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

http://www.compassproject.net/sadhana/teaching/readings/deftainsworth.pdf

DeFT: A conceptual framework for considering learning with multiple representations

Contents: J. Gale, Preface. Part I:Radical Constructivism and Social Constructionism. E. von Glasersfeld, A Constructivist Approach to Teaching. K.J. Gergen, Social Construction and the Educational Process. J. Shotter, In Dialogue: Social Constructionism and Radical Constructivism. J. Richards, Construct[ion/iv]ism: Pick One of the Above. Part II:Information-Processing Constructivism and Cybernetic Systems. F. Steier, From Universing to Conversing: An Ecological Constructionist Approach to Learning and Multiple Description. R.J. Spiro, P.J. Feltovich, M.J. Jacobson, R.L. Coulson, Cognitive Flexibility, Constructivism, and Hypertext: Random Access Instruction for Advanced Knowledge Acquisition in Ill-Structured Domains. K. Tomm, Response to Chapters by Spiro et al. and Steier. P.W. Thompson, Constructivism, Cybernetics, and Information Processing: Implications for Technologies of Research on Learning. Part III:Social Constructivism and Sociocultural Approaches. H. Bauersfeld, The Structuring of the Structures: Development and Function of Mathematizing as a Social Practice. J.V. Wertsch, C. Toma, Discourse and Learning in the Classroom: A Sociocultural Approach. C. Konold, Social and Cultural Dimensions of Knowledge and Classroom Teaching. J. Confrey, How Compatible Are Radical Constructivism, Sociocultural Approaches, and Social Constructivism? Analysis and Synthesis I: Alternative Epistemologies. M.H. Bickhard, World Mirroring Versus World Making: There's Gotta Be a Better Way. Part IV:Alternative Epistemologies in Language, Mathematics, and Science Education. R. Duit, The Constructivist View: A Fashionable and Fruitful Paradigm for Science Education Research and Practice. G.B. Saxe, From the Field to the Classroom: Studies in Mathematical Understanding. N.N. Spivey, Written Discourse: A Constructivist Perspective. T. Wood, From Alternative Epistemologies to Practice in Education: Rethinking What It Means to Teach and Learn. E. Ackermann, Construction and Transference of Meaning Through Form. D. Rubin, Constructivism, Sexual Harassment, and Presupposition: A (Very) Loose Response to Duit, Saxe, and Spivey. Part V:Alternative Epistemologies in Clinical, Mathematics, and Science Education. E. von Glasersfeld, Sensory Experience, Abstraction, and Teaching. R. Driver, Constructivist Approaches to Science Teaching. T. Wood, P. Cobb, E. Yackel, Reflections on Learning and Teaching Mathematics in Elementary School. P. Lewin, The Social Already Inhabits the Epistemic: A Discussion of Driver Wood, Cobb, and Yackel and von Glasersfeld. J. Becker, M. Varelas, Assisting Construction: The Role of the Teacher in Assisting the Learner's Construction of Preexisting Cultural Knowledge. E.H. Auerswald, Shifting Paradigms: A Self-Reflective Critique. Analysis and Synthesis II: Epsitemologies in Education. P. Ernest, The One and the Many. Analysis and Synthesis III: Retrospective Comments and Future Prospects. L.P. Steffe, Alternative Epistemologies: An Educator's Perspective. J. Gale, Epilogue.

教育中的建构主义 = Constructivism in Education

The learning and teaching of school algebra.

Emphasizing the developmental need for positive peer relationships, in this study the authors tested a social-contextual view of the mechanisms and processes by which early adolescents' achievement and peer relationships may be promoted simultaneously. Meta-analysis was used to review 148 independent studies comparing the relative effectiveness of cooperative, competitive, and individualistic goal structures in promoting early adolescents' achievement and positive peer relationships. These studies represented over 8 decades of research on over 17,000 early adolescents from 11 countries and 4 multinational samples. As predicted by social interdependence theory, results indicate that higher achievement and more positive peer relationships were associated with cooperative rather than competitive or individualistic goal structures. Also as predicted, results show that cooperative goal structures were associated with a positive relation between achievement and positive peer relationships. Implications for theory and application are discussed.

