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Lesson study supports teachers in learning about curriculum and effective teaching practices. We discuss a district-wide lesson study process used to explore and adopt a new curriculum.

The study explored the verb clauses and thematic development evident in curriculum materials and in transcripts of teachers planning lessons using the materials. A central argument is that though teacher characteristics influence the ways they plan lessons with curriculum materials, the materials themselves influence teachers’ planned lessons via the ways mathematics is construed in the materials. We used verb clause and thematic analysis to analyze the features of curriculum materials and teachers’ lesson planning using those materials. We found differences between the curriculum materials in terms of how they construed mathematical activity and found that teachers’ lesson plans roughly aligned with the features in the materials; this speaks to the impact of features of curriculum materials on teachers’ lesson planning. We also found small but perceptible differences between teachers from different curriculum backgrounds, suggesting an enculturating effect from the materials.

Mathematical modeling is a high-leverage topic, critical for college and career readiness, participation in STEM education, and civic engagement. Mathematical modeling involves connecting real-world situations, phenomenon, and/or data with mathematical models, and in this way applies across various STEM disciplines, including mathematics, engineering, and science. Although research has begun to explore mathematical modeling instruction in the elementary grades, questions remain about how to assess student learning at the elementary level. We addressed this need by designing an assessment of mathematical modeling competencies for students in grades 3 through 5. Informed by international research, our assessment includes a hybrid structure to assess mathematical modeling competencies holistically (as students engage in the complete modeling process) and atomistically (as students engage in different components of the modeling process, including making sense of phenomena and real-world situations, setting up and operating on mathematical models, and interpreting results in relation to the real-world context). We conducted student interviews, followed by two rounds of pilot testing to inform item development and ensure acceptable psychometric properties. The final assessment included 13 items (9 multiple choice, 3 open-response, and 1 complete modeling task). We describe our assessment development process, and provide sample assessment items and detailed coding rubrics. We summarize quantitative analyses which established high reliability and low standard error for our assessment, supporting its use for grades 3 to 5. Implications of our framework and assessment for mathematical modeling instruction and future research on STEM learning are discussed.

We studied how the distal policy mechanisms of curricular aims and objectives articulated in official curriculum documents influenced classroom instruction, and the factors that were associated wit...

This study explored how teachers engage and support students at a project-based STEM high school to identify key elements defining students’ learning experiences in a project-based environment. Interviews and surveys provided students’ and teachers’ perspectives of challenges and supports in the project-based learning process. The teachers’ support was personal and caring, they challenged students and provided responsive guidance and advice, responding directly to students’ interests while retaining high standards and expectations; we saw that teachers’ caring in an environment of high expectations was crucial for student success. Identification of three critical elements of the teachers’ supports and challenges (student-centered projects, a focus on twenty-first century competencies, and strong teacher-student relationships) led to recommendations for teacher preparation such as providing preservice teachers with authentic experiences, celebrating failure as part of learning, explicitly teaching about project-based learning pedagogy, and stressing the importance of caring relationships and twenty-first century competencies.

Mathematical modeling is a high-leverage topic, critical for participation in STEM education and civic engagement (Aguirre et al., Math Teach Educ 7(2):7–26, 2019). In the Mathematical Modeling with Cultural and Community Contexts (M2C3) project, we investigate culturally responsive, community-based approaches that support mathematical modeling with elementary students, particularly those historically marginalized in STEM. This study focused on how students drew upon their knowledge and experiences as they engaged in the mathematical modeling process. Using a qualitative case study design (Stake, The art of case study research. Sage, Thousand Oaks, 1995), we investigated elementary students (third and fifth graders) in four different classes as they engaged in a common modeling lesson – Upcycling Plastic Bags to Make Jump Ropes. Data sources included video-recorded lesson observations, teacher interviews and reflections, and student work. Through iterative cycles of analysis, we conducted within-case and cross case analyses. Across the four lessons, we found that students’ experiences and knowledge related to jump rope activities supported their engagement in each phase of the modeling process. Specifically, students leveraged experiences to identify important quantities and relationships, to make assumptions, to analyze and interpret the reasonableness of their solutions, and to inform model revisions. Our findings have important implications for mathematics modeling 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 e...

