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Learning fractions poses a challenge for many elementary school students, including applying fraction knowledge in novel contexts. For instance, there are substantial individual differences in students' tendency of spontaneous focusing on quantitative relations (SFOR), which is related to the development of rational number knowledge. In this study, 4th grade students (N = 129) took part in a quasi-experimental study comparing an intervention condition (n = 71) aimed at improving students' multiplicative relational reasoning and fraction knowledge with a control condition (n = 58) of business as usual fraction instruction. Five lessons of intervention activities were designed to promote students ability to recognize and describe multiplicative relations in their everyday surroundings. There was an overall positive effect on the students' mathematical knowledge. Students who participated in the intervention improved their ability to recognize and describe multiplicative relations embedded in pictures representing everyday situations. There were no significant differences in the development of fraction knowledge despite replacing five traditional fraction lessons. These findings provide further evidence that researchers and educators should continue to pay attention to issues surrounding students' spontaneous mathematical focusing tendencies.

Mathematics anxiety hinders students' mathematical achievement already in primary school, but research on its effects beyond whole number knowledge is limited. The main aim of the current study is to examine how state and trait mathematics anxiety relate to performance across five tasks that are relevant for the development of mathematics in primary school, including a measure of adaptive expertise with school mathematics. These include mathematical tasks with non-symbolic quantities, whole numbers, and rational numbers. The participants were 406 primary school students attending the 5th grade (N = 188) and 6th grade (N = 218). Our results showed that state anxiety varies across task type. Furthermore, students' self-evaluated state and trait mathematics anxiety had varying negative relations with performance depending on the task type. In particular, we found that mathematics anxiety may limit students' adaptive expertise with rational numbers, even after controlling for other relevant mathematical skills. Overall, our results indicate that existing accounts on the role mathematics anxiety plays in school mathematics should expand to consider differences across task type and measures of anxiety.

This study investigated the effect of children's first formal school years on mathematical skill profiles, measured by a variety of arithmetical skills and Spontaneous Focusing On Numerosity (SFON) tasks. By using person-centered approach the aim was to investigate whether the amount of formal schooling is associated with mathematical skills in the same way for all children, or, whether the associations differ according to the children's mathematical skill profiles. Data was analyzed from 652 4-7-year-old children from four European countries with different school entrance ages. A person-centered approach with latent profile regression analyses on four-factor score variables identified six mathematical skill profiles with both qualitative and quantitative differences. The results revealed significant, but small effects of the amount of schooling on mathematical profiles when chronological age and country-specific school entrance age were controlled for. Educational implications of the findings emphasize regarding the heterogeneity in children's mathematical skill profiles and the potentially different effects of starting formal schooling across different profiles.

Background: Adaptive expertise is a highly valued outcome of mathematics curricula. One aspect of adaptive expertise with rational numbers is adaptive rational number knowledge, which refers to the ability to integrate knowledge of numerical characteristics and relations in solving novel tasks. Even among students with strong conceptual and procedural knowledge of rational numbers, there are substantial individual differences in adaptive rational number knowledge. Aims: We aimed to examine how a wide range of domain-general and mathematically specific skills and knowledge predicted different aspects of rational number knowledge, including procedural, conceptual, and adaptive rational number knowledge. Sample: 173 6th and 7th grade students from a school in the southeastern US (51% female) participated in the study. Methods: At three time points across 1.5 years, we measured students' domain-general and domain-specific skills and knowledge. We used multiple hierarchal regression analysis to examine how these predictors related to rational number knowledge at the third time point. Result: Prior knowledge of rational numbers, general mathematical calculation knowledge, and spontaneous focusing on multiplicative relations (SFOR) tendency uniquely predicted adaptive rational number knowledge, after taking into account domain-general and mathematically specific skills and knowledge. Although conceptual knowledge of rational numbers and general mathematical achievement also predicted later conceptual and procedural knowledge of rational numbers, SFOR tendency did not. Conclusion: Results suggest expanding investigations of mathematical development to also explore different features of adaptive expertise as well as spontaneous mathematical focusing tendencies.

