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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.

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.

Cited By :1 Export Date: 10 February 2021 Correspondence Address: McMullen, J.; Department of Teacher EducationFinland; email: jake.mcmullen@utu.fi The present study aims to examine inter-individual differences in adaptive number knowledge in secondary school students. Adaptive number knowledge is defined as a well-connected network of knowledge of numerical characteristics and arithmetic relations. Substantial and relevant qualitative differences in the strategies and expression of adaptive number knowledge have been found in primary school students still in the process of learning arithmetic. We present a study involving 879 seventh-grade students that examines the structure of individual differences in adaptive number knowledge with students who have completed one year of algebra instruction. Results of a latent profile analysis reveal a model that is similar than was previously found in primary school students. As well, arithmetic fluency and the development of arithmetic fluency are strong predictors of adaptive number knowledge latent profile membership. These results suggest that adaptive number knowledge may be characteristic of high-level performance extending into secondary school, even after formal instruction with arithmetic concludes. The present study aims to examine inter-individual differences in adaptive number knowledge in secondary school students. Adaptive number knowledge is defined as a well-connected network of knowledge of numerical characteristics and arithmetic relations. Substantial and relevant qualitative differences in the strategies and expression of adaptive number knowledge have been found in primary school students still in the process of learning arithmetic. We present a study involving 879 seventh-grade students that examines the structure of individual differences in adaptive number knowledge with students who have completed one year of algebra instruction. Results of a latent profile analysis reveal a model that is similar than was previously found in primary school students. As well, arithmetic fluency and the development of arithmetic fluency are strong predictors of adaptive number knowledge latent profile membership. These results suggest that adaptive number knowledge may be characteristic of high-level performance extending into secondary school, even after formal instruction with arithmetic concludes. © 2019, PsychOpen. All rights reserved.

The number of available educational games has enormously grown and it is difficult for users to identify which games are pedagogically effective among the multitude of options in app marketplaces. Recent studies on math games have highlighted the importance of (1) linking learning attributes and gameplay and (2) game design that supports students' deliberate practise. Using these as a measure for pedagogical quality, we investigated 109 math games found on Google Play Store (n=61) and iOS App Store (n=48). Furthermore, monetization solutions, data use permissions, target age group and type of mathematical content were retrieved from the apps. The analysis showed that only 11,0% of the games integrated learning attributes with gameplay and 12,0% of the games contained tasks which support learners' deliberate practice. The most commonly featured math subject was arithmetic and the games were targeted mostly to early childhood (6-12 years). Finally, games following curricula content and recommended design principles were difficult to locate using the search tools of the app marketplaces.

Classroom professional vision is a teaching skill that refers to the ability of teachers to rapidly notice information in class and engage in knowledge-based reasoning about the noticed information. Knowledge-based reasoning includes three interrelated processes: description, explanation, and prediction. The present study aimed to examine how pre-service teachers, in-service teachers, and school principals differed in these three reasoning processes after viewing classroom photographs with varying presentation time and interactional complexity. A 3×2×4 factorial design was used. Teacher expertise (pre-service teachers vs. in-service teachers vs. school principals) was a between-group factor, presentation time (1 sec vs. 3 sec) and complexity (teacher vs. dyad vs. small group vs. whole class) were within-group factors. Analysis of verbal reports suggested that in-service teachers and school principals used significantly more episodic knowledge, content knowledge, and pedagogical content knowledge in their reasoning than pre-service teachers did. Explanations with mathematical content knowledge were more frequent for in-service teachers, for shorter rather than longer presentation times, and for photographs showing the teacher only. Explanations with pedagogical content knowledge were more frequent for in-service teachers, for shorter rather than longer presentation times, and for photographs showing a small group. Across time and complexity, school principals verbalized less frequently what they noticed. In-service teachers and school principals verbalized significantly more self-monitoring and more predictions of teacher actions than pre-service teachers. The study findings contribute to the growing body of evidence on classroom professional vision, teacher noticing, and visual teacher expertise, and provide initial evidence on expert teachers’ frequent metacognitive self-monitoring.

