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As content-specific educational coaches, elementary mathematics specialists (EMSs) have emerged as school-based professionals who are needs-driven and work closely with school stakeholders in regard to mathematics teaching and learning. While leading mathematics education organizations have identified the specialized knowledge and skills required for EMS positions, how to best prepare these individuals is knowledge that the field is still exploring. This paper first presents a theoretical model for EMS preparation that aligns an emerging coaching tool, the Decision-Making Protocol for Mathematics Coaching (Baker & Knapp, 2019, [DMPMC]) with the Professional Development Design Framework (Loucks-Horsley et al. in Designing professional development for teachers of science and mathematics, Corwin Press, 2010). The paper then presents a descriptive case study that examines the application of this model in an EMS preparation course. The findings indicate that assessing the coaching situation fostered administrative partnerships, revisiting goals increased specificity of anticipated outcomes, and applying research-informed practices increased EMS self-efficacy and advanced coaching agendas. Taken together, these findings suggest that integrating the DMPMC into an EMS preparation course led to positive changes in EMS candidate learning of professional development design. Notably, this is one of the first studies that documents the influence of a coaching education tool on EMS candidates’ professional development design.

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 multi-case study examines how three elementary mathematics methods instructors, in the same teacher education program, provide their prospective teachers with learning opportunities. Qualitative data were collected through interviews, classroom observations, and artifacts. The findings suggest that the instructors’ beliefs associated with teaching philosophies influence both the content that prospective teachers have the opportunity to learn (what) and the nature of the prospective teachers’ opportunities to learn (how). Through analytic induction, three assertions were developed to understand and explicate: similarities in opportunities to learn, differences in opportunities to learn, and perceptions about the purpose of the methods courses across the three cases. Specifically, the first assertion examines how all three methods instructors focused on developing conceptual understanding and combating mathematical misconceptions for which prospective teachers most often experience opportunities to learn through representations and approximations. The second and third assertions place more emphasis on differences across the cases based upon observed instructor actions and their beliefs. This study is significant because it helps us gain a deeper understanding about prospective teachers’ opportunities to learn within one teacher education program, and therefore, may point toward what can be done in the future to better prepare teachers in elementary mathematics education and the development of ambitious instruction. Additionally, this study unpacks how prospective teachers in the same teacher education program may have varying experiences and thus varied access to opportunities to learn.

Professional noticing of children’s mathematical thinking (Jacobs et al. in J Res Math Educ 41(2):169–202, 2010) is a construct that is used frequently to understand the ways in which teachers attend to, interpret, and respond to children’s mathematical thinking. In this paper, we present our analysis of prospective elementary mathematics teachers’ (PTs’) responses to a pedagogical activity designed to engage them in an approximation (Grossman et al. in Teach Coll Rec 111(9): 2055–2100, 2009) of professional noticing of children’s mathematical thinking in the moment of teaching, with a focus on developing responses which extend the student’s thinking. PTs were provided two different examples of student work and a series of questions designed to elicit information about their ability to professionally notice student work. We present data and analysis of 73 PTs’ abilities to engage in the individual skills associated with noticing as well as their abilities to use the skills cohesively in concert. Our results demonstrate the type of representation used determined PTs’ success in all three facets of noticing, with a particular effect on interpreting. Further, our focus on a specific type of response move, along with the introduction of a scaffold for that response move, made a difference in PTs’ ability to successfully respond to student work when compared to a prior study with the same PTs (Tyminski et al. 2014). We unpack these results along with suggestions for pedagogical practice as well as directions for future research.