In this quasi-experimental study we examined how the Count How Many (CHM) intervention program can support multilingual children's early numeracy and oral language skills. The program is aimed at promoting spontaneous focusing on numerosity (SFON) and early numeracy skills in 3- to 5-year-old children who attend day care. We examined the effects of the CHM intervention with existing, but unanalyzed data, of 16 multilingual children who participated in the intervention, which consists of 6 weeks of intensive training followed by a 5-month rehearsal phase. We matched two monolingual participants with each multilingual participant by age, SFON, and cardinality-related skills for each multilingual child. One of the matched children participated in the CHM intervention, while the other took part in an early literacy program, Let's Read and Talk. Children's early numeracy and oral language skills were measured at pretest, posttest, and delayed posttest. Multilingual children's SFON tendency, cardinality skills, number sequence production abilities, and story comprehension skills developed at a similar rate as monolingual participants during the follow-up period. The results suggest that it is possible to enhance SFON tendency and cardinality-related skills in multilingual children before school age. Furthermore, the time spent supporting early numeracy skills does not take away from language learning. However, more research is needed in larger populations to determine the applicability to broader segments of national and global societies.

Adaptive expertise is a highly sought after, but difficult to achieve, outcome of mathematics education. Many teaching methods appear to support the development of adaptive expertise only in a small proportion of students. Game-based learning environments may be useful for supporting adaptive expertise. Therefore, we carried out a quasi-experimental classroom intervention to examine the possibility of using a game-based learning environment to promote adaptive rational number knowledge, a potential indicator of adaptive expertise in the domain of rational numbers. The Number Line Elaboration and Exploration learning environment relied on the increasing elaboration of the number line analogy as a means for students to explore connections between multiple aspects of rational number knowledge. Our results show the game-based learning environment was successful in promoting adaptive rational number knowledge. These results provide directions for the development and examination of how learning environments may be able to support adaptive expertise. [ABSTRACT FROM AUTHOR]

Serious games for learning have received increased attention in recent years. However, empirical studies on students' gaming experiences throughout the developmental process of serious games and discussions regarding game design are missing. The aims of the present study were to analyze students' gaming experiences while playing four consecutive versions of the Number Navigation Game (NNG)--a mathematical game-based learning environment focusing on flexibility and adaptivity with whole-number arithmetic; and to provide an extensive review of the NNG developmental and design process over 3 years with focus on how and why the design decisions were made, and how those choices affected students' gaming experiences. The study employed a mixed-methods design of quantitative and qualitative research. The Game Experience Questionnaire about eight core game experience dimensions was answered by different groups of students at primary schools in Finland in three different experiments after students played four versions of the NNG from 2014 to 2016. Six semi structured interviews related to students' game experiences, preferences and game features of the latest version of NNG were conducted. Overall, results indicate that improvement in game's usability and clarity in the user interface has positive impacts on students' game experiences. Furthermore, there seems to be a clear advantage in having better aesthetics and value in improving extrinsic elements that could contribute to maintain players' enthusiasm and situational interest in serious games.

In this study we investigated the effects of two naturalistic 2- to 4-year-old children's intervention programs aimed at promoting children's Spontaneous Focusing On Numerosity (SFON) and early numerical skills. The study consisted of a quasi-experimental, pretest-posttest design with a delayed posttest and an active control group participating in the Let's Read and Talk program. All conditions had 6 weeks of intensive training followed by a 4-month rehearsal phase, when intervention activities were integrated into normal day care. The results of both numerical interventions in the whole group level show positive, small- to medium-sized long-term effects on cardinality-related skills from pretest to delayed posttest. The SFON tendency increased more from pretest to posttest in both studies but the group differences vanished in the delayed posttest. The children in the Count, how many programs developed more in SFON tendency from pretest to posttest, while the better development of SFON from pretest to posttest was significant only for the low group in the One, two, how many programs. There were no group differences in vocabulary or story comprehension skills. Educational implications suggest that combining SFON and cardinality-related skill training at day care results in developmentally effective activities for early educators and children.

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.

Spontaneous focusing on quantitative relations (SFOR) has been shown to be a strong predictor of rational number conceptual development in late primary school. The present study outlines an intervention program that examines the possibilities to enhance late primary school students' SFOR tendency. The intervention program harnessed mobile technology in order to allow students to explore and identify quantitative relations in their everyday environment, including situations outside of the classroom. A total of 38 thirteen-year-olds from two classrooms participated in the seven-week long quasi-experimental study. One classroom spent five lessons over five weeks participating in activities which involved uncovering, defining and describing multiplicative relations in their everyday surroundings. In comparison to a business-as-usual control group, results show the intervention to be successful in enhancing SFOR tendency. These results suggest that it is possible to utilize mobile technologies to enhance students' awareness of the possibilities to use quantitative relations as explicit targets of focusing and reasoning in nonexplicitly mathematical situations.