This contribution to curriculum ergonomics focuses on teachers’ perceptions of problem solving and rigor in the Common Core State Standards for Mathematics, the curricular aims and objectives component of the official curriculum in the U.S. Our analyses involve 89 middle school mathematics teachers (MSMTs) using two different types of district-adopted textbooks: thinking device (TD) and delivery mechanism (DM). We found that MSMTs’ perceptions of problem solving and rigor in the Common Core varied by textbook type. Consequently, while this paper examines teachers’ perceptions of curricular aims and objectives, it was apparent that another component of the official curriculum, teachers’ district-adopted textbooks, played a strong role in influencing these perceptions. The implications of these results are discussed.

Elementary mathematics curriculum materials can serve as a lever for instructional change. In this paper, we promote a particular kind of instructional change: supporting teachers in learning to in...

We conceptualize curriculum ergonomics as a field that studies the interactions between users and curriculum materials. We identify five themes that a curriculum ergonomics lens brings into sharper focus: (1) teachers’ relationships with and capacity to use curriculum resources; (2) alignment between design intentions and patterns of curriculum use; (3) ways in which curriculum resources influence instruction; (4) ways in which curriculum features are purposefully designed to achieve an educative purpose; and (5) the dissolution of boundaries between design and use. We first summarize the literature in curriculum ergonomics and then show how key themes from that literature inform the discussion of curriculum ergonomics.

To support a growth mindset in students, consider components involving cognitive, social, and emotional aspects so that students can work within their zone of productive stuggle.

We report findings on teachers’ noticing of features in the teacher resources of mathematics curriculum programs. Based on prior analysis, we selected teachers using one of two curriculum types: delivery mechanism or thinking device. The participating teachers and the curriculum programs aimed to align with the Common Core Standards for Mathematics, and thus, they ostensibly held a common aim for instruction. We analyzed 147 lesson planning interviews with 20 middle school mathematics teachers. We found that teachers attended to similar features of teacher resources; however, patterns for interpreting and planning decisions varied based on teachers’ orientations and curriculum type.

This study examines Middle School Mathematics Teachers’ (MSMTs') (N = 1,241) perceptions of the Common Core State Standards for Mathematics (CCSSM) and its impact on the instructional environment. A total of eight factors appear in the data. These factors include professional support, teachers’ use of district-adopted and non-district-adopted curricular resources, influence of CCSSM assessment and teacher evaluation on the instructional environment, influence of the CCSSM on classroom instruction, teachers’ planning practices, and perceived rigor of the CCSSM. The data suggest that teachers’ use of digital resources located online was disconnected from their district-adopted resources suggesting a lack of curricular coherence. MSMTs note that the CCSSM had caused them to incorporate more reform-oriented practices such as exploration and productive struggle into their daily instruction. MSMTs also perceive that the CCSSM includes new content that is more rigorous than previous state standards. Sampled MSMTs state that while state assessments will measure CCSSM content, they are less likely to include more complex problems or the standards for mathematical practice. Teachers are more likely to read teacher resources than student textbook activities online and to use digital resources for remediation instead of inquiry activities. Over one-third of MSMTs wanted more CCSSM professional development.

Eight middle school mathematics teachers’ perceptions and uses of curriculum materials and the Common Core State Standards for Mathematics (CCSSM) were investigated. Adapting a noticing framework and models of dialogic instruction and direct instruction, teachers’ noticing practices with curriculum materials and the CCSSM when planning, enacting, and reflecting on lessons were examined. Teachers who were committed to implementing the CCSSM and who were using one of two substantively different curriculum programs were purposefully selected. Data sources included multiple forms of interviews and classroom observations. The teaching evidenced three distinct noticing patterns. These patterns indicated that teachers’ curriculum materials were associated with how teachers perceived and enacted the CCSSM. Teaching with a curriculum program that was designed as a thinking device prioritized the Standards for Mathematical Practice of CCSSM evidenced noticing that was consistent with dialogic instruction. Teaching with a curriculum program that was designed as a delivery mechanism prioritized the Content Standards of CCSSM and evidenced noticing consistent with direct instruction. Findings indicated that the designated curriculum and contributed to differing interpretations of CCSSM and served as a lens for noticing. However, a dialogic curriculum program was not sufficient to support dialogic approaches in practice. One pattern showed teachers planning dialogic lessons, but the lesson enactments were not consistent with teachers’ plans, with evidence that the teachers were not aware that their practices differed from dialogic approaches. Implications for research and practice are discussed.

