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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 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.]

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 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.]

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.]

This study reports an analysis of inductive reasoning of Mexican middle school mathematics teachers, when solving tasks of generalization of a quadratic sequence in the context of figural patterns. Data was collected from individual interviews and written answers to generalization tasks. Based on Cañadas and Castro's inductive reasoning model, we found that most of the teachers followed four stages to obtain a general rule: observation of particular cases, search of patterns, conjecture formulation and generalization. [For the complete proceedings, see ED629884.]

This research project focuses on identifying the difficulties that children between 6-7 years-old have in learning natural numbers, when working a teaching model with a Von Neumann formal mathematical basis. In this study, we present the analysis of the experimentation with a first-grade class in the construction of zero and one numbers, contrasting the results with a case of the clinical interview. Our methodological theoretical framework is the Local Theoretical Model (LTM) and its four components: Formal, Cognitive, Communication and Teaching. With the theoretical contribution of each component, the categories of analysis are designed to observe and explain difficulties in the use of the Mathematical Sign System (MSS) involved in the construction of the first natural numbers, using iteration and the recursive process. [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.]

In this study we show the models that students of administration and accounting careers built when performing an activity close to the real life, with the use of Excel. The representations and ideas exhibited by the students when they used the software are analyzed. The methodology was qualitative. The theoretical framework was the Models and Modeling Perspective. The results show that the use of the tool allowed the dynamic exploration of mathematical concepts such as: covariation, rate of change and exponential function. The students formulated conjectures, arguments, explanations and justifications. [For the complete proceedings, see ED606556.]

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 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.]

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.]

This proceedings of the annual conference of the International Group for the Psychology of Mathematics Education (PME) includes the following papers: "The Knowledge of Cats: Epistemological Foundations of Mathematics Education" (R.B. Davis) and "PME Algebra Research: A Working Perspective" (E. Filloy); "Some Misconceptions in Calculus: Anecdotes or the Tip of an Iceberg?" (M. Amit & S. Vinner); "Difficultes Cognitives et Didactiques dans la Construction de Relations entre Cadre Algebrique et Cadre Graphique" (M. Artigue); "Unbalance and Recovery: Categories Related to the Appropriation of a Basis of Meaning Pertaining to the Domain of Physical Thinking" (R. Cantoral); "On Difficulties with Diagrams: Theoretical Issues" (T. Dreyfus & T. Eisenberg); "The Two Faces of the Inverse Function: Prospective Teachers' Use of 'Undoing'" (R. Even); "Intuitive Processes, Mental Image, and Analytical and Graphic Representations of the Stationary State: A Case Study" (R.M. Farfan & F. Hitt); "The Role of Conceptual Entities in Learning Mathematical Concepts at the Undergraduate Level" (G. Harel & J. Kaput); "Mathematical Concept of Formation in the Individual" (L. Lindenskov); "Pupils' Interpretations of the Limit Concept: A Comparison Study between Greeks and English" (J. Mamona-Downs); "Infinity in Mathematics as a Scientific Subject for Cognitive Psychology" (R.N. Errazuriz); "Organizations Deductives et Demonstration" (L. Radford); "The Teaching Experiment 'Heuristic Mathematics Education'" (A. Van Streun); "The Understanding of Limit: Three Perspectives" (S.R. Williams);"Self Control in Analyzing Problem Solving Strategies" (G. Becker); "Influences of Teacher Cognitive/Conceptual Levels on Problem-Solving Instruction" (B.J. Dougherty); "Can Teachers Evaluate Problem Solving Ability?" (F.O. Flener & J. Reedy); "Teacher Conceptions about Problem Solving and Problem Solving Instruction" (D.A. Grouws, T.A. Good, & B.J. Dougherty); "Math Teachers and Gender Differences in Math Achievement, Math Participation and Attitudes Towards Math" (H. Kuyper & M.P.C. van der Werf); "Teaching Students to be Reflective: A Study of Two Grade Seven Classes" (F.F. Lester & D.L. Kroll); "Students' Affective Responses to Non-Routine Mathematical Problems: An Empirical Study" (D.B. McLeod, C. Craviotto, & M. Ortega); "Accommodating Curriculum Change in Mathematics: Teachers' Dilemmas" (R. Nolder); "Teachers' Characteristics and Attitudes as Mediating Variables in Computer-Based Mathematics Learning" (R. Noss, C. Hoyles, & R. Sutherland); "Teachers' Perceived Roles of the Computer in Mathematics Education" (J. Ponte); "Mathematics Process as Mathematics Content: A Course for Teachers" (D. Schifter); "Psychological/Philosophical Aspects of Mathematical Activity: Does Theory Influence Practice?" (R. Scott-Hodgetts & S. Lerman); "A Web of Beliefs: Learning to Teach in an Environment with Conflicting Messages" (R.G. Underhill). Includes 27 poster presentation abstracts and a listing of author addresses. (MKR)

