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Relationships Between Teachers’ Interactions with Learner Errors and Learners’ Mathematical Identities
Abstract
Engaging with learner errors in mathematics classrooms is an important aspect of teacher pedagogy which can support learner identification with mathematics. This study examined the opportunities that teachers provided for the construction of learner identities based on how the teachers spoke about and interacted with learner errors in two secondary school mathematics classrooms. Data were collected over two years in the form of videotaped lessons, field notes, photographed learner notebooks and audiotaped interviews with participants, and were analyzed qualitatively. Two ways in which teachers spoke about errors are discussed: errors are contagious and errors are lessons. How the teachers spoke about and interacted with errors were informed by their views of learners and their reasoning in mathematics, which shaped the identities they offered learners. We show that while learners’ experiences of their teachers’ approaches to errors informed their identification with mathematics to some extent, the learners also played a role in the construction of their mathematical identities.
... The selection of the teachers and learners was purposeful. Of the five teachers who participated in the study, only these two teachers offered all learners "identities of affiliation" with the classroom community, meaning that they supported all learners to learn mathematics and become full members of their classroom community (see Gardee 2019Gardee , 2021Gardee & Brodie, 2021, 2022 for more detailed analyses). In Mr Moyo's classroom, the two learners were Shane and Jack, who were good friends and were identified by Mr Moyo as high-performing learners. ...
... Field notes and photographs of learners' notebooks provided context and supplemented the recordings. In both classrooms, the teachers taught using traditional, textbook-based pedagogies, but provided learners with opportunities to engage with others in almost every lesson (see Gardee 2019Gardee , 2021Gardee & Brodie, 2021, 2022. ...
... Some questions asked to the teacher participants were: how do you see [name] as a learner in your classroom?; and how does [name] participate in the classroom? All questions were followed by further probing questions, aimed at supporting participants to give detailed explanations (see Gardee, 2019Gardee, , 2021Gardee & Brodie, 2021;2022). ...
The absence of discussions of identity that consider both subjective/personal and social aspects of identity is an important concern in mathematics education research. This paper proposes a framework to analyse the mathematical identities offered to and constructed by learners during peer interactions, by considering personal and social identities. To exemplify the framework, we discuss how two pairs of secondary school learners, who were engaged with similarly by their teachers and who performed similarly academically, interacted with each other and offered and constructed micro-identities during moments of a lesson, and macro-identities across lessons. We show that when learners were offered identities by a peer as higher, lower or equal status, they constructed their identities by affiliating with or resisting the identities offered by their peer. Their interactions and the identities that they constructed were informed by their personal identities, including their emotions when learning mathematics, and their social identities, including their choices to occupy certain roles in the classroom community.
... Errors can provide opportunities for learning if the reasoning and underlying errors can be identified and explored as part of learning activities (Brodie, 2014). Alternatively, certain error-handling practices could also influence students to develop unproductive mathematical identities (Gardee & Brodie, 2021). Due to several dimensions of the role of errors for students' learning, research has examined different strands of teachers' error-handling practices in mathematical teaching, such as how student errors can be treated as an opportunity for learning in the classroom (Ingram et al., 2015), how teachers from different cultures handle errors differently (Santagata, 2005;Schleppenbach et al., 2007), and how teachers' beliefs and knowledge influence their error-handling practices during whole-class discussions (Bray, 2011). ...
... We, therefore, distinguish between slips and conceptual errors. As mentioned in the introduction and then further clarified in the methodology, this study focused on conceptual errors, which might be considered either misconceptions, or yet incoherent knowledge pieces that are productive in some situations but not in others (Gardee & Brodie, 2021;Scheiner, 2020). According to a knowledgein-structures' perspective, students' misconceptions can be characterized as systematic patterns of errors (Scheiner, 2020). ...
... Discussing errors with the students will give teachers access to students' current thinking (Gardee & Brodie, 2015). Students might understand and learn better when errors are discussed in the classroom, rather than being corrected or avoided (Gardee & Brodie, 2021). If teachers avoid or ignore the error, the students will not have access to understand why their errors are incorrect. ...
Multiple studies have been conducted regarding teachers’ error-handling practices, and how errors can be treated as opportunities for learning, albeit in the context of whole-class discussions. The aim of the present research is to continue to investigate teachers’ error-handling practices as they occur in different phases of maths lessons: introduction of the task, when students are working alone, and when students are working in pairs and finally, as part of the whole-class discussion. The study included 51 lessons from twelve teachers. A cross-case analysis was made across the individual teacher cases to look for similarities and differences between different teachers’ error-handling practices across the lesson phases in order to create teaching profiles with similar handling of student errors across the lesson phases. Five error-handling teaching profiles were identified; correcting errors throughout all phases, correcting errors during students’ work while few errors are brought up in whole class, correcting errors during students’ work while using a variety of practices in whole class, ignoring errors while using some of them in whole class, and discussing and explaining errors.
