Journal of Mathematics Teacher Education

Published by Springer Nature
Online ISSN: 1573-1820
Print ISSN: 1386-4416
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Recent publications
  • Sarah LillySarah Lilly
  • Kristen N. BiedaKristen N. Bieda
  • Peter A. YoungsPeter A. Youngs
Given that early career elementary teachers face pressure to plan pedagogy that provides equitable opportunities to learn mathematics for understanding, it is important to consider their mathematics lesson planning practices. This study uses interview data to compare the planning practices of four early career teachers (ECTs) who consistently rated highly on an observational measure of high-quality mathematics instruction with those of three ECTs who consistently rated lower. ECTs who scored highly addressed planning challenges in interconnected ways that were responsive to students’ needs, demonstrated more agency in representing mathematics content in their planning, and were more likely to maintain curricular rigor when differentiating. We consider implications for efforts to help ECTs plan lessons that are likely to result in high-quality instruction.
 
Teachers’ use of moves that enact epistemological frames
Teachers’ use of moves that enact positional frames
This interpretive cross-case study investigates complexity in the ways teachers frame mistakes and the reasons behind their framing, challenging the assumption in the literature that productive beliefs about errors generate productive error-handling practices, while unproductive beliefs result in unproductive practices. The study draws on interviews and classroom observations with three secondary mathematics teachers who were selected using purposive sampling. Analysis examines how the teachers framed mistakes in their classroom practice and the instructional moves that enacted each frame. We paid particular attention to epistemological frames, which concern the value of mistakes in the construction of mathematical knowledge, and positional frames, which concern the roles that students are authorized or obligated to take to address errors, researching how teachers’ epistemological and positional framing of errors were related. The results suggest that teachers invoked four primary frames related to errors: a epistemological frame of errors-as-resources, a positional frame of students-as-capable, an epistemological frame of errors-as-deficiencies, and an positional frame of students-as-incapable. We discuss barriers to using mistakes as resources for learning and implications for mathematics teacher education.
 
This study explores secondary mathematics teachers’ perceptions of the experiences that contributed to their capacities to understand mathematical modeling and to facilitate students’ modeling experiences. The retrospective research methods and transformative learning theory frame used in the study honor teachers as adult learners and value their perspectives while providing a way to study the complexity of learning to model and to teach modeling. Data analysis identified triggers and knowledge dilemmas that challenged and prompted teacher learning as well as opportunities to resolve dilemmas through rational discourse and critical reflection. Patterns in teacher-identified meaningful learning experiences reveal a trajectory with strands that address aspects of doing and teaching mathematical modeling: mathematics, social aspects of learning, real-world contexts, student thinking, and curriculum. Results of this study provide a holistic view of learning to do and teach mathematical modeling, complementing studies of designed professional learning interventions that out of necessity target specific parts of the modeling process. The results both support and challenge common teacher education content and practices. The study illustrates the usefulness of retrospective methods to understand teachers as lifelong learners.
 
A 12-episode constructivist teaching experiment with two pairs of elementary preservice teachers was conducted to examine how they reason distributively and proportionally. Specifically, I studied how prospective teachers reason with ratios as multiplicative comparisons. Prospective teachers solved different problems in one specific context (Step Problems) that require to reason with ratios as multiplicative comparisons, and represented their pictures during the episodes and the follow-up interviews. Second-order models of four prospective teachers were made. In this study I found that reasoning with ratios as multiplicative comparisons in that specific context of problems was challenging for prospective teachers and all four had considerable difficulty. Only one of the four prospective teachers fully reasoned with ratios as multiplicative comparisons. The findings of this study provide teacher educators with a new perspective on prospective teachers’ quantitative understanding of proportional reasoning, a domain as we know little about, and has implications for the mathematical preparation of prospective elementary teachers.
 
The implementation of large-scale intervention and development projects is often problematic, and the impacts of such projects usually fall somewhat short of what was expected. Additionally, the rationalities of intervention projects are not carried over into classroom teaching as directly as expected. This problem is generally known, but comprehensive explanations continue to elude the research community at large. Using the theory of cultural-historical activity theory (CHAT), we propose that the heart of the problem lies in the expansive learning process that teachers undergo. This process is driven by unrecognised contradictions in terms of cultural and historical origin, which are fundamentally different from the processes governing the projects. We analyze two cases taken from two large Danish professional development projects. In each case, we focus on a teacher as part of two activity systems (‘the project’ and ‘the classroom’) and how the contradictions within and between these shape learning through epistemic actions. The results indicate the importance of making these contradictions apparent and accessible to everyone in the activity systems. Because of these various contradictions, the agency conferred upon teachers leads to unintended outcomes.
 
The proposed model with standardized path coefficients (Note: CPB: constructivist pedagogical beliefs; TPB: traditional pedagogical beliefs; PEU: perceived ease of use; PU: perceived usefulness; ATT: attitude toward technology; BIU: behavioral intention to use; single-line arrow: positive influence; double-line arrow: negative influence)
Pedagogical beliefs are a critical factor in terms of integrating technology into teaching, but very few technology acceptance models (TAMs) have considered them. Hence, this study aims to extend the TAM by incorporating pre-service teachers' conception of teaching and learning. The revised model examined the influence of pre-service mathematics teachers' constructivist and traditional pedagogical beliefs on their technology acceptance through perceived ease of use, perceived usefulness, attitude toward technology, and behavioral intention to use. Survey data were collected from 714 pre-service mathematics teachers in Turkey and analyzed through path analysis. The results showed that pre-service mathematics teachers' pedagogical beliefs were more constructivist-oriented than traditional oriented , and constructivist beliefs had a significant influence on the components of the TAM. On the other hand, pre-service teachers' traditional-oriented beliefs did not influence their perceived usefulness of and attitudes toward technology but had positive effects on perceived ease of use. Implications for pre-service mathematics teacher education were discussed.
 
