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.     

“This is the First Time I’ve Done This”: Exploring secondary prospective mathematics teachers’ noticing of students’ mathematical thinking

This mixed methods study investigates the ways in which secondary mathematics prospective teachers acquire skills needed to attend to, interpret, and respond to students’ mathematical thinking and the ways in which their perceived strengths and weaknesses influence their skills when this type of formalized training is not part of their program. These skills (attending, interpreting, and responding) are defined as teachers’ professional noticing of students’ thinking. Results indicate that seniors respond to students’ thinking in significantly different ways from juniors and sophomores. Converging the data highlighted inconsistencies in how participants’ were making sense of students’ mathematical thinking, as well as in participants’ self-identified strengths and weaknesses.     

Prediction assessments: Using video-based predictions to assess prospective teachers’ knowledge of students’ mathematical thinking

In order to evaluate the effectiveness of an experimental elementary mathematics field experience course, we have designed a new assessment instrument. These video-based prediction assessments engage prospective teachers in a video analysis of a child solving mathematical tasks. The prospective teachers build a model of that child’s mathematics and then use that model to predict how the child will respond to a subsequent task. In this paper, we share data concerning the evolution and effectiveness of the instrument. Results from implementation indicate moderate to high degrees of inter-rater reliability in using the rubric to assess prospective teachers’ models and predictions. They also indicate strong correlation between participation in the experimental course and prospective teachers’ performances on the video-based prediction assessments. Such findings suggest that prediction assessments effectively evaluate the pedagogical content knowledge that we are seeking to foster among the prospective teachers.     

Providing written feedback on students’ mathematical arguments: proof validations of prospective secondary mathematics teachers

Mathematics teachers play a unique role as experts who provide opportunities for students to engage in the practices of the mathematics community. Proof is a tool essential to the practice of mathematics, and therefore, if teachers are to provide adequate opportunities for students to engage with this tool, they must be able to validate student arguments and provide feedback to students based on those validations. Prior research has demonstrated several weaknesses teachers have with respect to proof validation, but little research has investigated instructional sequences aimed to improve this skill. In this article, we present the results from the implementation of such an instructional sequence. A sample of 34 prospective secondary mathematics teachers (PSMTs) validated twelve mathematical arguments written by high school students. They provided a numeric score as well as a short paragraph of written feedback, indicating the strengths and weaknesses of each argument. The results provide insight into the errors to which PSMTs attend when validating mathematical arguments. In particular, PSMTs’ written feedback indicated that they were aware of the limitations of inductive argumentation. However, PSMTs had a superficial understanding of the “proof by contradiction” mode of argumentation, and their attendance to particular errors seemed to be mediated by the mode of argument representation (e.g., symbolic, verbal). We discuss implications of these findings for mathematics teacher education.     

Exploring prospective teachers’ noticing of students' understanding through micro-case videos

This paper explores the nature of prospective teachers’ noticing of students’ understanding as they analyze and discuss middle school students’ understandings of trapezoids in micro-case videos in the context of geometry. In this exploratory study, the data were obtained from eight prospective middle school mathematics teachers through individual video analysis, reflection papers, and group discussions. The results indicated that the use of purposeful micro-case video designs based on prospective teachers’ background knowledge of quadrilaterals allowed them to be productive in video analyses and discussions. In individual video analyses, prospective teachers attended to various mathematical elements to identify students’ responses but did not always use them to make interpretations of each student’s understanding of trapezoid. In the group discussions of the micro-case videos, in contrast, prospective teachers could provide alternative interpretations of students’ understanding by identifying links between the mathematical elements in students’ responses and the characteristics of students’ understandings. In the group discussions, they provided more detailed and specific instructional actions to support each student’s understanding of trapezoid than their individual video analyses. This study suggests practical implications for teacher education programs on how to use video cases (e.g., firstly, working individually and then having group discussions about the videos) to explore prospective teachers’ professional noticing skills. Considering prospective teachers’ background knowledge of related mathematical contents, this study can also inspire future studies on how to design effective videos about students’ mathematical understanding.

     

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.     

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.     

The development of elementary pre-service teachers’ professional noticing of students’ thinking through adapted Lesson Study

The purpose of the study was to investigate how Lesson Study influences pre-service teachers’ learning to professionally notice students’ mathematical thinking. The initial and last video-recorded discussions were analysed using a qualitative method. Findings suggest that Lesson Study can be useful means to improve pre-service teachers’ noticing expertise in reviewing and planning lessons. However, to maximise the effect of Lesson Study on developing pre-service teachers’ noticing expertise, several modifications need to be made to the Lesson Study protocol, which are discussed.     

Preservice teachers’ articulated noticing through pedagogies of practice

Elementary preservice teachers at six universities engaged in a task that provided them opportunities to articulate their professional noticing within video representations, written decompositions, and animated approximations of practice. The preservice teachers’ written accounts indicated that a majority attended to students or student thinking; however, when asked to illustrate their noticing through animation, focus shifted to the classroom teacher. Findings indicate the extent to which preservice teachers articulated specific mathematics concepts within and across pedagogies of practice and highlight the critical importance for selecting and utilizing multiple types of tasks to better understand preservice teacher noticing. Implications for eliciting and supporting preservice teacher noticing are discussed.     

