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These proceedings are a written record of the research presented at the 44th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (PME-NA) held in Nashville, Tennessee, and virtually. This year's conference theme is "Critical Dissonance and Resonant Harmony." The aim of this theme is to reflect not only the time and place that of gathering, but also the time and place in which to conduct academic work. Dissonance can be jarring to experience, whereas harmony can be pleasing. Nashville -- Music City -- which is no stranger to both dissonance and harmony. The papers accepted comprised 101 research reports, 140 brief research reports, 84 Posters, and 16 Working Groups or Research Colloquia. [Individual papers are indexed in ERIC.]

Technology to support collaborative learning has come a long way. Interactive tabletops support collaboration when correctly integrated in activity design. While these devices can now, in principle, be purchased by the general public, wide adoption in schools is hindered by their high cost and lack of mobility. In this paper, we analyze the potential technologies that could replace such devices. After developing a dozen prototypes to further test the potential of the most promising technologies, we found one robust, affordable and accurate solution: SPART (on-Surface Positioning for Augmented RealiTy) allows augmenting any flat surface (images etc.) with tablets or smartphones, addressing both affordability and mobility. This paper focuses on the technological innovations that were designed for SPART. [For the full proceedings, see ED659933.]

Access to body-based resources has been shown to augment cognitive processes, but not all movements equally aid reasoning. Interactive technologies, like dynamic geometry systems (DGS), potentially amplify the link between movement and geometric representation, thereby deepening students' understanding of geometric properties. This study investigated the influence of access to a DGS on students' geometric reasoning and gesture production compared to the impact of only spontaneous gestures. Access to simulations did not significantly increase the likelihood of students producing correct mathematical insights compared to spontaneous gestures. However, when controlling gesture production, DGS access became a significant predictor for transformational proof production. We also found that access to DGS influenced the type of gestures students produced during geometric reasoning tasks. This study underlines the vital role of gestures in geometric reasoning and suggests that while DGS offers benefits, spontaneous gestures remain a crucial embodied resource for exploring geometric concepts. [This paper was published in: "Proceedings of the 18th International Conference of the Learning Sciences - ICLS 2024," edited by R. Lindgren et al., International Society of the Learning Sciences, 2024, pp. 131-38.]

Mathematics curricula are often the primary resource used in the teaching and learning of mathematics--in particular, they represent mathematical knowledge for teachers and students. Yet, there seems to be a discrepancy between the curricular materials written for teachers and students (e.g., more variety in the semiotic choices in text provided for teachers than for students). This paper illustrates a way to compare the language written for the teacher and the student in curricula. Drawing on the social semiotic theory known as systemic functional linguistics (SFL), we examine what meaning potentials can be associated with the exchanges of knowledge proposed in mathematical text. Examining a Unit on Similarity in an online mathematics curriculum widely used in the United States, the paper explores how the semiotic systems in text presented for the teacher and student may impact the students' opportunity to learn and pose challenges in the instructional exchange between teacher and student. [For the complete proceedings, see ED657822.]

The interactional role of language use in the mathematics classroom is explored in the last few decades. This paper adds to the knowledge base by exploring different ways of talking that children and their teacher use in a New Zealand primary geometry English-medium classroom. Bakhtin's concept of speech genres is used for an analysing transcript of one audiovisually recorded whole-class interaction episode from a geometry lesson in a Year 5/6 classroom. The analysis suggests that the teacher and children use various genres to participate in classroom interactions. The use of several genres provides insights into what children and the teacher construct as mathematical in the real sense. The paper finished with a few implications for teaching and research.

Research studies are abundant in pointing at how the transition from additive to multiplicative thinking acts as a core challenge for students' understanding of proportionality. This said, we have yet to understand how this transition can be supported, and there remains significant questions to address about how students experience it. Recent work on proportional reasoning has pointed to a type of strategy, called "relative", that appears to be lodged right between additive and multiplicative ways of thinking. This sort of "in-between" strategy raises significant interest and motivates further analysis. In this paper, I explore several of these relative strategies engaged in by a 13-year-old student, Marie, during a series of individual interviews. The analysis outlines several dimensions that can inform as much the transition from additive to multiplicative thinking than proportional reasoning itself. [For the complete proceedings, see ED658295.]

