Using Student Video Cases to Assess Pre-service Elementary Teachers’ Engineering Teaching Responsiveness

Our study addresses the need for new approaches to prepare novice elementary teachers to teach both science and engineering, and for new tools to measure how well those approaches are working. This in particular would inform the teacher educators of the extent to which novice teachers are developing expertise in facilitating their students’ engineering design work. One important dimension to measure is novice teachers’ abilities to notice the substance of student thinking and to respond in productive ways. This teacher noticing is particularly important in science and engineering education, where students’ initial, idiosyncratic ideas and practices influence the likelihood that particular instructional strategies will help them learn. This paper describes evidence of validity and reliability for the Video Case Diagnosis (VCD) task, a new instrument for measuring pre-service elementary teachers’ engineering teaching responsiveness. To complete the VCD, participants view a 6-min video episode of children solving an engineering design problem, describe in writing what they notice about the students’ science ideas and engineering practices, and propose how a teacher could productively respond to the students. The rubric for scoring VCD responses allowed two independent scorers to achieve inter-rater reliability. Content analysis of the video episode, systematic review of literature on science and engineering practices, and solicitation of external expert educator responses establish content validity for VCD. Field test results with three different participant groups who have different levels of engineering education experience offer evidence of construct validity.     

Professional noticing of coordinated mathematical thinking

Over the past three decades, research and policy in many geographic regions has promoted a shift from direct, lecture-oriented mathematics instruction to inquiry-based, dialogic forms of instruction. While theory and research support dialogic instructional approaches, some have noted that the complexities of dialogic teaching make it difficult for teachers to implement. One mechanism by which teachers can improve their decision-making practices in dialogic classrooms is learning to notice (i.e. becoming aware of learners’ processes). While research has contributed frameworks for understanding how teachers notice individual learners’ mathematical thinking, there is little conceptualization regarding how teachers notice group processes in mathematics classrooms, which is integral to dialogic instruction. We offer a noticing framework termed professional noticing of coordinated mathematical thinking that describes how teachers notice group activity in mathematics classrooms. Professional noticing of coordinated mathematical thinking is conceptualized as a bi-dimensional process: noticing groups’ mathematical activity and noticing groups’ coordinated activity. Teachers must become aware of how groups approach the mathematical and collaborative nature of a task, since both of these aspects inform whether learners develop opportunities to learn in groups. The framework describes noticing practices integral to dialogic instruction and promotes inquiry for future research related to teaching moves in dialogic classrooms.     

Mathematics teacher educator noticing: examining interpretations and evidence of students’ thinking

Noticing is a skill that is not overtly observable yet is consequential to effective mathematics instruction. Researchers have found that prospective and practicing teachers can learn to notice, but little focus has been given to those who teach teachers to notice. The purpose of the study was to characterize mathematics teacher educators’ noticing and their ability to interpret students’ thinking and connect interpretations to evidence. Participants in the study included 16 mathematics teacher educators who took part in a course designed to support noticing. Results indicate the mathematics teacher educators noticed at varying degrees and improved their noticing and incidence of connections between interpretations and evidence. Findings indicated that 19% of participants had no shift in their noticing because they were at the highest level of noticing to begin with (Robust with Strong Evidence), which was considered advanced noticing. Twenty-five percent of the participants did not shift in their noticing at all and remained at Limited, which is considered an intermediate level of noticing. The remaining 56% of the participants improved their noticing. The results of the study reveal that at the end of the course a majority of the participants were able to connect interpretations with evidence. These findings are important because they describe mathematics teacher educators’ interpretations and evidence as they notice.

     

Characterizing pivotal teaching moments in beginning mathematics teachers’ practice

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

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

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

Multiple representations as sites for teacher reflection about mathematics learning

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.     

Assessing students' understanding of inquiry: What do prospective science teachers notice?