This case study explored Prospective Elementary Mathematics Specialists’ (PEMSs) pedagogical practices during a K-5 Mathematics Endorsement (K-5 ME) program, specifically focusing on the authentic residency with mentor sessions. Participants included nine PEMSs, with qualitative and quantitative data collected via individual interviews, professional portfolios, and observations of classroom practices. Upon completion of the K-5 ME program, all participants were implementing instructional practices learned about in the program in substantial ways. Participants were honing their skills for orchestrating effective discourse, and the application of those skills became more meaningful as they saw the impact it was having on student learning. This implementation and documentation program coursework was rigorous as PEMSs worked to develop themselves as mathematics teachers and teacher leaders. The collaborative support during the authentic residency and mentor sessions seem to have been significant for PEMSs in supporting implementation of newly learned instructional practices. Ultimately, PEMSs completed the program feeling more prepared as elementary mathematics teachers and teacher leaders.

This study compares web usage data with interviews from 41 participants, who are members of an online professional development site called the Everyday Mathematics Virtual Learning Community (VLC), to explore how elementary school teachers learn from classroom video. Web usage data reveal that the commentary surrounding video posted to the VLC is sparse and surface level, possibly indicating a lack of serious attention to the videos. Interview data, however, indicate that participants felt they learned from this resource. Participants reported that the videos provided them with the opportunity to view and reflect on model lessons, plan curricula, and consider student thinking, among other learning outcomes. Participants also identified key factors that prevented them from posting comments to the site to convey their learning. These results can be used to understand not only how teachers perceive their own learning from classroom video, but also to redesign online professional development experiences to promote expression of that learning.

The purpose of this study is to investigate the impact of teacher education by examining instructional visions of graduates from a teacher education program, 2 or 3 years post-graduation. An instructional vision is a teacher’s idealized image of teaching practice, encompassing aspirations and hopes of what could occur in her classroom. We conducted an inductive analysis of participants’ self-reports of: (1) their instructional visions for teaching elementary mathematics, (2) whether their visions changed over time, (3) whether teacher education influenced changes in their visions, and (4) what their teacher education program promoted as an ideal instructional vision. We administered an open-ended questionnaire to 81 graduates from two cohorts of the same teacher education program. Using latent class analysis, we identified three profiles. Two profiles (89% of the participants) appeared to align with teacher education, suggesting more than one manifestation of the impact of teacher education. Our results indicated alignment between instructional visions and teacher education experiences with respect to teaching mathematics for conceptual understanding and engaging students in productive struggle. A contribution of this study is an awareness that teacher education programs could impact the instructional visions of their graduates in more than one way, with different subsets of intended visions remaining as residue. We conjectured that experiencing instruction that aligned with these instructional visions as learners while in teacher education influenced the development of these graduates’ visions.

A critical role of mathematics teacher education is to equip teachers with understandings and ambitious practices that support effective mathematics teaching for students from diverse backgrounds, specifically connecting to children’s mathematical thinking (CMT) and children’s linguistic, cultural, and family funds of knowledge (CFoK). Drawing on data from a larger research project, TEACH Math, this longitudinal case study uses the lens of authority to examine one teacher’s, Sena’s, understandings and practices related to CMT and CFoK over a 4-year period, i.e., through mathematics methods, student teaching, and early career teaching. Findings identify changes made in understandings and practices related to CMT and CFoK, with a focus on the development of recognition and realization rules. Emphasis is placed on how authority was shared in the classroom and how this may have impacted one teacher’s development of two very different learning-to-teach trajectories for connecting to CMT and CFoK. Implications for teacher educators focused on supporting novice teachers in developing ambitious mathematics teaching practices are discussed.

As teacher education shifts to focus on teaching beginners to do the work of teaching, assessments need to shift to focus on assessing practice. We focus on one teaching practice, eliciting student thinking, in the context of elementary mathematics. We describe assessments in two contexts (field and simulation). For each assessment, we describe the eliciting of three prospective teachers what could be seen about the skills of group of prospective teachers (N = 44). We report on how three prospective teachers had differing opportunities to demonstrate their skills in the context of the field assessment, but similar opportunities in the context of the simulation assessment. Although both contexts make important contributions, in each case, contributions are counterbalanced are by significant challenges. Although the authors do not argue for one assessment context over the other, they offer insights into the affordances and challenges of each so that teacher educators can make responsible decisions.

