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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.

Rational number knowledge is a crucial feature of primary school mathematics that predicts students' later mathematics achievement. Many students struggle with the transition from natural number to rational number reasoning, so novel pedagogical approaches to support the development of rational number knowledge are valuable to mathematics educators worldwide. Digital game-based learning environments may support a wide range of mathematics skills. NanoRoboMath, a digital game-based learning environment, was developed to enhance students' conceptual and adaptive rational number knowledge. In this paper, we tested the effectiveness of a preliminary version of the game with fifth and sixth grade primary school students (N = 195) using a quasi-experimental design. A small positive effect of playing the NanoRoboMath game on students' rational number conceptual knowledge was observed. Students' overall game performance was related to learning outcomes concerning their adaptive rational number knowledge and understanding of rational number representations and operations.

The aim of this study was to investigate differences in student teachers' and experienced teachers' professional vision in natural settings and to elicit clues of the relation of in-the-moment noticing and instruction quality of students' understanding of rational number concept. Rational number concept challenges both students and teachers because of natural number bias that learning of rational numbers is vulnerable to. Accurate professional vision and adequate instructions are needed to enhance students' understanding of rational number concept. Mobile eye-tracking technique enables video recording of natural teaching situations from a teacher's perspective with more specific information of teacher's in-the-moment noticing. Combined with cued retrospective reporting, this approach can gather more explicit evidence of teachers' professional vision and instructions. Results indicated that both student teachers and experienced teachers attended to mathematical and fraction-related aspect similarly but differed in interpreting and instructing students' fraction understanding. Student teachers made more advanced interpretations but their instructions were less adequate, whereas among experienced teachers, it was just the opposite. Furthermore, student teachers made more attempts to shared attention when using fraction understanding non-supporting instructions, whereas experienced teachers' attempts to shared attention were related to fraction understanding supporting instructions. Results indicate student teachers' difficulty to transfer pedagogical content knowledge from noticing to actions and experienced teachers to have more enhanced in-the-moment professional vision and its application to teaching. Practical implications for teacher training as well as methodological decisions of in-the-moment professional vision studies in natural settings are discussed.

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 word-problem (WP) item characteristics, individual differences in text comprehension and arithmetic skills, and their relations to mathematical WP-solving. The participants were 891 fourth-grade students from elementary schools in Finland. Analyses were conducted in two phases. In the first phase, WP characteristics concerning linguistic and numerical factors and their difficulty level were investigated. In contrast to our expectations, the results did not show a clear connection between WP difficulty level and their other characteristics regarding linguistic and numerical factors. In the second phase, text comprehension and arithmetic skills were used to classify participants into four groups: skilful in text comprehension but poor in arithmetic; poor in text comprehension but skilful in arithmetic; very poor in both skills; very skilful in both skills. The results indicated that WP-solving performance on both easy and difficult items was strongly related to text comprehension and arithmetic skills. In easy items, the students who were poor in text comprehension but skilful in arithmetic performed better than those who were skilful in text comprehension but poor in arithmetic. However, there were no differences between these two groups in WP-solving performance on difficult items, showing that more challenging WPs require both skills from students.

The significance of socially shared metacognitive regulation (SSMR) in collaborative learning is gaining momentum. To date, however, there is still a paucity of research of how SSMR is manifested in asynchronous computer-supported collaborative learning (CSCL), and hardly any systematic investigation of SSMR's functions and evolution across different phases of complex collaborative learning activities. Furthermore, how individual students influence group regulatory effort is not well known and even less how they participate in SSMR over the entire collaborative learning process. The multi-method, in-depth case study presented in this article addresses these gaps by scrutinizing the participation of a small group of students in SSMR in asynchronous computer supported collaborative inquiry learning. The networked discussion, consisting of 640 notes, was used as baseline data. The sets of notes, which formed nine SSMR threads, were identified and their functions analyzed. Several analytical methods, including social network analysis, were used to investigate various aspects of individual participation. The findings show that some SSMR threads lasted over an extended period, and they sometimes intertwined or overlapped. Furthermore, SSMR threads were found to play different functions, mainly inhibiting the perceived inappropriate direction of the ongoing cognitive process. Finally, SSMR was found in all phases of the process -- but with some variation. The use of different analytical methods was critical as this provided a variety of complementary insights into students' participation in SSMR. The value of using multiple, rigorous analytical methods to understand SSMR's significance over the entire course of an asynchronous CSCL activity is discussed.

