Taylor series play a ubiquitous role in calculus courses, and their applications as approximants to functions are widely taught and used everywhere. However, it is not common to present the students with other types of approximations besides Taylor polynomials. These notes show that polynomials construed to satisfy certain boundary conditions at…

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,…

This paper aims to broaden the scope of the construct concept image by addressing the mathematically oriented notion of "solution of an algebraic equation." The notion solution of an algebraic equation is not mathematical per se, but it is most prevalent in mathematical communications and in mathematics education. The paper describes…

This article describes "Build It!--The Rectangle Game" task that uses the context of a game to develop mathematical generalizations based on strategy. The underlying mathematics in this game-based task is for students to discover factors and prime and composite numbers through 100. The playful use of "The Rectangle Game"…

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…

Mistakes are a window into students' mathematical understanding. Taking the time to analyze mistakes can inform you of where students are in their thinking. They can also guide you with respect to how to proceed with instruction. Introduced here are two categories of mistakes: meaty mistakes and minor mistakes. Meaty mistakes demonstrate gaps in…

The mathematics performance of students identified with emotional and behavioral disorders (EBD) is poor. Students identified with EBD require individualized instruction aligned with their unique academic and behavioral needs. One way that teachers can efficiently identify core mathematics deficits and provide individualized instruction is through…

Mistakes can be a source of frustration for teachers and students in mathematics classrooms because they reveal potential misunderstandings or a lack of learning. However, increasing evidence shows that making mistakes creates productive pathways for learning new ideas and building new concepts (Boaler 2016; Borasi 1996). Learning through…

This is a study of my dramatized narrative based on a true story of what took place in a mathematics class in an international school setting. My narrative includes the exploration of a mathematical "mistake" and the discovery of newly emergent problems. The drama, commentary, and reflections provide a close examination of the conduct…

Research has shown that word problem difficulty is influenced by various linguistic, numerical and general variables, but results in this area are inconsistent. This paper focuses on the question of how the order of numerical data, the context, the position of the unknown transformation, and the length of the statement in an additive word problem,…

This article endeavours to define how the application of incorrect or wrong solutions to mathematical equations or problems can be used to help stimulate students to perform a critical analysis of the mathematics being applied or the context of the equations used. The context of the application is discussed together with the specific learning…

This paper describes a framework for identifying, classifying, and coding student proofs, modified from existing proof-grading rubrics. The framework includes 20 common errors, as well as categories for interpreting the severity of the error. The coding scheme is intended for use in a classroom context, for providing effective student feedback. In…

Middle-grades teachers and students can have different perspectives on the value of discussing students' mathematical mistakes, despite various classroom evidence that such discussions can help foster strong conceptual understanding. Some teachers consider student mistakes to be an opportunity to correct errors in individual student thinking.…

Teachers often promote care in doing calculations, but for most students a single mistake rarely has major consequences. This article presents several real-life events in which relatively minor mathematical errors led to situations that ranged from public embarrassment to the loss of millions of dollars' worth of equipment. The stories here…

The Common Core State Standards for Mathematics (CCSSM) (CCSSI 2010) outlines ambitious goals for fraction learning, starting in third grade, that include the use of the number line model. Understanding and constructing fractions on a number line are particularly complex tasks. The current work of the authors centers on ways to successfully…