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

导航数学过渡：学生对课程和教育学根本变化的调整主要调查者：密歇根州立大学的杰克·史密斯，当学生从课堂上走出来时，他们可以体验到数学过渡，因为在课堂上，一种关于做和思考数学的观点被解释和实践，而另一些观点占主导地位的观点截然不同。这种期望的不一致可能会给学生带来严重的挑战，他们必须理解并适应这些期望。考虑到受NCTM标准启发的课程和教学实践的不均衡实施，数学过渡现在更有可能发生。寻找和研究这些过渡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方面而有很大的不同：(1)学习的基本对象和过程(方程式和符号处理程序与函数、多重表示法和每种表示法中的解决方案)；(2)向学生呈现的典型问题(操纵符号的命令与上下文中关于不同数量的问题)；以及(3)课堂学习的组织(教师陈述和个人解决问题与这些结合小组工作和全班讨论)。这个项目将开发一个数学过渡(即，主要因素及其如何相互作用)的有效概念化，并分析高中生和大学生通过这些过渡的过程。将讨论三个主要问题：(1)成功的(和不成功的)数学转换的特征是什么？(2)学生如何驾驭它们？(3)什么样的资源(对学生来说是“内部的”和“外部的”)支持更成功的转换？四个学校地点(两所高中和两所大学)将参加。在两个地点(1个高中和1个大学)，课程和教学实践从学习方程及其解、多问题练习、个人学习和教师演示转变为学习多重表示中的函数、口头和数字“答案”、上下文问题的工作以及小组和整个小组工作。在其他两个地点，这些条件是相反的。该项目将选择100名学生(每个站点25名)，并通过两年的数学跟踪他们的进步。它将评估和分析他们在未来职业和教育目标、数学成绩、内容学习、对数学和作为学习者的自己的信念、日常经验和对工作的反应以及适应不同期望的策略方面的变化。该项目将制作整个样本的经验和学习的综合肖像，以及一组具有代表性的详细案例研究，以阐明学生经验的动态。它将试图将其发现展示给教师、家长、管理人员以及研究人员。

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.