David C. Geary - US grants

A dynamic process model of skill acquisition in elementary mathematics that can accommodate individual differences is proposed. Four factors governing skill acquisition were identified and include the development of; (l) counting knowledge; (2) skill at using counting algorithms to solve arithmetic problems; (3) long-term memory representations of number facts; and, (4) working memory resources. The process model represents the dynamic relationships among these factors and serves as the basis for design of the proposed studies. Experiment 1 will provide a comprehensive longitudinal assessment of the development of each of these four skills for groups of normal and mathematically disabled (MD) children, and a comprehensive neuropsychological assessment of the numerical skills of individual MD children. Experiment 2 will test the hypothesis that cross-cultural differences in the learning of elementary mathematics are related to national differences in numerical working memory resources and mathematics instruction. Finally, a computational system, following the production-system framework, of the process model will be refined so that individual differences in skill acquisition can be modeled. These experiments will enable an empirical test of hypotheses generated by the process model and at the same time provide invaluable information on the source of mathematical ability and achievement differences in children. Empirical verification of the model will greatly advance our understanding of the development of number skills in children, because the model subsumes extant developmental models (e.g., the strategy choice model) and can accommodate individual differences, whether the differences are manifested as a mathematical disability, or multinational achievement differences. More practically, these studies will provide a solid foundation from which sensitive techniques for the assessment and early identification of mathematical learning problems could be developed. Such techniques should help to identify the locus of the learning problem and therefore could lead to the development of more effective remedial education programs in mathematics. Finally, the studies will provide unique information on the components of mathematical skill that most sharply differentiate the achievement of East Asian and American children. These, in turn, could be targets for mathematics education in the United States.

DESCRIPTION: Research on average-IQ children with poor math achievement, that is, math disabled (MD) children, has revealed several sources of their lower than expected (based on IQ) achievement. These children have a poor conceptual understanding of counting, use immature counting procedures to solve arithmetic problems, commit many procedural errors, and have difficulties remembering basic arithmetic facts. The poor counting knowledge of these children appears to contribute to their use of immature counting procedures and their procedural errors, although it is not currently known if this poor counting knowledge extends beyond second grade. Poor working memory resources also appear to contribute to these procedural deficits and it has been argued that these working memory deficits are due to a slow counting speed. The source of the fact-retrieval deficits of these children is unclear, but appears to be related to a combination of difficulties in inhibiting irrelevant associations from entering working memory and from difficulties in lexical access (e.g., retrieving information from semantic memory). Recent theoretical and methodological advances suggest that these working memory and retrieval deficits are more complex than previously conceptualized. Individual differences in working memory appear to be related to ease of inhibiting irrelevant associations, speed of articulating words, speed of scanning the contents of working memory, and ease of regenerating phonetic and semantic traces that are decaying in working memory. The latter, in theory, is related to MD children's difficulties in lexical access and the associated fact-retrieval deficits. The proposed research will be the first to: 1) assess the counting knowledge of first, third and fifth grade MD children and normal controls and examine the relations among counting span (an index of working memory), counting knowledge and procedural competencies in arithmetic; 2) use recent theoretical and methodological advances to provide an assessment of each of the above described components of working memory in first, third and fifth grade MD children and normal controls and relate these competencies to the working memory, counting procedure, and fact-retrieval deficits of MD children; 3) use negative priming methods to test the hypothesis that the fact-retrieval deficit of MD children is related, in part, to difficulties in inhibiting irrelevant associations from entering working memory; and 4) systematically compare the counting and arithmetic competencies of MD children and children with low-IQ scores and low math achievement scores.

