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1.
In this study we investigate a strategy for engaging high school mathematics teachers in an initial examination of their teaching in a way that is non-threatening and at the same time effectively supports the development
of teachers’ pedagogical content knowledge [Shulman (1986). Educational Researcher, 15(2), 4–14]. Based on the work undertaken by the QUASAR project with middle school mathematics teachers, we engaged a group
of seven high school mathematics teachers in learning about the Levels of Cognitive Demand, a set of criteria that can be
used to examine mathematical tasks critically. Using qualitative methods of data collection and analysis, we sought to understand
how focusing the teachers on critically examining mathematical tasks influenced their thinking about the nature of mathematical
tasks as well as their choice of tasks to use in their classrooms. Our research indicates that the teachers showed growth
in the ways that they consider tasks, and that some of the teachers changed their patterns of task choice. Further, this study
provides a new research instrument for measuring teachers’ growth in pedagogical content knowledge.
An earlier version of this paper was presented at the American Educational Research Association Annual Meeting, New Orleans,
LA, April 2002. 相似文献
2.
Eva Thanheiser Dana Olanoff Amy Hillen Ziv Feldman Jennifer M. Tobias Rachael M. Welder 《Journal of Mathematics Teacher Education》2016,19(2-3):123-148
Mathematical task design has been a central focus of the mathematics education research community over the last few years. In this study, six university teacher educators from six different US institutions formed a community of practice to explore key aspects of task design (planning, implementing, reflecting, and modifying) in the context of comparing fractions using reasoning and sense-making. By presenting results of their implementation of two tasks with 63 prospective elementary teachers across three institutions and their reflective analysis of the implementation, the authors highlight the importance of collecting and analyzing data and reflecting on this analysis to inform the redesign of tasks. The authors also found that considering different types of tasks (problem solving vs. problem posing) helps illuminate different aspects of prospective elementary teachers' understanding, which can inform task redesign. Finally the authors contribute to the knowledge base on reasoning strategies for comparing fractions and prospective elementary teachers’ knowledge of these strategies. 相似文献
3.
Melissa D. Boston 《Journal of Mathematics Teacher Education》2013,16(1):7-31
This investigation describes secondary mathematics teachers’ learning and instructional change following their participation in a professional development workshop, the Enhancing Secondary Mathematics Teacher Preparation Project (ESP) (2004–2005), specifically focused on the selection and implementation of cognitively challenging mathematical tasks. Data consist of a pre/post-assessment of teachers’ knowledge of the cognitive demands of mathematical tasks and videotaped discussions and written artifacts from the professional development sessions. A mixed methods approach was used to identify connections between teachers’ learning and their experiences in the ESP workshop. Results indicate that ESP teachers developed new ideas about the influence of mathematical tasks on students’ learning. Increases in teachers’ knowledge of the cognitive demands of mathematical tasks were closely linked to ideas represented in frameworks and discussions from the ESP workshop and to teachers’ experiences in solving challenging mathematical tasks as learners. 相似文献
4.
In this article we elaborate a conceptualisation of mathematics for teaching as a form of applied mathematics (using Bass's idea of characterising mathematics education as a form of applied mathematics) and we examine implications of this conceptualisation for the mathematical preparation of teachers. Specifically, we focus on issues of design and implementation of a special kind of mathematics tasks whose use in mathematics teacher education can support the development of knowledge of mathematics for teaching. Also, we discuss broader implications of the article for mathematics teacher education, including implications for mathematics teacher educators' knowledge for promoting mathematics for teaching. 相似文献
5.
Rebecca McGraw Kathleen Lynch Yusuf Koc Ayfer Budak Catherine A. Brown 《Journal of Mathematics Teacher Education》2007,10(2):95-121
In this study, we consider the potential of multimedia cases as tools for teacher professional development. Specifically,
we examined online and face-to-face discussions that occurred within groups composed of pre-service mathematics teachers,
in-service mathematics teachers, mathematicians, and mathematics teacher educators. Discussions within these heterogeneous
groups tended to focus on issues of classroom implementation of the tasks shown in the multimedia case. Secondary foci of
discussion included task characteristics and appropriateness of tasks for engaging students in thinking about mathematical
concepts and processes. Analysis of contributions to discussions across group member type revealed differences that suggest
that the variety of backgrounds and experiences of group members can blend in ways that support rich and critical discussions
of mathematics, teaching, and learning. 相似文献
6.
Joan Ferrini-Mundy Gail Burrill William H. Schmidt 《Journal of Mathematics Teacher Education》2007,10(4-6):311-324
Improving mathematics education in the United States has taken many forms. Our work has focused on two aspects: the content knowledge of teachers and a well-articulated coherent curriculum. Our aim was teacher “capacity building” that is enabling teachers to teach to coherent and significant mathematical curricular goals and describe the implementation in a large-scale project based at Michigan State University. We highlight the design, structure and use of mathematics teacher learning tasks that were intended to improve teachers’ capacity to teach to these goals and note how the teachers’ perceptions of the structure and sequencing of mathematics itself affect the ways they organize mathematics in their teaching and the ways they teach. 相似文献
7.
