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In 12 audio taped sessions, three kindergarten children engaged algebra in a teaching for understanding, thematic project. Toni, Asa, and Cornel had one-on-one lessons dealing with simple natural numbers, patterns, and relationships. Along the way, each child studied one of Toni Morrison's Who's got game books to explore repetition patterns in well written literature. Then, each child "algebrafied" a Liberian folktalk and a Chinese folktale to explore number, pattern, and relationship in simple arrays. Finally, they engaged square natural numbers one to four. On the practitioner research side of the project, the inquiry was this: What happens when kids explore natural numbers in simple arrays? This descriptive inquiry used an ethnographic-like, participant observer model to collect data systematically (3 lessons per session times 12 sessions or 36 lessons in all). In brief, the study found that three kindergarten children made generalizations, added to a web of knowledge, used mathematical memory, and displayed flawed reasoning as a springboard for new mathematical knowledge--what Susan Jo Russell called "mathematical reasoning." The paper captured what counted for learning and what counted for research. These were two themes of the 29th Ethnography and Education Research Forum at the University of Pennsylvania, 1 March 2008. (Contains an annotated bibliography.)

This paper presents a conceptual model of assessment that defines six sequential, iterative steps: purpose identification; outcomes identification; measurement and evaluation design; data collection; analysis and evaluation; and decision making/action. The conceptual model of assessment closely parallels systematic scientific inquiry, especially the portion of inquiry that corresponds to qualitative research. Therefore, the paper argues that the conduct of assessment based on this model is qualitative research that often takes the form of both action research and institutional research. The paper reviews quantitative and qualitative research paradigms and defines assessment, action research, and institutional research. The conceptual model explains assessment in higher education at all levels of the academy and in regard to all of the academy's ends. Within the parameters of this conceptual model and the purposes of assessment, three primary foci of assessment (classroom assessment, program review, and institutional impact) are discussed. The paper posits that action research and institutional research are the forms of systemic inquiry that support the institution's internal information management and planning needs and the need to meet accountability mandates from external constituencies. It also discusses generalizability, suggesting that generalization is the responsibility of the reader and not the institutional researcher. (Contains 11 references.) (SM)

The Purposeful Algebraic Activity Project is a longitudinal study of the development of pupils' algebraic activity in the early years of their secondary schooling. Analysis of data from a spreadsheet-based teaching programme and from semi-structured interviews leads us to identify three features of the spreadsheet environment that appear to shape pupils' generalising: focus on calculations; use of notation; and feedback. We discuss how pupils' experience of generating spreadsheet formulae can potentially support pupils' generalising in a paper and pencil environment. [For complete proceedings, see ED489597.]

Students who take an online course, such as a MOOC, use the course's discussion forum to ask questions or reach out to instructors when encountering an issue. However, reading and responding to students' questions is difficult to scale because of the time needed to consider each message. As a result, critical issues may be left unresolved, and students may lose the motivation to continue in the course. To help address this problem, we build predictive models that automatically determine the urgency of each forum post, so that these posts can be brought to instructors' attention. This paper goes beyond previous work by predicting not just a binary decision cut-off but a post's level of urgency on a 7-point scale. First, we train and cross-validate several models on an original data set of 3,503 posts from MOOCs at University of Pennsylvania. Second, to determine the generalizability of our models, we test their performance on a separate, previously published data set of 29,604 posts from MOOCs at Stanford University. While the previous work on post urgency used only one data set, we evaluated the prediction across different data sets and courses. The best-performing model was a support vector regressor trained on the Universal Sentence Encoder embeddings of the posts, achieving an RMSE of 1.1 on the training set and 1.4 on the test set. Understanding the urgency of forum posts enables instructors to focus their time more effectively and, as a result, better support student learning. [For the complete proceedings, see ED630829.]

In this paper, we explore how students' algebraic noticing's and explanations changed across a two-year period with the introduction of designed instructional material. The data in this report is drawn from n=53 Year 7-8 students' responses to a free-response assessment task across two different years. Analysis focused on how students noticed and explained algebraic relationships in pairs of equivalent equations. Findings indicate that with the introduction of designed instructional material, there was a shift in student noticing of number properties to identify equivalence between pairs of equations. However, identifying the distributive property of multiplication and developing generalisations about the algebraic relationships remained challenging for students.

