From the principle of bijection to the isomorphism of structures: an analysis of some teaching paradigms in discrete mathematics

This paper is concerned with the teaching of Discrete Mathematics to university undergraduate students. Two to three decades ago this course became a requirement for math and computer science students in most universities world wide. Today this course is taken by students in many other disciplines as well. The paper begins with a discussion of a few topics that we feel should be included in the syllabus for any course in Discrete Mathematics, independent of the audience. We then discuss several potential models for teaching the course, depending upon the interests and mathematical background of the audience. We also investigate various educational links with other components of the curriculum, consider pedagogical issues associated with the teaching of discrete mathematics, and discuss some logistical and psychological difficulties that must be overcome. A special emphasis is placed on the role of textbooks.     

概率统计的多元化教学探讨

概率统计是一门具有独特思想方法的数学学科,我们在传统的理论教学基础上,以概率思想教育为主线,加强实践环节的教学和实际操作,注重提升学生在数学文化层面上的认识.从教学情况来看,收到了较好的教学效果.     

A Sino-German semi-virtual seminar in mathematics education

In summer 2006 the University of Education in Weingarten, Germany, and East China Normal University, Shanghai, performed a semi-virtual seminar with mathematics students on “Mathematics and Architecture”. The goal was the joint development of teaching materials for German or Chinese school, based on different buildings such as “Nanpu Bridge”, or the “Eiffel Tower”. The purpose of the seminar was to provide a learning environment for students supported by using information and communication technology (ICT) to understand how the hidden mathematics in buildings should be related to school mathematics; to experience the multicultural potential of the international language “Mathematics”; to develop “media competence” while communicating with others and using technologies in mathematics education; and to recognize the differences in teaching mathematics between the two cultures. In this paper we will present our ideas, experiences and results from the seminar.     

Mathematics standards and curricula in the Netherlands

This paper addresses the question of what mathematics Dutch students should learn according to the standards as established by the Dutch Ministry of Education. The focus is on primary school and the foundation phase of secondary school. This means that the paper covers the range from kindergarten to grade 8 (4~14 years olds). Apart from giving an overview of the standards, we also discuss the standards' nature and history Furthermore, we look at textbooks and examination programs that in the Netherlands both have a key role in determining the intended mathematics curriculum. In addition to addressing the mathematical content, we also pay attention to the way mathematics is taught. The domain-specific education theory that forms the basis for the Dutch approach to teaching mathematics is called “Realistic Mathematics Education” Achievement scores of Dutch students from national and international tests complete this paper. These scores reveal what the standards bring us in terms of students' mathematical understanding. In addition to informing an international audience about the Dutch standards and curricula, we include some critical reflections on them.     

Balancing mathematics education research and the NCTM standards

The release of the Principles and Standards for School Mathematics in the United States by the National Council of Teachers of Mathematics (NCTM) brought to the forefront the debate of whether research should determine the validity of the espoused Standards? Or conversely whether the Standards should influence the research agenda of the mathematics education community? How should university teacher educators address this issue? Should pre-service and practicing teachers blindly, accept the Standards as well as the research, or do we cultivate the critical thinking skills that will allow preparing teachers to resolve this dilemma? In this article a university mathematics educator and an idealistic pre-service elementary teacher try to resolve the dilemma of balancing the Standards with research and personal beliefs about the teaching and learning of mathematics.     

经济数学基础精品课程的建设与教学实践

介绍了安徽省精品课程《经济数学基础》课程建设的主要成果.其中包括教学内容的改革,教材和教辅材料的建设,教学方法和教学手段的改革以及实践性教学环节和教学科研的成果.     

高职数学教学方法探讨

从PBL教学法、个性化教学方法、开设数学实验法、采用模型教学法、自拟例题和练习的方法、渗透数学文化法等六个方面探讨高职数学的教学方法．     

关于“无穷小的比较”的教学研究

分析“无穷小的比较”相关教学内容在高等数学课程中的地位和作用,指出现行教材针对该部分内容的实际编写存在诸多问题,给出实际教学过程中需要补充和扩展的内容,并就这部分内容的教学设计与教学方法提出几点具体建议．     

工科数学教育特征研究

根据工科数学教育的现状,讨论了数学教育的基础性、全体性、主体性和可持续性等重要特征.提出了改革应兼顾上述特征才能收到较好成效的观点.     

“应用数学基础”课程改革的思考

本探讨了“应用数学基础”课程改革的必要性，从教师队伍建设、教学内容改革、教学手段与教学方法改革、课程教材建设和教学实战等五个方面提出了我们的想法。     

How to make mathematics relevant to first-year engineering students: perceptions of students on student-produced resources

Many approaches to make mathematics relevant to first-year engineering students have been described. These include teaching practical engineering applications, or a close collaboration between engineering and mathematics teaching staff on unit design and teaching. In this paper, we report on a novel approach where we gave higher year engineering and multimedia students the task to ‘make maths relevant’ for first-year students. This approach is novel as we moved away from the traditional thinking that staff should produce these resources to students producing the same. These students have more recently undertaken first-year mathematical study themselves and can also provide a more mature student perspective to the task than first-year students. Two final-year engineering students and three final-year multimedia students worked on this project over the Australian summer term and produced two animated videos showing where concepts taught in first-year mathematics are applied by professional engineers. It is this student perspective on how to make mathematics relevant to first-year students that we investigate in this paper. We analyse interviews with higher year students as well as focus groups with first-year students who had been shown the videos in class, with a focus on answering the following three research questions: (1) How would students demonstrate the relevance of mathematics in engineering? (2) What are first-year students' views on the resources produced for them? (3) Who should produce resources to demonstrate the relevance of mathematics? There seemed to be some disagreement between first- and final-year students as to how the importance of mathematics should be demonstrated in a video. We therefore argue that it should ideally be a collaboration between higher year students and first-year students, with advice from lecturers, to produce such resources.     

