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Using an Analytic Model to Gauge the Potential of Innovative Pedagogies of Approximation in Mathematics Teacher Education
Abstract
To leverage pedagogies of approximation to improve teacher education, educational researchers must first tackle the methodological question, "How can we determine the potential of pedagogies of approximation for supporting the growth of prospective teachers' knowledge and practices for teaching?" In this paper, we offer a model called the Dual Action Cycles of Approximations of Practice for identifying the pedagogical practices made available to pre-service teachers within various approximations of practice and describe its use to empirically investigate the potential of StoryCircles-a facilitated process of collaboratively representing a lesson using a multimedia storyboarding tool. We illustrate ways the model was used to make observations of pre-service teachers during StoryCircles. A key feature of the model is that it provides a bi-focal perspective on both the preactive and interactive phases of teaching, which help bring into focus the interdependent nature of these two phases. We close by suggesting that the development of new forms of approximation needs to be accompanied by research frameworks capable of investigating the potential of these various innovations.
... Anticipating a lesson (i.e., anticipating how a lesson may unfold) is one type of approximation of practice that, we argue, is particularly helpful for PT to experience the demands the discipline may make on the work of the teacher (Brown et al., 2022). The pedagogy of practice called approximation of practice, described by Grossman et al. (2009), has been a part of teacher education for decades and is often exemplified in rehearsals, live simulations, and digital simulations of various sorts. ...
... Reading or visualizing them by perusing a series of frames as a continuous story organizes all those actions and speech as a sequence of events taking place in time and participants can contrast this experience (e.g., hearing themselves say what they would say to students) against prior experiences in real classrooms (e.g., things they have heard teachers say to students). As they do this, they get a sense of what might need to be changed (e.g., if something they say is a mouthful or poorly phrased, they might realize that and change it; see Brown et al., 2022). Both in the sense of anticipating possible reactions from the students and practicing what they would look for, say, show, and do, we contend storyboarding a lesson or anticipating a lesson through scripting it multimodally is an approximation of practice. ...
... 81) asynchronously (the activity has been called "Leave Tracks") and for which they are asked to leave comments based on their observations or reactions to the storyline in order to help the group to eventually envision alternative trajectories for the lesson storyline. The StoryCircles process has been used with both PT (Brown et al., 2022) and practicing teachers in the understanding that both groups have had experiences in classrooms and in doing mathematics that can serve as learning resources. Most of the research to date, however, has concerned practicing teachers. ...
Addressing the need for practice-based teacher education to attend to context, we describe anticipations of a lesson as a case of approximations of practice that may anchor practice to its disciplinary context. After providing a general framework for how to think about anchoring practice in context we consider the StoryCircles process as an approximation of practice and illustrate the variability in anticipations of a lesson that can be generated by different StoryCircles. We use this variability to propose a conceptualization of lesson which is of particular value to practice-based teacher education.
Respect and listening are two issues that are complicated to research. This paper presents how respectful listening may constitute one aspect of a teacher’s role in child-centered learning. The analysis focus on a teacher’s reflections about events that took place after she and a colleague carried out a mathematics teaching unit on culturally diverse children’s understanding of ‘pattern’. The teacher observed situations that she found interesting and relevant for the children’s learning. She communicated with the researcher about this for some months. A closer look at the teacher’s reflections caused the research focus to change from what the teacher observed, to how she carried out the observations. So, the research focus is the teacher’s application of respectful listening skills when these observations were made. Our analysis reveals the outcomes of two situations. Situation 1 is about communication between the teacher and a child’s mother, while situation 2 is about communication between the teacher and a child.
Learning to name and notice students’ mathematical strengths is a challenging process requiring time and multiple iterations of practice for prospective teachers (PTs) to adopt. Mathematics teacher educators (MTEs) can approximate and decompose the complex practice of naming and noticing students’ mathematical strengths so PTs learn to teach mathematics while emphasizing what students know and can do. This study uses two tools MTEs can use to support PTs as they learn to name and notice students’ mathematical strengths: A LessonSketch experience, a digital platform with comic-based storyboards showing children engaged in a mathematics task, and a strengths-based sentence frame. Our study presents the findings from the 111 noticing statements from 18 PTs as they engaged in the LessonSketch digital experience and practiced making noticing statements about what children know about mathematics. The study found that after a sentence-frame intervention, the PTs are more likely to use strengths-based language and more likely to identify mathematical evidence in their noticing statements. Uncommitted language (statements that do not align with a strength- or deficit-based coding scheme), suggests a fruitful, yet complex space for supporting more PTs as they learn to name and notice students’ mathematical strengths. The paper concludes with implications for future research in teacher education.
We show how we used a national distribution of responses from 416 practicing teachers to items of a test of mathematical knowledge for teaching geometry to estimate changes in knowledge by a group of 11 practicing teachers who participated in a 2-year practice-based professional development programme. To draw a reliable interpretation of the change, we use multiple measurement models under different assumptions on the scale of MKT-G. We demonstrate how Diagnostic Classification Modelling was used to determine whether participants had grown. The participants’ gain (status change in achievement profile) was also examined in relationship with the amount of time participants spent during the PD, which was estimated using log data recorded by the online platform. We discuss the findings from these explorations, in the context of the broader problem of improving conceptualizations of MKT.
