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## Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity (2009) Center for Education (CFE)

### Citation Manager

. "2 Foundational Mathematics Content." Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity. Washington, DC: The National Academies Press, 2009.

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Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity

consist of rectangles. If the outer surface of a 3-D shape consists of flat surfaces, these are often called faces. For example, a long wooden building block has two faces at each end that are small rectangles and four faces around the middle that are long rectangles. Faces are joined along straight edges, and edges meet at points called vertices. Children might observe that some shapes (like that building block) have pairs of faces on opposite sides that are the same (congruent). Children might also observe that some shapes, like cylinders (like a pole or a can), cones (like a party hat), and spheres (like a ball), have outer surfaces that are not flat.

Although the outer surface of a 3-D shape is usually visible, unless one cuts the shape open, or the shape is made of clear plastic, or the shape is hollow and a face can be removed to look inside, one must usually imagine and visualize the inside. One exception is rooms, which are often (roughly) in the shape of a rectangular prism, and which one experiences from the inside. Distinguishing the inside of a 3-D shape from its outer surface is an especially important foundation for understanding the distinction between the surface area and volume of a shape in later grades.

##### Composing and Decomposing Shapes

Just as 10 ones can be composed to make a single unit of 10, shapes can also be composed to make new, larger shapes. And just as a 10 can be decomposed into 10 ones, so too shapes can be decomposed to make new, smaller shapes. Figure 2-5 presents a few examples of relationships among shapes obtained by composing and decomposing shapes based on equilateral triangles. Figure 2-6 shows relationships among shapes obtained by composing and decomposing rectangles.

Composing and decomposing 2-D shapes is an important foundation for understanding area in later grades. In particular, viewing rectangles as

FIGURE 2-5 Relationships among shapes based on equilateral triangles.

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 Front Matter (R1-R12) Summary (1-4) Part I: Introduction and Research on Learning (5-6) 1: Introduction (7-20) 2 Foundational Mathematics Content (21-58) 3 Cognitive Foundations for Early Mathematics Learning (59-94) 4 Developmental Variation, Sociocultural Influences, and Difficulties in Mathematics (95-120) Part II: Teaching-Learning Paths (121-126) 5 The Teaching-Learning Paths for Number, Relations, and Operations (127-174) 6 The Teaching-Learning Paths for Geometry, Spatial Thinking, and Measurement (175-222) Part III: Contexts for Teaching and Learning (223-224) 7 Standards, Curriculum, Instruction, and Assessment (225-288) 8 The Early Childhood Workforce and Its Professional Development (289-328) Part IV: Future Directions for Policy, Practice, and Research (329-330) 9 Conclusions and Recommendations (331-350) Appendix A: Glossary (351-358) Appendix B: Concepts of Measurement (359-362) Appendix C: Biographical Sketches of Committee Members and Staff (363-370) Index (371-386)