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We present the results of a research project on arithmetic-algebraic thinking that was carried out jointly by a team in Mexico and another in Quebec. The project deals with the concepts of variable and covariation between variables in the sixth grade at the elementary level and the first, second, and third years of secondary school--namely, children from 11 to 14 years old. We target secondary students (first year or K7) in this article. Our objective relates to the development of a gradual generalization in arithmetic-algebraic thinking in a socio-cultural approach to the learning of mathematics. We experimented with investigative situations using a paper-and-pencil approach and technology. We analyze the emergence, in this context, of a visual abstraction, the production of institutional and non-institutional representations, a sensitivity to contradiction, and, finally, the concepts of variable and of covariation between variables. [For the complete proceedings, see ED629884.]

This submission contains the Proceedings of the 2007 Annual Meeting of the Canadian Mathematics Education Study Group (CMESG), held at the University of New Brunswick in Fredricton, New Brunswick. The CMESG is a group of mathematicians and mathematics educators who meet annually to discuss mathematics education issues at all levels of learning. The aims of the Study Group are: to advance education by organizing and coordinating national conferences and seminars to study and improve the theories of the study of mathematics or any other aspects of mathematics education in Canada at all levels; and to undertake research in mathematics education and to disseminate the results of this research. These proceedings include plenary lectures, working group reports, topic session descriptions, new PhD reports, and summaries of ad hoc sessions and a panel discussion. Papers include: (1) Understanding Abstraction in Mathematics Education: Meaning, Language, Gesture, and the Human Brain (Rafael Nunez); (2) Mathematics Departments, New Faculty, and the Future of Collegiate Mathematics (T. Christine Stevens); (3) Outreach in Mathematics--Activities, Engagement, & Reflection (Veronique Hussin and Eric Muller); (4) Geometry, Space, and Technology: Challenges for Teachers and Students (Shelley Hunter, Donna Kotsopoulos, and Walter Whiteley); (5) The Design and Implementation of Learning Situations (Anna McQuillan, Fernando Hitt, and Luis Radford); (6) The Multifaceted Role of Feedback in the Teaching and Learning of Mathematics (Jerome Proulx and Florence Glanfield); (7) Mathematics Educational Neuroscience: Origins, Activities, and New Opportunities (Stephen R. Campbell); (8) Communicating the Excitement and Beauty of Mathematics (Malgorzata Dubiel); (9) Conception et experimentation de situations didactiques au prescolaire/primaire / Design and Experimentation of Didactical Situations in Kindergarten and Elementary School (Jacinthe Giroux) [Written in French]; (10) Cabri 3D: An Environment for Purposeful Mathematical Activity? (Kate Mackrell); (11) Equity and Opportunities to Learn in Cooperative Mathematical Work (Indigo Esmond); (12) (Elargir les) Connaissances mathematiques des enseignants de mathematiques du secondaire: une etude sur la formation continue / (Enlarging) Secondary-Level Mathematics Teachers' Mathematical Knowledge: An Investigation of Professional Development (Jerome Proulx) [Presented in both French and English]; (13) Un paradigme d'experimentation au laboratoire de sciences pour l'identification et l'optimisation statistique d'un modele algebrique par interaction visuo-graphique (Georges Touma) [Written in French]; (14) What Factors Influence Understanding of Written Arithmetic Problems? (Dominic Voyer); (15) Probabilities / Probabilites (Egan Chernoff and Annie Savard); (16) First Steps toward an Archeology of Gesture in Graphing (Susan Gerofsky); (17) Function Modelling Using Secondary Data from Statistics Canada's E-STAT Database (Jennifer Hall); (18) Explanation and Proof in Mathematics and Mathematics Education (Gilla Hanna, Ella Kaye, and Riaz Saloojee); (19) Exploring Our Embodied Knowing of the Gauss-Bonnet Theorem: Barn-Raising and Endo-Pentakis-Icosi-Dodecahedron (Eva Knoll); (20) Mathematical Biography (John Grant McLoughlin); (21) This Is Mathematics; This Isn't Mathematics; But That... I'm Not So Certain About: The Possible "Emancipation" of Secondary School Mathematics from the Bonds of "Real" Mathematics (Craig Newell); and (22) What Courses Could or Should Mathematics Departments Offer to Graduate Programs in Mathematics Education? (France Caron, Morris Orzech, and Elaine Simmt). Appended are: (1) Working Groups at Each Annual Meeting; (2) Plenary Lecture at Each Annual Meeting; and (3) Proceedings of Annual Meetings. Individual papers contain references, figures, and tables. [Abstract modified to meet ERIC guidelines. For the 2006 proceedings, see ED529562.]

