Too often, high school algebra students encounter topics in a discrete, disconnected fashion. In the authors' first years of teaching, they found students to be adept at solving problems they were currently studying; for instance, they could solve quadratics during a quadratics unit. Too often, however, students struggled with that same content…

In this article, the authors present the use of digital tape diagrams--implemented using GeoGebra (2001) dynamic mathematics software--as a support to static diagrams to pose and solve proportional reasoning tasks. They show how dynamic sketches can allow students to explore standard tasks in proportional reasoning in ways that emphasise…

The article opens with a Geoboard Triangle Quest in this form: "How many noncongruent triangles can be constructed on a 4 × 4 geoboard? How do you know? Justify your answer with significant supporting work." The use of advanced digital technologies as tools for problem solving receives much attention in the methods classes if the…

A transition from "doer" to "teacher" for prospective teachers requires them to reorient from thinking about how they do mathematics to engaging with students and their work, understanding student representations, and planning instruction accordingly. To scaffold a transition, we developed a five-step mathematics as teacher…

During the past few years, several of the authors have incorporated student problem posing as a regular instructional feature in their classrooms. When they offer their students the opportunity to construct their own problems, particularly during the course of an entire school year, they create many novel tasks. Student-created tasks not only…

Mathematics educators frequently extol the virtues of inquiry-based instruction to classroom teachers. Although research suggests that inquiry benefits learners by allowing them to make sense of the mathematics they encounter, particularly through discourse with peers, to assume that all (or even most) teachers successfully create and implement…

Successful teacher preparation programs provide learning experiences that help teacher candidates make the shift from "student" to "teacher." In this paper we present research on the implementation of a process for providing candidates such experiences. Utilizing the Mathematics as Teacher Heuristic (MATH) process, prospective…

This paper presents the results of a study designed to facilitate teacher candidates' shift from students of mathematics to teachers of mathematics. We develop a series of tasks each of which encourages candidates to reconsider rich tasks from increasingly teacher-centric points of view. The results show that using authentic high school student…

Solving true problems requires persistence. The National Council of Teachers of Mathematics (NCTM) states that "problem solving means engaging in a task for which the solution method is not known in advance. In order to find a solution, students must draw on their knowledge, and through this process they will often develop new mathematical…

In this paper are discussed recent efforts to provide preservice mathematics teachers with opportunities to connect elementary teaching methods and content with the content and methods of secondary school mathematics. Through an in-depth exploration of the game, Shut the Box, preservice elementary and secondary mathematics teachers thoughtfully…