Barlow et al. (2018) discussed three types of mistakes worthy of inspection: procedural errors, inappropriate solution processes, and misconceptions. Here, the authors focus on procedural errors, as these often led the teachers in their professional development project to limit their inspection of mistakes to correcting. Despite this narrow focus,…

One of the greatest conceptual difficulties faced by middle level mathematics students is developing a rich understanding of irrational numbers that includes recognizing that irrational numbers are truly real numbers with an exact value and an exact place on the number line. Developing a deep conceptual understanding of irrational numbers is…

How we should teach mathematics has long been debated by advocates of different instructional approaches. These debates can lead to tensions that are difficult to navigate for those who are new to (or even veterans of) the profession. Educators often propose a "balance" of different approaches as a solution. Charles Munter considers…

The authors describe a lesson--"You Decide"--which challenges students but also provides opportunities for success for those who may struggle. They show how this lesson has been helpful for teachers in revealing some misconceptions that often exist in primary students' thinking. In this article, they share data on the apparent relative…

This article is the first in a series about teaching statistics. The authors discuss the role of statistics and the difference between mathematics and statistics.

Quadratic equations are a notorious topic for the challenge it provides to students in secondary mathematics. Despite this, there is limited research, particularly in the Australian context, that explains why such challenges persist. This article details the causes of Year 11 students' difficulties in solving quadratic equations. Observing…

Determining what students know and understand, as well as what misconceptions they have, is essential to planning and providing targeted intervention and support. In this article, a teacher uses formative assessment interviews to uncover evidence of students' understandings and to plan targeted instruction in a mathematics intervention class. The…

Fragile understanding is where new learning begins. Students' understanding of new concepts is often shaky at first, when they have only had limited experiences with or single viewpoints on an idea. This is not inherently bad. Despite teachers' best efforts, students' tenuous grasp of mathematics concepts often falters with time or when presented…

In this paper, we propose an elementary proof of Niven's Theorem in which the tangent function will have a primary role.

Students studying statistics often misunderstand what statistics represent. Some of the most well-known misunderstandings of statistics revolve around null hypothesis significance testing. One pervasive misunderstanding is that the calculated p-value represents the probability that the null hypothesis is true, and that if p < 0.05, there is…

The fundamental theorem of calculus (FTC) plays a crucial role in mathematics, showing that the seemingly unconnected topics of differentiation and integration are intimately related. Indeed, it is the fundamental theorem that enables definite integrals to be evaluated exactly in many cases that would otherwise be intractable. Students commonly…

Examples are an essential part of mathematics teaching and learning, used on a daily basis to teach and practice content. Yet, selecting good examples for teaching is complex and challenging. This article presents ideas to consider when selecting examples, drawn from a research study with algebra 2 teachers.

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…

Students often view their role as that of a replicator, rather than a creator, of mathematical arguments. We aimed to engage our students more fully in the creation process, helping them to see themselves as legitimate proof creators. In this paper, we describe an instructional activity (i.e., the "group proof activity") that is…

This article describes several ways to connect probability to other topics within mathematics, over a range of year levels, and across the curriculum. It includes a description of common issues and misconceptions, and practical learning activities to address them.