Models in Science (Stanford Encyclopedia of Philosophy)

Models in Science (Stanford Encyclopedia of Philosophy): "Models in Science
First published Mon Feb 27, 2006

Models are of central importance in many scientific contexts. The centrality of models such as the billiard ball model of a gas, the Bohr model of the atom, the MIT bag model of the nucleon, the Gaussian-chain model of a polymer, the Lorenz model of the atmosphere, the Lotka-Volterra model of predator-prey interaction, the double helix model of DNA, agent-based and evolutionary models in the social sciences, or general equilibrium models of markets in their respective domains are cases in point. Scientists spend a great deal of time building, testing, comparing and revising models, and much journal space is dedicated to introducing, applying and interpreting these valuable tools. In short, models are one of the principal instruments of modern science.

Philosophers are acknowledging the importance of models with increasing attention and are probing the assorted roles that models play in scientific practice. The result has been an incredible proliferation of model-types in the philosophical literature. Probing models, phenomenological models, computational models, developmental models, explanatory models, impoverished models, testing models, idealized models, theoretical models, scale models, heuristic models, caricature models, didactic models, fantasy models, toy models, imaginary models, mathematical models, substitute models, iconic models, formal models, analogue models and instrumental models are but some of the notions that are used to categorize models. While at first glance this abundance is overwhelming, it can quickly be brought under control by recognizing that these notions pertain to different problems that arise in connection with models. For example, models raise questions in semantics (what is the representational function that models perform?), ontology (what kind of things are models?), epistemology (how do we learn with models?), and, of course, in philosophy of science (how do models relate to theory?; what are the implications of a model based approach to science for the debates over scientific realism, reductionism, explanation and laws of nature?).

* 1. Semantics: Models and Representation
o 1.1 Representational models I: models of phenomena
o 1.2 Representational models II: models of data
o 1.3 Models of theory
* 2. Ontology: What Are Models?
o 2.1 Physical objects
o 2.2 Fictional objects
o 2.3 Set-theoretic structures
o 2.4 Descriptions
o 2.5 Equations
o 2.6 Gerrymandered ontologies
* 3. Epistemology: Learning with Models
o 3.1 Learning about the model: experiments, thought experiments and simulation
o 3.2 Converting knowledge about the model into knowledge about the target
* 4. Models and Theory
o 4.1 The two extremes: the syntactic and the semantic view of theories
o 4.2 Models as independent of theories
* 5. Models and Other Debates in the Philosophy of Science
o 5.1 Models and the realism versus antirealism debate
o 5.2 Model and reductionism
o 5.3 Models and laws of nature
o 5.4 Models and scientific explanation
* 6. Conclusion
* Bibliography
* Other Internet Resources
* Related Entries

1. Semantics: Models and Representation

Models can perform two fundamentally different representational functions. On the one hand, a model can be a representation of a selected part of the world (the ‘target system’). Depending on the nature of the target, such models are either models of phenomena or models of data. On the other hand, a model can represent a theory in the sense that it interprets the laws and axioms of that theory. These two notions are not mutually exclusive as scientific models can be representations in both senses at the same time.
1.1 Representational models I: models of phenomena

Many scientific models represent a phenomenon, where ‘phenomenon’ is used as an umbrella term covering all relatively stable and general features of the world that are interesting from a scientific point of view. Empiricists like van Fraassen (1980) only allow for observables to qualify as such, while realists like Bogen and Woodward (1988) do not impose any such restrictions. The billiard ball model of a gas, the Bohr model of the atom, the double helix model of DNA, the scale model of a bridge, the Mundell-Fleming model of an open economy, or the Lorenz model of the atmosphere are well-known examples for models of this kind.

A first step towards a discussion of the issue of scientific representation is to realize that there is no such thing as the problem of scientific representation. Rather, there are different but related problems. It is not yet clear what specific set of questions a theory of representation has to come to terms with, but whatever list of questions one might put on the agenda of a theory of scientific representation, there are two problems that will occupy center stage in the discussion (Frigg 2006). The first problem is to explain in virtue of what a model is a representation of something else. To appreciate the thrust of this question we have to anticipate a position as regards the ontology of models (which we discuss in the next section). It is now common to construe models as non-linguistic entities rather than as descriptions. This approach has wide-ranging consequences. If we understand models as descriptions, the above question would be reduced to the time-honored problem of how language relates to reality and there would not be any problems over and above those already discussed in the philosophy of language. However, if we understand models as non-linguistic entities, we are faced with the new question of what it is for an object (that is not a word or a sentence) to scientifically represent a phenomenon.

Somewhat surprisingly, until recently this question has not attracted much attention in twentieth century philosophy of science, despite the fact that the corresponding problems in the philosophy of mind and in aesthetics have been discussed extensively for decades (there is a substantial body of literature dealing with the question of what it means for a mental state to represent a certain state of affairs; and the question of how a configuration of flat marks on a canvass can depict something beyond this canvass has puzzled aestheticians for a long time). However, some recent publications address this and other closely related problems (Bailer-Jones 2003, Frigg 2006, Giere 2004, Suárez 2004, van Fraassen 2004), while others dismiss it as a non-issue (Callender and Cohen 2006, Teller 2001).

The second problem is concerned with representational styles. It is a commonplace that one can represent the same subject matter in different ways. This pluralism does not seem to be a prerogative of the fine arts as the representations used in the sciences are not all of one kind either. Weizsäcker's liquid drop model represents the nucleus of an atom in a manner very different from the shell model, and a scale model of the wing of an air plane represents the wing in a way that is different from how a mathematical model of its shape does. What representational styles are there in the sciences?

Although this question is not explicitly addressed in the literature on the so-called semantic view of theories, some answers seem to emerge from its understanding of models. One version of the semantic view, one that builds on a mathematical notion of models (see Sec. 2), posits that a model and its target have to be isomorphic (van Fraassen 1980; Suppes 2002) or partially isomorphic (Da Costa and French 2003) to each other. Formal requirements weaker than these have been discussed by Mundy (1986) and Swoyer (1991). Another version of the semantic view drops formal requirements in favor of similarity (Giere 1988 and 2004, Teller 2001). This approach enjoys the advantage over the isomorphism view that it is less restrictive and also can account for cases of inexact and simplifying models. However, as Giere points out, this account remains empty as long as no relevant respects and degrees of similarity are specified. The specification of such respects and degrees depends on the problem at hand and the larger scientific context and cannot be made on the basis of purely philosophical considerations (Teller 2001).

Further notions that can be understood as addressing the issue of representational styles have been introduced in the literature on models. Among them, scale models, idealized models, analogical models and phenomenological models play an important role. These categories are not mutually exclusive; for instance, some scale models would also qualify as idealized models and it is not clear where exactly to draw the line between idealized and analogue models.

Scale models. Some models are basically down-sized or enlarged copies of their target systems (Black 1962). Typical examples are wooden cars or model bridges. The leading intuition is that a scale model is a naturalistic replica or a truthful mirror image of the target; for this reason scale models are sometimes also referred to as ‘true models’ (Achinstein 1968, Ch. 7). However, there is no such thing as a perfectly faithful scale model; faithfulness is always restricted to some respects. The wooden model of the car, for instance, provides a faithful portrayal of the car's shape but not its material. Scale models seem to be a special case of a broader category of representations that Peirce dubbed icons: representations that stand for something else because they closely resemble it (Peirce 1931-1958 Vol. 3, Para. 362). This raises the question of what criteria a model has to satisfy in order to qualify as an icon. Although we seem to have strong intuitions about how to answer this question in particular cases, no theory of iconicity for models has been formulated yet.

