Connected Knowledge in Science and Mathematics Education

While the traditional meaning of connected knowledge is valuable in some school subjects, it does not address the main activities of knowledge acquisition in subjects such as physics and mathematics. The goal of this article is to analyze the relationships between the concepts “learning for understanding” and “connected knowledge”, a central theme in feminist epistemology. In learning for understanding, the learner forms multiple, intricate connections among the concepts she is studying in school, between school concepts and her everyday concepts, and between school concepts and their wider context. Viewing connected knowledge as tightly related to understanding has several important implications. It brings connected knowledge into the central learning activities that take place in school science and mathematics, and gives it a high status. It contributes to our understanding of gender‐related patterns in thinking; and it may form a unifying theoretical framework for many studies and projects in the field of gender fair education.     

Promoting teacher learning: a framework for evaluating the educative features of mathematics curriculum materials

Educative curricula, curriculum materials that intentionally foster teacher professional development, can serve as a site for teacher learning through their use in daily instructional practices. The present article introduces a framework, Teacher Learning Opportunities in Mathematics Curriculum Materials (TLO-Math), for designing and evaluating mathematics curriculum materials’ educative features according to seven theoretically based variables: (1) mathematics content knowledge for teaching, (2) teacher knowledge of student thinking in mathematics, (3) teacher knowledge of disciplinary discourse in mathematics, (4) teacher knowledge of assessment in mathematics, (5) teacher knowledge of differentiated instruction in mathematics, (6) teacher knowledge of technology use in mathematics, and (7) teacher knowledge of mathematical community. Each variable is illustrated with a definition, guiding questions, discipline-specific literature, and examples from two sets of elementary mathematics curriculum materials. The development of the TLO-Math framework is a critical first step for further study of the use of mathematics curriculum materials as sites for teacher learning.     

小学课程中的数学文化:内涵特点、主要内容与学习价值

数学文化是指数学的思想、精神、语言、方法、观点以及它们的形成与发展,还包括数学在人类生活、科学技术、社会发展中的贡献和意义,以及与数学相关的人文活动。与纯粹的数学知识相比较,数学文化体现了人文性与科学性交融、开放性与包容性并存、民族性与统一性共生,以及价值理性与工具理性互推的特点。适合小学生学习的数学文化内容主要有数学知识的发展历史、数学家的成长故事、数学的游戏活动、数学的生活应用和学科应用。小学生学习数学文化有助于他们理解数学知识、掌握数学思想方法、提高数学思维水平、加强数学应用意识和培养数学精神。     

RETHINKING FOOD UTILIZED AS MATERIALS FOR ART

ABSTRACT

Teaching mathematics in an early childhood program requires mathematical content knowledge and teacher self-efficacy, yet research has shown that early childhood educators often have negative attitudes towards mathematics and feel underprepared to teach mathematical concepts. The study reported here documents the reconceptualization of a graduate, preservice teacher education program, a program designed to address teacher anxiety and increase capacity to teach mathematics in a play-based early childhood setting. The study aimed to investigate: (1) the effectiveness of the mathematics component of the course in equipping teacher candidates to teach mathematics in early childhood, and (2) whether participation in the mathematics component of the course changed teacher candidates’ self-efficacy regarding mathematics. Findings show that both self-efficacy and content knowledge improved when teacher candidates had the opportunity to engage with play-based learning experiences that embed mathematical concepts. Furthermore, the focus on a learning trajectories approach supports the identification of developmental progression points in children’s emerging mathematical understanding, assisting with teacher candidates’ fine-grained observations, assessment of children’s learning, and authentic, individualized planning for learning.     

‘Walking in a Foreign and Unknown Landscape’: Studying the History of Mathematics in Initial Teacher Education

This article develops the argument that students in initial teacher education benefit in terms of who they are becoming from developing awareness of and engagement in the history of mathematics. Initially, current school mathematics practices in the UK are considered and challenged. Then the role of teachers’ relationship to mathematical subject knowledge and of teachers’ engagement in critical thinking are considered. Connections are made between these concerns and studying the history of mathematics in initial teacher education classrooms. I then draw on the perspectives and practices of the mathematics teacher educators at one institution to understand these connections better and to exemplify them. Issues of equity are threaded throughout.     

