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Developing prospective elementary teachers’ abilities to identify evidence of student mathematical achievement
Abstract
This study investigated the effects of a classroom intervention on prospective elementary teachers’ ability to evaluate evidence
of student achievement of mathematical learning goals. The intervention was informed by a framework for teacher education
which aims to provide prospective teachers (PTs) with the skills needed to systematically learn from their own teaching practice.
Prospective teachers (N=160) participated in an intervention aimed at addressing their misconceptions about evidence of student learning. Results
revealed that after the intervention, PTs were less likely to consider teacher behaviors to be evidence of student learning
and more likely to discount student responses that were irrelevant to a specified learning goal. However, PTs were still likely
to take procedural fluency as evidence of conceptual understanding and may have become overly skeptical of student understanding.
Implications of the study suggest new ways of developing prospective teachers’ ability to systematically study and improve
their teaching.
KeywordsProspective teachers–Evaluating teaching–Analyzing teaching–Analyzing student work–Learning to teach–Teacher preparation
... Previous research has shown that PTs can learn to evaluate evidence within targeted interventions (see, e.g. Santagata and Angelici 2010;Spitzer et al. 2011). This research indicates PTs who have been taught to evaluate evidence show positive change in other aspects of their teaching (Santagata and Yeh 2013). ...
... In addition, it seems that the longest interventions (those which comprise an entire semester-long university course) are generally the most effective, whereas after shorter interventions, PTs often still show significant gaps in their abilities to analyze evidence (see, e.g. Bartell et al. 2013;Spitzer et al. 2011). However, the practical considerations of teacher education indicate a need for short yet more effective interventions. ...
... Yeh and Santagata (2015) found that, prior to intervention, PTs often failed to construct a justified claim about student learning, instead writing general descriptive statements, often failing to support them with evidence. And Spitzer et al. (2011) suggest that elementary PTs' claims are often poorly aligned with the evidence, with PTs basing their claims on nonrevealing lesson features (such as students nodding or a teacher explanation) and irrelevant mathematics. PTs also made claims about children's conceptual understanding based only on evidence of their procedural fluency, a result also found by Bartell et al. (2013). ...
This study examined elementary and secondary prospective teachers’ (PTs’) abilities to analyze a classroom lesson in order to make claims about student thinking around specific mathematical learning goals based on relevant and revealing evidence. Previous research suggests PTs have some skills in analyzing evidence but apply them inconsistently. Our goal was to describe in more detail the strengths and weaknesses in PTs’ ability to analyze evidence of student thinking. Results indicate that PTs can make some appropriate claims about student learning in a lesson transcript, but more often make overly broad and general claims. PTs were able to support their claims with specific student work but often used poorly aligned evidence. PTs also often explicitly recognized the shortcomings of evidence from the lesson transcript, but then relied on that evidence to make claims about student thinking. Finally, PTs’ background, such as number of teacher education courses completed, does not appear to strongly influence their ability to make claims and support them with evidence, though secondary PTs were more likely to recognize the limitations of evidence than elementary PTs. These results have implications for teacher educators, pointing to the importance of designing interventions to help PTs look beyond the most visible and salient features of a lesson when analyzing student thinking.
... In several studies, the theoretical tool is a concept relating to student thinking (Table 3), which is at the heart of reform-oriented teaching. These studies aim to extend prospective teachers' interpretation of student work beyond the right/ wrong dichotomy, for example by looking for evidence for conceptual understanding (Bartell et al., 2013) and connecting learning goals to evidence from student work (Spitzer et al., 2011). Here participants analyse student work, without attempting to intervene (with the exception of Simpson and Haltiwanger (2017)). ...
... The results pinpoint two major challenges. First, prospective teachers' difficulties in making judgements based on evidence leads them to equate correct answers with conceptual understanding (Bartell et al., 2013;Spitzer et al., 2011). A decomposition of practice in which prospective teachers analysed student solutions of three types (with evidence of conceptual understanding, purely procedural, and procedural with surface features of conceptual understanding) was partially successful in addressing the problem, as prospective teachers made progress in recognising purely procedural solutions; however, they still let themselves be confused by surface features (e.g. a drawing) accompanying a purely procedural solution (Bartell et al., 2013). ...
... A decomposition of practice in which prospective teachers analysed student solutions of three types (with evidence of conceptual understanding, purely procedural, and procedural with surface features of conceptual understanding) was partially successful in addressing the problem, as prospective teachers made progress in recognising purely procedural solutions; however, they still let themselves be confused by surface features (e.g. a drawing) accompanying a purely procedural solution (Bartell et al., 2013). Secondly, prospective teachers tend interpret lack of evidence of conceptual understanding as lack of conceptual understanding (Bartell et al., 2013;Spitzer et al., 2011;Walkoe, 2015). Considering the consequences of this problem, Walkoe (2015) expresses concern that, if procedural solutions are seen as lack of conceptual understanding, teachers will put less effort in analysing them. ...
The first edition of this Handbook put forward a question on the role of theory as a tool in mathematics teacher education. In this chapter, in order to investigate the questions ‘why’, ‘what’, ‘how’, and ‘to what end’ theory is included in mathematics teacher education, we draw on literature published since that time. At system level, increased length of field experience at the expense of university-based courses seem to be the trend, raising a debate on what can be achieved and what gets lost through this approach in light of the envisaged role of research. Exploring these issues at programme level, the mathematics-specific affordances and limitations on connecting theory and practice emerge. Finally, studies on teaching experiments – learning experiences using explicitly theories as tools – reveal new layers of complexities, with programme-level issues resurfacing alongside new ones relating to the design and to the individual’s manner of engaging with the experience. Keeping in mind the limitations, the overall picture of the field is promising; much can be gained in mathematics teacher education through using theoretical tools.
... Only a few have looked at PSTs' noticing using student artifacts (e.g. Dick 2017;Fernandez et al. 2013), despite the fact that PSTs need assistance learning how to analyze student work (Bartell et al. 2013;Crespo 2000;Spitzer et al. 2011). ...
... Simply looking at student work does not ensure that PST learning will occur (Ball and Cohen 1999;Bartell et al. 2013). When PSTs begin to analyze student work, they tend to (a) describe the work as right or wrong without specifically attending to the mathematics present (Crespo 2000;Goldsmith and Seago 2011;Shaughnessy and Boerst this volume;Spitzer et al. 2011); (b) draw on personal experience (often their own traditional instruction) and/or knowledge to fill in gaps in children's work (Bartell et al. 2013;Goldsmith and Seago 2011;Spitzer et al. 2011); not know what to do with non-standard or surprising solutions (Bartell et al. 2013;Crespo 2000;Goldsmith and Seago 2011). Since PSTs must be taught how to analyze student work, the researchers contend that the professional noticing framework provides a lens for the PSTs to focus their analysis. ...
