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Preschoolers' repeating patterning knowledge is predictive of their concurrent and later math and numeracy knowledge, but strong experimental evidence is needed to determine if these relations are causal. The purpose of the current Study was to examine the causal effects of repeating patterning and numeracy tutoring on repeating patterning, numeracy, and general mathematics knowledge in the year before kindergarten (i.e., pre-K). Children in pre-K (N = 211) were randomly assigned to receive five sessions of researcher-delivered tutoring (a) on repeating patterns and numeracy or (b) on numeracy (and literacy as an active control), or received no tutoring and business as usual classroom instruction(control). Children who received tutoring in repeating patterning and numeracy improved in their repeating patterning knowledge the most. However, children's general math and numeracy knowledge improved similarly across conditions, and a specific aspect of numeracy emphasized during the tutoring did not improve. Children's repeating patterning knowledge is malleable, but this initial attempt to demonstrate causal links between repeating patterning and math knowledge was not successful. Results parallel mixed success in research training other skills, such as working memory or spatial skills, for improving mathematics knowledge. Findings are discussed in terms of the relations between patterning, numeracy, and general math knowledge in preschoolers. [This paper will be published in "Journal of Educational Psychology."]

The current study broadens our understanding of preschoolers' early math experiences with parents, recognizing that math knowledge and experiences are inclusive of numeracy as well as non-numeracy domains. Parents and preschoolers (N = 45) were observed exploring three domains of early mathematics knowledge (i.e., number, space, and pattern) during play in three activities (playing cards, building with blocks, and stringing beads, all with activity suggestions). Children were administered a broad math and numeracy measure, and individual measures of spatial and patterning skills concurrently and 7 months later. Dyads explored math broadly across most activities, but emphasized number more than space or patterning. In addition, there was more overall math exploration during card and bead play than block play, with the greatest parent support during card play. Parent support was not linked to children's skills, although children's exploration of space and patterns related moderately to their concurrent spatial and pattern skills. Overall, parents and young children explored a variety of early math domains in guided play contexts, with an emphasis on numeracy. Future work should aim to increase the breadth and rigor of individual concepts that parents and preschoolers explore during play. [This paper was published in "Journal of Experimental Child Psychology" v192 2020.]

Both recent evidence and research-based early mathematics curricula indicate that repeating patterns--predictable sequences that follow a rule--are a topic of major importance for mathematics development. The purpose of the current study was to help build a theory for how early repeating patterning knowledge contributes to early math development, focusing on development in children ages 4-6. The current study examined the relation between 65 preschool children's repeating patterning knowledge (via a fast, teacher friendly measure) and their end-of-kindergarten broad math and numeracy knowledge, controlling for verbal and visual-spatial working memory (WM) skills as well as end-of-pre-k broad math knowledge. Relations were also examined between repeating patterning and specific aspects of their numeracy knowledge--knowledge of the count sequence to 100 and the successor principle. Children's repeating patterning knowledge was significantly predictive of their broad math and general numeracy knowledge, as well as one specific aspect of their numeracy knowledge (counting to 100), even after controlling for verbal and visual-spatial WM skills. It remained a unique predictor of general numeracy knowledge and count to 100 after controlling for end-of-pre-K broad math knowledge. The relation between repeating patterning and mathematics may be explained by the central role that identifying predictable sequences based on underlying rules plays in both. Theories of math development and early math instruction standards should give greater attention to the role of children's repeating patterning knowledge. [This paper will be published in the "Journal of Experimental Child Psychology."]

Preschoolers' patterning skills are predictive of their concurrent and later math knowledge; however, it is unclear if patterning is only a proxy for general intelligence, or how it might support specific math skills. The current study examined the relation between 66 preschool children's patterning skills and their general cognitive abilities, including fluid reasoning, working memory, and spatial skill. Further, the link between patterning and general math knowledge and specific math skills (i.e., numeracy and shape knowledge) was examined. Children's patterning skills were significantly but only moderately linked to their performance on general cognitive ability measures (e.g., fluid reasoning). Further, beyond the effects of these general cognitive abilities, patterning significantly predicted both general math and numeracy knowledge, as well as verbal calculation and magnitude comparison. Thus, although patterning skills related to measures of general cognitive ability, patterning was a unique contributor to children's general math and numeracy knowledge and specific math skills. Theories of math development and early math standards should thus be modified to emphasize the role of children's patterning skills to their math development. [This paper will be published in "Journal of Cognition and Development."]

