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.     

谈数学概念教学

数学概念教学需要联系实际，引入数学概念，利用肯定、否定例证，剖析数学概念，抓住本质特征，强化数学概念，运用辩证观点，发展数学概念，利用比较与联系，深化数学概念。     

影响高等数学概念学习的主要因素与教学策略

分析了数学概念的特点及其学习的主要影响因素,认为不正确的概念意象和薄弱的抽象概括能力是阻碍学生数学概念学习的主要原因;提出了营造概念学习情境、计算机图形培育概念直觉、突出概念本质属性、强化概念之间普遍联系等教学策略。     

高等数学中一些重要概念间关系的辨析

数学概念是数学知识体系的核心部分,是构建整个数学大厦的基石,是数学推理论证的基础,也是学生学好高数的关键。但概念教学是高数教学中的一个难点,学生往往难以掌握,和学生辨析清高数相关概念间的关系对学生掌握概念有很大的帮助。如连续与间断、连续与一致连续、连续与可导及可导与可微等概念的教学都是如此,只有充分辨析清楚这些概念的本质,才能使学生真正掌握。     

Animated cell biology: a quick and easy method for making effective, high-quality teaching animations

There is accumulating evidence that animations aid learning of dynamic concepts in cell biology. However, existing animation packages are expensive and difficult to learn, and the subsequent production of even short animations can take weeks to months. Here I outline the principles and sequence of steps for producing high-quality PowerPoint animations in less than a day that are suitable for teaching in high school through college/university. After developing the animation it can be easily converted to any appropriate movie file format using Camtasia Studio for Internet or classroom presentations. Thus anyone who can use PowerPoint has the potential to make animations. Students who viewed the approximately 3-min PowerPoint/Camtasia Studio animation "Calcium and the Dual Signalling Pathway" over 15 min scored significantly higher marks on a subsequent quiz than those who had viewed still graphics with text for an equivalent time. In addition, results from student evaluations provided some data validating the use of such animations in cell biology teaching with some interesting caveats. Information is also provided on how such animations can be modified or updated easily or shared with others who can modify them to fit their own needs.     

Introducing Young Children to Mathematical Concepts: Problems with 'new' terminology

This paper explores the nature of the language used when teaching mathematics to young children. It proposes that an important part of the teaching of a mathematical concept is the introduction of specific terminology. Children may need to be taught new meanings for already familiar words. The timing of these introductions to new words or meanings is critical to their understanding of the concepts being taught. It will be argued that there are two aspects of the children's learning that need to be considered. First, their understanding of the concept being introduced, and secondly, their learning the appropriate word to describe that concept. By assessing children's understanding of new mathematical concepts through their own use of the terminology, the teacher can then negotiate new meanings with them through practical experiences, introducing new word meanings only when the concepts have been understood.     

概念教学中数学思想方法的渗透

数学概念的形成过程是一个归纳、概括、抽象的过程,因此概念的教学过程是一个探究的过程。数学概念从其形式上看,它是中学教学的表层知识,但是一个数学概念的背后往往蕴含着丰富的数学思想。在概念教学中注意数学思想方法的渗透,可以完成数学概念和数学思想方法的双重教学。     

In Physics Education,Perception Matters

Student difficulties in science learning are frequently attributed to misconceptions about scientific concepts. We argue that domain‐general perceptual processes may also influence students' ability to learn and demonstrate mastery of difficult science concepts. Using the concept of center of gravity (CoG), we show how student difficulty in applying CoG to an object such as a baseball bat can be accounted for, at least in part, by general principles of perception (i.e., not exclusively physics‐based) that make perceiving the CoG of some objects more difficult than others. In particular, it is perceptually difficult to locate the CoG of objects with asymmetric‐extended properties. The basic perceptual features of objects must be taken into account when assessing students' classroom performance and developing effective science, technology, engineering, and mathematics (STEM) teaching methods.     

Subject matter knowledge for teaching and the case of functions

Interest in teachers' subject matter knowledge has arisen in recent years. But most of the analysis has been general and not topic-specific. This paper shows how one may approach the question of teachers' knowledge about mathematical topics. It demonstrates the building of an analytic framework of subject matter knowledge for teaching a specific topic in mathematics and then uses the concept of function to provide an illustrative case of a paradigm for analyzing subject matter knowledge for teaching. The choice of the aspects, which form the main facets of the framework, was based on integrated knowledge from several bodies of work: the role and importance of the topic in the discipline of mathematics and in the mathematics curriculum; research and theoretical work on learning, knowledge and understanding of mathematical concepts in general and the specific topic in particular; and research and theoretical work on teachers' subject matter knowledge and its role in teaching. An application of the framework in the case of the concept of function is described and illustrated by anecdotes drawn from a study of prospective secondary teachers' knowledge and understanding of functions.Recipient of a Sir Charles Clore Post-Doctoral Fellowship.     

