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This paper discusses the determination of college faculty salary inequity and identifies the areas in which human judgment must be used in order to conduct a statistical analysis of salary equity. In addition, it provides some informed guidelines for making those judgments. The paper provides a framework for selecting salary equity models, based on four decision elements and the contributions of four fields of study (law, economics, statistics, and institutional research) as the basis for establishing criteria for selecting an appropriate salary equity model. The four decision elements include defining the group to be analyzed, determining the variables to be included, deciding what statistical model should be used, and determining which outcome statistics should be used for interpretation. Contributions from the field of law focus on pay equity, disparate treatment, and comparable worth. The contribution of the field of economics comes from human capital theory. The contribution of statistics is the use of multiple regression analysis in salary equity research and the issues of multicollinearity and statistical significance. The field of institutional research has contributed various studies on methods of indentifying underpaid employees. (Contains 75 references.) (GLR)

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 papers in this publication were developed from speeches and reactions presented at the first National Conference on Remedial Mathematics held at Kent State University in 1974. Papers focus on identifying and describing the remedial mathematics student, classroom diagnosis, clinical diagnosis, the diagnostic process, and promising procedures and directions in remediation. In addition to reactions to each paper, a conference summary is included. (MS)

The issues discussed in these four papers concern the validity and generalizability of classroom observation instruments. These issues have been studied and are reported here in an attempt to better define the limits to which classroom observation instruments can be used in researching relationships between teacher behavior and student outcome. The premise undergirding these investigations is that before consistent and positive process-product relationships can be found, investigators must be cognizant of the sources of variance which affect the validity and generalizability of their process measures and which, in turn, affect the credibility of their research findings. The four papers are: "Convergent and Discriminant Validity of Five Classroom Observation Systems: Testing the Model" by G. Borich, D. Malitz, C.L. Kugle, and M. Pascone; "Generalizability of Teacher Behaviors Across Classroom Observation Systems" by D. Calkins, G. Borich, M. Pascone, and C.L. Kugle; "Measuring Classroom Interactions: How Many Occasions Are Required to Measure Them Reliably?" and "Generalizability of Teacher Process Behaviors During Reading Instruction" both by O. Erlich and G. Borich. (RC)

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.]

Data on 174 Colorado school districts were used to measure districts' responses (as indicated by per-pupil expenditures) to their own fiscal capacity and to state aid that changes over time. Colorado's modified guaranteed tax base (GTB) formula was analyzed and a model constructed that took into account the formula's limits on district spending and its year-to-year changes in a district's state aid based on the district's tax revenues in the preceding year. Such changes comprise an "intertemporal price variable." Data were collected on such variables as district wealth (total and residential assessed value), income, state and federal aid, enrollment size and growth, percentage of minority pupils, and agricultural land prices. Using linear and log-linear correlation equations, researchers analyzed the variables' effects on districts' total and locally-derived expenditures and on the elasticity of these expenditures in relation to changes in other variables' values. Results indicate that Colorado's GTB formula has had significant effects neither on differences across districts in expenditures per student nor on the strong correlation between assessed valuation and expenditures. The results also carry implications for the construction of models of districts' fiscal responses. (RW)

This paper discusses discrepancies between the observable labor market and the idealized world which is assumed in theory. The proposed solutions are focused on the development of an empirical model applicable to data on prime-age males from the Michigan Panel Study of Income Dynamics, but the author notes that many of these issues are relevant for the construction of models for other groups. The discussion covers five basic areas: (1) The potential limitations of the simple labor supply model, based only on income-leisure tradeoffs, are discussed. (2) Sources of randomness in observed labor supply behavior and its consequences for the stimulation of systematic labor supply responses are considered, and the intertemporal variation are discussed. (3) The model is extended to accommodate earnings opportunities other than a simple constant wage rate, with increasing marginal income tax rates and overtime premiums being the major factors considered. (4) Demand-related factors that prevent workers from achieving marginal equilibrium at their marginal wage rates are explored, and criteria are suggested for the selection of a sample of workers who are less seriously affected by these problems. (5) The treatment of time lost due to unemployment and illness is discussed in the context of a model developed by Samuel Rea, which is also applied to time spent commuting to work. (Author/HD)

The STRATOP algorithm was developed to help planners and proponents find and test effectively designed choice objects and communication strategies. Choice objects can range from complex social, scientific, military, or educational alternatives to simple economic alternatives between assortments of branded convenience goods. Two classes of measured input data are used, one cognitive and the other affective. In addition, data on brand choice are needed to fit the parameters of the choice model. The STRATOP technique and the assorted preliminary analytical methods used modest amounts of standard data and yield very extensive findings, explicitly tailored to the needs of strategists and designers. Further experience is being accomulated with the expectation that the methods will find application in a number of areas involving significant social and economic choices among competing alternatives. (Author/JG)