This document contains the second volume of the proceedings of the 30th Annual Conference of the International Group for the Psychology of Mathematics Education. Conference presentations are centered around the theme "Mathematics at the Centre." This volume features 60 research reports by presenters with last names beginning between Abr and Dri: (1) The Odds of Understanding the Law of Large Numbers: A Design for Grounding Intuitive Probability in Combinatorial Analysis (Dor Abrahamson and Rose M. Cendak); (2) Imaginary-Symbolic Relations, Pedagogic Resources and the Constitution of Mathematics for Teaching in In-Service Mathematics Teacher Education (Jill Adler and Zain Davis); (3) Relationship between Pre-Service Mathematics Teachers' Teaching and Learning Beliefs and Their Practices (Hatice Akkoc and Feral Ogan-Bekiroglu); (4) Teachers' Awareness of Dimensions of Variation: A Mathematics Intervention Project (Thabit Al-Murani); (5) The Student Teacher and the Others: Multimembership on the Process of Introducing Technology in the Classroom (Nelia Amado and Susana Carreira); (6) Improving Student Teachers' Understanding of Fractions (Solange Amorim Amato); (7) Autodidactic Learning of Probabilistic Concepts through Games (Miriam Amit and Irma Jan); (8) Graduate Students' Processes in Generating Examples of Mathematical Objects (Samuele Antonini); (9) Reasoning in an Absurd World: Difficulties with Proof by Contradiction (Samuele Antonini and Maria Alessandra Mariotti); (10) Will Penelope Choose Another Bridegroom? Looking for an Answer through Signs (Ferdinando Arzarello, Luciana Bazzini, Francesca Ferrara, Ornella Robutti, Cristina Sabena, and Bruna Villa); (11) Motivation and Perceptions of Classroom Culture in Mathematics of Students across Grades 5 to 7 (Chryso Athanasiou and George N. Philippou); (12) Deductive Reasoning: Different Conceptions and Approaches (Michal Ayalon and Ruhama Even); (13) The Tendency to Use Intuitive Rules among Students with Different Piagetian Cognitive Levels (Reuven Babai); (14) Coming to Appreciate the Pedagogical Uses of CAS (Lynda Ball and Kaye Stacey); (15) Students' Conceptions of "m" and "c": How to Tune a Linear Function (Caroline Bardini and Kaye Stacey); (16) A Contradiction between Pedagogical Content Knowledge and Teaching Indications (Ibrahim Bayazit and Eddie Gray); (17) Identifying and Supporting Mathematical Conjectures through the Use of Dynamic Software (David Benitez Mojica and Manuel Santos Trigo); (18) Students Constructing Representations for Outcomes of Experiments (Palma Benko and Carolyn A. Maher); (19) Logarithms: Snapshots from Two Tasks (Tanya Berezovski and Rina Zazkis); (20) Trying to Reach the Limit--The Role of Algebra in Mathematical Reasoning (Christer Bergsten); (21) Semiotic Sequence Analysis--Constructing Epistemic Types Empirically (Angelika Bikner-Ahsbahs); (22) Service Teaching: Mathematical Education of Students of Client Departments (Erhan Bingolbali, John Monaghan, and Tom Roper); (23) Students' Thinking about the Tangent Line (Irene Biza, Constantinos Christou, and Theodossios Zachariades); (24) Habermas' Theory of Rationality as a Comprehensive Frame for Conjecturing and Proving in School (Paulo Boero); (25) Extending Students' Understanding of Decimal Numbers via Realistic Mathematical Modeling and Problem Posing (Cinzia Bonotto); (26) Different Media, Different Types of Collective Work in Online Continuing Teacher Education: Would You Pass the Pen, Please? (Marcelo C. Borba and Rubia B. A. Zulatto); (27) Reformulating "Mathematical Modelling" in the Framework of the Anthropological Theory of Didactics (Marianna Bosch, Fco. Javier Garcia, Josep Gascon, and Luisa Ruiz Higueras); (28) Students' Impressions of the Value of Games for the Learning of Mathematics (Leicha A. Bragg); (29) The Transition from Arithmetic to Algebra: To Reason, Explain, Argue, Generalize and Justify (Trygve Breiteig and Barbro Grevholm); (30) Resisting Reform Pedagogy: Teacher and Learner Contributions (Karin Brodie); (31) Manifestations