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This submission contains the Proceedings of the 2017 Annual Meeting of the Canadian Mathematics Education Study Group (CMESG), held at McGill University in Montreal, Quebec June 2-6. 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, panel discussions, working group reports, new PhD reports, and summaries of ad hoc and poster sessions. Papers include the following: (1) The Most Unglamorous Job of All: Writing Mathematics Exercises (Yvan Saint-Aubin); (2) 40+ Years of Teaching and Thinking about University Mathematics Students, Proofs, and Proving (Annie Selden); (3) 'Elder Talk'-A Revisionist Version (Joel Hillel); (4) Teaching First Year Mathematics Courses in Transition from Secondary to Tertiary (Ann Arden, Wesley Maciejewski, Nadia Hardy); (5) L'anxiété mathématique chez les futurs enseignants du primaire : à la recherche de nouvelles réponses à des enjeux qui perdurent / Elementary Preservice Teachers and Mathematics Anxiety: Searching for New Responses to Enduring Issues (Manion LeBlanc, Jamie Pyper, Jo Towers) [Written in French and English]; (6) Social Media and Mathematics Education (Judy Larsen, Egan Chernoff, Viktor Freiman); (7) Quantitative Reasoning in the Early Years / Le raisonnement quantitatif dans les premières années du parcours scolaire (Doris Jeannotte, Lynn McGarvey)[Written in English and French ]; (8) Social, Cultural, Historical and Philosophical Perspectives on Tools for Mathematics (Yasmine Abtahi, Susan Gerofsky, Jean Francois); (9) Compréhension approfondie des mathématiques scolaires / Deep Understanding of School Mathematics (Richelle Marynowski, Sarah Dufour, Peter Liljedhal); (10) Modelling Mathematical Modelling / Modéliser la modélisation mathématique (France Caron)[Written in French and English]; (11) Collective Learning: Re-thinking the Environment, Artifacts and Classroom Interactions (Joyce Mgombelo); (12) A Psychological View of Teaching Proof Construction (John Selden); (13) Things Kids Think With: The Role of the Physical Properties of Mathematical Tools in Children's Learning in the Context of Addition of Fractions (Yasmine Abtahi); (14) Examining Mathematics Anxiety Among Classroom Teachers (Atinuke Adeyemi); (15) Teaching Teachers: A Look Inside Professional Development (Melania Alvarez); (16) Be Innovative but Don't Be Wrong: Are 21st Century Students Experiencing 21st Century Mathematics? (Jennifer Godfrey Anderson); (17) College Foundational Mathematics: Can the Affordances of ICT Enhance Self-regulation Skill of Students? / Mathématiques fondamentales du collège : est-ce que les apports des « TIC » peuvent améliorer l'autorégulation des compétences des élèves? (Carol Carruthers)[Written in English and French]; (18) Influence of the Learning Environment on Student Test Performance in Undergraduate Mathematics Courses (Amenda Chow); (19) The Role of Oral Communication Strategies in Accessing and Assessing Mathematical Understanding: Case Studies of Primary School Teachers' Perceptions of Teaching Mathematics and Teaching Literacy (Celia Kutas Chisu); (20) Reciprocal Partnership: An Intervention to Enhance Mathematics Self-efficacy and Achievement of First- and Second-semester College Students (Kerry Kwan); (21) Understanding the Interactions within a New Teacher Learning Community Composed of First Time Participants and a Novice Facilitator (Terry Wan Jung Lin); (22) Valued Kinds of Knowledge and Ways of Knowing in Mathematics and the Teaching and Learning of Mathematics: A Worldview Analysis (Gale L. Russell); (23) Opening Space: Complexity Thinking, Classroom Discourse, and Mathematics Learning in the Elementary Classroom / La technologie du forum ouvert: complexité, le discours de classe, et l'apprentissage de mathématiques en école élémentaire (Evan Throop-Robinson)[Written in English and French]; (24) Investigating Mathematics Teachers' Knowledge for Teaching and Their Learning Trajectories (Zhaoyun Wang]; (25) What/How Can We Learn from the Deaf Mathematics Classroom? (Christins M. Krause); (26) The Mathematical Association of Tanzania (MAT) Experience in CMESG Conference (Said A. Sima); (27) Toward a Hybrid Model to Describe Creative Acts in Mathematics Learning Environments (Ayman Aljarrah); (28) Problem Drift: Imaging Emerging Curricular Significance (Nat Banting); (29) Incorporating Writing into the Undergraduate Mathematics Curriculum (Lauren Dedieu); (30) Informing Local Materials Used for Concept-Rich Instruction Practice in Mathematics Class with Pre-service Teachers in Tanzanian Context (Emmanuel, Deogratias); (31) Les jeux technologiques dans les cours de mathématiques au secondaire (Caitlin, Furlong) [Written in French]; (32) De jeunes chercheurs à la rencontre de la philosophie des mathématiques (Sabrina Heroux, Sarah Mathieu-Soucy, Laura Broley) [Provided in French]; (33) Examining Parent Perspectives of Multiple Strategies (Jennifer Holm, Lynn McGarvey, Lixin Luo, Janelle McFeeters, and Iris Yin); (34) La dialectique perfection/imperfection : une nouvelle manière de penser l'activité mathématique (Marie-Line L. LaMarche, Jean-Francois Maheux) [Provided in French]; (35) Understanding School Leaders' Discourse in Regard to Mathematics Achievement (Jhonel Morvan); (36) I SEE What You're Saying: Visualization and the Growth of Mathematical Images (Jennifer Plosz); (37) Students' Relationships With Mathematics Revealed Through Drawings (Jennifer Plosz, Jo Towers, Kori Czuy, Ayman Aljarrah, Andrew Vorhies and Miwa Takeuchi); (38) Teacher Professional Learning in Tanzania: Experiences of Mathematics Teacher Leaders (Calvin Zakaria Swai); (39) "Perimeter is easier to work with than area": Students' Language Use in Group Work (Miwa Akoi Takeuchi); and (40) Toward an Understanding of Mathematics Teachers' Participation in Professional Learning Networks (Xiong Wang). Individual papers provide referenced. [For the 2016 proceedings, see ED581045.]

