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The theme of the 39th proceedings of the North American Chapter of the International Group for the Psychology of Mathematics Education (PME-NA) conference was "Synergy at the Crossroads: Future Directions for Theory, Research, and Practice." The metaphor of crossroads was inspired by the conference venue--the historic Indianapolis Union Station, as well as by the state motto, a reference to how Indiana is connected to the rest of the United States. PME-NA 39 includes research presentations, discussion, and reflection focusing on four driving questions connecting to the metaphor of crossroads: (1) What have we learned from the routes we have traversed; what are potential routes for mathematics education research in the future; and what considerations are relevant as we make choices about future directions in mathematics education?; (2) How do we address issues of access and equity within mathematics education today?; (3) How can we lay the groundwork for future crossroads or intersections between theory, research, and practice?; and (4) What barriers within research traditions, educational policy, and teaching practice impede researchers', students', and teachers' success, and how can we work to overcome these barriers? The accepted proposals included 75 research reports, 142 brief research reports, 167 posters, and 13 working groups. Chapters include: (1) Plenary Papers; (2) Curriculum and Related Factors; (3) Early Algebra, Algebra, and Number Concepts; (4) Geometry and Measurement; (5) Inservice Teacher Education/Professional Development; (6) Mathematical Knowledge for Teaching; (7) Mathematical Processes; (8) Preservice Teacher Education; (9) Statistics and Probability; (10) Student Learning and Related Factors; (11) Teaching and Classroom Practice; (12) Technology; (13) Theory and Research Methods; and (14) Working Groups. Individual papers include references. [Individual plenary and research papers are available in ERIC. Some papers are written in both Spanish and English.]

A web-based activity and testing system (WATS) has features such as adaptive problem sets, videos, and data-driven tools for instructors to monitor and scaffold student learning. Central to WATS adoption and use are questions about the implementation process: What constitutes "good" implementation and how far from "good" is "good enough"? This proceedings paper reports on and illustrates work to provide structure for such examination. The context is a study about implementation that is part of a state-wide randomized controlled trial examining student learning in community college algebra when a particular WATS suite of tools is used. Discussion questions for conference participants dug into the distinctions among intended, enacted, and achieved curriculum and the processes surrounding these as well as the challenges and opportunities in researching fidelity of implementation in the community college context, particularly the role of instructional practice as a contextual component of the research. [This paper was published in the "Proceedings of the 19th Annual Conference on Research in Undergraduate Mathematics Education," T. Fukawa-Connelly, N. Infante, K. Keene, and M. Zandieh (Eds), 2016.]

The purpose of this study was to examine the ways that teachers use Connected Classroom Technology (CCT) to potentially support achievement on translation problems that require moving between algebraic representations. Four mathematics classrooms were chosen based on their gain scores on pre- and post-test Algebraic translation problems. Two classrooms with the highest and the lowest gain scores were chosen among the classrooms with pre-test scores that were below 50%. This study used video-recorded observational data and found that teachers in effective classrooms created environments wherein students used multiple representations simultaneously and translated between representations through discussion. In contrast, teachers in less effective classrooms fostered environments wherein students used representations independently and missed opportunities to translate representations through discussion. [For the complete proceedings, see ED583608.]

In this paper, we explore how students' algebraic noticing's and explanations changed across a two-year period with the introduction of designed instructional material. The data in this report is drawn from n=53 Year 7-8 students' responses to a free-response assessment task across two different years. Analysis focused on how students noticed and explained algebraic relationships in pairs of equivalent equations. Findings indicate that with the introduction of designed instructional material, there was a shift in student noticing of number properties to identify equivalence between pairs of equations. However, identifying the distributive property of multiplication and developing generalisations about the algebraic relationships remained challenging for students.

