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Eighteen papers on instructional technology, computerized testing, and computer assisted test construction (CATC) presented at the 1976 Association for Educational Data Systems (AEDS) convention are included here. Two papers discuss computer assisted instruction in calculus and teacher education courses. The use of computers in theoretical mathematics, school media centers, and individualized instruction programs is presented in four papers. Goal programing in education is explained and the uses of the hand-held calculator for education are reviewed. Faculty rating policies for mathematics students are analyzed. Eight articles examine aspects of computerized testing and CATC. They include an overview of computers and testing, the use of computerized quiz grading, interactive computerized testing, descriptions of SOCRATES, ALLCOMBS, CREAM, and the Classroom Teacher Support System. (CH)

Ambiguity of language can form a barrier to understanding mathematical concepts. The word 'capacity' is problematic as it holds a variety of meanings and might not be understood in school mathematics, to be an aspect of volume. We investigated pre-service primary school teachers' interpretations of 'capacity' when given the definitions of 'volume' and 'capacity' provided in the Australian Curriculum: Mathematics. Of the 72 participants, 20 appeared to hold the conception that the volume of a liquid is its capacity. We suggest reducing ambiguity by defining 'capacity' more clearly or by excluding this word from formal mathematics terminology.

For the past six years the National Assessment of Educational Progress has sponsored a national Conference on Large-Scale Assessment, designed to promote and improve communications among educational assessment personnel in State Departments of Education and other agencies. This volume contains most of the papers that were accepted for presentation at the half-day formal paper session. The 11 papers included here are: (1) "The State Agency as a Resource in Local Needs Assessment" by Paula T. Brictson; (2) "Establishing Criterion Levels for Judging the Acceptability of Assessment Results" by Iris Weiss and Larry Conaway; (3) "N-Abels--A Manageable Technique for Monitoring the Acquisition of Essential Learning Skills" by Harriet A. Egertson and Hugh A. Harlan; (4) "A Process for Developing, Implementing and Following Through on an Assessment Program in Fifth- and Eighth-Grade Mathematics" by Max Morrison; (5) "Educational Quality Assessment Follow-Up Survey of the 1974 Assessment" by Joyce S. Kim; (6) "Hypothesis-Testing in Large-Scale Assessment" by Frank W. Rivas; (7) "A Plan for Utilization of Assessment Data by Local Education Agencies" by John A. Jones and Charles D. Oviatt; (8) "ACT Test Data and Program Assessment for Large School Districts" by Robert Cramer; (9) An Example of the Use of Multiple Matrix Sampling Procedures in a Local District Assessment Program" by Carl D. Novak; (10) "Measurement Problems and Issues Related to Applied Performance Testing" by James R. Sanders; and (11) "Symposium on: Large-Scale Assessment Reporting and Usage: Delaware and Georgia as Exemplars" by Robert Bigelow and Hervey Scudder. Primary type of information provided by report: Procedures (Theoretical) (Training).

In Australia, all teachers are expected to explicitly attend to numeracy in the subjects they teach. Pre-service teachers, therefore, need to begin to develop effective numeracy teaching strategies but there is a limited research base to inform the design of courses that address this need. This paper reports on findings from a study investigating the impact of one such course. The paper draws on data from two pre-service teachers: two course tasks completed at the beginning of the course and interviews conducted at the end of the course. Data were analysed using the 21st Century Numeracy Model. Findings suggest there were changes to the pre-service teachers' understanding of numeracy and their capacity to embed numeracy in the subjects they will teach.

Engagement in mathematics education is an important factor in a successful student experience. This paper reports findings from a study of Year 5 student perceptions of engagement during a two-week, inquiry-based learning (IBL), problem-posing investigation. The study triangulated data from semistructured interviews, video observations and student work samples to understand the student's perspectives of IBL; however, this paper reports the interview data. The findings indicate that most of the students were behaviourally, emotionally, and cognitively engaged during the IBL investigation, and that the investigation provided an opportunity for students to experience levels of competence, relatedness, and autonomy need satisfaction.

