Making the transition to formal proof

This study examined the cognitive difficulties that university students experience in learning to do formal mathematical proofs. Two preliminary studies and the main study were conducted in undergraduate mathematics courses at the University of Georgia in 1989. The students in these courses were majoring in mathematics or mathematics education. The data were collected primarily through daily nonparticipant observation of class, tutorial sessions with the students, and interviews with the professor and the students. An inductive analysis of the data revealed three major sources of the students' difficulties: (a) concept understanding, (b) mathematical language and notation, and (c) getting started on a proof. Also, the students' perceptions of mathematics and proof influenced their proof writing. Their difficulties with concept understanding are discussed in terms of a concept-understanding scheme involving concept definitions, concept images, and concept usage. The other major sources of difficulty are discussed in relation to this scheme.This article is based on the author's doctoral dissertation completed in 1990 at the University of Georgia under the direction of Jeremy Kilpatrick.     

Supporting higher levels of reflection among teacher candidates: a pedagogical framework

An integrated mathematics and science methods course was designed to focus on the knowledge, skills, and beliefs of teacher candidates. Teacher candidates were involved in experiences that would prompt them to consider the influence of their experiences on their beliefs, the influence of their beliefs on their instructional decisions and the impact of those decisions on their effectiveness as teachers. Three teacher candidates who exhibited high levels of reflection and who made significant changes in the ways that they thought about mathematics and the teaching/learning process were chosen for study. Two themes related to this cognitive change emerged: a more complex, sophisticated understanding of the nature of mathematics; and an increased focus on children's thinking to guide instruction. Three program components having a major impact on these students' reconceptualization were reflection on past experiences, engagement in mathematical problem‐solving and opportunities to act on new beliefs.     

Concept mapping in research on mathematical knowledge development: Background,methods, findings and conclusions

The theoretical background and different methods ofconcept mapping for use in teaching and in research on learning processes are discussed. Two mathematical projects, one on fractions and one on geometry, are presented in more detail. In the first one special characteristics of concept maps were elaborated. In the second one concept mapping allowed students' individual understanding to be monitored over time and provided information about students' conceptual understanding that would not have been obtained using other methods. Regarding the students' individual concept maps in more detail there were some additional findings: (i) The characteristics of the maps change remarkably from fourth grade to sixth grade; (ii) There is some evidence that prior knowledge related to some mathematical topics plays a very important role in students' learning behaviour and in their achievement; (iii) Concept maps provide information about how individual students relate concepts to form organised conceptual frameworks; (iv) Long-term difficulties with specific concepts are able to be traced. These findings are discussed with regard to results of other studies as well as to their implications for the teaching of mathematics in the classroom.     

高等代数教学中数学思维若干性质的两重性

在师范院校数学系基础课程中,高等代数作为数学系最重要的基础课之一,在建构学生知识体系中起着重要的基础作用。高等代数的学习是比较困难的。学生课听得懂,但习题无从下手。因为它的抽象性,学生还不习惯、也没有这方面的思维基本训练。分析问题的原因,主要是学生的学习习惯,尤其是数学思维方法存在着缺陷。具体说对数学思维个性品质的两重性认识不够。本文通过高等代数教学,对存在着的七种优秀思维品质所具有的两重性进行了剖析,旨在教学中应引起足够的重视。     

小学数学教师的学科知识：专家与非专家教师的对比分析

经采用问卷测查法，考察了32名小学数学专家与非专家教师。的学科知识。结果表明，两类教师在数学知识与数学学科本质的理解方面表现出明显的差异。与非专家教师相比，专家教师对数学知识具有深刻的理解，包括深层的概念理解与结构化的知识组织。专家教师倾向于用“问题解决”的观点看待数学学科与学生的数学学习，而非专家教师则更倾向于“掌握知识”的观点。     

论数学思想方法的教学

数学思想是数学的灵魂,也是学好数学的重要武器。因此,数学思想方法是学生形成良好认知结构的纽带,是知识转化为能力的桥梁,是培养学生的数学观念,形成良好思维素质的关键。     

Factors that influence teachers learning to do interviews to understand students' mathematical understandings

Understanding students' understanding of mathematical ideas can inform mathematics teaching, and task-based interviews are one way in which teachers can learn more about their students' understandings. The CIME project was designed to empower mathematics teachers to use interviews to understand their students' mathematical understandings as well as to prepare teachers to use technology-intensive curricula. This study examined the influences on three high school mathematics teachers as they learned to use task-based interviews to understand students' mathematical understandings. The areas of teacher knowledge and conceptions that influenced the teachers we studied were: teachers' mathematical understandings and knowledge of technology and the perceived importance of curriculum topics; teachers' views of knowing mathematics; teachers' perceptions of students' characteristics and needs; and teachers' perceptions of interviewing and the role of questioning in interviews. This revised version was published online in August 2006 with corrections to the Cover Date.     