As practice-based teacher education (PBTE) has become more prevalent, debates about its contribution have emerged. Critics of PBTE question whether emphasizing practice will support a technocratic approach to teacher education rather than promoting instruction that is responsive to students’ ideas. This qualitative case study was motivated by an interest in understanding whether and in what ways practice-based approaches to teacher learning can support teachers in practicing responsiveness as opposed to practicing decontextualized moves. To this end, we investigated how early-career teachers in a practice-based professional development program were supported to approximate teaching practices. We focused on the extent to which approximations of practice supported teachers to hone their skill at being responsive to students’ ideas. Findings revealed characteristics of approximations of practice that support teachers in developing their capacity to enact responsive instruction. These findings have implications for program design, teacher educator pedagogy, and future research.
This paper highlights the second phase of a multi-university research project focusing on improving secondary prospective mathematics teachers’ abilities to respond to student thinking. This work is based on an Interview Module, created by the authors, which showed gains in prospective teachers’ (PSTs) abilities to attend to and interpret student thinking but did not show the same gains in their abilities to respond to student thinking. The Interview Module originally included a pre–post-video, readings, discussion, and one-on-one interview with a secondary student (Lesseig et al. in Math Teach Educ 5(1):29–46, 2016). This second phase adds a take-home assignment based on solutions from the secondary students who were interviewed in Phase 1. This take-home assignment was designed to specifically improve PSTs’ abilities to respond to student thinking. Results indicate that gains were made in their ability to respond, as was intended, but, unexpectedly, greater gains were also found in their abilities to attend to and interpret student thinking compared with Phase 1.
Curriculum materials are integral to the mathematics classroom, yet little is known about how teachers read curriculum in order to envision enactment. Through an assignment involving an approximation of actual teaching practice, we investigated prospective teachers’ documentational genesis (Pepin et al. in ZDM 45:929–943, 2013) from curriculum materials to their visions of enactment based on those materials. In mathematics methods courses, we assigned prospective teachers (N = 33) to one of two widely adopted curricula to generate a lesson plan and corresponding animation. We analyzed data from the assignment, termed lesson planimation, to reveal how prospective teachers use curriculum for the purposes of enactment. Prospective teachers drew upon curricular elements to different extents, with goals ranging from concern for learning mathematical ideas to student enjoyment of math class. Implications for theory, curriculum, and instruction are discussed.
The incorporation of animation in mathematics teacher education courses is one method for transforming practices and promoting practice-based education. Animation can be used as an approximation of practice that engages preservice teachers (PSTs) in creating classroom scenes in which they select characters, regulate movement, and construct conversations to generate playable animations (Grossman et al., 2009). We focus on the role of animation as a practice-based expression to denote who, what, and how PSTs professionally notice within a mathematics lesson (Jacobs, Lamb, & Philipp, 2010). The animation provides a mechanism for analysis that transfers PSTs’ concepts of teaching to a perceptible reality (Lowe & Schnotz, 2008). We examined the affordances and constraints of using animations as a means of supporting PSTs’ development of noticing.
Numerous studies have been conducted on the effectiveness of video as an effective means of reflective practice in pre-service Teacher Education. However, only few studies have explored preservice teachers' own perceptions in this regard in the field of ELT and none of these was related to primary level. To address this gap, multiple forms of qualitative data were triangulated. Participants were found to consider the use of video combined with guided reflection and peer dialogue to have a great potential in helping them form links between theory and practice and bring a heightened awareness of their teaching practices, especially in the areas ofclassroom language, error correction and student-centred activities. The study calls for more research which provides clear evidence of any changes in student teachers' teaching practices over a period of time and of possible factors which may hinder such changes.
Background/Context
Research shows that teachers’ understandings of students, subject, and teaching influence their classroom practice. Additionally, teachers’ colleagues have a role in shaping individuals’ approaches to teaching and their responses to reform.
Focus of Study
To understand how interactions with colleagues support teachers’ informal learning, I examined teachers’ collegial conversations in a highly collaborative teacher community.
Setting
This research took place within a collaborative, improvement-oriented urban high school mathematics department that showed evidence of increasing access to and rigor in its curriculum.
Participants
Six high school mathematics teachers and the researcher as a participant observer worked together in a group focusing on detracking ninth-grade algebra classes.
Research Design
This 2-year ethnographic study went inside the teacher community, with the researcher teaching alongside the teachers to gain access to their interactions and workings.
Data Collection and Analysis
Approximately 100 hours of teachers’ collaborative conversations were observed and recorded through audio- and videotape and field notes. Using a situated learning framework and sociolinguistic analyses, I examined teacher-to-teacher talk that constituted episodes of pedagogical reasoning (EPRs) to understand how collegial conversations provide resources for teacher learning.
Findings
Across EPRs, two forms of discourse were important sites for representing, and sometimes learning about, teaching practice, which I call teaching replays and teaching rehearsals. Extended replays or rehearsals supported a re-visioning routine, interactions in which teachers elaborated, reconsidered, or revised their understanding of complex teaching situations while providing particular, emotionally involving accounts of the classroom. The examination of these interactions further specifies conditions that support teachers’ collegial learning.
