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This is an extended conference proceedings report [Long Paper] based on a shorter contributed report at the same conference. A variety of computerized learning platforms exist. In mathematics, most include sets of problems to complete. Feedback to users ranges from a single word like "Correct!" to offers of hints and partially- to fully-worked examples. Behind-the-scenes design of such systems also varies--from static dictionaries of problems to responsive programming that adapts assignments to users' demonstrated skills within the computerized environment. This report presents background on digital learning contexts and early results of a mixed-methods study that included a cluster randomized controlled trial design. The study was in community college algebra classes where the intervention was a particular type of web-based activity and testing system. [This paper was published in: Weinberg, A., Rasmussen, C., Rabin, J., Wawro, M., and Brown, S. (Eds.), "Proceedings of the 20th Annual Conference on Research in Undergraduate Mathematics Education," p638-645. San Diego, CA (2017).]

This paper reports on a study of choices about the use of a computer algebra system (CAS) or pen-and-paper (p&p) by a class of seven Year 11 Mathematical Methods (CAS) students as they completed a calculus worksheet. Factors that influenced students' choices are highlighted by comparing and contrasting the use of CAS and p&p between students. Teacher expectation of students' use of CAS and p&p reveals that, even in a small class, the students' use of CAS and p&p sometimes differed from what was expected. The analysis here indicates that there are a variety of factors that influence students' decisions, including speed of calculation and accuracy of p&p work.

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.]

Instructors are a chief input into the higher education production process, yet we know very little about their role in promoting student success. This is in contrast to elementary and secondary schooling, for which ample evidence suggests teacher quality is an important determinant of student achievement. Whether colleges could improve student and institutional performance by reallocating instructors or altering personnel policies hinges on the role of instructors in student success. In this paper we measure variation in postsecondary instructor effectiveness and estimate its relationship to overall and course-specific teaching experience. We explore this issue in the context of the University of Phoenix, a large for-profit university that offers both online and in-person courses in a wide array of fields and degree programs. We focus on instructors in the college algebra course that is required for all BA degree program students. We find substantial variation in student performance across instructors both in the current class and subsequent classes. Variation is larger for in-person classes, but is still substantial for online courses. Effectiveness grows modestly with course-specific teaching experience, but is unrelated to pay. Our results suggest that personnel policies for recruiting, developing, motivating, and retaining effective postsecondary instructors may be a key, yet underdeveloped, tool for improving institutional productivity. [This paper was prepared for the NBER Conference "Productivity in Higher Education" held on June 1, 2016.]

This paper discusses two "GeoGebra" applets, "Radical" and "Parabola," that are designed to strengthen the conceptual understanding of specific topics in algebra. The design and pedagogical basis of the applets are presented. The integration of the applets in teaching Grade 9 mathematics in a partner high school in the Philippines is then discussed. Finally, we report feedback gathered from students and teachers during this integration. Their feedback indicates the potential of these applets for improving students' learning of algebra. [For the full proceedings, see ED659933.]

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.]

This paper is a part of a larger research study that we are conducting to develop a framework consisting of the data-driven best practices around teaching and learning algebra in K-12 classrooms. The purpose of this paper is to develop a usable definition for algebra interventions for the second stage of screening within the larger study, which will be done by reviewing seminal or systematic reviews in the algebra field across grades K-12. Combining the algebra definitions with the type of interventions produces three questions to ask during the second stage of screening to decide if a study is an algebra intervention. [For the complete proceedings, see ED658295.]

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 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.]

This paper provides a case study account of a preservice secondary mathematics teacher's thinking while engaging in slope tasks using dynagraphs. The data included audio recordings and screen captures of a small group of preservice teachers engaging with these tasks, with our analysis focusing on the case of Robin. Despite familiarity using slope to measure steepness and determine relationships in a Cartesian plane setting, results indicate Robin struggled to reason about these same uses of slope when presented with a dynagraph. Using the APOS-Slope framework as a lens, the analysis suggests that Robin is limited to thinking of slope as an Action when using slope to measure steepness and determine relationships, relying heavily on shape thinking in the Cartesian plane. Implications for further research are provided. [For the complete proceedings, see ED658295.]

The positive relationship between students' attitudes toward mathematics and mathematics achievement is well documented. Yet there is a worsening problem of low appeal of mathematics especially at the secondary level. Therefore, in this paper we focus on three high school students who report a strong dislike of mathematics. By analyzing student surveys, interviews, and lesson observation data, we examined how some mathematical lessons improved these students' experiences (i.e., their aesthetic dimensions). We found that while student preferences varied, each student was interested in lessons that centered them as sense-makers and in which the content unfolded with suspense. Such lessons led to positive aesthetic responses such as surprise, curiosity, and satisfaction. Thus, lessons can be designed in which even students with the most negative views of mathematics can find mathematical concepts interesting. [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.]

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.

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.

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 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.]

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.]