David J. Purpura

Early literacy and early numeracy: The value of including early literacy skills in the prediction of numeracy development

The purpose of this study was to examine whether early literacy skills uniquely predict early numeracy skills development. During the first year of the study, 69 3- to 5-year-old preschoolers were assessed on the Preschool Early Numeracy Skills (PENS) test and the Test of Preschool Early Literacy Skills (TOPEL). Participants were assessed again a year later on the PENS test and on the Applied Problems and Calculation subtests of the Woodcock-Johnson III Tests of Achievement. Three mixed effect regressions were conducted using Time 2 PENS, Applied Problems, and Calculation as the dependent variables. Print Knowledge and Vocabulary accounted for unique variance in the prediction of Time 2 numeracy scores. Phonological Awareness did not uniquely predict any of the mathematics domains. The findings of this study identify an important link between early literacy and early numeracy development.

Working memory and language: skill-specific or domain-general relations to mathematics?

Childrens early mathematics skills develop in a cumulative fashion; foundational skills form a basis for the acquisition of later skills. However, non-mathematical factors such as working memory and language skills have also been linked to mathematical development at a broad level. Unfortunately, little research has been conducted to evaluate the specific relations of these two non-mathematical factors to individual aspects of early mathematics. Thus, the focus of this study was to determine whether working memory and language were related to only individual aspects of early mathematics or related to many components of early mathematics skills. A total of 199 4- to 6-year-old preschool and kindergarten children were assessed on a battery of early mathematics tasks as well as measures of working memory and language. Results indicated that working memory has a specific relation to only a few-but critically important-early mathematics skills and language has a broad relation to nearly all early mathematics skills.

Informal Numeracy Skills The Structure and Relations Among Numbering, Relations, and Arithmetic Operations in Preschool

Validating the structure of informal numeracy skills is critical to understanding the developmental trajectories of mathematics skills at early ages; however, little research has been devoted to construct evaluation of the Numbering, Relations, and Arithmetic Operations domains. This study was designed to address this knowledge gap by examining the structure of these three numeracy skill domains and examining the relations among these domains. Three hundred ninety-three children participated in the study (51.7% girls, 55.7% White, 33.8% African American, and 10.5% other). Results indicated that the relations among the informal numeracy skills were best explained by a three-factor model that included Numbering, Relations, and Arithmetic Operations factors, and this factor structure was the same in both younger and older preschool children.

Foundations of mathematics and literacy: The role of executive functioning components.

The current study investigated the relations between the three cognitive processes that comprise executive functioning (EF)-response inhibition, working memory, and cognitive flexibility-and individual components of mathematics and literacy skills in preschool children. Participants were 125 preschool children ranging in age from 3.12 to 5.26years (M=4.17years, SD=0.58). Approximately 53.2% were female, and the sample was predominantly Caucasian (69.8%). Results suggest that the components of EF may be differentially related to the specific components of early mathematics and literacy. For mathematics, response inhibition was broadly related to most components. Working memory was related to more advanced mathematics skills that involve comparison or combination of numbers and quantities. Cognitive flexibility was related to more conceptual or abstract mathematics skills. For early literacy, response inhibition and cognitive flexibility were related to print knowledge, and working memory was related only to phonological awareness. None of the EF components was related to vocabulary. These findings provide initial evidence for better understanding the ways in which EF components and academic skills are related and measured. Furthermore, the findings provide a foundation for further study of the components of each domain using a broader and more diverse array of measures.

The nonlinear relations of the approximate number system and mathematical language to early mathematics development.

Both mathematical language and the approximate number system (ANS) have been identified as strong predictors of early mathematics performance. Yet, these relations may be different depending on a childs developmental level. The purpose of this study was to evaluate the relations between these domains across different levels of ability. Participants included 114 children who were assessed in the fall and spring of preschool on a battery of academic and cognitive tasks. Children were 3.12 to 5.26 years old (M = 4.18, SD = .58) and 53.6% were girls. Both mixed-effect and quantile regressions were conducted. The mixed-effect regressions indicated that mathematical language, but not the ANS, nor other cognitive domains, predicted mathematics performance. However, the quantile regression analyses revealed a more nuanced relation among domains. Specifically, it was found that mathematical language and the ANS predicted mathematical performance at different points on the ability continuum. These dual nonlinear relations indicate that different mechanisms may enhance mathematical acquisition dependent on childrens developmental abilities.

Can Computer-Assisted Discovery Learning Foster First Graders’ Fluency With the Most Basic Addition Combinations?