Mathematics for teaching and deep subject knowledge: voices of Mathematics Enhancement Course students in England

This article reports an investigation into how students of a mathematics course for prospective secondary mathematics teachers in England talk about the notion of ‘understanding mathematics in depth’, which was an explicit goal of the course. We interviewed eighteen students of the course. Through our social practice frame and in the light of a review of the literature on mathematical knowledge for teaching, we describe three themes that weave through the students’ talk: reasoning, connectedness and being mathematical. We argue that these themes illuminate privileged messages in the course, as well as the boundary and relationship between mathematical and pedagogic content knowledge in secondary mathematics teacher education practice.     

数学建模的专业教学实践与认识

开设数学建模课,一方面可以培养学生的数学建模能力,提高学生把专业理论知识运用于实践的能力;另一方面,数学建模课的学习却在一定程度上影响了专业课的学习。因而如何将数学建模思想和方法融入到大学专业主干课程的教学中,是当前数学和专业教学体系、内容、方法改革面临的一个新课题。     

初中学生数学学习过程分析

初中学生的知识结构、认知特点、年龄特征决定其在数学学习过程中有自身的发展规律和模式。初中数学教学要有效地促进学生数学学习,就必须在充分分析影响学生数学学习的各种因素的基础上,针对学生数学学习业绩具有影响力的本质因素进行教学。学生学习数学知识的认知心理特征和由不同性质的学习内容所决定的不同学习方式,是影响学生的数学学习最本质的因素。文章就初中学生的认知心理和知识结构两方面,对初中学生的数学学习过程进行分析。     

A Cognitive Analysis of Problems of Comprehension in a Learning of Mathematics

To understand the difficulties that many students have with comprehension of mathematics, we must determine the cognitive functioning underlying the diversity of mathematical processes. What are the cognitive systems that are required to give access to mathematical objects? Are these systems common to all processes of knowledge or, on the contrary, some of them are specific to mathematical activity? Starting from the paramount importance of semiotic representation for any mathematical activity, we put forward a classification of the various registers of semiotic representations that are mobilized in mathematical processes. Thus, we can reveal two types of transformation of semiotic representations: treatment and conversion. These two types correspond to quite different cognitive processes. They are two separate sources of incomprehension in the learning of mathematics. If treatment is the more important from a mathematical point of view, conversion is basically the deciding factor for learning. Supporting empirical data, at any level of curriculum and for any area of mathematics, can be widely and methodologically gathered: some empirical evidence is presented in this paper.     

Metacognition in basic skills instruction

Metacognition is increasingly recognized as important to learning. This article describes self-regulatory processes that promote achievement in the basic skills of reading and mathematical problem solving. Self-regulatory behaviors in reading include clarifying one's purpose, understanding meanings, drawing inferences, looking for relationships, and reformulating text in one's own terms. Self-regulatory behaviors in mathematics include clarifying problem goals, understanding concepts, applying knowledge to reach goals, and monitoring progress toward a solution. The article then describes the author's experiences integrating metacognition with reading and mathematics instruction and highlights students' reactions to learning to think metacognitively.     

高等数学教学中创设情境的探究

现今大学生普遍认为高等数学难学,学习高等数学的兴趣不高。要改变这一现状就要努力探究如何利用学生已有的认知结构,挖掘数学思想、利用数学史实和知识间的辩证关系、利用具体实例抽象出数学概念,通过在学生认知冲突等方面创设情境,让学生在情境中感性地探索、发现、理解和掌握高等数学的内容、思想和本质。     

Ability stereotyping in mathematics

Ability is a concept central to the current practice of mathematics teaching. However, the widespread view of mathematics learning as an ordered progression through a hierarchy of knowledge and skill, mediated by the stable cognitive capability of the individual pupil, can be sustained only as a gross global model, and is of limited value in describing and understanding the particular cognitive capabilities of individual pupils in order to plan, promote and evaluate their learning. In effect, individual pupils, and groups of pupils, are subject to ability stereotyping; characterisation in terms of a summary global judgement of cognitive capability, associated with overgeneralised and stereotyped expectations of mathematical behaviour, and stereotyped perceptions of an appropriate mathematics curriculum.     

