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...sented a greater quantity (“more”). Because of the low expected frequencies, a Fisher Exact Probability test was used to test whether there was a significant difference between the conditions. There was no difference Table 2 Quantity judgments for each condition of Experiment 2 Mapping direction Number reporting “more” for both judgments Number reporting “less” for both judgments Number of inconsistent judgments Total number in group Increases mapped up 16 1 1 18 Increases mapped down 11 2 7 20 1168 M. Gattis / Cognitive Development 17 (2002) 1157–1183 (P = .33, Fisher Exact Probability test, Siegel, 1956). Thus, whereas the rate judgments in Experiment 1 corresponded to slope, the quantity judgments in Experiment 2 corresponded to height. 4. Experiment 3 The results of Experiments 1 and 2 are consistent with the hypothesis that spatial reasoning involves an alignment of structurally similar relations between concepts and space. In both experiments, children’s judgments of the value of a conceptual variable represented in a diagram corresponded in level of complexity to a structurally appropriate spatial cue. In Experiment 1, children’s judgments of rate, a second-order variable, corresponded t...

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...ince steeper hills lead to faster rates of travel downhill, the reverse is equally true for uphill travel, when steeper hills lead to slower rates of travel. 1.2. Mapping relational structure A second explanation for mapping consistencies is that mappings between concepts and space are based on general constraints governing the mapping process, rather than or in addition to specific associations. An example of such a general constraint is the tendency observed in analogical mapping to map two concepts based 1160 M. Gattis / Cognitive Development 17 (2002) 1157–1183 on structural similarities (Gentner, 1983). When asked to compare two problems or concepts, adults tend to map elements of one concept or problem to elements of the other, to map the relations between elements in one concept the relations between elements in the other concept, and to map relations between relations in one concept to relations between relations in the other concept. This sensitivity to relational structure emerges in early childhood. Children are sensitive to the relational structures of perceptual analogy tasks by the age of 4, and will align the relational structures to choose “matches” to complete a analogy by the a...

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... a single dimension, or first-order variables. In all three experiments, children’s judgments corresponded to the height of the line. 6. General discussion These experiments examined the origins of a remarkable human ability — the capacity to reason with conceptual information presented spatially. Whereas many aspects of reasoning are understood well enough to build computer models of human performance, no current theory of reasoning explains why humans are so good at reasoning spatially. Previous work has tended to focus on why spatial representations facilitate recognition and search (e.g., Larkin & Simon, 1987), and left unaddressed the question of why spatial representations are so powerful for reasoning and inference. The four experiments presented in this paper investigated an hypothesized constraint on reasoning that could explain why spatial reasoning is fast and flexible, but also yields intelligent inferences. This constraint is the mapping of relational structure between concepts and space. Six- and 7-year-old children with no formal training in graphing were given a brief orientation about the elements of graphs, and asked to make judgments about the value represented by one of two sloping ...

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... the view that young children are capable of reasoning about functional relationships is found in a variety of studies with young children. Piaget and his colleagues concluded from several studies that preschool children have an intuitive “logic of functions” which is qualitative in nature and is the basis of mature functional reasoning (Piaget, Grize, Szeminska, & Bang, 1968/1977). More recent developmental studies also indicate that by the age of five, children have a basic understanding of time, distance, and speed, and are able to integrate two of those dimensions to reason about a third (Halford, 1993; Wilkening, 1981). 6.2. Spatial reasoning and graphing conventions The results reported here suggest that spatial reasoning is influenced by general constraints in reasoning, and that these constraints precede the learning of graphing conventions. The children in these experiments had no formal instruction in graphing, and but when asked to interpret the meaning of function lines in graph-like diagrams, answered in highly consistent ways. These consistencies were not necessarily correct according to all the rules of graphing, but they do reflect a basic principle of graphing, which is that co...

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...ked to compare two problems or concepts, adults tend to map elements of one concept or problem to elements of the other, to map the relations between elements in one concept the relations between elements in the other concept, and to map relations between relations in one concept to relations between relations in the other concept. This sensitivity to relational structure emerges in early childhood. Children are sensitive to the relational structures of perceptual analogy tasks by the age of 4, and will align the relational structures to choose “matches” to complete a analogy by the age of 6 (Kotovsky & Gentner, 1996). If the tendency to map corresponding relational structures extends beyond semantic concepts and analogical reasoning to include spatial reasoning as well, it could explain children’s tendency to align height and quantity (as reported by Tversky et al., 1991), which are both first-order variables because they are relations between elements, and adults’ tendency to align slope and rate (as reported by Gattis & Holyoak, 1996), which are both second-order variables because they are relations between relations. The following experiments investigated the hypothesis that sensitivity to relational s...

