Learning Over Time

The driving forces behind mathematics learning trajectories is the need to understand how children actuallylearn and make sense of mathematics-how they progress from prior knowledge, through intermediateunderstandings, to the mathematics target understandings-and how to use these insights to improveinstruction and student learning. In this book, readers will come to understand what learning trajectories are,the research and methodology that are necessary for developing them, and gain insight into potentialapplications of learning trajectories.A synthesis and research outcome in their own right, learning trajectories provide detailed description ofinstructionally-grounded development of mathematical concepts and reasoning from the perspective of student learning, and, overall, building ondecades of accumulated experience in mathematics education research. However, their greater importance may lie in their potential as frameworks thatcontribute an unprecedented coherence across classroom instruction, professional development, standards, and assessment, by focusing squarely onconceptual understanding and reasoning instead of assessment-driven procedural knowledge. This potential was sufficiently compelling as anorganizing framework to have been cited as a basis for the Common Core mathematics standards, the new mathematics learning expectations that arenow consistent across most of the United States. (Among the conference attendees were the writers of the Common Core State Standards forMathematics, at the beginning of the Standards drafting process.)This book is an outgrowth of a conference on learning trajectories, hosted in 2009 at North Carolina State University, which examined research onlearning trajectories. An overarching message of the chapters in this volume is that learning trajectories, by focusing on how children's mathematicalreasoning develops, are coming into their own as a rigorous underpinning for both instruction andaccountability. Some of the learning scientists featured in this volume have played major roleslearning trajectories' evolution--from small-scale day-to-day conjectures by individual teachers, tosystematic research endeavors that teachers and scientists alike can use to interpret standards, planinstruction, and formatively assess student work. The work in this volume will be of interest tomathematics educators, teachers, and professional development specialists.