Idealized models. An idealization is a deliberate simplification of something complicated with the objective of making it more tractable. Frictionless planes, point masses, infinite velocities, isolated systems, omniscient agents, or markets in perfect equilibrium are but some well-know examples. Philosophical debates over idealization have focused on two general kinds of idealizations: so-called Aristotelian and Galilean idealizations.

Aristotelian idealization amounts to ‘stripping away’, in our imagination, all properties from a concrete object that we believe are not relevant to the problem at hand. This allows us to focus on a limited set of properties in isolation. An example is a classical mechanics model of the planetary system, describing the planets as objects only having shape and mass, disregarding all other properties. Other labels for this kind of idealization include ‘abstraction’ (Cartwright 1989, Ch. 5), ‘negligibility assumptions’ (Musgrave 1981) and ‘method of isolation’ (Mäki 1994).

Galilean idealizations are ones that involve deliberate distortions. Physicists build models consisting of point masses moving on frictionless planes, economists assume that agents are omniscient, biologists study isolated populations, and so on. It was characteristic of Galileo's approach to science to use simplifications of this sort whenever a situation was too complicated to tackle. For this reason it is common to refer to this sort of idealizations as ‘Galilean idealizations’ (McMullin 1985); another common label is ‘distorted models’.

Galilean idealizations are beset with riddles. What does a model involving distortions of this kind tell us about reality? How can we test its accuracy? In reply to these questions Laymon (1991) has put forward a theory which understands idealizations as ideal limits: imagine a series of experimental refinements of the actual situation which approach the postulated limit and then require that the closer the properties of a system come to the ideal limit, the closer its behavior has to come to the behavior of the ideal limit (monotonicity). But these conditions need not always hold and it is not clear how to understand situations in which no ideal limit exists. We can, at least in principle, produce a series of table tops that are ever more slippery but we cannot possibly produce a series of systems in which Planck's constant approaches zero. This raises the question of whether one can always make an idealized model more realistic by de-idealizing it. We will come back to this issue in section 5.1.

Galilean and Aristotelian idealizations are not mutually exclusive. On the contrary, they often come together. Consider again the mechanical model of the planetary system: the model only takes into account a narrow set of properties and distorts these, for instance by describing planets as ideal spheres with a rotation-symmetric mass distribution.

Models that involve substantial Galilean as well as Aristotelian idealizations are sometimes referred to as ‘caricatures’ (Gibbard and Varian 1978). Caricature models isolate a small number of salient characteristics of a system and distort them into an extreme case. A classical example is Ackerlof's (1970) model of the car market, which explains the difference in price between new and used cars solely in terms of asymmetric information, thereby disregarding all other factors that may influence prices of cars. However, it is controversial whether such highly idealised models can still be regarded as informative representations of their target systems (for a discussion of caricature models, in particular in economics, see Reiss 2006).

At this point we would like to mention a notion that seems to be closely related to idealization, namely approximation. Although the terms are sometimes used interchangeably, there seems to be a substantial difference between the two. Approximations are introduced in a mathematical context. One mathematical item is an approximation of another one if it is close to it in some relevant sense. What this item is may vary. Sometimes we want to approximate one curve with another one. This happens when we expand a function into a power series and only keep the first two or three terms. In other situations we approximate an equation by another one by letting a control parameter tend towards zero (Redhead 1980). The salient point is that the issue of physical interpretation need not arise. Unlike Galilean idealization, which involves a distortion of a real system, approximation is a purely formal matter. This, of course, does not imply that there cannot be interesting relations between approximations and idealization. For instance, an approximation can be justified by pointing out that it is the ‘mathematical pendant’ to an acceptable idealization (e.g. when we neglect a dissipative term in an equation because we make the idealizing assumption that the system is frictionless).

Analogical models. Standard examples of analogical models include the hydraulic model of an economic system, the billiard ball model of a gas, the computer model of the mind or the liquid drop model of the nucleus. At the most basic level, two things are analogous if there are certain relevant similarities between them. Hesse (1963) distinguishes different types of analogies according to the kinds of similarity relations in which two objects enter. A simple type of analogy is one that is based on shared properties. There is an analogy between the earth and the moon based on the fact that both are large, solid, opaque, spherical bodies, receiving heat and light from the sun, revolving around their axes, and gravitating towards other bodies. But sameness of properties is not a necessary condition. An analogy between two objects can also be based on relevant similarities between their properties. In this more liberal sense we can say that there is an analogy between sound and light because echoes are similar to reflections, loudness to brightness, pitch to color, detectability by the ear to detectability by the eye, and so on.

Analogies can also be based on the sameness or resemblance of relations between parts of two systems rather than on their monadic properties. It is this sense that some politicians assert that the relation of a father to his children is analogous to the relation of the state to the citizens. The analogies mentioned so far have been what Hesse calls ‘material analogies’. We obtain a more formal notion of analogy when we abstract from the concrete features the systems possess and only focus on their formal set-up. What the analogue model then shares with its target is not a set of features, but the same pattern of abstract relationships (i.e. the same structure, where structure is understood in the formal sense). This notion of analogy is closely related to what Hesse calls ‘formal analogy’. Two items are related by formal analogy if they are both interpretations of the same formal calculus. For instance, there is a formal analogy between a swinging pendulum and an oscillating electric circuit because they are both described by the same mathematical equation.

A further distinction due to Hesse is the one between positive, negative and neutral analogies. The positive analogy between two items consists in the properties or relations they share (both gas molecules and billiard balls have mass), the negative analogy in the ones they do not share (billiard balls are colored, gas molecules are not). The neutral analogy comprises the properties of which it is not known yet whether they belong to the positive or the negative analogy (do gas molecules obeying Newton's laws of collision exhibit an approach to equilibrium?). Neutral analogies play an important role in scientific research because they give rise to questions and suggest new hypotheses. In this vein, various authors have emphasized the heuristic role that analogies play in theory construction and in creative thought (Bailer-Jones and Bailer-Jones 2002; Hesse 1974, Holyoak and Thagard 1995, Kroes 1989, Psillos 1995, and the essays collected in Hellman 1988).

Phenomenological models. Phenomenological models have been defined in different, though related, ways. A traditional definition takes them to be models that only represent observable properties of their targets and refrain from postulating hidden mechanisms and the like. Another approach, due to McMullin (1968), defines phenomenological models as models that are independent of theories. This, however, seems to be too strong. Many phenomenological models, while failing to be derivable from a theory, incorporate principles and laws associated with theories. The liquid drop model of the atomic nucleus, for instance, portrays the nucleus as a liquid drop and describes it as having several properties (surface tension and charge, among others) originating in different theories (hydrodynamics and electrodynamics, respectively). Certain aspects of these theories—though usually not the complete theory—are then used to determine both the static and dynamical properties of the nucleus.

Concluding remarks. Each of these notions is still somewhat vague, suffering from internal problems, and much work needs to be done to tighten them. But more pressing than these is the question of how the different notions relate to each other. Are analogies fundamentally different from idealizations, or do they occupy different areas on a continuous scale? How do icons differ from idealizations and analogies? At the present stage we do not know how to answer these questions. What we need is a systematic account of the different ways in which models can relate to reality and of how these ways compare to each other.
1.2 Representational models II: models of data

Another kind of representational models are so-called ‘models of data’ (Suppes 1962). A model of data is a corrected, rectified, regimented, and in many instances idealized version of the data we gain from immediate observation, the so-called raw data. Characteristically, one first eliminates errors (e.g. removes points from the record that are due to faulty observation) and then present the data in a ‘neat’ way, for instance by drawing a smooth curve through a set of points. These two steps are commonly referred to as ‘data reduction’ and ‘curve fitting’. When we investigate the trajectory of a certain planet, for instance, we first eliminate points that are fallacious from the observation records and then fit a smooth curve to the remaining ones. Models of data play a crucial role in confirming theories because it is the model of data and not the often messy and complex raw data that we compare to a theoretical prediction.

