Development of three-dimensional object completion in infancy

Three-dimensional (3D) object completion was investigated by habituating 4- and 6-month-old infants (n= 24 total) with a computer-generated wedge stimulus that pivoted 15 degrees , providing only a limited view. Two displays, rotating 360 degrees , were then shown: a complete, solid volume and an incomplete, hollow form composed only of the sides seen during habituation. There were no reliable preferences for either test display by 4-month-olds. At 6 months, infants showed a reliable novelty preference for the incomplete test display. Infants in a control group (n= 24) not habituated to the limited-view wedge preferred neither test display. By 6 months, infants may represent simple objects as complete in 3D space despite a limited perspective. Possible mechanisms of development of 3D object completion are discussed.     

On mathematical understanding: perspectives of experienced Chinese mathematics teachers

Researchers have long debated the meaning of mathematical understanding and ways to achieve mathematical understanding. This study investigated experienced Chinese mathematics teachers’ views about mathematical understanding. It was found that these mathematics teachers embrace the view that understanding is a web of connections, which is a result of continuous connection making. However, in contrast to the popular view which separates understanding into conceptual and procedural, Chinese teachers prefer to view understanding in terms of concepts and procedures. They place more stress on the process of concept development, which is viewed as a source of students’ failures in transfer. To achieve mathematical understanding, the Chinese teachers emphasize strategies such as reinventing a concept, verbalizing a concept, and using examples and comparisons for analogical reasoning. These findings draw on the perspective of classroom practitioners to inform the long-debated issue of the meaning of mathematical understanding and ways to achieve mathematical understanding.     

Uses of video in understanding and improving mathematical thinking and teaching

This article characterizes my use of video as a tool for research, design and development. I argue that videos, while a potentially overwhelming source of data, provide the kind of large bandwidth that enables one to capture phenomena that one might otherwise miss; and that although the act of taping is in itself an act of selection, there is typically enough shown in a video that it rewards multiple watching and supports the kinds of arguments over data that are essential for theory testing and replication. In pragmatic terms, video presents phenomena in ways that have an immediacy that is tremendously valuable. I discuss ways in which videos help students and teachers focus on phenomena that might otherwise be very hard to grapple with. This article begins with a brief review of my uses of video, almost 40 years ago, for research and development in problem solving. It then moves to the discussion of very fine-grained research on learning and decision making. The bulk of the article is devoted to a discussion of the teaching for robust understanding (TRU) framework, which was derived in large measure from the extensive review of classroom videotapes, and which serves as the basis for an extensive program of pre-service and in-service professional development. The professional development relies heavily on the use of videos to convey the key ideas in TRU, and to help teachers plan and review instruction.     

Developing understanding of mathematical modeling in secondary teacher preparation

This study examines the evolution of 11 prospective teachers’ understanding of mathematical modeling through the implementation of a modeling module within a curriculum course in a secondary teacher preparation program. While the prospective teachers had not previously taken a course on mathematical modeling, they will be expected to include modeling as part of the school curriculum under current state standards. The module consisted of readings, analysis of the Common Core State Standards, carefully designed modeling activities, individual and group work, discussion, presentations, and reflections. The results show that while most prospective teachers had misconceived definitions of mathematical modeling prior to the module, they developed the correct understanding of modeling as an iterative process involving making assumptions and validating conclusions connected to everyday situations. The study reveals how the prospective teachers translated the modeling cycle into practice in the context of a carefully designed open-ended problem and the strong connections between modeling activities and promoting mathematical practices.     

A legacy of apartheid: The case of mathematical education in south Africa

Post-independence South Africa faces many consequences of apartheid. One legacy has been the creation of a population of school leavers for whom under-achievement in mathematics has become the norm. This paper evaluates (a) the extent and causes of under-performance in mathematics in the general population; (b) the resourcing of mathematical education, and (c) the mathematics curriculum. Evidence is provided to show that in addition to personal and social losses created by under-performance in mathematics, economic wastage has resulted from having neglected to create the conditions for the development of an appropriately numerate population. The paper offers proposals for the improvement of mathematical education through an extension of the democratic process.     

The role of strategy-based motivation in mathematical problem solving: The case of learner-generated drawings

This study investigated the role of strategy-based motivation (SBM) in solving real-world geometry problems. Students from 19 classes (N = 437) were assigned to the strategy training or control condition. Before the treatment, students were asked about their SBM (i.e., self-efficacy, cost, and value). After the treatment, they were instructed to generate drawings and solve problems. The results revealed that self-efficacy expectations and cost (but not value) regarding the drawing strategy predicted drawing quality and performance. The relation between SBM and performance was mediated by drawing quality. SBM did not moderate the effects of strategy training on drawing quality or performance. The results emphasize the importance of SBM in the context of strategy training for predicting the quality of strategy use and problem-solving performance. The results are consistent with assumptions of expectancy-value and learner-generated drawing theories. One practical implication is that attention should be given to SBM in mathematics lessons.     

