1 Introduction

The particular cognitive, cultural, and practical potential of the natural sciences make them a basic component of education, alongside mathematics and the humanities. Research has, however, shown unsatisfactory results with regard to the achievement of goals in science and mathematics education and revealed waning student interest in learning or pursuing a career in these disciplines, particularly in western countries. This has provoked concerned discussion about education in the STEM subjects (science, technology, engineering, and mathematics) in the political-financial and educational spheres at national and international levels (see e.g., in Kelley and Knowles 2016; Holmlund et al. 2018; Martín-Páez et al. 2019).

In the political/financial sphere, there has been an increase in funding to improve education in STEM subjects, driven by the recognition that STEM education (and a STEM-skilled workforce) is a critical factor for economic competitiveness and for dealing with the big challenges (environmental and energy crises) in the contemporary context of socio-political globalization. In the educational field, there have been a series of curricular and policy reforms focusing on the overarching goal of ensuring that all tomorrow’s citizens are scientifically, mathematically, and technologically literate (AAAS 1993; NRC 2012; NGSS Lead States 2013). Among these reforms and innovations, one proposal for improving STEM education that has increasingly gained ground in recent decades is that the traditional separate teaching of these subjects be abandoned in favor of an integrated STEM education contextualized in real-world problems and applications, in line with the learning theories that support the situation, contextualization, and application of knowledge learning (see e.g., Kelley and Knowles 2016; Moore et al. 2014; Dare et al. 2018).

Although the term “SΤΕΜ Education” is generally applied to the teaching of just one of the STEM disciplines (see e.g., Bybee 2013; Holmlund et al. 2018; Dare et al. 2018), most of the recent research papers focus on an integrated STEM education approach, which would be best effected as an interdisciplinary project through inquiry-based learning and engagement in argumentation and reasoning practices (Holmlund et al. 2018; Kelley and Knowles 2016; Martín-Páez et al. 2019; McDonald 2016).

Numerous papers, conferences, journals, policy documents, and professional organizations have examined several aspects of both degrees of integrationFootnote 1 and ways of interconnecting the several STEM subjects, and have reviewed and commented on the results of STEM interventions (Martín-Páez et al. 2019; Bybee 2013; McDonald 2016; Dare et al. 2018). They have also noted problems and limitations associated with the integrated approach (e.g., Kelley and Knowles 2016; Dare et al. 2018) and the related research that remains to be conducted, such as the lack of a single definition for the integrated STEM subjects, the complexity of making crosscutting STEM connections, inadequate teacher knowledge incorporating all STEM fields, and the lack of materials and instructional and assessment support and guidance (Holmlund et al. 2018; Dare et al. 2018; Martín-Páez et al. 2019).

These articles propose various definitionsFootnote 2 for an integrated STEM education and develop frameworks relating, e.g., to the aims and contextualization of STEM education and the degree and manner of interconnection of the several STEM disciplines (e.g., Cunningham and Kelly 2017; Pleasants and Olson 2019; Chalmers et al. 2017; Johnson et al. 2016). There is an important body of STEM literature dealing particularly with the nature and goals of engineering practices and their incorporation into and role in STEM education (e.g., interconnection and contextualization of the several STEM disciplines through engineering design) (Martin-Paez et al. 2019; Moore et al. 2014; Bryan et al. 2016; Chalmers et al. 2017).

A shared finding and concern of almost all these articles is the existence of multiple conceptions of STEM education, which results in ambiguity and uncertainty as to its aims and the means of its achievement (see e.g., in McDonald 2016; Martín-Páez et al. 2019; Kelley and Knowles 2016; Dare et al. 2018; Holmlund et al. 2018). For the purposes of this article, we think that more important than a final definition is an overarching conception, which by analogy with the definition of science education would mean that (integrated) STEM education comprises three dimensions, content knowledge, procedural knowledge, and epistemological knowledge, each of which should incorporate knowledge, practices, and epistemologies of the integrated STEM disciplines.

Most of the existing studies focus on the first two aspects (see in Martin-Paez et al. 2019; Kelley and Knowles 2016), while the related epistemology and its educational relevance have only recently begun to appear, particularly in certain studies focusing on the nature of engineering knowledge and practices in comparison with, in particular, those of the natural sciences (Peters-Burton 2014; Chalmers et al. 2017; Pleasants and Olson 2019; Antink-Meyer and Brown 2019).

In this paper, we focus on the third dimension, the epistemology (or the nature of STEM more broadly), and concretely on the epistemology of the common STEM practices of modeling and argumentation. With nature of STEM we mean here, as with the definition of the nature of science (NOS) in science education, a broader notion than epistemology, encompassing in addition the social dimension of the STEM disciplines, i.e., their internal traditions and their interactions with society. Science education has a long research history of defining and teaching the nature of science (NOS) that can be utilized for promoting analogous issues in the conception and teaching of an integrated STEM epistemology. In the NOS field, it is for example broadly accepted that the NOS includes the nature of scientific methods and knowledge as well as the social dimension of science, and that an effective NOS teaching requires, along with and beyond students’ engagement with scientific practices that key-NOS ideasFootnote 3 be explicitly addressed (see e.g., Lederman et al. 2014; Holton 2014; Clough and Olson 2008; Khisfhe and Abd-El-Khalick 2002; Gilbert and Justi 2016).

While there are, of course, accounts of the nature and specific habits of working of the several STEM disciplines individually, what is now required is a conception of the nature of STEM taken and taught as a whole, and a definition of its key ideas, which could establish whether and why an integrated STEM epistemology can offer deeper and superior epistemological knowledge (and STEM literacyFootnote 4) than the sum of the epistemologies of its parts (see e.g., Peters-Burton 2014). Peters-Burton sees the nature of engineering and nature of technology as “the study of humans influencing the world” and the nature of science and mathematics as “about humans understanding the mechanisms in nature” (p. 100), and suggests that these two perspectives can serve as a basis for the conception and scope of the nature of STEM. We consider that an integrated STEM epistemology should properly integrate the epistemologies of the individual STEM disciplines, i.e., that carefully considers both their differences and their convergences, leading thus to a higher level of epistemological knowledge that reflects contemporary interdisciplinary research without conflating the nature and contribution of the participating individual STEM fields.

Epistemological knowledge, on which we focus here, supports both students’ understanding and practicing scientific practices, such as scientific reasoning and model-based inquiring (Cunningham and Kelly 2017; He et al. 2020). For example, research suggests that NOS teaching fosters the acquisition of skills and epistemologies that students need in order to cope with the demands and the socio-scientific issues of the age (see e.g., Clough and Olson 2008; Lederman et al. 2014; Zeidler et al. 2019). Argumentation and modeling are basic processes in all STEM fields, and consequently they are also fundamental to the teaching and learning of STEM knowledge, practices, and epistemologies. Moreover, as cross-disciplinary practices, they enable cross-STEM connections to be made, and may thus effectively link the several disciplines (Duschl and Bismack 2016; Kelley and Knowles 2016; Peters-Burton 2014).

The papers that advocate a practice-based STEM integration emphasize the similarities inherent in the common practices of the various STEM fields, but also warn of the existence of significant differences which, if ignored or underestimated, could lead to conflation of the STEM disciplines, misconceptions about their several natures, or ill-informed decisions about STEM pathways and careers (e.g., Pleasants and Olson 2019; Chalmers et al. 2017; Cunningham and Kelly 2017). They signal the importance of students learning what it is to think like a STEM professional and developing the requisite habits of mind, and suggest that “[c]reating a STEM learning environment can be accomplished by examining the nature of each discipline and considering what is alike and what is different about the core content areas integrated into STEM learning experience” (Moore et al. 2016, p. 9). As a rule, however, these differences are merely labeled as fundamental, without further analysis, while the focus usually remains on the similarities.