A mathematical model of the Ph.D. scientist labor market demonstrates that subsidies of graduate training can be more cost effective than higher wages if the supply of doctorates is substantially more responsive to $1000 of early subsidy than to higher future wages with a present discounted value of $1000. Whether this is true is an unsettled empirical matter. There are three reasons for targeting subsidies at Ph.D. scientists: (1) As a condition of taking a job they demand freedom to do basic research and publish their results. (2) Because of their special knowledge and loyalty to professional values, hiring scientists and engineers contains an extra risk that trade secrets will be stolen or that one of them will be a "whistle blower." (3) From the firm's point of view these factors reduce the scientist's productivity. They do not from society's point of view, so an externality is created by the employment of scientists. (Author/LBH)

This submission contains the Proceedings of the 2011 Annual Meeting of the Canadian Mathematics Education Study Group (CMESG), held at Memorial University of Newfoundland in St. John's, Newfoundland. The CMESG is a group of mathematicians and mathematics educators who meet annually to discuss mathematics education issues at all levels of learning. The aims of the Study Group are: to advance education by organizing and coordinating national conferences and seminars to study and improve the theories of the study of mathematics or any other aspects of mathematics education in Canada at all levels; and to undertake research in mathematics education and to disseminate the results of this research. These proceedings include plenary lectures, working group reports, topic session descriptions, new PhD reports, and summaries of ad hoc and poster sessions. Papers include: (1) Pattern Composition: Beyond the Basics (Chris K. Palmer); (2) The Pair-Dialogue Approach in Mathematics Teacher Education (Pessia Tsamir and Dina Tirosh); (3) Mathematics Teaching and Climate Change (Richard Barwell, Stewart Craven, and David Lidstone); (4) Meaningful Procedural Knowledge in Mathematics Learning (Wes Maciejewski, Joyce Mgombelo, and Annie Savard); (5) Emergent Methods for Mathematics Education Research: Using Data to Develop Theory / Méthodes émergentes pour les recherches en didactique des mathématiques : Partir des données pour développer des théories (Olive Chapman, Souleymane Barry, and P. Janelle McFeetors); (6) Using Simulation to Develop Students' Mathematical Competencies--Post Secondary and Teacher Education (Eric Muller, Jean-Philippe Villeneuve, and Philippe Etchecopar); (7) Making Art, Doing Mathematics / Créer de l'art; faire des maths (Eva Knoll and Tara Taylor); (8) Selecting Tasks for Prospective Teachers in Mathematics Education (Marie-Pier Morin, Christian Bernèche, and Ralph Mason); (9) How to Prepare a Public Lecture? First Questions, Then Execution??? (Yvan Saint-Aubin); (10) Warm Hands Taking Cold Mathematics (David Wagner); (11) Opportunities to Learn IN and THROUGH Professional Development: An Analysis of Curriculum Materials (Jenny Sealy Badee); (12) Challenging Our Beliefs and Practices in Secondary Mathematics Education (Lorraine Baron); (13) Pattern Rules, Patterns and Graphs: Analyzing Grade 6 Students' Learning of Linear Functions Through the Processes of Webbing, Situated Abstractions, and Convergent Conceptual Change (Ruth Beatty); (14) Learning Mathematics for the Workplace: An Activity Theory Study of Pipe Trades Training (Lionel Lacroix); (15) Transforming Mathematics Education for Mi'kmaw Students through Mawikinutimatimk (Lisa Lunney Borden); (16) Coming to Know Mathematics: Views of Two Teacher Mathematicians (Veda Abu-Bakare); (17) What Does 'Better' University Mathematics Instruction Look Like? (Mary Beisiegel and Asia Matthews); (18) Exploring Variability in a Dynamic Computer-Based Environment (George Ekol); (19) Virtual Mathematics Marathon: A Mathematical Game For All Children (Margo Kondratieva and Viktor Freiman); (20) Early Childhood Mathematics Education (Donna Kotsopolous and Joanne Lee); (21) The Challenges of Mathematics In-Service (Susan Oesterle); (22) High School Mathematics Students' Trajectories: Tracking or Differentiating for Success? (Ralph T. Mason and P. Janelle McFeetors). Appended are: (1) Working Groups at Each Annual Meeting; (2) Plenary Lectures at Each Annual Meeting; and (3) Proceedings of Annual Meetings. Appendixes contain: (1) Working Groups at Each Annual Meeting; (2) Plenary Lectures at Each Annual Meeting; and (3) Proceedings of Annual Meetings. Individual papers contain references, tables, and figures. [For the 2010 proceedings, see ED529564.]

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.]

