算术应用题的理解 ——成功和不成功的问题解决者的比较外文翻译资料

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算术应用题的理解

——成功和不成功的问题解决者的比较

原文作者Mary Hegarty, Richard E. Mayer, and Christopher A. Monk

单位University of California, Santa Barbara

摘要：有人提出：当解决一个算术题的时候，一个不成功问题解决者的解决方案主要是基于他们从问题中挑选的数字和关键词（直接翻译策略）；而一个成功的问题解决者基于情景建构模型，并且在模型的基础上确定问题解决计划。（问题模型化战略）。这个假说在以下两个实验中得到验证。在实验1中，我们主要关注成功的问题解决者和不成功的问题解决者在文字和数字陈述方面做了比较。在实验2中，对成功和不成功的问题解决者在记忆单词的意思和文字题确切措辞进行了程度检测。

关键词：问题解决者； 解决计划； 问题模型化策略；直接翻译策略

为什么有些学生能成功解决应用题而另一些不能？为了回答这个问题，我们从一个成熟的观察开始，幼儿园到成年期的许多学生难以解决包含关系陈述文字的算术问题，就是那些表达两个变量之间数量关系的句子。(Hegarty, Mayer, amp; Green, 1992; Lewis amp; Mayer, 1987; Riley, Greeno, amp; Heller, 1983; Verschaffel, De Corte, amp; Pauwels, 1992)。例如，附录A显示了一个成功的和不成功的解决者关于两个商店直接的价格声明的两步骤解决方案。我们将此作为不一致问题的视角，因为关系的关键字（如“少”）预示了一个不恰当的算术运算（减法而不是加法），而在一致的问题中，第二个问题中陈述的关系词预示了所需的运算（例如，“多”预示着加法）。有相当一部分大学生可以被称为不成功的问题解决者。他们用错误的运算操作不一致问题，但执行正确的一致问题(Hegarty et al., 1992; Lewis, 1989; Lewis amp; Mayer, 1987; Verschaffel et al., 1992)。我们认为这一发现的证据表明，问题的理解过程在算术应用题的解决方案中扮演重要的角色。

在这篇文章中，我们比较了在不一致问题中犯错误和未饭错误的问题解决者的阅读进程，并且我们把这两组当做成功的问题解决者和不成功的问题解决者的对照。我们假设，当面对一个算术的故事类问题，不成功的问题解决者会从选择问题中的数字和关键字开始，并且将他们的解决计划建立在在这些——我们称之为直接翻译策略的一个过程。相比之下，我们假设一个成功的问题解决者会从试图构建一个描述问题中情景的中心模型，并且计划把他们的问题解决基于模型之上——我们称之为问题模型策略的一个过程。我们的目标是检验假设——不成功的问题解决者更倾向于使用直接翻译策略，而成功的解决问题更有可能使用问题模型策略。我们承认，直接翻译的策略可能是不成功解决问题的原因之一，而问题模型策略可能是成功解决问题的原因之一。

基本原理

数学问题解决的领域正成长为一个令人兴奋的，让研究人员进行对问题解决的认知研究领域(Campbell, 1992; Mayer, 1989, 1992; Schoenfeld, 1985, 1987)。虽然，创造问题解决的一半理论——基于一般问题解决的启发式方法——是上世纪70年代的主要目标(Newell amp; Simon, 1972)，更多最近的专业研究指出，特定知识领域和一个完整地解决问题的过程起至关重要的作用(Chi, Glaser, amp; Farr, 1988; Ericsson amp; Smith, 1991; Smith, 1991; Sternberg amp; Frensch, 1991)。我们的目标是提供一种由成功和不成功的问题解决者在解决代数问题时的过程中发展出来的领域专门性策略的描述，以及这些策略如何导致个体表现差异。

除了其理论意义，由于在科学和数学教育在美国的地位，数学问题解决这个话题具有重要的实际意义。尽管提高学生的数学类问题解决能力的研究是六大国家教育目标之一，但是有令人不安的证据表明，美国学生目前跟不上其他工业化的同龄人(Lapointe, Mead, amp; Phillips, 1989; McKnight et al., 1987; Robitaille amp; Garden, 1989)。比如，在一个综合性研究中，美国样本中表现最好班级的学生取得了比日本样本中分数最差班级更低的分数，并且进一步的研究显示，美国学生在数学问题解决上表现尤其差(Stevenson amp; Stigler, 1992; Stigler, Lee, amp; Stevenson, 1990。关于成功的问题解决者如何理解应用题的基础研究可以有利于这个国家层面问题的解决。