This study examines the impact of the Primarily Math Elementary Mathematics Specialist program on K-3 teachers’ mathematical content knowledge for teaching, attitudes toward learning mathematics, and beliefs about mathematics teaching and learning. Three cohorts of teachers participating in the program were compared to a similar group of non-participating teachers. Teacher outcomes were measured longitudinally across 5 years. Participating teachers showed changes in their knowledge, attitudes, and beliefs in line with program goals immediately after completion of coursework. Moreover, these changes were sustained in subsequent years, following program completion. Relative to the comparison group, participants demonstrated greater gains in knowledge as well as greater improvements in attitudes and beliefs. Implications of these results for professional development design, implementation, and evaluation are discussed.

Many in the field of mathematics education call for elementary schools to have elementary mathematics specialists (EMSs) who provide needed mathematical expertise and support for children and teachers. EMSs serve as a reasonable, immediate alternative to the challenges generated by elementary teachers needing improved mathematical knowledge for teaching in the classroom. However, limited inquiry has explored how to best prepare EMSs and how program features and learning activities influence their development. This mixed-method study identifies some of the interrelated benefits from a K-5 Mathematics Endorsement Program designed to prepare EMSs through examining changes in mathematical beliefs, specialized content knowledge (SCK), and classroom teaching practices during the program. Data (n = 32) were collected over the 2-semester program via belief surveys, a content knowledge assessment, observations of teaching practices, and individual interviews from elementary teachers participating in the program. The findings show some changes in beliefs can be made relatively quickly, other shifts in beliefs take more time and continued support, and changes in SCK and adoption of various aspects of standard-based pedagogy require considerably greater opportunities to learn. The described program features and learning experiences provided a context for these changes and offer considerations for EMS preparation programs.

Mathematical habits of prospective teachers affect problem comprehension and success and expose their beliefs about mathematics. Prospective elementary teachers (PSTs) (n = 121) engaged in a problem solving activity each week in class. Data were collected from PSTs enrolled in an undergraduate elementary mathematics methods course at a Southeastern State University over multiple semesters (six semesters, seven classes). PSTs’ solution methods for one intentionally misleading mathematics problem were analyzed using a convergent parallel mixed methods content analysis. Two-thirds of PSTs misunderstood the problem scenario and directly translated numbers from the problem text. PSTs who answered correctly used a problem model strategy to comprehend the scenario and were more likely to use multiple models, draw a diagram, and draw a diagram before using another model. However, a large number of PSTs who answered incorrectly also used multiple models and drew diagrams. Self-correction was not common (8 of 121), because their equations did not provide feedback or support comprehension. Three kinds of imprecision also affected problem comprehension and were evident in both correct and incorrect solutions. Intentionally misleading problems helped PSTs see consequences of their mathematical habits and highlighted the importance of sense making and precision when creating problem models.

This article addresses the problem of how opportunities to learn in teacher education programs influence future elementary mathematics teachers’ knowledge. This study used data collected for the Teacher Education and Development Study in Mathematics (TEDS-M). TEDS-M measured the mathematics content knowledge (MCK) and the mathematics pedagogical content knowledge (MPCK) of future teachers in their final year in teacher preparation programs. The purpose of this study is to explore whether elementary teaching candidates’ MCK and MPCK are associated with their opportunities to learn in mathematics courses and mathematics methods courses in five countries. The results showed that opportunities to learn in some teacher preparation components are more important than in other components.

