A theory of mathematics education

Although the van Hiele theory (van Hiele, 1986) has been most widely recognised for its role in explicating the levels of thinking associated with the learning of geometry, it has been developed as a general theory of mathematics education. Growing from the concerns of teachers, it does not stop at the description of "levels of thinking," but seeks to provide a basis for understanding the movement between these levels, and the role of the teacher in assisting such progression. As such, the theory goes beyond the SOLO taxonomy (which is essentially descriptive) and beyond, too, the concerns of Piaget, who deliberately distanced himself from the question of how students may be encouraged to progress from level to level. His was a developmental theory, holding that such progression was largely independent of the influence of instruction; he referred to such concerns disparagingly as "the American question," but in fact it was the Dutch van Hieles who appear to have made significant progress in addressing it.

In his recent work (1986), Pierre van Hiele describes a theory of mathematics education arising from the study of two fundamental concepts - structure and insight. Although reluctant to specify a definition for the first, van Hiele admits that it may be broadly thought of as a "network of relations" (van Hiele, 1986, p. 49) in which commonalities are recognised across all types of events and perceptions. In everyday life, we recognise structure in going through daily routines, at work and home; structures are apparent in the patterns of nature and man; continuing a sequence of numbers is a recognition of structure, as is the recognition that the symbol ( x + 2 )² may be seen as a sign to expand the given expression and produce a new equivalent one. Insight, in this context, is a recognition of structure - we know what to do when we experience such insight, and it is precisely an absence of such insight which leaves so many school students at a loss as to know what to do with a given algebraic expression, equation or problem.

Van Hiele distinguishes between rigid and feeble structures (van Hiele, 1986, pp. 19-23), strongly reminiscent of Wood's principle of uncertainty (Wood, 1986). Consider, for example, a student presented with the expression,

x² - ( x + h )².

The more likely response for a student of at least moderate algebraic facility is to attempt to expand and simplify the expression. The recognition of the requirement to expand the squared part of the expression may be thought of as a relatively rigid structure. Recognising that such an expression provides an opportunity for factorisation, as a "difference of two squares," however, is likely in most students, to be a relatively feeble structure. Some prompting may be required for students to recognise this structure, even when they quickly recognise it in a case such as x² - 4. The dominant strategy of algebra instruction in the past has centred around the development of rigid structures through repetition, seeking to "automate" student responses to algebraic prompts. Such learning, however, is likely to occur at a very superficial level, and is relatively easily exposed when students encounter an exceptional case. As explained by Confrey, this relates closely to the Vygotskian notion of "pseudoconcept" (Confrey, 1993),

... acknowledging that children often use words before they have grounded its [sic] meaning in conceptual operations. Vygotsky suggests that this use of language that runs ahead of cognitive depth is an important part of learning - and describes a key mechanism in how adults teach children to advance to higher levels of cognitive thought. (p. 50)

This recognition of the central role of language in the learning process is a common theme throughout the works of both Vygotsky and van Hiele.

To van Hiele, true learning is that which students achieve through their own efforts, efforts which involve them in experiencing what he terms a "crisis of thinking" (van Hiele, 1986, p. 43). Similar to the Piagetian notion of disequilibrium, and very close to Doll's "perturbation" (Doll, 1986, p. 15), van Hiele sees such a crisis as necessary for students to achieve a higher level of thinking. While teachers may be successful in having students "mimic" the responses of a higher level, unless the learner has struggled with the material personally, no cognitive gain has been made. The cognitive "safety nets" (described by Tobin and Fraser, 1988) which are a feature of many mathematics classrooms are attempts by students (and by their teachers) to reduce the cognitive load of the material to be learnt; such efforts in van Hiele's view, must be carefully controlled, since meaningful learning involves transition to a higher level of thinking, and this can only occur by going beyond the present state. The links with Vygotsky's zone of proximal development are apparent, where "the only good learning is that which is in advance of development" (Vygotsky, 1987, p. 89).

This is the point at which the theories of learning described here coincide. For all their various forms and distinct priorities, the common ground is the perceived need for challenge. The teacher does not encourage learning by predigesting the material; rather, the learner must be an active participant in the process of creating meaning through interacting with that which is to be learnt in a context which supports exploration, verbalisation and activity.

The ways in which the levels of thinking proposed in the van Hiele theory (van Hiele, 1986, p. 53) complement those of the SOLO taxonomy have been described in detail elsewhere (Burger and Shaughnessy, 1986, Jurdak, 1991, Olive, 1991). The van Hiele levels begin, not with the level of action proposed as the sensori-motor mode of the SOLO taxonomy, but with the level of visualisation or recognition (Hoffer, 1981), corresponding to the global, intuitive thinking associated with ikonic thought. Next is the level of analysis, or the descriptive level (van Hiele, 1986, p. 53), corresponding closely to the concrete-symbolic mode of the SOLO taxonomy. This is followed by a level alternatively labelled abstraction (Burger and Shaughnessy, 1986), ordering (Hoffer, 1981, p. 14) or, simply, the theoretical level (van Hiele, 1986), easily recognised as encompassing formal modes of thought. Although the literature describes successive levels (commonly as deduction and rigour) van Hiele himself appears more inclined to view these as logical extensions of the theoretical level (van Hiele, 1986, p. 53) which, once achieved, experience a phenomenon he describes as level reduction (van Hiele, 1986, p. 53). Although the objects of thought may involve successively higher levels of abstraction, the actual mode or style of thinking remains essentially the same. The correspondences which occur with the SOLO taxonomy enable the two models to be considered as logically compatible; the different ways in which each illuminates each style of thought, however, makes the synthesis proposed here attractive.

The theory of van Hiele, in addition to describing levels of thinking, offers an important addition. This is the notion of stages of learning as means by which the learner may be assisted to seek higher cognitive ground. Five such stages are specified (van Hiele, 1986):

1. In the first stage, that of information, pupils get acquainted with the working domain.

2. In the second stage, that of guided orientation, they are guided by tasks (given by the teacher, or made by themselves) with different relations of the network to be formed.

3. In the third stage, that of explicitation, they become conscious of the relations, they try to express them in words, they learn the technical language of the subject matter.

4. In the fourth stage, that of free orientation, they learn by general tasks to find their own way in the network of relations.

5. In the fifth stage, that of integration, they build an overview of all they have learned of the subject, of the newly formed network of relations now at their disposal. (pp. 53-54)

These stages of learning are significant in providing a framework for instruction aimed to develop understanding of the material or skills to be learnt. The signicant role of language (and, by implication, social interaction) is clearly consistent with the Vygotskian approach to learning described above. A third consistent theoretical base is found in the neo-Piagetian model developed by the Australians, Biggs and Collis, which has become known as the SOLO Taxonomy. Each of these three theoretical positions illuminates the object of focus from a different perspective, highlighting aspects which each alone might fail to accentuate. If Vygotsky painted with broad brush strokes, then the SOLO Taxonomy provides the fine detail by which we might better understand the learning process for individuals.

Vygotsky's contribution. The SOLO Taxonomy.