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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.]

Prisms VR, a secondary math learning application, allows for users to see, manipulate, and engage with mathematical concepts in an embodied way in Virtual Reality (VR) environment. We examine cases in which mathematics teachers and middle school students worked through Prisms and reflected upon their experiences. Findings indicate that VR environments can help teachers find alternative ways of presenting algebraic concepts in a hands-on, interactive, and embodied manner; and these findings were confirmed based on the student experience. VR allows teachers to make connections through scaffolding and personalized experiences, while also aiding students in the exploration and meaning-making. [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. 1610-13.]

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.

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.]

The use of online learning objects in teaching algebra is examined in this research, emphasizing its benefits, such as accessibility, flexibility, interactive involvement, differentiated instruction, quick feedback, and links to real-world situations. Strategies include conceptual comprehension, interactive practice, individualized learning, real-world applications, and collaborative learning. It is investigated how to enhance cognitive processes, including mathematical reasoning, problem-solving abilities, metacognition, and learning transfer. Motivational variables such as personalization, collaboration, relevance, authenticity, goal-setting, and progress tracking are examined. The link between using online learning objects to teach algebra and fostering a growth mindset in math is reviewed, with techniques stressing effort, perseverance, constructive criticism, and a supportive learning environment. There is also research into neuroscientific topics, including multimodal learning, cognitive load theory, active learning techniques, efficient feedback, and customization. In conclusion, including online learning objects in algebra instruction may increase students' mathematical skills, boost learning opportunities, and build a positive learning environment that encourages growth mindsets and unlocks students' potential in algebra. [For the full proceedings, see ED652261.]

This study describes students' learning obstacles in solving early algebra problems requiring functional thinking ability. To reach this aim, qualitative research was conducted in this study. Participants of this study were 39 ninth graders and a mathematics teacher at one of the lower secondary schools in Bandung, Indonesia. The data were collected through the written test about early algebra problems, interviews, and document study. The findings revealed that fewer students achieve the correspondence level in their functional thinking ability. Many of them are on covariation or recursive patterns level. The variety of students' functional thinking levels in solving the problem is influenced by their previous learning experiences with early algebra, mainly functions. By exploring students' learning experiences, this study shows that students have some learning obstacles, including ontogenic obstacles due to students' lack of prerequisite knowledge about the concept of variables, didactical obstacles due to the teacher's teaching implementation focusing solely on the operational rather than the structural conception of functions, and epistemological obstacles due to students' limited knowledge in the concept of variables and functions. Therefore, the identified learning obstacles can be one of the references when developing a lesson design about functions for enhancing students' functional thinking ability. [For the complete proceedings, see ED655360.]

We share results from a one-year early algebra classroom intervention. One kindergarten, one first-grade, and one second-grade classroom participated in the intervention and three classrooms of the same grade levels served as control sites. The intervention addressed the structure of even and odd numbers, mathematical equivalence and equations, properties of arithmetic, the representation of varying unknown quantities, and functional thinking. Students in the control classrooms received their regular mathematics instruction. We compare growth in algebraic thinking across the two groups as measured by an online assessment completed by 40 students per condition. Findings suggest that students who participated in the early algebra lessons experienced more overall growth in their understanding of early algebra concepts as compared to control students.

Integrating visual representations in an interactive learning activity effectively scaffolds performance and learning. However, it is unclear whether and how "sustaining" or "interleaving" visual scaffolding helps learners solve problems efficiently and learn from problem solving. We conducted a classroom study with 63 middle-school students in which we tested whether sustaining or interleaving a particular form of visual scaffolding, called anticipatory diagrammatic self-explanation in an Intelligent Tutoring System, helps students' learning and performance in the domain of early algebra. Sustaining visual scaffolding during problem solving helped students solve problems efficiently with no negative effects on learning. However, in-depth log data analyses suggest that interleaving visual scaffolding allowed students to practice important skills that may help them in later phases of algebra learning. This paper extends scientific understanding that sustaining visual scaffold does not over-scaffold student learning in the early phase of skill acquisition in algebra. [This paper was published in: "Proceedings of the 44th Annual Conference of the Cognitive Science Society," edited by J. Culbertson et al., Cognitive Science Society, 2022, pp. 1751-58.]

