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5 The Mathematical Knowledge Children Bring to School
Pages 157-180

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From page 157...
... Preschoolers' mathematical thinking rests on a combination of conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition. During the last 25 years, developmental psychologists and mathematics educators have made substantial progress in understanding the ways in which these strands interact.
From page 158...
... The Swiss psychologist lean Piaget developed a task based in part on this definition that has been widely used to assess whether children understand the critical importance of this one-to-one correspondence in defining numerosity.~ In this task, children are shown an array like the one below, which might represent candies. They are then asked a question like the following: Are there more light candies, the same number of dark and light candies, or more dark candies?
From page 159...
... Whether and how this early sensitivity to number affects later mathematical development remains to be shown, but children enter the world prepared to notice number as a feature of their environment. Much of what preschool children know about number is bound up in their developing understanding and mastery of counting.
From page 160...
... The first three principles define rules for how one ought to go about counting; the last two define circumstances under which such counting procedures should apply.
From page 161...
... Other studies indicate that much of children's conceptual understanding of counting follows (and may be based on) an initial mastery of conventional counting procedures.~° An intermediate view is that conceptual and procedural knowledge of counting develop interactively, with small changes in one contributing to small changes in the other.
From page 162...
... The ability of young preschool children to follow counting principles in their own counting and to focus on them in evaluating the counting of others is also quite vulnerable to situational variations. 16 The controversy about the relation between how understanding of counting principles develops and how conventional counting ability is acquired echoes issues that emerge throughout children's later mathematics learning.
From page 163...
... Language and Early Mathematical Development One aspect of counting that preschool children find particularly difficult is learning the number names. Learning a list of number names up to 100 is a challenging task for young children.
From page 164...
... 164 ADDING IT UP Box 5-1 Number Names in Chinese, English, anal Spanish a. One to ten Arabic numeral ~ ~ 4 5 Chinese (spoken)
From page 165...
... THE MATHEMAT/CAL KNOWLEDGE CHILDREN BR/N6 TO SCHOOL 165 6 7 8 9 10 flu qi ba jiu shi six seven eight nine ten seis siete ocho nueve diez 16 17 18 19 20 shi liu shi qi shi ba sixteen seventeen eighteen diez y seis diez y siete Example san shi qi thirty-seven trenta y siete shi jiu er shi nineteen twenty diez y ocho diez y nueve vein te
From page 166...
... Thus the name for 16 in Spanish, dies y seis (literally ten and six) , follows the same basic structure as Arabic numerals and Chinese number names (starting with the tens value and then naming the ones value)
From page 167...
... Research on children's acquisition of number names suggests that U.S. children learn to recite the list of English number names through at least the teens as essentially a rote-learning task,23 though occasional errors such as "fiveteen" suggest that some children notice the structure of the counting words for 13 through 19 that is partially obscured by linguistic modifications.24 When first counting above 20, American preschoolers often produce idiosyncratic number names, indicating that they fail to understand the base-10 structure underlying larger number names; for instance, they might count "twenty-eight, twenty-nine, twenty-ten, twenty-eleven, twenty-twelve." This kind of mistake is extremely rare for Chinese children and indicates that the base-10 structure of number names is more accessible for learners of Chinese than it is for children learning to count in English.
From page 168...
... Nevertheless, these effects have implications for learning Arabic numerals and thus for understanding the principal symbol system used in school mathematics. As with other aspects of mathematics, counting requires combining a conceptual understanding of the nature of number with a fluent mastery of procedures that allow one to determine how many objects there are.
From page 169...
... The diversity of strategies that children show in early arithmetic is a feature of their later mathematical development as well. In some circumstances the number of different strategies children show predicts their later learning.32 The fact that children are inventing their own diverse strategies for doing arithmetic creates its own educational issues, however, as teachers need to be able to help children understand why some strategies work and others do not and to help them move on to advanced strategies.
From page 170...
... "~.38 Four- and 5-year-olds do begin to use and often their knowledge to answer correctly the Piagetian number task presented fragile above involving equivalent sets of candies, and later they recognize without understand~n that childregn counting that the sets have the same number of candies.39 bri ng to Most preschool children enter school with an initial understanding of proschool and cedures (e.g., counting, addition, subtraction) that forms the basis for much to make it Of their later mathematics learning, although they have limited ability to generrel jab~ree alize that knowledge and to understand its importance.
From page 171...
... Extensive research in the learning of mathematics and other domains has shown that children who attribute success to a relatively fixed ability are likely to approach new tasks with a performance rather than a learningorientation, which causes them to show less interest in putting themselves in challenging situations that result in them (at least initially) performing poorly.43 Preschoolers generally enter school with a learning orientation, but already by first grade a sizable minority react to criticism of their performance by inferring that they are not smart rather than that they just need to work harder.44 Most preschoolers enter school interested in mathematics and motivated to learn it.
From page 172...
... Equity anal Reme~liation Most U.S. children enter school with mathematical abilities that provide a strong base for formal instruction in mathematics.
From page 173...
... For example, the Rightstart program consists of a set of games and number-line activities aimed at providing children needing remedial assistance with an understanding of the relative magnitudes of numbers. Twenty minutes a day over a three- to four-month period in kindergarten was successful in bringing these children's mathematical knowledge up to a level commensurate with their peers, gains that persisted through the end of first grade.53 Another intervention is aimed at ensuring that Latino children understand the base-10 structure of number names, something that, as noted above, U.S.
From page 174...
... Furthermore, not all children enter school with the intuitive understanding of number described above and assumed by the elementary school curriculum. Recent research suggests that effective methods exist for providing this basic understanding of number.
From page 175...
... THE MATHEMAT/CAL KNOWLEDGE CHILDREN BR/N6 TO SCHOOL '75 5. Steffe, von Glasersfeld, Richard, and Cobb, 1983, p.
From page 176...
... Child Development, 54, 695-701. Baroody, A
From page 177...
... (1984~. How children learn mathematics (4th ode.
From page 178...
... (1994~. Rightstart: Providing the central conceptual prerequisites for first formal learning of arithmetic to students at risk for school failure.
From page 179...
... Journal of Educational Psychology, 85, 24-30. National Center for Education Statistics.
From page 180...
... (1997~. Change in children's competence beliefs and subjective task values across the elementary school years: A 3-year study.


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