This paper introduces the concept of logical consistency in students' thinking in mathematical contexts. We present the Logical in-Consistency (LinC) instrument as a valuable assessment tool designed to examine the prevalence and types of logical inconsistencies among undergraduate students' evaluation of mathematical statements and accompanying…

Analogy has played an important role in developing modern mathematics. However, it is unclear to what extent students are granted opportunities to productively reason by analogy. This article proposes a set of lessons for introducing topics in ring theory that allow students to engage with the process of reasoning by analogy while exploring new…

Recursive reasoning is a powerful tool used extensively in problem solving. For us, recursive reasoning includes iteration, sequences, difference equations, discrete dynamical systems, pattern identification, and mathematical induction; all of these can represent how things change, but in discrete jumps. Given the school mathematics curriculum's…

Comparison is a powerful and important way that we learn. To support teachers in the use of comparison in their instruction, the authors developed an instructional routine called compare and discuss multiple strategies (CDMS). Similar to other instructional routines, CDMS is a structured, specific, repeatable minilesson that teachers can use to…

The mathematics classroom is particularly vulnerable to these judgments of perfectionism, with endless evidence of students and teachers believing that mathematics is based on an ultimate truth or a single, objective, unique answer. School mathematics still favors students' participation in rote procedures, memorization, and using only a few…

Mathematics in its nature is exploratory, giving learners a chance to view it from different perspectives. However, during most of their schooling, South African learners are rarely exposed to mathematical explorations, either because of the lack of resources or the nature of the curriculum in use. Perhaps, this is due to teachers' inability to…

While participating as a mentor teacher in a professional development project, Alison Vickery, a middle school teacher, developed a strategy: claim-rule-connection (CRC). The "claim" was the answer or response to the question; the "rule" was the theorem, fact, or proof; and the "connection" was an explanation of how…

Mathematical argumentation is generally thought to be the paradigm of cogent reasoning. The concept of mathematical proof thus seems to be associated with necessity and enforcement, but not with freedom; however, in various ways a reference to freedom is also needed to understand the phenomenon of mathematical proof, for example, to distinguish it…

Duncan Symons and Derek Holton discuss the different types of mathematical reasoning and what each of these might look like in the classroom. By suggesting language that can be used to describe the different methods of reasoning, they hope to provide teachers with the tools they need to better recognise and assess student reasoning.

This article examines a common proportional reasoning problem used in schools, often referred to as the Orange Juice task. The authors show how these six strategies described by Nikula (e.g., Unitizing, Norming, etc.) and one additional strategy can be used to either solve or make progress in the Orange Juice task. The article presents work from…

Many students share a certain amount of discomfort when encountering proofs in geometry class for the first time. The logic and reasoning process behind proof writing, however, is a vital foundation for mathematical understanding that should not be overlooked. A clearly developed argument helps students organize their thoughts and make…

Mathematics education research has focused on developing teachers' knowledge or other visible aspects of the teaching practice. This paper contributes to conversations around making a teacher's thinking visible and enhancing a teacher's pedagogical reasoning by exploring the use of pedagogical documentation. In this paper, we describe how a…

Should proof by induction be reserved for higher levels of mathematical instruction? How can teachers show students the nature of mathematics without first requiring that they master algebra and calculus? Proof by induction is one of the more difficult types of proof to teach, to learn, and to understand. Thus, this article delves deeper into…

Early experiences of reasoning while playing games of strategy are foundational for future proofs that students will be expected to build using conventionally structured arguments. But how did game playing in school occur? How can educators be sure that mathematical reasoning is going on? The authors investigated these questions to understand how…

Many mathematics departments have instituted transition-to-proof courses for second semester sophomores to help them learn how to construct proofs and to prepare them for proof-based courses, such as abstract algebra and real analysis. We have developed a way of getting students, who often stare at a blank piece of paper not knowing what to do, to…