The traditional technique for converting repeating decimals to common fractions can be found in nearly every algebra textbook that has been published, as well as in many precalculus texts. However, students generally encounter repeating decimal numerals earlier than high school when they study rational numbers in prealgebra classes. Therefore, how…

It is difficult but important for mathematics teachers to help students see themselves as capable mathematics learners, that is, to help each student develop a positive mathematics identity. A student's mathematics identity is a powerful influence on his or her engagement and participation in mathematics. Students act in ways that are consistent…

Classroom teachers try to provide opportunities for students to practice and use the algebra skills they are learning in ways that are nonroutine. They also want to help students connect the big ideas of math with the skills they are learning as part of the balance between understanding concepts and procedures. Math games can be used to accomplish…

Students enrolled in a middle school prealgebra or algebra course often struggle to conceptualize and understand the meaning of radical notation when it is introduced. For example, although it is important for students to approximate the decimal value of a number such as [square root of] 30 and estimate the value of a square root in the form of…

It is beneficial for students to discover intuitive strategies, as opposed to the teacher presenting strategies to them. Certain proportional reasoning tasks are more likely to elicit intuitive strategies than other tasks. The strategies that students are apt to use when approaching a task, as well as the likelihood of a student's success or…

Empirical arguments rely on examples without necessarily addressing all cases. Students should be skeptical of empirical evidence and should seek more secure arguments for generalizations, such as those that explain why a generalization is true for all cases. Generalizing on the basis of patterns in data is an important mathematical practice;…

The mean can be interpreted as a fair-share value and as a balance point. Standards documents, including Common Core State Standards for Mathematics (CCSSM) (CCSSI 2010), suggest focusing on both interpretations. In this article, the authors propose a sequence of five activities to help students develop these understandings of the mean, and they…

As students develop algebraic reasoning in grades 5 to 9, they learn to recognize patterns and understand expressions, equations, and variables. Linear functions are a focus in eighth-grade mathematics, and by algebra 1, students must make sense of functions that are not linear. This article describes how students worked through a classroom task…

How does one look for mathematical structure? Finding structure is a challenging yet accessible activity for all students. This article describes a lesson in which seventh graders engaged with mathematical structure. The setting was a seventh-grade prealgebra classroom in a suburban school. The classroom teacher was co-author Tatyana Finkelstein.…

A solid understanding of the structure and function of cells can help establish the foundation for learning advanced concepts in the biological sciences. The concept of the cell is introduced in middle school life science courses and is continued at the undergraduate level in college (NRC 2012; Reece et al. 2014). Cells are introduced to students…

"Mathematical modeling of the physical world should be the central theme of physics instruction" (Hestenes 1987). Science teachers know this, but because of standardized state testing, many math teachers and some science teachers feel the need to focus on memorizing facts that will be on a multiple-choice test (Volante 2004). As a…

Engineering-based modeling activities provide a rich source of meaningful situations that capitalize on and extend students' routine learning. By integrating such activities within existing curricula, students better appreciate how their school learning in mathematics and science applies to problems in the outside world. Furthermore, modeling…

One of the most common household polyhedra is a cereal box. For over 100 years, American companies like Kellogg's™ have packaged cereal in containers shaped like rectangular prisms. Why is a rectangular prism the most commonly used solid for holding cereal? Would another design work equally or more efficient? This article describes an…

Mathematical modeling, a focus of the Common Core State Standards for School Mathematics (CCSSI 2010) and one of the Standards for Mathematical Practice, is generally considered to be the process of exploring a real-world situation and making sense of it using mathematics (Lesh and Zawojewski 2007). Teachers need to create opportunities for…

The rapid growth of digital technology, including the worldwide adoption of mobile and embedded computers, places new demands on K-grade 12 educators and their students. Young people should have an opportunity to learn the technical knowledge of computer science (e.g., computer programming, mathematical logic, and discrete mathematics) in order to…