... Simply looking at student work does not ensure that PST learning will occur (Ball and Cohen 1999;Bartell et al. 2013). When PSTs begin to analyze student work, they tend to (a) describe the work as right or wrong without specifically attending to the mathematics present (Crespo 2000;Goldsmith and Seago 2011;Shaughnessy and Boerst this volume;Spitzer et al. 2011); (b) draw on personal experience (often their own traditional instruction) and/or knowledge to fill in gaps in children's work (Bartell et al. 2013;Goldsmith and Seago 2011;Spitzer et al. 2011); not know what to do with non-standard or surprising solutions (Bartell et al. 2013;Crespo 2000;Goldsmith and Seago 2011). Since PSTs must be taught how to analyze student work, the researchers contend that the professional noticing framework provides a lens for the PSTs to focus their analysis. ...
Teaching is a complex endeavor which requires teachers to make decisions based on children’s thinking. Pre-service teachers (PSTs) training to be future teachers need experiences to increase their ability to notice, understand, and analyze children’s responses/work to make sense of children’s mathematical thinking. This chapter examines elementary PSTs’ skills to recognize, identify, and make instructional decisions in their teacher preparation programs when provided with opportunities to engage in noticing practices in their mathematics methods courses. Qualitative analysis using open coding was used to analyze data sets collected from three different universities across the U.S. to find commonalities and themes. Results of the study provide insight into the effectiveness of the instructional activities and PSTs’ next-step instructional decisions.
... Pre-Service Teachers' Learning 5 of procedural steps as evidence of conceptual understanding. ( Bartell, Webel, Bowen, & Dyson;2012;Spitzer et al., 2010). For example, Morris (2006) found that PSTs were willing to accept a student's correct use of the area formula of a triangle as evidence of conceptual understanding rather than evidence of procedural knowledge. ...
... The reviewed studies provide some evidence that the skills for systematic analysis of practice can be taught. However, the majority of the existing literature focuses on the development of PSTs' ability to analyze student thinking ( Bartell, et al., 2012;Morris, 2006;Spitzer et al., 2010), while only a few focus on the ability to generate hypotheses about the effectiveness of teaching. This skill, we would argue, is necessary if teachers want to learn from their current practice to inform future practices. ...
... However, these responses were inconsistent and scarce. The majority of pretest responses displayed misconceptions consistent with findings from prior studies ( Davis, 2006;Morris, 2006;Spitzer et al., 2010) and provides additional evidence on this under-studied skill. Most PSTs displayed responses consisting of a clip summary or made claims using teacher-based evidence. ...
This study examines the development of a specific sub-skill for studying and improving teaching—the generation of hypotheses about the effects of teaching on student learning. Two groups of elementary preservice teachers (PSTs) were compared: one group that attended a typical mathematics-methods course and one that attended a course integrating analysis skills for learning from teaching. Data consist of PSTs’ comments on video clips of mathematics instruction administered before and after course completion. Findings reveal that PSTs at the beginning of the program struggled to generate hypotheses with relevant evidence, often equating teacher behavior or student correct answers as evidence of student understanding. After course participation, PSTs who attended the course with integrated analysis skills significantly improved in their ability to generate hypotheses based on student evidence whereas their counterparts continued to display difficulties. Implications for teacher education and future research are considered.
... PSTs often assess teaching effectiveness based on teacher strategies rather than on student responses (Hiebert & Stigler, 2000;Morris, 2006;Santagata, Zannoni, & Stigler, 2007). Even when the attention centers on students, over-attribution of conceptual understanding often occurs as PSTs equate correct answers or the completion of a set of procedural steps as evidence of conceptual understanding (Bartell, Webel, Bowen, & Dyson, 2013;Spitzer et al., 2010). For example, Morris (2006) found that PSTs were willing to accept a student's correct use of the area formula of a triangle as evidence of conceptual understanding rather than evidence of procedural knowledge. ...
... The reviewed studies provide some evidence that the skills for systematic analysis of practice can be taught. However, the majority of the existing literature focuses on the development of PSTs' ability to analyze student thinking (Bartell et al., 2013;Morris, 2006;Spitzer et al., 2010), whereas only a few focus on the ability to generate hypotheses about the effectiveness of teaching. This skill, we would argue, is necessary if teachers want to learn from their current practice to inform future practices. ...
... It has become a goal of many teacher education programs to help teachers become more deliberate in their practices. A growing body of literature suggests that PSTs can learn to attend to evidence of student thinking (Bartell et al., 2013;Morris, 2006;Spitzer et al., 2010). The ability to reason about the relation between teaching and learning has been less investigated. ...
This study examines the development of a specific sub-skill for studying
and improving teaching – the generation of hypotheses about the effects of teaching on student learning. Two groups of elementary pre-service
teachers (PSTs) were compared: one group who attended a typical
mathematics methods course and one who attended a course integrating
analysis skills for learning from teaching. Data consists of PSTs’ comments
on video clips of mathematics instruction administered before and after
course completion. Findings reveal that PSTs at the beginning of the
program struggled to generate hypotheses with relevant evidence, often
equating teacher behavior or student correct answers as evidence of
student understanding. After course participation, PSTs who attended the
course with integrated analysis skills significantly improved in their ability
to generate hypotheses based on student evidence while their
counterparts continued to display difficulties. Implications for teacher education and future research are
considered.
... PSTs often assess teaching effectiveness based on teacher strategies rather than on student responses (Hiebert & Stigler, 2000;Morris, 2006;Santagata, Zannoni, & Stigler, 2007). Even when the attention centers on students, over-attribution of conceptual understanding often occurs as PSTs equate correct answers or the completion of a set of procedural steps as evidence of conceptual understanding (Bartell, Webel, Bowen, & Dyson, 2013;Spitzer et al., 2010). For example, Morris (2006) found that PSTs were willing to accept a student's correct use of the area formula of a triangle as evidence of conceptual understanding rather than evidence of procedural knowledge. ...
... The reviewed studies provide some evidence that the skills for systematic analysis of practice can be taught. However, the majority of the existing literature focuses on the development of PSTs' ability to analyze student thinking (Bartell et al., 2013;Morris, 2006;Spitzer et al., 2010), whereas only a few focus on the ability to generate hypotheses about the effectiveness of teaching. This skill, we would argue, is necessary if teachers want to learn from their current practice to inform future practices. ...
... It has become a goal of many teacher education programs to help teachers become more deliberate in their practices. A growing body of literature suggests that PSTs can learn to attend to evidence of student thinking (Bartell et al., 2013;Morris, 2006;Spitzer et al., 2010). The ability to reason about the relation between teaching and learning has been less investigated. ...
Although emerging consensus exists that practice -based approaches to teacher preparation assist in closing the distance between university coursework and fieldwork experiences and in assuring that future teachers learn to implement innovative research-based instructional strategies, little empirical research has investigated teacher learning from this approach. This study examines the impact of a video and practice-based course on pre-service teachers’ mathematics classroom practices and analysis of their own teaching. Two groups of elementary pre-service teachers participated in the study—one attended the course and one did not. Findings reveal that the course assisted participants in making student thinking visible and in pursuing it further during instruction and in conducting evidence-based analyses of their own teaching. Conclusions discuss the importance of teaching these skills systematically during teacher preparation.
... Morris et al. (2009) found that K-8 PTs focused primarily on students' explicit and visible actions instead of the concepts underlying those actions and that the correctness of students' responses also positively influenced PTs' analyses of students' mathematical understanding. In other words, PTs overestimated students' mathematical understanding when students correctly used procedures (e.g., the area formula) or gave correct answers; they deemed procedural understanding as evidence of conceptual understanding (Morris 2006;Spitzer et al. 2011). ...
