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Learning trajectories (LTs) can inform teaching by contributing a variety of pedagogical and content-related insights and strategies. They support sequencing topic introduction and development. Because ideas evolve gradually and often rely on a careful introduction of new representations, operations, cases of numbers, structures, and definitions, LTs can help teachers avoid overwhelming students with prematurely formal concepts. They shed light on where an idea comes from and how it allows students to see certain situations differently. They support recognizing and leveraging diverse student ideas and shed light on student misconceptions and errors. Discussions about content and pedagogy often reside within grade bands (elementary, middle, and high school). In the authors' work with teachers on LTs, they have witnessed very productive exchanges across grade bands, which can strengthen vertical coherence. To illustrate this, they describe three LTs: (1) equipartitioning, (2) ratio and proportion, and (3) exponential functions in the article. These three LTs share the territory often referred to as multiplicative space (Harel and Confrey 1994; Vergnaud 1988), which includes the operations of multiplication, division, and exponentiation. Competency in multiplicative reasoning opens the door to a myriad of applications (involving ratio and proportion, percentages, density, similarity, rate, etc.).

This study investigated the process of instructional change required to translate data on student progress along learning trajectories (LTs) into relevant instructional modifications. Researchers conducted a professional development session on ratio LTs, which included analyzing 3 years of district-level data from Math-Mapper 6-8, a digital LT-based diagnostic assessment application, with fifteen 6th and 7th grade teachers. Teachers subsequently conducted a lesson study to enact what they had learned, allowing researchers to study how teachers used data on student progress along ratio equivalence LTs to design, implement, and evaluate the lesson study. Researchers applied a framework for LT-based data-driven decision making to analyze video data of the lesson study activities. Teachers successfully scanned data reports to pinpoint the LT levels at which to target modified instruction. In one instance, they focused too narrowly on a single item resulting in excessive lesson time on tasks on graph literacy external to the LT. In the other, their data interpretation was overly general and resulted in the design and implementation of a sequence of tasks that reversed the order implied in the LT and relied on the use of more sophisticated strategies from subsequent LTs. Results suggest a need for more data interpretation skills, a deep understanding of the learning theory underpinning LTs, and more precision in teacher discourse around LTs.

Fully articulating validation arguments in the context of classroom assessment requires connecting evidence from multiple sources and addressing multiple types of validity in a coherent chain of reasoning. This type of validation argument is particularly complex for assessments that function in close proximity to instruction, address the fine granularity of learning trajectories (LTs), have multiple stakeholders, and are delivered digitally with a quick turn-around for formative assessment purposes. This article describes a validation framework for classroom assessment and uses it to illustrate a validation argument addressing one of several purposes for the assessments, the use of class-level data by individual teachers. The argument concerns the use of a middle-grades digital learning system, Math-Mapper 6-8, which contains LT-based diagnostic assessments. The argument is structured as a set of six claims that examine the assessment structure, the identification and treatment of non-conforming items, the analysis of student data, and the analysis of teachers' interpretations of data. The article stresses the critical role of scrutiny and debate among learning scientists, psychometricians, and practitioners in the validation process.

The paper describes how designers used the construct of learning trajectories to create a tool, Math-Mapper 6-8, to help scaffold curricula toward increased learner-centered coherence. It defines "learner-centered curricular coherence" as "an organizational means to promote a high likelihood that each learner traverses one of many possible paths to understand target disciplinary ideas in a curriculum." The tool's features, including its learning map, diagnostic assessments, and reporting system, are tied to its underlying foundation in learning trajectories. Three preliminary studies of the implemented tool's effects are reported to provide insight to its influence on curricular sequencing, students' patterns of performance in early algebra, and student responses to the assessments.

This study compared the effects of an integrated reform-based curriculum to a subject-specific curriculum on student learning of 19,526 high school algebra students. Using hierarchical linear modelling to account for variation in student achievement, the impact of the reform-based "Core-Plus Mathematics" curricular materials on student test scores is compared to the subject-specific curriculum. Findings from this study indicate that students enrolled in integrated mathematics outperformed subject-specific students on an Algebra I exam (highly aligned with content), and performed equally on an Algebra II exam (poorly aligned). High minority students in high-need schools demonstrated higher performance when they were enrolled in integrated mathematics.

Design research continues to gain prominence as a significant methodology in the mathematics education research community. This overview summarizes the origins and the current state of design research practices focusing on methodological requirements and processes of theorizing. While recognizing the rich variations in the foci and scale of design research, it also emphasizes the fundamental core of understanding and investigating learning processes. That is why the article distinguishes two archetypes of design research, one being focused on curriculum innovations, one being focused on developing theories on the learning processes, which is the main focus of the thematic issue. For deepening the methodological discussion on design research, it is worth it to distinguish aims and quality criteria along the archetypes and elaborate achievement and challenges for the future.