Units coordination, defined by Steffe (1992) as the mental distribution of one composite unit (i.e., a unit of units) "over the elements of another composite unit" (p. 264) is a powerful tool for modeling students' mathematical thinking in the context of whole number and fractional reasoning. This paper proposes extending the idea of a numerical unit to an algebraic unit and a covariational unit. Evidence to support this extension is taken from two qualitative studies conducted with middle-grades students. Results suggest the coordination of two unknowns in algebraic contexts is limited for students who assimilate with composite units. Additionally, in the context of covariational reasoning, results describe a new partitioned unit that students who assimilate with composite units may apply. Implications for engaging all learners are discussed. [For the complete proceedings, see ED658295.]

A central goal of lesson-centered professional development programs (PD) for mathematics teachers is to learn by constructing an artifact, for example, by designing and improving a lesson plan together. That leads to the questions, what does it mean, for mathematics teachers, to improve a lesson? And how can improvements be accounted for in the analysis of the resulting artifacts, especially when these are multimodal? This paper lays the groundwork for answering such questions by drawing on empirical data from a lesson-centered PD program for secondary geometry teachers. We show how semiotic choices were made to convey that the teacher would need to support students when geometry instruction moves from a construction task to a proof, by (1) addressing students' confusion; and (2) creating a shared language to discuss diagrams. We relate these findings to teachers' professional growth and the conference theme. [For the complete proceedings, see ED657822.]

For centuries, there has been a debate about the role of undergraduate education in society. Some have argued that universities should focus on practical skills and knowledge to prepare students for the workforce, while others have supported the idea that universities should prioritize providing a broad understanding of disciplinary knowledge and practices. In this paper, we leverage data collected from 32 interviews to explore how instructors of the undergraduate geometry course for teachers (GeT) talk about the various tensions they experience in their work. Three distinct ways of talking about tensions emerged from the data: the tension as a dilemma that needs to be managed, the tension as a place to take sides, the tension as an opportunity to reframe aspects of the work. In closing we draw connections between these patterns in the data and the two perspectives about the role of undergraduate mathematics courses in preparing PTs for the work of teaching. [For the complete proceedings, see ED657822.]

South Australia's PISA performance has been in constant decline since 2003 with the proportion of PISA participants meeting the Australian national proficient standard dropping from 73% (in 2003) to 50% (in 2018). In contrast, Singapore is a consistently strong performer. To better understand student readiness in answering PISA questions, this paper reports a curriculum comparison between Australia and Singapore for Years 9 and 10 in Measurement and Geometry. The findings highlight the similarities and differences in the topics covered in both countries' curricula and raise questions about potential implications for student outcomes in PISA.

Building from the evidence-based learning progression in geometric reasoning from the RMFII project, this paper presents data from students' solutions to three problems in geometry and measurement situations to identify key components needed to nurture reasoning. To show emerging analytical reasoning students must coordinate multiple pieces of information and demonstrate cognitive flexibility in their use of visualisation, diagrams, language, and symbols.

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 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.]

Researchers have reported that preservice mathematics teachers' (PMTs') conceptions for radian angle measure are dominated by conceptions of degrees, memorization, and calculational strategies, and are always expressed in terms on [pi]. In this report, I unpack the mathematical meanings of a PMT, Henry (pseudonym), as he engaged in three task-based interviews involving radian angle measure. In the first interview, Henry described radians procedurally, emphasizing memorized special angles written in terms of [pi]. By the end of the final interview, Henry described radians conceptually, including generalizing special angles. I conclude by highlighting the importance of tasks designed to build and support conceptual understanding. [For the complete proceedings, see ED658295.]

We conducted a multimedia survey experiment with a nationally representative sample (n = 405) of secondary geometry teachers. Participants were shown storyboard depictions of instructional episodes and asked to rate the appropriateness of the (hypothetical) teacher's actions using a Likert-like response format. We analyzed participants responses using ANOVA. The purpose of the experiment was to investigate how secondary geometry teachers expect students to communicate when presenting proofs during class. Our results (1) replicated findings from a prior investigation of what teachers expect when students present proofs and (2) investigated how geometry teachers reacted to instructional practices that attempted to steer student presentations of proofs toward disciplinary communication practices. [For the complete proceedings, see ED658295.]

