This article presents the Snail problem, a relatively simple challenge about motion that offers engaging extensions involving the notion of infinity. It encourages students in grades 5-9 to connect mathematics learning to logic, history, and philosophy through analyzing the problem, making sense of quantitative relationships, and modeling with…

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…

When students begin work with fraction division in fifth grade or sixth grade, they bring with them experiences from whole-number division. Many students think that a division problem should lead to a quotient that is smaller than the dividend. It is also common for students to believe that the dividend should be larger than the divisor. Many, if…

In the elementary grades, students learn procedures to compute the four arithmetic operations on multidigit whole numbers, often by being shown a series of steps and rules. In the middle grades, students are then expected to perform these same procedures, with further twists. The Reasoning and Proof Process Standard suggests that students need to…

Division of fractions is often considered the most mechanical and least understood topic in elementary school. Enacting fraction division tasks in meaningful ways requires that teachers know not only "how" fraction division works but also "why" it works. The authors have created materials to help preservice teachers develop that knowledge. To…

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…

Dividing by fractions is considered by many to be one of the most complicated procedures in elementary mathematics. The computations are not only complicated but also challenging to explain in the context of a word problem. In this article, the author discusses the role of the divisor in the concept of division by fractions. She examines three…

"Mathematical habits of mind" include reasoning by continuity, looking at extreme cases, performing thought experiments, and using abstraction that mathematicians use in their work. Current recommendations emphasize the critical nature of developing these habits of mind: "Once this kind of thinking is established, students can apply it in the…

Many students and adults feel that algebra is merely the shuffling of symbols. The three interrelated concepts of variable, expression, and equation are central to beginning algebra, and in recent years, helping students understand the idea of a variable has been emphasized. Although graphing calculators help students solve equations, it is also…

Infinity and infinitely small numbers pique the curiosity of middle school students. From a very young age, many students are intrigued, interested, and even fascinated by extremely large or extremely small numbers or quantities. This article describes an activity that takes this curiosity about infinity into the domain of adding the numbers of an…

Many mathematics educators believe that it is insufficient for students' knowledge of the division of fractions to be limited to the invert-and-multiply algorithm, arguing that students' learning of mathematics must go beyond rote memorization of procedures. Teachers ought to carefully consider what students need to learn beyond the algorithmic…

This article explores how conceptualization of absolute value can start long before it is introduced. The manner in which absolute value is introduced to students in middle school has far-reaching consequences for their future mathematical understanding. It begins to lay the foundation for students' understanding of algebra, which can change…

This article shares an assessment tool that uses student-written word problems to provide meaningful information regarding the depth of mathematical understanding. Students' word problems representing 6 [division] 1/2 were classified using the scoring categories described in the NAEP. By categorizing the problems, students' levels of understanding…

In the 1970s, the movement to the metric system (which has still not completely occurred in the United States) and the advent of hand-held calculators led some to speculate that decimal representation of numbers would render fractions obsolete. This provocative proposition stimulated Zalman Usiskin to write "The Future of Fractions" in 1979. He…

This article discusses two sequences of activities that were developed to support middle school students' and preservice teachers' construction of algorithms for dividing fractions. One sequence is intended to promote understanding of the common-denominator algorithm; the other sequence is intended to promote understanding of the…