Relationships between knowledge, learning and transfer

Mutual bootstrapping of conceptual knowledge

Wagner (2006) resonates with my own intuitive beliefs about learning, knowledge and transfer. In the development of my own conceptual

knowledge, I am aware of asense of a hand-over-hand process whereby the growth of span of relevance and of the applicability of situations to knowledge resources (and vice versa) which constitutes conceptual development, is characterised by a small change in one dimension facilitating a small change in another dimension, after which the first dimension is enabled to grow or change a little more. Wagner describes

how his main subject revealed her incremental development of a particular concept; bootstrapping different aspects of the concept:

“Within the active knowledge framework, Maria expected

representativeness and accuracy to be relevant explanatory ideas, but the role they served in the particular problem, and the role they served in co-ordination with other ideas in the framework, had yet to be clarified. These incremental moves are a hallmark of a knowledge-in-pieces perspective on the growth of conceptual understanding, as well as the transfer-in-pieces perspective on how knowledge of a principle develops to span more and more situations.” (p62)

I find the work of diSessa (1988), diSessa & Wagner (2005) and Wagner (2006) very appealing. It provides a framework for identifying the different aspects of the learning process and offers a well drawn rationale for characterising the relationships between knowledge and learning and transfer, taking account of what I call “micro-transfer”. Though other theories have offered models (often hierarchical) for moving

understanding and knowledge from that of the particular to that of something more generic that might be inserted into a new, structurally similar situation, they have not succeeded, for me, in filling in the gaps – the major shortcoming of abstractionist, or encapsulation theories is in their failure to meaningfully (for me) address the question of perception of similarity and, therefore, relevance. It has been clear to me that

application of knowledge in a new situation requires some understanding of the relationship between the existing knowledge and the demands of the new situation as well as some notion of correspondence or mapping between contextual attributes old and new. Wagner (2006) highlights,

“the in-separability of the perception of structure in a problem from the knowledge of the principles needed to solve it.” (p61)

“seeing complex structures within a given situation depends on our having complex expectations – not necessarily ready-made

structures previously stored that are retrieved in their entirety, but complex associations of descriptive and explanatory knowledge resources that have proven to be mutually supportive in other circumstances. These associations are learned.” (Wagner 2006; p63)

Mason (2002) also acknowledges co-evolution of an individual’s capacity to recognise connections between resources with the development of some experience with those resources,

”Perception and preparedness to be able to perceive emerge together as the result of perceiving and preparing to perceive.” (p229)

To recap, transfer may be understood as follows:-

• Transfer requires an incremental growth of span of relevance;

• Transfer enables an incremental growth of span of relevance;

• These processes together constitute a “hand-over-hand” growth of interpretive knowledge (includes descriptive, explanatory, readout, co-ordination);

• Transfer is dependent on perception of similarity, which is dependent on development of span of relevance;

• Transfer does not require general (abstract) expression; to illustrate this point, Wagner (2006) states that “We expect no single understanding of the law of large numbers to be applicable anywhere and everywhere, rather different combinations of distributed elements may support its recognition and use in different circumstances” (i.e. is sensitive to context and takes account of it p56). This is a key idea, vital for shaping

How does abstraction relate to transfer?

In considering what transfer is, what it looks like and what it entails, it is clear that abstraction, as opposed to abstracting, is not necessary for transfer (of Class B or Class C) to occur. Interestingly, Maria did develop and express,

“an increasingly invariant understanding of a statistical principle as she incrementally constructed an interpretive knowledge frame that widened the span of phenomena to which she understood the law of large numbers to apply. …. Maria’s attempts at stating a generalised principle took place only in the aftermath of the development of interpretive knowledge frames at a much finer level of detail.”

Therefore,

“ .. abstraction was a consequence of transfer and the growth of understanding, not the cause of it.” (Wagner 2003 p72)

It was argued previously that the perception of relevance emerges in the relationship between the situation and the individual’s interpretive

knowledge that frames the situation. Since the “individual’s interpretive knowledge that frames the situation” might be more simply referred to as the learner’s understanding of the situation, it is clear that understanding is related to both transfer and abstraction. A learner with deep

understanding is likely to perceive similarities with the greatest span of relevant knowledge resources. Therefore, we see that understanding supports transfer and transfer contributes to understanding. Eventually, when sufficient examples have been experienced and understood at some level (facilitated through recognition of similarities), generalisation across situations will become possible – i.e. when multiple instances of transfer have occurred, abstraction is enabled; this in turn will deepen understanding. This model locates abstraction (rather than abstracting) as an outcome of (Class C) transfer, rather than a prerequisite for it.

