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Although student covariation reasoning has been explored in depth to improve understanding of the correspondence between variables, research has focused on studying existing reasoning about variables in Cartesian representations. The working method had a qualitative approach, with a descriptive exploratory scope, the spontaneous representations that the participants evidenced under the level of covariational reasoning of the variables present in three contextualized situations were explored, posed to a population of third-grade middle school students. The students argued with concrete and abstract drawings the general behavior of the variables. At low levels of reasoning they used pictorial representations, at higher levels they used graphical diagrams and tables. The exploratory study shows a relationship between the type of spontaneous representations and the student's level of covariational reasoning, as well as the rigor of the description of the problem.

The Titanic disaster is a topic that continues to fascinate. As the importance of analytics continues to grow in industry, data literacy skills have become increasingly important in business education. This project allows students to use the passenger data from the Titanic to build their data literacy skills using an engaging, experiential topic. The project requires students to extract, transform, describe, analyze, and draw conclusions regarding the factors that impacted survival on the Titanic. The project can be deployed using various application software and tools. We describe how the assignment can be completed using Excel, Tableau, and Python using the Pandas library.

Learning activities related to students' real-life experiences may create meaningful learning. Integrating traditional games into the physics classroom can reconstruct local knowledge into scientific knowledge. The use of technology can affect student learning achievement. One of the technologies currently attracting attention is Augmented reality (AR). This study investigates the impact of AR-integrated traditional games Nglarak Blarak on students' critical thinking and graphical representation skills. In this study, a quasi-experimental design was used, consisting of 62 10th junior high school students. Class A completed the "work and energy" concept of their physics learning using AR technology, while class B used PowerPoint and textbooks. Class A students had higher critical thinking and graphical representation skills scores than those in class B. In addition, it was found that the critical thinking and graphical representation skills of students in class A who used AR technology showed a significant correlation. The results of this study can be used as a reference for integrating technology and indigenous knowledge in physics learning.

The capacity to generate prediction is indispensable in daily existence, particularly amidst the swift transformations that are occurring on a global scale. Therefore, this study aimed to analyze the levels of prediction ability among mathematics students when presented with data in graphs. A qualitative approach was adopted, involving 37 mathematics students, using taskbased tests and interviews as data collection techniques. The results showed that the ability of most students to make a prediction based on the COVID-19 graph was at a multi-structural level of 35.14%. This level was characterized by students making predictions based on the trends of the pandemic graph patterns, but they tended to overlook the overall patterns. The prediction generated at the unistrutural, multistructural, relational, and extended abstract levels was considered reasonable because of the graph or data provided. These findings indicated the existence of predictive reasoning conducted by students in making predictions of problems related to the COVID-19 graph. The insights gained into the prediction ability prompted teachers to enhance graphic literacy instruction, to equip students with the skills needed to thrive in the 21st century.

Causal inference is a central topic in education research, although oftentimes it relies on observational studies, which makes causal identification methodologically challenging. This manuscript introduces causal graphs as a powerful language for elucidating causal theories and an effective tool for causal identification analysis. It discusses graphical criteria for causal identification, which provide principled approaches for removing bias and assessing causal identification given a causal theory. Through illustrative examples, this manuscript demonstrates the application of causal graphs and adjustment criterion for covariate selection in the context of education research, exemplifying their key advantages particularly in scenarios where randomized experiments are impractical. This manuscript aims to acquaint researchers with causal graphs as an effective tool for causal inference, thereby facilitating theory-based causal inquiries in applied education research.

Nowadays, mathematics teachers in K-12 strive to promote their students' mathematical knowledge and computational thinking (CT) skills. There is an increasing need for effective CT-embedded mathematics learning material and a better understanding of students' views toward them. In this work, we present the results of a research study, which included the design of a six-lesson learning activity aimed at fostering 16- to 17-year-old secondary students' CT skills in calculus lessons using the dynamic mathematics software "GeoGebra." Our goal was to investigate how students experienced the CT-embedded calculus lessons with "GeoGebra" and what challenges they faced during their interaction with the learning material and software. We collected and analyzed data from students' code in "GeoGebra," workbooks, semi-structured interviews, and questionnaires. Our findings suggest that most students mastered using CT concepts in calculus activities to a satisfactory degree and could reason about their computational solutions using "GeoGebra" and the generated graphs. Students' understanding of the mathematical content knowledge introduced was essential to complete the lesson series successfully and unnoticed gaps in prior knowledge emerged. Our study shows that students appreciate the CT-embedded calculus lessons and "GeoGebra's" exploratory approach to mathematics problems when provided with appropriate support. We conclude that an integrated approach to mathematics education and CT is viable and can contribute not only to fostering CT but also to increasing interest in mathematics.