/pdf/promoting-early-adolescents-achievement-and-peer-2u5jxeov6f.pdf

Promoting early adolescents' achievement and peer relationships: the effects of cooperative, competitive, and individualistic goal structures.

This review of the introductory instructional substance of functions and graphs analyzes research on the interpretation and construction tasks associated with functions and some of their representations: algebraic, tabular, and graphical. The review also analyzes the nature of learning in terms of intuitions and misconceptions, and the plausible approaches to teaching through sequences, explanations, and examples. The topic is significant because of (a) the increased recognition of the organizing power of the concept of functions from middle school mathematics through more advanced topics in high school and college, and (b) the symbolic connections that represent potentials for increased understanding between graphical and algebraic worlds. This is a review of a specific part of the mathematics subject mailer and how it is learned and may be taught; this specificity reflects the issues raised by recent theoretical research concerning how specific context and content contribute to learning and meaning.

https://www.math.ksu.edu/~bennett/onlinehw/qcenter/lzs.pdf

Functions, Graphs, and Graphing: Tasks, Learning, and Teaching

In this paper we present a study conductedwithin the framework of an in-serviceprofessional development program for junior andsenior high school mathematics teachers. Thefocus of the study is the analysis of processesencountered by the staff members, as members ofa community of practice, which contributed totheir growth as teacher educators. We offer athree-layer model of growth through practice asa conceptual framework to think about becominga mathematics teacher educator, and illustratehow our suggested model can be adapted to thecomplexities and commonalities of theunderlying processes of professionaldevelopment of mathematics teacher educators.

Professional Development of Mathematics Teacher Educators: Growth Through Practice

The paper is a reflective account of the design and implementation of mathematical tasks that evoke uncertainty for the learner. Three types of uncertainty associated with mathematical tasks are discussed and illustrated: competing claims, unknown path or questionable conclusion, and non-readily verifiable outcomes. One task is presented in depth, pointing to the dynamic nature of task design, and the added value stimulated by the uncertainty component entailed in the task in terms of mathematical and pedagogical musing.

Seizing the Opportunity to Create Uncertainty in Learning Mathematics

This article deals with 12th-grade students' conceptions of a mathematical definition. Their conceptions of a definition were revealed through individual and group activities in which they were asked to consider a number of possible definitions of four mathematical concepts: two geometric and two analytic. Data consisted of written responses to questionnaires and transcriptions of videotaped group discussions. The findings point to three types of students' arguments: mathematical, communicative, and figurative. In addition, two types of reasoning were identified surrounding the contemplation of alternative definitions: for the geometric concepts, the dominant type of reasoning was a definition-based reasoning; for the analytic concepts, the dominant type was an example-based reasoning. Students' conceptions of a definition are described in terms of the features and roles they attribute to a mathematical definition.

Students' Conceptions of a Mathematical Definition.

The main goal of the study reported in our paper is to characterize teachers’ choice of examples in and for the mathematics classroom. Our data is based on 54 lesson observations of five different teachers. Altogether 15 groups of students were observed, three seventh grade, six eighth grade, and six ninth grade classes. The classes varied according to their level—seven classes of top level students and six classes of mixed—average and low level students. In addition, pre and post lesson interviews with the teachers were conducted, and their lesson plans were examined. Data analysis was done in an iterative way, and the categories we explored emerged accordingly. We distinguish between pre-planned and spontaneous examples, and examine their manifestations, as well as the different kinds of underlying considerations teachers employ in making their choices, and the kinds of knowledge they need to draw on. We conclude with a dynamic framework accounting for teachers’ choices and generation of examples in the course of teaching mathematics.

https://link.springer.com/content/pdf/10.1007%2Fs10649-008-9140-6.pdf

Orit Zaslavsky

Papers

Functions, Graphs, and Graphing: Tasks, Learning, and Teaching

Professional Development of Mathematics Teacher Educators: Growth Through Practice

Seizing the Opportunity to Create Uncertainty in Learning Mathematics

Students' Conceptions of a Mathematical Definition.

Characteristics of teachers’ choice of examples in and for the mathematics classroom