关于中学生数学观的思考

中学生的数学观总体上是朴素的、肤浅的,有合理的认识也有不合理的认识。文章从三个方面分析了中学生具有这种数学观的原因,并在此基础上展望了中学生具有怎样的数学观才算是合理的:即从动态的、易谬主义数学观,静态的、绝对主义数学观,工具主义数学观和文化主义数学观的不同层面,全方位的认识数学。     

数学教科书中的多元文化问题

数学观是数学教育的核心一环。在我国数学教育中占主导地位的是以知识客观性、普遍性和价值中立性为主要特征的柏拉图主义数学观。数学文化观对该数学观进行了重要补充。新课程改革十分关注学生的文化背景,现行教科书也在很多方面体现了数学文化多样性,但也存在一些可以完善的方面。在教科书编制过程中,应完善教科书编制者的数学观,挖掘数学多元文化素材,并以显性与隐性相结合的方式将数学的文化多样性呈现在教科书中。     

数学观的嬗变及其对数学教育的启示

数学观是人们对数学的总体看法。数学观是不断演变的。数学观的发展经历了从绝对主义数学观到可误主义数学观以及建构主义数学观三个阶段。数学观的演变给数学教育极大的启示,在数学教育中,我们应该把握数学观的深刻内涵,领悟数学是一种文化,理解数学知识的本质,把握数学思想方法,鉴赏数学美和追求数学精神。     

数学认识论的历史及其发展趋势

从数学基础研究的角度看，数学认识论的发展经历了从数学基础主义的“确证”观到庞加菜、皮亚杰等人的数学的“发现”的认识观，直至数学社会学理论中所强调的“确证”与“发现”相结合的数学认识观。数学观有由“绝对主义”向“可误主义”或“拟经验主义”转变的发展趋势。     

数学观的层面分析

无论从数学教育的角度还是从数学创造的角度,对数学观的研究都有必要进行层面分析,从社会层面看,数学观的形成与民族的经济文化相关联;从数学的本体层面,数学观是对数学本质的理性思考;从数学教育上看,主要表现为功利性与素质性两种观念.     

建构主义学习理论在小学数学教育教学中的应用

数学哲学现代发展的一个重要特点就是,数学观已发生了革命性的变化;传统的、静态的、绝对主义的数学观逐渐为动态的、易谬的（或者说,经验与拟经验的）数学观所取代。随着新的数学观的形成就意味着数学教育教学观也要发生相应的变化,从本质上来说,数学教学就是一种数学活动,教学不应看成一系列静态的现成结论的传授,而应该是一个动态的、活动的、创造性的过程。对     

论数学心理观的科学基础

数学心理观是适应我国心理科学发展的形势要求，在当代联结主义认知心理学空前的数学研究及其应用的时代背景下提出来的。当代脑科学、人工智能技术以及数学哲学、实验心理学等多学科的发展历程充分证明，数学心理观作为心理“量”属性的基本观念、理论心理学元理论的新内涵，安全具有现实的科学基础。我国新世纪的心理科学不仅应在方法论上确立心理的量化观念，而且应在认识论上确立心理的“量”性本质的数学心理观。     

认知主义学习观与情境主义学习观

认知主义学习观与情境主义学习观是影响当今教育的两种主要的学习观.由于两种学习观在什么是知识、怎样理解动机与迁移等理论方面存在分歧,从而,产生了不同的教育环境,形成了不同的课程设计以及不同的评价方法.两种学习观既有对立的方面,但也有统一的地方.     

数学是什么？——初中数学教材中“数学文化”之分析

“数学观”即对“数学是什么”的认识。数学的文化观是其中的一种解释，即认为“数学是一种文化”。本文希望从数学文化的角度剖析新教材对这一特点的呈现：（1）数学是人类文化的创造性活动；（2）猜想也是数学；（3）数学内容是丰富多彩的；（4）数学是身边的科学；（5）数学问题的答案不是唯一的。     

知识观转型中教师主体性的回归

"照着讲"和"接着讲"体现了教师不同的知识观.理性主义知识观和经验主义知识观陷入了客体对主体的牵制和绝对主义,因而是一种权威主义知识观;建构主义知识观和后现代主义知识观则倡导主体建构和批判反思,因而是一种批判主义知识观.从权威主义知识观向批判主义知识观的转型,消失了的教师的主体性也重新得到了回归.     

