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"Wheel spinning" is the phenomenon in which a student fails to master a Knowledge Component (KC), despite significant practice. Ideally, an intelligent tutoring system would detect this phenomenon early, so that the system or a teacher could try alternative instructional strategies. Prior work has put forward several criteria for wheel spinning and has demonstrated that wheel spinning can be detected reasonably early. Yet the literature lacks systematic comparisons among the multiple wheel spinning criteria, features, and models that have been proposed, across multiple evaluation criteria (e.g., earliness, precision, and generalizability) and datasets. In our experiments, we constructed six wheel spinning detectors and compared their performance under two different wheel spinning criteria with three datasets. The results show that two prominent criteria for wheel spinning diverge substantially, and that a Random Forest model has the most consistent performance in early detection of wheel spinning across datasets and wheel spinning criteria. In addition, we found that a simple model overlooked by previous research (Logistic Regression trained on a single feature) is able to detect wheel spinning at an early stage with decent performance. This work brings us closer to unifying strands of prior work on wheel spinning (e.g., understanding how different criteria compare) and to early detection of wheel spinning in educational practice. [This paper was published in: Proceedings of the 12th International Conference on Educational Data Mining Montreal, QC, Canada, July 2-5, 2019.]

Systems of equations are an important topic in school mathematics. However, there is limited research examining productive ways of supporting students' understandings of systems of equations. In this paper, we first present a conceptual analysis of potential ways students may leverage their quantitative and covariational reasoning to graph systems of relationships. We then describe results from a design experiment in which we examined the potential of supporting middle-grades students in reasoning in ways compatible with this conceptual analysis. We highlight two different ways of reasoning students engaged in as they compared the relative magnitudes of two quantities with respect to a third quantity and leveraged this reasoning to graphically represent two relationships on the same coordinate system. We draw implications from these results for the teaching and learning of systems of equations. [For the complete proceedings, see ED606556.]

This paper provides evidence on how elementary school students from a Mexican public school move from using an operational sense, expressed in computational strategies, to a kind of relational thinking based on structure sense ideas and expressed by number decomposition. Even though the results are somewhat preliminary, they illustrate how students can leave behind their computational strategies and develop a more sophisticated mathematical reasoning regarding equivalence of numerical expressions and equalities; however the strategies they developed tended to be based on "ad hoc decomposition of one side and comparison with the initial form of the other side," rather than on compensation [For the complete proceedings, see ED606556.]

We study the usage of a self-guided online tutoring platform called Algebra Nation, which is widely by middle school and high school students who take the End-of-Course Algebra I exam at the end of the school year. This article aims to study how the platform contributes to increasing students' exam scores by examining users' logs over a three year period. The platform under consideration was used by more than 36,000 students in the first year, to nearly 67,000 by the third year, thus enabling us to examine how usage patterns evolved and influenced students' performance at scale. We first identify which Algebra Nation usage factors in conjunction with math overall preparation and socioeconomic factors contribute to the students' exam performance. Subsequently, we investigate the effect of increased teacher familiarity level with the Algebra Nation on students' scores across different grades through mediation analysis. The results show that the indirect effect of teacher's familiarity with the platform through increasing student's usage dosage is more significant in higher grades. [This paper was published in: "The 9th International Learning Analytics & Knowledge Conference (LAK19)" (Tempe, AZ, March 4-8, 2019). New York, NY: Association for Computing Machinery (ACM). (978-1-4503-6256-6).]

Research in both the New Zealand and international contexts identifies the need for New Zealand classrooms to foster culturally responsive and mathematical practices that align with Påsifika students' cultural values, backgrounds, interests and experiences. The aim of this study was to investigate how the use of culturally responsive tasks along with cultural practices can foster Påsifika students' participation and engagement in mathematics. Eleven Year 5 and 6 students used cameras to document their out-of-school mathematical experiences. These photographs were used to design culturally relevant tasks that were implemented in the classroom. In this paper, we analyse the students' perspectives and understandings of mathematics embedded within their lives beyond the classroom.