在建立问题解决的认知理论的时候，区分卷入问题重述建构进程和卷入问题解决的进程(Mayer, 1992)。数学学习的认知研究有时候强调问题解决的过程，比如计算过程和问题解决方法(Anderson, 1983; Siegler amp; Jenkins, 1989)。我们在现今研究中一个同样重要的目标就是开发一个问题解决者理解问题，也就是建立问题重述的方法途径(Hinsley, Hayes, amp; Simon, 1977; Kintsch amp; Greeno, 1985; Mayer, 1982; Nathan, Kintsch, amp; Young, 1992; Reed, 1987)。通过集中注意在理解进程，我们没有理由希望减少其他数学问题解决中的认知技巧所占角色的分量（比如计算过程）。我们学习问题理解过程的动机源于越来越多的证据表明大多数问题解决者在建立一个有用的问题理解模型上遇到的困难比构造关键解决还多(Cardelle-Elawar, 1992; Cummins, Kintsch, Reusser, amp; Weimer, 1988; Dossey, Mullis, Lindquist, amp; Chambers, 1988; Robitaille amp; Garden, 1989; Stern, 1993)。

在这个研究中采用的特殊途径是为了观察那些我们假设具备基本数学问题解决基本的学生们的理解进程以及那些拥有被高度训练过技能的大学生。这个方法容许我们专注于问题理解中的根本困难，即那些不能归属于拙劣的计算技能、缺少基本阅读理解技巧以及缺啥基本知识或者对应用题不熟悉所造成的困难。通过学习问题理解中一个即使在成人人口中都坚挺的困难，我们可以确定，当儿童首先学到问题解决技巧时，认知过程可能需要更过的注意力。所以，虽然我们的参与对象是成年人，但我们的理论可能对年轻的学习者来说也一样关键。比如，Riley(1983)和Verschaffel(1992)已经建立了小学生在解决包含关联性陈述的应用题的第一步遇到更大困难。

我们方法的另一个独特的特点时，我们检测学生制定数学问题的解决计划时眼睛的定位，这容许我们获得对理解进程的内在属性的深刻见解(De Corte, Verschaffel, amp; Pauwels, 1990; Hegarty et al., 1992; Littlefield amp; Rieser, 1993; Verschaffel et al., 1992)。

应用题中两个理解战略

我们对比以往研究者提出的理解数学应用题两个主要的途径(Hegarty以及其他人, 1992)：一个走捷径的方法和一个基于复杂问题模型的有意义方法。这个走捷径的方法，我们指的是直接翻译，问题解决者试图选择问题中的数字和关键的关系醒词语（比如“多”和“少”），并且发展一个包含了复合问题中由关键词发动，用于数学计算的数字的解决方案（举例来说，如果关键词是“多”就用加法，如果关键词是“少”就用剪法）。如此，问题解决者试图把问题陈述关键命题直接翻译成一系列能带来答案的运算，并且不去建立一个问题中的情景描述的有效代表。

有意义方法，我们指的是问题模型方法，问题解决者把问题陈述翻译成一个描述问题中情景的中心模型。这个问题模型与这样宁可基于目标相似性的表述而不是基于命题陈述的文本区分开。这个中心模型后来成为建立解决计划的基础。

直接翻译策略在一些文献调查中和不太成功的问题解决者的解决法类似。直接翻译也被叫做“先计算后思考” (Stigler et al., 1990, p. 15)、关键词方法(Briars amp; Larkin, 1984)、数字挖掘(Littlefield amp; Rieser, 1993)。专家新手研究揭示了初学者更有可能把注意力集中在计算一个定量的答案而非应用题（例如在物理中），然而专家们最初更倾向于把依赖定性的理解放在寻找解决问题中的定量术语之前(Chi et al., 1988; Smith, 1991; Steinberg amp; Frensch, 1991)。同样的，在数学问题上的跨国研究表明，用直接翻译策略可能要为美国儿童相对于日本儿童的糟糕表现负责。美国儿童比日本儿童更可能从事走捷径的方法而非应用题，并且美国学校的教导比日本学校更可能会强调计算准确的数值答案，而不是理解问题(Stevenson amp; Stigler, 1992-Stigler et al., 1990)。

直接翻译法制造了工作记忆中的最低需求，并且他并不取决于问题类型的广博知识。然而，当情景中描述的隐性信息是解决的关键时，直接翻译会致使不正确的答案，因为使用直接翻译法的学生未能描绘这个情景。在这篇文章里，我们提出直接翻译方法占两步比较问题错误的一定比例，如附录A所示。

外文文献出处：Journal of Educational Psychology 1995, Vol.87, No. 1, 18-32

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Comprehension of Arithmetic Word Problems: A Comparison of Successful and Unsuccessful Problem Solvers