Angularity is a persistent quantity throughout K-12+ school mathematics, and many studies have shown that individuals often conflate angularity with linear attributes (e.g., the length of an angle model's sides). However, few studies have examined the productive ways in which students might reason about angularity while attending to linear attributes like side lengths. Leveraging data from a yearlong teaching experiment with ninth-grade students, I present four mental operations that students indicated for altering lengths while preserving angularity. Additionally, I consider implications of these mental operations for teaching and research. [For the complete proceedings, see ED658295.]

What mathematics is and how to teach it are questions that mathematics educators constantly confront. A challenge identified in mathematics education is supporting students to see mathematics as normal human activity. Mathematics educators' viewpoints of mathematics determine whether they can recognize and exploit learning opportunities for student. This study presents the boundary crossing collaboration between two female educators within informal learning settings. Drawing on a boundary crossing learning perspective, I explored two female educators' mathematics experiences and interrogated their boundary crossing experiences as sites for humanizing mathematics. Based on this exploration, I address the possibilities for humanizing mathematics by facilitating boundary crossing between mathematics education and informal education. [For the complete proceedings, see ED658295.]

This study explores volitional and affective responses to instructional activities aimed at developing geometric and spatial vocabulary of a neurodivergent student. Using teaching experiment methodology across 15 instructional sessions, we observed how the student responded to games, direct instruction, and vocabulary support embedded in spatial reasoning activities. A narrative microgenetic analysis explored how these activities were associated with evidence of engagement and confidence in learning. We describe how embedding vocabulary in the student's own mathematical activity was supportive of developing vocabulary, engagement, and confidence, while games and direct vocabulary instruction were not. [For the complete proceedings, see ED658295.]

Understanding fraction as a quantity has been identified as a key developmental understanding. In this study, students in Grades 5, 8, and 11 were asked to compare the areas of two halves of the same square--a rectangle and a right triangle. Findings from this study suggest that students who understand fraction as a quantity use reasoning related to a generalization, whereas students who understand fraction as an arrangement use reasoning related to visualization, computation, or characteristics of the specific shapes involved. Knowing the reasoning exhibited by students can inform both teachers and mathematics curriculum writers in the creation of and planning for instructional tasks. [For the complete proceedings, see ED658295.]

Numerous studies (Anwar et al., 2016; Carpenter et al., 1980; Huang & Witz, 2013; Outhred & Mitchelmore, 1996; Machaba, 2016; Van de Walle, Karp & Bay-Williams, 2014; Winarti et al., 2012) have shown that students need help understanding the concept of area and perimeter. This study aimed to investigate what students know about finding the area and perimeter of rectangles through problem-solving. The participants for the study were five 4th Graders from a public school located in a mid-size suburb setting outside the Chicago metropolitan area. Participants were asked to solve five problem-solving tasks, after which an interview was conducted for three out of the five participants. Results showed that students did not understand the concept of the area and perimeter of rectangles. Implications for the study, as well as future research, were also discussed. [For the complete proceedings, see ED658295.]

A digital wall is a tool for students to structure and register their online work on mathematical problem-solving activities that involve the coordinated use of digital technologies. How could students use such digital wall to understand mathematical concepts and to develop problem-solving competencies? The aim of this study is to analyze and document the extent to which the students' use of the wall became a powerful tool to share, discuss, and refine their problem-solving approaches. To this end, the development of the sessions was structured around principles that foster an inquiry method that privileges the participants' formulation of questions; the systematic and coordinated use of digital technologies, and a support system for students to face technical and content questions. Results indicate that the use of a digital wall became a powerful tool for the participants to report and share their ideas and problem-solving approaches to refine and extend their solution methods. [For the complete proceedings, see ED658295.]

Immersive spatial diagrams (three-dimensional diagrams rendered with immersive stereoscopic displays) offer learners the opportunity to extend practices of diagramming in school geometry into a human-scale, spatial context. Learners can use two or more hands to inscribe simultaneously with digital spatial painting tools, a spatial analog of the input affordances of multi-touch tablets. In this study, we describe how learners coordinate their use of multi-hand spatial painting tools to inscribe diagrams. In particular, with human-scale spatial diagrams, learners' embodied inscriptions can center them within the diagram (e.g., with one's torso as an axis) and realize mathematical relationships (e.g., an arm sweeping a circle as a radius). Our analysis suggests that the design of spatial diagramming environments should consider the opportunities for embodied connections afforded by large-scale, collaborative, and multi-handed interactions. [For the complete proceedings, see ED658295.]