Gaming the system, a behavior in which learners exploit a system's properties to make progress while avoiding learning, has frequently been shown to be associated with lower learning. However, when we applied a previously validated gaming detector across conditions in experiments with an algebra tutor, the detected gaming was not associated with learning, challenging its construct validity. Our iterative exploratory data analysis suggested that some contextual factors that varied across and within conditions might contribute to this lack of association. We present a latent variable model, "item response theory-based gaming detection" (IRT-GD), that accounts for contextual factors and estimates latent gaming tendencies as the degree of deviation from normative behaviors across contexts. Item response theory models, widely used in knowledge assessment, account for item difficulty in estimating latent student abilities: students are estimated as having higher ability when they can get harder items correct than when they only get easier items correct. Similarly, IRT-GD accounts for contextual factors in estimating latent gaming tendencies: students are estimated as having a higher gaming tendency when they game in less commonly gamed contexts than when they only game in more commonly gamed contexts. IRT-GD outperformed the original detector on three datasets in terms of the association with learning. IRT-GD also more accurately revealed intervention effects on gaming and revealed a correlation between gaming and perceived competence in math. Our approach is not only useful for others wanting to apply a gaming assessment in their context but is also generally applicable in creating robust behavioral measures. [For the full proceedings, see ED623995.]

Problem solving and reasoning are two key components of becoming numerate. Reports obtained from international assessments show that Australian students' problem solving ability is in a long-term decline. There is little evidence that teachers are embracing problem solving as part of the classroom routine. In this study, we analyse 598 Year 7 to 10 students' responses to a measurement task using Sfard's commognition framework. Four implications lead to recommendations on how to support curriculum, assessment and pedagogical alignment.

This study reports on a professional learning (PL) initiative aimed at establishing a community of practice, through teacher working groups in which teachers can explore and develop their algebraic pedagogical content knowledge (PCK). Here we report on teachers' solutions to three differently represented algebraic problems and explore what the nature of their solutions tells us about their algebraic reasoning and their PCK. The findings showed that most participants favoured only one solution and provided useful insights for the value of teacher working groups in PL activities to develop teachers' algebraic reasoning, understanding, and extend their range of problem-solving strategies.

Asynchronous, online mathematics teacher professional development (PD) was designed to align with research on teacher professional learning as well as to support Communities of Inquiry (e.g., Garrison et al., 2000). The intervention included two actively facilitated formats and a structured independent condition, where facilitation was integrated into the design of the intervention. Participants' responses to intervention activities were analyzed using indicators of Garrison et al.'s Community of Inquiry framework, seeking to understand the ways in which the intervention enabled participant learning across facilitation formats. Analysis has implications for building the CoI framework into subsequent online asynchronous mathematics teacher PD as a way to increase teacher learning, build community, and effectively scale interventions. [For the complete proceedings, see ED630210.]

This study shares two frameworks for analyzing teacher actions that support students in generalizing and examines how those frameworks align with teacher questioning. One classroom teaching episode focused on the mathematical activity of generalizing is shared to illustrate effective generalizing promoting practices. We found several patterns of productive and unproductive generalizing promoting actions and questioning. Repeating generalizing promoting actions in succession were needed to produce student generalizations. Priming actions that set up for later generalizing promoting were helpful when students struggled to identify and state generalizations. Connection questions promoted generalizing, but justification and concept questions did not. Further research will explore the additional strategies to support teachers in fostering student-created generalizations. [For the complete proceedings, see ED630210.]

