Memory in mathematical understanding

Whatever cognitive processes are involved in understanding mathematics, it is clear that one of them is learning. No one is born with an understanding of measure theory, abstract algebra or general topology; the very name mathematics means that which is to be learned (Boyer, 1968). One of the outcomes of learning is remembered knowledge. Indeed it is our contention that memory plays an essential role in the understanding of mathematics, However, what it is that is remembered by students who understand mathematics in contrast to those who do not is by no means a trivial question. In fact, we would like to suggest that there is a gap in this connection between recent developments in memory research and the theory and practice of mathematics education. A second purpose of the present article is to survey briefly the orgins of this gap.     

数学建模与数学实验

讨论了数学建模与数学实验的步骤、工具和目的，着重阐述3个观点：数学建模的核心步骤是建模，数学模型的求解隶属于数学，而非数学建模；数学建模的核心工具是数学，计算机作为数学之工具间接地用于数学建模；就数学实验自身而言，其主要目的是辅助做数学；就教学而言，数学实验的主要目的是培养学生运用计算机研究、学习数学的能力。     

数学教育学浅议

数学教育学不是简单的教育学加数学.正确地认识理解数学教育学与教育学的关系及其规律,用以指导数学教育,对提高数学教育质量至关重要.     

试论数学的分类讨论思想

分类讨论是高中数学重要的思想方法和解题策略,也是教学的重点和难点,更是高考的热点.本文论述了数学分类讨论的原则、步骤,并从六个方面阐述了高中数学中哪些内容需要分类讨论以及怎样分类讨论.     

Construction of mathematical knowledge through whole-class discussion

A word problem that requires addition of fractions with different denominators was presented to 11 classes of 4th- or 5th-graders, 10 years of age. It had three answer alternatives: adding the denominators and numerators separately, transforming fractions into decimals before adding them, and the standard, most appropriate solution. Students in each class were required to choose an alternative themselves, state their reason for the choice, and discuss which alternative would be right. After whole-class discussion, they were asked to choose an alternative again and nominate the student who had offered the most plausible idea. Then students in six classes were informed which alternative was the most appropriate, whereas those in five classes were not. Finally, the students, irrespective of the presence/absence of feedback, were required to solve the initial problem without the alternatives as the post-test, and two new problems as the transfer test.The results showed that (a) the students could offer more or less plausible arguments for or against each alternative, which in most classes included the correct explanation; (b) both vocal and silent students could write a mathematical expression for the post-test problem and manipulated it correctly, incorporating other students' ideas presented during the discussion, with or without the teacher's feedback, though their generalization was rather limited; and (c) the students could recognize and memorize reasonable explanations offered by other students in the discussion.     

Mathematical induction: A pedagogical discussion

It is observed that many students have difficulty in producing correct proofs by the method of mathematical induction. The notion of a correct proof by this method is analysed mathematically. Subsequently, the topic is analysed into behavioural skills and subjected to a conceptual analysis. Common misconception of induction are considered, with recommendations for their remediation. Finally, criteria for the analysis and evaluation of textbook treatments of induction are evolved and applied to a selection of texts.     

Mathematical discussion and perspective drawing in primary school

The aim of this paper is to analyse the functions of semiotic mediation in a long term teaching experiment on the plane representation of three-dimensional space by means of perspective drawing, that has been tried out from grade 2 to grade 5 in three different classrooms within the research project Mathematical Discussion. On the one side, drawing has a functional role in the overall development of the child; on the other side, perspective drawing has a phenomenological role in the genesis of modern geometry. The experiment aims at connecting (1) pupils' spatial experiences to the development of the geometry of three-dimensional space and (2) pupils' drawing experiences to the geometry of two-dimensional space, up to the mastery of early geometrical strategies of plane representation of space. Classroom activity alternates individual problems and classroom discussions orchestrated by the teacher. The paper is divided into several parts: after a brief introduction containing some contextual information (§§ 1, 2), the problem of the social construction of knowledge is addressed and some theoretical constructs mainly borrowed from the Vygotskian school are elaborated (§ 3); then two analyses of the experiment are made, according to the motives of activity (§ 4) and to the sequence of actions (§ 5); finally the role of semiotic mediation in the whole experiment is analysed (§ 6); in the final section (§ 7) some results are recapitulated and compared with the literature on the teaching and learning of geometry and the function of semiotic mediation is discussed with reference to the other distinctive features of the teaching experiment.     