... This study was built on the theoretical foundation of assimilation theory, which states that meaningful learning occurs when students assimilate, or anchor, new concepts into their existing prior knowledge structure (Ausubel, 1968;Ausubel and Robinson, 1969;Ausubel, Novak and Hanesian, 1978;Gardee and Brodie, 2021). This theory frames the world around us as a web of interconnected thoughts and ideas. ...
The Dynamic and Interactive Mathematical Expressions (DIME) Map system automatically generates DIME maps, which are personalizable and manipulable concept maps that allow students to interact with the mathematical concepts contained in any portable document format (PDF) textbook or document. A teacher can automatically upload a PDF textbook chapter and retrieve a DIME map of the contained mathematically based concepts. The DIME map is interactive and manipulable and can be used to interactively navigate the PDF textbook chapter. Our goal was to investigate the relationship between use of DIME maps and student learning outcomes, including self-efficacy and ability to understand and recall connections between physics concepts. We implemented a pretest/posttest to determine if student self-efficacy and connections in knowledge increased after participation in a summer camp physics class. We additionally conducted student interviews to better understand how changes in these two factors may have occurred. We then used multivariate analysis of variance and thematic analysis, finding and investigating positive effects of students using DIME maps, namely growth in self-efficacy and connections in knowledge. Based on our findings, we conclude that DIME maps can be valuable learning tools for students that have positive effects on both cognitive and affective learning outcomes.
... 1,2 Mole ratio related errors include failing to identify limiting reagents, 3 misusing coefficients, 4 failing to recognize the need to use proportion, 5 conflating ratio and amount, and applying the mole ratio as a mass ratio. 1 Such errors are prevalent among students throughout the world and even among teachers in developing world contexts, such as South Africa. 5 Error analysis can provide valuable insight into the thinking that causes the error, 6 pedagogy which may effectively address the error, 7,8 and, more fundamentally, inferences that can be made about the nature of knowledge and its acquisition, i.e., epistemology. ...
Presenters’ pedagogical choices are strongly affected by whether they view knowledge as coherent or fragmented. This mixed-methods survey study seeks to contribute to epistemological discussions and associated pedagogical suggestions in the context of mole ratio related mistakes evident in stoichiometry calculations performed by in-service teachers in a developing world country. Physical sciences teachers in South Africa were asked to solve four stoichiometry calculation questions, and their written solutions (361 sets) were analyzed for consistency of various types of mole ratio related mistakes. The study found high levels of inconsistencies in each observed mistake, supporting the view that most participants’ knowledge lay closer to the fragmented end of the coherent–fragmented epistemological continuum. Evidence is presented for differential cognitive load, overgeneralization, and intuitive heuristics as possible explanations for these mistakes. Pedagogical suggestions are made for addressing such errors in in-service chemistry teacher workshops about stoichiometry calculations, although these may only be applicable where teacher subject matter knowledge tends to be low, such as within the developing world.
... In hindsight, Jordanian universities have started to develop special programs for preparing teachers in line with the recommendations of the Jordanian educational reform movement. These programs have sought to provide student teachers with appropriate knowledge, skills and tendencies that would contribute to the preparation of the teacher (Makovec, 2018;Fraihat et al., 2022;Gardee & Brodie, 2022), because there are essential things considered to ensure a high level for the teacher to participate in developing the teaching and learning 427 http://ijlter.org/index.php/ijlter process (Makovec, 2018;Verde & Valero, 2021). ...
We are very happy to publish this issue of the International Journal of Learning, Teaching and Educational Research. The International Journal of Learning, Teaching and Educational Research is a peer-reviewed open-access journal committed to publishing high-quality articles in the field of education. Submissions may include full-length articles, case studies and innovative solutions to problems faced by students, educators and directors of educational organisations.
To learn more about this journal, please visit the website http://www.ijlter.org.
We are grateful to the editor-in-chief, members of the Editorial Board and the reviewers for accepting only high quality articles in this issue.
We seize this opportunity to thank them for their great collaboration. The Editorial Board is composed of renowned people from across the world. Each paper is reviewed by at least two blind reviewers.
We will endeavour to ensure the reputation and quality of this journal with this issue.
