Representation and working memory in early arithmetic

“In preschoolers, the research of Bisanz and colleagues (Bisanz et al., 2005; Klein & Bisanz, 2000; Rasmussen & Bisanz, 2005) shows that the cognitive resources that children recruit to solve particular problems change over fairly short developmental time windows.”

“…memory in mathematical development has indicated that the spatial span task (also known as Corsi or block span) is a useful index of visualspatial working memory in young children (Bull et al., 2008; Holmes et al., 2008; McKenzie et al., 2003; Rasmussen & Bisanz, 2005; Simmons et al., 2008).”

“…been shown in previous work to be precursors of skills development in their respective domains (VSWM and number skills: LeFevre, Fast, et al., 2010; Rasmussen & Bisanz, 2005; Rasmussen, Ho, & Bisanz, 2003; phonological awareness and literacy skills: Hatcher & Hulme, 1999; Hoien, Lundberg,…”

“Recall that children with better VSWM also exhibit better math performance (Alloway & Passolunghi, 2011; Gathercole & Pickering, 2000; Geary et al., 2007; Kyttälä et al., 2003; Meyer et al., 2010; Raghubar et al., 2010; Rasmussen & Bisanz, 2005).”

“To understand (non)numerical relations between quantities (Levels II and III), preschoolers’ visual–spatial abilities should take on special importance because the nonverbal representation of quantities seems important to solve such tasks (cf. Rasmussen & Bisanz, 2005).”

“Converse evidence comes from a study by Rasmussen and Bisanz (2005) who found that the visuospatial component of WM did not contribute significantly to verbal problem solving among preschoolers and 1st graders.”

“…line estimation (Geary, Hoard, ByrdCraven, Nugent, & Numtee, 2007; Thompson et al., 2013), and nonverbal problem solving (LeFevre et al., 2010; Rasmussen & Bisanz, 2005), as well as better overall math performance (Alloway & Passolunghi, 2011; Dulaney, 2014; Gathercole & Pickering, 2000;…”

“…inhibition and/or shifting abilities (Blair & Razza, 2007; Bull, Espy, & Wiebe, 2008; Van der Sluis et al., 2007), while others did not or only partially support these findings (Censabella & Noël, 2008; Espy et al., 2004; Lee et al., 2011; Rasmussen & Bisanz, 2005; Van der Sluis et al., 2004).”

“…of number and number operations (Clements & Sarama, 2011), hints in the literature suggested that both EF (Espy et al., 2004; Mazzocco & Kover, 2007) and spatial skills (Ansari et al., 2003; Rasmussen & Bisanz, 2005) might play a foundational role in children’s success in the mathematics arena.”

“…to perform better on nonverbal rather than verbal arithmetic tasks and that the visuo-spatial sketchpad capacity is the best predictor of these abilities in this age group (Levine, Jordan & Huttenlocher, 1992; McKenzie, Bull & Gray, 2003; Rasmussen & Bisanz, 2005; Simmons, Chris & Horne, 2008).”

“…processing (Ga-PC in CHC taxonomy) has been reported to predict arithmetic achievement (e.g., Leather & Henry, 1994; Psychology in the Schools DOI: 10.1002/pits Rasmussen & Bisanz, 2005) and is associated with MD and low math achieving children with fact fluency deficits (Chong & Siegel, 2008).”

“Much of this evidence arises from the fact that those who perform better on spatial tasks tend to perform better on tests of mathematical ability (Holmes, Adams, & Hamilton, 2008; Rasmussen & Bisanz, 2005).”

“Finally, several studies have shown that working memory is related to story problems with elementary school children (Fuchs et al., 2005; Rasmussen & Bisanz, 2005), and it is likely to be similar with younger children who have fewer cognitive resources.”

“Rasmussen and Bisanz (2005) reported that VSSP scores predicted unique variance in preschoolers’ performance on nonverbal mathematics problems, suggesting they may use a mental model for arithmetic that requires the VSSP.”

“, [2,11,33,37,62]).”

“However, Rasmussen and Bisanz (2005) also used only one measure of VSSP capacity (the same one as used here).”

“This is consistent with the view that during the basic stage of acquiring arithmetical understanding, children use an internalized representation of a problem, or a mental model, for arithmetic that requires VSWM (Rasmussen & Bisanz, 2005).”

“…including relatively small samples and a limited age range (Andersson & Lyxell, 2007; De Smedt et al., 2009; Henry & MacLean, 2003; Holmes & Adams, 2006; Kyttälä et al., 2010; Meyer et al., 2010; Rasmussen & Bisanz, 2005), or lacking a measure of verbal working memory (Van der Ven et al., 2013).”

“…strong relationships have been found between later maths (dis)abilities and working memory (e.g., Bull, Espy, & Wiebe, 2008; Raghubar, Barnes, & Hecht, 2010; Rasmussen & Bisanz, 2005; Swanson, 2012; Toll, Van der Ven, Kroesbergen, & Van Luit, 2011; Van der Ven, Kroesbergen, Boom, & Leseman, 2012).”

