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The topics of correlation and linear regression constitute a complex and subtle system of statistical and mathematical ideas whose teaching-learning raises numerous practical and theoretical problems. In this research paper, the patterns of reasoning that students exhibit, under the approach of informal inferences when they face problems of correlation and regression line are identified. To achieve this, activities were implemented in two stages: in the first stage, two problems (one of estimation and other of best-fit line) were applied to be solved using pencil and paper; the second stage incorporates the use of the Fathom software. [For the complete proceedings, see ED629884.]

From mathematics education, a great variety of studies have focused on analyzing the presence of digital technology from different points of view. Those who work with proof and argumentation have resisted presenting results with a technological component, due to the idea of formality associated with those terms, which has increased the existing tension between the theoretical and empirical nature of mathematics. Taking into consideration that with the presence of environments such as dynamic geometry software (DGS) the student can access to dynamic representations, we seek to understand the impact of the interaction with these representations on the arguments proposed by the students. For this, we carried out an exploratory and descriptive study whose results show that the interaction with dynamic representations provides students with validation tools different from those on paper. [For the complete proceedings, see ED630210.]

In this paper, we present the development of an investigation on the promotion of covariational reasoning in high school students (14-15 years old) in Mexico. The study consists of designing and applying a sequence of didactic activities that simulate a real situation virtually. The activities are organized through a Hypothetical Learning Trajectory supported by digital technology and elements of Cuevas-Pluvinage didactics. The activities were evaluated according to the levels of covariation proposed by Carlson and colleagues, categorizing students' achievements and difficulties for each level of understanding. The results show that the activities favor students' progress by moving from the context situation to the different representations, establishing the relationship between the variables, and identifying their functional dependence. [For the complete proceedings, see ED630210.]

In this study we analyze and contrast approaches that future high school math teachers show when solving word problems with the use of pencil and paper and later, with the use of a Dynamic Geometry System (DGS). The resources, representations, strategies and forms of mathematical reasoning that the participants exhibit when they use GeoGebra in the process of problem solving are analyzed. The results show that the use of the tool favors the dynamic exploration of the concepts involved, the formulation of conjectures and the search for different arguments to validate the solution. [This paper is provided in both Spanish and English. For the complete proceedings, see ED606531.]

Throughout the years, Mathematical Modeling has gained international attention, not only in research but also in the development of curricula and its applications in the classroom. However, systematic literary surveys are scarce. In this paper, we present some findings from a survey of 485 international publications related to different aspects of mathematical modeling. While the number of Latin-American publications is rather small compared to the number of international publications, the vitality of the currently discussed themes in this region and their innovative perspectives testify to the international relevance of this developed work on mathematical modeling. [This paper is provided in both Spanish and English. For the complete proceedings, see ED606531.]

In this paper characterizations of mental objects for fractions of middle school students (from 12 to 14 years old) with absenteeism problems and low academic performance, are described. A test was designed as part of a research whose general purpose is to contribute in the building up of better mental objects for fractions through a teaching sequence. Items for the test were structured according to the results of a didactic phenomenology of fractions and the curricular content proposed for the last years of elementary school. Results indicate that students have a greater success in issues related to continuous model partitions phenomena, whereas they are less successful using fractions on the number line. Students cannot transfer their knowledge from an area model to a linear model to identify or represent fractions. [This paper is written in both Spanish and English. For complete proceedings, see ED581294.]

In this paper we describe an educational proposal for prospective secondary school mathematics teachers in Mexico, whose aim was to contribute to the development of their didactic-mathematical knowledge about linear variation through mathematical tasks, the development of the competence of identifying primary mathematical objects and the reflection on hypothetical cases of teaching. The proposal is supported within both the Onto-semiotic Approach (EOS) and the Mathematics Teacher's Didactic-Mathematical Knowledge and Competence model. The proposal follows a design research methodology based on EOS. The results show the complexity related to the development of the competence of identifying primary mathematical objects, they also highlight the influence that competence has on the development of the epistemic and cognitive facets of prospective teachers' didactic-mathematical knowledge on linear variation. [For the complete proceedings, see ED629884.]