The development of students' mathematical problem-solving skills is contingent upon the approaches and methods employed by primary school teachers. This research endeavors to scrutinize the effectiveness of primary school teachers in their roles within the problem-solving process, with particular attention directed toward their inquiry techniques, instructional approaches, and responses to student errors. Employing as a explanatory mixed-design research, firstly quantitative data were obtained from the primary school teachers (N=116) and analyzed by applying the questionnaire form developed by the researchers; additionally, qualitative data were obtained by conducting focus group interviews with a subset of participants (N=6) based on the information obtained from these data. Thereby, teachers get the opportunity to share their ideas in the process. Second phase of the research aimed to develop the active roles of the participants in the problem-solving process. The research outcomes underscore that teachers frequently gravitate toward questioning techniques and teaching methods that promote a nuanced comprehension of the problem. There also appears to be an underutilization of inquiries pertaining to prediction, generalization, and collaborative group work. Furthermore, the research unveils an array of strategies employed by teachers to rectify student errors. These encompass commonly employed techniques such as question-answer, exemplification, and lecture responses, irrespective of the specific type of question posed. [This paper was published in: "EJER Congress 2023 International Eurasian Educational Research Congress Conference Proceedings," Ani Publishing, 2023, pp. 476-492.]

Recent research seeks to develop more comprehensive learner models for adaptive learning software. For example, models of reading comprehension built using data from students' use of adaptive instructional software for mathematics have recently been developed. These models aim to deliver experiences that consider factors related to learning beyond performance in the target domain for instruction. We investigate the extent to which generalization is possible for a recently developed predictive model that seeks to infer students' reading comprehension ability (as measured by end-of-year standardized test scores) using an introductory learning experience in Carnegie Learning's MATHia intelligent tutoring system for mathematics. Building on a model learned on data from middle school students in a single school district in a mid-western U.S. state, using that state's end-of-year English Language Arts (ELA) standardized test score as an outcome, we consider data from a school district in a south-eastern U.S. state as well as that state's end-of-year ELA standardized test outcome. Generalization is explored by considering prediction performance when training and testing models on data from each of the individual school districts (and for their respective state's test outcomes) as well as pooling data from both districts together. We conclude with discussion of investigations of some algorithmic fairness characteristics of the learned models. The results suggest that a model trained on data from the smaller of the two school districts considered may achieve greater fairness in its predictions over models trained on data from the other district or both districts, despite broad, overall similarities in some demographic characteristics of the two school districts. This raises interesting questions for future research on generalizing these kinds of models as well as on ensuring algorithmic fairness of resulting models for use in real-world adaptive systems for learning. [This paper was published in: "Proceedings of the 16th International Conference on Educational Data Mining," edited by M. Feng et al., International Educational Data Mining Society, 2023, pp. 207-16.]

The purpose of this paper is to highlight issues related to students' personal inferences that arise when students verbally explain their justification for calculus statements. We conducted clinical interviews with three undergraduate students who had taken first-semester calculus but had not yet been exposed to formal proof writing activities through undergraduate mathematics courses. We analyzed these students' verbal justification of four statements, each of which described a different relationship between two quantified variables in calculus contexts. In this paper, we document students' personal inferences that were evoked in their justifications and discuss how they are similar to or different from the logical inferences that have been accepted and practiced by the mathematics community for mathematical arguments. [For the complete proceedings, see ED630210.]

According to the document, Dr. Andrew Brimmer did an excellent job of identifying emerging economic concerns. Dr. Brimmer's characterization of economics as a tool kit can help young people examine important social questions using principles of economics as the tool for analysis. One way to build an economics tool kit is by placing more stress on the basic premises of economics. These premises can be thought of as a set of generalizations about economic behavior which students can use to analyze current concerns such as the economics of national health insurance or the economics of pricing natural resources. The tool kit should provide students with answers to their questions. Three values derived from the discipline of economics might be worthy of inclusion in a citizenship education program. These values are as follows: (1) individuals are responsible for their own actions; (2) private property builds freedom and personal responsibility; and (3) people should not impose unwanted costs on others. Conference participants were asked to examine the papers presented and identify examples of how the ideas could translate into good teaching or be developed into sound instructional material. More specific examples are needed to help translate these powerful new ideas into meaningful classroom instruction. (GEA)

