Exploring the impact of a fraction sense intervention in authentic school environments: An initial investigation.

A solid understanding of fractions is the cornerstone for acquiring proficiency with rational numbers and paves the way for learning advanced mathematical concepts such as algebra. Fraction difficulties limit not only students' educational and vocational opportunities but also their ability to solve everyday problems. Students who exit sixth grade with inadequate understanding of fractions may experience far-reaching repercussions that lead to lifelong avoidance of mathematics. This article presents the results of a randomized controlled trial focusing on the first two cohorts of a larger efficacy investigation aimed at building fraction sense in students with mathematics difficulties. Teachers implemented an evidence-informed fraction sense intervention (FSI) within their sixth-grade intervention classrooms. The lessons draw from research in cognitive science as well as mathematics education research. Employing random assignment at the classroom level, multilevel modeling revealed a significant effect of the intervention on posttest fractions scores after controlling for pretest fractions scores, working memory, vocabulary, proportional reasoning, and classroom attentive behavior. Students in the FSI group outperformed their counterparts in the control group, with noteworthy effect sizes on most fraction measures. Challenges associated with carrying out school-based intervention research are addressed.

Predicting mathematics achievement from subdomains of early number competence: Differences by grade and achievement level.

This study investigated the relative importance of three subdomains of early number competence (number, number relations, and number operations) in predicting later mathematics achievement in cross-sequential samples of pre-kindergarten (pre-K), kindergarten, and first-grade children (n = 150 at each grade). Ordinary least-squares (OLS) regression analyses showed that each subdomain predicted mathematics achievement at each grade level, controlling for the other two subdomains as well as background variables. All the subdomains explained a significant amount of variance in later mathematics achievement. Unconditional quantile regression analyses examined relations between number competencies and mathematics achievement at quantiles representing low (0.2), intermediate (0.5) and high (0.8) achievement. The subdomain of number operations was highly related to mathematics achievement for high achievers. For low achievers, future mathematics achievement was most highly related to number and number relations abilities in the pre-K sample and to number relations abilities in the kindergarten and first-grade samples. Findings highlight the unique importance of all three subdomains of early number competence for later mathematics achievement, but they show that some of the relations are contingent on achievement level.

Magnitude, numerosity, and development of number: Implications for mathematics disabilities.

Leibovich et al. challenge the prevailing view that non-symbolic number sense (e.g., sensing number the same way one might sense color) is innate, that detection of numerosity is distinct from detection of continuous magnitude. In the present commentary, the authors' viewpoint is discussed in light of the integrative theory of numerical development along with implications for understanding mathematics disabilities.

Pathways to fraction learning: Numerical abilities mediate the relation between early cognitive competencies and later fraction knowledge.

The current study investigated the mediating role of number-related skills in the developmental relationship between early cognitive competencies and later fraction knowledge using structural equation modeling. Fifth-grade numerical skills (i.e., whole number line estimation, non-symbolic proportional reasoning, multiplication, and long division skills) mapped onto two distinct factors: magnitude reasoning and calculation. Controlling for participants' (N=536) demographic characteristics, these two factors fully mediated relationships between third-grade general cognitive competencies (attentive behavior, verbal and nonverbal intellectual abilities, and working memory) and sixth-grade fraction knowledge (concepts and procedures combined). However, specific developmental pathways differed by type of fraction knowledge. Magnitude reasoning ability fully mediated paths from all four cognitive competencies to knowledge of fraction concepts, whereas calculation ability fully mediated paths from attentive behavior and verbal ability to knowledge of fraction procedures (all with medium to large effect sizes). These findings suggest that there are partly overlapping, yet distinct, developmental pathways from cognitive competencies to general fraction knowledge, fraction concepts, and fraction procedures.

Development of fraction concepts and procedures in U.S. and Chinese children.

We compared knowledge of fraction concepts and procedures among sixth and eighth graders in China and the United States. As anticipated, Chinese middle school children had higher knowledge of fraction concepts and procedures than U.S. children in the same grades, and the difference in procedural knowledge was much larger than the difference in conceptual knowledge. Of particular interest, national differences in knowledge of fraction concepts were fully mediated by differences in knowledge of fraction procedures, and differences between the knowledge of Chinese and U.S. children were most pronounced among the lowest achieving children within each country. Based on these and previous results, a theoretical model of the mutually facilitative interaction between conceptual and procedural knowledge of fractions is proposed and discussed.

