A Model of the Development and Use of Early Childhood Mathematics Learning Trajectories

• In Event: Learning Trajectories as Boundary Objects for Psychometricians, Learning Scientists, and Practitioners in Mathematics Education

Abstract

Research-based learning trajectories can contribute to the development, refinement, and evaluation of policies, standards, curriculum, teaching practices, and professional development, as well as to the collaboration of groups engaged in those activities, including those who play distinct roles. In doing so, they can help maintain the coherence and connections. In curriculum development, for example, using learning trajectories ensures a common theoretical and empirical base for the goals, assessments, instructional activities, and teaching practices. They also serve as boundary objects—concepts and structures that help connect previously disparate groups—such as between curriculum developers at different levels (of age of students, or of administrative hierarchies). The integration of the research corpuses makes the LT construct particular useful. For example, specificity of goals helps those wishing to write standards as well as do the developmental progressions. Both are also invaluable for those creating assessments, especially those that are to serve as formative assessments.
Coherence is also supported, because practice, research, and policy efforts can use and contribute to improvements in the same learning trajectories. All LTs are hypothetical and should be re-conceptualized and re-created by all concerned, including researchers and teachers. Indeed, teachers’ instantiation is based on more intimate knowledge of the particular students involved—their culture, language, knowledge, learning preferences, and engagement in certain task types or contexts. The teacher must construct models of children’s mathematics as they interact with children around the instructional tasks and thus alter their own knowledge of children and future instructional strategies and paths. Thus, the LTs in practice are always emergent and teachers’ unique implementations not only help their students but can contribute to refinements and extensions of the original LTs. This is less a caveat regarding use of LTs and more an argument for their use in professional development. LTs empower teachers with a well-formed and specific set of expectations about children’s ways of learning and a likely pace along a path that includes major mathematical ideas and objectives. But they also respect an benefit from teachers’ professionalism.
The benefits of synergism between the three components of LTs also can produce novel research results, even within the local theoretical fields of psychology and pedagogy. The enactment of an effective, complete, learning trajectory may actually alter developmental progressions or expectations previously established by psychological studies, because they open up new paths for learning and development. Such an enactment based on the fine-grain cognitive analysis of the developmental progression and the similarly detailed analysis of the instructional tasks provides a more elaborated theoretical base for curriculum and instruction than is often available and may also open instructional approaches or avenues not previously considered.
This presentation will present a model of the development and use of LTs based on coordinated interdisciplinary research ranging from cognitive science to scale-up. The instantiation of this model will illustrate the hypothesized benefits.

Authors

• Douglas H. Clements, University of Denver

• Julie Sarama, University of Denver