Friendships and Group Work in Linguistically Diverse Mathematics Classrooms: Opportunities to Learn for English Language Learners

This ethnographic study examined students’ opportunities to learn in linguistically diverse mathematics classrooms in a Canadian elementary school. I specifically examined the contextual change of group work, which influenced opportunities to learn for newly arrived English language learners (ELLs). Based on analyses of video-recorded interactions, this study revealed a shift in these ELLs’ opportunities to learn from when they worked with teacher-assigned peers to when they worked with friends. In both settings, ELLs tended to be positioned as novices. However, when working with friends, they accessed a wider variety of work practices. In friend groups, ELLs were occasionally positioned as experts and had more opportunities to raise questions and offer ideas. In contrast, when working with teacher-assigned peers, ELLs tended to remain in the position of being helped. In some teacher-assigned groups, interactions were characterized as authoritative, and ELLs’ contributions and ideas were rejected or neglected without relevant justifications or mathematical authority established by their peers. The findings contribute to ongoing discussions on group work and friendship in linguistically diverse classrooms.     

Opportunities to Learn in the Budapest Mathematics Classroom

This paper examines the nature of the problems posed, the processes by which their solutions were made public, their structural contribution to a lesson and the opportunities they offered for the learning of mathematics in Budapest, Hungary. The problems were derived from observations, the processes of which were informed by various comparative studies, of 94 mathematics lessons in the 10–14 age range. Problems, the analysis and categorisation of which were informed by the literature, tended to fall into two categories – single response with high levels of cognitive demand and multiple response with high levels of cognitive demand. Structurally lessons fell into a well practised pattern of classroom behaviour. They began with a public review of homework, comprising two or three multiple response problems which were solved collaboratively. This was followed by a warm-up period involving the oral setting and mental solving of a few single response problems. Lastly, the main body of the lesson involved several cycles in which a multiple response problem was posed, attempted individually before solutions were shared publicly. Lessons ended with homework being set. The discussion of the findings is framed by Hungary's variable performance on recent international tests of mathematical attainment.     

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.     

Mathematics and Science Learning Opportunities in Preschool Classrooms

Research Findings: The present study observed and coded instruction in 65 preschool classrooms to examine (a) overall amounts and (b) types of mathematics and science learning opportunities experienced by preschool children as well as (c) the extent to which these opportunities were associated with classroom and program characteristics. Results indicated that children were afforded an average of 24 and 26 min of mathematics and science learning opportunities, respectively, corresponding to spending approximately 25% of total instructional time in each domain. Considerable variability existed, however, in the amounts and types of mathematics and science opportunities provided to children in their classrooms; to some extent, this variability was associated with teachers' years of experience, teachers' levels of education, and the socioeconomic status of children served in the program. Practice or Policy: Although results suggest greater integration of mathematics and science in preschool classrooms than previously established, there was considerable diversity in the amounts and types of learning opportunities provided in the preschool classrooms. Affording mathematics and science experiences to all preschool children as outlined in professional and state standards may require additional professional development aimed at increasing preschool teachers' understanding and implementation of learning opportunities in these 2 domains in their classrooms.     

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。     

Playful Talk: Negotiating Opportunities to Learn in Collaborative Groups

This case study examines the role of playful talk in negotiating the “how” of collaborative group work in a 6th-grade science classroom. Here we develop and test a Vygotsky-derived hypothesis that postulates playful talk as a mechanism for identity exploration and group status negotiation. Our findings indicate that students utilized the playful talk genre as a means to (a) position themselves and others as more or less competent within the group in order to create or foreclose opportunities to learn, (b) maintain and strengthen bonds that were developing within the group toward the goal of achieving higher levels of coordination and opportunities to learn, and (c) pivot toward a self-selected identity aimed sometimes at affecting one’s status within the group. Taken together, these 3 playful talk functions allowed the group to manage tensions that arose as a result of periods of low coordination and to open opportunities to learn for lower status group members. This study contributes to researchers’ understanding of affective aspects of collaborative learning, which are theorized as foundational to success in such endeavors.     

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.     

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.     

Quality in Kindergarten Classrooms: Observational Evidence for the Need to Increase Children's Learning Opportunities in Early Education Classrooms

Research Findings: Using observational data gathered in 730 kindergarten classrooms in 6 states, the present study focuses on the quality of children's learning opportunities in kindergarten classrooms. Findings show that overall, children experience moderate to low levels of quality in the areas of classroom organization and instructional support in kindergarten. Results are also presented in comparison to children's experiences in the pre-kindergarten year. These comparisons indicate that kindergarten children spend a greater proportion of the school day in language arts, math, and whole-group instruction and less time in centers than they do in pre-kindergarten. An examination of predictors of kindergarten classroom quality indicated that program characteristics (i.e., adult–child ratio, length of school day) and teacher psychological variables (i.e., beliefs and depressive symptoms) were stronger predictors of classroom quality than were teacher experience and educational background. Practice or Policy: Findings are discussed in terms of implications for children's academic and social development in kindergarten as well as for kindergarten teacher preparation and development.     

Exploring Opportunities for Argumentation in Modelling Classrooms

On several levels it can be said that the act of modelling in science is inherently an argumentative act. That is, in virtually all aspects of modelling, from developing a question to judging between competing models that might answer that question, an individual is engaged in persuasive acts. Those acts may be private or public. They may be mental, written or oral, but they are about judging ideas and making sense of them; convincing oneself or others that the ideas and ways of looking at and explaining a phenomenon are useful. These acts are what scientists find exciting. They are what make science intellectually interesting and challenging. Inviting students into this practice is one way to help them learn both the content and process of science. This paper introduces a framework that is attentive to the research on how people learn while simultaneously pushing for curriculum and instruction that engages students in elements of the practice of science. We explore how this framework can be used to foster argumentation by describing the theoretical underpinnings of the framework and using classroom examples to illustrate the utility of the framework for promoting argumentation.     

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.     

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.     

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。     

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。     

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?     

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.