... In hindsight, Jordanian universities have started to develop special programs for preparing teachers in line with the recommendations of the Jordanian educational reform movement. These programs have sought to provide student teachers with appropriate knowledge, skills and tendencies that would contribute to the preparation of the teacher (Makovec, 2018;Fraihat et al., 2022;Gardee & Brodie, 2022), because there are essential things considered to ensure a high level for the teacher to participate in developing the teaching and learning process (Makovec, 2018;Verde & Valero, 2021). Teacher-preparation programs accompanying the educational development process have been characterized by the inclusion of the so-called education field-training, which is a transitional stage between academic and behavioral education and the practicality of the teaching profession in order to bridge the gap between theory and practice (Verde & Valero, 2021;Zega & Lase, 2021). ...
The purpose of this study was to investigate the degree of effectiveness of the supervisory performance of cooperative teachers in improving the professional preparation of student teachers in the specialization of early childhood education.The participants were 155 student teachers enrolled in the education field-training program at Yarmouk University in Jordan. To achieve the purpose of the study, a questionnaire was developed, after insuring its validity and reliability. The findings showed that, according to the means and standard deviations of the questionnaire items, cooperative teachers did not fulfill their supervisory roles in improving the professional preparation of student teachers during the education field-training. The findings also showed that there were no statistically significant differences in the effectiveness of the supervisory performance of cooperative teachers due to gender, academic qualification and teaching experience. This indicates that these variables do not affect the cooperative teachers' supervisory role in improving student teachers' preparation. This study contributes to a better comprehension of the supervisory performance of cooperative teachers in improving the professional preparation of student teachers. It is suggested that the teacher education program at Yarmouk University should pay attention to improve student teachers' preparation during the education field-training phase.
... In hindsight, Jordanian universities have started to develop special programs for preparing teachers in line with the recommendations of the Jordanian educational reform movement. These programs have sought to provide student teachers with appropriate knowledge, skills and tendencies that would contribute to the preparation of the teacher (Makovec, 2018;Fraihat et al., 2022;Gardee & Brodie, 2022), because there are essential things considered to ensure a high level for the teacher to participate in developing the teaching and learning process (Makovec, 2018;Verde & Valero, 2021). Teacher-preparation programs accompanying the educational development process have been characterized by the inclusion of the so-called education field-training, which is a transitional stage between academic and behavioral education and the practicality of the teaching profession in order to bridge the gap between theory and practice (Verde & Valero, 2021;Zega & Lase, 2021). ...
The purpose of this study was to investigate the degree of effectiveness of the supervisory performance of cooperative teachers in improving the professional preparation of student teachers in the specialization of early childhood education.The participants were 155 student teachers enrolled in the education field-training program at Yarmouk University in Jordan. To achieve the purpose of the study, a questionnaire was developed, after insuring its validity and reliability. The findings showed that, according to the means and standard deviations of the questionnaire items, cooperative teachers did not fulfill their supervisory roles in improving the professional preparation of student teachers during the education field-training. The findings also showed that there were no statistically significant differences in the effectiveness of the supervisory performance of cooperative teachers due to gender, academic qualification and teaching experience. This indicates that these variables do not affect the cooperative teachers' supervisory role in improving student teachers' preparation. This study contributes to a better comprehension of the supervisory performance of cooperative teachers in improving the professional preparation of student teachers. It is suggested that the teacher education program at Yarmouk University should pay attention to improve student teachers' preparation during the education field-training phase.
Empirical research has demonstrated that mathematical errors originate from various causes that may have different implications for the learning process. Mathematical errors are an international phenomenon as all teachers from different cultural background must handle them every day during teaching-and-learning processes. When errors arise, teachers must determine whether and how to deal with them and whether the error should be corrected. Although errors and how they are addressed by teachers are of high pedagogical importance, teachers’ approaches to handling student errors remain underexplored. The study reported herein examines the ways in which mathematics teachers handle student errors in various situations, and whether any connection between the error type and the manner in which it is corrected may be reconstructed. The study participants comprised 13 secondary mathematics teachers in a large city in northern Germany. Each teacher was video-recorded over two 90-min-lessons, and the data were subsequently evaluated using qualitative content analysis. The results indicate that teachers’ approaches to correcting mathematical student errors diverged considerably depending on the phase of the lesson during which the error occurred. In particular, during the class discussion phase, a high proportion of the corrections were performed by the students’ classmates, whereas corrections executed during the group work, partner work, or individual work phases were primarily performed by students who had made the errors and the teachers. Given that the teacher's approach to handling errors exerts a significant influence on teaching-and-learning processes, further research is required to identify the most prolific approaches to handling errors in order to foster students’ mathematical understanding.