This paper explains and compares the epistemology of two common STEM practices: modeling and argumentation. More concretely, we make a comparative analysis of the features and functions of modeling and argumentation in mathematics and in the natural and engineering sciences, based on related accounts developed by philosophers, scientists, and educators in each discipline. From this analysis, we identify differences and similarities that can contribute to a solid basis for the perception especially of an integrated STEM epistemology, i.e., which ideas correctly reflect the nature of STEM as a whole and should be taught. In other words, the comparative study is intended to contribute to the research towards the development of an integrated STEM epistemology; this will strengthen the epistemological aspect and related educational benefits of integrated STEM education, which are unrepresented in the research.

In our analysis, we identify and elaborate the differences in modeling and reasoning in mathematics and the natural and engineering sciences before proceeding to their commonalities. This differentiated approach to STEM epistemology presents a more accurate and authentic picture of contemporary major STEM research programs and can show, without conflation of their epistemological identity, how the several disciplines contribute to their effectuation. In other words, our interest centers on an integrated STEM epistemology that can both differentiate and correlate the discrete epistemological features of the several STEM disciplines and reveal the benefits that students may derive from understanding these various ways of thinking and knowing, as well as when each is most appropriate, and the potential gain from combining them, which can best be attained through integrated STEM units. A differentiated epistemological conception of STEM practices (here, modeling and argumentation) reveals the interrelated diversity of ways of thinking and reasoning in the STEM disciplines and can convey and nurture mindsets and skills useful for future STEM professionals’ effective participation and communication in contemporary interdisciplinary research projects and, more generally, for enabling future citizens to cope with socio-scientific issues. (Socio-scientific issues and their relation with STEM is a wide and complex field that goes beyond the scope of this paper.)

Moreover, while papers dealing with a practice-oriented integration approach tend to focus narrowly on the application of STEM practices to real-world problems and the designing of solutionsFootnote 5 (a pragmatic approach which might imply a technocratic and career-focused perspective, one might say), our focus on the nature of STEM aims more at the intellectual benefits deriving from this knowledge. The object, in other words, is to show the diversity of ways of inquiring and reasoning in the STEM disciplines and how they interrelate, leading thus to a conception of STEM as a system with a characteristic way of approaching things and problems, a characteristic way of thinking, acting, and viewing the world. Also, while we accept the argument that STEM education can make an important contribution to dealing with the big global challenges of today, we are critical of the argument that relates the necessity of improving it solely to strengthening the competitiveness of national economies.

In Sections “Models and Theories in Mathematics—Philosophical Views” and “Models and Theories in the Empirical Sciences,” we describe models and modeling, focusing on the features specific to each STEM discipline, and discuss some of their differences and commonalities. In Sections “Comparing Arguments in Mathematics and the Empirical Sciences” and “Views on Scientific Argumentation,” we look at and compare the kinds of arguments that are used in the development and validation of knowledge in these fields, based on traditional and contemporary philosophical views. In Section “Mathematics and Science in the Field of Mathematical Modeling,” we describe mathematical modeling, which is the product of the collaborative interaction of mathematics and science, and show how this combination expanded the potentialities of scientific research. Some conclusions from these comparisons and analyses are tabulated at the end of Sections “Modeling in the Engineering Sciences” and “Mathematics and Science in the Field of Mathematical Modeling” and correlated with teaching in Section “Some Suggestions for Conveying Epistemological Ideas in Teaching.” In Section “Summary and Conclusions,” we summarize the main points of the analysis in relation to the conception of an integrated STEM epistemology and its educational benefits.

2 Models and Theories in Mathematics—Philosophical Views

The theory of models was initially strongly connected with logic but later developed views more closely reflecting the actual praxis of mathematics and the empirical sciences. These initial views nonetheless laid the foundations for standard views in both the philosophy of mathematics and the philosophy of science.

A standard definition of the concept of the model in mathematical logic is that given by Tarski: “A possible realization in which all valid sentences of a theory T are satisfied is called a model of T” (Tarski 1953, p. 11). Theories here mean axiomatic, deductively connected systems of linguistic statements, such as the various mathematical theories (e.g., Euclidean geometry, set theory, and number theory), while an example of possible realization is given in Suppes (1960): “For example, a possible realization of the theory of groups is any ordered couple whose first member is a set and whose second member is a binary operation [e.g. addition, multiplication, etc.] on this set.” In sum, “a theory here is a linguistic entity consisting of a set of sentences [the syntactic approach to theories] and models are non-linguistic entities in which the theory is satisfied” (Suppes 1960, p. 290).

In this conception, a model is a set-theoretical entity, i.e., a set of objects with precisely defined features and operations, which fulfills the axioms of the theory. In logic and pure mathematics, then, models provide an interpretation or instantiation of formalized (non-interpreted) axiomatic systems/theories (the instantial view) (see e.g., Giere 1999b). More specifically, a model provides an interpretation of a formal axiomatic system through the substitution of meaningful elements (e.g., numbers) for its formal terms, which leads to true, valid statements. For example, a model in Euclidian geometry is the interpretation of planes (or lines) as a set of solutions of linear equations, where e.g., a specific plane is interpreted as the solution of the eq. 2x + 3y + 4z = 0.

Much research in mathematics involves the construction and use of models, e.g., to investigate and demonstrate the consistency and completeness of a theory, or to establish representation theorems (Suppes 1960). Mathematics, however, is also involved in the solution of theoretical and practical problems in other fields as well, and thus in empirically based modeling, and specifically in the mathematical modeling of the natural and social sciences (see Section “Mathematics and Science in the Field of Mathematical Modeling”).

These accounts were of interest to and had a significant influence on the philosophy of science, since they were directly connected with its own analyses of the nature of scientific theories and models. Suppes (1960), for example, argued that the concept of the model in Tarski’s sense, i.e., as a set-theoretical entity, a set of objects that satisfies the axioms or principles of a theory, can be applied without distortion to the models of the empirical sciences. In other words, the concept of the model is fundamentally the same in the empirical sciences as in mathematics, since “the physical model may be simply taken to define the set of objects in the set-theoretical model” (p. 290): for example, in the case of classical particle mechanics, “[w]e can simply take the set of particles to be, in the case of the solar systems, the set of planetary bodies” (p. 291).Footnote 6 Suppes made it clear that the application of Tarski’s concept to scientific models is confined to evolved scientific theories whose formulation can satisfy “the standards of exactness and clarity customary in the axiomatic formulation of theories in pure mathematics” (p. 294), i.e., to scientific theories which can be reconstructed as axiomatic systems, like mathematical theories.

Within that framework, one might say that the essential difference (at least in the context of the standard views) between models in the empirical sciences and in mathematics is that in pure mathematics models are basically abstract entities (related e.g., to numbers, geometrical points and lines) used to instantiate or interpret formal systems, whose development is not committed to any relation with the real world, while in the empirical sciences (such as the natural and engineering sciences, statistics, psychology, or sociology) models are simplified representations of aspects of the real world intended to explain and predict phenomena, and their relation to the real world is of critical importance.

More recent work on model theory introduces the notion of structure (like the groups of group theory), for a more generalized definition of models. Models are now structures in which the axioms of the theory are true: “It would, therefore, be misleading to refer to the models of contemporary mathematical model theory exclusively as interpretive or instantial models” (Giere 1999b). We must also point out that the field of mathematics has also produced accounts that criticize the traditional view of models “as just instantiations of formal structures, useful at best for purposes of presentation and communication to the non-specialist (for instance learners), but inessential to (pure) mathematics itself, which was conceived as a strictly deductive system of statements” (Glas 2002, p. 95) and explore the relation between modeling and argumentation in mathematics and the general theories of models and arguments (see Section “Contemporary Accounts of Argumentation in Mathematics”).

3 Models and Theories in the Empirical Sciences

3.1 Philosophical Views on Scientific Models and Theories—Initial Conceptions and Developments

As for mathematics, so for the empirical sciences philosophical analyses of theories and models were initially strongly influenced by logic but have since evolved into something closer to the praxis of real science. The philosophy of science moved from the statement view, which focused on a logical analysis of theories, to historically oriented approaches that criticized the statement view as not reflecting real scientific practice (e.g., Duhem 1991; Kuhn 1970, 1996) and to the semantic or model-based view, which is now a basic framework in historical and sociological studies of science as well.