An analysis is presented of the basic instructional productivity formula, PR = TL x CS, where PR is the index of productivity, TL is the index reflecting teaching load, and CS is the class-size index. Objectives of the analysis are to: extend the basic instructional productivity formula to include recognition of different types of instruction; indicate that the formula may be applied at any level of organizational or temporal aggregation; specify alternate measures of faculty-count data for use on the formula; specify how the course data of the formula may be credit, hour, course, or section measures; suggest other factors and options for consideration in developing the course measures of the formula; and suggest additional extensions and applications of the resulting algebra. (SW)

This paper demonstrates how to apply Machine Learning (ML) techniques to analyze student interaction data collected in an online mathematics game. We examined: (1) how different ML algorithms influenced the precision of middle-school students' (N = 359) performance prediction; and (2) what types of in-game features were associated with student math knowledge scores. The results indicated that the Random Forest algorithm showed the best performance in predicting posttest math knowledge scores among the seven algorithms employed. Out of 37 features included in the model, the validity of the students' first mathematical transformation was the most predictive of their math knowledge scores. Implications for game learning analytics and supporting students' algebraic learning are discussed based on the findings.

Computer-assisted learning platforms (CALPS) increasingly include gamified elements to improve student outcomes by enhancing their engagement with content. Although evidence exists that gamified programs increase engagement and learning outcomes, there is little causal research on what programmatic mechanisms drive the effect between engagement and learning. In the following paper, we explore this relationship through a method of causal moderation known as fully latent principal stratification. Using data from a large-scale randomized control trial assessing gamified and traditional CALP systems' effects on algebraic knowledge, we estimate the impact of using the gamified CALP on students who engage with one of its key gamification elements--replaying a problem after a suboptimal attempt. The gamified CALP asks students to manipulate algebraic expressions from start to goal states and provides feedback based on the efficiency of these manipulations, allowing students to replay the problems when their efficiency can be improved. We find that the effect of gamification is greater for students with a higher propensity to replay problems. This finding suggests that gamification elements that provide students with opportunities to retry problems are driving the game's efficacy and provide evidence for a scalable mechanism of gamification that can improve students' learning. [This paper was published in: "Proceedings of the Tenth ACM Conference on Learning @ Scale (L@S '23), July 20-22, 2023, Copenhagen, Denmark," ACM, 2023, pp.164-173.]

In this paper, we explore student solutions to a free response mathematical assessment task which had opportunities for students to notice structural properties in the context of number systems. In total, 308 students aged between 10 years to 13 years participated in the study. Their responses were analysed to determine whether they noticed algebraic structures in a task using equivalent equations. Findings indicate that students were able to recognise equation pairs that drew on both the associative and distributive properties. A limited number of students were able to notice the general structure and draw on number properties to support their claims, moving beyond using algorithmic thinking.

This paper will elaborate five levels of algebraic generalisation based on an analysis of students' responses to Reframing Mathematical Futures II (RMFII) tasks designed to assess algebraic reasoning. The five levels of algebraic generalisation will be elaborated and illustrated using selected tasks from the RMFII study. The five levels will be matched against the eight zones identified in the RMFII study supported by its Rasch analysis. We identify two shifts where students' capacity to generalise appear difficult to navigate. The first being where students move from noticing and describing regularities to formalising these regularities into verbal or symbolic expressions. The second is where students use their understanding of equivalence based on relational thinking to write and recognise equivalent algebraic expressions.

Participation structures are important in relation to who gets equitable access to mathematics within classrooms premised on co-construction of mathematical reasoning. This paper takes a strength-based focus to explore how two teachers extended their Pasifika students' known repertoires of practice to encompass others, which supported them to better engage in mathematics. The data illustrates the importance of teachers drawing on task contexts and student ways of knowing and being to facilitate student engagement in reasoned discourse and argumentation. The teacher actions provide a model which other teachers could draw on when teaching mathematics to students from predominantly collectivist groupings.

The transition between arithmetic and algebraic thinking is challenging for students. One notable difficulty for students is understanding algebraic symbols--pronumerals. Researchers are exploring pedagogical approaches in seeking to address this issue. The current paper is contributing to this body of literature by illustrating how an online card matching game-based learning activity supports students' understanding of pronumerals.

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.]

