Learning Mathematics for Teaching: From a Teachers' Seminar to the Classroom

A successful practice grounded in the principles that guide the current mathematics education reform effort requires a qualitatively different and significantly richer understanding of mathematics than most teachers currently possess. However, it is not as clear how teachers' mathematical understandings develop and how those understandings affect instruction. This paper explores two avenues for K-6 teachers' mathematical development, (a) engagement in inquiry into mathematics itself, and (b) investigation of children's mathematical thinking, illustrating how the need for these two kinds of investigations arises in classroom situations and how they can be pursued in a professional development setting.     

Cognitive Strategy Instruction for Mathematical Word Problem-solving of Students with Mathematics Disabilities in China

This study investigated the impact of cognitive strategy instruction (CSI) on mathematical word problem solving of students with mathematics disabilities. A sample of fourth-grade students in a Chinese primary school was divided into a treatment group (75 students) and a comparison group (75 students). The sample consisted of students with mathematics disabilities only, students with both mathematics and reading disabilities, as well as average- and high-achieving students. Results showed that students at all ability levels (except high-achieving students) in the treatment group outperformed significantly their counterparts in the comparison group; the intervention effect was stronger for students with mathematics disabilities only than for those with both mathematics and reading disabilities. The present study indicates that CSI is a contextually and pedagogically appropriate model that has a strong potential to improve mathematical word problem solving.     

免费师范生数学专业素质的调查研究

数学知识、数学能力和数学情意是影响数学免费师范生数学专业素质的主要因素.数学免费师范生的数学情意素质具备程度较高,数学能力素质具备程度较低;数学专业素质在性别和年级之间存在差异;数学专业素质与数学知识、数学能力和数学情意均存在显著的正相关;数学知识、数学能力和数学情意对数学免费师范生的数学专业素质的形成和发展有不同程度的影响。     

Images of mathematics held by university teachers of mathematics education

This article presents data about views of mathematics collected through a national survey of teachers of mathematics education in Canadian universities. The results are compared to those obtained through a similar survey of university teachers of mathematical sciences carried out earlier.     

What makes a problem mathematically interesting? Inviting prospective teachers to pose better problems

School students of all ages, including those who subsequently become teachers, have limited experience posing their own mathematical problems. Yet problem posing, both as an act of mathematical inquiry and of mathematics teaching, is part of the mathematics education reform vision that seeks to promote mathematics as an worthy intellectual activity. In this study, the authors explored the problem-posing behavior of elementary prospective teachers, which entailed analyzing the kinds of problems they posed as a result of two interventions. The interventions were designed to probe the effects of (a) exploration of a mathematical situation as a precursor to mathematical problem posing, and (b) development of aesthetic criteria to judge the mathematical quality of the problems posed. Results show that both interventions led to improved problem posing and mathematically richer understandings of what makes a problem ‘good.’     

On mathematics teaching for future engineers—in the digital era

Current changes, especially the wide application of information technology, in all fields of our life, mean that mathematical knowledge becomes necessary in almost every domain. It implies new expectations for mathematical education. An urgent need of a new mathematical literacy for all—also a new mathematical literacy for engineers—is evident. It is necessary to consider a process of mathematics learning at tertiary level from the epistemological perspective and to investigate students’ ways of mathematical thinking. This epistemological knowledge is especially indispensable when students use information technology. In this article current requirements regarding mathematical education are discussed, especially those for future engineers. Analysis of examples of learning elementary statistics, using graphing calculators as supporting tools, leads to the formulation of essential aims for mathematics educators concerning mathematics teaching for future engineers.     

A comparison of Hungarian and English teachers' conceptions of mathematics and its teaching

This paper reports on a statistical study of English and Hungarian teachers' conceptions of mathematics and its teaching. A questionnaire was developed and distributed to teachers of mathematics in 200 English and40 Hungarian schools teaching children in the 11–14 age range. Factor analyses identified four conceptions of mathematics and five of mathematics teaching. These were compared with those yielded by an earlier study involving the same English teachers and found to be consistent indicating the existence of similar conceptions in different educational systems. Differences and similarities in the strengths with which those conceptions are held were suggestive of both global and national conceptual traditions. The significant similarity to emerge concerned teachers from both countries sharing, with similar strengths, a general conception of mathematics teaching incorporating the teaching of mathematical skills, a variety of classroom approaches including investigations and problem-solving, and a recognition that mathematics provides an essential lifetool. Multi-dimensional scaling indicated that English teachers have their perspectives informed by two underlying, and possibly conflicting, traditions– pedagogic relevance and mathematical utility. The Hungarians appeared concerned only with notions of pedagogic relevance – those practices perceived to facilitate effective learning of a subject which is untainted by utilitarian perspectives. This revised version was published online in July 2006 with corrections to the Cover Date.     