The statement view sees scientific theories as sets of (linguistic) statements (basically axioms and principles) and axioms as immediate descriptions and universal laws of the real world (see e.g., in Suppe 1977; Giere 1999a), and it sought an axiomatic reconstruction of the scientific theories, which would permit them to be tested logically. The statement view was criticized, with various arguments (see e.g., in Suppe 1977), as not reflecting real science: for example, not all scientific theories are susceptible to axiomatic reconstruction, nor can a deductive process always be applied to, and provide an adequate basis for, the justification or choice of scientific hypotheses and models (see Section “Philosophical Views on Argumentation in the Empirical Sciences”).

In comparison with the statement view’s linguistic-syntactic approach to theories, the semantic or model-based view focused on models and their relation to theories and the real world, recognizing them as the basic tools of scientific research and reasoning. In this view, theories consist basically of a set or sets of models, while axioms and principles are not understood as universal statements, as in the statement view, but rather as providing the basis and the rules for the construction of models (Giere 1988, 1999a). Some basic accounts of this view are given by, e.g., Suppe 1977; van Fraassen 1980; Cartwright 1983; Giere 1988, 1999a; Morrison and Morgan 1999; Knuuttila 2011; Nersessian 2008).

The semantic view’s conceptions of models, especially in its early phase, are largely similar to the model-theoretic conceptions of mathematical logic. A model is defined as any structure in which the axioms of a theory are true (Suppe 1977). In structuralist analysis, for example, the models of a theory are defined as abstract structures which can contain and be applied alike to abstract entities like numbers or to physical objects; in this sense, the content of a theory is more than its empirical applications. For the scientist, however, the object is to include (structures of) physical systems as instances of the application of these abstract structures by correlating their (abstract) symbols with physical properties of the physical systems (Grandy 1992).

In sum, in the semantic view, theoretical models have been defined as abstract structures that fulfill the basic equations and principles of the theories plus some additional specific functions that specialize the general theory for application to specific categories of real systems and phenomena.Footnote 7 This conception means that theories are true and precisely applicable to models but less so for the real world, since models contain a series of abstractions, idealizations, and adaptations of the theory in order to make it applicable to the very complex real systems; models thus have a mediating role between theories and the real world (see in Suppe 1977; Giere 1999a; Morrison and Morgan 1999). The interpretation of the relation of scientific theories to the world through the mediating role of models is a major contribution of the model-based view to the understanding of scientific knowledge.

We distinguish different types of models according to their nature and function in the processes of the acquisition and transmission of knowledge. Two basic categories are mental models, which are internal, personal, and incomplete representations of things and functions in the world (Johnson-Laird 1983; Vosniadou 1996), and expressed models, which are presented to others for interaction, criticism, and the transmission of knowledge and ideas. Cognitive psychology argues that the processes of learning, reasoning, and decision-making are essentially carried out through the construction and manipulation of mental models. Under the influence of these positions many science education accounts have been developed in which the learning process and conceptual change are considered to be achieved through the progressive development of intermediate models directed towards the creation of the target model, a concept that implies the involvement of models and modeling in instruction as most suitable for facilitating this (for the students, not easy) cognitive outcome (Clement 2000; Greca and Moreira 2000).

Expressed models are made public in a variety of forms, including material models (scale models), visual models (pictures, drawings, diagrams), and more abstract models like the theoretical or conceptual models discussed above. Education scientists have investigated the nature of these models and their use in science teaching for presenting information and for reinforcing students’ abilities to reason, inquire, explain, and communicate ideas (e.g., Gilbert and Justi 2016; Clement and Ramirez 2008). Gilbert and Justi (2016) describe the characteristics of these models and compare their capacity to represent particular aspects of a represented system: diagrams, for example, provide qualitative information about the (models of) modeled systems, while graphs provide quantitative information about the temporal or spatial evolution of their variables.

Theoretical models are the most basic and useful, because they can address complex problems and because the further specialization and elaboration of their structure yields more realistic models or new models for representing more cases of the phenomenon. (For example, the very idealized model of the harmonic oscillator yields the more specialized models of, for example, damped or driven oscillation). They also yield more exact quantitative predictions, which make possible safer experimental tests and technological applications.

The construction of theoretical models is guided by the fundamental theories relevant to the modeled systems, e.g., classical mechanics, quantum mechanics, or the theory of evolution; they are thus mathematical models (see Section “Mathematics and Science in the Field of Mathematical Modeling”) if the theory is mathematicalized. The relation of models to theories described above provides a powerful conception of scientific modeling, especially when well-established theories for the systems/phenomena to be modeled and investigated already exist. This is not, however, always the case, and in research, where there are as yet inadequate theoretical accounts for the phenomena studied, models often precede the theory and contribute to its elaboration (see e.g., Nersessian 2008; Morisson and Morgan 1999).

The semantic view’s initial conceptions of models and their relation to theory became progressively more concrete and practice-oriented, a development mediated, e.g., by Giere (Giere 1988, 1999a, 1999b, 2010). Giere (1999b) thinks, for example, that while the aforementioned instantial conception of models is very useful, particularly “in the study of formal logic and the foundations of mathematics,” it is not the best concept for scientific models. He proposed replacing it with a representational understanding of scientific models, which “takes models not primarily as providing a means for interpreting formal systems, but as tools for representing the world” (p. 44). Giere introduced more generally a shift to the practical use and functionality of models, which was reinforced by many other accounts that criticize, complement and extend the semantic view, emphasizing a relative independence of models from theory and their involvement in multiple scientific practices beyond that of representation, for example, simplifying, explaining, abstracting, arguing, predicting, and designing experiments (e.g., Morrison and Morgan 1999; Knuuttila 2011; Gilbert and Justi 2016).

3.2 Basic Features of Models and Implications for Scientific Realism

The model-based view is now a basic framework also in historical and sociological studies of science, which have further developed and extended the model-based approach from their own perspectives. Based on fundamental accounts of models and modeling, drawn mainly from the philosophy of science, we summarize the basic features and functions of scientific models:

Models are idealized representations of real systems and processes, which means that they represent real systems with regard to certain aspects only and to a certain degree of accuracy (e.g., suns with point masses); they are conceptual and perspectival, which means that their development is based on our theoretical approaches to and conceptions of the world (e.g., a deterministic conception of processes or a particle conception of matter or a quantified or continuous energy exchange approach) and reflect the specific perspective of their underlying theory (e.g., the electromagnetic perspective in physics or an evolutionary perspective in biology); they are context-dependent, which means that the construction and choice of a model depends on how well it meets the objectives and the accuracy requirements of a specific investigation which explains the simultaneous existence of multiple models for the same phenomenon (for example, depending on a specific study of the behavior of matter the researcher will choose either the particle or the quantum mechanical model of matter); and they mediate the application of the theories to complex real world systems, because the real systems are too complex and the theories too general and abstract for direct application.

Models are intended to provide explanations and predictions about phenomena that are compatible with existing theoretical knowledge and empirical data, and in order to fulfill this function modeling is based on a continuous interaction between theorizing, experimenting, and justificatory reasoning (see e.g., Nola 2004). All the different kinds of models share these features, to a greater or lesser degree.

The perception of scientific knowledge essentially as models, i.e., as idealized and perspectival representations of the real world, is directly related to the fundamental theme of scientific realism, the relation of scientific knowledge to reality. The debate on scientific realism is a major issue in the philosophy of science and is also central to science educationFootnote 8 relating to the question of which fundamental epistemological conception should underlie the teaching of the nature of science, but it is beyond the scope of this article and therefore will be touched on very summarily.

