What they notice in video: a study of prospective secondary mathematics teachers learning to teach

Most teacher preparation programs have embraced the use of video as an effective methodology for developing teachers’ noticing skills. This study focused on learning about what secondary mathematics prospective teachers (PSTs) were able to notice when viewing video of their own co-teaching, particularly in a microteaching setting that consisted of peers. PSTs documented their observations on an observation tool while re-watching their video and then identified and ranked their top three observations. The ranked noticing statements were analyzed based on a grounded theory approach. Overall, PSTs’ ranked observations were more likely to attend to students and had a strong focus on mathematics and student learning. Ranked observations equally demonstrated both broad and specific understanding of video moments and often made suggestions that something they noticed could be improved in the implementation stage, versus improvements in planning or changes in themselves. Results support PSTs’ use of video for developing noticing skills in teacher education programs.     

Pre-service teachers in mathematics: Struggling with the Standards

The purpose of this paper is to describe a two-year investigation of pre-service teachers in mathematics. Of particular interest is the degree to which the NCTM (1989) Curriculum and evaluation standards for school mathematics, the NCTM (1991) Professional standards for teaching mathematics, and other reform agendas impact pre-service teachers' thinking and practices. The beliefs, perceptions, philosophies, and classroom actions of four cohorts of student teachers (n = 44) were examined throughout this 2year study. Of particular interest were the pressure student teachers felt to implement the Standards in their teaching, their perceptions of their own teaching with respect to the Standards, the content of their lessons, and the impact of cooperating teachers on the development of the student teachers. Data was collected through a variety of sources, organized and analyzed in the spirit of ethnographic work. Data was generated through classroom observations, pre- and post-lesson conferences, lesson plans, seminar sessions, survey questionnaires, and informal conversations. A minimum of four classroom observations were made for each participant. Analysis of the data revealed general agreement among the student teachers in their regard for the Standards. Although they appeared to affirm the goals, content and recommendations contained therein, they almost uniformly suggest that they lack the tools to implement the Standards appropriately in the classroom. The consistency in the types and content of lessons observed appeared to validate this concern. An overwhelming majority of the lessons observed bore little or no resemblance to the values so highly espoused by the student teachers. Student teachers perceived notable pressure from the education program to include the Standards in their teaching practices, while feeling little pressure from cooperating teachers. They also point to cooperating teachers as the most significant influence on their teaching.     

Studying mathematics problem-solving classrooms. A comparison between the discourse of in-service teachers and student teachers

In the article we compare the approaches of 3 in-service teachers and 3 student teachers when they tried to solve a verbal arithmetic problem in the classroom. Each interaction was studied using a System of Analysis that takes into account the cognitive processes involved in the solution of a mathematic problem and describes the interaction at different levels showing what is done and to what degree teachers and/or pupils are responsible for what is done. The results of the study suggest that both groups of teachers are different in how they direct the student’s attention toward the essential aspects implied in the resolution of word problem. On the one hand, the in-service teachers guaranteed students’ understanding of the problem before dealing with the solution, while students teachers only did so when pupils committed errors. On the other hand, the in-service teachers allowed a high level of student participation, while student teachers took a more prominent role so children’s participation was lower.     

Examining the supervision of mathematics student teachers through analysis of conference communications

Student teaching is often a capstone experience in the preparation of mathematics teachers. Thus, it is essential to better understand key aspects of the experience. We conducted a qualitative study of post-lesson conferences led by supervisors (classroom cooperating teachers and a university supervisor) working with mathematics student teachers. Analysis of conference communications revealed differences in the types and content of communications in conferences led by the cooperating teachers and by the university supervisor. Cooperating teachers tended toward evaluative supervision that lacked a focus on the mathematics of the lessons while the university supervisor tended toward educative supervision, guiding student teachers to reflect on and learn from their own classroom experiences including the mathematics of their lessons. Differences are discussed, and suggestions concerning the supervision of student teachers are made along with recommendations for further research.     

Peer collaboration: the relation of regulatory behaviors to learning with hypermedia

Peer collaboration is a pedagogical method currently used to facilitate learning in classrooms. Similarly, computer-learning environments (CLEs) are often used to promote student learning in science classrooms, in particular. However, students often have difficulty utilizing these environments effectively. Does peer collaboration help students learn with these environments? Little research looking closely at face-to-face peer collaboration with computer learning environments exists. Utilizing a social-cognitive theoretical framework, this study investigated the relation between the conceptual-knowledge learning and the collaborative regulatory behaviors of students working with a peer as they studied about the human circulatory system using a hypermedia CLE. Fifty-four high-school students from the East Coast of the United States were audiotaped to identify the collaborative regulatory behaviors they evidenced as they studied. Results revealed significant correlations among students’ proportion of categories of regulatory behaviors and their learning gains (from pretest to posttest). Moreover, qualitative analyses revealed particular behaviors that larger-gain collaborative pairs engaged in to a greater extent than smaller-gain pairs as they learned with the hypermedia environment.     

