Phonological awareness and RAN contribute to Chinese reading and arithmetic for different reasons.

The present study investigated how phonological awareness and rapid automatized naming (hereafter, RAN), simultaneously contributed to Chinese reading and arithmetic fluency. Specifically, we proposed a new hypothesized mechanism that processing speed would mediate the relations of RAN with Chinese reading and arithmetic fluency. One hundred and forty-five Chinese children at the fifth grade were administered with a battery of measures, including three phonological processing measures, character reading, and whole number computation, as well as nonverbal IQ, and vocabulary knowledge. Path analyses revealed that phonological awareness and RAN were uniquely related to character reading and arithmetic fluency, while phonological memory was not significantly correlated to either character reading or arithmetic fluency, after controlling for age, nonverbal IQ, and vocabulary knowledge. Further analysis indicated that processing speed demonstrated a mediating effect on the importance of RAN in character reading, rather than in arithmetic fluency. Results underscore the potential importance of phonological awareness and RAN in character reading and arithmetic fluency, and the mediating role of processing speed in RAN to promote Chinese character reading fluency.

Chinese parents' support of preschoolers' mathematical development.

Research has documented the critical role played by the early home environment in children's mathematical development in Western contexts. Yet little is known about how Chinese parents support their preschoolers' development of math skills. The Chinese context is of particular interest because Chinese children outperform their Western counterparts in math, even early in development. The current study sought to fill this gap by examining a sample of 90 families of 4- and 5-year-olds from mainland China. Parental support-as measured by the frequency of parent-child engagement in home activities as well as parent number talk-and parents' role in children's numeracy skills were investigated. Results indicate wide variation among parents in both types of support. Frequency of engagement in formal numeracy activities, including counting objects and reading number story books, was related to children's knowledge of cardinality. A principal components analysis did not identify informal numeracy activities as a distinct home activity component, likely due to the infrequent occurrences of game-like numeracy activities among the Chinese families. Instead, a structured activity component emerged (e.g., playing musical instruments) and was positively related to children's arithmetic skills. Diversity, but not quantity, of parent number talk was related to children's symbolic magnitude understanding. The distinctive relationships between specific parental measures and child outcomes speak to the need for nuanced identification of home environment factors that are beneficial to particular math competencies. The findings also suggest cultural variations in the mechanisms that support children's mathematical development, highlighting the merits of investigating this topic in non-Western contexts.

Turning Potential Flexibility Into Flexible Performance: Moderating Effect of Self-Efficacy and Use of Flexible Cognition.

This study examined the relationship between two types of mathematical flexibility - potential flexibility, which indicates individuals' knowledge of multiple strategies and strategy efficiency, and practical flexibility, which refers to individuals' flexible performances when solving math problems. Both types of flexibility were assessed in the domain of linear equation solving. Furthermore, two types of beliefs - self-efficacy and use of flexible cognition (UFC) - were investigated as potential moderators between potential and practical flexibility. 121 8th grade students from China took part in this study. Results indicate that potential flexibility positively predicted practical flexibility. Additionally, self-efficacy and UFC might moderate the relationship between these two types of flexibility, suggesting that potential flexibility may lead to different degrees of practical flexibility depending on different levels of beliefs. Implications of these findings for research on mathematical flexibility and for educational practice are discussed.

Measures of Potential Flexibility and Practical Flexibility in Equation Solving.

Researchers interested in mathematical proficiency have recently begun to explore the development of strategic flexibility, where flexibility is defined as knowledge of multiple strategies for solving a problem and the ability to implement an innovative strategy for a given problem solving circumstance. However, anecdotal findings from this literature indicate that students do not consistently use an innovative strategy for solving a given problem, even when these same students demonstrate knowledge of innovative strategies. This distinction, sometimes framed in the psychological literature as competence vs. performance-has not been previously studied for flexibility. In order to explore the competence/performance distinction in flexibility, this study developed and validated measures for potential flexibility (e.g., competence, or knowledge of multiple strategies) and practical flexibility (e.g., performance, use of innovative strategies) for solving equations. The measures were administrated to a sample of 158 Chinese middle school students through a Tri-Phase Flexibility Assessment, in which the students were asked to solve each equation, generate additional strategies, and evaluate own multiple strategies. Confirmatory factor analysis supported a two-factor model of potential and practical flexibility. Satisfactory internal consistency was found for the measures. Additional validity evidence included the significant association with flexibility measured with the previous method. Potential flexibility and practical flexibility were found to be distinct but related. The theoretical and practical implications of the concepts and their measures of potential flexibility and practical flexibility are discussed.

The Effects of Schema-Based Instruction on the Proportional Thinking of Students With Mathematics Difficulties With and Without Reading Difficulties.