The proposed research will provide a 5-year longitudinal study of the development of arithmetic-procedural competencies and number estimation skills in children with a learning disability in mathematics (MD) and normal peers. It is well established that during the solving of simple arithmetic problems (e.g., 6+7) MD children 1) rely on developmentally immature strategies, such as finger counting;2) frequently commit counting errors;3) use immature counting procedures [e.g., they count both addends rather than stating the cardinal value of one addend and counting-on a number of times equal to the value of the other addend];4) and, have difficulties retrieving basic facts from long-term memory. The focus of our currently funded research is to identify the basic cognitive systems, such as working memory, that underlie these arithmetic deficits. We have found, for instance, a strong relation between working memory and MD children's reliance on finger counting and counting errors, and a relation between their use of immature counting procedures and knowledge of counting concepts. In other words, MD children have a lower working memory capacity than their normal peers, and this group difference in working memory explains part of the group differences in use of finger counting and counting errors. Group differences in knowledge of counting principles, in contrast, is unrelated to use of finger counting or counting errors, but explains part of the group difference in the type of procedure used during counting. The proposed research will provide the first longitudinal assessment of these relations between MD children's arithmetic deficits and underlying cognitive systems, and will expand the assessments to include more complex arithmetic problems than have been used in most previous studies. The proposed research will also be the first to assess MD children's ability to use spatial representations (i.e., a mental number line) to make number estimations, and assess how the form of the representation changes with formal schooling. The goals are to document the developmental progression of arithmetic and number estimation competencies in MD children, and to identify - based on the theoretical rationale described in the Background and Significance Section - the cognitive systems (e.g., working memory) that underlie MD children's developmental delays and deficits in these competencies.

This project is a five-year longitudinal study designed to examine early foundations of formal mathematical learning. Approximately, 250 children in Missouri will be assessed twice per year from preschool through first grade. Specific competencies to be measured include number, number relations, and number operations as well as language, executive function, attention, IQ, and social behavior. Symbolic and non-symbolic quantitative skills are considered. Data collected on this project will link to the Missouri Longitudinal Study of Mathematical Development and Disability.

Competence in arithmetic and basic algebra has been shown to be strongly related to employability, wages, and on-the-job productivity. Children who begin school behind their peers in mathematical competencies tend to stay behind throughout their schooling. The goal of this project is to conduct a longitudinal study with at-risk 3 year olds to improve the field?s understanding of the development of early numeracy development in young children and its relation to school mathematics outcomes. The project hopes to shed light on how domain general executive functions, nonverbal intelligence, and verbal intelligence interact with quantitative knowledge to lead to math achievement at the end of kindergarten.

Modified Project Summary/Abstract Section Competence with algebra is the foundation for the mathematics that facilitates reasoned decision making in many aspects of day-to-day life, including medical treatment choices (Reyna et al. 2009), is demanded in high-paying technology and science fields (National Mathematics Advisory Panel 2008), and contributes to employability and wages in many blue collar occupations (Bynner 1997). Yet, only 39% of US high school graduates have the algebraic competencies needed for these endeavors. Improving the algebraic skills of US students will yield individual and wider social benefits, but this goal has been difficult to achieve. One impediment is our piecemeal understanding of algebraic learning and cognition that has resulted in both practical and theoretical costs. Practically, one student may have difficulty due to an undervaluation of the importance of mathematics, another may have poorly developed prerequisite skills, and a third may confuse current learning with prior learning (proactive inhibition). The reasons for algebraic deficits in these students are very different and consequently the interventions to address them would differ as well. No current theory exists that integrates cognitive and non-cognitive processes on algebra readiness and learning, an integration that is needed for the development of measures to better assess individual differences in algebraic learning. In this proposal we develop such a model and use it to organize a comprehensive and deep longitudinal study of individual and gender differences in readiness for algebra and the learning of specific, core algorithmic and spatial-related algebraic competencies. Carefully chosen measures of prerequisite skills, visuospatial memory and cognition, complex spatial ability, proactive interference, and inherent sensitivity to magnitude will be administered to 450 7th graders. Their attitudes about mathematics and their math anxiety, as well as their attentive behavior in mathematics and English (as a contrast) classrooms will be annually assessed in 7th to 9th grade. At the beginning and end of 9th grade, participants will be assessed using psychometric and cognitive measures of algebraic competence generally and competence with algorithmic and spatial-dependent aspects of algebra in particular. The breadth and depth of measures will enable testing of nuanced hypotheses regarding the contributions of different cognitive and memory systems to different aspects of algebraic learning; the dynamic interactions between these systems and developmental change in math attitudes and anxiety as related to engagement in the mathematics classroom and algebraic learning and cognition; and, the cognitive and non-cognitive factors that contribute to gender differences in certain aspects of algebra, specifically girls? disadvantage in spatial aspects of algebra and their undervaluation of mathematics. The combination will significantly advance our understanding of algebraic learning and cognition, and lead to the development of better assessment measures and targeted interventions to address individual and gender differences in algebraic learning.

