Coordinating Social and Individual Aspects of Generalizing Activity: A Multi-tiered 'Focusing Phenomena' Study

教育的一个中心和持久目标是提供超越最初学习的具体条件的学习经验。然而，数学研究和国家评估表明，学生在真实的世界应用中的表现往往很差，许多学生毕业时无法将学校数学与工作或日常环境联系起来。此外，由于迁移结构的理论问题，研究人员在支持学习泛化方面的进展有限。帮助学生有效地概括学习经验的尝试之所以没有预期的那么成功，原因之一可能是现有的解释没有充分考虑概括活动的社会和个人方面。聚焦现象的构建是为了解释社会环境的特征如何影响学生在数学上关注的内容，以及这些特征如何反过来影响学生概括他们的学习经验的特定方式（Lobato Ellis，2002 a，2002 b; Lobato，Ellis，Munoz，2003）。聚焦现象是课堂环境中可观察到的特征，当各种特征争夺学生的注意力时，这些特征会定期将学生的注意力引向某些（数学）属性或特征。探索性比较工作（Ellis Lobato，2004年）表明，有可能生成一个概况，这是指一组对比的聚焦现象（在各种教学处理中），在概念上与一组相关的学生对给定主题的概括联系在一起。在拟议的研究中，配置文件是在一个更严格和系统的方式比可能在探索性的工作，最终开发预测模型，将描述如何改变聚焦现象的性质影响相关的个人概括的性质。为了实现这一目标，一个复杂的研究设计，包括一组综合研究被用来调查三个研究问题：1。Profile.学生如何在一系列教学处理中概括他们的学习经验（在行动者导向迁移视角中的“迁移”）的各种方式在概念上与课堂中建立的各种聚焦现象相关？2.层之间的联系。（a）教师的内容知识和课程目标，（B）课堂上出现的聚焦现象，（c）学生在这些课堂上概括他们的学习经验的方式之间的关系是什么？3.概括的内容和特征。学生对一次函数和二次函数的概括的数学内容和特征是什么？随着时间的推移，学生们发展出更复杂、更有力的概括的轨迹是什么？智力优势。拟议中的研究的智力价值在于发展的轮廓连接教师的知识，在课堂上的现象，并与学生的概括的概念关系。这些特征提供了一个有用的对比，将与富有成效的学生概括相关的焦点类型与那些无意中提供不太强大的学生概括的焦点类型进行了对比。这项研究也有助于正在进行的转移辩论，进一步发展聚焦现象作为一种转移机制。最近两次由NSF资助的关于转移的会议（Lobato，2004; Mestre，2003）以及Schoenfeld（1999）AERA总统演讲中将转移列为下一个十年研究的六个重要研究挑战之一，都表明了对转移的新兴趣。通过展示学生从教学中获得的持久概念如何受到教学实践中许多微妙和往往是无意的方面的影响，这些概况的发展可以为教师和学生带来好处。正如学生对任何一个数学题目都有一系列的替代概念和错误概念一样，教师也可以引导学生注意一系列可能的数学焦点。对这些多重焦点保持敏感可以帮助教师识别潜在的陷阱，并提出生成性的替代方案。这项研究也有助于实施理论表明，即使当课程材料是好的，实施保真度的条件得到满足，许多微观功能的教学可以发挥作用，工作破坏改革。结果可以帮助实现NRC报告（2004年）的挑战，参加一个教学计划，现有的教学实践，和程序评估应该考虑的特定学生分组的特点之间的复杂的相互作用。

A central and enduring goal of education has been to provide learning experiences that generalize beyond the specific conditions of initial learning. However, research studies and national assessments in mathematics indicate that students often perform poorly on real world applications and many students graduate unable to connect school mathematics to work or everyday settings.Furthermore, researchers' progress in supporting the generalization of learning has been limited due to theoretical problems with the transfer construct. One reason why attempts to help students productively generalize their learning experiences have not been as successful as anticipated may be because existing accounts do not adequately account for both social and individual aspects of generalizing activity. The construct of focusing phenomena was developed to account for the ways in which features of social environments influence what students attend to mathematically and how these features in turn affect the particular ways in which students generalize their learning experiences (Lobato & Ellis, 2002a, 2002b; Lobato, Ellis, & Munoz, 2003). Focusing phenomena are observable features of classroom environments that regularly direct students' attention toward certain (mathematical) properties or regularities when a variety of features compete for students' attention. Exploratory comparative work (Ellis & Lobato, 2004) suggests that it is possible to generate a profile, which refers to a contrasting set of focusing phenomena (across a variety of instructional treatments) linked conceptually with an associated set of students' generalizations for a given topic. In the proposed study, profiles are developed in a more rigorous and systematic waythan was possible in the exploratory work, eventually developing predictive models that will describe how changing the nature of focusing phenomena affects the nature of the associated individual generalizations. To accomplish this goal, a complex research design comprised of a set of integrated studies is utilized to investigate three research questions:1. Profile. How are the various ways in which students generalize their learning experiences ("transfer" in the actor oriented transfer perspective) across a range of instructional treatments related conceptually to the various focusing phenomena that get established in the classrooms?2. Links among Tiers. What are the relationships among: (a) teachers' content knowledge and lesson goals, (b) the focusing phenomena that emerge in their classrooms, and (c) the ways in which students in these classrooms generalize their learning experiences?3. Content and Character of Generalization. What is the mathematical content and the character of the generalizations students construct about linear and quadratic functions? What are the trajectories as students develop more sophisticated, powerful generalizations over time?Intellectual Merit. The intellectual merit of the proposed study lies in the development of profiles of conceptual relationships connecting teachers' knowledge, focusing phenomena in classrooms, and associated student generalizations. The profiles provide a useful contrast of the types of foci that are related to productive student generalizations with those that unwittingly afford less powerful student generalizations. This research also contributes to the ongoing transfer debate by further developing focusing phenomena as a transfer mechanism. Renewed interest in transfer is demonstrated by two recent NSF-funded conferences on transfer (Lobato, 2004; Mestre, 2003) and by the inclusion of transfer as one of six important research challenges for the next decade of research in Schoenfeld's (1999) AERA presidential address.Broad Impact. The development of the profiles can result in benefits for teachers and their students, by demonstrating how the durable concepts that students take away from instruction are influenced by many subtle and often unintentional aspects of teaching practices. Just as there is a range of alternative conceptions and misconceptions that students construct for any given mathematical topic, there is a range of possible mathematical foci to which teachers can direct students' attention. Being sensitive to these multiple foci can help teachers identify potential traps and suggest generative alternatives. This research also contributes to implementation theory by demonstrating that even when curricular materials are good and the conditions of implementation fidelity are met, many micro-features of instruction can come into play that work to undermine reform. The results can help fulfill the challenge of the NRC report (2004) to attend to the complex interactions among key features of an instructional program, existing instructional practices, and characteristics of particular student subgroups that program evaluations should consider.