Abstract
Educational Standards provide a statement of educational competency goals. How to integrate such goal statements with the instructional core, in ways that promote curricular and instructional coherence and continuity of student learning, is a perennial challenge. In the United States, the Common Core State Standards for Mathematics, or CCSS-M, have been widely adopted, and are claimed to be based on research on learning in general and on learning trajectories in particular. The relationships, however, are tacit and incompletely, and sometimes controversially, articulated. This paper describes a body of work that associates the first nine grades of Standards (K-8) to eighteen learning trajectories and, for each learning trajectory, unpacks, interprets, and fills in the relationships to standards with the goal of bringing the relevant research to teachers (TurnOnCCMath.net). The connections are made using a set of descriptor elements, comprised of conceptual principles, coherent structural links, student strategies, mathematical distinctions or models, and bridging standards. A more detailed description of the learning trajectory for equipartitioning (EQP) shows the detailed research base on student learning that underpins a particular learning trajectory. How curriculum materials for EQP are designed from the learning trajectory completes the analysis, illustrating the rich connections possible among standards, descriptors, an elaborated learning trajectory, and related curricular materials.
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Notes
Author Confrey served on the National Validation Committee for the Common Core State Standards. She analyzed drafts of the CCSS-M for alignment with the learning trajectories research base and provided corresponding feedback to the CCSS-M writers.
Educators recognize the importance of prior knowledge and of identifying clear targets for learning. A major challenge, however, lies in identifying and evaluating intermediate states of proficiency and understanding their role in moving students forward in their thinking. To describe these intermediate states, teachers and researchers must recognize or invent meaningful distinctions; vocabulary terms for these tend to exhibit properties that are both cognitive and mathematical, such as partitive vs. quotative division, which later simply collapse to “division.” We refer to these as “meaningful distinctions.” In addition, “big ideas” (conceptual principles within a LT) may depend on earlier models corresponding to different schemes governing recognition of situations in the real world. These are typically captured as a “generalization” that encapsulates multiple meanings for experts, but tends to obscure the distinctions and models that students need to grapple with as they learn. Students need ample opportunity to explore such distinctions and models before moving to a generalization, to understand implications and adaptability of a generalization for its many referents and applications.
Confrey (2009) introduced equipartitioning as a more general term for the operation of splitting, in which equal shares are produced. It is distinct from partitioning, which can refer to breaking or segmenting into unequal parts, such as a room partition.
In our professional judgment, fractions and division should be instructionally related earlier than 5th grade. However, mutual accommodation of empirically derived LTs and LTs as Standards and descriptors required compromises. Nonetheless, we would recommend that teachers build the link earlier.
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Acknowledgments
This material is based upon work supported by the National Science Foundation (DRL-0758151 and DRL-073272) and Qualcomm. Any opinions, findings, conclusions, or recommendations expressed in these materials are those of the authors and do not necessarily reflect the views of the National Science Foundation or other funders. The authors wish to acknowledge the contributions of K. Nguyen, S. Varela, N. Monrose, Z. Yilmaz, and L. Neal to the development of the LPPSync software, curriculum writing, data analysis, and/or the conduct of the teaching experiment itself.
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The term curriculum can refer to a framework of standards, a scope and sequence of detailed objectives, or a set of organized materials. The term in the title refers to the full array of meanings; other uses in the paper specify the reference.
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Confrey, J., Maloney, A.P. & Corley, A.K. Learning trajectories: a framework for connecting standards with curriculum. ZDM Mathematics Education 46, 719–733 (2014). https://doi.org/10.1007/s11858-014-0598-7
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DOI: https://doi.org/10.1007/s11858-014-0598-7