如何在五年制师范生的数学教学中渗透数学思想方法

数学思想是数学知识的精髓,是知识转化为能力的桥梁。传授数学思想方法,就是教学生学数学、用数学的意识,这样才能使学生终生受益于数学教材,为他们将来从事的小学数学教育打下坚实的基础。而在数学教学中渗透常用的数学思想方法,这对于我们训练学生掌握解决问题的能力,提高学生综合从师素质能力非常重要。     

论数学思维能力中的分类思想

数学思想是数学知识和方法在更高层次上的抽象与概括,分类是一种重要的数学思想,分类思想是根据数学本质属性的相同点和不同点,将数学研究对象分为不同种类的一种数学思想。分类以比较为基础,比较是分类的前提,分类是比较的结果。分类应在同一标准下具有完备性和互斥性,不同的类之间的逻辑关系是＂或＂,因此,分类的结果是集合运算的＂并＂,通过分类思想的学习,培养学生思维的条理性,缜密性,提高学生的思维能力。初中数学中,分类问题总体归结为两类,涉及数与代数、空间与图形,因此,分类思想是安徽省中考每年必考的核心思想方法。     

The effects of IMPROVE on mathematical knowledge, mathematical reasoning and meta-cognition

The purpose of the present study is to examine the effects of IMPROVE, a meta-cognitive instructional method, on students' mathematical knowledge, mathematical reasoning and meta-cognition. Participants were 81 students who studied a pre-college course in mathematical. Students were randomly assigned into one of two groups and groups were randomly assigned into one of two conditions: IMPROVE vs. traditional instruction (the control group). Both groups were exposed to the same learning materials, solved exactly the same mathematical problems, and were taught by the same experienced teacher. The IMPROVE students were explicitly trained to activate meta-cognitive processes during the solution of mathematical problems. The control group was exposed to traditional instruction with no explicit exposure to meta-cognitive training. Results indicate that the IMPROVE students significantly outperformed their counterparts on both mathematical knowledge and mathematical reasoning. In addition, the IMPORVE students attained significantly higher scores then the control group on the three measures of meta-cognition: (a) general knowledge of cognition; (b) regulation of general cognition; and (c) domain-specific meta-cognitive knowledge. The theoretical and practical implications are discussed.     

Pathways to word problem solving: The mediating roles of schema construction and mathematical vocabulary

When solving word problems, many children encounter difficulties in making sense of the information and integrate it into a meaningful schema. This is the fundamental phase on which subsequent problem solution depends. To better understand the processing underlying this fundamental phase, this study examined the roles of schema construction and knowledge of mathematical vocabularies in word problem solving. The participants were 139 Chinese third graders studying in Hong Kong. Path analysis showed that there were two kinds of pathways to word problem solving: language-related and number-related. In particular, reading fluency was related to word problem solving in two mediated language-related pathways: one via schema construction, the other via knowledge of mathematical vocabularies. In the number-related pathway, arithmetic concept was related to word problem solving via knowledge of mathematical vocabularies. These findings highlight the specific roles of schema construction and mathematical vocabulary in word problem solving, thereby providing useful implications of how best to support children in understanding and integrating the information from the problem.     

试论高中生思维自立性的培养

思维自立性是指在自己解决所遇到的个人基本生活问题中形成的综合性的非人际的思维特征,对于提高学生综合素质具有重要意义。当前高中生思维自立性欠缺的主要原因是:思维懒惰,不想思考;学业至上,无暇思考;当局者迷,不知思考。应从激发动力、制定计划、细化目标三个方面加强高中生思维自立性的培养。     

Elementary school teachers' knowledge and beliefs regarding non-routine problems

This study focuses on the knowledge exhibited by 30 elementary school in-service and pre-service teachers in solving non-routine mathematical problems and on their beliefs regarding these kinds of problems. Interviews were used to reveal teachers' knowledge and beliefs. The findings indicated that these teachers had difficulty in solving non-routine problems and that their ability to solve these problems was influenced by their professional backgrounds. Most of the teachers, although failing to solve the given problems, expressed their willingness to give such problems to their students in class, explaining that such problems are important for students to learn how to solve as they help develop mathematical thinking and the skill of solving problems in everyday life. However, the teachers were unwilling to include such problems in examinations.     

