Table 1 Roles, functions, strategies and recommendations for using models and modelling in pursuit of authentic STEM education and literacy

S

Gilbert (2004)

A model can exist in at least five modes of representation: concrete (e.g. 3D physical models), verbal (e.g. spoken or written description of model entities), symbolic (e.g. chemical formulae), visual (e.g. diagrams and animations), gestural (e.g. bodily representation of model entities).

Understanding what a model is, understanding the entities that a particular model represents and how these entities interact with each other, mentally visualizing models, displaying visual literacy skills associated with interpreting models, understanding analogy and metaphor in relation to describing model components, and understanding how a model can be used.

To promote an authentic model-based teaching approach, science teachers need to show learners what representational ‘entities’ constitute a model; demonstrate the scope and limitations of different models; select, develop or change model usage depending on the topic taught; and, design meaningful modelling activities that involve learners’ active model construction.

Integrating models and modelling can increase the authenticity of STEM curricula by explicitly training science teachers in the nature and role of models. Integrating models and modelling in STEM curricula can be promoted through pupils learning to use models, revise models, reconstruct models, and construct models de novo.

S

Justi and Gilbert (2002)

Modelling is the formation, expression, testing and revision of models. Scientific models can comprise consensus models (used in research) or historical models (replaced by revised models over time).

Models and modelling play a central role in learning science and learning how to do science. The modelling process includes ascertaining the purpose of the model, which often includes producing a mental model of the phenomenon, and deciding in what mode of representation (e.g. visual, verbal or mathematical) to express the model. While “testing” whether a model satisfies its intended purpose, the modeller deduces the scope and limitations of the model.

Promote the following modelling strategies during teaching: convey the purpose of a particular model or modelling activity, provide a meaningful experience of the phenomenon being modelled during any practical work, explicitly describe the source from which a particular model arises, support the mental visualization of a particular model, and show how different modes of representation making up different models of a phenomenon are related.

Pre- (and in-) service STEM education should focus heavily on unpacking the nature of a ‘model’, how to use different models in different contexts (transfer), following the historical sequence of model development in a certain topic area as a means of cognitive reconstruction in modelling, and providing skills for evaluating the strengths and limitations of models.

SD&T Davies and Gilbert (2003)

A model represents an object, idea, system, event, or process in different modes of representation. In science education modes often include concrete, visual, verbal, mathematical, or gestural representations. In technology education modes often include iconic (e.g. a sketch), analogue (e.g. simulation), and symbolic (e.g. mathematical) models.

Two roles of modelling are modelling ideas in the mind (communicating with oneself) and modelling ideas in the world (communicating with others). The modelling process includes having experience of the phenomenon or problem, formulating suitable metaphors and analogies to express the model, visualizing the outcome of the modelling process, producing a representation of the model, and evaluating the scope and limitations of the produced model.

Teach modelling strategies for learners to develop a mental image of a model, realise that a model can exist in multiple modes, ascertain that models comprise entities for conveying concepts, understand that a model helps to make predictions or solve a problem, take cognisance that a model’s value is determined by a community, comprehend that a model is open to modification, and discover that models exist as a historical sequence of different expressions.

Implementing models and modelling to promote authentic learning that embraces both science and technology education requires realising the importance of the learner’s context. This means that approaches and curricula need to be purposeful, cogent, and personally meaningful. Doing so will allow learners to perceive relationships between science and technology content while applying knowledge in finding solutions to real problems.

SD&T

Gilbert et al. (2000)

Models are ubiquitous representations of phenomena that often include: concrete entities that are smaller (e.g. car diagram) or larger (e.g. molecule drawing) than the represented phenomenon; abstractions (e.g. force depictions); coordinated concrete and abstract entities; and, representations of an idea, system, event, or process.

Modelling components include mental models (cognitive representations), expressed models (available for others to interpret), consensus models (expressed models that gain acceptance), scientific models (tested expressed models that become predictive tools), historical models (exist in a context and perhaps later displaced), curricular models (historical models in curricula), teaching models (aid interpreting historical and curricular models), and hybrid models (coordinate scientific, historical or curricular models).

Teaching models and modelling contributes to science learning since mental modelling is central to understanding; expressing and testing models reflects the ‘doing’ of science and understanding science relies on interpreting scientific and historical models. Modelling in science and technology both involve a developmental cycle where iterative changes are associated with the produced model, a fitness for purpose where a certain specification is envisioned, and a visualization of the intended outcome of the process.

Models and modelling should be viewed by educators and curriculum developers as a valuable bridge between science and technology, which in turn, may promote authentic STEM education and an opportunity to develop activities to find solutions to, and make informed decisions, about real-world issues by integrating both science and technology concepts.

SEM

Zawojewski et al. (2008); Diefes-Dux, Hjalmarson, Miller, and Lesh (2008).

Models for making sense of or learning about complex systems consist of hardware (one-way cause-and-effect relationships), software (recursive interactions, more than the ‘sum of the parts’), and wetware (neurochemical, fuzzy interactions in complex, dynamic systems).

Principles for model-eliciting activities include model construction, reality, self-assessment, model documentation, model shareability and re-usability, and producing an effective prototype.

Students should be provided with an opportunity to experience how mathematical models come to be and interrogate the trade-offs involved in developing a mathematical model, including assessing the limitations and strengths of different models.

