Involvement of the Working Memory Model in the Number Strand

The involvement of working memory in general mathematical skills has been extensively discussed throughout this thesis with regard to both the previous literature (Introduction - Chapter 2) and its context with the overall mathematics ability that has been focussed upon in this thesis (Chapter 5). From this discussion it is understood that working memory has an influence in general (Adams & Hitch, 1998; Bull & Espy, 2006; Bull et al., 1999; Geary et al., 2004;

Huttenlocher, Jordan, & Levine, 1994; Imbo & Vandierendonck, 2007b; Pennington, 2006;

Pennington & Willis, 2004, 2006) and some specific mathematics abilities including addition, subtraction, multiplication and problem solving (Adams & Hitch, 1997; Geary et al., 1987;

Mabbott & Bisanz, 2003; Ostad, 1998; Swanson & Beebe-Frankenberger, 2004; Swanson &

Sachse-Lee, 2001), but exactly how the working memory model relates to the UK mathematics curriculum, and in particular the individual mathematics strands is not apparent. A clearer picture is evident when considering some specific maths competencies, particularly those such as

addition and subtraction (Adams & Hitch, 1997; Imbo & Vandierendonck, 2007b), but in terms of the basic cognitive principles underpinning mathematics as identified by the Number Strand and curricular mathematics very little is known.

6.2.1 Associations between working memory and Number

Holmes and Adams (2006) tested UK curricular “Number & Algebra” which is primarily number knowledge and counting, but also includes understanding of the four key number operations (add, subtract, multiply, and divide), recognition of number patterns and sequences, and dealing with fractions and decimals, as well as using the related vocabulary to solve problems. Under this curricular structure, and as a consequence of the age of participants the “Number & Algebra”

strand in the Holmes and Adams study goes much deeper into more complex processing aspects of Number than the way it is defined by the current study; however this research is the closest to a precursor that is known, and deriving some information from this research is useful. From the

92 Holmes and Adams paper, first considering correlates it is noted that “Number & Algebra”

correlates with NV-STM and CE-CWM after age was statistically controlled for. Holmes and

Adams subsequent regression analyses identified that their VSSP (NV-STM) model accounted for 3%

of the variance in performance on this Strand, and their CE model accounted for 12% of the variance in overall scores on this Strand. The Holmes and Adams data substantiates previous general findings that the central executive is an important predictor of children’s mathematics in children aged 7-10 years old (Bull et al., 1999; Bull & Scerif, 2001). Across both age groups, CE predicted a significant amount of unique variance on “Number and Algebra”. However Holmes and Adams performed principal components analysis that suggested that tasks measuring central executive loaded on both the WM and mathematics factors, potentially indicating that the CE measure in the study is interrelated to a more general resource such as intelligence (Fry & Hale, 2000).

Approximation and number transcoding are both important skills for children to master and these competencies would fall under the remit of the Number Strand. Working memory has been demonstrated to have an impact upon approximation in 7 year olds and pre-schoolers

respectively (Caviola, Mammarella, Cornoldi, & Lucangeli, 2012; Xenidou-Dervou, van Lieshout, &

van der Schoot, 2013) and Camos (2008) showed a consistent relationship among error rates, working memory capacity, and the quantity of rules in a study with second grade children, where children with a low working memory span performed poorly on all these aspects of transcoding. A subsequent study by Moura et al (2013) gave evidence that the influence of working memory on number transcoding is somewhat selective, with influence of working memory being stronger for effects that reflect the complexity of Arabic numerals and that involve “online” manipulations of numerical units. Whilst these studies are not directly referencing the Number Strand, or indeed the UK curriculum, this is indicative that working memory is influential in key aspects of

development of number.

Geary (1993) discussed the importance of nonverbal short-term memory in early number competencies and asserted a specific subtype of MD characterised by visuospatial deficits. The

93 sorts of mathematics deficits arising from VSSP problems were issues such as column

misalignment and failure to handle simple place value. Both of these problems are elements clearly specified in the Number Strand (DfEE, 1999, n.d.-b). Furthermore, Mclean and Hitch (1999) also verified that children with mathematics deficits (aged 9 years old) do indeed perform

significantly more poorly on measures of visuospatial sketchpad synchronous with the ideas put forward by Geary .

With reference to typically developing children, once youngsters attend formal preschool and primary education they are often helped to represent number using external tokens, such as building blocks, or toys (DfES, 2001). This is in part to make learning a more fun and engaging pursuit, however there is also evidence that the use of concrete material in learning aids effective learning (Ball, 1992). Crucially the DfES advise that concrete visual and tactile support in primary mathematics is an effective way to aid learning in children with dyslexia or dyscalculia who require extra learning support (DfES, 2001). The use of concrete materials as aids to memory might assist children by means of representing number physically and thus reducing the working memory resources necessary to represent that data in short-term memory, however it has been shown in some research that children with poor working memory tend not to use aids (Gathercole

& Alloway, 2004). Additionally, in the absence of concrete visual and/or tactile support, it can be noted that the human body is a very efficient and natural means for supporting the physical representations and development of early number skills, and children will unsurprisingly utilise parts of the body to facilitate mathematics understanding and latterly, mathematical processing (Hunting, 2003).

In considering a role for complex working memory in the Number Strand, CE-CWM would typically be identified as being involved in higher order mathematical operations where simultaneous storage and processing is necessary, such as addition or multiplication, and far less so in the early lower order mathematical learning. However there is a suggestion that CE-CWM has a role to play in the commitment to, and retrieval of, early number facts to LTM (Kaufmann, 2002; Zuber, Pixner, Moeller, & Nuerk, 2009). These cognitive theorists have driven two key hypotheses regarding the

94 starting place for a CE-CWM deficit in mathematics, suggesting two potential forms of retrieval deficit. One idea involves an uncomplicated deficit in the ability to retrieve correct number facts from a semantics based long-term memory archive. The second idea posits that the deficits result from disruption to the retrieval process brought about by problems arising from the functioning of inhibitory mechanisms. By way of an example, think about solving a simple addition problem such as 2 + 4. Children with a CE-CWM deficit might be inclined to retrieve 3 or 5 in place of 2 or 4, and sum the incorrect number fact; or they could retrieve both 3 and 5. The rationale being, that these numbers are the next numbers in the counting sequence and thus closely associated with the counting string, and as such inhibiting the retrieval of them is likely to be more difficult.