浅谈教学中增强学生数学应用意识的意义及途径

长期以来 ,受传统教育思想的影响 ,数学教学存在严重缺陷。课堂上不讲知识的实际来源和应用方法 ,“掐头去尾烧中段” ,把能力培养局限于解答考题 ,造成学生动手能力差 ,应用意识弱的现状。随着科技的发展、技术的进步 ,数学的应用日益广泛 ,教学中如何增强学生的数学应用意识是一个迫切需要解决的问题。1　增强学生数学应用意识的意义1 1　符合数学学科自身特点数学的特点之一是应用广泛性 ,这是数学生命力之所在 ,也是数学内容虽然高度抽象却仍能蓬勃发展的基础。马克思就曾预言 :“一门科学只有成功地运用数学后 ,才能达到完善的地步。…     

Synergies between theoretical approaches to mathematics education with technology: A case study through a cross-analysis methodology

Mathematics education as a research domain is characterized by a plurality of theoretical approaches. Acknowledging the existence of such diversity and the risks of an excessive theoretical fragmentation does not mean to search for a unifying theory but to urge the community to develop strategies for coping with this diversity. This article is meant to show the potential of a “cross-analysis” methodology for establishing connections between different theoretical approaches to mathematics education with technology. Within the frame of the ReMath European Project, two Teaching Experiments were realized, centred on the use of a same ICT tool—Casyopée. Two distinct theoretical approaches shaped both the Teaching Experiments design and their enactments: the Theory of Didactical Situations and the Theory of Semiotic Mediation. The two Teaching Experiments have then been analysed from both theoretical points of view. In this article we will provide some examples drawn from this cross-analysis that show the synergy which can be established between the aforementioned theoretical approaches. Beyond contributing to a deeper understanding of the observed “didactical phenomena”, that synergy allows establishing connections between the two approaches that lead to their reciprocal enrichment.     

Using a video-based curriculum to develop a reflective stance in prospective mathematics teachers

Although video cases are increasingly being used in teacher education as a means of situating learning and developing habits of reflection, there has been little evidence of the outcomes of such use. This study investigates the effects of using a coherent video-case curriculum in a university mathematics methods course by addressing two issues: (1) how the use of a video-case curriculum affects the reflective stance of prospective teachers (PTs); and (2) the extent to which a reflective stance developed while reflecting on other teachers’ practice transfers for reflecting on one’s own practice. Data sources include videotapes of course sessions and PTs’ written work from a middle school mathematics methods course that used a video-case curriculum as a major instructional tool. Both qualitative and quantitative analytical methods were used, including comparative and chi-square contingency table analyses. The PTs in this study showed changes in their level of reflection, their tendency to ground their analyses in evidence, and their focus on student thinking. In particular, they began to analyze teaching in terms of how it affects student thinking, to consider multiple interpretations of student thinking, and to develop a more tentative stance of inquiry. More significantly, the reflective stance developed via the video curriculum transferred to the PTs’ self-reflection in a course field experience. The results of this study speak to the power of using a video-case curriculum as a means of developing a reflective stance in prospective mathematics teachers.

Looking through the lens of a teacher's life: The power of prototypical stories in understanding teachers' instructional decisions in mathematics

The focus of this narrative inquiry was to examine how teachers describe influences on their instructional decisions in mathematics. Using a lens provided by pivotal or prototypical stories teachers told about their past experiences, connections were suggested between these past experiences and their instructional decisions. These connections helped to illustrate how instructional decisions may be shaped by priorities and beliefs established earlier in life, as well as by current contexts. Further research on the use and characteristics of prototypical stories is suggested.     

Novice teachers learning through participation in a teacher study group

Using Lave and Wenger’s framework of legitimate peripheral participation in communities of practice, this case study explores the experiences of three novice teachers engaged with more experienced teachers in a teacher study group during their first year of teaching. The study illustrates how, over time, the novices moved from more peripheral to more central participation as revealed through changes in their talk and feelings about participating. The study highlights the importance of legitimacy and peripherality provided by the more experienced teachers and identifies the ways in which the facilitator scaffolded and mediated the novices’ participation and learning.     