Conclusions
Theoretically, this article illustrates a process of learning as recontextualiza-tion, as the teachers work between general teaching principles and specific occurrences in their classrooms. Practically, by highlighting the work that teachers do to make sense of innovative practices, this analysis provides a description of how collegial conversations can support teachers’ informal learning, supporting the development of professional communities.
In recent years there have been calls for ambitious mathematics teaching which places student thinking and reasoning at the centre of instruction. Drawing on a larger study concerning implementation of practice-based pedagogies within our initial teacher education mathematics programme, this paper explores the range of opportunities for prospective teachers to practise and explore the role of professional noticing within rehearsal activities. We illustrate how the rehearsal process served to highlight components of professional noticing, namely, making students’ thinking visible, eliciting and responding to student thinking, and connecting to mathematical ideas. In looking at exemplars of each of these components, we illustrate how the mathematics teacher educators’ coaching moves—prompted by their professional noticing of prospective teachers’ learning—supported learning. We conjecture that the learning community’s collective exchanges within the rehearsals affirmed more than the ‘desired’ teaching orientation towards professional noticing; these efforts also made explicit the ‘how’, ‘why’ and to ‘what effect’ such an orientation has on diverse mathematics learners.
Pre-service TEFL programs are designed to prepare candidate teachers of English language for their future careers. To this end they provide students with a wide variety of techniques and strategies for enabling them to acculturate into their profession. An effective way of creating a professional learning community is to use a process of peer-teaching. The present study attempts to examine the impact of peer-teaching application on pre-service EFL teachers' understandings of methodology course content and its effect on their learning and teaching experiences. The main goals of this study are to find out how teacher candidates view peer-teaching and how they think their understanding of course content is affected when peer-teaching technique is implemented. The study cohorts 38 pre-service teachers enrolled in a 14 week methodology course at a large, public university in Istanbul. Data were collected qualitatively via observations, interviews, peer-evaluation papers and self-analysis reports. All interview transcripts and written samples were inductively analyzed. Findings suggest that peer-teaching exercise is found valuable by teacher candidates in that it helps them to increase learning and to be reflective and at the same time provides the students the opportunity of self-reflection which later might lead to development of their teaching and evaluation skills. Öz: İngiliz Dili Eğitim Programları, İngilizce öğretmeni adaylarını gelecekteki mesleklerine hazırlamak üzere yapılandırılmış programlardır. Bu amaç doğrultusunda söz konusu programlar öğrencilere mesleki bilgi ve becerileri kazandırmak için çeşitli teknik ve stratejiler sunarlar. Öğrencilerin mesleki donanımı kazanmalarında etkili yollardan biri de akran öğretimi sürecidir. Bu çalışmada akran öğretimi uygulamasının İngilizce öğretmen adaylarının yöntem derslerinin içeriğini anlama ve kendi öğrenme ve öğretme deneyimleri üzerindeki etkisinin araştırılması amaçlanmıştır. İstanbul'da bir devlet üniversitesinde yürütülen bu nitel çalışmaya yöntem dersine kayıtlı 38 İngilizce öğretmeni adayı katılmış, çalışma 14 hafta sürmüştür. Veriler gözlem, görüşme, akran değerlendirme formları ve öz değerlendirme raporları yoluyla elde edilmistir. Tüm görüşme kayıtları ve yazılı belgeler tümevarımsal yöntemle çözümlenmiştir. Bulgular, akran öğretimi tekniğinin öğrenciler tarafından etkili bulunduğunu göstermiştir. Çalışmaya katılan öğrenciler bu surecin kendi öğretim tekniklerini ve değerlendirme becerilerini geliştirdiğini öne sürmüşlerdir. Anahtar Sözcükler: akran-öğretimi, öğretmen adayları, öz değerlendirme, akran değerlendirmesi
This study used text-related, video-based case materials to assess teacher candidates’ readiness to communicate with families. Participants (N = 141) rated their efficacy for home–school communication and then responded to a description of a classroom-based challenge regarding one student’s behavioral and academic performance. Next, they evaluated two videos, each capturing how a teacher addressed the challenge in a parent–teacher conference. Cases offered contrasting models of communication effectiveness along two dimensions: structuring and responsiveness. Finally, candidates chose which model did the better job and justified their choice. Findings revealed that candidates had high self-efficacy for communicating with families but generated a small number and range of strategies for dealing with the situation; could discriminate between the models’ effectiveness; and their reasons for choosing one model as best centered on their valuing of structuring or responsiveness and their conceptions of partnership. Content validity and reliability assessments of the research materials are described.
This study investigates the nature of pedagogical expertise by comparing the planning, teaching, and postlesson reflections of three student teachers (two secondary and one elementary) with those of the cooperating teachers with whom they were placed. Participants were observed teaching mathematics for 1 week of instruction and were interviewed prior to and following each lesson. Differences in the thinking and actions of these experts and novices were analyzed by perceiving teaching both as a complex cognitive skill and as improvisational performance Novices showed more time-consuming, less efficient planning, encountered problems when attempts to be responsive to students led them away from scripted lesson plans, and reported more varied, less selective postlesson reflections than experts. These differences were accounted for by the assumptions that novices' cognitive schemata are less elaborate, interconnected, and accessible than experts' and that their pedagogical reasoning skills are less well developed. We offer several recommendations for student teaching based on this analysis.