This proposal was submitted in response to EHR Core Research (ECR) program announcement NSF 15-509. The ECR program of fundamental research in STEM education provides funding in critical research areas that are essential, broad and enduring. EHR seeks proposals that will help synthesize, build and/or expand research foundations in the following focal areas: STEM learning, STEM learning environments, STEM workforce development, and broadening participation in STEM. The ECR program is distinguished by its emphasis on the accumulation of robust evidence to inform efforts to (a) understand, (b) build theory to explain, and (c) suggest interventions (and innovations) to address persistent challenges in STEM interest, education, learning, and participation.

High school algebra is the gateway to a career in science, technology, engineering, and mathematics (STEM), and can influence employability and wages in many non-STEM occupations. Students who struggle with or fail high school algebra have compromised occupational prospects, and nations that do not produce mathematically competent citizens may compromise their economic growth. Much is known about the factors that contribute to students' difficulties with arithmetic learning and interventions are being developed to address these difficulties. Little is known, however, about why some students struggle with algebra. Accordingly, the project will follow at risk students (including for example, those with dyslexia) from 7th grade through high school algebra and assess their prerequisite knowledge (e.g. fractions skills), cognitive systems (e.g., memory), attitudes and reactions to mathematics (e.g. math anxiety) and their attentiveness in math classrooms. The comprehensive evaluation of these students will allow us to identify the factors that influence difficulties in learning different aspects of algebra and risk of failing algebra more generally. The results will provide unique scientific insights into the cognitive and motivational influences on students' understanding and learning of algebra and identify areas for intervention with at-risk students. The results will also be used to develop a screening measure for the early identification of at-risk students and to identify specific areas for targeted intervention. The measure will be made freely available to interested school districts throughout the United States.

The project is a 7th to 9th grade longitudinal assessment of the prerequisite (e.g. fractions), cognitive (e.g. working memory), and non-cognitive (e.g. math anxiety) factors that dynamically influence students' algebraic learning and cognition, with a focus on students with learning disabilities in mathematics. The study will provide the most comprehensive assessment of the development of algebra competence ever conducted and is organized by an integrative model of cognitive and non-cognitive influences on students' engagement in math classrooms and on the learning of procedural and spatial-related aspects of algebra. The focus on students at risk for failing high school algebra is informed by research on the number and arithmetic deficits of these students, providing continuity with previous work, and a strong a priori framework for assessing their most likely difficulties in learning algebra; specifically, we developed novel measures that assess different aspects of procedural algebra (e.g. memory for the structure of algebra equations) and spatial-related algebra (e.g. recognizing how common functions map to coordinate space) that will allow for the study of different types of learning deficits and a determination of how more basic cognitive abilities, such as visuospatial working memory, may underlie these deficits. Prior cognitive studies of at-risk students have largely ignored the contributions of non-cognitive factors, such as math anxiety, on their learning or how their learning difficulties change attitudes about and reactions to mathematics (e.g. increasing math anxiety). The proposed research will address this important oversight and integrate these non-cognitive factors with assessments of teacher-rated student engagement in pre-algebra and algebra classrooms (and language arts classrooms as a contrast) and how engagement in the classroom influences the learning of procedural and spatial-related algebra. The research will also provide a thorough analysis of cognitive and non-cognitive influences on algebraic learning and cognition more generally, and thus inform general educational practices. In all, the proposed research will provide a comprehensive model for the study algebraic learning and cognition generally, and will provide a comprehensive assessment of associated deficits of learning disabled students and students at risk for failing high school algebra. The research will also make available revised or newly developed cognitive measures of procedural and spatial-related algebra skills that should facilitate future cognitive science and educational studies of algebra learning.