Mathematics as part of technology

Competence in dealing with fundamental problems connected with mathematical model building requires three different forms of knowledge. Mathematical knowledge itself, technological knowledge about how to develop a model, and reflective knowledge relevant for evaluation of the model building process. We find that reflective knowledge cannot be reduced to technological knowledge, so that it is important for mathematical education that is to be consistent with a critical pedagogy to provide opportunities for development of that type of knowledge.A general conceptual framework for identification of reflective knowledge is presented via a structural (synchronic) and a developmental (diachronic) perspective of a mathematical model.The synchronic perspective includes relationships between the model, its object in reality, a complex of theories, a complex of interests, and a conceptual framework or system mediating the connection between model and object. The diachronic perspective includes the components: problem identification, structure of argumentation, basis for critique, and space of possible actions, all of them sensitive to the application of mathematics. In that sense mathematics is not a neutral tool in a technological investigation, a fact which mathematical education has to reflect.     

Mathematical education and democracy

Is it possible to develop the content and form of mathematical education in such a way that it may serve as a tool of democratization in both school and society? This question is related to two different arguments. The social argument of democratization states: (1) Mathematics has an extensive range of applications, (2) because of its applications mathematics has a “society-shaping” function, and (3) in order to carry out democratic obligations and rights it is necessary to be able to identify the main principles of the development of society. The pedagogical argument of democratization states: (1) Mathematical education has a “hidden curriculum”, (2) the “hidden curriculum” of mathematical education in a traditional form implants a servile attitude towards technological questions into a large number of students, and (3) we cannot expect any development of democratic competence in school unless the teaching-learning situation is based on a dialogue and unless the curriculum is not totally determined from outside the classroom. The social argument implies that we must aim at “empowering material” which could constitute a basis for reflective knowledge i.e. knowledge about how to evaluate and criticize a mathematical model, while the pedagogical argument implies that we must aim at “open material” leaving space for decisions to be taken in the classroom. Will it become possible to create materials at the same time open and empowering? To answer this question we have to analyse the concept ‘democratic competence’, which can be related to ‘reflective knowledge’ characterized by a specific object of knowledge and a specific way of knowledge production. The ultimate aim will be to unify these characteristics in an epistemological theory of mathematical education. This paper is a revised version of \ldDemocratization and Mathematical Education\rd, R. 88-33 Department of Mathematics and Computer Science, Aalborg University Centre.     

Conceptualizing Teachers' Ways of Knowing

This article addresses issues related to the ways teachers learn mathematics and the teaching of mathematics and the relevance of those ways to their professional development. Preservice teachers' understanding of school mathematics lacks sophistication, a situation that needs to be addressed in mathematics teacher education programs. What is critical is the means by which they encounter and explore the mathematics they will be teaching. Fundamentally, their mathematical experiences need to be congruous with the kind of teaching we would expect of a reflective, adaptive teacher. The article contains both practical and theoretical considerations of how these experiences might be structured. Theoretical orientations for conceptualizing teachers' belief structures are offered as a foundation for conceptualizing teachers' ways of knowing. The moral dimension of teacher education is considered as a backdrop for understanding how teachers come to know.This revised version was published online in September 2005 with corrections to the Cover Date.     

课堂教学视野下的反思性数学学习

数学课程标准对学生的反思性数学学习给予了很高的关注,但当前数学课堂教学中学生反思性学习的现实状况却不容乐观。这主要表现为：教师的课堂权威挤占了学生进行自我反思的空间;学生在数学学习中的自我反思能力水平普遍偏低。通过分析课堂教学视野下反思性数学学习的具体内涵,可以发现数学课堂教学中学生进行反思性学习的重要意义主要在于发展学生的数学思维能力、促成个体化的数学学习方式和促进学生自主性的发展。     