Automatic short answer grading is an important research direction in the exploration of how to use artificial intelligence (AI)-based tools to improve education. Current state-of-the-art approaches use neural language models to create vectorized representations of students responses, followed by classifiers to predict the score. However, these approaches have several key limitations, including i) they use pre-trained language models that are not well-adapted to educational subject domains and/or student-generated text and ii) they almost always train one model per question, ignoring the linkage across question and result in a significant model storage problem due to the size of advanced language models. In this paper, we study the problem of automatic short answer grading for students' responses to math questions and propose a novel framework for this task. First, we use MathBERT, a variant of the popular language model BERT adapted to mathematical content, as our base model and fine-tune it on the downstream task of student response grading. Second, we use an in-context learning approach that provides scoring examples as input to the language model to provide additional context information and promote generalization to previously unseen questions. We evaluate our framework on a real-world dataset of student responses to open-ended math questions and show that our framework (often significantly) outperform existing approaches, especially for new questions that are not seen during training. [For the full proceedings, see ED623995.]

This paper will elaborate five levels of algebraic generalisation based on an analysis of students' responses to Reframing Mathematical Futures II (RMFII) tasks designed to assess algebraic reasoning. The five levels of algebraic generalisation will be elaborated and illustrated using selected tasks from the RMFII study. The five levels will be matched against the eight zones identified in the RMFII study supported by its Rasch analysis. We identify two shifts where students' capacity to generalise appear difficult to navigate. The first being where students move from noticing and describing regularities to formalising these regularities into verbal or symbolic expressions. The second is where students use their understanding of equivalence based on relational thinking to write and recognise equivalent algebraic expressions.

Learning progressions have become an important construct in educational research, in part because of their ability to inform the design of coherent standards, curricula, assessments, and instruction. In this paper, I discuss how a learning progressions approach has guided our development of an early algebra innovation for the elementary grades and provide examples of how this approach can help challenge a settled mathematics learning status quo about the kind of algebra students can learn, when they can learn it, and how all students can be successful. Empirically derived learning progressions are an important part of designing early algebra innovations that can open new curricular pathways for teaching and learning algebra, creating accessible and effective avenues of learning for all students. [For the complete proceedings, see ED630210.]

