Fostering Instructor Knowledge of Student Thinking Using The Flipped Classroom

Abstract

In this article, we share a model of flipped instruction that allowed us to gain a window into our students’ mathematical thinking. We depict how that increased awareness of student thinking shaped our mathematics instruction in productive ways. Drawing on our experiences with students in our own classrooms, we show how flipped instruction can be used to design experiences that help students make sense of mathematics during class sessions.     

Learning to Learn From Benchmark Assessment Data: How Teachers Analyze Results

Although interim assessments are currently promoted as a mechanism for improving teaching and student learning, we know little about how teachers use this data to modify instruction. This article presents findings from a larger study on teachers’ use of interim assessment information in elementary mathematics. We address the following questions: (a) How do the Philadelphia teachers in our sample analyze benchmark assessment results, (b) how do they plan instruction based on these results, and (c) what are their reported instructional responses to such results? To answer these questions, we interviewed all 3rd- and 5th-grade teachers in five average- and above-average-performing elementary schools three times during the 2006–07 school year. We found that although the teachers in our study used interim assessment results to gain information about students’ learning in mathematics, teachers did not use interim assessments to make sense of students’ conceptual understanding. Furthermore, teachers’ tendency to interpret student errors as procedural missteps was paralleled by a trend toward procedural instructional responses.     

Building the wall brick by brick: one prospective teacher’s experiences with mathematics anxiety

Mathematics education researchers have investigated mathematics anxiety in prospective elementary teachers. While many of these studies have focused on the bodily sensations and emotions of mathematics anxiety, particularly those felt in assessment situations, opportunities remain to investigate how prospective elementary teachers interpret their experiences with mathematics anxiety and connect them over time to compose personal histories of mathematics anxiety. Currently, over 90 % of elementary teachers in US schools are women, and women have been shown to suffer more from mathematics anxiety than do men. In this article, I analyze how one woman prospective elementary teacher described, explained, and related her experiences of mathematics anxiety across her personal narratives of learning mathematics as a K-12 student and of learning to teach mathematics as a college student in a teacher preparation program. My research demonstrates that experiences of mathematics anxiety may persist beyond assessment situations to influence women prospective elementary teachers’ larger mathematical histories. I also show that women prospective elementary teachers may interpret mathematics anxiety as specific fears (e.g., loss of opportunities for social participation) and may develop particular coping strategies related to those fears. Finally, I point out that while a coping strategy may be used consistently across K-12 mathematics learning and undergraduate teacher preparation, and may even offer a woman prospective elementary teacher some relief from mathematics anxiety, it may also limit her mathematics learning and professional development. To conclude, I present implications of my research for mathematics teacher educators.     

试析初中数学翻转课堂教学策略

翻转课堂是指学生能在课前利用互联网、电子信息技术等手段自主学习,课堂上与教师、同学互相探讨合作交流解决疑问的新型教学方式。信息时代,翻转课堂在教学中的应用场景越来越丰富,本文以初中数学教学为例,阐述开展翻转课堂教学的必要性,并就如何在数学教学中使用翻转课堂辅助课程教学表达几点看法。     

Bergmann,Jonathan & Sams,Aaron (2014) Flipped learning. ISTE (Washington & Eurospan,London) isbn 978‐1‐56484‐344‐9 227 pp £18.50 https://www.iste.org/resources/product?ID=3014

This deceptively simple book aims to show how flipped learning optimises class time, by delivering direct instruction by video for students to use outside of class, freeing up teachers to individualise their learning in the classroom. It is essentially a “book of stories” for teachers shared by practising teachers who have enthusiastically embraced this model. I strongly recommend that you buy Flipped learning if it is in your area of interest. Marie Martin     

The relationship among elementary teachers’ content knowledge,attitudes, beliefs,and practices