The construction of a data model can be extremely complicated. It requires sophisticated statistical techniques and raises serious methodological as well as philosophical questions. How do we decide which points on the record need to be removed? And given a clean set of data, what curve do we fit to it? The first question has been dealt with mainly within the context of the philosophy of experiment (see for instance Galison 1997 and Staley 2004). At the heart of the latter question lies the so-called curve fitting problem, which is that the data themselves do not indicate what form the fitted curve should take. Traditional discussions of theory choice suggest that this issue is settled by background theory, considerations of simplicity, prior probabilities, or a combination of these. Forster and Sober (1994) point out that this formulation of the curve fitting problem is a slight overstatement because there is a theorem in statistics due to Akaike which shows (given certain assumptions) that the data themselves underwrite (though not determine) an inference concerning the curve's shape if we assume that the fitted curve has to be chosen such that it strikes a balance between simplicity and goodness of fit in a way that maximizes predictive accuracy. (Further discussions of data models can be found in Chin and Brewer 1994, Harris 2003, and Mayo 1996).
1.3 Models of theory

In modern logic, a model is a structure that makes all sentences of a theory true, where a theory is taken to be a (usually deductively closed) set of sentences in a formal language (see Bell and Machover 1977 or Hodges 1997 for details). The structure is a ‘model’ in the sense that it is what the theory represents. As a simple example consider Euclidean geometry, which consists of axioms—e.g. ‘any two points can be joined by a straight line’—and the theorems that can be derived from these axioms. Any structure of which all these statements are true is a model of Euclidean geometry.

A structure S = <U, O, R> is a composite entity consisting of (i) a non-empty set U of individuals called the domain (or universe) of S, (ii) an indexed set O (i.e. an ordered list) of operations on U (which may be empty), and (iii) a non-empty indexed set R of relations on U. It is important to note that nothing about what the objects are matters for the definition of a structure—they are mere dummies. Similarly, operations and functions are specified purely extensionally; that is, n-place relations are defined as classes of n-tuples, and functions taking n arguments are defined as classes of (n+1)-tuples. If all sentences of a theory are true when its symbols are interpreted as referring to either objects, relations, or functions of a structure S, then S is a model of this theory.

Many models in science carry over from logic the idea of being the interpretation of an abstract calculus. This is particularly pertinent in physics, where general laws—such as Newton's equation of motion—lie at the heart of a theory. These laws are applied to a particular system—e.g. a pendulum—by choosing a special force function, making assumptions about the mass distribution of the pendulum etc. The resulting model then is an interpretation (or realization) of the general law.
2. Ontology: What Are Models?

There is a variety of things that are commonly referred to as models: physical objects, fictional objects, set-theoretic structures, descriptions, equations, or combinations of some of these. However, these categories are neither mutually exclusive nor jointly exhaustive. Where one draws the line between, say, fictional objects and set-theoretical structures may well depend on one's metaphysical convictions, and some models may fall into yet another class of things. What models are is, of course, an interesting question in its own right, but, as briefly indicated in the last section, it has also important implications for semantics and, as we will see below, for epistemology.
2.1 Physical objects

Some models are straightforward physical objects. These are commonly referred to as ‘material models’. The class of material models comprises anything that is a physical entity and that serves as a scientific representation of something else. Among the members of this class we find stock examples like wooden models of bridges, planes, or ships, analogue models like electric circuit models of neural systems or pipe models of an economy, or Watson and Crick's model of DNA. But also more cutting edge cases, especially from the life sciences, where certain organisms are studied as stand-ins for others, belong to this category.

Material models do not give rise to any ontological difficulties over and above the well-known quibbles in connection with objects, which metaphysicians deal with (e.g. the nature of properties, the identity of objects, parts and wholes, and so on).
2.2 Fictional objects

Many models are not material models. The Bohr model of the atom, a frictionless pendulum, or isolated populations, for instance, are in the scientist's mind rather than in the laboratory and they do not have to be physically realized and experimented upon to perform their representational function. It seems natural to view them as fictional entities. This position can be traced back to the German neo-Kantian Vaihinger (1911), who emphasized the importance of fictions for scientific reasoning. Giere has recently advocated the view that models are abstract entities (1988, 81). It is not entirely clear what Giere means by ‘abstract entities’, but his discussion of mechanical models seems to suggest that he uses the term to designate fictional entities.

This view squares well with scientific practice, where scientists often talk about models as if they were objects, as well as with philosophical views that see the manipulation of models as an essential part of the process of scientific investigation (Morgan 1999). It is natural to assume that one can manipulate something only if it exists. Furthermore, models often have more properties than we explicitly attribute to them when we construct them, which is why they are interesting vehicles of research. A view that regards models as objects can easily explain this without further ado: when we introduce a model we use an identifying description, but the object itself is not exhaustively characterized by this description. Research then simply amounts to finding out more about the object thus identified.

The drawback of this suggestion is that fictional entities are notoriously beset with ontological riddles. This has led many philosophers to argue that there are no such things as fictional entities and that apparent ontological commitments to them must be renounced. The most influential of these deflationary accounts goes back to Quine (1953). Building on Russell's discussion of definite descriptions, Quine argues that it is an illusion that we refer to fictional entities when we talk about them. Instead, we can dispose of these alleged objects by turning the terms that refer to them into predicates and analyse sentences like ‘Pegasus does not exist’ as ‘nothing pegasizes’. By eliminating the troublesome term we eschew the ontological commitment they seem to carry. This has resulted in a glaring neglect of fictional entities, in particular among philosophers of science. Fine (1993), in a programmatic essay, draws attention to this neglect but does not offer a systematic account of how fictions are put to use in science.
2.3 Set-theoretic structures

An influential point of view takes models to be set-theoretic structures. This position can be traced back to Suppes (1960) and is now, with slight variants, held by most proponents of the semantic view of theories. Needless to say, there are differences between different versions of the semantic view (van Fraassen, for instance, emphasizes that models are state-space structures); a survey of the different positions can be found in Suppe (1989, Ch. 1). However, on all these accounts models are structures of one sort or another (Da Costa and French 2000). As models of this kind are often closely tied to mathematized sciences, they are sometimes also referred to as ‘mathematical models’. (For a discussion of such models in biology see Lloyd 1984 and 1994.)

This view of models has been criticized on different grounds. One pervasive criticism is that many types of models that play an important role in science are not structures and cannot be accommodated within the structuralist view of models, which can neither account for how these models are constructed nor for how they work in the context of investigation (Cartwright 1999, Downes 1992, Morrison 1999). Another charge held against the set-theoretic approach is that it is not possible to explain how structures represent a target system which forms part of the physical world without making assumptions that go beyond what the approach can afford (Frigg 2006).
2.4 Descriptions

A time-honored position has it that what scientists display in scientific papers and textbooks when they present a model are more or less stylized descriptions of the relevant target systems (Achinstein 1968, Black 1962).