This proceedings of the annual conference of the International Group for the Psychology of Mathematics Education (PME) contains the following research papers: "The Construct Theory of Rational Numbers: Toward a Semantic Analysis" (M. Behr & G. Harel); "Reflections on Dealing: An Analysis of One Child's Interpretations" (G. Davis); "About Intuitional Knowledge of Density in Elementary School" (J. Gimenez); "Understanding the Multiplicative Structure: Concepts at the Undergraduate Level" (G. Harel & M. Behr); "A Contextual Approach to the Teaching and Learning of Mathematics: Outlining a Teaching Strategy that Makes Use of Pupil's Real World Experiences and Strategies, and the Results of the First Teaching Experiment of Project" (T.O. Keranto); "On Children's Mathematics Informal Method" (F.L. Lin & L.R. Booth); "A Case Study of the Role of Unitizing Operations with Natural Numbers in the Conceptualization of Fractions" (A. Ludlow); "Constructing Fractions in Computer Microworlds" (J. Olive & L.P. Steffe); "Proportional Reasoning: From Shopping to Kitchens, Laboratories, and Hopefully, Schools" (A.D. Schliemann & V.P. Magalhaes); "The Fraction Concept in Comprehensive School at Grade Levels 3-6 in Finland" (T. Strang); "Critical Decisions in the Generalization Process: A Methodology for Researching Pupil Collaboration in Computer and Non Computer Environments" (L. Healy, C. Hoyles, & R.J. Sutherland); "'Scaffolding' a Crutch or a Support for Pupils' Sense-Making in Learning Mathematics" (B. Jaworski); "The Role of Mathematical Knowledge in Children's Understanding of Geographical Concepts" (R.G. Kaplan); "Speaking Mathematically in Bilingual Classrooms: An Exploratory Study of Teacher Discourse" (L.L. Khisty, D.B. McLeod, & K. Bertilson); "The Emergence of Mathematical Argumentation in the Small Group Interaction of Second Graders" (G. Krummheuer & E. Yackel); "Potential Mathematics Learning Opportunities in Grade Three Class Discussions" (J.J. Lo, G.H. Wheatley, & A.C. Smith); "Certain Metonymic Aspects of Mathematical Discourse" (D.J. Pimm); "Inverse Relations: The Case of the Quantity of Matter" (R. Stavv & T. Rager); "The Development of Mathematical Discussion" (T. Wood); "Estrategias y Argumentos en el Estudio Descriptivo de la Asociacion Usando Microordenadores" (J.D. Godino, C. Batanero, & A.E. Castro); "Computerized Tools and the Process of Modeling" (C. Hancock & J. Kaput); "Examples of Incorrect Use of Analogy in Word Problems" (L. Bazzini); "Children's Pre-concept of Multiplication: Procedural Understanding" (C. Beattys, N. Herscovics, & N. Nantais); "The Kindergartners' Knowledge of Numerals" (J.C. Bergeron); "An Analysis of the Value and Limitations of Mathematical Representations Used by Teachers and Young Children" (G.M. Bouton-Lewis & G.S. Halford); "A Study on the Development of Second-Graders' Ability in Solving Two-Step Problems" (F. Cao); "Understanding the Division Algorithm from New Perspectives" (D.W. Carraher); "Negative Numbers Without the Minus Sign" (T.N. Carraher); "Learning Difficulties Behind the Notion of Absolute Value" (I. Chiarugi, G. Fracassina, & F. Furinghetti); "Le Role de la Representation dans la Resolution des Problemes Additifs" (R.F. Damm); "Using a Computerized Tool to Promote Students''Number Sense' and Problem Solving Strategies" (N. Hativa, V. Bill, S. Hershkovitz, & I. Machmandrow); "Children's Understanding of Compensation, Addition and Subtraction in Part/Whole Relationships" (K.C. Irwin); "Factors Affecting Children's Strategies and Success in Estimation" (C. Morgan); "Assessment in Primary Mathematics: The Effects of Item Readability" (J.A. Mousley); "Social Sense-Making in Mathematics: Children's Ideas of Negative Numbers" (S. Mukhopadhyay, L. B. Resnick, & L. Schauble); "Children's Pre-Concept of Multiplication: Logico Physical Abstraction" (N. Nantais & N. Herscovics); "Building on Young Children's Informal Arithmetical Knowledge" (A.I. Oliver, A. Murray, & P. Human); "From Protoquantities to Number Sense" (L.B. Resnick, S. Lesgold, & V. Bill); "Prospective Elementary Teachers' Knowledge of Division" (M.A. Simon); "Relative and Absolute Error in Computational Estimation" (J. Sowder, L. Sowder, & Z. Markovits); "A Child Generated Multiplying Scheme" (L. Steffe); and "Salient Aspects of Experience with Concrete Manipulatives" (P. Thompson & A.G. Thompson). Includes a listing of author addresses. (MKR)