This submission contains the Proceedings of the 2012 Annual Meeting of the Canadian Mathematics Education Study Group (CMESG), held at Laval University in Québec City, Québec. The CMESG is a group of mathematicians and mathematics educators who meet annually to discuss mathematics education issues at all levels of learning. The aims of the Study Group are: to advance education by organizing and coordinating national conferences and seminars to study and improve the theories of the study of mathematics or any other aspects of mathematics education in Canada at all levels; and to undertake research in mathematics education and to disseminate the results of this research. These proceedings include plenary lectures, a panel discussion, working group reports, topic session descriptions, new PhD reports, and summaries of ad hoc and poster sessions. Papers include: (1) Towards an Understanding of Ethical Practical Action in Mathematics Education: Insights from Contemporary Inquiries (Margaret Walshaw); (2) Old and New Mathematical Ideas from Africa: Challenges for Reflection (Paulus Gerdes); (3) Cooda, Wooda, Didda, Shooda: Time Series Reflections on CMESG/GCEDM (William Higginson); (4) Panel: What is Fundamental Mathematics for Learners? (Elaine Simmt, Darien Allan, Ralph Mason, Ruth Beatty, Peter Taylor, and Hélène Paradis); (5) Numeracy: Goals, Affordances, and Challenges (France Caron and Peter Liljedahl); (6) Diversities in Mathematics and their Relation to Equity (Beverly Caswell and David Wagner); (7) Technology and Mathematics Teachers (K-16) / La technologie et l'enseignant mathématique (K-16) (Chantal Buteau and Nathalie Sinclair); (8) La preuve en mathématiques et en classe / Proof in Mathematics and in Schools (David Reid and Denis Tanguay); (9) The Role of Text/books in the Mathematics Classroom / Le rôle des manuels scolaires dans la classe de mathématiques (Peter Appelbaum and Susan Gerofsky); (10) Preparing Teachers to Develop Algebraic Thinking in Primary and Secondary School / Préparer les enseignants au développement de la pensée algébrique au primaire et au secondaire (Hassane Squalli, Chris Suurtamm, and Viktor Freiman); (11) Collaboration Between Research in Mathematics Education and Teaching Mathematics: Case Study of Teaching Infinity in Calculus (Miroslav Lovric); (12) Dialogue sur la lecture de textes historiques dans la classe de mathématiques / Dialogue on Reading Original Texts in the Mathematics Classroom (Louis Charbonneau and David Guillemette) [Written in French]; (13) Teaching Toward Equity in Mathematics (Beverly Caswell); (14) Inequalities in the History of Mathematics: From Peculiarities to a Hard Discipline (Elena Halmaghi); (15) The Study of On-line Situations of Validation Experienced by 13- and 14-year-old Students With and Without the Aid of an Electronic Forum (Manon Leblanc); (16) Institutional Acculturation of the Researcher, Teacher, and Secondary 1 Students with Learning Difficulties in Problem Situations Involving Rational Numbers / Les effets d'une démarche d'acculturation sur l'action didactique conjointe de l'enseignant, des élèves et du chercheur, dans l'enseignement/apprentissage des nombres rationnels auprès d'élèves en difficultés d'apprentissage (Geneviève Lessard) [Written in French]; (17) Mathematics Education: An Aporetic of Epistemology, Language and Ethics (Jean-François Maheux); (18) Diverse Perspectives on Teaching "Math for Teachers": Living the Tensions (Susan Oesterle); (19) Conversations Held and Roles Played During Mathematics Teachers' Collaborative Design: Two Dimensions of Interaction (Armando Paulino Preciado Babb); (20) The Ordinary Yet Extraordinary Emotions and Motives of Preservice Mathematics Teachers (Oana Radu); (21) Élaboration et analyse d'une intervention didactique, co-construite entre chercheur et enseignant, visant le développement d'un contrôle sur l'activité mathématique chez les élèves du secondaire / Development and Analysis of a Didactic Intervention, Co-constructed Between Researcher and Teacher For the Development of a Control of the Mathematical Activity Among High School Students (Mireille Saboya) [Written in French]; (22) Publishing in the Journal of Mathematics Teacher Education (Olive Chapman and Margaret Walshaw); (23) Is It Possible to Measure the Effectiveness of a Specific Approach to Teaching Foundations Mathematics in a Post-Secondary Setting? / Est-il possible de mesurer l'efficacité d'une approche spécifique à l'enseignement mathématiques fondations dans un cadre de post-secondaire? (Taras Gula); (24) Challenges in Supporting Mathematics Teachers to Develop Their Teaching Practices (Lionel Lacroix); (25) Rapport sur le ad hoc éthique et éducation mathématique / Report on the Ethics and Mathematics Education Ad Hoc (Jean-François Maheux); (26) Reading Biographies and Autobiographies of Mathematicians: What Do They Tell Us About the Subject of Mathematics? (Veda Roodal Persad); (27) Online Environments for Mathematics Sharing and Collaboration (Geoffrey Roulet); (28) Ad Hoc Session on Planning for the Next Canadian Math Education Forum (CMEF) to be Held in May 2014 (Peter Taylor); (29) Preservice Elementary Teachers' Beliefs Toward Mathematics and Mathematics Teaching (Sean Beaudette, Alexandra Penn, and Geoffrey Roulet); (30) Gearing Up For Grade 9: A Learning Object (Laura Broley); (31) Students' Perceptions of the Role of Theory and Examples in College Level Mathematics (Dalia Challita and Nadia Hardy); (32) Une expérimentation de pratiques gagnantes en enseignement des mathématiques / An Experiment with Successful Practices in Mathematics Teaching (Lucie Deblois); (33) Blended Mathematical Collaboration Using a Wiki, Geogebra and Jing (Jill Lazarus and Geoffrey Roulet); (34) Some Things Technologies Can Tell Us About Technologies: An Instrumented Analysis of Two Successive Mathematics Curricula (Jean-François Maheux and Fabienne Venant); (35) Investigating the Teaching Practices of a Group of Mathematics Graduate Students (Tod L. Shockey and Sibylle Weck-Schwarz); (36) Mathematics For Young Children: Exploring What is Possible in Early Mathematics Education? (Diane Tepylo, Joan Moss, Catherine Bruce, Tara Flynn, Diana Chang, and Zachary Hawes); and (37) Mise à l'essai d'une situation d'enseignement-apprentissage en lien avec le métier du scénographe pour favoriser l'engagement mathématique des élèves du 1er cycle du secondaire (Josianne Trudel) [Written in French]. Appended are: (1) Working Groups at Each Annual Meeting; (2) Plenary Lectures at Each Annual Meeting; and (3) Proceedings of Annual Meetings. Individual papers contain references, tables, and figures. [For the 2011 proceedings, see ED547245.]

