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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

or in terms of a rotation (see National Research Council, 2001, Box 3-1). The associative property of multiplication can be illustrated by decomposing a 3-D array (or box shape built of blocks) in different ways (one way is shown in Figure 2-7). The distributive property is illustrated by viewing objects as grouped in two different ways (see National Research Council, 2001, Box 3-1).

The properties of multiplication can be illustrated with arrays and rectangles, and they are also visible in the multiplication tables, which contain many relationships and have important structure. One structural aspect of the multiplication tables is their diagonal symmetry. This diagonal symmetry corresponds with the commutative property of multiplication, namely that a × b = b × a for all numbers a and b. Recognizing this symmetry allows children to learn multiplication facts more efficiently. In other words, once they know the upper right-hand triangular portion of the multiplication tables in around third grade, they can fill in the rest of the table by reflecting across the diagonal (see Figure 2-10).

Patterns associated with horizontal or vertical shifts (slides) can also be seen in the multiplication tables. For example, the entries in two columns are related by the column that is associated with the amount of shift between the columns (see Figure 2-10). This structural relationship corresponds with the distributive property.

##### Connections in Measurement and Number: Fractions

Once children encounter measurement situations, the possibility of fractions arises naturally. Fractions can be shown well in the context of

FIGURE 2-10 Symmetry and relationships in the multiplication table.

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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)