Proportional reasoning is both complex and layered, making it challenging to define. Lamon (1999) identified characteristics of proportional thinkers, such as being able to understand covariance of quantities; distinguish between proportional and nonproportional relationships; use a variety of strategies flexibly, most of which are nonalgorithmic,…

An interesting change has been occurring in the author's middle school mathematics classes. Some students have become creative, free-thinking inventors. If they have had little experience with an operation, they invent strategies of their own. Sometimes these strategies have been well informed and supported by strong mental math skills and a good…

Most future teachers are familiar with number patterns that represent an arithmetic sequence, and most are able to determine the general representation of the "n"th number in the pattern. However, when they are given a visual representation instead of the numbers in the pattern, it is not always easy for them to make the connection between the…

The concept of "function" spans elementary school through high school. When introducing the topic to elementary school and middle school students, the authors ask them to work with a variety of examples and representations of functions, since a formal definition of a function is rather abstract. They give the most accessible geometric functions…

It is important to use rich problem-solving tasks to develop students' understanding of important mathematical ideas (NCTM 2000). Rich tasks are complex, nonalgorithmic, and nonroutine, requiring reasoning, problem solving, and communication of thinking. Evidence has shown that using rich tasks in student-centered classrooms is especially…

Students appear to be able to see number relationships and patterns but have difficulty recognizing the visual properties of shapes, especially if the shapes are in different positions. Their difficulty in the visual and spatial realm is often linked to a lack of drawing experience and possibly undeveloped fine-motor skills. The author, as a…

Too often, the experiences that middle school students have in math do not connect to what they experience in elementary school. Although middle school teachers often have a sense of the content that students have learned in elementary school, the school structure and other factors can inhibit communication between elementary school and middle…

To focus on mathematical reasoning and what makes a good argument, the authors developed an assignment that requires college students to submit a book of mathematical reasoning as an assessment during the semester. To begin, the authors looked for questions and tasks that lend themselves to developing mathematical arguments and justifications and…

This article describes how to make an origami paper box and explores the algebra, geometry, and other mathematics that unfolds. A set of origami steps that transforms the paper into an open box can hold mathematical surprises for both students and teachers. An origami lesson can engage students in an open-ended exploration of the relationship…

Using the structure of a metacognitive journal, this article describes the author's discovery of an unusual method for adding fractions after carefully considering a student's response. (Contains 3 figures.)

The NCTM calls for the use of rich tasks that encourage students to apply their own reasoning to problem situations. When students work through algebraic generalization tasks, their reasoning often elicits a variety of strategies (Lannin 2003; Stacey 1989; Swafford and Langrall 2000). Challenges for teachers include facilitating student awareness…

Recognizing schemes, which are different from strategies, can help teachers understand their students' thinking about fractions. Using Steffe's advanced fraction schemes, the authors describe a progression of development that upper elementary and middle school students might follow in understanding fractions. Each scheme can be viewed as a…

Mathematical argumentation is an important skill. It leads to the process of proof, which is one area that students are being asked to master by the end of secondary school. Encouraging explanation and justification in math class allows this skill to develop. Explaining and justifying their ideas forces students to think deeply about mathematics…

Algebra is, among other things, a shorthand way to express quantitative reasoning. This article illustrates ways for the classroom teacher to convert algebraic solutions to verbal problems into conversational solutions that can be understood by students in the lower grades. Three reasonably typical verbal problems that either appeared as or…

This article focuses on the Thinking Through a Lesson Protocol (TTLP) that is intended to facilitate the design of lessons based on cognitively challenging mathematics tasks. Included is a discussion of the key features of the TTLP, suggestions on ways in which the protocol can be used as a tool for collaborative lesson planning, and a discussion…