Partial states of knowledge construction:

Schemas, models – where do they fit in?

Within the field of cognitive science, the terms “schema” and “model” abound. However, it is not always clear exactly what it is that these terms are being used to describe. Generally, these terms are intended to

describe the way we represent (internally and externally) what is in our heads. “Schema” is often used to label a kind of mental map; “model” is sometimes equated with some sort of analogue or metaphorical

Fischbein (1987) set out 3 requirements for an effective model: “comprehensiveness”, “obviousness” and “correctness”.

Chinnappan (1998) focused on schemas and mental models and considered the nature of,

“possible interactions between the state of organisation of available geometric knowledge and the accessing of that knowledge during problem solving”. (p214)

He judged that,

“attention to the qualitative aspects of knowledge development and utilisation has the potential to improve current levels of

understandings about why some mathematics students experience difficulties in applying previously learnt knowledge.” (p214);

a view I have previously expressed. He goes on to consider the relevance and appropriateness of certain models and representations in different areas of mathematics.

There are some aspects of Chinnapan’s view that might be seen to parallel the findings of diSessa and Wagner (2005), as I shall now describe.

Chinnapan (1998) was interested in the relationship between the quality of children’s knowledge base and their ability to access and make

effective use of that knowledge. He considers whether more able children are more likely to use a greater number of different schemas and more often than lower ability children. This, of course, implies that more able children might have a more sophisticated relational knowledge base than their less able peers. Relational knowledge is that which is rich in

connections which, is, of course, the way that diSessa and Wagner (and others) characterise knowledge and conceptual growth. Clearly,

Chinnappan also views the extent to which knowledge is relational as key to “ability”. He sees mental models as images or representations of what

exists/occurs in the mind of the learner. Chinnappan believes that there are “key concepts that anchor other concepts” (p203) and that,

“Two characteristics are important to understanding geometric schemas; organisation and spread. Organisation refers to the establishment of connections between ideas, whereas spread refers to the extent of those connections.” (p203)

There are similarities, even parallels, here with diSessa & Wagner (2005); for example transfer and alignment correspond at some level with the idea of “organisation”, including scope for complexity and sophistication. The term schema suggests mapping of elements from one case to another; this implies a focus on structure which, I have argued, is not appropriate for describing learning. However, perhaps schemas are an appropriate description of what learners are able to construct through repeated experience? Schwarz et al (2005) think so.

Schemas and models contribute to what Schwarz et al describe as “efficiency” – i.e. that through repeated opportunities to work with similar tools to solve similar problems, developing “replicative” and “applicative” knowledge, schemas and models are developed and readily utilised. They go on to stress, however, that the development of interpretive knowledge that equips learners to learn, also increases efficiency, Schwarz et al find that,

“ … enhanced learning does indeed occur when people have an opportunity to develop the interpretive knowledge that prepares them to learn.” (p11)

They also stress that learners need to interact and to access additional resources, obtaining feedback. This, clearly, is in stark contrast to much of the transfer research that has been based on “Sequestered Problem Solving” (SPS) i.e. in which,

“Tests of the ‘direct application’ view [that] typically place people in sequestered environments, where they have no access to

‘contaminating’ information sources other than what they have learned previously, and where they receive no chances to learn by trying out an idea and revising as necessary.” (p5)

There is clearly a conflict and tension in the design and management of research into learning wherein the methodology is based on SPS: if learners are denied opportunities for feedback and revision, they will not be able to show learning.

Partial states; grey areas

Wilensky (1993) believes, like Piaget, that interaction and familiarity with a concept lead the learner to make more and more connections between other experiences and the new concept. However, in contrast to Piaget (who was, after all, considering processes and outcomes on a grand scale), Wilensky is concerned more about learning at the level of the individual and believes that concreteness is,

".. not a property of an object but rather a property of a person's relationship to an object" (p198)

and that, as the relationship becomes stronger, it becomes more concrete. Concreteness is, therefore, something to which a learner aspires, rather than from which he/she develops. Abstraction, using Wilensky’s terminology, occurs through concretisation.

Wilensky also sees conceptual growth as augmentation of connections and that this may facilitate abstraction. Wilensky’s notion of concreteness also suggests a continuum – i.e. partial states, rather than a have/have not model.

diSessa & Wagner (2005) also feel that much educational research in this area is guilty of over simplifying the process of learning. They point out that,

“ … we should expect no sharp line between “having” and “not having” a concept.” (p6)

They go on to state that,

.... "states of partial construction are much more important to describe” (p6)

and emphasise that with a “complex knowledge system” perspective, it is necessary to acknowledge all the grey areas, the intermediate states. diSessa & Wagner believe that there is a need to characterize partial constructions, particularly the early phases in the construction of a true co-ordination class.

