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Adding + It Up: Helping Children Learn Mathematics 5 THE MATHEMATICAL KNOWLEDGE CHILDREN BRING TO SCHOOL Children begin learning mathematics well before they enter elementary school. Starting from infancy and continuing throughout the preschool period, they develop a base of skills, concepts, and misconceptions about numbers and mathematics. The state of children’s mathematical development as they begin school both determines what they must learn to achieve mathematical proficiency and points toward how that proficiency can be acquired. Chapter 4 laid out a framework for describing mathematical proficiency in terms of a set of interwoven strands. That framework is useful in thinking about the skills and knowledge that children bring to school, as well as the limitations of preschoolers’ mathematical competence. Applying the framework to research on preschoolers’ mathematical thinking also provides a good example of the way in which the strands of proficiency are interwoven and interdependent. 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. In this chapter we describe the current state of knowledge concerning the proficiency that children bring to school, some of the factors that account for limitations in their mathematical competence, and current understanding about what can be done to ensure that all children enter school prepared for the mathematical demands of formal education.
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Adding + It Up: Helping Children Learn Mathematics Preschoolers’ Mathematical Proficiency Conceptual Understanding The most fundamental concept in elementary school mathematics is that of number, specifically whole number. To get a sense of both the difficulty of the concept and how much of it is taken for granted, try to define what a whole number is. One common conception of whole number says that two sets have the same numerosity (same number of members) if and only if each member of one set can be paired with exactly one member of the other (with no members left over from either set). If one set has members left over after this pairing, then that set has a greater numerosity (more items in it) than the other does. This definition allows one to decide whether two sets have the same number of items without knowing how many there are in either set. The Swiss psychologist Jean 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.1 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? Most preschoolers recognize that the sets have the same amount of candy, based on the one-to-one alignment of the individual pieces. Next, the child watches the experimenter spread out the items in one set, which alters the spatial alignment of the pieces: Shown this diagram, many children younger than 5 years assert that there are more of whichever kind of candy is in the longer row (the light candies in this example). Piaget argued that a true understanding of number requires an ability to reason about the effects of transformations that is beyond the capacity of preschool children. It was not uncommon several decades ago for educators aware of Piaget’s findings and his claims to make assertions such as
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Adding + It Up: Helping Children Learn Mathematics the following: “Children at different stages cannot learn the same content. They cannot learn about number, for example, until they reach the concrete operational stage [roughly ages 7 to 11, according to Piaget].”2 Preschool children in fact know quite a bit about number before they enter school. Research over the last 25 years, however, suggests that preschool children in fact know quite a bit about number before they enter school. Much of that knowledge is tied up with their understanding of counting. Even for preschoolers, the act of counting a set of objects is not entirely a rote activity but is guided by their mathematical understanding. Counting and the Origins of the Number Concept Babies show numerical competence almost from the day they are born,3 and some infants younger than six months have shown they can perform a rudimentary kind of addition and subtraction.4 These abilities suggest that number is a fundamental component of the world children know. 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. Counting a set of objects is a complex task involving thinking, perception, and movement, with much of its complexity obscured by familiarity. Consider what you need to do to count a set of objects: The items to be counted must be identified and distinguished from items not to be counted, as well as from those that have already been counted. Items are counted by pairing each one with some sort of verbal representation (typically a number name). An indicating act is needed that pairs each object in space with a word said in time. Finally, you need to understand that counting results in a number that represents how many things are in the set that was counted. Competent counting requires mastery of a symbolic system, facility with a complicated set of procedures that require pointing at objects and designating them with symbols, and understanding that some aspects of counting are merely conventional, while others lie at the heart of its mathematical usefulness. We discuss issues related to competent counting, including the learning of number names, in the section on procedural fluency below. In this section, we discuss children’s understanding of the conceptual aspects of counting. This separation is somewhat artificial because counting is a good example of the way in which the different strands of mathematical proficiency are interwoven.
