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

Ulric Neisser (Chair) Gwyneth Boodoo Thomas J. Bouchard, Jr. A. Wade Boykin Nathan Brody Stephen J. Ceci Diane E Halpern John C. Loehlin Robert Perloff Robert J. Sternberg Susana Urbina Emory University Educational Testing Service, Princeton, New Jersey University of Minnesota, Minneapolis Howard University Wesleyan University Cornell University California State University, San Bernardino University of Texas, Austin University of Pittsburgh Yale University University of North Florida

/pdf/intelligence-knowns-and-unknowns-10r4cnx354.pdf

Intelligence: Knowns and unknowns.

Despite a century's worth of research, arguments surrounding the question of whether far transfer occurs have made little progress toward resolution. The authors argue the reason for this confusion is a failure to specify various dimensions along which transfer can occur, resulting in comparisons of "apples and oranges." They provide a framework that describes 9 relevant dimensions and show that the literature can productively be classified along these dimensions, with each study situated at the intersection of various dimensions. Estimation of a single effect size for far transfer is misguided in view of this complexity. The past 100 years of research shows that evidence for transfer under some conditions is substantial, but critical conditions for many key questions are untested.

/pdf/when-and-where-do-we-apply-what-we-learn-a-taxonomy-for-far-3ic2fxpips.pdf

When and where do we apply what we learn?: A taxonomy for far transfer.

The situative perspective shifts the focus of analysis from individual behavior and cognition to larger systems that include behaving cognitive agents interacting with each other and with other subsystems in the environment. The first section presents a version of the situative perspective that draws on studies of social interaction, philosophical situation theory, and ecological psychology. Framing assumptions and concepts are proposed for a synthesis of the situative and cognitive theoretical perspectives, and a further situative synthesis is suggested that would draw on dynamic-systems theory. The second section discusses relations between the situative, cognitive, and behaviorist theoretical perspectives and principles of educational practice. The third section discusses an approach to research and social practice called interactive research and design, which fits with the situative perspective and provides a productive, albeit syncretic, combination of theory-oriented and instrumental functions of research.

/pdf/the-situativity-of-knowing-learning-and-research-dw9hfgyit3.pdf

The situativity of knowing, learning, and research.

In this article, we make an attempt to operationalize the notion of identity so as to justify the claim about its potential as an analytic tool for investigating learning. According to our definition, identity is a set of reifying, significant, endorsable stories about a person. The subsequent analysis of the dynamics of narratives makes it clear that identities, even if individually told, are products of a collective storytelling. Our main claim is that learning may be thought of as closing the gap between actual identity and designated identity, two particular sets of reifying significant stories about the learner, endorsed by this learner. The theoretical substantiation of this assertion is accompanied by vignettes from a study in which mathematical learning practices of a group of 17 year old immigrant students from the former Soviet Union newly arrived in Israel were compared to those of native Israelis.

Telling Identities: In Search of an Analytic Tool for Investigating Learning as a Culturally Shaped Activity

An analysis of everyday use of mathematics by working youngsters in commercial transactions in Recife, Brazil, revealed computational strategies different from those taught in schools Performance on mathematical problems embedded in real-life contexts was superior to that on school-type word problems and context-free computational problems involving the same numbers and operations Implications for education are examined

Mathematics in the streets and in schools

Preface Series foreword 1. What is street mathematics? 2. Arithmetic in the streets and in schools 3. Written and oral arithmetic 4. Situational representation in oral and written mathematics 5. Situational and mathematical relations: A study on understanding proportions 6. Reversibility and transfer in the schema of proportionality 7. Reflections on street mathematics in hindsight References Index.

Street mathematics and school mathematics

Algebra instruction has traditionally been postponed until adolescence because of historical reasons (algebra emerged relatively recently), assumptions about psychological development (“developmental constraints” and “developmental readiness”), and data documenting the difficulties that adolescents have with algebra. Here we provide evidence that young students, aged 9–10 years, can make use of algebraic ideas and representations typically absent from the early mathematics curriculum and thought to be beyond students’ reach. The data come from a 30–month longitudinal classroom study of four classrooms in a public school in Massachusetts, with students between Grades 2–4. The data help clarify the conditions under which young students can integrate algebraic concepts and representations into their thinking. It is hoped that the present findings, along with those emerging from other research groups, will provide a research basis for integrating algebra into early mathematics education.

Arithmetic and Algebra in Early Mathematics Education

We examine issues that arise in students’ making of generalizations about geometrical figures as they are introduced to linear functions. We focus on the concepts of patterns, function, and generalization in mathematics education in examining how 15 third grade students (9 years old) come to produce and represent generalizations during the implementation of two lessons from a longitudinal study of early algebra. Many students scan output values of f(n) as n increases, conceptualizing the function as a recursive sequence. If this instructional route is pursued, educators need to recognize how students’ conceptualizations of functions depart from the closed form expressions ultimately aimed for. Even more fundamentally, it is important to nurture a transition from empirical generalizations, based on conjectures regarding cases at hand, to theoretical generalizations that follow from operations on explicit statements about mathematical relations.

Early algebra and mathematical generalization

Previous studies have demonstrated that children use oral calculation procedures not taught in school. The present study provided evidence for situational variables that strongly influence the tendency to use such procedures. It also provided a qualitative analysis of the oral mathematics used by Brazilian third graders. Concrete problem situations were powerful elicitors of oral computation procedures, whereas computation exercises tended to elicit school-learned computation algorithms. Oral computation procedures involved the use of two reliably identifiable routines, decomposition and repeated grouping, that revealed the children's solid understanding of the decimal system. In general, the children were far more successful in using oral mathematics than written mathematics. An understanding of children's oral procedures may be useful in developing more successful programs for elementary mathematics instruction.

Analúcia D. Schliemann

Papers

Mathematics in the streets and in schools

Street mathematics and school mathematics

Arithmetic and Algebra in Early Mathematics Education

Early algebra and mathematical generalization

Written and Oral Mathematics.