Model of a teacher's decision-making during interaction with students
Illustration of solution methods
Often, mathematics teachers do not incorporate whole-class discourse of students’ various ideas and solution methods into their teaching practice. Particularly complex is the in-the-moment decision-making that is necessary to build on students’ thinking and develop their collective construction of mathematics. This study explores the decision-making patterns of five experienced Dutch mathematics teachers during their novice attempts at orchestrating whole-class discourse concerning students’ various solution methods. Our goal has been to unpack the complexity of their in-the-moment decision-making during whole-class discourse through lesson observations and stimulated recall interviews. We investigated teacher decision-making adopting a model that combines two perspectives, namely (1) we explored student-teacher interaction with regard to building on student thinking and (2) we explored how the teachers based decisions during such interaction upon their own personal conceptions and interpretation of student thinking. During these novice attempts at orchestrating whole-class discourse, the teachers created many situations for students to articulate their thinking. We found that at certain instances, teachers’ in-the-moment decision-making resulted in opportunities to build on student thinking that were not completely seized. During such instances, the teachers’ decision-making was shaped by the teachers’ own conceptions of the relevant mathematics and by teacher conceptions that centered around student understanding and mathematical goals. Our findings suggest that teachers might be supported in their novice attempts at whole-class discourse by explicit discussion of the mathematics and of their conceptions with regard to student understanding and mathematical goals.
 
Sample questions from 1st and 2nd interviews
To better understand the critical early years of teaching and teacher identity development, this study explored the mathematics teacher identity of two early career middle school mathematics teachers and the influence of their working communities on its development. Emel and Berk graduated from the same reform-oriented teacher education program and were working in fundamentally different working communities that could be characterized as reform-unsupportive (Emel) and reform-supportive (Berk). They were interviewed three times and were observed extensively in their working communities. Findings revealed two different teacher identities: Emel exhibited a mathematics teacher identity that was more reform-oriented than non-reform-oriented, and Berk exhibited a non-reform-oriented mathematics teacher identity. Their working communities had influenced their mathematics teacher identity in ways that were both theoretically expected (positive effects of supportive conditions and negative effects of unsupportive conditions) and surprising (positive effects of negative conditions and no effects of positive conditions). The findings are discussed in detail with implications for teacher education programs and supports for early career teachers.
 
Studies of facilitators of professional development (PD) for mathematics teachers have been increasing in order to improve their preparation for conducting PD. However, specifications of what facilitators should learn often lack a conceptualization that captures facilitators’ expertise for different PD content. In this article, we provide a framework for facilitator expertise that is in line with current conceptualizations but makes explicit the content-related aspects of such expertise. The framework for content-related facilitator expertise combines cognitive and situated perspectives and allows unpacking different components at the PD level and the classroom level. Using two illustrative cases of different PD content (probability education in primary school and language-responsive mathematics teaching in secondary school), we exemplify how the framework can help to analyze facilitators’ practices in content-related ways in a descriptive mode. This analysis reveals valuable insights that support designers of facilitator preparation programs to specify what facilitators should learn in a prescriptive mode. We particularly emphasize the importance of working on content-related aspects, unpacking the PD content goals into the content knowledge and pedagogical content knowledge elements on the classroom level and developing facilitators’ pedagogical content knowledge on the PD level (PCK-PD), which includes curricular knowledge, as well as knowledge about teachers’ typical thinking about a specific PD content. Situated learning opportunities in facilitator preparation programs can support facilitators to activate these knowledge elements for managing typical situational demands in PD.
 
Percentage of responding institutions offering these mathematics content courses
Percentage of institutions including the listed mathematics content
Percentage of schools reporting teaching by instructional approach
Number of credits offered and required
This article reports on a second national survey of higher education institutions in the USA to answer the question “Who teaches mathematics content courses for prospective elementary teachers, and what are these instructors’ academic and teaching backgrounds?” and addresses valuable information not collected with the first survey conducted in 2010. We surveyed 1740 institutions and a faculty member from each of 413 institutions (23.7%) participated in the survey. The survey results demonstrate that the majority of these institutions are not meeting the recommendations of the Conference Board of Mathematical Sciences (2012) and the Association of Mathematics Teacher Educators (2017) for prospective elementary teachers to take at least 12 semester-hour credits of mathematics content designed specifically for them. The data do indicate that there is movement toward more activity-based approaches for these courses as compared to the 2010 survey. Additionally, there is an increase in these courses having instructors with doctorates in mathematics education as well as an increase in instructors having grades 7–12 teaching experience. Most instructors for these courses do not have elementary teaching experience and have likely not had opportunities to think deeply about the important ideas in elementary mathematics. While most institutions still do not provide training and/or support for these instructors, formalized support and training appear to be increasing since the 2010 survey.
 
This research investigates how a lesson study (LS) on designing and implementing challenging tasks impacts Vietnamese high school mathematics teacher knowledge and beliefs. Its contribution highlights cultural considerations when adopting LS to the forefront to contextualize the impacts. The results show that the teachers developed their specialized content knowledge by attending to students’ mathematics and creating cognitive conflicts building on student responses. The teachers changed their curriculum knowledge from implementer to transformer, improved knowledge about content and students attending to difficulties and misconceptions, and enhanced their knowledge about content and teaching in ways they designed, sequenced, and evaluated approaches that fit student learning. Finally, they changed their beliefs about mathematics to a comprehensive view of knowledge, mathematical proficiency, and sophisticated beliefs of teaching and learning. Discussion about the essence of LS when adopting it to different cultures is included.
 