Understanding equivalence is fundamental to STEM disciplines, but many students struggle with the concept. We present a novel method for students to explore ideas of equivalence where students dynamically transform expressions from initial states to goal states in the web-based app Graspable Math. The structure of the goal state activities implies that any initial and goal state pair represent the same quantity (or varying quantity). We propose that for students, the physical experience of moving algebraic objects and observing how the initial state transforms into the goal state helps generalize notation mechanics. In fall of 2019, we will test this supposition in a randomized control trial of 1500 students in which student performance in pre- and post-tests will be compared to an online problem set control. [This paper was published in: "Proceedings of the forty-first annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education," edited by S. Otten et al., University of Missouri, 2019, pp. 1738-43.]

Through a theoretical framework emphasizing the importance of fidelity of implementation (FOI), this paper explores how 3rd and 4th grade teachers implemented an early algebra intervention, and the extent to which the FOI related to student learning. The data for this report are taken from the first two years of an experimental research project. Videotaped classroom observations, our primary measure of FOI, were coded by adding to and adapting the Mathematical Quality of Instruction (MQI) instrument, and student performance was measured by overall score (correctness) on an algebra assessment. Results revealed a significant positive relationship between teachers' implementation and their students' performance. [This paper was published in: Galindo, E., & Newton, J., (Eds.), "Proceedings of the Thirty-Ninth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education" (pp. 299-302). Hoosier Association of Mathematics Teacher Educators.]

The results of educational research studies are only as accurate as the data used to produce them. Drawing on experiences conducting large-scale efficacy studies of classroom-based algebra interventions for community college and middle school students, I am developing practice-based data cleaning procedures to support scholars in conducting rigorous research. The paper identifies common sources of data errors in mathematics education research and offers a framework and related data cleaning process designed to address these errors. [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," p129-140. San Diego, CA (2017).]

Learners' choices as to whether and how to use visual representations during learning are an important yet understudied aspect of self-regulated learning. To gain insight, we developed a "choice-based" intelligent tutor in which students can choose whether and when to use diagrams to aid their problem solving in algebra. In an exploratory classroom study with 26 students, we investigated how learners choose diagrams and how their choice behaviors relate to learning outcomes. Students who "proactively" chose to use diagrams achieved higher learning outcomes than those who "reactively" used diagrams when they made incorrect attempts. This study contributes to understanding of self-regulated use of visual representations during problem solving. [This paper was published in: "ICLS2022 Proceedings," International Society of the Learning Sciences, 2022, pp. 1649-52.]

One pedagogical technique that promotes conceptual understanding in mathematics learners is self-explanation integrated with worked examples (e.g., Rittle-Johnson et al., 2017). In this work, we implemented self-explanations with worked examples (correct and erroneous) in a software-based Intelligent Tutoring System (ITS) for learning algebra. We developed an approach to eliciting self-explanations in which the ITS guided students to select explanations that were conceptually rich in nature. Students who used the ITS with self-explanations scored higher on a posttest that included items tapping both conceptual and procedural knowledge than did students who used a version of the ITS that included only traditional problem-solving practice. This study replicates previous findings that self-explanation and worked examples in an ITS can foster algebra learning (Booth et al., 2013). Further, this study extends prior work to show that guiding students towards conceptual explanations is beneficial. [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. 3466-72.]

Proportional reasoning problems can be solved using algebraic reasoning. Therefore, making connections between proportional reasoning and algebraic thinking is important for solving problems. This study examined K-8 teachers' problem-solving strategies as they worked out a real world multi-step problem that involved proportional reasoning and algebraic thinking. The findings revealed that many teachers found this problem challenging. Particularly, some teachers had difficulty figuring out how to translate the variables into an algebraic equation. Some teachers who used variables as labels tended to engage in additive reasoning. They had difficulty representing the proportional problem context algebraically and solving the problem for the unknown quantity. Implications for further research are discussed.

