The open access version can be found on MIT’s website.

When thinking about a child learning about the concept of equivalence with a balance scale, there are many things to consider. How does the presence of the balance scale and the blocks influence a child’s understanding of equivalence? How do the physical and interactive features of the blocks and balance scale allow for certain actions? Actions that may (or may not) lead to better understanding or performance.
Is it true that “Movement Matters” for learning? We focus on how people learn with manipulatives, which are physical objects that can be used to represent concepts. 
We focus on the possibilities for action that manipulatives offer, and we argue that considering manipulatives with this lens can offer new insights into when and why manipulatives are effective as a tool for learning. To learn effectively with manipulatives, learners must be able to connect the objects and/ or their actions on the objects to the target concepts.
We argue that movements–including actions with manipulatives–do matter for learning. By considering manipulatives’ perceptual and interactive features, we can gain insights into learning as well as design more effective instruction.
We propose that the most effective manipulatives have transparent links to the target concepts, and that these links may emerge either from perceptual features of the objects themselves or from the objects’ affordances for action. Some of these connections may be natural or obvious to learners, and others may require instructional support.

Therefore, when using manipulatives during instruction, educators should consider the perceptual features of the objects, the actions the objects afford, and how those perceptual features and actions relate to the target concepts.
To learn more, see the corresponding chapter of Movement Matters (MIT Press, 2022).

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