This document contains the second volume of the proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education. Conference papers are centered around the theme of "Learners and Learning Environments." This volume features 43 research reports by presenters with last names beginning between Adl and Fre: (1) Working with Learners' Mathematics: Exploring a Key Element of Mathematical Knowledge for Teaching (Jill Adler, Zain Davis, Mercy Kazima, Diane Parker, and Lyn Webb); (2) A Comparison between Teachers' and Pupils' Tendency to Use a Representativeness Heuristic (Thekla Afantiti-Lamprianou, Julian S. Williams, and Iasonas Lamprianou); (3) Purposeful Task Design and the Emergence of Transparency (Janet G. Ainley, Liz Bills, and Kirsty Wilson); (4) A Developmental Model for Proportional Reasoning in Ratio Comparison Tasks (Silvia Alatorre and Olimpia Figueras); (5) Referential and Syntactic Approaches to Proof: Case Studies from a Transition Course (Lara Alcock and Keith Weber); (6) Teachers' Beliefs about Students' Development of the Pre-Algebraic Concept of Equation (Vassiliki Alexandrou-Leonidou and George N. Philippou); (7) Developing Students' Understanding of the Concept of Fractions as Numbers (Solange Amorim Amato); (8) Multiple Representations in 8th Grade Algebra Lessons: Are Learners Really Getting It? (Miriam Amit and Michael N. Fried); (9) Reform-Oriented Teaching Practices: A Survey of Primary School Teachers (Judy Anderson and Janette Bobis); (10) The Genesis of Signs by Gestures: The Case of Gustavo (Ferdinando Arzarello, Francesca Ferrara, Ornella Robutti, and Domingo Paola); (11) Students' Experience of Equivalence Relations: A Phenomenological Approach (Amir H. Asghari and David Tall); (12) How Series Problems Integrating Geometric and Arithmetic Schemes Influence Prospective Secondary Teachers' Pedagogical Understanding (Leslie Aspinwall, Kenneth L. Shaw, and Hasan Unal); (13) Dealing with Learning in Practice: Tools for Managing the Complexity of Teaching and Learning (Sikunder Ali Baber and Bettina Dahl); (14) Situations of Psychological Cognitive No-Growth (Roberto R. Baldino and Tania C. B. Cabral); (15) Good CAS Written Records: Insight from Teachers (Lynda Ball and Kaye Stacey); (16) Developing Procedure and Structure Sense of Arithmetic Expressions (Rakhi Banerjee and K. Subramaniam); (17) Struggling with Variables, Parameters, and Indeterminate Objects, or How to Go Insane in Mathematics (Caroline Bardini, Luis Radford, and Cristina Sabena); (18) Exploring How Power is Enacted in Small Groups (Mary Barnes); (19) A Framework for the Comparison of PME Research into Multilingual Mathematics Education in Different Sociolinguistic Settings (Richard Barwell); (20) Vygotsky's Theory of Concept Formation and Mathematics Education (Margot Berger); (21) Preservice Teachers' Understandings of Relational and Instrumental Understanding (Kim Beswick); (22) The Transformation of Mathematics in On-Line Courses (Marcelo C. Borba); (23) Using Cognitive and Situated Perspectives to Understand Teacher Interactions with Learner Errors (Karin Brodie); (24) Identification of Affordances of a Technology-Rich Teaching and Learning Environment (TRTLE) (Jill P. Brown); (25) The "A4-Project": Statistical World Views Expressed through Pictures (Michael Bulmer and Katrin Rolka); (26) A Whole-School Approach to Developing Mental Computation Strategies (Rosemary Callingham); (27) A Comparison of Perceived Parental Influence on Mathematics Learning among Students in China and Australia (Zhongjun Cao, Helen Forgasz, and Alan Bishop); (28) Using Word Problems in Malaysian Mathematics Education: Looking beneath the Surface (Kah Yein Chan and Judith Mousley); (29) Constructing Pedagogical Knowledge of Problem Solving: Preservice Mathematics Teachers (Olive Chapman); (30) Revisiting a Theoretical Model on Fractions: Implications for Teaching and Research (Charalambos Y. Charalambous and Demetra Pitta-Pantazi); (31) Students' Reflection on Their Sociomathematical Small-Group Interaction: A Case Study (Petros Chaviaris and Sonia Kafoussi); (32) Investigating Teachers' Responses to Student Misconceptions (Helen L. Chick and Monica K. Baker); (33) Studying the Distribution of Responsibility for the Generation of Knowledge in Mathematics Classrooms in Hong Kong, Melbourne, San Diego and Shanghai (David Clarke and Lay Hoon Seah); (34) Indigenous and Non-Indigenous Teaching Relationships in Three Mathematics Classrooms in Remote Queensland (Tom J. Cooper, Annette R. Baturo, and Elizabeth Warren); (35) Exploring the Strategies Used by Grade 1 to 3 Children through Visual Prompts, Symbols and Worded Problems: A Case for a Learning Pathway for Number (Ty Corvell Cranfield, Cally Kuhne, and Gary Powell); (36) Primary Students' Knowledge of the Properties of Spatially-Oriented Diagrams (Carmel Diezmann); (37) A Conceptual Framework for Studying Teacher Preparation: The Pirie-Kieren Model, Collective Understanding, and Metaphor (Maria A. Droujkova, Sarah B. Berenson, Kelli Slaten, and Sue Tombes); (38) Mathematical Modelling with 9-Year-Olds (Lyn English and James Watters); (39) Exploring "Lesson Study" in Teacher Preparation (Maria L. Fernandez); (40) Child-Initiated Mathematical Patterning in the Pre-Compulsory Years (Jillian Fox); (41) The Tacit-Explicit Nature of Students' Knowledge: A Case Study on Area Measurement (Cristina Frade); (42) Teachers as Interns in Informal Mathematics Research (John M. Francisco and Carolyn A. Maher); and (43) Exploring Excellence and Equity within Canadian Mathematics Classrooms (George Frempong). (Individual papers contain references.)

An intelligent textbook may be defined as an interaction layer between the text and the student, helping the student master the content in the text. The Mobile Fact and Concept Training System (MoFaCTS) is an adaptive instructional system for simple content that has been developed into an interaction layer to mediate textbook instruction and so is being transformed into the Mobile Fact and Concept Textbook System (MoFaCTS). In this paper, we document the several terms of the logistic regression model we use to track performance adaptively. We then examine the contribution of each component of our model when it is fit to 4 semesters of Anatomy and Physiology course practice data. Following this documentation of the model, we explain how it is applied in the MoFaCTS system to schedule performance by targeting practice for each item at an optimal efficiency threshold. [This paper was published in the CEUR workshop proceedings (Vol. 2895).]

Bayesian Knowledge Tracing, a model used for cognitive mastery estimation, has been a hallmark of adaptive learning research and an integral component of deployed intelligent tutoring systems (ITS). In this paper, we provide a brief history of knowledge tracing model research and introduce pyBKT, an accessible and computationally efficient library of model extensions from the literature. The library provides data generation, fitting, prediction, and cross-validation routines, as well as a simple to use data helper interface to ingest typical tutor log dataset formats. We evaluate the runtime with various dataset sizes and compare to past implementations. Additionally, we conduct sanity checks of the model using experiments with simulated data to evaluate the accuracy of its EM parameter learning and use real-world data to validate its predictions, comparing pyBKT's supported model variants with results from the papers in which they were originally introduced. The library is open source and open license for the purpose of making knowledge tracing more accessible to communities of research and practice and to facilitate progress in the field through easier replication of past approaches. [For the full proceedings, see ED615472.]

Prospective secondary mathematics teachers in Australia are typically taught by mathematics educators and mathematicians who work in different faculties and seldom collaborate--a situation that can lead to conflicting views about how to teach mathematics. This paper reports on findings from semi-structured interviews with three mathematics teacher educators in a hybrid mathematics/mathematics education department. Valsiner's zone theory is used to analyse how their beliefs, institutional context and professional learning opportunities shape their identities as MTEs. Findings reveal that the MTEs had developed similar beliefs within supportive institutional context but drew on different sources of professional learning.

