Solving many of the pressing issues facing the world today will require a deep and integrated understanding of science, technology, engineering, and mathematics (STEM). To prepare today’s K-12 students to tackle these challenges, STEM education must create opportunities to learn disciplinary content while inventing actionable solutions to messy, interdisciplinary problems. Learning frameworks, such as Project-Based Learning (PBL), Design-Based Learning (DBL), and Entrepreneurial-Based Learning (EBL), could support this reconceptualization of STEM education. New approaches are needed that leverage and integrate what works from these frameworks to better prepare students for success post-schooling. This means leveraging frameworks that emphasize practices and ways of thinking that support students to build and justify solutions that create value for users, while also creating a need for disciplinary content knowledge. This is especially necessary for mathematics, a discipline that is often treated insufficiently in interdisciplinary STEM activities. This paper introduces the Design & Pitch (D&P) Challenges in STEM Learning Framework, a novel learning framework that leverages features of PBL, DBL, and EBL, situating math learning within entrepreneurial pitch competitions. It describes the D&P Learning Framework and explores how each contributing learning framework combines to enhance students’ work, focusing their mathematical reasoning, while also empowering them to invent relevant solutions to authentic problems.

The question posed for this conference was "what are our current understandings of the theoretical foundations of mathematics education?" To address this question, I first asked myself, "Why it is important to answer such a question?" Is it primarily an issue of philosophical investigation? Or, is it, rather, a call for some clarification about the proliferation of theories in mathematics education and their interrelationships, reexamining each theory in light of multiple theories. This second query interests me, because increasingly as I extend the scope of my research to see constructivism enacted in classrooms and schools, I need broader theoretical constructs. And, I need to reexamine of my own assumptions and their warrant.

This paper reports on a five-day study exploring six middle school students' developing understanding of algorithms and functions during the Building Algorithms challenge, an entrepreneurial Design & Pitch Challenge in STEM. Students proposed businesses and built automated spreadsheet algorithms to inform users' decision making by assigning ratings to objects (e.g. videos, music, racetracks) based on their preferences. Part of a larger NSF study of students' participation in entrepreneurial experiences with mathematical connections, the study used a design-based research methodology. Qualitative data (video of teams working, team interviews, and daily work samples) were analysed to answer the research question: How do students demonstrate functional reasoning as they build entrepreneurial solutions to the Building Algorithms challenge? Three themes emerged that describe how the challenge and its entrepreneurial framing enhanced students' engagement with key processes of function building, including: (1) identifying personally meaningful functional relationships; (2) defining and operationalising authentic variables and function rules; and (3) making sense of writing, testing, and refining function rules. This study contributes to the field's understanding of functions, demonstrating how entrepreneurship and a math-focused design challenge can elicit and enhance specific and targeted middle grades mathematics content.

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.

AbstractLearning trajectory (LT)-based diagnostic formative assessments answer Wiggins’s call for authentic assessments that do not “lose sight” of the learner (, p. 712). Based on empirical patterns in learning, LTs are structured in terms of increasingly sophisticated levels of thinking towards a target mathematical concept, and thus provide teachers a roadmap to proficiency on those concepts. This chapter reports on a design study, with sixth-grade classes at a diverse middle school, that investigated five mathematics teachers’ implementation of an LT-based diagnostic formative assessment tool and an associated curriculum on ratio reasoning. Results show that students achieved moderate post-test scores with significant positive learning gains, equitably distributed across sub-groups. Students expressed enthusiasm for revising and resubmitting answers. Although the teachers in this study did not fully leverage the structures of the LTs in the learner-centred way anticipated by the tool’s designers during data reviews, the chapter gives insights into potential additional components. Additional components needed to scaffold learner-centred assessment practices include a more explicit model of learner-centred instruction, a framework for LT-based data-driven decision-making, and a design for short- and long-feedback cycles for data use, including one involving collective data reviews and instructional planning by groups of teachers.

This study reports how a validation argument for a learning trajectory (LT) is constituted from test design, empirical recovery, and data use through a collaborative process, described as a “trading zone” among learning scientists, psychometricians, and practitioners. The validation argument is tied to a learning theory about learning trajectories and a framework (LT-based data-driven decision-making, or LT-DDDM) to guide instructional modifications. A validation study was conducted on a middle school LT on “Relations and Functions” using a Rasch model and stepwise regression. Of five potentially non-conforming items, three were adjusted, one retained to collect more data, and one was flagged as a discussion item. One LT level description was revised. A linear logistic test model (LLTM) revealed that LT level and item type explained substantial variance in item difficulty. Using the LT-DDDM framework, a hypothesized teacher analysis of a class report led to three conjectures for interventions, demonstrating the LT assessment’s potential to inform instructional decision-making.

