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

教育的一个核心和持久的目标是提供超越最初学习的特定条件的学习经验。然而，研究研究和国家数学评估表明，学生在实际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生活联系起来。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他们的学习经验的尝试没有像预期的那样成功的原因之一可能是因为现有的账户没有充分地考虑到概括活动的社会和个人方面。聚焦现象的概念是为了解释社会环境的特征如何影响学生在数学上注意什么，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&amp；Ellis，2002b；Lobato，Ellis，&amp；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，它会有规律地将学生的注意力引导到某些(数学)性质或规律上。探索性比较工作(Ellis&Amp；Lobato，2004)表明，可以生成一种简档，它指的是一组对比鲜明的聚焦现象(跨越各种教学方法)，在概念上与学生对给定主题的一组相关概括联系在一起。在拟议的研究中，以比探索性工作更严格和更系统的方式开发了轮廓，最终开发了预测模型，该模型将描述聚焦现象的性质变化如何影响相关个体概括的性质。为了实现这一目标，采用由一组综合研究组成的复杂研究设计来调查三个研究问题：1.概况。学生如何在一系列教学处理中概括他们的学习经验(在以行动者为导向的迁移视角下的“迁移”)，这些方法在概念上与课堂上形成的各种聚焦现象有关？2.层次之间的联系。(A)教师的教学知识和教学目标，(B)课堂上出现的聚焦现象，以及(C)学生在课堂上概括学习经验的方式之间有什么关系？3.概括的内容和特点。学生关于线性函数和二次函数所构造的推广的数学内容和性质是什么？随着时间的推移，学生发展出更复杂、更强大的概括的轨迹是什么？智力上的价值。这项研究的智力价值在于发展了概念关系的轮廓，连接了教师的知识，关注了课堂上的现象，以及相关的学生概括。这些配置文件提供了一个有用的对比，对比了与富有成效的学生概括相关的焦点类型，以及那些无意中提供了不太有力的学生概括的焦点类型。这项研究还通过进一步发展聚焦现象作为一种转移机制，为正在进行的转移辩论做出贡献。最近两次由NSF资助的关于转移的会议(Lobato，2004；Mestre，2003)以及在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表明了对转移的重新兴趣。通过展示学生从教学中吸取的持久概念如何受到教学实践中许多微妙且往往是无意的方面的影响，这些简介的开发可以为教师和他们的学生带来好处。正如学生对任何给定的数学主题都有一系列的替代概念和错误概念一样，教师也可以将学生的注意力引向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。这项研究还为实施理论做出了贡献，证明了即使当课程材料良好并且满足实施忠诚度的条件时，教学的许多微观特征也可以发挥作用，从而破坏改革。这些结果有助于完成NRC报告(2004)中的挑战，即关注教学计划的关键特征、现有的教学实践以及计划评估应考虑的特定学生亚群的特征之间的复杂交互作用。

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.