Breaking the habit: engineering students’ understanding of mathematical creativity

ABSTRACT

In higher education, engineering students have to be prepared for their future jobs, with knowledge but also with several soft skills, among them creativity. In this paper, we present a study carried on with 128 engineering undergraduate students on their understanding of mathematical creativity. The students were in the first year of different engineering first degrees in a north-eastern Portuguese university and we analysed the content of their texts for the question ‘What do you understand by mathematical creativity?’. Data collection was done in the first semester of the academic years 2014/2015 and 2016/2017 in a Linear Algebra course. The results showed that ‘problem solving’ category had the majority of the references in 2014/2015, but not in the academic year 2016/2017 were ‘involving mathematics’ category had the majority. This exploratory study pointed out for ‘problem solving’ and ‘involving mathematics’ categories and gave us hints for teaching mathematics courses in engineering degrees.     

Pre-service teachers' representations of children's understanding of mathematical concepts: conflicts and conflict resolution

The purpose of this study was to investigate the extent to which research experience could enhance pre-service teachers' understanding of children's knowledge of mathematical concepts. A group of five pre-service teachers designed and conducted a study of children's understanding of fractions. As participant-observers throughout their study we gathered evidence of change in pre-service teachers' representations of children's knowledge of fractions. The focus of our observations was on the conflicts generated by the gaps between the pre-service teachers' research findings and their representations of the children's knowledge of fractions. Results indicate that research experience is fruitful in developing an inquisitive disposition in pre-service teachers and in sensitizing future teachers to children's knowledge of a mathematical concept. Pre-service teachers became familiar with the research literature regarding children's understanding of fractions, they questioned the typical instructional sequence of teaching fractions and they scrutinized methods of assessing children's knowledge of mathematical topics.     

Gender equity in mathematical achievement: the case of China

Gender differences in achievement in mathematics, a traditionally male-stereotyped subject, have long been a concern for many educators around the world. Gender differences in mathematical achievement have decreased in recent decades, especially in Western countries, and become small or insignificant in large-scale tests, such as the Programme for International Student Assessment (PISA). The situation in China has not yet been studied. The recent PISA report lists China B-S-J-G (representing Beijing–Shanghai–Jiangsu–Guangdong) as an educational system with no significant gender difference in mathematical achievement. Based on a secondary analysis of PISA 2015 mathematics data from China B-S-J-G, this study more deeply scrutinized gender differences in Chinese students’ mathematical performance, emphasizing societal factors, namely students’ socioeconomic status, school level, school type, school location, and socioeconomic status at school level. This analysis revealed important differences within the overall picture. Most importantly, significantly more boys than girls scored in the top tier of mathematics achievement. At the lower- and upper-secondary school levels, boys performed significantly better than girls, with the achievement difference increasing at the upper-secondary level. Furthermore, this study found that, on average, Chinese (B-S-J-G) girls achieved significantly lower average scores on the PISA 2015 mathematics test than boys in the same school. Overall, students’ individual characteristics and school characteristics need to be separated and both taken into account to examine the role of gender in mathematical achievement, which has not been thoroughly investigated in the past.     

Students understanding and participation using ULearn: dynamics case

In this paper, ULearn (University of Surrey learning on-line) and e-learning techniques are used to enhance the students’ understanding and participation in Dynamics modules for an undergraduate programme in the School of Engineering, University of Surrey. The Dynamics modules are taught at Levels 2 and 3 and contain several mechanisms and vibrating systems. The modules consist of several chapters, each of which has its own folder, or content file, in ULearn. In addition to the discussion board and chat room, on-line simulations have been made available and integrated within ULearn. In order to evaluate students’ understanding and encourage the student participation in ULearn, timed on-line assessments, in the form of multiple-choice quizzes, have been designed. The feedback of the students has indicated that the use of ULearn was of great help to improve their understanding of the Dynamics modules.     