This paper presents a case study that is part of a research studying how university statistics teachers integrate digital tools in their practice. The case study is of a teacher who claims to follow constructionist ideas for her teaching of statistics, using the R programming language. The teacher developed a series of activities with R, aiming to promote the understanding of statistical concepts. We observed 11 of her classes, in a three-week period. These classes were analyzed using as framework the notion of Instrumental Orchestration and Papert's constructionist principles. We identified four ways in which this teacher orchestrates her class and the constructionist elements present in them. [For the complete proceedings, see ED629884.]

In this paper a didactic proposal is presented which consists of a methodology aimed at secondary level mathematics teachers, the aim of this proposal is to provide the teacher with a tool to design teaching sequences in order to take them to the classroom; These designs have the particularity of incorporating digital technology for the development of the sequence and its implementation. The methodology was developed based on the articulation of: the didactic structure of Díaz-Barriga, the teaching method ACODESA of Hitt and the curricular developments of Taba. The methodology has been tested with a group of 11 teachers in a course-workshop of 40 hours, the results obtained were three didactic sequences elaborated by three teams of teachers in a technological context (using Geogebra), where it is perceived that it is possible to design using this methodological proposal, however, teachers presented some difficulties during the process of articulation with technology. [For the complete proceedings, see ED629884.]

This paper presents the design and analysis of a Model-Eliciting Activity (MEA), which aims to support the refinement of the conceptual system associated with directly proportional distribution problems (example: linear function, percentage, proportion, variation, etc.). The situation explores a context that is close to the student, encouraging the construction of a generalizable mathematical model that can be transferred to diverse contexts. For the design, the principles of the Models and Modeling perspective (Lesh and Doerr, 2003) were considered. The target population were students enrolled in the first semesters of a university degree in Accounting. For the analysis, we considered the construction of models (performed in teams); the mathematical representations within the process observed (such as diagrams and tabular organizations); as well as the individual solutions of the students. [For the complete proceedings, see ED629884.]

In this paper we report the results obtained when implementing Activities related to the visualization of geometric representations in paper-and-pencil and technological environments with students1 grade 11. This is a qualitative research supported by both the representation theory and the visualization of mathematical objects using technological aid. We discuss how the use of technology promotes learning mathematical concepts through the analysis of static and dynamic geometric figures. Our results show that the technological tool contributes to the visualization of geometric representations arisen from working with Activities; however, paper-and-pencil as a working tool becomes necessary to give rise to conjectures about the meaning of abstract concepts linked to these representations. [This paper is presented in both English and Spanish. For the complete proceedings, see ED583608.]

In this study we document how a mathematical idea, shared by a student, changes the development of the class discussion. Our work characterizes the flexible and open dynamics in mathematics instruction that a teacher adopts to give her students control of their own mathematical ideas. Our work argues that this flexibility is necessary for teachers to engage in instructional practices that respond to students' mathematical thinking. To demonstrate the teacher's flexibility we use the analogy of soccer, relying on the literature of Game Sense coaching approach, which suggests that the coach (the teacher in our context) is positioned to manipulate the practice environment with the purpose of facilitating learning. [This paper is provided in both Spanish and English. For the complete proceedings, see ED606556.]

In this study, the elements of the design and results of the implementation of a Massive Open Online Course (MOOC) based on the problem solving and the use of digital technology are analyzed and discussed. The results show that the design of the activities, the intervention of the MOOC design team in the discussion forums and the interaction among the participants allowed and favored the creation of a collaborative environment in problem solving. Participants shared and discussed their mathematical ideas, as part of a virtual learning community, which allowed them to formulate conjectures, explore and search properties to sustain relationships, and communicate results. [This paper is provided in both Spanish and English. For the complete proceedings, see ED606531.]

