Course Description In this course, we will explore the question of the actual and potential connections between democracy and education. Our focus of attention will be placed on a critical examination of democratic theory and its implications for the civic education roles and contributions of teachers, adult educators, community development practitioners, and community organizers. We will survey and deal critically with a range of competing conceptions of democracy, variously described as classical, republican, liberal, radical, marxist, neomarxist, pragmatist, feminist, populist, pluralist, postmodern, and/or participatory. Using narrative inquiry as a means for illuminating and interpreting contemporary practice, we will analyze the implications of different conceptions of democracy for the practical work of civic education.

Democracy and education.

The power of feedback.

25. Multiple Imputation for Nonresponse in Surveys. By D. B. Rubin. ISBN 0 471 08705 X. Wiley, Chichester, 1987. 258 pp. £30.25.

Multiple Imputation for Nonresponse in Surveys

A wide variety of media can be used in learning, including distance learning, such as print, lectures, conference sections, tutors, pictures, video, sound, and computers. Any one instance of distance learning will make choices among these media, perhaps using several.

Multimedia in Learning

미국의 “전국 수학 교사 협의회”(National Council of Teachers of Mathematics, NCTM)는 1989년부터 〈학교 수학의 교육과정과 평가 규준〉(1989), 〈수학 가르침(교수)의 전문성 규준〉(1991), 〈학교 수학의 평가(시험) 규준〉(NCTM, 1995), 〈학교 수학의 원리와 규준〉(2000)을 출판하여 미국의 수학 교육 의 전망(목표, 나아갈 길)과 규준(실행 지침)을 제시하였다. 수학 교사들로 구성된 미국의 NCTM은 학생, 학부모, 학교 행정가 등 많은 사람들과 힘을 합하여 모든 학생들에게 수준 높은 수학 교육을 받을 수 있는 여건(환경, 기회)을 조성하는 데 구심점의 역할을 하였다. 한편 많은 관련 단체들은 여러 배경과 능력을 가진 학생들이 전문성을 지닌 교사(특수 교사를 일컫는 밀이 아니다. 수학 교과를 이해하고 수학의 전문성과 특수성을 가르칠 수 있는 일반 교사를 일컫는 말이다.)로부터 미래를 대비해 평등하고, 진취적이며, 지원이 잘 이루어지고, 공학 도구(IT)가 잘 갖춰진 환경에서 중요한 수학적 아이디어를 이해하면서 학습할 수 있는 수학 교실(미국에서는 우리나라처럼 수학 교사가 수학 시간에 학생의 방(교실: Homeroom)에 찾아가지 않고 학생들이 선생의 방(수학 교실: Classroom)을 찾아온다. 전형적인 수학 교실의 사진은 2쪽에 나와 있다.)을 만들기 위해 함께 힘썼다. NCTM에서 출간한 여러 규준들은 우리나라의 제6차와 제7차 교육과정에도 큰 영향을 미쳤다. 이 글에서는 NCTM(2000)에서 제시한 학습 원리를 간단히 살펴본 다음 이를 중심으로 현재 미국 수학 교육의 교수ㆍ학습 이론의 동향을 살펴본다.

미국 NCTM의 Principles and Standards for School Mathematics에 나타난 수학과 교수,학습의 이론

The purpose of this article is to describe how, within the Dutch approach to mathematics education, called Realistic Mathematics Education (RME), models are used to elicit students' growth in understanding of mathematics. First some background information is given about the characteristics of RME related to the role of models in this approach. Then the focus is on the use of the bar model within a longitudinal trajectory on percentage that has been designed for Mathematics in Context, a curriculum for the U.S. middle school. The power of this model is that it develops alongside both the teaching and the students: from a drawing that represents a context related to percentage to a strip for estimation and reasoning to an abstract tool that supports the use of percentage as an operator.

