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Free play, play controlled by the player(s), is an essential and positive determinant of children’s social, physical, and emotional health. Ensuring opportunities for dynamic free play in rich phys...

Tablet computers (tablets) are positioned to be powerful, innovative, effective, and motivating research and assessment tools. We addressed two questions critical for evaluating the appropriateness of using tablets to study number-line estimation, a skill associated with math achievement and argued to be central to numerical cognition. First, is performance with paper and pencil comparable with performance on a tablet? Second, is comparability affected by students’ preference for one method of presentation? Thirty-two students in Grade 6 estimated targets on a number line; half estimated with paper and pencil and half with a tablet. For both presentation methods, students’ performance was comparable. Students liked both presentation conditions equally but, when asked to choose, most students preferred the tablet. Preference did not influence comparability of results across presentation methods. Finally, students’ reasons for their preferences were explored, along with implications for using tablet applications in research and educational assessment.

The principle of inversion, that a + b − b must equal a, is a fundamental property of arithmetic, but many children fail to apply it in symbolic contexts through 10 years of age. We explore three hypotheses relating to the use of inversion that stem from a model proposed by Siegler and Araya (2005). Hypothesis 1 is that greater calculational skill is related to greater use of inversion. Hypothesis 2 is that greater attentional skill is related to greater use of inversion. Hypothesis 3 is that the relation between attentional skill and the use of inversion is particularly strong among children with high skill in calculation. We found suggestive evidence for Hypothesis 2 and clear evidence for Hypothesis 3, indicating that for children who are strong at calculation, attentional flexibility is related to use of inversion.

Background Increasing access to health and social services through service-integration approaches may provide a direct and sustainable way to improve health and social outcomes in low-income families. Methods We did a community-based randomized trial evaluating the effects of two service-integration practices (healthy family lifestyle and recreational activities for children) among low-income families in Alberta, Canada. These two practices in combination formed four groups: Self-Directed (no intervention), Family Healthy Lifestyle, Family Recreation, and Comprehensive (Family Healthy Lifestyle plus Family Recreation programs). The primary outcome was the total number of service linkages. Results We randomized 1168 families, 50 % of which were retained through the last follow-up visit. The number of service linkages for all three intervention groups was not significantly different from the number of linkages in the Self-Directed group (Comprehensive 1.15 (95 % CI 0.98–1.35), Family Healthy Lifestyle 1.17 (0.99–1.38), and Family Recreation 1.12 (0.95–1.32) rate ratios). However, when we explored the number of linkages by the categories of linkages, we found significantly more healthcare service linkages in the Comprehensive group compared to the Self-Directed group (1.27 (1.06–1.51)) and significantly more linkages with child-development services in the Family Healthy Lifestyle group compared to the Self-Directed group (3.27 (1.59-6.74)). The monthly hours of direct intervention was much lower than the assigned number of hours (ranging from 5 to 32 % of the assigned hours). Conclusions Our findings are relevant to two challenges faced by policymakers and funders. First, if funds are to be expended on service-integration approaches, then, given the lack of intervention fidelity found in this study, policymakers need to insist, and therefore fund a) a well-described practice, b) auditing of that practice, c) retention of family participants, and d) examination of family use and outcomes. Second, if child-development services are widely required and are difficult for low-income families to access, then current policy needs to be examined. Trial registration ClinicalTrials.gov, NCT00705328. Registered on 24 June 2008. Electronic supplementary material The online version of this article (doi:10.1186/s13063-016-1444-8) contains supplementary material, which is available to authorized users.

A model of the relations among cognitive precursors, early numeracy skill, and mathematical outcomes was tested for 182 children from 4.5 to 7.5 years of age. The model integrates research from neuroimaging, clinical populations, and normal development in children and adults. It includes 3 precursor pathways: quantitative, linguistic, and spatial attention. These pathways (a) contributed independently to early numeracy skills during preschool and kindergarten and (b) related differentially to performance on a variety of mathematical outcomes 2 years later. The success of the model in accounting for performance highlights the need to understand the fundamental underlying skills that contribute to diverse forms of mathematical competence.

The goal of this study was to examine the relation between mathematics and working memory in young children with Fetal Alcohol Spectrum Disorders (FASD). Children with FASD and comparison children (4 to 6 years old) completed standardized tests of mathematics and working memory. Children with FASD showed impairments on mathematics (applied problems and quantitative concepts) as compared to comparison children. The FASD group also performed less well than the comparison group on measures of phonological working memory. Math performance was highly correlated with phonological working memory in both groups and with central executive working memory only in the FASD group. The mathematics deficits in children with FASD appear to be associated with underlying working memory impairments.

