Improper Use of Physics‐Related Context in High School Mathematics Problems: Implications for Learning and Teaching

This article discusses potential pedagogical difficulties arising from using physics‐related contexts in high‐school mathematics problems. It is suggested that such problems should not require any external knowledge of physics by the students; meanwhile, the problems should not contradict the physics knowledge that the students may already have. Several examples of recently published textbook problems are presented along with the discussion of the context pitfalls. A simple technique (SCAN list) is proposed for evaluating the physics‐related mathematics problems.     

Key Developmental Understandings in Mathematics: A Direction for Investigating and Establishing Learning Goals

Although mathematics educators seem to agree on the importance of teaching mathematics for understanding, what they mean by understanding varies greatly. In this article, I elaborate and exemplify the construct of key developmental understanding to emphasize a particular aspect of teaching for understanding and to offer a construct that could be used to frame the identification of conceptual learning goals in mathematics. The key developmental understanding construct is based on extant empirical and theoretical work. The construct can be used in the context of research and curriculum development. Using a classroom example involving fractions, I illustrate how focusing on key developmental understandings leads to particular, potentially useful types of pedagogical thinking and directions for inquiry.     

Key Developmental Understandings in Mathematics: A Direction for Investigating and Establishing Learning Goals

Although mathematics educators seem to agree on the importance of teaching mathematics for understanding, what they mean by understanding varies greatly. In this article, I elaborate and exemplify the construct of key developmental understanding to emphasize a particular aspect of teaching for understanding and to offer a construct that could be used to frame the identification of conceptual learning goals in mathematics. The key developmental understanding construct is based on extant empirical and theoretical work. The construct can be used in the context of research and curriculum development. Using a classroom example involving fractions, I illustrate how focusing on key developmental understandings leads to particular, potentially useful types of pedagogical thinking and directions for inquiry.     

Exploring Mathematics in Imaginative Places: Rethinking What Counts as Meaningful Contexts for Learning Mathematics

This paper explores what happens when students engage with mathematical tasks that make no attempt to be connected with students' everyday life experiences. The investigation draws on the work of educators who call for a broader view of what might count as real and relevant contexts for studying mathematics. It investigates students' experiences with two imaginative tasks and reports on the students' intellectual and emotional engagement. This engagement is examined and described in terms of the character and quality of the class and group discussions generated. Findings suggest that students can indeed engage productively with mathematics when it is explored in imaginative settings and that such contexts can help students support and sustain their engagement with the mathematics in the task.     

A Review of the Integration of Science and Mathematics: Implications for Further Research

Building on the earlier analysis by Berlin (1991) , this paper reviews various studies on integrating mathematics and science in the 1990s and provides some implications for further research. The areas identified for further exploration include comparison of the nature of mathematics and science, epistemological debates in mathematics and in science education, the bases used to emphasize science over mathematics or vice versa, empirical evidence of effectiveness of integration, connections between teacher education programs for integration and teachers' subsequent classroom teaching practices, perceptions of integration on the part of teacher educators, contextual difficulties in implementing integrated approaches and possible solutions, and rationales of integrating mathematics and science through technology. In order to help all students become scientifically literate, which most reform documents call for, more focused attention on integration of curriculum and instruction is necessary.     

Mathematics Learning and Participation as Racialized Forms of Experience: African American Parents Speak on the Struggle for Mathematics Literacy

This article draws on 3 ethnographic and participant observation studies of African American parents and adults from 3 northern California communities. Although studies have shown that African American parents hold the same folk theories about mathematics as other parents, stressing it as an important school subject, few studies have sought to directly examine their beliefs about constraints and opportunities associated with mathematics learning for both themselves and their children. I argue that, as they situate the struggle for mathematical literacy within the larger contexts of African American, political, socioeconomic, and educational struggle, these parents help reveal that mathematics learning and participation can be conceptualized as racialized forms of experience. As they attempt to become doers of mathematics and advocates for their children's mathematics learning, discriminatory experiences have continued to subjugate some of these parents, whereas others—as demonstrated in their oppositional voices and behaviors—resisted their continued subjugation based on a belief that mathematics knowledge, beyond its role in schools, can be used to change the conditions of their lives. The characterization of mathematics learning as racialized experience put forth in this article contrasts with culture-free and situated perspectives of mathematics learning often found in the literature. As a result of their experiences with oppression in this society, the concept of race has historically played a major role in the lives of African Americans. Although race has dubious value as a scientific classification system, it has had real consequences for the life experiences and life opportunities of African Americans in the United States. Race is a socially constructed concept which is [a] defining characteristic for African American group membership. (Sellers, Smith, Shelton, Rowley, & Chavous, 1998, p. 18)     