The Indian Institute of Science is proud to host the fully in-person sixteenth iteration of the International Conference on Educational Data Mining (EDM) during July 11-14, 2023. EDM is the annual flagship conference of the International Educational Data Mining Society. The theme of this year's conference is "Educational data mining for amplifying human potential." Not all students or seekers of knowledge receive the education necessary to help them realize their full potential, be it due to a lack of resources or lack of access to high quality teaching. The dearth in high-quality educational content, teaching aids, and methodologies, and non-availability of objective feedback on how they could become better teachers, deprive our teachers from achieving their full potential. The administrators and policy makers lack tools for making optimal decisions such as optimal class sizes, class composition, and course sequencing. All these handicap the nations, particularly the economically emergent ones, who recognize the centrality of education for their growth. EDM-2023 has striven to focus on concepts, principles, and techniques mined from educational data for amplifying the potential of all the stakeholders in the education system. The spotlights of EDM-2023 include: (1) Five keynote talks by outstanding researchers of eminence; (2) A plenary Test of Time award talk and a Banquet talk; (3) Five tutorials (foundational as well as advanced); (4) Four thought provoking panels on contemporary themes; (5) Peer reviewed technical paper and poster presentations; (6) Doctoral students consortium; and (7) An enchanting cultural programme. [Individual papers are indexed in ERIC.]

In response to the challenge of geometry's abstract and less engaging nature for students, this study explored the potential of connecting geometry to cultural elements, specifically geometric ornaments found in various cultures worldwide. Geometric ornaments, laden with cultural and spiritual significance, serve as a bridge between mathematics education and cultural contexts. The current research focused on in-depth interviews with 10 school students chosen randomly from two ninth-grade classes from an Arab middle school who were engaged in activities involving analyzing of geometric properties of ornaments, constructing ornaments using a compass and straightedge, and problem-solving exercises related to these ornaments. The qualitative data were analyzed to answer how the students perceived the learning of geometry in the context of geometric ornaments. The findings revealed a significant positive view of the students towards geometry after engaging in these activities. [For the full proceedings, see ED656038.]

"Geometry--Do" is a textbook about plane geometry. It will be divided into two volumes, "Geometry without Multiplication: White through Red Belt," and "Geometry with Multiplication: Blue through Black Belt. "The white- and yellow-belt chapters are neutral geometry; the remainder of "Volume One" and all of "Volume Two" is Euclidean geometry. It is primarily intended to teach geometry from the ground up, starting with the postulates and citing only already-proven theorems. It trains mathletes for competition, but it is not the usual grab-bag of unproven theorems chosen haphazardly and solely because they appeared in past exams. The early chapters prepare students for jobs in construction, architecture, surveying, graphic arts, and military defense. The later chapters teach geometry needed by engineers and military officers. In this lecture, the White Belt chapter is presented. I will address these people: Pure Mathematicians: Moise derides the "lighthearted use of the word let." I prove the crossbar theorem and other foundations not usually taught in high school, and I discuss Hilbert's "Foundations of Geometry." High-School Teachers: Randomly assigning letters to points is what makes geometry confusing. I have special symbols for midpoints, perpendicular feet, and infeet (where the angle bisector cuts the opposite side of a triangle) and exfeet. Administrators: I present clear distinctions between "Geometry--Do" and "Common Core" with examples that concerned parents can understand. Construction Workers: I invent the Aguilar A-Frame, give detailed instructions on squaring abasement foundation wider than a tape measure without exiting the rectangle, and discuss how building with wood differs from steel construction. Military Officers: I discuss troop positioning along a frontier that is plagued with cross-border raids, which assumes that friendly and enemy troops move at the same speed, and a parabola is the set of points equidistant from the focus and the directrix. [For the complete proceedings, see ED655360.]

Many scholars have argued that mathematics classrooms often offer narrow conceptions of mathematical excellence, recognizing only some kinds of thinking and some kinds of people as valuable, and conflating mathematical aptitude with overall intelligence. Play offers the potential to disrupt such classroom mathematical practices, by offering new and broader ways to exercise agency, and, relatedly, more expansive visions of who is seen as mathematically capable. Offering an in-depth analysis of the participation of two students as they engage in mathematics in their Kindergarten class, whole group rug time and small group play centers, we investigate how different activity structures create space for students to exercise agency in ways that demonstrate multiple forms of competence, creating liberating mathematical spaces. [For the complete proceedings, see ED658295.]

Dramatic differences in children's math knowledge at school entry are thought to originate in the Home Math Environment (HME), where parents and caregivers are the primary provider of experiences that influence children's early math knowledge development. Little is known about what informs parent decision-making around "mathematical parenting" (i.e., parents' cognitions, motivations, and behaviors that impact and influence child math development in the HME). This study uses the RESET Framework and survey instrument to investigate parents' mathematical parenting perceptions (n = 847) across the domains of Role, Expectations, Skills, Efficacy, and Time. Parent self-reports of early childhood math knowledge and of shared math activity are also examined to shed light on the factors that influence mathematical parenting of 4-5-year-old children in the home. [For the complete proceedings, see ED658295.]

ChatGPT and Bard, two chatbots powered by Large Language Models (LLMs), are propelling the educational sector towards a new era of instructional innovation. Within this educational paradigm, the present investigation conducts a comparative analysis of these groundbreaking chatbots, scrutinizing their distinct operational characteristics and applications as depicted in current scholarly discourse. ChatGPT emerges as an exemplary tool in task-oriented textual interactions, while Bard brandishes unique features such as Text-To-Speech (TTS) functionality, which enhances accessibility and inclusive education, as well as integration with Google Workspace applications. This research critically examines their utilization in various spheres such as pedagogy, academic research, Massive Open Online Courses (MOOCs), mathematics, and software programming. Findings accentuate ChatGPT's superior efficacy in content drafting, code generation, language translation, and providing clinically precise responses, notwithstanding Bard's significant potential encapsulated in its exclusive features. Furthermore, the study traverses' crucial ethical aspects, including privacy concerns and inherent bias, underscoring the profound implications of these Artificial Intelligence (AI) technologies on literature and advocating against the indiscriminate reliance on such models. [For the full proceedings, see ED636095.]

Mechanical reasoning is crucial for many engineering fields, yet undergraduate engineering students struggle to understand discipline-specific formalisms from their courses that model mechanical concepts. The current investigation observed undergraduate engineering students' speech during mechanical reasoning and the benefits of attending to students' speech for assessing their mechanical reasoning and learning. Results from discourse analysis of two undergraduate engineering student dyads enrolled in a Mechanics of Materials course revealed students' mechanical reasoning through both course-specific and non-course-specific speech. Students generated speech that described functional or dynamic behaviors, as well as structure that described the static features. Students with highly developed formal knowledge of mathematics and physics often reason about mechanical systems in ways not captured traditional formalisms has implications for instruction and assessment.