Goodwin’s notion of professional vision suggests that learning to see in professionally relevant ways includes appropriating the visual practices within a domain. This observational study aimed to analyze how experts communicate these visual practices to novices to help them make meaning of domain-specific representations. Informed by a sociocultural perspective and founded on conversation analysis and ethnomethodology, video-recorded discourse and interaction between one expert in radiology and four laypeople were analyzed. The findings indicate three visual practices the medical expert uses to teach the novices how to see: highlighting, rotating, and zooming. The qualitative analyses suggest that learning to see professionally can be described as the mastering of expert practices in a focal domain. Implications for visual expertise research are discussed.

Developing adaptive expertise with arithmetic problem solving is a much desired aim of primary school mathematics education. However, there are very few practical tools for teachers that would aid reaching this complex mathematical learning goal. The aim of the present study was to test the effects of a game-based learning environment in supporting primary school students' adaptive number knowledge and related arithmetic skills. Participants were 1168 students in grades four, five, and six. Classes were randomized in two conditions: in the experimental group regular mathematics teaching was enriched with gameplay using the Number Navigation Game (NNG), and in the control group students continued according to their regular math curriculum. An experimental design with pre- and post-test was used to measure students' adaptive number knowledge, arithmetic fluency, and pre-algebra knowledge. Overall, results showed that the experimental group outperformed the control group on adaptive number knowledge and math fluency. Results showed varying effects of the training in different grade levels, with more pronounced improvement of students' adaptive number knowledge in grade five. Game performance was related to the experimental group's post-test scores even after controlling for pre-test scores and grade. Results suggest that the NNG is effective in enhancing different types of arithmetic skills and knowledge in different grades of primary school education and can provide teachers with a practical and flexible tool to extend their regular classroom practice.

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. [ABSTRACT FROM AUTHOR]

Spontaneous Focusing On quantitative Relations (SFOR) has been found to predict the development of rational number conceptual knowledge in primary students. Additionally, rational number knowledge has been shown to be related to later algebra knowledge. However, it is not yet clear: (a) the relative consistency of SFOR across multiple measurement points, (b) how SFOR tendency and rational number knowledge are inter-related across multiple time points, and (c) if SFOR tendency also predicts algebra knowledge. A sample of 140 third to fifth graders were followed over a four-year period and completed measures of SFOR tendency, rational number conceptual knowledge, and algebra knowledge. Results revealed that the SFOR was relatively consistent over a one-year period, suggesting that SFOR is not entirely context-dependent, but a more generalizable tendency. SFOR tendency was in a reciprocal relation with rational number conceptual knowledge, each being uniquely predictive of the other over a four-year period. Finally, SFOR tendency predicted algebra knowledge three-years later, even after taking into account non-verbal intelligence and rational number knowledge. The results of the present study provide further evidence that individual differences in SFOR tendency may have an important role in the development of mathematical knowledge, including rational numbers and algebra.

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.

Traditionally measured skills with arithmetic are not related to later algebra success at levels that would be expected given the close conceptual relation between arithmetic and algebra. However, adaptivity with arithmetic may be one aspect of arithmetic competences that can account for additional variation in algebra attainment. With this in mind, the present study aims to present evidence for the existence and relevance of a newly acknowledged component of adaptivity with arithmetic, namely, adaptive number knowledge. In particular, we aim to examine whether there are substantial individual differences in adaptive number knowledge and to what extent these differences are related to arithmetic and pre-algebra skills and knowledge. Adaptive number knowledge is defined as the well-connected knowledge of numerical characteristics and relations. A large sample of 1065 Finnish late primary school students completed measures of adaptive number knowledge, arithmetic conceptual knowledge, and arithmetic fluency. Three months later they completed a measure of pre-algebra skills. Substantial individual differences in adaptive number knowledge were identified using latent profile analysis. The identified profiles were related to concurrent arithmetic skills and knowledge. As well, adaptive number knowledge was found to predict later pre-algebra skills, even after taking into account arithmetic conceptual knowledge and arithmetic fluency. These results suggest that adaptive number knowledge is a relevant component of mathematical development, and may help account for disparities in algebra development.