The aim of this study was to determine the role of semiotics in assisting young Indigenous students to engage with and identify the general structure of growing patterns. The theoretical perspective of semiotics underpinned the study. Data are drawn from two Year 3 students, including analysis of pretest questions and two conjecture-driven lessons. Results indicate that particular semiotic signs (iconic signs) contribute to how young Indigenous students attend to, and identify the structure of growing patterns.

This document contains the fourth volume of the proceedings of the 28th annual conference of the International Group for the Psychology of Mathematics Education. Conference presentations are centered around the theme "Inclusion and Diversity". This volume features 64 research report papers: (1) Situated or Abstract: The Effect of Combining Context and Structure on Constructing an Additive (Part-Part-Whole) Schema (Irit Peled and Ruth Meron); (2) Using Graphical Profiles to Study the Learning and Teaching of Mathematics (Dvora Peretz); (3) The Role of Number in Proportional Reasoning: A Prospective Teacher's Understanding (Axelle C. Person, Sarah B. Berenson, and Paula J. Greenspon); (4) Learning to Use CAS: Voices from a Classroom (Robyn Pierce and Kaye Stacey); (5) Technical School Students' Conceptions of Tangent Lines (Marcia Maria Fusaro Pinto and Valeria Guimaraes Moreira); (6) Elementary School Students' Mental Representations of Fractions (Demetra Pitta-Pantazi, Eddie M. Gray, and Constantinos Christou); (7) A Structural Model for Problem Posing (M. Pittalis, C. Christou, N. Mousoulides, and D. Pitta-Pantazi); (8) Some Undergraduates' Experiences of Learning Mathematics (Hilary Povey and Corinne Angier); (9) Normalising Geometrical Constructions: A Context for the Generation of Meanings for Ratio and Proportion (Georgos Psycharis and Chronis Kynigos); (10) The Sensual and the Conceptual: Artefact-Mediated Kinesthetic Actions and Semiotic Activity (Luis Radford, Serge Demers, Jose Guzman, and Michele Cerulli); (11) A Sociocultural Account of Students' Collective Mathematical Understanding of Polynomial Inequalities in Instrumented Activity (Ferdinand Rivera and Joanne Rossi Becker); (12) Infinity as a Multi-Faceted Concept in History and in the Mathematics Classroom (Ornella Robutti, Ferdinando Arzarello, and Maria G. Bartolini Bussi); (13) Successful Undergraduate Mathematicians: A Study of Students in Two Universities (Melissa Rodd and Margaret Brown); (14) Bilingual Lessons and Mathematical World Views--A German Perspective (Katrin Rolka); (15) The Use of Models in Teaching Proof by Mathematical Induction (Gila Ron and Tommy Dreyfus); (16) Reflecting on Prospective Elementary Teachers' Mathematics Content Knowledge: The Case of Laura (Tim Rowland, Peter Huckstep, and Anne Thwaites); (17) The Competent Use of the Analytic Method in the Solution of Algebraic Word Problems. A Didactical Model Based on a Numerical Approach with Junior High Students (Guillermo Rubio and Rafael Del Valle); (18) Language and Concept Development in Geometry (Frode Ronning); (19) The Role of Gestures in Conceptualisation: An Exploratory Study on the Integral Function (Cristina Sabena); (20) Psychological Aspects of Genetic Approach to Teaching Mathematics (Ildar S. Safuanov); (21) Routines in Non-Routine Problem Solving Processes (Wolfgang Schloeglmann); (22) Teacher Guidance of Knowledge Construction (Baruch Schwarz, Tommy Dreyfus, Nurit Hadas, and Rina Hershkowitz); (23) Types of Student Reasoning on Sampling Tasks (J. Michael Shaughnessy, Matt Ciancetta, and Dan Canada); (24) Didactic Model--Bridging a Concept with Phenomena (Beba Shternberg and Michal Yerushalmy); (25) Elaborating the Teacher's Role--Towards a Professional Language (Dianne Siemon, Jo Virgona, Maria Lasso, Vanessa Parsons, and Juli Cathcart); (26) Making the Connection: Procedural and Conceptual Students' Use of Linking Words in Solving Problems (Adrian Simpson and Nora Zakaria); (27) Children's Conceptual Understanding of Counting (Hannah Slovin and Barbara J. Dougherty); (28) Adding Fractions Using "Half" as an Anchor for Reasoning (Alina Galvao Spinillo and Maria Soraia Silva Cruz); (29) Persistence of Decimal Misconceptions and Readiness to Move to Expertise (Vicki