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

The conditions of learning

The World's a Stage@@@Structure and Insight: A Theory of Mathematics Education

Predicting children's mathematical performance in grade one by early numeracy

Research on Visualization in Learning and Teaching Mathematics: Emergence from Psychology

Recent educational research has turned increasing attention to the structural development of young students’ mathematical thinking. Early algebra, multiplicative reasoning, and spatial structuring are three areas central to this research. There is increasing evidence that an awareness of mathematical structure is crucial to mathematical competence among young children. The purpose of this paper is to propose a new construct, Awareness of Mathematical Pattern and Structure (AMPS), which generalises across mathematical concepts, can be reliably measured, and is correlated with general mathematical understanding. We provide supporting evidence drawn from a study of 103 Grade 1 students.

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Awareness of pattern and structure in early mathematical development

The development of patterning strategies during the year prior to formal schooling was studied in 53 children from 2 similar preschools. One preschool implemented a 6-month intervention focusing on repeating and spatial patterns. An interview-based Early Mathematical Patterning Assessment (EMPA) was developed and administered preand postintervention, and again following the 1st year of formal schooling. The intervention group outperformed the comparison group across a wide range of patterning tasks at the postand follow-up assessments. Children from the intervention group demonstrated greater understanding of unit of repeat and spatial structuring, and most were also able to extend and explain growing patterns 1 year later. In contrast, most of the comparison group treated repeating patterns as alternating items and rarely recognized simple geometrical patterns. The findings indicate a fundamental link between patterning and multiplicative reasoning through the development of composite units.

https://www.jstor.org/stable/10.5951/jresematheduc.42.3.0237

Assessing the Development of Preschoolers' Mathematical Patterning

In this study, an intuitive model was defined as an internal mental structure corresponding to a class of calculation strategies. A sample of female students was observed 4 times during Grades 2 and 3 as they solved the same set of 24 word problems. From the correct responses, 12 distinct calculation strategies were identified and grouped into categories from which the children’s intuitive models of multiplication and division were inferred. It was found that the students used 3 main intuitive models: direct counting, repeated addition, and multiplicative operation. A fourth model, repeated subtraction, only occurred in division problems. All the intuitive models were used with all semantic structures, their frequency varying as a complex interaction of age, size of numbers, language, and semantic structure. The results are interpreted as showing that children acquire an expanding repertoire of intuitive models and that the model they employ to solve any particular problem reflects the mathematical structure they impose on it. Several recent studies have shown that students can solve a variety of multiplicative problems long before formal instruction on the operations of multiplication and division. For example, Kouba (1989) found that 30% of Grade 1 and 70% of Grade 2 students could solve simple equivalent group problems and Mulligan (1992) found a steady increase in success rate on similar problems from over 50% at the beginning of Grade 2 to nearly 95% at the end of Grade 3. More recently Carpenter, Ansell, Franke, Fennema, and Weisbeck (1993) found that even kindergarten students could

https://www.nctm.org/Publications/journal-for-research-in-mathematics-education/1997/Vol28/Issue3/jrme1997-05-309a_pdf/

Young children's intuitive models of multiplication and division

The focus of this article is the strategies young children use to solve rectangular covering tasks before they have been taught area measurement. One hundred fifteen children from Grades 1 to 4 were observed while they solved various array-based tasks, and their drawings were collected and analyzed. Children's solution strategies were classified into 5 developmental levels; we suggest that children sequentially learn 4 principles underlying rectangular covering. In the analysis we emphasize the importance of understanding the relation between the size of the unit and the dimensions of the rectangle in learning about rectangular covering, clarify the role of multiplication, and identify the significance of a relational understanding of length measurement. Implications for the learning of area measurement are addressed.

https://www.jstor.org/stable/pdfplus/749749.pdf

Young Children's Intuitive Understanding of Rectangular Area Measurement.

Responses to word problems involving rates of change were collected on four occasions during and after 24 hours of concept-based calculus instruction given to a group of first-year university students, all of whom had studied calculus in secondary school. The number of students who could symbolize rates of change in noncomplex situations increased dramatically. However, there was almost no increase in the number who could symbolize rates of change in complex items or in items that required modeling a situation using algebraic variables. Detailed analysis revealed three main categories of error, in all of which variables are treated as symbols to be manipulated rather than as quantities to be related. We surmise that the students had developed an "abstract-apart" concept of a variable rather than the "abstract-general" concept that is needed for the successful study of calculus. Changes in technology, the qualifications of teachers, and the mathematical competence of students have led many to question the role of traditional calculus courses in the curriculum. There is, in particular, much concern about the large numbers of students taking calculus and the rote, manipulative learning that takes place (Cipra, 1988; Steen, 1988; White, 1990). The value of skill-based calculus courses has also come under fire because computers and calculators now perform most (if not all) of the manipulative procedures taught in such courses (Steen, 1988; Tall, 1987a). Several studies (Heid, 1988, 1989; Hickernell & Proskurowski, 1985; Judson, 1990; Palmiter, 1991) have shown how using the computer as a tool for performing the procedures of calculus and algebra can free students to explore applications. Other studies (Hsaio, 1984/85; Tall, 1986) have used graphics to explore secants and tangents to curves in order to build up a sound concept of a derivative. The general tendency is for less emphasis on skills and greater emphasis on underlying concepts. Research into the understanding of calculus has shown a whole spectrum of concepts that cause problems for students. In particular, student difficulties with the

/pdf/conceptual-knowledge-in-introductory-calculus-35yalwfed0.pdf

Michael Mitchelmore

Papers

Awareness of pattern and structure in early mathematical development

Assessing the Development of Preschoolers' Mathematical Patterning

Young children's intuitive models of multiplication and division

Young Children's Intuitive Understanding of Rectangular Area Measurement.

Conceptual Knowledge in Introductory Calculus.