《线性代数（非数学专业）》整体教学的实践与认识

首先明确了《线性代数（非数学专业）》整体教学的目的和实践的过程,其次从学生构建《线性代数》知识、技能和思想方法的角度总结了《线性代数》整体教学实践的一些体会,最后指出《线性代数》整体教学应把握数学观念,更好地将启发式教学与问题解决结合起来．     

Matrix multiplication and transformations: an APOS approach

This study presents a contribution to research in undergraduate teaching and learning of linear algebra, in particular, the learning of matrix multiplication. A didactical experience consisting on a modeling situation and a didactical sequence to guide students’ work on the situation were designed and tested using APOS theory. We show results of research on students’ activity and learning while using the sequence and through analysis of student’s work and assessment questions. The didactic sequence proved to have potential to foster students’ learning of function, matrix transformations and matrix multiplication. A detailed analysis of those constructions that seem to be essential for students understanding of this topic including linear transformations is presented. These results are contributions of this study to the literature.     

From the didactical triangle to the socio-didactical tetrahedron: artifacts as fundamental constituents of the didactical situation

Research on the use of artifacts such as textbooks and digital technologies has shown that their use is not a straight forward process but an activity characterized by mutual participation between artifact and user. Taking a socio-cultural perspective, we analyze the role of artifacts in the teaching and learning of mathematics and argue that artifacts influence the didactical situation in a fundamental way. Therefore, we believe that understanding the role of artifacts within the didactical situation is crucial in order to become aware of and work on the relationships between the teacher, their students and the mathematics and, therefore, are worthwhile to be considered as an additional fundamental aspect in the didactical situation. Thus, by expanding the didactical triangle, first to a didactical tetrahedron, and finally to a ??socio-didactical tetrahedron??, a more comprehensive model is provided in order to understand the teaching and learning of mathematics.     

Integer number sense and notation: A case study of a student with a mathematics learning disability

Although approximately 6% of students have a mathematics learning disability (MLD) also known as dyscalculia, little is known about how MLD impacts students beyond basic arithmetic. In this study we focused on one mathematical topic foundational to algebra – integer operations – and conducted a videotaped design experiment with one student with MLD. Through 14 teaching episodes we explored the ways in which standard mathematical tools (e.g., symbols, representations) were inaccessible and evaluated the design of alternative tools. Our detailed retrospective analysis revealed that the student had an unconventional understanding of integer quantities and symbolic notation, which resulted in issues of accessibility and persistent difficulties. Deliberate attempts to address inaccessibility revealed nuances in the student’s understanding, and suggests that both number sense and notational issues needed to be addressed in tandem. Implications for instruction are discussed.     

Transparent Rule Based CAS to Support Formalization of Knowledge

The complexity of computer algebra systems hinders many students to develop an adequate mental model of such a system. As a result, they are often suspicious about the results and the didactical benefit is limited. The paper suggests that it is possible to design a transparent version of a computer algebra system that gives students a transparent access to the inner working of such a system. Didactical uses of such a system are discussed.     

Designing a duo of material and digital artifacts: the pascaline and Cabri Elem e-books in primary school mathematics

This paper focuses on a duo of artifacts, constituted by a physical artifact and its digital counterpart. It deals with the theoretically and empirically underpinned design process of the digital artifact, the e-pascaline developed with Cabri Elem technology, in reference to a physical artifact, the pascaline. The theoretical frameworks of the instrumental approach and the theory of semiotic mediation together with the analysis of teaching experiments with the pascaline support the design in terms of continuity and discontinuity between the two artifacts. The components of the digital artifact were chosen from among the components of the physical artifact that are the object of instrumental genesis by the students and that are analyzed as having a semiotic potential that contributes to didactical aims. Components instrumented by students which had inadequate semiotic potential were eliminated. With the resulting duo, each artifact adds value to the use of the other artifact for mathematical learning.     

Improving Algebra Preparation: Implications From Research on Student Misconceptions and Difficulties

Through historical and contemporary research, educators have identified widespread misconceptions and difficulties faced by students in learning algebra. Many of these universal issues stem from content addressed long before students take their first algebra course. Yet elementary and middle school teachers may not understand how the subtleties of the arithmetic content they teach can dramatically, and sometimes negatively, impact their students' ability to transition to algebra. The purpose of this article is to bring awareness of some common algebra misconceptions, and suggestions on how they can be averted, to those who are teaching students the early mathematical concepts they will build upon when learning formal algebra. Published literature discussing misconceptions will be presented for four prerequisite concepts, related to symbolic representation: bracket usage, equality, operational symbols, and letter usage. Each section will conclude with research‐based practical applications and suggestions for preventing such misconceptions. The literature discussed in this article makes a case for elementary and middle school teachers to have a deeper and more flexible understanding of the mathematics they teach, so they can recognize how the structure of algebra can and should be exposed while teaching arithmetic.     