The theoretical construct of teacher noticing has allowed mathematics teacher educators to examine teacher thinking and practice by looking at the range of activities that teachers notice in the classroom. Guided by this approach to the study of teacher thinking, the central goal of this exploratory study was to identify what prospective science teachers notice when evaluating evidence of student understanding in another teacher's inquiry‐based unit. Our results are based on the qualitative analysis of 43 prospective teachers' evaluations of assessment evidence presented to them in the form of a video case and associated written artifacts. Analysis of our data revealed two major categories of elements, Task‐General and Task‐Specific, noticed by our study participants. Task‐General elements included attention to learning objectives, independent student work, and presentation issues and they often served to guide or qualify the specific inquiry skills that were evaluated. Task‐Specific elements included the noticing of students' abilities to perform different components of an investigation. In general, study participants paid attention to important general and specific aspects of student work in the context of inquiry. However, they showed preferential attention to those process skills associated with designing an investigation versus those practices related to the analysis of data and generation of conclusions. Additionally, their interpretations of assessment outcomes were largely focused on the demonstration of general science process skills; much less attention was paid to the analysis of the epistemological validity or scientific plausibility of students' ideas. Our results provide insights into the design of meaningful learning experiences for prospective teachers that elicit, challenge, and enrich their conceptions of student understanding in the context of inquiry. © 2013 Wiley Periodicals, Inc. J Res Sci Teach 50:189–208, 2013     

Culture and ideology in mathematics teacher noticing

This paper responds to the burgeoning literature on mathematics teacher noticing, arguing that its cognitive orientation misses the cultural and ideological dimensions of what and how teachers notice. The author highlights Goodwin’s concept of professional vision as a way of bringing analyses of culture and power into studies of teacher noticing. The case of a high school algebra teacher who learned to notice the mathematical strengths of students from marginalized groups is used to illustrate how this might be done. The teacher’s noticing involved not only cognitive processes like attending to, interpreting, and deciding how to respond to students’ thinking, but also managing dominant ideologies that position students—especially students from non-dominant communities—as mathematically deficient rather than as sense-makers whose ideas should form the basis for further learning. The paper advances the field’s capacity for understanding the challenges that teachers face as they attempt to notice in ways that are ambitious as well as equitable.     

Characterizing a highly accomplished teacher’s noticing of third-grade students’ mathematical thinking

This study investigated a highly accomplished third-grade teacher’s noticing of students’ mathematical thinking as she taught multiplication and division. Through an innovative method, which allowed for documenting in-the-moment teacher noticing, the author was able to explore teacher noticing and reflective practices in the context of classroom teaching as opposed to professional development environments. Noticing was conceptualized as both attending to different elements of classroom instruction and making sense of classroom events. The teacher paid most attention to student thinking and was able to offer a variety of rich interpretations of student thinking which were presented in an emergent framework. The results also indicated how the teacher’s noticing might influence her instructional decisions. Implications for both research methods in studying noticing and teacher learning and practices are discussed.     

Understanding Teacher Educators and Their Pre-Service Teachers through Multi-Media Case Studies of Practice

The challenges facing those who seek to prepare mathematics teachers are well established in the literature. Most of the research to date has focused on the perceptions and understandings of pre-service teachers, but not on the perceptions and understandings of teacher educators. In this study, we explore how four teacher educators understand their pre-service secondary teachers as the pre-service teachers attempt to make sense of teaching through the investigation of a multimedia case study of practice. We found that the teacher educators adopted two different implementation strategies: one strategy tended to be open-ended and exploratory; the other was more focused on the teacher educators' goals of anticipating student understanding and developing mathematical content knowledge for teaching. We also found that, in using the case study, teacher educators elicited pre-service teachers' thinking about the complexities of the teacher's role in small group work, about the value of explicitly revealing the teacher's reflections on the lessons, about the role of planning and preparation, and about the limits of pre-service teachers' abilities to understand and appreciate students' thinking and to extend lesson ideas. Both teacher educators and their pre-service teachers gained perspectives on the role of a teacher's mathematical content knowledge. These results imply that multimedia case studies of practice can serve as vehicles for revealing the knowledge and practice of teacher educators, as they engage in supporting the professional development of pre-service teachers.     