The move to leveraging technology to support the preparation of teachers has steadily increased over the past decade, with a sudden and rapid transition taking place in response to COVID-19 in 2020. In this chapter, we aim to explore the implications for preparing preservice mathematics teachers to use equitable teaching practices through virtual field experiences (VFEs) and highlight how equitable instruction may be affected by this shift to more technologically mediated methods. In particular, we take up the question of whether VFEs provide adequate opportunities to develop equitable mathematics teaching practices, drawing on exploratory work conducted in the Mursion™ environment.KeywordsTechnologySimulationVirtual field experiencesMath educationPreservice teachersEfficacyEquityEquitable instructional practicesCommunity of practiceVirtual students
In this conceptual chapter, we illustrate how a professional-development innovation called contingency cards can be used to elicit teachers’ subject-specific pedagogical decisions in an online learning environment. Contingency cards are representations of practice that can be used by facilitators to occasion conversations about realistic contingencies teachers might face in the context of their practice. Using the instructional situations of constructing a figure and doing proof drawn from the secondary geometry course in the USA (Herbst 2006), we illustrate how such contingency cards can be used to support teachers to interact about the subject-specific demands of teaching. Drawing on data from an online, practice-based professional development aimed to support teachers in learning to facilitate problem-based lessons, we demonstrate ways that geographically distant teachers can be supported in learning from both practice and one another through the use of contingency cards. More specifically, we illustrate various kinds of subject-specific decisions our teachers needed to grapple with when considering how to (1) elicit, (2) select, and (3) respond to students’ mathematical contributions. In particular, we share decisions that practicing teachers were asked to consider regarding how a whole class discussion about an open, novel task we call the Tangent Circle Problem could unfold. Finally, we share contingency card activities that could be used to elicit teachers' discussion about such decisions.KeywordsSubject-specific instructional practicesDiscursive movesPractical rationalitySecondary geometryProofConstructionNorms of instructionSecondary algebraContingency cardsProblem-based tasks
Deficit discourse and framing of multilingual students and their academic potential is a persistent challenge in mathematics education research and practice. It is a challenge for teachers of all experience levels to learn how to use asset-based language when working with students who are learning mathematics in a non-dominant language. This article reports on a study that explored how prospective teachers (PTs) position multilingual students and their primary language when learning mathematics. The participants are elementary and secondary PTs who were introduced to and had the opportunity to practice asset-based discourse to describe students’ mathematical strengths. Findings illustrate that after these learning opportunities PTs’ positioning of multilingual students shifted from using deficit and uncommitted discourse to asset-based. A closer look, however, showed no explicit mention of the value of the student’s primary language for learning mathematics. Only a few PTs did so implicitly. This study shows it is possible for PTs to shift deficit language, but that more explicit support is needed to help them communicate the value of language/linguistic diversity in the learning of mathematics.
Students’ contributions that are not yet mathematically complete, precise, or correct are crucial for learning through classroom discussions. We aim to support teacher candidates (TCs) to see students’ ideas as evidence of sensemaking and to respond in ways that keep student thinking central to the discussion. Although approximations of practice are a promising pedagogy for supporting TCs’ learning, more research is needed to link features of approximations and TCs’ opportunities to learn. Using data from a multiyear collaboration across two institutions, we investigate the use of coached rehearsals and how the design feature of “planted errors” contributed to multiple components of teacher learning in part through the impact on key features of approximations of practice. Our findings illustrate the available opportunities for TCs to develop skill with a complex practice. These results offer implications for the design of approximations of practice and approaches to studying their role in teacher development.
Approximations of practice provide opportunities for teacher candidates (TCs) to engage in the work of teaching in situations of reduced complexity. A problem of practice for teacher educators relates to how to represent student voice in approximations to engage TCs with interactive practices in meaningful ways. In this article, we share an analysis of our use of “planted errors” in coached rehearsals with secondary mathematics TCs focused on the practice of responding to errors in whole-class discussion. We highlight how different iterations of the planted errors affect the authenticity of how student voice was represented in the rehearsals and the resulting opportunities for TC learning. We offer design considerations for coached rehearsals and other approximations of practice.
OPEN ACCESS LINK: http://www.learntechlib.org/p/216903/
ABSTRACT: We describe how we further developed StoryCircles to support teacher learning online during COVID-19. In StoryCircles, a facilitator gathers teachers to collectively represent a lesson through iterative phases of scripting, visualizing, and arguing about alternatives. We share new innovations to the StoryCircles process that have helped us overcome common challenges, such as supporting teachers in anticipating elements of the lesson prior to implementation and intervening on teachers’ learning with instructional practices that may be novel for the group. Our work has implications for teacher educators across the world who are committed to supporting teachers to learn in, from, and for the complexities of actual classroom practice but are facing the very real challenges necessitated by times of extreme societal disruption—having to cancel field experiences or offer teacher education courses in blended contexts.
