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.

Linear generalising problems are questions which require students to observe and use a linear pattern of the formf(n)=an+b withb≠0 This study reports responses of students aged between 9 and 13 to these questions, documenting the mathematical models that they select, the strategies used in implementing them and the explanations they give Substantial inconsistency of choice of model is observed; students who began a question correctly frequently adopted a simpler but incorrect model for more difficult parts of the question Students who had undertaken a course in problem solving implicitly used a linear model more frequently and consistently and their explanations more often related the spatial patterns and the number patterns They seemed to understand the relationship between the data and the generalising rule more completely

Finding and Using Patterns in Linear Generalising Problems.

The Mathematics and Technology Attitudes Scale (MTAS) is a simple scale for middle secondary years students that monitors five affective variables relevant to learning mathematics with technology The subscales measure mathematics confidence, confidence with technology, attitude to learning mathematics with technology and two aspects of engagement in learning mathematics The paper presents a model of how technology use can enhance mathematics achievement, a review of other instruments and a psychometric analysis of the MTAS It also reports the responses of 350 students from 6 schools to demonstrate the power of the MTAS to provide useful insights for teachers and researchers ‘Attitude to learning mathematics with technology’ had a wider range of scores than other variables studied For boys, this attitude is correlated only with confidence in using technology, but for girls the only relationship found was a negative correlation with mathematics confidence These differences need to be taken into account when planning instruction

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A scale for monitoring students¿ attitudes to learning mathematics with technology

Research studies have found that the majority of students up to age 15 seem unable to interpret algebraic letters as generalised numbers or even as specific unknowns Instead, they ignore the letters, replace them with numerical values, or regard them as shorthand names The principal explanation given in the literature has been a general link to levels of cognitive development In this paper we present evidence for specific origins of misinterpretation that have been overlooked in the literature, and which may or may not be associated with cognitive level These origins are: intuitive assumptions and pragmatic reasoning about a new notation, analogies with familiar symbol systems, interference from new learning in mathematics, and the effects of misleading teaching materials Recognition of these origins of misunderstanding is necessary for improving the teaching of algebra

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Students' understanding of algebraic notation: 11-15

PISA, the OECDâ€™s international program of assessment of reading,scientific and mathematical literacy (www.oecd.org/pisa), aims to assess the ability of 15 year olds to use the knowledge and skills that have acquired at school in real world tasks and challenges. It also uses questionnaires to gather data on studentsâ€™ attitudes to learning and the conditions of schooling. Since 2000, PISA has tested the mathematical literacy of 15 year old students from many countries around the world. This paper describes the design of the PISA assessments, discusses mathematical 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. Results for Indonesia are compared with the OECD average and with a selection of countries, addressing issues of overall achievement, equity, and classroom environment. Keyword: PISA View, Mathematical Literacy, Indonesia Â  DOI : http://dx.doi.org/10.22342/jme.2.2.746.95-126

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Kaye Stacey

Papers

Finding and Using Patterns in Linear Generalising Problems.

A scale for monitoring students¿ attitudes to learning mathematics with technology

Students' understanding of algebraic notation: 11-15

The PISA view of mathematical literacy in Indonesia

Learning the Algebraic Method of Solving Problems