Building on student mathematical thinking in whole-class discourse: exploring teachers’ in-the-moment decision-making,interpretation, and underlying conceptions

Often, mathematics teachers do not incorporate whole-class discourse of students’ various ideas and solution methods into their teaching practice. Particularly complex is the in-the-moment decision-making that is necessary to build on students’ thinking and develop their collective construction of mathematics. This study explores the decision-making patterns of five experienced Dutch mathematics teachers during their novice attempts at orchestrating whole-class discourse concerning students’ various solution methods. Our goal has been to unpack the complexity of their in-the-moment decision-making during whole-class discourse through lesson observations and stimulated recall interviews. We investigated teacher decision-making adopting a model that combines two perspectives, namely (1) we explored student-teacher interaction with regard to building on student thinking and (2) we explored how the teachers based decisions during such interaction upon their own personal conceptions and interpretation of student thinking. During these novice attempts at orchestrating whole-class discourse, the teachers created many situations for students to articulate their thinking. We found that at certain instances, teachers’ in-the-moment decision-making resulted in opportunities to build on student thinking that were not completely seized. During such instances, the teachers’ decision-making was shaped by the teachers’ own conceptions of the relevant mathematics and by teacher conceptions that centered around student understanding and mathematical goals. Our findings suggest that teachers might be supported in their novice attempts at whole-class discourse by explicit discussion of the mathematics and of their conceptions with regard to student understanding and mathematical goals.

     

Supporting teacher responsiveness in assessment for learning through disciplined noticing

Abstract

Assessment for learning (AfL) practices are commonly recommended as effective classroom strategies for providing teachers with information about student understanding. For teachers, the substantive potential of these AfL practices to inform student learning actions depends on what teachers notice and select as a focus and how they interpret and act on the information they have. This paper suggests research on teacher professional noticing has something to offer in understanding how teacher AfL attention and actions are framed in the moment. It explains noticing as a responsive act that invites action that is an inclusive, dynamic and purposeful response to evidence of student ideas. Noticing is what enables a teacher to act on the fly because it informs and underpins possible actions. Three frames of noticing are introduced to guide noticing and action – curriculum connoisseurship, cultural and community connectedness, and collaborative ways of working. Through consideration of classroom events using these frames, teachers become better prepared to choose to respond in the moment creatively. The three frames provide teachers and researchers with a conceptual language to articulate and sharpen their AfL practices as part of a flexible and rigorous responsiveness to their students’ learning.     

Audible conversational affordances and constraints of verbalizing professional noticing during prospective teacher lesson study

Lesson Study, which assumes multiple variations, is a collaborative method that provides an opportunity for teachers to notice professionally. We analyzed how audible conversational components afforded and constrained the verbalization of professional noticing of a lesson study team. The team, comprised of six prospective teachers, a classroom teacher, and a university facilitator, participated in weekly lesson study cycles in a teacher education program. Results indicate that a structured lesson study process afforded professional noticing as participants attended to and analyzed students’ thinking. However, noticing was constrained by shifts in the conversation content and lengthy segments of individuals exerting expertise. Findings indicate that the facilitator played a crucial role in increasing the incidence of professional noticing and also contributed to the decline of noticing. Implications for teacher education programs are included.     

Professional primary school teacher identity development: a pursuit in line with an unexpressed image

The results presented in this article are taken from a case study of novice primary school mathematics teachers’ professional identity development from the perspective of the teachers themselves. The empirical material was collected through self-recordings, observations and interviews. The results show how the professional identity development of these novice teachers becomes a pursuit in line with their image of a primary school teacher. To develop a sense of themselves as primary school teachers they need to establish their own criteria - individual (including graduation and personal knowledge) and social (the ability to work in one school, have colleagues and have a class of their own for which they do the planning and teaching). These criteria are shown to be both a precondition for and a part of professional identity development. The novice teachers’ image of what it means to be a primary school teacher directs their actions and becomes the goal of their professional identity development. Because of its high impact, student and novice teachers’ image of primary school teachers ought to be made visible in both teacher education and teacher induction.     