In this 3-year longitudinal study, we examined the interplay of children's motivational orientations, spontaneous focusing on numerosity (SFON) and number sequence and arithmetical skills from kindergarten (at age 6) to grade 1 (at age 7). We also examined the direct and indirect contributions of motivation and SFON to the acquisition of arithmetical skills from grades 1 to 2 (at age 8). The study involved 136 Finnish-speaking, typically developing children. Teacher ratings were used to assess the children's motivational orientations (i.e., task orientation, task avoidance orientation and social dependence orientation) in kindergarten and grade 1. Number sequence skills and SFON were evaluated in kindergarten, and arithmetical skills were assessed in kindergarten, grade 1 and grade 2. A theoretical model of the relationships among the variables was proposed, and the associations were analysed using path analysis. The results showed that task orientation, social dependence orientation and task avoidance orientation were each unique predictors of later arithmetical achievement. Furthermore, together with arithmetical skills in grade 1 and counting abilities in kindergarten, they helped to predict arithmetical achievement in grade 2. Motivational orientations did not explain the significant contributions of SFON to later arithmetical skills. The study suggests that both motivational orientations and SFON require attention and scaffolding in educational contexts, such as assessment and teacher-child interaction from early on.

A Word Problem Enrichment programme (WPE) has been found to increase student word problem solving performance when facing non-routine and application problems. However, it is unknown if the WPE has an impact on student beliefs about word problem solving and how the WPE works for students with different motivation in learning mathematics. This study investigated the impact of the WPE on student beliefs about word problem solving by using Latent Profile Analysis and Structural Equation Modelling to analyse relations among the different cognitive, motivation, and belief factors. A total of 170 fourth- and sixth-grade students from elementary schools participated. Results showed that the effects of WPE are various depending on students' initial motivation level. The impacts of the WPE on student beliefs were found only in students with a low initial motivation level, while its impacts on student problem-solving performance were found only in students with a high initial motivation level.

Background: Young children's spontaneous focusing on numerosity (SFON) as measured by experimental tasks is related to their mathematics achievement. This association is hypothetically explained by children's self-initiated practice in number recognition during everyday activities. As such, experimentally measured SFON should be associated with SFON exhibited during everyday activities and play. However, prior studies investigating this assumed association provided inconsistent findings. Aims: We aimed to address this issue by investigating the association between kindergartners' SFON as measured by two different experimental tasks and the frequency of their number-related utterances during a typical picture book reading activity. Sample: Participants were 65 4- to 6-year-olds in kindergarten (before the start of formal education). Methods: Kindergartners individually participated in two sessions. First, they completed an action-based SFON Imitation task and a verbal SFON Picture task, with a short visuo-motor task in between. Next, children were invited to spontaneously comment on the pictures of a picture book during a typical picture book reading activity. Results: Results revealed a positive association between children's SFON as measured by the Picture task and the frequency of their number-related utterances during typical picture book reading, but no such association for the Imitation task. Conclusions: Our findings indicate that children with higher SFON as measured by a verbal experimental task also tend to focus more frequently on number during verbal everyday activities, such as picture book reading. In view of the divergent associations between our SFON measures under study with everyday number activities, the current data suggest that SFON may not be a unitary construct and/or might be task-dependent.

Background: Research has shown that mathematics anxiety negatively correlates with primary school mathematics performance, including fraction knowledge. However, recently no significant correlation was found between fraction arithmetic performance and state anxiety measured after the fraction task. One possible explanation is the natural number bias (NNB), a tendency to apply natural number reasoning in fraction tasks, even when this is inappropriate. Students with the NNB may not realize they are answering incorrectly. Aims: The aim is to examine whether a misconception, namely the NNB, can influence students' fraction state anxiety. Sample: The participants were 119 fifth‐ and sixth‐grade students categorized as belonging to an NNB group (n = 60) or a No‐NNB group (n = 59), according to their NNB‐related answering profile on a fraction arithmetic task. Methods: Group differences were examined for state anxiety and performance on a fraction and a whole number arithmetic task and self‐reported trait mathematics anxiety. Results: The NNB group reported lower fraction state anxiety than the No‐NNB group, but there was no significant difference in trait mathematics anxiety. Furthermore, the NNB group reported lower fraction state anxiety than whole number state anxiety, while the opposite was true for the No‐NNB group. Conclusion: The present study suggests that students' perceptions of their own performance influence their state anxiety responses, and students with a NNB may not be aware of their misconception and poor performance. Not taking into account qualitative differences in low performance, such as misconceptions, may lead to misinterpretations in state anxiety‐performance relations. [ABSTRACT FROM AUTHOR]