This view has not been subject to explicit criticism. However, some of the criticisms that have been marshaled against the syntactic view of theories equally threaten a linguistic understanding of models. First, it is a commonplace that we can describe the same thing in different ways. But if we identify a model with its description, then each new description yields a new model, which seems to be counterintuitive. One can translate a description into other languages (formal or natural), but one would not say that one hereby obtains a different model. Second, models have different properties than descriptions. On the one hand, we say that the model of the solar system consists of spheres orbiting around a big mass or that the population in the model is isolated from its environment, but it does not seem to make sense to say this about a description. On the other hand, descriptions have properties that models do not have. A description can be written in English, consist of 517 words, be printed in red ink, and so on. None of this makes any sense when said about a model. The descriptivist faces the challenge to either make a case that these arguments are mistaken or to show how to get around these difficulties.
2.5 Equations

Another group of things that is habitually referred to as ‘models’, in particular in economics, is equations (which are then also termed ‘mathematical models’). The Black-Scholes model of the stock market or the Mundell-Fleming model of an open economy are cases in point.

The problem with this suggestion is that equations are syntactic items and as such they face objections similar to the ones put forward against descriptions. First, one can describe the same situation using different co-ordinates and as a result obtain different equations; but we do not seem to obtain a different model. Second, the model has properties different from the equation. An oscillator is three-dimensional but the equation describing its motion is not. Equally, an equation may be inhomogeneous but the system it describes is not.
2.6 Gerrymandered ontologies

The proposals discussed so far have tacitly assumed that a model belongs to one particular class of objects. But this assumption is not necessary. It might be the case that models are a mixture of elements belonging to different ontological categories. In this vein Morgan (2001) suggests that models involve structural as well as narrative elements (‘stories’, as she calls them).
3. Epistemology: Learning with Models

Models are vehicles for learning about the world. Significant parts of scientific investigation are carried out on models rather than on reality itself because by studying a model we can discover features of and ascertain facts about the system the model stands for; in brief, models allow for surrogative reasoning (Swoyer 1991). For instance, we study the nature of the hydrogen atom, the dynamics of populations, or the behavior of polymers by studying their respective models. This cognitive function of models has been widely acknowledged in the literature, and some even suggest that models give rise to a new style of reasoning, so-called ‘model based reasoning’ (Magnani and Nersessian 2002, Magnani, Nersessian and Thagard 1999). This leaves us with the question of how learning with a model is possible.

Hughes (1997) provides a general framework for discussing this question. According to his so-called DDI account, learning takes place in three stages: denotation, demonstration, and interpretation. We begin by establishing a representation relation (‘denotation’) between the model and the target. Then we investigate the features of the model in order to demonstrate certain theoretical claims about its internal constitution or mechanism; i.e. we learn about the model (‘demonstration’). Finally, these findings have to be converted into claims about the target system; Hughes refers to this step as ‘interpretation’. It is the latter two notions that are at stake here.
3.1 Learning about the model: experiments, thought experiments and simulation

Learning about a model happens at two places, in the construction and the manipulation of the model (Morgan 1999). There are no fixed rules or recipes for model building and so the very activity of figuring out what fits together and how it does so affords an opportunity to learn about the model. Once the model is built, we do not learn about its properties by looking at it; we have to use and manipulate the model in order to elicit its secrets.

Depending on what kind of model we are dealing with, building and manipulating a model amounts to different activities demanding a different methodology. Material models seem to be unproblematic as they are commonly used in the kind of experimental contexts that have been discussed extensively by philosophers of science (e.g. we put the model of a car in the wind tunnel and measure its air resistance).

Not so with fictional models. What constraints are there to the construction of fictional models and how do we manipulate them? The natural response seems to be that we answer these questions by performing a thought experiment. Different authors (e.g. Brown 1991, Gendler 2000, Norton 1991, Reiss 2003, Sorensen 1992) have explored this line of argument but they have reached very different and often conflicting conclusions as to how thought experiments are performed and what the status of their outcomes is (for details see the entry on thought experiments).

An important class of models is of mathematical nature. In some cases it is possible to derive results or solve equations analytically. But quite often this is not the case. It is at this point where the invention of the computer had a great impact, as it allows us to solve equations which are otherwise intractable by making a computer simulation. Many parts of current research in both the natural and social sciences rely on computer simulations. The formation and development of stars and galaxies, the detailed dynamics of high-energy heavy ion reactions, aspects of the intricate process of the evolution of life as well as the outbreak of wars, the progression of an economy, decision procedures in an organization and moral behavior are explored with computer simulations, to mention only a few examples (Hegselmann et al. 1996, Skyrms 1996).

What is a simulation? Simulations characteristically are used in connection with dynamic models, i.e. models that involve time. The aim of a simulation is to solve the equations of motion of such a model, which is designed to represent the time-evolution of its target system. So one can say that a simulation imitates a (usually real) process by another process (Hartmann 1996, Humphreys 2004).

It has been claimed that computer simulations constitute a genuinely new methodology of science or even a new scientific paradigm (Humphreys 2004, Rohrlich 1991, Winsberg 2001 and 2003, and various contributions to Sismondo and Gissis 1999). Although this contention may not meet with univocal consent, there is no doubt about the practical significance of computer simulations. When standard methods fail, computer simulations are often the only way to learn something about a dynamical model; they help us to ‘extend ourselves’ (Humphreys 2004), as it were. In situations in which the underlying model is well confirmed and understood, computer experiments may even replace real experiments, which has economic advantages and minimizes risk (as, for example, in the case of the simulation of atomic explosions). Computer simulations are also heuristically important. They may suggest new theories, models and hypotheses, for example based on a systematic exploration of a model's parameter space (Hartmann 1996).

But computer simulations also bear methodological perils. They may provide misleading results because due to the discrete nature of the calculations carried out on a digital computer they only allow for the exploration of a part of the full parameter space; and this subspace may not reveal certain important features of the model. The severity of this problem is somehow mitigated by the increasing power of modern computers. But the availability of more computational power also may have adverse effects. It may encourage scientists to swiftly come up with increasingly complex but conceptually premature models, involving poorly understood assumptions or mechanisms and too many additional adjustable parameters (for a discussion of a related problem in the context of individual actor models in the social sciences see Schnell 1990). This may lead to an increase in empirical adequacy—which may be welcome when it comes, for example, to forecasting the weather—but not necessarily to a better understanding of the underlying mechanisms. As a result, the use of computer simulations may change the weight we assign to the various goals of science. So it is important not to be carried away with the means that new powerful computers offer and to thereby place out of sight the actual goals of research.
3.2 Converting knowledge about the model into knowledge about the target

Once we have knowledge about the model, this knowledge has to be ‘translated’ into knowledge about the target system. It is at this point that the representational function of models becomes important again. Models can instruct us about the nature of reality only if we assume that (at least some of) the model's aspects have counterparts in the world. But if learning is tied to representation and if there are different kinds of representation (analogies, idealizations, etc.), then there are also different kinds of learning. If, for instance, we have a model we take to be a realistic depiction, the transfer of knowledge from the model to the target is accomplished in a different manner than when we deal with an analogue, or a model that involves idealizing assumptions.

What are these different ways of learning? Although numerous case studies have been made of how certain specific models work, there do not seem to be any general accounts of how the transfer of knowledge from a model to its target is achieved (this with the possible exception of theories of analogical reasoning, see references above). This is a difficult question, but it is one that deserves more attention than it has gotten so far.
4. Models and Theory

One of the most perplexing questions in connection with models is how they relate to theories. The separation between models and theory is a very hazy one and in the jargon of many scientists it is often difficult, if not impossible, to draw a line. So the question is: is there a distinction between models and theories and if so how do they relate to one another?