This study investigated 481 in-service elementary teachers’ level of mathematical content knowledge, attitudes toward mathematics, beliefs about the effectiveness of inquiry-based instruction, use of inquiry-based instruction and modeled the relationship among these variables. Upper elementary teachers (grades 3–5) were found to have greater content knowledge and more positive attitudes toward mathematics than primary teachers (grades K-2). There was no difference in teachers’ beliefs about effective instruction, but primary level teachers were found to use inquiry-based instruction more frequently than upper elementary teachers. Consistent with Ernest’s [Ernest (1989). The knowledge, beliefs and attitudes of the mathematics teacher: A model. Journal of Education for Teaching, 15(1), 13–33] model of mathematics teaching, content knowledge, attitudes, and beliefs were all found to be related to teachers’ instructional practice. Furthermore, beliefs were found to partially mediate the effects of content knowledge and attitudes on instructional practice. Content knowledge was found to be negatively related to beliefs in the effectiveness of inquiry-based instruction and teachers’ use of inquiry-based instruction in their classrooms. However, overall, teachers with more positive attitudes toward mathematics were more likely to believe in the effectiveness of inquiry-based instruction and use it more frequently in their classroom. Teacher beliefs were found to have the strongest effect on teachers’ practice. Implications for the goals and objectives of elementary mathematics methods courses and professional development are discussed.

Teaching the partial products algorithm with the concrete representational abstract sequence and the strategic instruction model

Elementary standards include multiplication of single-digit numbers and students advance to solve complex problems and demonstrate procedural fluency in algorithms. The ability to illustrate procedural fluency in algorithms is dependent on the development of understanding and reasoning in multiplication. Development of multiplicative reasoning provides the foundation for advanced mathematics and algebraic reasoning. For students who struggle in mathematics, instruction in multiplication algorithms should ensure conceptual understanding so that students have a foundation for success in advanced mathematics. The concrete representational abstract (CRA) sequence addresses conceptual understanding and the strategic instruction model (SIM) supports procedural knowledge. The current pilot study combined these methods to teach elementary students the partial products algorithm. Twelve students in grades four and five participated in the study, receiving instruction from teachers in their school during an intervention period. Within a pre-experimental design, using pre- and postintervention data, students showed a significant change in performance. The article will describe and show how teachers implemented the CRA-SIM interventions and discuss implications for practice.     

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 representations,decomposition, and approximations of practices to support prospective elementary mathematics teachers’ practice of organizing discussions

This paper examines a series of instructional activities that provide prospective elementary teachers with an opportunity to engage in one of the more difficult practices to learn within mathematics teaching—organizing a mathematical discussion. Within a mathematics methods course, representations and decomposition of practice built from the Five Practices framework (Smith and Stein in Five practices for orchestrating productive mathematics discussions. Corwin Press, Thousand Oaks, CA, 2011) were implemented and studied to examine how prospective elementary teachers set goals, selected and sequenced available student work, and planned questions within a mathematical discussion. We examined prospective elementary teachers’ strengths and weaknesses in these facets through an approximation of practice set in a lesson context familiar to the prospective elementary teachers. Our results demonstrated that although prospective elementary teachers set varying goals for a discussion, their pedagogical choices in planning their discussion tended to be consistent with the goals they have set. These results support the focused development of prospective elementary teachers’ goal setting as an implication for mathematics teacher educators.     

Mathematics professional development for elementary teachers: Building trust within a school-based mathematics education community

Despite increased attention in professional development to building communities of teachers, few studies have investigated the development of trust in these communities. This paper presents issues related to trust in a professional development project at an elementary school. The project aimed for school-based elementary teachers and university-based mathematics educators to work together to improve mathematics instruction at the school by developing a mathematics education community of learners, where learners were conceived of as both the teachers and the mathematics educators. This paper addresses the following question: What factors in the project supported the development of trust among mathematics educators and teachers as the community was formed? More specifically, the paper is about one aspect of trust, namely, building school-based elementary teachers’ trust in the university-based mathematics educators. We point to three factors that helped the development of trust and we use care theory to further understand and discuss these factors in the context of in-service teacher education initiatives.     