Enabling the Practice of Mathematics Teachers in Context: Toward a New Equity Research Agenda

In this article, I address the need for a more clearly articulated research agenda around equity issues by proposing a working definition of equity and a focal point for research. More specifically, I assert that rather than pitting them against each other, we must coordinate (a) efforts to get marginalized students to master what currently counts as “dominant” mathematics with (b) efforts to develop a critical perspective among all students about knowledge and society in ways that ultimately facilitate (c) a positive relationship between mathematics, people, and equity on the planet. I make this argument partly by reviewing the literature on (school) contexts that engage marginalized students in mathematics. Then, I argue that the place that holds the most promise for addressing equity is a research agenda that emphasizes enabling the practice of teachers and that draws more heavily on design-based and action research, thereby redefining what the practice of mathematics means along the way. Specific research questions are offered.     

Enabling the Practice of Mathematics Teachers in Context: Toward a New Equity Research Agenda

In this article, I address the need for a more clearly articulated research agenda around equity issues by proposing a working definition of equity and a focal point for research. More specifically, I assert that rather than pitting them against each other, we must coordinate (a) efforts to get marginalized students to master what currently counts as "dominant" mathematics with (b) efforts to develop a critical perspective among all students about knowledge and society in ways that ultimately facilitate (c) a positive relationship between mathematics, people, and equity on the planet. I make this argument partly by reviewing the literature on (school) contexts that engage marginalized students in mathematics. Then, I argue that the place that holds the most promise for addressing equity is a research agenda that emphasizes enabling the practice of teachers and that draws more heavily on design-based and action research, thereby redefining what the practice of mathematics means along the way. Specific research questions are offered.     

高等数学教学渗透研究性学习的研究与实践

近年来,研究性学习逐步成为高等教育教学改革的一个热点.在高等数学教学中实施研究性学习,可以有效改变学生多年来养成的应试性学习习惯,有利于培养学生的创新精神,提高学生的数学素养.做好研究性学习与传统教学的融合与协调,是高等数学教学中开展研究性学习的重要环节之一.我们在多年教学实践的基础上,形成了在传统教学模式中渗入研究性学习的课程教学理念,提出了研究性学习与传统教学融合与协调的教学新模式,并积极探索开展研究性学习的有效途径.教学实践表明,我们的做法既能发挥传统课堂教学的高效率,有助于改变学生的学习方式,提高其自主学习能力和创新能力     

Mathematics Education & Digital Technologies: Facing the Challenge of Networking European Research Teams

This paper introduces the IJCML Special Issue dedicated to digital technologies and mathematics education and, in particular, to the work performed by the European Research Team TELMA (Technology Enhanced Learning in Mathematics). TELMA was one of the initiatives of the Kaleidoscope Network of Excellence established by the European Community (IST-507838—2003–2007) to promote the joint elaboration of concepts and methods for exploring the future of learning with digital technologies. TELMA addressed the problem of fragmentation of theoretical frameworks in the research field of mathematics education with digital technologies and developed a methodology based on the in field cross-experimentation of educational digital environments for maths. Six European research teams engaged in cross-experimentation in classrooms as a means to begin to develop a common language and to analyse the intertwined influence played, both explicitly and implicitly, by different contextual characteristics and theoretical frames assumed as reference by the diverse teams participating in the initiative.     