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Adding + It Up: Helping Children Learn Mathematics As children learn to count, their thinking changes in a way that shapes their concept of number. Counting is not simply reciting the number word sequence. There must be items to count; and there must be a procedure to make each utterance of a number word correspond with one of the items to be counted.5 At first, these items are perceptual; they might be, for example, beads, marbles, fingers, taps, steps, or drumbeats. The child must not only be able to perceive the items but also to conceive of them as individual things to be counted. Later, children become able to count sets of things (e.g., “how many different colors of buttons are there?”) as well as items that may not be readily perceivable.6 The counter must always create a mental representation of the items that are counted. This process of creation is clearly demonstrated when a child appears to count specific items in a situation where no such items are visible, audible, or tangible. Counting in the absence of perceivable objects is the culmination of a rather intricate developmental process. The process includes the progressive development of an ability to create unit items to be counted, first on the basis of conscious perception of external objects and then on the basis of internal representations.7 Early research on children’s understanding of the mathematical basis for counting focused on five principles their thinking must follow if their counting is to be mathematically useful:8 One-to-one: there must be a one-to-one relation between counting words and objects; Stable order (of the counting words): these counting words must be recited in a consistent, reproducible order; Cardinal: the last counting word spoken indicates how many objects are in the set as a whole (rather than being a property of a particular object in the set); Abstraction: any kinds of objects can be collected together for purposes of a count; and Order irrelevance (for the objects counted): objects can be counted in any sequence without altering the outcome. 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.
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Adding + It Up: Helping Children Learn Mathematics Understanding Counting and Mastering it The relation between children’s conceptual understanding of counting and their mastery of conventional counting remains controversial. According to one viewpoint,9 children’s emerging understanding of these counting principles organizes and motivates their acquisition of conventional counting procedures. 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.10 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.11 One reason it has been hard to resolve contrasting claims about how children come to understand the conceptual basis for counting is that preschoolers’ performance when they count is often quite variable, as it is with most other tasks.12 The many errors preschoolers make when counting could indicate that they fail to understand the importance of the counting principles. The variability of their performance makes fundamentally ambiguous the task of inferring their knowledge of principles from their behavior. A child’s difficulty in managing the complex processes involved in counting could mask a real understanding of its conceptual basis. One way of circumventing the ambiguity of children’s counting behavior involves asking them to judge the adequacy someone else’s counting rather than perform the activity themselves. For example, asked to judge the accuracy of counting by a puppet who counted either correctly, incorrectly, or unconventionally (e.g., starting from an unusual starting point but counting all of a set of items), 3- to 5-year-olds demonstrated very good performance. Three-year-olds showed perfect acceptance of correct counting, 96% acceptance of unconventional but correct counting, and 67% rejection of real errors. Four-year-olds were better than 3-year-olds at rejecting true errors.13 Presented with a larger set of counting strategies to judge, children in a later study did not perform quite as well.14 In fact, 3-year-olds’ acceptance of unconventional correct counting was actually higher than that of 4-year-olds, suggesting that some of the acceptance of unconventional correct counting came from a blanket acceptance of the puppet’s performance. Finally, and most relevant to the relation between counting skill and judgment of another’s counting, the only children who failed to meet a criterion of 75% correct in rejecting the puppet’s counting errors also failed to meet the same criterion in their own counting. Thus, children’s own counting activity might form the basis for their judgments of what constitutes successful counting.
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Adding + It Up: Helping Children Learn Mathematics There are also important limits on children’s ability to use counting in problem solving. Several studies have found that children 3 years and younger have a great deal of difficulty in using counting to produce sets of a given numerosity, even when that numerosity is well within their counting range.15 Taken as a whole, these studies indicate that variations in the context in which children are asked to judge another’s counting can have a great effect on their acceptance of deviations from conventional counting and of errors that violate the counting principles. 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. Nevertheless, two points are clear. First, both aspects of counting are important developmental acquisitions. Second, by the time they enter kindergarten, most U.S. children understand the rules that underlie counting, can perform conventional counting correctly with sets of objects greater than 10, and can use counting to solve some simple mathematical problems. Procedural Fluency Procedural fluency refers to the ability to perform procedures flexibly, accurately, and efficiently. As we noted in Chapter 4, procedural fluency makes it possible for children to use mathematics reliably to solve problems and generate examples to test their mathematical ideas. Procedural Fluency and Counting In the case of counting, the difficulties young children have in fluently performing the complex activities required to count a set of objects accurately are a major obstacle to their mathematical development. For example, when asked to count increasingly longer row of up to 30 objects, 90% of- to -year-olds made some kind of violation of the one-to-one correspondence between pointing and objects or between pointing and saying the number words, although these errors were made on only 6% of the sets of objects counted.17 Directives to “try hard” or “be careful” decreased errors substantially. Thus, effort and concentration are important aspects of accurate counting.