We offer a framework for analyzing teachers' practice of responding to students' mathematical contributions. This work builds on a framework developed by secondary mathematics teachers that we then augmented with one developed by linguists to create a modified framework that addresses pedagogic responding moves in detail. We show, through in-depth transcript analysis, along with commentary by a teacher and researcher, how the modifications support teachers and researchers to better understand how the dual focus on particular moves and their functional organization shapes discourse to support learning. The resulting framework is situated in the effort s toward building a common language of practice useful for enabling teachers to see and intervene on their own practice while supporting researchers to build theory about how teachers' instructional practices shape classroom discourse.
This study draws on a theoretical model for teacher assessment literacy to investigate the process of how secondary mathematics pre-service teachers (PSTs) might develop theirs through experiencing an approximation of practice. Sixty PSTs participated in designing and then refining assessment criteria for a rich open-ended quadratics task using real student example responses. Data analysis examined variations in the PSTs’ attention to quadratic features and mathematical language as assessment criteria for the task throughout the activity. This paper discusses evidence suggesting that the collaborative analysis of example student responses can make certain, but not all, assessment criteria salient to PSTs.
Teacher educators understand that the preparation of teachers needs to be rooted in the practice of teaching. This understanding, paired with the advancement in digital technologies capable of delivering practice-based teaching experiences, requires that those charged with preparing teachers consider how to best to position these technologies within their programs. This article positions virtual field experience platforms as on-ramps to professional practice and provides guidance for examining the features and capabilities of such platforms to inform their selection and use within teacher preparation programs.
Calls for a "practice-based" approach to teacher education have become common in scholarship on teacher education, and preservice-teaching (PST) mathematics programs are increasingly heeding this call. Practice-based teacher educa tion (PBTE) moves beyond standard approaches to teacher education in which PSTs learn about teaching in ways they are then expected to apply in practice and toward an ap proach that provides PSTs opportunities to gain experience in particular core practices in ways that approximate enact ment in the classroom. A growing body of research suggests that teachers' responses, including the questions they ask, can help students' develop content knowledge and proficiency in mathematics and science practices in the classroom. Howev er, despite evidence that PSTs can notice students' thinking in various activities in their preparation programs, it is not clear that they are sufficiently well-prepared to propose quality re sponses before entering the classroom. In this paper, we de scribe two different approaches that we have taken to provide support for quality teacher questioning in the LessonSketch environment. From our results, we develop a hypothesis that a pedagogical approach that primes novices to notice model questioning can support a stance of focusing on the substance of students' thinking and probing rather than guiding stu dents' thinking in their proposed questions. Calls for a "practice-based" approach to teacher education have become common in scholarship on teacher education, and preservice-teaching (PST) mathematics programs are increasingly heeding this call. Practice-based teacher education (PBTE) moves beyond standard approaches to teacher education in which PSTs learn about teaching in ways they are then expected to apply in practice and toward an approach that provides PSTs opportunities to gain experience in particular core practices in ways that approximate enact-ment in the classroom. A growing body of research suggests that teachers' responses, including the questions they ask, can help students' develop content knowledge and proficiency in mathematics and science practices in the classroom. However , despite evidence that PSTs can notice students' thinking in various activities in their preparation programs, it is not clear that they are sufficiently well-prepared to propose quality responses before entering the classroom. In this paper, we describe two different approaches that we have taken to provide support for quality teacher questioning in the LessonSketch environment. From our results, we develop a hypothesis that a pedagogical approach that primes novices to notice model questioning can support a stance of focusing on the substance of students' thinking and probing rather than guiding students' thinking in their proposed questions.
There is growing interest in the field of education for leveraging emerging digital technologies to support teachers' learning in online or blended settings. This paper builds on Clarke and Hollingsworth's (2002) Interconnected Model of Professional Growth by investigating an alternative instantiation of professional experimentation. In particular, we examine the StoryCircles model of professional development (Herbst & Milewski, 2018), which ushers teachers into a simulated type of professional experimentation to support teacher growth through the design and improvement of lessons using storyboards. In that context, we investigate how StoryCircles enable teachers to experiment professionally in a virtual space. Focusing on the experiences of two secondary mathematics teachers, we illustrate how the StoryCircles processes of scripting and argumentation were associated with teacher growth. We discuss how the Clarke and Hollingsworth (2002) Interconnected Model of Professional Growth can be useful for the design and study of simulated professional experimentation.
Partnering with families in ways that support children’s learning is a skill that is central to high-quality teaching. Parent–teacher conferences are a key site for teachers to learn how to leverage family resources in service of instruction. To know how to support the development of well-started beginners, teacher education (TE) programs must design substantive opportunities for interns to learn to bridge students’ work in schools on academic content with their home and community-based experiences; they must also be able to assess teacher candidates’ capacities to enact this learning. The authors report on their efforts to design a simulated parent–teacher conference, used here as a pre-assessment, as a means for assessing interns’ capacities to interact with and learn from diverse families to better support the needs of students. They refer to these skills as “partnering practices.”
