Mathematics coaching and instructional reform: Individual and collective change

Mathematics coaching initiatives are being implemented in schools and districts across the country, guided by the notion that these initiatives will foster individual teacher’s learning and thereby support system-wide instructional improvement in mathematics. This paper explores the evolving roles that mathematics coaches played in a system-wide instructional improvement effort focused on elementary mathematics education in a medium-sized suburban school district. Using social network analysis and qualitative analysis of interviews, we argue that coaches facilitated teachers’ implementation of a new mathematics curriculum by acting as brokers, first as intermediaries between the district office and schools, then as catalysts for collective inquiry. Further, we show how coaches’ work was both enabled and constrained over time by various organizational dimensions at the school and district levels. Overall, our findings suggest that district and school leaders should think beyond the roles and responsibilities of individual coaches, and consider how to support coaches as participants in system-wide networks focused on continuous learning and instructional improvement.     

Developing collective capacity to improve mathematics instruction: Coaching as a lever for school-wide improvement

Mathematics coaching, as typically practiced in US schools tends to be responsive and individually focused work in which coaches respond to invitations from individual teachers to help them improve their teaching. But what does the work of coaching look like if it is organized instead to engage teachers collectively in service of school-wide improvement? This is the question we take up in this paper through examining the case of one school-based elementary mathematics coach whose work more closely aligned with emerging findings in the field of instructional improvement about the power of coaching for school-wide reform. The coach helped to dramatically transform a recent history of poor performance and deficit-oriented narratives pertaining to the school and its children. Through a fine-grain analysis, we illustrate the coach’s work implicated in supporting groups of teachers to come to mutual understanding around and further development of shared high-quality instructional practices. The components of coaching that help support collective capacity are discussed.     

Effectively coaching middle school teachers: A case for teacher and student learning

In this paper we report findings from a two-year, large-scale research project that describes the work of middle school mathematics specialists (also referred to as mathematics coaches or instructional coaches) who served in 10 school districts. We use mixed methods to describe how mathematics specialists spent their time supporting teachers and how these supports contributed to meaningful changes that teachers made in their instructional practices. We also report results that correlate student achievement scores with whether or not teachers were highly engaged with the mathematics specialists. We coordinate these quantitative results with findings from several case studies to illustrate the qualitatively different ways that mathematics specialists supported teachers’ ongoing work with their students. We also account for why some teachers participated more fully than others. Importantly, because mathematics specialists’ work was situated in different school settings with different demands, resources and administrative supports, these constraints and affordances contributed in part to how they could effectively support teachers’ work with their students.     

Principal and coach as partners

This paper explores the roles and responsibilities of mathematics coaches and principals. It suggests that principals need to equilibrate power by treating the coach as a partner, even if the coach is subordinate in the educational hierarchy. It posits that coaching would benefit all parties if they worked together across roles to enhance the capacity of educators at every level in the school hierarchy—teacher, coach, principal—to design, analyze, and implement mathematics instruction that results in deeper student learning. It briefly describes the model of Content Coaching. It questions some of the prevailing norms of interaction among educators, and it offers coaching scenarios that demonstrate that the initial design and implementation of coaching initiatives can lead to resistance and misunderstanding of the power and benefits of coaching for all educators and administrators. It goes on to challenge some of our assumptions regarding potential barriers (e.g. confidentiality) in relationships among coaches, teachers, and principals, and suggests that teacher learning goals be public and collaboratively engaged. It posits that positional authority is not a particularly strong lever for improving teacher practice, particularly if the principal lacks facility or interest in mathematics. It argues that if we want to enhance mathematics education in this country (as evidenced by robust student understanding of mathematics and its application in the world), there is great power in working collaboratively and inclusively, using coaching to support educators at every level.     