This study examined the effect of schema-based instruction (SBI) on the proportional problem-solving performance of students with mathematics difficulties only (MD) and students with mathematics and reading difficulties (MDRD). Specifically, we examined the responsiveness of 260 seventh grade students identified as MD or MDRD to a 6-week treatment (SBI) on measures of proportional problem solving. Results indicated that students in the SBI condition significantly outperformed students in the control condition on a measure of proportional problem solving administered at posttest (g = 0.40) and again 6 weeks later (g = 0.42). The interaction between treatment group and students' difficulty status was not significant, which indicates that SBI was equally effective for both students with MD and those with MDRD. Further analyses revealed that SBI was particularly effective at improving students' performance on items related to percents. Finally, students with MD significantly outperformed students with MDRD on all measures of proportional problem solving. These findings suggest that interventions designed to include effective instructional features (e.g., SBI) promote student understanding of mathematical ideas.

Developing procedural flexibility: are novices prepared to learn from comparing procedures?

BACKGROUND: A key learning outcome in problem-solving domains is the development of procedural flexibility, where learners know multiple procedures and use them appropriately to solve a range of problems (e.g., Verschaffel, Luwel, Torbeyns, & Van Dooren, 2009). However, students often fail to become flexible problem solvers in mathematics. To support flexibility, teaching standards in many countries recommend that students be exposed to multiple procedures early in instruction and be encouraged to compare them. AIMS: We experimentally evaluated this recommended instructional practice for supporting procedural flexibility during a classroom lesson, relative to two alternative conditions. The alternatives reflected the common instructional practice of delayed exposure to multiple procedures, either with or without comparison of procedures. SAMPLE: Grade 8 students from two public schools (N= 198) were randomly assigned to condition. Students had not received prior instruction on multi-step equation solving, which was the topic of our lessons. METHOD: Students learned about multi-step equation solving under one of three conditions in math class for about 3 hr. They also completed a pre-test, post-test, and 1-month-retention test on their procedural knowledge, procedural flexibility, and conceptual knowledge of equation solving. RESULTS: Novices who compared procedures immediately were more flexible problem solvers than those who did not, even on a 1-month retention test. Although condition had limited direct impact on conceptual and procedural knowledge, greater flexibility was associated with greater knowledge of both types. CONCLUSIONS: Comparing procedures can support flexibility in novices and early introduction to multiple procedures may be one important reason.

The Cognitive Predictors of Computational Skill with Whole versus Rational Numbers: An Exploratory Study.

The purpose of the present study was to explore the 3(rd)-grade cognitive predictors of 5th-grade computational skill with rational numbers and how those are similar to and different from the cognitive predictors of whole-number computational skill. Students (n = 688) were assessed on incoming whole-number calculation skill, language, nonverbal reasoning, concept formation, processing speed, and working memory in the fall of 3(rd) grade. Students were followed longitudinally and assessed on calculation skill with whole numbers and with rational numbers in the spring of 5(th) grade. The unique predictors of skill with whole-number computation were incoming whole-number calculation skill, nonverbal reasoning, concept formation, and working memory (numerical executive control). In addition to these cognitive abilities, language emerged as a unique predictor of rational-number computational skill.

Relations among conceptual knowledge, procedural knowledge, and procedural flexibility in two samples differing in prior knowledge.

Competence in many domains rests on children developing conceptual and procedural knowledge, as well as procedural flexibility. However, research on the developmental relations between these different types of knowledge has yielded unclear results, in part because little attention has been paid to the validity of the measures or to the effects of prior knowledge on the relations. To overcome these problems, we modeled the three constructs in the domain of equation solving as latent factors and tested (a) whether the predictive relations between conceptual and procedural knowledge were bidirectional, (b) whether these interrelations were moderated by prior knowledge, and (c) how both constructs contributed to procedural flexibility. We analyzed data from 2 measurement points each from two samples (Ns = 228 and 304) of middle school students who differed in prior knowledge. Conceptual and procedural knowledge had stable bidirectional relations that were not moderated by prior knowledge. Both kinds of knowledge contributed independently to procedural flexibility. The results demonstrate how changes in complex knowledge structures contribute to competence development.

It pays to compare: an experimental study on computational estimation.

Comparing and contrasting examples is a core cognitive process that supports learning in children and adults across a variety of topics. In this experimental study, we evaluated the benefits of supporting comparison in a classroom context for children learning about computational estimation. Fifth- and sixth-grade students (N=157) learned about estimation either by comparing alternative solution strategies or by reflecting on the strategies one at a time. At posttest and retention test, students who compared were more flexible problem solvers on a variety of measures. Comparison also supported greater conceptual knowledge, but only for students who already knew some estimation strategies. These findings indicate that comparison is an effective learning and instructional practice in a domain with multiple acceptable answers.