Expanded instructional triangle for engaging students in mathematical practice. Note.
Adapted from Cohen et al. (2003)
The Evolution of Meghan’s Counterexamples. Note. The first image shows the counterexample that Meghan drew during the first interjection, after being prompted by the TE to draw a counterexample. The second image shows the counterexample that Meghan drew after Samantha suggested to draw the equilateral triangle without the bottom. The third image is the final counterexample that Meghan drew after Samantha prompted her with “Don’t draw the bottom.”
This study investigates the potential of rehearsal interjections to provide opportunities for novice teachers and teacher educators to discuss topics related to teaching the practice of mathematical defining. Through analysis of video recordings of seven elementary novice teachers’ rehearsals about geometric definitions, we identified the problems of practice that initiated rehearsal interjections, the topics discussed during rehearsal interjections, and relations between initiating problems of practice and topics discussed. We found that initiating problems of practice focused overwhelmingly on pedagogical issues, with most related to aspects specific to the teaching of mathematical defining. Likewise, discussions during interjections tended to focus on definitional pedagogical topics. Although epistemic topics were mentioned, they were only conveyed implicitly and at times in conflicting manners. Moreover, few opportunities arose for novices to make sense of student thinking about definitions and the mathematics of shape. Our results illustrate ways in which the goal of improving pedagogy, although important, can overshadow learning of other aspects for teaching mathematical practice.
 
Screenshot of the Teacher Responding Tool interface. A student explanation is shown in the blue box (top left) and the teacher can respond by writing in the bottom left box. The right column presents three recommendations for how the teacher might respond to the student explanation. These recommendations change when a different student explanation is displayed. In this example, the teacher has responded to the student using some text taken from the recommendations and some they wrote themselves
Screenshot of the visualization, data table, and prompt for Task A. When a student moves the scale factor slider, the visualization changes dynamically by increasing or decreasing in size and by changing the numerical values shown for the dimensions, volume, or surface area. The wrap-unwrap slider wraps or unwraps the net of the prism
Violin plots comparing the initial and revised explanations for variables scores (left) and relationships scores (right)
Shifts in variable and relationship scores from initial to revised
This study examines technology-enhanced teacher responses and students’ written mathematical explanations to understand how to support effective teacher responding and the centering of students’ mathematical ideas. Although prior research has focused on teacher noticing and responding to students’ mathematical ideas, few studies have explored the revisions that students make to their written explanations after teacher responding and very few explore this in authentic classroom contexts. Four high school geometry teachers and thirty of their students participated in project-based tasks examining the relationships between scale factor and the dimensions, surface area, and volume of a rectangular prism. The teachers’ written feedback and the students’ written explanations were coded and scored. Results show that student explanations improved significantly after students received feedback about their mathematical ideas. Furthermore, results indicate that the teacher feedback may be more effective if it focuses on mathematical relationships between variables and that additional feedback about the variables alone may have little impact. Results contribute to an understanding of how eliciting students’ written mathematical explanations might support effective teacher responding in classroom contexts.
 
Initiating teacher learning: a conceptual model
Social media is an increasingly prevalent informal learning site for mathematics teachers. These platforms offer an emotionally and philosophically supportive space for teachers who seek to address oppressive teaching practices within their school communities. In this paper, we examine interactions in a Facebook group with over 14,000 members to understand how a social media platform can be used by teachers to develop professionally. In our analysis of interactions within the Facebook group, we found teachers often use this space to seek support in combating the negative effects of tracking while also seeking support to better implement mixed ability learning opportunities. Through comments, community members encourage rehumanizing mathematics classrooms by promoting practices of creative insubordination (Gutiérrez in Teach Excell Equity Math 7(1):52–60, 2016).
 
How can teachers refine their strategies for purposefully engaging students in mathematical discussions when students are working in groups and the teacher enters an ongoing group conversation? In three educational design research cycles, four teachers collaborated with a researcher for one year to analyse, design and evaluate strategies for engaging students in small-group mathematical discussions. The idea of noticing (Mason in Researching your own practice: the discipline of noticing, RoutledgeFalmer, London, 2002; Sherin et al. in Mathematics teacher noticing: seeing through teachers’ eyes, Taylor & Francis, New York, 2011) was used to organize the findings—by paying attention to aspects in the mathematical discussions and interpreting the interactions, teachers could together refine their own actions/responses to better support students’ work. The Inquiry Co-operation Model of Alrø and Skovsmose (Dialogue and learning in mathematics education: intention, reflection, critique, Kluwer Academic Publishers, Dordrecht, 2004) was used as a theoretical base for understanding qualities in mathematical discussions. Ehrenfeld and Horn’s (Educ Stud Math 103(7):251–272, 2020) model of initiation-entry-focus-exit and participation was for interpreting and organizing the findings on teachers’ actions. The results show that teachers became more aware of the importance of explicit instructions and their own role as facilitators of mathematical questions to students, by directing specific mathematical questions to all students within the groups. In this article, by going back and forth between what happened in the teachers’ professional development group and in the classrooms, it was possible to simultaneously follow the teachers’ development processes and what changed in students’ mathematical discussions.
 
When teaching fractions, teachers make instructional decisions about if, when, and how to use the many different types of fraction models and manipulatives. In this study, we sought insights into their pedagogical reasoning with fraction representations via their preferences, both for solving tasks themselves and for teaching (in general and for specific fraction concepts and operations). Nearly 200 practising Australian primary teachers participated in an online survey and we drew on a Fraction Schemes theorisation to analyse quantitative and qualitative data. A majority of teachers indicated a personal preference for the set model for four out of five schemes; for one scheme most teachers preferred the circle model. Their reasons suggested that the nature of each task in a scheme and the specific fractions involved, played a role in influencing their preferences. With respect to teaching fractions, the teachers also indicated a high level of preference for teaching with the set model in general, and secondly for the rectangle model. Their preferences, except for number lines, were not found to be associated with the teachers’ nominated year level. We found that a high personal preference for a set model was associated with a preference for teaching with the same model in general, but not for teaching with the matching manipulative (counters or chips). The teachers indicated a high level of preference for teaching with the fraction bars manipulative for several fraction concepts, but this was not associated with a personal preference for linear models. Implications for further research are discussed.
 