Design research studies provide significant opportunities to study new innovations and approaches and how they affect the forms of learning in complex classroom ecologies. This paper reports on a two-week long design research study with twelve 2nd through 4th graders using curricular materials and a tablet-based diagnostic assessment system, both designed around a learning trajectory on equipartitioning. A learning trajectory is a conceptual model of how students move from naïve to more sophisticated understandings as they engage with a carefully sequenced set of tasks. The equipartitioning construct describes how students come to understand the ideas involved in sharing fairly an evenly divisible collection, a single shape or multiple shapes. The paper is organized around the three phases of design research: planning, conduct, and retrospective analysis Cobb et al. ("Educ Res" 32(1):9-13, 2003). It illustrates how the conjectures of the study are subjected to testing and revision, based on the students' and teachers' behaviors during the study, and how interpretations and theories evolve during the different phases.

Briggs and Peck make a compelling case for creating new, more intuitive measures of learning, based on creating vertical scales using learning trajectories (LT) in place of "domain sampling." We believe that the importance of creating measurement scales that coordinate recognizable landmarks in learning trajectories with interval scales cannot be understated. As we have learned from the previous decades of high-stakes assessment, when tests are required to serve multiple purposes, such as improving systemic accountability and providing instructional guidance, one purpose often suffers. These high-stakes tests have provided weak instructional guidance, producing unintentional outcomes such as excessive focus on "bubble kids" without sufficient information on how to promote deep learning (e.g., Horn, Kane, & Wilson, 2015). And in spite of the field's best efforts, we are still without a set of measurement scales that communicate how different areas of the scales are related to qualitatively different states of children's coming to know big ideas. Our own current research program is devoted to a similar goal. Instead of end-of-year, large-scale testing, however, we target periodic intrayear diagnostic uses of such scales without high-stakes consequences for teachers or students. We are designing our measures to inform teachers and students about student thinking and learning. In this response to the Briggs and Peck paper, we wish to comment on 2 broad issues related to the paper: (1) the modeling theories that underlie the relationships between measures and the phenomena they are designed to index and (2) the details and challenges we have experienced in building such vertical scales and their implications for developing high-stakes assessments based on learning trajectories.

This article analyzes the design decisions of a team developing diagnostic assessments for a learning trajectory focused on rational number reasoning. The analysis focuses on the design rationale for key decisions about how to develop the cognitive assessments and related validity arguments within a fluid state and national policy context. The study draws on ethnographic methods adapted from science, technology, and society studies to document key rationales for decisions. For this team, concerns about the validity of both the assessments and the hypothetical trajectory, anticipated uses of the assessments, and the available distribution of resources and expertise to different project activities were all considerations for significant design decisions throughout the project. The study findings suggest that success in the design of trajectories-based assessments depends on teams' attention to balancing core design activities with engagement in the external policy environment, balancing precision with utility for diagnosis in defining the levels of a trajectory, and balancing the goals of supporting and assessing student learning.

Recent work by researchers has focused on synthesizing and elaborating knowledge of students' thinking on particular concepts as core progressions called learning trajectories. Although useful at the level of curriculum development, assessment design, and the articulation of standards, evidence is only beginning to emerge to suggest how learning trajectories can be utilized in teacher education. Our paper reports on two studies investigating practicing and prospective elementary teachers' uses of a learning trajectory to make sense of students' thinking about a foundational idea of rational number reasoning. Findings suggest that a mathematics learning trajectory supports teachers in creating models of students' thinking and in restructuring teachers' own understandings of mathematics and students' reasoning. (Contains 2 tables and 1 figure.)

In this article, we propose a theoretical connection between research on learning and research on teaching through recent research on students' learning trajectories (LTs). We define learning trajectory based instruction (LTBI) as teaching that uses students' LTs as the basis for instructional decisions. We use mathematics as the context for our argument, first examining current research on LTs and then examining emerging research on how mathematics teachers use LTs to support their teaching. We consider how LTs provide specificity to four highly used frameworks for examining mathematics teaching, namely mathematical knowledge for teaching, task analysis, discourse facilitation practices, and formative assessment. We contend that by unifying various teaching frameworks around the science of LTs, LTBI begins to define a theory of teaching organized around and grounded in research on student learning. Thus, moving from the accumulation of various frameworks into a reorganization of the frameworks, LTBI provides an integrated explanatory framework for teaching. (Contains 4 figures and 1 table.)