Covariational reasoning and the creation and interpretation of graphs of covariational situations are important skills in math and science. Unfortunately, research shows that students often struggle to make meaningful connections between graphs and the covariational situations they represent. Educational activities designed to help students overcome this struggle tend to use either student-generated or automatically-generated graphs, and have students either act out covariational situations or more passively observe them. In this paper, we present the design of a tool and task that enabled two students to simultaneously embody both the creation of a graph and the covariational actions that the graph represents. Through a process of collaborative instrumentation, the students made meaningful connections between their motions and the embodied traces they created as they reasoned about the covarying quantities of height and time. [For the complete proceedings, see ED630210.]

In this paper we examine how facilitators' prior experiences as mathematics teachers frame their work when facilitating a practice-based professional development (PD) for the first time. We focus on the experiences of a novice facilitator of StoryCircles, a professional learning process in which teachers collectively script and visualize a problem-based lesson, arguing about their rationales for different decisions connected to discussions of students' work. We situate the challenges the facilitator encountered by identifying the expectations that were not met while she was facilitating and offer possible connections to the PD design. We close by suggesting a perspective to account for facilitation challenges. [For the complete proceedings, see ED630210.]

This paper describes the conceptions about complex numbers that a group of university students has, these were built from the application of an activity sequence centered on these numbers. This sequence is based on the APOS theory, some aspects of semiotic representation theory, and the use of digital technology. Particularly, both the general results of a pretest and a posttest are shown and compared. Additionally, the example of a student is analyzed to show evidence of how the mental structures and mechanisms that define the students' conceptions are built through the implementation of the sequence. The results show how the activity sequence allowed students to coordinate algebraic and geometric processes on complex numbers to improve their conceptions. [For the complete proceedings, see ED630210.]

This paper explores how a professional learning community (PLC) redesigns high school mathematics lessons towards a shared commitment. We describe the nature of a PLC's collective curricular vision to illuminate how teachers can come to new understandings as a group in order to shift the ways students experience mathematics. Using the curricular noticing framework (attending, interpreting, and responding), we analyzed the meetings of a PLC with six teachers as they individually presented lessons to be redesigned with a focus on the group's shared commitment. Findings indicate three ways ideas were introduced that led to expansive responses, which suggests this analytic approach could identify ways in which a PLC can work towards new curricular decisions. [For the complete proceedings, see ED630210.]

Learning progressions have become an important construct in educational research, in part because of their ability to inform the design of coherent standards, curricula, assessments, and instruction. In this paper, I discuss how a learning progressions approach has guided our development of an early algebra innovation for the elementary grades and provide examples of how this approach can help challenge a settled mathematics learning status quo about the kind of algebra students can learn, when they can learn it, and how all students can be successful. Empirically derived learning progressions are an important part of designing early algebra innovations that can open new curricular pathways for teaching and learning algebra, creating accessible and effective avenues of learning for all students. [For the complete proceedings, see ED630210.]

The National Council of Teachers of Mathematics (NCTM) has been emphasizing the importance of teachers' pedagogical communication as part of mathematical teaching and learning for decades. Specifically, NCTM has provided guidance on how teachers can foster mathematical communication that positively impacts student learning. A teacher may have different academic goals towards what needs to be achieved in a classroom, which require a variety of discourse-based tools that allow students to engage fully in mathematical thinking and reasoning. Accountable or academically productive talk is one such approach for classroom discourse that may ensure that the discussions are coherent, purposeful and productive. This paper discusses the use of a transformer model for classifying classroom talk moves based on the accountable talk framework. We investigate the extent to which the classroom Accountable Talk framework can be successfully applied to one-on-one online mathematics tutoring environments. We further propose a framework adapted from Accountable Talk, but more specifically aligned to one-on-one online tutoring. The model performance for the proposed framework is evaluated and compared with a small sample of expert coding. The results obtained from the proposed framework for one-on-one tutoring are promising and improve classification performance of the talk moves for our dataset. [For the full proceedings, see ED623995.]

