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Having better knowledge of fractions is causally related to the ability to learn algebra, so what happens when teaching and learning about fractions is disrupted, as was the case during the COVID-19 pandemic? In this study, we examine how educational disruption caused by a pandemic differentially impacted students' fraction learning relative to students who were learning other mathematics content during that time. This study provides results from a cross-sequential project examining various facets of mathematics knowledge for students in 4th-10th grades over three years (2021, 2022, 2023; N=903 students). We investigate differences in fractions and algebra knowledge based on students' grade levels across cohorts to determine if there are particular periods at which students' learning was differentially affected by the disruption. Individual differences in students' self-regulation, self-efficacy, and personality will also be explored as potential buffers.

The current study assessed whether adding worked examples with self-explanation prompts focused on making connections between mathematical principles, procedures, and concepts of rational numbers to a curriculum focused on invented strategies improves pre-algebra students' fraction number line acuity, rational number concepts and procedures. Finally, the study assessed whether individual differences in prior knowledge of rational numbers moderate the effect of the textbooks. The experimental textbooks resulted in significantly greater improvements overall in fraction number line estimation skills than control textbooks. Further, example-supplemented textbooks were more effective than control textbooks at improving understanding of rational number concepts for those with low prior conceptual knowledge. Both textbooks were equally effective at improving procedural skills.

The efficacy of a research-based fraction sense intervention for sixth graders with or at risk for mathematics difficulties (N = 52) was examined. The intervention aimed to build understanding of fraction magnitudes on the number line. Key concepts were taught with a narrow range of denominators to develop deep understanding. The intervention was centered on a visual number line in the meaningful context of a color run race. Students were randomly assigned to the fraction sense intervention (n = 25) or a business-as-usual control group (n = 27). Students in the intervention condition received 21 lessons in small groups (45 min each) during their regular mathematics intervention period. Students in the intervention group performed significantly better than those in the control group on a measure of fraction number line estimation and a more general measure of fraction concepts, both at immediate posttest and delayed posttest, with large effect sizes; lesser effects were shown for fraction arithmetic. [For the corresponding grantee submission, see ED603994.]

The present study examined the effect of children's enrollment in U.S. dual-language immersion (DLI) programs in first grade on English development across five years, using the Early Childhood Longitudinal Study, Kindergarten Class of 2011 (ECLS-K:2011) database. Propensity score matching was used to create comparable groups of DLI and non-DLI students based on students' kindergarten reading performance and a series of student-, family-, and school-level characteristics. Growth curve models demonstrate that first-grade DLI enrollment had a positive effect on children's English reading growth from Grade 1 to 5. Children who enrolled in DLI experienced greater improvements by Grade 5. We also found that first-grade teacher judgment was related to children's initial reading performance but not their reading growth. Implications related to DLI programs, teacher practices, and bilingual educational policies are discussed.

The present study examines the effectiveness of incorporating worked examples with prompts for self-explanation into a middle school math textbook. Algebra 1 students (N = 75) completed an equation-solving unit with textbooks either containing the original practice problems or in which a portion of those problems were converted into a combination of correct, incorrect, and incomplete examples. Students completed pre- and posttest measures of algebraic feature knowledge, equation-solving skills, and error anticipation. Example-based textbook assignments increased students' equation-solving skills and their ability to anticipate errors one might make when solving problems. Differences in students' anticipation of various types of errors are also examined. Error anticipation, a particular form of "negative knowledge," is a potentially important skill that relates to algebraic feature knowledge and equation-solving skills.

Recent studies have established that students knowledge about fractions is predictive of their readiness, performance, andlearning in Algebra (Booth & Newton, 2012; Booth, Newton, & Twiss-Garrity, 2013). However, it is yet unknown whetherthe relationship between fractions and algebra is causal; that is, would improving students’ knowledge of fractions causeimprovements in their ability to perform in and learn Algebra? The present study examines the impact of improvingfraction computation and fraction magnitude knowledge in real world classrooms on middle school students’ learningof key concepts and problem-solving techniques in Algebra. Individual differences in the impact of improved fractionknowledge will also be investigated and discussed.

