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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.

The aim of this study is to analyze and document the extent to which high school teachers rely on a set of technology affordances to articulate epistemological and cognitive actions in problem solving approaches. Participants were encouraged to construct dynamic representations of tasks and always to look for different ways to identify and support mathematical relations. To this end, three interrelated geneses (instrumental, semiotic, discursive) proposed in the Mathematical Working Space frame are used to document ways in which the participants constructed dynamic configuration of mathematical tasks as a means initially to explore invariants and eventually formulate conjectures based on empirical arguments. Technology affordances that involve dragging objects, quantifying parameters, graphing loci and using sliders were important for the participants to articulate semiotic and discursive geneses. Thus, the participants' approaches that appeared individual, small group and plenary discussions contributed to identifying initial conjectures that later were analyzed via empirical, geometric or algebraic arguments. In addition, the participants recognized that a working space to understand mathematical ideas and to solve problems should foster mathematical discussions beyond formal settings. The use of digital technologies plays an important role in both representing and exploring mathematical tasks and in following the discussion outside the formal class. As a result, the participants outlined a possible route to implement a route for learners to integrate activities to account for instrumental, semiotic, and discursive genesis.

Triamcinolone acetonide (TA) is a powerful anti-inflammatory drug used in the treatment of inflammatory ocular disorders; however, its poor aqueous solubility and ocular anatomical barriers hinder optimal treatment. The aim of this work was to obtain triamcinolone acetonide nanocrystals (TA-NC) to improve ocular corticosteroid therapy. Self-dispersible TA-NC were prepared by the bead milling technique followed by spray-drying, exhaustively characterized and then evaluated in vivo in an ocular model of endotoxin-induced uveitis (EIU). Self-dispersible TA-NC presented an average particle size of 257 ± 30 nm, a narrow size distribution and a zeta potential of −25 ± 3 mV, which remained unchanged for 120 days under storage conditions at 25 °C. In addition, SEM studies of the TA-NC showed uniform and spherical morphology, and FTIR and XRDP analyses indicated no apparent chemical and crystallinity changes. The subconjunctival administration of TA-NC in albino male white rabbits showed no clinical signs of ocular damage. In vivo studies proved that treatment with self-dispersible TA-NC alleviated the inflammatory response in the anterior chamber and iris in EUI rabbit eyes. Dispersible TA-NC are a promising approach to obtaining a novel nanometric TA formulation for ocular disorders. [ABSTRACT FROM AUTHOR]

With an empirical study, we explore the process of designing modeling tasks for teaching mathematics in a Dynamic Geometry environment. We address the research questions: What are the characteristics of the tasks that pre-service teachers design to teach mathematics in digital and modeling environments? What are the activities like that these tasks encourage in students? By means of a qualitative methodology, we characterize the tasks designed by secondary pre-service mathematics teachers and identify the activities that these tasks encourage in students. By analyzing the process of designing and planning a class, we can determine how the characteristics of the modeling tasks are modified based on the software used. In this context, the software also acts as a medium that reflects aspects of the models involved, the objectives of the tasks, and the activities intended for the students. The results contribute to the discussion regarding task design, where technology is a means of supporting the exploration, understanding, and study of different phenomena. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Textbooks are an essential resource that teachers use to organize and guide their students' learning of mathematics. In general, textbooks help students review and extend their knowledge of the themes that they study in their courses. Digital technology developments are transforming the ways in which textbooks are designed since this technology fosters the inclusion of interactive activities where students can consult videos to review and extend concepts, access and discuss online materials and share and discuss ideas or problem solving approaches within a learning community that include their peers or other students. In this article, we present a framework that aims to help teachers design interactive textbooks via the identification of main problem solving episodes such as understanding the problem statement and the construction of dynamic representations, looking for different ways to solve the problem, extending the initial problem, and communicating results. [ABSTRACT FROM AUTHOR]

This is the second part of a study published in a previous issue of this Journal whose main goal was to analyze and document the types of knowledge that university students exhibit to deal with fundamental issues that they had studied in a first ordinary differential equation course. Here, we focus on analyzing and discussing the following research questions: (ii) How do students make sense of, interpret and deal with the concept of solution to an ODE? (iii) What systems of representation do they use to represent and explore the information embedded in those questions in order to answer them? And, to what extent do the students privilege the use of certain type of representation? [ABSTRACT FROM AUTHOR]

This study documents the extent to which university students utilize diverse representations and mathematical processes to interpret and respond to a set of questions that involves fundamental concepts in the study of differential equations. Results indicate that students' idea to solve a differential equation is reduced to the application of proper solution methods to a certain type of equation differential expressions. Thus, instructional activities should promote the students' use of several representation systems in which they can reflect on the various aspects associated with the concept itself, the solution methods, procedures, and the corresponding meaning and connections among those representations. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper analyses the answers of a group of first-year university Mathematics students to a questionnaire, with the aim of determining the difficulties they have when carrying out non-routine tasks related to improper integrals. The questionnaire consisted of nine questions including not only calculus tasks and determining the convergence of given improper integrals, but also intuitive questions and some paradoxical results too (for example, a figure with an infinite longitude which closes the same area as the unit circumference, or an infinite figure with a finite volume). In particular, the students were asked to interpret most of the results they had obtained. Answers given by the students to each of the questions were categorized, which allowed some initial conclusions to be reached regarding this research. From analyses carried out, one can conclude that there are students who have difficulties in articulating the different systems of representation, and have problems in connecting and relating this knowledge as a generalization of previous concepts, such as definite integrals, series and sequences. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]