Teacher Actions to Maximize Mathematics Learning Opportunities in Heterogeneous Classrooms

The basic unit of school based mathematics teaching is the lesson. This article is a contribution to understanding teacher actions that facilitate successful lessons, defined as those that engage all students, especially those who may sometimes feel alienated from mathematics and schooling, in productive and successful mathematical thinking and learning. An underlying assumption is that lessons can seek to build a sense in the students that their experience has elements in common with the rest of the class and that this can be done through attention to particular aspects of the mathematical and socio-mathematical goals. We examine three teacher actions that address the mathematical goals: using open-ended tasks, preparing prompts to support students experiencing difficulty, and posing extension tasks to students who finish the set tasks quickly; as well as actions that address the socio-mathematical goals by making classroom processes explicit. To illustrate and elaborate these actions, we describe a particular lesson taught to a heterogeneous upper primary (age 11–12) class.     

抓住契机建设世界文化大国

文化建设是中国现代化的内涵之一，那么，抓住契机，建设文化大国，应当成为当前文化建设的重要内容。建设文化大国过程中，既要注意借鉴学习外国先进文化，亦要重视和发展民族、民俗文化。不能再走“建设性破坏”之路。     

Teaching Mathematics in Multilingual Classrooms

In this paper we present the way in which language issues have become a relevant factor in research which aims to study the socio-cultural aspects of mathematics education in classrooms with a high percentage of immigrant students. Our research on language issues focuses on two aspects, namely the language as a social tool within the mathematics classroom and the language as a vehicle in the construction of mathematical knowledge. We introduce our problem within this area and provide a rationale for our research methodology, not avoiding its drawbacks,but rather giving examples of the kinds of difficulties we encountered. The paper highlights the integrated nature of the social, cultural and linguistic aspects of mathematics teaching and learning, and illustrates the fact that, even if the mathematical language can be considered universal, the language of ‘doing mathematics within the classroom’ is far from being universal. This revised version was published online in July 2006 with corrections to the Cover Date.     

Adapting Reform Ideas in Different Mathematics Classrooms: Beliefs Beyond Mathematics

The goals of this study are to understandelementary school teachers' beliefs andpractices and to unveil factors that influencethe way teachers adapt mathematics reformrhetoric when trying to adopt it. In theresearch, I searched for beliefs beyondmathematics that influence teachers' decisionsand choices for teaching mathematics. Workingwith children from different socioeconomicbackgrounds, teachers interpret reform indifferent ways. Based on their concept of students' needs, teachers select which partsof the reform documents are appropriate fortheir students. While children from uppersocioeconomic backgrounds experience problemsolving, those from lower socioeconomicbackgrounds undergo rote learning. Because notall children have the opportunity to learn thesame quality mathematics, the emerging concernof this study is the issue of equity inmathematics teaching.     

Opportunities to Learn and Achievement in Mathematics in a Sample of Sixth Grade Students in Lima, Peru

In recent years, the mathematics achievement of Peruvian students has been evaluated in three national and two international studies. The results in all cases suggest very poor learning. A similar situation is found in many developing countries. In this study, we analyzed the opportunities to learn (OTL) mathematics of sixth grade students from 22 public schools in Lima, Peru. OTL were defined in this study as curriculum coverage, cognitive demand of the tasks posed to the students, percent of mathematical exercises that were correct and quality of feedback. These variables were coded in the workbooks and notebooks of the students, which were gathered at the end of the school year (at the same time, the mathematics achievement test was administered). The results show that: (a) regarding OTL, less than half of the exercises available in the workbooks were solved, teachers overemphasize some topics of the national curriculum ({i.e.} related to Number and Number Sense), they pose tasks that are at very low levels of cognitive demand, and it is common to find mistakes in the students' answers to problems that have no feedback (or even worse, the feedback is wrong); (b) students in relatively poorer, multigrade classrooms have less OTL; (c) OTL, as defined earlier, is positively associated with achievement.A preliminary version of this paper received the Research Medal in the category Knowledge, Education andTechnology at the Annual Meeting of the Global Development Network (GDN) in Cairo, in January 2003.     

数学课堂教学实现新课标的策略

实现新课标,数学课堂教学应采取:联系现实生活,注重生活体验;创设问题情境,引导科学探究;鼓励学生思维,促进个性发展;变革学习方式,着重培养能力;重视教学过程,建构数学知识等五个主要策略.     

学生社团:大学生学习的特殊课堂

社团相对于课堂教学、党团活动、社会实践,社团具有平等交互性、自愿自主性、个性化、创新性等特点。高校辅导员应指导大学生充分利用社团活动的特点和优势,并与第一课堂、党团活动和社会实践的学习相结合,通过建立社团与第一课堂、党团活动和社会实践的联动机制,打造大学生社团学习的新课堂。     

Working in the Mathematics Frame: Maximizing the Potential to Learn From Students' Mathematics Classroom Discussions

The quality of students' mathematics classroom discussions is an important factor in determining mathematics classroom learning outcomes. Good mathematics classroom discussions provide an opportunity for ideas to be shared and developed, but not all mathematics classroom discussions produce these learning opportunities. In this paper I discuss three constructs (subject, purpose, and frame) and how they can be used to analyze and characterize the quality of students' mathematics classroom discussions in terms of the mathematics learning potential that those discussions provide. This revised version was published online in July 2006 with corrections to the Cover Date.     

激发中职学生数学学习兴趣的几点做法

数学是一门逻辑性强、十分抽象的基础学科,常常让学生感觉枯燥无味,从而影响学习积极性和主动性.中等职业学校数学教师要因势利导,唤醒学生对数学学习的兴趣,树立学好数学的信心.     