“…young children use the VSSP to link nonverbal quantity representations with number words when enumerating objects (Krajewski & Schneider, 2009) and that they use nonverbal mental models for calculation (Huttenlocher, Jordan, & Levine, 1994; Krajewski & Schneider, 2009; Rasmussen & Bisanz, 2005).”

“…may be less important for calculation than sometimes proposed, though they are likely involved in certain aspects of early mathematical development (Rasmussen & Bisanz, 2005) and in domains of math other than calculation (Barnes et al., 2002; Geary, 1996; Holmes & Adams, 2006; Kyttala & Lehto,…”

“This has been proposed in mathematics research with children (e.g., Rasmussen & Bisanz, 2005).”

“Rasmussen and Bisanz (2005) report that phonological loop performance is correlated with both preschool and school-age children’s performance on all types of arithmetic problems.”

“…in spatial short-term and working memory (Geary et al., 2004; Holmes, Adams & Hamilton, 2008; McKenzie, Bull & Gray, 2003; McLean & Hitch, 1999; Rasmussen & Bisanz, 2005; Simmons, Singleton & Horne, 2008; Swanson & Sachse-Lee, 2001) and broader executive functions (Bull, Espy & Wiebe, 2008;…”

“For example, mental images are more easily represented and preserved in working memory than abstract expressions in mathematics (Rasmussen and Bisanz 2005).”

“All these processes involve working memory (Passolunghi & Pazzaglia, 2004 ; Rassmussen, & Bisanz, 2005 ; Swanson et al, 2008 ).”

“Rasmussen and Bisanz (2005) observed that several working-memory measures together accounted for a substantial proportion of the variability in arithmetic performance (R 2 ¸ 0.40).”

“However, these relationships may be indirect or vary with development/type of problem in the case of arithmetic (Lee, Ng, Ng & Lim, 2004; Rasmussen & Bisanz, 2005; see also Durand, Hulme, Larkin & Snowling, 2005, for contrary evidence).”

“…in counting ability (Kyttälä, Aunio, Lehto, van Luit, & Hautamäki, 2003) and multidigit operations (Heathcote, 1994) in nonverbal problem solving (Rasmussen & Bisanz, 2005) and in mathematics ability in children aged 10 years (Maybery & Do, 2003) and 11 and 14 years (Jarvis & Gathercole, 2003).”

“…he have altogether?’) were examined in children ages 4–6 years old, non-symbolic problems were found to be easier for the younger children (Huttenlocher et al., 1994; Levine et al., 1992), but this difference was diminished for the older children (Levine et al., 1992; Rasmussen & Bisanz, 2005).”

“…single math domain measures that have dominated prior work (i.e., counting, number identification, numbermagnitude, or simple operations; Bull et al., 2008; Clark et al., 2010; Fitzpatrick et al., 2014; Holmes & Adams, 2006; Rasmussen & Bisanz, 2005; Verdine et al., 2014; J. A. Welsh et al., 2010).”

“…data, (3) investigate a predictive model of strategy use based on RTs, (4) determine a RT cutoff threshold that maximizes the differentiability between the two strategies of counting and retrieval, and (5) compare strategy use in typically and low achieving children using all of these methods.”

“For instance, both the phonological loop and the central executive are related to mathematical problem solving ability (Rasmussen & Bisanz, 2005).”

“…28, pp.119-129 Metadata Record: https://dspace.lboro.ac.uk/2134/20085 Version: Accepted version Publisher: c© Elsevier Inc. Rights: This work is made available according to the conditions of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) licence.”

“Visuo-spatial memory skills also uniquely predict performance in nonverbal problems, such as sums presented with blocks, in pre-school children (Rasmussen & Bisanz, 2005).”

“Similarly to us, others have reported that the link between spatial STM and math varies with age/school grades and is less robust than links between math and WM tasks (Holmes & Adams, 2006; Rasmussen & Bisanz, 2005).”

“…in children's ability to count (Kyttälä, Aunio, Lehto, van Luit, & Hautamaki, 2003), performance in multi-digit operations (Heathcote, 1994) and nonverbal problem-solving (Rasmussen & Bisanz, 2005), and mathematical achievement (Bull et al., 2008; Jarvis & Gathercole, 2003; Maybery & Do, 2003).”