Identitas dalam penelitian pendidikan semakin banyak dimanfaatkan khususnya dalam mendeskripsikan dan mengeksplor kegigihan seorang peserta didik dalam memahami suatu materi. Kaitannya dengan konteks matematika, Identitas merupakan hasil deskripsi dari bagaimana seorang peserta didik dipandang sebagai seseorang yang ‘memiliki matematika’ (math person) oleh orang lain ataupun diri individu itu sendiri. Dengan demikian, siswa dan guru membutuhkan peningkatan identitas matematika agar mendapatkan karakter matematika yang kuat dan abadi dalam mengaplikasikan konsep-konsep matematika. Artikel ini menyajikan systematic literature review (SLR) tentang identitas matematika yang diterbitkan dalam lima tahun terakhir. Penelitian ini telah mengumpulkan informasi dan hasil studi peneliti dari seluruh dunia untuk memberikan gambaran umum tentang penelitian identitas matematika dalam pembelajaran matematika di sekolah. Pertanyaan kunci dalam penelitian ini adalah bagaimana artikel ini didistribusikan berdasarkan tahun publikasi, jenjang pendidikan, metode, kerangka konseptual identitas matematika, dan ukuran tingkatan identitas matematika. Penelitian ini berpedoman pada Preferred Reporting Items for Systematic Review and Meta-Analysis (PRISMA) yang menghasilkan artikel penelitian untuk ditinjau secara sistematis. Hasil penelitian menunjukkan bahwa studi terkait identitas matematika pada tahun 2019 dan 2020 merupakan masa berkembangnya publikasi artikel terkait identitas matematika di jurnal internasional yang bereputasi, para peneliti lebih tertarik untuk melakukan penelitian identitas matematika di tingkat sekolah dasar, sebagian besar studi tentang identitas matematika menerapkan metode kualitatif menggunakan wawancara sebagai instrumen kunci dalam mengungkap dan mendeskripsikan identitas matematika seseorang.
Identity research in mathematics education has become increasingly prominent over the past two decades. In the last few years, there have been several reviews of identity literature in the field of mathematics education generally, or specifically focused on mathematics learner identities or mathematics teacher identity. We begin our paper by summarizing the key findings of these reviews, pointing to various categorizations proposed by their authors, and the key critiques raised therein. We then report on our more recent review of identity literature in mathematics education published in the top twenty mathematics education journals over the past 5 years (47 articles in total). This review enables us firstly to speak descriptively about the extent to which the field is evolving (or not) in relation to earlier findings (dominant regions, focus of research, methods and perspectives used). Secondly, following deeper analysis of the papers, we speak to the way in which the field is addressing (or not) key critiques raised by reviews. Our analysis points to several theoretical, methodological and empirical absences and challenges requiring further engagement. We further highlight key absences in the research, particularly a lack of cross-national comparative studies and studies that connect learner and teacher identities. We then provide a summary discussion and analysis of the contributions of the present special issue papers and engage with how these take the field forward. Finally, we engage with the implications of our analysis and recommendations for future directions for research.
In this article, we demonstrate how I poems, part of the Listening Guide method, can help researchers to better understand the multiple voices that might comprise a student’s mathematical identity. We provide an analysis of one student’s interview as an example of this method. We argue that the use of the Listening Guide method illuminates the complexity of students’ mathematical identities, especially their fluid, negotiated, and sometimes contested character. In the example shared here, we show how others’ discourses become part of a student’s developing mathematical identity, as well as highlight how the addition of drawings enhances the Listening Guide method.
This paper presents the results of a systematic review of the empirical literature, reported in research journals, on Mathematics Learner Identity (MLI). In the mathematics education research field, arguments have been made that the conceptualization of Mathematics Identity (MI) is inconsistent and this makes the literature as a whole incoherent. This study aims to summarize how the concept has been employed and to develop an integrative model for analyzing concept-definitions and concept-operationalizations of MLI. An analysis of 69 papers revealed 3 main dimensions that characterized how identity is conceptually defined in any research literature (social/subjective, enacted/representational and change/stability) and 5 main categories that describe how the literature has implemented these dimensions operationally (identity as individual attributes; identity as narratives; identity as a relationship with specific practices; identity as ways of acting; and identities as afforded and constrained by local practices). An emphasis on representational aspects of identities, particularly during higher education, and enacted and practice-related identities during primary and secondary education research was found. The discussion shows how the field would be clarified if studies made their choices of conceptualization clear in terms of these dimensions and categories, and if research as a whole becomes more aware of restrictions/limitations in identity research, particularly at different points of the educational trajectory of students.
