CAREER: Fostering "Mathematical Practices" In Urban Classrooms: Focus On Representation And Justification

数学实践的学习，即成功的数学学习者在做数学时所参与的具体过程，如证明和各种表示的使用，对于获得高数学成绩和了解数学理论一样重要。然而，这些实践在学校数学中的作用充其量只是外围的。许多学生的数学经验仅限于记忆和实践事实和程序，很少强调这些实践的发展。在多样化的城市环境中尤其如此，那里的学生传统上数学表现不佳。然而，最近的改革努力正在呼吁数学教学的转变，以促进所有学生这种实践的发展。如果这些练习要成为学校数学课程中有意义的一部分，我们必须提高我们对学生这些练习的发展以及促进这种发展的条件的理解。虽然研究为我们提供了一些有价值的见解，但对于这些实践如何作为一个整体和随着时间的推移而发展，特别是在具有挑战性的教学环境中，我们知之甚少。我提议进行一项研究来解决这个问题。本研究将追踪中学生表征和辩护数学实践能力的发展，并找出影响这种发展的条件。我将首先对大量城市学生的数学实践概念进行广泛的分析。随后将使用各种仪器和课堂观察，对六年级至八年级学生的发展进行重点和详细的分析。目标是超越当前条件的记录，并追踪和理解学生能力随时间的变化。此外，本研究将分析影响这些变化的课堂教学实践。为了实现这一目标，学生的教师将参与并行的专业发展活动，旨在提高他们对中学生数学推理发展和促进这种发展的课堂实践的理解，并将与学生并行研究。知识价值。拟议的工作呼吁在新兴领域开展新的研究项目，包括那些关注数学实践发展的领域。该研究将产生一个概念框架，将数学学习的社会和认知视角与城市学校背景和中学生数学实践发展（不仅仅是技能和事实）领域急需的工作协调起来，以及如何为支持这一点的教师提供专业发展。其结果将是更深入地了解支持学生在城市数学课程中学习的教学法形式和内容。更广泛的影响。我希望拟议的工作能够在中学数学的学习和教学中开辟新的途径，特别是在数学实践的教学和学习中，以与数学教育界整体进步趋势产生共鸣的方式。更好地理解与学生对这些实践形成概念相关的过程，可以为课程开发和教学提出建议。本研究将建议如何在中学数学中更有效地纳入表征和证明的教学，并将提供一些关于在更高水平上抑制有效数学推理发展的认知和社会障碍的本质的见解。拟议的研究设置位于一个主要的城市地区，那里有不同的学生群体，传统的数学教学并没有证明是富有成效的。因此，了解如何培养数学上成功的学生，以及如何在此背景下支持教师，可以支持不同学生群体的学习。调查结果将广泛传播，以增进对该领域的了解，并协助教师开展日常活动。总的来说，该项目的活动将直接影响到纽约多样化的城市社区，并将指导任职前教师和研究生，他们是代表性不足的群体的成员。Finbarr C. Sloane博士。项目主任REC: ROLE & IERI

The learning of mathematical practices, that is, the specific processes that successful mathematics learners engage in when doing mathematics, such as the use of justifications and various representations, is as important in achieving high performance in mathematics as it is to know the theory of mathematics. Yet, the role of these practices in school mathematics has been peripheral at best. Many students' experience with mathematics is limited to the memorization and practice of facts and procedures and little emphasis is placed on the development of these practices. This is particularly true in diverse, urban environments where students have been traditionally under-performing in mathematics. However, recent reform efforts are calling for a shift of mathematics instruction to foster the development of such practices in all students. If these practices are to become a meaningful part of school mathematics programs, we must advance our understanding of students' development of these practices and the conditions that promote that development. While research has provided us with valuable insights on several of these practices, there has been little understanding of how these practices develop as a whole and over time especially in challenging instructional settings.I am proposing a study that will address this topic. This study will trace the development of middle school students' competencies in the mathematical practices of representation and justification and will identify conditions that influence that development. I will begin first with a broad analysis of a large number of urban students' conceptions of mathematical practices. This will be followed by a focused and detailed analysis of the development of 6th to 8th grade students, using a variety of instruments and classroom observations. The goal is to go beyond documentation of the current conditions, and to trace and understand changes in students' competencies over time. Furthermore, the study will analyze classroom-teaching practices that influence these changes. To achieve this, the students' teachers will participate in concurrent professional development activities designed to enhance their understandings of middle school students' development of mathematical reasoning and classroom practices that promote that development, and will also be studied in parallel with their students.Intellectual Merit. The proposed work addresses calls for new research programs in emerging areas, including those that attend to the development of mathematical practices. The study will lead to conceptual frameworks that coordinate social and cognitive perspectives on mathematics learning with much-needed work in the urban schools' context and in the area of middle school students' development of mathematical practices (and not only skills and facts) and how to engage in professional development for teachers that supports this. The result will be deeper insight into the forms of pedagogy and content that support student learning in urban mathematics programs.Broader Impacts. I expect the proposed work to open new avenues in the learning and teaching of middle school mathematics, in particular, in the teaching and learning of mathematical practices, in ways, which resonate with overall progressive trends in the mathematics education community. A better understanding of the processes associated with students' developing conceptions of these practices can lead to recommendations for curriculum development and instruction. This study will suggest how instruction on representation and justification might be more effectively incorporated in middle school mathematics and will provide some insights on the nature of the cognitive and social obstacles that inhibit the development of efficient mathematical reasoning in more advanced levels. The proposed research setting is situated in a major urban area with a diverse student population for whom traditional mathematics instruction has not proven fruitful. Thus, understanding how to build mathematically successful students and how to support teachers within this context can support learning across multiple and diverse student populations. Findings will be broadly disseminated to enhance the field's understanding and to assist teachers in their everyday activities. Overall, the activities of the project will directly impact the diverse urban community of New York and will mentor pre-service teachers and graduate students who are members of underrepresented groups.Finbarr C. Sloane Ph.D.Program Director REC: ROLE & IERI