In common parlance, the terms ‘model’ and ‘theory’ are sometimes used to express someone's attitude towards a particular piece of science. The phrase ‘it's just a model’ indicates that the hypothesis at stake is asserted only tentatively or is even known to be false, while something is awarded the label ‘theory’ if it has acquired some degree of general acceptance. However, this way of drawing a line between models and theories is of no use to a systematic understanding of models.
4.1 The two extremes: the syntactic and the semantic view of theories

The syntactic view of theories, which is an integral part of the logical positivist picture of science, construes a theory as a set of sentences in an axiomatized system of first order logic. Within this approach, the term model is used in a wider and in a narrower sense. In the wider sense, a model is just a system of semantic rules that interpret the abstract calculus and the study of a model amounts to scrutinizing the semantics of a scientific language. In the narrower sense, a model is an alternative interpretation of a certain calculus (Braithwaite 1953, Campbell 1920, Nagel 1961, Spector 1965). If, for instance, we take the mathematics used in the kinetic theory of gases and reinterpret the terms of this calculus in a way that they refer to billiard balls, the billiard balls are a model of the kinetic theory of gases. Proponents of the syntactic view believe such models to be irrelevant to science. Models, they hold, are superfluous additions that are at best of pedagogical, aesthetical or psychological value (Carnap 1938, Hempel 1965; see also Bailer-Jones 1999).

The semantic view of theories (see e.g. van Fraassen 1980, Giere 1988, Suppe 1989, and Suppes 2002) reverses this standpoint and declares that we should dispense with a formal calculus altogether and view a theory as a family of models. Although different version of the semantic view assume a different notion of model (see above) they all agree that models are the central unit of scientific theorizing.
4.2 Models as independent of theories

One of the most perspicuous criticisms of the semantic view is that it mislocates the place of models in the scientific edifice. Models are relatively independent from theory, rather than being constitutive of them; or to use Morrison's (1998) slogan, they are ‘autonomous agents’. This independence has two aspects: construction and functioning (Morgan and Morrison 1999).

A look at how models are constructed in actual science shows that they are neither derived entirely from data nor from theory. Theories do not provide us with algorithms for the construction a model; they are not ‘vending machines’ into which one can insert a problem and a models pops out (Cartwright 1999, Ch. 8). Model building is an art and not a mechanical procedure. The London model of superconductivity affords us with a good example of this relationship. The model's principal equation has no theoretical justification (in the sense that it could be derived from electromagnetic or any other fundamental theory) and is motivated solely on the basis of phenomenological considerations (Cartwright et al. 1995). Or, to put it another way, the model has been constructed ‘bottom up’ and not ‘top down’ and therefore enjoys a great deal of independence from theory.

The second aspect of the independence of models is that they perform functions which they could not perform if they were a part of, or strongly dependent on, theories.

Models as complements of theories. A theory may be incompletely specified in the sense that it imposes certain general constraints but remains silent about the details of concrete situations, which are provided by a model (Redhead 1980). A special case of this situation is when a qualitative theory is known and the model introduces quantitative measures (Apostel 1961). Redhead's example for a theory that is underdetermined in this way is axiomatic quantum field theory, which only imposes certain general constraints on quantum fields but does not provide an account of particular fields.

While Redhead and others seem to think of cases of this sort as somehow special, Cartwright (1983) has argued that they are the rule rather than the exception. On her view, fundamental theories such as classical mechanics and quantum mechanics do not represent anything at all as they do not describe any real world situation. Laws in such theories are schemata that need to be concretized and filled with the details of a specific situation, which is a task that is accomplished by a model.

Models stepping in when theories are too complex to handle. Theories may be too complicated to handle. In such a case a simplified model may be employed that allows for a solution (Apostel 1961, Redhead 1980). Quantum chromodynamics, for instance, cannot easily be used to study the hadron structure of a nucleus, although it is the fundamental theory for this problem. To get around this difficulty physicists construct tractable phenomenological models (e.g. the MIT bag model) that effectively describes the relevant degrees of freedom of the system under consideration (Hartmann 1999). The advantage of these models is that they yield results where theories remain silent. Their drawback is that it is often not clear how to understand the relationship between the theory and the model as the two are, strictly speaking, contradictory.

A more extreme case is the use of a model when there are no theories at all available. We encounter this situation in all domains, but it is particularly rampant in biology and economics where overarching theories are often not to be had. The models that scientists then construct to tackle the situation are sometimes referred to as ‘substitute models’ (Groenewold 1961).

Models as preliminary theories. The notion of models as substitutes for theories is closely related to the notion of a developmental model. This term has been coined by Leplin (1980), who pointed out how useful models were in the development of early quantum theory and is now used as an umbrella notion covering cases in which models are some sort of a preliminary exercises to theory.

A closely related notion is the one of probing models (also ‘study models’ or ‘toy models’). These are models which do not perform a representational function and which are not expected to instruct us about anything beyond the model itself. The purpose of these models is to test new theoretical tools that are used later on to build representational models. In field theory, for instance, the so-called φ4-model has been studied extensively not because it represents anything real (it is well-known that it doesn't) but because it serves several heuristic functions. The simplicity of the φ4-model allows physicist to ‘get a feeling’ for what quantum field theories are like and to extract some general features that this simple model shares with more complicated ones. One can try complicated techniques such as renormalization in a simple setting and it is possible to get acquainted with mechanisms—in this case symmetry breaking—that can be used later on (Hartmann 1995). This is true not only for physics. As Wimsatt (1987) points out, false models in genetics can perform many useful functions, among them the following: the false model can help to answer questions about more realistic models, provide an arena for answering questions about properties of more complex models, ‘factor out’ phenomena that would not otherwise be seen, serve as a limiting case of a more general model (or two false models may define the extreme of a continuum of cases in which the real case is supposed to lie), or it can lead to the identification of relevant variables and the estimation of their values.
5. Models and Other Debates in the Philosophy of Science

The debate over scientific models has important repercussions for other debates in the philosophy of science. The reason for this is that traditionally the debates over scientific realism, reductionism, explanation, and laws of nature were couched in terms of theories, because only theories were acknowledged as carriers of scientific knowledge. So the question is whether, and if so how, discussions of these matters change when we shift the focus from theories to models. Up to now, no comprehensive model-based accounts of any of these issues have been developed; but models did leave some traces in the discussions of these topics.
5.1 Models and the realism versus antirealism debate

It has been claimed that the practice of model building favors antirealism over realism. Antirealists point out that truth is not the main goal of scientific modeling. Cartwright (1983), for instance, presents several case studies illustrating that good models are often false and that supposedly true theories might not help much when it comes to understanding, say, the working of a laser.

Realists deny that the falsity of models renders a realist approach to science impossible by pointing out that a good model, thought not literally true, is usually at least approximately true. Laymon (1985) argues that the predictions of a model typically become better when we relax idealizations (i.e. de-idealize the model), which he takes to support realism (see also McMullin 1985, Nowak 1979 and Brzezinski and Nowak 1992).

Apart from the usual complaints about the elusiveness of the notion of approximate truth, antirealists have taken issue with this reply for two (related) reasons. First, as Cartwright (1989) points out, there is no reason to assume that one can always improve a model by adding de-idealizing corrections. Second, it seems that the outlined procedure is not in accordance with scientific practice. It is unusual that scientists invest work in repeatedly de-idealizing an existing model. Rather, they shift to a completely different modeling framework once the adjustments to be made get too involved (Hartmann 1998). The various models of the atomic nucleus are a case in point. Once it has been realized that shell effects are important to understand various phenomena, the (collective) liquid drop model has been put aside and the (single-particle) shell model has been developed to account for these findings. A further difficulty with de-idealization is that most idealizations are not ‘controlled’. It is, for example, not clear in what way one has to de-idealize the MIT-Bag Model to eventually arrive at quantum chromodynamics, the supposedly correct underlying theory.