Developing models and using mathematics are two key practices in internationally recognized science education standards, such as the Next Generation Science Standards (NGSS). However, students often struggle with these two intersecting practices, particularly when developing mathematical models about scientific phenomena. Formative performance-based assessments designed to elicit fine-grained data about students' competencies on these practices can be leveraged to develop embedded AI scaffolds to support students' learning. In this paper, we present the design and initial classroom test of virtual labs that automatically assess fine-grained sub-components of students' mathematical modeling competencies based on their actions within the learning environment. We describe how we leveraged underlying machine-learned and knowledge-engineered algorithms to trigger scaffolds, delivered proactively by a pedagogical agent, that address students' individual difficulties as they work. Results show that the students who received automated scaffolds for a given practice on their first virtual lab improved on that practice for the next virtual lab on the same science topic in a different scenario (a near-transfer task). These findings suggest that real-time automated scaffolds based on fine-grained assessment can foster students' mathematical modeling competencies related to the NGSS.

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.]

Developing models and using mathematics are two key practices in internationally recognized science education standards such as the Next Generation Science Standards (NGSS, 2013). In this paper, we used a virtual performance-based formative assessment to capture students' competencies at both "developing" and "evaluating" mathematical models in science inquiry contexts aligned with the NGSS (2013). Results show that model development and evaluation competencies are correlated, but students who demonstrate proficiency with model development often struggle with evaluation. Nuanced data illustrate how components of modeling competencies differ and how they may be related.

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.]

The Departmental Enrollment Projection Model (DEPM), which is based on the assumption that enrollments are essentially influenced by economic and social forces, is described. Under the model's approach, the department, rather than the institution as a whole, is seen as the focal point of both the enrollment projection and, by implication, the planning strategies. The pool of students individual departments are now concerned with is that of potential students, in the relevant higher-education region, contemplating formal training. Quality of education, placement prospects, and relative tuition are factors which can now be directly related to enrollment and controlled in the attainment of institutional and departmental goals. Labor market and education market factors form the economic foundations of the model. The current version of the DEPM has been developed as a recursive system of four behavioral, one trend, and one policy or control equations for each program or department of instruction. Restrictively, each department's direct enrollments are assumed to be independent from the other departments. Equations used in the model are presented and discussed. The data concepts and sources of information used in the model are summarized. The effect of specification, exogenous variables, and policy variations are also described. (SC)

The development of a predictive equation capable of explaining a significant percentage of enrollment variability at Florida International University is described. A model utilizing trend analysis and a multiple regression approach to enrollment forecasting was adapted to investigate enrollment dynamics at the university. Four independent variables were used in a stepwise linear regression analysis to explain variance in the total university full-time-equivalent enrollment. Three of the four variables used are presently employed in business forecasting models to account for changes caused by trends, seasonal variations, and cyclical variations. The fourth variable was introduced to account for the opening and operation of a new campus. The model accounts for 99.6 percent of the variation observed in total university enrollments for the past 20 quarters and provides estimates of the enrollment within one percent of the enrollments actually observed. The model also has the potential for making reasonably accurate and reliable short-range enrollment forecasts. The model demonstrated that changes in economic conditions in the county, reflected by changing unemployment rates, correlate significantly and positively with changes in total university enrollment. Total university enrollment did not change from its second to its sixth year of operation except for those changes that could be accounted for by changes in unemployment in the county. The opening and operation of the new campus increased total university enrollment by about six percent. (SC)

A search for better predictors of incomes of high school and college graduates is described. The accuracy of the prediction, implicit in the work of John R. Walsh of Harvard University, that the income differences in a given year are good indicators of income differences in future years, was tested by applying standard statistical procedures to data describing the annual incomes of men 25 years old and older during the period 1956 to 1975. Addition of the gross national product in the year prior to the previous equation yields much improved results. Application of Richard B. Freeman's three-equation recursive adjustment model indicates that the starting salaries of college graduates are not good correlates of income differences. Several sets of data from the American Council on Education "Fact Book on Higher Education" were then entered into a series of multiple correlation and regression studies. Studies of the 184 independent variables identified four that, taken together, yield significantly better predictions of future income differences. These factors were: (1) male population 22 years old, 13 years prior; (2) male college graduates at the B.A. and first-professional degree level/male population 29 years old, 11 years prior; (3) male college graduates at the B.A. and first-professional degree level, 14 years prior; and (4) male college graduates at the B.A. and first-professional degree level/male population 29 years old, 14 years prior. However, no theory to explain the connections among these variables is apparent. (SC)

The use of data on longitudinal student attendance patterns to determine variances, and hence confidence bounds, on student enrollment forecasts, in addition to finding the forecasts themselves, is demonstrated. The formulation of the enrollment model based on longitudinal student attendance patterns is described step by step, presenting the equations used in the model and some sample data obtained through application of the model. Applications of various equations are demonstrated. The model is analyzed assuming that the new admissions each period follow a Poisson distribution. The enrollment permits the development of statements about the approximate behavior of the conditional distribution of the total number of students in attendance at a given time. It is assumed that entering cohorts contain large numbers of students. Equations relative to this function are presented and discussed. Sample data on the forecasts and confidence intervals for each semester enrollment are presented. The model provides a measure of the error that could be present for each forecast. (SC)

