Attending to Precision: Mathematical Definitions in Courses for Pre-Service and Practicing Teachers

In this article we address the Common Core State Standard for Mathematical Practice: Attend to Precision. We describe work in mathematics content courses for pre-service and inservice teachers focused on developing teachers’ understanding of mathematical definitions. We discuss teachers’ work in developing definitions for students at various levels of mathematics, in generalizing a definition in order to extend the domain to which it may be applied, and in correcting teachers’ misconceptions. The work discussed in this article is consistent with the recommendations of the Conference Board of the Mathematical Sciences for the mathematics education of teachers. In explicit response to contemporary mathematics education research, the author presents constructive assessment prompts which lend themselves to formal study by mathematics education researchers.     

数学本质观的历史考察

由于数学在现代科技中的重要作用，数学教育始终能够反映一个时代的社会理念和数学学科的本质。从历史的纵向角度考察数学本质观的演变，并从哲学角度分析其中的继承和发展的辩证关系，对当前数学教育改革和数学课程设置有启示意义。     

Causal Connections Between Mathematical Language and Mathematical Knowledge: A Dialogic Reading Intervention

The acquisition of early mathematical knowledge is critical for successful long-term academic development. Mathematical language is one of the strongest predictors of children's early mathematical success. Findings from previous studies have provided correlational evidence supporting the importance of mathematical language to the development of children's mathematics skills, but there is limited causal evidence supporting this link. To address this research gap, 47 Head Start children were randomly assigned to a mathematical language intervention group or a business-as-usual group. Over the course of eight weeks, interventionists implemented a dialogic reading intervention focused on quantitative and spatial mathematical language. At posttest, students in the intervention group significantly outperformed the students in the comparison group not only on a mathematical language assessment, but on a mathematical knowledge assessment as well. These findings indicate that increasing children's exposure to mathematical language can positively affect their general mathematics skills. This study is an important first step in providing causal evidence of the importance of early mathematical language for children's general mathematical knowledge and the potential for mathematical language interventions to increase children's overall mathematics abilities.     

We Talk About It,But Do They Get It?

This article reports on the last of a series of iterative research studies involving students with learning disabilities in reform mathematics classrooms at the intermediate grade levels. This study reports the findings from a larger, year‐long case study that focused on ways to include students with learning disabilities and other students who are at risk for special education services in classwide discussions of problem solving. The data reported in this article detail the changes in teacher and student discourse over a nine‐week period in one classroom. Sources of data for this study included videotapes, audiotapes, and informal interviews with the teacher, a paraprofessional, and students. A quantitative analysis of the results indicates clear patterns of change in teacher and student discourse. Nonetheless, intentional efforts to include target students in the whole‐class discussions yielded instructional dilemmas that are underdescribed in the mathematics reform literature. Findings from this study have implications for special educators interested in mathematical problem solving, as well as math reformers who value the role of classroom discourse in daily instruction.     

The development of a mathematics teacher’s professional identity during her first year teaching

This study is focused on a period that poses several challenges for the development of mathematics teachers’ professional identities and agency: their first year of teaching in schools. During this period, beginning mathematics teachers confront tensions and contradictions among the principles, ideals, and experiences encountered during pre-service education and the demands, and restrictions of their teaching practice in schools. This article approaches this topic by developing an interpretative case study centered on one novice mathematics teacher, Sol. The aim is to describe and understand the development of Sol’s professional identity and agency during her first teaching year. Considering identity development as a diachronic phenomenon, we carry out a narrative analysis of the research data. The findings show that Sol developed her professional identity and agency through a process that gathered together the teaching practices possible inside her school, the positions she could negotiate as a newcomer inside the institution, and the cultural practices and discourses embodied during pre-service education. The results bring to the forefront the profound and tense interactions between the intimate and personal terrain of mathematics teachers and the social and cultural world of the schools where they work. Sol’s case also contributes to understanding the role that a robust pre-service education can play in the development of beginning teachers’ professional identities and in the possibility they could become agents of transformation in their schools.     

Development of Arithmetical Thinking: Evaluation of Subject Matter Knowledge of Pre-Service Teachers in Order to Design the Appropriate Course

One of the key courses in the mathematics teacher education program in Israel is arithmetic, which engages in contents which these pre-service mathematics teachers (PMTs) will later teach at school. Teaching arithmetic involves knowledge about the essence of the concept of “number” and the development thereof, calculation methods and strategies. properties of operations on different sets of numbers, as well as the properties of the numbers themselves. Hence, the question arises: how to educate PMTs in order to supplement their mathematical knowledge with the required components? The present study explored the development of arithmetic thinking among pre-service teachers intending to teach mathematics at elementary school. This was done by matching the van Hiele theory of the development of geometric thinking to arithmetic. Analysis of findings obtained both in the present study and in many studies of geometry teaching indicates that this approach to considering the learners’ level of thinking development might lead to meaningful learning in arithmetic course for PMTs.     