Worked examples have been shown to improve student learning in algebra. However, less is known about how to design worked examples to support student learning in online settings. We explore how college students react to worked examples that vary in their degree of extensiveness and dynamicness. In an online, within-subjects study, 109 college students viewed six worked example presentations: (1) static concise; (2) static extended; (3) sequential concise; (4) sequential extended; (5) dynamic history; and (6) dynamic no history. Students rated the helpfulness of each worked example and explained their rating. We found that students rated the static concise presentation as the most helpful and the dynamic no history presentation as the least helpful example. Responses were coded by researchers for common themes and revealed insights that may inform how researchers and teachers design worked examples for online environments. [For the complete proceedings, see ED630210.]

Research on dyscalculia has focused almost exclusively on elementary-aged students' deficits in speed and accuracy in arithmetic calculation. This case study expands our understanding of dyscalculia by documenting how one college student with dyscalculia understood algebra during a one-on-one design experiment. A detailed case study of 19 video recorded sessions revealed that she relied upon unconventional understandings of algebraic quantities and notation, which led to persistent difficulties. This exploratory case study provides new insights into the character of difficulties that arose and persisted for one student with dyscalculia in the context of algebra and suggests the utility of documenting the persistent understandings that students with dyscalculia rely upon, particularly in understudied mathematical domains, like algebra. [For the complete proceedings, see ED630210.]

Interest-driven activities, such as mathematical play, can support student agency, motivation, and engagement, and can foster dispositions that reflect authentic disciplinary engagement. However, the bulk of research on mathematical play investigates the mathematics that emerges in young children's natural play or in informal spaces such as video games. We introduce the term "playful math" to highlight the potential of playifying classroom-based activities, and we explore the nature of students' activity when engaged in playful math tasks in a teaching experiment. Our findings show that playful math tasks increased students' agency, authority, investment, and goal selection, as well as encouraged the development of creative, challenging ideas. We present a case of two students' playful engagement in the form of an Explore-Strategize Cycle and discuss implications of playful math for student engagement. [For the complete proceedings, see ED630210.]

Despite significant research exploring students' quantitative reasoning, few studies have explored the semiotic processes that mediate its development. In this report, we present a case study to show how one student constructed a semiotic chain for a quantity as he worked with a mathematical task. Importantly, we connect frameworks for quantitative reasoning and semiotics to make sense of this process. Our findings show how our case student constructed a sign for a chunk of change in a triangle to support his later construction of the quantity of amount of change of area. We also describe how the case student leveraged these signs to bolster his development of the quantity of total area. We emphasize the role of artifacts, such as physical manipulatives, a digital applet, and a diagram, in this process. Finally, we discuss the implications of this analysis for future studies that explore students' constructions of quantity. [For the complete proceedings, see ED630210.]

Isomorphism and homomorphism appear throughout abstract algebra, yet how algebraists characterize these concepts, especially homomorphism, remains understudied. Based on interviews with nine research-active mathematicians, we highlight new sameness-based conceptual metaphors and three new clusters of metaphors: sameness/formal definition, changing perspectives, and generalizations beyond algebra. Implications include a way to articulate a conceptual purpose for homomorphism beyond its relationship to isomorphism: namely, as a tool for changing perspectives when problem-solving. [For the complete proceedings, see ED630210.]

This research reports on the teacher language and gesture that contributed to shifts in thinking about the equal sign and equations observed in twenty kindergarteners who took part in an early algebra intervention. Our analysis revealed ways in which the teacher used language and gesture to support students in moving from describing and working with the equal sign operationally (i.e., as a signal to compute) to describing the symbol as indicating the equivalence of two amounts and successfully working with equations of various forms. We detail four kinds of language and two kinds of gesture specifically related to mathematical equivalence that we believe contributed to students' growth. [For the complete proceedings, see ED630210.]