The main question concerning the relation between scientific claims and the real world is how far scientific theories, and especially their theoretical, non-observable entities, and properties (e.g., photons, quarks, and spin), are related to something real or whether their function is exhausted in the internal structures and aims of the theories. The two main opposing views (both with a variety of versions) are those of scientific realism on the one hand and empiricism and social constructivism on the other. Scientific realism holds that scientific theories and laws can, at least sometimes, reveal truths about the natural world and that truth is the ultimate aim of science (see e.g., Popper 1959; Boyd 1983, 1992; Devitt 1991; Musgrave 1988). The empiricists, on the other hand, perceive theories (especially those concerning the microscopic world) essentially as useful research instruments, e.g., for providing successful predictions and technically applicable knowledge, while social constructivism argues that the impact of internal and external social influences on the methodology of science (methodological traditions, conventions, hierarchies, etc., and external social and political factors) is sufficiently powerful to affect the evaluation of scientific knowledge in terms of truth and objectivity. The response of scientific realism, particularly to the important constructivist arguments and accounts, was the development of more realistic and sophisticated conceptions, which recognize the constructive element of scientific activity (the existence of constructivist episodes in the history of science, e.g., when racist theories appeared as biologically conditioned). It argued, however, that science has developed mechanisms to ensure objectivity in scientific knowledge, since scientists modify or radically change their theories in order to bring them into accordance with the outer world.

In the framework of the model-based view, the two basic conceptions that have been developed are van Fraassen’s constructive empiricism and Giere’s perspectival realism. Van Fraassen (1980) holds that the purpose of science is to formulate empirically adequate theories, i.e., theories that correctly describe the observable aspects of the world; in other words, only the empirical substructure of theoretical models, that relating to observable phenomena, needs to match those phenomena. Theoretical entities and their properties, postulated for the modeling of unobservable areas of the world, could in fact really exist, but that is something we cannot know with empirical certainty, nor is science committed to proving their existence. Perspectival realism, on the other hand, as Giere (2006) explains, “makes room for constructivist influences in any scientific investigation. The extent of such influence can be judged only on a case-by-case basis, and then far more easily in retrospect than during the ongoing process of research” (p.15). Giere also rejects “all claims to absolute truths” of strongly objectivist realism (p. 16); he also makes it clear that perspectival realism does not mean a silly relativism, in the sense that “every perspective is regarded as good as any other” (p. 13), which would imply that a comparative evaluation of perspectives is therefore meaningless.

The similar questioning of the fundamental epistemological conception of the relation of scientific claims to the real world and the related arguments that have developed in science education usually reject extreme social-constructivist positions in favor of more moderate and sophisticated realistic views (e.g., Matthews 1994; Koponen 2007; Tala 2011; Wong and Hodson 2009): for example, Koponen (2007) argues that a combination of empirical reliability (in van Fraassen’s sense) and minimal realism (rather than a starkly realistic interpretation of models) is sufficient to give an authentic image of models in physics education. In another article (Develaki 2019), we have suggested that perspectival realism (in Giere’s sense) can meet the considerations and objectives of science education for an epistemologically solid and pedagogically adequate basis for supporting an understanding of NOS and familiarizing students with scientific practices.

The relation of knowledge to reality, of scientific claims to the real world, is also an element in the comparison of science and mathematics, since it is crucial for the empirical sciences but less so for mathematics (see Sections “Models and Theories in Mathematics—Philosophical Views” and “Mathematics and Science in the Field of Mathematical Modeling”).

3.3 Modeling in the Engineering Sciences

Engineering and technology are fields that particularly rely on constructing, evaluating, and improving models. Engineering modeling involves all the different kinds of models: material/scale models (physical prototypes), theoretical-mathematical models, and computer simulation models; and models and modeling should therefore be basic epistemological topics in engineering education.

With regard to modeling in the engineering sciences, the methodology of the construction of models does not differ substantially from that of the natural sciences, insofar as a fundamental part of this work concerns the study, e.g., from the point of view of physics, chemistry or biology, of the properties and interactions of the systems constructed or designed. “Somewhat like the models developed by scientists, engineering models support the exploration of a system in order for predictions to be made and understanding to be developed. They are refined, reflect existent evidence, and promote clarification in existing knowledge.” (Antink-Meyer and Brown 2019, p. 548).

Where the difference in modeling practice between the natural and the engineering sciences is fundamental, however, is in terms of the purpose for which the models are constructed, which in the first case is the acquisition of knowledge or explanation of the phenomena modeled and in the second is intervention in the phenomena (Boon 2011; Cunningham and Kelly 2017; Antink-Meyer and Brown 2019). “Significant differences between the two practices lie in the role attributed to phenomena of interest and the epistemic purposes of their scientific modeling. In the natural sciences, phenomena and their descriptions and scientific models are usually considered as the gateway to knowing how the world is. In the engineering sciences, these elements are firstly crucial for intervening in the world.” (Boon 2011, p. 3).

Engineering modeling does have also some distinguishing characteristics, among them the great specialization and adaptation of the scientific and mathematical knowledge used in the specific system modeled, that is, the endeavor to construct the most realistic model possible, which requires additional engineering knowledge and techniques. “Technologies do not exist as abstractions, and an overreliance on reductionism can cause inattention to important real-world complexities” (Pleasants and Olson, 158; see also Antink-Meyer and Βrown 2019).

One consequence of this pursuit is the more frequent use of scale models/prototypes in engineering, which means a time-consuming and laborious iterative process of building and experimenting with multiple versions (Pleasants and Olson 2019; Cunningham and Kelly 2017 Antink-Meyer and Βrown 2019); this, however, has become much easier through the use of software and the construction of computer simulation models. More concretely, engineers frequently have to solve problems for which there is no corresponding scientific theory and are thus forced to experiment with the construction of multiple scale models with varied design parameters until they achieve a good, workable solution to the problem (Cunningham and Kelly 2017). Today, running a computer simulation program has replaced the laborious, time-consuming, and costly process of constructing multiple physical prototypes. Computer simulations enable the quick running of, and immediate feedback for, numerous modified or alternative simulation models, thus permitting their comparative evaluation. Knowledge of the methodology and epistemology of computer simulations (see Develaki 2019) is an important element of engineering training and STEM education more generally.

Another significant difference in modeling practice between engineering and the empirical sciences concerns the criteria governing the final choice between alternative design models/solutions, as we shall see in Section “Argumentation in the Engineering Sciences.”

In Table 1, we summarize some basic points from the comparative description of the models in mathematics and the empirical sciences that were discussed in this and the preceding sections “Models and Theories in Mathematics—Philosophical Views”, “Philosophical Views on Scientific Models and Theories—Initial Conceptions and Developments,” and “Basic Features of Models and Implications for Scientific Realism.”

Table 1  A comparative outline of the concept of ‘model’ in mathematics and in the empirical sciences
Full size table

4 Comparing Arguments in Mathematics and the Empirical Sciences

4.1 Some Elements of the Argumentation Theory

Argumentation is inherent in every human activity: in thinking and doing, in decision-taking and communicating. Improving students’ ability to argue is a fundamental object in every school subject, and one which is promoted in science education especially through familiarization with scientific argumentation. “Argumentation is a critically important discourse process in science and it should be taught and learned in the science classroom as part of scientific inquiry and literacy” (Erduran et al. 2015, p. 1).

Toulmin (1958) offered a structural, domain-independent schema for the argument, which analyzes the argument into 6 elements (data, claim, warrant, backings, qualifiers, and rebuttals), upon which the strength and soundness of an argument can be assessed. Toulmin’s schema is widely used, for it is broad enough to give it a great theoretical compass, but it is criticized in terms of the practicality of its application in the analysis and assessment of the arguments that are used in various specific fields. In science education, for example, it has been found inadequate for the assessment of the quality of students’ arguments, and more extended conceptions are proposed that take into account the domain-specific aspects of the argumentation, i.e., the correctness of the content of the argument and the argumentation practices and norms of the specific domain within which the argumentation is developed, as well as other perspectives such as the dialogical and persuasive elements of the argumentation (e.g., Erduran et al. 2004; Sampson and Clark 2008; Bricker and Bell 2008; Osborne 2010; Böttcher and Meisert 2011).

Context has a significant influence on the intentions of the argumentation and the kinds of arguments used, and more specific accounts have been elaborated for different fields, e.g., the political, the legal, the medical, and the academic (see e.g., in Aberdein 2009; Voss and van Dyke 2001; van Eemeren 2016; Erduran and Jimenez-Aleixandre 2007). In science, for example, argumentation means the coordination of evidence and theory in order to support or refute explanatory conclusions and their underelying models and hypotheses (Suppe 1998; Erduran et al. 2015); in philosophy, an argument “is typically defined as a connected series of claims intending to establish some conclusion, e.g., a sequence of claims with an inference, i.e., a logical move, to a conclusion/contention.” (Davies et al. 2019, p.12).