The didactical Use of Models in Realistic Mathematics Education : An Example from a Longitudinal Trajectory on Percentage

The intention of this study was to clarify students' difficulties in solving context-based mathematics tasks as used in the Programme for International Student Assessment (PISA). The study was carried out with 362 Indonesian ninth- and tenth-grade students. In the study we used 34 released PISA mathematics tasks including three task types: reproduction, connection, and reflection. Students' difficulties were identified by using Newman's error categories, which were connected to the modeling process described by Blum and Leiss and to the PISA stages of mathematization, including (1) comprehending a task, (2) transforming the task into a mathematical problem, (3) processing mathematical procedures, and (4) interpreting or encoding the solution in terms of the real situation. Our data analysis revealed that students made most mistakes in the first two stages of the solution process. Out of the total amount of errors 38% of them has to do with understanding the meaning of the context-based tasks. These comprehension errors particularly include the selection of relevant information. In transforming a context-based task into a mathematical problem 42% of the errors were made. Less errors were made in mathematical processing and encoding the answers. These types of errors formed respectively 17% and 3% of the total amount of errors. Our study also revealed a significant relation between the error types and the task types. In reproduction tasks, mostly comprehension errors (37%) and transformation errors (34%) were made. Also in connection tasks

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Difficulties in solving context-based PISA mathematics tasks: An analysis of students' errors

Many researchers have investigated flexibility of strategies in various mathematical domains. This study investigates strategy use and strategy flexibility, as well as their relations with performance in non-routine problem solving. In this context, we propose and investigate two types of strategy flexibility, namely inter-task flexibility (changing strategies across problems) and intra-task flexibility (changing strategies within problems). Data were collected on three non-routine problems from 152 Dutch students in grade 4 (age 9–10) with high mathematics scores. Findings showed that students rarely applied heuristic strategies in solving the problems. Among these strategies, the trial-and-error strategy was found to have a general potential to lead to success. The two types of flexibility were not displayed to a large extent in students’ strategic behavior. However, on the one hand, students who showed inter-task strategy flexibility were more successful than students who persevered with the same strategy. On the other hand, contrary to our expectations, intra-task strategy flexibility did not support the students in reaching the correct answer. This stemmed from the construction of an incomplete mental representation of the problems by the students. Findings are discussed and suggestions for further research are made.

Exploring strategy use and strategy flexibility in non-routine problem solving by primary school high achievers in mathematics

Realistic Mathematics Education

Based on the findings of an error analysis revealing that Indonesian ninth- and tenth-graders had difficulties in solving context-based tasks, we investigated the opportunity-to-learn offered by Indonesian textbooks for solving context-based mathematics tasks and the relation of this opportunity-to-learn to students’ difficulties in solving these tasks. An analysis framework was developed to investigate the characteristics of tasks in textbooks from four perspectives: the type of context used in tasks, the purpose of context-based tasks, the type of information provided in tasks, and the type of cognitive demands of tasks. With this framework, three Indonesian mathematics textbooks were analyzed. Our analysis showed that only about 10 % of the tasks in the textbooks are context-based tasks. Moreover, at least 85 % of these tasks provide exactly the information needed to solve them and do not leave room for students to select relevant information by themselves. Furthermore, of the context-based tasks, 45 % are reproduction tasks requiring performing routine mathematical procedures, 53 % are connection tasks requiring linking different mathematical curriculum strands, and only 2 % are reflection tasks, which are considered as tasks with the highest level of cognitive demand. A linkage between the findings of the error analysis and the textbook analysis suggests that the lacking opportunity-to-learn in Indonesian mathematics textbooks may cause Indonesian students’ difficulties in solving context-based tasks. Based on the results of this study, recommendations are given for improving the opportunities-to-learn to solve context-based tasks as well as for doing further research on this topic.

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Marja van den Heuvel-Panhuizen

Papers

The didactical Use of Models in Realistic Mathematics Education : An Example from a Longitudinal Trajectory on Percentage

Difficulties in solving context-based PISA mathematics tasks: An analysis of students' errors

Exploring strategy use and strategy flexibility in non-routine problem solving by primary school high achievers in mathematics

Realistic Mathematics Education

Opportunity-to-learn context-based tasks provided by mathematics textbooks