On the Dirichlet's box principle

In this note, we will focus on several applications on the Dirichlet's box principle in Discrete Mathematics lesson and number theory lesson. In addition, the main result is an innovative game on a triangular board developed by the authors. The game has been used in teaching and learning mathematics in Discrete Mathematics and some high schools in Hong Kong.     

Curriculum Materials Supporting Problem-Based Teaching

The vision for school mathematics described by the National Council of Teachers of Mathematics (NCTM) suggests a need for new approaches to the teaching and learning of mathematics, as well as new curriculum materials to support such change. This article discusses implications of the NCTM standards for mathematics curriculum and instruction and provides three examples of lessons from problem-based curricula for various grade levels. These examples illustrate how the teaching of important mathematics through student exploration of interesting problems might unfold, and they highlight the differences between a problem-based approach and more traditional approaches. Considerations for teaching through a problem-based approach are raised, as well as reflections on the potential impact on student learning.     

Response to key issues raised in the Post-14 mathematics inquiry

This article is a detailed response to the issues raised by the Post-14 Mathematics Inquiry in the UK. It aims to debate some of the central issues in mathematics teaching in the UK, including recruitment and retention of mathematics teachers, the curriculum content, national assessment, teaching resources (including ICT) and national strategies and policy (including inspection). Throughout, we have tried to base our recommendations on evidence and experience from the many teachers and tutors we work with as well as on our own experience. We have not hesitated to make what could be seen as controversial recommendations, but we believe a fundamental rethink of education policy and practice is needed if mathematics teaching and learning is to improve. We have also considered the impact that a proposed ‘National Centre for Excellence in Mathematics Education’ might have on the situation, although we doubt that it can have a marked long-term impact in the current UK situation.     

Contextual approach in teaching mathematics: an example using the sum of series of positive integers

In this paper, a contextual approach to teaching Mathematics at the pre-university level is recommended, and an example is illustrated. A context in the form of a real common mathematical problem is presented to the students. Different approaches to tackle the problem (from topics within and outside the syllabus) can be elicited from students. The insight obtained from the various methods of solving the problem can be used to deepen students’ learnt concepts and to enhance concepts to be learnt later in the curriculum.     

高等数学课堂教学与一题多解

高等数学是理工科院校一门十分重要的公共基础课程,在自然科学、工程技术、生命科学、社会科学、经济管理等众多领域有着广泛的应用．对待作为高等数学课堂上的主体——学生,教师应当积极正面评价学生,激发学生的学习兴趣,启发学生主动思考,全面培养学生解决数学问题的能力,并在课堂教学中贯彻这一思想．课堂教学也表明：学生积极主动的应用多种思路解决一道题远比老师灌输性的介绍一道题的多种解法要更具有积极意义．     

《数学实验》课程建设及分层次教学与实践

主要介绍在本科一年级讲授《高等数学》课程时同步分层次开展《数学实验》课程教学的教学模式、教学内容、教学方法与改革、课程考核办法,以及我们对《数学实验》课程的认识和建设过程.     

Traveling down the road: from cognitive neuroscience to mathematics education … and back

In this commentary paper to the special issue on “Cognitive Neuroscience and Mathematics Education”, we reflect on the connection between cognitive neuroscience and mathematics education from an educational research point of view. The current issue highlights that cognitive neuroscience offers a series of tools, methodologies and theories to investigate cognitive processes that take place during mathematical thinking and learning. This might complement and extend our knowledge that has been obtained on the basis of behavioral data only, the common approach in educational research. At the same time, we note that the existing neuroscientific studies have investigated mathematical performance in relative isolation from the educational context. The characteristics of this context have, however, a large influence on mathematical performance and its correlated brain activity, an issue that should be addressed in future research. We contend that traveling back and forth from cognitive neuroscience to mathematics education might yield a better understanding of how mathematical learning takes place and how it can be influenced.     

高等农业院校开设数学应用课程——“数学建模”的实践与体会

鉴于数学在农业应用方面不断普遍化、深入化,而农林院校在数学应用教育方面的落后,结合我校开设数学类应用课程的实践,具体探讨农业院校如何开设数学应用课程——"数学建模"的教学内容和教学方式,并分析了实际教学效果及对学生的影响.     

解决高等数学教学问题的七点措施——以衡阳师范学院为例

本文简要分析了当前高校高等数学教学所面临的几个突出问题,在此基础上结合衡阳师范学院高等数学教学实际,从七个方面阐述了如何加强高等数学教学以提高教学质量.