“This is the First Time I’ve Done This”: Exploring secondary prospective mathematics teachers’ noticing of students’ mathematical thinking

This mixed methods study investigates the ways in which secondary mathematics prospective teachers acquire skills needed to attend to, interpret, and respond to students’ mathematical thinking and the ways in which their perceived strengths and weaknesses influence their skills when this type of formalized training is not part of their program. These skills (attending, interpreting, and responding) are defined as teachers’ professional noticing of students’ thinking. Results indicate that seniors respond to students’ thinking in significantly different ways from juniors and sophomores. Converging the data highlighted inconsistencies in how participants’ were making sense of students’ mathematical thinking, as well as in participants’ self-identified strengths and weaknesses.     

Word problems and mathematical reasoning-A study of children''s mastery of reference and meaning in textual realities

The mastery of word problems is seen as an important test of mathematical ability. When solving such problems, students supposedly go beyond rote learning and mechanical exercises to apply their knowledge to realistic problem situations in which mathematical reasoning becomes an important instrument for making concrete judgements. Research shows that performance on word problems is often surprisingly poor. Non-realistic, and even logically inconsistent, answers to word problems are often accepted by students, and there are many signs that students seldom make so-called realistic considerations when applying their mathematical knowledge to real world events. The study reported is a follow-up of the work by Verschaffel, De Corte, and Lasure (1994) in which the difficulties students have in making realistic considerations were clearly illustrated. In the present study, students (10–12 years of age) worked in groups, and the tasks given (estimating distances) were introduced as part of a general discussion of how to calculate distances to school. Results show that the participants were clearly able to entertain different assumptions regarding how to measure distances, and they make distinctions between alternative options when discussing, for instance, the distance between two villages as indicated on a road sign on the one hand, and when talking about the shortest possible distance on the other. It is argued that the problem of what constitutes a realistic consideration when solving word problems is far from simple but has to be understood in context.     

The mathematical discourse of 13-year-old partnered problem solving and its relation to the mathematics that emerges

This paper, written within a discursive perspective, explores the co-shaping of public and private discourse, and some of the circumstances under which one occasions the other, in the evolution of mathematical thinking by pairs of 13-year-olds. The discourse of six pairs of students, engaged in interpreting and graphing problem situations involving rational functions, was analyzed by means of recently developed methodological tools. The nature of the mathematics that emerged for each pair was found to be related to several factors that included the characteristics of the interpersonal object-level utterances both before and after the solution path had been generated, the degree of activity of the personal channels of the interlocutors, and the extent to which the thoughts of participants were made explicit in the public discourse. The analysis of the discursive interactions provided evidence that adolescents within novel problem situations can experience some difficulty in making their emergent thinking available to their partners in such a way that the interaction be highly mathematically productive for both of them. This revised version was published online in July 2006 with corrections to the Cover Date.     

‘I like it instead of maths’: how pupils with moderate learning difficulties in Scottish primary special schools intuitively solved mathematical word problems

This study by Lio Moscardini of the University of Strathclyde shows how a group of 24 children in three Scottish primary schools for pupils with moderate learning difficulties responded to word problems following their teachers' introduction to the principles of Cognitively Guided Instruction (CGI). CGI is a professional development programme in mathematics instruction based on constructivist principles developed at the University of Wisconsin‐Madison. The study found that the sample group of pupils were able to develop their understanding of mathematical concepts through actively engaging in word problems without prior explicit instruction and with minimal teacher adjustments. The pupils' conceptual understandings demonstrated by their solution strategies within CGI activities were generally not consistent with classroom records of assessment. The results were encouraging in illustrating the capacity of the sample group of pupils with moderate learning difficulties to reveal their mathematical thinking and considers the importance of this insight for instructional decision making.     