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.

Learning outcomes as a concept has encountered a revival since the beginning of the Bologna process in 1999. The concept itself has a longer history with its roots in the behaviourist tradition of the 1960s. The goal of this review is to study how the historical roots of learning outcomes are noted in current research articles since the launch of the Bologna process and whether the concept of learning outcomes is used critically or uncritically. The review of 90 articles shows that the behaviourist tradition is still evident in the 21st century research with 29% of the articles directly and 11% indirectly referring uncritically to the respective publications or to the behaviourist epistemology. Only a minority of the articles, i.e. 8%, was found to be critical towards the behaviourist meaning of learning outcomes.

Background. Serious games are often used in formal school contexts, in which students’ lack of control over the playing situation may have repercussions on any motivational gains. Aims and Method. The first aim was to investigate to what extent n = 579 fifth grade students in Mexico who received a mathematics serious game played it voluntarily. Then, we explored how students who played voluntarily ( n = 337) differed from those who did not by either gender or pre-test mathematical skills or motivation. The second aim was to find out whether two play contexts, the group of voluntary players and a second group consisting of students playing at school as a compulsory part of their regular mathematics lessons ( n = 482), differed in game experience, game performance, and cognitive and motivational outcomes. Results. Students from the volunteer group who played had higher pre-test mathematical skills and math interest than those who did not play. Students in this group did not otherwise differ. Compared to students from the volunteer group who played, students in the school group played for longer, completed more tasks, and enjoyed playing the game more. However, their advanced mathematical skills did not improve as much. Conclusion. Motivation did not improve regardless of play context, suggesting serious games should be implemented for their learning content rather than because they are assumed to be motivating.

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.

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.

Adaptivity with arithmetic problem solving is a key aspect in the long-term development of mathematical skills, and the knowledge of numerical characteristics and relations is a core component of adaptivity with whole-number arithmetic. The present studies represent the first attempt to investigate the nature of the adaptive number knowledge, referring to the well-connected knowledge of numerical characteristics and arithmetic relations between numbers, which underlies adaptivity with arithmetic. A new measurement tool, the Adaptive Number Knowledge Task, was developed to capture individual differences in primary school students' and adults' number knowledge. Three groups of participants – 3rd to 5th graders, university students, and 6th graders – completed the measure, which required participants to produce arithmetic sentences equaling a target number by using any combination of four to five given numbers and arithmetic operations. These studies revealed substantial individual differences in the quantity and quality of participants' adaptive number knowledge. Adaptive number knowledge was found to be related to arithmetic fluency and knowledge of arithmetic concepts in 6th graders. Results suggest that the Adaptive Number Knowledge Task was able to capture individual differences in recognizing and using numerical relations in whole-number arithmetic problem solving, and that these differences may help explain differences in adaptivity with arithmetic.

Li and Baroody presented a study in which they investigate children’ spontaneous attention to exact quantity without acknowledging how previous studies of spontaneous focusing on numerosity (SFON) are related to their concept and methods. The authors’ reaction to our pointing this out makes it is clear that SFON research has had foundational role in the development of Baroody and his colleagues spontaneous attention to number (SAN) idea. Thus the authors must acknowledge the SFON literature as the substantial part of their theoretical and methodological construct. The latest definition of SAN states this undeniably. “SAN is the attentional process that underlies exact number recognition initiated by the mathematical thinking triggered by SFON (e.g., attention to “threeness” directed by conceptual knowledge of “three” activated by SFON).” Furthermore, in their response, Dr Baroody and Dr Li admit that “An awareness of the SFON literature did help us explicitly recognize that, given the ambiguous in...

