This study aims to examine how teachers’ mathematical and pedagogical knowledge develop as they learn to use a multi-representational technological tool, the TI-Nspire handheld device and computer software. It is conducted as an enquiry into the learning trajectories of a group of secondary mathematics teachers as they begin to use the device with a focus on their interpretations of mathematical variance and invariance. The research is situated within an English secondary school setting and it seeks to reveal how teachers’ ideas shape, and are shaped by, their use of the technology through a scrutiny of the lesson artefacts, semi-structured interviews and lesson observations. Analysis of the data reveals the importance of the idea of the ‘hiccup’; that is the perturbation experienced by teachers during lessons stimulated by their use of the technology, which illuminates discontinuities within teachers’ knowledge. The study concludes that the use of such a multi-representational tool can substantially change the way in which both the teachers and their students perceive the notions of variance and invariance within dynamic mathematical environments. Furthermore, the study classifies the types of perturbations that underpin this conclusion. The study also contributes to the discourse on the design of mathematical problems and their associated instrumentation schemes in which linked multiple representations offer a new environment for developing mathematical meanings. This thesis makes an original contribution to understanding what and how teachers learn about the concept of mathematical variance and invariance within a technological environment.

Quantifying Uncertainty and Analyzing Numerical Trends (QUANT) 1 is a yearlong professional
development program for high school mathematics teachers that is designed to develop their statistical
proficiency for teaching. The approach used to develop such proficiency is technological pedagogical
content knowledge in the areas of measurement, data collection, data analysis, probability, and
statistics, combined with a classroom implementation focus on selecting, setting up, and enacting cognitively demanding tasks. This paper describes the QUANT program, its aims, the role of technology in the program, and the results from a series of exploratory investigations to measure and evaluate the program’s effectiveness. These studies have involved a total of 23 practicing teachers, and they have been used to refine and shape this ongoing professional development effort.

An interpretive review of 14 experimental and quasi-experimental studies identified these 9 attributes of effective professional development.
• Address student learning needs
• Incorporate hands-on technology use
• Be job-embedded
• Have application to specific curricula
• Address knowledge, skills and beliefs
• Occur over time
• Occur with colleagues
• Provide technical assistance and support to teachers
• Incorporate evaluation

This paper details the motivation, background, and analysis for studying the effect that an Increasing Achievement on Algebra Assessment (IAAA) workshop for a group of Florida high school teachers had on student pe/jormance in statewide testing. The main focus ofthe workshop was to provide participating teachers with both instruction and activities related to problem-solving techniques using the TI-83 Plus handheld graphing calculator that the teachers could then use in an effort to better prepare their students for state-wide testing. Overall, students (n = 328) who were taught by IAAA-trained teachers between consecutive annual statewide assessments had a greater gain in test scores than students (n = 202) who were taught by non-trained teachers. In particular, the data indicates a significant increase in scores for students taught by IAAA-trained teachers in a general mathematics course, as compared to no significant increase in scores for students taking that same course taught by non-trained teachers.

In mathematics education teachers experience a constant lack of time to properly instruct their students. In the Netherlands the contact time for mathematics in secondary education during the last fifteen years again declined. Mathematics is also perceived as difficult by students. This research focuses on the question: how can we better utilize contact time in mathematics education?
Meta-analyses of learning outcomes, such as those described by Hattie (2009), show that feedback is one of the most powerful single tools for improving learning achievements. In this study we explore the possibility of graphing calculators (GR), connected to the teacher computer through the use of a wireless network, to improve the feedback in mathematics education. First, students received immediate feedback on their worked out mathematics assignments GR and second, the teacher, usually in the next lesson, gave feedback on the work of the students, supported by an analysis of that work through the system. This study focused primarily on the development of 'data literacy' among students, while the 'algorithmic skills' were not forgotten.
In four stages, a prototype of the intervention designed, tested, evaluated and adjusted in nine groups of students. The mathematics teachers and their students are generally enthusiastic about the results. They for instance recommend to spend half of each lesson working this way. Though, the teachers explicitly state that they have experienced a tough workload while mastering this way of teaching. The study makes the conditions to be met before the method can be successful explicit.