A further antirealist argument, the ‘incompatible models argument’, takes as its starting point the observation that scientists often successfully use several incompatible models of one and the same target system for predictive purposes (Morrison 2000). These models seemingly contradict each other as they ascribe different properties to the same target system. In nuclear physics, for instance, the liquid drop model explores the analogy of the atomic nucleus with a (charged) fluid drop, while the shell model describes nuclear properties in terms of the properties of protons and neutrons, the constituents of an atomic nucleus. This practice appears to cause a problem for scientific realism. Realists typically hold that there is a close connection between the predictive success of a theory and its being at least approximately true. But if several theories of the same system are predictively successful and if these theories are mutually inconsistent, they cannot all be true, not even approximately.

Realists can react to this argument in various ways. First, they can challenge the claim that the models in question are indeed predictively successful. If the models aren't good predictors, the argument is blocked. Second, they can defend a version of perspectival realism (Giere 1999, Rueger 2005) according to which each model reveals one aspect of the phenomenno in question, and when taken together a full (or fuller) account emerges. Third, realists can deny that there is a problem in the first place because scientific models, which are always idealized in one way or another and therefore strictly speaking false, are just the wrong vehicle to make a point about realism. Finally, one can urge that all representation, the everyday no less than the scientific, involves idealization so that trading in suitable idealizations is what it is to know about an independent objective world (Teller 2004).
5.2 Model and reductionism

The multiple-models problem mentioned in the last section raises the question of how different models are related. Evidently, multiple models for the same target system do not generally stand in a deductive relation as they often contradict each other. Given that most of these models seem to be indispensable to the practice of science, a simple picture of the organization of science along the lines of Nagel's (1961) model of reduction or Oppenheim and Putnam's (1958) pyramid picture does not seem plausible.

Some have suggested (Cartwright 1999, Hacking 1983) a picture of science according to which there are no systematic relations that hold between different models. Some models are tied together because they represent the same target system, but this does not imply that they enter into any further relationships (deductive or otherwise). We are confronted with a patchwork of models, all of which hold ceteris paribus in their specific domains of applicability (see also the papers collected in Falkenburg and Muschik 1998).

Some argue that this picture is at least partially incorrect because there are various interesting relations that hold between different models or theories. These relations range from controlled approximations over singular limit relations (Batterman 2004) to structural relations (Gähde 1997) and rather loose relations called stories (Hartmann 1999; see also Bokulich 2003). These suggestions have been made on the basis of cases studies (for instance of so-called effective quantum field theories, see Hartmann 2001) and it remains to be seen whether a more general account of these relations can be given and whether a deeper justification for them can be provided (e.g. within a Bayesian framework).
5.3 Models and laws of nature

It is widely held that science aims at discovering laws of nature. Philosophers, in turn, have been faced with the challenge of explicating what laws of nature are. According to the two currently dominant accounts, the best systems approach and the universals approach, laws of nature are understood to be universal in scope, meaning that they apply to everything that there is in the world. This take on laws does not seem to square with a view that assigns models a center stage in scientific theorizing. What role do general laws play in science if it is models that represent what is happening in the world and how are models and laws related?

One possible response is to argue that laws of nature govern entities and processes in a model rather than in the world. Fundamental laws, on this approach, do not state facts about the world but hold true of entities and processes in the model. Different variants of this view have been advocated by Cartwright (1983, 1999), Giere (1999), and van Fraassen (1989). Surprisingly, realists about laws do not seem to have responded to this challenge and so it remains an open question whether (and if so how) a realistic understanding of laws and a model-based approach to science can be made compatible.
5.4 Models and scientific explanation

Laws of nature play an important role in many accounts of explanation, most prominently in the deductive-nomological model and the unification approach. Unfortunately, these accounts inherit the problems that beset the relationship between models and laws. This leaves us with two options. Either one can argue that laws can be dispensed with in explanations, an idea which is employed in both van Fraassen's (1980) pragmatic theory of explanation and approaches to causal explanation such as Woodward's (2003). According to the latter, models are tools to find out about the causal relations that hold between certain facts or processes and it is these relations that do the explanatory job. Or one can shift the explanatory burden on models. A positive suggestion along these lines is Cartwright's so-called ‘simulacrum account of explanation’, which suggests that we explain a phenomenon by constructing a model that fits the phenomenon into the basic framework of a grand theory (1983, Ch. 8). On this account, the model itself is the explanation we seek. This squares well with basic scientific intuitions but leaves us with the question of what notion of explanation is at work (see also Elgin and Sober 2002).
6. Conclusion

Models play an important role in science. But despite the fact that they have generated considerable interest among philosophers, there remain significant lacunas in our understanding of what models are and of how they work.
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Related Entries

laws of nature | scientific explanation | scientific realism | scientific unity | thought experiments
Acknowledgments

We would like to thank Nancy Cartwright, Paul Humphreys, Julian Reiss, Elliott Sober, Chris Swoyer, and Paul Teller for helpful comments and suggestions on a draft of this entry."

Câmera Typhoon

O incrível vídeo acima foi feito com a câmera Typhoon HD4, que consegue capturar até 200.000 frames por segundo e custa mais de US$ 100 mil. O vídeo foi feito para promover a série South Pacific, e mostra o surfista Dylan Longbottom surfando uma onda de mais de 3 metros e altura.

A Typhoon HD4 tem um sensor CMOS com sensibilidade de 1000 ASA, uma resolução de 1.280 × 1.024 pixels e filma em 720p com até 1000 frames por segundo. Ela também tem 4GB de memória interna, que são suficientes para 3.5 segundos em 1000 FPS ou 7 segundos em 720p, e usa lentes Nikon de 14mm e 17-35mm.

Saiba mais na Science Media.

Via Gadget Lab.

Seed Bomb

Seed Bomb é um conceito para uma “bomba” feita em plástico biodegradável que ao ser lançada solta sementes para reflorestar áreas devastadas.

Segundo os designers Hwang Jin wook, Jeon You ho, Han Kuk il e Kim Ji myung, as bombas seriam lançadas por um bombardeiro, e trariam várias cápsulas em seu interior.

Cada cápsula teria um solo artificial com as sementes que seriam plantadas no local da queda. O solo artificial seria suficiente para a planta se manter sozinha, enquanto a própria umidade da planta não derrete o plástico.

Via Yanko Design.

The Theory Underlying Concept Maps and How to Construct and Use Them[1]

realfgpneto Institute for Human and Machine Cognition

The Theory Underlying Concept Maps and How to Construct and Use Them[1]

Joseph D. Novak & Alberto J. Cañas

Florida Institute for Human and Machine Cognition

Pensacola Fl, 32502

www.ihmc.us

Technical Report IHMC CmapTools 2006-01 Rev 2008-01

(click here for a printable PDF version: small filesize (998K) or high quality 6.6MB))

Introduction

Concept maps are graphical tools for organizing and representing knowledge. They include concepts, usually enclosed in circles or boxes of some type, and relationships between concepts indicated by a connecting line linking two concepts. Words on the line, referred to as linking words or linking phrases, specify the relationship between the two concepts. We define concept as a perceived regularity in events or objects, or records of events or objects, designated by a label. The label for most concepts is a word, although sometimes we use symbols such as + or %, and sometimes more than one word is used. Propositions are statements about some object or event in the universe, either naturally occurring or constructed. Propositions contain two or more concepts connected using linking words or phrases to form a meaningful statement. Sometimes these are called semantic units, or units of meaning. Figure 1 shows an example of a concept map that describes the structure of concept maps and illustrates the above characteristics.