— Fetal alcohol spectrum disorders (FASD) may cause a range of physical, mental, behavioral, and learning problems in children whose mothers consumed alcohol during pregnancy. One area of particular weakness for these children is mathematics, with deficits apparent even in studies that control for many potentially confounding variables and IQ. Deficits related to FASD are evident on many different components of mathematics, from standardized achievement measures to specific mathematics tests, and occur among young children, adolescents, and even adults. Children with FASD tend to have more difficulty with mathematics than with other cognitive areas, and mathematics is most highly correlated with the amount of prenatal alcohol exposure. To determine why math is such a significant area of weakness for children with FASD, research must explore how these children perform on different components of mathematics, the brain areas implicated in their mathematics deficits, the cognitive factors that underlie these deficits, and the profile of strengths and weaknesses of these children. Research also is needed on mathematical skills of preschool children with FASD and how math abilities develop in later years. Finally, studies are imperative on the efficacy of interventions designed to improve the math skills of children with FASD.

The goal of this project was to examine the profile of executive function (EF) deficits and age-related differences among children with FASD. Twenty-nine children with FASD (8 to 16 years of age) completed 8 tests from the Delis-Kaplan Executive Function System (D-KEFS). They had difficulty on many components of EF including cognitive flexibility, inhibition, some measures of verbal fluency, abstract thinking, deductive reasoning, hypothesis testing, problem solving, and concept formation. A distinctive profile emerged with performance being poorest on the card sorting test and relatively high on category fluency, design fluency, and the tower test, indicating relative strengths on some visual-spatial EF tasks. Older children with FASD showed more difficulty (relative to the norm) on some verbal tests of EF than younger children with FASD, suggesting that difficulty on some verbal EF tasks appears to become more pronounced with increasing age.

Children’s numerical competence in kindergarten is highly predictive of their acquisition of mathematics in Grade 1 and Grade 2, suggesting that experiences at home before schooling are important in understanding how numeracy develops. In this study, the mathematical skills of 146 children in Kindergarten, Grade 1, and Grade 2 were correlated with the frequency with which parents reported informal activities that have quantitative components such as board and card games, shopping, or cooking. Effect sizes were consistent with research relating home literacy experiences to children’s vocabulary. The present research supports claims about the importance of home experiences in children’s acquisition of mathematics.

The authors explored the relationship between measures of self-belief, behavioural development, and academic achievement in Canadian Aboriginal children. Standardized measures of intelligence are unable to consistently predict academic achievement in students from indigenous populations. Exploring alternative factors that may be both predictive and amenable to improvements with interventions is therefore important in order to address the growing educational disparity in Canadian Aboriginal children. In this study, significant correlations were found between the Self-Perception Profile for Children rating of behavioural conduct and close friendships, the Behavior Assessment Scales for Children ratings of leadership and study skills, and the Wechsler Individual Achievement Test measures of academic achievement. A school environment that provides opportunities for developing social skills and creating friendships as well as culturally appropriate interventions that support the development of leadership and study skills may provide Canadian Aboriginal children with the tools they need to achieve academically.

The principle of inversion, that a + b − b must equal a, is a fundamental property of conventional arithmetic. Exploring how children use and understand the principle of inversion can provide important insights about the development of mathematical thinking and about ways of optimizing instruction. Research on children's use and understanding of inversion has been focused primarily on whether they use inversion, with much less attention placed on what this understanding comprises and how it develops. To remedy this situation, we propose a framework in which understanding inversion is represented in terms of a matrix of possibilities. This framework is useful for highlighting the diverse ways in which children can show their understanding, for describing individual differences, for tracking changes in understanding, and for prompting investigations on the mechanisms that contribute to conceptual development.

Children's failure on equivalence problems (e.g., 5 + 4 = 7 + _) is believed to be the result of misunderstanding the equal sign and has been tested using symbolic problems (including " = "). For Study I (N = 48), we designed a nonsymbolic method for presenting equivalence problems to· determine whether Grade 2 children's difficulty is due to the presence of symbols or to a more fundamental misunderstanding of equivalence. Children's superior performance on nonsymbolic versus symbolic problems suggests that children fail to map their understanding of equivalence onto problems presented with the symbols of arithmetic. For Study 2 (N = 32), we implemented a within-subject design to assess whether experience with nonsymbolic problems would facilitate performance on symbolic problems. This hypothesis was confirmed. Exposure to nonsymbolic problems may have enabled children to map their successful concepts and strategies to symbolic equivalence problems

Knowledge and skill in multiplication were investigated for late elementary-grade students with mathematics learning disabilities (MLD), typically achieving age-matched peers, low-achieving age-matched peers, and ability-matched peers by examining multiple measures of computational skill, working memory, and conceptual knowledge. Poor multiplication fact mastery and calculation fluency and general working memory discriminated children with MLD from typically achieving age-matched peers. Furthermore, children with MLD were slower in executing backup procedures than typically achieving age-matched peers. The performance of children with MLD on multiple measures of multiplication skill and knowledge was most similar to that of ability-matched younger children. MLD may be due to difficulties in computational skills and working memory. Implications for the diagnosis and remediation of MLD are discussed.

