谈谈无穷级数中蕴涵的数学思想方法

以无穷级数这一章教学内容为例,探讨教学内容中蕴涵的＂极限＂＂、比较＂＂、转化＂等数学思想方法,使学生在学习知识的同时掌握数学思想方法,更好地领会数学内容和数学精神,从而提高学生的文化素质和学习能力。     

数学教学中培养学生阅读能力的实验与思考

数学阅读是数学学习的基础环节之一，它常常被许多中学生所忽视．数学教学中要针对不同课型，讲授不同的阅读方法；提出作业要求，培养良好的阅读习惯；推荐课外阅读书籍，加强阅读指导；及时组织阅读后的交流，提高阅读兴趣等等培养学生阅读能力，以更好地实施素质教育．     

Multicultural Student-to-Student Dialogue and the Generation of Creative Ideas

Both participation in dialog and exposure to others from different cultures have been theorized to increase the ability of an individual to be creative. This study focused on the multicultural student-to-student dialog that took place in one graduate level course. The purpose of this study was to explore how participation in this dialog might influence student creativity, specifically how the students generated creative ideas for their final course papers. No connection was found between participation in the multicultural student-to-student dialog and the generation of creative ideas for their final papers. However, insight was gained into the creative process used by students in generating their final paper ideas.     

Role of teacher characteristics and school resources in early mathematics learning

The authors investigated the degree to which school-level teacher characteristics and resources are related to the mathematics learning of kindergarten children using a sample drawn from a large US database. Kindergarten-level teacher characteristics were operationalised as collective teacher efficacy, teacher experience and teacher preparation; kindergarten resources were operationalised as efforts that schools make to reach out to parents, classrooms with mathematics manipulatives, and classroom technology. Kindergarten students learned more in mathematics over a year in schools where teachers’ collective efficacy was high. School resources yielded a range of significant yet mixed results in young children’s mathematics achievement. Notably, younger students and students entering kindergarten with lower levels of mathematical skills benefited greatly from classroom resources. Overall, the presence of classroom technology as a school-level resource was positively related with older kindergarten students’ learning. Based upon the findings, organisational and policy implications are discussed.     

文化源导入与大学英语教学

要想在大学英语教学中培养学生运用英语的交际能力, 就应在语言教学中进行一定文化导入, 而了解一个民族文化的总体特征将更能增强学生对此文化的敏感性和洞察力。本文引入文化源的概念, 尝试以文化源头导入的方式为先导, 来进行大学英语教学, 并结合文本用霍夫斯特德所提出的文化模式来进行说明。     

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.     

Knowledge,beliefs and pedagogy: how the nature of science should inform the aims of science education (and not just when teaching evolution)

Lisa Borgerding’s work highlights how students can understand evolution without necessarily committing to it, and how learners may come to see it as one available way of thinking amongst others. This is presented as something that should be considered a successful outcome when teaching about material that many students may find incompatible with their personal worldviews. These findings derive from work exploring a cause célèbre of the science education community—the teaching of natural selection in cultural contexts where learners feel they have strong reasons for rejecting evolutionary ideas. Accepting that students may understand but not commit to scientific ideas that are (from some cultural perspectives) controversial may easily be considered as a form of compromise position when teaching canonical science prescribed in curriculum but resisted by learners. Yet if we take scholarship on the nature of science seriously, and wish to reflect the nature of scientific knowledge in science teaching, then the aim of science education should always be to facilitate understanding of, yet to avoid belief in, the ideas taught in science lessons. The philosophy of science suggests that scientific knowledge needs to be understood as theoretical in nature, as conjectural and provisional; and the history of science warns of the risks of strongly committing to any particular conceptualisation as a final account of some feature of nature. Research into student thinking and learning in science suggests that learning science is often a matter of coming to understand a new viable way of thinking about a topic to complement established ways of thinking. Science teaching should then seek to have students appreciate scientific ideas as viable ways of making sense of the currently available empirical evidence, but should not be about persuading students of the truth of any particular scientific account.     

Development of a Framework to Characterise the Openness of Mathematical Tasks

Educators usually mean different constructs when they speak of open tasks: some may refer to pure-mathematics investigative tasks while others may have authentic real-life tasks in mind; some may think of the answer being open while others may refer to an open method. On the other hand, some educators use different terms, e.g. open and open-ended, to mean the same construct, while others distinguish between these terms. It is difficult to hold a meaningful discussion or to define clearly an area of research on open tasks if the idea of what constitutes the construct of openness is vague. Moreover, what students learn depends on the types of tasks that they are given, and different kinds of tasks place differing cognitive demands on students. Thus, the objectives of this article are to clarify the types of mathematical tasks and develop a framework to characterise their openness based on five task variables: goal, method, task complexity, answer and extension; and to discuss how different types of tasks and openness may affect student learning. The openness framework can help teachers to design or select more appropriate tasks to cater to students with different abilities in order to develop in them various kinds of mathematical thinking processes, and it can also make it easier for researchers to study the interaction between different types of openness and student learning.     