例谈直觉思维与数学解题

数学教学是数学思维活动的教学,直觉思维在数学思维活动中有着特殊的地位和作用。文章通过对直觉猜想、直觉洞察、直觉类比、数形结合、直觉归纳和审美直觉这六个方面举例论证直觉思维在数学解题活动中的作用。同时也分析了运用直觉思维解题需要注意的问题,并介绍了调控这些问题所必需掌握的知识,如数学观念、学习本质以及逻辑思维与直觉思维的互补作用。     

Number sense and the calculating child: Measure,multiplicity and mathematical monsters

ABSTRACT

Children and animals of all kinds are said to develop some degree of number sense. The search for ‘number neurons’ and neural correlates of computational thinking aims to identify biological primitives to explain the emergence of number sense. This work typically looks for the sources of number sense in organisms, but one might extend this search and study the possibility of a calculating matter more generally. Such a speculative project explores the implications of the non-human turn within the posthumanities. In this paper, I draw primarily on the work of Vicky Kirby and Gilles Deleuze in order to focus on becoming-monster through calculation. I show how calculation, as a machinic and empirical act that both serves and troubles images of mathematical truth, has always played a unique role in the production of mathematical monsters. I then discuss calculating children who participate in abacus clubs and annual abacus competitions, calculating at inhuman rates with imaginary abacuses. I argue that a new materialist philosophy of immanence demands a radically new approach to number sense.     

数学建模对高职院校学生职业能力的培养

在高职数学教学中融入数学建模思想,能激发学生学习数学的兴趣和积极性,体验数学在实际中的应用。通过数学建模这个载体,培养学生的表达、独立思考、自学、创新意识、团队协作、联想归纳、分析解决问题及计算机应用等方面能力,提高学生的综合素养。     

Mathematica在常微分方程实验中的应用

通过实验阐述用Mathematica求解各类常微分方程的输入格式和应注意的问题,使常微分方程的解法更直观、简便和高效,充分说明用Mathematica进行数学实验,有利于激发学生学习数学的兴趣,培养学生建立数学模型、使用计算机解决实际问题的能力.     

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.     

解析几何中的最值问题

解析几何是高中数学的重要内容,其主要特点是综合性强,在解题中几乎处处涉及函数与方程、不等式、三角等内容.因此,在教学中应重视对数学思想、方法进行归纳提炼,如方程思想、函数思想、参数思想、数形结合的思想、对称思想、整体思想等思想方法,达到优化解题思维、简化解题过程的目的.本文通过对一些典型例题的分析和解答,归纳了解析几何中常见的解决最值问题的思想方法,总结了解答典型例题的具体规律,并提供了一些常用的解题方法、技能与技巧.     

The role of a children's mathematical thinking experience in the preparation of prospective elementary school teachers

Historically, content preparation and pedagogical preparation of teachers in California have been separated. Recently, in integrating these areas, many mathematics methodology instructors have incorporated children's thinking into their courses, which are generally offered late in students’ undergraduate studies. We have implemented and studied a model for integrating mathematical content and children's mathematical thinking earlier, so that prospective elementary school teachers (PSTs) engage with children's mathematical thinking while enrolled in their first mathematics course. PSTs’ work with children in the Children's Mathematical Thinking Experience (CMTE) may enhance their mathematical learning. Preliminary study results indicate that the sophistication of CMTE students’ beliefs about mathematics, teaching, and learning increased more than the sophistication of beliefs held by students enrolled in a reform-oriented early field experience and that experiences considering children's mathematical thinking provided PSTs with increased motivation for learning mathematics.     

例谈数学期望在生活中的应用

数学期望是随机变量的重要数字特征之一,文章解析了数学期望在日常生活中的应用,如求职决策问题,投资问题,彩票问题等,从而不断激发学生学习数学的积极性和主动性,让学生在兴趣中学习探索,并应用于生活,让数学改变生活。     

中职数学教学中数学思想的挖掘与渗透点滴谈

数学教学在向学生展示获取知识、技能及解决问题的思维过程的同时,不仅要善于＂挖掘＂教材中蕴含的数学思想和数学方法,还要使数学思想方法的渗透自然地贯穿于整个教学过程中.