Development of an Augmented Reality Game to Teach Abstract Concepts in Food Chemistry

One of the most pressing issues for many land grant institutions is the ever increasing cost to build and operate wet chemistry laboratories. A partial solution is to develop computer‐based teaching modules that take advantage of animation, web‐based or off‐campus learning experiences directed at engaging students’ creative experiences. We used the learning objectives of one of the most difficult topics in food chemistry, enzyme kinetics, to test this concept. Students are apprehensive of this subject and often criticize the staid instructional methods typically used in teaching this material. As a result, students do not acquire a useful background in this important subject. To rectify these issues, we developed an interactive augmented reality application to teach the basic concepts of enzyme kinetics in the context of an interactive search that took students to several locations on campus where they were able to gather raw materials and view videos that taught the basics of enzyme kinetics as applied to the production of high fructose corn syrup (HFCS). The students needed this background to prepare for a mock interview with an HFCS manufacturer. Students and instructors alike found the game to be preferable to sitting in a classroom listening to, or giving, a PowerPoint presentation. We feel that this use of gaming technology to teach difficult, abstract concepts may be a breakthrough in food science education and help alleviate the drain on administrative budgets from multiple wet labs.     

Using “Slowmation” to Enable Preservice Primary Teachers to Create Multimodal Representations of Science Concepts

Research has identified the value of students constructing their own representations of science concepts using modes such as writing, diagrams, 2-D and 3-D models, images or speech to communicate meaning. ??Slowmation?? (abbreviated from ??Slow Animation??) is a simplified way for students, such as preservice teachers, to make a narrated animation using a combination of modes. In this study, 13 preservice primary teachers learned how to create a slowmation during a two-hour class in a science methods course and then created one about an allocated science topic as an assignment. The research question that guided this study was, ??What are the preservice teachers?? perceptions of making a slowmation and how was the science concept represented in the animation??? Data included pre and post individual interviews, concept maps constructed during the interviews and the animations as artifacts. Three case studies provide a window into the perceptions of preservice teachers making a slowmation and show how they represented their concept. Slowmation is a new form of student-generated representation which enables them to use their own technology to construct a narrated animation as a multimodal representation to explain a science concept.     

The evolution of an approach for using analogies in teaching and learning science

An important contribution to effective teaching and learning can be made by teachers' understanding of the central topics in each subject area and knowing how to transform their content knowledge into knowledge for teaching. One aspect of this knowledge is the use of analogies which can effectively communicate concepts to students of particular backgrounds and prerequisite knowledge. Indeed, analogies are considered to be an important component in the repertoire of effective teachers. However, research about teachers' use of analogies in science lessons provides little guidance about the optimum approaches that may be taken by preservice teachers, novice teachers, experienced teachers or reluctant analogy users. This paper describes the evolution of an approach for using analogies in science teaching that addresses both findings from the research literature and recognises the needs of practising teachers. Specializations: learning and teaching science concepts, technology education.     

高校教学用演播室的设计与建设

全国高等院校由于新闻、影视动画等相关专业的开设及教学和科研的需要,纷纷在学校内建设教学用影视演播室。由于演播室是开展新闻制作、影视动画、文艺编导、播音与主持等专业的重要教学装备之一,是学生学习相关课程制作必不可少的实验场地。以现代传媒观念和多视角、多场景、多机位、多功能的开放式演播理念,将艺术性和实用性相结合对演播室科学合理地创意、设计和配置,使演播室成为一个多视角、多场景、多机位、多功能的开放式演播室,满足开放式、多功能的使用要求。     

Proof and application

This paper outlines an epistemological conception which attempts to relate the formal aspects of mathematical proof to its pragmatic dimensions. In addition to the key concept of application, the paper makes use of several concepts from the domain of analytical philosophy, to present a view of proof that might best be categorized as a dialectical one. A number of implications for teaching are discussed.     