The principle of inversion—that a + b − b must equal a—requires a sensitivity to the relation between addition and subtraction that is critical for understanding arithmetic. Use of inversion, albeit inconsistent, has been observed in school-age children, but when use of a computational shortcut based on inversion emerges and how awareness of the inversion principle relates to other mathematical or numerical skills remain unclear. Two possibilities were explored in 3-year-olds by adapting a method used previously with older children involving the addition and subtraction of blocks differing in length. These children were significantly more accurate on inversion than standard problems, this difference was observed even in children who did not count well, and performance did not differ between formats that afforded qualitative or quantitative solutions. Thus, 3-year-olds appear to develop an early sensitivity to quantitative inversion.

This study assessed the feasibility of a randomized controlled trial to evaluate the Big Brothers Big Sisters (BBBS) community match programs at the national level. Following a standardized protocol, 71 intake families (parents and children ages 7–14 years) and 30 adult mentors were recruited over 12 months from two BBBS agencies. Families were randomly assigned to the BBBS program ( n = 39) or a waiting list control ( n = 32). Information on children's behavior and psychosocial outcomes was obtained from children and parents using face-to-face interviews and questionnaires administered after enrolling in the study (but prior to knowledge of assigned condition) and at 12 months follow-up. Tests of equivalence revealed non-significant group differences on most baseline outcomes. Twelve families dropped out prior to the post-test assessment. ANCOVA results revealed beneficial program effects for five outcomes (child self-reports): symptoms of emotional problems, symptoms of social anxiety (fear of negative peer evaluations and generalized social anxiety and distress), teacher social support, and social skills (self-control). The study demonstrated the feasibility of an RCT to evaluate the BBBS programs. However, the design may be challenged ethically given the evidence of positive benefits and the requirement that control group children remain unmatched for a minimum of 12 months.

The development of conceptual and procedural knowledge about counting was explored for children in kindergarten, Grade 1, and Grade 2 (N = 255). Conceptual knowledge was assessed by asking children to make judgments about three types of counts modeled by an animated frog: standard (correct) left-to-right counts, incorrect counts, and unusual counts. On incorrect counts, the frog violated the word-object correspondence principle. On unusual counts, the frog violated a conventional but inessential feature of counting, for example, starting in the middle of the array of objects. Procedural knowledge was assessed using speed and accuracy in counting objects. The patterns of change for procedural knowledge and conceptual knowledge were different. Counting speed and accuracy (procedural knowledge) improved with grade. In contrast, there was a curvilinear relation between conceptual knowledge and grade that was further moderated by children's numeration skills (as measured by a standardized test); the most skilled children gradually increased their acceptance of unusual counts over grade, whereas the least skilled children decreased their acceptance of these counts. These results have implications for studying conceptual and procedural knowledge about mathematics.

What is differential bundle functioning and how is this different from differential item functioning? Can test specifications be used to identify and aid in the interpretation of differential bundle functioning? How can differential bundle functioning lead to an improved understanding of why groups perform differently on achievement tests?

Progress has been made in developing statistical methods for identifying DIF items, but procedures to aid with the substantive interpretations of these items have lagged behind. To overcome this problem, Roussos and Stout (1996) proposed a multidimensionality-based DIF analysis paradigm. We illustrate and evaluate an application of this framework as it applied to the study of gender differences in mathematics. Four characteristics distinguish this study from previous research: the substantive analysis was guided by past research on the content and cognitive-related sources of gender differences in mathematics achievement, as presented in the taxonomy by Gallagher, De Lisi, Holst, McGillicuddy-De Lisi, Morely, and Cahalan (2000); the substantive analysis was conducted by reviewers who were highly knowledgeable about the cognitive strategies students use to solve math problems; three statistical methods were used to test hypotheses about gender differences, including SIBTEST, DIMTEST, and multiple linear regression; and the data were from a curriculum-based achievement test developed with the goal of minimizing obvious, content-related gender differences. We show that the framework can lead to clearly interpretable results and we highlight both the strengths and weaknesses of applying the Roussos and Stout framework to the study of group differences.

Age-related change and patterns of individual differences in children's knowledge and skill in multiplication were investigated for students in Grades 4 and 6 (approximately ages 9 and 11, respectively) by examining multiple measures of computational skill, conceptual knowledge, and working memory. Regression analyses revealed that indexes reflecting probability of retrieval and special problem characteristics overshadow other, more general indexes (problem size and frequency of presentation) in predicting solution latencies. Some improvement in the use of conceptual knowledge was evident between Grades 4 and 6, but this change was neither strong nor uniform across tasks. Finally, patterns of individual differences across tasks differed as a function of grade level. The findings have implications for understanding developmental change and individual differences in mathematical cognition.

