How Emergent Models May Foster the Constitution of Formal Mathematics

This article deals with the role that so-called emergent models can play in the process of constituting formal mathematics. The underlying philosophy is that formal mathematics is something that is, or should be, constituted by the students themselves. In the instructional design theory for realistic mathematics education, models always have been employed to foster a process in which formal mathematics is reinvented by the students themselves. This article describes how the use of models became more and more explicated over time and developed into the notion of emergent models. The design of an instructional sequence, which deals with flexible mental computation strategies for addition and subtraction up to 100, is taken as an instance for elaborating what is meant by emergent models and what role they play in fostering the constitution of formal mathematics. The analysis shows that there are 3 interrelated processes. First. at a more holistic level, there is a global transition in which “the model” initially emerges as a model of informal mathematical activity and then gradually develops into a model for more formal mathematical reasoning. Second, the transition from “model of” to “model for” involves the constitution of anew mathematical reality that can be denoted formal in relation to the original starting points of the students. Third, in the series of instructional activities, there is not 1 model, but the model actually is shaped as a series of signs, in which each new sign comes to signify activity with a previous sign in a chain of signification.     

How Informal Out-of-School Mathematics Can Help Students Make Sense of Formal In-School Mathematics: The Case of Multiplying by Decimal Numbers

This article presents a teaching experiment on the relationship between informal out-of-school and formal in-school mathematics, and the ways each can inform the other in the development of abstract mathematical knowledge. This study concerns the understanding of some aspects of the multiplicative structure of decimal numbers. It involved a series of classroom activities in upper elementary school, using suitable cultural artifacts and interactive teaching methods. To create a substantially modified teaching/learning environment, new sociomathematical norms (Yackel & Cobb, 1996) were also introduced. The focus was on fostering a mindful approach toward realistic mathematical modeling, which is both real-world based and quantitatively constrained sense-making (Reusser & Stebler, 1997). In addition, procedures not commonly used in ordinary teaching activities, such as estimation and approximation processes, were also introduced.     

元认知在高等数学教学中的作用及培养途径

元认知是认知主体对自身认知活动的认知,大学生的元认知能力对高等数学教学具有很大的影响作用.在高等数学教学中应通过多种途径,把培养和发展大学生的元认知能力作为一项重要教学任务来完成.     

Linearly Stratified Models for the Foundations of Nonstandard Mathematics

Assuming the existence of an inaccessible cardinal, transitive full models of the whole set theory, equipped with a linearly valued rank function, are constructed. Such models provide a global framework for nonstandard mathematics.     

作物垄作的数学模型

应用高等数学中对面积的曲面积分等方法,建立了垄作种植中半椭圆形、抛物线型和三角形垄的数学模型,比较了不同垄形、垄宽、垄高在增加单位土地表面积和突出地面垄体体积的效果.     

Mathematics and poetry: How wide the gap?

Despite the differences between mathematics and poetry, there are common threads enough to show that essential elements of mathematics are humanistic. In fact, the only way to avoid seeing anything poetic in mathematics is to read its language in a purely formalistic fashion, paying attention to denotation and the laws of logic, and to nothing else.     

如何提高高等数学课堂教学的质量

数学的课堂教学过程是在教师的传授和指导下进行学习、掌握数学知识、技能、思想、方法的一种认识过程.影响数学课堂教学质量的因素是多方面的,本文仅就提高高等数学课堂教学质量谈几点个人的认识.     

大学数学教学中数学思想的培养

在大学数学教学中引入数学历史人物的生平、研究问题以及研究方法的介绍对培养大学生正确的数学思想有着重要的意义。     

浅谈在高等数学教学中如何将基本概念形象化

高等数学中的基本概念具有高度抽象性，是教师讲授的难点，也是学生学习的重点，而基本概念的掌握是学生学习理解高等数学的关键，也是培养学生学习兴趣的前提．本总结了多种方法，将基本概念形象化变为可“听”，可“看”，可“摸”，降低了基本概念的教与学难度，提高了学生学好高等数学的兴趣。     

Implications of Informal Education Experiences for Mathematics Teachers' Ability to Make Connections Beyond Formal Classroom

The Common Core Standard for Mathematical Practice 4: Model with Mathematics specifies that mathematically proficient students are able to make connections between school mathematics and its applications to solving real‐world problems. Hence, mathematics teachers are expected to incorporate connections between mathematical concepts they teach and their applications to solving problems arising in real‐world situation. Clearly, it is assumed that the teachers themselves are able to make such connections. On the other hand, research shows that mathematics teachers find it difficult to make those connections. In this paper, we present the results of the study that investigated the ways in which exploring mathematics in informal sites, and in particular science museum, assist teachers with making connections between school mathematics and its applications in real world.     

How Grades 1–7 Teachers Assess Mathematics and How They Use the Assessment Data

The purpose of this study was to explore mathematics assessment methods employed by teachers and ways teachers used assessment data. The sample for the study consisted of 398 mathematics teachers in Grades 1 to 7 in a state in Australia. The teachers completed a Likert‐scale survey. Using Scheffe pair‐wise comparisons and correlation coefficients to examine the relationships between grade level, use of assessment data, and use of assessment techniques, the researchers found several significant differences in teaching and assessment cultures across Grades 1 to 7.     

Biomimicry of Social Foraging Bacteria for Distributed Optimization: Models,Principles, and Emergent Behaviors

In this paper, we explain the social foraging behavior of E. coli and M. xanthus bacteria and develop simulation models based on the principles of foraging theory that view foraging as optimization. This provides us with novel models of their foraging behavior and with new methods for distributed nongradient optimization. Moreover, we show that the models of both species of bacteria exhibit the property identified by Grunbaum that postulates that their foraging is social in order to be able to climb noisy gradients in nutrients. This provides a connection between evolutionary forces in social foraging and distributed nongradient optimization algorithm design for global optimization over noisy surfaces.     

浅谈高等数学中函数概念的整体把握

从结构、层次、特性等 8个方面对函数的本质及内外联系进行了概括 ,并对高等数学的教学提出了自己的看法