Assessment Theory and Research for Classrooms: From Taxonomies to Constructing Meaning in Context

Why should we take a broader view of assessment theory and research when constructing measures of school achievement? How might we incorporate a learner's awareness of his or her own learning strategies into the assessment process? How could an assessment developer combine the use of psychological constructs and varying contexts for learning in the assessment of competence in mathematics?     

新数学教学模式:从“学会”到“会学”

本文针对五年制高职学生的特点．结合自己的数学教学实践．就如何为学生架起从“学会”到“会学”的桥梁．谈一些个人的体会。     

Opportunities and Challenges of Implementing Instructional Games in Mathematics Classrooms: Examining the Quality of Teacher-Student Interactions During the Cover-up and Un-cover Games

This study explored the design and implementation of the Cover-up and Un-cover games, two manipulative-based fraction games, in 14 fifth-grade classrooms. We examined how the fraction concepts were integrated into the game design and explored the nature of teacher-student interactions during games using lesson videos. Our examination showed that interactions focused on game progress, rules, and turn taking with little exploration of strategies or the underlying mathematics. To compare the quality of teacher-student interactions during games with interaction quality during other instructional activities, we coded 5 videotaped lessons from each of the 14 classrooms. Statistical results from a dependent t test indicated that teacher-student interactions were of statistically significant lower quality during games than during non-game segments with similar instructional purpose. Teachers might benefit from additional curriculum support and training to implement games as rich mathematical learning opportunities.     

Optimizing Early Mathematics Experiences for Children from Low-Income Families: A Study on Opportunity to Learn Mathematics

Both the National Council of Teachers of Mathematics and the National Association for the Education of Young Children recognize that well-designed opportunity to learn mathematics can help improve mathematics achievement of students from low-income families and from minority backgrounds. Using data from a nationally representative sample, the Early Childhood Longitudinal Study-Kindergarten cohort, this study empirically examined a subset of children from low-income families to determine whether African American and Caucasian students have differential opportunity to learn mathematics and the extent to which opportunities to learn predict gains in mathematics achievement at kindergarten. Results indicated African American kindergartners have differential opportunity to learn mathematics than their Caucasian peers from low-income families. African American students were found to have received, on average, more reported instructional time spent on mathematics, higher use of math manipulatives, worksheets, textbooks and chalkboard work than their Caucasian peers. Moreover, greater opportunity to learn mathematics predicted higher mathematics achievement for both African American and Caucasian students from low-income families, specifically, opportunity to learn skills such as telling time, using measurement tools accurately, estimating quantities, and knowing the value of coins and cash.     

如何让学生乐学小学数学

在数学课堂教学的过程中,数学教师要想提高效率,激发学生学习数学的兴趣并收获知识,就应该实施快乐教学,让学生喜欢数学。接下来结合教学经验,谈谈在小学数学教学的过程中如何实施快乐教学。     

Learning to Learn to Teach: An ``Experiment' Model for Teaching and Teacher Preparation in Mathematics

This paper describes a model for generating andaccumulating knowledge for both teaching andteacher education. The model is applied firstto prepare prospective teachers to learn toteach mathematics when they enter theclassroom. The concept of treating lessons asexperiments is used to explicate theintentional, rigorous, and systematic processof learning to teach through studying one's ownpractice. The concept of planning teachingexperiences so that others can learn fromone's experience is used to put into practicethe notion of contributing to a sharedprofessional knowledge base for teachingmathematics. The same model is then applied tothe work of improving teacher preparationprograms in mathematics. Parallels are drawnbetween the concepts emphasized for prospectiveteachers and those that are employed byinstructors who study and improve teacherpreparation experiences. In this way, parallelsalso are seen in the processes used to generatean accumulating knowledge base for teaching andfor teacher education.     

谈谈怎样学好高等数学

学习高等数学需要掌握一定的方法和技巧。首先是对高等数学的主要内容和基本要求要心中有数,其次应着重理解重要的概念,尤其是区分好高等与初等数学的联系与不同点,掌握相关的性质。最后是学以致用。     

On Enhancing Teachers' Knowledge by Constructing Cases in Classrooms

This study was designed to enhance teachers' knowledge by constructing cases as part of a school-based professional development project in Taiwan. Cases, involving episodes and issues from real classroom events, were constructed collaboratively by a school-based team consisting of the researcher and four classroom teachers. The process of constructing cases, characterization of teachers' understanding of cases, and their skills for case writing were developed in the course of the study. In the process of constructing these cases, teachers improved their awareness of and their competence in dealing with the difficulties students encountered in the learning of mathematics; they enhanced their pedagogical content knowledge and their ability to reflect on classroom practices.     

一份极有价值的关于华人数学教育的著作--评《华人如何学习数学:局内者之研究》

《华人如何学习数学:局内者之研究》是一本从不同情景和角度出发,研究关于华人学生的数学教学的指南.全书分4个部分,20章,574页.它的出版表明数学教育可以从对相关国家在数学测试中的各自表现提出一些简单的问题进入一些更具挑战性和更有意义的研究领域.     

Investigating Forms of Children's Writing in Grade 7 Mathematics Classrooms

Recent changes in mathematics curricula, both in South Africa and elsewhere, have begun to change the overwhelmingly symbolic nature of mathematics in schools (in the sense of use of mathematical symbolism), promoting more use of the oral and written language. Engaging students in `Writing-to-Learn' activities in mathematics classrooms has been identified and claimed by various mathematics education researchers as having a positive impact on the learning of mathematics. In this paper, I report on a piece of research, which is part of a broader study, on forms of mathematical writing and written texts produced by learners in grade 7 (12–13-year-olds) classes in six junior high schools in KwaZulu-Natal, in South Africa.