Steinle and Kaye Stacey); (30) Molly and Equations in A2: A Case Study of Apprehending Structure (Nada Stehlikova and Adrian Simpson); (31) Image--Metaphor--Diagram: Visualisation in Learning Mathematics (Rudolf Straber and Gert Kadunz); (32) Images of Fractions "as" Processes and Images of Fractions "in" Processes (Marek Sulista, Jan Herman, Lucia Ilucova, Veronika Kremsova, Jiri Pribyl, Janka Ruppeldtova, Adrian Simpson, Nada Stehlikova, and Michaela Ulrychova); (33) Describing Elements of Mathematics Lessons that Accommodate Diversity in Student Background (Peter Sullivan, Judy Mousley, and Robyn Zevenbergen); (34) Orchestrating Mathematical Proof through the Use of Digital Tools (Rosamund Sutherland, Federica Olivero, and Marnie Weeden); (35) School-Based Community of Teachers and Outcomes for Students (Paola Sztajn, Martha Alexsaht-Snider, Dorothy Y. White, and Amy Hackenberg); (36) Thinking through Three Worlds of Mathematics (David Tall); (37) The Effect of Students' Roles on the Establishment of Shared Meanings During Problem Solving (Konstantinos Tatsis and Eugenia Koleza); (38) Integrating CAS Calculators into Mathematics Learning: Partnership Issues (Michael O. J. Thomas and Ye Yoon Hong); (39) The Development of Structure in the Number System (Noel Thomas); (40) Orientations to Numeracy: Teachers' Confidence and Disposition to Use Mathematics Across the Curriculum (Steve Thornton and John Hogan); (41) Efficiency and Adaptiveness of Multiple School-Taught Strategies in the Domain of Simple Addition (Joke Torbeyns, Lieven Verschaffel, and Pol Ghesquiere); (42) Supporting Teachers on Developing Teaching Norms Based on Children's Learning Mathematics (Wen-Huan Tsai); (43) Prospective Teachers' Images and Definitions: The Case of Inflection Points (Pessia Tsamir and Regina Ovodenko); (44) Fostering Mathematical Meaning via Scientific Inquiry: A Case Study (Ron Tzur, Rita A. Hagevik, and Mary E. Watson); (45) Students Building Isomorphisms (Elizabeth B. Uptegrove and Carolyn A. Maher); (46) How Do High School Students Interpret Parameters in Algebra? (Sonia Ursini and Maria Trigueros); (47) Causes Underlying Pre-Service Teachers' Negative Beliefs and Anxieties about Mathematics (Liisa Uusimaki and Rod Nason); (48) Fractions in Adult's Elementary School: The Lucina (Marta Elena Valdemoros-Alvarez); (49) Students' Overreliance on Proportionality: Evidence from Primary School Pupils Solving Arithmetic Word Problems (Wim Van Dooren, Dirk De Bock, An Hessels; Dirk Janssens, and Lieven Verschaffel); (50) From Functions to Equations: Introduction of Algebraic Thinking to 13 Year-Old Students (Petros Verikios, Vasiliki Farmaki, and Nikos Nikoudatos); (51) Critical Awareness of Voice in Mathematics Classroom Discourse: Learning the Steps in the "Dance of Agency" (David Wagner); (52) Metaphors and Cultural Models Afford Communication Repairs of Breakdowns between Mathematical Discourses (G. D. Wake and J. S. Williams); (53) Generalising Arithmetic: Supporting the Process in the Early Years (Elizabeth Warren); (54) A Framework for Describing the Processes that Undergraduates Use to Construct Proofs (Keith Weber); (55) The Nature of Spontaneity in High Quality Mathematics Learning Experiences (Gaye Williams); (56) Spreadsheet Generalising and Paper and Pencil Generalising (Kirsty Wilson, Janet Ainley, and Liz Bills); (57) A Comparison of a Visual-Spatial Approach and a Verbal Approach to Teaching Mathematics (Pamela Woolner); (58) Differentiation of Students' Reasoning on Linear and Quadratic Geometric Number Patterns (Kai-Lin Yang and Fou-Lai Lin); (59) Map Construction as a Context for Studying the Notion of Variable Scale (Nikoleta Yiannoutsou and Chronis Kynigos); (60) Understanding How the Concept of Fractions Develops: A Vygotskian Perspective (Kaori Yoshida); (61) Secondary Mathematics Teachers' Knowledge Concerning the Concept of Limit and Continuity (T. Zachariades and E. Mastorides); (62) Characteristics of Mathematical Problem Solving Tutoring in an Informal Setting (Orit Zaslavsky and Iris Zodik); (63) Making Sense of Irrational Numbers: Focusing on Representation (Rina Zazkis and Natasa Sirotic); and (64) Numeracy Practices of Young Workers (Robyn Zevenbergen and Kelly Zevenbergen). (Individual papers contain references.)