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.

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.

Teacher professional vision as a concept is gaining importance in research on teaching, and recently models for studying teacher professional vision and student self-regulated learning (SRL) have been proposed. There are interview and video intervention studies investigating teacher professional vision for SRL, but no real-life classroom research so far. This study investigated the role of student SRL behaviour, as it was reported by students themselves and teachers, in teacher attention distribution as part of teacher professional vision. Ten teachers and their 158 students at high school level in Lithuania took part in the research. The first step of the study resulted in identifying four student SRL-profiles, which differed based on student level of SRL and the extent to which teacher and student assessments coincided: mixed lower-regulated, mixed higher-regulated, systematic lower-regulated, systematic higher-regulated. The profiles demonstrated only a partial overlap in teacher and student judgement of student SRL. The second step of the study explored whether scores of students’ SRL from student and teacher reports were related to teachers’ distribution of visual attention in one lesson. The results showed that only one teacher rating scale of student information-seeking behaviour had a slight correlation with teacher attention. The results imply rather bottom-up trends in teacher attention to students in the classroom when it comes to SRL. Besides, the study results highlight the not directly observable nature of SRL processes and imply a difficulty for teachers to assess student SRL. [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.

Difficulties with rational numbers have been explained by a natural number bias, where concepts of natural numbers are inappropriately applied to rational numbers. Overcoming this difficulty may require a radical restructuring of previous knowledge. In order to capture this development, we examined third- to fifth-grade students' understanding of the size of rational numbers across a 1-year period. On three occasions, students answered a test about the size of decimals and fractions. Latent transition analysis allowed profiling five different rational number size concepts and mapping students' developmental trajectories during the early learning of this topic. Development from natural number-based reasoning to mathematically correct understanding was almost nonexistent during this time period. Instead, students appear to move through intermediary concepts. Locating these temporary and mediating number concepts supports the idea of a slow and gradually developing conceptual change in rational numbers.

The traditional classroom setting presents challenges when it comes to strengthening adaptive expertise in mathematics education through deliberate practice. This study aimed to investigate whether the Number Navigation Game (NNG) could help promote deliberate practice and whether students’ performance in the game was related to their development of Adaptive Number Knowledge, perceived challenge, flow, and math interest. NNG is a gamebased learning environment that requires students to progress by solving increasingly complex arithmetic problems, which is crucial for promoting adaptive number knowledge. Game performances of 214 Finnish students were analyzed and compared to the best possible performance for each game level. A growth mixture model based on the students' relative performance levels was used to gain insight into how students' game performance changed throughout the game, and how this related to their knowledge gains, perceived challenge, math motivation, and flow. There were four different profiles of students' game performance. The largest profile consisted of students who steadily improved their performance in the game, despite initially having lower-than-average performance. This group experienced lower levels of flow but achieved larger learning gains than the other groups, suggesting that their engagement may be more aligned with deliberate practice. [ABSTRACT FROM AUTHOR]