Li and Baroody present a study in which they investigate toddlers’ spontaneous attention to exact quantity without acknowledging how previous studies of spontaneous focusing on numerosity (SFON) are related to their concept and methods. In this commentary requested by the European Journal of Developmental Psychology, we argue that the concept and the methods of spontaneous attention to exact quantity in the study of Li and Baroody clearly arise from previous research on SFON, as the authors have previously noted in their paper published in 2008. It is highly questionable whether their approach can be theoretically or methodologically dissociated from the previous research on SFON tendency to the extent that it is necessary to use an alternative name for the concept in their study.

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. (PsycINFO Database Record (c) 2016 APA, all rights reserved)

Research indicates that expert performance is domain specific and hardly transfers to novel tasks or domains. However, due to technological changes in dynamic work settings, experts sometimes need to adapt and transfer their skills to new task affordances. The present mixed method study investigates whether eye movement modeling examples (EMME) can promote adaptive expertise in medical image diagnosis. Performance, eye tracking, and think-aloud protocol data were obtained from nine medical experts and fourteen medical students. Participants interpreted dynamic visualizations before (baseline) and after (retention, transfer) viewing an expert model's eye movements. Findings indicate that studying eye movement modeling examples had positive effects on performance, task-relevant fixations, and the use of cognitive and metacognitive comprehension strategies. Effects were stronger for the retention than for the transfer task. Medical experts benefitted more from the modeling examples than did medical students. Directions for future research and implications for related domains are discussed.

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 educati...

The purpose of the study was to investigate how teachers adopt information and communication technology (ICT) as a pedagogical tool, what consequences the intensive use of ICT has for teaching practices, and how the use of ICT diffuses in a teacher community. The study combines longitudinal data and results of three sub‐studies conducted in the same school during the years 1994–2001. Results of the study indicated that the intensive use of ICT has several effects on the teacher community. Almost all teachers acquired the necessary technical ICT skills, and the majority of teachers used ICT with students. The pedagogical practices varied, depending on the teacher's interest and pedagogical thinking; several teachers adopted advanced teaching methods. The important elements for successful implementation of ICT were the key persons (an appreciated ICT teacher, a devoted principal, and a group of innovative teachers), university support, reflective practices among the teacher community and the overall support...

This article presents a theoretical and empirical analysis of social interactions in computer-based learning environments. A special aim was to explore how technology can be used to support reciprocal understanding between a teacher and a student. In the belief that a computer-based learning environment creates an optimal situation for apprenticeship, we conducted studies on three different technology-based learning environments in Finland and the United States focusing on analyzing learning and social interaction. In the first study, control technology (LegoTClogo) and cognitive apprenticeship-based learning were used to promote modem technological thinking and problem solving skills in seventh grade students ( N = 16). The second study was aimed at investigating sixth grade students' ( N = 16) collaborative planning and design of Web-based knowledge bases and documents. The third study dealt with the possibilities of supporting student-student and student-teacher interaction in higher education with the help of ALEL ( N = 11), a software program developed for teaching methodology and statistical inference to university students. Detailed qualitative data (videotaping, tape recording, and interviewing) related to the working processes and teaching-learning interactions were collected during these three experiments. The results of the content analyses point to ways technology can improve task-related social interaction and provide multiple opportunities for students to negotiate meanings concerning different abstract phenomena. From an instructor viewpoint, modest scaffolded assistance seemed to enhance the learning activity in each of these settings. Through these studies, it is clear that various technology tools can support the teacher and the student to “turn toward” a joint task and facilitate their reciprocal understanding of the situation.

The purpose of this paper is to describe methods for analyzing the CSCL process by using logfiles and other quantitative measures. We present methodological approaches that would help CSCL teachers, tutors and researchers to assess the participation activity and system level features of social interaction that take place during CSCL.