This study articulates the epistemological support of reflective practices for the professional practice of crafting mathematics problems in realistic contexts. The data were collected from 15 middle school pre-service mathematics teachers in two semesters as they were generating problems and engaging in three reflective practices, which served to elicit their knowledge-in-action. The prospective teachers’ reflections indicated that the knowledge-in-action for crafting realistic mathematics problems encompassed various but particular forms of the content; that is, pedagogical, curricular, and interdisciplinary forms of the content. Thus, particularly for the professional practice of crafting realistic mathematics problems, the prospective teachers identified epistemological supports that blended not only content and pedagogy but also other knowledge dimensions, which have crucial implications for mathematics teacher education.
 
This paper focuses on the discourse of teachers and instructors in a TEAMS (Teaching Exploratively for All Mathematics Students) professional development (PD) setting and on the discursive mechanisms that afford such learning. We conceptualize learning in this PD as a change in a teacher's pedagogical discourse—from alignment with Delivery Pedagogical Discourse (DPD) to alignment with Explorative Pedagogical Discourse (EPD). This change in pedagogical discourse involves commognitive conflicts between the DPD and EPD. Data for the study include 8 videotaped sessions from a 2-year TEAMS PD for middle school mathematics teachers. Results show that conflict revolved around four themes. The most common was about students' abilities and its relation to instruction. The others were: Who is responsible for constructing mathematics knowledge in class? Limited view of the EPD, and what is considered ordinary instruction? We also identified four levels of conflict, ranging from commognitive conflict that is implicit to the participants, to conflict that is made explicit and reflected upon. Finally, results show movement from implicit conflict towards explication of conflict during the two years of the PD.
 
The instructional triangle
Means for pre-lesson item scores across all coach–teacher pairs from cycles 1 through 5
The past decade has witnessed a strong, standards-based call for improving what mathematics is taught and how it is taught. In the USA, districts have hired instructional coaches to help teachers shift their teaching from algorithm-based instruction to instruction that is more student-centered and conceptually focused. The purpose of this study was to contribute to the field’s understanding of (a) the specific coaching practices that help teachers enact more conceptual-based forms of instruction; and (b) how coaches learn to enact those practices. Using a design-based implementation research approach, we trained coaches using a particular model for one-on-one coaching (Content-Focused Coaching); the coaches then worked with teachers to plan lessons aligned with the coaching model. Data consisted of videotapes of pre-lesson conferences that were transcribed and coded according to the model. Analyses of 32 coaches’ practice over a 2-year period suggest that each of the three components of our coaching model (attention to student thinking, pedagogy, and mathematics) demonstrated statistically significant improvement over time. An illustrative analysis of five coaching sessions of one coach revealed a progression over five sessions from planning discussions that stayed at the level of general strategies to more specific conversations about teaching a particular task and then to deeper discussions that integrate attention to mathematical concepts, student thinking, and pedagogical moves. We view this delineation of coach learning as an important first step in laying the groundwork for the design of future coach training.
 
A textbook-adaptation approach to daily Kyouzaikenkyuu [Modified from Melville (2017)]
A curriculum-development approach to daily Kyouzaikenkyuu [Modified from Melville (2017)]
Kyouzaikenkyuu (translated as instructional materials research) is said to be a crucial part of successful Japanese lesson study. Kyouzaikenkyuu is described as the planning portion of the research lesson during lesson study. Kyouzaikenkyuu is also said to be done on a daily basis by Japanese teachers; however, there is very little written about this process in the literature. This study takes an in-depth look into what Japanese junior high mathematics teachers do during their daily kyouzaikenkyuu, and why they do it. Through interviews, observations, and participation in kyouzaikenkyuu, we explain the process and key ideas that many Japanese mathematics teachers go through to prepare their lessons. We document two different approaches to kyouzaikenkyuu. We also found that for some teachers there is a large difference between what they do during kyouzaikenkyuu for a research lesson compared to the kyouzaikenkyuu of a daily lesson. If teachers outside of Japan wish to engage in kyouzaikenkyuu and lesson study, these approaches are a good place to start.
 
Journey of the mathematics problem during the letter writing initiative. Shaded regions represent activity in primary school
Pre-problem appraisal and reflection on penpal response
Ruth’s post-intervention problem
Grace’s post-intervention problem
Responding to mathematical problems is a core activity in classrooms. The problems that teachers select determine the mathematical content, processes and nature of mathematical inquiry occurring in classrooms and thereby contribute to the development of mathematical skills and dispositions. Selecting, designing or reformulating mathematical problems is a critical skill, then, for prospective and practising teachers. This study explores the influence of a mathematical letter writing initiative in developing the problem posing skills of 28 prospective primary teachers. We examine the characteristics of mathematical problems designed by prospective teachers, and their understandings of what constitutes a good mathematical problem, prior to and following completion of a 12-week letter writing initiative with 10–11-year-old children. Analysis of the data reveals the benefits of engaging in the initiative as evidenced in improvements in several problem characteristics. There was an increase in the number of multiple approach and multiple solution problems and in the level of cognitive demand of problems posed. The challenge of posing non-traditional problems, alongside the competing demands of building in opportunities for success, may have diminished participants’ ability to evaluate and attend to the cognitive demand of problems.
 
Instruments designed to measure teachers’ knowledge for teaching mathematics have been widely used to evaluate the impact of professional development and to investigate the role of teachers’ knowledge in teaching and student learning. These instruments assess a mixture of content knowledge and pedagogical content knowledge. However, little attention has been given to the content alignment between such instruments and curricular standards, particularly in regard to how content knowledge and pedagogical content knowledge items are distributed across mathematical topics. This article provides content maps for two widely used teacher assessment instruments in the United States relative to the widely adopted Common Core State Standards. This common reference enables comparisons of content alignment both between the instruments and between parallel forms within each instrument. The findings indicate that only a small number of items on both instruments are designed to capture teachers’ pedagogical content knowledge and that the majority of these items are focused on curricular topics in the later grades rather than in the early grades. Furthermore, some forms designed for use as pre- and post-assessment of professional development or teacher education are not parallel in terms of curricular topics, so estimates of teachers’ knowledge growth based on these forms may not mean what users assume. The implications of these findings for teacher educators and researchers who use teacher knowledge instruments are discussed.
 