Reading Stories to Learn Math: Mathematics Vocabulary Instruction for Children with Early Numeracy Difficulties.

The present study involved examining whether a storybook reading intervention targeting mathematics vocabulary, such as "equal," "more," and "less," and associated number concepts would increase at-risk children's vocabulary knowledge and number competencies. Children with early numeracy difficulties (N = 124) were recruited from kindergarten classes in four schools. Participants were randomly assigned to one of three groups: a storybook number competencies (SNC) intervention, a number sense intervention, or a business-as-usual control. Interventions were carried out in groups of four children over 8 weeks (24 thirty-minute sessions). Findings demonstrated that the SNC intervention group outperformed the other groups on measures of mathematics vocabulary, both in terms of words that were closely aligned to the intervention and those that were not. There was no effect of the SNC intervention, however, on general mathematics measures, suggesting a need to provide the mathematics vocabulary work along with more intensive instruction in number concepts.

A Kindergarten Number-Sense Intervention With Contrasting Practice Conditions for Low-Achieving Children.

The efficacy of a research-based number-sense intervention for low-achieving kinder-gartners was examined. Children (N = 126) were randomly assigned to 1 of 3 conditions: a number-sense intervention followed by a number-fact practice session, an identical number-sense intervention followed by a number-list practice session, or a business-as-usual control group. The interventions were delivered in a small-group setting over 24 half-hour lessons. Both intervention groups performed better than controls on measures of number sense, arithmetic fluency, and general mathematics calculation achievement at immediate posttest. However, the number-fact practice condition gave children an additional advantage over the number-list practice condition on the outcomes at delayed posttest 8 weeks later. The number-fact practice condition was especially effective for producing gains in English learners.

Sources of individual differences in children's understanding of fractions.

Longitudinal associations of domain-general and numerical competencies with individual differences in children's understanding of fractions were investigated. Children (n = 163) were assessed at 6 years of age on domain-general (nonverbal reasoning, language, attentive behavior, executive control, visual-spatial memory) and numerical (number knowledge) competencies; at 7 years on whole-number arithmetic computations and number line estimation; and at 10 years on fraction concepts. Mediation analyses controlling for general mathematics ability and general academic ability revealed that numerical and mathematical competencies were direct predictors of fraction concepts, whereas domain-general competencies supported the acquisition of fraction concepts via whole-number arithmetic computations or number line estimation. Results indicate multiple pathways to fraction competence.

Domain-general mediators of the relation between kindergarten number sense and first-grade mathematics achievement.

Domain-general skills that mediate the relation between kindergarten number sense and first-grade mathematics skills were investigated. Participants were 107 children who displayed low number sense in the fall of kindergarten. Controlling for background variables, multiple regression analyses showed that both attention problems and executive functioning were unique predictors of mathematics outcomes. Attention problems were more important for predicting first-grade calculation performance, whereas executive functioning was more important for predicting first-grade performance on applied problems. Moreover, both executive functioning and attention problems were unique partial mediators of the relationship between kindergarten and first-grade mathematics skills. The results provide empirical support for developing interventions that target executive functioning and attention problems in addition to instruction in number skills for kindergartners with initial low number sense.

Profiles of preschoolers' numerical abilities across quantity representations.

Children's early numerical abilities shape their trajectories for math learning throughout schooling, and task quantity representation (e.g., nonsymbolic vs. symbolic) affects their reasoning about numerical concepts. The role of quantity representation in early numerical ability has typically been studied using variable-centered approaches. The present study builds on past work by using latent profile analysis as a person-centered approach to investigate heterogeneity in U.S. preschoolers' (N = 200, Mage = 4 years, 6 months) numerical abilities across nonsymbolic, verbal symbolic, and written symbolic representation types. The aim was to determine whether numerical ability indicators across the three representation types would result in empirically distinct ability profiles and whether preschool ability profile would predict variance on a standardized assessment of math achievement a year later, in kindergarten. We found evidence of four distinct preschool number ability profiles: (a) consistently low; (b) consistently high; (c) intermediate, with an advantage on nonsymbolic items; and (d) intermediate, with an advantage on verbal symbolic items. Although children in the consistently low and consistently high profiles performed reliably lower and higher, respectively, on the assessment of kindergarten math achievement, the two intermediate profiles performed similarly. The results reveal heterogeneity in preschool numerical ability across quantity representations and suggest that proficiency with either nonsymbolic or verbal representations may be a helpful foundation for building future math achievement. (PsycInfo Database Record (c) 2024 APA, all rights reserved).