Knowing, Insight Learning, and the Integrity of Kinetic Movement

Psychologists, philosophers, and educators have traditionally interpreted the phenomenon of insight learning as the result of the sudden comprehension of abstract/conceptual ideas. The present article shows that such phenomenon may also follow and emerge from the kinetic movements of the human body; that is, we conceptualize insight learning as a post-kinetic phenomenon. Further, it is suggested that kinetic movement constitutes the ground of all human knowing. To illustrate this innovative conceptualization of insight learning, we present the analysis of an exemplary classroom episode taken from a two-year longitudinal video-based ethnographic project. Our project is concerned with elementary students?? mathematical knowing and learning. In the episode, which was selected among other structurally similar examples, three children are sorting geometrical objects. The evidence shown is interpreted as support for the theory of mathematics in the flesh, a radical approach to embodied cognition. In contrast to other embodiment/ enactivist theories in the field of mathematics education, we suggest that the kinetic movement of the human body constitutes a necessary condition for the emergence of abstract mathematical knowledge, and more specifically for the emergence of geometrical insight.     

Mathematics, computers, and young children as a research-oriented learning environment for a teacher candidate

The advent of computer technology in the classroom raised the issue of its appropriate use by teachers and their students alike. It has been recommended that teacher education programs provide more opportunities for teacher candidates’ use of technology including teaching their own technology-enhanced lessons. With a goal of integrating scholarship into student teaching, a teacher candidate enrolled in a graduate program in childhood education carried out a technology-enhanced research project within a professional development school. Examining the impact of the project on the teacher candidate, this article describes how one’s pedagogical content knowledge and technological competence can be developed through a research-oriented teaching experience. The article also demonstrates the emergence of a community of practice that shares the goal of providing learning spaces for the teacher candidate and young children in the context of mathematics enrichment with computers.     

Not Your Usual Maths Course: critical mathematics education for adults

This article describes an action research project that investigated which features of critical theory were useful for teaching everyday mathematics in an evening course for adults. Paulo Freire's philosophy of education and Jürgen Habermas’ theory of knowledge interests are among the main influences on exponents of critical mathematics education such as Ole Skovsmose and Marilyn Frankenstein. Through the use of dialogical processes and critical reflection, students engage in social and political issues in their lives or communities and, as a result of their increased consciousness and mathematical learning, take “transforming action”. This study highlights the value of learning in a positive environment, through dialogical processes and critical questioning. Participants of the study were able to overcome barriers to learning mathematics; engage in everyday issues involving mathematics; and make changes to the way they dealt with mathematics, as a result of increased critical consciousness.     

Learning by Exploration in Mathematics Courses: a programme for student teachers

In order to adapt teacher education to new demands in mathematics classrooms, it is necessary to change the courses in mathematics at the university. Teachers’ beliefs about mathematics, learning and teaching has great impact on their teaching. At the University of Göteborg, a co‐operative project has been conducted in order to design a programme based on problem solving in courses taken by prospective Comprehensive School teachers (grade 4‐‐9). The main purpose of the project has been to make student teachers more reflective about mathematics as such, about learning and teaching. Another purpose of the project has been to use a teaching method in a university course‐‐a method which could be applied in a school classroom. The student teachers have worked co‐operatively in small groups of 3‐4 students and the educators role has been that of a facilitator. A preliminary evaluation indicates that student teachers have developed an insight into the complexity of learning and teaching, even though there are variations in this respect. However they still have difficulties in applying the method to teaching mathematics at school.     

CDIO教育模式在学生课外科技竞赛中的应用与探讨

探讨基于CDIO工程教育理念进行学生课外科技活动,使学生在工程实际环境中学习专业知识的同时培养个人能力、职业能力和态度,并提出与之相适应的学生学习评估方法。以"2010中国机器人大赛暨RoboCup公开赛"为例,阐述了CDIO工程教育模式的实现过程。基于CDIO的竞赛训练可提高大学生的创新能力、实践能力与综合素质。     