To support online learners at a large scale, extensive studies have adopted machine learning (ML) techniques to analyze students' artifacts and predict their learning outcomes automatically. However, limited attention has been paid to the fairness of prediction with ML in educational settings. This study intends to fill the gap by introducing a generic algorithm that can orchestrate with existing ML algorithms while yielding fairer results. Specifically, we have implemented logistic regression with the Seldonian algorithm and compared the fairness-aware model with fairness-unaware ML models. The results show that the Seldonian algorithm can achieve comparable predictive performance while producing notably higher fairness. [This paper was published in: "LAK21: 11th International Learning Analytics and Knowledge Conference (LAK21), April 12-16, 2021, Irvine, CA, USA," ACM, 2021.]

Bridging the gap between mathematical learning at home and school has been an issue for education research for decades (Galindo & Sheldon, 2012). Expectations for mathematics do not often align for teachers and parents (Posey-Maddox & Hayley-Lock, 2016) and a limited view of what counts as mathematics persists. What needs more attention is the meaningful mathematical learning that happens at home but is rarely seen as mathematics. Parents frequently struggle in supporting their children's mathematical learning, but that struggle becomes productive when parents are recognized as mathematically capable. This paper shows how two mothers shift their perspectives of what counts as mathematics and recognize the rich content in their current interactions with young children. Making such connections between mathematics and parent action can strengthen the relationship between at-home and school learning. [For the complete proceedings, see ED630060.]

Several researchers have promoted reimagining functions and graphs more quantitatively. One part of this research has examined graphing "conventions" that can at times conflict with quantitative reasoning about graphs. In this theoretical paper, we build on this work by considering a widespread convention in mathematics teaching: putting related, derived graphical objects (e.g., the graphs of a function and its inverse or the graphs of a function and its derivative) on the same set of axes. We show problems that arise from this convention in different mathematical content areas when considering contextualized functions and graphs. We discuss teaching implications about introducing such related graphical objects through context on separate axes, and eventually building the convention of placing them on the same axis in a way that this convention and its purposes become more transparent to students. [For the complete proceedings, see ED630060.]

In this paper we address the question, how do quantitative reasoning and covariational reasoning present as students build structural conceptions of real-world situations. We use data from an exploratory teaching experiment with an undergraduate STEM major to illustrate the explanatory roles quantitative reasoning and covariational reasoning play in, (a) coordinating more than two interdependent quantities, (b) conceiving of real-world situations in more than one way, (c) constructing networks of quantitative relationships, and (d) creating a mathematical expression. We make the case that looking at mathematical model construction through the lens of quantitative reasoning and covariational reasoning may provide insights into students' mathematical decisions as they structure complex real-world scenarios. [For the complete proceedings, see ED630060.]

Consistency of learning behaviors is known to play an important role in learners' engagement in a course and impact their learning outcomes. Despite significant advances in the area of learning analytics (LA) in measuring various self-regulated learning behaviors, using LA to measure consistency of online course engagement patterns remains largely unexplored. This study focuses on modeling consistency of learners in online courses to address this research gap. Toward this, we propose a novel unsupervised algorithm that combines sequence pattern mining and ideas from information retrieval with a clustering algorithm to first extract engagement patterns of learners, represent learners in a vector space of these patterns and finally group them into groups with similar consistency levels. Using clickstream data recorded in a popular learning management system over two offerings of a STEM course, we validate our proposed approach to detect learners that are inconsistent in their behaviors. We find that our method not only groups learners by consistency levels, but also provides reliable instructor support at an early stage in a course. [This paper was published in: "LAK21: 11th International Learning Analytics and Knowledge Conference (LAK21), April 12-16, 2021, Irvine, CA, USA." ACM, 2021, pp. 226-236.]

In contrast to simple feedback, which provides students with the correct answer, elaborated feedback provides an explanation of the correct answer with respect to the student's error. Elaborated feedback is thus a challenge for AI in education systems because it requires dynamic explanations, which traditionally require logical reasoning and knowledge engineering to generate. This study presents an alternative approach that formulates elaborated feedback in terms of long-form question answering (LFQA). An off-the-shelf LFQA system was evaluated by human raters in a 2x2x2x2 ablation design that manipulated the context documents given to the LFQA model and the post-processing of model output. Results indicate that context manipulations improve performance but that postprocessing can have detrimental results. [This paper was published in: "Proceedings of the Eighth ACM Conference on Learning @ Scale," 2021, pp. 27-36.]

In the mathematical community, two notions of "function" are used: the set-theoretic definition as a univalent set of ordered pairs, and the Bourbaki triple. These definitions entail different interpretations and answers to mathematical questions that even a secondary student might be prompted to answer. However, mathematicians and mathematics educators are often not explicit about which definition they are using. This paper discusses these parallel usages and the related implications for the field of mathematics education. [For the complete proceedings, see ED629884.]

With the increased number of MOOC offerings, it is unclear how these courses are related. Previous work has focused on capturing the prerequisite relationships between courses, lectures, and concepts. However, it is also essential to model the content structure of MOOC courses. Constructing a precedence graph that models the similarities and variations of learning paths followed by similar MOOCs would help both students and instructors. Students can personalize their learning by choosing the desired learning path and lectures across several courses guided by the precedence graph. Similarly, by examining the precedence graph, instructors can 1) identify knowledge gaps in their MOOC offerings, and 2) find alternative course plans. In this paper, we propose an unsupervised approach to build the precedence graph of similar MOOCs, where nodes are clusters of lectures with similar content, and edges depict alternative precedence relationships. Our approach to cluster similar lectures based on PCK-Means clustering algorithm that incorporates pairwise constraints: Must-Link and Cannot-Link with the standard K-Means algorithm. To build the precedence graph, we link the clusters according to the precedence relations mined from current MOOCs. Experiments over real-world MOOC data show that PCK-Means with our proposed pairwise constraints outperform the K-Means algorithm in both Adjusted Mutual Information (AMI) and Fowlkes-Mallows scores (FMI). [For the full proceedings, see ED607784.]