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Adding + It Up: Helping Children Learn Mathematics The difficulty preschoolers have in coordinating the process of keeping track of objects and counting them seems to be a universal characteristic of learning to count, with children in different cultures showing comparable rates of recounting or skipping objects.18 Large differences across languages have been found in a second key aspect of procedural fluency in the preschool period, the mastery of the set of number names used in the child’s native language. 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. Furthermore, the structure of the number names in a language is a major influence on the difficulties children have in learning to count correctly. These difficulties have important implications for the initial learning of mathematics in elementary school. The number names used in a language provide children with a readymade representation for number. Counting principles are universal and so do not differ between languages, but number names do differ in sound and structure across languages and influence children’s learning to count. Linguistic structure of number names. Names for numbers have been generated according to a bewildering variety of systems.19 The Hindu-Arabic system for representing the whole numbers is clearly a base-10 system, with 10 basic symbols (the digits 0–9). These may be freely combined, with the place of a digit indicating the power of 10 that it represents.20 The Hindu-Arabic system is a useful reference point in describing number-naming schemes for two reasons. First, it is a widely used system for writing numbers. Second, it is as consistent and concise as a base-10 system could be. Box 5–1 shows how spoken names for numbers are formed in three languages: English, Spanish, and Chinese. All of these languages use a base-10 system, but the languages differ in the clarity and consistency with which the base-10 structure is reflected in the number names. As the first section of the figure shows, representations for numbers from 1 to 9 consist of an unsystematically organized list. There is no way to predict that 5 or five or wu come after 4, four, and si, respectively, in the Arabic numeral, English, and Chinese systems.
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Adding + It Up: Helping Children Learn Mathematics Box 5–1 Number Names in Chinese, English, and Spanish a. One to ten Arabic numeral 1 2 3 4 5 Chinese (spoken) yi er san si wu English one two three four five Spanish uno dos tres cuatro cinco b. Eleven to twenty Arabic numeral 11 12 13 14 15 Chinese (spoken) shi yi shi er shi san shi si shi wu English eleven twelve thirteen fourteen fifteen Spanish once doce trece catorce quince c. Twenty to ninety-nine Language Rule Chinese (spoken) Decade name (unit name+shi)+unit name English Decade name [(twen, thir, for, fif, six, seven, eight, nine) +-ty]+unit name Spanish Decade name (veinte, treinta, cuarenta, cincuenta, sesenta, setenta, ochenta, noventa)+and (y)+unit name
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Adding + It Up: Helping Children Learn Mathematics 6 7 8 9 10 liu 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 shi jiu er shi sixteen seventeen eighteen nineteen twenty diez y seis diez y siete diez y ocho diez y nueve veinte Example san shi qi thirty-seven trenta y siete
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Adding + It Up: Helping Children Learn Mathematics Names for numbers above 10 diverge in interesting ways among these different languages, as the second part of Box 5–1 demonstrates. The Chinese number-naming system maps directly onto the Hindu-Arabic number system used to write numerals. For example, a word-for-word translation of shi qi (17) into English produces ten-seven. English has unpredictable names for 11 and 12 that bear only a historical relation to one and two.21 Whether the boundary between 10 and 11 is marked in some way can be very significant because this boundary can offer the first clue that number names are organized according to a base-10 system. The English names for numbers in the teens beyond 12 do have an internal structure, but it is obscured by phonetic modifications of many of the elements used in the first 10 numbers (e.g., ten becomes -teen, three becomes thir-, and five becomes fif-). Furthermore, the order of word formation reverses the place value, unlike the Hindu-Arabic and Chinese systems (and the English system above 20), naming the smaller value before the larger value. Spanish follows the same basic pattern for English to begin the teens, although there may be a clearer parallel between uno, dos, tres and once, doce, trece than between one, two, three and eleven, twelve, thirteen. The biggest difference between Spanish and English is that after 15 the number names in Spanish abruptly take on a different structure. Thus the name for 16 in Spanish, diez 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), rather than the structures of the number names in English from 13 to 19 and the names in Spanish from 11 to 15 (starting