... However, her student initially counted from numbers not in the problems (e.g., solving 10-3 by counting up from 1), which she did not address in her reflection. Although Spitzer et al. (2011) found that PTs overestimated children's mathematical thinking by using unrelated evidence, our results suggest they also make these conclusions by omitting relevant evidence. While working on two subtrahend missing subtraction problems (5 − __ = 3 & 10 − __ = 4), Jessica changed her pattern of first problem-posing and second follow-up: strategy and used four problem-posing: repeat questions to handle her student's misconception that getting to 3 and 4 mean the answer is 3 and 4 for the subtraction problems. ...
A challenge for prospective teachers (PTs) is to determine what students know about a topic through asking appropriate questions and being thoughtful about the wording of these questions so as to capture and reframe students' spontaneous mathematical thinking and eventually unriddle the fuzzy boundary of students' complex thinking. This study examined four PTs' efforts to elicit kindergarteners' subtraction strategies and make conclusions about their subtraction understanding. Drawing on PTs' plans for interviewing kindergarteners on subtraction problems, interview transcripts, and reflection papers, the results suggest that PTs provided effective scaffolds, adding context to numerical problems or explaining mathematical terms and symbols as needed. However, they avoided asking problems with subtrahends greater than five, numbers above ten, and missing starts or missing subtrahends, limiting their ability to draw targeted conclusions about the students' strengths and needs. Using effective questions allowed one PT who posed a limited variety of problems to make stronger conclusions about her student's subtraction understanding, while asking a broader variety of problems helped another PT who used limited questions make conclusions about her student's subtraction understanding. Based on these results, mathematics teacher educators could leverage their PTs' strengths to either encourage multiple interviews, each targeting different problem types, or one interview with a broader variety of problem types. The results of this study further highlight the need for PTs to move beyond just asking students to explain their strategies and have them justify or represent their strategies as well.
... Teacher educators use video for various reasons. Some educators use it to improve teacher pedagogical content knowledge (Borko, Koellner, Jacobs, & Seago, 2011;Santagata, 2009;Seago, Mumme, & Branca, 2004); others to model reform-minded teaching practices (Kellogg & Kersaint, 2004;Santagata & Guarino, 2011); others to develop refl ection and noticing skills (Jansen & Spitzer, 2009;Star & Strickland, 2008;Stockero, 2008;Spitzer et al., 2011; Purpose(s) Types Viewing Modalities Assessment ...
Die technologischen Fortschritte der vergangenen Jahre haben den Zugang zu Unterrichtsvideos stark vereinfacht und dadurch deren Verfügbarkeit merklich gesteigert. Es wird daher zunehmend wichtig, Modelle bereitzustellen, die sowohl die Lehrerinnen- und Lehrerbildung als auch die Forschung im Umgang mit Videos anleiten. Im vorliegenden Beitrag wird ein entsprechendes Modell vorgestellt, das vier Komponenten aufweist, die in den folgenden Fragestellungen zum Ausdruck kommen: (1) Worin besteht das Ziel, wenn Lehrpersonen mit Videos lernen? (2) Welche Arten von Videos sind für dieses Ziel am besten geeignet? (3) Welche Formen des Videoeinsatzes und der Anleitung vermögen dieses Ziel am effektivsten zu unterstützen? (4) Wie kann festgestellt werden, ob das Ziel auch erreicht wurde? Jede dieser vier Komponenten wird anhand von bereits vorliegenden Forschungserkenntnissen und deren Diskussion illustriert. Das Fazit am Ende des Beitrags unterstreicht die Bedeutung eines systematischen Einsatzes von Unterrichtsvideos. Denn nur auf dieser Basis können Anhaltspunkte zusammengetragen werden, die es erlauben, die Forschung zum Lernen von Lehrerinnen und Lehrern voranzutreiben und verbesserte Lernumgebungen zu entwickeln.
... Teacher educators use video for various reasons. Some educators use it to improve teacher pedagogical content knowledge (Borko, Koellner, Jacobs, & Seago, 2011;Santagata, 2009;Seago, Mumme, & Branca, 2004); others to model reform-minded teaching practices (Kellogg & Kersaint, 2004;Santagata & Guarino, 2011); others to develop refl ection and noticing skills (Jansen & Spitzer, 2009;Star & Strickland, 2008;Stockero, 2008;Spitzer et al., 2011; Purpose(s) Types Viewing Modalities Assessment ...
Die technologischen Fortschritte der vergangenen Jahre haben den Zugang zu Unterrichtsvideos stark vereinfacht und dadurch deren Verfügbarkeit merklich gesteigert. Es wird daher zunehmend wichtig, Modelle bereitzustellen, die sowohl die Lehrerinnen- und Lehrerbildung als auch die Forschung im Umgang mit Videos anleiten. Im vorliegenden Beitrag wird ein entsprechendes Modell vorgestellt, das vier Komponenten aufweist, die in den folgenden Fragestellungen zum Ausdruck kommen: (1) Worin besteht das Ziel, wenn Lehrpersonen mit Videos lernen? (2) Welche Arten von Videos sind für dieses Ziel am besten geeignet? (3) Welche Formen des Videoeinsatzes und der Anleitung vermögen dieses Ziel am effektivsten zu unterstützen? (4) Wie kann festgestellt werden, ob das Ziel auch erreicht wurde? Jede dieser vier Komponenten wird anhand von bereits vorliegenden Forschungserkenntnissen und deren Diskussion illustriert. Das Fazit am Ende des Beitrags unterstreicht die Bedeutung eines systematischen Einsatzes von Unterrichtsvideos. Denn nur auf dieser Basis können Anhaltspunkte zusammengetragen werden, die es erlauben, die Forschung zum Lernen von Lehrerinnen und Lehrern voranzutreiben und verbesserte Lernumgebungen zu entwickeln.
... Callejo & Zapatera, 2017). The need for modifications has stemmed from new findings about teacher noticing, such as the difficulty in separating attending and interpreting (Castro Superfine, Fisher, Bragelman, & Amador, 2017) and the inclusion of evidence as a form of productive noticing (Phelps-Gregory & Spitzer, 2018;Spitzer, Phelps, Beyers, Johnson, & Sieminski, 2011). The variation in frameworks used has given rise to questions about methodological decisions. ...
This systematic literature review highlights consistencies and differences in methodological processes of research focused on prospective teacher noticing. Six-hundred-eleven articles were initially identified; 43 of those were analyzed in depth. Of the anaylzed articles, researchers primarily used one of two frameworks for noticing (i.e. Professional Noticing or Learning to Notice). Despite some methodological similarities, researchers used different time durations for learning experiences and measured noticing at different intervals. Analysis of study results indicate that many researchers found mixed or neutral participant outcomes for developing noticing. Researchers using the Learning to Notice Framework reported positive results more often than researchers using the Professional Noticing Framework.