We report on an effort to vet a list of 10 student learning objectives (SLOs) for geometry courses taken by prospective geometry teachers. Members of a faculty online learning community, including mathematicians and mathematics educators who teach college geometry courses taken by prospective secondary teachers developed this list in an effort to reach a consensus that might satisfy various stakeholders. To provide feedback on the final list of 10 SLOs, we constructed and collected responses to a survey in which 121 college geometry instructors ranked a set of potential SLOs, including the 10 proposed SLOs as well as 11 distractors. The 10 SLOs were, for the most part, among the highest ranked by the sample. [For the complete proceedings, see ED658295.]

The study describes how the common representations are coordinated. I analyzed a popular geometry textbook using semiotics and a pragmatic approach to capture the variety of representations into categories and to use descriptive statistics to narrow the focus to the most common representations and coordinations. The major findings are: (1) exposing which representations are most often coordinated like written language; (2) some of the mechanisms in coordination use numbers, point names, and textbook gestures, which include color, arrows, font changes, etc. [For the complete proceedings, see ED658295.]

The Forty Fifth Annual meeting of the North American chapter of the International Group for the Psychology of Mathematics Education was held PME-NA 45 in Reno, Nevada, October 1-4, 2023. The conference theme is "Engaging All Learners." Math learning should be a joyful experience for all students. When students are engaged and inspired, they are motivated to learn. Instruction that targets the learning needs and interests of our students makes it possible for students to excel in learning math. Participants in the conference explored how to create conditions to support learning that build on student engagement and interest in addition to other research engaged by the PME-NA community. The specific conference theme questions explored as part of the conference was: (1) How can we engage all students to learn math content by building on their interest and motivation to learn? (2) How do we design learning environments that take students and learning into account? (3) What are the design features of tools and curricula design features considering student engagement and interest in supporting learning? (4) How do we build partnerships with schools and the community to support student engagement and math learning? and (5) What research agendas should we pursue to ensure that all students reach their potential by paying attention to engagement and learning needs? The acceptance rate for Research Report was 45%, the acceptance rate for brief research reports was 70%. The acceptance rate for posters was 90%. Note: some papers were accepted in alternate format than originally proposed. The total number of participants who submitted proposals as co-authors was 1083. [Individual papers are indexed in ERIC.]

Limited literature addresses how elementary preservice teachers (PSTs) can advance their thinking about hierarchical geometric relationships. Informed by a commognitive perspective, we investigated this phenomenon as a matter of discursive change, focusing on word use, visual mediators, and narratives. We report on a teaching experiment involving a PST whom we call Mariah. Mariah's progress seemed to be driven primarily by metalevel changes in her word use and interpretations of diagrams. These changes supported shifts in her narratives regarding hierarchical geometric relationships. Instructor moves related to these discursive changes involved brokering between multiple discourse communities. Our analysis reveals nuances of communication that we do not see highlighted in the literature on PST education. [For the complete proceedings, see ED657822.]

Opportunities for students to share their thinking with the class--which I refer to as "work-sharing practices--require a profound shift in who is positioned with intellectual authority in mathematics classrooms. This study explores work-sharing practices in one sixth grade mathematics classroom through an interactional lens. Video analysis revealed three types of work-sharing interactions along a continuum of distributions of intellectual authority, ranging from the presenting student holding authority to the teacher holding authority. Notably, in the center of the continuum were instances in which the student was initially positioned with authority, but that authority shifted to the teacher in the next moment, largely based on correctness of the student's work. Findings suggest the need to deepen our understanding of authority dynamics in work-sharing interactions, as the field works to center students' thinking. [For the complete proceedings, see ED657822.]

The aim of this study is to characterize ways of reasoning and arguing that first year university mathematics students exhibit in problem-solving activities from a course that emphasizes the importance of formulating conjectures and the search for different ways to support or validate them. The use of a Dynamic Geometry System in the representation of problems and in the formulation of conjectures or relationships that are important in the solution processes is highlighted. In this context, students have the opportunity to look for different ways to argue and support the relevance and validity of the conjectures. Results indicate that students extend their ways of reasoning so that allows them to move from empirical to formal arguments within problem solutions. [For the complete proceedings, see ED657822.]

