ECER 2000 – European Conference on Educational Research
University of Edinburgh, 20-23 September 2000
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Streitlien, Aase:
Learning through interaction
Programme for Research on Mathematics Education
Telemark Centre for Educational Research
Institute of Teacher Education and School Development (ILS), University of Oslo, Norway
Paper presented at the European Conference on Educational Research, Edinburgh, 20-23 September 2000
Abstract
There are a variety of theoretical approaches for studying learning in school. In mathematics educational research there is a discussion going on about epistemology and different frameworks such as radical constructivism, social constructivism and sociocultural approaches, what the differences between them are and if they are compatible. In this paper I reflect on how view of learning has impact on my study of children’s learning in school. The epistemological questions are: How do we gain knowledge, and what does it take to learn?
Keywords: Learning, knowledge, mathematics education, epistemology, philosophy of education
The author’s address: Telemark Centre for Educational Research
Lærerskoleveien 35, 3679 Notodden, Norway
E-mail:
Phone: + 47 35 02 66 89, fax: + 47 35 02 66 98
Introduction
The purpose of this paper is to focus on some epistemological questions with respect to studying children’s learning Epistemology has to do with the question of gaining knowledge, but also with the question of the truth and validity of the knowledge gained.
There are a variety of approaches for studying children’s learning. Different approaches will influence the focus of interest, how a study is designed and how the findings are interpreted. In mathematics educational research there is a discussion going on about epistemology and theoretical approaches such as radical constructivism, social constructivism and sociocultural approaches, what the differences between them are and if they are compatible. In this discussion we can find two basically ideas about learning that to some extent reflect the discussion – the idea of learning as interaction between the individual human subject and objects, and learning as interaction between subjects.
The background for my reflections on children’s learning is my doctoral thesis with the working title Creating mathematics through language and interaction. The aims of this project are to describe learning processes in mathematics of pupils in primary school by analysing the phenomena of language and communication. My study concerns the function of language in developing mathematical understanding. I will refer to the research context of this project as I assume that epistemological perspectives that are raised also concern learning in general.
About learning
If we take a review into the field of mathematics education and research on learning, we find in particular two prevailing theories of learning – one individual-oriented approach based on Piaget´s theories, and one sociocultural approach based on Vygotsky.
The Piagetian tradition represents a rational perspective on human thinking and communication. Piaget aimed at creating a theory for the cognitive development of mankind, the genesis of scientific knowledge, which is why he adopted the name of genetic epistemology for his enterprise. He turned to psychology for the development of a biologically oriented theory of nature and origins of knowledge structures, the general principles underlying all human reasoning. According to Boden (1979), psychology was only one of Piaget’s approaches to the “machinery” of human cognition, along with biology and philosophical epistemology.
Development is in his theory viewed as a process from inside. Perceptions are always directed and accompanied by schemes of action. This involves reciprocity between the child and its environment. A basic idea in Piaget’s theory is that children are natural learners: They accommodate to environmental demands; they assimilate what the environment has to offer.
Children learn because their minds are made to learn. When they cannot fully assimilate new events, cognitive conflict is created and the acquisition of new knowledge reduces the “disequilibrium”.
According to Piaget the learner is regarded as an active constructor of knowledge, taking an active role in the learning process and using previous experiences to develop knowledge. Piaget is focusing on the cognitive development of the individual learner and not so much on the social and cultural aspects of learning. Piaget considers the constructing individual as the prime factor in a learning process. Knowledge arises at the interface between subject andobjects as Piaget demonstrates in his numerous studies (e.g. Piaget 1976).
Within sociocultural approaches knowledge is not viewed as something entirely constituted on the spot, but rather something that is re-created and re-produced in the situation. The social semiotic psychology was initiated by Vygotsky and later on followed up by neo-Vygotskyan “activity theory” (e.g.Wertsch and Rogoff). A central idea in Vygotsky´s theory is the insistence on finding the genesis of meaning in the combination of the socio-cultural genesis and the situational “micro”-genesis. With respect to learning, Vygotsky puts primacy on the social and cultural aspects. His theory is based on the idea that human beings develop cultural identity by participating in cultural activities and by doing this, acquire the instituted meaning of the activity. The basic unit of analysis is mediated action, i.e. action operating through “mediational means” (Wertsch 1985). According to Vygotsky, cultural tools and signs mediate the relationship between individual human subjects and objects in the environment.
The tool’s function is to serve as the conductor of human influence on the object of activity; it is externally oriented; it must lead to changes in objects. It is a means by which human external activity is aimed at mastering, and triumphing over nature. The sign, on the other hand, changes nothing in the object of a psychological operation. It is a means of internal activity aimed at mastering oneself; the sign is internally oriented (Vygotsky 1978:55).