Challenges of critical colleagueship: examining and reflecting on mathematics teacher study group interactions

This article examines mathematics teacher collegiality by focusing on both the ways in which teachers interacted as critical colleagues in a long-term professional development project and the evolving role of the teacher–educator–researcher as the facilitator of this project. The professional development collaboration comprised two phases: one focused on reading classroom discourse literature and one focused on supporting each other through cycles of action research related to mathematics classroom discourse. Lord’s (1994) critical colleagueship framework is used to examine how a study group of middle-grades (ages 11–16) mathematics teacher–researchers took (or did not take) a more critical stance toward their own teaching practice and that of their colleagues. We found that challenging interactions were related to instances in which the teachers interacted as critical colleagues and were marked by particular features including the use of particular words and the use of personal experience as a form of evidence. We present the ways in which we came to understand what it might look like to scrutinize one’s practice and findings related to the development of this type of collegiality across the two different phases of this project. We end with a section in which the teacher–educator–researcher who facilitated the professional development project reflects on the ways in which the analysis caused her to reconsider both the nature of argumentation in mathematics study group settings and what implications this has with respect to her own practice as a facilitator.     

Experiencing equivalence but organizing order

The notion of equivalence relation is arguably one of the most fundamental ideas of mathematics. Accordingly, it plays an important role in teaching mathematics at all levels, whether explicitly or implicitly. Our success in introducing this notion for its own sake or as a means to teach other mathematical concepts, however, depends largely on our own conceptions of it. This paper considers various conceptions of equivalence, in history, in mathematics today, and in mathematics education. It reveals critical differences in the notion of equivalence at different points in history and a meaning for equivalence proposed by mathematicians and mathematics educators that is at variance with the ways that learners may think. These differences call into question the most popular view of the subject: that the mathematical notion of equivalence relation is the result of spelling out our experience of equivalence. Moreover, the findings of this study suggest that the standard definition of an equivalence relation is ill-chosen from a pedagogical point of view but well-crafted from a mathematical point of view.     

More than meets the eye: patterns and shifts in middle school mathematics teachers’ descriptions of models

Modeling is a major topic of interest in mathematics education. However, the field’s definition of models is diverse. Less is known about what teachers identify as mathematical models, even though it is teachers who ultimately enact modeling activities in the classroom. In this study, we asked nine middle school teachers with a variety of academic backgrounds and teaching experience to collect data related to one familiar physical phenomenon, cooling liquid. We then asked each participant to construct a model of that phenomenon, describe why it was a model, and identify whether a variety of artifacts representing the phenomenon also counted as models during a semi-structured interview. We sought to identify: what do mathematics teachers attend to when describing what constitutes a model? And, how do their attentions shift as they engage in different activities related to models? Using content analysis, we documented what features and purposes teachers attended to when describing a mathematical model. When constructing their own model, they focused on the visual form of the model and what quantitative information it should include. When deciding whether particular representational artifacts constituted models, they focused on how the representations reflected the system under study, and what purposes those representations could serve in further understanding that system. These findings suggest teachers may have multiple understandings of models, which are active at different times and reflect different perspectives. This has implications for research, teacher education, and professional development.     

Becoming a Teacher Educator: A Self-Study of the Use of Inquiry in a Mathematics Methods Course

This article details the self-study of a beginning teacher educator in her first experience in teaching a mathematics methods course. The transition from teacher to teacher educator is explored through the experience of a course focused on inquiry. Inquiry is embedded within the course from two perspectives: mathematical inquiry and teaching as inquiry. The framework of teaching as inquiry is an important part of both the self-study and the mathematics methods course. Implications of the transition from teacher to teacher educator as well as the role of inquiry in mathematics teacher education are discussed.     

Mathematical and Reading Difficulties as Predictors of School Achievement and Transition to Secondary Education

This study investigated whether mathematical and reading difficulties and self-reported learning problems predicted school achievement in the ninth grade, at the age of 16, and how these difficulties further explained the transition either to upper secondary academic education or to vocational education. The sample of the present study comprised one age group of ninth-grade adolescents (n?=?592; 300 girls, 292 boys) in a middle-sized Finnish city. These students completed tests of mathematics, reading comprehension and decoding. Participants were also asked to assess their learning problems in school work. Results demonstrated that mathematical and reading difficulties strongly predicted school achievement in the ninth grade and, through school achievement, also predicted the transition to different tracks in secondary education. The role of self-reported learning difficulties in this prediction was significant, but less significant than that of mathematical and reading difficulties. Parents' education did not play a major role in this prediction.     