... II), wurde die Aufmerksamkeit von Lehramtsstudierenden auf spezifische Unterrichtsmomente und -aspekte fokussiert und anschließend die studentischen Betrachtungsweisen diskutiert. Van Es und Sherin (2002) (Spitzer et al., 2010). Eine weitere "video-enhanced intervention" (Santagata & Yeh, 2014, S. 500) von Bartell und Kolleg*innen (2012) führte zu einem ähnlichen Befund, wobei die Analysefähigkeit der Studierenden hinsichtlich der Lernverläufe der Schüler*innen optimiert werden konnte. ...
Um sich dem Anforderungsprofil und damit der Professionalität von Lehrenden nähern zu können, sollen im Folgenden die zentralen Merkmale einer Profession benannt werden (vgl. 2.1). Hiernach werden Differenzen aufgezeigt zwischen Lehrexpert*innen, folglich „Fachleute[n]“ für das Lehren und Lernen (Bromme, 2008, S. 159) und solchen, die entsprechend der an Lehrexpert*innen angelegten Kriterien als Nicht-Expert*innen klassifiziert werden (vgl. 2.2.1). Der Expertiseansatz in der Lehrer*innenbildungsforschung hat insbesondere Unterschiede bezüglich des Reflexionsverhaltens und der Wissens- und Denkstrukturen zu Tage gefördert.
... Based on the research of [14] that to know the analysis of the ability of prospective teachers in analyzing students' understanding can be used the results of student work which is the result of student thinking. This is in accordance with research conducted using student's work. ...
... Öğrenci yanıtlarının ve dolayısı ile düşünce süreçlerinin bilinmesinin öğretimde kullanılacak yönergeler açısından önemli olduğu söylemek yanlış olmaz. Öğretim sürecinde uygun kararların verilebilmesi için, öğretmenlerin öğrencilerinin matematiği nasıl anladığını bilmesinin gerekliliği daha önceki çalışmalarda da vurgulanmıştır (Bartell, Webel, Bowen & Dyson,2013;Fernandez, Llinares & Valls, 2013;Wilson, Mojica & Confrey, 2013;Spitzer, Phelps, Beyers, Johnson & Sieminski, 2011). ...
Abstract
The knowledge that the mathematics teachers should have is a privileged knowledge beyond the basic concepts and concepts. One of the components of this knowledge is noticing students’ mathematical thinking. In particular, in the field of algebra, it is necessary for the teacher to have this kind of awareness in order to make sense of pat-tern generalization processes and to identify sources of errors. The aim of this study is to investigate the awareness of mathematics teacher candidates about students pattern generalization processes. For this purpose, students’ real answers consisting of different types of approaches and strategies were used as the data collection tool. This study employs the qualitative research methods and the participants are 35 prospective teachers taking the “teach-ing algebraic concepts” course in mathematics teaching department at a state university in 2015 – 2016 academic year. According to the research findings, prospective teachers can be considered successful in recognising students’ visible strategies; nevertheless the explanations provided by the prospective teachers on finding out the reasons that lie beneath the student mistakes are unsatisfactory.
Keywords: prospective mathematics teacher, student knowledge, pattern generalization
... Studies have also shown that, despite the challenges associated with shifting from their own actions to consider student learning outcomes, pre-service teachers can learn to become more reflective practitioners (Spitzer, Phelps, Beyers, Johnson, & Sieminski, 2011). Pre-service teachers who develop a capacity for reflective thinking become more aware of the assumptions on which their teaching decisions and actions are based (Yost, Sentner, & Forlenza-Bailey, 2000) and are better able to make connections between theory and practice (Ward & McCotter, 2004 The Use of Video One means of becoming more reflective is through the use of video. ...
... In this vein, it is enunciated that teachers should gradually improve their practice over time by engaging in systematic analysis of the effects of instruction on student learning. ( Spitzer et al. 2011, p. 68) [It] highlights the importance of teachers and school and system leaders increasingly taking responsibility for improving the enactment of the sequence, and for drawing on the underlying principles in various aspects of their practice. (Cobb and Jackson 2015, p. 1029) ...
... Researchers generally agree that teachers need to understand mathematics in ways needed specifically for teaching in order to be effective in the classroom (Ball et al. 2008). And while some researchers have demonstrated how preservice teachers can develop mathematical knowledge for teaching in the context of certain courses (e.g., Spitzer et al. 2011;Jansen and Spitzer 2009;Bartell et al. 2013; Morris et al. 2009), missing from this body of work is research about mathematics content courses in particular. Indeed, the purview of content courses is largely to develop preservice teachers' mathematical knowledge. ...
This descriptive case study examines six mathematics teacher educators’ (MTEs) perspectives on their design of content courses for elementary preservice teachers. By focusing on MTE’s design goals and considerations for their mathematics content courses, the means by which they achieved these course design goals, and the challenges they encountered as they carried out their design goals, this study adds to the image of mathematics content courses, and sheds light on the work of MTEs. Our findings indicate that MTEs design content courses that take a learner-centered approach to instruction and that in designing learner-centered content courses, MTEs face a variety of challenges. Implications for the professional development of MTEs are discussed.
... Their findings suggest that students were more able to recall before and after professional development the details of children's strategies (attending) than to interpret children's mathematical thinking; furthermore, PTs demonstrated significant growth in all three components. Spitzer et al. (2011) investigated prospective elementary teachers' ability to evaluate evidence of student achievement of the mathematical learning goal. PTs were less likely to consider teacher behaviors to be evidence of student learning and more likely to discount student responses that were irrelevant to a specified learning goal. ...
The aim of this research is to characterize profiles of the teaching competence “noticing students’ mathematical thinking” in the context of the pattern generalization. Prospective primary teachers were asked to describe and interpret the answers of three primary students to three linear pattern generalization problems. Five profiles for this competence have been identified. Prospective teachers named various mathematical elements to describe the students’ answers but did not always use them to interpret the understanding of pattern generalization of each student. Furthermore, this teaching competence ranges from the PPTs who could not recognize the understanding of the primary students to those capable of recognizing degrees of primary students’ understanding. Our findings allow one to generate descriptors of the development of this teaching competence and provide information for the design of interventions in teacher education addressed to support the recognition of evidence of students’ mathematical understanding.
... The findings of such studies suggest that most teachers are able to shift the focus of their reflections from classroom management issues or their own teaching to student understanding after participating in long-term professional development programs. Adding to what teacher educators know about teachers' reflections on classroom instruction, some studies have explored how teachers attend and respond to student thinking during professional development or teacher education programs by using video clips and written excerpts of student work (Goldsmith and Seago 2011;Jacobs et al. 2011;Son and Crespo 2009;Spitzer et al. 2011). Similar to the findings of the studies on video clubs, researchers observed that overtime and through participating in such programs, prospective and practicing teachers improved in their skills of productively attending to and making sense of student thinking after their initial difficulties. ...
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.
... Consistent across these models is a focus on defining a clear learning goal, using evidence from practice -such as, student work or video of 9 DEVELPOING PSTS' ABILITY TO ATTEND, ANALYZE, AND RESPOND classroom interactions -to inform analysis, and examining the cause-effect relationship between particular teaching practices and student learning. Teaching is framed then as an experiment with the teacher engaged in a methodical approach to examine and learn from practice (Spitzer, Phelps, Beyers, Johnson, & Sieminski, 2011). ...