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...lts asked to make rate judgments about line graphs also make the mappings, “steeper equals faster” and “shallower equals slower,” even when the graph is constructed in such a way that this simple mapping leads to an incorrect answer (see Gattis & Holyoak, 1996). 6.3. Spatial reasoning in educational contexts In contrast to the facility for spatial reasoning demonstrated by young children in these experiments without any prior graphing instruction, educational researchers have documented numerous failures in graphing performances of school children subsequent to extensive graphing instruction (Leinhardt, Zaslavsky, & Stein, 1990; McDermott, Rosenquist, & VanZee, 1987). Leinhardt et al. (1990) review many studies of school children’s performance in graph interpretation and construction, in which they note a myriad of common errors in prediction, classification, translation and scaling tasks. These two contrasting pictures of children’s graphing abilities pose an apparent paradox. If spatial reasoning, including reasoning with graphs, is governed by fundamental cognitive constraints, why do young children and even college students often perform poorly on tests of graphing skills? As Leinhardt et al. point out, most tas...

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...s in one concept to relations between relations in the other concept. This sensitivity to relational structure emerges in early childhood. Children are sensitive to the relational structures of perceptual analogy tasks by the age of 4, and will align the relational structures to choose “matches” to complete a analogy by the age of 6 (Kotovsky & Gentner, 1996). If the tendency to map corresponding relational structures extends beyond semantic concepts and analogical reasoning to include spatial reasoning as well, it could explain children’s tendency to align height and quantity (as reported by Tversky et al., 1991), which are both first-order variables because they are relations between elements, and adults’ tendency to align slope and rate (as reported by Gattis & Holyoak, 1996), which are both second-order variables because they are relations between relations. The following experiments investigated the hypothesis that sensitivity to relational structure influences spatial reasoning by examining how young children with no experience with graphs map first-order and second-order conceptual relations to spatial relations when reasoning with graph-like diagrams. Examining how young children with no graphi...

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...vel of a concept at the top of the page, so that increases moved in an upward direction. This mapping preference, Tversky et al. pointed out, corresponds to a metaphorical association between “up” and “more,” and “down” and “less”: Tversky et al.’s results fit nicely with Handel, De Soto, and London’s (1968) report that adults asked to assign word pairs to spatial locations on a cross-like diagram preferred to assign quantity to the vertical, placing “more” at the upper vertical end and “less” at the lower vertical end. A second consistency in the mapping of concepts to space is the report of Gattis and Holyoak (1996) that adults assigned “faster” to “steeper” in a graphical reasoning task. In several experiments, adults judged the relative rate of two continuous linear variables using simple line graphs. Graph-like diagrams were constructed which varied the assignments of variables to axes, the perceived cause–effect relation between the variables, and the causal status of the variable being queried. Across all of these conditions, a single factor seemed to account for reasoning performance. People were more accurate at rate judgments when the variable being queried was assigned to the vertical axis, so t...

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...al reasoning and children’s understanding of functional relationships An important assumption of the structure-mapping interpretation of the current results is that 6- and 7-year-old children are capable of reasoning about functional relationships. Support for the view that young children are capable of reasoning about functional relationships is found in a variety of studies with young children. Piaget and his colleagues concluded from several studies that preschool children have an intuitive “logic of functions” which is qualitative in nature and is the basis of mature functional reasoning (Piaget, Grize, Szeminska, & Bang, 1968/1977). More recent developmental studies also indicate that by the age of five, children have a basic understanding of time, distance, and speed, and are able to integrate two of those dimensions to reason about a third (Halford, 1993; Wilkening, 1981). 6.2. Spatial reasoning and graphing conventions The results reported here suggest that spatial reasoning is influenced by general constraints in reasoning, and that these constraints precede the learning of graphing conventions. The children in these experiments had no formal instruction in graphing, and but when asked to interpret the meaning...

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...ildren’s graphing abilities pose an apparent paradox. If spatial reasoning, including reasoning with graphs, is governed by fundamental cognitive constraints, why do young children and even college students often perform poorly on tests of graphing skills? As Leinhardt et al. point out, most tasks studied in educational settings involve either formal knowledge (i.e., the definition of a function), specific experience (i.e., translation between the algebraic notational system and the Cartesian coordinate system), or construction, which is a more difficult cognitive process than interpretation (Bates, 1993; Savage-Rumbaugh, 1993). In contrast, the task used in these experiments was a qualitative interpretation task. Leinhardt et al. note that qualitative interpretation tasks are rare in the mathematics curriculum, but are easier than most graphing tasks, in part because they rely on processing of global features. Some researchers argue that qualitative interpretive tasks ought to be the initial step in graphing instruction (Bell & Janvier, 1981), and the results reported here indicate that qualitative interpretation tasks may indeed be an excellent introduction to graphing for young children. A...