Schwarz et al (2005), in exploring issues of efficiency and innovation and the balance of these 2 aspects of knowledge, emphasise the need for efficiency in that,

“ … if people confronted with a new complex problem, have solved aspects of it before, this helps make these sub-problems routine and easy to solve. This frees attentional bandwidth and enables people to concentrate on other aspects of the new situation that may require non-routine adaptation.” (p30)

They also explain that efficiency is insufficient for innovation and that both are required if learners are to continue to learn and solve new problems. Schwarz et al note that,

“ … innovation is often preceded by a sense of disequilibrium that signals that certain processes or ways of thinking (e.g. previously learned routines) are not quite working properly …” (p32)

Pratt & Noss (2002) acknowledge the importance of the grey areas when they explore the notion of situated abstractions. They found that recently constructed situated abstractions might be called upon in new situations in which similarities are recognised, but that children will, initially, attempt to use other long-established internal resources. (This resonates with the “disequilibrium” noted by Schwarz et al (2005)). This is because

resources are “brought to mind” according to a priority order that is established and modified over time, according to feedback regarding the

“success” of resources utilised. diSessa (1993) offers the related notions of cueing priority and reliability priority: cueing priority refers to the

likelihood that a resource will be activated as potentially useful in a

situation; reliability priority is established according to feedback regarding the usefulness of the resource on previous occasions, taking account of other resources also activated. Thereby, high cueing priority and high reliability priority (diSessa referred to these together as “structured priorities”) take time to develop, and new resources can only have low reliability priority (and will not, therefore, be utilised in novel situations) until they are tried and tested.

Pratt & Noss (2002) put forward a theoretical model in which meanings constructed in one setting might also be valuable in another setting. They observed and analysed the way children made sense of the effects of using a variety of computational devices that simulate everyday situations familiar to the children, but offering enhanced functionality in the virtual world. Pratt & Noss believe that there is a distinction between abstraction and de-contextualisation which is generally overlooked in the literature. They point out that,

“A central issue is the extent to which mathematical abstraction depends on decontextualization … “ (p454)

They acknowledge the differences between macro- and micro-level research and sought to illuminate the ways that the findings of research both macro- and micro- might be related and therefore reconciled. They attempt to achieve this by elaborating the relationship between

mathematical abstraction and de-contextualisation. Pratt & Noss maintain that,

“situated abstraction is observable as more or less tacitly articulated invariance of relations, framed within the situation itself”. (p457)

“Situated abstractions emerge during activity as internal resources that serve as relatively general devices for making sense of

situations that arise within a setting.” (p456)

Pratt & Noss go on to say that situated abstractions are,

“ … types of knowledge that enable learners to reflect on the structures within a setting a make sense of phenomena that hold true across it.” (p456)

They show that situated abstractions “are expressed in a language [ …] that remains embedded in the situation in which it was constructed”. (p456)

This analysis (Pratt & Noss, 2002) suggested that children:

• will, when making sense of devices, articulate situated abstractions of the way they work;

• will, when encountering superficially new situations, initially attempt to use long-established internal resources for making sense of such situations, rather than situated abstractions recently constructed;

• will, subsequently, employ recently constructed mental resources as long as:

a) feedback from the system emphasises the lack of explanatory power of the long-established resources (increases cueing priority), and

b) there is sufficient similarity between the old and new contexts. Pratt & Noss’s model contributes to my view that learning is not about detachment from contextual features but development of increasingly rich and intricate networks of attachments comprising aspects of experience of, and within, those features. Therefore, if abstraction depends on decontextualization, it follows that learning cannot be dependant on abstraction. This point is becoming increasingly clear to me: that

abstraction (of abstract notions) might be an outcome of learning that, in itself, might enable advanced functioning at high levels within the domain,

but it is not the cause of earlier stages of concept development. Therefore, I do not believe that abstraction (understood as decontextualization) is necessary for learning.

Salomon & Perkins (1989) offer a view which acknowledges different models of transfer:

“ .. we argue that transfer is not at all a unitary phenomenon. Rather, transfer can occur by different routes dependant on different mechanisms and combinations of mechanisms.” (p115)

They propose 2 types of transfer – “high-road” and “low-road”, whereby the former,

“ … occurs by intentional mindful abstraction of something from one context and application in a new context.”

And the latter,

“ .. depends on extensive, varied practice and occurs by the automatic triggering of well-learned behaviour in a new context.” (p113)

This acknowledgement of different kinds of transfer is most helpful: it accommodates and validates the range of behaviours and outcomes that research has observed. Moreover, it might provide a way forward in that it might provoke future work in this field to clarify its aims and match these to appropriate methodologies and theoretical frameworks.