Professional learning communities as a form of teacher development have been in existence internationally for some time now and more recently in South Africa. Although strong claims have been made for their influence on teacher practices, very few research studies have investigated these claims. This paper presents a case study that connects teacher-learning activities in a professional learning community to shifts in teaching practices. Four high school mathematics teachers’ practices were analysed before, during and after their participation in a professional learning community. The results show that two teachers made modest shifts in three of the five key dimensions of their lessons and one of the two teachers sustained these shifts. The two other teachers did not make major shifts in their teaching. The shifts are linked to learning activities in the professional learning community, which involved the teachers in conversations about learners’ errors and learners’ learning needs, and designing lessons to address these needs. Findings from the study indicate how teacher-learning activities in a professional learning community can support shifts in teachers’ teaching and explain why changes differ among teachers.
The interest in teachers’ discourses and vocabularies has for a long time been studied under the rubric of knowledge, most notably teachers’ professional knowledge. This interest can be traced back to Shulman’s distinction between different kinds of teacher knowledge and Schwab’s interest in the role of practical reasoning and judgement in teaching. Within the research, a distinction can be found between a more narrow approach that focuses on teachers’ propositional or theoretical knowledge and a more encompassing approach in which teachers’ knowledge is not only the knowledge for teachers generated elsewhere, but also the knowledge of teachers. This is the ‘stock of knowledge’ gained from a range of sources and experiences, including teachers’ ongoing engagement with the practice of teaching itself. In this paper, we focus on the role of teachers’ talk in their achievement of agency. We explore how, in what way and to what extent such talk helps or hinders teachers in exerting control over and giving direction to their everyday practices, bearing in mind that such practices are not just the outcome of teachers’ judgements and actions, but are also shaped by the structures and cultures within which teachers work.
This paper examines the literature on identity within mathematics education published in journals over the past two decades. It analyses the theoretical underpinnings, research methods and definitions of identity, providing a critique rather than a summary of the literature. A total of 188 articles from 85 different journals are reviewed in the sample. This review finds support for common complaints of this research area as lacking in definitions of the concept of identity and suggests that the writing in this topic is at times theoretically incompatible. Furthermore, the work in this field may be coming from two distinct paradigms. Identity may be seen as an action and fit within a sociological frame or it may be seen as an acquisition, fitting within a psychological framing. Defining identity as something we do, as an action, and in particular as performative is promoted in this paper. Finally, suggestions are made for future directions in identity research.
How errors are dealt with in a mathematics classroom is important as it can either support or deny learner access to mathematical knowledge. This study examines how a teacher, who participated in a professional development programme that focused on learner errors, engaged with mathematical errors in her Grade 9 classroom. Data were collected over a twoyear period in the form of videotapes and were analysed qualitatively. Our findings illustrate that this teacher dealt with four types of mistakes: slips, errors derived from misconceptions, language-related errors and errors that occurred from the incorrect usage of the calculator. She dealt with these by correcting, probing or embracing them. We found that, over time, this teacher dealt with more errors and corrected and embraced errors in different ways. We recommend that teachers use their professional knowledge to decide when, why and how it is appropriate to correct, probe or embrace errors in light of their knowledge of the content and their learners.
The mathematics education field, including prospective teacher education program, has seen a continuous effort to change teaching practices to be more cognitively demanding, conceptually oriented and student centred. Our goal in this study was to examine how certain underlying assumptions about mathematical learning, as reflected in a skilled instructor's discourse, align with opportunities to learn. The data included a set of fully transcribed 11 lessons from an introductory algebra course. The method of analysis was built upon the communicational (commognitive) framework and included discerning between the instructor's mathematizing and identifying talk. This framework was extended to quantify the instructor's identifying talk over the whole set of lessons. Our findings showed that at the surface level, the instruction in the class seemed to align with ''ex-plorative'' goals. On a deeper level, however, it was more aligned with ''ritual'' goals that are concerned with producing narratives about people, not about mathematics.
In this paper I share some key principles and examples from the Data Informed Practice Improvement Project. The project develops an innovative model of mathematics teacher development, with three main strands: teachers work with data from their classrooms, they use this data to understand and engage with learner errors in mathematics, and they do this collectively in professional learning communities, with facilitation from members of the project team. I describe each of these strands and present three examples that show how the strands work together to support teacher learning in professional communities. These examples illuminate a key achievement of the project: a focus on learner errors in professional learning communities can develop powerful conceptual knowledge of mathematics among teachers at the same time as developing teachers’ knowledge for teaching and teaching practices.
