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

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

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

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

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

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

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

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

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

Verbal protocols were collected from math-anxious and math-comfortable college students while they solved algebra problems. These protocols were then examined for differences in problem-solving processes. Differences occurred in the use of two basic strategies: generating values and symbolic transformations. Math-anxious students generated more values; math-comfortable students made more symbolic transformations. The data suggest that for the math-anxious students the real problem is not numerical; rather, it is the inability to apply symbolic procedures. (MP)

Prior research shows that self-explanation promotes understanding by helping learners connect new knowledge with prior knowledge. However, despite ample evidence supporting the effectiveness of self-explanation, an instructional design challenge emerges in how best to scaffold self-explanation. In particular, it is an open challenge to design self- explanation support that simultaneously facilitates performance and learning outcomes. Towards this goal, we designed anticipatory diagrammatic self-explanation, a novel form of self-explanation embedded in an Intelligent Tutoring System (ITS). In our ITS, anticipatory diagrammatic self-explanation scaffolds learners by providing visual representations to help learners predict an upcoming strategic step in algebra problem solving. A classroom experiment with 108 middle-school students found that anticipatory diagrammatic self-explanation helped students learn formal algebraic strategies and significantly improve their problem-solving performance. This study contributes to understanding of how self-explanation can be scaffolded to support learning and performance.

Research has shown that tape diagrams are beneficial for algebra learning. However, it is unclear whether certain visual features of tape diagrams have implications for learning. We investigated, with undergraduate students and math teachers, whether tape diagrams with different visual features (color, presence of outer lines, and position of the constant) differentially support reasoning about equations and whether people have preferences for certain visual features. Variations in visual features did not affect students' or teachers' reasoning accuracy; but each group displayed systematic preferences for most visual features considered. Future research should examine the effects of these visual features on performance while solving equations.

We apply an advanced data visualization technique, "Sankey diagram," to explore how middle-school students (N = 343) solved problems in a game-based algebraic notation tool. The results indicate that there is a large variation in the types of students' strategies to solve the problems, with some approaches being more efficient than others. The findings suggest that Sankey diagrams can be used both in research and practice to unpack our understanding of variability in mathematical problem-solving. [This paper was published in: "Proceedings of the 15th International Conference of the Learning Sciences--ICLS 2021," 2021, pp. 887-888.]

"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.]