Fraction knowledge and algebraic skill are closely linked. Algebra is a gatekeeper for advanced courses (Booth & Newton, 2012; Brown & Quinn, 2007). This study uses the person-centered approach of latent profile analysis to examine individual differences in middle schoolers' (N = 350) algebra performance at the end of the year (EOY). The relative importance of a range of fraction skills is considered in predicting the likelihood of displaying a particular profile of EOY algebra knowledge, measured by feature knowledge, equation encoding, equation solving, and word-problem skills. Notably, fraction number line estimation did not predict algebra performance on any of the four measures. Fraction arithmetic at the beginning of the year is most predictive of displaying the highest performing EOY algebra profile.

Prior research highlights a positive effect of incorrect worked examples on mathematics learning. Yet the mechanisms underlying these benefits are unclear. To investigate potential mechanisms of the benefits of various worked example types, we examined process data from a previously published classroom-based experiment. More specifically, we analyzed students' explanations made while explaining worked examples in three varying example conditions as well as students' problem-solving errors made when solving problems. These data operationalize two potential mechanisms: a reduction of misconceptions (i.e., fewer targeted conceptual errors), and an increase in principled algebra knowledge (i.e., explanations focusing on principles underlying procedures). Mediation analyses revealed both as important mechanisms of varying effects. A reduction of misconceptions explained greater benefits of all three worked example conditions, compared to a problem-solving control, on an algebra concepts posttest. More principled explanations of procedures explained the benefits of incorrect worked examples on problem-solving at posttest compared to the two other example conditions. These findings help explain differential findings in prior work by example type and may elucidate potential avenues for errorful instruction. [ABSTRACT FROM AUTHOR]

The effectiveness of an experimental middle school fraction intervention was evaluated. The intervention was centered on the number line and incorporated key principles from the science of learning. Sixth graders (N = 51) who struggled with fraction concepts were randomly assigned at the student level to the experimental intervention (n = 28) or to a business-as-usual control who received their school's intervention (n = 23). The experimental intervention occurred over 6 weeks (27 lessons). Fraction number line estimation, magnitude comparisons, concepts, and arithmetic were assessed at pretest, posttest, and delayed posttest. The experimental group demonstrated significantly more learning than the control group from pretest to posttest, with meaningful effect sizes on measures of fraction concepts (g = 1.09), number line estimation as measured by percent absolute error (g = -0.85), and magnitude comparisons (g = 0.82). These improvements held at delayed posttest 7 weeks later. Exploratory analyses showed a significant interaction between classroom attentive behavior and intervention group on fraction concepts at posttest, suggesting a buffering effect of the experimental intervention on the normally negative impact of low attentive behavior on learning. A number line--centered approach to teaching fractions that also incorporates research-based learning strategies helps struggling learners to make durable gains in their conceptual understanding of fractions. [For the corresponding grantee submission, see ED595952.]

Although findings from cognitive science have suggested learning benefits of confronting errors (Metcalfe, 2017), they are not often capitalized on in many mathematics classrooms (Tulis, 2013). The current study assessed the effects of examples focused on either common mathematical misconceptions and errors or correct concepts and procedures on algebraic feature knowledge and solving quadratic equations. Middle school algebra students (N = 206) were randomly assigned to four conditions. Two errorful conditions either displayed errors and asked students to explain or displayed correct solutions and primed students to reflect on potential errors by problem type. A correct example condition and problem-solving control group were also included. Studying and explaining common errors displayed in incorrect examples improved equation-solving ability. An aptitude-by-treatment interaction revealed that learners with limited understandings of algebraic features demonstrated greater benefits. Theoretical implications about using examples to promote learning from errors are considered in addition to suggestions for educational practice.

Students hold many misconceptions as they transition from arithmetic to algebraic thinking, and these misconceptions can hinder their performance and learning in the subject. To identify the errors in Algebra I which are most persistent and pernicious in terms of predicting student difficulty on standardized test items, the present study assessed algebraic misconceptions using an in-depth error analysis on algebra students' problem solving efforts at different points in the school year. Results indicate that different types of errors become more prominent with different content at different points in the year, and that there are certain types of errors that, when made during different levels of content, are indicative of math achievement difficulties. Recommendations for the necessity and timing of intervention on particular errors are discussed.