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. 相似文献
8.
9.
Colleen Vale Alasdair McAndrew Siva Krishnan 《Journal of Mathematics Teacher Education》2011,14(3):193-212
A professional learning program for teachers of junior secondary mathematics regarding the content and pedagogy of senior
secondary mathematics is the context for this study of teachers’ mathematical and pedagogical knowledge. The analysis of teachers’
reflections on their learning explored teachers’ understanding of mathematical connections and their appreciation of mathematical
structure. The findings indicate that a professional learning program about senior secondary mathematics can enable practicing
teachers to deepen and broaden their knowledge for teaching junior secondary mathematics and develop their practice to support
their students’ present and future learning of mathematics. Further research is needed about professional learning approaches
and tasks that may enable teachers to imbed and develop awareness of structure in their practice. 相似文献
10.
Amy E. Ryken 《Journal of Mathematics Teacher Education》2009,12(5):347-364
This documentary account situates teacher educator, prospective teacher, and elementary students’ mathematical thinking in
relation to one another, demonstrating shared challenges to learning mathematics. It highlights an important mathematics reasoning
skill—creating and analyzing representations. The author examines responses of prospective teachers to a visual representation
task and, in turn, their examination of school children’s responses to mathematical tasks. The analysis revealed the initial
tendency of prospective teachers to create pictorial representations and highlights the importance of looking beyond the pictures
created to how prospective teachers use mathematical models. In addition, the challenges prospective teachers face in moving
beyond a ruled-based conception of mathematics and a right/wrong framework for assessing student work are documented. Findings
suggest that analyzing representations helps prospective teachers (and teacher educators) rethink their teaching practices
by engaging with a culture of teaching focused on reading for multiple meanings and posing questions about student thinking
and curriculum materials. 相似文献
11.
AbstractThis article describes reflections of two mathematicians and a mathematics teacher educator who collaborated on the development and implementation of courses (probability and statistics connections, number concept connections, and middle school mathematics methods) for middle school mathematics preservice teachers. The instructors of the courses, two in mathematics and one in mathematics education, worked together to more explicitly link course materials, assignments, and the pedagogical approaches. Collectively, the courses were designed to address the five components of preservice teachers’ mathematical knowledge for teaching (PT-MKT), and to model effective teaching practices. Using their collective experiences co-planning and implementing these course adjustments were made in the subsequent year. The instructors were pleased by their implementation and student outcomes in all three courses.We describe how each component of the PT-MKT framework was approached in these courses and discuss challenges experienced by the instructors, who were part of a larger effort to develop and implement a middle school teacher preparation program. The information shared is based on data collected as part of a program evaluation effort, and is bolstered by the instructors’ recollection of events. Overall, the instructors enhanced the curricula and their instructional practices and found that the attention placed on developing PT-MKT support the mathematical development of middle school mathematics preservice teachers. 相似文献
12.
In this study, we report on what types of mathematical knowledge for teaching teachers (MKTT) mathematics teacher educators (MTEs) use and develop when they work together and reflect on their teaching in a Community of Practice while helping prospective primary teachers (PTs) generate their own mathematical knowledge for teaching in learning mathematics via problem solving. Two novice MTEs worked with an experienced MTE and reflected on the process of learning to teach via problem solving and supporting PTs in developing deep understandings of foundational mathematical ideas. Taking a position of inquiry as stance, we examined our experiences teaching mathematics content courses for PTs via problem solving. We found that all of the MTEs used and developed some MKTT through (a) understanding and deciding on the mathematical goals of both the individual lessons and the two-course sequence as a whole, (b) choosing and facilitating tasks, and (c) using questions to scaffold PTs learning and engage them in mathematical processes such as making conjectures, justifying their reasoning, and proving or disproving conjectures. 相似文献
13.
Irene Biza Elena Nardi Theodossios Zachariades 《Journal of Mathematics Teacher Education》2007,10(4-6):301-309
Research often reports an overt discrepancy between theoretically/out-of context expressed teacher beliefs about mathematics and pedagogy and actual practice. In order to explore teacher knowledge in situation-specific contexts we have engaged mathematics teachers with classroom scenarios (Tasks) which: are hypothetical but grounded on learning and teaching issues that previous research and experience have highlighted as seminal; are likely to occur in actual practice; have purpose and utility; and, can be used both in (pre- and in-service) teacher education and research through generating access to teachers’ views and intended practices. The Tasks have the following structure: reflecting upon the learning objectives within a mathematical problem (and solving it); examining a flawed (fictional) student solution; and, describing, in writing, feedback to the student. Here we draw on the written responses to one Task (which involved reflecting on solutions of $ {\left| x \right|} + {\left| {x - 1} \right|} = 0 $ ) of 53 Greek in-service mathematics teachers in order to demonstrate the range of teacher knowledge (mathematical, didactical and pedagogical) that engagement with these tasks allows us to explore. 相似文献
14.