Developmental predictors of fraction concepts and procedures.

Developmental predictors of children's fraction concepts and procedures at the end of fourth grade were investigated in a 2-year longitudinal study. Participants were 357 children who started the study in third grade. Attentive behavior, language, nonverbal reasoning, number line estimation, calculation fluency, and reading fluency each contributed uniquely to later conceptual understanding of fractions. Number line estimation, attentive behavior, calculation fluency, and working memory made unique contributions to acquisition of fraction arithmetic procedures. Notably, number line estimation made the largest independent contribution in both models. The results suggest that although there is considerable shared variance among the predictors, both general and number-related competencies are uniquely important for explaining why some children struggle with fractions.

Tracking informal fraction knowledge and its correlates across first grade.

Young children have informal knowledge of fractions before learning about fraction symbols in school. In the current study, we followed 103 children in the Mid-Atlantic United States from the fall to the spring of first grade to characterize the development of individual differences in early informal fraction knowledge, as well as its relation to other mathematical and cognitive skills. Most children in our sample showed some early fraction knowledge at the beginning of first grade, especially with nonsymbolic fractions and halving, and this knowledge improved over the school year without explicit instruction in fractions. However, there were large individual differences in early fraction knowledge at the start of first grade, which explained significant variance in math achievement at the end of first grade, even when controlling for whole number knowledge and a variety of cognitive skills. Start-of-year whole number knowledge, but not spatial scaling or proportional reasoning, also predicted early end-of-year fraction knowledge. These data can inform activities for learning in the early years to foster both early fraction and integer knowledge in parallel, which may better prepare students for later formal instruction in fractions. (PsycInfo Database Record (c) 2023 APA, all rights reserved).

Building Kindergartners' Number Sense: A Randomized Controlled Study.

Math achievement in elementary school is mediated by performance and growth in number sense during kindergarten. The aim of the present study was to test the effectiveness of a targeted small group number sense intervention for high-risk kindergartners from low-income communities. Children were randomly assigned to one of three groups (n = 44 in each group): a number sense intervention group, a language intervention group, or a business as usual control group. Accounting for initial skill level in mathematical knowledge, children who received the number sense intervention performed better than controls at immediate post test, with meaningful effects on measures of number competencies and general math achievement. Many of the effects held eight weeks after the intervention was completed, suggesting that children internalized what they had learned. There were no differences between the language and control groups on any math-related measures.

The Importance of Number Sense to Mathematics Achievement in First and Third Grades.

Children's symbolic number sense was examined at the beginning of first grade with a short screen of competencies related to counting, number knowledge, and arithmetic operations. Conventional mathematics achievement was then assessed at the end of both first and third grades. Controlling for age and cognitive abilities (i.e., language, spatial, and memory), number sense made a unique and meaningful contribution to the variance in mathematics achievement at both first and third grades. Furthermore, the strength of the predictions did not weaken over time. Number sense was most strongly related to the ability to solve applied mathematics problems presented in various contexts. The number sense screen taps important intermediate skills that should be considered in the development of early mathematics assessments and interventions.

Early math matters: kindergarten number competence and later mathematics outcomes.

Children's number competencies over 6 time points, from the beginning of kindergarten to the middle of 1st grade, were examined in relation to their mathematics achievement over 5 later time points, from the end of 1st grade to the end of 3rd grade. The relation between early number competence and mathematics achievement was strong and significant throughout the study period. A sequential process growth curve model showed that kindergarten number competence predicted rate of growth in mathematics achievement between 1st and 3rd grades as well as achievement level through 3rd grade. Further, rate of growth in early number competence predicted mathematics performance level in 3rd grade. Although low-income children performed more poorly than their middle-income counterparts in mathematics achievement and progressed at a slower rate, their performance and growth were mediated through relatively weak kindergarten number competence. Similarly, the better performance and faster growth of children who entered kindergarten at an older age were explained by kindergarten number competence. The findings show the importance of early number competence for setting children's learning trajectories in elementary school mathematics.

Identifying Fraction Measures as Screeners of Mathematics Risk Status.

