Search

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.

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.