of Affordances of a Technology-Rich Teaching and Learning Environment (TRTLE) (Jill P. Brown); (32) Types of Representations of the Number Line in Textbooks (Alicia Bruno and Noemi Cabrera); (33) Educational Neuroscience: New Horizons for Research in Mathematics Education (Stephen R. Campbell); (34) Variability in a Probability Context: Developing Pre-Service Teachers' Understanding (Daniel L. Canada); (35) Implementing a Reform-Oriented Mathematics Syllabus: A Survey of Secondary Teachers (Michael Cavanagh); (36) Student's Modelling with a Lattice of Conceptions in the Domain of Linear Equations and Inequations (Hamid Chaachoua, Marilena Bittar, and Jean-Francois Nicaud); (37) Using Reading and Coloring to Enhance Incomplete Prover's Performance in Geometry Proof (Ying-Hao Cheng and Fou-Lai Lin); (38) Aspects of Teachers' Pedagogical Content Knowledge for Decimals (Helen Chick, Monica Baker, Thuy Pham, and Hui Cheng); (39) Collaborative Action Research on Implementing Inquiry-Based Instruction in an Eighth Grade Mathematics Class: An Alternative Mode for Mathematics Teacher Professional Development (Erh-Tsung Chin, Yung-Chi Lin, Yann-Tyng Ko, Chi-Tung Chien, and Hsiao-Lin Tuan); (40) Routine and Novel Mathematical Solutions: Central-Cognitive or Peripheral-Affective Participation in Mathematics Learning (Mei-Shiu Chiu); (41) The Role of Self-Generated Problem Posing in Mathematics Exploration (Victor V. Cifarelli and Jinfa Cai); (42) A Longitudinal Study of Children's Mental Computation Strategies (Barbara Clarke, Doug M. Clarke, and Marj Horne); (43) Assessing Fraction Understanding Using Task-Based Interviews (Doug M. Clarke, Michal Sukenik, Anne Roche, and Annie Mitchell); (44) Evaluation of a Teaching Concept for the Development of Problem Solving Competences in Connection with Self-Regulation (Christina Collet and Regina Bruder); (45) Developing Probability Thinking in Primary School: A Case Study on the Constructive Role of Natural Language in Classroom Discussions (Valeria Consogno, Teresa Gazzolo, and Paulo Boero); (46) Collaboration with Teachers to Improve Mathematics Learning: Pedagogy at Three Levels (Tom J. Cooper, Annette R. Baturo, and Edlyn J. Grant); (47) "Aim High--Beat Yourself": Effective Mathematics Teaching in a Remote Indigenous Community (Tom J. Cooper, Annette R. Baturo, Elizabeth Warren, and Edlyn J. Grant); (48) Development of Children's Understanding of Length, Area, and Volume Measurement Principles (Margaret Curry, Michael Mitchelmore, and Lynne Outhred; (49) Mathematics-for-Teaching: The Cases of Multiplication and Division (Brent Davis, Elaine Simmt, and Dennis Sumara); (50) Generative Concept Images (Gary E. Davis and Catherine A. Pearn); (51) Developmental Assessment of Data Handling Performance Age 7-14 (Pauline Davis, Maria Pampaka, Julian Williams, and Lawrence Wo); (52) The Effect of Different Teaching Tools in Overcoming the Impact of the Intuitive Rules (Eleni Deliyianni, Eleni Michael, and Demetra Pitta-Pantazi); (53) Investigating Social and Individual Aspects in Teacher's Approaches to Problem Solving (Fien Depaepe, Erik De Corte, and Lieven Verschaffel); (54) Maths Avoidance and the Choice of University (Pietro Di Martino and Francesca Morselli); (55) Primary Students' Reasoning about Diagrams: The Building Blocks of Matrix Knowledge (Carmel M. Diezmann); (56) Integrating Errors into Developmental Assessment: "Time" for Ages 8-13 (Brian Doig, Julian Williams, Lawrence Wo, and Maria Pampaka); (57) Vygotsky's Everyday Concepts/Scientific Concepts Dialectics in School Context: A Case Study (Nadia Douek); (58) Creating Mathematical Models with Structures (Katherine Doyle); (59) Mechanisms for Consolidating Knowledge Constructs (Tommy Dreyfus, Nurit Hadas, Rina Hershkowitz, and Baruch Schwarz); and (60) Reconciling Factorizations Made with CAS and with Paper-and-Pencil: The Power of Confronting Two Media (Paul Drijvers, Carolyn Kieran, Andre Boileau, Fernando Hitt, Denis Tanguay, Luis Saldanha, and Jose Guzman). (Individual papers contain references.)