Figure 1. A concept map showing the key features of concept maps. Concept maps tend to be read progressing from the top downward.

(click on an image for a larger view)

Another characteristic of concept maps is that the concepts are represented in a hierarchical fashion with the most inclusive, most general concepts at the top of the map and the more specific, less general concepts arranged hierarchically below. The hierarchical structure for a particular domain of knowledge also depends on the context in which that knowledge is being applied or considered. Therefore, it is best to construct concept maps with reference to some particular question we seek to answer, which we have called a focus question. The concept map may pertain to some situation or event that we are trying to understand through the organization of knowledge in the form of a concept map, thus providing the context for the concept map.

Another important characteristic of concept maps is the inclusion of cross-links. These are relationships or links between concepts in different segments or domains of the concept map. Cross-links help us see how a concept in one domain of knowledge represented on the map is related to a concept in another domain shown on the map. In the creation of new knowledge, cross-links often represent creative leaps on the part of the knowledge producer. There are two features of concept maps that are important in the facilitation of creative thinking: the hierarchical structure that is represented in a good map and the ability to search for and characterize new cross-links.

A final feature that may be added to concept maps is specific examples of events or objects that help to clarify the meaning of a given concept. Normally these are not included in ovals or boxes, since they are specific events or objects and do not represent concepts.

Concept maps were developed in 1972 in the course of Novak’s research program at Cornell where he sought to follow and understand changes in children’s knowledge of science (Novak & Musonda, 1991). During the course of this study the researchers interviewed many children, and they found it difficult to identify specific changes in the children’s understanding of science concepts by examination of interview transcripts. This program was based on the learning psychology of David Ausubel (1963; 1968; Ausubel et al., 1978). The fundamental idea in Ausubel’s cognitive psychology is that learning takes place by the assimilation of new concepts and propositions into existing concept and propositional frameworks held by the learner. This knowledge structure as held by a learner is also referred to as the individual’s cognitive structure. Out of the necessity to find a better way to represent children’s conceptual understanding emerged the idea of representing children’s knowledge in the form of a concept map. Thus was born a new tool not only for use in research, but also for many other uses.

 

Psychological Foundations of Concept Maps

The question sometimes arises as to the origin of our first concepts. These are acquired by children during the ages of birth to three years, when they recognize regularities in the world around them and begin to identify language labels or symbols for these regularities (Macnamara, 1982). This early learning of concepts is primarily a discovery learning process, where the individual discerns patterns or regularities in events or objects and recognizes these as the same regularities labeled by older persons with words or symbols. This is a phenomenal ability that is part of the evolutionary heritage of all normal human beings. After age 3, new concept and propositional learning is mediated heavily by language, and takes place primarily by a reception learning process where new meanings are obtained by asking questions and getting clarification of relationships between old concepts and propositions and new concepts and propositions. This acquisition is mediated in a very important way when concrete experiences or props are available; hence the importance of “hands-on” activity for science learning with young children, but this is also true with learners of any age and in any subject matter domain.

In addition to the distinction between the discovery learning process, where the attributes of concepts are identified autonomously by the learner, and the reception learning process, where attributes of concepts are described using language and transmitted to the learner, Ausubel made the very important distinction between rote learning and meaningful learning. Meaningful learning requires three conditions:

1. The material to be learned must be conceptually clear and presented with language and examples relatable to the learner’s prior knowledge. Concept maps can be helpful to meet this condition, both by identifying large general concepts held by the learner prior to instruction on more specific concepts, and by assisting in the sequencing of learning tasks though progressively more explicit knowledge that can be anchored into developing conceptual frameworks.

2. The learner must possess relevant prior knowledge. This condition can be met after age 3 for virtually any domain of subject matter, but it is necessary to be careful and explicit in building concept frameworks if one hopes to present detailed specific knowledge in any field in subsequent lessons. We see, therefore, that conditions (1) and (2) are interrelated and both are important.

3. The learner must choose to learn meaningfully. The one condition over which the teacher or mentor has only indirect control is the motivation of students to choose to learn by attempting to incorporate new meanings into their prior knowledge, rather than simply memorizing concept definitions or propositional statements or computational procedures. The indirect control over this choice is primarily in instructional strategies used and the evaluation strategies used. Instructional strategies that emphasize relating new knowledge to the learner’s existing knowledge foster meaningful learning. Evaluation strategies that encourage learners to relate ideas they possess with new ideas also encourage meaningful learning. Typical objective tests seldom require more than rote learning (Bloom, 1956; Holden, 1992). In fact, the worst forms of objective tests, or short-answers tests, require verbatim recall of statements and this may be impeded by meaningful learning where new knowledge is assimilated into existing frameworks, making it difficult to recall specific, verbatim definitions or descriptions. This kind of problem was recognized years ago in Hoffman’s (1962) The Tyranny of Testing.

As noted above, it is important to recognize that because individuals vary in the quantity and quality of the relevant knowledge they possess, and in the strength of their motivation to seek ways to incorporate new knowledge into relevant knowledge they already possess, the rote-meaningful distinction is not a simple dichotomy but rather a continuum. Creativity can be seen as a very high level of meaningful learning, and we will discuss this further. These ideas are shown in Figure 2.

Figure 2. Learning can vary from hightly rote to highly meaningful. Creativity results from very high levels of meaningful learning.

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People often confuse rote learning and meaningful learning with teaching approaches that can vary on a continuum from direct presentation of information (which may be conceptually obscure or conceptually explicit) to autonomous discovery approaches where the learner perceives the regularities and constructs her/his own concepts. Both direct presentation and discovery teaching methods can lead to highly rote or highly meaningful learning by the learner, depending on the disposition of the learner and the organization of the instructional materials. These distinctions are shown in Figure 3. There is the mistaken notion that “inquiry” studies will assure meaningful learning. The reality is that unless students possess at least a rudimentary conceptual understanding of the phenomenon they are investigating, the activity may lead to little or no gain in their relevant knowledge and may be little more than busy work. In fact, the research basis for support of widely recommended inquiry learning is largely absent (Mayer, 2004; Kirschner et al., 2006; Sweller et al., 2007).

Figure 3. The Rote-Meaningful learning continuum is not the same as the Reception-Discovery instructional continuum.

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One of the powerful uses of concept maps is not only as a learning tool but also as an evaluation tool, thus encouraging students to use meaningful-mode learning patterns (Mintzes et al., 2000; Novak, 1990; Novak & Gowin, 1984). Concept maps are also effective in identifying both valid and invalid ideas held by students, and this will be discussed further in another section. They can be as effective as more time-consuming clinical interviews for identifying the relevant knowledge a learner possesses before or after instruction (Edwards & Fraser, 1983).

Another important advance in our understanding of learning is that the human memory is not a single “vessel” to be filled, but rather a complex set of interrelated memory systems. Figure 4 illustrates the memory systems of the human mind, and interactions with inputs from our affective and psychomotor inputs.

 

Figure 4. Key memory systems of the brain all interact when we are learning.

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While all memory systems are interdependent (and have information going in both directions), the most critical memory systems for incorporating knowledge into long-term memory are the short-term and “working memory.” All incoming information is organized and processed in the working memory by interaction with knowledge in long-term memory. The limiting feature here is that working memory can process only a relatively small number of psychological units (five to nine) at any one moment (Miller, 1956).