The aim of this investigation is to explore the preservice teachers' reasoning about variation (variability or spread) when they analyze data in situations that involve risk. In particular, in this communication the responses to two problems of a questionnaire administered to 96 preservice teachers are reported. The problems are of comparing groups of data in situations of risk: stakes in games and the life expected after medical treatments. The questionnaire was applied before the preservice teachers began a course of statistical information processing and the results showed the difficulty found by students to interpret variation in this type of contexts. For these results it is necessary to reflect on the instruction of future teachers about the meanings of measures of center dispersion and dispersion to contribute to an improvement in their academic training. [This paper is written in both Spanish and English. For complete proceedings, see ED581294.]

We present results of the analysis of knowledge used by a secondary school mathematics teacher in her classroom practice. This knowledge takes shape and is displayed as specific teaching strategies in the management of her class when she incorporates Computer Algebra Systems. Based on observations of regular classes, we find that her knowledge (mathematical, pedagogical and technological) is put into action at various moments during the lesson and through a variety of teaching strategies. These strategies depend on many factors, such as the planned objective of the lesson, the specific moment in the class, student participation, and the use of technological tools. Within such complexity, the teacher applies her teaching strategies in a flexible way, and manages to control and even modify the course of her class. [This paper is written in both Spanish and English. For complete proceedings, see ED581294.]

The number line is a model that is used to measure, count, order and even operate, which requires a symbolic interpretation. Then, we investigate the conceptions of 72 in service secondary teachers, when they manage the model of the number line associated with order, spatial location and the relative position between numbers and marks. In a workshop carried out through Internet, we found that the participants consider that the usefulness of the model is in the manipulation, and they do not see the need to modify it, despite the requirements of the problem, taking for granted its usefulness and the control that resources could give them, such as the change of the numerical scale. They don't even realize the symbolic properties to interpret the numerability. [For the complete proceedings, see ED630210.]

In this report, we present an advance of doctoral research. We explore teaching alternatives that promote proportional reasoning in Mexican students between 14 and 15 years old with the support of digital technology. We designed a sequence of activities that pretends to signify the concepts of ratio and proportion in their diverse representations and from the perspective of linear functions. For the design of the tasks, elements of the Realistic Mathematics Education framework and Cuevas-Pluvinage Didactics were used. The learning objectives and proposed activities were organized through a hypothetical learning trajectory. Subject to presenting the group results, we randomly selected a student to present the analysis of the results achieved. We found that the tasks promoted proportional reasoning. [For the complete proceedings, see ED630210.]

In previous publications the authors of this paper identified specific relations between comprehension and convincement. On the basis of Grounded Theory, this research analyzes the changes that arise in said relations as a response to changing conditions. For their study, the authors analyze an interaction, at a distance, between a tutor and a student. At an initial level, a micro-analysis is undertaken--resorting to the Toulmin Model--which describes and interprets the relations that arise there between comprehension and convincement. From a more general standpoint, at a second level the authors contribute explanations concerning those relations and their changes. Finally, the authors suggest proposals for educational practice. [Written in both Spanish and English. For the complete proceedings, see ED583608.]

Existing technologies pose challenges for the education system related to what contents, strategies and skills students should learn and what types of teaching scenarios should be considered in learning. In this study, prospective high school teachers were encouraged to systematic use various digital technologies in a problem solving course during one semester. Results indicate that the participants developed and implemented a set of strategies and ways of thinking that complement and extend the approaches based on the use of pencil and paper. In addition, the use of technology allowed the participants extend mathematical discussion beyond formal settings. [Written in Spanish and English. For the complete proceedings, see ED583608.]

