高等数学中行列式知识教学探析

行列式是高等代数课程里基本而重要的内容之一,是为了求解线性方程组而引入的,在线性代数和其他数学领域以及工程技术中有着广泛的应用,懂得如何计算行列式显得尤为重要。行列式知识教学首先要合理引入概念,在清楚掌握行列式性质的前提下,灵活地选择方法将之应用于具体问题。     

高职学校数学教学中融入数学史的重要性

数学史融入数学教学既受到了越来越多的数学教育实践者的关注,也得到了数学教育研究者的重视。研究"数学教学中融入数学史"这个问题的目的,从数学的角度来看,是要深化对数学的认识,同时要更好地促进数学未来的发展;而从教学的角度来看,希望借由数学史与数学教育的互动,提升数学教师的教学质量与学生的学习成效。     

Procedural knowledge in mathematics: the role of the curriculum

Over the last few years, the information processing model of cognition has become increasingly prominent in the field. With this model, and other related research in cognitive science, there is a greater appreciation for the role of instruction in learning theory. One of the model's central tenets is that, as learners become more competent, they move from declarative to procedural states of knowledge. Researchers have concerned themselves with the efficiency of this transition, as well as with any systematic misconceptions that develop along the way. A carefully developed curriculum addresses both of these concerns. This article will focus on three principles of curriculum development: the nature of examples, explicitness, and parsimony. Recently conducted research that supports these principles will also be discussed. It is suggested that special educators strongly consider these principles when selecting or developing curricular material for students with learning disabilities.     

The PISA mathematics regime: knowledge structures and practices of the self

Within mathematics education research, the responses to the Programme for International Student Assessment’s (PISA’s) international testing regime tend to accept its framework and results as necessary points of reference, even when offering a critical reinterpretation or challenging national policy discourses based on PISA. In this article, we offer a different approach to the critique of PISA, drawing on theoretical tools provided by Bernstein and Foucault, to ask what the PISA regime achieves. Our understanding of this achievement encompasses both the production of knowledge structures and the production of students, teachers and other agents as subjects. We propose that the theoretical approach we offer provides a methodological entry point into analysis of the texts comprising the PISA mathematics regime. Analysis of a single PISA item is used to illustrate the insights that may be gained from such a theoretical lens. Such insights into the logic of PISA have the potential to allow us better to understand and hence contest the role that PISA and other large-scale assessment regimes may play in global and local policy discourses.     

The importance of gender in the assessment of historical knowledge

Current efforts to assess higher education have highlighted the need for better evaluation of student outcomes and educational needs. Several recent studies have called for an increase in the use of tests as part of assessment activities. However, assessment research has not evaluated whether tests are an appropriate technique for measuring student outcomes, particularly when women or minority students are involved. Using data gathered from a 1992 historical knowledge exam, this study illustrates how the assessment of women's academic achievements can be distorted by using tests as measures of student outcomes. At the same time, the results indicate a need to rethink how we teach history at the university level. Consequently, this paper suggests that the use of tests as an assessment tool should be limited to questions about curricular shortcomings and not be expanded as a means to evaluate student proficiency.     

Elevate or relegate? The relative importance of mathematics

The value of any subject on a crowded, contested and compulsory curriculum is, or at least should be, open to debate. More importantly, when different subjects are prioritized over others, the justification for the ranking of such subjects should also be submitted to sustained enquiry. Mathematics enjoys a prestigious place in the English National Curriculum, elsewhere, and perhaps everywhere. There have been several responses, over the last few years, to the question of not only why pupils should be compelled to learn mathematics but also for how long this should continue. This article takes a broad and critical look, from a largely philosophical standpoint, at some of the dominant themes that underlie the most recurrent claims that have been made in defence of the privileged place of mathematics in school.     

论文化背景知识在英语教学中的重要性

语言是文化的载体,文化依托语言得以传承和发展;文化又影响着语言,对于语言的形成和发展有着至关重要的作用。要学好一门语言,不能仅仅依靠词汇的累积和语法知识的背诵,对其所扎根的文化也要有深刻的了解,英语学习也是如此。教师应当深刻地认识到这一点,将文化背景知识的讲解充分融入到日常的教学活动当中去,才能够使英语教学发挥出应有的效果。     

The meaning of mathematics instruction in multilingual classrooms: analyzing the importance of responsibility for learning

In the multilingual mathematics classroom, the assignment for teachers to scaffold students by means of instruction and guidance in order to facilitate language progress and learning for all is often emphasized. In Sweden, where mathematics education is characterized by a low level of teacher responsibility for students’ performance, this responsibility is in part passed on to students. However, research investigating the complexity of relations between mathematics teaching and learning in multilingual classrooms, as well as effect studies of mathematics teaching, often take the existence of teachers’ responsibility for offering specific content activities for granted. This study investigates the relations between different aspects of responsibility in mathematics teaching and students’ performance in the multilingual mathematics classroom. The relationship between different group compositions and how the responsibility is expressed is also investigated. Multilevel structural equation models using TIMSS 2003 data identified a substantial positive influence on mathematics achievement of teachers taking responsibility for students’ learning processes by organizing and offering a learning environment where the teacher actively and openly supports the students in their mathematics learning, and where the students also are active and learn mathematics themselves. A correlation was also revealed between group composition, in terms of students’ social and linguistic background, and how mathematics teaching was performed. This relationship indicates pedagogical segregation in Swedish mathematics education by teachers taking less responsibility for students’ learning processes in classes with a high proportion of students born abroad or a high proportion of students with low socio-economic status.     

