The theory of affordances

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

What is Qualitative Interviewing? is an accessible and comprehensive ‘what is’ and ‘how to’ methods book. It is distinctive in emphasizing the importance of good practice in understanding and undertaking qualitative interviews within the framework of a clear philosophical position. Rosalind Edwards and Janet Holland provide clear and succinct explanations of relevant philosophies and theories of how to know about the social world, and a thorough discussion of how to go about researching it using interviews. A series of short chapters explain a range of interview types and practices. Drawing on their own and colleagues’ experiences Edwards and Holland provide real research examples as informative illustrations of qualitative interviewing in practice and the use of creative interview tools. They discuss the use of new technologies as well as tackling enduring issues around asking and listening, and power dynamics in research. Written in a clear and accessible style the book concludes with a useful annotated bibliography of key texts and journals in the field. What is Qualitative Interviewing? provides a vital resource for both new and experienced researchers across the social science disciplines.

What is Qualitative Interviewing

Bayesian Reasoning. Introduction to Bayesian Networks. Inference in Bayesian Networks. Bayesian Network Applications. Bayesian Planning and Decision-Making. Bayesian Network Applications II. Learning Bayesian Networks I. Learning Bayesian Networks II. Causality vs. Probability. Knowledge Engineering with Bayesian Networks I. Knowledge Engineering with Bayesian Networks II. Application Software.

Bayesian artificial intelligence

The authors propose that conceptual and procedural knowledge develop in an iterative fashion and that improved problem representation is 1 mechanism underlying the relations between them. Two experiments were conducted with 5th- and 6th-grade students learning about decimal fractions. In Experiment 1, children's initial conceptual knowledge predicted gains in procedural knowledge, and gains in procedural knowledge predicted improvements in conceptual knowledge. Correct problem representations mediated the relation between initial conceptual knowledge and improved procedural knowledge. In Experiment 2, amount of support for correct problem representation was experimentally manipulated, and the manipulations led to gains in procedural knowledge. Thus, conceptual and procedural knowledge develop iteratively, and improved problem representation is 1 mechanism in this process.

https://psych.wisc.edu/alibali/home/Publications_files/Rittle-JohnsonSieglerandAlibali2001.pdf

Developing Conceptual Understanding and Procedural Skill in Mathematics: An Iterative Process.

Understanding that mathematics is founded on reasoning and is not just a collection of rules to apply is an important message to convey to students. Here we examined the reasoning presented in seven topics in nine Australian eighth-grade textbooks. Focusing on explanatory text that introduced new mathematical rules or relationships, we classified explanations according to the mode of reasoning used. Seven modes were identified, making a classification scheme which both refined and extended previous schemes. Most textbooks provided explanations for most topics rather than presenting “rules without reasons” but the main purpose appeared to be rule derivation or justification in preparation for practise exercises, rather than using explanations as thinking tools. Textbooks generally did not distinguish between the legitimacies of deductive and other modes of reasoning.

Modes of reasoning in explanations in Australian eighth-grade mathematics textbooks

The OECD PISA international survey of mathematical literacy for 2012 is based on a new Framework and has several new constructs. New features include an improved definition of mathematical literacy; the separate reporting of mathematical processes involved in using mathematics to solve real world problems; a computer-based component to assess mathematical literacy as it is likely to be encountered in modern workplaces; and new questionnaire items targeting mathematics. Procedures for quality assurance that arise in the preparation of an assessment for use in many countries around the world are illustrated with some items and results from the 2011 international field trial. The paper will provide background for the interpretation of the results of the PISA 2012 survey, which are to be published in December 2013.

The International Assessment of Mathematical Literacy: PISA 2012 Framework and Items

National curriculum guidelines advise teachers to begin algebra with the study of patterns leading to their description as algebra rules relating two variables. Traditional approaches are based on the use of algebraic letters to stand for specific but unknown numbers. In this paper we show that a pattern-based approach does not automatically lead to better understanding of functional relationships and algebraic rules. Approximately 1200 students in Years 7 to 10 in 10 schools were tested on recognising, using and describing rules relating two variables. Fourteen students were interviewed. Students saw a variety of patterns, many of which were not helpful for algebra. The critical steps in moving from a table to an algebraic rule are outlined.

The effect of different approaches to algebra on students' perceptions of functional relationships

PISA, the OECD’s international program of assessment of reading, scientific and mathematical literacy, aims to assess the knowledge and skills that students have acquired at school and their ability to use them in everyday tasks and challenges. It also uses questionnaires to gather data on students’ attitudes to learning and the conditions of schooling. Since 2000, PISA has tested the scientific and mathematical literacy of 15 year old students from around the world. By 2009 there are over 70 participating countries. This paper describes the design of the PISA assessments, discusses mathematical and scientific literacy and reports on a selection of results from the PISA assessments, associated surveys and related analyses to give a flavour of the information that has resulted from this major international initiative.

Kaye Stacey

Papers

Modes of reasoning in explanations in Australian eighth-grade mathematics textbooks

The International Assessment of Mathematical Literacy: PISA 2012 Framework and Items

The effect of different approaches to algebra on students' perceptions of functional relationships

Mathematical and Scientific Literacy around the World.

Assessing Mathematical Literacy