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.     

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.     

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.     

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.     

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。     

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。     

The etiology of mathematical self-evaluation and mathematics achievement: understanding the relationship using a cross-lagged twin study from age 9 to 12

The genetic and environmental origins of individual differences in mathematical self-evaluation over time and its association with later mathematics achievement were investigated in a UK sample of 2138 twin pairs at ages 9 and 12. Self-evaluation indexed how good children think they are at mathematical activities and how much they like those activities. Mathematics achievement was assessed by teachers based on UK National Curriculum standards. At both ages self-evaluation was approximately 40% heritable, with the rest of the variance explained by non-shared environment. The results also suggested moderate reciprocal associations between self-evaluation and mathematics achievement across time, with earlier self-evaluation predicting later performance and earlier performance predicting later self-evaluation. These cross-lagged relationships were genetically rather than environmentally mediated.     

数学模型与数学实验教学改革探索

数学模型与数学实验教学如何适应现代数学科学的迅猛发展,满足社会对高素质数学人才的需求,是高等理工科院校共同面临的课题。从教学观念的改变、教学内容的更新、教学模式的优化以及教学考核机制的完善等方面进行研讨,主旨在为数学模型与数学实验教学改革探索有效的解决途径。     

数学教育的实践探索与数学方法论

本文简要回顾了 2 0世纪 80年代以来 ,我国数学教育界部分学者、教师自觉学习与应用数学方法论指导数学教学方法改革的历史足迹 ,其中“贯彻数学方法的教育方式 ,全面提高学生素质”的数学教育实验 (简称MM教育方式实验 )已从无锡拓展到湖北、天津、北京、新疆等十余个省市。这对进一步推进数学素质教育提供了重要启示。     

The influence of theoretical mathematical foundations on teaching and learning: a case study of whole numbers in elementary school

This paper examines the existence and impact of theoretical mathematical foundations on the teaching and learning of whole numbers in elementary school in France. It shows that the study of the New Math reform –which was eventually itself replaced in the longer term – provides some keys to understanding the influence of mathematical theories on teaching and learning. The paper studies changes related to place value, a notion that was deeply impacted by the introduction of numeration bases other than ten in 1970, and their subsequent removal in the 1980s. What the author terms ‘numeration units’ (ones, tens, hundreds, thousands, etc.) and ‘powers-of-ten written in figures’ (1, 10, 100, 1000, etc.) are key tools for describing and understanding changes. The author identifies two theories that have formed the basis for place value teaching in the twentieth century, and examines some aspects of their influence. The paper also addresses epistemological issues in the relation between academic mathematics and school mathematics, and highlights the role of units in the teaching of basic arithmetic.     

Developing mathematical knowledge for teaching in a methods course: the case of function

This study describes teacher learning in a teaching experiment consisting of a content-focused methods course involving the mathematical knowledge for teaching function. Prospective and practicing teachers in the course showed growth in their ability to define function, to provide examples of functions and link them to the definition, in the connections they could make between function representations, and to consider the role of definition in mathematics and the K-12 classroom. Written assessments, interview data, and class discourse analyses illustrate how the course supported the development of mathematical knowledge that built on individual teachers’ prior knowledge as well as the development of a stronger collective understanding of function.     

数学教育的价值与数学教育改革

历史上每一次重大的数学教育改革 ,无不关涉到数学教育价值的抉择与建构 ;数学教育的“训练价值”与“实用价值”、“知识价值”与“能力价值”一直处在“钟摆”与“嬗变”之中 ;新世纪的数学教育、尤其是数学课程改革 ,应该把握住数学的本质特征和根本特点 ,在提高学生的数学素质 (数学知识、技能和思想、方法 ,数学概括、抽象和推理、证明能力等 )的基础之上 ,促进学生理性和非理性充分、和谐地发展。     

Benefits of practicing 4 = 2 + 2: nontraditional problem formats facilitate children's understanding of mathematical equivalence

This study examined whether practice with arithmetic problems presented in a nontraditional problem format improves understanding of mathematical equivalence. Children (M age = 8;0; N = 90) were randomly assigned to practice addition in one of three conditions: (a) traditional, in which problems were presented in the traditional operations on left side format (e.g., 9 + 8 = 17); (b) nontraditional, in which problems were presented in a nontraditional format (e.g., 17 = 9 + 8); or (c) no extra practice. Children developed a better understanding of mathematical equivalence after receiving nontraditional practice than after receiving traditional practice or no extra practice. Results suggest that minor differences in early input can yield substantial differences in children's understanding of fundamental concepts.