Various intervention programs for fostering at-risk students’ understanding of basic concepts (such as place value understanding or meanings of multiplication and division) have been developed and evaluated. However, little is known about the teacher expertise required to enact these intervention programs, and how this teacher expertise can be promoted. The article suggests a conceptual model for teacher expertise for fostering at-risk students’ understanding based on three recurrent jobs: (a) specify learning content (in basic concepts), (b) monitor students’ learning progress (in basic concepts) and (3) enhance students’ understanding (of basic concepts). Mastering these jobs with productive teaching practices involves four orientations in particular (conceptual rather than procedural orientation, diagnostic rather than syllabus-led orientation, communicative rather than individualistic orientation, and long-term rather than short-term orientation) as well as detailed pedagogical content categories for unpacking relevant knowledge elements. The paper reports on the professional development program Mastering Math which aims at promoting this expertise and its evaluation using a pre–post-design. For 95 participating teachers, the practices for specifying goals and monitoring and enhancing at-risk students’ understanding were captured in self-reports and situated in vignette-based activities for eliciting diagnostic judgments. Teachers’ development across different aspects of their expertise from the beginning and the end of the 1-year PD reveals the first quantitative evidence that the PD was effective in promoting growth of expertise. Whereas specifying and monitoring practices had substantially developed, the enhancement practices were hindered by a persistent short-term orientation.
 
This study is an examination of 79 elementary prospective teachers’ (PSTs’) capacity for recognizing the core ideas involved in modeling fraction addition problems and their difficulties in solving and presenting the process of fraction addition using area, length, and set models. PSTs completed a written task in which they represented the process of solving a set of fraction addition problems of varying complexity in terms of the sizes of the addends, the sums, and the denominators using the three models. The PSTs’ responses to the task were analyzed and clustered to identify the most salient and recurring patterns of erroneous approaches to modeling. While the PSTs recognized the core ideas involved in modeling fraction addition problems, their actual work samples demonstrated several areas for improvement. Some PSTs had invalid answers and representations that were indicative of their misunderstanding of key concepts. Some PSTs had correct answers, but their model did not support sense-making. Additionally, some PSTs’ representations did not clearly incorporate the salient properties of the area, length, and set models. Our analysis generated implications for mathematics teacher educators regarding what PSTs need to learn, develop further, unlearn, and refine in order to effectively teach uses of representation in elementary classroom.
 
Data collected during the 1-day and 2-day lesson scenarios
Framework for selecting strategies for whole-class discussions
Four strategies shared in Ms. Henry’s 1-day scenario discussion
Four strategies shared in Ms. Henry’s 2-day scenario discussion
Tara’s strategy from Ms. Henry’s 1-day scenario discussion
This study focused on the practice of selecting student strategies for whole-class discussions with teachers working toward a vision of instruction that is responsive to students’ mathematical thinking. Selecting strategies is one of the 5 practices identified in Smith and Stein's (5 practices for orchestrating productive mathematics discussions, National Council of Teachers of Mathematics, 2018) seminal work on orchestrating productive mathematics discussions, but I argue that this practice has been under-appreciated and under-researched, and implementation remains challenging for many teachers. To address these issues, I explored the selecting of student strategies with two sets of upper elementary school teachers engaged in professional development focused on the teaching and learning of fractions. One set consisted of 3 teachers who had demonstrated expertise in teaching that is responsive to students’ mathematical thinking, and I explored the selecting of strategies in these teachers’ classrooms. The other set consisted of 30 teachers with varying levels of expertise in this type of teaching, and I examined their selecting of strategies during professional development activities. Findings led to the creation of a two-level framework that showcases teachers’ criteria for selecting strategies for whole-class mathematics discussions. The framework, which extends and adds a coherent structure to selection criteria currently identified in the literature, highlights considerations related to the mathematics of the strategy, the author of the strategy, and the class engagement with the strategy. The framework, which is presented through a case, was designed to not only provide guidance for the purposeful selection of strategies but also to be accessible and useful to teachers at any phase of their development in being responsive to students’ mathematical thinking.
 
Comparing feedback types with longitudinal changes
In this study, we sought to identify how feedback about classroom observations affected novice university mathematics instructors’ (UMIs) teaching practices. Specifically, we examined how a Red–Yellow–Green feedback system (RYG feedback) affected graduate student instructor (GSI) scores on an observation protocol (GSIOP). The protocol was developed specifically for this population, and both the GSIOP and RYG feedback were used within a peer mentoring program for GSIs, wherein novice GSIs were mentored by more experienced GSIs. Mentors observed novices’ classrooms using the GSIOP and provided RYG feedback as part of observation–feedback cycles. We analyzed 100 sets of scores, each collected over the course of a semester containing on average three observation–feedback cycles. Analyzing the semester-long datasets longitudinally provided insight into what types of feedback informed and influenced observed teaching. After qualitatively coding the feedback provided to the GSIs by their mentors along multiple dimensions, we found certain forms of feedback were more influential for observable changes in GSIs’ teaching. For example, pedagogical feedback that included contextualization (context and focal events) demonstrated a more positive change in GSIOP score than feedback that lacked contextualization. Our results suggest that contextual formative feedback has a positive change to student-focused and teacher-focused observations.
 