These proceedings result from a symposium designed to provide a forum for the consideration of major social issues confronting science today. Participants (including scientists at different stages of career development from undergraduate concentrator to Nobel laureate) discussed issues related to the scientist's responsibilities as scientist and citizen, the nature and limitation of objectivity, and the inclusion of women as well as men in the world of science tomorrow. The papers presented at the symposium and included in this document are: (1) "Meritocracy and Marginality: Women in Science Today and Tomorrow" (Jonathan R. Cole); (2) "Status of Women in the Physical Sciences, Mathematics and Engineering" (Rhonda Hughes); (3) "Critical Filters in Science Careers" (Bonnie Spanier); (4) "Objectivity in Clinical Trials" (Sonja M. McKinlay); and (5) "The Cult of Objectivity in the Physical Sciences" (Jill C. Bonner). Also included is a report, "Ethical Choices for Science Panel," a summary of panel discussions which centered on Everett Mendelsohn's discussion of "Nuclear Energy, Weapons, and Scientists" and Miriam Schweber's report on "Ethics and DNA Research." These discussions focused on the dichotomy that exists between the acquisition of scientific knowledge and its subsequent application. (JN)

In test equating, one critical equating property is the group invariance property which indicates that the equating function used to convert performance on each alternate form to the reporting scale should be the same for various subgroups. To mitigate the impact of disrupted learning on the item parameters during the COVID-19 pandemic, a pre-equated approach was frequently implemented for student scoring in the spring 2021 administration of large-scale assessments. This study examines how much the group invariance property was satisfied for various subgroups when pre-equating was applied to large-scale assessments in spring 2021. While most studies for invariance property have been for post-equating, this study investigated the invariance property with pre-equating. Due to the potential impact on live calibration and equating, many large-scale assessment programs have opted to reuse previously administered forms and applied pre-equating rather than developing new forms and post-equating in 2021. This study included grades 6 and 8 ELA and Math from a spring 2021 large-scale assessment program and examined the invariance property for four subgroups: gender, SES, ethnicity, and LEP. Three different scoring methods were studied: unrounded equated scores, rounded equated scores, and MLE as IRT pattern scoring. As evaluation criteria, RMSD and REMSD indices were included (Dorans & Holland, 2000). The effect size was also used for evaluating the equating function differences between each subgroup's categories and the population. Invariance property is often studied when post-equating is performed with alternate forms under common (i.e., anchor) item design. In this case, the quality and number of anchor items between alternate forms influence the equating results and invariance property. In general, if anchors are close to a minimal test version, equating results can be better with many anchor items. Pre-equating design fully satisfies the requirement of the maximum number of anchor items because all items can be used as anchor items. Therefore invariance property can be better achieved than post-equating with some common items under the same condition. RMSD and REMSD values for ELA, where students' performances were not much impacted by the pandemic, showed the same results. However, there were some deviations in the invariance property for Math where students' performances appeared to be impacted.

The purpose of the current research study was to investigate the extent to which student performance was influenced by the disruption to learning that occurred due to the COVID-19 pandemic. To explain observed differences in performance, we compared outcomes from a state-wide summative testing program between students who were administered the test in 2019 and 2021. Because the background and demographic characteristics of the two testing populations differed on several variables, including race/ethnicity, locale, and subgroup membership, propensity score matching was used to develop comparable groups for analyses. Independent samples t-tests were then conducted to assess whether there were significant differences in mean scaled scores and domain subscores between the two matched groups. In addition, the same procedures were used to develop and analyze differences for specific student subgroups (students with disabilities, economically disadvantaged students, and English language learners). Overall, results indicated that the 2021 test-takers earned lower scores than students who tested in 2019 for all subjects and grade levels, except English Language Arts (ELA) grade 8. Moreover, larger score differences were observed for mathematics than ELA. Lastly, results from subgroup analyses indicated that in mathematics all subgroups earned significantly lower scores in 2021 than 2019, whereas in ELA grades 5 through 7, English Language Learners and economically disadvantaged subgroups earned lower scores in 2021 than 2019. The results of this study suggest that after controlling for differences between the 2021 and 2019 populations, student performance was likely influenced by the disruption to learning due to pandemic-related factors.

This study investigated how motivational orientations and learning strategies predict ninth graders' non-routine mathematical problem-solving anxiety. Non-routine mathematical problem-solving anxiety classification and prediction were investigated through TwoStep cluster analysis, linear discriminant analysis, and logistic regression. 274 ninth graders participated in the study. The participants were clustered based on their problem-solving achievements and test anxiety levels: high-level and low-level problem-solving anxiety. Extrinsic goal orientation, rehearsal, and peer learning were significant classifiers. Intrinsic goal orientation, self-efficacy, rehearsal, and help-seeking were significant predictors for ninth graders' non-routine mathematical problem-solving anxiety.

Metacognition is a valuable tool for learning, since it is closely related to self-regulation and awareness of one's own affect. However, methods for automatically detecting and studying metacognition are scarce. Thus, in this paper we describe an algorithm for automatic detection of metacognitive language in writing. We analyzed text from the forums of two online, university-level science courses, which revealed common patterns of phrases that we used for automatic metacognition detection. The algorithm we developed exhibited high accuracy on expert-labeled metacognitive phrases (Spearman's rho = 0.878 and Cohen's kappa = 0.792), and provides a reliable, fast method for automatically annotating text corpora that are too large for manual annotation. We applied this algorithm to analyze relationships between students' metacognitive language and their academic performance, finding small correlations with course grade and medium-sized differences in metacognition across courses. We discuss how our algorithm can be used to advance metacognitive studies and online educational systems.

One of the essential problems, in educational data mining, is to predict students' performance on future learning materials, such as problems, assignments, and quizzes. Pioneer algorithms for predicting student performance mostly rely on two sources of information: students' past performance, and learning materials' domain knowledge model. The domain knowledge model, traditionally curated by domain experts, maps learning materials to concepts, topics, or knowledge components that are presented in them. However, creating a domain model by manually labeling the learning material can be a difficult and time-consuming task. In this paper, we propose a tensor factorization model for student performance prediction that does not rely on a predefined domain model. Our proposed algorithm models student knowledge as a soft membership of latent concepts. It also represents the knowledge acquisition process with an added rank-based constraint in the tensor factorization objective function. Our experiments show that the proposed model outperforms state-of-the-art algorithms in predicting student performance in two real-world datasets, and is robust to hyper-parameters. [For the full proceedings, see ED599096.]

One significant challenge in the field of measuring ability is measuring the current ability of a learner while they are learning. Many forms of inference become computationally complex in the presence of time-dependent learner ability, and are not feasible to implement in an online context. In this paper, we demonstrate an approach which can estimate learner skill over time even in the presence of large data sets. We use a rating system derived from the Elo rating system and its relatives, which are commonly used in chess and sports tournaments. A learner's submission of a course assignment is interpreted as a single match. We apply this approach to Coursera's online learning platform, which includes millions of learners who have submitted assignments tens of millions of times in over 3000 courses. We demonstrate that this provides reliable estimates of item difficulty and learner ability. Finally, we address how this scoring framework may be used as a basis for various applications that account for a learner's ability, such as adaptive diagnostic tests and personalized recommendations. [For the full proceedings, see ED599096.]