Unraveling the culture of the mathematics classroom: A video-based study in sixth grade

Changing perspectives on mathematics teaching and learning resulted in a new generation of mathematics textbooks, stressing among others the importance of mathematical reasoning and problem-solving skills and their application to real-life situations. The article reports a study that investigates to what extent the reform-based ideas underlying these mathematical textbooks impact the current teaching of mathematics. Two problem-solving lessons were videotaped in 10 sixth-grade classrooms and a coding scheme was developed to analyze these lessons with regard to three aspects of the classroom culture that are assumed to enhance students’ mathematical beliefs and problem-solving competencies: (1) the classroom norms that are established, (2) the instructional techniques and classroom organization forms, and (3) the set of tasks students are confronted with. Two instruments were administered to measure students’ beliefs about learning mathematical word problem solving, and to assess their problem-solving processes and skills. The results indicate that some reform-based aspects seemed to be easier to implement (e.g., a strong focus on heuristic skills, embedding tasks in a realistic context) than others (e.g., the use of group work, an explicit negotiation of appropriate social norms).     

费曼技巧在本科生电路课程中的运用与思考

随着信息时代的不断发展,课堂教学面临着前所未有的挑战.现今流行的MOOC以及翻转课堂并未侧重实现费曼技巧中教授他人的核心过程.本文以作者讲授的本科生电路课程教学现状为例,分析电路教学中存在的问题,并基于费曼技巧提出改进思路,通过引发新旧知识体系共鸣、开设匿名讨论群、实施学生主讲习题课教学等学习方式,使学生在教授他人的过程中深思内化知识,提高学生学习的积极性.     

A description and characterization of student activity in an open, online, mathematics help forum

Free, open, online, calculus forums are websites where students from around the world can post course-related queries that may be viewed and responded to by anonymous others. These sites are an emergent resource for students seeking help and have become a part of many students’ mathematical experience. The purpose of this paper is to introduce and describe the forums to the research community as a computer-mediated form of help seeking, and to briefly characterize the forms of student activity in one popular calculus help forum. Two hundred exchanges on limit and related rates were collected and examined for evidence that students are contributing ideas and proposals for action when they initiate the exchange and following intervention. The findings justify the need for further research since students are using the forums in a variety of ways and sometimes participate meaningfully. The forums have potential for testing student responses to pedagogical approaches.     

Learning in collaborative settings: students building on each other’s ideas to promote their mathematical understanding

The purpose of this study is to contribute insights into how collaborative activity can help promote students’ mathematical understanding. A group of six high school students (15- to 16-year olds) worked together on a challenging probability task as part of a larger, after-school, longitudinal study on students’ development of mathematical ideas in problem-solving settings. The students solved the problem and produced a valid justification of their solution. This study shows that collaborative activity can help promote students’ mathematical understanding by providing opportunities for students to critically reexamine how they make claims from facts and also enable them to build on one another’s ideas to construct more sophisticated ways of reasoning. Implications for classroom teaching and ideas for future research are also discussed. The study helps address a documented need for a better understanding of how mathematical learning evolves in social settings.     

The nature of student predictions and learning opportunities in middle school algebra

In this article, we describe how using prediction during instruction can create learning opportunities to enhance the understanding and doing of mathematics. In doing so, we characterize the nature of the predictions students made and the levels of sophistication in students’ reasoning within a middle school algebra context. In this study, when linear and exponential functions were taught, prediction questions were posed at the launch of the lessons to reflect the mathematical ideas of each lesson. Students responded in writing along with supportive reasoning individually and then discussed their predictions and rationale. A total of 395 prediction responses were coded using a dual system: sophistication of reasoning, and the mechanism students appeared to utilize to formulate their prediction response. The results indicate that using prediction provoked students to connect among mathematical ideas that they learned. It was apparent that students also visualized mathematical ideas in the problem or the possible results of the problem. These results suggest that using prediction in fact provides learning opportunities for students to engage in mathematical sense making and reasoning, which promotes students’ understanding of the mathematics that they learn.     

The views of Pasifika students in New Zealand about communicating mathematically

This paper focuses on the perspectives of Years 7 and 8 Pasifika students on mathematics learning, in particular their views about the communication of solution strategies with others (their peers), and their teacher. Pasifika students' ideas about the importance of communicating their mathematical reasoning and strategies to peers and teachers were analysed and themes identified. While most students were in agreement about the importance of explaining their thinking to others, there was considerable variation in the reasons they gave for their views. The findings are interpreted in relation to socio-cultural theory and recent writing about requirements for citizenship in the twenty-first century.     

