and epistemological understandings rather than mere computations, has implications for the design of early mathematics instruction as well. The approach emphasizes the importance of developing important domain-specific concepts and foundational epistemological ideas as a base from which to build in later grades.

Grades 3-5
Developing an Explicit Macroscopic Understanding of Matter

Work in grades K-2 to elaborate children’s understanding of specific materials prepares them to move up a level of abstraction and develop an initial macroscopic understanding of matter at this next age band. Students now can consider not just the salient properties that distinguish different kinds of materials, but ask the question of whether there are some properties that all material entities have in common. In this way, they can be led (with relevant instructional practices) to articulate a general concept of matter that was initially implicit in their notion of material kind. In so doing, a new causal nexus—matter—is developed as students come to realize that objects made of different materials “have weight and occupy space because they are made of something (pieces of stuff) that continues to exist, take up space, and have weight across a broad range of transformations” (Smith et al., 2004, p. 46). Some core ideas important to develop at this age band include understanding that:

  • Objects are made of matter that takes up space and has weight.

  • Solids, liquids, and air are forms of matter and share these general properties.

  • There can be invisible pieces of matter (i.e., too small to see with the naked eye).

  • Matter continues to exist when broken into pieces too tiny to be visible.

  • Amount of matter and weight are conserved across a range of transformations, including melting, freezing, and dissolving (Smith et al., 2004, p. 45).

Research has shown that elementary schoolchildren are beginning to develop an intuitive (abstract) notion that there is “an amount of stuff” in objects that can remain constant across changes in surface appearance. For example, in their classic conservation studies, which have been replicated many times by others, Piaget and Inhelder (1974) poured liquid from a short, fat container into a tall, thin one and asked children if there was the same amount of liquid in both containers or if one had more. Similarly, they



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