This document contains the third volume of the proceedings of the 28th Annual Conference of the International Group for the Psychology of Mathematics. Conference presentations are centered around the theme "Inclusion and Diversity". A total of 65 research reports are presented here: (1) A Teacher's Model of Students Algebraic Thinking About Equivalent Expressions (Jean E. Hallagan); (2) On Motivational Aspects of Instructor-Learner Interactions in Extra-Curriculum Activities (Stefan Halverscheid); (3) Development of Understanding and Self-Confidence in Mathematics, Grades 5-8 (Markku S. Hannula, Hanna Maijala, and Erkki Pehkonen); (4) Mathematics Teachers' Knowledge Base: Preliminary Results (Guershon Harel and Kien H. Lim); (5) The Role of Tool and Teacher Mediations in the Construction of Meanings for Reflection (Lulu Healy); (6) The Proving Process in Mathematics Classroom--Method and Results of a Video Study (Aiso Heinze); (7) Structure Sense in High School Algebra: The Effect of Brackets (Maureen Hoch and Tommy Dreyfus); (8) Explaining Variability in Retrieval Times for Addition Produced by Students with Mathematical Learning Difficulties (Sarah L. Hopkins and Michael J. Lawson); (9) Early Gender Differences (Marj Horne); (10) Perceptual and Symbolic Representations as a Starting Point of the Acquisition of the Derivative (Markus Hahkioniemi); (11) Implementation of a Model Using Authentic Investigative Activities for Teaching Ratio & Proportion in Pre-Service Teacher Education (Bat-Sheva Ilany, Yaffa Keret, and David Ben-Chaim); (12) Mathematicians and the Selection Task (Matthew Inglis and Adrian Simpson); (13) Insight into Pupils' Understanding of Infinity in a Geometrical Context (Darina Jirotkova and Graham Littler); (14) Textual Differences as Conditions for Learning Processes (Marit Johnsen Hoines); (15) Students' Gender Attitudes Towards the Use of Calculators in Mathematics Instruction (L. M. Kaino and E. B. Salani); (16) Teaching Arithmetic and Algebraic Expressions (K. Subramakhi Banerjee); (17) An Introduction to the Profound Potential of Connected Algebra Activities: Issues of Representation, Engagement and Pedagogy (James J. Kaput and Stephen J. Hegedus); (18) The Tension between Teacher Beliefs and Teacher Practice: The Impact of the Work Setting (M. Kerem Karaagac and John Threlfall); (19) What a Simple Task Can Show: Teachers Explore the Complexity of Children's Thinking (Heather Kelleher, Cynthia Nicol, and Carole Saundry); (20) Constructing Knowledge about the Bifurcation Diagram: Epistemic Actions and Parallel Constructions (Ivy Kidron and Tommy Dreyfus); (21) Experiencing Research Practice in Pure Mathematics in a Teacher Training Context (Eva Knoll, Paul Ernest, and Simon Morgan); (22) Enhancing Mathematical Literacy with the Use of Metacognitive Guidance in Forum Discussion (Bracha Kramarski); (23) The Impact of Developing Teacher Conceptual Knowledge on Students' Knowledge of Division (Janeen Lamb and George Booker); (24) Kinds of Arguments Emerging While Exploring in a Computerized Environment (Ilana Lavy); (25) Using Solo to Analyse Group Responses (Christine Lawrie and Chris Reading); (26) Connections between Qualitative and Quantitative Thinking about Proportion: The Case of Paulina (Elena Fabiola Ruiz Ledesma and Marta Elena Valdemoros Alvarez); (27) Towards High Quality Geometrical Tasks: Reformulation of a Proof Problem (Roza Leikin); (28) Mathematical Thinking & Human Nature: Consonance & Conflict (Uri Leron); (29) The Mathematics Pedagogical Values Delivered by an Elementary Teacher in Her Mathematics Instruction: Attainment of Higher Education and Achievement (Yuh-Chyn Leu and Chao-Jung Wu); (30) Visual Reasoning in Computational Environment: A Case of Graph Sketching (Allen Leung and King Wah Chan); (31) Elementary School Students' Use of Mathematically-Based and Practically-Based Explanations: The Case of Multiplication (Esther Levenson, Dina Tirosh, and Pessia Tsamir); (32) Mathematical Discovery: Hadamard Resurrected (Peter Liljedahl); (33)Supporting Teachers on Designing Problem-Posing Tasks as a Tool of Assessment to Understand Students' Mathematical Learning (Pi-Jen Lin); (34) Prospective Elementary School Teachers' Solution Strategies and Reasoning for a Missing Value Proportion Task (Jane Lo); (35) Introduction of Calculus in 12th Grade: The Role of Artefacts (Michela Maschietto); (36) The Critical Role of Institutional Context in Teacher Development (Kay McClain and Paul Cobb); (37) Students' Perceptions of Factors Contributing to Successful Participation in Mathematics (Andrea McDonough, Peter Sullivan, and Robyn Turner Harrison); (38) The Cognitive--Motivational Profiles of Students Dealing with Decimal Numbers and Fractions (Kaarina Merenluoto); (39) The Number Line as a Representation of Decimal Numbers: A Research with Sixth Grade Students (Niki Michaelidou, Athanasios Gagatsis, and Demetra Pitta-Pantazi); (40) Preservice Teachers Conceptions of Mathematics-Based Software (James Middleton, Terri Kurz, and H. Bahadir Yanik); (41) Helping Children to Model Proportionally in Group Argumentation: Overcoming the "Constant Sum" Error (Christina Misailidou and Julian Williams); (42) Abstraction in Mathematics and Mathematics Learning (Michael Mitchelmore and Paul White); (43) Reflective Symmetry in Construction and Proving (Takeshi Miyakawa); (44) Students' Improper Proportional Reasoning: The Case of Area and Volume of Rectangular Figures (Modestina Modestou, Athanasios Gagatsis, and Demetra Pitta-Pantazi); (45) Abstraction and Consolidation (John Monaghan and Mehmet Fatih Ozmantar); (46) Critical Sense in Interpretations of Media Graphs (Carlos Monteiro and Janet Ainley); (47) Between Affect and Cognition: Proving at University Level (Francesca Morselli and Fulvia Furinghetti); (48) An Aspect of Mathematical Understanding: The Notion of "Connected Knowing" (Judith Mousley); (49) Algebraic and Geometric Approach in Function Problem Solving (Nikos Mousoulides and Athanasios Gagatsis); (50) Children's Development of Structure in Early Mathematics (Joanne Mulligam, Anne Prescott, and Michael Mitchelmore); (51) On the Fragile, Yet Crucial, Relationship between Mathematicians and Researchers in Mathematics Education (Elena Nardi and Paola Iannone); (52) Students' Preference of Non-Algebraic Representations in Mathematical Communication (Dorit Neria and Miriam Amit); (53) Learning to See in Mathematics Classrooms (Cynthia Nicol and Sandra Crespo); (54) Students' Ways of Interpreting Aspects of Chance Embedded in a Dice Game (Per Nilsson); (55) The Impact of State-Wide Numeracy Testing on the Teaching of Mathematics in Primary Schools (Steven Nisbet); (56) "The Poetry of the Universe": New Mathematics Teachers' Metaphoric Meaning-Making (Andrew Noyes); (57) What Students Do When Hearing Others Explaining (Kazuhiko Nunokawa and Satoshi Ohzeki); (58) Changes in Teachers' Practices While Assuming New Leadership Roles (Jo Clay Olson); (59) Students' Structuring of Rectangular Arrays (Lynne Outhred and Michael Mitchelmore); (60) Construction of Mathematical Definitions: An Epistemological and Didactical Study (Cecile Ouvrier-Buffet); (61) Mathematical Abstraction through Scaffolding (Mehmet Fatih Ozmantar and Tom Roper); (62) The Use of Diagrams in Solving Non Routine Problems (Marilena Pantziara, Athanasios Gagatsis, and Demetra Pitta-Pantazi); (63) Designing for Local and Global Meanings of Randomness (Efi Paparistodemou and Richard Noss); (64) Integrating the History of Mathematics in Educational Praxis: An Euclidean Geometry Approach to the Solution of Motion Problems (Theodorus Paschos, Vassilike Farmaki, and Nikos Klaudatos); and (65) The Magic Circle of the Textbook--An Option or an Obstacle for Teacher Change (Leila Pehkonen). (Individual papers contain references.)