Isomorphism and homomorphism are topics central to abstract algebra, but research on mathematicians' views of these topics, especially with respect to sameness, remains limited. This study examines 197 mathematicians' views of how sameness could be helpful or harmful when studying isomorphism and homomorphism. Instructors saw benefits to connecting isomorphism and sameness but expressed reservations about homomorphism. Pedagogical considerations and the dual function-structure nature of isomorphism and homomorphism are also explored. [For the complete proceedings, see ED630060.]

The appropriateness of guess and check as a problem-solving strategy has been debated. This qualitative study examines the use of guess and check by middle-grades students to solve linear systems of equations. Students' reasoning is examined within the number sequences framework, which is based in part on students' units coordination. Only students at the fourth and fifth stages (out of five) correctly solved systems of equations algebraically; this is attributed to their operations on two- and three-level unit structures, and to a disembedding operation. Students at the third stage applied strategic guess and check methods, which is attributed to assimilating with composite units (i.e., units of units), but these students could not correctly use an algebraic method. For students at the second stage, guess and check was non-strategic, which is attributed to their construction of composite units in activity. Implications for instruction are discussed. [For the complete proceedings, see ED630060.]

Emergent graphical shape thinking (Moore & Thompson, 2015) is a way of reasoning that is critical across numerous STEM fields. However, evidence indicates that the underlying component ideas for emergent thinking are underdeveloped in school mathematics education (e.g., Thompson & Carlson, 2017), and few studies directly report on students' development of this way of thinking. We present the results of a teaching experiment conducted with eighth-grade students to support stable meanings for emergent graphical shape thinking. We focus on the in-the-moment meanings expressed by a pair of students as they engaged in a sequence of tasks that we conjecture could support stable meanings for constructing and interpreting graphs. [For the complete proceedings, see ED630060.]

We present an analysis of three 9th grade integrated mathematics lessons in which a group of teachers and researchers redesigned a sequence of lessons with the goal of engaging a linguistically diverse group of students in productive and powerful discussions (Herbel-Eisenmann et al. 2013). The three lessons were part of a design experiment. Two lessons were observed during regular school day instruction, and the other lesson was part of an after-school teaching experiment. Drawing on a sociocultural framework and methods of classroom discourse analysis (Cazden, 2001; Pierson, 2008), we analyze how the teachers in the three settings structured whole-class discourse to create opportunities for a multilingual group of students to participate in the discussion and to appropriate mathematical tools for thinking. [For the complete proceedings, see ED630060.]

The purpose of this study was to analyze how the systematic use of GeoGebra influences the ways of reasoning of high school students when they incorporate it in the resolution of word problems. The study was carried out with a group of 20 students who were studying Mathematics I. The results show that the students appropriated resources from the dynamic geometry system (GeoGebra) which, when used systematically, allowed them to implement strategies to represent geometrically the concepts involved in the problems, explore and analyze relationships between the elements of the dynamic models built and find the solutions. Thus, thanks to the exploration and analysis of relationships, it was possible to discuss with the students' concepts such as: ratio, domain, locus, variation. [For the complete proceedings, see ED630060.]

This study aims to describe how preservice secondary mathematics teachers (PSMTs) reason about different function representations. The study focuses on two PSMTs' reasonings across static and dynamic representations of functions. Sfard's (2008) Theory of Commognition guided our analysis. Findings indicate that while static representations restrict attention given to covariation, dynamic representations support PSMTs' reasoning about covariation including making connections to how covariation is represented in static graphs. [For the complete proceedings, see ED630060.]

Generalization is a critical component of mathematics learning, but it can be challenging to foster generalization in classroom settings. Teachers need access to better tools and resources to teach for generalization, including an understanding of what tasks and pedagogical moves are most effective. This study identifies the types of instruction, student engagement, and enacted tasks that support generalizing in the classroom. We identified three categories of Classroom Supports for Generalizing (CSGs): Interactional Moves, Structuring Actions, and Instructional Routines. The three categories operate at different levels to show how teachers, students, tasks, and artifacts work in interaction to mutually support classroom generalizing. [For the complete proceedings, see ED630060.]

