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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

requests (to them, each count is a correct response to the How many are there? question). Children making the other error (giving a number that differs from the last word) are understanding that the question How many are there? is a request for cardinal information about the whole set, but they do not yet understand that the cardinal information is given by counting, and, in particular, by the last word said in counting.

Verbal knowledge is also required for full competence in discriminating the use of individual number words for each thing counted versus the use of the final number word to refer to the whole set. Even children who gesture correctly to show their count meaning (gesture to one thing) or their cardinal meaning (gesture to the whole set) may struggle with correct verbal expressions (see Box 5-6). Mastering these is a later achievement that will be learned with modeling and practice.

BOX 5-6

Learning the Correct Counting Language

Learning the singular and plural forms that go with counting (single) and with cardinal (plural) references to objects takes some time. Here are typical examples of errors that children initially make while they are sorting out all of these conceptual and linguistic issues. After children counted a row of objects, they were asked a count-reference question and a cardinality-reference question (the order varied across children).The count-reference question was Is this the soldier (chip) where you said n? where n was the last word said by the child. The experimenter asked the question three times and pointed to the last item, the next-to-last item, and all the items in the row. The cardinal-reference question was Are these the n soldiers (chips)? The correct answer was always in the middle, because research indicated that young children have a strong bias toward choosing the last alternative. In the examples below, children spontaneously verbalized cardinality or counting references that disagreed with their gesture.

 Response to cardinality question: Those are five soldiers, said as child points to the last soldier. Response to cardinality question: This one’s the five chips, said as child points to the last chip. Response to cardinality question: This is the six soldiers, said as child points to each soldier (said six times). Response to cardinality question: This is the four chips, said as child points to the last chip. Response to cardinality question: This is where I said chip four, said as child’s hands gesture to all of the chips. Response to count question: All of these animals I said five.

SOURCE: Fuson (1988, p. 232).

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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)