Learning trajectories in mathematics: A foundation for standards, curriculum, assessment, and instruction

Abstract

Learning Trajectories in Mathematics: A Foundation for Standards, Curriculum, Assessment, and Instruction aims to provide:• A useful introduction to current work and thinking about learning trajectories for mathematics education • An explanation for why we should care about these questions • A strategy for how to think about what is being attempted in the field, casting some light on the varying, and perhaps confusing, ways in which the terms trajectory, progression, learning, teaching, and so on, are being used by the education community.Specifically, the report builds on arguments published elsewhere to offer a working definition of the concept of learning trajectories in mathematics and to reflect on the intellectual status of the concept and its usefulness for policy and practice. It considers the potential of trajectories and progressions for informing the development of more useful assessments and supporting more effective formative assessment practices, for informing the on-going redesign of mathematics content and performance standards, and for supporting teachers' understanding of students' learning in ways that can strengthen their capability for providing adaptive instruction. The authors conclude with a set of recommended next steps for research and development, and for policy. LearnInG TraJeCTOrIeS In MaTHeMaTICSA Foundation for Standards, Curriculum, Assessment, and instructionEstablished in 1985, CPRE unites researchers from seven of the nation's leading research institutions in efforts to improve elementary and secondary education through practical research on policy, finance, school reform, and school governance. CPRE studies alternative approaches to education reform to determine how state and local policies can promote student learning. The Consortium's member institutions are the University of Pennsylvania, Teachers College-Columbia University, Harvard University, Stanford University, the University of Michigan, University of Wisconsin-Madison, and Northwestern University.In March 2006, CPRE launched the Center on Continuous Instructional Improvement (CCII), a center engaged in research and development on tools, processes, and policies intended to promote the continuous improvement of instructional practice. CCII also aspires to be a forum for sharing, discussing, and strengthening the work of leading researchers, developers and practitioners, both in the United States and across the globe.To learn more about CPRE and our research centers, visit the following web sites: Visit our website at http://www.cpre.org or sign up for our e-newsletter, In-Sites, at insites@gse.upenn.edu.Research Reports are issued by CPRE to facilitate the exchange of ideas among policymakers, practitioners, and researchers who share an interest in education policy. The views expressed in the reports are those of individual authors, and not necessarily shared by CPRE or its institutional partners. About the Consortium for PoliCy reseArCh in eduCAtion (CPre)CPre reseArCh rePort series CPRE Research Report # RR-...

Design Decisions in Developing Learning Trajectories–Based Assessments in Mathematics: A Case Study

“…These upper anchors often reflect ideals of subject matter experts and/or policymakers, and researchers acknowledge that few students in typical schools may reach these levels of understanding (Daro et al, 2011;Gunckel et al, 2012). In certain instances, assessments of learning trajectories help to identify the gap between what is now and what could be, and in that sense they may be useful as amplifiers of messages related to what is important for students to learn.…”

“…Learning trajectories in mathematics and learning progressions in science refer to testable, empirically supported hypotheses about how student understanding develops toward specific disciplinary goals for learning (Clements & Sarama, 2004;Confrey, Maloney, Nguyen, Mojica, & Myers, 2009;Daro, Mosher, & Corcoran, 2011;Simon, 1995). Elsewhere, Confrey and colleagues (2009) defined a learning trajectory as a researcher-conjectured, empirically-supported description of the ordered network of constructs a student encounters through instruction (i.e.…”

Design research with a focus on learning processes: an overview on achievements and challenges

“…Next to those terms, the appellation "learning progression" is often used nowadays. However, this term comes with a different connotation, in that it primarily refers to an empirically grounded sequence of steps that describe how students are likely to reason as they move from naïve to more sophisticated reasoning (Daro, Mosher, and Corcoran 2011). Finally Stephan (2015) distinguishes between a classroom learning trajectory and an individual learning trajectory, highlighting that individual trajectories will vary while Confrey (2006) argues that a learning trajectory always contains an understanding of multiple pathways in relation to certain landmarks and obstacles.…”

Early mathematics assessment: validation of the short form of a prekindergarten and kindergarten mathematics measure

“…Learning trajectories (also known as developmental progressions) have played a substantial role in recent theory and research (Clements, Sarama, Spitler, Lange, & Wolfe, 2011;Confrey, Maloney, Nguyen, Mojica, & Myers, 2009;Wilson, 2009), as well in as the development of standards and curriculum (Daro et al, 2011). The ongoing work of The Common Core State Standards Initiative, a joint effort by the National Governors Association Center for Best Practices and the Council of Chief State School Educational Psychology 313…”