The Catalan Teaching Tool – A Learning Process Design for Supplemental Instruction and Learning Process Facilitation

ABSTRACTThis paper is based on a supplemental instruction technique aimed at disseminating a difficult concept at the undergraduate level. The software developed as a teaching tool attacks the problem from an intuitive perspective with the aid of visual tools. The tool, run as an experiment, conclusively proves that the original problem of calculating the number of paths from a point (0, 0) to any point (<i>n</i>, <i>n</i>) on a graph might be better understood. The moves can only be east and north in whole number increments and cannot exceed the <i>x</i> = <i>y</i> line at any point. A program is written to produce the graph. Incrementing the value of (<i>n</i>, <i>n</i>), the Catalan number series is produced. Another program is written using the formula which produces the Catalan numbers, and the growth of the series is determined. The teaching tool is created based on the algorithm. A robot is designed which follows the Catalan path is also demonstrated. A survey conducted using this teaching tool shows that it enhances the learning and understanding of Catalan numbers.

摘要
本文基于一项辅助教学技术，旨在面向本科阶段群体传授晦涩难懂的专业概念。本文开发的教学工具软件，借助可视化手段从直观视角切入该问题。该工具作为实验载体运行后，确凿证明：针对从坐标系中点(0,0)到任意点(n,n)的路径条数计算这一经典问题，借助本工具可获得更清晰的理解。此类路径仅能沿东向与北向以整数步长移动，且全程不得越过直线x=y。
本研究编写了一款程序以生成该类图形；逐步递增点(n,n)的坐标值，即可得到卡特兰数（Catalan number）序列。另一款程序则基于卡特兰数的通项公式编写完成，借此可分析该序列的增长特性。本教学工具基于上述算法开发完成，此外还展示了一款可沿卡特兰路径行进的机器人设计。基于该教学工具开展的问卷调查结果显示，其可有效提升学习者对卡特兰数的掌握与理解程度。