with the ones value and then naming the tens value). Above 20, all these number-naming systems converge on the Chinese structure of naming the larger value before the smaller one. Despite this convergence, the systems continue to differ in the clarity of the connection between the decade names and the corresponding unit values. Chinese numbers are consistent in forming decade names by combining a unit value and the base (ten). Decade names in English and Spanish generally can be derived from the name for the corresponding unit value, with varying degrees of phonetic modification (e.g., five becomes fif- in English, cinco becomes cincuenta in Spanish) and with some notable exceptions, primarily the special name for 20 used in Spanish. Psychological consequences of number names. Although all the number-naming systems being reviewed are essentially base-10 systems, they differ in the consistency and transparency with which that structure is reflected in the number names. Several studies comparing English-
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Adding + It Up: Helping Children Learn Mathematics and Chinese-speaking children demonstrate that the organization of number names does indeed play a significant role in mediating children’s mastery of this symbolic system.22 These studies have reported that (a) differences in performance on counting-related tasks do not emerge until children in both the United States and China begin learning the second decade of number names, sometime between 3 and 4 years of age; (b) those differences are generally limited to the verbal aspect of counting, rather than affecting children’s ability to use counting in problem solving or their understanding of basic counting principles; and (c) differences in the patterns of mistakes that children make in learning to count reflect the structure of the systems they are learning. 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. The relative complexity of English number names has other cognitive consequences. Speakers of English and other European languages face a complex task in learning to write Arabic numerals, one that is more difficult than that faced by speakers of Chinese.25 (For example, compare the mapping between name and numeral for twenty-four with that for fourteen in the two languages.) Speakers of languages whose number names are patterned after Chinese (including Korean and Japanese) are better able than speakers of English and other European languages to represent numbers using base-10 blocks and to perform other place-value tasks.26 Because school arithmetic algorithms are largely structured around place value, the finding of a relationship between the complexity of number names and the ease with which children learn to count has important educational implications. When learning to count, children must acquire a combination of conventional knowledge of number names, conceptual understanding of the mathematical principles that underlie counting, and ability to apply that knowledge in solving mathematical problems. Language differences during preschool
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Adding + It Up: Helping Children Learn Mathematics In addition to using counting to solve simple arithmetic problems, preschool children show understanding at an early age that written marks on paper can preserve and communicate information about quantity.35 For example, 3- and 4-year-olds can invent informal marks on paper, such as tally marks and diagrams, to show how many objects are in a set. But they are less able to represent changes in sets or relationships between sets, in part because they fail to realize that the order of their actions is not automatically preserved on paper. Adaptive Reasoning Adaptive reasoning refers to the capacity to think logically about the relationships among concepts and situations and to justify and ultimately prove the correctness of a mathematical procedure or assertion. Adaptive reasoning also includes reasoning based on pattern, analogy, or metaphor. Research suggests that young children are able to display reasoning ability if they have a sufficient knowledge base, if the task is understandable and motivating, and if the context is familiar and comfortable.36 In particular, preschool children can generate solutions to problems and can explain their thinking. Situations that require preschoolers to use their mathematical concepts and procedures in unconventional ways often cause them difficulty. For example, when preschool children are asked to count features of objects (e.g., the tines of forks) or subsets of objects (e.g., just the red buttons in a mixed set), they often cannot overcome their tendency to count all the separate objects.37 Another example of the limitations on preschoolers’ ability to generalize their mathematics is that they perform better in situations that require them to think about adding or subtracting actual objects (even if those objects are hidden from view in a box) than they do when simply asked an equivalent question (e.g., “What’s 3 and 5?”).38 Four- and 5-year-olds do begin to