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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).]

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.]

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.

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.]

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 purpose of this study was to examine the ways that teachers use Connected Classroom Technology (CCT) to potentially support achievement on translation problems that require moving between algebraic representations. Four mathematics classrooms were chosen based on their gain scores on pre- and post-test Algebraic translation problems. Two classrooms with the highest and the lowest gain scores were chosen among the classrooms with pre-test scores that were below 50%. This study used video-recorded observational data and found that teachers in effective classrooms created environments wherein students used multiple representations simultaneously and translated between representations through discussion. In contrast, teachers in less effective classrooms fostered environments wherein students used representations independently and missed opportunities to translate representations through discussion. [For the complete proceedings, see ED583608.]

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.

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.]

In this paper, I propose a new construct named "analytic equation sense" to conceptually model a desired way of reasoning that involves students' algebraic manipulations and use of equivalent expressions. Building from the analysis of two existing models in the field, I argue for the need for a new model and use empirical evidence to explain the new model. [For the complete proceedings, see ED630060.]

A historical review of mathematics textbooks suggests a canonical method to solving equations that teachers often see as "the" way to solve equations. In this paper, we examine data from a nationally-distributed sample of 524 secondary mathematics teachers who responded to scenario-based survey items that represent the instructional situation of solving equations. The items featured scenarios in which students presented non-canonical solution methods and asked participants to share how they would respond. Using a framework that draws on systemic functional linguistics, we describe the linguistic resources teachers used. While closed moves are frequently used to avoid discussion of non-canonical solutions, our results suggest that teachers find ways to make regular use of: (1) closed moves for accommodating non-canonical solutions; and (2) open moves when steering the conversation back to the canonical method. [For the complete proceedings, see ED630060.]

The capacity to recognise, represent, and reason about relationships between different quantities, that is, to think multiplicatively, has long been recognised as critical to success in school mathematics in the middle years and beyond. Building on recent research that found a strong link between multiplicative thinking and algebraic, geometrical, and statistical reasoning, this paper will describe the development and validation of two new assessment options for multiplicative thinking and discuss the significance of this for the teaching and learning of mathematics in the middle years of schooling.

This paper drills deeper into the documented effects of the Cognitive Tutor Algebra I and ASSISTments intelligent tutoring systems by estimating their effects on specific problems. We start by describing a multilevel Rasch-type model that facilitates testing for differences in the effects between problems and precise problem-specific effect estimation without the need for multiple comparisons corrections. We find that the effects of both intelligent tutors vary between problems-- the effects are positive for some, negative for others, and undeterminable for the rest. Next we explore hypotheses explaining why effects might be larger for some problems than for others. In the case of ASSISTments, there is no evidence that problems that are more closely related to students' work in the tutor displayed larger treatment effects. [For the full proceedings, see ED615472.]

This paper reports the factors that three Year 11 mathematics students identified as influencing their choice to use a Computer Algebra System (CAS) for solving selected problems, some of which were expected to be outside their pen-and-paper (p&p) range. In individual interviews, the students were presented with eleven algebra problems and were asked to indicate the items that they would choose CAS for, and to provide a rationale for their choices. Factors that influenced students' choices included their perception of the amount of p&p working required, their p&p facility, the complexity of the problem and the presence of words in the problem statements which were CAS commands (for example, solve).

In this paper we present the results of an investigation related to the developing of mathematical knowledge and skills by first semester university students when solving a Model Eliciting Activity [MEA] which involves quadratic function knowledge. This was a qualitative research. The theoretical framework was Models and Modeling Perspective [MMP]. The results show that the students used their mathematical knowledge and skills related to linear and quadratic functions to describe the situation; they moved from a quantitative cycle of understanding (associated with linear and quadratic behaviors), to an algebraic cycle of understanding (associated with quadratic behaviors). [For the complete proceedings, see ED629884.]

We present the results of a research project on arithmetic-algebraic thinking that was carried out jointly by a team in Mexico and another in Quebec. The project deals with the concepts of variable and covariation between variables in the sixth grade at the elementary level and the first, second, and third years of secondary school--namely, children from 11 to 14 years old. We target secondary students (first year or K7) in this article. Our objective relates to the development of a gradual generalization in arithmetic-algebraic thinking in a socio-cultural approach to the learning of mathematics. We experimented with investigative situations using a paper-and-pencil approach and technology. We analyze the emergence, in this context, of a visual abstraction, the production of institutional and non-institutional representations, a sensitivity to contradiction, and, finally, the concepts of variable and of covariation between variables. [For the complete proceedings, see ED629884.]

