Prior research showed that many secondary students fail to construct arguments that meet the standard of proof in mathematics. However, this research tended to use survey methods and only consider students presenting their perceived proofs in written form. The limited use of observation methods and the lack of consideration of students presenting…

The concept of "proof" has attracted considerable research attention over the past decades in part due to its indisputable importance to the discipline of mathematics and to students' learning of mathematics. Yet, the teaching and learning of proof is an instructionally arduous territory, with proof being recognized as a hard-to-teach…

Ambitious teaching is a form of teaching that requires a high level of teacher responsiveness to what students do as they actively engage with the subject matter. Thus, a teacher enacting ambitious teaching is often confronted with uncertainties about how to advance students' learning while also building on students' contributions. In…

In this article, we focus on a group of 39 prospective elementary (grades K-6) teachers who had rich experiences with proof, and we examine their ability to construct proofs and evaluate their own constructions. We claim that the combined "construction-evaluation" activity helps illuminate certain aspects of prospective teachers' and presumably…

There are currently increased efforts to make proof central to school mathematics throughout the grades. Yet, realizing this goal is challenging because it requires that students master several abilities. In this article we focus on one such ability, namely, the ability for "deductive reasoning," and we review psychological research to enhance…

A proof's potential to promote understanding and conviction is one of the main reasons for which proof is so important for students' learning of mathematics. Unless students realise the limitations of empirical arguments as methods for validating mathematical generalisations, they are unlikely to appreciate the importance of proof in mathematics.…

Although students of all levels of education face serious difficulties with proof, there is limited research knowledge about how instruction can help students overcome these difficulties. In this article, we discuss the theoretical foundation and implementation of an instructional sequence that aimed to help students begin to realize the…

This article is situated in the research domain that investigates what mathematical knowledge is useful for, and usable in, mathematics teaching. Specifically, the article contributes to the issue of understanding and describing what knowledge about proof is likely to be important for teachers to have as they engage students in the activity of…

The notion of assumptions permeates school mathematics, but instruction tends to highlight this notion only in the advanced grades. In this article, I argue that it is important for even young children to develop a sense of the role of assumptions in proving, and I investigate what it might mean and look like for instruction to promote this goal.…

Many researchers and curriculum frameworks recommend that the concept of "proof" and the corresponding activity of "proving" become part of students' mathematical experiences throughout the grades. Yet it is still unclear what "proof" means in school mathematics, especially in the elementary grades, and what role the teacher has in cultivating…

Despite increased appreciation of the role of "proof" in students' mathematical experiences across "all" grades, little research has focused on the issue of understanding and characterizing the notion of proof at the elementary school level. This paper takes a step toward addressing this limitation, by examining the characteristics of four major…

There is a growing effort to make "proof" central to all students' mathematical experiences across all grades. Success in this goal depends highly on teachers' knowledge of proof, but limited research has examined this knowledge. This paper contributes to this domain of research by investigating preservice elementary and secondary school…