We report the results of a study on informal covariate statistical reasoning conducted with 22 students (aged 16 and 18 years). We designed and implemented a task in a digital technology environment to introduce the line of best fit. The task design having elements that foresee misconceptions reported in the literature, and by focusing on four statistical ideas that we consider being central to the development of informal reasoning about the line of best fit. After having used the digital technology environment, students transitioned from viewing points of a scatterplot as individual points or fragmented into subsets to viewing the scatterplot as an aggregate from a mathematical mechanism that links them through the notion of distance from a point set to a right line. [For the complete proceedings, see ED630060.]

Backward transfer is when learning about a new topic influences an individual's prior ways of reasoning about a topic they previously learned about. This study looked at how quadratic functions instruction differentially influenced students' prior ways of reasoning about linear functions. Specifically, we compared students at three levels of reasoning about linear functions, low-, mid-, and high-level, using a pre/posttest design that bracketed a two-week quadratic functions math program. Results showed that students at different reasoning levels experienced different backward transfer effects, that particular mathematical reasoning processes were most involved in the effects, and that the effects spanned two dimensions of productiveness of mathematical reasoning. Results from this study are significant for better understanding the construct of backward transfer, and have implications for teaching quadratic functions. [For the complete proceedings, see ED630060.]

This study examines data from a field experiment investigating the effects of a personalized recommendation algorithm that proposes to students which videos to watch next, after they complete mini-assessments for algebra that available on the Math Nation intelligent virtual learning environment (IVLE). The end users of Math Nation are students enrolled in an Algebra 1 course in middle and high schools of the state of Florida, and the IVLE is used both during and out of school time. The objective of the developed recommendation algorithm is to increase student preparation to take the state-mandated End-of-Course (EoC) Algebra 1 assessment at the end of the school year. The algorithm is based on a Markov Decision Process framework that uses as input the students' responses to a series of mini-assessment tests. The current study randomly assigned 16,406 students to either treatment or control conditions, which were blind to both students and teachers. The results indicate that the effects of the recommendation algorithm depend on the level of usage of students, showing significant improvements on EoC test scores of students who have a moderate level of usage. However, there was no effect for low usage students. The study also shows that students practicing with the mini-assessments available on Math Nation, helps them improve by a small margin their performance on the End-of-Course test, irrespective of the usage level. Finally, the study provides insights on challenges posed for implementing personalized recommendation algorithms at a large scale, related both to student self-regulation and teacher orchestration of technology use in the classroom. [For the full proceedings, see ED615472.]

In Intelligent Tutoring Systems (ITS), methods to choose the next exercise for a student are inspired from generic recommender systems, used, for instance, in online shopping or multimedia recommendation. As such, collaborative filtering, especially matrix factorization, is often included as a part of recommendation algorithms in ITS. One notable difference in ITS is the rapid evolution of users, who improve their performance, as opposed to multimedia recommendation where preferences are more static. This raises the following question: how reliably can we use matrix factorization, a tool tried and tested in a static environment, in a context where timelines seem to be of importance. In this article we tried to quantify empirically how much information can be extracted statically from datasets in education versus datasets in multimedia, as the quality of such information is critical to be able to accurately make predictions and recommendations. We found that educational datasets contain less static information compared to multimedia datasets, to the extent that vectors of higher dimensions only marginally increase the precision of the matrix factorization compared to a 1-dimensional characterization. These results show that educational datasets must be used with time information, and warn against the dangers of directly trying to use existing algorithms developed for static datasets. [For the full proceedings, see ED615472.]

Facilitating productive mathematical argumentation is challenging; it is critical to develop a specific guiding vision of practices to help teachers learn to teach argumentation. However, what counts as acceptable classroom-based mathematical argumentation remains an open question. In this study, building on Habermas' theory of communicative action, we developed two analytic frameworks to examine questioning strategies used to support the validity of collective mathematical argumentation. Habermas' three components of rationality allowed us to focus on fine-grained rationality components of teacher questioning as well as teachers' intentions of asking these questions. The theory of validity claims was used to capture different forms of validating argumentation. The frameworks may help teachers to be aware of the types of questions that they are asking when aiming at supporting valid argumentation. [For the complete proceedings, see ED629884.]