We examine here some of the basic accounts of argumentation in mathematics and in science, mainly from the philosophies of those domains. From these analyses, we identify elements of argumentation common to both fields and differences between them, the former lying mainly in the development and elaboration of hypotheses and proposals and the latter affecting their acceptance or choice.

4.2 Arguments in Mathematics and in Science—Induction, Deduction, and Abduction

Argumentation in mathematics involves different processes and strategies, the most characteristic being mathematical proof (see in ICMI Study 19 (2009). The common view is that mathematical proof is typically based on formal deductive processes, although, as we shall see in the next section, this is frequently not the case (see e.g., Hanna and de Villiers 2012; Aberdein 2009). Formal deductive proof is, however, held to constitute complete and valid mathematical proof. In the traditional philosophy of mathematics, this science is basically concerned with rigorous proofs based on formal, logical deductive arguments (Aberdein 2009). Deductive arguments are the standard for acceptable proofs of statements in (pure) mathematics (Glas 2002).

Deduction and induction are standard argumentation forms in both mathematics and the empirical sciences. It must be made clear that induction and deduction are domain-independent inference schemes: that is, they are applied in the same way in any field regardless of the content of their premises and conclusions.

Deduction is the process in which, by following a series of well-defined inferential steps, specific statements (conclusions) are derived from an initial set of statements (premises) that are taken to be true; the truth of the premises guarantees the truth of the conclusion (Popper 1959; Lawson 2009; Uhden et al. 2012; Glas 2002). Thus, theorems, for example, are true, are considered proven to be true, because they are logically deduced from the axioms or other theorems of the theory.

Induction, on the other hand, is the inference of a general conclusion from a (limited) number of premises. In this case, the conclusion does not follow necessarily from the premises; the premises provide some evidence for the conclusion but not a guarantee. In mathematics, inductive conclusions are merely “conjectures” of little value if they cannot stand up to rigorous deductive proof (Popper 1959; Lawson 2009). We should add here that many authors interpret induction and its operation in mathematics in a subtler way, namely as a process that helps judge the possible truth of a hypothesis rather than simply as a progression from particular instances to generalized conclusions (see e.g., Lawson 2009; Mariotti 2006). Lawson (2009), for example, explains that, in Peirce’s sense (Peirce 1931–1958), certain observations concerning a phenomenon provide evidence for an explanatory hypothesis, conceived in some way, e.g., by abduction (see below), accounting for those observations, while further predictions deduced from this hypothesis, if confirmed, provide additional evidence for the hypothesis and thus increase the probability that it is correct. Hempel’s 1966 statement that “[w]hen a hypothesis is designed to explain certain observed phenomena, it will of course be so constructed that it implies their occurrence; hence, the fact to be explained will then constitute confirmatory evidence for it” (p. 370) is in the same vein. This approach also links induction, deduction, and abduction in a whole process of scientific argumentation.

Argumentation in the empirical sciences (which in many domains are increasingly mathematicalized) has commonalities with argumentation in mathematics, but there are also significant differences. First of all, in science, as in mathematics, the commonest and most typical modes of argumentation are inductive reasoning (inference from specific cases to general conclusions) and deductive reasoning (inference from certain premises to necessary (testable) conclusions).

In comparison with (pure) mathematics, where the premises and conclusions of induction and deduction concern the abstract entities of the mathematical construct, in the empirical sciences they concern entities and statements about the real world. Thus, induction in the empirical sciences means the inference of universal statements about a phenomenon or system from a limited number of specific observations and/or experiments (e.g., inferring the laws of gases from relevant observations with some gases).Footnote 9 Deduction in the empirical sciences means the derivation of specific conclusions (predictions) about a phenomenon from a general principle/hypothesis/model, which are then compared to empirical data to check the validity of the initial general principle/hypothesis/model from which they were deduced: for example, the value of the lunar period, which is calculated (deduced) from the law or principle of universal gravitation, is compared to the value determined by observations with a telescope. Deductive-based arguments are the most powerful tool for proving scientific claims and, where possible, are welcome not only because the predictions they furnish provide a basis for the evidence-based evaluation of hypotheses and models (which is very important when these concern non-observable micro- or macro systems and processes) but also because they spark further research and discoveries.

To summarize, we may say that while as a rule in mathematics induction leads to conjectures, which have little value unless they are rigorously proved, in science it leads to an acceptable, and initially at least legitimate, hypothesis or model that can continue to be explored; here, too, the desiderandum is, insofar as possible, deductive-based tests of these hypotheses and models which will lead to their validation and establishment.

Deduction in mathematics is used to infer true conclusions from true premises (e.g., theorems from axioms or other theorems): that is, the proof concerns the truth of the conclusion; deduction in science is used to draw conclusions that will provide evidence, supporting or otherwise, for the hypothetical premises from which they were deduced: that is, the proof concerns the truth of the premises. This last point is one on which we must comment a little more. In science, the premises are usually hypothetical, because most scientific research concerns hypotheses and theories that are in the process of being formulated and tested and the deductive proofs are developed to verify such cases through the predictions furnished by their models. If however the proving deductive process starts from models derived from well-established fundamental theories (such as e.g., classical mechanics or quantum theory), that is, if the premises are essentially the axioms of fundamental theories, then, as in mathematics, the deduction must (if the models are correct) lead to true conclusions: the predictions must agree with the empirical data. If this is not the case, then scientists will initially conclude that the problem lies not with the theory but with the models or with the way in which the predictions were derived and experimentally tested, and these will be revised, improved, re-examined, etc. Rarely do unverified predictions lead scientists to contemplate revising or rejecting a fundamental theory (see the example about the discovery of the planet Neptune below). If, in spite of all, the observed phenomena continue to be unpredictable-inexplicable or to increase in number, then the scientists begin to challenge the theories as unsatisfactory and propose changes to or replacements for them, in the spirit of Kuhn’s scientific revolution (Kuhn 1996).

While the deductive proof of statements in (pure) mathematics follows a formal deductive process that is bound by the logic of inferential rules but not by empirical facts, the deductive-based proof in science is a complex process of justification that seeks to provide evidence for the models examined and their underlying hypotheses and to evaluate that evidence (see Sections “Philosophical Views on Argumentation in the Empirical Sciences” and “Mathematics and Science in the Field of Mathematical Modeling”).

Alongside inductive and deductive reasoning, science also makes use of other types of argument that provide support in many cases, for example abductive reasoning. Charles Sanders Peirce (1931–1958) introduced the term “abduction” to characterize a type of inferential reasoning distinct from both induction and deduction and described it as the process of generating explanatory hypotheses for puzzling observations. Abduction is thus related to the discovery process of generating hypotheses, which are then subjected to further testing in order to be ultimately justified. Further elaborations of abduction addressed the fact that there can be several hypotheses capable of explaining the observed phenomena and the question of how one decides which of them is most likely to be true and thus merits further development (Harman 1965). The problem of choosing among alternative explanatory hypotheses (and on what criteria) led to the redefinition of abduction as “inference to the best explanation,” a conception introduced by Harman (1965), who also mentions some of the criteria that could direct the choice: “Presumably such a judgment will be based on considerations such as which hypothesis is simpler, which is more plausible, which explains more, which is less ad hoc, and so forth” (p. 89).

In sum, abductive reasoning is usually understood as inferring the “best” explanation from a set of alternatives and is used to suggest explanatory hypotheses which would not be possible by more standard reasoning processes (induction, abduction). Abductive argumentation may be based, for example, on analogies (e.g., a successful explanation for one situation is applied to a new situation as a tentative explanation of it), on thought experiments, or on good and plausible reasons that can make the abductive explanation the “best explanation” for a phenomenon.