Proofs and refutations in the undergraduate mathematics classroom

In his 1976 book, Proofs and Refutations, Lakatos presents a collection of case studies to illustrate methods of mathematical discovery in the history of mathematics. In this paper, we reframe these methods in ways that we have found make them more amenable for use as a framework for research on learning and teaching mathematics. We present an episode from an undergraduate abstract algebra classroom to illustrate the guided reinvention of mathematics through processes that strongly parallel those described by Lakatos. Our analysis suggests that the constructs described by Lakatos can provide a useful framework for making sense of the mathematical activity in classrooms where students are actively engaged in the development of mathematical ideas and provide design heuristics for instructional approaches that support the learning of mathematics through the process of guided reinvention.     

知识元与问题活性化设计研究

数学教学的目的是把科学形态的数学有效地转化为教育形态的数学知识.引入知识元和问题活性化等新概念,以中学数学为例,对新概念进行解析,对提高教师在知识传授过程中活性化知识的量和质是十分有益的.     

对提高中学数学教师数学修养的思考和尝试

提高中学数学教师的数学修养在塑造人的过程中具有不可替代的作用.在实际调查中我们发现:一方面,教师们对提高自身数学修养水平的热情很高;另一方面,在涉及到具体数学知识时,教师们又往往把自己的眼光仅局限在个人所讲授或熟悉的具体中学课程中,而对关系“较远”的内容则倾向于采取回避和应付的态度.提高教师自身的数学修养是解决目前数学教育中种种弊端的关键.在实际教学中可采用如下方法加强数学教育中数学类课程:(1)加强对数学内容的数学思想和方法论的讲解;(2)加强数学计算和论证的训练;(3)加强与中学数学课程的结合.     

数学全息方法及其在数学科研和教学中的应用

宇宙全息律作为客观世界的一种普遍规律,被反映到数学这种人们认识世界的基本工具上,形成数学科学的一种普遍规律,即数学全息律.以数学全息元为起点,利用数学全息律进行数学科研与教学的方法就是数学全息方法.利用周期研究时间函数和利用“生成元”研究数学结构等均是应用数学全息方法的生动事例;同时,数学全息方法还是促进数学创造的有力工具.数学全息方法在数学教学中也有着广泛的应用.     

数学教师的数学观和数学教育观

作为数学教师素质结构的先导性成分,数学观由数学知识观、数学本质观和数学价值观构成.数学教育观包括数学教育目的观、数学教育过程观和数学教育价值观3个主要部分.数学教师的数学观对其数学教育观有重要影响.数学教育文化观、数学教育哲学观和数学教育价值观之间有着复杂的互动关系.     

Klassenwiederholen in PISA-I-Plus: Was lernen Sitzenbleiber in Mathematik dazu?

This study analyses the effects which repeating a class has on ninth grade students’ development of mathematical competency. The following research questions were addressed: How many students repeat grades in the different types of schools? How do students who repeat a grade differ from those who do not in their performance and background characteristics? How much extra mathematics do students repeating a grade learn in one school year? What are the differences between various types of school? Can students with poor mathematics grades in particular profit from repeating a grade? The sample is a sub-sample of the PISA-I-Plus study and comprises N = 360 ninth grade students. The total sample of PISA-I-Plus is representative for all ninth/tenth grade students from the different school types in Germany. The data survey was carried out in the ninth grade and then repeated after the students had repeated a year. The results document differences in the amount of grade repeat quotas between types of school. Furthermore, compared to students not repeating, those repeating a grade had lower mathematics (d = 1.02) and german (d = 1.14) grades, a lower level of mathematical literacy (d = 0.51), and lower test results with regard to basic cognitive abilities (d = 0.32). In terms of the development of mathematical literacy, the students repeating a grade could improve by an average of 23 points (d = 0.27) on the PISA mathematics scale. However, the results identify 38 percent of students repeating a grade who do not make any significant improvement in mathematics or even get worse. A differentiation according to school types shows that students repeating a grade in integrated comprehensive secondary schools and in schools with several educational levels in particular do not, on average, show any noteworthy improvement in their mathematical literacy. The analysis of the school grades received in mathematics shows that students whose mathematics grades are unsatisfactory do not benefit more from repeating a grade than students whose mathematics performance has been rated as being “satisfactory” or better. The article concludes with a discussion of the possible consequences of changing the way in which repetitions of grades are dealt with.     