The process is based on gradually acquiring the cultural tools and signs symbolising the instituted meaning. This will prepare the individual human subject to act in a way that is relevant for the situation. The development of concepts takes place in a complex interaction between the psychological processes and the acquirement of the cultural language. The internalisation of cultural forms of behaviour involves the reconstruction of psychological activity on the basis of sign operation.
The difference between what e.g. a pupil is capable of in collaboration with the teacher and what a pupil is able to do by her/himself is by Vygotsky described as the proximal zone. There is variation in how the proximal zone is interpreted. We will often find the proximal zone “described as a kind of force field which the child carries around, whose dimensions must be determined by the teacher so that activities offered are within the child’s range” (Lerman 1998:71). According to Davydov and followers the proximal zone is created in the learning activity, which is a product of the task, the texts, the previous networks of experiences of the participants, the power relationships in the classroom, etc. The proximal zone is the classroom’s, not the child’s.
Vygotsky´s theory of learning attaches great meaning to language activities. Knowledge is socially constructed, and language is the medium for such construction. Language arises as a way for the child to communicate with others. “Later on, when the language has become internal mental talking, it will start organising the child’s ideas, that means, it has become an internal mental function“ (referred in Dale 1996:163).
According to Vygotsky, more sophisticated mental processes like thinking in concepts, will develop as a transformation of semiotic activity. The semiotic activity will start as an external social activity and then be reconstructed mentally.
In semiotic activities the children are exploring the relation between sign forms and their referents. By doing this they will extend and develop their cultural tools.
Vygotsky distinguished between spontaneous concepts that the child develops through informal interactions with people in everyday life, and the scientific concepts that the child learns at school. The spontaneous concepts may be inductive generalisations from everyday experiences of the child, but development of scientific concepts goes in the opposite direction: from the general to the concrete.
Neither Vygotsky nor Piaget believed in the possibility of verbal transmissions of knowledge. The mental functions, such as abstractions and discrimination, cannot be learned as recipes. They cannot be learned in a ready-made form. The meanings of words cannot be transmitted from one mind to another with the help of other words because every word is generalisation and different people may operate with different levels of generalisation as mental functions.
However, as noted by Wertsch (1991), there is at least one essential shortcoming in Vygotsky´s claims of internalisation. This is that, in his account of the social origins of individual mental functioning, Vygotsky tended to equate social with intermental functioning. This amounts to a reduction of the range of social phenomena one addresses when examining the social origins of individual mental functioning. Instead of recognising that intermental functioning is always situated with regard to cultural, historical and institutional setting, Vygotsky often treated it as if it always occurs in essentially the same form. Such assumptions are clearly antithetical to the goals of a sociocultural approach to mediated action, and they contradict Vygotsky´s expressed desire to understand human mental functioning as it occurs in various historical, cultural and institutional settings.
As we can see, learning in both these two traditions is not viewed as a simple response in the pupils to the teacher’s presentation of a topic. Learning is viewed as an active process in which the pupils construct their knowledge in their own way. The pupil is an operator in her/his own learning process.
The constructivist view of learning (Piaget) on the one hand and the sociocultural view on the other hand (Vygotsky) will influence teaching practise as well as focus of interest in educational research. Within the constructivist view the learner is constructing her/his own internal world of knowledge. Within the sociocultural tradition concepts are regarded as social acts and tools:
Consciousness is constituted in historical, sociocultural settings and cultural tools are internalised in the strong sense that the mental plane is formed in that process. Thus, cultural tools both transform the person and the world for that person, and these cultural tools precede the individual (Lerman 1998:74).
In many ways these two positions contrast each other in their differing emphasis. Brown (1994) claims that these two positions are “incompatible”, but suggests that this discrepancy can be transcended with a hermeneutic approach, in which the interpretative process is the prime category. Subjects and objects are both constituted through such processes and hence secondary categories.
Cobb & Bowers (1999), on their hand, see these views as complementary and that learning mathematics should be viewed as both a process of active individual construction and a process of enculturation into the mathematical practices of wider society.
They claim that the two views can be seen as background for each other and that the researcher as well as the teacher should alter focusing between the two, depending on the situation.
Others will argue that this is more complicated. According to Confrey (1995), there is a need of an alternative theory to Vygotsky and Piaget. “I predict that the new theory will establish a relatively distinct basis – one in which the diversity plays a more significant role, and in which the individuality of the child is tempered by the responsibility of community and culture” (p.225).
Perspectives on knowledge
If we see learning as a knowledge construction process, it is in my view impossible to have a view of learning without a view of knowledge. For a researcher as well as for a teacher, a main question is: How do we gain knowledge, and what is truth-value of the knowledge gained? Teaching methods emphasising delivering facts, wrong and right answers, procedures for problem solving and only one way of approaching a problem, will express a platonic or absolutistic view of knowledge. Platon viewed the world of ideas as an unchangeable world. Science for him consisted of a variety of absolutistic truths. According to Brown (1994) the platonic view has been largely responsible for the tradition of seeing mathematics as a body of knowledge to be discovered or encountered by the students. This view of mathematics tends to be associated with metaphors such as “delivering”, “receiving” and “ready made”, being used in relation to teaching, learning and content.