Why is evidence of transfer so elusive to researchers?

Co-ordination classes are, according to diSessa & Wagner (2005), those concepts which are complex and comprise multiple layers and

dimensions of meaning and relationships. It is this complexity that means that co-ordination classes are difficult concepts to learn.

Not all concepts are co-ordination classes but many of the more

problematic concepts (in maths and science learning) are (diSessa 1993). An alternative contribution to the field comes from Mason and Spence (1999) and it is interesting to contrast their ideas with those of diSessa and Wagner and others. Mason & Spence describe a framework that, like the contribution of diSessa and Wagner focuses on the earlier “precursor” stages of knowledge-building in order to illuminate the cognitive

mechanisms for “transfer” of that knowledge into new situations. They explain their view that

“Knowing-about, that is, knowing-that, -how, and –why forms the heart of institutionalised education: students can learn and be tested on it. But success in examinations gives little indication of whether that knowledge can be used or called upon when

required, which is the essence of “knowing-to”. Although knowing- to does of course depend on training in behaviour, it is based, as we shall see, in awareness. It has to do with the structure of attention.” (p138)

I feel that this latter point is an interesting one; that knowing-to is more than a behaviour that can be trained – it is about awareness and what is attended to (or is not). I agree with Mason & Spence (1999) that knowing- to is clearly not the same as reacting – i.e. it cannot be achieved simply by training.

Once knowing-to has occurred, the other aspects of knowing-about are enabled: without the trigger of knowing-to, all that is known-about is not accessible . After all,

“No-one can act if they are unaware of a possibility to act; no-one can act unless they have an act to perform.” (Mason & Spence 1999, p135)

It is interesting to note, at this point, that Mason & Spence seem to support the view that transfer (“knowing-to”) precedes abstraction (“knowing-about”). It is also clear that, in order to “know-to”, it is necessary to know something; this, I would argue, entails “knowing-

about”. Papert (1996) introduced what he called the “Power Principle”. He described how children working with LOGO were able to learn about angle by working with angle to construct shapes: they were learning by using; Papert posed the question, “What comes first, using it or getting it?” (p4).

Anecdotes of learners failing to re-use knowledge that they are thought to have learned are rife in schools the world over. This might be described using Mason & Spence’s (1999) parlance: that learners who have demonstrated that they know-about in some way (know-that, know-how and/or know-why) do not apply any of that knowledge in a situation in which it would facilitate them to access the problem, possibly solving it. Presumably, we might infer, this might be because they do not realise the relevance of the knowledge-about to the new situation and so do not use it – i.e. they do not know-to. In terms of diSessa and Wagner’s “CCT” lexicon, their span is not sufficiently developed.

Broudy 1977 (cited in Schwarz et al 2005) believes that there are 3 kinds of knowledge: “replicative”, “applicative” and “interpretive”. This provides a valuable framework with which to analyse the elements and demands of tasks and through which we might understand the reasons for an apparent lack of transfer. Transfer research is often based on

measurements of retention of skills and knowledge after a learning event. Replicative and applicative knowledgeare relatively straightforward to assess and these form the basis of a great deal of what are used as assessments of knowledge. However, replicating and applying old

by performance rather than through understanding. Tests and

assessments and research tasks that facilitate such “performance” after a learning episode may actually find transfer, if transfer is understood as “the degree to which a behaviour will be repeated in a new situation” (Detterman & Sternberg 1993, cited Schwarz et al 2005). This is the classic definition of transfer. If, however, we hold that transfer actually involves modification and adaptation of old knowledge to new situations, then the third kind of knowledge identified by Broudy - i.e. interpretive - is required and would need to be evident in research tasks. Interpretive knowledge is that which enables learners to interpret the content and context of new situations in order to select appropriate knowledge skills and understanding for addressing a new problem. Schwarz et al point out that research into transfer has not, traditionally, designed methodologies that enable interpretive knowledge to be used and/or observed.

Mason & Spence (1999) believe that,

“The state of sensitivity-awareness of the individual, combined with elements of the situation which metonymically trigger or

metaphorically resonate with experience, are what produce the sudden knowing-to act in the moment.” (p146)

This reference to “elements of the situation” suggests that this approach to understanding the mental processes involved in transfer of knowledge, allows for attention to contextual information, rather than focusing on structural similarity. I would suggest that, if we accept this account of the processes involved in transfer, it would be helpful to gain some

understanding of ways in which educators might effectively sensitise the “triggers” to which Mason & Spence refer. Or, to use Co-ordination Class Theory terminology, strategies for extending span, including testing alignment need to be explored and developed.