Mary Hegarty, Richard E. Mayer, and Christopher A. Monk

University of California, Santa Barbara

Abstract：It is proposed that when solving an arithmetic word problem, unsuccessful problem solvers base their solution plan on numbers and keywords that they select from the problem (the direct translation strategy), whereas successful problem solvers construct a model of the situation described in the problem and base their solution plan on this model (the problemmodel strategy). Evidence for this hypothesis was obtained in 2 experiments. In Experiment 1, the eye fixations of successful and unsuccessful problem solvers on words and numbers in the problem statement were compared. In Experiment 2, the degree to which successful and unsuccessful problem solvers remember the meaning and exact wording of word problems was examined.

Why are some students successful in solving word problems whereas others are unsuccessful? To help answer this question, we begin with the well-established observation that many students from kindergarten through adulthood have difficulty in solving arithmetic word problems that contain relational statements, that is, sentences that express a numerical relation between two variables (Hegarty, Mayer, amp; Green, 1992; Lewis amp; Mayer, 1987; Riley, Greeno, amp; Heller, 1983; Verschaffel, De Corte, amp; Pauwels, 1992). For example, Appendix A shows a successful and an unsuccessful solution to a two-step word problem containing a relational statement about the price of butter at two stores. We refer to this as an inconsistent version of the problem because the relational keyword (e.g., 'less') primes an inappropriate arithmetic operation (subtraction rather than addition), whereas in a consistent problem, the relational term in the second problem statement primes the required arithmetic operation (e.g., 'more' when the required operation is addition). A substantial proportion of college students, who could be called unsuccessful problem solvers, use the wrong arithmetic operation on inconsistent problems but perform correctly on consistent problems (Hegarty et al., 1992; Lewis, 1989; Lewis amp; Mayer, 1987; Verschaffel et al., 1992). We interpret this finding as evidence that problem comprehension processes play an important role in the solution of arithmetic word problems.

In this article, we compare the reading comprehension processes used by problem solvers who make errors on inconsistent problems with those of problem solvers who do not make errors on inconsistent problems, and we refer to these two groups as successful and unsuccessful problem solvers, respectively.1 We hypothesize that when confronted with an arithmetic story problem, unsuccessful problem solvers begin by selecting numbers and keywords from the problem and base their solution plan on these—a procedure we call the direct-translation strategy. In contrast, we hypothesize that successful problem solvers begin by trying to construct a mental model of the situation being described in the problem and plan their solution on the basis of this model—a procedure we call the problem model strategy. Our goal is to examine the hypothesis that unsuccessful problem solvers are more likely to use a direct translation strategy whereas successful problem solvers are more likely to use a problem-model strategy. We acknowledge that the direct-translation strategy might be just one source of unsuccessful problem solving and that the problem- model strategy might be just one source of successful problem solving.

Rationale

The domain of mathematical problem solving is becoming an exciting domain for researchers conducting cognitive studies of problem solving (Campbell, 1992; Mayer, 1989, 1992; Schoenfeld, 1985, 1987). Although the creation of a general theory of problem solving—that was based on general problem-solving heuristics—was a major goal in the 1970s (Newell amp; Simon, 1972), more recent research in the study of expertise points to the crucial role of domainspecific knowledge and processes in a complete account of problem solving (Chi, Glaser, amp; Farr, 1988; Ericsson amp; Smith, 1991; Smith, 1991; Sternberg amp; Frensch, 1991). Our goal is to provide an account of the domain-specific strategies that successful and unsuccessful problem solvers develop with practice on solving arithmetic problems and of how these strategies account for individual differences in performance.

In addition to its theoretical significance, the topic of mathematical problem solving has important practical implications for the status of science and mathematics education in the United States. Although improving mathematical problem-solving skills of students is one of the six national educational goals, there is disturbing evidence that American students are currently not keeping pace with their cohorts in other industrialized nations (Lapointe, Mead, amp; Phillips, 1989; McKnight et al., 1987; Robitaille amp; Garden, 1989). For example, in one comprehensive study, students in the highest performing classroom of those sampled in the United States scored lower than students in the lowest scoring classroom of those sampled in Japan, and follow-up studies revealed that American students are particularly weak in mathematical problem-solving performance (Stevenson amp; Stigler, 1992; Stigler, Lee, amp; Stevenson, 1990). Basic research on how successful problem solvers comprehend word problems could contribute to the solution of this national problem.

In building cognitive theories of problem solving, it is useful to distinguish between processes involved in constructing a problem representation and processes involved in solving a problem (Mayer, 1992). Cognitive research in mathematics learning sometimes emphasiz

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