Escuder, Ana

The theory of continued fractions goes possibly as far back as 300 B.C. to Euclid. Some
consider continued fractions as part of the "lost mathematics". I came across continued
fractions for the first time while taking a graduate math class and I was amazed as how
beautiful the representation of some numbers were using them and also how easy it was
to understand the theory by making pictures. As an educator, I want to share this
knowledge with my students and make it something that they could discover, manipulate,
and appreciate.
This paper is divided m three sections. The first deals with the basic theory and
definitions of continued fractions. The second shows how to use technology, especially
Dynamic Geometry Software to make the basic theory accessible to students in middle
and high school. The third section gives some reflections of my experience working with
students on continued fractions.

This study described, analyzed, and compared the internal and external factors that prevented or fostered the implementation of a cognitive tool, GeoGebra, in the mathematics practices of 12 middle school teachers who had completed a master's degree program in mathematics successfully. Through the application of a case study approach as a systematic method for the analysis of qualitative data, and under a social constructivist framework, the study examined different factors such as concerns of teachers; their beliefs about technology, mathematics as a subject, math teaching, and learning; external factors such as resources and school support; their TPACK development; and their instrumental orchestration approach through classroom observations. Among the major findings, the study revealed that the personal concerns of the teacher users of GeoGebra included the desire to continue learning the new features of the software, as well as the desire to connect themselves with others in common endeavors for the benefit of other teachers and, ultimately, the students... There was a consensus among the teacher users that they had to strike a balance between their professional goals and the available resources.