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Cyndi Edgington, P. Holt Wilson, Paola Sztajn and Jared Webb

Mathematics teacher educators play a critical role in translating research findings into frameworks that are useful for mathematics teachers in their daily practice. In this article, we describe the development of a representation that brings together four research-based learning trajectories on number and operations. We detail our design process, present the ways in which we shared this representation with teachers during a professional development project, and provide evidence of the ways teachers used this translation of research into a pedagogical tool to make sense of students' mathematics. We conclude with revisions to the representation based on our analysis and discuss the role of mathematics teacher educators in translating research findings into useful tools for teachers.

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Frederick A. Peck

In this article, I describe a learning trajectory for slope that emerged as our team 1 conducted a design-based research study in a high school algebra classroom. This study was motivated by a problem in our own practice. As teachers and

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Carmen S. Brown, Julie Sarama and Douglas H. Clements

Learning trajectories (routes, curves) in preschool and how they helped a teacher develop goals and objectives for her students' mathematical knowledge. Learning trajectories have three parts: a mathematical goal, a developmental path, and a set of activities matched to each of those levels. Activities and a teacher's explanation are included.

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Janka Szilágyi, Douglas H. Clements and Julie Sarama

This study investigated the development of length measurement ideas in students from prekindergarten through 2nd grade. The main purpose was to evaluate and elaborate the developmental progression, or levels of thinking, of a hypothesized learning trajectory for length measurement to ensure that the sequence of levels of thinking is consistent with observed behaviors of most young children. The findings generally validate the developmental progression, including the tasks and the mental actions on objects that define each level, with several elaborations of the levels of thinking and minor modifications of the levels themselves.

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Maria Blanton, Bárbara M. Brizuela, Angela Murphy Gardiner, Katie Sawrey and Ashley Newman-Owens

The study of functions is a critical route into teaching and learning algebra in the elementary grades, yet important questions remain regarding the nature of young children's understanding of functions. This article reports an empirically developed learning trajectory in first-grade children's (6-year-olds') thinking about generalizing functional relationships. We employed design research and analyzed data qualitatively to characterize the levels of sophistication in children's thinking about functional relationships. Findings suggest that children can learn to think in quite sophisticated and generalized ways about relationships in function data, thus challenging the typical curricular approach in the lower elementary grades in which children consider only variation in a single sequence of values.

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Nicole Panorkou and Jennifer L. Kobrin

This research study was designed to evaluate the extent to which professional development (PD) designed around a learning trajectory (LT) on geometric measurement of area was successful in helping teachers use the LT to conduct formative assessment. Six 3rd-grade teachers from the Midwest participated in 20 hours of PD centered on the LT. Data to evaluate the PD were obtained from a set of questionnaire prompts administered before and after teachers' participation in the PD. The results suggest that teachers increased their ability to elicit and interpret student thinking and use assessment results to make instructional decisions. We consider the design and evaluation of this PD to be valuable for future efforts aiming to use LTs to support teachers in their formative assessment practices.

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Douglas H. Clements, Julie Sarama, Mary Elaine Spitler, Alissa A. Lange and Christopher B. Wolfe

This study employed a cluster randomized trial design to evaluate the effectiveness of a research-based intervention for improving the mathematics education of very young children. This intervention includes the Building Blocks mathematics curriculum, which is structured in research-based learning trajectories, and congruous professional development emphasizing teaching for understanding via learning trajectories and technology. A total of 42 schools serving low-resource communities were randomly selected and randomly assigned to 3 treatment groups using a randomized block design involving 1,375 preschoolers in 106 classrooms. Teachers implemented the intervention with adequate fidelity. Pre- to posttest scores revealed that the children in the Building Blocks group learned more mathematics than the children in the control group (effect size, g = 0.72). Specific components of a measure of the quantity and quality of classroom mathematics environments and teaching partially mediated the treatment effect.

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Caroline B. Ebby and Marjorie Petit

Numerous research studies have shown that formative assessment is a classroom practice that when carried out effectively can improve student learning (Black and Wiliam 1998). Formative assessment is not just giving tests and quizzes more frequently. When assessment is truly formative, the evidence that is generated is interpreted by the teacher and the student and then used to make adjustments in the teaching and learning process. In other words, the formative assessment generates feedback, and that feedback is used to enhance student learning. Formative assessment is therefore fundamentally an interpretive process: It is less about the structure, format, or timing of the assessment and more about the function and use by both the teacher and student (Wiliam 2011). For teachers of mathematics, the heart of this process is making sense of and understanding student thinking in relation to content goals.

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Jennifer Suh, Sara Birkhead, Rachelle Romero Farmer, Terrie Galanti, Alexandrea Nietert, Tyler Bauer and Padmanabhan Seshaiyer

Working with a mathematics coach and university researchers in a K-4 lesson study, teachers increase their understanding of student abilities in a fair-share sandwich problem.

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Jinfa Cai, Anne Morris, Charles Hohensee, Stephen Hwang, Victoria Robison and James Hiebert

We concluded our November editorial (Cai et al., 2018b) with a promise to consider research paradigms that could bring us closer to the new world we have envisioned where research is intertwined with practice. We will call the paradigms we have in mind research pathways to avoid the range of complicated connotations often applied to the term paradigm. By research pathways in education, we mean the collection of assumptions that define the purposes of educational research, the principles that differentiate research from other educational activities, and the guidelines for how research should be conducted.