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While symbolic number processing is an important correlate for typical and low mathematics achievement, it remains to be determined whether children with high mathematics achievement also have excellent symbolic number processing abilities. We investigated this question in 64 children (aged 8 to 10), i.e., 32 children with persistent high achievement in mathematics (above the 90th percentile) and 32 average-achieving peers (between the 25th and 75th percentile). Children completed measures of symbolic number processing (comparison and order). We additionally investigated the roles of spatial visualization and working memory. High mathematics achievers were faster and more accurate in order processing compared to average achievers, but no differences were found in magnitude comparison. High mathematics achievers demonstrated better spatial visualization ability, while group differences in working memory were less clear. Spatial visualization ability was the only significant predictor of group membership. Our results therefore highlight the role of high spatial visualization ability in high mathematics achievement.

This study examined adults' frequent, efficient and adaptive use of direct subtraction (DS) and subtraction by addition (SBA) in mental multi-digit subtraction with the choice/no-choice method. Participants were offered subtractions in one choice condition (choice between DS and SBA) and two no-choice conditions (mandatory use of either DS or SBA). SBA was used as frequently as DS in the choice condition. DS was most accurate on subtractions with a large difference (e.g., 502 - 18), while SBA was fastest on subtractions with a small difference (e.g., 903 - 886). In general, participants were adaptive for task characteristics and their personal speed characteristics. We further analyzed task-based adaptivity on an individual level via a Latent Class Analysis. Results showed that two-thirds of the participants were adaptive to task characteristics, and that these adaptive participants were the most proficient in accuracy and speed in the choice condition. We further examined whether executive functions (updating, inhibition, shifting) were related to individual differences in strategy efficiency and task-based adaptivity. In line with our hypothesis, updating was related to strategy efficiency, such that participants with higher updating skills were more accurate. In contrast to our expectations, inhibition and shifting were not related to task-based strategy adaptivity. This study highlights adults' efficient and adaptive use of arithmetic strategies, and its association with their proficiency and executive functions.

Children start preschool with large individual differences in their early numerical abilities. Little is known about the importance of heterogeneous patterns that exist within these individual differences. A person-centered analytic approach might be helpful to unravel these patterns and the cognitive and environmental factors that are associated with them. We applied a person-centered approach to a 5-year longitudinal study (N = 410, 213 boys) conducted in Belgium from preschool to grade 3. Preschoolers (M[subscript age] = 58.14 months, SD[subscript age] = 3.51) were selected to represent the full range of socioeconomic backgrounds. We examined via Latent Profile Analysis the heterogeneous patterns that exist in preschoolers' early numerical development using measures of counting, numeral identification, comparison, ordering, and arithmetic abilities. We investigated the association between the derived numerical ability pathways, general cognitive factors (working memory, language, spatial ability) and the home math environment. We also evaluated the relation of these early numerical ability pathways to later mathematics achievement in grade 1 and 3. Four longitudinal pathways emerged: a low (15%), below-average (28%), above-average (44%), and a high numerical ability pathway (13%). Differences between the four pathways were mostly quantitative. Most of the general cognitive factors contributed to pathway membership, whereas the home math environment and socioeconomic status (SES) did not. The pathways differed in mathematics achievement in grade 1 and 3, and most of these differences remained when the covariates were considered. The results highlight the heterogeneity that is already present in preschoolers' numerical abilities and their predictive value for subsequent mathematics achievement.

Research showed that the capacity of making simple estimations begins to develop already at the age of five, but little is known about the early development of this estimation capacity and the strategies that underly it. The current study longitudinally followed the estimation capacity and strategies of 332 children from first to second grade of primary school by gathering strategy information through immediate trial-by-trial verbal reports on 14 addition items divided over six number size levels. Both in first and second grade children already used strategies that exhibited a basic understanding of computational estimation. For instance, their verbalizations suggested that they knew that a good estimate had to be close to the exact answer without necessarily coinciding with it. In addition to these rudimentary computational estimation strategies, we observed an increasing urge in those children to solve computational estimation problems by means of exact arithmetic. Implications for the earlier introduction of computational estimation in primary mathematics education are discussed.