This document is the fourth volume of the proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education. Conference papers are centered around the theme of "Learners and Learning Environments." This volume features 42 research reports by presenters with last names beginning between Mul and Wu: (1) Case Studies of Children's Development of Structure in Early Mathematics: A Two-Year Longitudinal Study (Joanne Mulligan, Michael Mitchelmore, and Anne Prescott); (2) A Case Study of How Kinesthetic Experiences Can Participate in and Transfer to Work with Equations (Ricardo Nemirovsky and Chris Rasmussen); (3) The Construction of Proportional Reasoning (Stephen Norton); (4) The Transition of a Secondary Mathematics Teacher: From a Reform Listener into a Believer (Jo Clay Olson and Karmen Kirtley); (5) Substantive Communication of Space Mathematics in Upper Primary School (Kay Owens); (6) Transforming Korean Elementary Mathematics Classrooms to Student-Centered Instruction (JeongSuk Pang); (7) The Effect of Improved Automaticity and Retrieval of Basic Number Skills on Persistently Low-Achieving Students (John Pegg, Lorraine Graham, and Anne Bellert); (8) Degrees of Freedom in Modeling: Taking Certainty out of Proportion (Irit Peled and Ronit Bassan-Cincinatus); (9) "I Know That You Don't Have to Work Hard": Mathematics Learning in the First Year of Primary School (Bob Perry and Sue Dockett); (10) Disentangling Mentors' Role in the Development of Prospective Teachers' Efficacy Beliefs in Teaching Mathematics (George Philippou and Charalambos Y. Charalambous); (11) Linear Functions and a Triple Influence of Teaching on the Development of Students' Algebraic Expectation (Robyn Pierce); (12) Engaging the Learner's Voice? Catechetics and Oral Involvement in Reform Strategy Lessons (Adrian J. Pinel); (13) Teaching Projectile Motion to Eliminate Misconceptions (Anne Prescott and Michael Mitchelmore); (14) An Investigation of a Preservice Teacher's Use of Representations in Solving Algebraic Problems Involving Exponential Relationships (Norma Presmeg and Rajeev Nenduradu); (15) On Embodiment, Artifacts, and Signs: A Semiotic-Cultural Perspective on Mathematical Thinking (Luis Radford, Caroline Bardini, Cristina Sabena, Pounthioun Diallo, and Athanase Simbagoye); (16) Generalization Strategies of Beginning High School Algebra Students (Joanne Rossi Becker and Ferdinand Rivera); (17) Synchronizing Gestures, Words and Actions in Pattern Generalizations (Cristina Sabena, Luis Radford, and Caroline Bardini); (18) Analyzing Student Modeling Cycles in the Context of a "Real World" Problem (Roberta Y. Schorr and Miriam Amit); (19) Negotiating about Perceived Value Differences in Mathematics Teaching: The Case of Immigrant Teachers in Australia (Wee Tiong Seah); (20) Development of Mathematical Norms in