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

### Citation Manager

. "5 The Teaching-Learning Paths for Number, Relations, and Operations." 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

3 + 2 and 5 = 4 + 1, giving children experiences with the meaning of the symbol as is the same number as and with algebraic equations with one number on the left. Initially children shift from seeing the total and then seeing the partners (addends), but with experience and fluency, they can simultaneously see the addend within the total. This is called embedded numbers: The two addends are embedded within the total. Such embedded numbers, along with the number word sequence skill of starting counting at any number, allow children to move to the second level of addition/subtraction solution procedures, counting on. Initially composed/decomposed number triads and even embedded number triads are constructed with small numbers using conceptual subitizing, but eventually counting is used with larger numbers to construct larger triads.

Many children from low-income backgrounds cannot initially solve such oral numerical problems, even with very small numbers (see Chapter 4). They need opportunities to learn and practice the Level 1 solution methods with objects and with fingers and experience composing/decomposing numbers to be able to see the addends (partners) hiding inside the small numbers 3, 4, 5. Such alternating focusing on the total and then on the partners (addends) will enable them to answer such oral numerical problems and also begin the learning path toward embedded numbers that is vital for the Level 2 addition/subtraction solution methods.

##### Step 3 (Kindergarten)

At this step, children extend cardinal counting and use math drawings as well as objects to solve situation, word, oral number word, written numeral, and which-is-more/less problems with totals ≤ 10 (see Box 5-10). Written work, including worksheets, is appropriate in kindergarten if it follows up on activities with objects or presents supportive visualizations. Children at these ages need practice that builds fluency after related experiences with objects to build mathematical understanding, and they need experience relating symbols for quantities to actual or drawn quantities.

Kindergarten children can extend their addition and subtraction problem solving to all problems with totals ≤ 10. Close to half of these problems have one addend of six or more. For these problems, knowing the 5-patterns using fingers for 6 through 10 can be helpful (5 + 1 = 6, 5 + 2 = 7, etc., to 5 + 5 = 10). All children can begin to make math drawings themselves, even for these larger numbers. This allows them to reflect on and discuss their solution methods. Math drawings involving circles or other simple shapes also enable more advanced children to explore problems with totals greater than ten. It is difficult to solve such problems with fingers until one advances to the general counting on solution methods (see Box 5-11, Level 2), which typically does not occur until Grade 1. Children

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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)