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

I A Theory of Affect for Mathematical Problem Solving.- 1 Affect and Learning: Causes and Consequences of Emotional Interactions.- 2 The Role of Affect in Mathematical Problem Solving.- 3 Describing the Affective Domain: Saying What We Mean.- II Studies of Learning.- 4 Affect in Schema Knowledge: Source and Impact.- 5 Aesthetic Influences on Expert Mathematical Problem Solving.- 6 Self-Confidence, Interest, Beliefs, and Metacognition: Key Influences on Problem-Solving Behavior.- 7 Information Technologies and Affect in Mathematical Experiences.- 8 Searching for Affect in the Solution of Story Problems in Mathematics.- III Studies of Teaching.- 9 Young Children's Emotional Acts While Engaged in Mathematical Problem Solving.- 10 Teaching Practices and Student Affect in Problem-Solving Lessons of Select Junior-High Mathematics Teachers.- 11 Affect and Problem Solving in an Elementary School Mathematics Classroom.- 12 Affective Factors and Computational Estimation Ability.- 13 Affective Issues in Teaching Problem Solving: A Teacher's Perspective.- IV Responses to the Theory.- 14 The Study of Affect and Mathematics: A Proposed Generic Model for Research.- 15 Psychological Conceptions of Mathematics and Emotion.- V Looking Back.- 16 Affect and Learning: Reflections and Prospects.- 17 Beliefs, Attitudes, and Emotions: New Views of Affect in Mathematics Education.- Author Index.

Affect + Cognition = Mathematical Problem Solving@@@Affect and Mathematical Problem Solving: A New Perspective

The topic of this paper is mathematical modelling or—as it is often, more broadly, called—applications and modelling. This has been an important topic in mathematics education during the last few decades, beginning with Pollak’s survey lecture (New Trends in Mathematics Teaching IV, Paris, pp. 232–248, 1979) at ICME-3, Karlsruhe 1976. By using the term “applications and modelling”, both the products and the processes in the interplay between the real world and mathematics are addressed. In this paper, I will try to summarize some important aspects, in particular, concerning the teaching of applications and modelling.

/pdf/quality-teaching-of-mathematical-modelling-what-do-we-know-215dvamef9.pdf

Quality Teaching of Mathematical Modelling: What Do We Know, What Can We Do?

Word problems are among the most difficult kinds of problems that mathematics learners encounter. Perhaps as a result, they have been the object of a tremendous amount research over the past 50 years. This opening article gives an overview of the research literature on word problem solving, by pointing to a number of major topics, questions, and debates that have dominated the field. After a short introduction, we begin with research that has conceived word problems primarily as problems of comprehension, and we describe the various ways in which this complex comprehension process has been conceived theoretically as well as the empirical evidence supporting different theoretical models. Next we review research that has focused on strategies for actually solving the word problem. Strengths and weaknesses of informal and formal solution strategies—at various levels of learners’ mathematical development (i.e., arithmetic, algebra)—are discussed. Fourth, we address research that thinks of word problems as exercises in complex problem solving, requiring the use of cognitive strategies (heuristics) as well as metacognitive (or self-regulatory) strategies. The fifth section concerns the role of graphical representations in word problem solving. The complex and sometimes surprising results of research on representations—both self-made and externally provided ones—are summarized and discussed. As in many other domains of mathematics learning, word problem solving performance has been shown to be significantly associated with a number of general cognitive resources such as working memory capacity and inhibitory skills. Research focusing on the role of these general cognitive resources is reviewed afterwards. The seventh section discusses research that analyzes the complex relationship between (traditional) word problems and (genuine) mathematical modeling tasks. Generally, this research points to the gap between the artificial word problems learners encounter in their mathematics lessons, on the one hand, and the authentic mathematical modeling situations with which they are confronted in real life, on the other hand. Finally, we review research on the impact of three important elements of the teaching/learning environment on the development of learners’ word problem solving competence: textbooks, software, and teachers. It is shown how each of these three environmental elements may support or hinder the development of learners’ word problem solving competence. With this general overview of international research on the various perspectives on this complex and fascinating kind of mathematical problem, we set the scene for the empirical contributions on word problems that appear in this special issue.

Word problems in mathematics education: a survey

We tested the interest and choice portion of social-cognitive career theory (SCCT; Lent, Brown, & Hackett, 1994) in the context of science, technology, engineering, and mathematics (STEM) domains. Data from 143 studies (including 196 independent samples) conducted over a 30-year period (1983 through 2013) were subjected to meta-analytic path analyses. The interest/choice model was found to fit the data well over all samples as well as within samples composed primarily of women and men and racial/ethnic minority and majority persons. The model also accounted for large portions of the variance in interests and choice goals within each path analysis. Despite the general predictive utility of SCCT across gender and racial/ethnic groups, we did find that several parameter estimates differed by group. We present both the group similarities and differences and consider their implications for future research, intervention, and theory refinement. (PsycINFO Database Record

Predictors of science, technology, engineering, and mathematics choice options: A meta-analytic path analysis of the social-cognitive choice model by gender and race/ethnicity.

In this study which was part of the DISUM-project, 224 ninth graders from 14 German classes from middle track schools (Realschule) were asked about their enjoyment, interest, value and self-efficacy expectations concerning three types of mathematical problems: intra-mathematical problems, word problems and modelling problems. Enjoyment, interest, value and self-efficacy were assessed before and after a ten-lesson teaching unit promoting modelling competency related to the topics “Pythagoras’ theorem” and “linear functions”. The study aimed to answer the following research questions: (1) Do students’ enjoyment, value, interest and self-efficacy expectations differ depending on the type of task? (2) Does the treatment of modelling problems in classroom instruction influence these variables? (3) Are there any differential effects for different ways of teaching modelling problems, including a “directive”, teacher-centred instruction and an “operative-strategic”, more student-centred instruction emphasising group work and strategic scaffolding by the teacher? The findings show that there were no differences in students’ enjoyment, interest, value and self-efficacy between the three types of tasks. However, teaching oriented towards modelling problems had positive effects on some of the student variables, with the student-centred teaching method producing the most beneficial effects.

Teaching methods for modelling problems and students’ task-specific enjoyment, value, interest and self-efficacy expectations

In a study within the DISUM research project, we investigated the role that the construction of situation models plays as an essential prerequisite for understanding a given mathematical modelling task, using a sample of 21 9th grade classes (N=416). Specific task characteristics, general mathematical competence, reading competence, and teacher interventions aiming at understanding the situation model were analyzed as crucial factors influencing students’ ability to solve modelling tasks. The results show that: (1) strategies for constructing an adequate situation model have a significant influence on modelling competence, (2) mathematical reading competence and intra-mathematical competence can explain almost one third of the variance of the performance on the modelling test, (3) teacher interventions may encourage students to adopt strategies facilitating the construction of situation models, but an increase of modelling competence requires separate strategy training.

The Role of the Situation Model in Mathematical Modelling—Task Analyses, Student Competencies, and Teacher Interventions

Zeitschrift fur Padagogik 49 (2003) 3, S. 400-412
Padagogische Teildisziplin: Schulpadagogik;

PISA und Allgemeinbildung

Empowerment durch Schulpraktika. Perspektiven wechseln in der Lehrerbildung

Unterrichtsformen, erlebte Selbständigkeit, Emotionen und Anstrengung als Prädiktoren von Schülerleistungen bei anspruchsvollen mathematischen Modellierungsaufgaben

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