Sheri-Lynn Skwarchuk

Pathways to Mathematics: Longitudinal Predictors of Performance.

A model of the relations among cognitive precursors, early numeracy skill, and mathematical outcomes was tested for 182 children from 4.5 to 7.5 years of age. The model integrates research from neuroimaging, clinical populations, and normal development in children and adults. It includes 3 precursor pathways: quantitative, linguistic, and spatial attention. These pathways (a) contributed independently to early numeracy skills during preschool and kindergarten and (b) related differentially to performance on a variety of mathematical outcomes 2 years later. The success of the model in accounting for performance highlights the need to understand the fundamental underlying skills that contribute to diverse forms of mathematical competence.

Do home numeracy and literacy practices of Greek and Canadian parents predict the numeracy skills of kindergarten children

Children’s experiences with early numeracy and literacy activities are a likely source of individual differences in their preparation for academic learning in school. What factors predict differences in children’s experiences? We hypothesised that relations between parents’ practices and children’s numeracy skills would mediate the relations between numeracy skills and parents’ education, attitudes and expectations. Parents of Greek (N = 100) and Canadian (N = 104) five‐year‐old children completed a survey about parents’ home practices, academic expectations and attitudes; their children were tested on two numeracy measures (i.e., KeyMath‐Revised Numeration and next number generation). Greek parents reported numeracy and literacy activities less frequently than Canadian parents; however, the frequency of home numeracy activities that involved direct experiences with numbers or mathematical content (e.g., learning simple sums, mental math) was related to children’s numeracy skills in both countries. For Greek children, home literacy experiences (i.e., storybook exposure) also predicted numeracy outcomes. The mediation model was supported for Greek children, but for Canadian children, the parent factors had both direct and mediated relations with home practices.

The role of executive attention in the acquisition of mathematical skills for children in Grades 2 through 4.

We examined the role of executive attention, which encompasses the common aspects of executive function and executive working memory, in childrens acquisition of two aspects of mathematical skill: (a) knowledge of the number system (e.g., place value) and of arithmetic procedures (e.g., multi-digit addition) and (b) arithmetic fluency (i.e., speed of solutions to simple equations such as 3+4 and 8-5). Children in Grades 2 and 3 (N=157) completed executive attention and mathematical tasks. They repeated the mathematical tasks 1 year later. We used structural equation modeling to examine the relations between executive attention and (a) concurrent measures of mathematical knowledge and arithmetic fluency and (b) growth in performance on these measures 1 year later. Executive attention was concurrently predictive of both knowledge and fluency but predicted growth in performance only for fluency. A composite language measure predicted growth in knowledge from Grade 2 to Grade 3. The results support an important role for executive attention in childrens acquisition of novel procedures and the development of automatic access to arithmetic facts.

Charting the role of the number line in mathematical development.

Individuals who do well in mathematics and science also often have good spatial skills. However, the predictive direction of links between spatial abilities and mathematical learning has not been firmly established, especially for young children. In the present research, we addressed this issue using a sample from a longitudinal data set that spanned 4 years and which includes measures of mathematical performance and various cognitive skills, including spatial ability. Children were tested once in each of 4 years (Time 1, 2, 3, and 4). At Time 3 and 4, 101 children (in Grades 2, 3, or 4 at Time 3) completed mathematical measures including (a) a number line task (0–1000), (b) arithmetic, and (c) number system knowledge. Measures of spatial ability were collected at Time 1, 2, or 3. As expected, spatial ability was correlated with all of the mathematical measures at Time 3 and 4, and predicted growth in number line performance from Time 3 to Time 4. However, spatial ability did not predict growth in either arithmetic or in number system knowledge. Path analyses were used to test whether number line performance at Time 3 was predictive of arithmetic and number system knowledge at Time 4 or whether the reverse patterns were dominant. Contrary to the prediction that the number line is an important causal construct that facilitates learning arithmetic, no evidence was found that number line performance predicted growth in calculation more than calculation predicted number line growth. However, number system knowledge at Time 3 was predictive of number line performance at Time 4, independently of spatial ability. These results provide useful information about which aspects of growth in mathematical performance are (and are not) related to spatial ability and clarify the relations between number line performance and measures of arithmetic and number system knowledge.

Knowledge of counting principles: How relevant is order irrelevance?

