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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
 Change minus: Four take away one is ? Put together/take apart: Three has ? and ? Step 3 (Kindergarten) Use conceptual subitizing and cardinal counting to solve situation, word, oral number word, and written numeral problems with totals ≤ 10. For word problems, model action with objects or fingers or a math drawing and count or see to solve; write an expression or equation. For oral or written numeral problems, use fingers, objects, or a math drawing to solve. Engage in learning the partners for 6, 7, 8, 9, 10. For relations, act out or show with objects or a drawing, then count or match to solve. Use =, ≠ symbols. Step 4 (Grade 1) Use Level 2 or Level 3 solution procedures: count on or use a derived fact method for problems with totals ≤ 18 and find subtraction as an unknown addend. Solve change plus problems by counting on to find the total 6 + 3 = ? Solve change minus problems by counting on to find the unknown addend 9 – 6 = ? is 6 + ? = 9. Solve put together/take apart problems by counting on to find the unknown addend 6 + ? = 9. Advanced first graders use Level 3 solution procedures: (a) doubles and doubles ± 1. (b) they experience make-a-ten methods: 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14 ; 14 – 8 is 8 + ? = 14, so 8 + 2 + 4 = 14, so ? = 6 (not all children master these in Grade 1). Solve comparison situations or determine how much/many more/less by counting or matching for totals ≤ 10, then for totals ≤ 18.

mathematical aspects—the numbers of things and the additive or subtractive operation in the situation. As we discuss each level, we also describe ways in which children can be helped to learn methods appropriate for that level and the prerequisite knowledge. Children need opportunities to relate strategies to actual objects or pictures of objects and to discuss and explain their thinking.

The solution methods at Level 1 use direct modeling of every object. In direct modeling children must carry out the actions in the situation using actual objects or fingers. Until around age 6, children primarily use such direct modeling to solve situations presented in objects, word problems

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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)