In this paper, we report an enquiry into elementary preservice teachers’ learning, as they engage in doing mathematics for themselves. As a group of researchers working in elementary initial teacher education in English universities, we co-planned and taught sessions on growing pattern generalisation. Following the sessions, interviews of fifteen preservice teachers at two universities focused on their expressed awareness of their approach to the mathematical activity. Preservice teachers’ prospective planning and post-teaching evaluations of similar activities in their classrooms were also examined. We draw on aspects of enactivism and the notion of reflective ‘spection' in the context of teacher learning, tracing threads between preservice teachers’ retro-spection of learning and pro-spection of teaching. Our analysis indicates that increasing sensitivity to their own embodied processes of generalisation offers opportunities for novice teachers to respond deliberately, rather than to react impulsively, to different pedagogical possibilities. The paper contributes a new dimension to the discussion about the focus of novice elementary school teachers’ retrospective reflection by examining how deliberate retrospective analysis of doing mathematics, and not only of teaching actions, can develop awarenesses that underlie the growth of expertise in mathematics teaching. We argue that engaging preservice teachers in mathematics to support deliberate retrospective analysis of their mathematics learning and prospective consideration of the implications for teaching, can enable more critical pedagogical choices.
 
Process for Formative Assessment Interviews and Model Building
Change in level of noticing from first to second Model Building, expressed in percent of total participants
Change in level of noticing and connections from first to second Model Building, expressed in percent of total participants
Student work used as evidence for the model
Evidence provided to support interpretations made
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.
 
A mathematical modeling process
(taken from Blum 2011, p. 18)
The effective teaching and learning of mathematics through mathematical modeling and connecting mathematics to the real world has gained rapid growth at various educational levels all over the world. The growth in modeling practices in the United States of America (US) and the international community is the result of the development and implementation of new mathematics standards and curricula across the world. In this article, we report on the development and use of a quantitative instrument aimed at assessing teachers of mathematics attitudes toward mathematical modeling practices. Based on the responses of 310 practicing teachers of mathematics from the US, the mathematical modeling attitude scale was evaluated using item analysis, exploratory factor analysis, confirmatory factor analysis, and other psychometric properties. The scale isolated four dimensions: constructivism, understanding, relevance and real-life, and motivation and interest. These 4 factors accounted for 59% of the variation in the 28-item measure. Cronbach’s alpha coefficient for the overall scale was .96, and that for the subscales was .93 for constructivism, .81 for understanding, .88 for relevance and real-life, and .89 for motivation and interest. The findings suggest a psychometrically useable, reliable, and valid scale for studying teacher’s attitudes toward mathematical modeling. We discuss the instrument’s merit for research and teaching, and implications for teacher education, professional development, and future research.
 
Task launch planning questions; adapted from Jackson et al. (2012)
Apples task description
Contextual features code frequencies
Mathematical relationships code frequencies
Common language code frequencies
Cognitively demanding tasks have been shown to support students’ understanding of mathematics. How a task is launched, or introduced, determines how students engage with the task, as well as the type of work the teacher engages in during the task implementation. The authors designed a unit focused on launching a task where prospective teachers (PSTs) planned and enacted a task launch as part of their methods of teaching secondary mathematics courses. Participants in the study spanned three institutions. Using a curricular noticing framework, this paper describes how the prospective teachers (PSTs) interpreted their task by identifying key contextual features, mathematical relationships, and common language, and then focuses on how they responded to their interpretations as they planned to elicit and develop their students’ understanding of those key aspects. This paper also describes how the PSTs responded to their interpretations in planning to maintain the cognitive demand of the task. Findings suggest that the prospective teachers valued student-centered approaches and used a range of pedagogical methods to launch cognitively demanding tasks. We also found the prospective teachers needed support identifying context in tasks and in determining how much to tell while maintaining the cognitive demand of the task. We conclude with implications and suggestions for methods courses on both launching a cognitively demanding task and on teaching practices more generally.
 
In response to recent trends in the field of mathematics education, real-world contexts are frequently found in curricula and classrooms across the world. Numerous scholars contend that, in addition to making instruction more engaging and relevant, rich, imaginable contexts can provide a semantic grounding that helps students deeply understand abstract mathematical ideas. To achieve this form of understanding, A. Thompson et al. (1994) describe the need for conceptually oriented teaching, aimed at supporting students in maintaining connection between numerical values and operations and the quantities and relationships to which they refer. We elaborate the construct of conceptually oriented teaching through an empirical analysis of two teachers’ instruction around a context-based ratio instructional sequence. More specifically, we differentiate between two types of explicit contextualization present within conceptually oriented explanations: contextualized quantities and contextualized operating. We identify the importance of a particular teacher move, pressing for contextualized operating, that aims to explicitly ground students’ use of representations to their common sense understanding of the problem context. We describe two ways that the teachers work to make the contextualization explicit in the classroom discourse and argue that this explicit contextualization is essential in order for all students to have the opportunity to experience the contexts as supportive of conceptual development.
 
An example of a Grade-6 challenging task on the sum of polygon angles and its associated enabler and extenders
An example of a Grade-1 challenging task on the commutative property of multiplication and its associated enablers and extenders
An example of a Grade-2 challenging task on comparing and ordering numbers and its associated extender
An example of a Grade-6 challenging task on multiplying integers with mixed numbers and its associated enabler
An example of a Grade-6 challenging task on multiplying integers with fractions and its associated enabler
Over the past decade, teaching mathematics ambitiously has received increased attention. In this paper, we argue that to materialize this vision in contemporary classes we need to understand how practicing teachers experiment with certain aspects of this teaching and what challenges they encounter. Toward this end, we focus on an aspect of teaching ambitiously—designing and using enablers and extenders—and examine how four elementary schoolteachers experimented with it in their practice while also participating in a video-club setting. Drawing on a corpus of data including lesson plans, videotaped lessons, pre- and post-lesson interviews, end-of-program interviews, and videotaped video-club sessions, and looking across the four cases, we sketch how these teachers worked with enablers and extenders and the challenges they faced. Our analysis helped identify certain components entailed in working with enablers and extenders during the phases of lesson planning and enactment; it also yielded a classification of observed and reported challenges encountered as teachers engage with this work. This mapping of work associated with designing and using enablers and extenders, along with the classification of challenges generated, can inform professional learning development attempts aiming to support teachers enact ambitious teaching by identifying and naming separate components of practice that merit consideration and by providing insights into the types of scaffolds needed to support teachers in teaching ambitiously.
 