试论算术中的代数思维:准变量表达式

在数学教学中,小学与中学(主要是初中)的衔接问题,仅仅依赖体制上的变革根本无法解决。而在新课程标准的推进当中,由于更加强调整体性和一贯性,所以,衔接问题的解决已刻不容缓。算术中的准变量(表达式)正是解决这一问题的强有力思想。准变量表达式既动摇了算术与代数之间的传统割裂,又在算术思维与代数思维之间起到了桥梁作用。而且,它还体现了新课程标准所倡导的算法多样化之精神。准变量(表达式)思想对改进义务教育阶段数学教学有着诸多切实的现实意义。     

A Case Study Exploring the Identity of an In-Service Elementary Science Teacher: a Language Teacher First

Teachers are central to providing high-quality science learning experiences called for in recent reform efforts, as their understanding of science impacts both what they teach and how they teach it. Yet, most elementary teachers do not enter the profession with a particular interest in science or expertise in science teaching. Research also indicates elementary schools present unique barriers that may inhibit science teaching. This case study utilizes the framework of identity to explore how one elementary classroom teacher’s understandings of herself as a science specialist were shaped by the bilingual elementary school context as she planned for and provided reform-based science instruction. Utilizing Gee’s (2000) sociocultural framework, identity was defined as consisting of four interrelated dimensions that served as analytic frames for examining how this teacher understood her new role through social positioning within her school. Findings describe the ways in which this teacher’s identity as a science teacher was influenced by the school context. The case study reveals two important implications for teacher identity. First, collaboration for science teaching is essential for elementary teachers to change their practice. It can be challenging for teachers to form an identity as a science teacher in isolation. In addition, elementary teachers new to science teaching negotiate their emerging science practice with their prior experiences and the school context. For example, in the context of a bilingual school, this teacher adapted the reform-based science curriculum to better meet the unique linguistic needs of her students.

     

An interactive problem-posing guiding approach to bridging and facilitating pre- and in-class learning for flipped classrooms

ABSTRACT

Educators have indicated that a good flipped learning design has the potential to enable teachers to use the class time effectively for conducting higher-order thinking activities and enhancing the interactions among peers and teachers. Several previous studies have also pointed out the importance of employing proper learning strategies, such that students’ learning performance would meet the expectations of both the pre- and in-class activities of flipped learning. In this study, an interactive problem-posing guiding strategy was proposed to guide students to understand the concepts of problem-posing by the use of multiple choice options before class, and to engage them in effective problem-posing activities during class. To examine the effectiveness of the proposed learning mode, a flipped learning system was developed and an experiment was conducted in a natural science learning activity in an elementary school. The participants were two classes of fifth graders of an elementary school, a total of 56 students, with one class as the experimental group and the other as the control group. The students in the experimental group adopted the interactive problem-posing guiding strategy in the flipped learning mode, while the control group used the conventional flipped learning mode. The experimental results confirmed that the flipped learning activity with the interactive problem-posing guiding strategy was more effective in terms of learning achievement, self-efficacy, and deep approaches to learning science in the natural science course than the conventional flipped learning method. It also confirmed the importance of integrating the stepwise learning strategy (e.g. guided problem-posing and collaborative problem-posing) into flipped learning to support students’ pre- and in-class learning.     

Use of comics to enhance students’ learning for the development of the twenty-first century competencies in the mathematics classroom

Abstract

This paper discusses the use of comics in teaching mathematics in the secondary mathematics classroom. We explicate how the use of comics in teaching mathematics can prepare students for the twenty-first century competencies. We developed an alternative teaching package using comics for two lower secondary mathematics topics. This alternative teaching package consists of (1) several sets of comic strips expounding all related mathematical concepts in a lively way; (2) tiered practice questions for learning reinforcement; and (3) a set of proposed lesson outlines with suggestions on how to use the comics for mathematics teaching. We also report how one of the teachers in our study used this teaching package in her mathematics lessons. Her lessons were video-recorded and 11 students were interviewed to help us understand how the mathematics comics lessons were enacted and the students’ perception of comics as instruction. We identified instances in which the teacher tweaked the provided resource to further enhance student learning and incorporated elements of the twenty-first century competencies during her lessons. Through selected student interviews, we also identified instances in which students commented on their gain from the new approach from the perspective of the twenty-first century competencies.     