Research on learner errors in mathematics education is beginning to focus on how teachers can learn to identify and engage with the reasoning behind these errors. Research on professional learning communities is beginning to show that they present powerful opportunities for on-going teacher collaboration and learning. In this paper, I bring the two areas of research together. Drawing on data from one professional learning community in the Data Informed Practice Improvement Project, I show how teachers in this community came to understand key concepts about learner errors and shifted their ways of talking about learner errors. I identify three important shifts that the teachers made in their learning about learner errors: from identifying to interpreting errors; from interpreting to engaging errors; and from focusing on learner errors to focusing on their own knowledge. I argue that these three shifts suggest a deepening of teachers’ thinking in relation to learner errors.
Our primary purpose in this article is to propose an interpretive scheme for analyzing the identities that students develop in mathematics classrooms that can inform instruc- tional design and teaching. We first introduce the key constructs of normative iden- tity and personal identity, and then illustrate how they can be used to conduct empir- ical analyses. The case on which the sample analysis focuses concerns a single group of middle school students who were members of two contrasting classrooms in which what it meant to know and do mathematics differed significantly. The resulting analyses document the forms of agency that students can legitimately exercise in partic- ular classrooms, together with how authority is distributed and thus to whom students are accountable, and what they are accountable for mathematically. In the final section of the article, we clarify the relation of the interpretive scheme to other current work on the identities that students are developing in mathematics classrooms.
The concept of agency has become widely used in learning research, especially in studies addressing professional and workplace learning, but also in policy discussion on how to promote individually meaningful careers and life-courses amid rapid changes in working life. The purpose of this article is to provide a critical review of the multidisciplinary concept of agency, and to suggest a fruitful conceptualization of professional agency at work. The following questions are addressed: (i) How have the ontological characteristics and manifestations of agency been understood? (ii) How have the relationships between the social and individual aspects of agency been understood? We examined previous studies and discussions on agency in the fields of education and social sciences, looking selectively also at psychology and gender studies. We identified four major research traditions in which notions of agency were prominent: (i) the social science tradition, (ii) the post-structural tradition, (iii) the socio-cultural approach, and (iv) the identity and life-course approach. Analyses within and across these traditions brought out a range of understandings and manifestations; thus agency might be viewed merely as rational and intentional activity, or else it might be seen from a temporally broad perspective, covering subjects’ ontogenetic development, and encompassing discursive, practical, and embodied relations with the world. Analysis of the relationships between individual and social/contextual elements revealed assumptions ranging from analytical inseparability to separateness, and in case of analytical separateness assumptions of strong or weak contextual influence. Based on our review, we suggest a conceptualization of professional agency from a subject-centered socio-cultural perspective. This takes individual agency and social context to be analytically separate, but mutually constitutive, and in complex ways highly interdependent. The suggested conceptualization is summarized in terms of seven propositions.
The interpretive framework for the study of learning introduced in this article and called commognitive is grounded in the assumption that thinking is a form of communication and that learning mathematics is tantamount to modifying and extending one's discourse. These basic tenets lead to the conclusion that substantial discursive change, rather than being necessitated by an extradiscursive reality, is spurred by commognitive conflict, that is, by the situation that arises whenever different interlocutors are acting according to differing discursive rules. The framework is applied in 2 studies, one of them featuring a class learning about negative numbers and the other focusing on 2 first graders learning about triangles and quadrilaterals. In both cases, the analysis of data is guided by questions about (a) features of the new mathematical discourse that set it apart from the mathematical discourse in which the students were conversant when the learning began; (b) students' and teachers' efforts toward the necessary discursive transformation; and (c) effects of the learning–teaching process, that is, the extent of discursive change actually resulting from these efforts. One of the claims corroborated by the findings is that school learning requires an active lead of an experienced interlocutor and needs to be fueled by a learning-teaching agreement between the interlocutor and the learners.