The role of a children's mathematical thinking experience in the preparation of prospective elementary school teachers

Historically, content preparation and pedagogical preparation of teachers in California have been separated. Recently, in integrating these areas, many mathematics methodology instructors have incorporated children's thinking into their courses, which are generally offered late in students’ undergraduate studies. We have implemented and studied a model for integrating mathematical content and children's mathematical thinking earlier, so that prospective elementary school teachers (PSTs) engage with children's mathematical thinking while enrolled in their first mathematics course. PSTs’ work with children in the Children's Mathematical Thinking Experience (CMTE) may enhance their mathematical learning. Preliminary study results indicate that the sophistication of CMTE students’ beliefs about mathematics, teaching, and learning increased more than the sophistication of beliefs held by students enrolled in a reform-oriented early field experience and that experiences considering children's mathematical thinking provided PSTs with increased motivation for learning mathematics.     

Decentering: A construct to analyze and explain teacher actions as they relate to student thinking

Mathematics educators and writers of mathematics education policy documents continue to emphasize the importance of teachers focusing on and using student thinking to inform their instructional decisions and interactions with students. In this paper, we characterize the interactions between a teacher and student(s) that exhibit this focus. Specifically, we extend previous work in this area by utilizing Piaget’s construct of decentering (The language and thought of the child. Meridian Books, Cleveland, 1955) to explain teachers’ actions relative to both their thinking and their students’ thinking. In characterizing decentering with respect to a teacher’s focus on student thinking, we use two illustrations that highlight the importance of decentering in making in-the-moment decisions that are based on student thinking. We also discuss the influence of teacher decentering actions on the quality of student–teacher interactions and their influence on student learning. We close by discussing various implications of decentering, including how decentering is related to other research constructs including teachers’ development and enactment of mathematical knowledge for teaching.     

Reification of instructional materials as part of the process of developing problem‐based practices in mathematics education

This teacher development study closely examined a teacher's practice for the purpose of understanding how she selected and implemented instructional materials, and correspondingly how these processes changed as she developed her problem‐based practice throughout a school year. Data sources included over 20 hours of planning and analysis meetings with the teacher and 27 video‐taped lessons with discussions before and after each lesson. Through qualitative analysis we examined the data for: students' cognitive demand for curricular materials the teacher selected and implemented; teacher's beliefs and practices for students' engagement in mathematical thinking; and teacher's and students' communication about mathematics during instruction. We found that the teacher shifted her views and use of instructional materials as she changed her practice towards more problem‐based approaches. The teacher moved from closely following her traditional, district‐adopted textbook to selecting problem‐based tasks from outside resources to build a curriculum. Simultaneously, she changed her practice to focus more on students' engagement in mathematical thinking and their communication about mathematics as part of learning. During this shift in practice, the teacher began to reify instructional materials, viewing them as instruments of her practice to meet students' needs. The process of shifting her views was gradual over the school year and involved substantial analysis and reflection on practice from the teacher. Implications include that teachers and teacher educators may need to devote more attention and support for teachers to use instructional materials to support instruction, rather than materials to prescribe instruction. This use of instructional materials may be an important part of transforming practice overall.     

Using video analysis to support prospective K-8 teachers’ noticing of students’ multiple mathematical knowledge bases

As part of a larger research project aimed at transforming preK-8 mathematics teacher preparation, the purpose of this study was to examine the extent to which prospective teachers notice children’s competencies related to children’s mathematical thinking, and children’s community, cultural, and linguistic funds of knowledge or what we refer to as children’s multiple mathematical knowledge bases. Teachers’ noticing supports students’ learning in deep and meaningful ways. Researchers designed and enacted a video analysis activity with prospective teachers in their mathematics methods course. The activity served as a decomposition of practice in order to support prospective teachers in engaging in an approximation of the practice of noticing. Our findings showed that prospective teachers evidenced noticing of mathematics teaching and learning as early as the mathematics methods course. We also found that the prompts and structure of the activity supported prospective teachers by increasing their depth of noticing and their foci in noticing, moving from attending primarily to teacher moves (and merely describing what they saw) to becoming aware of significant interactions (and interpreting effects of these interactions on learning). Implications for teacher educators interested in designing and enacting activities to support noticing are discussed.     

PROFESSIONAL NOTICING PRACTICES OF NOVICE MATHEMATICS TEACHER EDUCATORS

The focus on professional noticing in mathematics education has recently gained increased interest as researchers work to understand how and what is noticed and how this translates into practice. Much of this work has focused on the professional noticing practices of inservice teachers and preservice teachers, with less attention focused on those educating teachers. This research explores how novice mathematics teacher educators professionally notice as they engage in teaching experiments and create models of student’s mathematical thinking. Findings indicate the novice teacher educators are including some evaluative comments in their professional noticing practices but lack in-depth interpretive analysis about student thinking and rarely make connections between student’s thinking and the broader principles of teaching and learning. These findings provide evidence for the importance of supporting teacher educators with developing their abilities to professionally notice.     