This article examines a new training design for continuing professional development that aims to support the learning of the novel knowledge and skills needed in emerging professional fields by interconnecting academic and workplace settings. The training design is based on using two advisors, one from working life and the other from an academic context. The article examined whether participants' personal orientation to adaptive expertise predicts the success of a guidance process. The interconnection of workplace and academic contexts was expected to occur through guidance practices. In addition, the features underlying the most successful guidance relationships were analysed. Data were collected by conducting repeated semi-structured interviews with 18 course participants, eight academic advisors and eight workplace advisors in the context of a 1-year energy efficiency training programme. The results indicated that a trainee's personal orientation towards adaptive expertise is a significant component in successful guidance processes. An interconnection of workplace and academic knowledge and practices was hardly found in the guidance provided by each participant's academic and workplace advisors. The feature underlying the most successful guidance relationships are related at the personal, dyad and context levels. An excellent match between the expert profiles of the learner and the advisor appears to be especially critical for successful guidance and powerful knowledge exchange in emerging fields. However, finding matching advisors is often challenging. Many problems are presumably solved if these "right persons" can be found and if the trainees are themselves oriented to utilise the novel resources provided to them by the advisors.

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.

In this commentary we take a critical look at the various studies being reported in this issue about the relationship between cognitive neuroscience and mathematics, from a mathematics education viewpoint. After a discussion of the individual contributions, which we have grouped into three categories--namely neuroscientific studies of (a) children's numerical magnitude representation, (b) arithmetical thinking, and (c) more advanced mathematical thinking--and which nicely document the scientific progression being made within the domain of educational neuroscience applied to the domain of mathematics education during the last 5 years, we point to some general caveats that should be considered in future research.

The purpose of the present investigation was to examine how a collective knowledge-creation-oriented approach to doctoral education is being adopted in research within the field of education. The authors interviewed nine leaders of national centres of excellence in science research and 12 education professors whose research communities cultivate collective practices of doctoral education. In order to examine how the collective practices differ from the individual ones, the authors also interviewed nine education professors from well-known Finnish and other European universities engaged in supervising traditional monograph theses. Three principal features of the collective model (CM) of doctoral education were distinguished: (1) pursuit of collectively shared research objects; (2) pursuit of externally reviewed co-authored journal articles; and (3) focus on collective supervision. The results indicated that the principal features of the CM can be productively implemented in research in the field of education despite certain challenges and constraints addressed in this report.

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.

This seven-year longitudinal study examined how children's spontaneous focusing on numerosity (SFON), subitizing based enumeration, and counting skills assessed at five or six years predict their school mathematics achievement at 12 years. The participants were 36 Finnish children without diagnosed neurological disorders. The results, based on partial least squares modeling, demonstrate that SFON and verbal counting skills before school age predict mathematical performance on a standardized test for typical school mathematics in Grade 5. After controlling for nonverbal IQ, only SFON predict school mathematics. Subitizing-based enumeration skills have an indirect effect via number sequence skills and SFON on mathematical performance at 12 years. Early mathematic skills do not predict reading skills at 12 years. Children's early numerical skills, including SFON, before school age are important contributors to substantially later success in school mathematics.

In order to address the requirements of future education in different fields of academic professional activity, a model called Academic Apprenticeship Education was initiated in Finland in 2009. The aim of this article is to analyse the development of expert networks in the context of a 1-year Academic Apprenticeship Education model in the field of energy efficiency, which is a new and rapidly developing knowledge-intensive field. We examined the creation of networking ties among all course participants, the process of networking in small groups, and individual participants' networking activity. Data was collected by administering a social networking questionnaire in the beginning and at the end of the training to all course participants (n = 87) and analysed using social network analysis and repeated measures ANOVA. In addition, semi-structured interviews were conducted with organizers of the training to examine how the operational practices of the training supported networking. The results indicated that there was little change in the networking ties among all course participants. However, those small groups that were able to communicate appeared to create internal linkages. At the individual level, more new ties emerged for private sector actors than for public sector actors. In conclusion, we propose that a consolidated educational model should be created for the Academic Apprenticeship Education model in general. The quality of education might be better assured if the current ad hoc networks were not the only way to organise knowledge exchange among participants.

