© 1996: The University of Newcastle: Faculty of Education
The Piagetian model of development and learning underpins the constructivist approach and has dominated our thinking about teaching and learning for more than three decades. Its limitations, however, are becoming increasingly apparent as more is revealed concerning the ways in which individuals learn, and particularly the importance of social and cultural influences upon learning. Piaget's approach treats the individual in a largely decontextualised way, directing attention to the cognitive processes of individuals in isolation from their learning environments. In order to adequately consider the individual within the context of the learning environment (and particularly in relation to the use of external tools and the interaction with others) it is necessary to look beyond neo-Piagetian theories; to look, instead, at the work initiated by the Soviet psychologist, Lev Semanovich Vygotsky (1962, 1978, 1987). These theories consider learner as inseparable from context, particularly the social and cultural context in which learning occurs. Vygotsky believed that all higher cognitive processes are acquired initially through social interaction - occurring on an inter-personal level before they become internalised to occur on an intra-personal level.
Both Piaget and Vygotsky began their studies in the early years of this century. While Piaget's life and work spanned most of this century, Vygotsky died in 1934, at only 38 years of age. His work was largely unknown outside his own country until the 1960s; even within the Soviet Union, it was suppressed for many years. Over the past three decades, however, Vygotsky's writings have assumed growing importance, particularly in the development of wholistic theories of language learning, but increasingly in fields such as mathematics learning and teaching (Zepp, 1989, Manning and Payne, 1993, Confrey, 1993b). His notion of a Zone of Proximal Development has proved appealing in a wide variety of contexts, and offers much in the present exploration of the way in which the creation of an appropriate learning environment may assist the growth of understanding.
In seeking to understand something of the contribution of Vygotsky in the present context, it is appropriate to begin with the notion of "tools." He began his work, Mind in Society (Vygotsky, 1987) with a quotation (in Latin) from Sir Francis Bacon, translated by Bruner (1986) as:
Neither hand nor mind alone , left to itself, would amount to much. And what are these prosthetic devices that perfect them? (p. 72)
The additional tools to which Vygotsky appears to be referring are, most importantly, thought and language, those means by which we are recognised as most uniquely human. In opposition to Piaget, Vygotsky places language as the precursor to thought, claiming that it is only through the use of language that the higher mental processes may develop and become operational. Language is social in origin, developed through interaction with others, and, in Vygotsky's view, serves two primary purposes - self-direction and communication. This perception of language as a tool which aids thought is a fundamental feature of Vygotsky's view. He believed that the higher mental processes are mediated by language, first observed as egocentric speech in children, which then becomes internalised, developing into thought. (Zepp, 1989, p. 30-32)
Mathematics shares with language these twin characteristics: it is, at once, both cognitive tool and means of communication (Confrey, 1993, p. 50). In its role as tool, mathematics may be perceived primarily as a means of effecting some outcome; as Confrey points out, the image of mathematics as tool links it with action, a significant aspect often overlooked. The potential for computer technology to assume an active mediating role in supporting mathematical thinking, learning and practise, is relevant in the current context. The potential role of goal-directed action may be exemplified by comparing the interface offered by the Macintosh computer algebra package, Theorist, with that of other packages of the same type. Most computer algebra tools allow the immediate solution of equations, for example, through a general "Solve" command; in some cases, the intermediate steps of the equation-solving process are supported by allowing operations to be carried out on both sides of the equation. Theorist is unique in allowing the user to physically manipulate the terms and elements of algebraic expressions and equations. Solving an equation such as
3 / (x - 1) = 2
may be achieved by physically selecting the denominator, x - 1, and, using the mouse, dragging it across to the right-hand side of the equation, to automatically produce
3 = 2 ( x - 1 )
This may be expanded, and the x term isolated by similar manipulations. The program offers the choice of alternative techniques which, in some ways, may be preferable pedagogically (involving performing the same operation to both sides of the equation, producing equivalent equations). At the same time, many students and teachers solve equations in exactly this way, involving either overt or covert physical manipulation of the terms. The role of action in higher mathematical processes is likely to be significant, but as yet remains largely unexplored. This recent development of computer software which simulates and supports such physical involvement invites such exploration.