Research Findings: Shared-picture book reading can stimulate children's mathematical development. Evidence of learning-supportive characteristics in picture books is limited in this domain. A first step is systematically analyzing the occurrence of domain-specific features in publicly available picture books. We analyzed the occurrence of general book features, features of numbers and quantitative language, and features of sets in 100 mathematical and 45 matched non-mathematical picture books written in Dutch. The results indicated variety among picture books, but more than 50% of the mathematical picture books focused on the 1-10 number range and ascending counting, and less than 20% paid attention to the counting principles and other numeracy skills. Quantitative language was included in 75% of the mathematical picture books, but infrequently across pages and with little variety in wording. Mathematical picture books with a story hardly differed from picture books without a story regarding the presence of domain-specific features. Finally, fewer non-mathematical picture books included mathematical content and generally on less pages compared to mathematical picture books. Practice or Policy: These findings stress the importance for picture book developers and teachers and parents to, respectively, design and select picture books in view of literature regarding children's mathematical development.

Recent studies demonstrated that the adult-preschooler interaction during shared book reading (SBR) contributes to its effectiveness (Mol et al., 2008). The level of abstraction, or complexity, of the mathematical questions adults formulate during SBR serves as an indicator of the interaction quality. We aimed to investigate the chance of spontaneously formulating a mathematical question and the level of abstraction of the mathematical questions future preschool teachers propose to formulate during SBR, and their association with teachers' professional knowledge and beliefs, and type of picture book. Participants were 111 future preschool teachers. We investigated their chance of formulating a mathematical question and the level of abstraction of their mathematical questions using a video-based instrument and distinguished between two types of picture books, namely, mathematical and non-mathematical picture books. We additionally assessed their (1) mathematical content knowledge, (2) mathematical pedagogical content knowledge, and (3) beliefs about mathematics in general and about the teaching and learning of mathematics, with three online questionnaires. Data were analyzed using multilevel analyses. Results revealed that mathematical picture books increase the likelihood of formulating a mathematical question and provoked more abstract mathematical questions compared to non-mathematical picture books. There were no significant associations between teachers' professional knowledge and beliefs and the dependent variables. Our findings point to the importance of adequately selecting picture books to stimulate mathematical preschoolers' development via SBR and also call for further investigations on the learning-supportive picture book characteristics and teacher characteristics.

The success or failure of education systems in promoting student problem-solving skills depends on attitudinal, political, and pedagogical variables. Among these variables, the design of mathematics textbooks is thought to partially explain why students from high-achieving countries show better problem-solving ability in international assessments. In the current study, we delved into this question and compared the frequency and characteristics of arithmetic word problems (AWPs) contained in primary school math textbooks in two countries with different levels of performance in international assessments--Singapore and Spain. In our analyses, we focused on (1) the quantity of arithmetic word problems, (2) the variety of problems in terms of their additive or multiplicative structures and semantic-mathematical substructures, and (3) the quantity and nature of illustrations that were presented together with arithmetic word problems. Although a larger proportion of AWP activities was found in Singaporean textbooks, the results showed a similar variety of AWPs in both Singaporean and Spanish math textbooks. Furthermore, in both countries, math textbooks emphasized the structures classified as (additive) combine 1 and (multiplication) simple rate in AWPs. Notably, the Singaporean textbook contained a larger percentage of illustrations that reflected the semantic-mathematical structures of the problems and helped students learn how to solve AWPs (e.g., bar models). The findings are discussed in light of theories that posit that textbooks constitute a fundamental part of the teaching-learning process in the classroom.

The present study longitudinally investigated proportional reasoning abilities in early elementary school before the start of its instruction. Three aims were put forward: (a) distinguishing the different developmental states in young children's understanding of missing-value proportional situations, (b) investigating how children transition through these states, and (c) exploring possible predictors that explain individual differences in young children's development of proportional reasoning abilities. We longitudinally investigated 5- to 8-year-olds' (n = 315) proportional reasoning abilities in a fair-sharing missing-value proportional reasoning task. First, results showed that the development of proportional reasoning already starts at a very early age and is still ongoing when children are in their third year of elementary school. Second, latent class analysis revealed five different early states of proportional reasoning. The understanding of one-to-many correspondence was identified as an essential stepping-stone toward success in more difficult proportional reasoning problems with many-to-many correspondences. Third, exploratory analyses revealed that the large individual differences in children's development of proportional reasoning abilities were associated with socioeconomic status, language, spatial abilities, and numerical abilities. Theoretical, methodological, and educational implications are discussed.