This chapter provides an overview of the mathematics modules developed in the Teachers Empowered to Advance Change in Mathematics (TEACH Math) project. We designed these modules to support preK-8 prospective teachers in learning to integrate mathematics, children’s mathematical thinking, and their cultural, linguistic, and community-based funds of knowledge in mathematics instruction. We first discuss the process by which we designed and researched these modules in our university preK-8 mathematics methods courses. We then describe each module, the goals, and the activities within each module. The three modules are Mathematics Learning Case Study, Community Mathematics Exploration, and Classroom Practices. We conclude with considerations for mathematics teacher educators for using the modules in their mathematics methods courses.

This study was conducted at an innovative science, technology, engineering, and mathematics high school, providing a rich contextual description of the teaching and learning at the school, specifically focusing on problem solving and inquiry approaches, and students' motivation, social interactions, and collaborative work. Data were collected through interviews with teachers and students, classroom observations, school meetings, school design and planning efforts, student surveys, and student standardized test scores. We saw that the teaching and learning at the school were typically inquiry and problem based and that student motivation, collaboration, and social interactions were strong due to the environment of inquiry and problem-based teaching and learning.

Make these small adjustments to your syllabus and watch spaces open to connect to children's multiple mathematical knowledge bases.

Research, theory, and the Common Core State Standards apply to lesson planning'a process similar to planning a road trip.

Mathematics teacher educators (MTEs) designed and studied a video analysis activity intended to support prospective teachers (PSTs) in learning to notice equitable instructional practices. PSTs from 4 sites (N= 73) engaged in the activity 4 to 5 times during the semester, using a set of 4 “lenses” to analyze teaching and learning as shown in videos. In an earlier analysis of this activity, we found that PSTs increased their depth and expanded their foci in noticing equitable instructional practices (Roth McDuf_ e et al., 2013). In this analysis, we shift the focus to our work as MTEs: We examine our decisions and moves in facilitating the video analysis activity with a focus on equity, and we discuss implications for other MTEs.

As part of a larger research project aimed at transforming preK-8 mathematics teacher preparation, the purpose of this study was to examine the extent to which prospective teachers notice children’s competencies related to children’s mathematical thinking, and children’s community, cultural, and linguistic funds of knowledge or what we refer to as children’s multiple mathematical knowledge bases. Teachers’ noticing supports students’ learning in deep and meaningful ways. Researchers designed and enacted a video analysis activity with prospective teachers in their mathematics methods course. The activity served as a decomposition of practice in order to support prospective teachers in engaging in an approximation of the practice of noticing. Our findings showed that prospective teachers evidenced noticing of mathematics teaching and learning as early as the mathematics methods course. We also found that the prompts and structure of the activity supported prospective teachers by increasing their depth of noticing and their foci in noticing, moving from attending primarily to teacher moves (and merely describing what they saw) to becoming aware of significant interactions (and interpreting effects of these interactions on learning). Implications for teacher educators interested in designing and enacting activities to support noticing are discussed.

Teacher educators need an awareness of orientations that prospective teachers (PSTs) hold about children's families and communities, so that they can support PSTs in developing positive perspectives and addressing deficit perspectives. Orientations expressed by 20 PSTs while in preK-8 mathematics methods courses are examined. Results from interviews indicate that PSTs recognize the importance of connecting with parents, understanding home and community practices, and building on these practices to support children's mathematical learning. They also, however, exhibit inconsistent perspectives, at times indicating a lack of understanding as to why some families appear to be less able to support students' academic efforts.