This paper examines a series of instructional activities that provide prospective elementary teachers with an opportunity to engage in one of the more difficult practices to learn within mathematics teaching—organizing a mathematical discussion. Within a mathematics methods course, representations and decomposition of practice built from the Five Practices framework (Smith and Stein in Five practices for orchestrating productive mathematics discussions. Corwin Press, Thousand Oaks, CA, 2011) were implemented and studied to examine how prospective elementary teachers set goals, selected and sequenced available student work, and planned questions within a mathematical discussion. We examined prospective elementary teachers’ strengths and weaknesses in these facets through an approximation of practice set in a lesson context familiar to the prospective elementary teachers. Our results demonstrated that although prospective elementary teachers set varying goals for a discussion, their pedagogical choices in planning their discussion tended to be consistent with the goals they have set. These results support the focused development of prospective elementary teachers’ goal setting as an implication for mathematics teacher educators.

This paper describes a study of observed relationships between the design of a preservice elementary mathematics methods course with accompanying field practicum and changes in the extent to which participating prospective teachers identified themselves with the mathematics reform movement after becoming practicing teachers. The curriculum of the course with its embedded field practicum experiences was designed to support prospective teachers in interweaving methods coursework and pedagogical instruction with classroom practice. University and elementary school structures were interwoven by conducting weekly mathematics methods course sessions at a school site followed immediately by related practicum experiences in classrooms in the school. The interweaving of conceptualizations about mathematics teaching and learning was facilitated as the methods instructor provided professional development for classroom teachers simultaneously with the methods course, encouraging the teachers to create “reform-friendly” classroom environments. Survey data from 68 prospective teachers involved in the project were analyzed to identify and describe relationships between course/practicum experiences and the degree to which they as practicing teachers, 1–3 years following the course/practicum, identified with a reform mathematics perspective, as well as to examine factors influencing those relationships.

Prospective elementary teachers must understand fraction division deeply in order to meaningfully teach this topic to their future students. This paper explores the nature of the subject content knowledge of fraction division possessed by a group of Taiwanese prospective elementary teachers at the beginning of their mathematics methods course. The findings provide preliminary evidence that many prospective Taiwanese elementary teachers have developed the knowledge package of fraction division as described by Ma (Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States. Lawrence Erlbaum Associates, Mahwah, 1999). The nature of various strategies used by these teachers provides further illustration of a secure common content knowledge that can serve as a benchmark for the development of mathematics courses for prospective teachers. However, the findings also show that the tasks of representing fraction division, through either word problems or pictorial diagrams, are challenging even for those highly proficient in elementary and middle school mathematics. The broader implications of this research for the international community are discussed, and recommendations for elementary teacher education programs are presented.

This article reports on a national survey of higher education institutions in the United States to answer the question, “Who teaches mathematics content courses for prospective elementary teachers, and what are these instructors’ academic and teaching backgrounds?” We surveyed 1,926 institutions, and a faculty member from each of 825 institutions (42.8 %) participated in the survey. The survey results demonstrate that the majority of institutions are not meeting the recommendations of the Conference Board of the Mathematical Sciences (The mathematical education of teachers. American Mathematical Society, Providence, RI, 2001), the National Council of Teachers of Mathematics (Professional standards for teaching mathematics. National Council of Teachers of Mathematics, Reston, VA, 2005), and the National Council on Teacher Quality (Greenberg and Walsh 2008) for prospective elementary teachers to take at least nine credits of mathematics content designed specifically for them. Additionally, 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, and most institutions do not provide training and/or support for these instructors.

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

This study investigated 481 in-service elementary teachers’ level of mathematical content knowledge, attitudes toward mathematics, beliefs about the effectiveness of inquiry-based instruction, use of inquiry-based instruction and modeled the relationship among these variables. Upper elementary teachers (grades 3–5) were found to have greater content knowledge and more positive attitudes toward mathematics than primary teachers (grades K-2). There was no difference in teachers’ beliefs about effective instruction, but primary level teachers were found to use inquiry-based instruction more frequently than upper elementary teachers. Consistent with Ernest’s [Ernest (1989). The knowledge, beliefs and attitudes of the mathematics teacher: A model. Journal of Education for Teaching, 15(1), 13–33] model of mathematics teaching, content knowledge, attitudes, and beliefs were all found to be related to teachers’ instructional practice. Furthermore, beliefs were found to partially mediate the effects of content knowledge and attitudes on instructional practice. Content knowledge was found to be negatively related to beliefs in the effectiveness of inquiry-based instruction and teachers’ use of inquiry-based instruction in their classrooms. However, overall, teachers with more positive attitudes toward mathematics were more likely to believe in the effectiveness of inquiry-based instruction and use it more frequently in their classroom. Teacher beliefs were found to have the strongest effect on teachers’ practice. Implications for the goals and objectives of elementary mathematics methods courses and professional development are discussed.