Elementary mathematics specialists in “departmentalized” teaching assignments: Affordances and constraints

In this article, we describe the experiences of three Elementary Mathematics Specialists (EMS) who were part of a larger project investigating the impact of EMS certification and assignment (self-contained or “departmentalized”) on teaching practices and student achievement outcomes. All three of the teachers were “departmentalized,” in the sense that each was responsible for teaching mathematics to at least two groups of students, and accordingly, did not teach all subjects as would a typical self-contained elementary teacher. Each teacher had recently earned an Elementary Mathematics Specialist certificate through completion of a 24-credit, graduate-level program designed to build pedagogical content knowledge and leadership capacity in mathematics. Through a series of observations and interviews over the course of one school year, we examined how the teachers described and navigated specific affordances and constraints they encountered in their particular contexts. Common affordances included opportunities to revise and learn from instruction, and constraints included reduced flexibility introduced by the need to schedule multiple classes of mathematics. Despite these common features, we found important differences between the three models of departmentalization, which we describe as team approach, class swap, and grade-level mathematics teacher. For example, some of the models provided more opportunities for collaboration while others made it difficult for teachers to address potential inequities in learning opportunities across sections. Despite the constraints of their respective models, we found evidence of the EMS-certified teachers drawing on professional expertise in mathematics to meet student needs.     

Students’ ratings of teacher practices

In this paper, we explore a novel approach for assessing the impact of a professional development programme on classroom practice of in-service middle school mathematics teachers. The particular focus of this study is the assessment of the impact on teachers’ employment of strategies used in the classroom to foster the mathematical habits of mind and mathematical self-efficacy of their students. We describe the creation and testing of a student survey designed to assess teacher classroom practice based primarily on students’ ratings of teacher practices.     

Assessing views of coaching via a video-based tool

Mathematics coaches represent a unique group of didacticians, or individuals who work with practicing teachers. Twenty-eight mathematics coaches participated in this exploratory study, which used video viewing to examine the coach–teacher dynamic. To gather data about participants’ views of effective coaching practices, we developed the Video Assessment of Coaching instrument, which provided coaches with opportunities to express their views of effective practice and implementation. The participants expressed views of effective coaching that often did not align with those of coaching authors. The significance of this research lies in its efforts to document the views that mathematics coaches develop as practitioners, as an early step in the examination of the relationships between the views of coaches and coaches’ effectiveness in improving teacher practice, knowledge, and attitudes.     

Examining Secondary Mathematics Teachers' Opportunities to Develop Mathematically in Professional Learning Communities

To make progress toward ambitious and equitable goals for students’ mathematical development, teachers need opportunities to develop specialized ways of knowing mathematics such as mathematical knowledge for teaching (MKT) for their work with students in the classroom. Professional learning communities (PLCs) are a common model used to support focused teacher collaboration and, in turn, foster teacher development, instructional improvement, and student outcomes. However, there is a lack of specificity in what is known about teachers’ work in PLCs and what teachers can gain from those experiences, despite broad claims of their benefit. We discuss an investigation of the work of secondary mathematics teachers in PLCs at two high schools to describe and explicate possible opportunities for teachers to develop the mathematical knowledge needed for the work of teaching and the ways in which these opportunities may be pursued or hindered. The findings show that, without pointed focus on mathematical content, opportunities to develop MKT can be rare, even among mathematics teachers. Two detailed images of teacher discussion are shared to highlight these claims. This article contributes to the ongoing discussion about the affordances and limitations of PLCs for mathematics teachers, considerations for their use, and how they can be supported.     

The development of instructional guidelines for elementary mathematical writing

Mathematical writing recently has been defined as writing to reason and communicate mathematically. But mathematics instructional resources lack guidance for teachers as to how to implement such writing. The purpose of this paper is to describe how methods of design-based research were used to develop an instructional resource when one does not currently exist. Thirty-four participants—including teachers, mathematics coaches, mathematics curriculum developers, literacy coaches, a mathematician, and academics in elementary mathematics education, mathematics education, writing education, and science education—participated in a multi-step process to recommend, revise, and confirm instructional guidelines for elementary mathematical writing. The development process began with thirty-two recommendations from science writing and language arts writing. Through multiple cycles of feedback, five instructional guidelines and related considerations and techniques for implementation of elementary mathematical writing emerged.     