Can math concepts be experienced through the sensory modality of balance? Balance Board Math (BBM) is a set of pedagogical math activities designed to instantiate mathematical concepts through stimulation to the vestibular sense: an organ in the inner ear that detects our bodily balance and orientation. BBM establishes the different ways children spontaneously rock and move as the basis for inclusively exploring mathematical concepts together across diverse sensory profiles. I describe two activity sets where students explore focal concepts by shifting their balance on rockable balance boards: "the Balance Number Line," using analog materials to foster understandings of the number line and negative numbers, and "Balance Graphing," using sensors and a digital display to foster exploration of functions and graphing concepts, including the parameters of trigonometric functions and function addition. I outline proposed ways that engaging with concepts through balance-activating movement can change learners' mathematical thinking and learning.

This article describes an activity and resource from The New York Times that can be used to help learners cultivate critical statistical literacy. Critical statistical literacy involves understanding, interpreting, and questioning statistical information to make informed decisions (Casey et al., 2023; Franklin et al., 2015; Weiland, 2017). It is a vital skill that needs to be learned and reinforced with students early and often. The authors find What's Going On in This Graph? (WGOITGraph?), a collaboration between the American Statistical Association (ASA) and The New York Times Learning Network (NYT LN), to be a useful, accessible, effective online resource and share an example implementation.

Biology is perhaps the most complex of the sciences, given the incredible variety of chemical species that are interconnected in spatial and temporal pathways that are daunting to understand. Their interconnections lead to emergent properties such as memory, consciousness, and recognition of self and non-self. To understand how these interconnected reactions lead to cellular life characterized by activation, inhibition, regulation, homeostasis, and adaptation, computational analyses and simulations are essential, a fact recognized by the biological communities. At the same time, students struggle to understand and apply binding and kinetic analyses for the simplest reactions such as the irreversible first-order conversion of a single reactant to a product. This likely results from cognitive difficulties in combining structural, chemical, mathematical, and textual descriptions of binding and catalytic reactions. To help students better understand dynamic reactions and their analyses, we have introduced two kinds of interactive graphs and simulations into the online educational resource, Fundamentals of Biochemistry, a LibreText biochemistry book. One is available for simple binding and kinetic reactions. The other displays progress curves (concentrations vs. time) for simple reactions and complex metabolic and signal transduction pathways. Users can move sliders to change dissociation and kinetic constants as well as initial concentrations and see instantaneous changes in the graphs. They can also export data into a spreadsheet for further processing, such as producing derivative Lineweaver-Burk and traditional Michaelis-Menten graphs of initial velocity (v[subscript 0]) versus substrate concentration.

A growing body of literature suggests manipulating some graphing conventions can affect visual analysis and adherence to graphing conventions varies widely in single case design publications. Yet use of graphing conventions by the wider population of behavior analysts is largely unexplored. The current study examined relations among types of graphing conventions (established versus unestablished functions), demographic variables (e.g., education level, primary job setting), and likelihood of revision ratings reported by 631 board certified behavior analysts (BCBAs, BCaBAs, and BCBA-Ds). Statistically significant differences in ratings were detected across types of conventions and demographic categories. Master's level BCBAs in clinical roles were more likely to revise conventions that affect visual analysis than conventions whose functions are unknown, whereas there were no differences between types of conventions for doctoral level BCBAs in academic roles. We discuss these findings in the context of audience control of graphing conventions.

This study surveyed 183 college and senior high school students' graph-interpretation competence in scientific and daily contexts. Specifically, whether students' graph interpretation in scientific and daily contexts differed across educational levels was investigated. Furthermore, the questions of whether students' graph interpretation competence in scientific contexts might predict their graph interpretation in daily contexts were explored. Other factors including age and gender were also considered and examined to discern significant factor(s) predicting graph interpretation competence in daily contexts. Results indicated that there were statistically significant differences across educational levels in terms of graph interpretation competence in both scientific and daily contexts. Furthermore, the students' graph interpretation competence was related between scientific and daily contexts. The important factors predicting students' graph interpretation in daily contexts include age and graph-interpretation in scientific literacy. Implications are discussed.