State mandated tests have taken center stage for assessing student learning and for holding teachers and students accountable for achieving adequate progress. What types of early knowledge predict performance on these tests, especially among low-income children who are at risk for poor performance? We report on a longitudinal study of 519 low-income American children from ages 5-12 with a focus on mathematics performance. We found that nonsymbolic quantity knowledge and repeating patterning knowledge at the end of preschool were reliable predictors of performance on standards-based high-stakes tests across three different grade levels (4th-6th grade), over and above other math and academic skills. Further, these effects of preschool math knowledge were partially mediated through symbolic mapping and calculation knowledge at the end of first grade. These findings suggest that nonsymbolic quantity knowledge and repeating patterning knowledge prior to formal schooling are valuable indicators of low-income children's performance on high-stakes state math tests in the middle grades. [The paper was published in "Journal of Educational Psychology" v111 n3 p402-413 2019.]

Because math knowledge begins to develop at a young age to varying degrees, it is important to identify foundational cognitive and academic skills that might contribute to its development. The current study focused on two important, but often overlooked skills that recent evidence suggests are important contributors to early math development: patterning and spatial skills. We assessed preschool children's repeating patterning skills, spatial skills, general cognitive skills and math knowledge at the beginning of the pre-kindergarten year. We re-assessed their math knowledge near the end of the school year, with complete data for 73 children. Children's repeating patterning and spatial skills were related and were each unique predictors of children's math knowledge at the same time point and seven months later. Further, repeating patterning skills predicted later math knowledge even after controlling for prior math knowledge. Thus, although repeating patterning and spatial skills are related, repeating patterning skills are a unique predictor of math knowledge and growth. Both theories of early math development and early math standards should be expanded to incorporate a role for repeating patterning and spatial skills. [This article will be published in "Early Childhood Research Quarterly."]

The goal of the current study is to develop a more complete understanding of the early "home math environment," encompassing both numeracy and non-numeracy aspects of that environment. Parents of preschoolers (n = 63) were surveyed about their support of three components of early mathematics knowledge (i.e., numeracy, spatial, and pattern) as well as parents' math-related beliefs about themselves and their children. Children were administered a broad math knowledge assessment which included a numeracy subscale, and individual measures of spatial and patterning skills in the fall (concurrently). Broad math knowledge was measured again in the spring of the preschool year. Parents indicated providing some support of early math development through numeracy, spatial, and patterning activities, with a stronger emphasis on numeracy than pattern and space. Parents' child-specific ability beliefs were related to their numeracy, pattern, and broad math support, while their parent-specific ability beliefs were related to their spatial support. Parent support was rarely linked to child skills, except that numeracy support related to concurrent numeracy knowledge. Findings suggest that although parents do support a broad range of early math skills at home, parents tend to prioritize supporting early numeracy. Parents' beliefs, especially about their child's academic abilities, may influence components of the early home math environment, but future research is needed to better understand the relations between parent's academic beliefs and the home math environment they create. [This paper will be published in "Early Childhood Research Quarterly."]

The focus of the present synthesis reflects the research on programs, practices, and policies intended to improve mathematics outcomes funded through the Institute of Education Sciences' (IES's) National Center for Education Research (NCER) and National Center for Special Education Research (NCSER). The authors were asked to review those published articles or book chapters that had emerged from projects funded by IES through NCER (U.S. Department of Education 2013) and NCSER (U.S. Department of Education 2012). Specifically, they were asked to review articles from peer-reviewed journals and book chapters from funded projects that were published from January 1, 2002, to June 30, 2014. In forming the synthesized contributions, the authors focused on publications that reported students' mathematics knowledge as an outcome, and the evidence needed to support a contribution generally had to consist of more than one publication from one grant. Thus, the report is not a general synthesis of research in the field, but rather what has been learned to date through review of peer-reviewed publications of 69 IES-funded grants on the topic. The authors identified 28 specific contributions that IES-funded research made to support mathematics learning and teaching from kindergarten through secondary school. The publication organizes the contributions by topic and grade level: (1) Improving Mathematics Learning: Whole Numbers, Operations, and Word Problem Solving in Elementary School; and Fractions and Algebra in the Middle Grades; and (2) Development and Evaluation of Teacher Professional Development Approaches. Each section describes the contributions IES-funded researchers are making in these areas and discusses the projects behind the contributions. Contains a combined list of references and project publications.

Early mathematics knowledge is a strong predictor of later academic achievement, but children from low-income families enter school with weak mathematics knowledge. An Early Math Trajectories model is proposed and evaluated within a longitudinal study of 517 low-income American children from age 4 to 11. This model includes a broad range of math topics, as well as potential pathways from preschool to middle-grades mathematics achievement. In preschool, nonsymbolic quantity, counting and patterning knowledge predicted fifth-grade mathematics achievement. By the end of first grade, symbolic mapping, calculation and patterning knowledge were the important predictors. Further, the first-grade predictors mediated the relation between preschool math knowledge and fifth-grade mathematics achievement. Findings support the Early Math Trajectories model among low-income children. [At time of submission to ERIC this article was in press with "Child Development."]