This research involved the observation and audio-recording of 30 ninth-grade mathematics students recommended as being outstanding by their mathematics teachers. Each student was presented with five different problem situations for which he or she was to make mathematical conjectures. During the interview, the students thought aloud and all of their manipulative moves were recorded. The primary focus of this paper is an analysis of creativity factors exhibited by the five highest-scoring students on a particular geometric problem. The comparisons and analyses showed that these five students had much in common even though each student was highly individualistic. (MP)

STEM education is one of the primary priorities in the Malaysian Education Blueprint (PPPM) 2013-2025, in line with the relevance of STEM at the global level. In keeping with the educational needs of 21st-century learning, the use of technology in education is also highly encouraged. Unfortunately, the Malaysian government's goal of producing one million experts in Science, Technology, Engineering, and Mathematics (STEM) by 2020 was not met because the 60:40 STEM: non-STEM policy failed to meet the goal. Students' interest in different STEM disciplines has been linked to their interest in integrated STEM in previous studies. Students' lack of interest in STEM fields appears to have an impact on their low mathematics achievement at the international assessment level, including in the field of geometry. Geometry is one of the most important branches of mathematics, and spatial thinking goes hand in hand with it. This paper presents the results of a critical review as well as a brief overview of ongoing research targeted at developing a model for the formation of interest among secondary school students in STEM through geometry learning based on augmented reality (AR) technology and visual-spatial abilities. The first part of the paper provides an in-depth review of these matters based on rather limited resources. The review revealed various difficulties and limitations in piquing students' interest in STEM and mathematics. The second part describes research being conducted by the researchers to examine the implementation of using AR in geometry learning based on visual-spatial abilities. For the pilot and main stages of the study, pre-and post-test, observations, and semi-structured interviews will be conducted. The data will be analysed quantitatively and qualitatively, and the theory of change approach will be applied. The findings will be used to develop a model for the formation of interests among students as a result of geometry learning. The model aims to provide some relevant information for technology-assisted mathematics instruction, and it aligns with national STEM policy goals of increasing the number of students selecting STEM disciplines in secondary and higher education. [ABSTRACT FROM AUTHOR]

Borel conjectured that all irrational numbers are normal in any integral base α. For each positive number ξ and integer α greater than 1, ξ is normal in base α if and only if the sequence ξαn (n = 0,1,...) is uniformly distributed modulo 1. In this paper we survey not only the digit of algebraic irrational numbers in integral base but also the fractional parts of geometric progressions whose common ratios are algebraic numbers greater than 1. In our main results, we give new lower bounds for the number of digit changes in the binary expansions of algebraic irrational numbers. [ABSTRACT FROM AUTHOR]

In this paper, we propose a novel limited feedback approach for MIMO systems with transmit beamforming over time-varying channels, where the optimal beamforming vector of the current frame is selected from the son-codewords of the codewords neighboring to previous beamforming codeword. The proposed approach exploits the channel temporal correlation to improve the CSI quantized precise. The simulation results show that the proposed approach can efficiently improve the BER performance relative to the common feedback approach, and the feedback rate can be further reduced under some conditions. Moreover, the effect of Doppler frequency shift on the performance of the proposed approach is analyzed. [ABSTRACT FROM AUTHOR]