In this paper we consider the problem of modelling when students end their session in an online mathematics educational system. Being able to model this accurately will help us optimize the way content is presented and consumed. This is done by modelling the probability of an action being the last in a session, which we denote as the End-of-Session probability. We use log data from a system where students can learn mathematics through various kinds of learning materials, as well as multiple types of exercises, such that a student session can consist of many different activities. We model the End-of-Session probability by a deep recurrent neural network in order to utilize the long term temporal aspect, which we experimentally show is central for this task. Using a large scale dataset of more than 70 million student actions, we obtain an AUC of 0.81 on an unseen collection of students. Through a detailed error analysis, we observe that our model is robust across different session structures and across varying session lengths. [For the full proceedings, see ED599096.]

Clustering of educational data allows similar students to be grouped, in either crisp or fuzzy sets, based on their similarities. Standard approaches are well suited to identifying common student behaviors; however, by design, they put much less emphasis on less common behaviors or outliers. The approach presented in this paper employs fuzzing clustering in the identification of these outlier behaviors. The algorithm is an iterative one, where clustering is applied, outliers identified, the data restricted to the outliers, and the process repeated. This approach produces a clustering that is crisp between each iteration and fuzzy within. It arose as a consequence of trying to cluster student progress trajectories in an adaptive learning platform. Included are results from applying the repeated fuzzy clustering algorithm to data from multiple courses and semesters at the University of Central Florida, (N=5,044). [For the full proceedings, see ED599096.]

Personalized education systems recommend learning contents to students based on their capacity to accelerate their learning. This paper proposes a personalized exercise recommendation system for online self-directed learning. We first improve the performance of knowledge tracing models. Existing deep knowledge tracing models, such as Dynamic Key-Value Memory Network (DKVMN), ignore exercises' concept tags, which are usually available in tutoring systems. We modify DKVMN to design its memory structure based on the course's concept list, and explicitly consider the exercise-concept mapping relationship during students' knowledge tracing. We evaluated the model on the 5th grade students' math exercising dataset in TAL, one of the biggest education groups in China, and found that our model has higher performance than existing models. We also enhance the DKVMN model to support more input features and obtain higher performance. Second, we use the model to build a student simulator, and use it to train an exercise recommendation policy with deep reinforcement learning. Experimental results show that our policy achieves better performance than existing heuristic policy in terms of maximizing the students' knowledge level. To the best of our knowledge, this is the first time that deep reinforcement learning has been applied to personalized mathematic exercise recommendation. [For the full proceedings, see ED599096.]

Short forms of psychometric scales have been commonly used in educational and psychological research to reduce the burden of test administration. However, it is challenging to select items for a short form that preserve the validity and reliability of the scores of the original scale. This paper presents and evaluates multiple automated methods for scale short form creation based on metaheuristic optimization algorithms that incorporate validity criteria based on internal structure and relationships with other variables. The ant colony optimization (ACO) algorithm, tabu search (TS), simulated annealing (SA) and genetic algorithm (GA) are examined using confirmatory factor analysis (CFA) of scales with one factor, three factor, and bi-factor factorial structure. The results indicate that SA created short forms with best model fit for scales with one and three factor structures, but ACO was able to obtain highest reliability. For scales with bi-factor structure, SA provide short forms with best model fit, but TS obtained highest reliability. Overall, the SA algorithm is recommended because it produced consistently best model fit and reliability that was only slightly lower than the ACO or TS algorithms. [For the full proceedings, see ED599096.]

This paper will cover the design of a learning environment as a classroom regarding the Quadratic Equations, Functions and Graphs. The goal of the learning environment offered in the paper is to design a classroom where students will enjoy the process, use their skills they already have during the learning process, control and plan their learning process, and have the right of free choice of which way they can learn easily. Besides, it is alleged that in the paper, students will be more engaged, motivated and self-confident in this learning environment in terms of theories and approaches. Besides, students will have computers and graph calculators in the classroom, and some applications and programs will be provided such as Cabri, Maple, Derive, etc. [For the complete proceedings, see ED579395.]

In this paper, we provide an overview of the "Maths Inside" project, funded by the Australian Maths and Science Partnership Program (AMSPP). The overall aim of the AMSPP is to improve uptake and participation of students in mathematics and science at secondary and tertiary levels. In this research project, we aim to improve student interest in mathematics and support mathematics teachers in their professional learning, through provision of rich and investigative learning resources, including video case studies of CSIRO scientists and mathematicians. Data collection on the outcomes of the project is ongoing and will be reported in subsequent papers.

This study examines the COVID-19 shift to online instruction and its impact on a charter school system (CSS) 3rd-10th grade students' academic achievement through the lens of noncognitive factors. We recruited 693 students and utilized qualitative and quantitative analyses. We found that students' NWEA math and ELA scores continued to increase although almost ninety percent of them completed 2020-2021 school year online. Second, students' self-efficacy, academic engagement, and growth mindset scores significantly explained some of the variance in students' NWEA math and ELA scores. Lastly, students indicated that they liked the comfort, family presence, safety, and personalization of the online learning most. They disliked the lack of social component of learning, technical, and focus and engagement problems most.

The Home Learning Environment (HLE) focuses on everyday learning habits in families to support children's competency development. In this study, we used multitrait-multimethod (MTMM) analyses to compare two theoretical dimensions and three methods for assessing the HLE and their associations with linguistic and mathematical competencies of kindergarten children. Our sample consisted of N = 190 children (M[subscript age] = 64 months, SD = 4.4). The MTMM matrix showed a substantial effect of common methods and indicated a one-dimensional HLE construct. A children's book title recognition test was the best predictor of children's linguistic and numeracy competencies. Even when controlling for child and family characteristics, the HLE was significantly related to both children's mathematical and linguistic competencies in a structural equation model.

The problem of development of algorithmic competence of students -- future Computer Science teachers as a component of information competence is investigated. The aim of the study is to identify the conditions for effective algorithmization and programming learning, involving a modular representation of the content, blended learning, allocated in accordance with the B. Bloom's taxonomy classes of problems in Computer Science and Methods of its Teaching and the use of electronic learning resources. This paper specifies the concept of algorithmic competence of university students--students' readiness to design algorithms and programs, their use in professional activities in the process of Computer Science teaching, electronic learning resources (ELR) design, self-education in the field of Computer Science. Classes of problems for algorithmic competence development in accordance with B. Bloom's taxonomy (knowledge, comprehension, application, analysis, synthesis and evaluation) are identified. Classes of problems correspond to ELR of a certain structure and content. The efficiency of ELR use in students' algorithmic competence development is statistically confirmed. [For the complete proceedings, see ED608557.]

