Students who otherwise seem unreachable through traditional approaches to algebra require some alternative teaching methods. So do teachers who seek to add elements of freshness and innovation to their classrooms or who simply appreciate variety. This article offers some unconventional techniques for teaching a few conventional algebra topics.…

The ability to represent numerical quantities in symbolic form is a necessary foundation for mathematical competence. Variables are particularly important symbolic representations for learning algebra and succeeding in higher mathematics, but the mechanisms of how students link a variable to what it represents are not well understood. Research…

This article introduces an optimization task with a ready-made motivating question that may be paraphrased as follows: "Are you smarter than a Welsh corgi?" The authors present the task along with descriptions of the ways in which two groups of students approached it. These group vignettes reveal as much about the nature of calculus students'…

We provide several constructions, both algebraic and geometric, for determining the ratio of the radii of two circles in an Apollonius-like packing problem. This problem was inspired by the art deco design in the transom window above the Shadek Fackenthal Library door on the Franklin & Marshall College campus.

This study examined ways in which students make use of a graphing calculator and how use relates to comfort and understanding with mathematical symbols. Analysis involved examining students' words and actions in problem solving to identify evidence of algebraic insight. Findings suggest that some symbols and symbolic structures had strong…

The purpose of this study was to investigate the effect of technology on one student's solution methods when solving mathematical tasks that included algebra word problems. The student was presented with context tasks in both paper-pencil and computer environments. The results of this study revealed that the student's solution methods differed in…

Principles and Standards for School Mathematics (NCTM 2000) states, "Technology is essential in teaching and learning mathematics; it influences the mathematics that is taught and enhances students' learning." The focus on reasoning and sense making with technology in the lesson presented in this article will enable students to do more…

As highlighted by the National Math Panel Report (2008), consistent results from laboratory studies have demonstrated that interleaving worked examples with problems to solve improves learning for novices. The purpose of this work is to create materials and tests that can be used flexibly in classrooms and which employ worked examples interleaved…

Students' struggles with algebra are well documented. Prior to the introduction of functions, mathematics is typically focused on applying a set of arithmetic operations to compute an answer. The introduction of functions, however, marks the point at which mathematics begins to focus on building up abstractions as a way to solve complex problems.…

This article presents a design experiment in which we explore new structures for classroom collaboration supported by a classroom network of handheld graphing calculators. We describe a design for small group investigations of linear functions and present findings from its implementation in three high school algebra classrooms. Our coding of the…

Students can use geometric representations of numbers as a way to explore algebraic ideas. With the help of these representations, students can think about the relations among the numbers, express them using their own words, and represent them with letters. The activities discussed here can stimulate students to try to find various ways of solving…

Using algebraic habits of mind as a framework, and focusing on thinking about functions and how they work, we examined the relationship between 18 pre-service middle school teachers' ability to use the features of the algebraic thinking (AT) habit of mind "Building Rules to Represent Functions" and their ability to recognize and…

This paper reports on different underlying structures of the derivative schema of three undergraduate students that were considered to be at the trans level of development of the derivative schema (action-process-object-schema). The derivative schema is characterized in terms of the students' ability to explicitly transfer the relationship between…

Learning algebra is difficult for many students in part because of an emphasis on the memorization of abstract rules. Algebraic reasoners across expertise levels often rely on perceptual-motor strategies to make these rules meaningful and memorable. However, in many cases, rules are provided as patterns to be memorized verbally, with little overt…

The teaching and learning of mathematics cannot be understood without considering the resolution of word problems. These kinds of problems not only connect mathematical concepts with language (and therefore with reality) but also promote the learning related to other scientific areas. In primary school, problems are solved by using basic…