Seeing language learning inside the math: Cognitive analysis yields transfer
Seeing language learning inside the math: Cognitive analysis yields transfer
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从数学中看到语言学习：认知分析产生迁移
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发表时间：
2010
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通讯作者：
Elizabeth Mclaughlin
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作者：
K. Koedinger;Elizabeth Mclaughlin
从数学中审视语言学习：认知分析产生迁移
肯尼斯·R·科丁格（koedinger@cmu.edu）
卡内基梅隆大学人机交互研究所，美国宾夕法尼亚州匹兹堡福布斯大道5000号，邮编15217
伊丽莎白·A·麦克劳克林（mimim@cs.cmu.edu）
卡内基梅隆大学人机交互研究所，美国宾夕法尼亚州匹兹堡福布斯大道5000号，邮编15217
（赫弗南和科丁格，1997；1998）。表1展示了符号化问题的示例，这些问题要求学生将一个应用题转化为一个代数表达式。语言学习过程与这项任务之间明显的潜在联系在于学习阅读和理解应用题。虽然这种学习对小学生来说确实是一个重大挑战（卡明斯、金茨克、罗伊瑟和魏默，1988），但我们过去的数据提供的证据表明，理解应用题对大多数初学代数的学生来说不再是一个主要的障碍。这一说法可以通过与外语翻译的类比来说明：将一个应用题翻译为代数就像将英语翻译为希腊语。对于一个说英语的人来说，翻译成希腊语的困难不在于理解英语，而在于生成希腊语。同样，在初级代数课程中，年龄较大的学生面临的挑战更多地在于能够用代数方式表达理解，即使用代数语言，而不是理解应用题所使用的英语。
有一个迹象表明，对代数应用题的理解对初学代数的学生来说不是一个主要障碍，这来自赫弗南和科丁格（1998）的数据，数据显示学生能够更准确地解决应用题（当给定自变量或“x”变量的值时，求出因变量或“y”变量的值）（63%正确），而他们将一个应用题符号化（写出一个关于x和y的方程）的正确率则低得多（18%正确）。由于解决问题需要理解题目，这种表现差异表明，符号化对学生来说存在问题，其困难超出了句子理解的要求。第二个迹象与被编程用来解决应用题的人工智能系统所遇到的困难形成对比，即理解题目中所描述的数量之间的算术关系（鲍勃罗，1968）。我们设计了一些问题，在这些问题中，对这种关系的自然隐含描述（例如，“林德奎斯特女士教62个女孩。林德奎斯特女士教b个男孩。”）被补充（赫弗南和科丁格，1997）或替换（科丁格、阿里巴利和内森，2008）为明确描述（例如，“林德奎斯特女士教的学生数量等于男孩数量加上女孩数量。”），这对程序来说更容易处理。然而，我们发现提供这样的明确描述并没有
摘要
实现和理解学习的有效迁移需要对要迁移的隐性知识和技能进行仔细分析。我们展示了一个实验，该实验测试了这种分析的一个微妙预测。实验得出结论，学生学习将代数应用题转化为符号表达式的一个关键困难在于学习这些表达式的语法。我们假设，要求学生将一个代数表达式代入另一个表达式的练习将增强学生的代数语法知识。这个假设导致了一个反直觉的预测，即通过练习看起来不同的代入练习，而不是通过练习看起来更相似的应用题，可能更好地增强将应用题符号化的学习。我们报告了一项涉及303名中学生的实验比较，该比较支持这一预测。我们讨论了让学习者将一种统一的抽象形式外化并获得关于它的互动反馈可能是增强迁移的重要因素。
关键词：认知任务分析；迁移；语法学习；数学教育
引言
人类在拥有一种用于学习的语言之前就开始学习语言。使这种惊人壮举成为可能的学习过程，比如在没有明确指导的情况下通过经验学习语法结构的能力，是否对其他类型的学习任务有用呢？一旦儿童掌握了语言，在语言学习中所使用的认知功能是否就不再有用了呢？例如，当学生学习像代数这样复杂的学术课程时，用于语言学习的大脑部分是否就毫无用处了呢？或者，在语言学习中所使用的一些相同的隐性学习机制是否可能对学习数学和科学有用呢？本文并不旨在为这些问题提供确凿的答案，然而，它确实提供了一个令人信服的证明，即语法学习过程在学习数学中可能是重要的。学生可能在没有明确意识的情况下参与这种学习，而且这种隐性学习在学术学习中可能比通常所认识到的更为普遍（例如，阿里巴利和戈尔丁 - 梅多，1993；兰迪和戈德斯通，2007）。在早期的工作中，我们对“符号化”这一重要任务领域进行了认知任务分析，即使用代数符号对问题情境或“应用题”进行建模的能力
Seeing Language Learning inside the Math: Cognitive Analysis Yields Transfer Kenneth R. Koedinger (koedinger@cmu.edu) Human-Computer Interaction Institute, Carnegie Mellon University 5000 Forbes Avenue, Pittsburgh, PA 15217 USA Elizabeth A. McLaughlin (mimim@cs.cmu.edu) Human-Computer Interaction Institute, Carnegie Mellon University 5000 Forbes Avenue, Pittsburgh, PA 15217 USA (Heffernan & Koedinger, 1997; 1998). Table 1 shows examples of symbolization problems, which ask students to translate a story problem into an algebraic expression. The obvious potential connection between language learning processes and this task is in learning to read and comprehend story problems. While such learning is indeed a significant challenge for elementary students (Cummins, Kintsch, Reusser, & Weimer, 1988), our past data provided evidence that comprehending story problems is no longer a major sticking point for most beginning algebra students. This claim can be illustrated by an analogy to foreign language translation: Translating a story problem to algebra is like translating English to Greek. For an English speaker, the difficulty in translating to Greek is not comprehending the English, but generating the Greek. Similarly, the challenge for older students in a beginning algebra course is much less in understanding the English in which the story problems are written and more in being able to express that understanding algebraically, that is, in the language of algebra. One indication that comprehension of algebra story problems is not a major sticking point for beginning algebra students comes from