Two related case studies of secondary mathematics teachers examine the roles and conditions helpful in initiating, directing, and/or supporting teachers' own professional development. Using multiple data sources from school-based and professional settings, we applied analytic induction to identify patterns of similarities and differences in teachers' experiences and developmental trajectories. Findings revealed how the teachers served as brokers in their own development process and how attitude, attention, and awareness played important roles in teacher learning. As the teachers initiated learning processes, they demonstrated awareness of broader educational contexts that support learning by identifying external policies and priorities that aligned with teachers' attitudes. They then used these external priorities as boundary objects to broker for teachers' learning, including other teachers' learning. They attended to learning needs in directing a plan for learning activities, rather than using externally mandated goals for changing practice (e.g., an external focus on improving students' tests scores). These cases suggest that in structuring professional learning, opportunities to self-identify learning needs should be part of the process, and external supports can become more powerful when teachers play a brokering role in relating these supports to their immediate contexts. Further implications for professional development processes are explored.

This study examines the ways prospective elementary teachers (PSTs) made connections to children’s mathematical thinking and children’s community funds of knowledge in mathematics lesson plans. We analyzed the work of 70 PSTs from across three university sites associated with an instructional module for elementary mathematics methods courses that asks PSTs to visit community settings and develop problem solving mathematics lessons that connect to mathematical practices in these settings (Community Mathematics Exploration Module). Using analytic induction, we identified three distinct levels of connections to children’s mathematical thinking and their community funds of knowledge evidenced in PSTs’ work (emergent, transitional, and meaningful). Findings describe how these connections reflected different points on a learning trajectory. This study has implications for understanding how PSTs begin to connect to children’s mathematical funds of knowledge in their teaching, a practice shown to be effective for teaching diverse groups of children.

Research repeatedly documents that teachers are underprepared to teach mathematics effectively in diverse classrooms. A critical aspect of learning to be an effective mathematics teacher for diverse learners is developing knowledge, dispositions, and practices that support building on children’s mathematical thinking, as well as their cultural, linguistic, and community-based knowledge. This article presents a conjectured learning trajectory for prospective teachers’ (PSTs’) development related to integrating children’s multiple mathematical knowledge bases (i.e., the understandings and experiences that have the potential to shape and support children’s mathematics learning—including children’s mathematical thinking, and children’s cultural, home, and community-based knowledge), in mathematics instruction. Data were collected from 200 PSTs enrolled in mathematics methods courses at six United States universities. Data sources included beginning and end-of-semester surveys, interviews, and PSTs’ written work. Our conjectured learning trajectory can serve as a tool for mathematics teacher educators and researchers as they focus on PSTs’ development of equitable mathematics instruction.

Thread small changes seamlessly into high-level reasoning tasks to reach all students.

What mathematics must teachers understand in order to teach elementary school mathematics? Historically, the answer to that question was that they needed to know only the mathematics concepts and procedures they taught. Research on the teaching and learning of mathematics challenges that myth and indicates that the role and substance of mathematics knowledge needed for teaching has expanded. In addition to a profound understanding of fundamental mathematics content (Ma 1999), teachers need deep knowledge about how students learn particular math concepts and processes and, correspondingly, which teaching approaches and strategies are most effective in meeting students' learning needs. Ball, Hill, and Bass (2005) refer to this kind of knowledge as mathematics knowledge for teaching. For example, to determine whether a child's unique approach for adding fractions can be generalized or how an area model might be used to develop the partial-product algorithm for multiplying two-digit numbers, teachers need a form of mathematical knowledge different from other professionals (e.g., engineers). Aligned with this research, the Teaching Principle identifies the necessity for teachers to develop deep knowledge and understanding of mathematics to meet the needs of their students today and in the future (NCTM 2000).