Abductive reasoning has always been a fundamental tool of scientific methodology (Psillos 2002; Lawson 2009; Adúriz-Bravo and Pinillos 2019). For example, the hypothesis (Leverrier 1846) that there was another as yet unobserved planet that was responsible for the deviation of Uranus’ orbit from that predicted by Newton’s gravitational theory was considered a better hypothesis than the alternative, namely that Newton’s long-established and successful theory was false, and indeed the former hypothesis was proved correct with the discovery of the planet Neptune. Creating alternative hypotheses and choosing between them is also a basic element of inquiry-based classroom activities. Students’ engagement in stating and evaluating alternative hypotheses/proposals/models promotes a higher level of reasoning in STEM classes. Abductive argumentation is proposed as an appropriate and effective strategy for supporting students’ reasoning abilities and more generally for teaching science and nature of science (see e.g., Adúriz-Bravo and Pinillos 2019; Lawson 2009; Motta and Pappalardo 2012; Svoboda and Passmore 2013; Kwon et al. 2006).

4.3 Contemporary Accounts of Argumentation in Mathematics

The traditional view that mathematical reasoning is based solely on rigorous deductive argument has been challenged in recent accounts, which, drawing from the history and the real practice of mathematics, argue that other informal forms of argumentation also take place in the “context of discovery” of mathematics and beyond (e.g., Aberdein 2009; Glas 2002).

As mentioned in Section “Models and Theories in Mathematics—Philosophical Views,” Glas (2002) noted the role of modeling and other non-deductive forms of reasoning in the development of important conceptual innovations, based on Klein’s model-based practices, which led to his contribution to the development of the different geometries. Although Klein acknowledged the importance of axiomatization for providing the body of mathematics with the rigorous logical skeleton needed to make intuitive notions more precise and to provide a solid logical basis for already established theories, he did not conceive of mathematics primarily as a deductively connected system of propositions but construed it holistically as a collection of interconnected models and its method as consisting essentially of imaginative modeling and model-based reasoning, “all the way ‘up’ to the axioms and all the way ‘down’ to its remotest social and cultural implications” (cited in Glas 2002, p.100).

The view that deductive proof is supreme has also been challenged in modern mathematics. Aberdein (2009) thinks that the assumption of the truth of axioms undermines the basis of formal deductive proof in mathematics, since axioms have not been formally deduced from some premises. Axioms are postulated and assumed to be true because of their plausibility or intuitive wide acceptance, but they also must have the potential to axiomatically organize the knowledge content of the field (Glas 2002). Khait (2005) noted that “there is a trend in the contemporary philosophy of mathematics that emphasizes the possibility that many mathematical facts cannot be proved and just happen to be true (…). This is a further development of the famous Gödel Theorem that states that there exist true statements concerning natural numbers that cannot be proved in principle” (p.143).

Aberdein (2009) observes that the traditional view of mathematics ignores much of mathematical practice and argues, with examples, that in real practice not all mathematical proofs are strict, formally valid, logical derivations and that “beyond the characteristic practice of proof, mathematical argumentation also includes arguments as described in the general argumentation theory” (p. 3), informal arguments like model-based reasoning, use of analogies, probabilistic reasoning, thought experiments, and so on.

In the literature of mathematics education, too, a distinction is drawn between argumentation and proof. Argumentation in mathematics is approached as a broader practice that, apart from formal deduction, also includes other kinds of arguments, each with a different object, and their role and utilization in teaching is also examined. It is pointed out, for example, that “[u]nderstanding the relationship between argumentation (a reasoned discourse that is not necessarily deductive but uses arguments of plausibility) and mathematical proof (a chain of well-organized deductive inferences that uses arguments of necessity) may be essential for designing learning tasks and curricula that aim at teaching proof and proving.” (ICMI Study 19, p. xxii); and that “argumentation consists in whatever rhetorical means are employed in order to convince somebody of the truth or the falsehood of a particular statement. On the contrary, proof consists in a logical sequence of implications that derive the theoretical validity of a statement” (Mariotti 2006, p. 182).

Distinctions are also made between different forms of proving (e.g., verbal, visual, or formal) and between different kinds of mathematical proof (empirical proofs and more formal, deductive-based proofs): “Empirical proofs, characterized by showing that the conjecture is true only in one or a few examples taken from a larger set of examples and assuming that the conjecture is also true in all other examples in the set. Deductive proofs, consisting of chains of logical implications connecting the hypothesis to the statement of the conjecture, and characterized by the decontextualization of the ideas presented” (Fiallo and Gutierrez 2017, p. 147).

The distinction between argumentation and proof also recognizes a cognitive and epistemological rupture between the two. Doing argumentation (in the above broader sense) and doing mathematical proof presuppose different cognitive capabilities and also perception of the differences between the semantic and the formal-logical level of argumentation in mathematics. On the classroom level, this means a shift from argumentation and empirical proof to formal mathematical proof; this represents a major mental achievement (to overcome that rupture), which could explain students’ difficulty in understanding and making deductive proofs. Some lay great weight on the crucial importance of this cognitive distinction for the transition from argumentation to formal deductive proof, while others see a continuity between the two rather than a dichotomy and propose ways and teaching interventions to accomplish it (see e.g., in Mariotti 2006; Fiallo and Gutierrez 2017).

5 Views on Scientific Argumentation

5.1 Philosophical Views on Argumentation in the Empirical Sciences

As described in Section “Philosophical Views on Scientific Models and Theories—Initial Conceptions and Developments,” the philosophy of science moved from the statement view to the historically oriented and the model-based approach, each of which implied corresponding conceptions of scientific reasoning and argumentation. Viewing theories as statements, as linguistic entities that must be either true or false, the statement view bases the validation of scientific claims on terms of truth and correspondence and formally correct arguments (Giere 2001). In this framework, the logical empiricist view is that the justification of scientific theories is (and should be) based on a hypothetic-deductive model, mainly in the form of modus tollens: if a prediction is proved to be false, the general hypothesis/theory from which it is deduced must be also false, which means that critical experiments should be designed to eliminate the false hypotheses and theories among competing propositions (which however in practice is rarely possible). This explains the weight given to falsificatory procedures (see e.g., Popper 1959) and crucial experiments/tests, while the case of positive evidence is under-represented.

The hypothetic-deductive model of logical empiricism was criticized as simplified and not reflecting the complexity of testing and especially of choosing theories in real science. Showing a prediction to be false can be due to various other factors apart from the hypothesis/model itself, e.g., an error in the experimental process or mistaken auxiliary assumptions used in deducing the predictions (Duhem 1991). Moreover, historical-philosophical and sociological studies of science have shown that in many cases scientists assess the available empirical data differently depending on the paradigm within which they are working, and that in addition to empirical and logical checks an important role in their choice of theories is played by more pragmatic criteria and conventions.

The model-based view, which followed the statement view, takes the above criticisms into account and offers a more pragmatic interpretation of scientific research and argumentation, addressing problems and insufficiencies that emerged in the framework of the statement view. The model-based view regards theories not as linguistic statements but essentially as sets of models, which are seen as the main means for research and knowledge production, and therefore it is the adequacy of the model that is the main object of the checks and justifications. Models, being idealized and conceptual entities, cannot, in the strict sense, be true for the real world. The evaluation or choice of model is based here on the more moderate and pragmatic concepts of similarity or fit, i.e., how similar the models are to, or how well they fit, the real systems.

The similarity of the models to the real systems cannot be observed directly (e.g., in the case of models of microscopic or macroscopic phenomena); what is checked is how far the predictions of the models match the empirical data for the modeled real system (Giere 2001). In the model-based view, therefore, deductive processes also play a basic role (deduction of predictions), as in the hypothetic-deductive conception of logical empiricism; however, deductive reasoning is interpreted here more pragmatically than in logical empiricism, taking into account also socio-psychological criteria (social ties, pursuit of fame) (Lakatos 1970; Kuhn 1996; Giere 1999a) as well as the practical and conventional criteria that scientists usually apply (in addition to empirical and logical tests) when they evaluate their models (e.g., the interpretive, unifying, and predictive force or the simplicity of a model). This means that the evidence yielded by comparison of the model’s predictions to the data does not necessarily lead to unequivocal conclusions for the evaluation of the model, as we shall explain in Section “Mathematics and Science in the Field of Mathematical Modeling.”