试论机械运动的数学认识

从科学方法论的角度,阐明数学方法在对机械运动认识的几次飞跃中的重要地位,并对机械运动的一些属性——运动状态的描述、运动规律的表达、运动本质的刻画、运动规律的揭示、运动优化的判据——的数学认识进行初步探讨。     

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.     

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.     

数学建模与数学实验课程的改革与实践

数学专业开设的＂数学建模与数学实验＂课程在教学过程中以多媒体教学为手段,并借助于MATLAB等数学软件培养学生的实践能力、创新意识。在教学内容、教学手段、考核方式、实践训练等方面进行改革尝试,促使学生掌握数学建模的基本理论,并且使学生能够学以致用,在数学建模、软件设计、团队协作等各个方面得到了锻炼。     

Mathematical language and mathematical abilities in preschool: A systematic literature review

It is well established that early general language during preschool is critical for children's mathematical abilities. In an attempt to further characterize this association between language and mathematics, an increasing number of studies show that one specific type of language, namely mathematical language or the key linguistic concepts that are required for performing mathematical activities, is even more critical to children's mathematical abilities. The purpose of this systematic review is to summarize the evidence on mathematical language and mathematical abilities. We focus on preschool children as nearly all of the existing work has been done at this age. We first explain how mathematical language has been defined across studies, and report how it has been evaluated in studies in preschool. Next, we present the results of our systematic review. Following the PRISMA guidelines and after a critical appraisal, we ended with a set of 18 papers that were all of sufficient methodological quality. In these studies, mathematical language was defined as terms that are about numbers and operations on numbers (e.g., nine), but also included linguistic terms that do not directly refer to numbers, yet are important to understand mathematical concepts (i.e., quantitative and spatial terms such as fewest and middle, respectively). Some of these studies evaluated children's performance on mathematical language tasks, while others evaluated the mathematical language input provided to the child by their (educational) environment (teachers/parents/interventionists). Mathematical language correlated positively with children's mathematical abilities, concurrently and longitudinally. It also directly affected children's mathematical abilities, as was shown by intervention studies. We discuss potential directions for future research and highlight implications for education, arguing for more support for teachers and parents to improve the use of mathematical language in the classroom and in home settings.     

Building middle school students' understanding of proportional reasoning through mathematical investigation

This article describes and shares an innovative pedagogical practice that holds promise in contributing to the teaching and learning of proportions in middle school. The teaching and learning of mathematics with understanding framework was used as a vehicle to help 21 seventh grade students reason proportionally. The findings of this unit suggest that the classroom culture, which encouraged the students to make connections between their existing and new ideas and reflecting and communicating their thinking, contributed to their emerging understanding of proportions. The use of an authentic and non-routine task involving liquid measurements also heightened their interest, curiosity and enthusiasm, thereby contributing to their excitement about the mathematics they were learning.     

Mathematical Subject Classification在数学文献信息分类中的应用

美国《数学评论》的分类体系《数学主题分类表》(Mathematical Subject Classification)，是国际数学界影响最大使用最广的数学专业分类表。数学工作者可以使用美国数学会在网站上提供的Mathematical Subject Classification为自己的数学论文标引分类号，也可从该分类途径检索网上的数学信息。     

数学教学中要重视数学思维的训练

本论述了数学的思维过程，说明了在数学教学中要加强数学思维训练，培养学生的解题能力。     

数学建模思想融入高等数学教学的研究与探讨

分析高等数学教学中存在的问题和数学建模思想融入高等数学教学的意义,给出将数学建模思想融入高等数学教学的具体措施和教学案例。