Many studies in the past decades have focused on low and typical mathematics achievers, yet little is known about children with high mathematics achievement, particularly at a young age. The current study aimed to fill this gap and started from the early work of Krutetskii (1976) as a theoretical lens to study the characteristics of high mathematics achievers in primary school. Krutetskii's work was extended with more recent research on mathematical cognition. A simultaneous investigation of mathematics-specific abilities and general motivational and cognitive factors allowed us to examine their unique contributions to high mathematics achievement. Participants were 162 children, that is, 81 high mathematics achievers and 81 average achievers (8- to 10-year-olds). Children completed measures assessing their mathematical cast of mind (attention to number), flexible mental calculation (adaptive number knowledge, strategy variety), and striving for elegance (use of varying strategies for multidigit arithmetic). We also measured children's need for cognition, spatial visualization ability, and working memory. There were significant group differences on all tasks, except for the attention to number task. A logistic regression analysis revealed that strategy variety, need for cognition, and spatial visualization ability were significant predictors of group membership. These data suggest that strategy variety, need for cognition, and spatial visualization ability might be critical characteristics of high mathematics achievers in primary school. The identification of such characteristics might be an important first step in creating supportive educational environments for these children.

Several studies have shown that children do not only erroneously use additive reasoning in proportional word problems, but also erroneously use proportional reasoning in additive word problems. Traditionally, these errors were contributed to a lack of calculation and discrimination skills. Recent research evidence puts forward an additional explanation, namely, children's relational preference (i.e., in tasks where both, additive and multiplicative reasoning, are appropriate, some children have a preference for additive relations, while others have a preference for multiplicative relations). Children's relational preference offers a unique explanation for erroneous word-problem solving, after taking into account computation and discrimination skills in 8- to 12-year-olds. However, it is still unclear whether relational preference is also associated with word-problem solving at an earlier age, before the start of formal instruction in word-problem solving. A task measuring children's relational preference as well as three additive and three proportional word problems was administered to a large group (n = 343) of 6- to 7-year-olds. Results show that relational preference is also associated with word-problem solving behavior at this young age: an additive preference is related with better performance on additive word problems but also with more erroneous additive reasoning in proportional word problems. Similarly, a multiplicative preference is related with better performance on proportional word problems but not yet with more erroneous proportional reasoning in additive word problems. The latter is possibly due to the low number of proportional errors that were made in the additive word problems at this young age. The implications of these findings for further research and educational practice are discussed.

Computational estimation is seen as an important mathematical competence. Little is known, however, about the mathematical skills that are predictive of early computational estimation development. The current study longitudinally followed a group of about 350 children at four time points: second (K2, 4-year-olds) and third grades of kindergarten (K3, 5-year-olds) and first (P1, 6-year-olds) and second (P2, 7-year-olds) grades of primary school. The computational estimation task was administered in two variants: a nonverbal variant, in which the problems were presented with manipulatives and children also answered using manipulatives was administered in K3 and P1; and a verbal variant, in which the problems were presented with Arabic numerals and children had to answer verbally was administered in P1 and P2. Furthermore, children's basic numerical skills and exact and approximate arithmetic skills were assessed in K2 and K3, respectively. Path analysis showed a positive autoregressive relationship between the verbal variants of the computational estimation task but not between the nonverbal ones. Basic numerical skills were important predictors for computational estimation at all time points. Approximate arithmetic positively contributed to nonverbal estimation, while exact arithmetic positively predicted verbal estimation. In sum, solid basic numerical and arithmetic skills support children when performing computational estimation. Future intervention research should further unravel the causal contribution of each of these basic numerical and arithmetic skills.

Children's spontaneous focusing on Arabic number symbols (SFONS) has been identified as a relevant component of their early mathematical development. This study investigated whether SFONS is a separate construct from spontaneous focusing on numerosity (SFON) and examined whether it is uniquely related to numerical abilities and mathematics achievement. Participants were 159 kindergartners (4-5-year-olds) who completed measures of SFONS, SFON, numerical abilities, mathematics achievement, spatial ability, and language ability. Confirmatory factor analysis revealed that a two-factor model representing a separate SFONS and SFON factor best fitted the data, indicating that SFONS is separate from SFON. Correlation and regression analyses showed that SFONS was associated with numerical abilities and mathematics achievement. These associations -- except for verbal counting -- remained after controlling for age, parental education, spatial ability, language ability, and SFON. These findings suggest that SFONS is a unique component of early mathematical development that deserves attention in early mathematics education.