By highlighting the curriculum modifications that lead to maintaining, or enhancing, the mathematical quality of an algebra lesson introducing the substitution method for solving systems of equations from an algebra textbook, we present an analysis of how a teacher and her students impact how the mathematical ideas unfold across the lesson and how they are experienced. Using a narrative-based analytical approach to write the stories of the written and enacted lessons, we found key similarities and differences in the lessons. In comparing the mathematical plots, we found evidence of how the teacher and students alter the unfolding story with the incorporation of more jamming than seen in the text and more questions developed based on the students' needs and their responses. [For the complete proceedings, see ED630210.]

This study aimed at exploring Senior High School Mathematic Teachers, Prospective Teachers and Student's conceptual misunderstanding on Senior High School algebra with an intent to uncover the errors they make as a result of conceptual misunderstanding. A test consisting of fourteen (14) tasks was used for data collection. A sample of 210 consisting of sixty (60) prospective senior high school mathematics Teachers from mathematics education department of University of Education Winneba forty (40) SHS mathematics teachers and one hundred and ten (110) senior high school students from four (4) selected senior high schools in Ashanti region of Ghana. The study employed convenience, purposive and simple random sampling as sampling techniques and descriptive survey design as the research design. The data collection tools used were test and semi structured interview guide. Constructivism and behaviourism theories were employed as the theoretical frame work for the study. The study identified seven (7) categories of conceptual misunderstanding in Senior High School algebra among the prospective teachers and the students' whiles six of these seven were also found among the teachers. The seven conceptual misunderstanding identified were on algebraic variables, algebraic expressions, algebraic equations and algebraic word problems. The study recommends that teachers, prospective teachers and students should be aware of the existence of conceptual misunderstanding in teaching and learning of algebraic concept. The study also recommends that, heads of schools should organize workshops and refresher courses for mathematics teachers on sensitive topics like conceptual misunderstandings in mathematics. [For the complete proceedings, see ED631021.]

Previous research has shown that providing video recommendations to students in virtual learning environments implemented at scale positively affects student achievement. However, it is also critical to evaluate whether the treatment effects are heterogeneous, and whether they depend on contextual variables such as disadvantaged student status and characteristics of the school settings. The current study extends the evaluation of a novel video recommendation system by performing an exploratory search for sources of heterogeneity of treatment effects. This study's design is a multi-site randomized controlled trial with an assignment at the student level across three large and diverse school districts in the southeast United States. The study occurred in Spring 2021, when some students were in regular classrooms and others in online classrooms. The results of the current study replicate positive effects found in a previous field experiment that occurred in Spring 2020, at the onset of the COVID-19 pandemic. Then, causal forests were used to investigate the heterogeneity of treatment effects. This study contributes to the literature on content sequencing systems and recommendation systems by showing how these systems can disproportionally benefit the groups of students who had higher levels of previous algebra ability, followed more recommendations, learned remotely, were Hispanic, and received free or reduced-price lunch, which has implications for the fairness of implementation of educational technology solutions.

Knowledge space theory (KST) is a mathematical framework for modeling and assessing student knowledge. While KST has successfully served as the foundation of several learning systems, recent advancements in machine learning provide an opportunity to improve on purely KST-based approaches to assessing student knowledge. As such, in this work we compare the performance of an existing KST-based adaptive assessment to that of a newly developed version--with this new version combining the predictive power of a neural network model with the strengths of existing KST-based approaches. Using a cluster randomized experiment containing data from approximately 140,000 assessments, we show that the new neural network assessment engine improves on the performance of the existing KST version, both on standard classification metrics, as well as on measures more specific to the student learning experience. [For the full proceedings, see ED623995.]

Any advanced class in Science, Technology, Engineering, and Mathematics fields requires prerequisite knowledge. Typically, different students will have different levels of knowledge in these prerequisite areas. A prerequisite (Linear Algebra for Machine Learning course) was implemented as an interactive online course using Jupyter Notebooks and nbgrader and compared with traditional classroom mode. Post-assessment test shows that traditional class provides a better level of understanding. However, a survey shows a preference by students and instructors for interactive implementation compared to traditional class. [For the full proceedings, see ED639633.]