Relevance for Complex Systems Knowledge
STEM education comprises three dimensions, content knowledge, procedural knowledge, and epistemological knowledge. This paper talks about modeling and argumentation as a common STEM practice, since both 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, they enable cross-STEM connections to be made, and may this effectively link the several disciplines. This paper compares those two practices in each discipline to show the diversity of ways of inquiring and reasoning and how they interrelate.
Models and theories in Mathematics
A model is a set of objects with precisely defined features and operations, which fulfills the axioms of the theory. Models provide an interpretation of formalized axiomatic systems/theories and their interpretation through the substitution of meaningful elements for its formal terms leads to true and valid statements.
The essential difference between models in sciences and in mathematics is that in pure mathematics models are basically abstract entities used to instantiate or interpret formal systems, whose development is not committed to any relations with the real world. However, in the empirical sciences models are simplifies representations of aspects of the real world intended to explain and predict phenomena and their relation to the real world.
Models and theories in the Empirical Sciences
In Science, models are:
- Idealized representations of real systems and processes: they represent real systems with regard to certain aspects only and to a certain degree of accuracy
- Conceptual and perspectival: their development is based on our theoretical approaches to and conceptions of the world and reflect the specific perspective of their theory
- Context-dependent: 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
- 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.
In the Engineering Sciences, modeling involves all the different kinds of models: material/scale models (Physical prototypes), theoretical-mathematical models, and computer simulation models. As in science, 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. Where the difference between the natural and the engineering sciences 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. It is also important how the final choice of the model is made in each discipline.
The standard arguments used both in mathematics and in the empirical sciences are deduction and induction. Deduction is the process of deriving inferences from a set of initial statements that are taken to be true by following the correct inferential steps, whereupon the conclusions must necessarily also be true. In mathematics is used to infer true conclusions from true premises. In science, the premises are in most cases models of theories that are still being elaborated and are thus hypothetical, and it is the conclusions that will provide evidence for the hypotheses and models tested. Thus, in science the proof usually concerns the truth of the premises. On the other hand, induction is the inference of general conclusions from a number of particular premises. 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.
In engineering and technology, the construction and evaluation or selection of products and design solutions requires evidence-based arguments similar to those used in science. The difference, however, is that the decision concerning the final choice of a design or solution must satisfy additional criteria and constraints connected with such factors as reliability and risk, cost and marketability, esthetics, cultural traditions, harmful side effects, and environmental pollution.
The aim of 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.