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...young children are capable of reasoning about functional relationships is found in a variety of studies with young children. Piaget and his colleagues concluded from several studies that preschool children have an intuitive “logic of functions” which is qualitative in nature and is the basis of mature functional reasoning (Piaget, Grize, Szeminska, & Bang, 1968/1977). More recent developmental studies also indicate that by the age of five, children have a basic understanding of time, distance, and speed, and are able to integrate two of those dimensions to reason about a third (Halford, 1993; Wilkening, 1981). 6.2. Spatial reasoning and graphing conventions The results reported here suggest that spatial reasoning is influenced by general constraints in reasoning, and that these constraints precede the learning of graphing conventions. The children in these experiments had no formal instruction in graphing, and but when asked to interpret the meaning of function lines in graph-like diagrams, answered in highly consistent ways. These consistencies were not necessarily correct according to all the rules of graphing, but they do reflect a basic principle of graphing, which is that conceptual dimension...

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...n between the algebraic notational system and the Cartesian coordinate system), or construction, which is a more difficult cognitive process than interpretation (Bates, 1993; Savage-Rumbaugh, 1993). In contrast, the task used in these experiments was a qualitative interpretation task. Leinhardt et al. note that qualitative interpretation tasks are rare in the mathematics curriculum, but are easier than most graphing tasks, in part because they rely on processing of global features. Some researchers argue that qualitative interpretive tasks ought to be the initial step in graphing instruction (Bell & Janvier, 1981), and the results reported here indicate that qualitative interpretation tasks may indeed be an excellent introduction to graphing for young children. Acknowledgments This research was supported by the Max Planck Society. I thank Tylor Hagerman, Marija Kulis, Denise Parks, and Felicitas Wiedermann for their assistance in preparing and running these experiments, the teachers and children of the Grundschulen Bad-Soden-Strasse, Bayernplatz, Farinelli-Strasse, KlenzeStrasse, Simmern-Strasse, and Torquato-Tasso-Strasse in Munich for their M. Gattis / Cognitive Development 17 (2002) 1157–1183 1177 p...

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...tory training tasks whose purpose was to teach children the elements of graphs, followed by a final judgment task which provided the dependent variable for the experiments. In the first and second training tasks children learned to map discrete values of a M. Gattis / Cognitive Development 17 (2002) 1157–1183 1161 dimension across first horizontal and then vertical lines using a procedure based on that of Tversky et al. (1991). In the third training task the children learned to coordinate values from the horizontal and vertical lines “to make a story,” with a procedure similar to that used by Bryant and Somerville (1986). The children in effect learned to map a function, but were taught to think of that function as a story represented by a line. In the final phase of the experiment, the children were asked to judge the rate of an event represented by a particular line. 2.1. Method 2.1.1. Participants Eighty-four first graders (43 girls and 41 boys) from two elementary schools in Munich, Germany participated in Experiment 1. Children were 6–8 years old (mean age: 6–10; range: 6–0 to 8–0; S.D.: 6 months). Ages for children in each of the four experimental groups were as follows: Time-Up (mean age: 6–10; range: ...

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...pped together. Young children thus appear to distinguish relations between elements from relations between relations, and to map concepts to spatial representations accordingly. This finding accords with Tversky’s (1995) observation that many graphic depictions involve a mapping of elements to elements and relations to relations, and that young children asked to create notational systems observe a similar rule. Recent studies of diagrammatic reasoning by adults provide further evidence that interpretations of novel diagrams rely on a mapping of elements to elements and relations to relations (Gattis, 2001, 2002). 6.1. Spatial reasoning and children’s understanding of functional relationships An important assumption of the structure-mapping interpretation of the current results is that 6- and 7-year-old children are capable of reasoning about functional relationships. Support for the view that young children are capable of reasoning about functional relationships is found in a variety of studies with young children. Piaget and his colleagues concluded from several studies that preschool children have an intuitive “logic of functions” which is qualitative in nature and is the basis of mature fun...

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...hing abilities pose an apparent paradox. If spatial reasoning, including reasoning with graphs, is governed by fundamental cognitive constraints, why do young children and even college students often perform poorly on tests of graphing skills? As Leinhardt et al. point out, most tasks studied in educational settings involve either formal knowledge (i.e., the definition of a function), specific experience (i.e., translation between the algebraic notational system and the Cartesian coordinate system), or construction, which is a more difficult cognitive process than interpretation (Bates, 1993; Savage-Rumbaugh, 1993). In contrast, the task used in these experiments was a qualitative interpretation task. Leinhardt et al. note that qualitative interpretation tasks are rare in the mathematics curriculum, but are easier than most graphing tasks, in part because they rely on processing of global features. Some researchers argue that qualitative interpretive tasks ought to be the initial step in graphing instruction (Bell & Janvier, 1981), and the results reported here indicate that qualitative interpretation tasks may indeed be an excellent introduction to graphing for young children. Acknowledgments This rese...