Mason & Spence conclude that,

“Knowing is not a simple matter of accumulation. It is rather a state of awareness, of preparedness to see in the moment.” (p151)

They go on to consider whether knowing-to can be prepared for (a more appropriate term than taught or trained). They believe that, even if

metaphors are deliberately provided, children’s uptake and use of them is highly variable; that attempts to “implant” the metaphors and images are unsuccessful and that children need to actively and personally take on board metaphors that might be suggested from within or outside of themselves.

Some of the ideas of Schwarz, Bransford & Sears (2005) resonate with the work of Mason & Spence. Schwarz et al describe how much transfer research has been based on a “sequestered problem solving” (SPS) approach to assessing transfer of knowledge and learning and they posit that this only facilitates the measurement of certain types of knowledge since it looks for “direct application” of old knowledge in a new situation. They propose that this type of research neglects and is blind to a range of modifications and adaptations to old knowledge that might facilitate

problem-solving in the future rather than in the test situation. This “preparation for future learning” (PFL) is a more helpful view of what transfer actually entails since it extends the range of situations where it might be evident.

Wagner (2003;2006) can be seen to have implanted metaphors (or prepared Maria for learning). He designed a sequence of learning

activities for Maria that exposed her to certain ideas. It would seem then that attempts to “design-in” exposure to relevant and potentially helpful metaphors might help knowledge to develop, although Mason & Spence (1999) thought it not worthwhile. This tension in the findings from different studies is not disconcerting: I believe that work in this field, with a sharp focus on the minutiae of conceptual change and growth, is only beginning to develop and that findings from one worker do not necessarily predict outcomes in other settings, even where they appear to be similar. I think transfer might be different for different knowledge domains. diSessa &Wagner (2005) assume this might be the case:

“We, ….., do not presume transfer is homogenous with respect to kind of knowledge or with respect to other such dimensions. Nonetheless, it is useful to examine transfer in a particular, reasonably well-elaborated case.” (p139)

This is what Wagner chose to do when he focused on one undergraduate student for his (2003;2006) study.

Research at this micro-level, focusing on an individual in one setting in order to discover at least how that individual thinks and learns will, I believe, help us to develop experience and understanding of those particular subjects in those particular settings. If a model of learning for transfer is built on the idea that understanding of multiple examples of particular situations is what leads ultimately to generalisation and deeper understanding then it is appropriate that micro-level research is the way forward. That is to say, an accumulation of knowledge about how

individuals are able to effect conceptual growth might be the most appropriate way forward if we are to begin to understand learning and transfer in a way that might subsequently contribute to the development of more macro-level theories.

Research into the development of knowledge and understanding presents significant challenges in the 21st century. The theories which dominated the field for most of the latter part of the 20th century have been rigorously challenged. We have now been shown that an abstract understanding in/of mathematics, stripped of any context-based

references, is not necessarily an appropriate goal for teachers and learners (or researchers) in primary schools. It is no longer abstraction which is the key objective for mathematics education; there is now a bigger challenge. Research in this area must develop strategies for changing its focus, perhaps for zooming-in on individual instantiations of learning and transfer. I am happy to adopt some of the terminology introduced by diSessa and Wagner (2005) as I summarise some of their ideas which, I believe, are thorough and meaningful; I believe, also, that

they reflect, develop and represent realistically and accurately what learners actually have and do:

• in cases where a co-ordination class is well prepared, almost by definition, subjects will be able to “transfer” that knowledge to any related problem within a sensible range;

• mismatch of contextual characteristics will prevent prior knowledge being used but so will underdeveloped readout strategies and naïve or flawed co-ordination knowledge of a concept;

• transfer research has found failure because it is looking for Class A transfer where it is unlikely to be found – i.e. knowledge is unlikely to be sufficiently prepared;

• Class C is a frequent, “blind” (understandably) process involved in the extended preparation that is required for Class A transfer to be enabled;

• investigating Class C “depends strongly on our ability to see particular knowledge in action even if it does not show up as context-transcending success.” (diSessa & Wagner 2005; p 148).

Some of these points will be discussed in later chapters in relation to findings from my study.