No abstract

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.

Number sense has been considered as one of the most important mathematical notions to be addressed in school mathematics in the 21st century. In this paper, we identify how students of a public middle school, located in a rural area in Mexico, showed several aspects of number sense by performing tasks involving arithmetic operations in a shopping context. Students invented mental and written calculation strategies that provide evidence of understanding regarding numbers and operations. Students used fantasy money as a representation system to decompose quantities in non-standard ways, or referred to strategies employed by unschooled people. Compensation strategies that involve modifying the quantities before applying the written algorithm for subtraction, in order to perform a subtraction without regrouping, also appeared. That is, it was identified a transfer from mental calculation strategies to a paper and pencil scenario for facilitating calculations during problem solving, which is an indicator of student' creativity and number sense. Additionally, the proposed tasks provided evidence that contextualized tasks could promote the understanding of decimal numbers.

En este artículo se caracteriza una perspectiva de la resolución de problemas como aproximación didáctica. Particularmente se identifican aquellos aspectos de una comunidad de aprendizaje que favorecen el desarrollo de un aprendizaje con entendimiento, a partir de las contribuciones de diversos matemáticos y educadores matemáticos, entre los que se encuentran George Polya, Paul Halmos, Jaques Hadamard, Alan Schoenfeld, James Hiebert y Richard Lesh.

If Ω/F is a Galois extension with Galois G and μ(Ω) denotes the group of roots of unity in Ω, we use the group Z1 (G,μ(Ω)) of crossed homomorphisms to study radical extensions inside Ω. Furthermore, we characterise cubic radical extension, and we provide an example to show that this result can not extended for higher degree extensions.