In the current study, we adopted the Pathways to Mathematics model of LeFevre et al. (2010). In this model, there are three cognitive domains—labeled as the quantitative, linguistic, and working memory pathways—that make unique contributions to children’s mathematical development. We attempted to refine the quantitative pathway by combining children’s (N = 141 in Grades 2 and 3) subitizing, counting, and symbolic magnitude comparison skills using principal components analysis. The quantitative pathway was examined in relation to dependent numerical measures (backward counting, arithmetic fluency, calculation, and number system knowledge) and a dependent reading measure, while simultaneously accounting for linguistic and working memory skills. Analyses controlled for processing speed, parental education, and gender. We hypothesized that the quantitative, linguistic, and working memory pathways would account for unique variance in the numerical outcomes; this was the case for backward counting and arithmetic fluency. However, only the quantitative and linguistic pathways (not working memory) accounted for unique variance in calculation and number system knowledge. Not surprisingly, only the linguistic pathway accounted for unique variance in the reading measure. These findings suggest that the relative contributions of quantitative, linguistic, and working memory skills vary depending on the specific cognitive task.

The popular print media constitute a major source of new information about scientific research for the public and for members of the scientific community outside their areas of expertise. Despite the potential importance of media reports to scientific literacy and public awareness of science, little is known about the content of these articles. We sampled the popular print media (e.g., publications such as those sold at a convenience store or supermarket) and found that the majority of articles about scientific research were in the form of news briefs. We analyzed and compared (a) the content of these news briefs, (b) advice given by experts about how to read media reports about science critically, and (c) university students' requests for information as they evaluated brief reports about research. Some marked discrepancies were found. For example, much of the information that experts advised readers to attend to or that students spontaneously requested were infrequently available in news briefs. Our find...

The study of early mathematical development provides important insights into young children's emerging academic competencies and, potentially, a basis for adapting instructional methods. We presented nonverbal forms of two- and three-term arithmetic problems to 4-year-olds to determine (a) the extent to which certain informationprocessing demands make some problems more difficult than others and (b) whether preschoolers use arithmetic concepts spontaneously when solving novel problems. Children's accuracy on simple arithmetic problems (a + b and a - b) was strongly related (r^sup 2^ = .88) to representational set size, the maximum number of units that need to be held in working memory to solve a given problem. Some children also showed spontaneous use of procedures based on the arithmetic principle of inversion when solving problems of the form a + b - b. These results highlight the importance of identifying information-processing and conceptual characteristics in the early development of mathematical cognition. The study of mathematical development prior to formal schooling provides important insights into young children's emerging competencies and, potentially, a basis for adapting instructional methods for later schooling (Ginsburg, Klein, & Starkey, 1998). Much of the work on this topic has been focused on the development of early mathematical activities such as counting (e.g., Gelman & Gallistel, 1978) and quantitative comparison (e.g., Mix, 1999). Relatively little research has been devoted to preschoolers' arithmetic, despite a growing body of research on arithmetic in older children (Geary, 1994) and the fact that even infants show some sensitivity to transformations involving addition and subtraction (Simon, 1997; Wynn, 1992). Because competence in arithmetic is an important goal of early schooling, research on the development of arithmetic-related skills prior to and during early schooling is important for understanding and optimizing the transitions children undergo as academic knowledge is acquired. Part of the difficulty in studying arithmetic in preschoolers is that they tend to do very poorly when problems are presented in the conventional, symbolic format used with older children and adults. When problems are presented in a manner that is less abstract and less verbal, however, performance tends to be higher (Hughes, 1986). Levine, Jordan, and Huttenlocher (1992), for example, found that 4to 6-year-old children performed much better on problems presented using a nonverbal format than they did on problems presented verbally. Their nonverbal method consisted of several steps. First, an experimenter presented an array of objects and then asked the child to construct an identical array. Next, the experimenter covered his or her array with a cardboard box and, as the child watched, changed the quantity of the hidden array by adding or removing objects through holes in the box. Finally, the child was asked to make his or her array match the experimenter's concealed array. At no point was the child required to verbalize a quantity. Using this method, these same researchers found that social class differences among 5-year-olds on verbal arithmetic problems were eliminated when nonverbal problems were used Jordan, Huttenlocher, & Levine, 1992) and that children as young as 2.5 years calculated solutions with some success when problems were presented nonverbally (Huttenlocher, Jordan, & Levine, 1994). Thus some of the difficulty that young children have in doing arithmetic stems from the words and symbols that are often used when arithmetic problems are presented. The use of nonverbal methods of presentation may facilitate performance by helping children establish an appropriate mental model of the transformations involved in adding and subtracting (Huttenlocher et al., 1994). Even when nonverbal methods are used, however, the performance of preschoolers is far from stellar. …

An important but underrepresented element in the growing body of work on informal science education is research designed to examine the nature and scope of children's science-related activities outside of school. We have begun to study children's activities related to science, nature, and technology by developing structured interviews for parents of middle class preschool and kindergarten children and for students in upper-elementary grades. These interviews can be used to construct profiles of children's exposure to science activities outside of school, such as watching television, reading, attending exhibits or events at community facilities, participating in experiments or demonstrations at home, and asking questions of parents. We describe how these interviews were developed, what kinds of information this type of research enabled us to obtain, and what lessons we have learned in the process. The level of extracurricular participation reported in a wide range of science-related activities was very high. Structured interviews can help teachers gain information about students' exposure to science-related learning activities in their home and community. This information can be used as a platform on which classroom instruction can be built. © 1997 John Wiley & Sons, Inc. Sci Ed81:651–662, 1997.

