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In this paper duality between real world phenomenon and mathematics will be discussed. This duality exists for many years which considerably remarked in the history of mathematics curriculum. One of the good potential for filling the gap between real world phenomenon and mathematical ideas would be modelling tasks which require performing the modelling cycle. In this paper after discussion about modelling in more details, two modelling cycles will be elaborated in the context of mathematical tasks that are related to everyday life. Main ideas behind different modelling cycles is starting point that would be from real world situation. Then gradually mathematical elements of real world phenomenon identified and mathematics problem will have shaped in a form that could be solved through mathematical problem solving techniques. Furthermore, it would be important phase that call interpretation of mathematics answer and check it in front of real world situation. This paper will be followed by discussion about learning theories that support the idea of modelling. Finally, two educational challenges (Design good modelling tasks and assessing of students performance) in mathematical modelling activity will be discussed. [For the complete proceedings, see ED655360.]

This paper explores use of a set of diagrammatic tools for representation and analysis of the moves a teacher makes implementing a mathematical modelling task. The focus here is on identifying what the teacher did so we can subsequently interrogate this, as to the why. Data include pre and post lesson teacher interviews and transcripts of a video and audio-recorded task implementation. The analytical tools developed, with one teacher and one task early in a three-year project were particularly useful in ascertaining what the teacher moves were as we subsequently sought to determine reasons for these.

The context for this paper is the "Synthetic Training Environment Experiential Learning -- Readiness" (STEEL-R) project [1], which aims to estimate individual and team competence using data collected from synthetic, semi-synthetic, and live scenario-based training exercises. In STEEL-R, the "Generalized Intelligent Framework for Tutoring" (GIFT) orchestrates scenario sessions and reports data as experience API (xAPI)statements. These statements are translated into assertions about individual and team competencies by the "Competency and Skills System" (CaSS). Mathematical models use these assertions to estimate the competency states of trainees. This information is displayed in a dashboard that enables users to explore progression over time and informs decisions concerning advancement to the next training phase and which skills to address. To test, tune, and demo STEEL-R, more data was needed than was available from real-world training exercises. Since the raw data used to estimate competencies are captured in xAPI statements, a component called DATASIM was added. DATASIM simulated training sessions by generating xAPI statements that conformed to a STEEL-R "xAPI Profile." This facilitated testing of STEEL-R and was used to create a demo that highlighted the ability to map data from multiple training systems to a single competency framework and to generate a display that team leaders can use to personalize and optimize training across multiple training modalities. This paper gives an overview of STEEL-R, its architecture, and the features that enabled the use of artificial data. The paper explains how xAPI statements are converted to assertions and how these are used to estimate trainee competency. This is followed by a section on xAPI Profiles and on the xAPI Profile used in STEEL-R. The paper then discusses how artificial data were generated and the challenges of modeling longitudinal development and team in these data. The paper ends with a section on future research. [For the full proceedings, see ED623995.]

This paper draws on numerous data sources to better understand the shift from additive to multiplicative thinking in years 4 to 9. Research studies that have used the Scaffolding Numeracy in the Middle Years assessment tasks have found that while students can be supported to move through the early and upper zones of the Learning and Assessment Framework for multiplicative thinking, it has been difficult to move students through Zone 4 at the same rate. A closer examination of item responses at this level reveal that a disposition to notice and work with relationships between quantities may explain this phenomenon.

Competency with mathematical modelling is increasingly important for career and informed and engaged participation in personal, civic and work life. In this paper we report on an aspect of a three-year longitudinal study that aimed to identify and describe enablers of mathematical modelling. Teacher interview data has been drawn upon to exemplify key features of a typology for instructional enablers of mathematical modelling. Findings highlight the importance of the didactical contract and socio-mathematical norms in promoting students' mathematical modelling competency, as well as teachers' anticipatory capabilities.