Radical constructivism (e.g. von Glaserfeld 1991) offers a view antithetical to Platonism. Glaserfeld has extended constructivism as a learning theory (originated from Piaget) to radical constructivism. His epistemology is based on two principles: First, that knowledge is not passively received, but actively built up by the cognising subject. Second, that the function of cognition is adaptive and serves the organisation of experimental world, not the discovery of ontological reality. Thus, we construct our theories of the world and our experiences. The process of this construction takes place in the mind of the individual human subject. This does not mean that radical constructivism denies reality, but we are not able to see reality as it is owing to the fact that we have no open access to reality. We therefore have to construct our own interpretations. In that way, there is no meaning in the idea of “the objective truth”. Our universe consists of a plenitude of descriptions rather than of an ontological world per se. Within radical constructivism, the influence of cultural traditions as language, sociocultural tools and the environment for learning might be neglected.
Post-structuralism presents a supplement to radical constructivism by emphasising the language aspect in knowledge construction. A connection can here be made with structuralism and its dominant focus on language. However, the post-structuralistic reinterpretation of the categories of language leads us to think about relationship between signifiers and signifieds in new ways. This approach assumes that any attempt to fix the meanings of a sign is doomed to fail, for there is always a surplus meaning – a “supplement” – which leaves the meanings of sign open to debate. In this tradition any curriculum is culturally defined. To get to know a subject matter is then viewed as part of the process of learning how to use language and handle language issues.
Post-structuralism moves away from seeing the world as something constructed by the individual human subject by recognising that this is always done in an inherited language.
In Foucault’s archaeology of knowledge (1972) it is emphasised that forms of knowing occurs in an environment full of historical leftovers.
Foucault was concerned about the complexities involved in representation, but he also investigated the ways in which knowledge is produced within a shared social context and within definite historical circumstances. Discourse is made up, then, of rules of conduct, established texts and institutions. Thus Foucault’s primary concern was the relationship between knowledge and the power relations of the location in question, rather than just establishing what representation means. Power is of major interest in constructing knowledge. His view of power is a shift away from the individualistic ways of viewing power to a notion of power that is not outside of an individual or invested in one individual to exert another; power is in the relation. Foucault starts from the premise that, primarily, knowledge and power work through language; that as we learn to speak we pick up basic knowledge and rules at the same time. He sees that these are organised forms of knowledge, which working together with their associated institutions, have significant effects on people and their possible actions. Foucault considers how knowledge is produced, and how it induces power through what he calls “discursive practices” in society. Foucault describes discursive practice as:
A body of anonymous, historical rules, always determined in the time and space that have defined a given period (Foucault 1972:117).
There is therefore no knowledge without a particular discursive practice; and any discursive practice may be defined by the knowledge that it forms. A discursive practice is constituted by the actions of members of community, their interactions with each other and the texts and communications and artefacts from within that practice.
Foucault argues that these discursive practices have profoundly shaped the structure of our society. In the context of classroom research, these theoretical notions allow us to look at the classroom (set within the wider mathematics education culture) as a discursive practice.
With respect to education, a problem of post-structuralism could be the downplaying of the meaning of a text beyond itself. “Another story can always be told”. Any phenomenon can be described in at least two different texts, and the texts can be referred merely to themselves. It is not possible to decide which one is the correct one, because the two texts cannot be compared. A text cannot be used with the intention of saying something about “reality”. Any text can be analysed according to itself. This might be a problem considering learning processes since the students are supposed to learn “something”.
Pragmatism tries to solve this problem by having in mind the relationship between the text, the “object” and interpretation. Pragmatism and interactionism is concerned with the way in which meanings and interpretations are the product of the “pragmatic concerns” of practical problems and purposes of social life. Interactionism treats social actors and their small face-to face interactions as the basis of all social life, so that the meaning of any concept or idea (personal, political, philosophical or scientific) can only be located in the experiential consequences, which it produces. On the basis of this, pragmatist James (1890) distinguished “knowledge about” from “knowledge of”. The first one, knowledge about, can be required through textbooks and is conveyed in abstract general principles, which can be learned and memorised.
Knowledge of, is acquired through experience in everyday life and can take an unconscious or tacit form. This treats the concept of the mind as a “thinking process” always in development, rather than a fixed thing. In this way, individuals can define objects and their context.
In summery, there are basically two ideas about knowledge that come into sight in the discussion above. First, we have the idea that true knowledge or received wisdom is an objective reflection of the world and that language represents this reality.