Media reports of scientific research are a pervasive and important source of new scientific knowledge. Evaluating conclusions found in those reports is an important form of scientific literacy. We examined the types of requests for information made by university students as they evaluated scientific news briefs. Students generated a variety of requests for information, focusing most often on how the research was conducted and why the results might have occurred. Fewer requests were made for information about what was found, who conducted the research, and where it was conducted. Least frequent were requests about related research. This pattern of findings may reflect emphasis of instruction in science classrooms. Requests appeared to be influenced by three dimensions of news briefs: plausibility of the conclusions; typicality (the degree to which the phenomena described are typical in biology, chemistry, and physics); and personal familiarity with the phenomena. Individual differences in age, number of science courses completed in university, and degree of belief in paranormal events were correlated with the types of requests made. Models of the evaluative process are needed to gain additional insights into how students' knowledge, dimensions of text, and personal characteristics interact in the course of learning. © 1997 John Wiley & Sons, Inc.

The Extension of the Interference Effect to Multiplication MIKE H. THIBODEAU, University of Alberta JO - ANNE LEFEVRE, Carleton University JEFFREY BISANZ, University of Alberta Abstract Using multiplication facts, this experiment demonstrated an interference effect in the number - matching task. Here, subjects verified the presence of a target number (e.g., 8) in a previously presented cue (e.g., 5 x 8) that was masked after 60 ms. The SOAs between cue and target were 100, 120, 220, and 350 ms. Subjects were slower to reject targets that were the product of the cue (e.g., 40) than to reject unrelated targets (e.g., 42), but this was true only at the 100 - and 120 - ms SOAs (i.e., the interference effect). This pattern is consistent with the interference effect found by LeFevre and colleagues using addition facts. Furthermore, the present result supports the interpretation that the interference effect previously found with addition facts was due to obligatory activation and not to automatic counting. LeFevre and colleagues (LeFevre, Bisanz, & Mrkonjic, 1988; LeFevre & Kulak, 1994; LeFevre, Kulak, & Bisanz, 1991) investigated the obligatory activation of addition facts by employing the number - matching task. In the number - matching task, subjects were shown pairs of numbers (e.g., 4 + 5). After a variable stimulus onset asynchrony (SOA), the pair of numbers was replaced by a target number (e.g., 4). Subjects were required to decide whether the target number was one of the numbers in the initial pair. Upon presentation of the number pair, LeFevre and colleagues hypothesized that subjects activate a set of possible targets that includes the actual numbers presented (e.g., 4, 5), as well as associated numbers (e.g., 6, 9). LeFevre et al. found that subjects were slower to respond "no" when the target was the sum of the pair (e.g., 4 + 5 and 9) than when the target was unrelated to the pair (e.g., 4 + 5 and 7). Activation of associated numbers, such as the sum, presumably makes the decision about the target more difficult, slowing response times. Such interference for sum targets occurred only at relatively short SOAs. At longer SOAs (greater than 400 ms, Neely, 1991), activation of potentially interfering information declined or was suppressed. Because successful completion of the number - matching task does not require mental arithmetic, LeFevre et al.(1988; 1991; LeFevre & Kulak, 1994) concluded that the interference effect on the sum trials can be attributed to obligatory activation, in that it occurs without intention (Posner & Snyder, 1975). However, the results are open to an alternative explanation. It could be argued that latencies for sum trials were significantly slower than latencies for unrelated trials as a result of an automatically - activated counting procedure rather than obligatory activation of the addition - fact network (Baroody, 1994). Thus, presentation of 5 + 2 might result in obligatory activation of a counting procedure, such that 7 is activated via counting 5, 6, 7 (see LeFevre, Sadesky, & Bisanz, 1996). LeFevre et al. (1988; 1991; LeFevre & Kulak, 1994) had a majority of trials with addends of 1 or 2 in their number - matching experiments, because they restricted their stimuli to problems with sums of 9 or less. In this view, obligatory activation of addition facts with small addends may reflect automatic counting rather than obligatory spreading activation in a number - fact network. Even if LeFevre et al.'s (1988; 1991; LeFevre & Kulak, 1994) results are due in part to activation of a counting procedure, results reported by Lemaire, Barrett, Fayol, and Abdi (1994, Experiment 1) suggest that obligatory activation also occurs on addition facts with addends larger than 1 or 2. Lemaire et al. used three types of cue - target relations in the number matching task: small - addend (e.g., 1 3 followed by 4), medium - addend (e. …

Researchers have assumed that adults solve simple arithmetic problems by retrieving answers from a network of stored facts. In 2 studies, undergraduates described their solutions of single-digit multiplication problems. They reported direct retrieval on approximately 80% of trials, but also reported rules (e.g., anything times 0 is 0), repeated addition (e.g., 2 × 4 = 4 + 4), number series (e.g., 3 × 5 = 5, 10, 15), and derived facts (e.g., 6 × 7 = [6 × 6] + 6). Participants were slower to retrieve problems that were most likely to be solved by nonretrieval procedures and faster to retrieve problems that were usually solved by retrieval. These results indicate that direct-retrieval models are incomplete accounts of adults' performance and support a continuing influence of learning and experience on the mental representation of simple multiplication problems.