Through implementing model-eliciting activities, students are potentially better positioned to understand the strengths and weaknesses of a conventional model, and better prepared to apply, adapt, and even create new mathematical models for novel and similar situations (e.g. across other STEM disciplines).

TE

de Vries (2013)

Models are a development of a simplified version of reality and can include concrete, conceptual, and formal/symbolic models.

Models support the development of theories and artefacts through manipulation (e.g. concrete models) and mental exploration (e.g. conceptual models, sketches). Models communicate theories and artefacts through educational models and procedural models.

Combining modelling and design activities promotes STEM learning since design connects scientific, technological, engineering and mathematical elements. Also, modelling activities provide a bridge between a practical situation and required mathematical analytical tools to model different versions of reality. This is particularly the case in problems when understanding reality (science) and manipulating reality (technology and engineering) is envisioned.

The encompassing facets of modelling allow it to be integrated at various points of pupil development in STEM curricula. For example, primary pupils can be provided with early experiences of modelling through concrete models. This could follow on to providing secondary pupils with formal aspects of modelling, which include nature, types and functions of models and modelling.

STE

Nia and de Vries (2017); de Vries (2013)

Models can be considered as “techno-scientific artefacts”. They have an intrinsic nature (material structure of different models and types), intentional nature (development and communication of knowledge and artefacts in science and technology), as well as an intrinsic-intentional interrelation.

The intrinsic and intentional nature of models can support building, revising, and communicating knowledge and artefacts related to pedagogical use (e.g. educational models); procedural use (e.g. CAD models); and decisional use (e.g. risk-mitigation models). The intrinsic-intentional interrelation of models informs design (users’ points of view), simplification (abstraction and idealization), iterativity (trial and error), and adequacy (judging appropriateness and effectiveness).

The intrinsic, intentional and intrinsic-intentional perspective of models can inform teaching strategies about how to talk about the nature and uses of models and modelling in science, technology and engineering.

The intrinsic, intentional and intrinsic-intentional perspective of models can be used to analyse curricula and policy documents on the integration of models and modelling. The perspectives can be seen as a benchmark of what a curriculum should include regarding models and modelling in the STEM disciplines.

ST

France et al. (2011); France (2018)

Models in science are central to knowledge building: they provide explanations and predictions. Multiple consensus models of a phenomenon illustrate the nature of science. Models in technology help develop technological knowledge by building and manipulating models. Models are used to understand design concepts and optimise prototypical and functional artefacts.

Functional modelling concerns the development of a design concept and prototyping of the realised outcome (often as a “thing”). Technological modelling provides epistemic strategies for ensuring that the technological outcome is “fit for purpose”. In technology, the purpose is a designed intervention where the outcome is judged by a successful function. In science, the purpose is to explain phenomena, where the explanation is judged by an ability to make sense of empirical evidence.

Developing students’ conceptual understanding of modelling allows for an enhanced individual modelling ability. In this way, modelling can be used as a pedagogical strategy to support technological practice in technology, as well as the learning of concepts in science.

Modelling can support the development of an understanding of the nature of technology and the nature of science both as separate disciplines, but also in relationship with one other. An understanding of models and modelling in technology and in science can provide the learner with the capacity to build bridges between the two disciplines.

M

Niss (2012)

A mathematical model can be defined as the indispensable combination of an extra-mathematical domain, a mathematical domain, and the mapping (translation) between the two.

Mathematical modelling is the selection, modification or construction of a mathematical model to describe an extra-mathematical domain. A mathematical modeller is the person or entity that introduces the model into the domain.

Two observations need to be noted for teaching practice. Firstly, there is no automatic transfer from mathematical skills to models and modelling skills. Secondly, effective teaching of models and modelling requires paying concerted attention to the design of activities, with adequate time for the activities to be realised.

Take advantage of models and modelling to support students’ concept formation and sense-making in mathematics, educational interventions for promoting authentic modelling skills and competencies need more than mere presentation of stereotypical examples on the assumption that this will automatically lead to transfer, modelling requires actively engaging and promoting a suite of skills.

M

Brady, Lesh, and Sevis (2015); Lesh, Amit, and Schorr (1997); Lesh et al. (2013)

A model can be seen as a system for describing, explaining, constructing, modifying, and predicting a series of experiences. Models help organise relevant information to generate or (re)interpret hypotheses about situations or events, explain how information is related, and make decisions based on cues and information.

Model-eliciting activities (MEAs) include the reality principle (is the situation authentic?), model construction principle (is the construction or modification of a model required?), self-evaluation principle (are there clear criteria for assessing the usefulness of the model?), model generalization principle (does the model apply to multiple situations?), and simple prototype principle (will the solution provide a useful prototype for interpreting other similar situations?).

In relation to models and modelling, mathematical knowledge and abilities can be developed along various dimensions that include from concrete to abstract, from specific to general, from global to refined, or from intuitions to formalisations. When students engage in model-eliciting activities to assess and monitor their own work using authentic tools, they induce the construction, modification and refinement of powerful conceptual models.

Implementing models and modelling in authentic learning programmes can help students construct, modify and refine conceptual models that are applicable not only to mathematics but also to other modelling adaptation activities in engineering and science.