This study investigated novice mathematics teachers participating in an online teacher education course focused on covariational reasoning and understanding the behavior of functions. The analysis centered on documenting the emergence of participants' sociomathematical norms for engaging in online asynchronous discussions. In this paper, we characterized participants' initial mathematical discourse and documented two emergent sociomathematical norms, namely "explaining why" and "emergent shape discourse." When participants "explained why," they used specific quantities or symbolic representations of functions to justify why function graphs have particular visual features. When participants engaged in "emergent shape discourse," they coordinated change between covarying quantities to justify why function graphs behave in certain ways. This study provides evidence that online settings can provide context for mathematics teachers engaging in legitimate collaborative mathematical activity and that activity can be enhanced by participation in discourse featuring specific sociomathematical norms. We discuss conjectures regarding the potential of reflective discussion activities paired with the Notice and Wonder Framework to support the emergence of generative sociomathematical norms. We also discuss potential relationships between characteristics of participants' mathematical discourse and their membership with the core and periphery of a social network.

Systematic reviews (SRs) have an important role in the healthcare decision-making practice. Assessing the overall confidence in the results of SRs using quality assessment tools, such as "A MeaSurement Tool to Assess Systematic Reviews 2" (AMSTAR 2), is crucial since not all SRs are conducted using the most rigorous methods. In this article, we introduce a free, open-source R package called "amstar2Vis" (https://github.com/bougioukas/amstar2Vis) that provides easy-to-use functions for presenting the critical appraisal of SRs, based on the items of AMSTAR 2 checklist. An illustrative example is outlined, describing the steps involved in creating a detailed table with the item ratings and the overall confidence ratings, generating a stacked bar plot that shows the distribution of ratings as percentages of SRs for each AMSTAR 2 item, and creating a "ggplot2" graph that shows the distribution of overall confidence ratings ("Critically Low," "Low," "Moderate," or "High"). We expect "amstar2Vis" to be useful for overview authors and methodologists who assess the quality of SRs with AMSTAR 2 checklist and facilitate the production of pertinent publication-ready tables and figures. Future research and applications could further investigate the functionality or potential improvements of our package.

The trace plot is seldom used in meta-analysis, yet it is a very informative plot. In this article, we define and illustrate what the trace plot is, and discuss why it is important. The Bayesian version of the plot combines the posterior density of [tau], the between-study standard deviation, and the shrunken estimates of the study effects as a function of [tau]. With a small or moderate number of studies, [tau] is not estimated with much precision, and parameter estimates and shrunken study effect estimates can vary widely depending on the correct value of [tau]. The trace plot allows visualization of the sensitivity to [tau] along with a plot that shows which values of [tau] are plausible and which are implausible. A comparable frequentist or empirical Bayes version provides similar results. The concepts are illustrated using examples in meta-analysis and meta-regression; implementation in R is facilitated in a Bayesian or frequentist framework using the bayesmeta and metafor packages, respectively.

It is widely recognized that we need to prepare students to think with data. This study investigates student interactions with digital data graphs and seeks to identify what might prompt them to shift toward using their graphs as thinking tools in the authentic activity of doing science. Drawing from video screencast data of three small groups engaged in sensor-based and computer simulation-based experiments in high school physics classes, exploratory qualitative methods are used to identify the student interactions with their graphs and what appeared to prompt shifts in those interactions. Analysis of the groups, one from a 9th grade class and two from 11th/12th grade combined classes, revealed that unexpected data patterns and graphical anomalies sometimes, but not always, preceded deeper engagement with the graphs. When shifts toward deeper engagement did occur, transcripts revealed that the students perceived the graphical patterns to be misaligned with the actions they had taken to produce those data. Misalignments between the physical, digital, and conceptual worlds of the investigations played an important role in these episodes, appearing to motivate students to revise either their experimental procedures or their conceptions of the phenomena being explored. If real-time graphs can help foster a sense in students that there should be alignments between their data production and data representations, it is suggested that pedagogy leverage this as a way to support deeper student engagement with graphs.

The nature of education has been transformed by technological advances and online learning platforms, providing educational institutions with more options than ever to thrive in a complex and competitive environment. However, they still face challenges such as academic underachievement, graduation delays, and student dropouts. Fortunately, by harnessing student data from institution databases and online platforms, it becomes possible to predict the academic performance of individual students at an early stage. In this study, we utilized knowledge graphs (KG), clustering, and machine learning (ML) techniques on data related to students in the College of Information Technology at UAEU. To construct knowledge graphs and visualize students' performance at various checkpoints, we employed Neo4j-a high-performance NoSQL graph database. The findings demonstrate that incorporating clustered knowledge graphs with machine learning reduces predictive errors, enhances classification accuracy, and effectively identifies students at risk of course failure. Additionally, the utilization of visualization methods facilitates communication and decision-making within educational institutions. The combination of KGs and ML empowers course instructors to rank students and provide personalized learning interventions based on individual performance and capabilities, allowing them to develop tailored remedial actions for at-risk students according to their unique profiles.

