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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.]

This study focuses on the undergraduate mathematics students' perceptions and applied techniques for the preparation for their final examination in Abstract Algebra. The results of this study suggest that the revision for the final examination involves, firstly, the review of the lecture notes, followed by the solution of the coursework together with the use of model solutions and the solution of the past papers. The order of the last two activities varies. An often-occurring revision technique involves, instead of a linear succession of the aforementioned activities, a 3-dimensional spiral approach towards revision, with the three activities interchanging until the students who apply it feel that they have achieved adequate object-level and metalevel learning.

This paper will focus on two students who depended on diagrammatic representations in both a Fraction Screening Test and in a subsequent Structured Interview. One student attempted to use diagrams, with limited success, to identify the correct relationships, and consequently struggled to generalise her strategies as she responded to the questions presented in the interview. The other student used diagrams more effectively, and was able to move from a reliance on diagrams to using a partially multiplicative solution strategy. When supported by strong number knowledge, diagrams are an effective means for helping to solve reverse fraction tasks but may hinder students attempts to generalise their thinking

Most community college students in the U.S. must complete at least one developmental class, such as elementary algebra, before they can enroll in a college-level mathematics course. Increasingly common in such courses is the use of a web-based activity and testing system (WATS). This report presents initial results of a mixed-methods study of elementary algebra learning among 510 students in the classes of 29 instructors across 18 community colleges. Instructors were randomly assigned to use a particular WATS (treatment condition) or their usual approach (control condition). The focal WATS had adaptive problem sets, hints, and videos. Treatment group instructors had access to online support for implementation. For the study, students completed common pre- and post-tests and instructors regularly provided information about their teaching practices. The early results reported here indicate that greater instructor fidelity to developer intentions regarding frequency of assignments are positively associated with greater student learning. [This paper was published in: A. Weinberg, C. Rasmussen, J. Rabin, M. Wawro, and S. Brown (Eds.), "Proceedings of the 21st Annual Conference on Research in Undergraduate Mathematics Education, San Diego, California," 2018.]

Language is often depicted as the sine qua non of mathematical thinking, a view buttressed by findings of language-of-training effects among bilinguals. These findings, however, have been limited to studies of arithmetic. Nothing is known about the potential influence of language on the ability to learn rules about the relations among variables (e.g., algebra). To test whether arithmetic and algebraic thinking differ, Chinese-English bilinguals were trained to solve arithmetic and algebra problems in either Chinese or English and then tested on new and old problems in both languages. For arithmetic problems, solution times were always longer for English than Chinese; in both languages, solution times dropped during training; after training, solution times continued to drop for old problems, but returned to pre-training levels for new problems. In contrast, for algebra problems, solution times did not differ across language; solution times dropped during training; after training, gains in speed were preserved for both old and new problems. These findings suggest that the contribution of language to mathematical thinking may be limited to the areas of mathematics that are learned by rote and not by rule. [This paper was published in: "Proceedings of the 40th Annual Conference of the Cognitive Science Society," edited by T. T. Rogers et al., Cognitive Science Society, 2018, pp. 918-923.]

We present results of the analysis of knowledge used by a secondary school mathematics teacher in her classroom practice. This knowledge takes shape and is displayed as specific teaching strategies in the management of her class when she incorporates Computer Algebra Systems. Based on observations of regular classes, we find that her knowledge (mathematical, pedagogical and technological) is put into action at various moments during the lesson and through a variety of teaching strategies. These strategies depend on many factors, such as the planned objective of the lesson, the specific moment in the class, student participation, and the use of technological tools. Within such complexity, the teacher applies her teaching strategies in a flexible way, and manages to control and even modify the course of her class. [This paper is written in both Spanish and English. For complete proceedings, see ED581294.]

This paper describes seven in-service teachers' interpretations of student statements about slope. The teachers interpreted sample student work, conjectured about student contributions, assessed the students' understanding, and positioned the students' statements in the mathematics curriculum. The teachers' responses provide insight into their Knowledge of Content and Students (KCS) and Knowledge of Content and Curriculum (KCC). Results suggest these teachers value academic terminology related to slope, have limited perspectives on slope in real world contexts, and struggle to describe the extension of slope to precalculus. [For complete proceedings, see ED581294.]

This paper uses multimodal discourse analysis to show that discursive form, in addition to words, gestures and sound dimensions of speech, is an important linguistic resource for expressing mathematical meaning. During a collaborative task, a student spoke an insight about an algebraic property five times over a few minutes, in slightly different ways. He consistently used repetition of grammar and words, a speech form known as poetic structure. These poetic structures were marked elaborately through discursive modes such as pause, intonation and gestures, suggesting that they form a meaning-making mode that is real to the speaker. [For complete proceedings, see ED581294.]

Generalization is a critical aspect of doing mathematics, with policy makers recommending that it be a central component of mathematics instruction at all levels. This recommendation poses serious challenges, however, given researchers consistently identifying students' difficulties in creating and expressing normative mathematical generalizations. We address these challenges by introducing a comprehensive framework characterizing students' generalizing, the Relating-Forming-Extending framework. Based on individual interviews with 90 students, we identify three major forms of generalizing and address relationships between forms of abstraction and forms of generalization. This paper presents the generalization framework and discusses the ways in which different forms of generalizing can play out in activity. [For complete proceedings, see ED581294.]

Algebra is an area of pressing national concern around issues of equity and access in education. Recent theories and research suggest that personalization of instruction can allow students to activate their funds of knowledge and can elicit interest in the content to be learned. This paper examines the results of a large-scale teaching experiment where 8th grade students posed, solved, and shared algebra problems related to their out of school interests in topics like sports, video games, and social networking. Results suggest that the teaching experiment improved both learning of and interest in algebra compared to "business as usual" instruction, particularly for those students who were struggling. Theoretical and practical implications are discussed. [For complete proceedings, see ED581294.]

Enacting equitable teaching practices and sustaining those practices continue to be challenges in most high schools (National Council of Teachers of Mathematics, 2018), particularly in an era of high-stakes testing. Just one year after Boaler and Staples (2008) reported on the successes of "Railside High," where equitable teaching practices resulted in improved mathematics achievement, those signature practices had all but disappeared (Louie & Nasir, 2014). This paper discusses the outcomes and implications of enacting equitable teaching practices at a school demographically similar to Railside. Unlike Railside, however, equitable practices were adopted as an explicit strategy for improving student achievement, demonstrating through this process how university consultants and school personnel can collaborate to make mathematics accessible and equitable for all students.

This study examined if and how thinking time, defined as the time from the start of the problem to the students' first action to solve, influences their problem-solving efficiency, defined as the number of steps taken to solve the problem, using Graspable Math, a web-based mathematical tool. We found that: (1) students who took longer to think about the problems solved individual problems with fewer steps; (2) thinking time significantly predicted step efficiency above and beyond students' algebra knowledge or their affective disposition towards mathematics; and (3) the relation between thinking time and step efficiency was not moderated by students' knowledge in algebra, math self-efficacy, or math anxiety.