This longitudinal cross-lagged panel study investigated the development of the structure of young children's spontaneous number focusing tendencies and their longitudinal associations with numerical abilities and mathematics achievement over the course of a 3-year period, that is, in the second and third year of kindergarten and in first grade of primary school in Flanders (Belgium). Participants included 177 children (50% boys) with a mean age of 4.35 years at the start of the study. Each school year, children completed measures of their spontaneous focusing on Arabic number symbols (SFONS), spontaneous focusing on numerosity (SFON), numerical abilities, and mathematics achievement. Confirmatory factor analyses demonstrated that SFONS and SFON were two separate, yet related, constructs across development. Cross-lagged panel models revealed that SFONS and SFON were moderately and differentially related to numerical abilities and mathematics achievement across development. Theoretical, methodological, and educational implications of these findings are discussed.

Young children's early repeating patterning abilities are important foundations for their later mathematical development. Prior studies on young children's repeating patterning abilities have been conducted exclusively in developed countries differing in economic, societal, and educational characteristics from developing countries. In this study, we analyzed the performance and the errors of 4- to 5-year-olds on three repeating patterning tasks, and we investigated the association between children's repeating patterning abilities and their early mathematical abilities in a developing country, Ecuador. A total of 60 preschoolers and 56 kindergartners completed a repeating patterning test and an early mathematics test at the end of the school year. Results first indicated rather low repeating patterning performances of both preschoolers and kindergartens, with large inter-individual differences in these performances. Kindergartners had significantly better patterning abilities than preschoolers. Second, error analyses indicated that children made different types of errors when solving the patterning tasks. The majority of errors were pattern-related errors, with preschoolers making more errors that were not pattern related than kindergarteners. Third, children's patterning abilities were positively associated with early mathematical abilities in both grades. These results complement current theoretical models of young children's mathematical development and inform educational policy and practices in this domain for developing countries.

The transition from informal to formal mathematics is an important episode in children's mathematical development. The current study investigated how young children's computational estimation performance and strategy use develops in this transitional period. The computational estimation performance of 350 children was assessed before the start of formal schooling (i.e., third grade of kindergarten) and again after the start of formal schooling (i.e., first grade of primary school) by means of a computational estimation addition task with manipulatives. Both children's answer construction and their counting behavior while constructing the answer were observed during task administration. Results showed an age-related increase in children's estimation accuracy as well as in their proportion of exact answers. Age-related changes in strategy use were also observed. Children demonstrated an increase in their counting behavior while constructing the answer, but no changes in the way the answer was constructed. In both grades, the answer was most often constructed by laying down all manipulatives immediately in one group. These results suggested that children can follow two pathways to solve the estimation problems: (1) relying on the visual representation of the addends without using counting and (2) using the verbal labels provided by the experimenters while using counting. More use of counting in first grade positively influenced children's estimation accuracy in this grade, suggesting that these children strive for more precision compared to children who do not count.

Since the early nineteenth century, western governments have expected history education to play a vital role in the formation of a national identity and the pursuit of national cohesion, by fostering shared knowledge and a shared (master)narrative of the national past. This article reports on a qualitative study that examines which narratives young adults construct about their national past, to what extent those narratives are underpinned by existing narrative templates, whether they reflect on the fact that the national past can be narrated in different ways, and to what extent they share a common reference knowledge. The study addresses the Flemish region of Belgium, a case characterized by a specific context of a nation state in decline, wherein diverse and often conflicting historical narratives coexist in popular historical culture and where the national past is almost absent from history education. A total of 107 first-year undergraduate history students were asked to write an essay on how they saw the national past. The influence of both history education and popular historical culture was reflected in the reference knowledge as well as in the (absence of) templates that students used to build their essays. Templates were not critically deconstructed, although some students nevertheless were able to discern and criticize existing "myths" in the national past.

