The term model has a wide range of uses: it can refer to many things, from a physical scale model to a set of abstract ideas. Here we shall examine the use that we make of this term in Educational Mathematics.

We shall use the term theoretical models, or simply models, without claiming that everything that receives the name of model may be a model in this sense. In fact, in this usage, models differ considerably from what receives the same name in other uses. Our aim in this book is to analyse how the various examples have certain common characteristics, which is why we call them models.

Here, first of all, we shall point out four characteristics.

The first characteristic is the fact that a theoretical model consists of a set of assumptions about some concept or system.

Firstly, it is necessary to distinguish theoretical models from diagrams, illustrations or physical models which, although sometimes useful to represent the model, must not be identified with the model itself. Secondly, it is true that sometimes, although not always, what is called a model also receives the name of theory.

This interchangeability of names is possible because, in such cases, the terms ‘model’ and ‘theory’ refer to the same set of assumptions, although the same things are not suggested about this set when we call it a model as when we call it a theory. Some of the differences, and also the reasons why not all models are called theories, must be analysed. The second characteristic has to do with this.

The second characteristic is the fact that a theoretical model describes a type of object or system by attributing to it what might be called an internal structure, a composition or mechanism that, when taken as a reference, will explain various properties of that object or system.

A theoretical model, therefore, analyses a phenomenon that exhibits certain known regularities by reducing it to more basic components, and not simply by expressing those regularities in quantitative terms or by relating the known properties to those of different objects or systems. Accordingly, the use of the term ‘theory’ in this sense is broader than that of ‘model’, because not all theories are formulated with the aim of providing structural analyses, which are typical of models.

The third characteristic is the fact that a theoretical model is considered as an approximation that is useful for certain purposes.

The value of a particular model can be judged from two different but related viewpoints: how well it serves the purposes for which it is employed, and the completeness and accuracy of the representation that it provides.

The fact that a theoretical model may be proposed as a way of representing the structure of an object or system for certain purposes explains why various models are often used alternately. This represents another difference between the use of the terms ‘model’ and ‘theory’. To propose something as a model of something is equivalent to suggesting it as a representation that provides at least some approximation to the real situation; furthermore, it means admitting the possibility of alternative representations that may be useful for different purposes. To propose something as a theory, however, is equivalent to suggesting that that something is governed by certain specified principles, and not just that it is useful for certain purposes to represent it as being governed by those principles or that those principles approximate to the principles that actually apply. Consequently, someone who proposes something as a theory is obliged to maintain that any alternative theories must be discarded or modified, or that they will only be valid in special cases.

Finally, the fourth characteristic is the fact that a theoretical model is often formulated and developed and perhaps even named on the basis of an analogy between the object or system that it describes and some other object or different system.

This implies a comparison in which one observes properties and principles that are similar in certain aspects, which fits in with the previous observation to the effect that theoretical models have the aim of providing a useful representation of a system: in order to provide such a representation, it is often helpful to establish an analogy between the system in question and some known system that is governed by rules or principles that are understood, and one supposes that some of those rules, or others like them, also govern the system that one is trying to describe with the model. Reasoning of this kind, based as it is on an argument by analogy, is never considered sufficient to establish the principles in question, but only to suggest that they may be considered as first approximations, subject to proof and subsequent modification. In each case, however, the model itself can be distinguished from any analogy on the basis of which it was developed.

Theoretical models can fulfil the same functions as theories: they can be used for purposes of explanation, prediction, calculation, systematisation, derivation of principles, and so on. The difference between the use of a model and the use of a theory does not lie in the kind of function for which it can be used, but in the way in which it fulfils that function. Theoretical models provide explanations; but these explanations are based on assumptions that may be simplified, and this condition must be borne in mind when one compares them with theories. It is often said of explanation and systematisation by means of a theory that they are more profound and penetrating, which reflects the belief that the principles that constitute a theory are more accurate than those of a model and take more known magnitudes into account.

The stability of phenomena of educational mathematics and the well-established replicability of the experimental designs that have been used to study them are such that we cannot fail to include these observations among the components that are important for any theoretical model for observation in Educational Mathematics. Thus we have the need to propose theoretical components that deal with different types of (1) teaching models, together with (2) models for the cognitive processes, both related to (3) models of formal competence that simulate the competent performance of an ideal user of an MSS, and (4) models of communication, to describe the rules of communicative competence, formation and decoding of texts, and contextual and circumstantial disambiguation.