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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. publishedVersion

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.

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.

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.

The present chapter provides an overview of the iterative development and testing of the Number Navigation Game (NNG), a game-based learning environment which aims to strengthen primary school students’ flexibility and adaptivity with arithmetic problem solving. The set of studies described in this chapter asked different questions using different methodologies through the phases of game development. However, the connecting principle was placing the integration of game features and mathematical content in the center of the process and developing valid and reliable means to operationalize the expected mathematical learning outcomes. Each step in the cycle served as a source of information to be used both in the development of the NNG and in the development of different types of measures of the mathematical learning outcomes. Conclusions of these steps are discussed in relation to the iterative game design process, the importance of using an integrated game design, and questions of measurement in the broader context of educational game design.

In this chapter, we describe challenges of early childhood math education and propose that enhancing children’s mathematically focused activities not only in formal educational contexts but also in more informal settings is one solution to these challenges. Spontaneous mathematical focusing tendencies are argued to affect the mathematically relevant aspects a child focuses their attention on. Doing so allows children to acquire self-initiated practice with the related mathematical skills and concepts, which, in turn, may facilitate the development of further mathematical cognition. The first intervention programs enhancing spontaneous focusing on numerosity (SFON) and quantitative relations (SFOR) may help us understand how the world becomes mathematical for children and leads to sustained improvements in children’s mathematical development.

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.

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.

Theories of adult learning can be studied from individual, social, and cultural perspectives. The individual viewpoint has dominantly focused on cognitive aspects, dating back to the ancient philosophers who set the grounding for our understanding of human knowledge production. Today, a well-developed understanding of adult learning processes considers learning as a complex system where learning processes on different levels interact but are not reducible to each other. The roles of memory, executive functions, and deliberate practice are elaborated in this chapter. Motivational, metacognitive, and regulation aspects have added to our understanding of the success of individual learning processes, complemented with the more recent notions of the importance of different types of conceptions and changes in them. Looking at learning processes from social and cultural perspectives have helped us to understand the situational and collaborative nature of knowledge building processes. In adult learning situations, such as in studying or working life contexts, the individual and social learning are intertwined, making use of participants prior understanding and resulting in collaborative knowledge building.

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.