This study investigated the accuracy of three fraction measures (i.e., fraction number line estimation accuracy, general fraction concepts, and fraction arithmetic) for screening fourth graders who might be at risk for mathematics difficulties. Receiver operating characteristic (ROC) curve analyses assessed diagnostic accuracy of the fraction measures for predicting which students would not meet state standards on the state mathematics test in fourth grade (n = 411), fifth grade (n = 362), and sixth grade (n = 304). A combined measure consisting primarily of fraction number line estimation items and general fraction concept items was the most accurate screener of risk status in fourth, fifth, and sixth grades (area under the curve [AUC] = .84, .81, and .85, respectively). To maximize efficiency for classroom use, the length of the combined screener was reduced using best subset automatic linear modeling. The study highlights the importance of fraction knowledge for predicting mathematics achievement more generally and validates an effective and practical screening tool for the intermediate grades.

Development of number combination skill in the early school years: when do fingers help?

Children's change over time in frequency of finger use on number combinations was examined in relation to their change in accuracy. Performance was tracked longitudinally over 11 time points, from the beginning of kindergarten (mean age = 5.7 years) to the end of second grade (n= 217). Accuracy in number combinations increased steadily during the time period while frequency of finger use declined. Correlations between finger use and accuracy decreased gradually, ranging from 0.60 in kindergarten to -0.15 at the end of second grade. Low-income children showed linear growth in frequency of finger use while middle-income children slowed down by second grade and even started to decline. Although girls and boys showed similar growth patterns in frequency and accuracy, boys used their fingers less often than girls and were more accurate. The findings indicate that finger use is most adaptive when children are first learning number combinations, but this benefit lessens over time.

Using kindergarten number sense to predict calculation fluency in second grade.

Children's number sense in kindergarten was used to predict their calculation fluency in second grade (N = 198). Using block entry regression, usual predictors of age, reading, memory, and verbal and spatial cognition were entered in the first block and number sense measures were added in the second block. Number sense measures contributed a significant amount of variance over and above the more general predictors (26%-42%). Uniquely predictive subareas were active memory for numbers, number knowledge, and number combinations, with number combinations standing out as the strongest single predictor. Number sense screening in kindergarten, using "at-risk" versus "not-at-risk" criteria, successfully ruled out 84% of the children who did not go on to have calculation fluency difficulties and positively identified 52% of the children who later showed fluency difficulties. The relation of early number skills to later calculation fluency has important implications for math screening and intervention.

Life on the Number Line: Routes to Understanding Fraction Magnitude for Students With Difficulties Learning Mathematics.

Magnitude understanding is critical for students to develop a deep understanding of fractions and more advanced mathematics curriculum. The research reports in this special issue underscore magnitude understanding for fractions and emphasize number lines as both an assessment and an instructional tool. In this commentary, we discuss how number lines broaden the concept of fractions for students who are tied to the more general part-whole representations of area models. We also discuss how number lines, compared to other representations, are a superior and more mathematically correct way to explain fraction concepts.

Development of fraction comparison strategies: A latent transition analysis.

The present study investigated the development of fraction comparison strategies through a longitudinal analysis of children's responses to a fraction comparison task in 4th through 6th grades (N = 394). Participants were asked to choose the larger value for 24 fraction pairs blocked by fraction type. Latent class analysis of performance over item blocks showed that most children initially exhibited a "whole number bias," indicating that larger numbers in numerators and denominators produce larger fraction values. However, some children instead chose fractions with smaller numerators and denominators, demonstrating a partial understanding that smaller numbers can yield larger fractions. Latent transition analysis showed that most children eventually adopted normative comparison strategies. Children who exhibited a partial understanding by choosing fractions with smaller numbers were more likely to adopt normative comparison strategies earlier than those with larger number biases. Controlling for general math achievement and other cognitive abilities, whole number line estimation accuracy predicted the probability of transitioning to normative comparison strategies. Exploratory factor analyses showed that over time, children appeared to increasingly represent fractions as discrete magnitudes when simpler strategies were unavailable. These results support the integrated theory of numerical development, which posits that an understanding of numbers as magnitudes unifies the process of learning whole numbers and fractions. The findings contrast with conceptual change theories, which propose that children must move from a view of numbers as counting units to a new view that accommodates fractions to overcome whole number bias. (PsycINFO Database Record