The design of the Cognitive Tutor Algebra I (CTA1) intelligent tutoring system assumes that students work through sections of material following a pre-specified order, and only move on from one section to the next after mastering the first section's skills. However, the software gives teachers the flexibility to override that structure, by reassigning students to different sections of the curriculum. Which students get reassigned? Does reassignment hurt student learning? Does it help? This paper used data from the treatment arm of a large effectiveness study of the CTA1 curriculum to estimate the effects of reassignment on students' scores on an Algebra I posttest. Since reassignment is not randomized, we used a multilevel propensity score matching design, along with assessments of sensitivity to bias from unmeasured confounding, to estimate the effects of reassignment. We found that reassignment reduces posttest scores by roughly 0.2 standard deviations--about the same as the overall CTA1 treatment effect--that unmeasured confounding is unlikely to completely explain this observed effect, and that the effect of reassignment may vary widely between classrooms. [For the full proceedings, see ED607784.]

This study aims to describe how subjects construct explicit warrant derived from implicit warrant when completing mathematical proof. This research was conducted on seventeen students of mathematics education study programs by providing tests on vector material in elementary linear algebra courses. The test results show that there are four students who can solve the evidentiary questions correctly using warrant, but there is only one student who can be the subject of research. The selection of subjects is based on students' ability to communicate verbally about the thought process carried out in constructing implicit warrant to explicit warrant when conducting proof. Data in this study were obtained from think aloud, tests, and interviews conducted by researchers to subjects. The results showed that the explicit warrant constructed by the subjects was obtained from an implicit warrant based on the thought process carried out. The subject constructs an explicit warrant derived from an implicit warrant through four steps, namely identifying problems,determining warrant, doing algebraic manipulation, and making conclusions. Warrant is a guarantor used to get the correct conclusions from a mathematical proof, thus further research needs to be done related to warrant. [This paper was published in: "AIP Conference Proceedings" v2215, p060005-1-060005-5, 2020. AIP Publishing (ISBN 978-0-7354-1968-1).]

The concept of equivalence can be elusive to students and can be confounded with unproductive understandings of the equals sign. Using the game-based digital algebraic notation system, From Here to There! (FH2T), students explore ideas of equivalence by dynamically transforming expressions or equations among mathematically equivalent states. In fall of 2019, 409 middle-school students completed a randomized control trial where they worked in either GM or an online problem set control over four half-hour sessions during their math block. We found that students in the FH2T condition showed enhanced performance over the control (p = 0.043, [partial eta-squared] = 0.010). We will describe the FH2T technology, describe our results, and discuss implications for the usefulness of digital environments in mathematics education. [This paper was published in: "ICLS 2020 Proceedings," ISLS, 2020, pp. 1657-1660.]

Many studies have shown that visual representations can enhance student understanding of STEM concepts. However, prior research suggests that visual representations alone are not necessarily effective across a broad range of students. To address this problem, we created a novel, scaffolded form of diagrammatic self-explanation in which students "explain" their problem-solving steps in the form of diagrams. We used contrasting cases to support students' sense-making between algebraic equations and diagrams in the self-explanation activity. We conducted a classroom experiment with 41 students in grades 5 and 6 to test the effectiveness of this strategy when embedded in an Intelligent Tutoring System for algebra. We found that scaffolded diagrammatic self-explanation enhanced conceptual knowledge for students who did not have prior knowledge of formal equation-solving strategies. The study is the first experimental study showing that visual representations can enhance conceptual knowledge in early algebra. [This paper was published in: "ICLS 2020 Proceedings," ISLS, 2020, pp. 35-42.]

When designing an instructional tool and using it in pedagogical activities, it is essential that designers and users understand what pedagogical affordances and constraints the tool provides to support its successful integration into targeted pedagogical activities. Toward this end, we developed "Pedagogical Affordance Analysis (PAA)." PAA involves analyzing teachers' Pedagogical Content Knowledge and/or Technological Pedagogical Content Knowledge to elicit pedagogical affordances and constraints that are specific to a given instructional goal. Information obtained through PAA can help in designing, refining, and/or evaluating instructional tools. We present a case study in which we used PAA to successfully design a visual representation for middle-school algebra. To the best of our knowledge, PAA is the only available systematic method that leverages teachers' pedagogical knowledge in identifying pedagogical affordances and constraints. PAA can be used across a wide range of existing tools and prototypes of to-be-designed tools. [This paper was published in: "ICLS 2020 Proceedings," ISLS, 2020, pp. 1561-64.]

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.]