterns and rules of language use;
f) mathematics is the theory of form and structure that arise within language.
This provides the basis for a social constructivist philosophy of mathematics. The concepts of mathematics are derived by abstraction
of previously constructed concepts, by negotiating meanings with others during discourse, or by some combination of these
means. The mathematics is a branch of knowledge which is indissolubly connected with other knowledge, through the web of language.
Language enables the formulation of theories about social situations and physical realities. Dialogue with other persons and interactions
with the physical world play a key role in refining these theories, which consequently are continually being revised to improve their "fit."
As a part of the web of language, mathematics thus maintains contact with the theories describing social and physical reality, and hence
indirectly, with the physical world. The "fit" of mathematical stuctures in areas beyond mathematics is continuosly being tested, and
mathematics is evolving to provide the patterns and solve the tensions that arise from this modeling enterprise. Thus, "the unreasonable effectiveness
of mathematics" is no miracle of coincidence. It is build in. It derives from empirical and linguistic origins and functions
of mathematics (Ernest, 1991).
Learning Mathematics based on Piaget's and Vygotsky's Theories
Piaget considers action and operations that are constructed through it, not language, as central to the development of intelligence.
Vygotsky adopts a somewhat different stance, he argued that the development origins of language and thought are separate. Although he ageed with
Piaget's stress on importance of activity as the basis for practical intelligence he argued that, around the third year of life,
language intersects with non-verbal thought to form the foundation for the development of verbal reasoning and self-regulation.
From this time on, language starts to play a fundamental, formative role in intellectual development. However, non-verbal thinking remains. Not
all symbolic activity requires language. Art, arithmetic, skills in sport and many other activities may proceed adaptively and
intelligently without the involment of verbal thought.
The fact that some forms of activity, including some ways of thinking, do not implicate language does not necessarily
imply that they are not influenced by communication and teaching, however.
Mathematics is difficult to learn and hard to teach. Perhaps one of the reasons for the popularity of Piaget's view on intellectual
development was the reassurance it seemed to offer in identifying children's natural capacities to construct the fundamental
conceptual basis for mathematical thinking. The objection, that instruction too often involves attention to procedures and a neglect
of conceptual understanding, can be seen as a critism of many approaches to the teaching of mathematics. But it also inspires
some hope that better methods can be invented. Piaget's emphasis on the importance of relevant activity and self-directed problem
solving as the proper development basis for more abstract conceptual understanding is shared by many educational theorists and teachers.
Next question is the importance of instruction, both informal and formal, at home and in school, in helping children to make
mathematical sense of their their experience. How active should teacher be, for example, in aiding a child in his problem-solving
and his conceptual construction? Vygotsky argues that instruction is a necessary requirement if a child's spontaneous activities
are to be transformed into symbolic, rational thinking. He shares Piaget's view that action is the starting place for the formation of abstract,
symbolic thinking (like that involved in solving mathematical equations, for example ) but does not agree with the notion
that the child is unable to grasp the conceptual relations between practical activity and more abstract levels of thinking
before a particular stage is reached. Piaget's view that different modes of representation only become available to children a specific stages.
Procedural Skills and Conceptual Understanding
The nature of the division between an ability to execute procedures and the achievement of a conceptual grasp of the purpose that such procedures
serve has proved, for many children, too wide to bridge. Can we identify and explain the common mistakes that children make?
Can we understand how and why these come about? Can we point to aspects of the process of teaching that probably fail, or
others that might succeed, in helping children to overcome their problem? Children appear to generalize what they learn in mathematics.
The trouble is that what they learnt or assimilated is often, at best, only partially correct. What they learn may be different from what
the teacher has tried to teach them. Children are usually systematic in their mistakes, using what are sometimes referred to as "buggy
algorithms" - procedures that yield consistent (but sometimes wrong) answers (Wood,1988). The problem stems from the fact that buggy
algorithms may give correct answers to some questions. Responses or ideas that receive occasional confirmation often yield the strongets habits.
Many children invent their own methods for answering math questions. For mathematically able children, this may confer benefits, since they
sometimes invent methods that are faster and more efficient than those taught by their teachers (Resnick, 1976).
Children do attempt to develop methods for answering questions and often generalize such methods to produce a form of "faulty logic."
Again, whatever they have been told does not get through to all children. The conclusion must be either that such children "lack something",
or that they have not been taught because the teaching methods used do not "bridge gap" between practical problem-solving,
intuitive understanding and symbolically evoked procedures.
Self-regulation and the "Zone of Proximal Development"
One of Vygotsky's theoretical arguments is that "self-regulation" is dicovered and perfected in the course of social and instructional interactions.
Self-regulation is usually a private, invisible and inaudible activity. Some children only needed very general and non-specific help in order to improve their
immediate problem-solving performance and to generalize what they learned to other, similar (sometimes more difficult) problems. In Vygotskian terms,
these children had "wide zones of proximal development." In other terms, what they could achieve with little expert guidance was far superior to
their unassisted efforts. They learned a lot with a little help. Other children needed more protracted and specific help in order to learn.