5.2 Argumentation in the Engineering Sciences

In engineering and technology, the construction and evaluation or selection of products and design solutions require kinds of argumentation similar to those in science (described in Sections “Arguments in Mathematics and in Science—Induction, Deduction, and Abduction” and “Philosophical Views on Argumentation in the Empirical Sciences”), since they also examine the physical and chemical properties of materials and systems. As in science, engineering design and argumentation is based on data and evidence (Kelley and Knowles 2016; Antink-Meyer and Brown 2019), but while the basic goal of the natural sciences is the acquisition of explanatory knowledge about natural phenomena, in engineering it is the development of realizable and applicable solutions to specific systems and problems, which affects the priorities in their handling of data: “Observations and evidence are tied to the nature of goals in engineering as well, but design specifications frame data collection and interpretation” (Antink-Meyer and Brown 2019, p. 553).

This fundamental difference bears most strongly on the way in which engineers judge a solution or product and choose between alternatives. In engineering, the decision concerning the choice of a design or solution is influenced by and must satisfy additional criteria, which moreover are imposed by people and agencies outside the scientific community. The criteria determining the choice of a solution must be compatible not only with the physical/scientific constraints relating to the properties of a system but also with additional context-dependent criteria, constraints and parameters connected with such factors as reliability and risk, cost and marketability, esthetics, cultural traditions, harmful side effects,Footnote 10 and environmental pollution (see e.g., in Pleasants and Olson 2019; Kelley and Knowles 2016; Cunningham and Kelly 2017; Antink-Meyer and Brown 2019). The arguments for the proposed solutions must balance these factors and establish priorities among them: “Engineers consider problems in context, make trade-offs between criteria and constraints, and assess implications of solutions” (Cunningham and Kelly 2017, p. 493).

Discussing models and arguments in an integrated STEM context permits comparison of their features and functions in engineering and the other STEM disciplines, which could help future engineers understand the priorities and working methods of the other fields and therefore lead to more effective collaboration with other experts in modern STEM projects.

6 Mathematics and Science in the Field of Mathematical Modeling

The interaction between science and mathematics has historical roots and an impressive effectiveness in scientific research (see e.g., Galili 2018; Kanderakis 2016; Quale 2011; Brush 2015). Various scientific fields (e.g., physics, quantum chemistry, biotechnology, geophysics and climate sciences, engineering and computer sciences, sociology, and psychology) make intensive use of mathematical language and methods. For example, the axiomatic method, as developed in mathematics, for organizing knowledge into a logical system in which the several pieces of knowledge can be logically deduced from a few fundamental propositions (the axioms) had a great impact on scientists (Quale 2011), who use it in their own way wherever possible (as described above for deductive reasoning in science). Conversely, genuine problems in, for example, physics motivated important conceptual developments in mathematics, such as calculus and vector analysis (see e.g., Uhden et al. 2012).

The field in which the methods of mathematical and scientific reasoning most closely interact and reinforce one another is that of mathematical modeling, which is used in many branches of the natural, engineering, and social sciences. The construction of mathematical models brings mathematics and science together in a combination that has led to a radical extension of scientific research methods.

Mathematical modeling follows a series of specific stages. The scientist’s first task is to determine the conception of the model, i.e., the theoretical approach (e.g., classical mechanical-deterministic or quantum mechanical-stochastic) that will permit his model to capture the intended aspects of the real system, and to define the variables, parameters and initial conditions for the interactions. The variables and relationships of the model are then transformed into mathematical symbols and relations (equations),Footnote 11 thus forming the mathematical structure of the model. These structures are then further manipulated mathematically, with additional constraints and hypotheses, in order to give them the sharpness and precision required for mathematical treatment: the mathematical formulation of a model “requires a certain level of mathematical sophistication and, more importantly, it requires the definition of vague concepts and loose relations in strict mathematical terms” (Motta and Pappalardo 2012, p.415; see also Kneubil and Robilotta 2015). At this point, the equations deriving from the given set of specifications can be solved.Footnote 12

For the mathematician, all solutions are valid and accepted as long as they are technically correct; the empirical data are irrelevant. For the scientist, however, this point marks the beginning of a complex and fascinating process: the physical interpretation of these solutions (the model’s predictions), their meaning and connection with the phenomenon modeled, and the evaluation of the model based on comparison of its predictions with the existing empirical data and theoretical knowledge.

In other words, deductive reasoning is fundamental to proving knowledge in both mathematics and science, but with a characteristic difference between the two fields. In (pure) mathematics, it is used in its formal sense, where the soundness of the argument is paramount: i.e., the premises must be true and the inferential steps logically correct; this will lead to true conclusions, which are independent of any empirical data. In science, on the other hand, the premises (as explained in Section “Arguments in Mathematics and in Science—Induction, Deduction, and Abduction”) are usually hypothetical and their truth is determined by the data, i.e., by whether the deduced conclusions, the system’s predictions, agree with the data or not (see also Section “Philosophical Views on Argumentation in the Empirical Sciences”).

Mathematical models, especially the more realistic and those of complex systems, usually include differential equations whose solution yields the evolution of the (variables and behaviors of the) modeled system in time and space. The solutions of the equations of a mathematical model, i.e., its predictions, can be divided into two categories: those (termed retrodictions) which can be compared straightaway with empirical data, that is, which concern characteristics and behaviors that have already been observed and that a good model must reproduce and therefore explain; and those which concern behaviors and phenomena not previously observed and/or unexpected (real predictions).Footnote 13

In the first case, the agreement or otherwise of the predictions with the data provides positive or negative evidence for the model. However, neither negative nor positive evidence necessarily leads to unequivocal conclusions for evaluation of the model, and any decision based on it cannot be taken (at least in an initial phase) as certain and absolute. Scientists are critical and cautious in both cases, whether the evidence for the model is positive or negative (agreement or non-agreement between the predictions and empirical and theoretical data), since a negative evidence for the model under examination can be due to a variety of factors other than the model itself (Section “Philosophical Views on Argumentation in the Empirical Sciences”); and a positive evidence for the model under examination (i.e., a successful prediction) does not necessarily mean that it is the best of all possible alternatives. Positive evidence for a model under examination is of critical importance for the evaluation of this model only if the possible alternative models do not yield the same successful prediction; when an alternative model also yields the same successful prediction, the evidence is inconclusive and scientists base their evaluation or choice of models also on pragmatic criteria and/or seek to acquire more conclusive data and evidence (see Giere 2001; Giere et al. 2006).

The second category of results, that of unexpected predictions, creates an interesting and challenging situation for scientists and fosters research. When serious account is taken of these predictions and they are investigated, they can lead to radical improvements in the models and theories and to the discovery of new phenomena. Many such historical and contemporary examples exist, among them the prediction and discovery of the planet Neptune, the discovery of new elements to complete the periodic table in chemistry, the discovery of the positron (anti-matter), and the discovery of the gravitational waves predicted by the general theory of relativity (see e.g., in Uhden et al. 2012; Brush 2015; Quale 2011).Footnote 14

In Table 2, we summarize some basic points from the comparative description of the arguments in mathematics and the empirical sciences that were discussed in this and the preceding sections “Arguments in Mathematics and in Science—Induction, Deduction, and Abduction,” “Contemporary Accounts of Argumentation in Mathematics,” “Philosophical Views on Argumentation in the Empirical Sciences,” and “Argumentation in the Engineering Sciences.”

Table 2 Α comparative outline of the concept of ‘argument’ in mathematics and in the empirical sciences
Full size table

7 Some Suggestions for Conveying Epistemological Ideas in Teaching

This article’s comparative study of the models and arguments used in mathematics and the empirical sciences is intended first of all to contribute to the research towards the development of an integrated STEM epistemology. Some elements of the background used for this analysis of STEM modeling and reasoning, for example concerning the statement and semantic approaches to scientific theory, are therefore intended for researchers and educators rather than students.