Adults' solution times to simple addition problems typically increase with the sum of the problems (the problem size effect). Models of the solution process are based on the assumption that adults always directly retrieve answers to problems from an associative network. Accordingly, attempts to explain the problem size effect have focused either on structural explanations that relate latencies to numerical indices (e.g., the area of a tabular representation) or on explanations that are based on frequency of presentation or amount of practice. In this study, the authors have shown that the problem size effect in simple addition is mainly due to participants' selection of nonretrieval procedures on larger problems (i.e., problems with sums greater than 10). The implications of these results for extant models of addition performance are discussed. Twenty years of research on mental arithmetic has shown that problems involving larger numbers (e.g., 9 + 6) are solved more slowly than problems involving smaller numbers (e.g., 3 + 4). Surprisingly, in spite of the wealth of empirical data and the extensive theoretical development on mental arithmetic, the problem size effect has eluded satisfactory explanation (Ashcraft, 1992; McCloskey, Harley, & Sokol, 1991; Widaman & Little, 1992). The goal of the present research was to test an explanation of the problem size effect in adults that has been used to account for the arithmetic performance of children (Ashcraft, 1992; Siegler, 1987). We hypothesized that variability in the selection of procedures to solve simple addition problems has a major impact on solution latencies and may account for a substantial portion of the problem size effect.

The influence of school- and age-related variables was examined separately on 2 tasks involving elementary quantitative skills: conservation of number and mental addition. Performance on these tasks was compared by using a cutoff design with 3 groups of kindergarten and Grade 1 children who differed in age but not amount of schooling (grade), in schooling but not age, or in both age and schooling. The effects of age and schooling were distinct. On conservation of number, performance improved as a function of age but not schooling. On mental arithmetic, accuracy improved with schooling rather than age, but children's use of various solution procedures (e.g., retrieval, counting) was not influenced by schooling. Thus, in the domain of elementary mathematical skills, the influence of schooling can be very specific, and age-related variables other than schooling play an important role in the development of elementary mathematical skills. Results illustrate the utility of the cutoff design for investigating instructional and developmental influences on cognitive development

Little is known about the development of mathematics anxiety in elementary school students. To address this gap in knowledge, the authors evaluated students in Grades 3 and 6 on measures of mathematics anxiety, school test anxiety, and attitudes toward mathematics to determine (a) whether different forms of mathematics anxiety exist, (b) whether mathematics test anxiety differs from school test anxiety, and (c) whether mathematics anxiety is related to different attitudes toward mathematics. Evidence was found for two distinct forms of mathematics anxiety: test and problem-solving anxiety. Mathematics test anxiety increased with age relative to mathematics problem-solving anxiety; this result demonstrated that children become more anxious about mathematics testing situations as they progress through school. Mathematics test anxiety was related, but not identical, to school test anxiety, and students in both grades were less anxious about math tests than about academic testing generally. Finally, ...

We examined the role of executive attention, which encompasses the common aspects of executive function and executive working memory, in children's acquisition of two aspects of mathematical skill: (a) knowledge of the number system (e.g., place value) and of arithmetic procedures (e.g., multi-digit addition) and (b) arithmetic fluency (i.e., speed of solutions to simple equations such as 3+4 and 8-5). Children in Grades 2 and 3 (N=157) completed executive attention and mathematical tasks. They repeated the mathematical tasks 1 year later. We used structural equation modeling to examine the relations between executive attention and (a) concurrent measures of mathematical knowledge and arithmetic fluency and (b) growth in performance on these measures 1 year later. Executive attention was concurrently predictive of both knowledge and fluency but predicted growth in performance only for fluency. A composite language measure predicted growth in knowledge from Grade 2 to Grade 3. The results support an important role for executive attention in children's acquisition of novel procedures and the development of automatic access to arithmetic facts.

Most children who are older than 6 years of age apply essential counting principles when they enumerate a set of objects. Essential principles include (a) one-to-one correspondence between items and count words, (b) stable order of the count words, and (c) cardinality-that the last number refers to numerosity. We found that the acquisition of a fourth principle, that the order in which items are counted is irrelevant, follows a different trajectory. The majority of 5- to 11-year-olds indicated that the order in which objects were counted was relevant, favoring a left-to-right, top-to-bottom order of counting. Only some 10- and 11-year-olds applied the principle of order irrelevance, and this knowledge was unrelated to their numeration skill. We conclude that the order irrelevance principle might not play an important role in the development of children's conceptual knowledge of counting.