Reasoning about numerical magnitudes is a key aspect of mathematics learning. Most research examining the relation of magnitude understanding to general mathematics achievement has focused on whole number and fraction magnitudes. The present longitudinal study (N = 435) used a 3-step latent class analysis to examine reasoning about magnitudes on a decimal comparison task in 4th grade, before systematic decimals instruction. Three classes of response patterns were identified, indicating empirically distinct levels of decimal magnitude understanding. Class 1 students consistently gave correct responses, suggesting that they understood decimal properties even before systematic decimal instruction. Class 2 students were accurate when a 0 immediately followed the decimal, but were inaccurate when a zero was added to the end of the decimal string, suggesting a partial understanding of place value; their performance was also negatively influenced by a whole number bias. Class 3 students showed misunderstanding of both place value and a whole number bias. Class membership accurately predicted 6th grade mathematics achievement, after controlling for whole number and fraction magnitude understanding as well as demographic and cognitive factors. Taken together, the findings suggest students may benefit from instruction that emphasizes decimal properties earlier in school. [For the corresponding grantee submission, see ED593867.]

The current study examined the effectiveness of self-explanation prompts, visual signaling cues, and a combination of the two features on middle school students' (N = 202) algebra learning. Also explored were the differential effects of features for students with faulty conceptual knowledge (evidenced by a higher prevalence of making errors during problem solving) on learning. That is, we assessed whether students who make prevalent conceptual errors predictive of algebra performance differentially benefit from design features. Participants were randomly assigned to complete 1 of 4 sets of worked example assignments supplemented with self-explanation prompts (n = 51), visual signaling cues (n = 49), both features (n = 51), or neither feature (n = 51). Worked examples supplemented with either self-explanation prompts or signaling cues led to greater learning from pre- to posttest in comparison to the worked example control, with practically meaningful effects. The effect of assignments supplemented with signaling cues was moderated by error prevalence. Those who made errors more frequently demonstrating misunderstanding of algebraic concepts (e.g., the meaning of a coefficient) benefited significantly more from visual signaling cues alone than self-explanation prompts alone or control assignments. These findings highlight the importance of considering differential effects of design features when used in combination or in isolation, particularly for struggling students.

Algebra I is considered a gatekeeper course for highereducation, high-paying jobs, and access to STEM careers, yetmany students find themselves struggling to learn algebra.Prior research links intrinsic motivation for learning mathwith mathematics achievement, particularly duringadolescence. The current study measured middle schoolstudents’ interest in algebra and their procedural skills acrossthe span of an algebra unit to determine whether students whoshow declines in algebraic problem-solving also show adecline in a particular type of intrinsic motivation – interest inalgebra. Pre-test and post-test scores were used to categorizeparticipants into those who showed declines in problem-solving skills and those who did not. Of the overall sample (N= 367), a group of 25 students showed declining skills overthe course of the unit. These students also showed significantdeclines in interest in mathematics from pre- to post-test incomparison to students who did not show procedural declines.Our findings support the relationship between performanceand motivation in the classroom, particularly in algebra class.Educational implications are discussed.

Research Findings: Families need access to early care and education programs, both to ensure parents' ability to work and support children's development. Subsidies through the Child Care and Development Fund (CCDF) are a set of policies aimed at assisting families living in poverty with accessing early education. However, the number of early education centers that accept subsidies is declining. Using observational data from the National Survey of Early Care and Education and state subsidy policies from the CCDF Policies Database, we use an innovative application of propensity score methods to estimate causality and provide actionable findings regarding the effect of state-specific policies on centers' subsidy participation. We create comparable groups of centers that accept subsidies and centers that do not, based on research-supported program- and community-level predictors of subsidy participation. Logistic regression models using the matched sample demonstrated state-specific subsidy policies impacted subsidy participation. Specifically, as the state subsidy reimbursement rate increased, centers were more likely to accept subsidies. Practice or Policy: Findings point to meaningful state-level actions and policies that may incentivize centers' subsidy participation. In general, state policies that increase revenue for centers accepting subsidies (e.g. reimbursement for child absences) may result in higher rates of subsidy participation. [ABSTRACT FROM AUTHOR]