The didactical tetrahedron as a heuristic for analysing the incorporation of digital technologies into classroom practice in support of investigative approaches to teaching mathematics

There have been various proposals to expand the heuristic device of the didactical triangle to form a didactical tetrahedron by adding a fourth vertex to acknowledge the significant role of technology in mediating relations between content, student and teacher. Under such a heuristic, the technology vertex can be interpreted at several levels from that of the material resources present in the classroom to that of the fundamental machinery of schooling itself. At the first level, recent research into teacher thinking and teaching practice involving use of digital technologies indicates that, while many teachers see particular tools and resources as supporting the classroom viability of investigative approaches to mathematics, the practical expressions of this idea in lessons vary in the degree of emphasis they give to a didactic of reconstruction of knowledge, as against reproduction. At the final level, examining key structuring features of teaching practice makes clear the scope and scale of the situational adaptation and professional learning required for teachers to successfully incorporate use of digital tools and resources in support of investigative approaches. These issues are illustrated through examining contrasting cases of classroom use of dynamic geometry in professionally well-regarded mathematics departments in English secondary schools.     

Indirect proof: what is specific to this way of proving?

The study presented in this paper is part of a wide research project concerning indirect proofs. Starting from the notion of mathematical theorem as the unity of a statement, a proof and a theory, a structural analysis of indirect proofs has been carried out. Such analysis leads to the production of a model to be used in the observation, analysis and interpretation of cognitive and didactical issues related to indirect proofs and indirect argumentations. Through the analysis of exemplar protocols, the paper discusses cognitive processes, outlining cognitive and didactical aspects of students’ difficulties with this way of proving.     

Problem solving as a challenge for mathematics education in The Netherlands

This paper deals with the challenge to establish problem solving as a living domain in mathematics education in The Netherlands. While serious attempts are made to implement a problem-oriented curriculum based on principles of realistic mathematics education with room for modelling and with integrated use of technology, the PISA 2003 results suggest that this has been successful in educational practice only to a limited extent. The main difficulties encountered include institutional factors such as national examinations and textbooks, and issues concerning design and training. One of the main challenges is the design of good problem solving tasks that are original, non-routine and new to the students. It is recommended to pay attention to problem solving in primary education and in textbook series, to exploit the benefits of technology for problem solving activities and to use the schools’ freedom to organize school-based examinations for types of assessment that are more appropriate for problem solving.     

线性代数教学中的归纳与演绎方法

针对线性代数概念多、内容抽象、逻辑性强等特点，介绍归纳和演绎方法在线性代数教学中的应用，并借助具体案例加以说明．     

“金课”背景下强化线性代数应用教学的实践与探索

线性代数课程具有高度的抽象性,再加上课时等多方面限制,使得学生学起来感觉非常枯燥.针对此问题,在线性代数课程中,适当增加应用的讲解,以增加学生对于相关知识应用的认知,提高学生综合利用线性代数知识解决问题的能力.通过调查问卷,分析了在线性代数课程中增加应用知识讲解的经验、存在的问题及应对措施.     

The learning and teaching of linear algebra: Observations and generalizations

This paper is about a teaching experiment (TE) with inservice secondary teachers (hereafter “participants”) in the theory of systems of linear equations. The TE was oriented within particular social and intellectual climates, and its design and implementation took into consideration a series of findings concerning the difficulties students have in linear algebra. The questions we set for this study were: (1) Did the participants in the particular TE climates construct viable knowledge in the theory of systems of linear equations? Our criteria for viable knowledge consist in evidence for the ability to (a) generate non-trivial conjectures, judged so subjectively by a mathematician, (b) prove such conjecture, and (c) move upward along the APOS conception levels. (2) What difficulties and insights did the participants experience as they constructed such knowledge?The potential contributions of our investigation into these questions to researchers and practitioners include (a) a detailed depiction of the participants’ achievements and challenges in dealing with theoretical questions concerning linear systems in an authentic learning environment and under a tutelage oriented in a particular constructivist perspective; and (b) a field-based hypothesis about the consequences of a particular learning environment vis-à-vis construction of knowledge in linear algebra.All of the participants had taken a linear algebra course as part of their undergraduate studies, on average 17 years prior to the TE, with an average grade of about 80%. Thus, a third question set for this study concerns retention. (3) What did the participants retain from their linear algebra courses vis-à-vis concepts, ideas, and problem solving pertaining to the theory of systems of linear equations, assuming they had constructed such knowledge during these courses?     

线性代数课堂中的“教”与“学”

结合教学实践经验,从人才培养的角度阐述线性代数课堂教学中的思维培养问题.指出课堂教学中要善于创设和营造和谐民主、积极向上、与学生心理相融的良好的课堂氛围;设置有利于学生参与认知的教学环节,通过采用灵活的教学方式,激发学生思考;尊重学生主体地位,让他们在教学活动中获得最大的情感体验;充分利用直观形象思维,教学中贯穿直观的几何形象,激发学生学习的兴趣,激发他们的求知欲.     

