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

Many innovative approaches to education such as problem-based learning (PBL) and inquiry learning (IL) situate learning in problem-solving or investigations of complex phenomena. Kirschner, Sweller, and Clark (2006) grouped these approaches together with unguided discovery learning. However, the problem with their line of argument is that IL and PBL approaches are highly scaffolded. In this article, we first demonstrate that Kirschner et al. have mistakenly conflated PBL and IL with discovery learning. We then present evidence demonstrating that PBL and IL are powerful and effective models of learning. Far from being contrary to many of the principles of guided learning that Kirschner et al. discussed, both PBL and IL employ scaffolding extensively thereby reducing the cognitive load and allowing students to learn in complex domains. Moreover, these approaches to learning address important goals of education that include content knowledge, epistemic practices, and soft skills such as collaboration and sel...

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Scaffolding and Achievement in Problem-Based and Inquiry Learning: A Response to Kirschner, Sweller, and Clark (2006)

Education is a hot topic. From the stage of presidential debates to tonight's dinner table, it is an issue that most Americans are deeply concerned about. While there are many strategies for improving the educational process, we need a way to find out what works and what doesn't work as well. Educational assessment seeks to determine just how well students are learning and is an integral part of our quest for improved education. The nation is pinning greater expectations on educational assessment than ever before. We look to these assessment tools when documenting whether students and institutions are truly meeting education goals. But we must stop and ask a crucial question: What kind of assessment is most effective? At a time when traditional testing is subject to increasing criticism, research suggests that new, exciting approaches to assessment may be on the horizon. Advances in the sciences of how people learn and how to measure such learning offer the hope of developing new kinds of assessments-assessments that help students succeed in school by making as clear as possible the nature of their accomplishments and the progress of their learning. Knowing What Students Know essentially explains how expanding knowledge in the scientific fields of human learning and educational measurement can form the foundations of an improved approach to assessment. These advances suggest ways that the targets of assessment-what students know and how well they know it-as well as the methods used to make inferences about student learning can be made more valid and instructionally useful. Principles for designing and using these new kinds of assessments are presented, and examples are used to illustrate the principles. Implications for policy, practice, and research are also explored. With the promise of a productive research-based approach to assessment of student learning, Knowing What Students Know will be important to education administrators, assessment designers, teachers and teacher educators, and education advocates.

Knowing What Students Know: The Science and Design of Educational Assessment

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.

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

Between 5% and 8% of school-age children have some form of memory or cognitive deficit that interferes with their ability to learn concepts or procedures in one or more mathematical domains. A review of the arithmetical competencies of these children is provided, along with discussion of underlying memory and cognitive deficits and potential neural correlates. The deficits are discussed in terms of three subtypes of mathematics learning disability and in terms of a more general framework for linking research in mathematical cognition to research in learning disabilities.

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Mathematics and Learning Disabilities

"Principles and Standards for School Mathematics" in the Classroom.

Geometry is an important part of the mathematics curriculum. However, students are not demonstrating strong conceptual knowledge of this subject. The research of Van Hiele and Van Hiele-Geldof has focused on the concept of thinking levels in geometry and the role of instruction in raising levels of thinking. This paper describes a field trial of a supplemental geometry unit intended to raise Van Hiele thinking levels in a group of 23 eighth-grade students by having them become more adept at using higher order thinking skills. Sample questions assessing particular Van Hiele thinking levels and attitudes toward geometry, as well as field-tested activities yielding the most positive results, are presented. Educators can benefit from this application of the Van Hiele model of geometric thinking, since the thought processes involved in learning geometry are explained, along with teaching techniques and tools for assessment. By having teachers become more aware of their students' cognitive skills, attitudes, and misconceptions, teaching practices and student achievement can be enhanced.

Enhancing geometric reasoning.

https://www.jstor.org/stable/10.5951/teacchilmath.19.9.0572

"We Do Care," Say Parents.