This is an exploratory study considering instructional methods in mathematics education. The study examined misconceptions arising from selected assignments undertaken by sixty-five students. The students were attending a calculus mathematical course unit on limits of functions, in different academic areas offered at selected tertiary institutions in Western Uganda. In this study, forms of expressing ideas on limits of functions are considered as far as they facilitate access to underlying mathematical principles. We explored the use and application of tools for representing mathematical ideas to enhance students' conceptual understanding and problem solving skills. A small-scale pilot of interventions using readily available GeoGebra dynamical computer software was applied. The diagnostic assignment on limits of functions was used for data collection. The main objective was to examine whether or not students preferred the application of multiple representations to the analytic approach. The semi-structured interview protocol was also conducted to probe further and correlate students' responses and their problem-solving abilities. The results showed that multiple representations-based instructions significantly changed students' understanding of the limits of functions. The results promise better access and understanding of more abstract mathematical concepts and may support students' problem-solving abilities. The semi-structured interviews conducted indicated that multiple representations supported and, therefore, enhanced students' understanding and solving of the limits of functions. This study highly recommends that mathematics educators should adapt multiple representations-based instructions to enhance students' critical thinking, and problem-solving.

The study of loops and spaces in mathematics has been the subject of much interest among researchers. In Part 1 of "The Theory on Loops and Spaces," published in the "Mathematics Teaching Research Journal," introduced the concept and the basic underlying idea of this theory. This article continues the exploration of this topic and aims to advance the understanding of the theory through observation and analysis of patterns. A systematic examination of intersection points and their numerical Sum is conducted, and the effect of the order of numbering on the analysis is analyzed. Furthermore, the physical implications of the theory are discussed, and the validity of the theory in the third dimension is confirmed through analysis. This article provides a solid foundation for the understanding of the elementary principles of Graph Theory and paves the way for the development of more advanced theorems in the field. Additionally, the article demonstrates how patterns in nature can be analyzed and expressed mathematically, offering a unique perspective on the interplay between mathematics and the natural world. The work is inspired by the video posted by mathematician Dr. James Tanton on his YouTube channel on September 27th, 2021. [For "The Theory on Loops and Spaces. Part 1," see EJ1350656.]

Several studies compared three different types of questions (conceptual, algorithmic, and graphical) across various topics, however, few focused specifically on gifted students. This study addressed this gap. The aim of the study, hence, was to determine whether there were notable differences in gifted students' performance in the three types of tests. The study involved 115 gifted students aged between 17 and 18 years old. They responded to the three different tests including 10 test items in each with the same content. Significant differences were observed in their performances between the conceptual versus algorithmic in favour of the conceptual test, the conceptual versus graphical in favour of the conceptual test, and the algorithmic versus graphical in favour of the algorithmic test. Moreover, the statistical analysis results revealed that the students performed markedly poorer performance on the graphical test in comparison to both the algorithmic and conceptual tests. These results suggest the need to enhance students' graphical skills to facilitate a better understanding of physics concepts. Proper steps should be taken to improve their proficiency in interpreting and analysing graphical representations.

A review of studies analyzing the statistical knowledge of primary schoolchildren (6-12 years old) is carried out. Based on a review in JCR/SSCI, Scopus, Eric, Google Scholar, Science Direct, World Scientific, Springer, and Wiley Online library, 18 articles (2003-2021) have been identified and analyzed based on two objectives: (1) to identify the different study approaches and (2) to analyze the elements of statistical knowledge. The results show that almost half of the investigations were carried out based on one of the following approaches: the Toulmin approach (TM), the statistical mathematical working space (SMWS), the structure of observed learning outcomes (SOLO) taxonomy and Curcio's graph reading levels (CGRL). It is concluded that CGRL is the most common approach and statistical graphs are the most analyzed statistical objects.

This study aims to determine the graphical literacy levels of middle school mathematics teachers. The participants of the study consisted of 33 middle school mathematics teachers working in different provinces. The data of the study were collected with the help of the graphical literacy test created by the researchers. The data were analyzed by content analysis method. As a result of the study, it was seen that the teachers were more successful in the questions that required reading the data in the graph. It has been revealed that teachers are unsuccessful in reading beyond data questions that require higher-level thinking. In addition, the teachers had the most difficulty creating the graphs. In other words, it was determined that the graphical literacy levels of the teachers were low. Therefore, it is thought that studies to improve the graphical literacy skills of teachers in in-service training programs will be important.