The epistemological ideas drawn from the analysis are intended mainly for use at high school level, although some of them may be more meaningful and accessible at lower college levels. There might be some question as to which of the epistemological ideas discussed here are appropriate for teaching, and at what level. In our view, the usual subject matter at each level of secondary education provides a basis for exemplifying and reflecting on the epistemological aspects of STEM reasoning and modeling discussed in the previous sections.

As a general principle, these epistemological ideas should be introduced in the classroom in a progressing degree of sophistication, following the level of the framework course material. As has been said, the research suggests that an effective teaching of epistemological ideas should be explicit and contextualized in scientific content, and as far as possible contextualized in curricular content and activities. In science and mathematics the curricula tend to be spiral, with basic topics revisited at higher levels in greater theoretical depth. Similarly, the degree of sophistication of epistemological discussions of models and arguments accompanying the course contents and activities should increase as the students advance through their school years, from the more empirical approaches to cognitive objects in the lower grades to more theoretical treatments in the higher ones. (Similar proposals are made in the ICMI Study 19 (2009) for teaching/introducing epistemological ideas in mathematics, and in Abd-El-Khalick 2012 for teaching domain-general NOS ideas in science education.)

For example, the progression in mathematics education from more informal argumentation and empirical (inductive-based) proofs to more formal deductive-based argumentation and proof would be progressively accompanied by discussion of the use and nature of the arguments used in these processes, of the distinction between stating and proving a conjecture, and of the difference between empirical and formal proof of conjectures. In science and engineering education, the progression from more qualitative models in the lower classes to more quantitative, mathematical models in the higher ones would be accompanied by discussions initially of more elementary and later of more advanced ideas about the features and functions of scientific models.

Based on the particle models of matter used in the early secondary school years (grades 7–9) to explain and predict thermal and electrical phenomena, for example, the students could be introduced to certain basic elements of the nature and functions of models: e.g., that they are idealized and perspectival (they focus only on certain aspects of real systems), that their function is explanatory and predictive (which is of particular importance for studying especially phenomena that are not accessible through direct sensory experience), and that their evaluation is evidence-based.

Similarly, based on the mathematical models of Newtonian mechanics (kinematics and planetary movement) and models of light used in upper secondary school classes (grades 10–11), the discussion could embrace the mediating and contextual character of models (e.g., the introduction of simplifications necessary especially for the mathematical elaboration of (kinematic) phenomena, and the co-existence and use of more than one models for the same phenomenon, such as geometric and wave models for light); while the oscillation models used in the final years of high school provide a basis for exemplifying and discussing the potentiality of theoretical-mathematical models for the study of more complex phenomena and for enabling the generation (through the further specification/elaboration of their mathematical structure) of more realistic models (e.g., models for motion with friction) or of new models (e.g., models for damped or driven oscillation).

In addition, a synopsis of the above models could lead to discussion of the epistemological status of the scientific theories, i.e., their interpreting and unifying potential: classical mechanics, for example, interprets and unifies several (models of) phenomena—linearly or curvilinearly moving bodies, oscillating pendulums, the movement of planets—within its scope.

Furthermore, comparing qualitative and quantitative/mathematical models demonstrates the effectiveness of combining mathematics and science. For example, the qualitative planetary model of the atom predicts/explains the discrete emission spectra of gases, while the mathematical equations of the model yield quantitative predictions, i.e., predict also the wavelengths of their spectral lines (e.g., the visible colored lines of hydrogen that the students can observe in the school laboratory.

These represent some ideas for designing epistemology-oriented integrated STEM units. The designing of such units is largely absent from the designing of integrated STEM units to date and requires further research. The availability to teachers of such material, focusing for example on the different but closely related STEM practices and their interaction in research, would greatly strengthen the epistemological aspect especially of an integrated approach of STEM education and the preparation of tomorrow’s STEM workforce.

8 Summary and Conclusions

The basic argument for integrating some or all of the STEM disciplines is that “the whole is more than the sum of the parts,” which means that an appropriate and purposeful integration of STEM disciplines in education can lead to higher levels of knowledge, abilities, and epistemologies than teaching them separately.

This paper deals with the epistemological aspect of STEM, and concretely the nature of their common practices. Based on traditional and more recent accounts of the philosophies of mathematics and of science, we give a comparative analysis of the nature of the basic crosscutting STEM practices of modeling and argumentation and identify and elaborate the core commonalities, intersections, and differences between the two practices in the domains of mathematics and the natural and engineering sciences as these emerge from this analysis. Apart from their importance in the teaching and learning of STEM topics, modeling and argumentation provide a powerful means of linking the STEM disciplines together in an integrated STEM approach.

The comparative analysis aims on the one hand to demonstrate the diversity of ways of thinking and reasoning that have developed within the STEM disciplines, and on the other to show that despite their differences these are akin and interrelated, so that when taken as a whole the STEM disciplines offer a characteristic way of approaching problems and solutions and in general of perceiving the world. This, we might say, is the ultimate aim of an integrated STEM epistemology and literacy.

The current literature affirms the existence of significant differences in the practices of the various fields and notes the importance of identifying these so as to avoid conflating the epistemological aspects of the several disciplines, but provides no satisfactory in-depth description of these differences covering all the related fields. In our analysis, we focus on identifying and describing the core differences distinguishing modeling and reasoning in the different STEM fields.

Very briefly, and in the framework of the traditional views, in pure mathematics models instantiate formal systems, whereas in science they are a basic research tool for the acquisition of knowledge of the natural world, designed to represent and explain aspects of the real world, while in engineering models serve more towards intervention in phenomena than explanation of them. Comparing the role played by inductive, deductive and abductive reasoning in science and mathematics, we find that in mathematics reasoning is mainly concerned with proofs whose essential characteristic is sound formal deductive argument unconnected with empirical data, in science with the evidence-based evaluation of its claims and the critical assessment of that evidence, while in engineering it must not only satisfy the scientific criteria but also factor in additional externally-posed parameters and constraints. A comprehensive summary of this comparison is presented in Tables 1 and 2, and how the resulting epistemological ideas could be conveyed in teaching is given in Section “Some Suggestions for Conveying Epistemological Ideas in Teaching.”

We also give an account of contemporary approaches that question the traditional view of the nature of mathematics and examine and identify areas of mathematical work that imply an attenuation of the differences between mathematics and science and show up the similarities in the modeling and reasoning practices of the two fields (Section “Contemporary Accounts of Argumentation in Mathematics”). With the description of mathematical modeling (Section “Mathematics and Science in the Field of Mathematical Modeling”), a very helpful means of comparison between mathematics and science, we demonstrate both the differences between the practices of mathematics and science and the effectiveness of their integration and use in scientific research.

Understanding these differences provides a solid basis for moving on to the commonalities and intersection of STEM discipline practices. In other words, our intention with this analysis is to contribute to a perception of the interrelated diversity of STEM practices and the intellectual and pragmatic benefits this perception confers, from the educational point of view, on future citizens and scientists. Awareness of the points of view of the various STEM fields permits a more authentic reflection of contemporary science and helps students develop different ways of thinking and problem-solving in scientific as well as broader contexts.

The differentiation and subsequent interrelation of STEM practices helps students understand both the context-dependence of reasoning and decision-taking and the perception of STEM as a whole whose parts can be used in combination to solve complex problems that are beyond the scope of any single field; more generally, to acquire an awareness of what it means to think and work from a STEM perspective. As we noted in the comparative analysis, mathematics is concerned with the precision of meanings and proofs, science with evidence-based justifications of claims and the critical evaluation of the evidence, and engineering with considering and balancing multiple constraints and interests. Familiarity with the various mindsets and practices of the STEM disciplines (a possibility best provided by integrated STEM projects/lessons/units) can improve the ability of future STEM professionals to work effectively together, understand and communicate with scientists in other fields, devise the innovative multi-perspective proposals and solutions required by today’s interdisciplinary STEM projects, and more generally improve the ability of future citizens to tackle the big challenges of our age. The analysis and comparison of modeling and reasoning in the various STEM disciplines naturally also concerns the teaching and learning of the epistemology of each separate discipline, but aims particularly at the conception and constitution of an integrated STEM epistemology for the achievement of STEM literacy.