In this paper, we present the development of an investigation on the promotion of covariational reasoning in high school students (14-15 years old) in Mexico. The study consists of designing and applying a sequence of didactic activities that simulate a real situation virtually. The activities are organized through a Hypothetical Learning Trajectory supported by digital technology and elements of Cuevas-Pluvinage didactics. The activities were evaluated according to the levels of covariation proposed by Carlson and colleagues, categorizing students' achievements and difficulties for each level of understanding. The results show that the activities favor students' progress by moving from the context situation to the different representations, establishing the relationship between the variables, and identifying their functional dependence. [For the complete proceedings, see ED630210.]

This theoretical paper describes how Community-based Mathematical Modeling can advance equity and cultivate civic empathy in elementary school settings. We provide a framework for community-based mathematical modeling instruction consisting of five goals: facilitating connections, fostering engagement, promoting rigor, cultivating civic empathy, and elevating justice. We illustrate how these goals work together to advance equity and cultivate civic empathy through classroom vignettes of community-based modeling lessons. Through this theoretical synthesis, implications for community-based mathematical modeling instruction will be discussed. [For the complete proceedings, see ED630210.]

The teaching quality evaluation of college teachers is an important indicator of school evaluation, performance assessment and teacher promotion. However, the existing evaluation methods are only limited to the formal evaluation of teaching behavior, and have not achieved objective, reasonable and scientific evaluation results. The purpose of this paper is to propose a [delta]S equation of the state of the teaching system from the point of view of the teaching system's operation process. On this basis, a Markov process evaluation method for teachers' teaching quality is established. The research results show that the application of the [delta]S equation of the teaching system to the evaluation of teaching behavior and teaching effect has its own characteristics, and the key lies in the reasonable selection of the [delta]S equation and the the combination of Markov analysis can effectively achieve the evaluation goal of teachers' teaching quality. [For the full proceedings, see ED638044.]

The integration of innovative interdisciplinary approaches to the ?-12 Curriculum aims to deepen students' knowledge and help them develop transversal skills. Applying an interdisciplinary lens, with a focus on pandemics, can help shape attitudes by means of inculcating the values of responsible global citizenship, and a high sense of personal and social responsibility. In the midst of the current lock-down due to Covid-19, a teaching approach to pandemics is presented, as an interdisciplinary connection between history and mathematics, based on the methodology of modeling in mathematics and the humanities. The main research questions posed: a) can modeling, as analyzed in the scientific literature, be used to interpret pandemics, e.g. in the case of the 'plague of Athens' (430 B.C.), as analyzed in the primary sources? b) Can the mathematical tools of statistical analysis be used to understand prevention measures through the centuries? Taking a longue durée perspective on history, students were asked to work on additional cases of pandemics across time and space, mobilizing both their mathematical and historical knowledge: process numerical data from primary sources, study maps, combine and compare elements of the past and the present using mathematical epidemiological models and real numerical data to study and predict the spread of infectious diseases. This paper presents the assessment of the effectiveness of this approach, conducted by means of closed and open questionnaires, administered in two phases (pre- and post-teaching) to a sample of 40 students aged 16-17 years. The results highlight statistics as a key tool for understanding real-world situations, and record the strengthening of students' knowledge in history, the raising of their critical thinking skills, as well as their enhanced ability to tackle real-world problems and understand responsible decision-making processes. Finally, the paper suggests that such good practices can prepare students for the complexity of globalized knowledge. [For the full proceedings, see ED626375.]

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.]

The role of mathematical modelling pertains several disciplines, both STEM and non-STEM, and various fields: education, academy, work, everyday and social life. Despite its importance, it is not uncommon to see university students facing difficulties with the use of Mathematics to create models, even when mathematical entities that play a role in facing a problem belong to the study programs of secondary schools, and should thus be familiar also to students without a specific background in Mathematics. Difficulties can arise in various phases of modelling: in the comprehension of the problem, in the translation into mathematical formulas, in the resolution process or even in the interpretation of the results. In this paper, we give an analysis of an online test taken by 75 non-STEM students. The 10 questions of the test focused on specific items in mathematical modelling. During the test, students had to write down the reason why they chose a specific answer. The test allowed us to find and categorize the common errors students make and the phase in which it happens, suggesting actions in order to prevent them. Results show percentages of errors and discuss students' arguments. [For the full proceedings, see ED621892.]