Chinese-speaking children have been shown to have an advantage over English-speaking children in a variety of mathematical areas, including counting. One possible explanation for the advantage in counting is that the Chinese number-naming system is relatively transparent, compared to English, in that number names typically are directly indicative of base-10 structure (e.g., 12 is named "ten-two" rather than "twelve"). To determine whether the transparency of the Chinese number-naming system influences counting in bilingual children, we tested 25 ChineseEnglish bilingual children between the ages of 3 and 5 years, both in English and in Chinese. Children were asked to count as high as they could (abstract counting) and also to count objects in small, medium, and large arrays (object counting). No evidence was found for transparency or for transfer from one language to the other. Instead, relative proficiency in the two languages influenced counting skill. These results are discussed in terms of linguistic and cultural variables that might account for cross-linguistic differences in counting. Resume Nous savons que les enfants d'expression chinoise ont un avantage sur les enfants d'expression anglaise dans divers domaines des mathematiques, y compris l'aptitude a compter. L'avantage des enfants d'expression chinoise peut s'expliquer par le fait que le systeme chinois de denomination des nombres est plus ou moins transparent, comparativement au systeme anglais, en ce sens que, generalement, le nom des nombres se rapporte directement a la structure en base 10 (par ex., le chiffre 12 se nomme « dix-deux » et non pas « douze »). Afin d'etablir si la transparence du systeme chinois de denomination des nombres a un effet sur l'aptitude a compter chez les enfants bilingues, nous avons teste, en utilisant l'anglais et le chinois, 25 enfants bilingues (chinois-anglais) de 3 a 5 ans. Les enfants etaient invites a compter le plus loin possible (numeration abstraite) et a compter les objets appartenant a des petits, des moyens et des grands ensembles (comptage d'objets). Nous n'avons observe aucun element probant mettant en evidence la transparence ou le transfert d'une langue a l'autre. Au contraire, nous avons constate que la maitrise relative des deux langues agissait sur l'aptitude a compter. Nous analysons les resultats a la lumiere des variables linguistiques et culturelles qui peuvent prendre en compte les differences interlinguistiques touchant l'aptitude a compter. Chinese-speaking children have been shown to have an advantage over English-speaking children in a variety of mathematical areas (Geary, 1996; Miller, Major, Shu, & Zhang, 2000; Miller, Smith, Zhu, & Zhang, 1995; Miller & Stigler, 1987; Muira, Okamoto, Vlahovic-Stetic, Kim, & Hann, 1999). For example, Miller et al. (1995) found better performance in Chinese-speaking than English-speaking 3-, 4-, and 5-year-old children on two counting tasks, abstract counting and object counting. In abstract counting children counted aloud from 1, and in object counting they counted the number of items in small (3-6 items), medium (7-10 items), and large (14-17 items) arrays. Chinese children at all three age levels were more accurate on large sets in the object counting task, and were able to count higher in the abstract counting task than English-speaking American children. Miller et al. (1995) suggested that this advantage is due at least in part to the Chinese number-naming system, which is more transparent than the English number-naming system. A comparison of Chinese and English number names illustrates the linguistic transparency of number names, particularly for the teen quantities in Chinese (Table 1). For example, in Chinese, whereas "ten-two" is clearly indicative of the quantitative composition of 12, in English, "twelve" is less so. Miller et al. argued that this difference in linguistic transparency may be a basis for higher performance in number-related skills in Chinese children. …

An important issue in the development of mathematical cognition is the extent to which children use and understand fundamental mathematical concepts. We examined whether young children successfully use the principle of inversion and, if so, whether they do so based on qualitative identity, length, or quantity. Twenty-four preschool children and 24 children in Grade 1 were presented with three-term inversion problems (e.g., 3+2−2) and standard problems of similar magnitude (e.g., 2+4−3). Problems were presented in three conditions to determine whether children used inversion at all and, if so, whether their decisions were based on quantitative or nonquantitative features of the problems. Both preschool and Grade 1 children showed evidence of using inversion in a fully quantitative manner, indicating that this principle is available in some form prior to extensive formal instruction in arithmetic.

Publisher Summary The ultimate measure of any particular approach is its utility in advancing theory, research, or application. This chapter describes induction as a central characteristic of cognition and knowledge acquisition, and a thorough explanation of its intricacies should be high on the list of goals of cognitive science. The cognitive-components approach has contributed to the development of precise, process-oriented theories, but typically it is narrow in scope and has failed to address the issue of constraints seriously. The pragmatic approach has expanded the scope of exploration considerably and has helped to identify the kinds of knowledge-based and cognitive constraints that are so critical for understanding how induction operates. With the exception of some recent attempts at computer simulation, however, it has been less successful at generating precise models of inductive reasoning. In principle, the two approaches are complementary and potentially synergistic. However, the vitality of research on inductive reasoning depends, in part, on how well the strengths of each approach can be integrated and on the degree to which research addresses the theoretical issues of cognitive science and the practical issues of human affairs.