“…& Lucangeli, 2012), mathematical skills (Alloway & Passolunghi, 2011), multi-digit operations (Heathcote, 1994), nonverbal problem solving (Rasmussen & Bisanz, 2005), mathematical achievement (Bull, Espy, & Wiebe, 2008; Jarvis & Gathercole, 2003; Maybery & Do, 2003; Passolunghi,…”

“…& Conway, 2005) such as reading comprehension (Borella & de Ribaupierre, 2014), arithmetic problem-solving (Passolunghi & Mammarella, 2012; Rasmussen & Bisanz, 2005), mental calculations (Caviola, Mammarella, Cornoldi, & Lucangeli, 2012; Mammarella et al., 2013), magnitude representation…”

“The visual–spatial component of working memory has been found to uniquely predict math achievement (Bull, Espy, & Wiebe, 2008; Hoard, Geary, Byrd-Craven, & Nugent, 2008; Rasmussen & Bisanz, 2005).”

“Word problem performance improves as children gain a greater ability to (a) understand underlying arithmetic operations (e.g., Rasmussen & Bisanz, 2005; Zamarian, López-Rolón, & Delazer, 2007), (b) distinguish between types of word problems on a basis of mathematical operations (e.g., Fuchs &…”

“…expectation to find a significant association between the visuospatial sketchpad and math performance; both cross-sectional (Holmes & Adams, 2006; Rasmussen & Bisanz, 2005) and longitudinal (De Smedt et al., 2009; van de Weijer-Bergsma et al., 2015) studies have shown that the importance of the…”

“…requires domain-general VS processes (de Hevia and Spelke, 2010; Dehaene et al., 2003); one central example for the support systems that link number and space is visuospatial WM (De Smedt et al., 2009; Geary and Hoard, 2001; Holmes and Adams, 2006; Meyer et al., 2010; Rasmussen and Bisanz, 2005).”

“After also controlling for cluster differences in general mathematics achievement and reading ability, only differences in symbolic numerical magnitude processing remained significant.”

“This finding supports research associating working memory and response inhibition measures with a range of preschool math abilities [19–22].”

“…596 arithmetic, some sort of visuo-spatial mental operation does take place, which is not, however, 597 adjunct to visuospatial WM. 598 But, apart from the preceding arguments, why did Rasmussen and Bisanz (2005) find 599 the VSSP playing a predominant role in nonsymbolic processing and we did not?”

“The role of the different WM components and their interactions when conducting a given cognitive task depend upon the characteristics of the task and the age of participants (Friso-van den Bos et al., 2013; Rasmussen & Bisanz, 2005; Simmons, Willis, & Adams, 2012; Xenidou-Dervou et al., 2015b).”

“Hecht et al. (2001) and Noel, Seron, and Trovarelli (2004) reported a relationship between phonological loop function and arithmetic in school age children, while Rasmussen and Bisanz (2005) found a similar relationship in preschoolers.”

“Rasmussen and Bisanz (2005) suggested that children’s arithmetic strategies are increasingly dependent on their phonological processing abilities.”

“…common when adding illustrations, can put extra cognitive load on students’ working memories, and split-attention effects can occur when students have to jump between text and illustration elements (Berends & van Lieshout, 2009; Rasmussen & Bisanz, 2005; Scheiter, Gerjets, & Catrambone, 2006).”

“Consistent with this, a number of studies have reported that phonological-processing abilities predict arithmetic impairment (Hecht, Torgesen, Wagner, & Rashotte, 2001; Leather & Henry, 1994; Rasmussen & Bisanz, 2005).”

“…words affect children’s addition and subtraction problem-solving procedures because when researchers have presented children with number-word problems, they have tended to remove access to concrete referents (Barth et al., 2006; Jordan et al., 1995; Levine et al. 1992; Rasmussen & Bisanz, 2005).”

“For example, in the domain of math, positive correlations between spatial ability and math performance have been found among preschoolers (Kyttälä et al., 2003; Rasmussen and Bisanz, 2005), middle childhood (Casey et al.”

“According to this model, while verbal skills (phonological awareness in particular) are paramount in stage 1, visual-spatial skills (visual working memory and nonverbal representation of numbers in particular) are important in stages 2 and 3 (e.g., Rasmussen and Bisanz 2005).”

“The counting Stroop task is designed to evaluate the underlying mechanism of cognitive interference between number and meaning (Bush et al. 1998), and the visual properties of the numbers to evaluate visual processing (Rasmussen & Bisanz, 2005).”

“It is suggested that visual-spatial STM is associated with performance on non-verbal arithmetic problems because young children use non-verbal mental models to solve them (Huttenlocher et al., 1994; Rasmussen & Bisanz, 2005).”

“…studies of the relation between visuospatial ability and performance on mathematical achievement and cognition measures has been mixed, once the central executive is controlled (Hegarty & Kozhevnikov, 1999; Holmes & Adams, 2006; Meyer, Salimpoor, Wu, Geary, & Menon, 2010; Rasmussen & Bisanz, 2005).”

“Visuo-spatial WM was hypothesised mainly to influence the knowing domain (see McLean & Hitch, 1999; Rasmussen & Bisanz, 2005).”

“This might be explained if it is considered that, at the beginning of school, learning is largely based on verbal codes to retain information and retrieve knowledge from memory [27].”