The present study explored 285 11th-grade students' preconceptions, misconceptions, and errors in solving mathematics tasks by graphical method. A descriptive-explorative study design was adopted. Cluster sampling was used to select students from sampled secondary schools in eastern and central Uganda. Students' paper and pen solution sketches together with the task-based interviews were used to identify students' preconceptions, misconceptions, and errors in linear programming (LP). Students' responses were analyzed thematically and interpreted as students' learning gaps. The results indicated that students lacked proficiency in relating basic algebraic concepts and procedures to the mathematical language with which LP is conveyed. Generally, most students could not adequately use their previous knowledge and connect it to the learning and solving of LP tasks. Besides applying wrong mathematical algebraic concepts, students had difficulties interpreting and writing correct models (inequalities) from LP word problems. Misconceptions and errors were common and peculiar to and in individual student's solution sketches especially in applying the concepts of equations and inequalities to graphically solve and optimize LP tasks. Students held extremely weak concept images of graphing equations, and inequalities and their linkage to optimizing feasible regions. This research provides insight into the learning of mathematics word problems(LP) and recommends that mathematics educators should effectively apply students' preconceptions, misconceptions, and errors as opportunities for enhancing students' LP conceptual changes. For mathematical proficiency, suitable learning approaches, methods, and strategies should be adapted to address specific individual student's flawed conceptual and procedural knowledge and understanding. These approaches may guide educators in helping students to construct the connections between the old and the new knowledge.

Official communications and media news in Greece about the COVID-19 pandemic very often involve a multitude of numerical concepts and mathematical relationships, which are presented in different forms and in various contexts. This paper reports selected findings of a survey among adults, which had two aims. First, to investigate their subjective assessments concerning the difficulties encountered in reading and understanding such numerical information, embedded in texts and presented by infographics and time-series graphs. And second, to investigate their assessments of the usefulness of such numerical information, presented in visual forms. A convenience sample of 310 adults responded to an online questionnaire. The main conclusion from the analysis of these data is that a considerable number of the participants, although relatively highly educated, assess themselves as having difficulties in reading and understanding numerical information concerning COVID-19. Namely, about 9% report difficulties with information included in textual forms, while regarding visual presentations the percentage ranges from 10% to 35%. Yet more than half of the adults surveyed assess the presentation of numerical information in visual forms as useful for comprehension. Furthermore, the distributions by gender, education, and age of participants' answers to our questionnaire generally show no substantial differences among them. These findings advise caution regarding communications or announcements that involve even simple numerical concepts and mathematical relationships concerning COVID-19 and health issues.

The study investigated the effects of a combination of Peer Discussion and Isomorphic Problems (PD-IPs) on high school students' graph understanding and conceptual learning. The study was conducted on kinematics and Newton's Laws of Motion with 67 high school students from two groups. The experimental group (EG) consisted of 34 students, while the control group (CG) combined with 33 students. The students in EG were taught using the PD-IPs approach, while those in CG were taught using the traditional method. The research data were collected using the Force Motion Achievement Test (FMAT), which was composed by selecting multiple-choice questions from some standardized tests (Force Concept Inventory, Mechanic Baseline Test, Test of Understanding Graphs in Kinematics). The FMAT includes 25 multiple-choice questions. 11 were related to graph understanding, and the rest were related to conceptual learning. The research showed that the PD-IPs approach had a more positive effect on students' understanding of graphs and conceptual learning than the traditional teaching method. The findings indicate that implementing the PD-IPs approach requires minimal effort while demonstrating the potential to evaluate and improve high school student's academic performance in physics education.

Graphic interpretation is as critical in physics education as problem-solving. However, we know that today's classes focus more on problem-solving. This study uses a survey to determine college students' graphic interpretation skills. The study consists of two phases. The first phase includes the development and statistical analysis of the survey. The second phase includes comparing and discussing the data resulting from the application of the developed survey. The research data were analyzed using both exploratory factor analysis and confirmatory factor analysis techniques. The survey on graphic interpretation skills, including the understanding and analysis processes, consisted of 17 items based on analysis results. The survey data were collected using purposive sampling from 113 college volunteers during the fall semester of 2022-2023 at Dokuz Eylul University in Turkey. The participants consisted of 57 geoscience students and 56 mining students. The survey results showed that the kinematic interpretation skills of mining engineering students were higher than those of geoscience students. These differences between geoscience and mining engineering students in cognitive, affective, and psychomotor behaviors were discussed.