Construction of Teacher Knowledge in Context: Preparing Elementary Teachers to Teach Mathematics and Science

The purpose of this study was to further the understanding of how preservice teachers construct teacher knowledge and pedagogical content knowledge of elementary mathematics and science in a school‐based setting and the extent of knowledge construction. Evidence of knowledge construction (its acquisition, its dimensions, and the social context) was collected through the use of a qualitative methodology. The methods course was content‐specific with instruction in elementary mathematics and science. Learning experiences were based on national standards with a constructivist instructional approach and immediate access to field experiences. Analysis and synthesis of data revealed an extensive acquisition of teacher knowledge and pedagogical content knowledge. Learning venues were discovered to be the conduits of learning in a situated learning context. As in this study, content‐specific, school‐based experiences may afford preservice teachers greater opportunities to focus on content and instructional strategies at deeper levels; to address anxieties typically associated with the teaching of elementary mathematics and science; and to become more confident and competent teachers. Gains in positive attitudes and confidence in teaching mathematics and science were identified as direct results of this experience.     

Learning study – promoting and hindering factors in mathematics teaching

This article focuses on presenting success factors for a group of teachers in carrying out a learning study in mathematics at their school. The research questions are: what are the actions of the school teaching community during development projects? What factors enable a group of teachers to carry out a learning study at their school? Activity theory provides a holistic framework to investigate relationships among the components present in a learning study. The results are based on analysis of interviews with teachers, students, principal organizers of schools and project coordinators, videotaped lessons, students’ tests and minutes taken at meetings of mathematics projects. The results show that the skills of facilitators, the time devoted to collaborative work, the link to learning theory and avoiding overly comprehensive content when teaching lessons are important promoting factors in mathematics teaching. The findings raise important questions about the way in which teacher work within universities.     

Exploring the Use of New Representations as a Resource for Teacher Learning

An important goal of mathematics education reform is to support teacher learning. Toward this end, researchers and teacher educators have investigated ways in which teachers learn about mathematical content, pedagogical strategies, and student thinking as they implement reform. This study extends such work by examining how one elementary school and one high school teacher learned from students' interpretations of new conceptually based representations contained in instructional materials aligned with the Principles and Standards for School Mathematics ( National Council of Teachers of Mathematics, 2000 ). Results indicated that teaching with new representations provided a rich context for teacher learning at both the elementary and high school level, and three dimensions were identified along which such learning occurred. The results suggest that pedagogical content knowledge with respect to representations is an important facet of teacher cognition that should be studied in greater depth.     

A study of early career teachers’ practices related to language and language diversity during mathematics instruction

The role of language in mathematics teaching and learning is increasingly highlighted by standards and reform movements in the US. However, little is known about teachers’, and especially early career teachers’ (ECTs) practices and understandings related to language in mathematics instruction. This multiple case study explored the language-related understandings and practices of six ECTs in diverse elementary classrooms. Using iterative cycles of analysis, we found that all ECTs regularly attended to students’ mathematical vocabulary use and development. Yet, there was variability in ECTs’ focus on how to teach mathematical vocabulary, expectations for students’ precise use of mathematical terminology, and the use of multiple languages during instruction. These findings indicate that ECTs need more targeted support during teacher preparation and early career teaching in order to better support all students’ language development in the mathematics classroom.     

An autoethnography of a (reluctant) teacher leader

Many believe teacher leaders can play a central role in meeting the needs of students in schools, but it is presumptuous to think that teachers intuitively know how to lead their colleagues or schools without any focused support. This paper describes how one middle school mathematics teacher learned to enact leadership through an informal role as a teacher leader. The results of this autoethnographic study also reveal how the teacher’s view of leadership changed during the year of the study. As that leadership identity transitioned, the researcher found that adopting a lead-by-example and lead-learner stance supported her work with colleagues. A presentation of findings related factors that supported and hindered the transition from mathematics teacher to that of teacher leader are shared. Factors that supported teacher leadership included maintaining a disposition of continuous learning, developing a community of practice with colleagues, and developing a systems view of leading. The experiences and factors that hindered the process of becoming a teacher leader included confusion about one’s leadership role, navigating the middle ground between colleague and leader, and the lack of communication with administration. The findings in this paper suggest several courses of action for supporting emergent teacher leaders.     