Opportunities to notice: Chinese prospective teachers noticing students’ ideas in a distance formula lesson

This paper investigates the noticing of six Chinese mathematics prospective teachers (PSTs) when looking at a procedural error and responding to three specific tasks related to that error. Using video clips of one student’s procedural error consisting of exchanging the order of coordinates when applying the distance formula, some variation was found in how PSTs attended to, interpreted, and responded to this error. A more important finding is represented by the inconsistent responses that individual PSTs provided to the three related tasks. This finding suggests that, to some extent, prior learning experience, beliefs, and orientations inform what PSTs notice. But the finding also suggests the centrality of selecting tasks that provide accurate representations of PSTs’ emerging professional noticing. Implications for teacher educators are discussed.     

Prospective elementary school teachers’ professional noticing of children’s early numeracy

The goal of this study is to develop the professional noticing abilities of prospective elementary school teachers in the context of the Stages of Early Arithmetic Learning. In their mathematics methods course, ninety-four prospective elementary school teachers from three institutions participated in a researcher-developed five-session module that progressively nests the three interrelated components of professional noticing—attending, interpreting, and deciding. The module embeds video excerpts of diagnostic interviews of children doing mathematics (representations of practice) to prepare the prospective teachers for similar work. The module culminates with prospective teachers implementing similar diagnostic interviews (approximations of practice) to gain experience in the three component skills of professional noticing. A pre- and post-assessment was administered to measure prospective teachers’ change in the three components. A Wilcoxon signed ranks test was conducted and found the prospective elementary school teachers demonstrated significant growth in all three components. Selected prospective elementary school teacher responses on the pre- and post-assessment are provided to illustrate sample growth in the prospective teachers’ abilities to professionally notice. These results, the first in an ongoing study, indicate the potential that prospective teachers can develop professional noticing skills through this module. Continued data collection and analysis from the ongoing study by these authors and future, longer-term emphasis on professional noticing for prospective teachers should be studied.     

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

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

The development of a mathematics teacher’s professional identity during her first year teaching

This study is focused on a period that poses several challenges for the development of mathematics teachers’ professional identities and agency: their first year of teaching in schools. During this period, beginning mathematics teachers confront tensions and contradictions among the principles, ideals, and experiences encountered during pre-service education and the demands, and restrictions of their teaching practice in schools. This article approaches this topic by developing an interpretative case study centered on one novice mathematics teacher, Sol. The aim is to describe and understand the development of Sol’s professional identity and agency during her first teaching year. Considering identity development as a diachronic phenomenon, we carry out a narrative analysis of the research data. The findings show that Sol developed her professional identity and agency through a process that gathered together the teaching practices possible inside her school, the positions she could negotiate as a newcomer inside the institution, and the cultural practices and discourses embodied during pre-service education. The results bring to the forefront the profound and tense interactions between the intimate and personal terrain of mathematics teachers and the social and cultural world of the schools where they work. Sol’s case also contributes to understanding the role that a robust pre-service education can play in the development of beginning teachers’ professional identities and in the possibility they could become agents of transformation in their schools.     

What Preservice Teachers and Knowledgeable Others Professionally Notice During Lesson Study

The purpose of this study was to understand what preservice teachers and knowledgeable others professionally notice as they engaged in repeated cycles of a modified version of lesson study, as a component of a field experience in a teacher education program. The study also centered on comparing the professional noticing practices of preservice teachers with other lesson study participants, including classroom teachers and university facilitators. Data analyzed included videos of weekly lesson study analysis meetings for seven weeks for each of four teams. Each team included six preservice teachers, a classroom teacher, and a university facilitator. Findings indicate that preservice teachers primarily noticed elements about the classroom environment and teacher pedagogy, but included instances of noticing centered on students' mathematical thinking. In contrast, classroom teachers and university facilitators, as knowledgeable others, typically noticed general events and were less focused on students' mathematical thinking. Analysis of noticing trends over the seven weeks indicated that noticing levels remained steady initially, dropped in the fourth and fifth week, and resumed original status in the final weeks. Results that the preservice teachers' noticing comments were at higher levels than the knowledgeable others are contrary to other research studies and indicate that incorporating lesson study with appropriate scaffolds into a field experience for preservice teachers may be a viable option for encouraging noticing of students' mathematical thinking.