Powerful tasks: A contribution to a high level of acting and reflecting in mathematics instruction

Mathematics instruction contains two conflicting demands: on the one hand, the demand for economical efficiency and for well-developed motorways and, on the other hand, the demand that pupils should investigate and discover for themselves and have the freedom to pave their own ways. It is argued that tasks with a certain richness and quality offer some steps towards a constructive handling of this dilemma. The author tries to develop some properties of powerful tasks and to sketch the structure and philosophy of one concrete system of powerful tasks for the concept of angle. The main part of this paper presents seven examples of powerful tasks: five from the system of tasks and two with regard to pupils' working with 2-D-graphic systems. The construction of powerful tasks is viewed as a valuable contribution to bringing the theory and practice of mathematics education closer together.     

Developing theories of mathematics education research: the ESM story

In this paper we present ananalysis of the articles in EducationalStudies in Mathematics since 1990. It ispart of a larger project looking at theproduction and use of theories of teachingand learning mathematics. We outline thetheoretical framework of our tool ofanalysis and discuss briefly some of themethodological difficulties we face. Wethen present our findings from the analysisof the journal and we also give one exampleof how we `read' an article, illustratingthe rules whereby criteria are applied. This revised version was published online in July 2006 with corrections to the Cover Date.     

Analysis of an instructional unit for level of cognitive demand

A section of the secondary chemistry curriculum was analyzed to determine the level of cognitive demand of the various aspects of the selected topic. Piagetian levels of thinking of 71 pupils were initially assessed by two group tests, a unit on the mole was taught, and guidelines were used to estimate the level of cognitive operations required by each concept and problem type in the unit. Results of a 23-item test were used to compare the estimated level of cognitive demand of each test item with the Piagetian cognitive level of pupils who were able to answer the item correctly. It was found that pupil cognitive level was positively associated with overall unit test score and with percent success on all test items. Predicted levels of cognitive demand were confirmed for eight items and were within one level for nine additional items.     

论数学史在数学教学中的作用

我国的数学教学一直注重形式演绎数学思维的训练,而忽视了对数学学科的思想体系、文化内涵的认识。学习数学史可以对学生进行爱国主义教育,可以培养学生学习数学的兴趣,可以加深学生对数学知识的理解,有利于培养合格的数学教师。     

Effects of relating mathematics to science instruction on the acquisition of quantitative science behaviors

Does it make sense to include mathematics instruction in science classes? Is there empirical support for Gagne's task analysis?     

学习和研究数学哲学和数学史

从初出茅庐漫无边际的自学 ,到确定以数学哲学和数学史为研究方向 ;从 2 0年的准备阶段 ,到集中力量进行研究 ;从论文不能发表到出书 ;从参加省级的学术会议到参加国际学术会议 ,作者学习和研究数学哲学和数学史的道路是不平坦的。     

关于高职数学教学中数学史的作用刍议

本文以当前数学史在高职数学教学中的现状问题为切入点,就其在高职数学教学教育中的具体作用与实施方法进行了研究,以期提升高职课堂教育质量。     

从“表达思想”到“表现个性”——中国现代文体观演变与现代文学的阶段性诉求

原指文章整体存在的古代汉语文体概念,在日本17世纪至近代的文化语境中,先后发生了两次由表示文章本体(体)到表示语言形式符号(用)的语义转化。19世纪末至20世纪初,以语言形式为实质内涵的日本近代文体观被引入中国,契合了同样以语言变革为主要目标的中国现代文学改良运动的整体文化情势。受日本近代文体观与西方文类学(Genology)、语体学(Stylistics)的双重影响,中国现代学界一方面将传统文类之体理解为文学作品的一般形式,同时又继续以文体译Style,表示文章中与思想情感相对的具体语言形式。就对应于西方Style的文体概念(具有不同特征的语言形式)而言,其内涵的分阶段呈现集中反映了中国现代文学不同发展阶段的诉求。在白话文学草创阶段,人们更关心文体表达思想情感的基本功能,因此突出其语言形式这层内涵。当现代文学渐臻成熟,便进而要求文体在表情达意的基础上还应表现作家的个性特征,因此文体概念的另一层内涵,即通过语言形式所表现的作者个性特征,得到作家和批评家的自觉关注。认识中国现代文体观的这一内在转变,可为中国现代文学史研究开启新的思路。     

Between authenticity and cognitive demand: Finding a balance in designing a video-based simulation in the context of mathematics teacher education

A key challenge for teachers is the on-the-fly assessment of student learning. Video-based simulations may provide a tool for measuring assessment skills and a basis for learning environments in teacher education. Based on the framework for teaching practice by Grossman et al. (2009), considerations for designing video-based simulations that balance authenticity and cognitive demand are derived. Results show that participants perceived the developed simulation as authentic, were mostly able to rank students according to their overall mathematical argumentation skills and showed potential for learning in their detailed assessment of students. Thus, results indicate the internal validity of the video-based simulation.     

From failure to success: Changing the experience of adult learners of mathematics

For the last few years, the author has been part of a team developing a mathematics course for students most of whom are women, many of whom are black and all of whom are attempting to gain entry to a teacher-training course by successfully completing a one year re-entry course. It is a requirement that teacher-training students should have attained a suitable standard in mathematics. Further Education colleges where re-entry courses are sited have a sad history of student failure in mathematics. The theoretical environment in which the development of this course took place is described in order to place the course in context. In particular, attention is drawn to the re-definition of mathematics which encourages student enquiry and experimentation in order to establish a basis for understanding the subject, and to the teaching/learning model which creates an environment of respect and confidence. The roles of students and staff in a learning environment of this kind are discussed. Particular attention is paid to the attitudes and feelings of the students and the effects on their expectations.     

Teacher education through the history of mathematics

In this paper I consider the problem of designing strategies for teacher education programs that may promote an aware style of teaching. Among the various elements to be considered I focus on the need to address prospective teachers’ belief that they must reproduce the style of mathematics teaching seen in their school days. Towards this aim, I argue that the prospective teachers need a context allowing them to look at the topics they will teach in a different manner. This context may be provided by the history of mathematics. In this paper I shall discuss how history affected the construction of teaching sequences on algebra during the activities of the ‘laboratory of mathematics education’ carried out in a 2 year education program for prospective teachers. The conditions of the experiment, notably the fact that our prospective teachers had not had specific preparation in the history of mathematics, allow us to outline opportunities and caveats of the use of history in teacher education.     

从数学活动论看数学教育思想的转变

从数学活动论的角度去阐述21世纪在素质教育意义下的数学教育思想或模式的转变。     

From behaviourism to cognitive behaviourism to cognitive development: Steps in the evolution of instructional design

The recent history of instructional technology is traced, starting with the work of Skinner, moving on to the task analytic approach of Gagné, and following through to contemporary efforts associated with the cognitive revolution. It is suggested that an understanding of the process of cognitive development may enable us to build on and improve earlier approaches, by adapting them more directly to students' current levels of cognitive development, and by ensuring that we do not overtax their information processing capabilities. To illustrate and support this claim, a number of recent instructional studies are cited, some of which have utilized classic developmental tasks, and some of which have utilized conventional classroom material.Presented at the conference for Educational Technology in the 80's Caracas, Venezuela, June 14–18, 1982.