This article uses a critical evaluation of research on student misconceptions in science and mathematics to articulate a constructivist view of learning in which student conceptions play productive roles in the acquisition of expertise. We acknowledge and build on the empirical results of misconceptions research but question accompanying views of the character, origins, and growth of students’ conceptions. Students have often been seen as holding flawed ideas, which are strongly held, interfere with learning, and which instruction must confront and replace. We argue that this view overemphasizes the discontinuity between students and expert scientists and mathematicians, making the acquisition of expertise difficult to conceptualize. It also conflicts with the basic premise of constructivism, that students build more advanced knowledge from prior understandings. Using case analyses we dispute some commonly cited dimensions of discontinuity and identify important continuities that have been previously ignored or underemphasized. We highlight elements of knowledge that serve both novices and experts, albeit in different contexts, and under different conditions. We provide an initial sketch a constructivist theory of learning that interprets students ’ prior conceptions as resources for cognitive growth within a “complex systems ” view of knowledge. This theoretical perspective aims to characterize the interrelationships among diverse knowledge elements rather than identify particular flawed conceptions; itMisconceptions Reconceived 3 emphasizes knowledge refinement and reorganization, rather than replacement, as primary metaphors for learning; and it provides a framework for understanding misconceptions as both flawed and
In this article, the authors make an attempt to operationalize the notion of identity to justify the claim about its potential as an analytic tool for investigating learning. They define identity as a set of reifying, significant, endorsable stories about a person. These stories, even if individually told, are products of a collective storytelling. The authors’ main claim is that learning may be thought of as closing the gap between actual identity and designated identity, two sets of reifying significant stories about the learner that are also endorsed by the learner. Empirical illustration comes from a study in which the mathematical learning practices of a group of 17-year-old immigrant students from the former Soviet Union, newly arrived in Israel, were compared with those of native Israelis.
Teacher learning through professional development is a complex process and is not yet well understood. Some features of professional
development programs are known to be important, such as a focus on learner needs, design of and reflection on classroom artefacts,
and the creation and sustaining of communities of support for teacher professional learning. In this paper, we describe the
workings of such communities in a teacher professional development program, which focused on learner errors in a well-researched
mathematical topic—the equal sign. Drawing on data from program sessions where teachers discussed their lesson designs and
reflections on their teaching with each other, we develop the notions of challenge and solidarity as important in developing
accountability conversations among teachers. We show how our program supported teachers to challenge each other and to build
solidarity with each other and in so doing to develop accountability to each other and the profession, for their practices
and their learning.
KeywordsProfessional development–Professional learning communities–Accountability
The field of mathematics education has seen an increase in research focused on the concept of identity, with many researchers focusing on the role of teacher pedagogies in shaping learners’ identities and their participation in and engagement with mathematics. In this paper, the actual and narrated practices of two teachers, which incorporate teachers’ pedagogical approaches and social relationships with learners, are analysed in relation to how they provided opportunities for the construction of learner identities. The paper draws on data that were collected over two years in the form of videotaped lessons, field notes and audiotaped semi-structured interviews with teachers and learners. Data were analysed using qualitative data analysis techniques. The analysis suggests a relationship between the teachers’ actual and narrated practices, with the teachers’ practices influencing the kinds of identities constructed by learners in their classrooms, in terms of learners’ enjoyment of learning mathematics and their willingness to participate in and become fully fledged members of their classroom community.
Scholars who take a situative perspective on mathematics identity have focused on different kinds of resources and mechanisms that are available for identity development, particularly for students from minoritized backgrounds. Although these approaches are generally not contradictory, they are not always consistent. Differences in what is foregrounded and backgrounded in these studies mean that at times it can be difficult to develop a comprehensive picture of identity as it develops in the mathematics classroom. Our goal in this paper is to connect these literatures with an eye towards offering a design heuristic for mathematics educators interested in accounting for students’ identity development in mathematics teaching and learning. To do so, we present a model of mathematical identities as they are realized in the classroom, articulate the elements involved in the development of such an identity, and then use the model to analyze approaches to intervene in classroom systems by focusing on different elements as starting points.
The belief that mathematics ability is a fixed trait is particularly common and may be a key reason for many students' disinterest and underperformance in mathematics. This study investigates how mathematics teaching practices might contribute to students' beliefs about mathematics ability being a fixed or malleable trait (mindset). Through a synthesis of existing literature and an analysis of data from classroom observations, this article presents a framework of teaching practices and identifies how varying implementations of these practices can be classified along a continuum from conveying fixed-mindset messages to conveying growth-mindset messages related to mathematics ability. © 2018 National Council of Teachers of Mathematics. All rights reservsed.