In this chapter we discuss some of the affordances and constraints of using online teaching simulations to support reflection on specific pedagogical actions. We share data from a research project in which we implemented multiple iterations of a set of simulated teaching experiences in an elementary mathematics methods course. In each experience, preservice teachers contrasted the consequences of different pedagogical choices in response to a particular example of student thinking. We share how their evaluations of their choices shifted within experiences at certain points, and their criteria for “good” questions began to evolve. We end with implications for how simulations can promote critical reflection on teaching practice.
The notion of practice-based teacher education has recently been offered to mitigate the long-standing tension between theory and practice in teacher education. In practice-based teacher education, preservice teachers study a core set of teaching practices by examining decompositions of practice and learn these practices through varied and multiple opportunities to approximate teaching. This chapter explores one way in which preservice teachers might engage in approximating practice through homework assignments. In particular, this chapter reports on a study comparing representations of practice produced in two media—via scripting classroom dialogue or representing them as storyboards. Over a one-semester secondary mathematics methods course, 13 preservice teachers created 412 representations of practice—182 in a text medium and 230 in a storyboard medium—to approximate five different mathematics teaching practices. This chapter discusses the results of the comparison between the content of these different representations and implications for mathematics teacher education. In particular, the storyboards included significantly more attention to the students in a classroom and to the mathematical representations as compared to the text medium. Yet, it was in the text medium where preservice teachers appeared to consider more alternative teaching moves. We also discuss the ways in which teacher education might maximize attention to these aspects by utilizing these features (student involvement and mathematical representations) in the prompts used to situate the work of storyboarding a response.
This article explores the ways in which a teacher educator uses digital technology to create a virtual field placement as a way to blur the boundaries between a university methods course and teacher candidates' field placements. After describing his goals for the course, the teacher educator provides a description of three Lesson Sketch experiences his teacher candidates complete in this virtual field placement site and how these experiences create opportunities for teacher candidates to learn to teach mathematics. The design process and choices of these virtual field placement experiences are explored via interviews with the first author. Reflecting on these Lesson Sketch experiences, all of the authors then explore affordances of virtual and hybrid placements as resources for supplementing real placements and bridging theory/practice divides in teacher education.
In this article, we report on efforts to develop a set of Web-based teaching simulations within the Lesson Sketch platform to support shifts in how preservice elementary teachers (PSTs) enact and evaluate their questioning practices in response to specific examples of students' mathematical thinking. The simulations included storyboard depictions of classroom situations, along with prompts for the PSTs to first analyze mathematical thinking and then construct, select, and analyze the effects of possible teacher questions. Participants included 54 PSTs across 5 sections of a mathematics content/methods class. Data were analyzed to document how PSTs enacted and reflected on their questioning practices in the context of these Lesson Sketch simulations. In this article, we focus on 2 storyboard depictions of classroom situations and describe how each appeared to provide different opportunities for PSTs to revise their ideas about questioning.
Although teacher education is the formal means by which novices are prepared for teaching, they come having already had significant experience in schools. Preservice teachers have formed habits of “teaching” which influence their learning to teach. This article reports a study of the specific knowledge of and skills with teaching practice that novices bring to teacher education with respect to one teaching practice, eliciting student thinking in elementary mathematics, and describes the use of a standardized teaching simulation to learn about novices’ skills. The findings reveal details about preservice teachers’ skills and habits of practice at the point that they enter formal teacher preparation. Preservice teachers’ ways of carrying out this particular practice are categorized into three distinct categories: (a) skills that need to be learned, (b) skills that can be built on, and (c) approaches that need to be unlearned.
This study addresses how rehearsals of teaching could support teachers in developing mathematical knowledge for teaching (MKT). The analysis of 30 rehearsals, from elementary methods courses in three teacher education programs, attends to what novice teachers and teacher educators did in rehearsal that enlisted MKT. The findings elaborate the tasks that enlisted MKT in practice and the work of the teacher educator in supporting novice teachers’ learning of MKT inside the interactive work of teaching.
Effective mathematics teaching involves eliciting and interpreting student thinking, and then using students' current understandings as a basis for instruction. Research indicates these skills are not innate but can be acquired through structured experiences. In this article, we describe the development and implementation of an interview module aimed at supporting secondary preservice teachers' ability to elicit and use evidence of student thinking. Analysis of preservice teachers' noticing of student thinking across components of the interview module demonstrated positive benefits of the assignment. We share our design considerations and results, and offer potential adaptations to the module for other mathematics methods instructors interested in using the module to develop secondary preservice teachers' ability to notice student thinking.
Organizing teacher learning around the study of mathematical tasks and associated student work is a version of practice-based professional development that has been used effectively with preservice teachers and inservice elementary school teachers of mathematics. In this chapter, we examine the research evidence regarding the use of student work in teacher education and professional development, and we consider the potential impediments to using such an approach with inservice secondary school teachers, given many facets of their work and their preparation that appear to mitigate against the effectiveness of such an approach. To explore the feasibility of this approach with secondary school mathematics teachers, we consider in some detail the use of student work on one mathematics task, adapted from the PISA mathematics assessment, within a particular professional development initiative involving teachers in grades 7–11. Our examination of this experience indicates that although student work is not a self-enacting tool for teacher learning, professional developers can engage secondary school mathematics teachers with student work in ways that afford powerful and potentially transformative learning opportunities.