A mathematics task using an interactive applet
Stage one core/periphery structure. *Blue = Core; Red = Periphery; Green = changing membership. (Color figure online)
New connections made by the periphery during week four. *Blue= Core; Maroon=Periphery; Red=Changed from periphery to core; Orange=Changed from core to the periphery; Red edges=New connections during week four. (Color figure online)
The instructor’s interactions during week four. *Red edges represent posts sent by the instructor. (Color figure online)
Online collaborative and content-focused professional development (PD) is becoming an increasingly important setting for supporting mathematics teachers’ professional learning. The purpose of this study was to better understand the process by which a community emerges in such a PD setting by examining how the cohesiveness of 21 mathematics teachers’ social network evolves and associated shifts in the quality of mathematics teachers’ mathematical discourse. We employed social network analysis (SNA) to examine the evolving cohesiveness of mathematics teachers’ social network and coding procedures to examine teachers’ mathematical discourse. A key finding was the documentation of an emergent divide between participation in the core and periphery during initial weeks of the PD and then a reduced divide and emergence of a social network that resembles a community. We argue that the instructor’s pattern of participation that included distributing their interactions across the subgroups while sending a common message regarding expectations for mathematical discourse in the PD may have contributed to the community formation process. We propose the Interaction Assessment Model, which outlines an approach for PD facilitators to use SNA as a feedback mechanism to differentiate facilitation of online collaborative and content-focused PD and build online communities.
 
Conceptual framework showing narrative arcs and mathematical orientations as a bridge between teacher’s experiences learning math as students and the aspects of their mathematics identities salient for teaching mathematics
Number of PST narratives demonstrating each narrative arc, grouped by valences
Percent of PSTs with each narrative arc in each cohort
Percent and number of PSTs in each cohort by mathematical orientation
This study explores narratives about critical moments in mathematics learning written by K-8 pre-service teachers’ (PSTs) in the United States over a 20-year period. These critical moments, such as a single memorable task, course, test, or comment from a teacher, had a powerful and sustained impact on PSTs’ mathematics identities, which they carried with them as they entered the teaching profession. We classified narratives using McCulloch et al. (Sch Sci Math 113(8):380–389, 2013) and Drake’s (J Math Teach Educ 9(6):579–608, 2006) categories and found a potentially new category, Taking the Reins. We also classified PSTs’ mathematics orientations as either relational (oriented toward creative problem solving and conceptual understanding) or instrumental (enacting rote procedures without meaning) (Skemp in Math Teach 77(1):20–26, 1976). Many PSTs identified a causal relationship between their mathematics orientations and narrative arcs: relational learning opportunities encouraged positive narratives, while instrumental learning opportunities either resulted in negative narrative arcs or positive but fragile mathematics identities that crumbled under minor stress. We found little variation over time in the nature and prevalence of the narrative arcs and mathematics orientations, suggesting that any changes in mathematics teaching practices over time were not reflected in students’ learning experiences.
 
Order and Percentages of Each Question Type among the Total Number of Teacher Questions (%) Note. Percentages may not add up to overall totals due to rounding. The last number in the top row represents the total number of teacher questions, for example, Sarah used 42 questions in total
Problem Landscape and Questioning Plane, Shown with the Four PTs' Placement
Base-8 chart
Jennifer’s part–part–whole problem showing five white apples
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.
 
Different dimensions to coordinate during problem posing for integer addition and subtraction
Elementary and middle school prospective teachers participated in semi-structured problem posing for integer addition and subtraction. The prospective teachers (n = 98) posed a variety of different stories, but this paper focuses on the temperature stories they posed. Results include descriptions of their posed temperature stories through the lens of the various dimensions (i.e., problem types, realism, consistency, correctness). Prospective teachers posed mainly state-translation-state problem types and rarely posed state-state-distance, state-state-translation, or translation-translation-translation problem types. They often changed the structure of their number sentences. Although they posed mostly realistic and mathematically correct temperature stories, the stories compromised realism or consistency in order to use state-translation-state problem types. Coordinating of the various dimensions (e.g., problem types, consistency) when problem posing requires flexibility with problem types. This work highlights the complexity of posing temperature stories, and coordinating the various dimensions highlights the need for prospective teachers to experience problem posing. Implications for problem posing with integers and temperature are extended to all contexts that inherently support translation and relativity. In the discussion, we coordinate the different problem types with various number sentences and dimensions. Unpacking the various dimensions illuminates prospective teachers’ thinking and offers a way of considering integers and contexts.
 
Data collection process
The first student's work in VC1
The second student's work in VC1
This qualitative study examined how prospective mathematics teachers attend to, interpret, and respond to student misconceptions through providing them a video-case-based professional development environment. A sample of 30 prospective teachers attending an elective course was asked to watch video cases about student misconceptions related to the concept of measurement. Then, class discussions were held on the misconceptions, reasons, and suggestions to remedy the misconceptions. Thereafter they individually wrote down their response to an open-ended question asking them how to remedy the misconceptions. Data obtained from the discussions and individual reports were analyzed using the content analysis technique with the framework of professional noticing of children’s mathematical thinking. Findings indicated that the professional development environment provided prospective teachers with an opportunity to attend to, interpret, and then decide how to respond to student misconceptions. We found that while prospective teachers generally understand how students think and why misconceptions arise, they mostly provided partial evidence for interpreting. Moreover, their suggestions for addressing mathematical misconceptions were generally based on conceptual understanding. Our findings suggested that in order to acquire skills in noticing student thinking, it is important that prospective teachers have sufficient opportunity to reflect on likely misconceptions in a professional context.
 