Teacher Learning and Mathematics Manipulatives: A Collective Case Study About Teacher Use of Manipulatives in Elementary and Middle School Mathematics Lessons

This collective case study analyzes the use of manipulatives in math lessons developed and taught by 4 groups of elementary teachers (K‐8) involved in lesson study as part of a professional development program. The study found that in three of four lessons studied manipulative use was turned into an end in and of itself, rather than a tool, and that in the fourth lesson manipulative use hindered rather than helped student learning. These problems with manipulative use by teachers in the lessons provide helpful guidance for planning of future professional development for math teaching. Our conclusion contains recommendations for successful implementation of manipulatives for both teachers and professional developers. Most importantly, we stress the need to emphasize the link between pedagogy and content, not the specific use of manipulatives.     

Exploring the Use of New Representations as a Resource for Teacher Learning

An important goal of mathematics education reform is to support teacher learning. Toward this end, researchers and teacher educators have investigated ways in which teachers learn about mathematical content, pedagogical strategies, and student thinking as they implement reform. This study extends such work by examining how one elementary school and one high school teacher learned from students' interpretations of new conceptually based representations contained in instructional materials aligned with the Principles and Standards for School Mathematics ( National Council of Teachers of Mathematics, 2000 ). Results indicated that teaching with new representations provided a rich context for teacher learning at both the elementary and high school level, and three dimensions were identified along which such learning occurred. The results suggest that pedagogical content knowledge with respect to representations is an important facet of teacher cognition that should be studied in greater depth.     

Controversies Around the Role of Statistical Tests in Experimental Research

Despite widespread use of significance testing in empirical research, its interpretation and researchers' excessive confidence in its results have been criticized for years. In this article, the logic of statistical testing in the Fisher and Neyman-Pearson approaches are described, some common misinterpretations of basic concepts behind statistical tests are reviewed, and the philosophical and psychological issues that can contribute to these misinterpretations are analyzed. Some frequent criticisms against statistical tests are revisited, with the conclusion that most of them refer not to the tests themselves but to the misuse of tests on the part of researchers. In accordance with Levin (1998a), statistical tests should be transformed into a more intelligent process that helps researchers in their work. Possible ways in which statistical education might contribute to the better understanding and application of statistical inference are suggested.     

Controversies Around the Role of Statistical Tests in Experimental Research

Despite widespread use of significance testing in empirical research, its interpretation and researchers' excessive confidence in its results have been criticized for years. In this article, the logic of statistical testing in the Fisher and Neyman-Pearson approaches are described, some common misinterpretations of basic concepts behind statistical tests are reviewed, and the philosophical and psychological issues that can contribute to these misinterpretations are analyzed. Some frequent criticisms against statistical tests are revisited, with the conclusion that most of them refer not to the tests themselves but to the misuse of tests on the part of researchers. In accordance with Levin (1998a), statistical tests should be transformed into a more intelligent process that helps researchers in their work. Possible ways in which statistical education might contribute to the better understanding and application of statistical inference are suggested.     

Behind the Masks: Identifying Students' Competencies for Learning Mathematics and Science in Urban Settings

Three mathematics and science educators reexamine and reflect on their teaching within the context of the American Association for the Advancement of Sciences (AAAS) and National Council of Mathematics' call to make math and science education accessible to all. The paper highlights the importance of teachers reflecting on their teaching practices in order to create opportunities for their students especially those in the urban setting. The educators argue that teachers' reflection on their teaching can cause them to recognize and validate their students' ways of knowing as they identify the students' hidden/concealed abilities that are often masked by their behaviors. The educators discuss their experiences and highlight the lessons that they learned about ways to prepare teachers to successfully teach math and science students in urban settings. Culturally responsive pedagogy and cultural competency are critical skills that teachers need to develop in order to teach all children, especially those in the math and science classroom in the urban setting.     

System of Equations for the Marangoni Boundary Layer in Media with Ladyzhenskaya Rheological Law

Doklady Mathematics - A system of equations describing boundary layers of nonlinear generalized Newtonian viscous fluids with the Ladyzhenskaya rheological law is studied. The well-posedness of the...