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Comparing Crosscutting Practices in STEM Disciplines

Modeling and Reasoning in Mathematics, Science, and Engineering

  • SI: Nature of STEM
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Abstract

In the literature, there are multiple definitions of integrated STEM education, resulting in ambiguity and uncertainty of application. A firm conception of integrated STEM education is needed to determine its content, procedures and epistemological knowledge, the latter being the focus of this article. While epistemological accounts exist for its several disciplines, an integrated STEM epistemology has not yet emerged. This article provides a comparative analysis of modeling and argumentation in mathematics, science, and engineering and notes the observed similarities, intersections, and differences of their practice in these fields. Emphasis is given to the differences, which have not been elaborated in current literature although their importance has been noted. In this context, we examine definitions and functions of models and arguments in science and mathematics. The difference in argumentation concerns mainly the validation of knowledge; in models, the differences pertain to their constitution and intended aim. We also describe mathematical modeling, which optimally illustrates the interplay and collaboration of mathematics and science and permits comparison of their respective practices. The article contributes to the issue of the interrelated diversity of STEM practices and the intellectual and pragmatic educational benefits this issue confers.

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Notes

  1. In the STEM literature, we may say that two basic directions are implied with regard to the degree and extent of the integration of STEM disciplines (see e.g., in Dare et al. 2018; Martín-Páez et al. 2019): the incorporation of STEM units or STEM projects into the teaching of each STEM field, which seems more achievable on the larger scale within the constraints of current curricula and teachers’ expertise; and a more global, more permanent unification of the teaching of STEM objects, which is more ambitious and would need additional research and assessment of the learning outcomes from its implementation (see e.g., in Kelley & Knowles 2016; Dare et al. 2018).

  2. Kelley & Knowles 2016 define integrated STEM education “as the approach to teaching the STEM content of two or more STEM domains, bound by STEM practices within an authentic context for the purpose of connecting these subjects to enhance student learning” (p.3). Bryan et al. 2016 define integrated STEM “as the teaching and learning of the content and practices of disciplinary knowledge which include science and/or mathematics through the integration of the practices of engineering and engineering design of relevant technologies” (p. 23). Martín-Páez et al. 2019 define STEM learning as “the integration of a number of conceptual, procedural, and attitudinal contents via a group of STEM skills for the application of ideas or the solving of interdisciplinary problems in real contexts” (p. 5).

  3. Certain “lists” of NOS ideas/items have been created, containing ideas such as that scientific knowledge is tentative, empirically based, theory-laden, and socio-culturally embedded (see e.g., McComas and Olson 1998). The lists have been criticized, e.g., that their NOS ideas present an unchanging and too generalized picture of science, ignoring the discipline-specific nature and the context-dependency of scientific work, and the discussion was then about whether domain-general or domain-specific NOS ideas are appropriate for NOS understanding and teaching (see in this regard, e.g., Gilbert and Justi 2016; Wong and Hodson 2009; Matthews 2012; Abd-El-Khalick 2012). One proposed alternative to NOS lists is the family resemblance approach (Irzik and Nola 2011), which presents science as a discrete system whose basic features are organized in two dimensions, that of “Science as cognitive-epistemic system of thought and practice” and that of “Science as social-institution system” (the professional, ethical, and social characteristics and standards of science).

  4. Definitions of STEM literacy are given, e.g., in Martín-Páez et al. 2019: “the capacity to identify and apply content from STEM knowledge areas to understand and resolve those problematic situations that cannot be concluded from a mono-disciplinary approach” (p. 2); and in McDonald 2016: “STEM literacy can be defined in numerous ways, including ‘Stem literacy is the ability to identify, apply, and integrate concepts from science, technology, engineering, and mathematics to understand complex problems and to innovate to solve them (…)” (p. 533).

  5. Kelley & Knowles (2016) acknowledge that: “Some necessary knowledge in mathematics and sciences that are theoretically focused may not provide authentic engineering design applications as well as common STEM practices limited by current technology” (p. 3).

  6. Suppes exemplified the set-theoretical concept of models with the case of classical particle mechanics: a model of classical particle mechanics (i.e., that satisfies Newton’s laws) is defined as a quintuple < P, T, s, m, f>, where P is a set of elements (e.g., bodies if one wants to associate the abstract structure with something real, and so on for the remaining symbols); T is an interval of real numbers (e.g., time/elapsed times); s is a function that ascribes to each element three real numbers for any time (e.g., the spatial coordinates of the position of the element); m is a function that ascribes a positive number to each element (e.g., mass); f is a force function that ascribes three real numbers to each element at a concrete t (the three components of the force vector). (Here some concrete functions, e.g., the second derivative of s (corresponding to the definition of velocity in Newtonian mechanics) are also defined) (see in Suppes 1960; Grandy 1992).