EXPLORING TEACHERS’ KNOWLEDGE AND PERCEPTIONS ACROSS MATHEMATICS AND SCIENCE THROUGH CONTENT-RICH LEARNING EXPERIENCES IN A PROFESSIONAL DEVELOPMENT SETTING

This paper examines upper elementary and middle school teachers’ learning of mathematics and science content, how their perceptions of their disciplines and learning of that discipline developed through content-rich learning experiences, and the differences and commonalities of the teachers’ learning experiences relative to content domain. This work was situated within a larger professional development (PD) program that had multiple, long-term components. Participants’ growth occurred in 4 primary areas: knowledge of content, perceptions of the discipline, perceptions about the learning of the discipline, and perceptions regarding how students learn content. Findings suggest that when embedded within an effective professional development context, content can be a critical vehicle through which change can be made in teachers’ understandings and perceptions of mathematics and science. When participants in our study were able to move beyond their internal conflicts and misunderstandings, they could expand their knowledge and perceptions of content and finally bridge to re-conceptualize how to teach that content. These findings further indicate that although teachers involved in both mathematics and science can benefit from similar overall PD structures, there are some unique challenges that need to be addressed for each particular discipline group. This study contributes to what we understand about teacher learning and change, as well as commonalities and differences between teachers’ learning of mathematics and science.     

Educational forms of initiation in mathematical culture

A review of literature shows that during the history of mathematics education at school the answer of what counts as ‘real mathematics’ varies. An argument will be given here that defines as ‘real mathematics’ any activity of participating in a mathematical practice. The acknowledgement of the discursive nature of school practices requires an in-depth analysis of the notion of classroom discourse. For a further analysis of this problem Bakhtin’s notion of speech genre is used. The genre particularly functions as a means for the interlocutors for evaluating utterances as a legitimate part of an ongoing mathematical discourse. The notion of speech genre brings a cultural historical dimension in the discourse that is supposed to be acted out by the teacher who demonstrates the tools, rules, and norms that are passed on by a mathematical community. This has several consequences for the role of the teacher. His or her mathematical attitude acts out tendencies emerging from the history of the mathematical community (like systemacy, non-contradiction etc.) that subsequently can be imitated and appropriated by pupils in a discourse. Mathematical attitude is the link between the cultural historical dimension of mathematical practices and individual mathematical thinking.     

Promoting equity in mathematics teacher preparation: a framework for advancing teacher learning of children’s multiple mathematics knowledge bases

Research repeatedly documents that teachers are underprepared to teach mathematics effectively in diverse classrooms. A critical aspect of learning to be an effective mathematics teacher for diverse learners is developing knowledge, dispositions, and practices that support building on children’s mathematical thinking, as well as their cultural, linguistic, and community-based knowledge. This article presents a conjectured learning trajectory for prospective teachers’ (PSTs’) development related to integrating children’s multiple mathematical knowledge bases (i.e., the understandings and experiences that have the potential to shape and support children’s mathematics learning—including children’s mathematical thinking, and children’s cultural, home, and community-based knowledge), in mathematics instruction. Data were collected from 200 PSTs enrolled in mathematics methods courses at six United States universities. Data sources included beginning and end-of-semester surveys, interviews, and PSTs’ written work. Our conjectured learning trajectory can serve as a tool for mathematics teacher educators and researchers as they focus on PSTs’ development of equitable mathematics instruction.     

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.     

Provoking contingent moments: Knowledge for ‘powerful teaching’ at the horizon

Background: Teacher knowledge continues to be a topic of debate in Australasia and in other parts of the world. There have been many attempts by mathematics educators and researchers to define the knowledge needed by teachers to teach mathematics effectively. A plethora of terms, such as mathematical content knowledge, pedagogical content knowledge, horizon content knowledge and specialised content knowledge, have been used to describe aspects of such knowledge.

Purpose: This paper proposes a model for teacher knowledge in mathematics that embraces and develops aspects of earlier models. It focuses on the notions of contingent knowledge and the connectedness of ‘big ideas’ of mathematics to enact what is described as ‘powerful teaching’. It involves the teacher’s ability to set up and provoke contingent moments to extend children’s mathematical horizons. The model proposed here considers the various cognitive and affective components and domains that teachers may require to enact ‘powerful teaching’. The intention is to validate the proposed model empirically during a future stage of research.

