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

There is widespread agreement that effective teachers have unique knowledge of students' mathematical ideas and thinking. However, few scholars have focused on conceptualizing this domain, and even fewer have focused on measuring this knowledge. In this article, we describe an effort to conceptualize and develop measures of teachers' combined knowledge of content and students by writing, piloting, and analyzing results from multiple-choice items.

/pdf/unpacking-pedagogical-content-knowledge-conceptualizing-and-1jmp4uiwdi.pdf

Unpacking Pedagogical Content Knowledge: Conceptualizing and Measuring Teachers' Topic-Specific Knowledge of Students

Focusing on expanding technical capabilities and new collaborative possibilities, we address 4 challenges for scientists who collect and use video records to conduct research in and on complex learning environments: (a) Selection: How can researchers be systematic in deciding which elements of a complex environment or extensive video corpus to select for study? (b) Analysis: What analytical frameworks and practices are appropriate for given research problems? (c) Technology: What technologies are available and what new tools must be developed to support collecting, archiving, analyzing, reporting, and collaboratively sharing video? and (d) Ethics: How can research protocols encourage broad video sharing and reuse while adequately protecting the rights of research participants who are recorded?

/pdf/conducting-video-research-in-the-learning-sciences-guidance-4dztdyeadv.pdf

Conducting Video Research in the Learning Sciences: Guidance on Selection, Analysis, Technology, and Ethics

The development of sub-disciplines within cognitive neuroscience follows common sense categories such as language, audition, action, memory, emotion and perception among others. There are also well-established research programmes into temporal perception, spatial perception and mathematical cognition that also reflect the subjective impression of how experience is constructed. There is of course no reason why the brain should respect these common sense, text book divisions and, here, we discuss the contention that generalized magnitude processing is a more accurate conceptual description of how the brain deals with information about time, space, number and other dimensions. The roots of the case for linking magnitudes are based on the use to which magnitude information is put (action), the way in which we learn about magnitudes (ontogeny), shared properties and locations of magnitude processing neurons, the effects of brain lesions and behavioural interference studies. Here, we assess this idea in the context of a theory of magnitude, which proposed common processing mechanisms of time, space, number and other dimensions.

The parietal cortex and the representation of time, space, number and other magnitudes

Background: Arithmetical skills are essential to the effective exercise of citizenship in a numerate society. How these skills are acquired, or fail to be acquired, is of great importance not only to individual children but to the organisation of formal education and its role in society. Method: The evidence on the normal and abnormal developmental progression of arithmetical abilities is reviewed; in particular, evidence for arithmetical ability arising from innate specific cognitive skills (innate numerosity) vs. general cognitive abilities (the Piagetian view) is compared. Results: These include evidence from infancy research, neuropsychological studies of developmental dyscalculia, neuroimaging and genetics. The development of arithmetical abilities can be described in terms of the idea of numerosity ‐ the number of objects in a set. Early arithmetic is usually thought of as the effects on numerosity of operations on sets such as set union. The child’s concept of numerosity appears to be innate, as infants, even in the first week of life, seem to discriminate visual arrays on the basis of numerosity. Development can be seen in terms of an increasingly sophisticated understanding of numerosity and its implications, and in increasing skill in manipulating numerosities. The impairment in the capacity to learn arithmetic ‐ dyscalculia ‐ can be interpreted in many cases as a deficit in the concept in the child’s concept of numerosity. The neuroanatomical bases of arithmetical development and other outstanding issues are discussed. Conclusions: The evidence broadly supports the idea of an innate specific capacity for acquiring arithmetical skills, but the effects of the content of learning, and the timing of learning in the course of development, requires further investigation. Keywords: Arithmetic, cognitive development, dyscalculia, numerosity, number, infants, child. ‘A child, at birth, is a candidate for humanity; it cannot become human in isolation.’ (Pieron, 1959)

/pdf/the-development-of-arithmetical-abilities-peje1t5qnh.pdf

The development of arithmetical abilities.

Part 1 How Children and Adults Use the Intuitive Rule "More A - More B": Equality Situations Inequality Situations Some Questions About the Use of This Rule. Part 2 Learning About the Intuitive Rule "Same A - Same B": Directly-Given Equality Logically-Deduced Equality Some Questions About the Use of This Rule. Part 3 The Nature of the Intuitive Rule "Everything Can Be Divided": Repeated Halving Decreasing Series Some Questions About the Use of This Rule. Part 4 Towards a Theory of Intuitive Rules: Theoretical Approaches to Students' Alternative Reasoning and Conceptions Some Questions About the Intuitive Rule Theory. Part 5 Using Knowledge About Intuitive Rules - Educational Implications: Overcoming the Effects of the Intuitive Rules - General and Specific Teaching Strategies Overcoming the Effects of the Intuitive Rules - How Mathematics and Science Educators Use Related Knowledge in Instruction.

How students (mis-) understand science and mathematics : intuitive rules

The intuition of infinity

In recent years much has been written about children's and adolescents' misconceptions concerning the operations needed to solve multiplication and division word problems (e.g., Bell, 1982; Bell, Fischbein, & Greer, 1984; Bell, Swan, & Taylor, 1981; Brown, 1981; Vergnaud, 1983). Fischbein, Deri, Nello, and Marino (1985) argued that the arithmetical operations generally remain linked to implicit primitive behavioral models. They noted that the primitive models for multiplication and division may be viewed as the source of misconceptions such as "the divisor must be a whole number" or "multiplication always makes bigger and division always makes smaller." The purpose of our study was to explore whether preservice elementary teachers have the same misconceptions. If preservice teachers hold these misconceptions, they are not likely to recognize the related errors students make. In fact, their instruction might inadvertently contribute to perpetuating the misconceptions. We attempted to determine whether preservice teachers select the correct operation when they are presented with problems having data that conflict with the implicit rules of the primitive behavioral models of multiplication and division. As teacher educators, we were also interested in noting whether the preservice teachers would exhibit other misconceptions and the extent to which such misconceptions were similar to those previously noted among children.

Preservice Teachers' Misconceptions in Solving Verbal Problems in Multiplication and Division

In this paper we examine the possibility of differentiating between two types of nonexamples. The first type, intuitive nonexamples, consists of nonexamples which are intuitively accepted as such. That is, children immediately identify them as nonexamples. The second type, non-intuitive nonexamples, consists of nonexamples that bear a significant similarity to valid examples of the concept, and consequently are more often mistakenly identified as examples. We describe and discuss these notions and present a study regarding kindergarten children’s grasp of nonexamples of triangles.

/pdf/intuitive-nonexamples-the-case-of-triangles-5c0zdlml5z.pdf

Intuitive Nonexamples: The Case of Triangles.

It is widely accepted today that teachers should be aware of, and knowledgeable about, students’ mathematical learning. It is believed that such awareness and knowledge signifi cantly contribute to various aspects of the practice of teaching. In this chapter, we critically examine this commonly held belief. We begin this chapter by interpreting what one might mean by teacher knowledge and understanding of students’ mathematical learning. Then we move to examining possible implications of such teacher knowledge on instruction. The third part of this chapter examines the assumption that teacher knowledge and understanding of students’ mathematical learning is essential for good teaching in light of different theoretical perspectives. The fourth part discusses pre- and in-service teacher education that focus on different aspects of students’ mathematical learning. Finally, we conclude this chapter by suggesting issues for further research.

Teacher knowledge and understanding of students’ mathematical learning and thinking

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