The Real Story behind Story Problems Effects of Representations on Quantitative Reasoning
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The Real Story behind Story Problems Effects of Representations on Quantitative Reasoning
1Story behind Story ProblemsThe Real Story behind Story Problems:Effects of Representations on Quantitative ReasoningKenneth R. Koedinger*Carnegie Mel The Real Story behind Story Problems Effects of Representations on Quantitative Reasoningllon UniversityMitchell J. NathanUniversity of ColoradoRUNNING HEAD: Story behind Story Problems^Contact Information:Ken KoedingerHuman-Computer Interaction InstituteCarnegie Mellon UniversityPittsburgh. PA 15213Phone: 412-268-7667Story behind Story ProblemsABSTRACTWe explored how differences in pro The Real Story behind Story Problems Effects of Representations on Quantitative Reasoningblem representations change both the performance and underlying cognitive processes of beginning algebra students engaged in quantitative reasoning. CThe Real Story behind Story Problems Effects of Representations on Quantitative Reasoning
ontrary to beliefs held by practitioners and researchers in mathematics education, we found that students were more successful solving simple algebra 1Story behind Story ProblemsThe Real Story behind Story Problems:Effects of Representations on Quantitative ReasoningKenneth R. Koedinger*Carnegie Mel The Real Story behind Story Problems Effects of Representations on Quantitative Reasoning of situated world knowledge facilitating problem solving performance, but rather a consequence of student difficulties with comprehending the formal symbolic representation of quantitative relations. We draw on analyses of students' strategics and errors as the basis for a cognitive process explana The Real Story behind Story Problems Effects of Representations on Quantitative Reasoningtion of when. why. and how differences in problem representation affect problem solving. In general, we conclude that differences in external represenThe Real Story behind Story Problems Effects of Representations on Quantitative Reasoning
tations can affect performance and learning when one representation is easier to comprehend than another or when one representation elicits more relia1Story behind Story ProblemsThe Real Story behind Story Problems:Effects of Representations on Quantitative ReasoningKenneth R. Koedinger*Carnegie Mel The Real Story behind Story Problems Effects of Representations on Quantitative Reasoningmplex skill acquisitionStory behind Story' Problems4INTRODUCTIONStory Problems Arc Believed to Be DifficultA commonly held belief about story problems at both the arithmetic and algebra levels is that they are notoriously difficult for Students. Support for this belief can be seen among a variety of The Real Story behind Story Problems Effects of Representations on Quantitative Reasoning populations including rhe general public, textbook authors, teachers, mathematics education researchers, and learning science researchers. For evidenThe Real Story behind Story Problems Effects of Representations on Quantitative Reasoning
ce that this belief is commonly held within the general public, ask your neighbor. More likely than not. he or she will express a sentiment toward sto1Story behind Story ProblemsThe Real Story behind Story Problems:Effects of Representations on Quantitative ReasoningKenneth R. Koedinger*Carnegie Mel The Real Story behind Story Problems Effects of Representations on Quantitative Reasoning Story Problems." and "Story Problems Galore." That many textbook authors believe in the greater difficulty of story problems is supported by an analysis of textbooks by Nathan. Long, and Alibali (2002). In 9 of the 10 textbooks they analyzed, new topics are initially presented through symbolic acti The Real Story behind Story Problems Effects of Representations on Quantitative Reasoningvities and only later are story problems presented, often as ••challenge problems". The choice of this ordering is consistent with the belief that symThe Real Story behind Story Problems Effects of Representations on Quantitative Reasoning
bolic representations are more accessible to students than story problems.More direct evidence of the common belief in the difficulty of story problem1Story behind Story ProblemsThe Real Story behind Story Problems:Effects of Representations on Quantitative ReasoningKenneth R. Koedinger*Carnegie Mel The Real Story behind Story Problems Effects of Representations on Quantitative Reasoningmost predicted that story problems would be harder for algebra students than marched equations. Nathan & Koedingcr (2000a) also surveyed 35 mathematics education researchers. The majority of these researchers also predicted that story problems would be harder for algebra students than matched equati The Real Story behind Story Problems Effects of Representations on Quantitative Reasoningons. In another study of 105 K-12 mathematics teachers. Nathan & Koedinger (2000b) found that significantly more teachers agree than disagree w ith stThe Real Story behind Story Problems Effects of Representations on Quantitative Reasoning
