In this paper we analyze and discuss students' performance in a CAS environment related to the simplification of rational expressions. Results indicate that if students have more initial paper-and-pencil techniques, the CAS environment spurs them to deeper theoretical reflections than for students who have fewer techniques. [For the complete…

It is common for both algebra and calculus instructors to use power functions of various degrees as well as exponential functions to examine and compare rates of growth. This can be done on a chalkboard, with a graphing calculator, or with a spreadsheet. Instructors often are careful to connect the symbolic and graphical (and occasionally the…

This multiple case study explores factors that shaped the curricular reasoning of mathematics faculty engaged in college algebra reform in community colleges. Overall, the study found that although proposed reform of college algebra is broad in scope and influenced by the student audience, competing influences emerge that can enable or constrain…

Technology permeates our daily lives; education has not been untouched. Liaw (2002) points out that "teachers, trainers, and instructional designers of computer-based or Web-based instruction would benefit by being more attentive to learners' perceptions toward Web-based environments." Reviewing the earlier research into student perceptions toward…

Determining how to provide good tutoring functions is an important research direction of intelligent tutoring systems. In this study, the authors develop an intelligent tutoring system with good tutoring functions, called "FUDAOWANG." The research domain that FUDAOWANG treats is junior middle school mathematics, which belongs to the objective…

Gabriel's Horn is a solid of revolution commonly featured in calculus textbooks as a counter-intuitive example of a solid having finite volume but infinite surface area. Other examples of solids with surprising geometrical finitude relationships have also appeared in the literature. This article cites several intriguing examples (some of fractal…

Pattern generalisation has become an important feature of mathematics classrooms around the globe. Sometimes these activities focus purely on given numerical terms, but the use of pictorial or figural patterns is now becoming part of the standard repertoire for such generalisation exercises. From a pedagogic point of view, the investigation of…

Math is everywhere! Mathematics is "a way of describing the world--a way of thinking, knowing, and problem-solving" (Virginia's Early Childhood Development Alignment Project 2008, 83). Infants and toddlers are natural mathematicians. Even without adult support, infants and toddlers use math concepts to make sense of their world. An important role…

Several interactive geometry software packages are available today to secondary school teachers. An example is The Geometer's Sketchpad[R] (GSP), also known as Dynamic Geometry[R] software, developed by Key Curriculum Press. This numeric based technology has been widely adopted in the last twenty years, and a vast amount of creativity has been…

Despite the proliferation of mathematics standards internationally and despite general agreement on the importance of teaching for conceptual understanding, conceptual learning goals for many K-12 mathematics topics have not been well-articulated. This article presents a coherent set of five conceptual learning goals for a complex mathematical…

The first part of the article focuses on deriving the essential properties of arithmetic sequences by appealing to students' sense making and reasoning. The second part describes how to guide students to translate their knowledge of arithmetic sequences into an understanding of linear equations. Ryota Matsuura originally wrote these lessons for…

Transfer is typically thought of as requiring individuals to "see" what is the same in the deep structure between a new target problem and a previously encountered source problem, even though the surface features may be dissimilar. We propose that experts can "see" the deep structure by considering the first-order interactions…

Has technology revolutionised the mathematics classroom, or is it still a device waiting to be exploited for the benefit of the learner? There are applets that will enable the user to solve complex equations at the push of a button. So, does this jeopardise other methods, make other methods redundant, or even diminish other methods in the mind of…

There is a growing movement toward the introduction of algebra in early grades. This is supported by an increasing number of research studies that have reported success in getting young students to "do" mathematics considered beyond their reach. Yet, the consensus is that more research is needed to provide insights into the processes by which…

This paper is a case study of the teaching of an undergraduate abstract algebra course, in particular the way the instructor presented proofs. It describes a framework for proof writing based on Selden and Selden (2009) and the work of Alcock (2010) on modes of thought that support proof writing. The paper offers a case study of the teaching of a…