Feedback is a ubiquitous learning tool that is theorized to help learners detect and correct their errors. The goal of this study was to examine the effects of feedback in a classroom context for children solving math equivalence problems (problems with operations on both sides of the equal sign). The authors worked with children in 7 second-grade and 7 third-grade classrooms from two suburban elementary schools (one public and one private) in Middle Tennessee. All second- and third-grade children were invited to participate, with a total of 274 children completing an initial screening measure. At the beginning of the intervention, children in all conditions received instruction on a correct problem-solving strategy with four math equivalence problems. Children were instructed on the commonly used equalize strategy, which involves adding the numbers on one side of the equal sign and then counting up from the number on the other side to get the same amount. The instructor demonstrated the procedure on all four problems. The study had a pretest-intervention-posttest design. The intervention occurred in children's classrooms during their normal mathematics instruction. The findings support the notion that learner characteristics may be as, if not more important than characteristics of the feedback itself. One figure is appended.

Comparison is a powerful tool that has been shown to improve learning in a variety of domains. In both laboratory studies and small-scale classroom studies, having learners compare and contrast worked examples has been shown to reliably lead to gains in students' knowledge. Comparison is also integral to "best practices" in mathematics education. Having students share solution procedures for a particular problem and then discuss the similarities and differences in the different procedures lies at the core of reform pedagogy in many countries throughout the world. Researchers' past empirical work has illustrated the potential benefits of comparison for students' learning of mathematics. Students who were shown two worked examples side-by-side and given the opportunity to compare and discuss similarities and differences between problems, solutions, and strategies achieved greater gains in conceptual knowledge, procedural knowledge, and flexibility, as compared to control students. The current intervention sought to build upon past research by scaling up materials to encourage comparison in the classroom throughout the academic year using a randomized control trial design. This report provides the results of a year-long experiment examining the impact of researcher-designed supplemental curriculum materials that "infused" comparison into the learning and teaching of Algebra I. The question guiding this research is as follows: What is the effect of the supplemental comparison curriculum on Algebra I students' knowledge? Data were collected from teachers and students in 57 public schools across the state of Massachusetts during the 2010-2011 school year. Suburban, urban, and rural schools were represented. Results suggest that, when implemented with sufficient dosage and instructional quality, use of the supplemental curriculum that "infused" comparison into the learning and teaching of Algebra I improved students' learning of mathematics. Future research will determine whether encouraging sufficient dosage and instructional quality of this supplemental curriculum in classrooms might lead to better student outcomes than business-as-usual classrooms. Tables and a figure are appended.

An emerging consensus suggests that guided discovery, which combines discovery and instruction, is a more effective educational approach than either one in isolation. The goal of this study was to examine two specific forms of guided discovery, testing whether conceptual instruction should precede or follow exploratory problem solving. In both cases, the problem solving is considered exploratory because children have not been told how to solve the problems; rather, children are expected to generate or apply their own problem solving procedures. The difference is in the timing of conceptual instruction. The authors worked with children from 12 second- and third-grade classrooms in two urban, public schools in Tennessee. Participants were 183 second- and third-grade children. The intervention had a conceptual instruction phase and an exploratory problem solving phase. The two phases were identical for all children; the only difference between conditions was the timing of the two phases (instruct-solve vs. solve-instruct). The authors used a pretest- intervention-posttest design followed by a two-week retention test. Children were randomly assigned to one of two conditions: instruct-solve or solve-instruct. Evidence suggests that guided discovery promotes deeper learning than discovery or instruction alone. Tables and figures are appended.

Assessing the effectiveness of educational interventions relies on quantifying differences between interventions groups over time in a between-within design. Binary outcome variables (e.g., correct responses versus incorrect responses) are often assessed. Widespread approaches use percent correct on assessments, and repeated measures analysis of variance (ANOVA) methods to detect differences between groups. However, this approach is not ideal, as in fact several assumptions are often violated when using this method that can result in less informative and at times biased and spurious findings (Dixon, 2008; Embretson, 1994). An alternative approach is to utilize item response models to detect differences between intervention groups over time. The benefits of item response methodology for intervention research are contrasted with repeated measures ANOVA approaches, using a longitudinal intervention dataset having a between-within design from elementary students learning about mathematical equivalence. The dependent measures of percent correct in repeated measures ANOVA approaches and item responses in item response models are contrasted, as well as the methods for quantifying differences between groups using repeated measures ANOVA approaches and item response models. Second and third grade students who scored below 75% correct on the pretest participated in a 20-minute one-on-one tutoring intervention that focused on mathematical equivalence problems. In conclusion, item response models offer many methodological advantages in the quantification of individual learning and group change over time compared to repeated measure ANOVA approaches based on percent correct outcomes. In particular, the generalized explanatory longitudinal item response model for multidimensional tests (Cho et al., in press) quantifies and tests for differences between intervention conditions, while utilizing the more informative and less problematic metrics of student performance. In addition to being methodologically more sound, these analyses can be performed using the open-source and free program R. Details of the model, as well as information how to run these analyses can be found in Cho et al. (in press). One drawback to preforming IRT analyses is that they do require more technical proficiency on the part of the data analyst than ANOVA approaches. Nevertheless, researchers should strive to adapt this more informative and less biased metric in the evaluation of intervention effectiveness. Generalized Explanatory IRT Model R Code and Select Results are appended. (Contains 3 figures and 1 footnote.