While preschool-aged children display some skills with quantitative relations, later learning of related fraction concepts is difficult for many students. We present two studies that investigate young children's tendency of Spontaneous Focusing On quantitative Relations (SFOR), which may help explain individual differences in the development of fraction knowledge. In the first study, a cross-sectional sample of 84 kindergarteners to third graders completed tasks measuring their spontaneous recognition and use of quantitative relations and then completed the tasks again with explicit guidance to focus on quantitative relations. Findings suggest that SFOR is a measure of the spontaneous focusing of attention on quantitative relations and the use of these relations in reasoning. In the second (longitudinal) study, 25 first graders completed measures of SFOR tendency and a measure of fraction knowledge three years later. SFOR tendency was found to predict fraction knowledge, suggesting that it plays a role in the development of fraction knowledge.

The focus of our study is social status among first graders. In particular, we will consider the relationship between acceptance and rejection, and how these are connected to three social behavioral traits: bullying, victimization, and social withdrawal. The data set is from peer nominations of 748 children from 49 classrooms in the southwest of Finland, collected by interviews. A Social Network Analysis technique was used to analyze the data. The results indicated a negative correlation between acceptance and rejection, bullying, and social withdrawal, but did not correlate victimization. Positive correlations, however, were observed between rejection and all social behavioral traits. Based on cluster analysis, the first graders can be divided into Liked Children (N?=?447), Popular Bullies (N?=?68), Unpopular Bullies (N?=?28), Marginalized Children (N?=?135) and High Risk Children (N?=?21). All Unpopular Bullies and almost all High Risk children were boys. Boys were also nominated more often than girls as Popular Bullies, whilst girls were more often reported as marginalized or as Liked Children.

The objective of this study is to elucidate new information on the possibility of leadership training through business computer-simulation gaming in a virtual working context. In the study, a business-simulation gaming session was organised for graduate students ("n"?=?26). The participants played the simulation game in virtual teams that were geographically dispersed and that were brought together by the use of technology. Before the gaming session, the team leaders were preselected and trained in how to operate the simulation game. Data consist of pre- and posttest questionnaires (the Multifactor Leadership Questionnaire measuring transformational, transactional and passive/avoidance leadership styles) and answers to open-ended questions. The results showed the difference in participants' opinions on leadership styles before and after the training. After the gaming sessions, team members scored lower in transformational and transactional scales than team leaders. Only team leaders' leadership styles correlated with game performance. However, shared leadership among team members was typical for most successful teams. Implications for leadership training are discussed.

The present study investigated the extent to which mothers' support for their children's sense of competence, autonomy and relatedness predicts their children's interest in math and reading, and also their mastery orientation, during the transition to primary school. One hundred fifty-two children were examined twice during their first grade year regarding their interests and mastery orientation (Time 1 and Time 2). Mothers filled in a diary and questionnaire measuring maternal support, also on two occasions (Time 1 and Time 2). Children's school performance in reading and math was tested at the beginning of the first grade (Time 1). The results showed that, after controlling for the autoregressor and level of performance, all of the maternal support variables predicted children's subsequent interest in mathematics but not in reading. Children's mastery orientation, in turn, was only predicted by mothers' support for autonomy. (Contains 1 table and 1 figure.)

Understanding how best to assess expertise, the situational variations of expertise, and distinctive qualities of expertise that arises from particular workplace experiences, presents an important challenge. Certainly, at this time, there is much interest in identifying standard occupational measures and competences, which are not well aligned with such variations and distinctiveness in performance. Therefore, this paper addresses the methodological issues posed by such a challenge through reviewing the expertise literature to identify ways forward. Based on the example of one target domain, medicine, and one criterion task, the comprehension of visualizations, the study identifies, elaborates, and evaluates assessments used to study qualitative changes in professional vision that occur as a result of extended periods of workplace learning experiences. It identifies the kinds of sample, materials, measures, and analysis methods used to assess the quality of expertise differences, as well as what elements of and differences in visual expertise are revealed in particular assessment processes. In doing this, the study seeks to illuminate how assessing the quality of expertise differences matured over the past 20 years, noting that strategies of scanning brain activity and tracking eye movements are now being used in ways that augment traditional approaches such as using verbal reports and observing representational practices. The findings demonstrate how the assessment of visual expertise has become more multidisciplinary over the past two decades. Implications for educational practice and future research directions on the development of professional vision are discussed.

