This paper describes a study that used a novel method to investigate conceptual difficulties with mathematical induction among two groups of undergraduate students: students who had received university-level instruction in formal mathematical induction, and students who had not been exposed to formal mathematical induction at the university level. The method used a ‘visual proof by induction’ – a simple image that is designed to demonstrate a theorem that would be formally proven using mathematical induction – to investigate students’ conceptualizations of mathematical induction. We report two major findings: first, the majority of students who were familiar with formal mathematical induction had difficulty using the image to justify the theorem, suggesting that their knowledge of the proof method was intimately linked to the algebraic method and thus largely procedural in nature. Second, students who had not studied formal mathematical induction generally used the image as the basis of a standard inductive generalization and did not recognize that the image could be used to establish the necessity of the theorem. Surprisingly, these students often expressed conceptualizations of natural number that were inconsistent with the formal characterization that forms the basis of formal mathematical induction. Implications for education are discussed.

本文描述了一项研究，该研究使用一种新颖的方法调查了两组本科生在数学归纳法方面的概念性困难：接受过正规数学归纳法的大学级教学的学生，以及未在此课程中接受过正规数学归纳法的学生。大学水平。该方法使用了“归纳的可视证明”（一种简单的图像，旨在证明可以用数学归纳法正式证明的一个定理）来研究学生对数学归纳法的概念。我们报告了两个主要发现：首先，大多数熟悉形式数学归纳法的学生难以使用图像来证明定理是正确的，这表明他们对证明方法的了解与代数方法密切相关，因此本质上是程序性的。其次，未学习过形式数学归纳法的学生通常将图像用作标准归纳概括的基础，并且不认识到图像可用于确定定理的必要性。令人惊讶的是，这些学生经常表达自然数的概念，这与构成形式数学归纳基础的形式特征不一致。讨论了对教育的影响。尚未学习过正式数学归纳法的学生通常将图像用作标准归纳概括的基础，并且不认识到图像可用于确定定理的必要性。令人惊讶的是，这些学生经常表达自然数的概念，这与构成形式数学归纳基础的形式特征不一致。讨论了对教育的影响。尚未学习过正式数学归纳法的学生通常将图像用作标准归纳概括的基础，并且不认识到图像可用于确定定理的必要性。令人惊讶的是，这些学生经常表达自然数的概念，这与构成形式数学归纳基础的形式特征不一致。讨论了对教育的影响。