IntroductionStudents perform best when family members are part of the learning process. For example, Jeynes (2013) points out the strong connection between voluntary expressions of family engagement and positive school outcomes. Engagement of this nature includes reading with one's child, setting high expectations for academic achievement, and parenting styles that can improve attendance (Sandefur & Wells, 1999), attitudes towards school (Centers for Disease Control and Prevention, 2012), and test scores (Strauss & Kohn, 2013).Math attitudes and achievement are particularly noted as benefiting from home influences. For example, family members who pose thought-provoking questions or break down a problem into smaller, more manageable pieces support children in organizing their thinking (Walker, Shenker, & Hoover-Dempsey, 2010). In addition, home environments that nurture children's self-confidence help minimize math anxiety levels and, in turn, positively influence children's academic success with math (Vukovic, Roberts, & Wright, 2013).Although family engagement clearly matters, opportunities for families to meaningfully engage in children's learning are sparse (Rivera & Waxman, 2011; Van Voorhis, 2011; Weiss, Lopez, & Rosenberg, 2010). In fact, the amount of discussion, rhetoric, and widespread agreement about the importance of family engagement far outweigh the actual degree of change in schools' practices.Contributing to the situation are prevailing school-centric family-school paradigms that portray educators dictating passive family member roles such as monitoring children's behavior, school attendance, homework completion, and television viewing. Oftentimes, family members are positioned only as consumers of services and information from schools, rather than as active collaborators in shaping learning environments (Evans & Radina, 2014). Some go as far as to say that school practices for involving family members serve more as a form of public relations, rather than a means for cultivating meaningful partnerships with families (Knight-Abowitz, 2011).Compounding these conditions is a consistent lack of teacher preparation. Even though federal and state policies call for equal partnerships between families and schools at a systemic level, attention to building capacity among all stakeholders to materialize such partnerships is lacking. Unfortunately, principals and teachers receive little training for engaging families despite the fact that family engagement is a high priority among school districts (Mapp & Kuttner, 2013).Minimal attention is given to developing teachers' understandings about family members' views on learning objectives, as well as skills for meaningfully involving family members (Remillard & Jackson, 2006). In addition, school leaders' attention is unfortunately and disturbingly directed away from homeschool relationship building due to the pressures of meeting state and national standards (Ferrara, 2009; Shirley & Evans, 2007).If these trends continue, researchers warn that self-efficacy and authentic engagement opportunities are at risk of being undermined, especially for low-income and minority families (Hoover-Dempsey et al., 2005). Given the reformed mathematics curriculum materials and family members' lack of familiarity with such materials, family members may resort to assisting children in ways that only mirror their own past learning environments, as opposed to that of current classrooms (Remillard & Jackson, 2006).In this article, the author responds to the need for equipping teachers with mindsets and skills for supporting families with math. Specifically described is how practitioner inquiry opportunities (conversations with family members about math) embedded into a mathematics methods course informed teachers' practices for interacting with family members.Findings from conversations facilitated by 72 teachers with 225 families of public and nonpublic elementary, middle, and high school students demonstrate how verbal dialogue enabled teachers to respond to circumstances surrounding the home. …

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Conversations with Family Members about Math.

This paper reports on a study concerning the effects of a 2‐year professional development initiative designed to improve teachers' methods of developing understanding of mathematical concepts and procedures. The initiative sought to accomplish this goal by engaging teachers in a learning environment consisting of modeled lessons, focus sessions, and hands‐on workshops. Pre and post analyses revealed significant changes in teaching methods and student achievement. Findings are shared to deepen the knowledge base and make it accessible to teacher educators and professional developers in order to inform their decisions about teacher education programs and professional development design.

Regina Mistretta

Papers

"Principles and Standards for School Mathematics" in the Classroom.

Enhancing geometric reasoning.

"We Do Care," Say Parents.

Conversations with Family Members about Math.

Mathematics instructional design: Observations from the field