Solving inductive reasoning problems in mathematics: not-so-trivial pursuit

✍ Scribed by Lisa A. Haverty; Kenneth R. Koedinger; David Klahr; Martha W. Alibali

• Publisher: Wiley (Blackwell Publishing)
• Year: 2000
• Tongue: English
• Weight: 313 KB
• Volume: 24
• Category: Article
• ISSN: 0364-0213
• DOI: 10.1016/s0364-0213(00)00019-7

✦ Synopsis

This study investigated the cognitive processes involved in inductive reasoning. Sixteen undergraduates solved quadratic function-finding problems and provided concurrent verbal protocols. Three fundamental areas of inductive activity were identified: Data Gathering, Pattern Finding, and Hypothesis Generation. These activities are evident in three different strategies that they used to successfully find functions. In all three strategies, Pattern Finding played a critical role not previously identified in the literature. In the most common strategy, called the Pursuit strategy, participants created new quantities from x and y, detected patterns in these quantities, and expressed these patterns in terms of x. These expressions were then built into full hypotheses. The processes involved in this strategy are instantiated in an ACT-based model that simulates both successful and unsuccessful performance. The protocols and the model suggest that numerical knowledge is essential to the detection of patterns and, therefore, to higher-order problem solving.

One of his teachers, apparently eager for a respite from the day's lessons, asked the class to work quietly at their desks and add up the first hundred whole numbers. Surely this would occupy the little tykes for a good long time. Yet the teacher had barely spoken, and the other children had hardly proceeded past "1 ϩ 2 ϩ 3 ϩ 4 ϩ 5 ϭ 15" when Carl walked up and placed the answer on the teacher's desk. One imagines that the teacher registered a combination of incredulity and frustration at this unexpected turn of events, but a quick look at Gauss's answer showed it to be perfectly correct. How did he do it?