In order to make sense of mathematics and to communicate mathematical ideas, it is essential to be able to assess and produce mathematical arguments, including formal proofs. Although student difficulty with proof has been well established in the literature (Chazan, 1993; Hart, 1994; Martin & Harel, 1989; Senk, 1985), research connecting teachers’ actions in the classroom to students’ understanding of proof is limited (Herbst, 2002). In this paper, we describe a classroom-based interpretive study in which we explore the relationship between teaching and understanding of proof in a proof-based geometry class. In particular, we address the following question: [ABSTRACT FROM AUTHOR]

This paper introduces the notion of pre-contrast functions and studies their geometric properties. A contrast function is an asymmetric squared distance like function. A pre-contrast function corresponds to the differential of a contrast function, however, it is not integrable in general. While a contrast function induces a statistical manifold, a pre-contrast function induces a statistical manifold admitting torsion. As an application of pre-contrast functions, geometry of non-conservative estimating functions in statistics is also studied. [ABSTRACT FROM AUTHOR]

This paper aims to advance understanding of instructional phenomena by focusing on the nature of an instructional situation. We argue that the decisions in managing a situation require a teacher to respond to norms by negotiating dispositions activated in the situation that could contradict each other. We illustrate this point examining the situation of installing theorems in high school geometry. ..PAT.-Unpublished Manuscript [ABSTRACT FROM AUTHOR]

Teachers must have an understanding of students' mathematical thinking in order to create meaningful learning opportunities. Literature on learning trajectories and geometric reasoning provide a framework for investigating the ways that teachers make sense of their students' thinking. This paper will report on two case studies that examined the experiential models of students' geometric thinking created by secondary teachers. ..PAT.-Unpublished Manuscript [ABSTRACT FROM AUTHOR]

In this paper, we examine elementary pre-service teachers' understanding of the phrase "is a special kind of" as in "a square is a special kind of rectangle." Data collected from 109 students indicated that they possess varying interpretations of the phrase. Our findings suggest that difficulty with shape categorization may be related to the meanings students attach to such phrases. ..PAT.-Unpublished Manuscript [ABSTRACT FROM AUTHOR]

Around 80 years ago, several authors (for instance H. Weyl, T.Y. Thomas, J. Douglas and J.H.C. Whitehead) studied the projective geometry of paths, using the methods of tensor calculus. The principal object of study was a spray, namely a homogeneous second-order differential equation, or more generally a projective equivalence class of sprays. At around the same time, E. Cartan studied the same topic from a different point of view, by imagining a projective space attached to a manifold, or, more generally, attached to a ‘manifold of elements’; the infinitesimal ‘glue’ may be interpreted in modern language as a Cartan projective connection on a principal bundle. This paper describes the geometrical relationship between these two points of view. © 2004 American Institute of Physics [ABSTRACT FROM AUTHOR]

A common activity in popular culture is the formation of top lists: the top 100 songs of all time, the 50 favorite movies, the top 100 movie stars and so on. These top lists axe then publicized on radio and on the inter-net. In this paper, which is part research, part expository and part computer simulation, we examine the mathematics of forming such lists in the situation where there are no measurable head-to-head comparisons and the compilation of the ranked list is formed by respondents' choices. There is an implied assumption that the compiled lists form a representative list of a true ranked list, statistical and therefore form a representative picture of a "true" ranked list. We construct statistical models for these list formations and define a true ranking in two different ways. We then prove several paradoxical results that show non-significant correlation between the true ranked list and the compiled list. For example, we show that there exist populations which contain an item that has a true ranking of number one but for which this item will not appear on any compiled list. These results are related to Arrow's impossibility theorem (see [1]) in the theory of elections. A computer simulation was conducted in conjunction with the study to determine whether these paradoxes are aberrations. The simulations show that these aberrations only occur with low probability and on average there is a positive but weak correlation between the compiled list and the true ranked list. [ABSTRACT FROM AUTHOR]

The period of the formation of the theory of generalized analytic functions, nowadays known as the Bers-Vekua theory, meets the beginning of scientific activity of Bogdan Bojarski together with his supervisor Ilia Vekua in Moscow. Bojarski's fundamental new approach to the solution of the system of elliptic partial differential equations on a plane, opened up a direct pathway to many important issues in the geometric theory of analytic functions and related boundary value problems. In this paper we give a short overview of important results of Bogdan Bojarski in the theory of generalized analytic functions and his point of view on the theory of boundary value problems. [ABSTRACT FROM AUTHOR]

