试论实分析教学实践中构建思维的培养

本文阐述了培养数学构建思维是培养创造性思维的一种体现。根据教学实践论述了构建函数是利于解证问题的心智活动；构建特定图形支持逻辑论证，是行之有效的方法。     

Mathematisches Denken in der Linearen Algebra

How can first years students learn to think and act mathematically by learning Linear Algebra? We want to present an approach that considers reflection of mathematical acting and its connections to general thinking to be an important part of learning. By understanding mathematics as a specific conventionalization of general thinking, patterns of general thinking can become the starting point for learning mathematics. This points out the specific contribution that mathematics can give to describe reality. By example of Linear Algebra, we discuss the common ground and differences between thinking in mathematics and in non-mathematical subjects. Based on this discussion, we analyse why and how these reflections can be objects of learning.     

概率论思维及其智力品质的培养

概率论思维是人脑和概率论研究对象交互作用并按照一般思维规律认识概率论内容的内在理性活动.它具有随机性、概括性、问题性、辐射性、指向性和创造性.提高概率论思维的效率及质量,必须从构筑知识平台,加强应用训练及强化批判意识等方面全面注意概率论思维智力品质的培养.     

引入自我调节发展大学生数学思维的研究

自我调节学习是按照现代教育理论发展起来的一种教育实践活动.分析了其理论基础和发展现状,针对大学生思维发展中辩证思维占优势、创造思维呈上升趋势等显著特点,结合高等数学自身以动态为主,高容量、深广度,较初等数学有质的飞跃的事实,分析了高等数学与初等数学的思维差异,阐明了自我调节运用于高等数学课程教学中的可能性和必要性,并通过教学实验研究,从获得的第一手材料证实了自我调节手段运用于高等数学教学中的可行性,从而也表明了它是促进学生数学思维发展的强有力的工具.     

Reflective abstraction in computational thinking

Computational thinking has become an increasingly popular notion in K-12 and college level education. Although researchers have accepted that abstraction is a central concept in computational thinking, they are quick to disagree on the meaning of it. A focus on reflective abstraction has led to the development of APOS Theory in Mathematics education. This has resulted in many cases of improved student learning in Mathematics (Arnon et al., 2013). Our main aim in this paper is to construct a theoretical bridge between computational thinking and APOS Theory and show that reflective abstraction can be used in the context of computational thinking.     

The idea of emergent property

Those who venture into systems thinking may find difficulties in identifying exactly to what the idea of emergent property refers and what its significance might be. Drawing upon the writings of four major systems thinkers, twelve aspects of emergent properties are identified. Simultaneously, four related epistemological tasks are made explicit. An overall result is that the idea of emergent property is a unifying epistemological concept. More generally, systems thinking may be understood as an epistemological theory, or at least as a theory whose strength lies in its epistemological aspects. Such an understanding is considered in view of the similar concerns and conceptual similarities which systems thinking shares with phenomenology, a consideration which yields two additional aspects of emergent properties. The correspondences seen to exist between systems thinking and phenomenology lead to the conclusion that the two fields taken together illuminate an untapped source for future interdisciplinary research.     

Defining and demonstrating an equivalence way of thinking in enumerative combinatorics

Counting problems offer opportunities for rich mathematical thinking, and yet there is evidence that students struggle to solve counting problems correctly. There is a need to identify useful approaches and thought processes that can help students be successful in their combinatorial activity. In this paper, we propose a characterization of an equivalence way of thinking, we discuss examples of how it arises mathematically in a variety of combinatorial concepts, and we offer episodes from a paired teaching experiment with undergraduate students that demonstrate useful ways in which students developed and leverage this way of thinking. Ultimately, we argue that this way of thinking can apply to a variety of combinatorial situations, and we make the case that it is a valuable way of thinking that should be prioritized for students learning combinatorics.     

Learning to notice students’ mathematical thinking through on-line discussions

The goal of this research is to characterize prospective mathematics teachers?? development of professional noticing of students?? mathematical thinking in on-line contexts. Specifically, we are interested in how the participation in on-line discussions, when prospective teachers solve specific tasks, supports the development of professional noticing of students?? mathematical thinking. Findings show that an aspect in which the on-line discussions, as an example of asynchronous collaborative communication interfaces, support this development is related to the role of writing; participating in an on-line discussion plays a significant role since the final written text is functional as regards the activity of interpreting students?? mathematical thinking collaboratively.     

Student interpretations of the terms in first-order ordinary differential equations in modelling contexts

A study of first-year undergraduate students′ interpretational difficulties with first-order ordinary differential equations (ODEs) in modelling contexts was conducted using a diagnostic quiz, exam questions and follow-up interviews. These investigations indicate that when thinking about such ODEs, many students muddle thinking about the function that gives the quantity to be determined and the equation for the quantity's rate of change, and at least some seem unaware of the need for unit consistency in the terms of an ODE. It appears that shifting from amount-type thinking to rates-of-change-type thinking is difficult for many students. Suggestions for pedagogical change based on our results are made.     