In response to the challenge of geometry's abstract and less engaging nature for students, this study explored the potential of connecting geometry to cultural elements, specifically geometric ornaments found in various cultures worldwide. Geometric ornaments, laden with cultural and spiritual significance, serve as a bridge between mathematics education and cultural contexts. The current research focused on in-depth interviews with 10 school students chosen randomly from two ninth-grade classes from an Arab middle school who were engaged in activities involving analyzing of geometric properties of ornaments, constructing ornaments using a compass and straightedge, and problem-solving exercises related to these ornaments. The qualitative data were analyzed to answer how the students perceived the learning of geometry in the context of geometric ornaments. The findings revealed a significant positive view of the students towards geometry after engaging in these activities. [For the full proceedings, see ED656038.]

"Geometry--Do" is a textbook about plane geometry. It will be divided into two volumes, "Geometry without Multiplication: White through Red Belt," and "Geometry with Multiplication: Blue through Black Belt. "The white- and yellow-belt chapters are neutral geometry; the remainder of "Volume One" and all of "Volume Two" is Euclidean geometry. It is primarily intended to teach geometry from the ground up, starting with the postulates and citing only already-proven theorems. It trains mathletes for competition, but it is not the usual grab-bag of unproven theorems chosen haphazardly and solely because they appeared in past exams. The early chapters prepare students for jobs in construction, architecture, surveying, graphic arts, and military defense. The later chapters teach geometry needed by engineers and military officers. In this lecture, the White Belt chapter is presented. I will address these people: Pure Mathematicians: Moise derides the "lighthearted use of the word let." I prove the crossbar theorem and other foundations not usually taught in high school, and I discuss Hilbert's "Foundations of Geometry." High-School Teachers: Randomly assigning letters to points is what makes geometry confusing. I have special symbols for midpoints, perpendicular feet, and infeet (where the angle bisector cuts the opposite side of a triangle) and exfeet. Administrators: I present clear distinctions between "Geometry--Do" and "Common Core" with examples that concerned parents can understand. Construction Workers: I invent the Aguilar A-Frame, give detailed instructions on squaring abasement foundation wider than a tape measure without exiting the rectangle, and discuss how building with wood differs from steel construction. Military Officers: I discuss troop positioning along a frontier that is plagued with cross-border raids, which assumes that friendly and enemy troops move at the same speed, and a parabola is the set of points equidistant from the focus and the directrix. [For the complete proceedings, see ED655360.]

This study investigated in-service teachers' conceptualization of the pure imaginary number, within the Cartesian form, "a" + "ib" where "a" and "b" are real numbers and "i" is the imaginary unit. As part of a larger design-based research study, in which a professional development (PD) program was designed to investigate five in-service teachers' conceptualization of different forms of complex numbers, we conducted pre and post written problem-solving sessions together with post PD interviews. Results showed that after PD all the participants defined "i" as one of the roots of the quadratic equation, x[superscript 2] + 1 = 0. They also could show geometrically as a point (0,1) on the complex plane. Although all the participants mentioned operator meaning of "i" as a 90-degree rotation when multiplied with b; only one of them mentioned the dilation meaning of b when multiplied with "i". Results suggested considering the pure imaginary part of the Cartesian form focusing on both the operator meanings of b and "i" is important for understanding the nature of complex numbers and the complex plane for teacher education and teacher content knowledge. These results further suggest that quantitative reasoning might lay a foundation for connecting different forms of complex numbers, including the unit, "i." [For the complete proceedings, see ED655360.]

Educational software reinforces the learning process in high school. This way, GEO-line software for learning geometry will strengthen teenagers' skillset between 15 and 16 years old while using this new application. The SCRUM and ADDE methodologies applied include the analysis, design, implementation, and evaluation phases as the core principles for software development. The technological toolset selected was the NEXTJS framework with server-side rendering technology. Strapi API CMS handled the content used on the model, view, controller (MVC) patterns, and REST architecture for communication and the PostgreSQL database. ISO/IEC 25010 metric was considered for the usability evaluation and applied with a pool of 80 students who were surveyed using the adapted USE and PSSUQ QUESTIONNAIRES with a Likert scale. The findings demonstrate that GEO-line software meets the satisfactory higher standards usability level by measuring pedagogical capacity, ease-of-use, and ease-of-learning parameters. This research turns the geometry learning reinforcement tool for students into a fun and operational way to increase the student's skillset. [For the complete proceedings, see ED655360.]