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

In this theoretical paper we compare the Piagetian perspective on knowledge construction to mathematical model construction, with the aim to understand how mathematical modeling enables learning of mathematics and learning of science, as is often claimed. We do this by examining data through two lenses: (1) examining the role of cognitive conflict as it arises during validation of a model; and (2) viewing model validation as a reflection on activity-effect relationship. We explain why we chose to look deeply into model validation specifically, present examples for each lens, and consider implications. [For the complete proceedings, see ED629884.]

The main purpose of the current study is modifying students' daily drinking water and managing waste water through modelling activities as context and modelling cycle as tool. In this study, 12 students (13-14 years old) from the lower secondary school were participated. These students were participated in two sessions (60 minutes) each week for three months. Three modelling activities had been designed for this study. These modelling activities were related to the amount of water consumption. Classroom activities were video recorded and used as data in this study. In some cases, interviews conducted and used as complementary data. Tension which happened during the implementation of these two activities analyzed with activity system as a theoretical framework. The results of this study show that students were able to manage and modify their daily water consumption through doing modelling activities. Results of this study also show that power of mathematics could help students to solve their real-world problems and help them to have better life. [For the full proceedings, see ED630948.]

The development of modeling skills in mathematics is essential for individuals to understand, describe, control, and predict phenomena around them. This article describes the results of an investigation to find out how an activity -- based on a Models and Modeling Perspective -- stimulates the mathematics modeling skills of undergraduate students who are in the first quarter of a business degree. As a result, it was shown that the MEA [model-eliciting activity] enabled students to exhibit, develop, and refine different modeling skills, such as: identification of variables, assumptions based on the real-life context, identification of patterns, and construction of mathematical representations. [For the complete proceedings, see ED630210.]

Literature typically describes mathematization, the process of transforming a real-world situation into a mathematical model, in terms of desirable actions and behaviors students exhibit. We attended to STEM undergraduate students' quantitative reasoning as they derived equations. Analysis of the meanings they held for arithmetic operations (+, -, ·, ÷) provided insight into how participants expressed real-world relationships among entities with arithmetic relationships among values. We extend the findings from K-12 literature (e.g., using multiplication to instantiate a rate) to STEM undergraduates and found evidence of new ways of justifying the usage of arithmetic operations (e.g., using multiplication to instantiate an amount). [For the complete proceedings, see ED630210.]

Using data from teaching experiments and theories from quantitative reasoning, we built second-order accounts of students' mathematics with regards to how they conceived rate of change through operating on existing quantities. In this report, we explain three different ways STEM undergraduates structurally conceive rate of change as they constructed mathematical models for real-world scenarios. [For the complete proceedings, see ED630210.]

Existing studies have defined and assessed disciplinary literacy, mathematical literacy, and general academic language. However, there is a need to define and assess mathematics discipline-specific language (MDL), particularly for elementary school teachers. Therefore, the purpose of this study was to develop a research instrument to assess the MDL of elementary school teachers. The final instrument developed through iterative analysis included 20 items on a 4-point Likert-like scale distributed between three distinct MDL categories: technical, symbolic, and visual. Instrument validity was confirmed using Confirmatory Factor Analysis with the set of 211 video recordings and corresponding lesson plans of mathematics lessons taught by pre-service elementary school teachers enrolled in a graduate special education program. [For the complete proceedings, see ED630210.]

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

Throughout the years, Mathematical Modeling has gained international attention, not only in research but also in the development of curricula and its applications in the classroom. However, systematic literary surveys are scarce. In this paper, we present some findings from a survey of 485 international publications related to different aspects of mathematical modeling. While the number of Latin-American publications is rather small compared to the number of international publications, the vitality of the currently discussed themes in this region and their innovative perspectives testify to the international relevance of this developed work on mathematical modeling. [This paper is provided in both Spanish and English. For the complete proceedings, see ED606531.]

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.]