The objective of this study was to examine the nature of preservice teachers’ evaluations of elementary mathematics problems using the Mathematical Tasks Framework (MTF), a model designed to discriminate among tasks according to their cognitive complexity. We also tested the relationship between mathematics content knowledge and problem length on the preservice teachers’ evaluations. Twenty-six undergraduate students enrolled in an elementary mathematics methods course at a large urban university were introduced to the MTF and cognitive complexity during a class lecture and were subsequently required to sort 32 mathematics problems according to the framework. Results demonstrated that overall, the preservice teachers had more difficulty accurately classifying problems considered to represent high levels of cognitive complexity compared to problems that were less complex. Those with strong mathematics content knowledge, as measured by a standardized test, were able to sort the problems more accurately than those with weaker content knowledge. Two open-ended items assessing content knowledge were not related to sorting performance. Finally, the preservice teachers were influenced by the surface characteristic of task length; the data indicated that the teachers tended to label short problems as less cognitively demanding and long problems as more so. Implications for preservice professional development include an increased emphasis on mathematics content knowledge as well as expert modeling of the identification of deep conceptual principles at the heart of the mathematics curriculum.

In the context of U.S. and world wide educational reforms that require teachers to understand and respond to student thinking about mathematics in new ways, ongoing learning from practice is a necessity. In this paper we report on this process for one teacher in one especially productive year of learning. This case study documents how Ms. Statz's engagement with children's thinking changed dramatically in a period of only a few months; observations and interviews several years later confirm she sustained this change. Our analysis focuses on the mathematical discussions she had with her students, and suggests this talk with children about their thinking in instruction served both as an index of change, and, in combination with other factors, as a mechanism for change. We identified four phases in Ms. Statz's growth toward practical inquiry, distinguished by her use of interactive talk with children. Motivating the evolution of phases were two sorts of mechanisms: scaffolded examination of her students' thinking; and asking and answering questions about individual students' thinking. Processes for generating and testing knowledge about children's thinking ultimately became integrated into Ms. Statz's instructional practices as she created opportunities for herself, and then students, to hear and respond to children's thinking.

Three elements of the ecology of a mathematics methodsfield experience were particularly important topromoting a shift in locus of authority thatprecipitated reflective thinking among the preserviceteachers: (a) an inquiry perspective, (b) the cohortgroup, and (c) school-university collaboration. Thearticle describes these three aspects of the ecologyof the field experience and provides the perspectivesof the mentor teacher, preservice teacher, and teachereducator. Also included are new challenges andproblems that arose with each aspect.

If the intellectual norms and values embedded in the mathematics education reform movement are to move beyond individual classrooms and significantly influence entire schools and districts, school and district administrators will need to become centrally, rather than peripherally, involved. This paper discusses the ways administrators' ideas about the nature of mathematics, learning, teaching, and school culture affect their interpretations of the nature and intent of the elementary mathematics reform movement and their thoughts about of how they might support it. In particular, administrators' views of parents' concerns, professional development for teachers, and of how new ideas move around in a school are discussed. I argue that administrators have well-formed ideas about mathematics, learning, and teaching, which influence their views of reform and their ideas of how to provide support. These ideas need to be taken into account if administrators are to be central actors in reform.