"Using Sentiment Analysis to Ease Students toward or around Macroanalysis" introduces teachers to the mathematical analytical method currently being promoted by Matthew Jockers which he calls macroanalysis. This method uses an algorithm called "R," originally designed by researchers in sociology. In many ways, macroanalysis presents an approach to literary analysis that might seem to many traditionalists to run counter to our training as close readers or advocates of specific literary theories. Instead of focusing on specific elements in a limited number of texts, macroanalysis uses a computer to consume a huge number of literary works--more than a single human being could read in a lifetime--and collect a statistically significant amount of data, a search which has been programmed into the algorithm by the researcher. This use of a computer to do the reading for the researcher results in a new methodology which Jockers calls "distant reading" (a term he borrows from Franco Moretti). After providing an introductory discussion of macroanalysis, I turn my focus onto making a case for the advantages of using graphs to aid in classroom discussions of reading assignments. In the introductory discussion, I point out that one serious advantage that arises from Jocker's macroanalysis is how the "R" algorithm produces a variety of graphs that provide visual reinforcement for the statistical data it generates. Consequently, in the second section of the essay, I provide a justification for why it would be advantageous in a college or even high school discussion of a reading assignment to privilege visual presentations of student responses to the assignment. Once I've established the nature of macroanalysis, and how challenging it is for the average English major, I offer an alternative approach to creating graphs to clarify student responses. This alternative approach is a blend of "sentiment analysis" and reader-response theory. Essentially this approach can be used to ease students toward the more serious graphing features of macroanalysis, or it can function as a self-contained form of analysis without shifting to the difficult mathematics characteristic of macroanalysis. To prepare the readers for the rather lengthy discussion of a graph produced by a group of Virginia Governor's School students, I first provide a simple graph that shows my responses to the opening scene of Flannery O'Connor's short story "A Good Man Is Hard to Find." Having clarified how sentiment analysis, with the mechanical help of Microsoft Excel, can produce an engaging graph of emotional and intellectual responses that can be easily shared with the class, I then move to the last half of my essay--a detailed discussion of an actual student graph derived from a reading of Helen Klein Ross's "What Was Mine" which was much more sophisticated than the graph I created for the O'Connor story. Even though the students' graph was overly ambitious, the discussion of its visual dynamics demonstrate how even a faulty graph can teach students how to become close readers and appreciate the importance of analysis that is based on specific elements of the literature being analyzed.

Eye tracking technology enables the visualisation of a problem solver's eye movement while working on a problem. The eye movement of experts has been used to draw attention to expert problem solving processes in a bid to teach procedural skills to learners. Such affordances appear as eye movement modelling examples (EMME) in the literature. This work intends to further this line of work by suggesting how eye gaze data can not only guide attention but also scaffold learning through constructive engagement with the problem solving process of another human. Inferring the models' problem solving process, be it that of an expert or novice, from their eye gaze display would require a learner to make interpretations that are rooted in the knowledge elements relevant to such problem solving. Such tasks, if designed properly, are expected to probe or foster a deeper understanding of a topic as their solutions would require not only following the expert gaze to learn a particular skill, but also interpreting the solution process as evident from the gaze pattern of an expert or even of a novice. This position paper presents a case for such tasks, which we call eye gaze interpretation (EGI) tasks. We start with the theoretical background of these tasks, followed by a conceptual example and representation to elucidate the concept of EGI tasks. Thereafter, we discuss design considerations and pedagogical affordances, using a domain-specific (chemistry) spectral graph problem. Finally, we explore the possibilities and constraints of EGI tasks in various fields that require visual representations for problem solving.

Many students persistently misinterpret histograms. Literature suggests that having students solve dotplot items may prepare for interpreting histograms, as interpreting dotplots can help students realize that the statistical variable is presented on the horizontal axis. In this study, we explore a special case of this suggestion, namely, how students' histogram interpretations alter during an assessment. The research question is: In what way do secondary school students' histogram interpretations change after solving dotplot items? Two histogram items were solved before solving dotplot items and two after. Students were asked to estimate or compare arithmetic means. Students' gaze data, answers, and cued retrospective verbal reports were collected. We used students' gaze data on four histogram items as inputs for a machine learning algorithm (MLA; random forest). Results show that the MLA can quite accurately classify whether students' gaze data belonged to an item solved before or after the dotplot items. Moreover, the direction (e.g., almost vertical) and length of students' saccades were different on the before and after items. These changes can indicate a change in strategies. A plausible explanation is that solving dotplot items creates readiness for learning and that reflecting on the solution strategy during recall then brings new insights. This study has implications for assessments and homework. Novel in the study is its use of spatial gaze data and its use of an MLA for finding differences in gazes that are relevant for changes in students' task-specific strategies.