This is the first comprehensive account of the Appraisal Framework. The underlying linguistic theory is explained and justified, and the application of this flexible tool, which has been applied to a wide variety of text and discourse analysis issues, is demonstrated throughout by sample text analyses from a range of registers, genres and fields.
In this article we examine an analytical framework generated by secondary mathematics teachers for tracking changes to their own instructional practices across time. We describe the journey of this group of teachers through professional development focused on improving instructional practice. In the midst of that experience, teachers struggled to find an analytical tool to examine one another's practices of responding to students' mathematical ideas and ultimately overcame this problem by considering the practitioner literature and their own experiences. We also describe how we adapted the framework to investigate its use for detecting shifts in teachers' practices, sharing findings obtained from its use. Lastly, we argue for this type of collaborative work with teachers as a means to develop common language for instructional practice.
Teaching is complex and relational work that involves teacher???s interactions with individual or multiple students around the subject matter. It has been argued that observation experiences (e.g. field placement or watching video clips) are not sufficient to help prospective teachers to develop knowledge of teaching. This study aims to identify, examine, and illustrate the ways in which comics-based representations of teaching facilitate prospective teachers??? learning to teach. Specifically, the author explored how the use of a technology supported lesson-sketching tool, Depict, enabled prospective secondary mathematics teachers to attend to mathematical interactions between teacher and students in instruction when anticipating the development of a lesson.
Drawing resources from Systemic Functional Linguistics, the author examined the ways in which anticipations of classroom interaction about a planned lesson differ when the anticipation was done using the Depict tool as compared with talking through the written lesson plan. Using case study methodology, the study investigated the aspects of the teaching work prospective teachers attended to when engaged in depicting a lesson, and observed the ways in which prospective teachers employed the graphic resources to support their lesson depiction.
The results reveal that prospective teachers using Depict tool to create comics-based lesson slideshows immersed themselves in classroom settings and demonstrated their capacity to incorporate detailed teacher instructional actions, student reactions and mathematical tasks in their lessons. The prospective teachers unpacked their planned discrete class activities and attended to the relational nature among teacher, students and mathematics in instruction.
The study indicates that the anticipation of a lesson, through creation of comics-based lesson depiction, could be a learning opportunity that approximates the interactive nature of teaching practice. The study suggests that comics-based representations of teaching can be seen as semiotic resources that mediate prospective teachers??? generation of teacher-student moment-to-moment class interactions, and facilitate their attention to instructional issues they have not previously been aware of. The study also implies that in order to engage prospective teachers in learning to do the work of teaching, teacher educators should consider directing prospective teachers??? attention to issues of temporality, multimodality and multivocality in instruction.
This chapter describes how the authors have utilized digital graphics and Web 2.0 technologies to design an information technology environment, LessonSketch. In LessonSketch teachers can learn about mathematical practice in instruction through the transaction of representations of practice. The authors describe the main features of LessonSketch, its collection of lessons, and its authoring tools, and illustrate what teacher educators have done with them.
In this book an experienced classroom teacher and noted researcher on teaching takes us into her fifth grade math class through the course of a year. Magdalene Lampert shows how classroom dynamics--the complex relationship of teacher, student, and content--are critical in the process of bringing each student to a deeper understanding of mathematics, or any other subject. She offers valuable insights into students and teaching for all who are concerned about improving the learning that happens in the classroom. Lampert considers the teacher's and students' work from many different angles, in views large and small. She analyzes her own practice in a particular classroom, student by student and moment by moment. She also investigates the particular kind of teaching that aims at engaging elementary school students in learning fundamentally important ideas and skills by working on problems. Finally, she looks at the common problems of teaching that occur regardless of the individuals, subject matter, or kinds of practice involved. Lampert arrives at an original model of teaching practice that casts new light on the complexity in teachers' work and on the ways teachers can successfully deal with teaching problems.
Make-believe play and self-regulation Early childhood is a vital time for laying the foundations of self-regulation, a set of capacities (often termed executive function in contemporary research) that enable children to engage in self-initiated, purposeful behavior in relatively challenging situations. By age five–six, the well-regulated child can wait for a turn, resist the temptation to grab a desired object from another child, spontaneously share a toy with a classmate, follow classroom routines (clean up independently after a play period), and willingly help an adult or peer with a task (Bronson, 2001; Blair, 2002; Berk, Mann and Ogan, 2006). Furthermore, when faced with a demanding activity – writing one's name, riding a bike, working a complicated puzzle, or controlling negative emotion – self-regulated children exert intentional effort, even when they would prefer to do something else. Young children who exhibit the capacities just mentioned have made substantial progress in gaining control ...