数学名题作为研究性学习的案例研究

数学名题在数学史融入中学数学教育中扮演着十分重要的角色,如何发挥数学名题的教育价值是中学数学教育面临的重要问题之一。数学名题案例研究是数学名题作为研究性学习的有效方法,应遵循经典性、趣味性和知识性三条原则。     

What Counts and How: Mathematics Teaching in Culturally, Linguistically, and Socioeconomically Diverse Urban Settings

This article presents findings from an ongoing study of urban teachers' efforts to embrace mathematics reform with student populations that are culturally, linguistically, and socioeconomically diverse (CLSD). We investigate the teacher's role in providing accessible and valuable mathematical learning opportunities to diverse students. Through narrative vignettes of practice and analyses of the personal and intellectual resources teachers draw on in CLSD contexts, we examine the challenges and possibilities two third-grade teachers face as they attempt such reform. One teacher's strengths were in making cultural connections with her students; the other's strengths were in pursuing complex and meaningful mathematics with her students. Building on our analysis, we offer a framework for examining the work of attending to mathematical and cultural issues simultaneously. Our findings suggest that such work is complex; however, teachers are seldom supported in their efforts to integrate these two perspectives. Our aim is to examine the dimensions of culturally relevant mathematics teaching and explore where the fields of mathematics and bilingual-bicultural education need to speak to one another.     

浅谈数学美育

随着当今数学发展 ,为了提高数学课堂教学的效果 ,培养更多的数学爱好者 ,数学教师不仅要完成向学生传授知识、培养能力、发展智力的任务 ,同时还要在数学学科的教育中进行美育渗透 ,让学生得到美的熏陶 ,激发学生的学习兴趣     

建构思维及其培养

数学是关于思维的科学,建构思维是指发现或发明数学概念、定理、公式和命题以及形成蕴涵于其中的数学思想、方法的个人认识.建构思维的培养有助于培养学生的创新精神,促进学生对数学知识的理解.建构思维包括外源建构思维、内源建构思维和辩证建构思维.     

检视新一轮数学课程改革的论争

新一轮数学课程改革在课程目标、课程内容、教学方式、教材编排方式和评价方式等方面发生了深刻的变革,这些变革促进了中小学数学教育的发展.不过,课程改革也出现了一些问题,引发了一些论争,如关于数学生活化的论争、关于数学"双基"的论争、关于学习方式转变的论争、关于几何课程改革的论争和关于教材编排方式的论争等.检视这些论争并对其进行理性思考,有助于推动课程改革健康发展.     

Incorporating out-of-school mathematics: from cultural context to embedded practice

Connecting students’ cultural and community mathematical practices to school mathematics is a critical issue in mathematics education. The goal of the study was to identify how teachers incorporate children’s cultural and out-of-school mathematics in instruction. Four related practices were identified, and three drew on children’s cultural or out-of-school experiences: (a) using these experiences as contexts for problems, (b) linking these experiences to school mathematics, and (c) identifying embedded mathematical practices prominent in these experiences. A fourth category, teacher initiated situated settings, focused on shared experiences using the classroom as a site of culture. Findings suggest that these practices represent varying levels of complexity and that use of this framework might support teachers in better relating students’ cultural and out-of-school experiences to mathematics.     

Preservice Teachers' Stories Of Mathematics Classrooms: Explorations Of Practice Through Fictional Accounts

This report describes four preservice secondary teachers' fictional accounts about mathematics classrooms. The narrative structures and meanings of the preservice teachers' stories, written while the preservice teachers were enrolled in a secondary methods course, were examined with the goal of gaining insight into the preservice teachers' emerging identities as mathematics teachers. Although past research has examined experienced teachers' stories, little research, if any, has focused on stories written by preservice mathematics teachers or on intentionally fictional stories about mathematics classrooms. This report uses methods of narrative analysis to develop understandings about preservice teachers' storied identities and specifies areas for further investigation into the role of narrative in mathematics teacher education.     

Interesting and difficult mathematical problems: changing teachers’ views by employing multiple-solution tasks

The study considers mathematical problem solving to be at the heart of mathematics teaching and learning, while mathematical challenge is a core element of any educational process. The study design addresses the complexity of teachers’ knowledge. It is aimed at exploring the development of teachers’ mathematical and pedagogical conceptions associated with systematic employment of multiple-solution tasks (MSTs) in a “problem-solving” course for prospective mathematics teachers (PMTs). Our attention to teachers’ mathematical conceptions focused on the development of PMTs’ problem-solving competences. Our attention to teachers’ meta-mathematical and pedagogical conceptions focused on changes in teachers’ views concerning the level of interest and level of difficulty of the mathematical tasks. We differentiated between the systematic and craft modes of professional development integrated in the course. Systematic mode involved problem-solving sessions and reflective discussions on collective solution spaces. Craft mode involved interviewing school students. The study demonstrates the effectiveness of MSTs for PMTs’ professional development.