MODELS AND SUPER-MODELS: WAYS OF THINKING ABOUT PROFESSIONAL KNOWLEDGE IN MATHEMATICS TEACHING[1]

Teaching about teaching is a complex process requiring knowledge about teaching as well as knowledge about teaching about teaching. We have published findings on research carried out over the last few years about teachers’ subject knowledge. This research led to the proposal of a model for thinking about subject knowledge which distinguishes between knowledge needed to pass an examination and knowledge needed to help someone else to come to know that knowledge. The first is necessary but not sufficient for the latter. This model for thinking about subject knowledge has led to proposals for similar models for thinking about other aspects of teacher knowledge and has more recently developed into a parallel model for thinking about teacher education.     

Perspectives on Advanced Mathematical Thinking

This article sets the stage for the following 3 articles. It opens with a brief history of attempts to characterize advanced mathematical thinking, beginning with the deliberations of the Advanced Mathematical Thinking Working Group of the International Group for the Psychology of Mathematics Education. It then locates the articles within 4 recurring themes: (a) the distinction between identifying kinds of thinking that might be regarded as advanced at any grade level, and taking as advanced any thinking about mathematical topics considered advanced; (b) the utility of characterizing such thinking for integrating the entire curriculum; (c) general tests, or criteria, for identifying advanced mathematical thinking; and (d) an emphasis on advancing mathematical practices. Finally, it points out some commonalities and differences among the 3 following articles.     

Acting with systems intelligence: integrating complex responsive processes with the systems perspective

The systems thinking literature assumes that the concept of a system is useful in management and organizational research. Ralph D. Stacey and his collaborators, however, have questioned this. They have presented the theory of complex responsive processes (CRP) as an alternative to systems thinking. We argue that systems thinking and the CRP perspective are complementary. The CRP illuminates many of the micro-behavioural, local interaction and creativity-related organizational phenomena whereas the systems perspective is useful for other purposes. CRP misses the mark in its criticism of systems thinking. The insights of CRP should and could be incorporated, not switched, with the systems perspective. The systems intelligence perspective, proposed by Hämäläinen and Saarinen, provides a framework to accomplish that. By integrating systems thinking and the CRP model we hope to provide a platform from which it is possible to appreciate the relative merits of the two apparently conflicting strands of thought.     

一道积分不等式的多种证法

通过对一道积分不等式提供多种不同的证法,希望能对学生创造性思维及发散性思维的培养、开阔解题思路、提高综合应用数学知识的能力等有所帮助,并使学生对证明不等式的常用方法有所了解.     

An Exploratory Study of College Students' Views of Mathematical Thinking in a Historical Approach Calculus Course

Heuristic training alone is not enough for developing one's mathematical thinking. One missing component is a mathematical point of view. This study reports findings regarding outcomes of a historical approach calculus course to foster Taiwanese college students' views of mathematical thinking. This study consisted of 3 stages. During the initial phase, 44 engineering majors' views on mathematical thinking were tabulated by an open-ended questionnaire, and 9 randomly selected students were invited to participate in follow-up interviews. Students then received an 18-week historical approach calculus course in which mathematical concepts were problematized to challenge their intuition-based empirical beliefs about doing mathematics. Near the end of the semester, all participants answered the identical questionnaire, and we interviewed the same students to pinpoint any shifts in their views on mathematical thinking. We found that participants were more likely to value logical sense, creativity, and imagination in doing mathematics. Further, students were leaning toward a conservative attitude toward certainty of mathematical knowledge. Participants' focus seemingly shifted from mathematics as a product to mathematics as a process.     

Scalar and vector line integrals: A conceptual analysis and an initial investigation of student understanding

This paper adds to the growing body of research happening in multivariable calculus by examining scalar and vector line integrals. This paper contributes in two ways. First, this paper provides a conceptual analysis for both types of line integrals in terms of how theoretical ways of thinking about definite integrals summarized from the research literature might be applied to understanding line integrals specifically. Second, this paper provides an initial investigation of students’ understandings of line integral expressions, and connects these understanding to the theoretical ways of thinking drawn from the literature. One key finding from the empirical part is that several students appeared to understand individual pieces of the integral expression based on one way of thinking, such as adding up pieces or anti-derivatives, while trying to understand the overall integral expression through a different way of thinking, such as area under a curve.     

如何在高等数学教学中培养学生的创新思维

创新是当今的时代精神.创新能力的培养是实施素质教育的重要目标之一.高等数学作为高等教育的重点基础课程,在训练和培养学生创新能力方面具有重要地位.如何在高等数学教学过程中培养学生的创新思维,提高创新能力是我们高等数学教学改革的重要任务.文章通过对当前教育形势的分析以及创新思维的特点的思考,从教学理念、教学模式以及教学内容三个方面讨论了在高等数学教学过程中学生的创新思维的培养问题.