  7. Giere (1988, 1999a) clarified and exemplified this conception for the case of classical mechanics: for example, the fundamental equation F = ma is valid for all the theoretical-kinematic models of Newtonian mechanics, but the function of the force F is specialized differently in each model, that is, as F = ct, or F = k/r2, or F = -Dx for the models of rectilinear and curvilinear motion and harmonic oscillation respectively. (F is the force, m is the mass, a is the acceleration, r is the radial distance, and k and D are constants.) (see also Develaki 2007).

  8. The model-based view had a great impact on science education, not only in relation to perceptions concerning the nature of scientific knowledge and methods but more generally as regards aims and teaching approaches, the designing of curricula, and teacher education. Many researchers have seen in it the possibility of a scientifically and pedagogically sound foundation for science education (e.g., Adúriz-Bravo & Izquierdo-Aymerich 2005; Adúriz-Bravo 2013; Ariza et al. 2016; Gilbert and Justi 2016; Develaki 2007, 2017). The research on model-based teaching suggests that students’ involvement in modeling activities can enhance the acquiring of knowledge, abilities and epistemologies that reflect real science (Schwarz & White 2005; Windschitl et al. 2008; Gilbert and Justi 2016), and describes modes of model-based inquiry teaching and learning, e.g., cycles of modeling activities for the construction, exploration, testing, and modification of models (e.g., Halloun 2004, 2007; Clement and Rea-Ramirez 2008; Oh and Oh 2011).

  9. Inferring general conclusions from specific instances is unjustified in logic, because the inductive conclusions have more content than the premises (Hume 1902; Popper 1959): they cover future instances/events that are not included in our present knowledge and observations. The philosophy of science, however, recognizes the heuristic and inventive function of induction in scientific practice (Stegmüller 1977).

  10. The harmful side effects and risks that often accompany applications of technology raise the question of the responsibility that engineers and technologists bear for the impacts of their solutions, designs, and products. As Pleasants & Olson 2019 mention, “Even though the ways that technologies affect society are difficult to predict, engineers must nevertheless consider potential consequences (…).” (p. 158). The responsibility for predicting and taking thought for the negative effects of the use of a technology is obviously shared among many sectors and agencies, politics, economy, philosophy, science, education, and citizens. STEM education is challenged to contribute to the development of students’ sensitivity and ability for anticipating and avoiding (as future scientists and/or citizens) technological solutions and applications with negative effects.

  11. Mathematical equations express relationships between abstract elements and can be used for the mathematical expression of relationships between the variables of scientific models “by replacing their symbols with the variables and initial conditions of the modeled system”; for example, the simple linear relationship expressed by the equation y = ax + b can be used in the mathematical formulation of a scientific model, e.g., for regular linear movement, by replacing the symbols y, a, x, and b with the position, speed, time (counted from zero), and initial position from a certain reference point of a moving body (see Giere 1999b). The choice of the appropriate equations in the modeling of a phenomenon is implied by the observation of patterns (for the phenomenon) or derived from the basic equations of the theories (where these exist) of the systems modeled.

  12. Ιn his analysis of Peirce’s (1931–1958) views of mathematics, Campos (2010) discusses a basic difference between mathematical and scientific reasoning and also their interrelation when mathematics enters scientific research. According to Peirce (1931–1958), in mathematical inquiry mathematicians frame a pure hypothesis (a hypothesis about an ideal state of things which is presumed to be true) without inquiring or caring whether it agrees with the actual facts or not, and then draw necessary (i.e. logically deduced) conclusions from that hypothesis. In science, on the other hand, hypotheses are about natural systems and their relations and cannot be true in the mathematical sense; they are also ‘too intricate’ or insufficient ‘for the mathematician to ascertain what their necessary consequences would be’. When mathematics is used in scientific inquiry, and concretely in mathematical modeling, the scientific hypotheses and models about real-world systems are transformed into ‘hypotheses that frame ideal states of things sufficiently clear and determinate for mathematical study.’ (Campos 2010, p. 426).

  13. The success of the use of mathematics in science implied a conviction among scientists that the mathematical equations of models will disclose all the possibilities, not only all the expected aspects of the modeled phenomenon but also unanticipated ones (Quale 2011; see also Kanderakis 2016; Brush 2015).

  14. Quale 2011 focuses on a third case of solutions to equations, which predict “unphysical” and counterfactual situations (incompatible with physical knowledge and experience), and examines scientists’ reactions and attitudes to such cases in relation to realist and antirealist/constructivist epistemologies about the nature of scientific knowledge (see Section “Basic Features of Models and Implications for Scientific Realism”).

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Develaki, M. Comparing Crosscutting Practices in STEM Disciplines. Sci & Educ 29, 949–979 (2020). https://doi.org/10.1007/s11191-020-00147-1

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