数感、数学效能感与数学成绩的关系研究

本研究以小学生为被试,探讨小学生数感、数学效能感发展特点以及数感、数学效能感和数学成绩之间的关系。研究结果如下:数感、数学效能感与数学成绩之间的相关均达到显著水平,且数感与数学成绩的关系较之数学效能感与数学成绩的关系更紧密;数感对数学成绩的预测力较强,数学效能感的预测力弱一些;数学效能感在数感和数学成绩之间起部分中介作用。     

Student teachers’ mathematics attitudes,authentic investigations and use of metacognitive tools

Based on findings from a semester-long study, this article examines the development of Samoan prospective teachers’ mathematical understandings and mathematics attitudes when investigating authentic contexts and applying working mathematically processes, mental computations and problem-solving strategies to find solutions of problems. The prospective teachers had enrolled for the second time (having failed their first attempt), in the first-year mathematics methods course of a 2-year Diploma of Education (Primary) programme. The group also included those enrolled in the Diploma of Education (Early Childhood and Special Needs) programmes, who recognizing their own limited understanding of mathematics would ordinarily shy away from opportunities for improvement. Given the negative mathematical and learning experiences, this group was ideal to engage in innovative and creative approaches that would make mathematics learning more meaningful and contextual in a Samoan environment. Only data from the attitudinal questionnaires and interviews are presented in this article. Main findings have implications for teaching and learning mathematics.     

Effects of metacognitive instruction on mathematics achievement and attitude towards mathematics of low mathematics achievers

An experiment was conducted to examine the effects of metacognitive instruction on mathematics achievement and attitude towards mathematics of low mathematics achievers at a middle school in the North‐West Province of South Africa. Forty standard (std) 7 pupils were identified whose non‐verbal general ability and previous mathematics achievements were significantly lower than those of other std 7 pupils. These subjects were randomly assigned to an experimental group and a control group. Metacognitive strategies in solving mathematical problems related to four mathematics topics were individually taught to the members of the experimental group, while the pupils in the control group were taught the four mathematics topics through the conventional method of teaching mathematics. The comparisons of pretest and posttest measures of general ability, metacognitive awareness, attitude towards mathematics, and mathematics achievement revealed that the posttest scores of all the four variables for the experimental group were significantly higher than those for the control group.     

小学课程中的数学文化:内涵特点、主要内容与学习价值

数学文化是指数学的思想、精神、语言、方法、观点以及它们的形成与发展,还包括数学在人类生活、科学技术、社会发展中的贡献和意义,以及与数学相关的人文活动。与纯粹的数学知识相比较,数学文化体现了人文性与科学性交融、开放性与包容性并存、民族性与统一性共生,以及价值理性与工具理性互推的特点。适合小学生学习的数学文化内容主要有数学知识的发展历史、数学家的成长故事、数学的游戏活动、数学的生活应用和学科应用。小学生学习数学文化有助于他们理解数学知识、掌握数学思想方法、提高数学思维水平、加强数学应用意识和培养数学精神。     

数学教育与教育数学

数学教育的现代发展,以及各国进行数学课程改革浪潮的涌现,提出了教育数学的研究课题。它似可与纯粹数学、应用数学并列为第三个研究领域。与“新数学”、“大众数学”相比,教育数学将能均衡好数学教育的“教育方面”与“数学方面”的关系和促进数学教育改革的健康发展。进行教育数学的研究,可先从数学现实、数学难点、数学新点等方面着手,然后再进行理论体系建设。     

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·数学史·数学教育

数学存在于历史之中,活的数学存在于活的数学史之中,活的数学教育讲授的是活的数学和活的数学史。活的数学由无数的数学故事构成,故事是先进的数学教育中不可缺少的养料和调味剂。