In this paper, we focus on two particularly problematic concepts in teaching mathematics: the complex unit i and angles. These concepts are naturally linked via De Moivre's theorem but are independently misused in numerous contexts. We present definitions, notations, and ways of speaking about these terms from mathematics education that are not…

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.

In this article, Jason Taff shares an approach that he presented to advanced seventh-grade prealgebra students. He begins by summarizing some of the shortcomings of equating the order of operations concept with the PEMDAS (often rendered mnemonically as "Please Excuse My Dear Aunt Sally") procedure with the hope of helping teachers at…

The perpetuation of mathematical rules that expire (Karp, Bush, and Dougherty 2014; 2015), or rules that are taught in previous grades that no longer hold true, suggests that many secondary students harbor misconceptions from their elementary and middle-grades mathematics experiences as they progress to mathematics classes that are more…

The ties between literacy and numeracy exist in the development of vocabulary and language for understanding mathematical concepts. Research indicates that explicit instruction in mathematics vocabulary supports success with mathematics problem solving (Biemiller 2009; Pierce and Fontaine 2009; Rubenstein and Thompson 2002) for native…

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…

Cara, a seventh-grade student with learning disabilities (LD) in mathematics, believes that the ratio 2:3 is equivalent to 4:5 because there is a difference of one between the two numbers in each ratio and there is a difference of two between corresponding numbers in the two ratios (2 + 2 = 4 and 3 + 2 = 5). This misconception affects her ability…

Ratios and proportions are foundational to student understanding across multiple topics in mathematics and science. In mathematics, they are central to developing concepts and skills related to slope, constant rate of change, and similar figures, which are all fundamental to algebraic concepts and skills. This article examines the importance of…

Many students struggle with algebra, from simplifying expressions to solving systems of equations. Students also have misconceptions about the meaning of variables. In response to the question "Can x + y + z ever equal x + p + z?" during a student interview, the student claimed, "Never . . . because p has to have a different value…

One area of algebraic thinking essential for students' success is a relational understanding of the equal sign. Research has indicated a positive correlation between students' relational understanding of the equal sign and their equation-solving performance, suggesting that students' early conception of the equal sign may affect their learning and…

Based on a study with six elementary school teachers and 115 students in grades 3, 4, and 5, this article discusses ways that teachers might support an understanding of variable as students explore pattern-finding activities. The practices emerged from interviews with teachers and a yearlong observation of teaching algebraic thinking during an…

"Modeling Mathematical Ideas" combining current research and practical strategies to build teachers and students strategic competence in problem solving.This must-have book supports teachers in understanding learning progressions that addresses conceptual guiding posts as well as students' common misconceptions in investigating and…

For students to be successful in algebra, they must have a truly conceptual understanding of key algebraic features as well as the procedural skills to complete a problem. One strategy to correct students' misconceptions combines the use of worked example problems in the classroom with student self-explanation. "Self-explanation" is the…

The statistics about community college developmental math level and completion rates can seem disheartening, especially for those who work with these students day-to-day and know their life stories. When it comes to teaching her students, the author chooses not to focus on the statistics. Instead, she pays attention to their varied career goals…

This article describes how the author has developed tasks for students that address the missed "essence of the matter" of algebraic transformations. Specifically, he has found that having students practice "perceiving" algebraic structure--by naming the "glue" in the expressions, drawing expressions using…