The present study aimed to analyze the direction of the associations between repeating patterning, growing patterning, and numerical ability. Participants were 410 children who were annually assessed on their repeating patterning, growing patterning, and numerical ability, at ages 4, 5, and 6 years (i.e., spring 2017, 2018, and 2019). A cross-lagged panel model identified bidirectional associations between all three abilities from ages 4 to 5 years while taking into account spatial skills. From ages 5 to 6 years, both patterning abilities predicted later numerical ability, but the reverse was no longer true. Associations between performances on both pattern types also disappeared. Results highlight the unique associations between repeating patterning, growing patterning, and numerical ability, above spatial skills.

Research distinguishes three types of arithmetic: exact arithmetic, computational estimation and approximate arithmetic. Little is, however, known about the interrelationship among these three arithmetic skills and the general cognitive and early numeracy skills that underlie these arithmetic skills. The current study investigates this interrelationship and underlying processes in 389 kindergartners. Exact arithmetic and computational estimation skills were assessed using a manipulative task, while approximate arithmetic was measured using a computerized task. Correlational analysis showed significant relations among the three arithmetic skills. At the same time, confirmatory factor analyses showed that they are different constructs. In addition, regression analysis showed that different general cognitive and early numeracy skills are predictive of the three arithmetic skills. Since the different arithmetic skills are related but distinct constructs and because they are driven by different processes, teaching and learning each of these skills ask for a different approach.

Early detection of and relevant information on children's mathematical difficulties is important to initiate targeted teaching and intervention. This study investigated the extent to which strategy use in single-digit addition provides additional predictive information about 61 grade one children's (6-year-old) mathematical achievement 3 years later that is not available from a standardised mathematics achievement test. Four predictors available in year one (arithmetic strategy use, mathematical achievement, non-verbal reasoning skills and sex) explained 54% of the variation in grade four mathematics achievement. Arithmetic strategy use was the most important single predictor of year four mathematics achievement (R2 = 30%) and explained an additional 12% variation if added to a model comprised by the three other year one predictors. This result suggests that systematically obtained measures of how young children solve single-digit arithmetic problems might provide useful information about their foundational number knowledge, which in turn may reveal how well they achieve later in school. [ABSTRACT FROM AUTHOR]

Background: Early patterning competence has recently been identified as an important precursor of mathematical development. Whereas the focus of this research has been on children's ability regarding repeating patterns, children might also differ in their spontaneous attention to patterns. Aims: The present study aimed to explore 4- to 5-year olds' Spontaneous Focusing On Patterns (SFOP) and its association with their patterning and mathematical ability. Sample: Participants were 378 children (M[subscript age] = 4 years 10 months; 191 boys) from 17 preschools. Methods: Spontaneous Focusing On Patterns was measured with a construction task in which children had to build a tower with 15 blocks of three different colours. The constructions of the children were grouped into three categories (i.e., pattern, random, and sorting). We additionally administered tasks assessing their patterning ability, mathematical ability, spatial ability, and visuospatial working memory. Results: When building a tower, 37% of the preschoolers spontaneously created a pattern, 49% made a random construction, and 14% sorted the blocks per colour. Preschoolers who spontaneously created a pattern had better patterning and mathematical ability than children in the random group. Group differences in patterning ability and spatial skills accounted for the difference in mathematical ability. Conclusions: The current data suggest that children's spontaneous attention to patterns is an important component of their mathematical ability. Children's spontaneous pattern constructions may provide opportunities to discuss and practice patterns in preschool settings or at home, but more research is required to further analyse the role of SFOP in early mathematical development.

The present study cross-sectionally investigated proportional reasoning abilities in 5- to 9-year-old children (n = 185) before they received instruction in proportional reasoning. This study addressed two important aspects of the development of proportional reasoning that remain unclear in the current literature: (1) the age range in which it develops and (2) the influence of the nature of the quantities (discrete or continuous) on children's performance. Three proportional reasoning tasks (i.e., one with two discrete quantities, one with a discrete and a continuous quantity, and one with two continuous quantities) were used. A two-step cluster analysis was conducted on the groups of children based on qualitative differences in understanding. Six different early stages of proportional reasoning were revealed, showing differences in understanding depending on the nature of the quantities involved and which quantity was unknown. The development of proportional reasoning starts at a very early age but it is not yet fully mastered at the age of 9.