They have "narrow zones of proximal development."
Piaget and Vygotsky: a brief comparison and summary
I will summarize some of the main ideas on the development of learning and thinking. On view of Piaget's theory, holds that
all children pass through a series of stages before they construct the ability to perceive, reason and understand in mature, rational terms.
In this view, teaching, whether through demonstration, explanation or asking questions, can only influence the course of intellectual
development if the child is able to assimilate what is said and done. Assimilation, in turn, is constrained by the child's stage
of development. This leads to a specific concept of learning "readiness" and, holds out many implications for design of curricula
and the timing of formal instruction.
A second perspective; introduced by Vygotsky, shares some important areas of agreement with Piagetian theory, particularly an emphasis on
activity as the basis for learning and for the development of thinking. However, it involves different assumptions about
the relationship between talking and thinking. It entails a far greater emphasis on the role of communication, social interaction
and instruction in determining the path of development.
The question is how and why children develop the intelligence to achieve understanding. What knowledge and experience are they drawing on and
and generalizing from in order to understand ?
For Piaget, the schemes of knowing that make possible the understanding and generalization of experience are rooted in actions on the world.
Appendix 1
The Development of Knowledge or Creativity by Chaos Theory
(Briggs & Peat, 1990, 1999)
1. Turbulence - Doubts and Uncertainties
The developing of knowledge is a creative process. Creativity can occur in a conversation when the turbulence of questioning
and exchange gives birth to a subtle, new understanding or a true way of expressing something. "The known" involves entering
into doubts and uncertainties and allowing our abstractions and mental constructions to die or to be transformed. When
this happen, creative insight self-organizes, catching us unaware with the shock or delight of the unexpected truth.
2. Bifurcation and Amplification
The literature of creativity is full of descriptions of some magical moment when the flux of the creator's perception shifts
and the chaos begins to self-organize - moments of the aha! A completed creative work is a record of the many small and large germs
and aha! that leapt into being as the individual pursued the creative activity. These moments of insight when we see or hear
something that would be meaningless, nonsensical, or trivial to someone else, but which seem to set in motion a significant change
in our perception, to get to the "truth" of our perception. They are bifurcation points. The images seemed somehow important
and became amplified and coupled with thoughts together. A new context was emerging. Old facts and questions became realigned
through the feedback, and more and more could be seen from a new perspective.
Piaget explained the process using concept of intellectual structures; the innate reflexes with which we are born. These
reflexes are initially isolated from one another but soon they are integrated together to form more powerful organizations of
knowledge. It is precisely this intrinsic ability to integrate and differentiate that is the secret of how structures of knowledge
continue to grow in complexity as well in flexibility. The continual integration and differentiation of physical actions and their mental
counterparts results in more elaborate, organized, and flexible cognitive structures.
3. The Open Flow
When we are being immersing in chaos, bifurcation happens and the moments of flow and exhilaration appear. The "flow" is
the period in creative process when self-consciousness disappears, and is total absorption in the activity. Our creative
moments are moments when we are in touch with our own authentic truth, when we experience of a unique presence in the world.
Piaget (1952) postulates the existance of two functional invariants, i.e., modes of action, which do not change with maturation or experience.
These two forms of functions are organization and adaptation. Organization refers to the tendency to integrate various experiences by
integrating parts into wholes and wholes into more comprehensive wholes. The emphasis here is on the abstraction of relations and
the relations between relations. As we interact with the world around us we tend to organize various parts of our experience into integrated wholes.
This is true of perception, memory, language, competence, mathematical skills, and other intellectual activity. We are
not passive with respect to our interaction with the world. We construct meaningful wholes from parts automatically.
In the theory of chaos this concept we call - the vortex. In a vortex, a constantly flowing cell wall separates inside from outside.
However, the wall itself is both inside and outside. The experience of a unique presence in the world is also often voupled with a sensation
of ourself as indivisible from the whole.
Fig 2. The process of creativity in theory of chaos.
| Turbulence |
Bifurcation Points |
Amplification |
Open Flow |
Vortex |
Appendix 2
Portrait of Mathematics
Two cognitive scientists: George Lakoff and Rafael Nunez gave a portrait of mathematics:
- Mathematics is a natural part of being human. It arises from our bodies, our brains, and our everyday experiences in the world.
Cultures evrywhere have some of mathematics.
- There is nothing mysterious, mystical, magical, or transcendent about mathematics. It is an important subject matter of scientific study.
It is a consequence of human evolutionary history, neurobiology, cognitive capacities, and culture.
- Mathematics is one of the greatest products of the collective human imagination. It has been constructed jointly by milions of dedicated
people over more than two thousand years, and is maintained by hundreds of thousands of scholars, teachers, and people who use it every day.