Guiding reinvention of conventional tools of mathematical logic: students’ reasoning about mathematical disjunctions

Motivated by the observation that formal logic answers questions students have not yet asked, we conducted exploratory teaching experiments with undergraduate students intended to guide their reinvention of truth-functional definitions for basic logical connectives. We intend to reframe the relationship between reasoning and logic by showing how logic emerges within students’ mathematical activity. This activity entails reflecting on and systematizing their own language use across diverse semantic content. We present categories of students’ untrained strategies for assessing the truth-values for mathematical disjunctions. Students’ initial reasoning heavily reflected content-specific and pragmatic factors in ways inconsistent with the norms and conventions of mathematical logic. Despite this, all student groups reinvented the standard truth-functional definition for simple disjunctions. We demonstrate how this learning depended upon particular forms of reasoning about logic. We also contrast various strategies for assessing quantified disjunctions and their different affordances in students’ mathematical activity.     

Working like real mathematicians: developing prospective teachers’ awareness of mathematical creativity through generating new concepts

This paper describes the experience of a group of 17 prospective mathematics teachers who were engaged in a series of activities aimed at developing their awareness of creativity in mathematics. This experience was initiated on the basis of ideas proposed by the participants regarding ways creativity of school students might be developed. Over a period of 6 weeks, they were engaged in inventing geometrical concepts and in the examination of their properties. The prospective teachers’ reflections upon the process they underwent indicate that they developed awareness of various aspects of creativity while deepening their mathematical and didactical knowledge.     

Forms of mathematical interaction in different social settings: examples from students’, teachers’ and teacher–students’ communication about mathematics

The study presented in this article investigates forms of mathematical interaction in different social settings. One major interest is to better understand mathematics teachers’ joint professional discourse while observing and analysing young students mathematical interaction followed by teacher’s intervention. The teachers’ joint professional discourse is about a combined learning and talking between two students before an intervention by their teacher (setting 1) and then it is about the students learning together with the teacher during their mathematical work (setting 2). The joint professional teachers’ discourse constitutes setting 3. This combination of social settings 1 and 2 is taken as an opportunity for mathematics teachers’ professionalisation process when interpreting the students’ mathematical interactions in a more and more professional and sensible way. The epistemological analysis of mathematical sign-systems in communication and interaction in these three settings gives evidence of different types of mathematical talk, which are explained depending on the according social setting. Whereas the interaction between students or between teachers is affected by phases of a process-oriented and investigated talk, the interaction between students and teachers is mainly closed and structured by the ideas of the teacher and by the expectations of the students.

Breaking with tradition: An investigation of an alternative instructional sequence designed to improve prospective teachers’ noticing skills

This experimental study aims to investigate the effect of an alternative instructional sequence designed to improve prospective teachers’ noticing skills. The participants consisted of two groups of prospective teachers (N = 28). To assess their noticing skills, the participants were asked to note critical events in the classroom and to provide detailed explanations about why they found these worth noting. The results indicate that the participants in experimental group showed higher levels of noticing skills compared to those in control group. The study presents a discussion of the success of the instructional approach in improving the noticing skills of future mathematics teachers.     

Supporting school teachers: primary teachers’ conceptions of their responses to diverse sexualities

Teachers experience a range of situations in the primary school context where students talk about, share ideas and use words and actions that might be described as non-heteronormative in character. This article reports on teachers’ experiences of the ways in which they respond to actions and events they see as non-heteronormative. It identifies six different ways in which teachers conceptualise their pedagogical responses to the situations they encounter: (1) being nonchalant, (2) avoiding a response, (3) being uncertain, (4) maintaining home and school boundaries, (5) protecting students’ well-being and (6) embracing non-heteronormative sexualities. Teachers’ responses to these events are important in that their actions may affect how students interpret and understand the world. Primary school teachers will benefit from support in knowing how to respond appropriately when faced with a scenario or situation that is non-heteronormative in character.     

In the eyes of the beholder: students’ and teachers’ judgments of students’ motivation

This study was designed to examine the nature and accuracy of teachers’ judgments about students’ motivation related to mathematics. Seventeen fourth- through sixth-grade teachers and 100 of their students participated in this study. Teachers were asked to rate six target students’ motivation four times during the school year. At each of these times students also rated their own motivation. Teachers’ ratings were more stable over time and across mathematics topics and were less differentiated across motivation dimensions than were students’ ratings. The findings suggest that teachers need to pay careful attention to evidence related to children's motivation that might contradict their perceptions and therefore support more effective interventions.     

Measuring pre-service teachers’ noticing competencies within a mathematical modeling context – an analysis of an instrument

Educational Studies in Mathematics - Teaching mathematical modeling is a demanding task. Thus, fostering teachers’ competencies in this regard is an essential component of teacher education....     

Supporting students’ mental health in schools: what do teachers want and need?

There is a growing expectation in the UK for teachers to have an understanding of common mental health problems in young people. This study aims to identify the training needs of secondary school teachers to enable them to adequately support and educate their students around mental health. Nine focus groups, each with between four to eight participants, were conducted with secondary school teachers in the UK. Discussions were centred on the needs and wants of teachers in terms of mental health training, provision and advice. Participants were also shown three online resources as an example of existing training. Thematic analysis was used to structure the data. Participants wanted training on how to identify and provide early support for students who are struggling, without taking on the perceived role of a therapist. Participants also emphasised the strong need for practical, interactive and expert-led training that provides resources that can be adapted to individual settings. Implications and recommendations are discussed.     

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.