This fourth and final volume of the 31st annual proceedings of the International Group for the Psychology of Mathematics Education conference presents research reports for author surnames beginning Na- through Zod-. Reports include: (1) Mathematically Gifted Students' Problem Solving Approaches on Conditional Probability (GwiSoo Na, DaeHee Han, KyungHwa Lee, and SangHun Song); (2) Students Engaged in Proving--Participants in an Inquiry Process or Executers of a Predetermined Script? (Talli Nachlieli and Patricio Herbst); (3) Exploring the Idea of Curriculum Materials Supporting Teacher Knowledge (Jihwa Noh and Ok-Ki Kang); (4) What Is the Price of Topaze? (Jarmila Novotna and Alena Hospesova); (5) Designing Unit for Teaching Proportion Based on Cultural-Historical Activity Theory: Process of Symbolizing through Collective Discourse (Minoru Ohtani); (6) Prototype Phenomena and Common Cognitive Paths in the Understanding of the Inclusion Relations between Quadrilaterals in Japan and Scotland (Masakazu Okazaki and Taro Fujita); (7) A Study of Gender Differences in Language Used by Parents and Children Working on Mathematical Tasks (Melfried Olson, Judith Olson, and Claire Okazaki); (8) Students' Motivation and Achievement and Teachers' Practices in the Classroom (Marilena Pantziara and George Philippou); (9) Using a Multiplicative Approach to Construct Decimal Structure (Irit Peled, Ruth Meron, and Shelly Rota); (10) If You Don't Listen to the Teacher, You Won't Know What to Do: Voices of Pasifika Learners (Pamela Perger); (11) Fostering Generalization in Connecting Registers of Dynamic Geometry and Euclidean Constructions (Norma Presmeg, Jeff Barrett, and Sharon McCrone); (12) Addressing the Issue of the Mathematical Knowledge of Secondary Mathematics Teachers (Jerome Proulx); (13) Abduction in Pattern Generalization (F. D. Rivera and Joanne Rossi Becker); (14) An Activity for Development of the Understanding of the Concept of Limit (Kyeong Hah Roh); (15) Early Access to Algebraic Ideas: The Role of Representations and of the Mathematics of Variation (Teresa Ceballos Rojano and Elvia Maximo Perrusquia); (16) The Role of Cognitive Conflict in Belief Changes (Katrin Rolka, Bettina Rosken, and Peter Liljedahl); (17) Factors Affecting Seventh Graders' Cognitive Perceptions of Patterns Involving Constructive and Deconstructive Generalizations (Joanne Rossi Becker and F. D. Rivera); (18) Mathematically Gifted Students' Spatial Visualization Ability of Solid Figures (HyunAh Ryu, YeongOk Chong, and SangHun Song); (19) Genetic Approach to Teaching Geometry (Ildar S. Safuanov); (20) On Primary Teachers' Assessment of Pupils' Written Work in Mathematics (H. Sakonidis and A. Klothou); (21) Qualities Co-Valued in Effective Mathematics Lessons in Australia: Preliminary Findings (Wee Tiong Seah); (22) Resolving Cognitive Conflict with Peers--Is There a Difference between Two and Four? (Hagit Sela and Orit Zaslavsky); (23) Explicit Linking in the Sequence of Consecutive Lessons in Mathematics Classrooms in Japan (Yoshinori Shimizu); (24) On the Teaching Situation of Conceptual Change: Epistemological Considerations of Irrational Numbers (Yusuke Shinno); (25) Posing Problems with Use the "What If Not?" Strategy in NIM Game (SangHun Song, JaeHoon Yim, EunJu Shin, and HyangHoon Lee); (26) Embodied, Symbolic and Formal Aspects of Basic Linear Algebra Concepts (Sepideh Stewart and Michael O. J. Thomas); (27) The Application of Dual Coding Theory in Multi-Representational Virtual Mathematics Environments (Jennifer M. Suh and Patricia S. Moyer-Packenham); (28) Interactions between Teaching Norms of Teacher's Professional Community and Learning Norms of Classroom Communities (Wen-Huan Tsai); (29) Analysis of a Learning Case: Jasmine (Zelha Tunc-Pekkan); (30) Elementary Teachers' Linguistic Inventions and Semantic Warrants for Mathematical Inferences (Janet G. Walter and Christine Johnson); (31) How Do Mentors Decide: Intervening in Practice Teachers' Teaching of Mathematics or Not (Chih-Yeuan Wang and Chien Chin); (32) Exploring an Understanding of Equals as Quantitative Sameness with 5 Year Old Students (Elizabeth Warren); (33) Classroom Teaching Experiment: Eliciting Creative Mathematical Thinking (Gaye Williams); and (34) Is a Visual Example in Geometry Always Helpful? (Iris Zodik and Orit Zaslavsky). (Individual papers contain references.) [For other volumes in the series, see ED499416, ED499417, and ED499419.]

Feedback on student answers and even during intermediate steps in their solutions to open-ended questions is an important element in math education. Such feedback can help students correct their errors and ultimately lead to improved learning outcomes. Most existing approaches for automated student solution analysis and feedback require manually constructing cognitive models and anticipating student errors for each question. This process requires significant human effort and does not scale to most questions used in homeworks and practices that do not come with this information. In this paper, we analyze students' step-by-step solution processes to equation solving questions in an attempt to scale up error diagnostics and feedback mechanisms developed for a small number of questions to a much larger number of questions. Leveraging a recent math expression encoding method, we represent each math operation applied in solution steps as a transition in the math embedding vector space. We use a dataset that contains student solution steps in the Cognitive Tutor system to learn implicit and explicit representations of math operations. We explore whether these representations can: (1) identify math operations a student intends to perform in each solution step, regardless of whether they did it correctly or not; and (2) select the appropriate feedback type for incorrect steps. Experimental results show that our learned math operation representations generalize well across different data distributions. [For the full proceedings, see ED615472.]