In a 9-month training experiment, 64 first graders with a risk factor were randomly assigned to computer-assisted structured discovery of the add-1 rule (e.g., the sum of 7 + 1 is the number after “seven” when we count), unstructured discovery learning of this regularity, or an active-control group. Planned contrasts revealed that the add-1 conditions were more effective than regular instruction/practice in promoting the learning of the add-1 rule. Contrary to the conclusions of Alfieri, Brooks, Aldrich, and Tenenbaum (2011) and Kirschner, Sweller, and Clark (2006), participants in the structured add-1 condition did not outperform those in the unstructured add-1 group on practiced and unpracticed n + 1 and 1 + n items at the posttests. The control participants did not exhibit evidence of learning a general near-doubles reasoning strategy (if 4 + 4 is 8 and 4 + 5 = 4 + 4 + 1, then the sum of 4 + 5 must be 9). The add-1, but not the active-control, participants achieved success, including transfer, because the former had mastered the developmental prerequisites for add-1 rule and the latter had not mastered the prerequisites for the near-doubles strategy.

Fostering At-Risk Kindergarten Children's Number Sense

A 9-month training experiment evaluated whether computer-assisted discovery learning of arithmetic regularities can facilitate kindergartners’ fluency with the easiest sums. After a pretest, kindergartners with at least one risk factor (n = 28) were randomly assigned to either a structured add-0/1 training condition, which focused on recognizing the n + 0/0 + n = n and the n + 1/1 + n = the number-after-n rules, or an active control group. Using pretest fluency as the covariate, ANCOVAs revealed that the structured add-0/1 group significantly outperformed the control group on both practiced and unpracticed (transfer) n + 0/0 + n and n + 1/1 + n items at the delayed posttest and had significantly larger gains in mathematics achievement. Key instructional implications include: Early intervention that targets discovering rules for adding with 0 and 1 and family-specific developmental prerequisites is feasible and more effective than typical classroom instruction in promoting fluency with such basic sums. Such rules may be a critically important bridge between informal and formal mathematics.

Examining the relations between executive function, math, and literacy during the transition to kindergarten: A multi-analytic approach.

The present study explored the bidirectional and longitudinal associations between executive function (EF) and early academic skills (math and literacy) across 4 waves of measurement during the transition from preschool to kindergarten using 2 complementary analytical approaches: cross-lagged panel modeling and latent growth curve modeling (LCGM). Participants included 424 children (49% female). On average, children were approximately 4.5 years old at the beginning of the study (M = 4.69, SD = .30) and 55% were enrolled in Head Start. Cross-lagged panel models indicated bidirectional relations between EF and math over preschool, which became directional in kindergarten with only EF predicting math. Moreover, there was a bidirectional relation between math and literacy that emerged in kindergarten. Similarly, LGCM revealed correlated growth between EF and math as well as math and literacy, but not EF and literacy. Exploring the patterns of relations across the waves of the panel model in conjunction with the patterns of relations between intercepts and slopes in the LGCMs led to a more nuanced understanding of the relations between EF and academic skills across preschool and kindergarten. Implications for future research on instruction and intervention development are discussed.

Using a Brief Preschool Early Numeracy Skills Screener to Identify Young Children With Mathematics Difficulties

Abstract. A critical component in enhancing academic success is identifying children at risk of later academic difficulties. Although significant efforts have been devoted to design effective assessment processes in elementary school, fewer efforts (particularly for mathematics) have been made for preschool. The focus of this study was to design and evaluate a brief early numeracy skills screening tool. Measure development and validation occurred in a two-stage process with diverse and distinct samples. In the first stage, 393 preschool children were assessed on a battery of early numeracy tasks. By use of an item response theory framework, 24 items that spanned the ability continuum were selected for inclusion in the brief measure. In the second stage, 129 preschool children were assessed on the brief measure, the Test of Early Mathematics Ability–Third Edition, and two literacy measures. The data resulted in acceptable psychometric properties and strong diagnostic accuracy. Theoretical and practical implications are discussed.

Early Numeracy Assessment: The Development of the Preschool Early Numeracy Scales

Research Findings: The focus of this study was to construct and validate 12 brief early numeracy assessment tasks that measure the skills and concepts identified as key to early mathematics development by the National Council of Teachers of Mathematics (2006) and the National Mathematics Advisory Panel (2008)—as well as critical developmental precursors to later mathematics skills noted in the Common Core State Standards (2010). Participants were 393 preschool children ages 3 to 5 years old. Measure development and validation occurred through 3 analytic phases designed to ensure that the measures were brief, reliable, and valid. These measures were 1-to-1 counting, cardinality, counting subsets, subitizing, number comparison, set comparison, number order, numeral identification, set-to-numerals, story problems, number combinations, and verbal counting. Practice or Policy: Teachers have extensive demands on their time, yet they are tasked with ensuring that all students’ academic needs are met. To identify individual instructional needs and measure progress, they need to be able to efficiently assess children’s numeracy skills. The measures developed in this study not only are reliable and exhibit evidence of validity but also are easy to use and can be utilized for measuring the effects of targeted instruction on individual numeracy skills.