Central to Vygotsky's view of cognition and learning is the social and cultural context of the learner. In particular, the learner is viewed as achieving higher cognitive ground through interaction with others, especially adults and knowledgeable peers. His zone of proximal development may be thought of as "the distance between the actual developmental level as determined through 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). Bruner explores the implications of this concept for tutoring and guided learning through his notion of "scaffolding" (Bruner, 1986, pp. 74-76).
Wood (1980, 1986) expands upon both Vygotsky's zone of proximal development and Bruner's notion of scaffolding to offer a model of learning based upon twin central principles of uncertainty and contingency. Wood observes that learning within a situation of uncertainty is always less effective than one in which the learner is able to recognise commonalities and familiar features. Motivation, task orientation, even the ability to remember particular features of the situation - all are likely to be reduced in unfamiliar situations, and much mathematics learning occurs within situations high in uncertainty. Support is required, then, in such situations of high uncertainty which will serve to alleviate these problems and so make the learning experience more effective.
Wood's second key principle defines the preferred nature of such support, requiring that the response of the teacher be contingent upon that of the child if optimal cognitive progression is to occur. Wood postulates five levels of increasing control which may be observed in a tutorial situation (that is, a learning situation involving interaction between the learner and a more capable other - the "tutor") (Wood, 1986, pp. 197-198). These range from minimal control - the tutor prompts the learner with a general question, such as "What might be done here?" - to highly controlled, in which the tutor actually demonstrates the steps needed to fulfil the requirements of the task:
- Level 0: No assistance
- Level 1: a general verbal prompt (what might you do here?)
- Level 2: specific verbal ("You might use your computer tools here.")
- Level 3: indicates materials ("Why not use a graph plotter?")
- Level 4: prepares materials (selects and sets up tool)
- Level 5: demonstrates use.
Wood's principle of contingency requires the tutor to decrease the level of control at each correct action of the learner, and to increase the level of control or intervention upon each error. This process of flexible scaffolding allows the learner to progress optimally across the zone of proximal development, with greater and lesser degrees of support as required. The critical principle in such learning is the promotion of autonomy and independence on the part of the learner. It is not difficult to support and scaffold learning; the challenge lies in doing this in such a way that the scaffolding is gradually removed, and the learner actually decreases the level of dependence upon the support structure as the learning sequence progresses. This is the primary goal of contingent learning. In the context of mother-child instructional situations, Wood cites research which supports such learning: "What we find is that the more frequently contingent a teacher is the more the child can do alone after instruction" (Wood, 1986, p. 198).
The structure of the learning environment is the focus for research by Valsiner (in Rogoff and Wertsch, 1984) which extends the study of the zone of proximal development. Exemplified by the adult-child learning experiences associated with the socialization of meals, Valsiner proposes two additional zones which serve to define more clearly some of the situational constraints which may act to support or impede progress across the zone of proximal development. The first of these constructs, called the zone of free movement (or ZFM) is based upon the observation that learning is facilitated by focussing the attention of the learner upon that which is to be achieved. This may be done by restricting the actions of the learner, or by defining a "zone of free movement" (Valsiner, 1984).
Within the field of objects and affordances related to them in the environment of the child, the zone of free movement (ZFM) is defined for the child's activities. The ZFM structures the child's access to different areas of the environment, to different objects within these areas, and to different ways of acting upon these objects. (pp. 67-68)
Defined conjointly with the zone of free movement is a zone of promoted action (ZPA). If the ZFM is effectively an "inhibitory mechanism" (Valsiner, 1984, p. 68) which functions to limit the actions of the learner within the structured environment, then within that zone exist "sub-zones" which are defined by those actions sought to be encouraged and learned. In the context of "meal time," these may involve the appropriate use of cutlery; in relation to the use of technology, the zone of promoted action will be defined by the appropriate use of available software tools to achieve mathematical goals.
Theory based upon the work of Vygotsky, then, offers much which may inform our development of the notion of a mathematics learning environment. Others, too, however, make significant contributions in this regard, particularly the theory developed initially by Dina van Hiele-Geldof and her husband, Pierre van Hiele. Since Dina's death in 1959, Pierre has continued to develop and extend their earlier joint work.
Introduction. The van Hiele Theory.
© 1996: The University of Newcastle: Faculty of Education