Navigating Mathematical Transitions: Students' Adjustments to Fundamental Changes in Cirriculum and Pedagogy

导航数学转换：学生对课程和教学法基本变化的调整首席研究员：杰克·史密斯，密歇根州立大学学生可以体验数学过渡，当他们从教室里做一个角度和思考数学的说明和实践，以其他相当不同的观点占主导地位。这种期望的不连续性会给学生带来严重的挑战，他们必须理解并适应它们。数学过渡现在更有可能考虑到由NCTM标准启发的课程和教学实践的不平衡实施。搜索和检查这些转变的两个重要地方是入门代数（初中和高中）和微积分。这些内容领域的课程材料和它们所建议的教学实践可以根据以下因素而发生巨大变化：（1）学习的基本目标和过程（方程式及符号处理程序与函数、多重表示法及每种表示法的解）;（2）向学生提出的典型问题（操纵符号的命令与上下文中有关变量的问题）;（3）课堂学习的组织（教师演示和个人解决问题与这些结合小组工作和全班讨论）。 该项目将开发数学转换的工作概念化（即，主要因素以及它们如何相互作用），并分析高中生和大学生通过它们的情况。三个主要问题将被解决：（1）成功的（和不成功的）数学过渡的特征是什么？(2)学生如何驾驭它们？以及（3）什么样的资源（学生的“内部”和“外部”）支持更成功的过渡？四所学校（两所高中和两所大学）将参加。在两个地点（1所高中和1所大学），课程和教学实践从方程及其解决方案的研究，许多问题的实践，个人研究和教师演示转向多种表征，口头和数字“答案”的功能研究，在上下文问题上的工作，以及小组和整个小组的工作。在其他两个地点，这些条件相反。该项目将选择100名学生（每个站点25名），并通过2年以上的数学跟踪他们的进步。它将评估和分析他们新兴的职业和教育目标，数学成绩，内容学习，对数学和自己作为学习者的信念，日常经验和对工作的反应，以及适应不同期望的策略的变化。该项目将产生一个综合的画像的经验和学习的整个样本和一组有代表性的详细的案例研究，以阐明学生的经验的动态。它将试图向教师、家长和管理人员以及研究人员展示其研究结果。

Navigating Mathematical Transitions: Students' Adjustments to Fundamental Changes in Curriculum and Pedagogy Principal Investigator: Jack Smith, Michigan State University Students can experience mathematical transitions when they move from classrooms where one perspective on doing and thinking about mathematics is illustrated and practiced to others where quite different perspectives dominate. Such discontinuities in expectations can seriously challenges students, who must make sense of and adjust to them. Mathematical transitions are now more likely given the uneven implementation of curricula and teaching practices inspired by the NCTM Standards. Two important places to search for and examine these transitions are introductory algebra (in middle and high school) and calculus. Curricular materials in these content areas and the teaching practices suggested by them can vary dramatically according to: (1) the fundamental objects and processes of study (equations and symbol manipulation procedures vs. functions, multiple representations, and solutions in each representation); (2) the typical problems presented to students (commands to manipulate symbols vs. questions about varying quantities in context); and (3) the organization of classroom learning (teacher presentation and individual problem-solving vs. these in combination with small group work and whole class discussion). This project will develop a working conceptualization of mathematical transitions (i.e., major factors and how they interact) and analyze high school and college students' passage through them. Three main questions will be addressed: (1) What are the characteristics of successful (and unsuccessful) mathematical transitions?; (2) How do students navigate them?; and (3) What kinds of resources ("internal" and "external" to the student) support more successful transitions? Four school sites (two high schools and two universities) will participate. At two sites (1 high school and 1 college, curricula and teaching practices shift from the study of equations and their solution, practice with many problems, individual study, and teacher presentation to the study of functions in multiple representations, verbal and numerical "answers," work on contextualized problems, and small group and whole group work. At the other two sites, these conditions are reversed. The project will select 100 students (25 per site) and track their progress through 2+ years of mathematics. It will assess and analyze the change in their emerging career and educational goals, mathematics achievement, content learning, beliefs about mathematics and themselves as learners, daily experience and reactions to their work, and strategies for adjusting to different expectations. The project will produce a composite portrait of the experience and learning of the entire sample and a representative set of detailed case studies to illuminate the dynamics of students' experience. It will attempt to present its findings to teachers, parents, and administrators, as well as researchers.