The study of three-dimensional objects is undertaken in the Mexican curriculum throughout all levels of basic education. However, few studies relate to the mathematics knowledge needed to teach those ideas. This report represents an approach to the study of such knowledge. The approach is based on an account, developed from class observations, of three six grade teachers and their use of mathematics knowledge in teaching their students to calculate the volume of prisms, in two schools in Mexico City. The results provide evidence related to some sub-domains of mathematics knowledge for teaching. While there are some strengths related to this knowledge, teachers also showed that there is room for improvement. This speaks to the need to develop training spaces (both for preservice and in-service teachers) where the analysis and discussions stimulate learning of both the geometric and pedagogical knowledge needed for teaching practice. [Funding for this project was provided by Conacyt/SEP/SEB. This paper is presented in both English and Spanish. For the complete proceedings, see ED583608.]

Interest in understanding mathematics teaching and learning phenomena, and to develop new and effective methodologies to teach Differential Calculus and Linear Algebra led me to look for ways to dialogue between APOS Theory and other mathematics education theories. This enterprise has facilitated a better understanding of them. Bridges between them have been constructed that enable to find possible points of contact and to expand their borders without changing their fundamental hypothesis. In this paper, a reflection on what a dialog between theories entails is illustrated and, through three examples, it shows how it promotes enriching experiences for researchers and stimulates students' creativity. Results also underline how positive outcomes in terms of students learning and in the development of effective teaching resources can be obtained. [Written in both Spanish and English. For the complete proceedings, see ED583608.]

The purpose of this study was to analyze how the systematic use of GeoGebra influences the ways of reasoning of high school students when they incorporate it in the resolution of word problems. The study was carried out with a group of 20 students who were studying Mathematics I. The results show that the students appropriated resources from the dynamic geometry system (GeoGebra) which, when used systematically, allowed them to implement strategies to represent geometrically the concepts involved in the problems, explore and analyze relationships between the elements of the dynamic models built and find the solutions. Thus, thanks to the exploration and analysis of relationships, it was possible to discuss with the students' concepts such as: ratio, domain, locus, variation. [For the complete proceedings, see ED630060.]

This article describes the results of an investigation based on a Models and Modeling Perspective [MMP]. We present the evolution of the models built by university students when solving a model development sequence designed to promote their learning of the exponential function. As a result, we observed that students' thinking was modified, expanded, and refined, as they developed different iterations of their models. Students' models evolved by creating, first, linear models that required direction; second, models where there was no dissociation between linear and exponential behavior; then, situated exponential models; and finally, sharable, and reusable exponential models. [For the complete proceedings, see ED630060.]

During their school life, students learn mathematical topics that can be considered essential for the understanding of the property of density in the set of real numbers. Therefore, we detected a need to design and elaborate a Hypothetical Learning Path to include topics to help promote the learning of this property. This report shows results of a first stage of an educational experiment as part of an ongoing research. It describes how through the trajectory, high school students are able to recognize ways of finding numbers in an interval using various semiotic representations. We also describe some difficulties that students had to recognize the non-existence of a successor in real numbers. [For the complete proceedings, see ED630060.]

In this research, quotient strategies and their influence on decision-making in situations that involve the comparison of probabilities are analyzed. In order to achieve this, classical probability situations modeled with urns were designed. In each situation, two urns with simple extraction, involving or not proportional relationships, were proposed to third grade secondary school students. The analysis was carried out through the categorization of the results based on the relationships established between the components of the urns (favorable, unfavorable and possible cases). The study concluded that when comparing probabilities, students not only resort to the quotient (ratio): favorable case numbers among the number of possible cases to make their choice, but to others where they also include unfavorable cases. [For the complete proceedings, see ED629884.]

Researchers, who have studied the understanding of the density property in the set of decimal numbers, have shown that the student uses the property of the discrete of natural numbers to solve tasks related to density. So, a restructuring of concepts is necessary, that is, a conceptual change from "the discrete" to "the dense". This report presents evidence from ten pre-service teachers from Mexico City that this change can initiate through the implementation of a didactic sequence. The pre-service teachers managed to visualize that a decimal number can be found in an interval. Consequently, they were able to conceive an infinity. However, several of them persisted with the idea of the existence of a successor in the set of decimal numbers. [For the complete proceedings, see ED629884.]