Many worthwhile mathematical activities can be generated by an overgeneralization. Some conjectures and their verification are described, with some exploration of historical procedures. (MNS)

A set of brief reflections about different aspects of mathematical errors and their roles in education is given. The nature and variety of errors and their positive use in learning mathematics are highlighted. (MNS)

Four different conceptions of error-making from a pedagogical perspective are developed. Using errors as springboards for learning is advocated, so that better understanding of content and of the nature of mathematics results. (MNS)

This is a report of an application of the interpretation of errors and springboards in a real-life situation. If and why we need to reduce fractions is considered. (MNS)

A common mistake in adding fractions involves the confusion betwen the concepts of fraction and ratio. A coherent system to operate with ratios is developed. (MNS)

Illustrated first is the case in which a wrong procedure (with fractions) leads to a correct result. Trying to justify why it works in this case and looking for similar patterns involved interesting algebraic considerations as well as use of computers. (MNS)

A series of common misconceptions with regard to infinite sets is considered. Several notions of infinite numbers proposed by different mathematicians are compared. It is argued that so-called errors should rather be called alternative conceptions. (MNS)

An overall schema of interpretation for programming instructors is given, so that the misconceptions students develop in programming can be determined more readily. Types of language-independent and -dependent bugs, how they can be identified, and how to help students overcome them are addressed. (MNS)

The nature of arithmetic disabilities in children is discussed first in this review. Then research studies are reported, classified by cognitive and neuropsychological studies and social competency studies. Finally, suggestions for remediation are provided. (MNS)