Most children who are older than 6 years of age apply essential counting principles when they enumerate a set of objects. Essential principles include (a) one-to-one correspondence between items and count words, (b) stable order of the count words, and (c) cardinality-that the last number refers to numerosity. We found that the acquisition of a fourth principle, that the order in which items are counted is irrelevant, follows a different trajectory. The majority of 5- to 11-year-olds indicated that the order in which objects were counted was relevant, favoring a left-to-right, top-to-bottom order of counting. Only some 10- and 11-year-olds applied the principle of order irrelevance, and this knowledge was unrelated to their numeration skill. We conclude that the order irrelevance principle might not play an important role in the development of childrens conceptual knowledge of counting.

Refining the quantitative pathway of the Pathways to Mathematics model.

In the current study, we adopted the Pathways to Mathematics model of LeFevre et al. (2010). In this model, there are three cognitive domains--labeled as the quantitative, linguistic, and working memory pathways--that make unique contributions to childrens mathematical development. We attempted to refine the quantitative pathway by combining childrens (N=141 in Grades 2 and 3) subitizing, counting, and symbolic magnitude comparison skills using principal components analysis. The quantitative pathway was examined in relation to dependent numerical measures (backward counting, arithmetic fluency, calculation, and number system knowledge) and a dependent reading measure, while simultaneously accounting for linguistic and working memory skills. Analyses controlled for processing speed, parental education, and gender. We hypothesized that the quantitative, linguistic, and working memory pathways would account for unique variance in the numerical outcomes; this was the case for backward counting and arithmetic fluency. However, only the quantitative and linguistic pathways (not working memory) accounted for unique variance in calculation and number system knowledge. Not surprisingly, only the linguistic pathway accounted for unique variance in the reading measure. These findings suggest that the relative contributions of quantitative, linguistic, and working memory skills vary depending on the specific cognitive task.

Development of Mathematical Knowledge in Young Children: Attentional Skill and the Use of Inversion.

The principle of inversion, that a + b − b must equal a, is a fundamental property of arithmetic, but many children fail to apply it in symbolic contexts through 10 years of age. We explore three hypotheses relating to the use of inversion that stem from a model proposed by Siegler and Araya (2005). Hypothesis 1 is that greater calculational skill is related to greater use of inversion. Hypothesis 2 is that greater attentional skill is related to greater use of inversion. Hypothesis 3 is that the relation between attentional skill and the use of inversion is particularly strong among children with high skill in calculation. We found suggestive evidence for Hypothesis 2 and clear evidence for Hypothesis 3, indicating that for children who are strong at calculation, attentional flexibility is related to use of inversion.

The Role of the Home Environment in Children’s Early Numeracy Development: A Canadian Perspective

Experiences that children have at home can establish a foundation for numeracy learning, and serve as an important transition toward school entry. However, Canadian parents and other caregivers do not often have a good understanding of numeracy learning, they may not be prepared to provide appropriate activities, and some may avoid numeracy activities because of their own negative views of mathematics. Accordingly, when parents and caregivers do focus on academic preparation, they typically emphasise literacy over numeracy activities. In this paper, we describe how children’s home experiences support numeracy learning, in preparation for school. Our research has shown that young children who are involved frequently in numeracy activities in both formal and informal contexts are better prepared for numeracy learning in school than their peers who have fewer numeracy experiences. These results support the view that parents and other caregivers should be encouraged to take an active interest in children’s early learning, and to help children to make appropriate connections between intuitive understandings of numeracy concepts and the formal knowledge that is emphasised in school. Our work has also shown that parents can use their children’s personal interests to foster numeracy knowledge. In this chapter, we summarise our findings, present two extreme cases, and provide recommendations to show how caregivers can be involved in providing stimulating numeracy opportunities for children.

Optimizing the Home Numeracy Environments of 3- to 6-Year-Old Children in the USA and Canada

Many children in the USA and Canada have access to a wide range of experiences that can support their early numeracy development. Nevertheless, children’s early numeracy knowledge varies considerably in these countries as a function of parents’ education, socioeconomic conditions, parents’ attitudes, beliefs, knowledge about mathematics, and preschool and school educators’ knowledge and experience. In this chapter we provide a selective overview of some of the factors that are related to children’s early numeracy experiences and suggest contexts and circumstances that may facilitate or hinder children’s learning. Parents’ provision of formal and informal opportunities that allow their children to experience number, space, and related concepts and their willingness to engage in meaningful activities may all influence the quality of children’s early experiences. We conclude with suggestions for further research to explore ways in which adults can enhance the home numeracy environments of preschool children.