Sources of evidence: Contingency is described in Rowland’s Knowledge Quartet as the ability to respond to children’s questions, misconceptions and actions and to be able to deviate from a teaching plan as needed. The notion of ‘horizon content knowledge’ (Ball et al.) is a key aspect of the proposed model and has provoked a discussion in this article about students’ mathematical horizons and what these might comprise. Together with a deep mathematical content knowledge and a sensibility for students and their mathematical horizons, these ideas form the foundations of the proposed model.

Main argument: It follows that a deeper level of knowledge might enable a teacher to respond better and to plan and anticipate contingent moments. By taking this further and considering teacher knowledge as ‘dynamic’, this paper suggests that instead of responding to contingent events, ‘powerful teaching’ is about provoking contingent events. This necessarily requires a broad, connected content knowledge based on ‘big mathematical ideas’, a sound knowledge of pedagogies and an understanding of common misconceptions in order to be able to engineer contingent moments.

Conclusions: In order to place genuine problem-solving at the heart of learning, this paper argues for the idea of planning for contingent events, provoking them and ‘setting them up’. The proposed model attempts to represent that process. It is anticipated that the new model will become the framework for an empirical research project, as it undergoes a validation process involving a sample of primary teachers.     

美国《中学数学教与学》教材的特点及其对中国数学教师教育的启示

美国《中学数学教与学》是初中年级数学教师教育"数学教学法"课程的教材,其特点表现为:单元主题以数学教学理论为线索呈现,单元内容以具体数学知识为载体展开并且以参与式活动的形式组织教学活动.该教材对我国的数学教师教育有如下启示:数学教师教育课程应以数学知识为载体学习数学教育理论;数学教师教育教材应突出教学方式的示范性,体现做中学,在数学教学活动中学习数学教学方法;数学教师教育应注重教师学科专业素质的提升.     

试论新课改下小学数学教学的创新

随着新课程改革的不断深入,小学数学教师也应转变传统的教学观念,创新数学教学方法,以达到新课程改革对小学阶段数学教学目标的新要求。小学阶段的数学教学能为学生未来更好地学习数学知识打好理论基础,在这一段时间内培养学生的良好学习习惯能使学生的学习达到事半功倍的效果。因此抓住这一关键期对学生进行教育,对帮助学生形成数学思维、提高学生运用数学知识解决实际问题的能力有重要作用。基于此,本文将研究在新课改背景下小学数学教学方法的几种创新形式。     

Characterizing pivotal teaching moments in beginning mathematics teachers’ practice

Although skilled mathematics teachers and teacher educators often “know” when interruptions in the flow of a lesson provide an opportunity to modify instruction to improve students’ mathematical understanding, others, particularly novice teachers, often fail to recognize or act on such moments. These pivotal teaching moments (PTMs), however, are key to instruction that builds on student thinking about mathematics. Video of beginning secondary school mathematics teachers’ instruction was analyzed to identify and characterize PTMs in mathematics lessons and to examine the relationships among the PTMs, the teachers’ decisions in response to them, and the likely impacts on student learning. These data were used to develop a preliminary framework for helping teachers learn to identify and respond to PTMs that occur during their instruction. The results of this exploratory study highlight the importance of teacher education preparing teachers to (a) understand the mathematical terrain their students are traversing, (b) notice high-leverage student mathematical thinking, and (c) productively act on that thinking. This preparation would improve beginning teachers’ abilities to act in ways that would increase their students’ mathematical understanding.     

Development of professional knowledge in learning to teach elementary science

The purpose of this research was to understand how preservice elementary teacher experiences within the context of reflective science teacher education influence the development of professional knowledge. We conducted a case analysis to investigate one preservice teacher's beliefs about science teaching and learning, identify the tensions with which she grappled in learning to teach elementary science, understand the frames from which she identified problems of practice, and discern how her experiences played a role in framing and reframing problems of practice. The teacher, Barbara, encountered tensions in thinking about science teaching and learning as a result of inconsistencies between her vision of science teaching and her practice. Confronting these tensions between ideals and realities prompted Barbara to rethink the connections between her classroom actions and students' learning and create new perspectives for viewing her practice. Through reframing, she was able to consider and begin implementing alternative practices more resonant with her beliefs. Barbara's case illustrates the value of understanding prospective teachers' beliefs, their experiences, and the relationship between beliefs and classroom actions. Furthermore, the findings underscore the significance of offering reflective experience as professionals early in the careers of prospective teachers. © 1999 John Wiley & Sons, Inc. J Res Sci Teach 36: 121–139, 1999