Although recent years have seen an increase in professional learning communities, use of video and lesson study groups, most teachers still work and learn in isolation. What they know is personal and remains private; little opportunity exists for most teachers to develop shared knowledge or language. The scale of the teaching force, and the rapid turnover of new teachers, makes this lack of shared knowledge an acute problem. This paper explores the potential of records of practice for developing collective professional knowledge about teaching and learning. We define and illustrate what we mean by records of practice and elaborate the meaning of the term collective professional knowledge. Three concrete examples of records of practice, originating in different contexts and times, are presented and their special features are analyzed and discussed. The paper clarifies that records are not in themselves professional knowledge, but constitute a valuable and unique resource for the study of practice and the generation of knowledge originating in and useful for practice. To illustrate varied approaches for studying practice, possible uses of records of practice for the generation of collective professional knowledge are described and analyzed. The paper concludes with questions about records of practice and their uses.
In recent years, a small but growing strand of research has investigated ways of focusing teachers’ professional education on “core” or “high leverage” practices of teaching. These efforts are easily conflated with other initiatives to develop “practice-focused” teacher education, raising questions about what these terms even mean. This article investigates what can be learned by comparing and contrasting teacher education focused on core practices with other approaches that might also be called “practice-based,” including those dating back to the 19th century. It focuses on three important periods in the history of teacher education: the heyday of the normal schools in the late 1800s, the period of scientific efficiency in the 1920s and 1930s, and the era of competency-based teacher education in the 1960s and 1970s.
[Link to the full version: http://www.citejournal.org/vol14/iss4/mathematics/article1.cfm] This paper builds on Grossman's notion of approximations of practice as scaled-down opportunities for preservice teachers to learn to teach by doing. The authors propose the use of media rich, collaborative web-authoring tools for preservice teachers to create, complete, or edit scenarios in which they practice particular activities of teaching, such as explaining a mathematics concept or reviewing students' work. The ways these environments can be used to fit the notion of approximations of practice are described, along with the authors’ experience using the web-based software Depict (in the LessonSketch platform) in the teaching of secondary mathematics methods. This use of multimedia scenarios combines the advantages of visual and video-based approaches to the study of practice with those approaches that ask the preservice teachers to create scenarios (e.g., lesson plays). The value of integrating this storyboarding web software in a larger environment where scenarios can be created collaboratively, annotated, and commented on in forums is presented.
This article describes how the writing of math letters can offer students a rich and genuine context for learning
mathematics. It uses examples from interactions between an adult and a young student to illustrate three specific mathematics communicative practices students engage in as they exchange letters -constructing and posing math problems, explaining and justifying, and reading and analyzing others’ explanations. It discusses how these practices are important and lead students to engage in mathematical inquiry.
A PREVIEW of this book can be found on Google Books at:
https://www.google.com/books/edition/Beyond_Formulas_in_Mathematics_and_Teach/E-GbNh2vi8cC?hl=en&gbpv=1&dq=daniel+chazan+beyond+formulas&printsec=frontcover
Background/Context
This study investigates how people are prepared for professional practice in the clergy, teaching, and clinical psychology. The work is located within research on professional education, and research on the teaching and learning of practice.
Purpose/Objective/Research Question/Focus of Study
The purpose of the study is to develop a framework to describe and analyze the teaching of practice in professional education programs, specifically preparation for relational practices.
Setting
The research took place in eight professional education programs located in seminaries, schools of professional psychology, and universities across the country.
Population/Participants/Subjects
Our research participants include faculty members, students, and administrators at each of these eight programs.
Research Design
This research is a comparative case study of professional education across three different professions—the clergy, clinical psychology, and teaching. Our data include qualitative case studies of eight preparation programs: two teacher education programs, three seminaries, and three clinical psychology programs.
Data Collection and Analysis
For each institution, we conducted site visits that included interviews with administrators, faculty, and staff; observations of multiple classes and field-work; and focus groups with students who were either at the midpoint or at the end of their programs.
Conclusions/Recommendations
We have identified three key concepts for understanding the pedagogies of practice in professional education: representations, decomposition, and approximations of practice. Representations of practice comprise the different ways that practice is represented in professional education and what these various representations make visible to novices. Decomposition of practice involves breaking down practice into its constituent parts for the purposes of teaching and learning. Approximations of practice refer to opportunities to engage in practices that are more or less proximal to the practices of a profession. In this article, we define and provide examples of the representation, decomposition, and approximation of practice from our study of professional education in the clergy, clinical psychology, and teaching. We conclude that, in the program we studied, prospective teachers have fewer opportunities to engage in approximations that focus on contingent, interactive practice than do novices in the other two professions we studied.
Current work to identify the core teaching practices that should be included in teacher education curriculum is a part of a long-standing tradition of reform in American teacher education. This article situates the proposals of Hiebert and Morris and the contemporary work to which it is linked within this historical tradition and identifies several issues that need to be addressed by this current work. These include the task of developing a system that unlike performance-based systems in the past is evidence-based, manageable, and sustainable, and that does not ignore important aspects of good teaching.
In talk about teacher preparation and professional development, we often hear the word practice associated with what, how, or when the learning of teaching is supposed to happen. In this article, four different conceptions of practice are investigated, and their implications for how learning teaching might be organized are explored. Rather than a comprehensive review of the literature, what is presented here is a set of ideas that draw on both past and present efforts at reform. The purpose of this essay is to provoke clarification of what we mean when we talk about practice in relation to learning teaching. The author draws on her own research on the work of teaching from the perspective of practice to represent the nature of the work and to speculate from various perspectives on how that work might be learned.