Literacy,achievement and success in a Māori tourism certificate programme: reading the world in order to read the word

ABSTRACT

This article critiques international assessment of adult literacy using research findings from students completing a Māori tourism certificate who achieved significant gains in assessment. It is argued that the focus of literacy assessments potentially forces educators to narrow their teaching and learning approaches, manoeuvring them into teaching toward singular or convergent literacy. This leads to utilising teaching and learning strategies drawn from the cultural and social capital of the dominant culture, which is problematic for students without abundance of such capital.

Blending Kaupapa Māori research theory with appreciative inquiry, research revealed that students made significant gains in assessment scores because their educators acknowledged and utilised ways in which they made sense and meaning of their world. Educators drew upon the social and cultural capital of students and engaged them as partners in culturally based teaching and learning processes. Relating to students’ multiple literacies, and the contexts with which they bring meaning to their world in this way, is referred to as drawing from divergent literacies.

This study proposes a need to develop and research alternative ways for improving convergent literacy outcomes. It is suggested that multiple interventions using multi-modal and 360° approaches need to occur simultaneously for best outcomes.     

Secondary School Students’ Construction and Use of Mathematical Models in Solving Word Problems

This study focussed on how secondary school students construct and use mathematical models as conceptual tools when solving word problems. The participants were 511 secondary-school students who were in the final year of compulsory education (15–16 years old). Four levels of the development of constructing and using mathematical models were identified using a constant-comparative methodology to analyse the student’s problem-solving processes. Identifying the general in the particular and using the particular to endow the general with meaning were the key elements employed by students in the processes of construction and use of models in the different situations. In addition, attention was paid to the difficulties that students had in using their mathematical knowledge to solve these situations. Finally, implications are provided for drawing upon student’s use of mathematical models as conceptual tools to support the development of mathematical competence from socio-cultural perspectives of learning.     

Students' Ideas about Reaction Rate and its Relationship with Concentration or Pressure

This cross‐sectional study identifies key conceptual difficulties experienced by upper secondary school and pre‐service chemistry teachers (N = 191) in the area of reaction rates. Students' ideas about reaction rates were elicited through a series of written tasks and individual interviews. In this paper, students' ideas related to reaction rate and its relationship with concentration or pressure are discussed. Evidence is presented to support the following claims. First, school students tended to use “macroscopic” modelling rather than using “particulate” and/or “mathematical” modelling. By contrast, undergraduates were more likely to provide explanations based upon theoretical models and entities within established chemical ideas. Nevertheless, second, they had conceptual difficulties in making transformation within and across different theoretical models. Finally, students did not generally use a scientifically acceptable concept of reaction rate across contexts. Although an acceptable concept may have been used in one context, incorrect ideas may, nonetheless, have been used in other contexts. However, undergraduates' responses were less affected by context. Several conceptual difficulties exhibited by school students persisted among undergraduates. Some possible implications for planning the curriculum and teaching are proposed in the light of the results.     

Using Concept Cartoons in Formative Assessment: Scaffolding students’ argumentation

The purpose of this study was to investigate how concept cartoons, together with other diagnostic and scaffolding tools, could be used in formative assessment, to stimulate talk and argumentation among students in small groups, as part of peer‐assessment and self‐assessment; and to provide diagnostic feedback about students’ misconceptions to the teacher for teaching towards conceptual change. Two classes of Primary 5 and 6 students worked in small groups to discuss the opposing viewpoints posed by the cartoon characters, using scaffolding tools to guide their discussions and to evaluate, challenge, and document each others’ ideas. Students also used drawings to depict their ideas. The conversation from one group was audio‐taped. These tools provided a record of students’ thinking in a form that was accessible to the teacher for monitoring and feedback purposes. Findings showed dialogic talk and interactive argumentation among students where they made their reasoning visible. Students’ assertions and questions had formative potential as they encouraged exploratory and reflective discourse by drawing upon each others’ ideas. The teacher’s discursive practices, as well as her role in designing scaffolding structures for supporting ‘assessment conversations’ when using concept cartoons and in devising strategies that take into account students’ conceptual and epistemic thinking, are emphasised.     

新课程背景下英语教学理念的转变

实现“以教师为中心”向“以学生为中心”教学角色的转变，以及实现“以结果为中心”向“以过程为中心”的评价方式的转变，是新课程环境下建构全新的英语教学理念，实现英语语言教学改革之必由之路。     

Learning-centred Teaching

Distinctions are drawn between different kinds of learning (such as ‘understanding’, ‘factual knowledge’ and ‘skills’) and between different kinds of learners (such as ‘holist’ versus ‘serialist’ learners, and between ‘visualizers’, ‘verbalizers’ and ‘oers’) so that teaching methods can be matched to them. Learning-centred teaching therefore involves adopting teaching methods, such as ‘student-centred learning’, ‘active learning’, ‘didactive teaching’, ‘problem-based learning’, etc., where they are appropriate, but not where they are inappropriate.