Linking university mathematics classroom environments to student achievement: The mediation of mathematics beliefs

This study examined the relationships between university mathematics classroom environments, mathematics beliefs, and achievement in the context of mainland China. As a type of affect in mathematics learning, mathematics beliefs including mathematics conceptions and efficacy were examined as mediators. Based on analysis of a sample of 1443 undergraduate students from a university in northern China, the study not only found that the mathematics classroom environment was associated with both affective and cognitive mathematics outcomes, but also confirmed the mediating role of mathematics beliefs. Specifically, a desirable classroom environment was related to students’ cohesive conceptions, enhanced self-efficacy, and higher mathematics achievement. In contrast, students’ perceived difficulty of learning mathematics was connected to their fragmented conceptions, reduced self-efficacy and lower mathematics achievement. Implications of the study are discussed.     

Undergraduate Students’ Conceptions of Mathematics: An International Study

In this paper, we report on an international study of undergraduate mathematics students’ conceptions of mathematics. Almost 1,200 students in five countries completed a short survey including three open-ended questions asking about their views of mathematics and its role in their future studies and planned professions. Responses were analysed starting from a previously-developed phenomenographic framework (Reid et al., 2003) which required only minor modification. Students’ conceptions of mathematics ranged from the narrowest view as a focus on calculations with numbers, through a notion of mathematics as a focus on models or abstract structures, to the broadest view of mathematics as an approach to life and a way of thinking. Broader conceptions of mathematics were more likely to be found in later-year students (p<0.001) and there were significant differences between universities (p<0.001). The information obtained from the study not only confirms previous research, but also provides a basis for future development of a monitoring questionnaire.     

The influences of conceptions of mathematics and self‐directed learning skills on university students' achievement in mathematics

This study tested the mediating role of self‐directed learning skills (SDL) between students’ conceptions of mathematics and their achievement in mathematics using a structural equation model. Data were collected using the “Conceptions of Mathematics Questionnaire” and the “Self‐Rating Scale of Self‐Directed Learning”, together with students’ achievement in mathematics. A sample of 440 first year university students at King Saud University participated in the study. The findings confirm the moderating role of students’ self‐directed learning skills between their conceptions of mathematics and their achievement in mathematics. Students who have a highly fragmented conception of mathematics tended to have low SDL skills, and in turn low mathematics achievement (partial mediation), whereas students who have a highly cohesive conception of mathematics tended to have high self‐directed learning skills, and in turn high mathematics achievement (full mediation). Mathematics educators should be aware that students’ conceptions of mathematics may be influential, but not sufficient to predict achievement in mathematics. Equipping students with appropriate conceptions of mathematics and self‐directed learning skills is key to enhancing their performance in mathematics.     

Conceptions of mathematics and how it is learned: The perspectives of students entering university

This paper reports results from an investigation to identify the conceptions of mathematics held by beginning university students and their approaches to the study of mathematics. Phenomenographic techniques were used to analyze responses to a questionnaire administered to approximately 300 students. An analysis of the results identified a structural relationship between students' conceptions of mathematics and their approaches to learning it, with the majority of students viewing mathematics as a necessary set of rules and procedures to be learned by rote. The results of this research have implications for the ways in which teaching and learning are constituted within universities.     

A comparison of Hungarian and English teachers' conceptions of mathematics and its teaching

This paper reports on a statistical study of English and Hungarian teachers' conceptions of mathematics and its teaching. A questionnaire was developed and distributed to teachers of mathematics in 200 English and40 Hungarian schools teaching children in the 11–14 age range. Factor analyses identified four conceptions of mathematics and five of mathematics teaching. These were compared with those yielded by an earlier study involving the same English teachers and found to be consistent indicating the existence of similar conceptions in different educational systems. Differences and similarities in the strengths with which those conceptions are held were suggestive of both global and national conceptual traditions. The significant similarity to emerge concerned teachers from both countries sharing, with similar strengths, a general conception of mathematics teaching incorporating the teaching of mathematical skills, a variety of classroom approaches including investigations and problem-solving, and a recognition that mathematics provides an essential lifetool. Multi-dimensional scaling indicated that English teachers have their perspectives informed by two underlying, and possibly conflicting, traditions– pedagogic relevance and mathematical utility. The Hungarians appeared concerned only with notions of pedagogic relevance – those practices perceived to facilitate effective learning of a subject which is untainted by utilitarian perspectives. This revised version was published online in July 2006 with corrections to the Cover Date.     