Heffernan and Koedinger’s (1998) data showing that students can solve story problems (produce a value for the dependent or “y” variable when a value for the independent or “x” variable is given) much more accurately (63% correct) than they can symbolize (write an equation relating x and y) a story problem (18% correct). Since solving requires comprehension of the story, the performance difference is suggestive that symbolizing is problematic for students in ways beyond the demands of sentence comprehension. A second indication presents a contrast with a difficulty experienced by Artificial Intelligence systems programmed to solve story problems, namely that of understanding the arithmetic relationships between quantities described in the story (Bobrow, 1968). We created problems where natural implicit descriptions of such relationships (e.g., “Ms. Lindquist teaches 62 girls. Ms. Lindquist teaches b boys.”) are supplemented (Heffernan & Koedinger, 1997) or replaced (Koedinger, Alibali, & Nathan, 2008) with explicit descriptions (e.g., “The number of students Ms. Lindquist teaches is equal to the number of boys plus the number of girls.”), which are much easier for a program to process. We found, however, that providing such explicit descriptions does not Abstract Achieving and understanding effective transfer of learning requires a careful analysis of the hidden knowledge and skills to be transferred. We present an experiment that tests a subtle prediction of such an analysis. It concluded that a critical difficulty in students’ learning to translate algebra story problems into symbolic expressions is in learning the grammar of such expressions. We hypothesized that exercises requiring students to substitute one algebraic expression into another would enhance students’ algebraic grammar knowledge. This hypothesis led to a counter-intuitive prediction that learning to symbolize story problems could be better enhanced through practice on dissimilar looking substitution exercises than through practice on more similar looking story problems. We report on an experimental comparison involving 303 middle school students that supports this prediction. We discuss how having learners externalize a uniform abstract form and get interactive feedback on it may be important factors in enhancing transfer. Keywords: cognitive task analysis; transfer; grammar learning; mathematics education. Introduction Humans learn language before they have a language to use to learn. Might the learning processes that make this amazing feat possible, like the capability to learn grammatical structures through experience without explicit instruction, be useful for other kinds of learning tasks? Once children have acquired language, are the cognitive functions employed in language learning no longer useful? For instance, as students take courses in complex academic topics, like algebra, does all that brain matter for language learning have nothing to do? Or is it possible that some of the same implicit learning mechanisms employed in language learning are useful for learning math and science? This paper does not aim to provide conclusive answers to these questions, however, it does provide a compelling demonstration that grammar learning processes may be important in learning mathematics. Students may engage in such learning without explicit awareness and such implicit learning may be more prevalent in academic learning than is generally recognized (e.g., Alibali & Goldin-Meadow, 1993; Landay & Goldstone, 2007). In earlier work, we performed a cognitive task analysis of the important task domain of “symbolization”, that is, the ability to model problem situations or “story problems” in algebraic symbols