Teaching approaches and assessment practices are key factors that contribute to the improvement of learner outcomes. The study on which this article is based, explored the methods used by KwaZulu-Natal (KZN) teachers in teaching and assessing mathematics and statistics. An instrument containing closed and open-ended questions was distributed to seventy-five KZN mathematics teachers from Grade Four upwards. Teachers were encouraged to write freely about the different teaching methods and assessments that they used in the classroom. The findings revealed that teachers were more likely to report a single method in teaching statistics, but more than one method for teaching mathematics topics. In terms of assessments, the teachers generally reported the use of a single method. We also found that teachers mostly focus on teacher-led instructional methods and formal assessments. Furthermore, the findings revealed that teachers' demographic factors such as gender, age, teaching experience, participation in professional development course and further studies are associated with the choice of a variety of teaching and assessment methods. It is recommended that professional development courses ought to focus on helping teachers to increase their repertoire of teaching and assessment strategies.
In this article, we explore the role of racialized narratives (e.g., "Asians are good at math") in the mathematics learning experiences of African American male students. Drawing on concepts from sociocultural theory and cultural psychology, we conceptualize racialized narratives as dynamic cultural artifacts, which students appropriate and deploy in processes of identification and positioning. Interview data suggest that students had deep knowledge of these narratives, and made sense of them in a way that linked perceptions of Asian Americans as a mathematically gifted "model minority" to perceptions of African American males as intellectually inferior. Despite this positioning, many students spoke of re-purposing racialized narratives in order to assume positions of mathematical competence. The learning sciences has long known that learning is deeply related to engagement; learning occurs when one is invested in the learning setting, and when one has access to the resources necessary to learn. Identity, then, is deeply intertwined with processes of learning, because identity speaks to one's sense of connection and belonging. In their groundbreaking book, Situated Learning, Lave and Wenger (1991) tell the story of a newcomer to the activity of meat-packing. He is charged with the task of wrapping the meat, and is sequestered in that task in a way that prevents him from learning how to cut the meat. They argue that this leads to him being unable to develop an identity as a meat-cutter, thus relegating him only to wrapping the meat and accepting a marginalized identity. This story illustrates the ways that we have typically thought about identity and learning in the sociocultural literature on learning. We have come to view identities as being rooted in and developed through the everyday learning practices that we have access to and engagement in (Wenger, 1998; Nasir & Hand, 2008).
Prologue Part I. Practice: Introduction I 1. Meaning 2. Community 3. Learning 4. Boundary 5. Locality Coda I. Knowing in practice Part II. Identity: Introduction II 6. Identity in practice 7. Participation and non-participation 8. Modes of belonging 9. Identification and negotiability Coda II. Learning communities Conclusion: Introduction III 10. Learning architectures 11. Organizations 12. Education Epilogue.
This collective case study examines the influence of 4 third-grade teachers' beliefs and knowledge on their error-handling practices during class discussion of mathematics. Across cases, 3 dimensions of teachers' error-handling practices are identified and discussed in relation to teacher beliefs and knowledge: (a) intentional focus on flawed solutions in class discussion, (b) promotion of conceptual understanding through discussion of errors, and (c) mobilization of a community of learners to address errors. Study findings suggest that, although teachers' ways of handling student errors during class discussion of mathematics are clearly linked to both teacher beliefs and teacher knowledge, some aspects of teacher response are more strongly linked to knowledge and others are influenced more by beliefs.
This study focuses on teacher learning of student thinking through grading homework, assessing and analyzing misconceptions. The data were collected from 10 teachers at fifth–eighth grade levels in the USA. The results show that assessing and analyzing misconceptions from grading homework is an important approach to acquiring knowledge of students’ thinking. By engaging in the inquiry process of the 4 steps of identifying errors, analyzing reasons for the errors, designing approaches for correction, and taking action for correction, the teachers made obvious progress in their knowledge of students’ thinking, understood the difficulties and challenges their students had in learning mathematics, and enhanced their pedagogical content knowledge.
The treatment of errors in mathematics classrooms has gained attention in recent years, with many researchers suggesting that errors should be used as starting points for student inquiry into mathematics. In the study reported in this article, we examined how teachers used discourse around errors to generate inquiry by looking at the treatment of mistakes in U.S. and Chinese elementary mathematics lessons. To do so, we videotaped 44 lessons from Chinese and U.S. first-grade (n = 15), and fourth- and fifth-grade (n = 29) classrooms and also interviewed the teachers of the lessons. We separated the lessons by the topic taught (place value or fractions) and analyzed them for frequency of students' errors and the types of teachers' responses to these errors. Results indicated that U.S. and Chinese students made errors at similar frequencies. However, the teachers in the 2 countries responded to errors differently. In particular, the U.S. teachers made more statements about errors than the Chinese teachers, who instead asked more follow-up questions about errors. Relying on qualitative analysis of teacher interview and in-class statements about errors, we shed light on both how teachers used errors for inquiry and what teachers believed about errors.
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