Developing mathematical understanding and students with emotional and behavioural disorders: a review of the literature

Recent shifts in focus on academic interventions for students diagnosed with emotional and behavioural disorders (EBD) create a need to evaluate existing interventional research in content areas such as mathematics. Literature reviews in the area of mathematics interventions for students with EBD have mostly focused on the outcomes and the rigour of existing studies with limited attention to the quality of the pedagogical methods used in those studies. The purpose of this conceptual review was to use a teaching-for-understanding lens to examine existing research interventions in mathematics for students with EBD. Teaching-mathematics-for-understanding requires pedagogical methods that aim to activate and capture students’ mathematical reasoning and thinking. An iterative electronic and hand search of the literature found 30 studies covering three decades (1985–2015) that met the criteria for inclusion in this review. Results of the analysis include: (1) mathematics assessments utilised ranged from very basic to comprehensive, with most studies applying basic assessments; (2) a majority of the studies focused on intervention strategies targeting procedural and rote knowledge; (3) most studies utilised quantitative inquiry methods; and (4) in more recent years, studies have increasingly examined conceptual mathematical understanding and incorporated qualitative inquiry methods.     

Reductionism versus emergentism: A framework for understanding conceptions of giftedness

In this article, I propose that various conceptions and theoretical models of giftedness can be understood along a continuum from the most reductionist to the most emergentist. Along with this continuum, I specify four levels of analysis based on the human functional hierarchy: the biological, operational (computational), intentional, and activity levels. I illustrate how reductionist and emergentist approaches provide contrasting views of intelligence and giftedness, with the former seeking basic elements and lower level explanations and the latter stressing complexity and higher level organization. I suggest that clearly articulating levels of analysis and principles for efforts of the reductionist or emergentist nature will enhance conceptual clarity and methodological rigor. Whether giftedness is understood in the reductionist or emergentist frameworks has many practical and policy implications for gifted education.     

小学数学估算教学案例思考

《课程标准》指出,"重视口算,加强估算,提倡算法多样化","能在解决问题的同时,选择合适的估算方法,培养估算习惯"。教师应把估算渗透于整个教学过程之中,有意识的创设估算的机会,培养学生对估算的兴趣,养成估算习惯,掌握估算策略,提高估算技能。     

Through a glass darkly: A naturalistic study of students' understanding of mathematical word problems

A father is now 20 years older than his son. In 8 years, the father's age will be 5 years more than twice the son's age. Find their present age.Word problems, such as this one, are a perennial source of difficulty for students of school mathematics. Unfortunately, very little is known about why they are problematic, mainly because so little is known about how students understand such problems or the strategies they use in their efforts to solve them. Traditional research into word problems has shed precious little light on this question owing, in no small part, to its almost singular preoccupation with results of pupils' activities—as expressed in some sort of test score, and to its tendency to all but ignore what students actually do when confronted with problems of this kind.This study was carried out as one facet of a larger research project designed to gain more insight into some of the ways in which students understand school mathematics. It focuses on the efforts of one pupil, a twelve-year-old girl in grade seven, to come to terms with solving word problems using an algebraic approach. Strategies associated with both the structured and the unstructured clinical interview were used in order to reveal what was involved in her attempts to make sense of the word problems in her grade seven mathematics textbook.Based on the information gained in the interview, a rational reconstruction of the student's problem-solving strategy is proposed, and compared with the strategies normally prescribed in contemporary school mathematics textbooks. What emerges from this comparison is the finding that, while there appear to be systematic and fundamental differences between the procedures prescribed by the text and those actually used by the pupil in working through certain problems, these differences are undetectable in the finished product; either in the answer itself or in the rough or finished work. What this suggests, among other things, is that if, as educators and/or researchers, we limit our attempts to understand how students go about learning to solve word problems (or how they approach any other part of the school mathematics curriculum, for that matter) to examining what they commit to paper, we are apt to be seriously misled concerning what they genuinely understand and what they fail to understand. In short, if we are to learn more about why pupils experience difficulties with word problems we must begin to pay serious attention to what they say and do as they work their way through them.This study was supported by a Research and Development Grant from the Faculty of Education, Queen's University.     

The role of motor experience in understanding action function: the case of the precision grasp

Recent evidence suggests adults and infants selectively attend to features of action, such as how a hand contacts an object. The current research investigated whether this bias stems from infants' processing of the functional consequences of grasps: understanding that different grasps afford different future actions. A habituation paradigm assessed 10-month-old infants' (N = 62) understanding of the functional consequences of precision and whole-hand grasps in others' actions, and infants' own precision grasping abilities were also assessed. The results indicate infants understood the functional consequences of another's grasp only if they could perform precision grasps themselves. These results highlight a previously unknown aspect of early action understanding, and deepen our understanding of the relation between motor experience and cognition.     

数学文化在数学教育中的意义

文章通过分析数学教育现有形势,简述了数学文化的内涵,并且着重分析了数学文化在数学教育中的作用和意义。