use their knowledge to answer correctly the Piagetian number task presented above involving equivalent sets of candies, and later they recognize without counting that the sets have the same number of candies.39 A major challenge of formal education is to build on the initial and often fragile understanding that children bring to school and to make it more reliable, flexible, and general. Most preschool children enter school with an initial understanding of procedures (e.g., counting, addition, subtraction) that forms the basis for much of their later mathematics learning, although they have limited ability to generalize that knowledge and to understand its importance. A major challenge of formal education is to build on the initial and often fragile understanding that children bring to school and to make it more reliable, flexible, and general.40
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Adding + It Up: Helping Children Learn Mathematics Productive Disposition In addition to the concepts and skills that underlie mathematical proficiency, children who are successful in mathematics have a set of attitudes and beliefs that support their learning. They see mathematics as a meaningful, interesting, and worthwhile activity; believe that they are capable of learning it; and are motivated to put in the effort required to learn. Reports on the attitudes of preschoolers toward learning in general and learning mathematics in particular suggest that most children enter school eager to become competent at mathematics. In a survey that examined a number of personality and motivational features relevant to success in mathematics, teachers and parents reported that kindergarteners have high levels of persistence and eagerness to learn (although teachers differed in their perceptions of children from different ethnic groups, as we discuss below).41 Children enter school viewing mathematics as important and themselves as being competent to master it. In one study, first graders rated their interest in mathematics on average at approximately 6 on a scale from 1 to 7 (with 7 being the highest).42 Children gave similar ratings to their competence in mathematics, with boys giving somewhat higher ratings for their mathematics competence than girls did, the opposite of the pattern for reading. One important factor in attaining a productive disposition toward mathematics and maintaining the motivation required to learn it is the extent to which children perceive achievement as the product of effort as opposed to fixed ability. 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 learning orientation, 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. The challenge to parents and educators is to help them maintain a productive disposition toward mathematics as they develop the other strands of their mathematical proficiency.
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Adding + It Up: Helping Children Learn Mathematics Limitations of Preschoolers’ Mathematical Proficiency In some circumstances, preschool children show impressive mathematical abilities that can provide the basis for their later learning of school mathematics. These abilities are, however, limited in a number of important ways. One of the most important limitations is that much of preschoolers’ understanding of number is constrained to sets of a certain size. Because the algorithms that preschoolers develop are based on counting and on their experience with sets of objects, they do not generalize to larger numbers. For example, preschool children can show a mastery of the concepts of addition and subtraction for very small numbers.45 But being able to predict the results of adding one to a number does not imply that children will be able to predict the results of adding two to the same number. This limitation is an important feature of preschool mathematical thinking and is an important way in which preschool mathematical proficiency differs from adult proficiency. Another important limitation is that preschoolers’ thinking about arithmetic is influenced heavily by the context of the problem. As stated above, the way in which a word problem is phrased can be the difference between success and failure. Furthermore, if children succeed, the strategy they use is a direct model of the story; they, in effect, act out the story to find the answer. They will need to make several advances in development before they realize that a few basic counting strategies can be used to solve a wide variety of word problems, that stories can be represented by written number sentences of the form a+b=c or a–b=c, and that many different stories can be represented by the same sentence. Equity and Remediation Most U.S. children enter school with mathematical abilities that provide a strong base for formal instruction in mathematics. These abilities include understanding the magnitudes of small numbers, being able to count and to use counting to solve simple mathematical problems, and understanding many of the basic concepts underlying measurement. For example, a large survey of U.S. kindergartners found that 94% of first-time kindergartners passed their Level 1 test (counting to 10 and recognizing numerals and shapes) and 58% passed their Level 2 test (reading numerals, counting beyond 10, sequencing patterns, and using nonstandard units of length to compare objects).46 A number of children, however, particularly those from low socioeconomic groups, enter school with specific gaps in their mathematical proficiency. For example, the survey of kindergartners found that while 79% of children whose