Mathematical reasoning is foundational to making sense of mathematics. Yet assessing mathematical reasoning can be challenging for teachers. This paper reports on a project where teachers taught two lessons with a specific focus on reasoning and came together with other teachers at their school to attempt to assess the reasoning of their students. Results, derived from an analysis of two post-lesson discussions about student work samples and associated completed assessment rubrics, provide insights into the challenges identified by 34 primary teachers at 4 Victorian government schools.

The Computational Thinking recently has been recognised as one of the basic knowledge to be developed since childhood. Coding and computers are not just programming, but tools that help students to develop problem solving skills and more deep understand of the way things work. For these reasons, great attention has been focused on this topic both from a pedagogical and technological point of view. In this paper, a first approach to Computational Thinking using Arduino is presented. To this end, some learning activities have been designed to introduce middle school students, without any experience in coding, to the process of building the algorithm from simple exercises to more complex tasks. The pilot test involved 25 subjects, many of them do not like to study mathematics, science and technology, but the results were promising. The approach was appreciated by the students and the results of the questionnaires confirmed the learning effectiveness too. [For the complete proceedings, see ED600498.]

This paper describes a multiple criteria approach based on a hybrid method of Collaborative Filtering (CF) and ContentBased Filtering (CBF) for discovering the most relevant criteria which could affect the elective course recommendation for university students. In order to determine which factors are the most important, it is proposed a genetic algorithm which automatically discovers the importance of the different criteria assigning weights to each one of them. We have carried out an in-depth study using a real data set with more than 1700 ratings of Computer Science graduates at University of Cordoba. We have used different proposals and different weights for each criterion in order to discover what is the combination of multiple criteria which provides better results. [For the full proceedings, see ED593090.]

The significant amount of information available on the web has led to difficulties for the learner to find useful information and relevant resources to carry out their training. The recommender systems have achieved significant success in the area of e-commerce, they still have difficulties in formulating relevant recommendations on e-learning resources because of the different characteristics of learners. Most of the existing recommendation techniques do not take these characteristics into account. This problem can be mitigated by including learner information in the referral process. Currently many recommendation techniques have cold start problems and classification problems. In this paper, we propose an ontology-based collaborative filtering recommendation system for recommending learners' online learning resources based on a decision algorithm (DA). In our approach, ontology is used to model and represent domain knowledge about the learner and learning resources. Our approach is divided into four parts: (a) the creation of an ontology for the representation of the learner's knowledge and learning resources (b) the calculation of the similarity of the assessments according to the ontology and the prediction for the learner concerned; (c) generating the K best items by the collaborative filtering recommendation engine and (d) applying the DA on the proposed items to generate the final recommendations for the targeted learner. [For the complete proceedings, see ED590269.]

To efficiently create books and other instructional content from videos and further improve accessibility of our course content we needed to solve the scene detection (SD) problem for engineering educational content. We present the pedagogical applications of extracting video images for the purposes of digital book generation and other shareable resources, within the themes of accessibility, inclusive education, universal design for learning and how we solved this problem for engineering education lecture videos. Scene detection refers to the process of merging visually similar frames into a single video segment, and subsequent extraction of semantic features from the video segment (e.g., title, words, transcription segment and representative image). In our approach, local features were extracted from inter-frame similarity comparisons using multiple metrics. These include numerical measures based on optical character recognition (OCR) and pixel similarity with and without face and body position masking. We analyze and discuss the trade-offs in accuracy, performance and computational resources required. By applying these features to a corpus of labeled videos, a support vector machine determined an optimal parametric decision surface to model if adjacent frames were semantically and visually similar or not. The algorithm design, data flow, and system accuracy and performance are presented. We evaluated our system using videos from multiple engineering disciplines where the content was comprised of different presentation styles including traditional paper handouts, Microsoft PowerPoint slides, and digital ink annotations. For each educational video, a comprehensive digital-book composed of lecture clips, slideshow text, and audio transcription content can be generated based on our new scene detection algorithm. Our new scene detection approach was adopted by ClassTranscribe, an inclusive video platform that follows Universal Design for Learning principles. We report on the subsequent experiences and feedback from students who reviewed the generated digital-books as a learning component. We highlight remaining challenges and describe how instructors can use this technology in their own courses. The main contributions of this work are: Identifying why automated scene detection of engineering lecture videos is challenging; Creation of a scene-labeled corpus of videos representative of multiple undergraduate engineering disciplines and lecture styles suitable for training and testing; Description of a set of image metrics and support vector machine-based classification approach; Evaluation of the accuracy, recall and precision of our algorithm; Use of an algorithmic optimization to obviate GPU resources; Student commentary on the digital book interface created from videos using our SD algorithm; Publishing of a labeled corpus of video content to encourage additional research in this area; and an independent open-source scene extraction tool that can be used pedagogically by the ASEE community e.g., to remix and create fun shareable instructional content memes, and to create accessible audio and text descriptions for students who are blind or have low vision. Text extracted from each scene can also used to improve the accuracy of captions and transcripts, improving accessibility for students who are hard of hearing or deaf.

Noticing children's mathematical thinking is foundational to teaching that is responsive to children's thinking. To better understand the range of noticing expertise for teachers engaged in multiyear professional development, we assessed the noticing of 72 upper elementary school teachers using three instructional scenarios involving fraction problem solving. Through a latent class analysis, we identified three subgroups of teachers that reflected different profiles of noticing expertise. Consideration was given to the noticing component skills of attending to children's strategy details, interpreting children's understandings, and deciding how to respond on the basis of children's understandings. We share theoretical and practical implications for not only the three profiles but also our choice to explore separately two versions of deciding how to respond (deciding on follow-up questions and deciding on next problems). [For the complete proceedings, see ED630060.]

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.]

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.]

People often view mathematics as abstract, cold, and irrelevant to real-life, and their school experiences influence such views. In this case study, we investigated the mathematics learning experiences of two women who participated in an afterschool girls STEM club 26 years ago. We explored their experiences in and out of school and how such experiences informed their images of mathematics. Data were collected from a survey, focus group interviews, and individual interviews. Using qualitative analysis, we learned that their school mathematics experiences influenced the participants' images of mathematics. The findings also revealed the participants' continuous and discontinuous learning experiences between school and out-of-school mathematics. This study suggests creating spaces to develop curricula that bridge the gap between school and out-of-school learning experiences. [For the complete proceedings, see ED630060.]