This study examined how a task-focused, year-long mathematics professional development program influenced elementary school teachers’ knowledge, beliefs, and practices. Participants completed 84 h of professional development over 13 months that were focused on exploring, modifying and implementing cognitively-demanding mathematical tasks. Using a multi-methods approach, teacher-participants completed pre- and post-measures of mathematical knowledge for teaching, teachers’ beliefs about teaching and learning mathematics, and teachers’ self-reports of enacted instructional practices. Further, three teacher-participants were randomly selected to be observed 3 times over the course of the school year. Data analyses indicated that the professional development had a statistically significant positive impact on participants’ mathematical knowledge for teaching, use of student-centered instructional practices, and beliefs towards mathematics as a subject area. Further, the observed teachers enacted some high-level mathematical tasks and questions, but these were more visible at the end of the study compared to the beginning of the study. Implications for future work are also shared. 相似文献
15.
Summative and Formative Assessments in Mathematics Supporting the Goals of the Common Core Standards
Alan H. Schoenfeld 《理论付诸实践》2015,54(3):183-194
Being proficient in mathematics involves having rich and connected mathematical knowledge, being a strategic and reflective thinker and problem solver, and having productive mathematical beliefs and dispositions. This broad set of mathematics goals is central to the Common Core State Standards for Mathematics.High-stakes testing often drives instructional practice. In this article, I discuss test specifications and sample assessment items from the two major national testing consortia and the prospects that their assessments will be positive levers for change.For more than 20 years, the Mathematics Assessment Project has focused on the development of assessments that emphasize productive mathematical practices, most recently creating formative assessment lessons (FALs) designed to help teachers build up student understandings through focusing on student thinking while engaging in rich mathematical tasks. This article describes our recent work. 相似文献
16.
Anderson Norton Andrea McCloskey Rick A. Hudson 《Journal of Mathematics Teacher Education》2011,14(4):305-325
In order to evaluate the effectiveness of an experimental elementary mathematics field experience course, we have designed
a new assessment instrument. These video-based prediction assessments engage prospective teachers in a video analysis of a
child solving mathematical tasks. The prospective teachers build a model of that child’s mathematics and then use that model
to predict how the child will respond to a subsequent task. In this paper, we share data concerning the evolution and effectiveness
of the instrument. Results from implementation indicate moderate to high degrees of inter-rater reliability in using the rubric
to assess prospective teachers’ models and predictions. They also indicate strong correlation between participation in the
experimental course and prospective teachers’ performances on the video-based prediction assessments. Such findings suggest
that prediction assessments effectively evaluate the pedagogical content knowledge that we are seeking to foster among the
prospective teachers. 相似文献
17.
There is an acknowledged gap between the theory presented in university preparation programmes and the reality of classroom practice that has resulted in many secondary mathematics pre-service teachers failing to implement university-endorsed teaching strategies. Using responses to a questionnaire and interviews, this qualitative study examined the factors that support or inhibit secondary mathematics pre-service teachers’ implementation of problem-solving tasks during professional experience. The results showed that even though the majority of pre-service teachers reported having beliefs compatible with using problem-solving tasks, the secondary students’ ability, preparation time, and the cooperating teacher were key factors that inhibited pre-service teachers’ implementation of problem-solving tasks. It is recommended that pre-service teachers regularly visit classrooms to observe the evolving implementation of problem-solving approaches. Furthermore, cooperating teachers should be required to attend professional development before the professional experience so they understand the goals of the university preparation programme and have the requisite skills and knowledge to support the implementation of problem-solving tasks in learning mathematics. 相似文献
18.
Technology-enhanced mathematics tasks were introduced to elementary pre-service candidates (n = 84) and in-service teachers (n = 38), who then, either in partners or small groups, created and taught inquiry-based lessons incorporating technology, with individual reflections. The lessons were coded using the following criteria: (a) The students themselves used the technology for inquiry learning, (b) technology was integral to the learning task, (c) the lesson focused on mathematics concepts—not the technology, and (d) the task would have been more difficult to accomplish without the technology. The lesson analysis revealed that, after instruction on inquiry learning and technology integration, each group achieved a high level of proficiency using these criteria. Further, the analysis assisted the instructors in identifying issues and concerns regarding implementation of technology in elementary mathematics instruction. 相似文献
19.
Mathematical tasks and tools, including tasks in the form of digital tools, are key resources in mathematics teaching and in mathematics teacher education. Even so, the ‘design’ of mathematical tasks is perceived in different ways: sometimes seen as something distinct from the teaching and learning process, and sometimes as integral to it. Whilst task design has often been carried out by designers or mathematicians (perhaps as textbook authors), the focus for this review article is research that has involved mathematics teachers as partners in the design of tasks. The article provides a state-of-the-art review of relevant literature and is presented under three headings that consider, in turn, the role of mathematical ‘tasks’; the nature of ‘task design’; and the notion of ‘partnerships for task design’ in mathematics education. Subsequently, we present current research that is providing new insights into tasks, task design, and task design partnership. Based on this, we argue that ‘task design’ needs to pay particular attention to what to design, which tools are necessary or beneficial, and under what conditions; digital tools and task resources offer particular affordances that traditional resources cannot provide; and not only do teachers benefit from being partners in task design (in terms of their professional learning) but without their involvement some aspects of task design would most likely be neglected. 相似文献
20.
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. 相似文献