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Adding + It Up: Helping Children Learn Mathematics mother had a bachelor’s degree passed the Level 2 test described above, only 32% of those whose mother had less than a high school degree could do so.47 The same survey found large differences between ethnic groups on the more difficult tests (but not on the Level 1 tasks) with 70% of Asian and 66% of non-Hispanic white children passing the Level 2 tasks, but only 42% of African American, 44% of Hispanic, 48% of Hawaiian Native or Pacific Islander, and 34% of American Indian or Alaska Native participants doing so.48 Other research has shown that children from lower socioeconomic backgrounds have particular difficulty understanding the relative magnitudes of single-digit whole numbers49 and solving addition and subtraction problems verbally rather than using objects.50 Overall, the research shows that poor and minority children entering school do possess some informal mathematical abilities but that many of these abilities have developed at a slower rate than in middle-class children.51 This immaturity of their mathematical development may account for the problems poor and minority children have understanding the basis for simple arithmetic and solving simple word problems.52 Several promising approaches have been developed to deal with this developmental immaturity in mathematical knowledge. 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. children in general find confusing.54 Performance at the end of a year-long intervention was at levels comparable to those reported for Asian children and substantially above those typically reported for nonminority children. Taken together, these results suggest that relatively simple interventions may yield substantial payoffs in ensuring that all children enter or leave first grade ready to profit from mathematics instruction. The kindergarten survey cited above reported smaller ethnic differences in factors related to productive disposition (persistence, eagerness to learn, and ability to pay attention) than in mathematical knowledge. There were, however, some noteworthy differences between the reports of teachers and parents for different ethnic groups. Parents reported high levels of eagerness to learn (e.g., 93% for non-Hispanic whites, 90% for non-Hispanic African Americans, and 90% for Hispanics), but teachers differed in their judgments
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Adding + It Up: Helping Children Learn Mathematics of eagerness (judging 78% of non-Hispanic whites, 66% of non-Hispanic African Americans, and 70% of Hispanics as eager to learn). Teachers and parents are, of course, judging children against different comparison groups, but the data at least raise the possibility that kindergarten teachers may be underestimating the eagerness of their students to learn mathematics. Preschool Children’s Proficiency For preschool children, the strands of mathematical proficiency are particularly closely intertwined. Although their conceptual understanding is limited, as their understanding of number emerges they become able to count and solve simple problems. It is only when they move beyond what they informally understand—to the base-10 system for teens and larger numbers, for example—that their fluency and strategic competencies falter. Young children also show a remarkable ability to formulate, represent, and solve simple mathematical problems and to reason and explain their mathematical activities. The desire to quantify the world around them seems to be a natural one for young children. They are positively disposed to do and understand mathematics when they first encounter it. Most U.S. children enter school with a basic understanding of number and number concepts that can form the foundation for learning school mathematics, but their knowledge is limited in some very important ways. Preschool children generally show a much more sophisticated understanding of small numbers than they do of larger numbers. They also have a great deal of difficulty in moving from the number names in languages such as English and Spanish to understanding the base-10 structure of number names and mastering the Arabic numerals used in school mathematics. 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. Notes 1. Piaget, 1941/1965. 2. Copeland, 1984, p. 12. In Piaget’s theory, children typically enter the concrete operational stage from about 7 to 11 years of age, when they can think in a logical way about the characteristics of real objects. 3. Antell and Keating, 1983. 4. Wynn, 1992a, 1992b.