This study investigates pre-service teachers' capacities to attend to, analyze, and respond to student thinking. Using a performance assessment of teacher competence, we compare two cohorts of science teacher candidates, one that participated in a video-based course designed to develop these skills and one that did not. Course participants demonstrate more sophisticated levels of attention to and analysis of student ideas. Analysis of the relationship among skills reveals that sophisticated analyses and responses to student ideas require high sophistication in attending to student ideas. However, high sophistication in attending to student ideas does not guarantee more sophisticated analyses or responses.
... This involves learning to identify and examine key routines such as effective questioning, design of open-ended mathematical problems, monitoring student work, leading a math discussion, and establishing a classroom discourse community (Chapin, O'Connor, & Anderson 2003;Hiebert & Wearne 1993;Hufferd-Ackles, Fuson, & Sherin 2004;Lampert 2001;Stein, Engle, Smith, & Hughes 2008;Stein, Smith, Henningsen, & Silver 2000). Use of the framework also requires the ability to reason about instructional strategies in terms of the extent to which they make student thinking visible and the ability to use evidence of student learning to reason about the effectiveness of teaching (Spitzer, Phelps, Beyers, Johnson, & Sieminski 2010). When reasoning about the impact of instructional decisions on student learning, PSTs need to be able to generalize the knowledge gained through the analysis of a particular teaching episode to more general hypotheses about teaching and learning that they can test again in future analyses or in their own teaching (Hiebert, Morris, Berk, & Jansen 2007). ...
... Shifting the attention from these features to student thinking and learning requires a different perspective on what is important to notice. Research on pre-service teachers finds that despite the difficulties related to learning this skill, pre-service teachers can shift the focus from themselves to students, learn to observe, record, and reflect on student learning, and use student thinking as evidence for accomplishing lesson goals (Levin, Hammer, & Coffey, 2009;Morris, 2006;Spitzer, Phelps, Beyers, Johnson, & Sieminski, 2011;van Es & Sherin, 2002). However, even with advances in helping pre-service teachers learn to effectively analyze student thinking, many do not make progress in Teachers and Teaching: theory and practice 3 this area (Santagata & Guarino, 2011), suggesting that this is a difficult skill to develop. ...
Learning to systematically analyze the relationship between teaching and student learning is an important but difficult skill to engender in teachers. In this study, we examine how pre-service teachers who were introduced to a framework for analyzing teaching in a video-based teacher education course drew on this tool to analyze their own practice after the conclusion of the course. We conceptualize the framework as a conceptual tool that scaffolds pre-service teachers to learn to attend to particular dimensions of teaching and learning and to analyze how their teaching influences student learning. Using the Portfolio Assessment for California Teachers-Teaching Event of 14 English language arts pre-service teachers, we conducted a qualitative analysis to examine the extent to which they applied this framework to analyze their own practice after the conclusion of the course, as well as different strategies they adopted as they analyzed their teaching practice. Findings suggest that pre-service teachers made progress in using the framework to study their teaching, but development of sub-skills for all four facets are needed to develop more productive analyses of teaching and student learning. This study has important implications for the design of teacher education that intends to support pre-service teachers in developing tools for learning to learn from their teaching practice.
... In this context, we have seen an emphasis on the need for teachers to base their teaching decisions on their students' learning characteristics of mathematical content (Franke & Kazemi, 2001). This has raised a number of questions about the way in which teachers notice the signs of students' understanding (An & Wu, 2012;Fernández, Llinares & Valls, 2012;Magiera, van den Kieboom & Moyer, 2013) and about the development of this skill (Bartell, Webel, Bowen & Dyson, 2013;Fernandez, Llinares & Valls, 2013;Wilson, Mojica & Confrey, 2013;Spitzer, Phelps, Beyers, Johnson & Sieminski, 2011). These research studies support the idea that teachers must know how students understand mathematical topics in order to make appropriate teaching decisions. ...
This research study examines the development of the ability of pre-service
teachers to notice signs of students’ understanding of the derivative concept. It analyses
preservice teachers’ interpretations of written solutions to problems involving the
derivative concept before and after participating in a teacher training module. The
results indicate that the development of this skill is linked to pre-service teachers’
progressive understanding of the mathematical elements that students use to solve
problems. We have used these results to make some suggestions for teacher training
programmes.
... We use the term «professionally» aware because this skill may not be innate in the math teacher. Research into the development of this awareness in teachers has shown how complex it is (An & Wu, 2012;Fernández, Llinares & Valls, 2011;Fernández, Llinares & Valls, 2013;Prediger, 2010;Prieto & Valls, 2010;Sánchez-Matamoros et al., 2012;Spitzer, Phelps, Beyers, Johnson & Sieminski, 2010). For example, faced with the same task as described earlier, (Figure 2) another future teacher remarked: ...
In this paper, I characterize the notion of «teaching competency» as being able to use knowledge in a pertinent way to carry out mathematics teaching tasks. One aspect of teaching competency is the teacher's «professional noticing» of students' mathematical thinking. This feature of teacher competency involves the cognitive ability to identify and interpret the salient features of the students' output in order to make informed decisions. This construct is illustrated using a context in which the prospective teachers are learning to interpret the students' answers to linear and nonlinear problems (in order to recognize the students' development of propor-tional reasoning). In this context, when prospective teachers use increasingly more explicit mathematics elements and features of the development of mathematical understanding to describe and interpret the students' mathematical thinking is considered as evidence of the development of professional awareness.
... This involves learning to identify and examine key routines such as effective questioning, design of open-ended mathematical problems, monitoring student work, leading a math discussion, and establishing a classroom discourse community (Chapin, O'Connor, & Anderson 2003;Hiebert & Wearne 1993;Hufferd-Ackles, Fuson, & Sherin 2004;Lampert 2001;Stein, Engle, Smith, & Hughes 2008;Stein, Smith, Henningsen, & Silver 2000). Use of the framework also requires the ability to reason about instructional strategies in terms of the extent to which they make student thinking visible and the ability to use evidence of student learning to reason about the effectiveness of teaching (Spitzer, Phelps, Beyers, Johnson, & Sieminski 2010). When reasoning about the impact of instructional decisions on student learning, PSTs need to be able to generalize the knowledge gained through the analysis of a particular teaching episode to more general hypotheses about teaching and learning that they can test again in future analyses or in their own teaching (Hiebert, Morris, Berk, & Jansen 2007). ...
Video is commonly used in teacher preparation programs. Teacher educators use video for various purposes. In this study, we
describe the Learning to Learn from Mathematics Teaching project. In this project, video is used to develop pre-service teachers’
(PSTs) orientations, knowledge and skills for analyzing and reflecting on mathematics teaching in ways that generate knowledge
for improvement. We discuss the ways we have used video in a course aimed at developing elementary PSTs’ abilities to learn
from teaching. In addition, we report on a study that investigated PSTs’ changes in lesson analysis abilities as a result
of participating in the course.
