The initial training of mathematics teachers in Spain can start in the final year of the mathematics degree when the students, and future teachers, select courses related to teaching the discipline. At that time, since over the next two years they will probably be certified to enter the profession, it is worth asking oneself what mathematical knowledge they should have and what is needed to teach at the secondary education level. This paper describes one case study in which a group of final-year students solves a task containing aspects of mathematical understanding that are present in solving problems using GeoGebra. This was done using a solution protocol intended to emphasize the different possibilities offered by GeoGebra, presenting three approaches for the solution: dynamic, numeric and algebraic. We noticed that the use of technology is key to bringing out some of aspect and to demonstrating an understanding of mathematics for teaching.

This study aims to document the extent to which the use of digital technology enhances and extends high school teachers’ problem-solving strategies when framing their teaching scenarios. The participants systematically relied on online developments such as Wikipedia to contextualize problem statements or to review involved concepts. Likewise, they activated GeoGebra’s affordances to construct and explore dynamic models of tasks. The Apollonius problem is used to illustrate and discuss how the participants contextualized the task and relied on technology affordances to construct and explore problems’ dynamic models. As a result, they exhibited and extended the domain of several problem-solving strategies including the use of simpler cases, dragging orderly objects, measuring objects attributes, and finding loci of some objects that shaped their approached to reasoning and solve problems.

Mathematical tasks are key ingredient to foster teachers and students’ development and construction of mathematical thinking. The use of distinct computational tools offers teachers a variety of ways to represent and explore mathematical tasks which often extends problem solving approaches based on the use of paper and pencil. We sketch a framework to characterize ways of reasoning that emerge as result of using computational technology to solve a task that involves dealing with variation phenomena.

To what extent does the process of solving textbook problems help students develop a way of thinking that is consistent with mathematical practice? Can routine problems be transformed into problem solving activities that promote students' mathematical reflection? These questions are used to outline and discuss features of an inquiry framework useful to deal with textbook problems in which the use of computational tools can help students formulate and pursue multiple approaches to the problems. In addition, a theoretical learning trajectory is sketched to help instructors or teachers prepare their lessons and instructional activities.

In this work, we examine a questionnaire for prospective secondary school teachers (students in their final year of Mathematics, graduates in Mathematics and graduates in other science subjects). We study the relationship between their own conceptions, beliefs and attitudes and the role the teacher set out in the ‘Ley de Ordenacion General del Sistema Educativo’ (the new law that orders the Spanish educational system) and we include a comparison of different students’ opinions. Our results show that there are no notable differences among these students. However, students’ notions of the role of the teacher contrast sharply with the role proposed in the current educational reforms.