Summary Understanding often is defined inconsistently, ambiguously, or narrowly, and consequently the relation between understanding and cognitive processing on mathematical tasks is not very clear. Many forms of behavior are related to understanding, and a framework is needed to describe these various forms in an integrated way. A “contextual space” is proposed for classifying different types of performance related to understanding elementary mathematical concepts. The two dimensions of this space are the type of activity involved (applying, justifying, and evaluating solution procedures) and the degree of generality with which these activities are exercised. These two aspects can be used to construct a “profile” that reflects the contexts in which an individual shows various forms of understanding. Acquisition of understanding can be described in terms of the sequences of profiles that emerge, and these sequences have implications for characterizing the mechanisms underlying changes in knowledge.

Mental arithmetic involves an intriguing interaction between (a) memory and problem - solving processes that are domain independent and (b) solution procedures and factual knowledge that are specific to the domain of arithmetic. Relations between domain - independent and domain - specific knowledge are of central concern to many developmental, cognitive, and instructional psychologists, and consequently the performance of children and adults on mental arithmetic tasks have come under increasing scrutiny. Despite this common interest, in recent years the focus of research on adults has diverged somewhat from that of research on children. Young children use a variety of solution methods, including retrieval, counting (on fingers or a mental number line), and transformation (e.g., converting 5 + 6 to 5 + 5 + 1 to take advantage of knowledge that 5 + 5 = 10). Consequently, recent research with children has been concentrated on the selection process, the process by which one of several available procedures is selected for use on specific problems (e.g., Siegler, 1987; Siegler & Jenkins, 1989). In contrast, it is commonly assumed that adults always or nearly always use fact retrieval to answer simple arithmetic problems. Thus research with adults has been focussed primarily on processes and knowledge structures involved in the retrieval of arithmetic facts, and the selection process has been virtually ignored. Two studies help to highlight the value of focussing on the selection process as a common and dynamic feature in the performance of children and adults.In one study, J. LeFevre, G. Sadesky, and I tested the common assumption that adults only use retrieval to solve simple addition problems. We asked adults to solve 100 addition problems with single - digit augends and addends. Following a method used in research with children (Siegler, 1987), adults also reported the solution procedure they used after each problem. As expected, retrieval was reported most frequently. Contrary to the common assumption, however, adults often used counting - based and transformation procedures, and the frequency with which these alternative procedures were used varied systematically as a function of problem characteristics, such as the magnitude of the augend and addend. Analyses of latencies confirmed the interpretation of self - report data. The prevalence of solution procedures other thanretrieval implies that adequate accounts of mental arithmetic in adults, as in children, must include the process by which diverse procedures are selected and implemented.Siegler (1988; Siegler & Shrager, 1984) has proposed a model for this selection process. …

An incidental learning procedure was used with 8-, 11-, and 19-year-olds to identify processing changes that might underlie a developmental shift in feature encoding. Presentation of each of 32 words was accompanied by a question designed to induce either acoustic or semantic encoding. Tests of recall and recognition followed an intervening task. Acoustic encoding was faster than semantic encoding at all ages, but this difference decreased with age. Semantic encoding resulted in greater recall and recognition accuracy than acoustic encoding only for adults. It is argued that decreases in the difference between acoustic and semantic encoding speeds could contribute to the encoding shift effect. The retention data are interpreted as support for the notion that the encoding shift effect may also be due, in part, to memory processes other than encoding. Changes in encoding speeds and in memory processes related to retrieval are considered as complementary aspects of memory development.

Two experiments were conducted to explore the characteristics of semantic processing in 8-, 10-, and 19-year-olds. In the first study subjects read through lists of words and responded to either a member or nonmember of a prespecified category. Semantic processing speed was found to increase with age even when motor response latencies were partialed out. The second study was designed to establish whether this increase in semantic processing speed was due to developmental changes in semantic organization and accessibility, to changes in the organization and execution of decision processes, or more simply to faster, nonsemantic encoding of words. Subjects read through lists of words and made decisions based on category membership and/or physical size attributes of the individual concepts. For 8-year-olds size and category decisions were made at approximately the same speed, whereas category information was processed more rapidly than size information for the older groups. A parallel, self-terminating model best accounted for the data at all age levels. These results are interpreted in terms of developmental changes in the organization and accessibility of semantic information, as well as invariance in certain characteristics of information processing.

Two experiments were conducted to identify the roles of three hypothesized procedures in the solution of simple number-series problems and to determine whether individual differences in these solution procedures are related to performance on a number-series subtest from a standardized test of intelligence. The three procedures are recognition of memorized series, calculation, and checking. Subjects verified whether number sequences formed rule-based series. True series included both memorized counting series (e.g., “5 10 1520”) and unfamiliar noncounting series (e.g., “14 710”). False series could not be described by simple rules. The results of Experiment 1 indicated that (1) counting series were verified more quickly than were noncounting series, and (2) partial counting information in false series facilitated rejection. In Experiment 2, reliable differences in recognition of memorized sequences and calculational efficiency were found between individuals who scored well on a standardized test of number-series completion and those who scored poorly. The results provide a basis for understanding how individual differences in knowledge influence performance on problems often used to assess inductive reasoning skill.