Efficacy of a learning trajectory approach compared to a teach-to-target approach for addition and subtraction

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Although basing instruction on a learning trajectory (LT) is often recommended, there is little direct evidence to support the premise of a “LT approach”—that to be maximally meaningful, engaging, and effective, instruction is best presented one LT level beyond a child’s present level of thinking. The present report serves to address the question: Is it necessary to teach each contiguous level of a LT or can instruction be similarly or more effective when skipping levels, provided the necessary exemplars are made? In a multimethod research study that included individual teaching experiments embedded inside of a quasi-experimental research design, one group of 13 kindergartners received instruction based on an empirically-validated LT for addition and subtraction (the “LT” treatment). The counterfactual, “skip” treatment (n = 12), received instruction focused mainly on levels at least two levels above their present level for the same amount of time as the LT treatment. More children in the LT treatment exhibited greater addition and subtraction learning during sessions and from pretest to posttest than children in the skip treatment. Implications for future study are discussed.

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Fig. 1

(adapted from Sarama & Clements, 2009)

Fig. 2


  1. 1.

    Note that the names of levels of the learning trajectory always end with “±”; however, because many similar terms appear in the problem type names, which are likewise capitalized in the literature, we also distinguish the learning trajectory names with boldface. The names were not elaborated for changed to keep consistency between this paper and the sources of the names (Clements & Sarama, 2014; Sarama & Clements, 2009, and


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This research was supported by the Institute of Education Sciences, U.S. Department of Education through Grant R305A150243. The opinions expressed are those of the authors and do not represent views of the U.S. Department of Education. Although the research is concerned with theoretical issues, not particular curricula, a small component of the intervention used in this research have been published by some of the authors, who could have a vested interest in the results. Researchers from an independent institution oversaw the research design, data collection, and analysis and confirmed findings and procedures. The authors wish to express appreciation to the teachers and students at the Ricks Center, Morgridge College of Education, University of Denver who participated in this research.

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Clements, D.H., Sarama, J., Baroody, A.J. et al. Efficacy of a learning trajectory approach compared to a teach-to-target approach for addition and subtraction. ZDM Mathematics Education (2020).

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  • Achievement
  • Curriculum
  • Early childhood
  • Instructional design/development
  • Learning trajectories
  • Learning environments
  • Mathematics education