atements like"Solving math problems presented in words should be taught only after students master solving the sameStory behind Story Problems5problem1Story behind Story ProblemsThe Real Story behind Story Problems:Effects of Representations on Quantitative ReasoningKenneth R. Koedinger*Carnegie Mel The Real Story behind Story Problems Effects of Representations on Quantitative Reasoningy problems is also reflected in the learning science literature. Research on story problem solving, at both the arithmetic (Carpenter, Kepner. Corbin. Lindquist. & Rcys. 1980; Cummins et al.. 1988: Kintsch & Greeno. 1985) and algebra levels (Clement. 1982; Nathan et al.. 1992; Paige & Simon. I960), The Real Story behind Story Problems Effects of Representations on Quantitative Reasoninghas emphasized the difficulty of such problems. For instance. Cummins and her colleagues (1988. p. 405) commented "word problems arc notoriously diffiThe Real Story behind Story Problems Effects of Representations on Quantitative Reasoning
cult to solve". They investigated first graders' performance on matched problems in story and numeric format for 18 different categories of one operat1Story behind Story ProblemsThe Real Story behind Story Problems:Effects of Representations on Quantitative ReasoningKenneth R. Koedinger*Carnegie Mel The Real Story behind Story Problems Effects of Representations on Quantitative Reasonings does John have less than Mary?" but were 100% correct on the matched numeric format problem. "6 - 2 = ?". They found performance on story problems was worse than performance on matched problems in numeric format for 14 of the 18 categories and was equivalent for the remaining 4 categories. Belief The Real Story behind Story Problems Effects of Representations on Quantitative Reasoningin the greater difficulty of story problems is also evident in the broader developmental literature. For instance. Geary (1994. p. 96) states "childreThe Real Story behind Story Problems Effects of Representations on Quantitative Reasoning
n make more ciTors when solving word problems than when solving comparable number problems."Although the research that Cummins and others (1988) perfo1Story behind Story ProblemsThe Real Story behind Story Problems:Effects of Representations on Quantitative ReasoningKenneth R. Koedinger*Carnegie Mel The Real Story behind Story Problems Effects of Representations on Quantitative Reasoningisticated domains, they continue to find word problems in those domains more difficult to solve than problems presented in symbolic format (e.g.. algebraic equations)" (p. 405). However, apart from our own studies reported here, this broader claim appears to have remained untested (cf.. Reed. 1998). The Real Story behind Story Problems Effects of Representations on Quantitative Reasoning We have not found prior experimental comparisons of solution correctness on matched algebra story problems and equations for students learning algebrThe Real Story behind Story Problems Effects of Representations on Quantitative Reasoning
a. In a related study.Mayer (1982a) used solution times to make inferences about the different strategics that well-preparedStory behind Story Problem1Story behind Story ProblemsThe Real Story behind Story Problems:Effects of Representations on Quantitative ReasoningKenneth R. Koedinger*Carnegie Mel The Real Story behind Story Problems Effects of Representations on Quantitative Reasoningems varied in complexity and accounted for these differences by the hypothesis that students use a goal-based “isolate" strategy on equations and a less memory-intensive "reduce" strategy on word problems. Overall, students took significantly longer to solve 1-5 step word problems (about 15 seconds) The Real Story behind Story Problems Effects of Representations on Quantitative Reasoning than matched equations (about 5 seconds) with no reliable difference in number of errors (7% for word problems. 4% for equations). Whereas Mayer's stThe Real Story behind Story Problems Effects of Representations on Quantitative Reasoning
udy focused on timing differences for well-practiced participants, the studies reported here focus on error differences for beginning algebra students1Story behind Story ProblemsThe Real Story behind Story Problems:Effects of Representations on Quantitative ReasoningKenneth R. Koedinger*Carnegie Mel1Story behind Story ProblemsThe Real Story behind Story Problems:Effects of Representations on Quantitative ReasoningKenneth R. Koedinger*Carnegie MelGọi ngay
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