The purpose of the study was to investigate Finnish elementary and high school students’ skills and practices of using the new information and communication technologies (ICT). Beliefs about the importance of ICT were also assessed. Five hundred and fifteen students responded to a self-report questionnaire. The students attended 25 schools that used ICT intensively and represented all provinces of Finland. From the analysis, there emerged three factors that represented these students’ relationships to ICT. Characteristic of the first factor was a belief that computer supported learning makes learning more meaningful and encourages one to make more efforts to study. Self-reported competence in using ICT was strongly loaded on the second factor, together with intensive reported use of ICT at home as well as networking with expert cultures and coaching of other people to improve their ICT skills. The third factor represented intensity of using ICT at school and appears to be determined more by the availability of equipment and the extent to which ICT is used in the school than by a student’s expertise in ICT.

IntroductionRational Number KnowledgeThere is a broad agreement in the literature that a good understanding of rational numbers is of critical importance for mathematics achievement in general and for performance in specific domains of the mathematics curriculum in particular (Siegler et al., 2012). For example, Siegler, Thompson, and Schneider (2011) found high correlations (all between .54 and .86) between three measures of fraction magnitude knowledge (0-1 fraction number line estimation, 0-5 fraction number line estimation, and 0-1 fraction magnitude comparison) and general mathematics achievement in upper elementary school learners. This finding was replicated by Torbeyns, Schneider, Xin, and Siegler (2015) in three countries from different continents. Similar findings emerged from a recent study of Siegler et al. (2012), who concluded that fifth graders' rational number understanding predicted their overall mathematics and algebra scores in high school, even after controlling for reading achievement, IQ, working memory, whole number knowledge, family income, and family education.Despite the critical importance of a good rational number knowledge, a large body of literature reported that children and even adults have a lot of difficulties dealing with various aspects of rational numbers (Bailey, Siegler, & Geary, 2014; Cramer, Post, & delMas, 2002; Li, Chen, & An, 2009; Mazzocco & Devlin, 2008; Merenluoto & Lehtinen, 2004; Vamvakoussi, Van Dooren, & Verschaffel, 2012; Vamvakoussi & Vosniadou, 2010; Van Hoof, Lijnen, Verschaffel, & Van Dooren, 2013). To give one example, more than one third of a representative sample of Flemish sixth graders did not reach the educational standards for rational numbers (Janssen, Verschaffel, Tuerlinckx, Van den Noortgate, & De Fraine, 2010).The difficulties learners have with rational number tasks are often - at least in part - attributed to the "natural number bias" (Vamvakoussi et al., 2012; see Ni & Zhou, 2005, for the closely related idea of "whole number bias"), which is the tendency to inappropriately use natural number properties in rational numbers tasks (Van Hoof, Vandewalle, Verschaffel, & Van Dooren, 2015). Before learners are introduced to rational numbers in the classroom, they have already formed an idea of what a number is. This idea is based on their experiences (both in daily life and in school) with natural numbers. Once the learners are then instructed about rational numbers, the properties of natural numbers are not always applicable anymore, leading to problems and misconceptions with rational numbers (Vamvakoussi & Vosniadou, 2010). This becomes apparent in learners' systematic mistakes, specifically in rational number tasks where reasoning purely in terms of natural numbers results in an incorrect solution - these tasks are called incongruent. At the same time, much higher accuracy levels are found in rational number tasks where reasoning in terms of natural numbers leads to a correct answer - these tasks are called congruent. The vast literature on this natural number bias reports three main aspects that elicit such systematic errors. The first aspect relates to the density of the set of rational numbers. While natural numbers are characterized by a discrete structure (one can always indicate which number follows a given number; for example after 13 comes 14), rational numbers are characterized by a dense structure (you cannot say which number comes next, because between any two given rational numbers are always infinitely many other rational numbers) (e.g., Merenluoto & Lehtinen, 2004). The second aspect relates to the size of rational numbers. Research indicates that errors in size comparison tasks are repeatedly made because students incorrectly assume that, as is the case with natural numbers, "longer decimals are larger, shorter decimals are smaller", or "that a fraction's numerical value always increases when its denominator, numerator, or both increase" (Mamede, Nunes, & Bryant, 2005; Meert, Gregoire, & Noel, 2010; Obersteiner, Van Dooren, Van Hoof, & Verschaffel, 2013; Resnick et al. …