Abstract

Mathematics undergraduates often encounter a variety of digital representations which are more idiosyncratic than the ones they have experienced in school and which often require the use of more sophisticated digital tools. This article analyses a collection of digital representations common to undergraduate dynamical systems courses, considers the significant ways in which the representations are interconnected and examines how they are similar or differ from those students are likely to have experienced at school. A key approach in the analysis is the identification of mathematical objects corresponding to manipulative elements of the representations that are most essential for typical exploratory tasks. As a result of the analysis, augmentations of familiar representations are proposed that address the gap between local and global perspectives, and a case is made for greater use of isoperiodic diagrams. In particular, these diagrams are proposed as a new stimulus for students to generate their own explorations of fundamental properties of the Mandelbrot set. The ideas presented are expected to inform the practice of teachers seeking to develop visually rich exploratory tasks which pre-empt some of the issues of instrumentation that mathematics undergraduates experience when introduced to new digital tools. The overarching aim is to address significant questions concerning visualization and inscriptions in mathematics education.

中文翻译：

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通往Mandelbrot集的另一种方法：连接本科生的特殊数字表示形式

摘要

数学专业的本科生经常遇到各种各样的数字表示形式，这些数字表示形式比他们在学校经历的数字表示形式更特殊，并且通常需要使用更复杂的数字工具。本文分析了本科生动力学系统课程共有的数字表示形式，研究了表示形式相互联系的重要方式，并研究了它们与那些学生可能在学校经历过的相似或不同之处。分析中的关键方法是识别与表示形式的操作元素相对应的数学对象，这些元素对于典型的探索性任务而言最重要。分析的结果是，建议增加熟悉的表示形式，以解决本地和全球视角之间的差距，并提出了更多使用等轴图的理由。尤其是，这些图被提议作为一种新的激励方式，供学生生成自己对Mandelbrot集的基本属性的探索。预期提出的想法将为寻求开发视觉上丰富的探索性任务的教师提供指导，这些任务可避免数学大学生在引入新的数字工具时所遇到的一些仪器仪表问题。总体目标是解决与数学教育中的可视化和题字有关的重大问题。预期提出的想法将为寻求开发视觉上丰富的探索性任务的教师提供指导，这些任务可避免数学大学生在引入新的数字工具时遇到的一些仪器仪表问题。总体目标是解决与数学教育中的可视化和题字有关的重大问题。预期提出的想法将为寻求开发视觉上丰富的探索性任务的教师提供指导，这些任务可避免数学大学生在引入新的数字工具时所遇到的一些仪器仪表问题。总体目标是解决与数学教育中的可视化和题字有关的重大问题。