This study investigates the embodied, symbolic, and formal reasoning of two fourth-year university students while exploring geometric reasoning about the Cauchy-Riemann equations with the aid of "Geometer's Sketchpad (GSP)." These students participated in a teaching activity designed to encourage shifts between embodied, symbolic, and…

Vectors have important applications both within and outside mathematics, but the concept of vectors is often taught to students in a less-than-engaging way, leading to students feeling inadequate and frustrated. This article describes the use of a mathematical microworld, "Driving with Vectors," to explore vectors using equitable…

Video games have emerged as a medium for learning by creating engaging environments, encouraging creative and deep thinking, and exposing learners to complex problems. Unfortunately, even though there are increasing examples of video games for many basic science and engineering concepts, similar efforts for higher level engineering concepts such…

Sequences are an important topic--not only in mathematics but also in the mathematics curriculum. Exploring sequences related to medial triangles offers learners opportunities to solve problems involving sequences within geometric contexts. Sequences related to medial triangles are also a rich source of beautiful and interesting geometric patterns…

Fractals are geometric objects used to describe irregular shapes that have a fractional dimension and commonly appear in nature. Although several proposals for the study of perfect fractals at the basic level are present in the literature, only few proposals for the study of real fractals exist, which does not seem reasonable considering the…

Right triangles play a significant role in mathematics. In this favorite lesson, the authors help students understand variant and invariant properties by considering relationships among angle measures and side lengths in right triangles. Students explore these relationships using interactive mathematics software, changing one angle and observing…

GeoGebra is a free tool that has the potential to change both how and what is taught in mathematics. GeoGebra allows teachers and students to explore various mathematical ideas either through the full applet (https://www.geogebra.org/graphing) or by sharing applets via GeoGebra's Materials site (https://www.geogebra. org/materials/). It has many…

The purpose of this article is to offer teaching ideas in the treatment of the definite integral concept and the Riemann sums in a technology-supported environment. Specifically, the article offers teaching ideas and activities for classroom for the numerical methods of approximating a definite integral via left- and right-hand Riemann sums, along…

In the authors' attempts to incorporate problem solving into their mathematics courses, they have found that student ambition and creativity are often hampered by feelings of risk, as many students are conditioned to value a produced solution over the actual process of building one. Eliminating risk is neither possible nor desired. The challenge,…

Calculus, perhaps more than other areas of mathematics, has a reputation for being steeped with procedures. In fact, through the years, it has been noticed of many students getting caught in the trap of trying to memorize algorithms and rules without developing associated concept knowledge. Specifically, students often struggle with the…

How can we, as teachers, encourage students to move from drawing geometric objects to constructing them and thereby build their mathematical reasoning skills? One way to encourage students to construct instead of draw shapes and to help students develop more robust understandings of geometric relationships and constructions is to use interactive…

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…

In this paper I describe how I have used the classic buried treasure problem with prospective and practicing mathematics teachers to enhance their problem solving abilities and disposition to integrate interactive geometry software (IGS) into the learning environment. I illustrate how IGS may be used as a strategic tool to gain insight into the…

Transformations are a central organizing idea in geometry. They are included in most geometry curricula and are likely to appear with even greater emphasis in the future, given the central role they play in the "Common Core State Standards" for K-12 mathematics. One of the attractions of geometry is the ability to draw and construct the…

Tangrams have sometimes been used as an extension activity intended only to keep faster students busy while others finished essential desk-work. Without adequate introduction, many find that tangrams are just an open-ended form of a jigsaw puzzle. Happily teachers have discovered that games provide an effective introduction to a new topic. In the…