基于结构思想的数学教学

结构思想是皮亚杰、布鲁纳、布尔巴基学派等发展起来的现代教育理论,数学结构思想是对教育层面上数学本质的认识与处理方式。运用数学结构思想进行中学数学教学,不仅能提高学生对知识掌握的效率,而且能使学生获得全面的数学素质。而在数学教学中渗透数学思想方法,能帮助学生真正认识数学的体质,提高他们分析问题和解决问题的能力。     

高中生数学反思能力培养的基本模式与实践探索

The reflective ability in mathematics is a highly individual mental process, which is engaged in the course of certain types of mathematics activity. This article describes the basic meaning, characteristics, and developmental features of reflectivity in mathematics. It is suggested that reflective ability in mathematics and its relations with other mathematical abilities helps improve students’ long term mathematical and creative capacities. The authors develop a trial reflective component to high school mathematics education in order to determine the degree of short-term mathematics improvement measured by examination success, as well as long-term benefits to the student, thus promoting mathematics curriculum reform.     

Didactics and History of Mathematics: Knowledge and Self-Knowledge

The basic assumption of this paper is that mathematics and history of mathematics are both forms of knowledge and, therefore, represent different ways of knowing. This was also the basic assumption of Fried (2001) who maintained that these ways of knowing imply different conceptual and methodological commitments, which, in turn, lead to a conflict between the commitments of mathematics education and history of mathematics. But that conclusion was far too peremptory. The present paper, by contrast, takes the position, relying in part on Saussurean semiotics, that the historian's and working mathematician's ways of knowing are complementary. Recognizing this fact, it is argued, brings us to a deeper understanding of ourselves as creatures that do mathematics. This understanding, which is a kind of mathematical self-knowledge, is then proposed as an alternative commitment for mathematics education. In light of that commitment, history of mathematics assumes an essential role in mathematics education both as a subject and as a mediator between the aforementioned ways of knowing.     

Adults’ Use of Mathematics and Its Influence on Mathematical Competence

The Programme for the International Assessment of Adult Competencies (PIAAC) has recently drawn additional attention to “mathematical literacy” as an important influential factor for individuals’ life chances. High levels of mathematical literacy have thereby been linked to using mathematics in daily and working life frequently. In this paper, based on the data from Germany, we focus on the construct “use of mathematics” in two ways: First, we analyze in depth how it can be utilized to describe different groups of adults. Second, we investigate its role as predictor of mathematical competence and mediator of other relevant background variables. Results show that three groups of adults can be distinguished that use mathematics differently in daily and working life. However, the construct can sensibly be described as unidimensional. In a path model, “use of mathematics” turns out to be the strongest predictor of mathematical competence. In addition, it mediates effects of the mathematical requirements of the job, duration of education, and gender.     

Core skills assessment to improve mathematical competency

Many engineering undergraduates begin third-level education with significant deficiencies in their core mathematical skills. Every year, in the Dublin Institute of Technology, a diagnostic test is given to incoming first-year students, consistently revealing problems in basic mathematics. It is difficult to motivate students to address these problems; instead, they struggle through their degree, carrying a serious handicap of poor core mathematical skills, as confirmed by exploratory testing of final year students. In order to improve these skills, a pilot project was set up in which a ‘module’ in core mathematics was developed. The course material was basic, but 90% or higher was required to pass. Students were allowed to repeat this module throughout the year by completing an automated examination on WebCT populated by a question bank. Subsequent to the success of this pilot with third-year mechanical engineering students, the project was extended to five different engineering programmes, across three different year-groups. Full results and analysis of this project are presented, including responses to interviews carried out with a selection of the students involved.     