MATLAB在“信号与系统”实验教学中的应用

为了克服"信号与系统"硬件实验系统干扰强、易故障等缺点,利用MATLAB的图形用户界面GUI开发"信号与系统"的相关仿真实验.通过仿真实验,能够帮助学生更加深刻地理解"信号与系统"课程中抽象的概念和复杂的数学运算,明显地提高了实验教学效果.     

Physics students' understanding of relative speed: A phenomenographic study

It is important that students of physics develop both quantitative and qualitative understanding of physical concepts and principles. Although accuracy and reliability in solving quantitative problems is necessary, a qualitative understanding is required in applying concepts and principles to new problems and in real-life situations. If students are not able to understand what underlies quantitative problem-solving procedures nor interpret the solution in physical terms, it is questionable whether they have developed an adequate understanding of physics. The research reported here is part of a larger phenomenographic study that is concerned with the assessment of physics students' understanding of some basic concepts and principles in kinematics. In this article students' understanding of the concept of relative speed is described. A variety of ways of understanding relative speed and of viewing a problem that dealt with this concept were uncovered. The results are used to suggest ways for teachers to proceed in assisting students to enhance their understanding of this concept. The teaching principles outlined concern both teaching relative speed, in particular, and teaching scientific concepts and principles, more generally.     

Programming-languages as a conceptual framework for teaching mathematics

Formal mathematical methods remain, for most high school students, mysterious, artificial and not a part of their regular intuitive thinking. The authors develop some themes that could lead to a radically new approach. According to this thesis, the teaching of programming languages as a regular part of academic progress can contribute effectively to reduce formal barriers. This education can also be used to enable pupils to access an accurate understanding of some key mathematical concepts. In the field of heuristic knowledge for technical problem solving, experience of programming is no less valuable: it lends itself to promote a discussion of relations between formal procedures and the comprehension of intuitive problem solving and provides examples for the development of heuristic precepts (formulating a plan, subdividing the complexities, etc.). The knowledge gained in programming can also be used for the discussion of concepts and problems of classical mathematics. Finally, it can also facilitate the expansion of mathematical culture to topics in biological and physical sciences, linguistics, etc. The authors describe a programming language called ‘Logo’ adapted to objectify an enduring framework of mathematical experimentation.     

High-School Chemistry Students' Performance and Gender Differences in a Computerized Molecular Modeling Learning Environment

Computerized molecular modeling (CMM) contributes to the development of visualization skills via vivid animation of three dimensional representations. Its power to illustrate and explore phenomena in chemistry teaching stems from the convenience and simplicity of building molecules of any size and color in a number of presentation styles. A new CMM-based learning environment for teaching and learning chemistry in Israeli high schools has been designed and implemented. Three tenth grade experimental classes used this discovery CMM approach, while two other classes, who studied the same topic in the customary approach, served as a control group. We investigated the effects of using molecular modeling on students' spatial ability, understanding of new concepts related to geometric and symbolic representations and students' perception of the model concept. Each variable was examined for gender differences. Students of the experimental group performed better than control group students in all three performance aspects. Experimental group students scored higher than the control group students in the achievement test on structure and bonding. Students' spatial ability improved in both groups, but students from the experimental group scored higher. For the average students in the two groups the improvement in all three spatial ability sub-tests —paper folding, card rotation, and cube comparison—was significantly higher for the experimental group. Experimental group students gained better insight into the model concept than the control group and could explain more phenomena with the aid of a variety of models. Hence, CMM helps in particular to improve the examined cognitive aspects of the average student population. In most of the achievement and spatial ability tests no significant differences between the genders were found, but in some aspects of model perception and verbal argumentation differences still exist. Experimental group females improved their model perception more than the control group females in understanding ways to create models and in the role of models as mental structures and prediction tools. Teachers' and students' feedback on the CMM learning environment was found to be positive, as it helped them understand concepts in molecular geometry and bonding. The results of this study suggest that teaching/learning of topics in chemistry that are related to three dimensional structures can be improved by using a discovery approach in a computerized learning environment.     

中小学生数学自主学习能力培养途径的研究

培养中小学生数学自主学习能力可以从创新校长和教师的教育观念、加强师资培训、更新教师的教学技能入手,而在数学课堂教学中,可恰当地运用：重视情感因素,鼓励学生质疑;挖掘数学学习素材,激发学习动机;注重学法指导;创设问题情境,提倡问题解决;重视榜样作用;有效利用各种信息资源来支持自主学习等策略。