an Eighth-Grade Japanese Classroom (Yasuhiro Sekiguchi); (21) Solving Additive Problems at Pre-Elementary School Level with the Support of Graphical Representation (Ana Coelho Vieira Selva, Jorge Tarcisio da Rocha Falcao, and Terezinha Nunes); (22) From the Everyday, through the Authentic, to Mathematics: Reflecting on the Process of Teaching Mathematics through the Everyday (Godfrey Sethole); (23) Personal Experiences and Beliefs in Early Probabilistic Reasoning: Implications for Research (Sashi Sharma); (24) Assimilating Innovative Learning/Teaching Approaches into Teacher Education: Why Is It so Difficult? (Atara Shriki and Ilana Lavy); (25) Student Thinking Strategies in Reconstructing Theorems (Tatag Yuli Eko Siswono); (26) A Comparison of How Textbooks Teach Multiplication of Fractions and Division of Fractions in Korea and in U.S. (Ji-Won Son); (27) Mathematical Knowledge of Pre-Service Primary Teachers (Beth Southwell and Marina Penglase); (28) Analysing Longitudinal Data on Students' Decimal Understanding Using Relative Risk and Odds Ratios (Vicki Steinle and Kaye Stacey); (29) Girls Journey toward Proportional Reasoning (Olof Bjorg Steinthorsdottir); (30) University Student Perceptions of CAS Use in Mathematics Learning (Sepideh Stewart and Michael O. J. Thomas); (31) Prospective Teachers' Understanding of Proof: What if the Truth Set of an Open Sentence Is Broader than that Covered by the Proof? (Andreas J. Stylianides, Gabriel J. Stylianides, and George Philippou); (32) Planning and Teaching Mathematics Lessons as a Dynamic, Interactive Process (Peter Sullivan, Robyn Zevenbergen, and Judy Mousley); (33) Teacher Factors in Integration of Graphic Calculators into Mathematics Learning (Michael O. J. Thomas and Ye Yoon Hong); (34) Students' Overreliance on Linearity: An Effect of School-Like Word Problems (Wim Van Dooren, Dirk De Bock, Dirk Janssens, and Lieven Verschaffel); (35) A Process of Abstraction by Representations of Concepts (N. C. Verhoef and H. G. B. Broekman); (36) Argumentation Profile Charts as Tools for Analysing Students' Argumentations (Jill Vincent, Helen Chick, and Barry McCrae); (37) Characterizing Middle School Students' Thinking in Estimation (Tanya N. Volkova); (38) Reviewing and Thinking the Affect/Cognition Relation (Margaret Walshaw and Tania Cabral); (39) Young Children's Ability to Generalise the Pattern Rule for Growing Patterns (Elizabeth Warren); (40) Consolidating One Novel Structure whilst Constructing Two More (Gaye Williams); (41) Spreadsheets, Pedagogic Strategies and the Evolution of Meaning for Variable (Kirsty Wilson, Janet Ainley, and Liz Bills); and (42) A Study of the Geometric Concepts of the Elementary School Students Who Are Assigned to the van Hiele Level One (Der-bang Wu and Hsiu-Lan Ma). (Individual papers contain references.)