Studying students' performance in learning differentiability is essential since it requires them to perform proceptual thinking. This case study research investigates students' proceptual thinking outcomes in learning the differentiability concept using Desmos Classroom Activities (DCA) based on the Three Worlds of Mathematics (TWM) framework. In this study, DCA facilitates students' learning of differentiability with a sensible approach and exposes them to the graphical exploration of non-differentiable functions. Students succeeded in concluding that continuity does not imply differentiability. With its graphical, computational, and symbol manipulation capabilities, DCA has facilitated students to perform proceptual thinking in learning differentiability through a richer learning experience. Many students (76%) could perform at the procept level when solving the differentiability problem. Also, twenty-two students (about 88%) had no limitations in the graphical representation. The result of this study confirms that DCA based on the TWM as a generic organizer can contribute to students' proceptual thinking outcomes in learning differentiability.

The Modular Object-Oriented Dynamic Learning Environment (Moodle), a type of Learning Management System (LMS), is growing globally. Across the world, the student population keeps growing, making it challenging for the existing physical infrastructure to accommodate all the students. Despite the enormous potential significance of Analytics Graphs, there needs to be more evidence regarding the utilisation of this Learning Analytics tool to significantly inform management on students' tracking capabilities and consequently improve their performance. This study demonstrates how Learning Analytics tools such as Analytics Graphs enable teachers and management to examine learners' activities in the LMS (e.g., Moodle) by utilising data from the system log to address the existing gap. This study focused on two fundamental questions. What is the usability of the Learning Management System Moodle through Analytics Graphs in the University of Applied Sciences context? Two, how can the Learning Management System Moodle through Analytics Graphs be adapted by educators and maximise its potential? The study employed a quantitative approach and tracked the available log data generated during teaching a marketing course at the University of Applied Sciences for five months (January to May 2022). The analytics graph plugin helps identify student profiles that enables the teachers to know the actual state of the students based on their responses to the activities. This study generates some interesting managerial implications for teachers, students, institutions (Universities) and the e-learning designer or Moodle administrator.

Multiple-choice questions (MCQs) are widely used in educational assessments and professional certification exams. Managing large repositories of MCQs, however, poses several challenges due to the high volume of questions and the need to maintain their quality and relevance over time. One of these challenges is the presence of questions that duplicate concepts but are formulated differently. Such questions can indeed elude syntactic controls but provide no added value to the repository. In this paper, we focus on this specific challenge and propose a workflow for the discovery and management of potential duplicate questions in large MCQ repositories. Overall, the workflow comprises three main steps: MCQ preprocessing, similarity computation, and finally a graph-based exploration and analysis of the obtained similarity values. For the preprocessing phase, we consider three main strategies: (i) removing the list of candidate answers from each question, (ii) augmenting each question with the correct answer, or (iii) augmenting each question with all candidate answers. Then, we use deep learning-based natural language processing (NLP) techniques, based on the Transformers architecture, to compute similarities between MCQs based on semantics. Finally, we propose a new approach to graph exploration based on graph communities to analyze the similarities and relationships between MCQs in the graph. We illustrate the approach with a case study of the Competenze Digitali program, a large-scale assessment project by the Italian government.

In course design, topic outline organization encompasses the structuring and sequencing of topics to be delivered in a learning environment. Recent studies in topic outline optimization revolve around massive open online courses (MOOCs) due to their abundance but not much has been studied on the traditional courses. This study investigates the organization of topic outlines in traditional introductory programming courses across Philippine higher education institutions (HEIs) and evaluates the viability of the synthesized alternative topic outlines. Course syllabi were collected from 16 HEIs. A topic precedence graph (TPG) model that provides a structured overview of the introductory programming was created via a semi-supervised pairwise constrained k-means (PCK-Means) clustering to structure the topics which produced 20 topic clusters with strong topic cohesion within the clusters. The TPG showed that HEIs tend to start the outline similarly, followed by core programming topics with varied sequences, and divergent ways of ending the outline. Two anomaly clusters were identified as having topic titles grouped that do not seem to have a unifying topic. Limitations of the clustering algorithm are identified where it cannot identify semantic meaning between words which may affect its applicability in situations where topic titles are named inconsistently. From the TPG, alternative optimal and comprehensive topic outlines were synthesized via greedy and DFS graph traversal algorithms. However, these alternative outlines performed very poorly when compared with the evaluators' (n=19) arrangement of topic outlines due to some prerequisite topics being discussed in the latter part already. Overall, this study introduces a method to incorporate computer science technologies in structuring topics across HEIs and aiding educators in topic outline design but more research is needed before it can be implemented in a real classroom setting.

