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…

It is critical for mathematics tasks to provide students with the opportunity to engage actively in reasoning, sense making, and problem solving so that they develop a deep understanding of mathematics. Learning mathematics while solving a problem can be like entering a dark room with a single small light. The objects are in the shadows, difficult…

Both the Common Core State Standards (CCSSI 2010) and the NCTM Process and Content Standards distinguish between Standards for Mathematical Practice (SMP) and standards for mathematical content. We believe this distinction is important and note that students often acquire knowledge of mathematical content without necessarily developing the related…

The goal of this article is to raise questions that will promote discussions among secondary mathematics teachers about the concept of inverse functions and how to motivate a more conceptual understanding of them in their classrooms. The authors share data to answer the following questions: (1) What meanings of inverse functions do high school…

Polynomials with rational roots and extrema may be difficult to create. Although techniques for solving cubic polynomials exist, students struggle with solutions that are in a complicated format. Presented in this article is a way instructors may wish to introduce the topics of roots and critical numbers of polynomial functions in calculus. In a…

Numerous studies have shown that mathematical knowledge is situational, meaning that students' abilities to transfer and apply skills depends on the conditions in which they were learned (Barab 1999; Boaler 2002, 2016). Given that the real world seldom presents problems that are confined to a single discipline (Barab 1999), it is imperative that…

Classroom culture is negotiated and established through both conversations and practices. Traditionally, teachers and researchers have focused primarily on the individual and social construction of mathematical content--that is, students' conceptual understanding and procedural skills--through mathematical actions and practices. This article…

Despite repeated discussions and practice, algebra students continue to make variable errors, in many ways, year after year. This same problem appeared thirty years ago in a list of common errors that math teachers today would immediately recognize, many involving exponents and distribution (Marquis 1988). Similar complaints even appeared in the…

The Common Core State Standards for Mathematics expect students to build on their knowledge of the number system, expressions and equations, and functions throughout school mathematics. For example, students learn that they can add something to both sides of an equation and that doing so will not affect the equivalency; however, squaring both…

Assessment and instruction are interwoven in mathematically rich formative assessment tasks, so employing these tasks in the classrooms is an exciting and time-efficient opportunity. To provide a window into how these tasks work in the classroom, this article analyzes summaries of student work on such a task and considers several students'…

In this article, the authors discuss the potential of the icon-based mathematical games, emoji math and mobile math, to promote student engagement with and understanding of algebra. They describe how these games serve as accessible entry points for algebraic thinking and that, in contrast to traditional symbolic algebra, the use of these…

The authors provide a dynamic Ferris wheel computer activity that teachers can use as an instructional tool to help students investigate functions. They use a student's work to illustrate how students can use relationships between quantities to further their thinking about functions.

In this article, Strayer and Matuszewski present a six-phase strategy that teachers can use to help students develop a conceptual understanding of inferential hypothesis testing through simulation. As Strayer and Matuszewski discuss the strategy, they describe each phase in general, explain how they implemented the phase while teaching their…

Unfortunately, many students learn about the concept of systems of linear equations in a procedural way. The lessons are taught as three discrete methods. Connections between the methods, in many cases, are not made. As a result, the students' overall understanding of the concept is very limited. By the time the teacher reaches the end of the…

Classroom culture is established through both conversations and practices. Traditionally in mathematics class, the focus is primarily on the latter; that is, students are shown what "doing mathematics" looks like, and then asked that they try it themselves. This article discusses three mathematical conversations that help bring…