Understanding the concept of area is a challenge for children. Beyond following procedures with formulas, studying area has been a source of confusion for students at all mathematical levels (Nitabach and Lehrer 1996; Thompson and Preston 2004). In fact, National Assessment of Educational Progress (NAEP) data from fourth and eighth graders suggest that many students have incomplete or superficial understanding of area (Martin and Strutchens 2000). Although students continue to struggle, the 2003 NAEP data shows that performance on area items is significantly improving, which may be due to an increased curricular focus on area concepts and the use of manipulatives and visualization (Blume, Galindo, and Walcott 2007).

This department features children's explorations in mathematics and presents teachers with open-ended investigations to enhance mathematics instruction. The tasks invoke problem solving and reasoning, require communication skills, connect various mathematical concepts and principles, and have been classroom tested.

The purpose of this study was to explore the effect of providing preservice teachers the opportunity to collect real data in a science methods inquiry investigation and using the data, design data displays in their mathematics methods course. The research questions focused on how preservice teachers' understandings of data displays, research design, and the specific content addressed improved when they used these displays to attempt to communicate the data they had collected themselves in their inquiry investigations. The 46 preservice teachers were given questionnaires at the beginning and end of the courses, twelve were interviewed both pre and post, all written work pertaining to data displays and the inquiry investigations was collected, methods class sessions were audio and videotaped, and the final data display and science investigation projects were photocopied. The findings show that by creating and scrutinizing their data displays, the preservice teachers were able to recognize the limitations of their inquiry investigation design. Through working with data in the context of inquiry projects of their own design, the preservice teachers realized meaningful connections and commonalities that exist in mathematics and science while strengthening their knowledge and skills in both disciplines.

Collecting, analyzing, and displaying data provide rich opportunities to connect mathematics and science concepts. However, mathematics and science teacher educators rarely work together to design tasks that connect mathematics and science. In this article, we describe collaboration between a mathematics teacher educator and a science teacher educator that included the design of an inquiry-based project for preservice elementary teachers to draw on the natural connections of these disciplines. We also discuss preservice teacher learning outcomes from the project and present recommendations for teacher educators.

This teacher development study closely examined a teacher's practice for the purpose of understanding how she selected and implemented instructional materials, and correspondingly how these processes changed as she developed her problem‐based practice throughout a school year. Data sources included over 20 hours of planning and analysis meetings with the teacher and 27 video‐taped lessons with discussions before and after each lesson. Through qualitative analysis we examined the data for: students' cognitive demand for curricular materials the teacher selected and implemented; teacher's beliefs and practices for students' engagement in mathematical thinking; and teacher's and students' communication about mathematics during instruction. We found that the teacher shifted her views and use of instructional materials as she changed her practice towards more problem‐based approaches. The teacher moved from closely following her traditional, district‐adopted textbook to selecting problem‐based tasks from outsi...

The integration of mathematics and science is a recommended pedagogical strategy made by major reform-based documents. The goal is to enhance learner understandings by recognizing the relationships between the two disciplines. Documented attempts to systematically enact this initiative in teacher preparation remain uncommon. As a result, an elementary science methods course instructor in collaboration with two mathematics education researchers conducted a practitioner-research study that examined his efforts to connect mathematics and science. The perceptions of two groups of interns (those in a specialized program that aimed to make connections between mathematics and science and those who were not in the program but who were concentrating in mathematics or science) were contrasted in regards to the following four elements: (1) an appropriate science learning environment for elementary students, (2) the extent to which their science methods instructor modeled good teaching of science, (3) the extent to which they observed their science methods instructor making connections to mathematics in his teaching, and (4) the rationale for and intent to make connections between science and mathematics in elementary teaching. We found that there were discernible differences between the two groups in their perceptions about the issues under study. An implication for practice is that caution should be exercised when attempting an integrated approach to a science methods course, since interns may perceive the discipline of mathematics in a limited manner.