This report proposes a framework for describing student analogical reasoning activities in abstract algebra that moves beyond the traditional literature-based treatment of analogical mapping. The Analogical Reasoning in Mathematics (ARM) framework captures the activities that students engage in when anticipating, creating, and reasoning from mathematical analogies. This considers activities along several dimensions including: inter/intra domain activity, foregrounded/backgrounded domain, and attention to similarity/difference. These dimensions are integrated with Gentner's (1983) analogical mapping framework to characterize student activity when they are presented with tasks where reasoning by analogy can productively support their mathematical investigations. By characterizing these activities, we can better develop tasks to support students in productively analogizing between mathematical domains. [For the complete proceedings, see ED629884.]

We report our findings and perspective to document the knowledge exhibited by three experienced high school teachers in their instructional decisions for lessons on the equation of an exponential function. We describe the nature of the mathematical ideas and connections teachers promoted in discourse and the decisions that supported the emergence and connections of the mathematics. Despite similarities in the structure of the mathematical activities, differences existed in the ideas that emerged in the three teachers' discussions regarding the relationship between the exponent value and the independent variable. We describe links between collections of teacher decisions to their influences on the mathematics discourse. [For the complete proceedings, see ED629884.]

We describe results of a long-term design experiment focused on promoting mathematics learning among multilingual ninth graders classified as English Learners. The intervention at a linguistically diverse public high school in the US focused on a unit introducing concepts related to linear rates of change. We analyze results from a curriculum-aligned pre- and post-unit assessment used to document student learning across each design cycle. The main result is that students in both the Pre-Intervention and Redesigned classrooms made gains on the pre- and post-unit assessments. However, student gains on the assessments were higher in the Redesigned classrooms than in the Pre-Intervention classrooms. Additionally, on the assessment in the Redesigned classes, students classified as English Learners made larger gains than their non-EL peers, and the majority of the gains occurred on conceptually-focused items. [For the complete proceedings, see ED629884.]

We describe how we supported middle-school students developing meanings for quadratic growth by bringing together three theoretical framings and designing a task sequence grounded in that framework. Specifically, this framework was assembled from the research on students' quantitative and covariational reasoning, Realistic Mathematics Education, and the theory of designing for mathematical abstraction. We present data from a teaching experiment to highlight how this task sequence supported students as they imagined constant increases in the amounts of change in one quantity, a kind of change they later understood to be a defining characteristic of quadratic growth. We conclude with a discussion of our findings and how our integration of the three frameworks proved useful to design a productive task sequence. [For the complete proceedings, see ED629884.]

These proceedings are a written record of the research presented at the 42nd annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (PME-NA) held in Mazatlán, Mexico, virtually beginning May 27, 2021 and in-person June 2-6, 2021. The conference was originally scheduled to take place October 14-18, 2020; it was postponed, due to the COVID pandemic. In accordance with the major goals of PME-NA that include promoting international contacts and stimulating interdisciplinary research, the theme was proposed as "Entre Culturas/Across Cultures". The goal of the theme was an academic exchange that would reflect the ample diversity of ways of teaching and learning of mathematics, and of the tools and communities involved in mathematics education; as well as an exploration of how the differences in cultures imply a need to consider how research results can be taken into account in varying contexts. The papers comprise 80 Research Reports, 190 Brief Reports, 147 Poster Presentations, 10 Working Groups, 2 Research Colloquia, and Plenary and special lectures. The program also includes a special tribute to Eugenio Filloy and a tribute to the PME-NA members who were affected by the COVID-19 pandemic. [These proceedings were produced with Cinvestav (Centro de Investigación y de Estudios Avanzados del IPN), México and AMIUTEM (Asociación Mexicana de Investigadores del Uso de Tecnología en Educación Matemática, A.C.), México. Individual papers are indexed in ERIC.]

