Getting students to think deeply about mathematical concepts is not an easy job, which is why we often use problem-solving tasks to engage students in higher-level mathematical thinking. Mathematical modeling, one of the mathematical practices found in the Common Core State Standards for Mathematics (CCSSM), is a type of problem solving that can…

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…

Pattern blocks are inexpensive wooden, foam, or plastic manipulatives developed in the 1960s to help students build an understanding of shapes, proportions, equivalence, and fractions (EDC 1968). The colorful collection of basic shapes in classic pattern block kits affords opportunities for amazing puzzle-like problem-solving tasks and for…

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…

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…

While some students are enjoying days at the park or at the beach, forty middle school students are in a classroom solving challenging mathematics problems. The second Math Camp for middle school students was offered in 2013 at a western university for students from local school districts. For three days, these students met from 10 a.m. to 3 p.m.…

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.

This article describes an exciting exploration-based activity in which students develop an alternative definition of factor that can help them solve problems like the one presented above. Students work in groups to collect data, analyze the data to make conjectures, and then spend a significant amount of time debating and justifying their…

A key characteristic of successful mathematics teachers is that they are able to provide varied activities that promote student learning and assessment. Web 2.0 applications can provide an assortment of tools to help produce creative activities. A Web 2.0 tool enables the student to enter data and create multimedia products using text, graphics,…

How many squares can be found on a typical 5 × 5 geoboard? Teachers who are unfamiliar with this question may wish to stop here and reflect a bit. The question can lead to wonderful student discourse: How can someone tell if something is a square? Should squares be counted that are the same size (but in a different location) or just different…

Dot arrays provide opportunities for students to notice structures like commutativity and distributivity, giving these properties an image that can be manipulated and explored. These images also connect to ways that we organize discrete objects in everyday life. This article describes how the authors developed an array of dot tasks that have been…

An important goal of school mathematics is to enable students to formulate, approach, and refine problems beyond those they have studied, allowing them to organize and consolidate their mathematical thinking. To achieve this goal, students should be encouraged to develop expertise in a variety of areas, such as problem solving, reasoning and…

As math teachers, the authors often encountered students who could ace a test but not explain their reasoning. This phenomenon was disturbing to them, and they fought for years to help students both understand mathematical concepts and develop meaning for them. Since their primary goal was to develop mathematically literate students, their…

The opening lines of "How the Grinch Stole Christmas!" (Geisel 1957) set the tone for an inspiring change-of-heart story about a grumpy creature that comes to value humanity and the Christmas spirit. Although the book is often considered inspirational for other reasons, the authors were particularly moved by Dr. Seuss's interesting conjecture…

Proportional reasoning is an important aspect of formal thinking that is acquired during the developmental years that approximate the middle years of schooling. Students who fail to acquire sound proportional reasoning often experience difficulties in subjects that require quantitative thinking, such as science, technology, engineering, and…