Background: Providing students with worked out problem solutions is a beneficial instructional technique in STEM disciplines, and studying examples that have been worked out incorrectly may be especially helpful for reducing misconceptions in students with low prior content knowledge. However, past results are inconclusive and the effects of incorrect worked examples alone or in combination with correct examples remains unclear. Objectives: We aim to address whether studying incorrect examples alone or in combination with correct examples can support the reduction of students' fraction misconceptions, operationalized as errors made with high confidence. Methods: After incorrectly solving a sampling problem, 130 students in 4th through 11th grade in the U.S. were randomly assigned to a condition in an online problem set focused on fraction equivalence. Students studied either single‐type worked examples (i.e., correct or incorrect; n = 49) or combination‐type worked examples (correct and incorrect; n = 41) or engaged in a problem‐solving control (n = 50). Results: Studying a combination of correct and incorrect worked examples was as effective as the problem‐solving control with feedback at improving fraction equivalence knowledge and reducing the rate of high‐confidence errors. Students in both the combination condition and the problem‐solving with feedback condition outperformed those who studied either correct or incorrect worked examples alone. Conclusions: Results support the inclusion of a combination of correct and incorrect worked examples when teaching students with low prior content knowledge. Studying a combination of example types within an online tutor helps to reduce misconceptions about fractions, a topic students commonly struggle with. A problem‐solving task with corrective feedback worked equally well. Lay Description: What is already known about this topic: Studying worked examples of mathematical solutions improves student learning.Incorrect examples, either alone or in combination with correct examples, are particularly effective for improving learning for students with low prior knowledge in the target content.Comparing correct and incorrect examples is also effective at improving problem‐solving but this effect may be specific to those with high prior knowledge.Still unknown are the impacts of these different combinations of examples on misconceptions in particular. What this paper adds: We show that studying a combination of correct and incorrect examples reduces math misconceptions.Problem‐solving with corrective feedback within an online tutor also has this effect. Implications for practice and/or policy: Using a combination of example types will help reduce students' misconceptions about fractions. [ABSTRACT FROM AUTHOR]

Procedural flexibility is an important skill for algebra. Although prior work has focused on measuring students' procedural flexibility using arithmetic problems, word problems may also capture students' flexibility because of their open-ended nature. To date, no published study has examined the use of word problems as another measure of procedural flexibility. The present study aims to establish flexibility on fraction word problems as a predictor of algebra learning and to determine whether flexibility demonstrated on fraction arithmetic versus fraction word problems differentially predicts two types of algebra learning (e.g., algebraic feature knowledge and algebra equation-solving). Middle-school students (N = 350) completed fraction arithmetic problems, fraction word problems, and algebra measures at the start and end of the school year. We coded fraction arithmetic and word problems for demonstrated procedural flexibility. Path models showed that overall flexibility significantly predicted both algebraic feature knowledge and equation-solving. When examining flexibility by type, procedural flexibility on arithmetic problems significantly predicted end-of-year algebraic feature knowledge and algebra equation-solving, whereas procedural flexibility on word problems significantly predicted algebraic feature knowledge. Procedural flexibility on arithmetic problems was still predictive of performance when accounting for fraction magnitude skill. These findings build on prior literature that establishes procedural flexibility as a crucial predictor of algebra learning and extends measures of procedural flexibility to include word problems. Educational Impact and Implications Statement: We found that students' ability to appropriately choose between problem-solving strategies, or their procedural flexibility, was predictive of their knowledge of algebraic features when measured by flexibility on both fraction arithmetic and word problems. Furthermore, students' overall procedural flexibility was predictive of their algebraic equation-solving skills. These results suggest that procedural flexibility with fractions is an important predictor of algebra learning and can be measured both through arithmetic and word problem measures. Practically, results suggest that procedural flexibility is a skill that should be given more consideration in classroom instruction, an area that may be explored in future research. [ABSTRACT FROM AUTHOR]