Contemporary theories of expertise development highlight the crucial role of deliberate practice in the development of high level performance. Deliberate practice is practice that intentionally aims at improving one's skills and competencies. It is not a mechanical or repetitive process of making performance more fluid. Instead, it involves a great deal of thinking, problem solving, and reflection for analyzing, conceptualizing, and cultivating developing performance. This includes directing and guiding future training efforts that are then fine-tuned to dynamically evolving levels of performance. Expertise studies, particularly in music and sport, have described early forms of deliberate practice among children. These findings are made use of in our analysis of the various forms of practice in school mathematics. It is widely accepted that mathematics learning requires practice that results in effortless conducting of lower level processes (such as quick and accurate whole number arithmetic with small numbers), which relieve cognitive capacity for more complex tasks. However, the typical training of mathematical skills in educational contexts can be characterized as drill-and-practice that helps automatize basic skills, but often leads to inert routine skills instead of adaptive and flexible number knowledge. In this article we summarize findings of studies which describe students' self-initiated, deliberate practice in learning number knowledge and intervention studies applying deliberate practice in mathematics teaching, including technology-based learning environments aimed at triggering practice that goes beyond mechanical repeating of number skills.

This study explored the development of Ecuadorian Kindergartners' spontaneous focusing on numerosity (SFON) during the kindergarten year, as well as the contribution of early numerical abilities to this development. One hundred Kindergartners coming from ten classrooms received two SFON tasks, one at the beginning and one at the end of the school year, and an early numerical abilities achievement test at the beginning of the school year. Results first demonstrated limited SFON development during the kindergarten year, with inter-individual differences and intra-individual stability of children's SFON tendency. Second, both children's SFON tendency and their early numerical abilities at the start of the kindergarten year were predictively related to their SFON tendency at the end of the year. Our results do not only add to our theoretical understanding of SFON in young children, but also inform educational policy and practices in the domain of early mathematics education in Ecuador, as they provide building blocks for optimizing the educational goals and curricula for kindergarten mathematics.

Many people have serious difficulties in understanding rational numbers, limiting their ability to interpret and make use of them in modern daily life. This also leads to later difficulties in learning more advanced mathematical content. In this study, novel tasks are used to measure 263 late primary school students' spontaneous focusing on quantitative relations, in situations that are not explicitly mathematical. Even after controlling for a number of known predictors of rational number knowledge, spontaneous focusing on quantitative relations is found to have a strong impact on students' learning of rational number conceptual knowledge. This finding opens new possibilities for developing pedagogical solutions to one of the most difficult challenges of mathematics education. The findings suggest that students' own focusing tendency and self-initiated practice may have on important role in the long-term development of complex cognitive skills.

This study investigated the relationship between kindergartners' Spontaneous Focusing on Numerosity (SFON) and their number-related utterances during numerical picture book reading. Forty-eight 4- to 5-year-olds were individually interviewed via a SFON Imitation Task and a numerical picture book reading activity. We expected differences in the frequency of number-related utterances during picture book reading between children with a higher SFON score, providing more number-related utterances, and children with a lower SFON score. Our results showed large inter-individual differences in both kindergartners' SFON and the frequency of their number-related utterances during picture book reading, yet SFON was not related to the frequency of number-related utterances. This unexpected result is discussed in terms of its scientific, methodological, and educational implications.

Recent evidence suggests that early natural number knowledge is a predictor of later rational number conceptual knowledge, even though students' difficulties with rational numbers have also been explained by the overuse of natural number concepts--often referred to as the natural number bias. Hannula and Lehtinen ("Learn Instr" 15:237-256, 2005) have shown that children's tendency to spontaneously focus on numerosity (SFON) predicts the development of natural number and arithmetic skills. The present study follows 36 children from the age of 6 years to the age of 12 years in order to determine how preschool SFON tendency and number sequence skills are related to rational number conceptual knowledge at the age of 12 years. The results show that children's SFON tendency before school age is a strong predictor of later rational numbers conceptual knowledge, even after controlling for preschool number sequence skills. This finding has implications for the understanding of how the transition from reasoning about natural number concepts to reasoning about rational numbers may be influenced by children's self-initiated practice with numbers in everyday situations.