This preliminary study examined whether distinct student profiles (N = 760) emerged based on their behavioral patterns in an online algebraic learning game. We applied k-means clustering analysis to clickstream data collected in the game and then examined how students' behavioral patterns varied across the clusters using data visualization. The results identified four groups of students based on their in-game behaviors, showing that there was a large variation in their behavioral patterns for engaging with the game.

A variety of computerized interactive learning platforms exist. Most include instructional supports in the form of problem sets. 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 varies as well --from static dictionaries of problems to "intelligent" and responsive programming that adapts assignments to users' demonstrated skills, timing, and an array of other learning theory-informed data collection within the computerized environment. This short paper presents background on digital learning contexts and describes the lively conversation with attendees at the conference session. The topics were the research design and early results of a cluster-randomized controlled trial study in community college elementary algebra classes where the intervention was a particular type of web-based activity and testing system. [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.]

This study on the mental solving of systems of linear equations is part of a larger research program, aimed to gain a better understanding of the potential of mental mathematics activities with topics/objects other than numbers. Through outlining the study details and the activities engaged with, the paper reports on the variety of strategies developed for mentally solving systems of linear equations. The analysis pays attention to specificities and particularities of these strategies, focusing on their economical, tailored, and spontaneous nature to solve the tasks. In reflecting on the context of mental mathematics, the paper closes with discussions of the potential and richness of these strategies for exploring systems of linear equations. [For the complete proceedings, see ED583989.]

The study reported on in this paper is an interview study conducted with 20 7th and 8th grade students whose purpose was to understand the generalizations they could make about non-linear meanings of multiplication (NLMM) and non-linear growth (NLG) in the context of solving combinatorics problems. The paper identifies productive challenges for the students, and thus fruitful areas where the students could generalize their reasoning about NLMM and NLG. [For the complete proceedings, see ED583989.]

In this paper, we address the variety of ways in which students conceive of binary operations and the metaphors they might leverage when working with binary operations in group theory. We use open-ended surveys paired with interviews to qualitatively explore student's conceptions of binary operation. Through this analysis, we identified three central metaphors (function, arithmetic, and structure), as well as a number of attributes related to binary operation that students perceive as critical. This work has implications for our treatment of binary operation in advanced mathematics, and the messages we send about operation in the K-12 setting. [For the complete proceedings, see ED606531.]

In this paper I explore eleven undergraduate students' comprehension of a proof taken from an undergraduate abstract algebra course. My interpretation of what it means to understand a proof is based on a proof comprehension model developed by Mejia-Ramos, et al. (2012). This study in particular examines the extent to which undergraduate students are able to summarize a proof using the proof's higher-level ideas. Additionally, eleven doctoral students in mathematics were asked to provide a summary of the same proof that the undergraduate students received. Undergraduates' holistic comprehension of the proof was then analyzed in light of summaries that the doctoral students provided. The main finding of the study is that undergraduates' comprehension of the proof was overall inadequate--notably, they demonstrated limited skills in summarizing a proof via the proof's key ideas. Moreover, undergraduates failed to recognize the scope of the method used in the proof. [For the complete proceedings, see ED606531.]

This paper examines the implementation of an instructional module on Preservice Elementary Teachers' (PSETs) professional noticing of children's mathematical thinking as defined by Jacobs, Lamb, and Philipp (2010). The module focuses on professional noticing skills through the content focus of early algebraic reasoning and uses complex video vignettes from whole class instruction in authentic elementary mathematics classrooms. It was found that two of the three components of professional noticing (attending and interpreting) showed statistically significant increases in a treatment group that did not occur in a comparison group. The deciding component remains a challenge that warrants further research. [For the complete proceedings, see ED606531.]

This paper presents strategies identified in Mexican rural school students in seeking structure on equivalence tasks that involve the equal sign and tasks that do not. The results arise from the pilot study of a research project on the structure of numbers and numerical operations -- a key aspect of early algebraic thinking. [For the complete proceedings, see ED606531.]