DIA VIII

There are points in my discussion in Chapter 6 at which I carried out further analyses of the accounts as data in their own right (secondary level, derived data):

• Stage 6a of my data analysis entailed analysis of types, or classes, of transfer according to a taxonomy put forward by diSessa & Wagner (2005) which I presented in Chapter 2: Literature Review. In order to analyse each transfer event (recorded as a row in the accounts for both children created at Stage 4 of my data analysis), it was necessary to clarify, within my own understanding, features of knowledge that are associated with different classes of transfer.

o Firstly, I concluded that re-use of any knowledge – old or new, effective or not – constitutes transfer of knowledge. This would mean that, where re-use of knowledge was not evident in any row in the accounts, it was judged that transfer had not occurred; and that transfer had occurred in all other rows. Such rows therefore represented transfer events.

o Next, I considered that a key distinction between transfer classes is the notion of preparedness of knowledge. My interpretation of preparedness of knowledge, as suggested by diSessa & Wagner, and its effect on transfer is that:-

o Confidence or lack of it is not an indicator of any particular class of transfer: confidence may or may not be evident in any class of transfer;

o Long-lived “-ness” does seem to influence the ways in which knowledge is likely to be transferred, in that knowledge that was only very recently

constructed is unlikely to be sufficiently aligned with other knowledge to be considered “well-prepared”. Class A transfer is effective and reliable re-use of knowledge – such reliability is achieved through multiple experiences with the knowledge. Therefore, Class A transfer is unlikely where very new

knowledge is re-used; Class C transfer might be of new or old knowledge.

o diSessa & Wagner describe Class A transfer as effective and unproblematic use of well-prepared knowledge. Therefore effectiveness is a necessary, though not sufficient, condition of Class A transfer. On the other hand, Class C transfer is not

determined by effectiveness, since it is not

necessarily productive nor unproductive. Invocation of some prior knowledge, old or new, is in itself Class C transfer; it could be argued that its effectiveness is in its potential to be productive. Identification of Class C transfer by an observer presumes the learner’s perception of relevance of a piece of knowledge, at least in its potential to be productive, even where it is found not to be so.

So, factors that determine preparedness and therefore transfer class are effectiveness and reliability of knowledge. Having excluded learner confidence and long-lived-ness as determinants of transfer class, it was possible to devise a key that would help in assessment of class of transfer where transfer was determined in accounts generated. This is shown in Figure 4.10. Findings from this analysis of transfer types are discussed in Chapter 6.

Evidence of re-use of knowledge? No Transfer is not evident Is it used effectively? Yes Is its effectiveness reliable, well-aligned/ commensurate with other

knowledge?

Yes Yes Class A

Is transfer likely to be reliably accomplished soon? No Class B Yes Class C No No Transfer is evident Start

Figure 4.10 Key to determine Transfer Class

• Stage 6b of my data analysis aimed to discover whether C’s and G’s ability to transfer was sensitive to different elements of the problem or task, (Wagner, 2006). Each row of the analysis grids was examined and where C or G demonstrated success or failure, I considered which facets of a problem or task were apparent at that point in our work. Wagner defined 3 facets of problems:

o problem type: the problem can be “distinguished by legitimate mathematics descriptors”;

o problem aspect; “any detail of a problem or problem situation that can be a focus of attention”;

o problem context: “the cover story in which the problem is embedded”. (p13)

Results of my analysis of problem type/aspect/context in relation to the boys’ success and failure in the tasks are shown in Chapter 6.

o Stage 6c of my data analysis was carried out to consider my two cases in relation to Bruno & Martinon’s (1996) work relating to transferences between dimensions when working with negative numbers. Each row, in each boy’s analysis grid, was analysed to consider whether there was evidence of transference between “Quantity: abstract”, “Quantity: contextual” and “Number line” dimensions. The number of occurrences in either direction was recorded to provide a representation of the extent of each boy’s ability to make connections between dimensions, evidence of richly associated conceptual resources and flexible re-use of those resources. Findings are shown in Chapter 6.

Retrospective note

Initially, accounts-for included explicit tracking of the growth of specific concepts – i.e. the analysis grids originally had 3 columns. However, the third column was abandoned when it became clear that it was not

possible to be at all confident about its content. This type of adjustment is what Robson (1993) calls “playing with the data”,

“ … case study design is flexible, with the final version evolving through interaction with the case … “playing with the data” at this intermediate stage may well assist in identifying themes which can form the basis for a workable descriptive framework. Even with a theoretical frame, initial exploration of this kind may give an early warning of its inadequacy, and perhaps lead to a beneficial recasting.” (p378)

“It should not be thought that this is an automatic process for getting at “the truth” about the case. It is an attempt to provide an integrated summary of what I know about it, but is necessarily more suggestive than definitive.” (p399)

I feel this is true of my analysis grids.