KNOWLEDGE TRANSFER IN BIOLOGY AND TRANSLATION ACROSS EXTERNAL REPRESENTATIONS: EXPERTS' VIEWS AND CHALLENGES FOR LEARNING

Recent curriculum reform promotes core competencies such as desired ‘content knowledge’ and ‘communication’ for meaningful learning in biology. Understanding in biology is demonstrated when pupils can apply acquired knowledge to new tasks. This process requires the transfer of knowledge and the subordinate process of translation across external representations. This study sought ten experts’ views on the role of transfer and translation processes in biology learning. Qualitative analysis of the responses revealed six expert themes surrounding the potential challenges that learners face, and the required cognitive abilities for transfer and translation processes. Consultation with relevant curriculum documents identified four types of biological knowledge that students are required to develop at the secondary level. The expert themes and the knowledge types exposed were used to determine how pupils might acquire and apply these four types of biological knowledge during learning. Based on the findings, we argue that teaching for understanding in biology necessitates fostering ‘horizontal’ and ‘vertical’ transfer (and translation) processes within learners through the integration of knowledge at different levels of biological organization.     

数学史在素质教育中的价值

数学史不仅是学习数学、认识数学的工具,也是教师用来引进一些数学思想与数学方法、使学生体会数学对人类文明发展作用的有力武器。数学史可以启迪学生的思想;数学史知识本身和数学家的可歌可泣的许多事迹,有利于对学生进行爱国主义教育;数学家的严谨态度和锲而不舍的探索精神,有利于培养学生良好的学习态度和激发创新精神。     

论中小学生的数学观

中小学生的数学观包括数学知识观、数学学习观和数学自我概念。它是通过学生自身数学实践活动经验、教师的教学目标和过程以及社会文化与学校文化传统三方面交互作用的过程形成的。它对学生的数学学习行为、学习策略、动机与情感都会产生重要影响 ,从而对良好数学学习成绩的获得有重要作用     

教师变量对小学生数学学习成绩影响的多水平分析

32名小学数学教师与这些教师所教班级的1691名学生参与了本研究。两个测量工具评价了教师的数学学科知识与学科教学知识,对教师的55节数学课进行了录像,并按照学习任务的认知水平与课堂对话的特点进行了编码,同时测查了学生的期末数学学习成绩。多水平分析结果表明:教师的学科教学知识、课堂学习任务的认知水平、课堂对话中教师提问问题的类型与对话的权威来源对学生的数学成绩具有显著的预测作用;而教师的学科知识对学生数学成绩的影响未达到显著性水平。     

高效数学教学行为的特征

高效数学教学行为与低效教学行为相比较应该凸显科学性、智慧性与艺术性等特征.其中,科学性是指数学教师在教心、导学与发挥数学的教育性方面更具有合理性,即能够恰当确定教学目标以及教学重点与难点,在数学认知方面重视促进学生的深刻理解与帮助学生建立良好的数学认知结构,在非认知方面促进激发学生的数学求知欲与求识欲,在元认知方面给予学生必要的数学学习方法指导,恰到好处发挥数学的教育性,让学生适时沐浴数学精神、思想与方法,获得理性的数学思维的教育.智慧性是指在选择教学内容以及教学方法等方面具有智慧,在调控教学节奏方面也显现着教学的智慧.艺术性是指在教学、形体与板书语言方面以及管理方面显现着艺术特征.     

Mathematical knowledge for teaching teachers: knowledge used and developed by mathematics teacher educators in learning to teach via problem solving

In this study, we report on what types of mathematical knowledge for teaching teachers (MKTT) mathematics teacher educators (MTEs) use and develop when they work together and reflect on their teaching in a Community of Practice while helping prospective primary teachers (PTs) generate their own mathematical knowledge for teaching in learning mathematics via problem solving. Two novice MTEs worked with an experienced MTE and reflected on the process of learning to teach via problem solving and supporting PTs in developing deep understandings of foundational mathematical ideas. Taking a position of inquiry as stance, we examined our experiences teaching mathematics content courses for PTs via problem solving. We found that all of the MTEs used and developed some MKTT through (a) understanding and deciding on the mathematical goals of both the individual lessons and the two-course sequence as a whole, (b) choosing and facilitating tasks, and (c) using questions to scaffold PTs learning and engage them in mathematical processes such as making conjectures, justifying their reasoning, and proving or disproving conjectures.