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Adding + It Up: Helping Children Learn Mathematics 5. Steffe, von Glasersfeld, Richard, and Cobb, 1983, p. 24. 6. Steffe, Cobb, and von Glasersfeld, 1988. 7. Steffe, 1994. 8. Gelman and Gallistel, 1978. 9. Gelman, Meck, and Merkin, 1986; Gelman, 1990, 1993. 10. Briars and Siegler, 1984; Frye, Braisby, Lowe, Maroudas, and Nicholls, 1989; Fuson, 1988; Fuson and Hall, 1983, Siegler, 1991, Sophian, 1988; Wynn, 1990. 11. Baroody, 1992a; Baroody and Ginsburg, 1986; Rittle-Johnson and Siegler, 1998. 12. Siegler, 1994. 13. Gelman and Meck, 1983. 14. Briars and Siegler, 1984. 15. Frye, Braisby, Lowe, Maroudas, and Nicholls, 1989; Miller, Smith, Zhu, and Zhang, 1995; Wynn, 1990. 16. Similar suggestions have been made by Baroody, 1992a, 1992b; Fuson, 1988, 1992; and Siegler, 1991. 17. Fuson, 1988, p. 73. 18. Miller, Smith, Zhu, and Zhang, 1995. 19. See Ifrah, 1985; and Menninger, 1969. 20. The so-called Hindu-Arabic numeration system is in some sense a misnomer because the Chinese numeration system has been a decimal one from the time of the earliest historical records. Because of the frequent contact between the Chinese and the Indians since the time of antiquity, there has always been some question of whether the Indians got their decimal system from the Chinese. Language has to be the product of its culture. So the fact that the names for numbers in Chinese, especially for the teens, reflect a base-10 system indicates that the decimal system has been in place in China all along. By contrast, the Hindu-Arabic system did not take root in the West until the sixteenth century, long after the names for numbers in the various Western languages had been set. The irregularities in the English and Spanish number names may perhaps be understood better in this light. 21. Menninger, 1969. 22. Miller, Smith, Zhu, and Zhang, 1995; Miller and Stigler, 1987. 23. Fuson, Richards, and Briars, 1982; Miller and Stigler, 1987; Siegler and Robinson, 1982. 24. Baroody, 1987a. 25. Fuson, Fraivillig, and Burghardt, 1992; Séron, Deloche, and Noël, 1992. 26. Miura, 1987; Miura, Kim, Chang, and Okamoto, 1988; Miura and Okamoto, 1989; Miura, Okamoto, Kim, Steere, and Fayol, 1993. 27. Miller, Smith, Zhu, and Zhang, 1995. 28. Huttenlocher, Jordan, and Levine, 1994. 29. Carpenter and Moser, 1984; Siegler, 1996; Siegler and Jenkins, 1989; Siegler and Robinson, 1982; see also Baroody, 1987b, 1989; and Fuson, 1992. 30. Siegler, 1987. 31. Siegler and Jenkins, 1989.
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Adding + It Up: Helping Children Learn Mathematics 32. Siegler, 1995. Alibali and Goldin-Meadow, 1993, showed that in learning to solve problems involving mathematical equivalence, students were most successful when they had passed through a stage of considering multiple solution strategies. 33. Carpenter, Ansell, Franke, Fennema, and Weisbeck, 1993; Riley, Greeno, and Heller, 1983; see also Fuson, 1992. 34. Riley, Greeno, and Heller, 1983. 35. Allardice, 1977; Ginsburg, 1989. 36. Alexander, White, and Daugherty, 1997, propose these three conditions for reasoning in young children. There is reason to believe that the conditions apply more generally. 37. Shipley and Shepperson, 1990. 38. Hughes, 1986; Jordan, Huttenlocher, and Levine, 1992. 39. Fuson, Secada, and Hall, 1983. 40. See Bowman, Donovan, and Burns, 2001, for a discussion of these ideas. 41. National Center for Education Statistics, 2000. 42. Wigfield, Eccles, Yoon, Harold, Arbreton, Freedman-Doan, and Blumenfeld, 1997. 43. Dweck, 1999; Heyman and Dweck, 1998. 44. Heyman, Dweck, and Cain, 1992. 45. For example, Jordan, Huttenlocher, and Levine, 1992. 46. National Center for Education Statistics, 2000. 47. National Center for Education Statistics, 2000. 48. National Center for Education Statistics, 2000. 49. Griffin, Case, and Siegler, 1994. 50. Jordan, Huttenlocher, and Levine, 1992. 51. Ginsburg, Klein, and Starkey, 1998. 52. Jordan, Levine, and Huttenlocher, 1995. 53. Griffin, Case, and Siegler, 1994. 54. Fuson, Smith, and Lo Cicero, 1997. References Alexander, P.A., White, C.S., & Daugherty, M. (1997). Analogical reasoning and early mathematics learning. In L.D.English (Ed.), Mathematical reasoning: Analogies, metaphors, and images (pp. 117–147). Mahwah, NJ: Erlbaum. Allardice, B. (1977). The development of written representations for some mathematical concepts. Journal of Children’s Mathematical Behavior, 1(4), 135–148. Alibali, M.W., & Goldin-Meadow, S. (1993). Gesture-speech mismatch and mechanisms of learning: What the hands reveal about a child’s state of mind. Cognitive Psychology, 25, 468–573. Antell, S.E., & Keating, D.P. (1983). Perception of numerical invariance in neonates. Child Development, 54, 695–701. Baroody, A.J. (1987a). Children’s mathematical thinking: A developmental framework for preschool, primary, and special education teachers. New York: Teachers College Press. Baroody, A.J. (1987b). The development of counting strategies for single-digit addition. Journal for Research in Mathematics Education, 18, 141–157.
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