Tablet-based AR technology: Impacts on students’ conceptions and approaches to learning mathematics according to their self-efficacy

Most studies of Augmented Reality (AR) in education have considered students’ learning outcomes and motivation. Previous studies have revealed that AR has the potential to help students learn abstract conceptions in mathematics. In this paper, a series of statistics and probability lessons using AR installed on tablets was designed and developed to examine the effect of the AR technology by comparing the conceptions and learning approaches of junior high school students with different levels of self-efficacy. A total of 101 students were divided into two groups based on their mathematics learning self-efficacy. The analysis of the results shows that AR applications in mathematics courses can help students with higher self-efficacy to pay closer attention to higher level conceptions. It can also help higher self-efficacy students to apply more advanced strategies when learning mathematics.     

The relationship of teachers' conceptions of mathematics and mathematics teaching to instructional practice

Case studies were conducted to investigate the conceptions of mathematics and mathematics teaching held by three junior high school teachers. Examination of the relationship between conceptions and practice showed that the teachers' beliefs, views, and preferences about mathematics and its teaching played a significant, albeit subtle, role in shaping their instructional behavior. Differences among the teachers in their conceptions and practices are explained followed by a discussion of properties of their conceptual systems.This paper is based on the author's doctoral dissertation which was directed by Professor Thomas J. Cooney of the University of Georgia and was partly supported by a grant from San Diego State University. I wish to express my appreciation to the teachers who participated in the study and their administrators.     

Elementary-age children’s conceptions about mathematics utility and their home-based mathematics engagement

Integrating multiple theoretical frameworks, the authors examined rising first- to fourth-grade students’ mathematics utility conceptions—their knowledge and beliefs about the usefulness of mathematics, home-based mathematics engagement, and grade-level differences in mathematics utility conceptions and home engagement. Most children viewed mathematics as heavily focused on low-level mathematics operations and as learned and used primarily in school. Older children showed more awareness of mathematics as part of daily living, but still viewed mathematics as mostly school-based—more so than their younger counterparts. Results suggest that awareness of mathematics in daily life may be associated with children’s mathematics utility value (perceived usefulness of mathematics). Although children engaged in activities at home with the potential to foster mathematics development, the frequency of engagement was not related to their awareness of mathematics in daily activities. Thus, there may be untapped opportunities for young children to connect the mathematics they learn in school to their daily life.     

数学教师的数学观和数学教育观

作为数学教师素质结构的先导性成分,数学观由数学知识观、数学本质观和数学价值观构成.数学教育观包括数学教育目的观、数学教育过程观和数学教育价值观3个主要部分.数学教师的数学观对其数学教育观有重要影响.数学教育文化观、数学教育哲学观和数学教育价值观之间有着复杂的互动关系.     

Building on student mathematical thinking in whole-class discourse: exploring teachers’ in-the-moment decision-making,interpretation, and underlying conceptions

Often, mathematics teachers do not incorporate whole-class discourse of students’ various ideas and solution methods into their teaching practice. Particularly complex is the in-the-moment decision-making that is necessary to build on students’ thinking and develop their collective construction of mathematics. This study explores the decision-making patterns of five experienced Dutch mathematics teachers during their novice attempts at orchestrating whole-class discourse concerning students’ various solution methods. Our goal has been to unpack the complexity of their in-the-moment decision-making during whole-class discourse through lesson observations and stimulated recall interviews. We investigated teacher decision-making adopting a model that combines two perspectives, namely (1) we explored student-teacher interaction with regard to building on student thinking and (2) we explored how the teachers based decisions during such interaction upon their own personal conceptions and interpretation of student thinking. During these novice attempts at orchestrating whole-class discourse, the teachers created many situations for students to articulate their thinking. We found that at certain instances, teachers’ in-the-moment decision-making resulted in opportunities to build on student thinking that were not completely seized. During such instances, the teachers’ decision-making was shaped by the teachers’ own conceptions of the relevant mathematics and by teacher conceptions that centered around student understanding and mathematical goals. Our findings suggest that teachers might be supported in their novice attempts at whole-class discourse by explicit discussion of the mathematics and of their conceptions with regard to student understanding and mathematical goals.