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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)

The Queensland Certificate of Education (QCE) system is Queensland's senior school qualification. To support the introduction of the system in 2019, existing senior syllabuses were redeveloped and a new senior assessment model was established, this included the implementation of a mandatory high-stakes assessment task, the Problem-solving and Modelling Task (PSMT) in all four mathematics syllabi. The PSMT required new skills from both students and teachers to manage complex, open-ended investigations. In this paper, we reflect on our school's approach to build teacher capability in designing PSMTs and supporting student engagement with PSMTs.

The revised Australian curriculum presents a new emphasis in primary schools on the process of Mathematical Modelling. A modelling focus brings close attention to confidence and capability with a potentially new problem-solving process, associated language, and pedagogical processes. This paper presents a Year 4 classroom modelling experience that arose as students planned their end-of-year party through Guided Mathematical Inquiry. Classroom video data captured two students working on vertical whiteboards as they formulated and solved a problem involving carrot sticks and dip. Findings reflect the students as doers of mathematics, engaged in productive struggle.

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.]

Another year of COVID-19 meant another virtual meeting but the Canadian Mathematics Education Study Group/Groupe Canadien d'Étude en Didactique des Mathématiques (CMESG/GCEDM) made the best of the situation and found ways to learn together and share connections. Having learned a lot from the 2021 virtual meeting, the executive was able to once again offer a virtual program that, although reduced from the normal offerings, ensured an opportunity for spirited discussions, meaningful learning, and celebrating the newest members. The 45th annual meeting took place May 27-29, 2022. These proceedings contain one plenary lecture, six working groups, and ten new PhD reports. Seventeen papers are included in these proceedings--two papers are written in both French and English; three papers are written in French; and the remainder are written in English.

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)

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)

This study elaborates on the pivotal roles of mathematical and statistical models in data-driven predictions in an integrated STEM context using the case of Year 4 students: (?) "a descriptive means" to describe the features of trends and variability of data and (?) "an explanatory means" to explain causal relationships behind data. These roles are linked to models in other STEM subjects (i.e., prototypes and scientific models) and the application and development of STEM content knowledge. The results contribute to a better understanding of the role of mathematics/statistics in STEM education.

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 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.

Our paper details the ways teachers understand and navigate equity-oriented dilemmas (Berlak & Berlak, 1981) when teaching mathematical modeling and how mathematics teacher educators can support teachers' learning of culturally responsive mathematics teaching. Using Zavala and Aguirre's (in press) framework for culturally responsive mathematics teaching, we explored the ways teachers describe and frame their choices when faced with dilemmas. Findings revealed that teachers identified dilemmas with Rigor and Support most often, followed by Knowledges and Identities. Dilemmas related to Power and Participation occurred far less frequently. Implications for teacher professional development are discussed. [For the complete proceedings, see ED658295.]

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 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.]

In this study, we examined one experienced mathematician's class practices, with particular attention to cognitive model described in genetic decomposition. Our findings indicate that students only had limited opportunities to be familiar with the first three steps in genetic decomposition, which may potentially lead students to answer limit tasks correctly, but not necessarily having a deep conceptual understanding behind those tasks. With limited opportunities, it will be challenging for students to overcome well-known learning challenges in limit. [For the complete proceedings, see ED658295.]

This research contributes to the need to identify and expand learning environments that encourage undergraduate students to develop collaborative work skills and apply their classroom knowledge to solve real-world problems. Using qualitative methods, we examine the effects of the interaction between two teams of students when solving a Model-eliciting activity, based on the theoretical framework of the Models and Modeling Perspective. Our analysis shows that the students had two opportunities for interaction, within the team and between the modeling teams. Through these interactions, the students refined their models in three directions: mathematical knowledge, interpretation and modeling of the phenomenon, and mathematical representations. The findings of this study emphasize the significant value of Model-eliciting activities in enhancing students' collaboration and modeling abilities. [For the complete proceedings, see ED658295.]

Mathematical modeling is a process in which students investigate authentic problems and everyday situations using mathematics. In doing so, they bring their multiple mathematical knowledge bases and cultural funds of knowledge into their solution strategies. During a task called "Abuelo's Birthday", 297 students in grades 3-5 decided how to split the costs of a gift "fairly" and justified their work using early notions of ratio and proportional reasoning. We argue that these young students were successful with a complex task due to the way the realistic context connected to students' lived experiences and funds of knowledge related to home and family life. We share rich examples of how students included elements outside of the task to justify and enhance their mathematical models and conclude with implications for the importance of modeling and non-routine tasks in elementary classrooms. [For the complete proceedings, see ED658295.]

There is growing recognition that mathematical modeling can be a lever for equity in elementary mathematics classrooms. This study focuses on the impact of a professional development program focused on culturally responsive mathematical modeling on 8 kindergarten through 2nd grade teachers' practices in modeling lessons. We use a project developed observation tool to evaluate two video recorded modeling lessons from each teacher (16 total). Findings focus on patterns in the strengths and challenges in primary grade teachers' practices for teaching modeling, including how teachers' practices align with culturally responsive teaching. We discuss implications of our findings for the design and refinement of professional development. [For the complete proceedings, see ED657822.]

The purpose of this study is to gain understanding of how secondary teachers conceptualize the relationships between mathematical modeling and problem solving. Eight secondary teachers participated in semi-structured, think-aloud individual interviews. Some conceptualizations include (a) modeling and problem solving are two distinct processes; (b) modeling is a subset of, or a tool for, problem solving; (c) the two processes share similar characteristics such as a real-life scenario but are different in terms of whether a single correct answer exists; (d) the two processes are inseparable, completely enmeshed in each other, and co-dependent; and (e) problem solving is a reduced process of modeling. Teachers' conceptualizations are related to their preferred instructional sequence and the types of problems or activities they'd rather use. [For the complete proceedings, see ED657822.]

Recently, abstracted quantitative structures (AQS), a construct from quantitative reasoning, has been offered as a means to conceptualize and study mathematization during mathematical modeling. Extending this theoretical work, we provide empirical evidence that an intervention targeting participants' AQS can assist in aligning modelers' models with normatively correct models. We report on a pre/post intervention study designed to elicit alignment between symbolic forms and AQS and alignment between AQS and modeling scenarios. We used the Sørenson-Dice coefficient and cluster analysis to identify shifts in student associations of symbolic forms with modeling scenarios. [For the complete proceedings, see ED657822.]

In this report, we present cases where students constructed new quantities through operating on quantities that does not fit the definitions of existing theories on quantitative operations. As a result, we identified five quantitative operators--operators that can be used on single qualities in order to transform the quantity to a new quantity--students used while constructing mathematical models for real-world scenarios. [For the complete proceedings, see ED657822.]

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). 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.

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.]

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 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 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 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 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 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)

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.]

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.]

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.]