In this paper, we investigate the links between prospective elementary teachers’ (PTs’) ability to notice student mathematical thinking, their mathematical knowledge for teaching (MKT) and their ability to decompose learning goals into component conceptual parts (Morris et al., 2009). Previously, researchers have proposed theoretical connections between MKT and teachers’ professional noticing, but empirical support for these relationships has been limited. Results of this mixed-methods study indicate that PTs who scored higher on an MKT assessment outperformed their peers in terms of attending to student thinking but had similar performance in terms of interpreting that thinking. However, PTs who were able to conceptually unpack a learning goal into subconstructs performed higher-quality interpretations of student thinking. We hypothesize that the skill of decomposing learning goals may allow PTs to apply their mathematics knowledge to successfully interpret student work. These results have implications for understanding how noticing and MKT may develop in relation with each other.
This paper shares findings from the study of a learning sequence designed to support prospective elementary teachers (PTs) in identifying evidence of conceptual understanding and procedural fluency. Conceptual understanding and procedural fluency are widely recognized as important to teaching and learning mathematics, and identifying evidence of each is emphasized on teacher candidates’ high-stakes performance assessments. As PTs practiced selecting evidence from student work samples, our analysis revealed successes as well as associated challenges. Our findings suggest areas for supporting PTs’ learning, as well as challenges stemming from the inherent overlap of these two knowledge types. This study raises important questions about the value of emphasizing PTs’ ability to cite evidence of conceptual understanding and procedural fluency, given their integrated nature.
This study focuses on the challenge of promoting significant mathematical discourse among preservice teachers. The study was conducted as part of an academic course that accompanies their practical training. Twenty-three math preservice teachers’ learning process was examined as a result of using an analytic model designed for discourse protocols’ analysis. Findings revealed that an active and dynamic process occurred, modifying teacher practice and developing critical reflective thinking among preservice teachers. The change occurred in two “ripples of influence”: (1) improving discourse to one promoting learning by demonstrating hypothetical scenarios and (2) perception of the role of teachers and class management.
This study investigated how Turkish mathematics teachers evaluate the effectiveness of classroom teaching in terms of improving students’ mathematical proficiency. To this purpose, teachers were asked to evaluate a mathematics lesson as presented them in a vignette. By means of cluster analysis, the participants’ evaluations of the lesson were described in five thematic dimensions, which could be further assembled into two overriding categories: students’ understanding of the subject, and teachers’ classroom practices. The overall aim of the current paper is to propose a preliminary model of the framework that Turkish mathematics teachers use to evaluate a mathematics lesson.
Large-scale, robust implementation of teacher data-driven decision making (DDDM) is a challenging endeavor, impeded by numerous organizational, and teacher, factors. One well-documented barrier to teacher DDDM is underdevelopment of teacher data literacy. This study examines common errors made by pre-service elementary teachers in the formulation of evidence-based claims about student cognition (e.g. student strengths and weaknesses relative to content mastery) in a United States of America context. Teachers in the sample commonly made several errors in the articulation of evidence-based claims based on classroom assessments, including providing nonspecific, irrelevant evidence for claims, and misinterpreting statistical evidence. Specific implications for pre-service teacher education and future research are discussed.
In order for teachers to improve over time, they must be proficient at collecting and analyzing evidence of student thinking and learning (Hiebert, Morris, Berk, & Jansen, J Teacher Educ 58(1):47–60, 2007). This specific type of diagnostic competence, which focuses on diagnosing student learning with the specific goal of studying and improving teaching, can be improved through interventions in teacher education (see, e.g., Spitzer, Phelps, Beyers, Johnson, & Sieminski, JMTE 14(1):67–87, 2011). In this chapter, we discuss the findings of previous interventions aimed at helping prospective teachers (PTs) learn to analyze student thinking. Then, we present a replication study using a classroom intervention to teach prospective elementary teachers (N = 23) to identify and evaluate evidence of student understanding. Results of this study and previous work show that diagnostic competence is a skill that is teachable through interventions. After the intervention described in this chapter, participants performed better on a measure of diagnostic competence. In particular, they improved their ability to distinguish evidence of student thinking from nonevidence, such as a teacher’s lecture. They were also more likely to recognize that students’ procedural work cannot be used to diagnose conceptual understanding. Results will be used to suggest key features of interventions to improve diagnostic competence.
Although diagnostic competence of teachers is regarded as a key component of successful teaching, there are many open questions regarding the structure, the development and the impact of diagnostic competence. This chapter presents an overview of different approaches to pinpoint diagnostic competence theoretically and to investigate it empirically: measuring judgment accuracy, assessing competences in diagnostic situations or analyzing judgment processes. These approaches are discussed with respect to their advantages, restrictions as well as some of their main findings and they are allocated within an overarching model of diagnostic competence as a continuum, comprising diagnostic dispositions, diagnostic thinking and diagnostic performance.
In this study, we focus on one preservice teacher’s noticing of students’ mathematical and scientific thinking with an emphasis on how the acts of attending and interpreting can influence decisions about pedagogical actions. The study centers on an innovative field experience approach that incorporates lesson study in order to emphasize students’ thinking and its impact. Consequently, we were interested in understanding how one teacher made decisions based on her noticing at three points in her career: preservice field experiences, student teaching, and her first-year teaching. We used a case study approach to focus on one preservice teacher. Findings indicate that scaffolding PSTs to notice students’ mathematical and scientific thinking influenced how she noticed and considered students’ thinking while teaching. Results further indicate that supporting the development of noticing during field experiences has a positive impact on a teacher when she was in her own classroom. The study provides a unique contribution to the field as it incorporates both the mathematics and science teaching practices of the same PST from her teacher education experience into her career.
This study investigates the potential of video analysis and a mathematical knowledge for teaching framework, the Knowledge Quartet (KQ), in mathematics teacher education programmes. It reports on the effectiveness of these tools in analysing and supporting secondary level pre-service mathematics teachers’ subject matter knowledge and pedagogical content knowledge. This paper describes how a videotaped lesson of one pre-service teacher, teaching a class of mature students, was analysed and makes comparisons between the teacher educators’ and the pre-service teacher's observations. Inter-rater reliability was investigated and a Kappa coefficient of .72 indicated substantial agreement between both coders. Findings are presented and implications of the use of video and the KQ for mathematics teacher education are drawn.
Although researchers and educators have suggested teaching prospective teachers (PTs) to investigate and learn from their own teaching over time, little research has investigated PTs’ beliefs about such an approach. This qualitative study examines prospective elementary teachers’ conceptions and values about systematically studying and improving their own teaching. Participants (N = 6) were students at a university in the USA enrolled in a semester-long mathematics methods course designed around one model for systematically improving teaching (learning-from-teaching [LFT], also known as lesson experiments). Interview results show that the PTs had a high level of procedural knowledge about the LFT skills and reported liking and intending to use all or part of the LFT model. However, results also reveal that four of the six PTs held multiple misconceptions and all six PTs placed less value on improving teaching than on other teaching goals. PTs’ conceptions and values about improving teaching may limit their ability to successfully learn from their own teaching, which has implications for how teacher educators teach PTs about lesson experiment skills such as writing learning goals or examining student learning.
This paper reports on a study conducted to explore prospective mathematics teachers' reflections on teaching practice at the secondary level through noticing key aspects of classroom interactions. The study used critical incidents taken from everyday classroom situations as a means to make the act of noticing more concrete. The participating prospective teachers were engaged in a number of different activities including observing, designing and teaching. The results indicate a progression of prospective teachers' noticing of classroom practice and development of teaching awareness marked by shifts in analysing and interpreting classroom events.
