The SOLO Taxonomy

Both pedagogical and mathematical thinking may be viewed as consisting of a range of elements, operating on different levels. The SOLO taxonomy provides valuable insights into the nature of such elements. Building upon the developmental learning theories of Piaget and Bruner, Biggs and Collis recognised that learners move through distinct modes of functioning, generally corresponding to the following age periods:

From Birth	Sensori-Motor
From around 18 months	Ikonic
From around 6 years	Concrete-symbolic (approx. K-Year 10)
From around 16 years	Formal (approx. Years 11 and 12 +)
From around 20 years	Post-Formal (University/professional practice)

Differing from classical stage theory, it is not suggested that each stage replaces the previous one, but that each adds to the available cognitive repertoire. In different situations, learners may "regress" to an earlier mode of functioning or utilise a higher cognitive function in the learning of a lower-order one, adopting a "multi-modal" approach to the task at hand (Biggs and Collis, 1991, Collis and Biggs, 1991). An example of the first situation (labelled "top-down" learning by the authors) would be the use of intuitive, visual methods in mathematical problem solving, where the ikonic mode is used to supplement the more usual concrete-symbolic approach. "Bottom-Up" learning may be illustrated by the use of higher-order practices in the learning of sensori-motor skills (such as thinking through the action of a golf swing, studying the style of expert players or learning the theory and techniques of art in order to improve in the ikonic aspects). Although much of secondary schooling may be recognised as occurring within the concrete-symbolic mode, the use of multi-modal strategies may be more extensive than previously realised (Biggs and Collis, 1991, Collis and Biggs, 1991). It is certainly common in areas such as music, which utilises sensori-motor, ikonic and concrete-symbolic elements in the learning process, and recent research suggests that the ikonic mode may be a powerful influence in mathematical problem solving (Collis, Watson and Campbell, 1992). At present, however, much of the focus of instruction in secondary schools lies within the concrete-symbolic domain. Even at the senior level, it is now believed that the end-point of instruction in most subjects will be at this level. Only in those areas in which the student is particularly competent (and likely to continue into tertiary study) is formal mode functioning likely to be observed with any degree of frequency (Collis and Biggs, 1992).

Some learners never reach the formal stage, at which the foci of interaction are theories and abstractions, rather than the more concrete objects of earlier stages; most do not achieve post-formal, which involves working with and extending theory systems themselves. With increased retention rates in the senior years of schooling, it is likely that increasing numbers of senior students will be operating throughout their studies at the concrete-symbolic level (Collis and Biggs, 1983). The preferred mode of operation for students has significant implications for learning and instruction (Collis and Biggs, 1991), and will become a focus for this investigation in the study of the representation and understanding by students and preservice teachers of algebra and learning interactions.

Further, Biggs and Collis suggest that, within each mode of functioning, learners display a consistent sequence or "learning cycle" when learning new tasks. This gives rise to the SOLO acronym, detailing the "Structure of the Observed Learning Outcomes." The theory postulates five structural levels through which learners pass (from Biggs and Collis, 1989):

Prestructural	The task is engaged, but the learner is distracted or misled by an irrelevant aspect belonging to a previous stage or mode.
Unistructural	The learner focuses on the relevant domain, and picks one aspect to work with.
Multistructural	The learner picks up more and more relevant or correct features, but does not integrate them.
Relational	The learner now integrates the parts with each other, so that the whole has a coherent structure and meaning.
Extended Abstract	The learner now generalises the structure to take in new and more abstract features, representing a higher mode of operation. (p. 152)

Using this framework it becomes possible to identify an individual's current level of operation for a particular task through a study of verbal and/or written responses. It thus provides a powerful tool for the assessment of student understanding of concepts, and for problem solving (Collis and Romberg, 1991). The taxonomy has also proved effective as a means of planning and developing curricula based on the cognitive characteristics of the learners.

In terms of pedagogy, the unistructural, multistructural and relational levels are recognised as the "target modes" for teaching; allowing for individual differences, it may be expected that all students should achieve one of these levels as a result of an effective learning experience. In the case of new work, it should be the teacher's objective to assist the students to move from a prestructural state (with no organised or coherent knowledge of the material) to one that is, ideally, relational. In practice, however, the more likely end result for instruction is multistructural, in which students and their teachers are satisfied to know "some things about an area." Relational understanding (in both the SOLO sense and that of Skemp, 1976) is frequently sacrificed for the demands of utility. The occurrence of an extended abstract response is not normally one that is anticipated as a direct result of instruction, but is more a function of the individual learner's ability to go beyond what has been taught.

Recent research suggests that, just as the linear sequence of modal development originally proposed has given way to a more complex multi-modal structure, so the cycle of levels within the modes may be more complex than originally anticipated. In particular, at least two Unistructural-Multistructural-Relational cycles now appear to exist within the concrete-symbolic mode, as observed across a range of mathematical topics in the junior years of high school (Pegg, 1992). This would help to explain quite distinct styles of thinking about complex mathematical objects (such as process and object conceptions of functions) while situated within a single cognitive domain. As more is revealed through research, the model of cognitive development offered by the SOLO Taxonomy assumes less of the linear, sequential pattern of its Piagetian origins, and more of a complex branching structure.

Figure 2: SOLO Taxonomy: Schematic Outline

Figure 2 (Pegg, 1992, p. 27) provides a schematic outline which relates modes, learning cycle, curriculum goals and suggested exit levels for schooling. Each mode of operation is associated with a particular "type of knowledge", as illustrated. That arising from the sensori-motor is likely to be tacit, unable to be articulated, as in the "feel" of a good golf swing. The ikonic mode produces knowledge which is intuitive, difficult to verbalise, and closely linked to visual and emotive aspects of the situation. The concrete-symbolic mode leads to knowledge that is declarative - not only knowing "how to," as in earlier modes, but being able to say "why," at least in terms of the concrete referents available. The theoretical knowledge which results from the later modes involves adopting theories and theory systems - complex networks of relations between ideas and concepts - as the objects of thinking.

The van Hiele Theory. Applying Theory to Practice.