Learning Mathematics through incorrect Problems

TL;DR

This paper introduces 'incorrect problems' that challenge students to develop critical thinking and pattern recognition skills in mathematics, demonstrated through examples and a study with engineering undergraduates.

Contribution

It proposes a novel teaching approach using incorrect problems to enhance mathematical exploration and reports on a study assessing student responses to such problems.

Findings

Students showed improved critical thinking skills.

Previous mathematical skills influence problem-solving success.

Incorrect problems motivate deeper mathematical exploration.

Abstract

In this article we describe special type of mathematical problems that may help develop teaching methods that motivate students to explore patterns, formulate conjectures and find solutions without only memorizing formulas and procedures. These are problems that either their solutions do not make sense in a real life context and/or provide a contradiction during the solution process. In this article we call these problems incorrect problems. We show several examples that can be applied in undergraduate mathematics courses and provide possible ways these examples can be used to motivate critical mathematical thinking. We also discuss the results of exposing a group of 168 undergraduate engineering students to an incorrect problem in a Differential Equations course. This experience provided us with important insight on how well prepared our students are to "out of the box" thinking and on the importance of previous mathematical skills in order to master further mathematical analysis to solve such a problem.