This study examined the views, and the retention of these views, of 19 preservice elementary teachers as they learned about nature of science (NOS). The preservice teachers participated in a cohort group as they took a science methods course during which they received explicit reflective instruction in nature of science. Through Views of Nature of Science version B (VNOS-B) surveys and interviews it was found that most preservice teachers held inadequate ideas of nature of science prior to instruction, but improved their views after one semester of instruction in the science methods course. However, 5 months after instruction, the graduate preservice teachers were again interviewed and it was found that several of the students reverted back to their earlier views. The results are interpreted through Perry's scheme, and implications and recommendations for elementary science teacher education are made. © 2005 Wiley Periodicals, Inc. J Res Sci Teach 43: 194–213, 2006

This study was conducted in a reform-based mathematics and science teacher education program in the USA, the Maryland Collaborative for Teacher Preparation(MCTP). The goal of the undergraduate program was to prepare upper elementary/middle level specialists in mathematics and science. One significant aspect of the MCTP was the expectation that the program's professors (in mathematics and science) would model a new vision of effective pedagogy based on reform-based recommendations. We determined, in general, that the program's mathematics and science content professors accepted the dual role of modeling effective instruction at the same time they were delivering content. However, this dual responsibility raised in their minds an ‘issue of appropriate balance’ between content and pedagogical foci in their courses. Previously, the professors' had not questioned a focus heavily tilted toward content coverage. We also determined that the program's teacher candidates believed that the mathematics and science professors modeled effective instruction. One of the primary reasons that the teacher candidates believed that their professors were modeling effective instruction was that the focus in the courses was primarily on conceptual understanding, not memorization. A major implication was that the professors' modeling of reform-based instruction prompted the teacher candidates to develop a new vision of mathematics and science teaching shaped by their professors' example.

In this case study I examine the reflectivepractices of two elementary pre-serviceteachers during their student teachinginternship. I extend current views ofreflective practice to create a framework for a`deliberate practitioner'. With this framework,I investigate the pre-service teachers'thinking with regard to reflective processesand how they use their pedagogical contentknowledge in their practices. My findingsindicate that the pre-service teachers usetheir pedagogical content knowledge inanticipating problematic events, and inreflecting on problematic events ininstruction. However, limits in pedagogicalcontent knowledge and lack of confidence impedethe pre-service teachers' reflection while inthe act of teaching. They were more likely toreflect on their practices outside of the actof teaching. Implications for teacher educatorsand pre-service teachers are discussed.

This case study was an investigation of the role of the institutional culture of a university in the process of changing to reform-based practices for two college mathematics professors. A framework is presented for identifying and analyzing institutional norms and policies that are present and those lacking in supporting faculty efforts toward reform. The primary categories considered in the framework are the teaching and learning context, the professional community, and the university's reward system. This framework is applied to the cases, and findings indicate that institutional norms in the forms of priorities for how time is allocated to professional responsibilities (as part of the teaching context), and colleagues and administrators who understood and shared the professors' goals for reform (as part of the professional community) strongly influenced the professors' efforts toward change. Additionally, a reward system that recognized efforts to improve teaching and learning was critical in establishing a culture that promoted change. However, despite the fact that both professors were in the same mathematics department, the nature of the influence of these factors was not uniform. The institutional culture indeed had multiple layers that required examination for each professor's individual context. The findings help highlight the importance of institutional context for both K-12 and college level teaching and learning. The nature of these influential factors and the institutional layers were discussed for each case, along with implications for other institutions.

MARCH 2003 Children’s literature involving mathematics creates a natural context for talking about mathematics (see Hellwig, Monroe, and Jacobs [2000]; Moyer [2000a, 2000b]). Because teachers and students are accustomed to using books for discussion in other subjects, such as social studies and language arts, using literature in mathematics might help teachers who are just beginning to incorporate mathematics discussions in instruction to create an environment that promotes discourse. Moreover, teachers who already use discourse in mathematics find that literature offers another opportunity for students to make meaning and build connections between mathematics and their lives (Austin 1998; Moyer 2000a). To explain how literature can serve as a fertile ground for fostering discourse communities in mathematics, we offer four examples of books to read aloud with a class and use in mathematical discussions. Although any of these books could be used to develop a mathematics activity, we are focusing on the mathematics discussions that might emanate from using such books in instruction. We have selected recent titles from various grade levels in order to expose readers to newer, high-quality literature for teaching and learning mathematics. A Pig Is Big