The paper reports on the design and validation argument for classroom assessments within a digital diagnostic assessment system built on learning trajectories (LTs). It consists of a learning map of nine big ideas, 25 relational learning clusters, and 62 LTs for grades 6-8 mathematics. Students take cluster assessments, and teachers use the data to adapt instruction. An ongoing validation process is presented with data for an algebra cluster. Validation among practitioners, learning scientists, and psychometricians is conceptualized as examining and adjusting inter level, intra-level, and construct-irrelevant variation in measures of item difficulty and deploying item response theory modeling followed by sequential regressions. Using data from 37,000 assessments collected over three years at 3 middle schools, 167 potentially non-conforming items of the 676 calibrated items (24 %) were identified and revised. The paper discusses how the trajectories and map were refined through a combination of data analysis and feedback from practitioners.

: This study applies a validation framework of cognition, implementation, and inference (Pellegrino, DiBello, & Goldman, 2016) to the use of a digital classroom assessment tool, Math-Mapper 6-8, for middle grades mathematics, built using "principled assessment design" (Nichols, Kobrin, Lai, & Koepfler, 2016). It reports on the results of a validation study of learning trajectories for student reasoning on the measurement of circles. A validation method for examining and refining item behavior along a learning trajectory using Rasch analysis and stepwise regression is illustrated as a means to improve learning trajectories at scale and over time.

Using an autobiographical approach, the author reports on over 30 years of designing software to promote learner-centered instruction. She describes what she learned about organizing her teams, first to emulate a scientific lab and, later, to leverage agile software development approaches. She recommends an agile approach as a means to plan and coordinate complex design activities and to emphasize active involvement by all members in the research process. The approach can improve team creativity and productivity.

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.

Curricular theory must evolve to keep pace with the implications of the design, use, and effects of deploying and adapting digital curricular resources, especially when placed within digital learning systems (DLS) with rapid feedback and analytic capacity. We introduce an “agile curriculum” framework describing how to use classroom assessment data to regulate teachers’ practices of iteratively adapting curricula. Our DLS, called Math-Mapper 6–8, is introduced as an example with its diagnostic assessments of students’ progress along learning trajectories. An exploratory video study of middle school teachers reviewing, interpreting, and acting on its data, both during instruction (short cycle feedback) and within professional learning communities (long cycle feedback) illustrates how an agile curriculum framework supports data-driven adjustments based on student learning.

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.

Statistics curricula and pedagogy are changing rapidly in response to a growing body of research findings involving students' reasoning processes, technology capability, attention to underpinning conceptual infrastructure, and new ways of statistical practice. Because many of the statistical ideas being considered are currently not in the curriculum, many researchers in statistics education have investigated students' reasoning processes through the use of learning trajectories in conjunction with design-based research methods. In this chapter, we outline the characteristics of learning trajectories and exemplify how learning trajectories have been used in three case studies in statistics education. Commonalities and differences across the learning trajectories are discussed as well as recommendations for future research.

SWE), and Celia Hoyles and Richard Noss (UK).Twenty six papers and thirteen posters were presented.Confrey (USA) authored a closing summary.The presentations are listed below (posters omitted due to space constraints); the text references these contributions.Topic Study Group 36's presentation began with the framework proposed by a prior ICME Topic Study Group (Watson) organized into the categories (or parameters) of Theory, Intentions, Likely Activity and Implementation.Over the course of the conference, additional components of a framework for TSG 36 emerged around (a) tasks, (b) learning environments, and (c) theory.Within the component of tasks, the topics addressed included how tasks are sequenced and structured (Gravemeijer, Brady et al., Goa et al.) what representations and tools were used in tasks (Thiel-Schneider, Johnson), what kinds of activities and actions

In “Tracing the Assessment Triangle With Learning Progression-Aligned Assessments in Mathematics,” Lai, Kobrin, and DiCerbo present a learning trajectory1 on the measurement of area, a topic typica...