The mathematical vitality of secondary mathematics graduates and prospective teachers: A comparative study

This article links two areas of contemporary interest in mathematics education. These are respectively; mathematical characteristics of prospective teachers, and the notion of levels of understanding.The study reported here considered questions relating to the mathematical appreciation of successful students at both senior secondary (university entrance) and tertiary level. The tertiary students were graduates embarking upon a post-graduate course of teacher-training.It was found that response patterns were stable across a variety of institutions, geographical regions (Australian States), and education systems. Mathematical malfunctions present among secondary graduates were found to exist in substantially the same proportions among the graduate teacher trainees. It was found that the type of understanding sought in this study was not enhanced merely by taking more mathematics courses. The problem of recycling of attitudes and mathematical misconceptions within the secondary teaching structure is addressed, and the question of the approach to the study of mathematics at both secondary and tertiary level is raised as a matter of continuing concern.     

Evaluating mental models in mathematics: a comparison of methods

Cognitive scientists investigate mental models (how humans organize and structure knowledge in their minds) so as to understand human understanding of and interactions with the world. Cognitive and mental model research is concerned with internal conceptual systems that are not easily or directly observable. The goal of this research was to investigate the use of Evaluation of Mental Models (EMM) to assess the mental models of individuals and groups in solving complex problems and to compare novices and experts models as bases for providing feedback to learners. This study tested a qualified web-based assessment tool kit, Highly Interactive Model-based Assessment Tools and Technologies (HIMATT), in an as yet untested domain—mathematics. In this study, university students and their mathematics instructors used two tools in HIMATT, Dynamic Evaluation of Enhanced Problem Solving (DEEP) and Text-Model Inspection Trace of Concepts and Relations (T-MITOCAR). The research questions include: Do novice participants exhibit common patterns of thoughts when they conceptualize complex mathematical problems? Do novices conceptualize complex mathematical problems differently from experts? What differences in DEEP and T-MITOCAR patterns and responses exist according to the measures of HIMATT? Findings suggest that EMM and HIMATT could effectively support formative assessment in a complex mathematical domain. Finally, this study confirms a common assumption of cognitive scientists that the tool being used could affect the tool user’s understanding of the problem being solved. In this case, while DEEP and T-MITOCAR led to somewhat different expert models, both tools prove useful in support of formative assessment.     

Knowing and teaching fractions: A cross-cultural study of American and Chinese mathematics teachers

Guided by Shulman, 1986 and Shulman, 1987 tripartate model of teacher expertise (subject matter knowledge (SMK), pedagogical content knowledge (PCK), and general pedagogical knowledge (GPK)), the present study examined 162 U.S. and Chinese 3rd grade mathematics teachers’ expertise in teaching fractions. Results show that U.S. teachers lag significantly behind Chinese teachers in SMK (concepts, computations, and word problems) and in some areas of PCK (e.g., such as identifying important points of teaching the fraction concepts and how to ensure students’ understanding). Although the Chinese teachers performed poorly in comparison to their U.S. counterparts on a test designed to measure GPK (e.g., psychological and educational theories and applications), its poor reliability calls these results into question. The implications of this study are discussed in terms of practical ideas for teacher training in the U.S. and China.     

特校教师保持心理健康的途径

特校教师存在的心理健康问题由来已久,但关于这个问题的研究却开展得极少。近两年来,随着素质教育的深人和新课程改革的推进,学生的心理健康教育已得到相当的重视,特校教师的心理健康问题也引起了关注。北京、上海、长沙等地的特教工作者开展了一些相关的调查和研究,虽然他们得出的数据和结论各有不同,但特校教师心理问题的存在是不争的事实,     

Teaching teachers to teach Boris: a framework for mathematics teacher educator pedagogical content knowledge

The notion of pedagogical content knowledge (PCK) was posited in the context of school teaching and the knowledge used by teachers teaching school students. It has been examined for a number of discipline areas, notably mathematics. There are, however, other teaching contexts, including those of teacher educators, whose students are pre-service teachers (PSTs). The content these teacher educators teach is not subject discipline knowledge (or not solely), but the PCK for teaching a subject discipline. What knowledge do teacher educators use as they teach PCK? This paper presents a framework for the PCK required of mathematics teacher educators as they work to develop PSTs’ PCK for teaching mathematics. The framework builds on existing research into PCK and categorises aspects of the work of teacher education. The framework’s usefulness is examined by studying the PCK used by the first author in building PSTs’ understanding of mathematics teacher PCK.     

Structuring professional dialogue through protocols: a study among elementary English teachers in Taiwan

This study focuses on the integration of protocols into elementary school English teacher’s professional dialogue and discusses the influence of protocols on teacher’s teaching and learning. Based on the analysis of documents, observations, and interviews, this study concludes that the introduction of protocols to elementary school English teachers led them to discuss their classroom practice and gain professional learning in instructional strategies and student learning during the professional dialogue. However, English teacher’s lack of experience in using protocols led to interruptions during the professional dialogue. Suggestions for effective protocol-based professional dialogue are provided.