The professional development of teachers is a task recognized by the European parliament and pursued by many institutions, both national and international ones. The support to teachers in STEM disciplines was one of the aims of the Erasmus+ SMART (Science and Mathematics Advanced Research for good Teaching) project, born in a European context, which developed as its main intellectual output two open online courses called Mathematical Modelling and Observing, Measuring and Modelling in Science, in order to help teachers in their continuous professional development. Both courses contain interactive problem-based materials ready to be downloaded, modified, used in the classroom and redistributed to the community. This paper gives an overview of the structure of the course Mathematical Modelling and analyzes its instructional quality, taking into account some measurables obtained in two years of operativity. [For the full proceedings, see ED621557.]

This study aims to explore seventh-grade students' understanding of negative integers as they engaged in mathematics history integrated model-eliciting activities in small groups. For this educational case study, we designed model-eliciting activities based on six design principles of the models-and-modeling perspective that incorporated history of negative integers. Both written data and video records of students were analyzed to elicit the facets of their models of negative integers. We found that students' thought that either daily life contexts or people's need drove the invention of negative integers. The findings also indicated students' reasoning on the evolvement of mathematics ideas by contribution of different culture, revealing the role of math history integration into the modeling process. In this sense, our study presents a unique approach in modeling literature. [For the complete proceedings, see ED630060.]

This article describes the results of an investigation based on a Models and Modeling Perspective [MMP]. We present the evolution of the models built by university students when solving a model development sequence designed to promote their learning of the exponential function. As a result, we observed that students' thinking was modified, expanded, and refined, as they developed different iterations of their models. Students' models evolved by creating, first, linear models that required direction; second, models where there was no dissociation between linear and exponential behavior; then, situated exponential models; and finally, sharable, and reusable exponential models. [For the complete proceedings, see ED630060.]

One promising approach for connecting undergraduate content coursework to secondary teaching is using teacher-created representations of practice. Using these representations effectively requires seeing teachers' use of mathematical knowledge in and for teaching (MKT). We argue that Rowland's (2013) Knowledge Quartet for MKT, in particular, the dimensions of Foundation and Contingency, is a fruitful conceptual framework for this purpose. We showcase an analytic framework derived from Rowland's work and our analysis of 85 representations of practice. These representations all featured geometry. We illustrate examples of combinations of "high" and "low" Foundation and Contingency, and show results of coding juxtaposed with performance on an instrument previously validated to measure MKT. We describe the potential for generalizing this framework to other domains, such as algebra and mathematical modeling. [For the complete proceedings, see ED630060.]

The purpose of this paper is to describe the processes utilised to develop an online learning module within the Opening Real Science (ORS) project--"Modelling the present: Predicting the future." The module was realised through an interdisciplinary collaboration, among mathematicians, scientists and mathematics and science educators that drew on the enquiry-based approach underpinning ORS as well as structuring devices and working practices that emerged during the course of the module development. The paper is a precursor to further research that will investigate the effectiveness of the module in terms of students' learning and attitudes as well as the module development team members' perspectives on the interdisciplinary collaboration that took place.

These proceedings are a written record of the research presented at the 42nd annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (PME-NA) held in Mazatlán, Mexico, virtually beginning May 27, 2021 and in-person June 2-6, 2021. The conference was originally scheduled to take place October 14-18, 2020; it was postponed, due to the COVID pandemic. In accordance with the major goals of PME-NA that include promoting international contacts and stimulating interdisciplinary research, the theme was proposed as "Entre Culturas/Across Cultures". The goal of the theme was an academic exchange that would reflect the ample diversity of ways of teaching and learning of mathematics, and of the tools and communities involved in mathematics education; as well as an exploration of how the differences in cultures imply a need to consider how research results can be taken into account in varying contexts. The papers comprise 80 Research Reports, 190 Brief Reports, 147 Poster Presentations, 10 Working Groups, 2 Research Colloquia, and Plenary and special lectures. The program also includes a special tribute to Eugenio Filloy and a tribute to the PME-NA members who were affected by the COVID-19 pandemic. [These proceedings were produced with Cinvestav (Centro de Investigación y de Estudios Avanzados del IPN), México and AMIUTEM (Asociación Mexicana de Investigadores del Uso de Tecnología en Educación Matemática, A.C.), México. Individual papers are indexed in ERIC.]