The current paper investigated the following research questions regarding measures of fidelity: (1) Is there a significant relationship between two different measures of fidelity of implementation: a survey of instructional practices and coded videos of classroom lessons? Does the strength of this relationship differ between treatment and control teachers?; and (2) Do these measures of fidelity of implementation differ in their predictive value regarding student outcomes? Data were collected from teachers and students in 57 public schools across the state of Massachusetts during the 2010-2011 school year. Suburban, urban, and rural schools were represented. Teachers in the treatment condition were asked to implement the intervention materials in their classrooms once or twice a week. Teachers in the control condition followed business-as-usual practices in their classrooms. (Contains 4 tables and 1 figure.)

In this study, the authors wanted to examine whether success on items testing basic equivalence knowledge, such as the meaning of the equal sign and ability to solve problems such as 3 + 5 = 4 + _, predicted success on items testing more advanced algebraic thinking (i.e. principles of equality and solving equations that use letter variables). This investigation is a follow up study to their initial efforts to design an instrument to measure children's understanding of equivalence (Rittle-Johnson, et. al. under review). This replication and extension with a new sample also provides evidence for the validity and generalizability of our instrument. They had two specific predictions about the relations between basic-level and advanced-level knowledge items. First, they expected that the relative difficulty of the two types of knowledge would be born out on the Rasch model. That is, they expected that their empirically derived difficulty scores would be higher for the advanced-level items than for the basic-level item. Second, they expected that performance on basic-level items could be used to predict performance on advanced-level items. (Contains 1 table and 1 figure.)

The focus of this research is to develop an initial framework for assessing and interpreting students' level of understanding of mathematical equivalence. Although this topic has been studied for many years, there has been no systematic development or evaluation of a valid measure of equivalence knowledge. A powerful method for accomplishing this task has been developed by Wilson (2005). Building upon item response theory, Wilson and his colleagues utilize an assessment instrument called a "construct map." A key feature of this tool is that it can assist educational researchers in making "criterion-referenced" interpretations of instructional materials and student performances. Thus, a construct map of mathematical equivalence can provide a coherent framework for interpreting findings from across elementary and middle school mathematics curricula. A written assessment was administered to ten second- through sixth grade classrooms (two per grade) in an urban, parochial school serving a working- to middle-class population. Participants were 181 second- through sixth grade students. This set was comprised of 39 second-grade students (17 girls and 21 boys), 43 third-grade students (26 girls and 17 boys), 39 fourth-grade students (16 girls and 22 boys), 34 fifth-grade students (17 girls and 17 boys), and 28 sixth-grade students (14 girls and 14 boys). The students were predominantly Caucasian. Utilizing the construct map and Wright map instruments developed by Wilson (2005), the authors developed a framework for understanding mathematical equivalence across elementary and middle school curricula. This framework can provide educational professionals with a tool for more accurately diagnosing students' current skill level and providing guidance for how to foster students' movement along the pathway toward greater mastery of this topic. Furthermore, the framework can provide educational professions with a means of identifying gaps or obstacles in curriculum design and suggest ways for remedying such problems. (Contains 2 tables and 2 figures.)

We compared alternative instructional strategies for integrating knowledge of decimal place value and regrouping concepts with procedures for adding and subtracting decimals. The first condition was based on recent research suggesting that conceptual and procedural knowledge develop in an iterative, hand over hand fashion. In this iterative condition, conceptual and procedural lessons were interleaved. The second condition followed the common ordering of conceptual lessons before procedural lesson (concepts-first condition). All lessons were presented via a computer-based intelligent tutoring system and 72 sixth-grade students participated. Students in the iterative condition made greater improvements in procedural knowledge and comparable improvements in conceptual knowledge, compared to the concepts-first condition. Students in both groups did better when problems were presented in money contexts rather than symbolically. Both the iterative ordering of lessons and presenting problems in money contexts reduced students' digit alignment errors when adding and subtracting decimals. (Author)