ICT integration in mathematics initial teacher training and its impact on visualization: the case of GeoGebra

This case study investigates the impact of the integration of information and communications technology (ICT) in mathematics visualization skills and initial teacher education programmes. It reports on the influence GeoGebra dynamic software use has on promoting mathematical learning at secondary school and on its impact on teachers’ conceptions about teaching and learning mathematics. This paper describes how GeoGebra-based dynamic applets – designed and used in an exploratory manner – promote mathematical processes such as conjectures. It also refers to the changes prospective teachers experience regarding the relevance visual dynamic representations acquire in teaching mathematics. This study observes a shift in school routines when incorporating technology into the mathematics classroom. Visualization appears as a basic competence associated to key mathematical processes. Implications of an early integration of ICT in mathematics initial teacher training and its impact on developing technological pedagogical content knowledge (TPCK) are drawn.     

Progressive, classical or balanced— A look at mathematical learning environments in Swiss-German lower-secondary schools

In a national supplement to TIMSS, lower-secondary school teachers (N=102) and their students (N=975) reported on mathematics instruction by means of a teacher questionnaire (teaching-learning methods, instructional sub-goals, facilitated student activities, achievement assessment, teacher role) and a student questionnaire (teachers' instructional proficiency, classroom climate). A cluster analysis performed on the ratings of teaching-learning methods yielded a solution with three clusters referred to as progressive, classical, and balanced learning environment. Cluster-related differences in facilitated student activities, achievement evaluation and preferred teacher role were found but not in instructional sub-goals. Students from different learning environments equally approved teachers' instructional proficiency and classroom climate and also had similar TIMSS mathematics scores. The results of this study provide empirical evidence that in addition to classical teacher-centered learning environments there seem to exist more diversified and studentcentered learning environments that address the needs for students to direct their own learning, communicate and work with others, and develop ways of dealing with complex problems. In line with the research literature it was also found that high mathematics achievement is not restricted to a certain type of learning environment.     

Answering the call by developing an online elementary mathematics specialist program

The Association of Mathematics Teacher Educators adopted Standards for Elementary Mathematics Specialists calling for structured preparation of math coaches, specialists, and instructional leaders across the country (AMTE, 2013). The purpose of this paper is to illustrate the structure and design of our fully online Elementary Mathematics Instructional Leader (EMIL) graduate program for inservice teachers aiming to answer the call. Our graduates are teacher leaders and coaches who are responsible for supporting effective mathematics instruction and student learning at the classroom, school, district, or state levels. We will review formative data from our first cohort of graduates, the benefits and drawbacks to providing online (and grant-funded) courses, and the impact of participant attrition. Our goal is to offer guidance and information to other organizations as they move to develop programs of their own.     

Mathematics teaching practices with technology that support conceptual understanding for Latino/a students

We analyze how three seventh grade mathematics teachers from a majority Latino/a, linguistically diverse region of Texas taught the same lesson on interpreting graphs of motion as part of the Scaling Up SimCalc study (Roschelle et al., 2010). The students of two of the teachers made strong learning gains as measured by a curriculum-aligned assessment, while the students of the third teacher were less successful. To investigate these different outcomes, we compare the teaching practices in each classroom, focusing on the teachers’ use of class time and instructional format, their use of mathematical discourse practices in whole-class discussions, and their responses to student contributions. We show that the more successful teachers allowed time for students to use the curriculum and software and discuss it with peers, that they used formal mathematical discourse along with less formal language, and that they responded to student errors using higher-level moves. We conclude by discussing implications for teachers and mathematics educators, with special attention to issues related to the mathematics education of Latinos/as.     

Examining and conceptualizing the relationship between teacher praise and the co-construction of mathematical competence in classrooms

In this study we examined how teacher praise varies across and within four middle school mathematics classrooms in relationship to mathematical competence. We then conceptualized how teacher praise contributes to the co-construction of normative identity: the class’ shared understanding of what counts as being a competent learner in a mathematics classroom. Findings revealed teachers rarely used person-based praise (e.g., “you’re smart”) and frequently gave generic praise (e.g., “good”). Each teacher’s praise patterns supported different co-constructions of mathematical competence. Although some teachers taught the same lessons or ascribed to similar pedagogical approaches, findings suggest teachers’ praise patterns may contribute to the co-construction of different normative identities, some more exclusive and others more inclusive. Findings indicate praise may be a low-stakes and potentially impactful teacher practice with implications for students’ understanding of what it means to be good at math.