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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 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.]

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.]

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.]

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.]

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.]

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.]

In this research we discuss relating the description of certain harmonic phenomena, with models that represent them, with the goal of promoting the understanding of the period, the frequency, and the amplitude of the trigonometric functions, particularly referring to sinusoidal functions; in this way the frontier between sinusoidal functions and one of its areas of application to physics is preserved. This is an ongoing research study where digital technology like a movement sensor is employed in order to generate graphics related to sinusoidal functions, in which the movement features followed by harmonic phenomena are identified. It is important to mention the use of an approach where it is possible to transit alternatively from the analysis of the phenomena to the mathematical model. It is conjectured that the way it approaches sinusoidal functions could help assign meaning to the model f(x) =Acos(?x+[phi]). [Written in both Spanish and English. For the complete proceedings, see ED583608.]

The articles included in this volume were selected as exemplary papers from the conference in Mexico. The goal of the forum was to provide an opportunity for researchers from the three branches of the Universidad Autonoma Metropolitana (UAM) to meet and discuss research projects currently in progress at the university. The works presented here reflect the wide interests and research priorities that are currently being investigated in the field of foreign language education. Papers include: "The Literary Text as a Reflective Tool and Active Reading Experience in the Foreign Language Classroom (Yvonne Cansigno Gutierrez, Luz Maria Reyna Malvaez Swain); "The Attitude of the University Student Toward Learning English and Its Impact on School Success: The Case of English Reading Classes at the UAM-Iztapalapa" (Susana Ibarra-Puig); "On the Threshold of Cross-Cultural Encounters" (Margaret Lee Zoreda); "Assessment in Foreign Languages: An Essential Part of the Teaching-Learning Process" (Margarita Elena Magana Sanchez, Mamoudou Si Diop, Amalia Tellez Salazar); "The Goal, Justification, and Basis of a Choral Poetry Workshop in the German Language, Designed for the UAM-Azcapotzalco" (Maria Clotilde Rivera Ochoa); and "Towards an Analysis of the Psycholinguistic Determinants of the Processing of Written Academic Discourse in a Foreign Language: The Case of Cognitive Style" (Javier Vivaldo Lima). (LH)

Career education should be part of every subject area on every grade level, from kindergarten through graduate school. Its components--self-awareness, educational awareness, career awareness, and curriculum infusion--can be included in language classes through special activities gleaned from career books, newspapers, and magazines; through judicious use of the regular classroom text; and by using the ancillary texts recommended. Important career activities described here include a Career Day and school and classroom bulletin boards for career information. Motivating activities are suggested for improving students' speaking, reading, and writing skills. Three texts dealing specifically with careers in foreign languages are mentioned, as well as a number of foreign language texts with information on numerous careers, which can be used in language and/or conversation courses at various levels. There is a sizable appendix containing an annotated bibiliography, guidelines for conducting a Career Day, and activities for each of the four aspects of career education. Some materials are given in both Spanish and English. (Author/AMH)

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)

This proceedings of the annual conference of the International Group for the Psychology of Mathematics Education (PME) includes the following research papers: "Children's Connections among Representations of Mathematical Ideas" (A. Alston & C.A. Maher); "Algebraic Syntax Errors: A Study with Secondary School Children" (A. Avila, F. Garcia, & T. Rojano); "The Development of Conceptual Structure as a Problem Solving Activity" (V. Cifarelli); "From Arithmetic to Algebra: Negotiating a Jump in the Learning Process" (A. Cortes, N. Kavafian, & G. Vergnaud); "Continuous Analysis of One Year of Science Students' Work in Linear Algebra, in First Year French University" (J.L. Dorier); "Avoidance and Acknowledgement of Negative Numbers in the Context of Linear Equations" (A. Gallardo);"Introducing Algebra: A Functional Approach in a Computer Environment" (M. Garancon, C. Kieran, & A. Boileau); "LOGO, to Teach the Concept of Function" (D. Guin & I.G. Retamal); "The Concept of Function: Continuity Image versus Discontinuity Image: Computer Experience" (F. Hitt); "Acquisition of Algebraic Grammar" (D. Kirshner); "Embedded Figures and Structures of Algebraic Expressions" (L. Linchevski & S. Vinner); "A Framework for Understanding What Algebraic Thinking Is" (R.L. Lins); "Developing Knowledge of Functions through Manipulation of a Physical Device" (L. de Lemos Meira); "Students' Interpretations of Linear Equations and Their Graphs" (J. Moschkovich); "An Experience to Improve Pupil's Performance in Inverse Problems" (A. Pesci); "Algebra Word Problems: A Numerical Approach for Its Resolution: A Teaching Experiment in the Classroom" (G. Rubio); "Children's Writing about the Idea of Variable in the Context of a Formula" (H. Sakonidis & J. Bliss); "Observations on the 'Reversal Error' in Algebra Tasks" (F. Seeger); "Generalization Process in Elementary Algebra: Interpretation and Symbolization" (S.U. Legovich); "Effects of Teaching Methods on Mathematical Abilities of Students in Secondary Education Compared by Means of a Transfer Test" (J. Meijer); "On Long Term Development of Some General Skills in Problem Solving: A Longitudinal Comparative Study" (P. Boero); "Cognitive Dissonance versus Success as the Basis for Meaningful Mathematical Learning" (N.F. Ellerton & M.A. Clements); "Time and Hypothetical Reasoning in Problem Solving" (P.L. Ferrari); "The Interplay between Student Behaviors and the Mathematical Structure of Problem Situations: Issues and Examples" (R. Herschkowitz & A. Arcavi); "Paradigm of Open-Approach Method in the Mathematics Classroom Activities: Focus on Mathematical Problem Solving" (N. Nohda); "Reflexions sur le Role du Maitre dans les Situations Didactiques a Partir du Cas de l'Enseignement a des Eleves en Difficulte" (M.J.P. Glorian); "Diagnosis and Response in Teaching Transformation Geometry" (A. Bell & D. Birks); "Children's Recognition of Right Angled Triangles in Unlearned Positions" (M. Cooper & K. Krainer); "The Role of Microworlds in the Construction of Conceptual Entities" (L.D. Edwards); "The Cognitive Challenge Involved in Escher's Potato Stamps Microworld" (R. Hadass); "Study of the Degree of Acquisition of the Van Hiele Levels by Secondary School Students" (A. Jaime & A. Gutierez); "Spatial Concepts in the Kalahari" (H. Lea); "Integrating LOGO in the Regular Maths Curriculum: A Developmental Risk or Opportunity?" (T. Lemerise); "Young Children Solving Spatial Problems" (H. Mansfield & J. Scott); "The Role of Format in Students' Achievement of Proof" (W.G. Martin); "L'influence des Aspects Figuratifs dans le Raisonnment des Eleves en Geometrie" (A. Mesquita); "Children's Understanding of Congruence According to the Van Hiele Model of Thinking" (L. Nasser); "Prospective Primary Teachers' Conceptions of Area" (C. Tierney, C. Boyd, & G. Davis); "Probability Concepts and Generative Learning Theory" (O. Bjorkqvist); "Some Considerations on the Learning of Probability" (A.M.O. Salazar); "Gambling and Ethnomathematics in Australia" (R. Peard); "Mathematization Project in Class as a Collective Higher Order Learning Process" (H.G. Steiner). Also includes a listing of author addresses. (MKR)

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)