- Mathematics is a system of human concepts that makes extraordinary use of the ordinary tools of human cognition. It is special in that it is stable, precise,
generalizable, symbolizable, calculable, consistent within each of its subject matters, universally available, and effective
for precisely conceptualizing a large number of aspects of the world as experience it.
- The effectiveness of mathematics in the world is a tribute to evolution and to culture. The effectiveness results from a combination of mathematical knowledge
and connectedness to the world. The connection between mathematical ideas and the world as human beings experience it occurs within human minds.
It is human beings who have created logarithmic spirals and fractals and who can “see” logarithmic spirals in snails and fractals in palm leaves.
- In the minds of those millions who have developed and sustained mathematics, conceptions of mathematics have been devised to fit the world
as perceived and conceptualized.
- Through the development of writing systems over millennia, culture has made possible the notational systems of mathematics.
It is the human capacity for conceptual metaphor that make possible the precise mathematization and sometimes even the arithmetization
of everyday concepts.
- Everything in mathematics is comprehensible – at least in principle. Since it makes use of general human conceptual capacities,
its conceptual structure can be analyzed and taught in meaningful terms.
- Human intelligence is multifaceted and that many forms of intelligence are vital to human culture. Mathematical intelligence is one of them.
- Mathematics is creative and open-ended. By virtue of the use of conceptual metaphors and conceptual blends, present mathematics can be extended to create new
forms by importing structure from one branch to another and by fusing mathematical ideas from different branches.
- Human conceptual systems are not monolithic. They allow alternative versions of concepts and multiple metaphorical perspectives of many important aspects of our live.
Mathematics allows for alternative visions and versions of concepts.
- Mathematics is a magnificent example of the beauty, richness, complexity, diversity, and importance of human ideas.
- Human beings have been responsible for the creation of mathematics, and we remain responsible for maintaining and extending it.
References
Briggs, J. & Peat F.D. (1999). Seven Life Lessons of Chaos, Harper Collins Publishers.
Briggs, J. & Peat F.D. (1990). Turbulent Mirror, perennial Library.
Cole, W. & Wertsch, J.V. (1996). Beyond the individual-social antimony in discussion of
Piaget and Vygotsky. Human Development, 39, pp 250-256.
Ernest, P. (1991). Social Constructivism as a philosophy of mathematics : Radical
constructivism rehabilitated? www.ex.ac.uk./~Pernest/soccon.htm.
Ernest, P. (1991). The Philosophy of Mathematics Education. The Falmer Press.
Halliday, M.A.K. (1993). Towards a language-based theory of learning. Linguistic and
Education, 5, 93-116.
Jacob, S.H. (1984). Foundations for Piagetian education, University Press of America.
Karplus, E.F. and Karlus, R. (1970). Intellectual Development Beyond Elementary
School, 1: Deductive Logic. School Science and Mathematics 70: 398-407.
Lakoff, G. ang Nunez, R.E. (2000). Where Mathematics Comes From: Basic Book, A Member of the Perseus Book Group.
Molle, M., Marshall, L., Lutzenberger, W., Pietrowsky, R., Fehm, H.L., Born, J. (1996). Enhanced dynamic complexity in the
human EEG during creative thinking. Neuroscience Letters 208, 61-64.
Piaget, J. (1952) The origins of intelligence in children. New York: International Universities Press.
Piaget, J. (1971). Science of education and the psychology of the child. New York: Viking Press.
Piaget, J. (1972). Principles of genetic epistemology. New York: Basic Books.
Resnick, L.B.(ed) (1976). The Nature of Intelligence. Erlbaum, Hillside, NJ
Taylor, P. (1996) Mythmaking and mythbreaking in the mathematics classroom, In:
Educational Studies in Mathematics 31, pp 151-173.
Von Glasersfeld, E. (1990). An exposition of constructivism: Why some like it radical. In
R.B. Davis, C.A. Maher and N. Noddings (Ed), Constructivist views on the teaching and
learning of mathematics (pp. 19-29). Reston, Virginia: National Council of Teachers of
Mathematics.
Von Glasersfeld, E. (1991). Radical constructivism in mathematics education.
Dordrecht, The Netherlands: Kluwer Academic Publishers
Vygotsky, L.S. (1978). Mind in society. Cambridge, MA: Harvard University Press.
Vygotsky, L.S. (1981). The genesis of higher mental functions In J.V. Wertsh (Ed.), The
concept of activity in Soviet Psychology. Armonk, NY: Sharpe.
Vygotsky, L.S. (1934/87). Thinking and speech. In R. W. Rieber & A. S. Carton (Eds)
The collected works of L.S. Vygotsky, Volume 1: Problem of general psychology. New
York : Plenum
Wells, G. Dialogic Inquiry in Education: Building on the Legacy of Vygotsky,
www.oise.utoronto.ca/~gwells/NCTE.html
Wood, D. (1988). How Children Think and Learn. Basil Blackwell.
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