Physical and Virtual Manipulatives: What Is “Concrete”?

  • Julie SaramaEmail author
  • Douglas H. Clements
Part of the Mathematics Education in the Digital Era book series (MEDE, volume 7)


We discuss research on both physical manipulatives and virtual manipulatives to provide a framework for understanding, creating, implementing, and evaluating efficacious manipulatives—physical, virtual, and a combination of these two. We provide a theoretical framework and a discussion of empirical evidence supporting that framework, for the use of manipulatives in learning and teaching mathematics, from early childhood through the elementary years. From this reformulation, we re-consider the role virtual manipulatives may play in helping students learn mathematics. We conclude that manipulatives are meaningful for learning only with respect to learners’ activities and thinking and that both physical and virtual manipulatives can be useful. When used in comprehensive, well planned, instructional settings, both physical and virtual manipulatives can encourage students to make their knowledge explicit, which helps them build Integrated-Concrete knowledge.


Mathematical Idea Number Word Direct Manipulation Algebraic Thinking Concrete Understanding 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



This paper was based upon work supported in small part by the Institute of Educational Sciences under Grant No. R305K05157, “Scaling Up TRIAD: Teaching Early Mathematics for Understanding with Trajectories and Technologies”; and Grant No. R305A120813, “Evaluating the Efficacy of Learning Trajectories in Early Mathematics,” and in part by the National Science Foundation under Grant No. DRL-1313695, “Using Rule Space and Poset-based Adaptive Testing Methodologies to Identify Ability Patterns in Early Mathematics and Create a Comprehensive Mathematics Ability Test”; and Grant No. DRL-1118745, “Early Childhood Education in the Context of Mathematics, Science, and Literacy.” Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the funding agencies.


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© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.University of DenverDenverUSA

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