Singaporean children are the best performers on international achievement tests in mathematics (i.e. the TIMSS). Their excellent results could be due at least partly to certain characteristics of the textbooks used there (Oates 2014). Therefore, these materials could be taken as a good benchmark to describe and evaluate aspects of the textbooks of other regions that could be improved. For this reason, the word problem-solving approaches proposed by primary education math textbooks from Spain were compared with those from Singapore based on the presence of the problem-solving steps that are most characteristic of a genuine word problem-solving approach. The results show that the Singaporean textbooks include much more reasoning than the Spanish textbooks, while they include fewer problem-solving steps related to strategies and checking. We conclude that Singaporean textbooks provide better scaffolding for high-quality learning of word problem solving than Spanish textbooks.

Young children show large individual differences in their tendency to focus spontaneously on numerical aspects (e.g., numerosities or Arabic number symbols) of their everyday environment. The origins of these individual differences are unclear. Given the role of the home numeracy environment (HNE) in children's early mathematical development and the assumed link between children's spontaneous number focusing tendencies and their numerical behavior in everyday life, it is plausible that children's spontaneous focusing tendencies are related to their HNE. The present study aimed to test this hypothesis by longitudinally investigating children's spontaneous focusing on numerosity (SFON) and spontaneous focusing on Arabic number symbols (SFONS) in relation to their HNE. Participants were 128 children (4- to 5-year-olds), who were followed from the second until the third year of Flemish kindergarten. In both kindergarten years, children completed a SFON and SFONS Picture task while their parents completed a home numeracy questionnaire. Correlation analyses and structural equation modeling revealed no significant associations between children's spontaneous number focusing tendencies and their HNE, neither in second nor in third year of kindergarten. This finding suggests that children's spontaneous number focusing tendencies are not per se related to their HNE. Various possible explanations for this unexpected finding are discussed and directions for further research on this relationship are suggested.

Previous research demonstrated that some children inappropriately solve multiplicative missing-value word problems additively, while others inappropriately solve additive missing-value word problems multiplicatively. Besides lacking skills, children's preference for additive or multiplicative relations has been shown to contribute to those errors. The present research, using a mixed-method approach, investigated the nature of upper primary school children's relational preference by empirically examining characteristics of intuitions that had been postulated previously. After administering a pre-test, selected children who preferred additive or multiplicative relations further participated in one of two studies using open problems for which both types of relations were appropriate: either a reaction time study (n = 110) in which children's acceptance behavior and reaction times were measured or a semi-structured individual interview study (n = 18) in which their answers, verbalizations, and conviction scores were collected. Results of both studies revealed that relational preference was perseverant and exerted a coercive effect on children's reasoning: Children mostly considered only the preferential type of relation as an appropriate answer in open problems and rejected alternative answers. Furthermore, relational preference appeared as immediate, self-evident, and certain: Children rejected the non-preferential answer more quickly than an irrelevant distractor of comparable size, experienced difficulties in justifying why they gave their preferential answer, and were very convinced of this preferential answer. While this characterization held for both relational preferences, it was especially prominent for the multiplicative one. These results not only have implications for research on and educational practice in multiplicative and additive reasoning but also for the measurement of relational preference.

This paper provides a survey of the research on the relationship between teachers' professional knowledge and their instructional quality in the domain of mathematics education. First, an overview is provided of the way in which mathematics professional knowledge has been conceptualized. Second, the relation between mathematics teachers' professional knowledge and instructional practice is outlined, with a special focus on the mediating role of teachers' professional beliefs and teachers' situation-specific knowledge and skills. Third, we describe the model of mathematics teachers' competence as an integrative model including knowledge, beliefs, situation-specific skills and observable classroom behavior. Fourth, we describe the current research on bridging teachers' professional knowledge and instructional quality, with special attention to recent issues with respect to the validation, development, and impact of professional competence. We end with some challenges for future research.

This paper presents the second phase of a larger research program with the purpose of exploring the possible consequences of a gap between what is done in the classroom regarding mathematical word problem solving and what research shows to be effective in this particular field of study. Data from the first phase of our study on teachers' self-proclaimed practices showed that one-third of elementary teachers from the region of Quebec require their students to follow a specific sequential problem-solving method, known as the 'what I know, what I look for' method. These results led us to hypothesize that the observed gap may have an impact on students' comprehension of mathematical word problems. The use of this particular method was the foundation for us to study, in the second phase, the effect of the imposition of this sequential method on students' literal and inferential understanding of word problems. A total of 278 fourth graders (9-10 years old) solved mathematical word problems followed by a test to assess their understanding of the word problems they had just solved. The results suggest that the use of this problem solving method does not seem to improve or impair students' understanding. From a more fundamental point of view, our study led us to the conclusion that the way word problem solving is addressed in the mathematics classroom, through sequential and inflexible methods, does not help students develop their word problem solving competence.