School mathematics in the era of globalization

This essay reviews the principles motivating contemporarycritical mathematics discourses. Drawing from varied critical discourses including ethno-mathematics, critical theory, post-structural theory, and situated and ecological cognition, the essay examines the pragmatics of critiques to the privileged role of school mathematics in the era of globalization. Critiques of modern school curricula argue that globalization practices linking education to technological and economic development are increasing, and the curriculum is being re-defined through discourses of privatization, national standards, and global competitiveness. Globalization has reinforced the utilitarian approach to school mathematics and the Western bias in the prevailing mathematics curricula, as well as helped to globalize pervasive mathematical ideologies. In most instances, a newfound status that mathematics is enjoying in this era of globalization is not well deserved, as school mathematics can no longer be considered culturally, socially, politically, nor economically neutral. In particular, school mathematics is increasingly critiqued as a cultural homogenizing force, a critical filter for status, a perpetuator of mistaken illusions of certainty, and an instrument of power. With such concerns it is becoming more evident that mathematics learning and education have implications for building just and democratic societies. As an African female scholar who is now living in Canada, I reflect on what the critical stance might mean for contexts with which I am familiar. I discuss the challenges of school mathematics with a view to improving curriculum and pedagogy so as to raise the awareness of teachers and learners to the questionable assumptions from which mathematics derives its prestige. The mathematics curriculum is central to cultivating values as well as fostering the conscientization of learners.     

试论弗莱登塔尔的数学教育思想及其启示

我国的基础教育正逐步由应试教育向素质教育全面推进 ,由此带来了教育观念、教育思想等方面的转变。荷兰数学家弗莱登塔尔指出 :数学教育应该是现实数学的教育 ;数学教育的目标应该是学会“数学化”。他的这些数学教育思想对我国数学素质教育有一定的启示。     

关于德国数学教育标准中的数学能力模型

为保障各联邦州数学教育质量的均衡发展,2003年底德国首次颁布全联邦性数学教育标准。这是一个较为典型的能力导向的教育标准,它提出学生应该发展的六大数学能力:(1)数学论证;(2)数学地解决问题;(3)数学建模;(4)数学表征的应用;(5)数学符号、公式以及技巧的熟练掌握;(6)数学交流。根据不同的认知要求这六大能力又分别被细化为三个能力水平。这个能力模型强调学生数学能力的可持续发展。     

Beyond motivation: history as a method for learning meta-discursive rules in mathematics

In this paper, we argue that history might have a profound role to play for learning mathematics by providing a self-evident (if not indispensable) strategy for revealing meta-discursive rules in mathematics and turning them into explicit objects of reflection for students. Our argument is based on Sfard’s theory of Thinking as Communicating, combined with ideas from historiography of mathematics regarding a multiple perspective approach to the history of practices of mathematics. We analyse two project reports from a cohort of history of mathematics projects performed by students at Roskilde University. These project reports constitute the experiential and empirical basis for our claims. The project reports are analysed with respect to students’ reflections about meta-discursive rules to illustrate how and in what sense history can be used in mathematics education to facilitate the development of students’ meta-discursive rules of mathematical discourse.     

Bilingualism and mathematical reasoning in English as a second language

This paper examines the ability of bilingual children to reason deductively in mathematics. In particular, the findings of a recent study of bilingual Punjabi, Mirpuri, Italian and Jamaican 11–13 year old children growing up in England are reported. It is found that first language competence is an important factor in the child's ability to reason in mathematics in English as a second language. This gives considerable support to theories which assert that a cognitively and academically beneficial form of bilingualism can only be achieved on the basis of adequately developed first language skills. However for both English monolingual and bilingual children knowledge of logical connectives in English is a crucial factor. It is suggested that published weaknesses in mathematics found among certain Asian and West Indian pupils may well be due to language factors. Furthermore there are strong cultural forces which predispose differential performance among boys and girls. The implications of the findings for a relevant mathematical education for bilingual children are discussed.     

Computer Programming

Summary

Computers can be powerful aids to mathematics teaching and learning. Changes brought about by the availability of these tools and the demands of an increasingly technological society impact curricular content and pedagogy in mathematics education as well as the very nature of mathematical thinking and understanding. This article presents ways in which technology is changing mathematics education, guidelines for appropriate technology use in the mathematics classroom, the impact of computers on mathematics learning, common uses of computers in mathematics education, and issues and concerns related to technology use in mathematics.