Many students who enroll in a first course in linear algebra major in STEM disciplines other than mathematics. Teachers who serve such students may find it difficult to provide authentic problems from these broader areas that ignite students' interest in linear algebra. In this paper, we highlight an interdisciplinary learning activity that engages students in using linear systems of equations to model the behavior of practical electric circuits. This exercise fits nicely into standard introductory linear algebra curricula and is designed to excite students majoring in engineering, physics, or applied mathematics. We also include references to a collection of open-access resources to support instructors who want to use this material in project-based, flipped-learning, inquiry-oriented, or independent-study environments.

This article presents a reflection on a teaching experience involving the use of the Brügner tangram, an interesting but little-known manipulative material. It details an activity conducted as part of an undergraduate mathematics education course for prospective primary school teachers. The main objective of this paper is to present the implementation of a sequence of activities designed to convey to future teachers the importance of systematically solving problems that involve searching for or constructing different cases. Specifically, participants are provided with the three pieces of the Brügner's tangram and assigned the task of identifying all possible convex polygons that can be constructed. Moreover, they are required to elucidate the methodological approach underpinning their exploratory process, with an emphasis on establishing relationships between the polygons. Graphs are introduced as one of the possible approaches for modelling the problem, offering a graphical representation that aids in the systematic search for solutions. This paper describes different activities involving the Brügner's tangram which has demonstrated its adaptability as an instructional resource. The teaching sequence adheres to the five-phase structure proposed within the framework of the van Hiele model, which is also part of the course.

Given the extensive use of online learning in educational settings, Knowledge Tracing (KT) is becoming increasingly essential. KT primarily aims to predict a student's future knowledge acquisition based on their past learning activities, thus enhancing the efficiency of student learning. However, the effective acquisition of dynamic and evolving student representations from their historical records presents a formidable challenge. This paper introduces a Knowledge Tracing methodology predicated on Dynamic Breadth Graph Convolutional Networks (DBGCN). DBGCN leverages the mechanisms of breadth graph convolutional networks to proficiently acquire representations of questions and knowledge points from dynamically constructed topological graphs. It employs student state information as an attention query vector to augment student representations, thereby partially mitigating the challenge of capturing the dynamic shifts in user states. The effectiveness of our proposed DBGCN method has been demonstrated through extensive experimentation.

This study examined college calculus instructors' preferences in solving two calculus tasks to examine college calculus instructors' use of important cognitive roots in understanding derivatives of function. Our results showed that only one instructor consistently uses cognitive roots while other instructors either focus on algebraic methods or expressing the constraints of students' knowledge, demonstrating their orientations and conflicts. These results potentially show what students experience when differentiation of function is taught.

Enzymes are nature's catalysts, mediating chemical processes in living systems. The study of enzyme function and mechanism includes defining the maximum catalytic rate and affinity for substrate/s (among other factors), referred to as enzyme kinetics. Enzyme kinetics is a staple of biochemistry curricula and other disciplines, from molecular and cellular biology to pharmacology. However, because enzyme kinetics involves concepts rarely employed in other areas of biology, it can be challenging for students and researchers. Traditional graphical analysis was replaced by computational analysis, requiring another skill not core to many life sciences curricula. Computational analysis can be time-consuming and difficult in free software (e.g., R) or require costly software (e.g., GraphPad Prism). We present Enzyme Kinetics Analysis (EKA), a web-tool to augment teaching and learning and streamline EKA. EKA is an interactive and free tool for analyzing enzyme kinetic data and improving student learning through simulation, built using R and RStudio's ShinyApps. EKA provides kinetic models (Michaelis-Menten, Hill, simple reversible inhibition models, ternary-complex, and ping-pong) for users to fit experimental data, providing graphical results and statistics. Additionally, EKA enables users to input parameters and create data and graphs, to visualize changes to parameters (e.g., K[subscript M] or number of measurements). This function is designed for students learning kinetics but also for researchers to design experiments. EKA (enzyme-kinetics.shinyapps.io/enzkinet_webpage/) provides a simple, interactive interface for teachers, students, and researchers to explore enzyme kinetics. It gives researchers the ability to design experiments and analyze data without specific software requirements.