Researchers have advocated building a knowledge base for teacher education based on practitioner knowledge. In this article, the authors, two teacher educators, present knowledge gained from their systematic study of two lessons to help prospective teachers learn to study teaching. They offer a detailed look at the lessons' successes and shortcomings as well as potential revisions. They also reflect on their use of a model for systematically studying lessons, sharing details about their learning during this process. This article attempts to provide a visible, tangible product for other teacher educators interested in helping prospective teachers examine evidence of student learning.
This study examined prospective teachers’ (PSTs) ability to recognize evidence of children’s conceptual understanding of mathematics in three content areas before and after an instructional intervention designed to support this ability. It also investigates the role PSTs’ content knowledge plays in their ability to recognize children’s mathematical understanding. Results of content knowledge assessments administered at the beginning of the study revealed that content knowledge did seem to support PSTs’ analyses of children’s understanding when the child’s response demonstrated understanding or demonstrated a misconception. Content knowledge did not seem to support PSTs’ analyses of children’s procedural responses, as many PSTs with good content knowledge initially characterized procedural solutions as evidence of conceptual understanding. Similarly, content knowledge did not seem to support PSTs’ analyses of children’s responses with features commonly associated with understanding but not evidence of understanding. After the instructional intervention consisting of three multifaceted lessons in which PSTs examined many examples of student thinking, they showed improved ability to analyze responses with conceptual features and no evidence of conceptual understanding and responses demonstrating procedural knowledge. Results suggest that content knowledge is not sufficient for supporting PSTs’ analysis of children’s thinking, and that building activities such as the intervention into content courses may help develop this ability. Implications for teacher education programs and future research are considered.
Mathematics and science education reforms encourage teachers to base their instruction in part on the lesson as it unfolds in the classroom, paying particular attention to the ideas that students raise. This ability to adapt instruction in the moment requires that teachers be able to notice and interpret aspects of classroom interactions that are key to reform teaching. This paper defines what it means to "notice" in the context of reform and describes a multimedia tool designed to help teachers learn to do so. The authors then report on a study in which six mathematics and science teachers seeking secondary teaching certification used the software to examine teaching. The results suggest that use of the software helped the teachers to develop new ways to analyze instruction. Specifically, the teachers began to identify particular events in their classroom interactions as noteworthy, to more frequently use specific evidence to discuss these events, and to provide their own interpretations of these events. This research adds to our understanding of teacher cognition and also has implications for those who are designing and implementing teacher education in the context of reform.
The authors propose a framework for teacher preparation programs that aims to help prospective teachers learn how to teach from studying teaching. The framework is motivated by their interest in defining a set of competencies that provide a deliberate, systematic path to becoming an effective teacher over time. The framework is composed of four skills, rooted in the daily activity of teach- ing, that when deployed deliberately and systematically, constitute a process of creating and test- ing hypotheses about cause-effect relationships between teaching and learning during classroom lessons. In spite of the challenges of acquiring these skills, the authors argue that the framework outlines a more realistic and more promising set of beginning teacher competencies than those of traditional programs designed to produce graduates with expert teaching strategies.
Results from the Third International Mathematics and Science Study (TIMSS) Video Study of Teaching show that many teachers in the United States believe they are changing the way they teach while they retain the core of traditional practice. Results from the video study, which also included teachers from Germany and Japan, along with comparisons of teacher development systems, help to explain the persistence of traditional teaching methods and point to a new model for improving teaching. After presenting results from the video study regarding teachers' perceptions of the effects of recent reforms on their practice, we propose a research and development system for improving teaching that builds on the Japanese process of lesson study. We describe the process as it works in Japan and outline the conditions that would enable such a process to function effectively in the United States.
A video-based program on lesson analysis for pre-service mathematics teachers was implemented for two consecutive years as
part of a teacher education program at the University of Lazio, Italy. Two questions were addressed: What can preservice teachers
learn from the analysis of videotaped lessons? How can preservice teachers’ analysis ability, and its improvement, be measured?
Two groups of preservice teachers (approximately 140 in total) participated in the program. A three-step lesson analysis framework
was applied to three lesson videos: (1) goal(s) and parts of the lesson; (2) student learning; and, (3) teaching alternatives.
Preservice teachers’ ability to analyze lessons was measured through an open-ended pre- and post-assessment. In the assessment,
preservice teachers were asked to mark and comment on events (in a lesson not included in the program) that they found interesting
for: teachers’ actions/decisions; students’ behavior/learning; and, mathematical content. A coding system was developed based
on five criteria: elaboration, mathematics content, student learning, critical approach, and alternative strategies. In both
studies, the ability to analyze instruction improved significantly on all five criteria. These data suggest promising directions
for the development of both an instrument to measure lesson analysis abilities and a model for teacher learning.
The study describes teachers' collective work in which they developed deeper understanding of their own students' mathematical thinking. Teachers at one school met in monthly workgroups throughout the year. Prior to each workgroup, they posed a similar mathematical problem to their students. The workgroup discussions centered on the student work those problems generated. This study draws on a transformation of participation perspective to address the questions: What do teachers learn through collective examination of student work? How is teacher learning evident in shifts in participation in discussions centered on student work? The analyses account for the learning of the group by documenting key shifts in teachers' participation across the year. The first shift in participation occurred when teachers as a group learned to attend to the details of children's thinking. A second shift in participation occurred as teachers began to develop possible instructional trajectories in mathematics. We focus our discussion on the significance of the use of student work and a transformation of participation view in analyzing the learning trajectory of teachers as a group.
This paper examines one model of professional development, the use of video clubs in which groups of teachers watch and discuss videotapes of their classrooms. Specifically, the paper investigates the learning that occurred as four middle-school mathematics teachers participated in a year-long series of video club meetings. Over time, discourse in the video clubs shifted from a primary focus on the teacher to increased attention to students’ actions and ideas. In addition, discussions of student thinking moved from simple restatements of students’ ideas to detailed analyses of student thinking. Furthermore, teachers began to reframe their discussions of pedagogical issues in terms of student thinking.
This study examined changes in the beliefs and instruction of 21 primary grade teachers over a 4-year period in which the teachers participated in a CGI (Cognitively Guided Instruction) teacher development program that focused on helping the teachers understand the development of children's mathematical thinking by interacting with a specific research-based model. Over the 4 years, there were fundamental changes in the beliefs and instruction of 18 teachers such that the teachers' role evolved from demonstrating procedures to helping children build on their mathematical thinking by engaging them in a variety of problem-solving situations and encouraging them to talk about their mathematical thinking. Changes in the instruction of individual teachers were directly related to changes in their students' achievement. For every teacher, class achievement in concepts and problem solving was higher at the end of the study than at the beginning. In spite of the shift in emphasis from skills to concepts and problem solving, there was no overall change in computational performance. The findings suggest that developing an understanding of children's mathematical thinking can be a productive basis for helping teachers to make the fundamental changes called for in current reform recommendations.