Computational activity is increasingly relevant in education and society, and researchers have investigated its role in students' mathematical thinking and activity. More work is needed within mathematics education to explore ways in which computational activity might afford development of mathematical practices. In this paper, we specifically examine the generalizing activity of undergraduate students who solved combinatorial problems in the context of Python programming. We demonstrate instances of generalizing in terms of Ellis et al.'s (2017) framework, and we argue that some opportunities were facilitated and supported by the computational setting in which the students worked. [For the complete proceedings, see ED629884.]

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

This study shares two frameworks for analyzing teacher actions that support students in generalizing and examines how those frameworks align with teacher questioning. One classroom teaching episode focused on the mathematical activity of generalizing is shared to illustrate effective generalizing promoting practices. We found several patterns of productive and unproductive generalizing promoting actions and questioning. Repeating generalizing promoting actions in succession were needed to produce student generalizations. Priming actions that set up for later generalizing promoting were helpful when students struggled to identify and state generalizations. Connection questions promoted generalizing, but justification and concept questions did not. Further research will explore the additional strategies to support teachers in fostering student-created generalizations. [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.]

The objective of this article is to describe types of mathematical reasoning evidenced by a middle school mathematics teacher, when answering two generalization questions in a figural pattern generalization task, related to quadratic sequences. Reasoning is delimited from teacher's arguments, reconstructed from a theoretical-methodological proposal that combines Peirce's definitions with some elements of Toulmin's argumentative model. The results show that the teacher evidenced abductive, inductive, and deductive reasoning, based on cognitive actions such as the decomposition of the figure, strategic counts, recognition of the behavior of the figural pattern, formulation, verification, and validation of conjectures. [For the complete proceedings, see ED630210.]

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.]

In this study, we explored enacted task characteristics (ETCs) that supported students' quantitative reasoning (QR). We employed a design-based methodology; we conducted a teaching experiment with eight secondary school students. Through ongoing and retrospective analyses, we identified ETCs which supported students' quantitative reasoning. The ETCs can set the infrastructure for students' QR when students are: (a) identifying changing attributes of the tasks or situations, (b) coordinating the change among quantities, and (c) making generalizations about quantitative relationships. ETCs play an important role in development of students' meaningful understanding when tasks are designed with focus on quantitative reasoning and representational fluency. [For the complete proceedings, see ED630060.]

Structural thinking skills should be developed as a prerequisite for a young person's future mathematical understanding and a teachers' understanding of mathematical structure is necessary to develop students' structural thinking skills. In this study, three secondary mathematics pre-service teachers (PSTs) learned to notice structural thinking through the CRIG framework of mathematical structure. The framework consists of Connections, Recognising patterns, Identifying similarities and difference, and Generalising and reasoning. I report here on how the CRIG framework helped the PSTs' notice structural thinking.

Measuring similarity of educational items has several applications in the development of adaptive learning systems, and previous research has already proposed a wide range of similarity measures. In this work, we provide an experimental evaluation of selected similarity measures using a large dataset. The used items are alternate-choice questions for the practice of English grammar for second language learners; the dataset contains thousands of items and over 10 million student answers. Our results provide warnings about the generalizability of results presented in EDM works: (1) the results vary significantly between knowledge components and (2) the size of available data is an important factor. [For the full proceedings, see ED615472.]

We explore how different components of an Automatic Short Answer Grading (ASAG) model affect the model's ability to generalize to questions outside of those used for training. For supervised automatic grading models, human ratings are primarily used as ground truth labels. Producing such ratings can be resource heavy, as subject matter experts spend vast amounts of time carefully rating a sample of responses. Further, it is often the case that multiple raters must come to a census before a final ground-truth rating is established. If ASAG models were developed that could generalize to out-of-sample questions, educators may be able to quickly add new questions to an auto-graded assessment without a continued manual rating process. For this project we explore various methods for producing vector representations of student responses including state-of-the-art representation methods such as Sentence-BERT as well as more traditional approaches including Word2Vec and Bag-of-words. We experiment with including previously untapped question-related information within the model input, such as the question text, question context text, scoring rubric information and a question-bundle identifier. The out-of-sample generalizability of the model is examined with both a leave-one-question-out and leave-one-bundle-out evaluation method and compared against a typical student-level cross validation. [For the full proceedings, see ED615472.]