Distribution of Item Difficulties in Norwegian and Slovak Adaptations
Scatter plot of IRT difficulty difference vs. DIF
Distribution of answers to item 15
Comparison of average θ of SK primary teachers and selected subgroups of NW teachers
Sally's method (Diagrams for Item 21)
The measures of mathematical knowledge for teaching developed at the University of Michigan in the U.S., have been adapted and used in studies measuring teacher knowledge in several countries around the world. In the adaptation, many of these studies relied on comparisons of item parameters and none of them considered a comparison of raw data. In this article, we take advantage of having access to the raw data from the adaptation pilot studies of the same instrument in Norway and Slovakia (149 practicing elementary teachers in Norway, 134 practicing elementary teachers in Slovakia) that allowed us to compare item parameters and teachers’ ability estimates on the same scale. Statistical analysis showed no significant difference in the mean scores between the Norwegian and the Slovak teachers in our samples and the paper provides further insight into the issues of cross-national adaptations of measures of teachers’ knowledge and the limitations of the methods commonly applied in the item adaptation research. We show how item adaptations can be refined by combining robust quantitative methods with qualitative data, how decisions on adaptation of individual items depend on context and purpose of the adaptation, and how comparability and heterogeneity of samples affects interpretation of the results.
 
Predicted mean scores on the ratios and proportional reasoning assessment by the use of each topic. Note The error bar for each topic indicates teachers’ scores within one standard deviation for the corresponding topic
Examining teachers' knowledge on a large scale involves addressing substantial measurement and logistical issues; thus, existing teacher knowledge assessments have mainly consisted of selected-response items because of their ease of scoring. Although open-ended responses could capture a more complex understanding of and provide further insights into teachers' thinking, scoring these responses is expensive and time consuming, which limits their use in large-scale studies. In this study, we investigated whether a novel statistical approach, topic modeling, could be used to score teachers' open-ended responses and if so, whether these scores would capture nuances of teachers' understanding. To test this hypothesis, we used topic modeling to analyze teachers' responses to a proportional reasoning task and examined the associations of the topics identified through this method with categories identified by a separate qualitative analysis of the same data as well as teachers' performance on a measure of ratios and proportional relationships. Our findings suggest that topic modeling seemed to capture nuances of teachers' responses and that such nuances differentiated teachers' performance on the same concept. We discuss the implications of this study for education research.
 
Coding scheme with examples and rationale
Coaches' opportunities for learning
Mathematics coaching is complex work, and coaches must be supported to become experts in mathematics, mathematics instruction, and mathematics coaching. Using video and interview data from 12 mathematics coaches and one district administrator in one public school district in the southeastern USA, this qualitative study explores how the performative aspect of one group of mathematics coaches’ doing the math routine opened up conversations about mathematics, mathematics instruction, and mathematics coaching. Findings indicate that as the coaches engaged in doing the math together, opportunities to discuss mathematics and mathematics instruction were opened up, while conversations about mathematics coaching rarely surfaced. In interviews, participants discussed the benefits and drawbacks of participating in doing the math. Implications for future research and practice are discussed.
 
This study aims to explore teacher noticing differences of the sampled US and Chinese elementary teachers from cross-cultural mathematics videos. A total of 34 expert teachers commented on 25 video clips online. We coded what and how teachers noticed from the videos both quantitatively and qualitatively. Findings reveal teachers’ strong interests and profound reflections, especially in the teaching domain including representations, communication, and teacher questioning/guide. Cross-cultural differences in teacher noticing were identified, which were discussed based on possible cultural influences. Implications in research and practice, teacher support, and methodology are discussed.
 
In this paper, we analyse a grade 8 (age 13–14) Japanese problem-solving lesson involving angles associated with parallel lines, taught by a highly regarded, expert Japanese mathematics teacher. The focus of our observation was on how the teacher used carefully planned board work to support a rich and extensive plenary discussion ( neriage ) in which he shifted the focus from individual mathematical solutions to generalised properties. By comparing the teacher’s detailed prior planning of the board work ( bansho ) with that which he produced during the lesson, we distinguish between aspects of the lesson that he considered essential and those he treated as contingent. Our analysis reveals how the careful planning of the board work enabled the teacher to be free to explore with the students the multiple alternative solution methods that they had produced, while at the same time having a clear overall purpose relating to how angle properties can be used to find additional solution methods. We outline how these findings from within the strong tradition of the Japanese problem-solving lesson might inform research and teaching practice outside of Japan, where a deep heritage of bansho and neriage is not present. In particular, we highlight three prominent features of this teacher’s practice: the detailed lesson planning in which particular solutions were prioritised for discussion; the considerable amount of time given over to student generation and comparison of alternative solutions; and the ways in which the teacher’s use of the board was seen to support the richness of the mathematical discussions.
 
Teacher education programs have a critical role in supporting prospective teachers’ connections between theory and practice. In this study, we examined three prospective secondary mathematics teachers’ discourses regarding collective argumentation during and after a unit of instruction addressing collective argumentation and ways they recontextualized their on-campus coursework (theory) into their student teaching (practice) as demonstrated by their support for students’ mathematical arguments during student teaching. Through a recursive process of coding data from interviews, reflections, and classroom discussions, we constructed descriptions of participants’ discourses about argumentation based in their coursework and identified three themes about collective argumentation in the prospective teachers’ discourses: the purposes of argumentation, the role of the teacher in argumentation, and characteristics of effective mathematical arguments. Analysis using extended Toulmin diagrams of classroom data from participants’ student teaching showed that these three interrelated themes were visible in the participants’ enactment of moves supporting collective argumentation. Our study opens spaces for future investigations into how teachers recontextualize their learning from coursework (theory) into their practice.
 
Top-cited authors
Shuhua An
  • California State University, Long Beach
Gerald Kulm
  • Texas A&M University
John Mason
  • University of Oxford & Open University
Barbara Jaworski
  • Loughborough University
Konrad Krainer
  • Alpen-Adria-Universität Klagenfurt