For decades, students have been encouraged to use "Please Excuse My Dear Aunt Sally" as a means of learning the order of operations. Teachers unfamiliar with the Aunt Sally mnemonic are perhaps more familiar with a mnemonic such as PEMDAS. Each mnemonic is intended to convey "parentheses, exponents, multiplication, division,…

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.…

The adoption of the Common Core State Standards for Mathematics (CCSSM) (CCSSI 2010) has caused a shift in the expectations for student learning, with implications for teaching. It has also introduced a new kind of standard focused on the way that students think about content in the form of the Standards for Mathematical Practice (SMP). The SMP…

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…

In the author's experience with this activity, students struggle with the idea of representativeness in probability. Therefore, this student misconception is part of the classroom discussion about the activities in this lesson. Representativeness is related to the (incorrect) idea that outcomes that seem more random are more likely to happen. This…

How can a simple dot--the decimal point--be the source of such frustration for students and teachers? As the author worked through her own frustrations, she found that her students seemed to fall into groups in terms of misconceptions that they revealed when talking about and working with decimals. When asking students to illustrate their thinking…

Many middle school students approach math with caution, often trepidation, and generally less-than-complete confidence. Several factors contribute to their views of what math actually is (a system of rules to follow and formulas to apply) as well as who might be good at it (people who can see the path to an answer and calculate quickly). A few say…

A story problem about borrowing money, presented in this article, may be represented with positive or negative numbers and thought about in different ways. The authors describe ideas related to integers (both positive and negative) and how students used them in relation to a story problem, and how they related these ideas to equations.

Researchers have extensively documented, and math teachers know from experience, that algebra is a "gatekeeper" to more advanced mathematical topics. Students must have a strong understanding of fundamental algebraic concepts to be successful in later mathematics courses. Unfortunately, algebraic misconceptions that students may form or…

Although discussions about inductive reasoning can be traced back thousands of years (Fitelson 2011), the implementation of the Standards for Mathematical Practice (SMP) within the Common Core State Standards (CCSSI 2010) is generating renewed attention to how students learn mathematics. The third SMP, "Construct viable arguments and critique…

Student errors are springboards for analyzing, reasoning, and justifying. The mathematics education community recognizes the value of student errors, noting that "mistakes are seen not as dead ends but rather as potential avenues for learning." To induce specific errors and help students learn, choose tasks that might produce mistakes.…

One of the many challenges that teachers face in mathematics classrooms is determining how much of the verbal and written explanations help students accomplish instructional goals. The challenge is greater in linguistically diverse classrooms because the explanations and multiple representations are not perceived uniformly by all students. Recent…

In the authors' examination of various instructional programs, they observed that most provide all the necessary data to solve proportion problems, employ compatible numbers that are usually unrealistic, present numbers (data) in the order in which they are to be manipulated, discuss contexts that cannot be easily replicated, and present…

Crime Scene for Mathematics Investigation (CSMI) was created as a small group, cooperative, self-selective grouping strategy to allow students to explore mathematics based on their own understanding and perceived readiness, much like the concept and application of literature circles in language arts. In essence, these self-chosen small groups met…

Middle school is a crucial transition period for students as they move from concrete to algebraic ways of thinking. This article describes a sequence of instruction geared toward helping prospective middle school instructors teach the topic of percentages.