This submission contains the Proceedings of the 2002 Annual Meeting of the Canadian Mathematics Education Study Group (CMESG), held at Queen's University in Kingston, Ontario. The CMESG is a group of mathematicians and mathematics educators who meet annually to discuss mathematics education issues at all levels of learning. The aims of the Study Group are: to advance education by organizing and coordinating national conferences and seminars to study and improve the theories of the study of mathematics or any other aspects of mathematics education in Canada at all levels; and to undertake research in mathematics education and to disseminate the results of this research. These proceedings include plenary lectures, working group reports, topic session descriptions, new PhD reports, and summaries of ad hoc and special anniversary sessions. Papers include: (1) Toward a Practice-Based Theory of Mathematical Knowledge for Teaching (Deborah Ball and Hyman Bass); (2) The Experimental Mathematician: The Pleasure of Discovery and the Role of Proof (Jon Borwein); (3) Mathematics and the Arts (Susan Gerofsky, Nathalie Sinclair, and Brent Davis); (4) Philosophy for Children and Mathematics (Richard Pallascio and Elaine Simmt); (5) The Arithmetic/Algebra Interface: Implications for Primary and Secondary Mathematics / Articulation arithmetique/algebre: Implications pour l'enseignement des mathematiques au primaire et au secondaire (Nadine Bednarz and Lesley Lee); (6) Mathematics, the Written and the Drawn (David Pimm and Nick Jackiw); (7) Des cours de mathematiques pour les futurs (et actuels) maitres au secondaire / Types and Characteristics Desired of Courses in Mathematics Programs for Future (and In-Service) Teachers (Walter Whiteley, Peter Taylor, and Harry White) [Presented in both French and English]; (8) High School Mathematics Teachers' Perspectives of Mathematical Word Problems (Olive Chapman); (9) What are Critical Online Experiences for Mathematics Teachers and Students? (George Gadanidis); (10) Socioeconomic Gradients in Mathematics Achievement: Findings for Canada from the Third International Mathematics and Science Study (George Frempong); (11) Light and Shadows (in) Knowing Mathematics and Science (Kathleen Nolan); (12) Supporting Student Efforts to Learn with Understanding: An Investigation of the Use of "JavaSketchpad" Sketches in the Secondary Geometry Classroom (Margaret P. Sinclair); (13) Conceptualizing Limit in Calculus (Why is It Still a Problem?) (Allan Brown); (14) Amplifying Mathematical Intelligence Using Web-Based Interactive Activities (George Gadanidis); (15) Grades 5-6 Teachers' Algebra Teaching Beliefs and Practices (George Gadanidis); (16) Researching the Effect of Interactive Applets in Mathematics Teaching (George Gadanidis); (17) Creativity and the Psychology of Mathematical Invention (Peter Liljedahl); (18) Logical Reasoning and Mathematical Games (Ralph Mason and Janelle McFeetors); (19) Collective Mathematical Thinking (Immaculate Namukasa); (20) Proofs and Refutations on the Web: Mathematics Environments for Grades 7 & 8 (Geoffrey Roulet); (21) Being in a Mathematical Place: Immersion in Mathematical Investigation (David Wagner); (22) Reflections on 25 Years with CMESG (Eric Muller); (23) Reflexions d'un mathematicien sur le GCEDM (Frederic Gourdeau) [Written in French]; (24) A Historical Perspective on Mathematics Education Research in Canada: The Emergence of a Community (Carolyn Kieran); (25) Lessons from the Past, Questions for the Future (Brent Davis); and (26) Lessons from the Past, Questions for the Future: meditation sur theme impose (Roberta Mura) [Written in French]. Appended are: (1) Working Groups at Each Annual Meeting; (2) Plenary Lectures at Each Annual Meeting; and (3) Proceedings of Annual Meetings. Individual papers contain references, figures, and tables. [Abstract modified to meet ERIC guidelines. For the 2001 proceedings, see ED472094.]

This study contributes to a growing body of research on the development of elementary teacher's content knowledge of mathematics. Individual clinical interviews were conducted with preservice elementary teachers (N=21) enrolled in a professional development course called "Foundations of Mathematics for Teachers." An instrument that allowed for the flexibility to probe and clarify participant understandings of elementary number theoretical concepts was employed. Questions were designed to clarify participants' understandings of procedures and concepts relating to divisibility and to investigate their abilities to make connections and inferences from them. A constructivist-oriented phenomenological analysis of reflective abstraction was adapted as a framework for interpreting data acquired in the interviews. Data analysis supports the general claim that teacher's content knowledge is "weak" and teacher's conceptual understanding is "insufficient" at times to teach arithmetic even in the elementary grades. Results provide a preliminary basis of a descriptive theory for the development of mental constructions involving elementary numbers concepts, their properties and relationships. An illustration of the model used to guide analysis and excerpts from the interviews are included. (Contains 16 references.) (LL)