The viability of Freeman's Recursive Adjustment Model as a management tool for explaining and predicting enrollments and the job market is examined using macro and micro cases. To establish a relationship between college attendance and the college job market, Freeman employed a 3-equation model to analyze the causative supply and demand determinants. Freeman's equations and published and unpublished data are used to ascertain the goodness of fit for the entire United States, the entire state of Oregon, the University of Oregon, and two major departments at the University of Oregon (business and psychology). Freeman's model is presented, its application is discussed, and the method utilized to test Freeman's model is described. The study results suggest that Freeman's model for explaining college enrollments and graduate job market could be effectively employed by university administrators and institutional researchers to help analyze the major determinants in college attendance. The model might be more valid for the larger macro entities, such as national, state, or university, than it is for the smaller micro entities. The study results indicate that the degree of fit for the model on the micro level will vary considerably from university department to university department and on the general availability of the required data. (SC)

The use of subjective probability as a theoretical model for enrollment forecasting is proposed, and the results of an application of subjective probability to enrollment forecasting at the University of Toledo are reported. Subjective probability can be used as an enrollment forecasting technique for both headcount and full-time equivalent using the following steps: (1) assume the various subpopulations are normally distributed; (2) determine the mean and standard deviations of these subpopulations; (3) sum the means and variances of the subpopulations to obtain the mean and variance of the university forecast; and (4) perform a linear transformation on the headcount distribution to obtain the full-time equivalent distribution. Major advantages of this method are: it allows the study of homogeneous subpopulations; it allows comparisons with other methods, since the variability is known; and it allows probability statements regarding various headcount and full-time equivalent outcomes. Application of the subjective probability method to enrollment forecasting at the University of Toledo resulted in a difference of 0.4 percent between the actual and the forecasted headcount enrollment and a difference of 0.3 percent between actual and forecasted full-time equivalent enrollment. In comparison with the conventional forecasting techniques also employed in the test study, the forecasting errors using the subjective probability were the smallest. (SC)

A least squares approach to estimating the reliability of a measure consisting of more than three content homogeneous or congeneric parts is proposed. The advantages of this method over a more indirect approach in which certain parts of a measure are combined to use Kristof's or Feldt's coefficients are examined. One hundred four-part tests were created by randomly assigning certain items from a 48-item mathematics achievement test to each of the four parts. Data on 400 examinees who took the test were divided into five subsets. Seven estimates of reliability were computed for 100 random splits for each subset of data. Some of the main results included: (1) there is a consistent order in the magnitudes of the means of the coefficients, regardless of sample size; (2) values of the average of the Kristof coefficients are greater than unity when sample size is small; (3) the proposed coefficient has smaller standard deviation and range than the Kristof coefficient; and (4) the Feldt coefficient tends to have smaller standard deviation and range than the proposed coefficient, and depends on the assumption that the error variance of a part is proportional to its length. (Author/RL)

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.]

Many national science frameworks (e.g., Next Generation Science Standards) argue that developing mathematical modeling competencies is critical for students' deep understanding of science. However, science teachers may be unprepared to assess these competencies. We are addressing this need by developing virtual lab performance assessments that assess these competencies in science inquiry contexts. Through our design processes, we developed a method for validating the assessments that takes advantage of the unique opportunities afforded by collecting log data. Here, we describe this method and demonstrate its utility by analyzing students' competencies with one example sub-practice of mathematical modeling, "plotting controlled data generated from a simulation." [This is a paper in: M. M. Rodrigo et al. (Eds.): AIED 2022, LNCS 13356, (pp. 488--491), 2022.]

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.]

The analysis of log data generated by online educational systems is an important task for improving the systems, and furthering our knowledge of how students learn. This paper uses previously unseen log data from Edulab, the largest provider of digital learning for mathematics in Denmark, to analyse the sessions of its users, where 1.08 million student sessions are extracted from a subset of their data. We propose to model students as a distribution of different underlying student behaviours, where the sequence of actions from each session belongs to an underlying student behaviour. We model student behaviour as Markov chains, such that a student is modelled as a distribution of Markov chains, which are estimated using a modified k-means clustering algorithm. The resulting Markov chains are readily interpretable, and in a qualitative analysis around 125,000 student sessions are identified as exhibiting unproductive student behaviour. Based on our results this student representation is promising, especially for educational systems offering many different learning usages, and offers an alternative to common approaches like modelling student behaviour as a single Markov chain often done in the literature. [For the full proceedings, see ED596512.]