Linking reform-oriented experiences to teacher identity: The case of an elementary mathematics teacher

Novice teachers often have difficulty transferring what they learn in teacher education programs to classroom practice. This is especially true for elementary school teachers who are expected to teach mathematics with reform-oriented methods. The purpose of this longitudinal case study was to examine the experience of one novice elementary school teacher over a 3-year period, and understand the factors that supported her to enact a reform-oriented practice in mathematics as she transitioned from being a preservice to inservice teacher. Influential mathematics education models, her commitment to learning, and school-based contexts affected her teaching identity and practices. To conclude, an argument is made for teacher education experiences to explicitly address mathematics teacher identity to support the enactment of reform-oriented practices.     

高校教师翻转课堂教学模式的需求性分析

翻转课堂通过颠倒传统课堂的方式让学生参与学习,在信息化的环境中,课堂教学被转移到课堂之外,学生可在线自由地安排学习时间,并进行学习交流和合作.为使翻转课堂能有效的实施,老师必须具备一系列的技术技能、理论知识和教学方法、策略的指导,通过此研究,一个基于网络环境下的教学模式为有兴趣实施翻转课堂的老师提供了信息,结果表明此种教学模式对于绝大多数的参与教师来说是愿意接受和使用的,并能在实际运用过程中提升其知识水平和自信度,于此同时在线学习社区的建立,对初用教师的教学指导,以及广泛地分享实际教学经验也是推动其发展的重要支撑因素.     

The purpose, design and use of examples in the teaching of elementary mathematics

This empirical paper considers the different purposes for which teachers use examples in elementary mathematics teaching, and how well the actual examples used fit these intended purposes. For this study, 24 mathematics lessons taught by prospective elementary school teachers were videotaped. In the spirit of grounded theory, the purpose of the analysis of these lessons was to discover, and to construct theories around, the ways that these novice teachers could be seen to draw upon their mathematics teaching knowledge-base in their lesson preparation and in their observed classroom instruction. A highly-pervasive dimension of the findings was these teachers’ choice and use of examples. Four categories of uses of examples are identified and exemplified: these are related to different kinds of teacher awareness.

The Flipped Classroom: Primary and Secondary Teachers’ Views on an Educational Movement in Schools in Sweden Today

The aim of this study is to contribute to an increased understanding of the flipped classroom movement. A total of 7 teachers working in school years 4–9 and who both actively flipped their classrooms and had been early adopters in this movement were interviewed. Two research questions were posed: “What characterizes flipped classroom instruction according to the teachers?” and “What objectives do the flipped classroom meet according to the teachers?” Regarding the first research question, a characteristic of a flipped classroom was “the flip,” a task to be accomplished outside the classroom before class. In relation to the second research question we found three objectives: Student activity in class; Educational change; Being part of a digital learning community.     

The Impact of a State Performance-Based Assessment and Accountability Program on Mathematics Instruction and Student Learning: Evidence From Survey Data and School Performance

The purpose of this study was to examine the impact of the Maryland School Performance Assessment Program (MSPAP) and the Maryland Learning Outcomes (MLOs) on mathematics classroom instruction and assessment practices, professional development, and student learning. The data sources included questionnaires for principals, mathematics teachers, and students, as well as student performance on MSPAP over a 5-year period. Ninety elementary and middle schools in Maryland participated in the study. The results indicate that principals and teachers tended to support MSPAP as a tool for making changes in instruction, teachers were making some positive changes in mathematics instruction because of MSPAP (based on the questionnaire data), and the schools for which teachers reported that MSPAP had a greater impact on their mathematics instruction had greater MSPAP performance gains in mathematics over the 5 years.     

Teacher perceptions of an online tutoring program for elementary mathematics

This study explores elementary teacher perceptions related to the implementation of an online tutoring program. Teachers were surveyed regarding factors that affected use of the online tutoring program as a supplement to mathematics instruction. Results indicated that teachers overwhelmingly reported positive views of the training and support opportunities provided, as well as confidence in their level of knowledge and the ease of use of the program. Teachers also indicated strong support of the mathematical content and instructional value of the program. Each of these factors increases the likelihood that online tutoring programs such as the one implemented in this study will continue to be used among teachers as a supplement to mathematics instruction.