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The conceptual framework of mathematical modeling (e.g., Lesh & Doerr, 2003) is a vital area in mathematics education research, and its implementation has potential for deeply involving children in integrated and meaningful learning. In mathematical modeling learners are active agents in content-integrated, real-world problem solving. This emphasis on integrating multiple content areas to answer big questions, the pursuit of mathematical modeling, descends from Dewey’s work. We present the definition, principles, and design of modeling practices for readers who may be familiar with early childhood curriculum but less so with using modeling for learning. We explore the application of mathematical modeling to early childhood classrooms and its compatibility with early childhood pedagogies and philosophies. Young children may often be underestimated, assumed to be unable to pose big questions that can be answered through activity, experience, and data; but we discuss how young children can be engaged in problems through mathematical modeling. Finally, as preservice teacher educators, we discuss preparing preservice and in-service teachers for modeling in their classrooms. We offer examples and guidance for early childhood teachers to engage in authentic practice – meeting children where their interests are and creating integrated problem-solving experiences.

This chapter presents the survey of selected linear and mixed integer programming multi-objective portfolio optimization. The definitions of selected percentile risk measures are presented. Some contrasts and similarities of the different types of portfolio formulations are drawn out. The survey of multi-criteria methods devoted to portfolio optimization such as weighting approach, lexicographic approach, and reference point method is also presented. This survey presents the nature of the multi-objective portfolio problems focuses on a compromise between the construction of objectives, constraints, and decision variables in a portfolio and the problem complexity of the implemented mathematical models. There is always a trade-off between computational time and the size of an input data, as well as the type of mathematical programming formulation with linear and/or mixed integer variables.

In his discussion of early lexical development in children Barrett (1995) notes that utterances can be of the following types: an expression of an internal state, a response to a specific context, a social-pragmatic utterance and a referential utterance. A referential utterance can be thought of in the following way: the thing referred to, a mental representation, a word representation and a word sound. An utterance may refer to an object, an action, an attribute or an event. Some utterances are used as the names of classes of objects while other utterances are used as the proper names of individual objects. Looking at this in abstract we might say that, in the early years of childhood, language is used to refer to elements, sets, functions and relations – in other words to the mathematical structures which were discussed in Chapter 2. Of course although early language is used to refer to mathematical objects, the character of the language itself takes the form of ordinary language.

Set theory provides a foundational approach to mathematics, and mathematics provides an abstract way of looking at social reality. The first section presents some of the elementary concepts of set theory. The second section presents a variety of examples of social reality and shows how the abstract features of reality can be modelled by set theory. The third section shows how set theory can provide a way of looking at the accounts of social reality presented in humanities disciplines such as history and literature. The fourth section briefly indicates how set theory and the concept of a structure provide a foundational approach to mathematics. The fifth section looks at the debates surrounding realism and, albeit warily, espouses mathematical social science realism.

The subject matter of psychology can be illustrated by addressing the questions, ‘who am I?’, ‘what do I spend my time doing?’ and ‘reflecting on my life, where have I come from and where am I going?’. I am a human being and also more generally an animal. I am a biological entity and also a psychological entity. I have an identity and a personality. I am an individual and also a member of a social system. I am similar to other human beings but also different. In some respects I am normal and in other respects abnormal.

The versions of positivism that are critically assessed in Bruce Caldwell’s Beyond Positivism bear two dominant sets of implications. One is that knowledge growth is monistic in nature; the other is that science has a specific deductivist structure. Caldwell focuses mainly on the former and its critics. I argue here that the second set of implications always did, and still does, perhaps more than ever, warrant critical attention.

In this chapter, we present our ongoing efforts in developing and sustaining interdisciplinary STEM undergraduate programs at North Carolina A&T State University (NCA&T) – a state-supported HBCU and National Science Foundation (NSF) Historically Black Colleges and Universities Undergraduate Program (HBCU-UP) Institutional Implementation Project grantee. Through three rounds of NSF HBCU-UP implementation grants, a concerted effort has been made in developing interdisciplinary STEM undergraduate research programs in geophysical and environmental science (in round 1), geospatial, computational, and information science (in round 2), and mathematical and computational biology (in round 3) on NCA&T campus. We first present a brief history and background information about the interdisciplinary STEM undergraduate research programs developed and sustained at NCA&T, giving rationales on how these programs had been conceived, and summarizing what have been achieved. Next we give a detailed description on the development of undergraduate research infrastructure including building research facilities through multiple and leveraged funding sources, and engaging a core of committed faculty mentors and research collaborators. We then present, as case studies, some sample interdisciplinary research projects in which STEM undergraduate students were engaged and project outcomes. Successes associated to our endeavor in developing undergraduate research programs as well as challenges and opportunities on implementing and sustaining these efforts are discussed. Finally, we discuss the impact of well-structured undergraduate research training on student success in terms of academic performance, graduation rate and continuing graduate study, and summarize many of the learnings we have gained from implementation and delivery of undergraduate research experiences at HBCUs.

This chapter presents application of multi-criteria mathematical programming models by integer and mixed-integer programming for optimal allocation of workers among supporting services in a hospital. The services include logistics, inventory management, financial management, operations management, medical analysis, etc. The optimality criteria of the problem are minimization of operational costs of supporting services subject to some specific constraints. The constraints represent specific conditions for resource allocation in a hospital. The overall problems are formulated as assignment models, where the decision variables represent the assignment of people to various jobs. Numerical examples are presented. Some computational results modeled on a real data from a hospital in Poland are reported.