This study was designed to identify the types of understandings, skills, and beliefs that affect pre-service teachers' evaluations of students' mathematical arguments in classroom contexts. Thirty-four pre-service teachers read a transcript of a third grade lesson in which the students were expected to prove a generalization. To investigate whether pre-service teachers evaluate students' arguments in a consistent way across different classroom contexts, pre-service teachers' evaluations of the responses were examined in two experimental conditions. In the first condition, one student made a valid argument that proved why the generalization was true, and in the second condition, this student's response was omitted from the transcript. Findings included the following: 1) Pre-service teachers' evaluations of students' inductive arguments differed dramatically across conditions; 2) Pre-service teachers rarely used logical validity as a criterion for evaluating arguments; 3) Pre-service teachers exhibited a wide variety of conceptions about the relationships among mathematical proof, explaining why something is true in mathematics, and inductive arguments, and these conceptions affected their evaluations of students' arguments; 4) Many pre-service teachers were able to distinguish between student responses that did and did not explain why a generalization was true; and 5) Pre-service teachers used their own knowledge to fill in "holes" in students' arguments which led to inappropriate evaluations of students' arguments and understanding. Implications for teacher preparation programs are discussed.
Based on rapid advances in what is known about how people learn and how to teach effectively, this important book examines the core concepts and central pedagogies that should be at the heart of any teacher education program. This book was edited in collaboration with Pamela LePage, Karen Hammerness, and Helen Duffy. It is the result of the National Academy of Education's Committee's work on teacher education. It was written for teacher educators in both traditional and alternative programs, university and school system leaders, teachers, staff development professionals, researchers, and educational policymakers, the book addresses the key foundational knowledge for teaching and discusses how to implement that knowledge within the classroom. This book recommends that, in addition to strong subject matter knowledge, all new teachers have a basic understanding of how people learn and develop, as well as how children acquire and use language, which is the currency of education. In addition, the book suggests that teaching professionals must be able to apply that knowledge in developing curriculum that attends to students' needs, the demands of the content, and the social purposes of education: in teaching specific subject matter to diverse students, in managing the classroom, assessing student performance, and using technology in the classroom. The ideas and suggestions outlined in this book have far-reaching implications for educational policy, classroom practice, and staff development and will go a long way toward informing the next generation of teachers. (PsycINFO Database Record (c) 2012 APA, all rights reserved)
This study investigated the learning-from-practice skills that pre-service teachers possess when they enter teacher preparation programs in the United States. Two subskills were hypothesized to represent, at least in part, what is required to learn from practice: (1) the ability to collect evidence about students’ learning in order to analyze the effects of instruction, and (2) the ability to use the analysis to revise the instruction. Because it seems likely that different teaching situations and contexts reveal these learning-from-practice skills in different ways and to different degrees, this study examined the skills that pre-service teachers exhibited under two experimental conditions. Thirty pre-service teachers were asked to analyze the effects of a videotaped mathematics lesson on student learning, to support their analysis with evidence, and to use their analysis to revise the lesson. Based on the results, it appears that many entry level pre-service teachers can carry out a cause-effect type of analysis of the relationships between specific instructional strategies and students’ learning, and can use this analysis to make productive revisions to the instruction. However, prospective teachers’ ability to collect evidence that supports their analysis appears to be less developed. In addition, the type of analysis that prospective teachers carried out about the effects of instruction on students’ learning differed dramatically across the two experimental task conditions.
Video has assumed an increasingly prominent role in teacher education, particularly in the form of the viewing of videotaped
class lessons by preservice teachers. Yet there is little research that confirms whether preservice teachers attend to the
aspects of the video(s) that teacher educators anticipate or desire. This article explores this issue and reports on the impact
of video viewing as a means to improve teachers’ ability to be observers of classroom practice. We utilized a pre- and post-test
design to measure the quantity and type of classroom events that preservice mathematics teachers noticed before and after
a teaching methods course where improving observation skills was an explicit goal. The results of the pre-assessment suggest
that preservice teachers generally do not enter teaching methods courses with well-developed observation skills. The post-assessment
indicates that the course led to significant increases in preservice teachers’ observation skills, particularly in teachers’
ability to notice features of the classroom environment, mathematical content of a lesson, and teacher and student communication
during a lesson.
Although video cases are increasingly being used in teacher education as a means of situating learning and developing habits
of reflection, there has been little evidence of the outcomes of such use. This study investigates the effects of using a
coherent video-case curriculum in a university mathematics methods course by addressing two issues: (1) how the use of a video-case
curriculum affects the reflective stance of prospective teachers (PTs); and (2) the extent to which a reflective stance developed
while reflecting on other teachers’ practice transfers for reflecting on one’s own practice. Data sources include videotapes
of course sessions and PTs’ written work from a middle school mathematics methods course that used a video-case curriculum
as a major instructional tool. Both qualitative and quantitative analytical methods were used, including comparative and chi-square
contingency table analyses. The PTs in this study showed changes in their level of reflection, their tendency to ground their
analyses in evidence, and their focus on student thinking. In particular, they began to analyze teaching in terms of how it
affects student thinking, to consider multiple interpretations of student thinking, and to develop a more tentative stance
of inquiry. More significantly, the reflective stance developed via the video curriculum transferred to the PTs’ self-reflection
in a course field experience. The results of this study speak to the power of using a video-case curriculum as a means of
developing a reflective stance in prospective mathematics teachers.
This paper describes a model for generating andaccumulating knowledge for both teaching andteacher education. The model is applied firstto prepare prospective teachers to learn toteach mathematics when they enter theclassroom. The concept of treating lessons asexperiments is used to explicate theintentional, rigorous, and systematic processof learning to teach through studying one's ownpractice. The concept of planning teachingexperiences so that others can learn fromone's experience is used to put into practicethe notion of contributing to a sharedprofessional knowledge base for teachingmathematics. The same model is then applied tothe work of improving teacher preparationprograms in mathematics. Parallels are drawnbetween the concepts emphasized for prospectiveteachers and those that are employed byinstructors who study and improve teacherpreparation experiences. In this way, parallelsalso are seen in the processes used to generatean accumulating knowledge base for teaching andfor teacher education.
Strong claims have been made about the potential of lesson study, a Japanese form of professional development in which teachers collaboratively plan and examine actual lessons. We have explored these claims by asking a group of US teachers to engage in lesson study with the support of Japanese teachers. Our findings suggest that to benefit from lesson study teachers will first need to learn how to apply critical lenses to their examination of lessons. We describe three such lenses (e.g. the researcher lens) and their role in making lesson study powerful. We also discuss the implications of these findings for other professional development efforts in which teachers attempt to learn from concrete examples of practice.
Libro de metodología cualitativo para investigación en las ciencias sociales. La utilización de la computadora, el uso de datos y la recolección de los mismos. Se describen detalladamente numerosos métodos de datos y análisis.
Developing practice, developing practitioners: Toward a practices-based theory of professional development Teaching as the learning profession: Handbook of policy and practice
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The contribution of settings to school improvement and school change: A case study Culture and context in human behavior change: Theory, research, and applications
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Adding it up: Helping children learn mathematics Mathematics Learning Study Committee, Center for Education, Division of Behavioral and Social Sciences and Education
- National Research Council