Transformations in the Visual Representation of a Figural Pattern

Multiple representations of a given mathematical object/concept are one of the biggest difficulties encountered by students. The aim of this study is to investigate the impact of the use of visual representations in teaching and learning algebra. In this paper, we analyze the transformations from and to visual representations that were performed by 18 students (aged between 10 and 13) in a task designed to explore a figural pattern. The data were collected from an audio recording of the class, the students’ work, and the teacher’s notes about each lesson. The results confirm that visual representations are important. However, visual treatments of any kind of representation are decisive, since they give students other possibilities for seeing and understanding tasks, continuity and flexibility in their activities, and the ability to make conversions between representations. The creative realization of visual treatments is necessary, and the teacher has a significant role in helping students to learn how to do this.     

A didactical situation for the enhancement of meta-analogical awareness

The aim of this study was to propose a didactical situation for the confrontation of the epistemological obstacle of linearity (routine proportionality) and consequently for the enhancement of meta-analogical awareness. Errors caused by students’ spontaneous tendency to apply linear functions in various situations are strong, persistent and do not disappear with traditional instruction. The effects of a didactical situation on the way students perceive and handle proportional and non-proportional relations were examined. The situation consisted of four parts which referred to the situations of action, formulation, validation and institutionalisation and was presented as a game to four twelve-year students of different abilities. The results showed the potential of the application of a didactical situation towards enhancing students’ meta-analogical awareness and therefore their ability to discern and handle linear and non-proportional relations.     

Teaching Algebraic Expressions to Young Students: The Three‐day Journey of “a+ 2”

This classroom scholarship report is based on the teaching experience using Davydov's mathematics curriculum, which was developed in the former Soviet Union. While “from arithmetic to algebra” is the normally accepted instructional sequence in school mathematics, Davydov's curriculum is laid out “from algebra to arithmetic,” focusing on algebraic thinking from the very beginning of the elementary grades. The purpose of this report is not to provide a definitive conclusion about which curriculum or sequence is better nor to address which instructional strategy is right in all circumstances. Rather, it is to explore how primary grade students develop their own conceptual understanding while confronting difficulties met within a specific context. This report provides actual classroom episodes from working with a group of first graders and describes dynamic interactions between the teacher and children while they discuss the use of algebraic expressions and understand the meaning behind them.     

A third grader's way of thinking about linear function tables

This paper is inscribed within the research effort to produce evidence regarding primary school students’ learning of algebra. Given the results obtained so far in the research community, we are convinced that young elementary school students can successfully learn algebra. Moreover, children this young can make use of different representational systems, including function tables, algebraic notation, and graphs in the Cartesian coordinate grid. In our research, we introduce algebra from a functional perspective. A functional perspective moves away from the mere symbolic manipulation of equations and focuses on relationships between variables. In investigating the processes of teaching and learning algebra at this age, we are interested in identifying meaningful teaching situations. Within each type of teaching situation, we focus on what kind of knowledge students produce, what are the main obstacles they find in their learning, as well as the intermediate states of knowledge between what they know and the target knowledge for the teaching situation. In this paper, we present a case study focusing on the approach adopted by a third grade student, Marisa, when she was producing the formula for a linear function while she was working with the information of a problem displayed in a function table containing pairs of inputs-outputs. We will frame the analysis and discussion on Marisa's approach in terms of the concept of theorem-in action (Vergnaud, 1982) and we will contrast it with the scalar and functional approaches introduced by Vergnaud (1988) in his Theory of Multiplicative Fields. The approach adopted by Marisa turns out to have both scalar and functional aspects to it, providing us with new ways of thinking of children's potential responses to functions.     

Introducing students to geometric theorems: how the teacher can exploit the semiotic potential of a DGS

Since their appearance new technologies have raised many expectations about their potential for innovating teaching and learning practices; in particular any didactical software, such as a Dynamic Geometry System (DGS) or a Computer Algebra System (CAS), has been considered an innovative element suited to enhance mathematical learning and support teachers’ classroom practice. This paper shows how the teacher can exploit the potential of a DGS to overcome crucial difficulties in moving from an intuitive to a deductive approach to geometry. A specific intervention will be presented and discussed through examples drawn from a long-term teaching experiment carried out in the 9th and 10th grades of a scientific high school. Focusing on an episode through the lens of a semiotic analysis we will see how the teacher’s intervention develops, exploiting the semiotic potential offered by the DGS Cabri-Géomètre. The semiotic lens highlights specific patterns in the teacher’s action that make students’ personal meanings evolve towards the mathematical meanings that are the objective of the intervention.