Over the years, the role and extent of the basic sciences in medical curricula have been challenged by research on clinical expertise, clinical teachers, and medical students, as well as by the development and diversification of the medical curricula themselves. The aim of this study was to examine how prior knowledge of basic histology and histopathology among students predicts early learning of diagnostic pathology. Participants ("N" = 118, representing 91% of the full student cohort) were medical students at the University of Turku, Finland. Data were collected during two preclinical courses that students attended in their first and second years of medical school. The measurements included tests on biomedical and clinical knowledge and a performance test in diagnostic pathology. Second-year performance on the diagnostic pathology examinations was predicted by the students' prior knowledge of histology, but not by the students' prior knowledge of histopathology. Although earlier research has demonstrated similar results in studies with shorter longitudinal designs, the present study demonstrates that the effect remains even if there is a considerably long time delay (a year) between the measurements, thus confirming the long-term value of basic science studies in the preclinical phase.

Negative expertise is conceptualised as the professional's ability to avoid errors during practice due to certain cognitive agencies. In this study, negative knowledge (i.e. knowledge about what is wrong in a certain context and situation) is conceptualised as one such agency. This study compares and investigates the negative knowledge of elder care nurses at three different levels of professional experience. Thereby, various forms of negative knowledge--declarative, procedural, self-reflective and vicarious--are investigated. Moreover, the specificity of negative knowledge is compared across different levels of professional experience. Verbal data was collected from a prompting task study of 37 elder care nurses. These were prompted 20 diagnoses of varying typicality for the professional field. The nurses were asked what was critical to pay attention to and what ought to be avoided in case of these diagnoses. The study's results reveal a significant superiority of highly tenured elder care nurses in vicarious negative knowledge, as well as in highly specified negative knowledge. Procedural and procedural negative knowledge (PNK)--which are the most important facets in quantitative terms--show an approximately parallel developmental pattern among the different groups. For declarative knowledge, a U-shaped intermediate effect was discovered. It is concluded that PNK has the most immediate error-preventive function. With regards to future research, the great specificity of highly tenured elder care nurses' negative knowledge discovered in this study is discussed here as one possible explanation as to why they are able to intentionally avoid errors (i.e. their negative expertise).

This study investigated how metacognition appears as a socially shared phenomenon within collaborative mathematical word-problem solving processes of dyads of high-achieving pupils. Four dyads solved problems of different difficulty levels. The pupils were 10 years old. The problem-solving activities were videotaped and transcribed in terms of verbal and nonverbal behaviours as well as of turns taken in communication (N = 14 675). Episodes of socially shared metacognition were identified and their function and focus analysed. There were significantly more and longer episodes of socially shared metacognition in difficult as compared to moderately difficult and easy problems. Their function was to facilitate or inhibit activities and their focus was on the situation model of the problem or on mathematical operations. Metacognitive experiences were found to trigger socially shared metacognition. (Contains 3 figures and 8 tables.)

The aim of this 2 year longitudinal study was to explore whether children's individual differences in spontaneous focusing on numerosity (SFON) in kindergarten predict arithmetical and reading skills 2 years later in school. Moreover, we investigated whether the positive relationship between SFON and mathematical skills is explained by children's individual differences in spontaneous focusing on a non-numerical aspect. The participants were 139 Finnish-speaking children. The results show that SFON tendency in kindergarten is a significant domain-specific predictor of arithmetical skills, but not reading skills, assessed at the end of Grade 2. In addition, the relationship between SFON and number sequence skills in kindergarten is not explained by children's individual differences in their focusing on a non-numerical aspect that is, spatial locations. (Contains 3 tables.)