Word problems are among the most difficult kinds of problems that mathematics learners encounter. Perhaps as a result, they have been the object of a tremendous amount research over the past 50 years. This opening article gives an overview of the research literature on word problem solving, by pointing to a number of major topics, questions, and debates that have dominated the field. After a short introduction, we begin with research that has conceived word problems primarily as problems of comprehension, and we describe the various ways in which this complex comprehension process has been conceived theoretically as well as the empirical evidence supporting different theoretical models. Next we review research that has focused on strategies for actually solving the word problem. Strengths and weaknesses of informal and formal solution strategies--at various levels of learners' mathematical development (i.e., arithmetic, algebra)--are discussed. Fourth, we address research that thinks of word problems as exercises in complex problem solving, requiring the use of cognitive strategies (heuristics) as well as metacognitive (or self-regulatory) strategies. The fifth section concerns the role of graphical representations in word problem solving. The complex and sometimes surprising results of research on representations--both self-made and externally provided ones--are summarized and discussed. As in many other domains of mathematics learning, word problem solving performance has been shown to be significantly associated with a number of general cognitive resources such as working memory capacity and inhibitory skills. Research focusing on the role of these general cognitive resources is reviewed afterwards. The seventh section discusses research that analyzes the complex relationship between (traditional) word problems and (genuine) mathematical modeling tasks. Generally, this research points to the gap between the artificial word problems learners encounter in their mathematics lessons, on the one hand, and the authentic mathematical modeling situations with which they are confronted in real life, on the other hand. Finally, we review research on the impact of three important elements of the teaching/learning environment on the development of learners' word problem solving competence: textbooks, software, and teachers. It is shown how each of these three environmental elements may support or hinder the development of learners' word problem solving competence. With this general overview of international research on the various perspectives on this complex and fascinating kind of mathematical problem, we set the scene for the empirical contributions on word problems that appear in this special issue.

Children's spontaneous focusing on Arabic number symbols (SFONS) has been identified as a relevant component of their early mathematical development. This study investigated whether SFONS is a separate construct from spontaneous focusing on numerosity (SFON) and examined whether it is uniquely related to numerical abilities and mathematics achievement. Participants were 159 kindergartners (4-5-year-olds) who completed measures of SFONS, SFON, numerical abilities, mathematics achievement, spatial ability, and language ability. Confirmatory factor analysis revealed that a two-factor model representing a separate SFONS and SFON factor best fitted the data, indicating that SFONS is separate from SFON. Correlation and regression analyses showed that SFONS was associated with numerical abilities and mathematics achievement. These associations -- except for verbal counting -- remained after controlling for age, parental education, spatial ability, language ability, and SFON. These findings suggest that SFONS is a unique component of early mathematical development that deserves attention in early mathematics education.

Children's own spontaneous mathematical activities are crucial for their mathematical development. Mathematical thinking and learning does not only occur in explicitly mathematical situations, such as the classroom. Those children with higher tendencies to recognize and use mathematical aspects of their everyday surroundings, both within the classroom and without, appear to have an advantage in learning formal mathematical skills and knowledge. In this introduction to the special issue, we provide an overview of the existing literature on spontaneous mathematical focusing tendencies. We then provide a brief overview of the contributions to the special issue.

Adults' simple arithmetic performance is more efficient when operands are presented in Arabic digit (3 + 5) than in number word (three + five) formats. An explanation provided is that visual familiarity with digits is higher respect to number words. However, most studies have been limited to single-digit addition and multiplication problems. In this article, we examine to what extent format effects can be found in the context of arithmetic word problems, in which visual familiarity is reduced (Manuel had 3 marbles, and then Pedro gave to him 5). Participants with high and low arithmetic fluency solved addition and subtraction word problems in which operands were presented in both formats. The overall results showed an advantage for digits operands relative to words operands. In addition, the format effects were more evident for subtraction and low-skilled participants. These results were interpreted in terms of more rapid access of digits to numerical magnitude.