The development of beginning mathematics teacher pedagogical content knowledge

Recent research efforts (Schmidt et al. in The preparation gap: teacher education for middle school mathematics in six countries, MSU Center for Research in Mathematics and Science Education, 2007) demonstrate that teacher development programs in high-performing countries offer experiences that are designed to develop both mathematical knowledge and pedagogical knowledge. However, identifying the nature of the mathematical knowledge and the pedagogical content knowledge (PCK) required for effective teaching remains elusive (Ball et al. in J Teacher Educ 59:389–407, 2008). Building on the initial conceptual framework of Magnusson et al. (Examining pedagogical content knowledge, Kluwer, Dordrecht, pp 95–132, 1999), we examined the PCK development for two beginning middle and secondary mathematics teachers in an alternative certification program. The PCK development of these two individuals varied due to their focus on developing particular aspects of their PCK, with one individual focusing on assessment and student understanding, and the other individual focusing on curricular knowledge. Our findings indicate that these individuals privileged particular aspects of their knowledge, leading to differences in their PCK development. This study provides insight into the specific aspects of PCK that developed through the course of actual instructional practice, providing a lens for future research in this area.     

Large‐scale mathematics assessment: looking globally to act locally

In this paper, I will argue that it is possible to use data from large‐scale international and national mathematics assessment programmes, whose attention is on summative achievement, to provide formative information that informs teachers about the effects of their classroom practice. However, to have impact on, and be useful for, classroom practitioners, these achievement data need to be reworked and re‐presented in ways that are plausible, provide a basis for inferences about practice, and be appropriate for the intended audience. This paper examines achievement‐focused assessment programmes in terms of their aims and approaches, and develops the argument that formative assessment possibilities are present, within these programmes, although usually hidden. Examples are drawn from several sources to support this argument, and demonstrate a variety of approaches that have been taken in the past. Suggestions for further action are made.     

Children's intuitive mathematics: the development of knowledge about nonlinear growth

This study is concerned with the development of children's intuitive understanding of nonlinear processes. The ability to estimate linear and exponential growth was examined in 7-, 9-, 11-, and 13-year-old children and adults (N=160). Whereas linear growth was judged correctly at all ages, estimations of exponential growth were in line with mathematically correct values only in 13-year-olds and adults. However, 9-year-olds already judged the result of exponential growth as being significantly higher than that of linear growth, and even a remarkable proportion of 7-year-olds showed such discrimination between the two types of functions. Results point to the existence of an early intuitive knowledge about the characteristics of nonlinear growth, long before those functions are taught in school.     

Truth and the renewal of knowledge: the case of mathematics education

Mathematics education research must enable adjustment to new conditions. Yet such research is often conducted within familiar conceptualisations of teaching, of learning and of mathematics. It may be necessary to express ourselves in new ways if we are to change our practices successfully, and potential changes can be understood in many alternative, sometimes conflicting, ways. The paper argues that our entrapment in specific pedagogic forms of mathematical knowledge and the styles of teaching that go with them can constrain students’ engagement with processes of cultural renewal and changes in the ways in which mathematics may be framed for new purposes, but there are some mathematical truths that survive the changing circumstances that require us to update our understandings of teaching and learning the subject. In meeting this challenge, Radford encountered a difficulty in framing notions of mathematical objectivity and truth commensurate with a cultural–historical perspective. Following Badiou, this paper distinguishes between objectivity, which is seen necessarily as a product of culturally generated knowledge, and truth, as glimpsed beyond the on-going attempt to fit a new language that never finally settles. Through this route, it is shown how Badiou’s differentiation of knowledge and truth enables us to conjure more futuristic conceptions of mathematics education.     

Beyond words: the importance of gesture to researchers and learners

Gesture has privileged access to information that children know but do not say. As such, it can serve as an additional window to the mind of the developing child, one that researchers are only beginning to acknowledge. Gesture might, however, do more than merely reflect understanding-it may be involved in the process of cognitive change itself. This question will guide research on gesture as we enter the new millennium. Gesture might contribute to change through two mechanisms which are not mutually exclusive: (1) indirectly, by communicating unspoken aspects of the learner's cognitive state to potential agents of change (parents, teachers, siblings, friends); and (2) directly, by offering the learner a simpler way to express and explore ideas that may be difficult to think through in a verbal format, thus easing the learner's cognitive burden. As a result, the next decade may well offer evidence of gesture's dual potential as an illuminating tool for researchers and as a facilitator of cognitive growth for learners themselves.