This document contains the second volume of the proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education. Conference papers are centered around the theme of "Learners and Learning Environments." This volume features 43 research reports by presenters with last names beginning between Adl and Fre: (1) Working with Learners' Mathematics: Exploring a Key Element of Mathematical Knowledge for Teaching (Jill Adler, Zain Davis, Mercy Kazima, Diane Parker, and Lyn Webb); (2) A Comparison between Teachers' and Pupils' Tendency to Use a Representativeness Heuristic (Thekla Afantiti-Lamprianou, Julian S. Williams, and Iasonas Lamprianou); (3) Purposeful Task Design and the Emergence of Transparency (Janet G. Ainley, Liz Bills, and Kirsty Wilson); (4) A Developmental Model for Proportional Reasoning in Ratio Comparison Tasks (Silvia Alatorre and Olimpia Figueras); (5) Referential and Syntactic Approaches to Proof: Case Studies from a Transition Course (Lara Alcock and Keith Weber); (6) Teachers' Beliefs about Students' Development of the Pre-Algebraic Concept of Equation (Vassiliki Alexandrou-Leonidou and George N. Philippou); (7) Developing Students' Understanding of the Concept of Fractions as Numbers (Solange Amorim Amato); (8) Multiple Representations in 8th Grade Algebra Lessons: Are Learners Really Getting It? (Miriam Amit and Michael N. Fried); (9) Reform-Oriented Teaching Practices: A Survey of Primary School Teachers (Judy Anderson and Janette Bobis); (10) The Genesis of Signs by Gestures: The Case of Gustavo (Ferdinando Arzarello, Francesca Ferrara, Ornella Robutti, and Domingo Paola); (11) Students' Experience of Equivalence Relations: A Phenomenological Approach (Amir H. Asghari and David Tall); (12) How Series Problems Integrating Geometric and Arithmetic Schemes Influence Prospective Secondary Teachers' Pedagogical Understanding (Leslie Aspinwall, Kenneth L. Shaw, and Hasan Unal); (13) Dealing with Learning in Practice: Tools for Managing the Complexity of Teaching and Learning (Sikunder Ali Baber and Bettina Dahl); (14) Situations of Psychological Cognitive No-Growth (Roberto R. Baldino and Tania C. B. Cabral); (15) Good CAS Written Records: Insight from Teachers (Lynda Ball and Kaye Stacey); (16) Developing Procedure and Structure Sense of Arithmetic Expressions (Rakhi Banerjee and K. Subramaniam); (17) Struggling with Variables, Parameters, and Indeterminate Objects, or How to Go Insane in Mathematics (Caroline Bardini, Luis Radford, and Cristina Sabena); (18) Exploring How Power is Enacted in Small Groups (Mary Barnes); (19) A Framework for the Comparison of PME Research into Multilingual Mathematics Education in Different Sociolinguistic Settings (Richard Barwell); (20) Vygotsky's Theory of Concept Formation and Mathematics Education (Margot Berger); (21) Preservice Teachers' Understandings of Relational and Instrumental Understanding (Kim Beswick); (22) The Transformation of Mathematics in On-Line Courses (Marcelo C. Borba); (23) Using Cognitive and Situated Perspectives to Understand Teacher Interactions with Learner Errors (Karin Brodie); (24) Identification of Affordances of a Technology-Rich Teaching and Learning Environment (TRTLE) (Jill P. Brown); (25) The "A4-Project": Statistical World Views Expressed through Pictures (Michael Bulmer and Katrin Rolka); (26) A Whole-School Approach to Developing Mental Computation Strategies (Rosemary Callingham); (27) A Comparison of Perceived Parental Influence on Mathematics Learning among Students in China and Australia (Zhongjun Cao, Helen Forgasz, and Alan Bishop); (28) Using Word Problems in Malaysian Mathematics Education: Looking beneath the Surface (Kah Yein Chan and Judith Mousley); (29) Constructing Pedagogical Knowledge of Problem Solving: Preservice Mathematics Teachers (Olive Chapman); (30) Revisiting a Theoretical Model on Fractions: Implications for Teaching and Research (Charalambos Y. Charalambous and Demetra Pitta-Pantazi); (31) Students' Reflection on Their Sociomathematical Small-Group Interaction: A Case Study (Petros Chaviaris and Sonia Kafoussi); (32) Investigating Teachers' Responses to Student Misconceptions (Helen L. Chick and Monica K. Baker); (33) Studying the Distribution of Responsibility for the Generation of Knowledge in Mathematics Classrooms in Hong Kong, Melbourne, San Diego and Shanghai (David Clarke and Lay Hoon Seah); (34) Indigenous and Non-Indigenous Teaching Relationships in Three Mathematics Classrooms in Remote Queensland (Tom J. Cooper, Annette R. Baturo, and Elizabeth Warren); (35) Exploring the Strategies Used by Grade 1 to 3 Children through Visual Prompts, Symbols and Worded Problems: A Case for a Learning Pathway for Number (Ty Corvell Cranfield, Cally Kuhne, and Gary Powell); (36) Primary Students' Knowledge of the Properties of Spatially-Oriented Diagrams (Carmel Diezmann); (37) A Conceptual Framework for Studying Teacher Preparation: The Pirie-Kieren Model, Collective Understanding, and Metaphor (Maria A. Droujkova, Sarah B. Berenson, Kelli Slaten, and Sue Tombes); (38) Mathematical Modelling with 9-Year-Olds (Lyn English and James Watters); (39) Exploring "Lesson Study" in Teacher Preparation (Maria L. Fernandez); (40) Child-Initiated Mathematical Patterning in the Pre-Compulsory Years (Jillian Fox); (41) The Tacit-Explicit Nature of Students' Knowledge: A Case Study on Area Measurement (Cristina Frade); (42) Teachers as Interns in Informal Mathematics Research (John M. Francisco and Carolyn A. Maher); and (43) Exploring Excellence and Equity within Canadian Mathematics Classrooms (George Frempong). (Individual papers contain references.)