The ability to connect mathematically is one of the abilities that students must have in learning mathematics at school, with a mathematical connection, students will find it easier to solve math problems. This study aims to analyze the mathematical connection ability of students in solving ethnomatematic based transformation geometry problems. The research method used is a qualitative descriptive approach. The subjects of this study were three students of class XII MA Daarurrahman, one student with high initial ability, one student with moderate initial ability, and one student with low initial ability. The categorization of initial abilities here is based on the results of daily tests of geometric transformation material and also recommendations from mathematics subject teachers. The data collection technique used in this study was a test to determine the connection ability of students and then to be interviewed to find out more about the results of the test. The research data were validated using triangulation techniques.The results of this study indicate (1) the mathematical connection ability of students with high initial abilities is in the sufficient category and is able to meet 2 indicators of 4 indicators, namely being able to find connections from one procedure to another, using mathematics in other fields of study or in everyday life - day (2) the mathematical connection ability of students with moderate initial abilities is included in the good category to meet 3 of the 4 indicators, namely being able to understand the relationship between concepts and procedures, the relationship between mathematical topics and other field topics, using mathematics in other fields of study or in everyday life - day (3) Mathematical connection ability of students with low initial ability only meets 1 indicator out of 4 indicators, namely using mathematics in other fields of study or in everyday life. [ABSTRACT FROM AUTHOR]

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In this study we applied a discursive perspective of learning (Sfard, 2008) to a sequence of 21 geometry mini-lessons taught in a fourth grade classroom. From this perspective, learning is defined as changes in mathematical discourse. We first characterize and then compare discourse from the beginning and the end of the mini-lesson sequence. We identify shifts in the discourse that occurred during the sequence. We then discuss how the characteristics of the students’ geometric discourse informed task design and instruction. This perspective provided a useful means for linking instruction to student learning in an operationalized manner. [ABSTRACT FROM AUTHOR]

Definitions are fundamental to the work of geometry, but many students struggle with understanding terms, and fail to learn the role of definitions in mathematics. We explore how ten college-level students created and evaluated definitions for common geometric terms. Using the idea of concept image and definition, we hoped to learn about the links between how students think about geometric terms, how they define them, and how they evaluate definitions. We found that despite demonstrating strengths in their understanding, none of them evaluated definitions by constructing a biconditional argument. We suggest ways of addressing the issue. [ABSTRACT FROM AUTHOR]

Reform efforts in the U.S. have placed an emphasis on reasoning and proving. Yet, previous geometry textbook analyses have found limited opportunities for reasoning and proving and lack of opportunities to engage students in high levels of rigor. This analysis investigated how purported Common Core aligned textbooks address standards about proving geometric theorems. Like previous studies conducted with pre-Common Core textbooks, the findings for opportunities to construct proofs and engage students in high levels of rigor were less than expected. These findings suggest that publishers and practitioners need to ensure that students are given adequate opportunities to construct proofs and gain conceptual understanding. [ABSTRACT FROM AUTHOR]

We examine responses from a national sample of high school mathematics teachers to a questionnaire, which had been developed to study teachers' recognition of a system of hypothesized norms that stipulate that geometry proof problems are to be posed using a diagrammatic register. We report on the psychometric properties of the questionnaire, as well as the relationship between these mathematics teachers' mathematical knowledge for teaching geometry (MKT-G) and their stances on breaching those norms. Although Herbst et al. (2013) hypothesized that the system consisted of five distinguishable sub-norms, the factor structure of the questionnaire suggested that two of those norms might not truly be distinguishable. We also found a positive and significant relationship between teachers' MKT-G and their stances on breaching two of the determined components of the system of norms. [ABSTRACT FROM AUTHOR]

In the current study, we examined the contributions of shape and pattern knowledge at four early time points to formal geometry knowledge in sixth grade within a longitudinal sample of over 500 low-income children. Shape knowledge at the beginning of pre-k predicted geometry in sixth grade, over and above general math and cognitive skills. However, at the end of pre-k, kindergarten and first grade, pattern knowledge was a unique predictor of later geometry, and shape knowledge was not. Results highlight the important roles of two important, but understudied components of math knowledge -- early shape and pattern skills -- and their contributions to the development of formal geometry knowledge. [ABSTRACT FROM AUTHOR]