This means that relationships among two or three concepts are about the limit of working memory’s processing capacity. For example, if a person is presented with a list of 10-12 letters or numbers to memorize in a few seconds, most will recall only 5 to 9 of these. However, if the letters can be grouped to form a know word, or word-like unit, or the numbers can be related to a phone number or something known, then 10 or more letters or numbers can be recalled. In a related test, if we give learners 10-12 familiar but unrelated words to memorize in a few seconds, most will recall only 5-9 words. If the words are unfamiliar, such as technical terms introduced for the first time, the learner may do well to recall correctly two or three of these. Conversely, if the words are familiar and can be related to knowledge the learner has in her/his cognitive structure, e.g. months of the year, 12 or more may be easily recalled.

It should be noted that retention of information learned by rote still takes place in long term memory, as does information learned meaningfully; the difference is that in rote learning, there is little or no integration of new knowledge with existing knowledge resulting in two negative consequences. First knowledge learned by rote tends to be quickly forgotten, unless much rehearsed. Second, the knowledge structure or cognitive structure of the learner is not enhanced or modified to clear up faulty ideas. Thus misconceptions will persist, and knowledge learned has little or no potential for use in further learning and/or problem solving (Novak, 2002).

Therefore, to structure large bodies of knowledge requires an orderly sequence of iterations between working memory and long-term memory as new knowledge is being received and processed (Anderson, 1992). We believe one of the reasons concept mapping is so powerful for the facilitation of meaningful learning is that it serves as a kind of template or scaffold to help to organize knowledge and to structure it, even though the structure must be built up piece by piece with small units of interacting concept and propositional frameworks. Many learners and teachers are surprised to see how this simple tool facilitates meaningful learning and the creation of powerful knowledge frameworks that not only permit utilization of the knowledge in new contexts, but also the retention of the knowledge for long periods of time (Novak, 1990; Novak & Wandersee, 1991). There is still relatively little known about memory processes and how knowledge finally gets incorporated into our brain, but it seems evident from diverse sources of research that our brain works to organize knowledge in hierarchical frameworks and that learning approaches that facilitate this process significantly enhance the learning capability of all learners (Bransford et al., 1999; Tsien, 2007).

Obviously, our brains store more than concepts and propositions. While the latter are the principal elements that make up our knowledge structures and form our cognitive structure in the brain, we pause briefly to discuss other forms of learning. Iconic learning involves the storage of images of scenes we encounter, people we meet, photos, and a host of other images. These are also referred to as iconic memories (Sperling, 1960; 1963). While the alphanumeric images Sperling used in his studies were quickly forgotten, other kinds of images are retained much longer. Our brains have a remarkable capacity for acquiring and retaining visual images of people or photos. For example, in one study (Shepard, 1967) presented 612 pictures of common scenes to subjects, and later asked which of two similar pictures shown was one of the 612 seen earlier? After the presentation the subjects were 97% correct in identifying picture they had seen. Three days later, they were still 92% correct, and three months later they were correct 58% of the time. This and many other studies have shown that humans have a remarkable ability to recall images, although they soon forget many of the details in the images. Considering how often we look at pennies, it is interesting that the subjects asked to draw a penny in a study by Nickerson and Adams (1979) omitted more than half of the features or located them in the wrong place. We believe that integrating various kind of images into a conceptual framework using concept mapping software like CmapTools (described below) could enhance iconic memory, and we hope research on this will be done.

Human’s ability to recall sounds is also remarkable. The learning and recall of sounds is also referred to as archic memory. Consider the musician who can play hundreds of songs without reading any music. Again we are dealing with memories that are not coded as concepts or propositions. Studies by Penfield and Perot (1963), among others, indicate that regions of our brain that are activated when we hear sounds are the same regions that are active when we recall sounds. While we can locate regions of the brain that are active in learning or recall of information using positron emission tomography (PET) scans, the specific mechanisms by which neurons store this information is not known. A full discussion of memory mechanisms is beyond the scope of this document.

There are obvious differences between individual’s abilities, and some of these have been explored by Gardner (1983). He has proposed a Theory of Multiple Intelligences. His work has received much attention in education and has served to draw attention to the broad range of differences in human abilities for various kinds of learning and performance. It is good that schools are recognizing that there are important human capabilities other than the recall of specific cognitive information so often the only form of learning represented in multiple-choice tests used commonly in schools and corporations. One reason we encourage the integration of the broad range of activities represented in our New Model for Education is to provide opportunities for these other abilities to be represented and expressed. Nevertheless, we seen the organizing opportunities afforded by associating the various activities with an explicit knowledge structure as very beneficial. Time will tell if future research studies will support this claim.

While it is true that some students have difficulty building concept maps and using these, at least early in their experience, this appears to result primarily from years of rote-mode learning practice in school settings rather than as a result of brain structure differences per se. So-called “learning style” differences are, to a large extent, derivative from differences in the patterns of learning that students have employed varying from high commitment to continuous rote-mode learning to almost exclusive commitment to meaningful mode learning. It is not easy to help students in the former condition move to patterns of learning of the latter type. While concept maps can help, students also need to be taught something about brain mechanisms and knowledge organization, and this instruction should accompany the use of concept maps. The information in the above paragraphs should become part on the instructional program for skillful use of concept maps. The information provided in this document could be part of this instruction. Other ideas for improving instruction to achieve understanding of the subject is available elsewhere (Mintzes et al., 1998).

To illustrate how difficult it can be for individuals to modify their ideas, especially if they learn primarily by rote, we cite the example of interviews done by the Private Universe Project (PUP) at Harvard University (Schneps, 1989). The staff of PUP interviewed 23 Harvard graduates, alumni and faculty, asking each “Why do we have seasons?” Only eleven concepts, properly organized are needed to understand why we have seasons, and one arrangement of these concepts is shown in Figure 5. The PUP interviewers found that 21 of the 23 interviewed could not explain why we have seasons, a topic that is taught repeatedly in school. Included in this group was a graduate who had recently taken a course in the Physics of Planetary Motion, who also believed erroneously that seasons were caused by the earth moving closer to the sun in summer and further away in the winter. In fact, the earth is slightly closer to the sun when it is winter in Massachusetts, rather than in summer. The primary reason we have seasons in latitudes away from the equator is due to the tilt of the earth on its axis toward the sun in summer resulting in longer days and more direct radiation, thus greater heating. In winter, the axis of the earth points away from the sun, thus resulting in shorter days and less intense radiation. What is interfering with these 21 Harvard people is confusion with the common experience that when we are closer to a fire or lamp, the heat is more intense than when we are further away. Thus, these people have failed to recognize that this same phenomenon is not operating to give seasons on Earth. They are transferring knowledge from one context to another, but incorrectly. This is commonly observed in many, many examples of “misconceptions” in every field of study. The only solution to the problem of overcoming misconceptions is to help learners learn meaningfully, and using concept maps can be very helpful.  (For more information on misconceptions in science and mathematics see Novak (2002), and: www.mlrg.org).

Figure 5. One representation of the knowledge structure required required for understanding why we have seasons.

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Epistemological Foundations of Concept Maps

As indicated earlier, we defined concept as a perceived regularity (or pattern) in events or objects, or records of events or objects, designated by label. It is coming to be generally recognized now that the meaningful learning processes described above are the same processes used by scientists and mathematicians, or experts in any discipline, to construct new knowledge. In fact, Novak has argued that new knowledge creation is nothing more than a relatively high level of meaningful learning accomplished by individuals who have a well organized knowledge structure in the particular area of knowledge, and also a strong emotional commitment to persist in finding new meanings (Novak, 1977, 1993, 1998). Epistemology is that branch of philosophy that deals with the nature of knowledge and new knowledge creation. There is an important relationship between the psychology of learning, as we understand it today, and the growing consensus among philosophers and epistemologists that new knowledge creation is a constructive process involving both our knowledge and our emotions or the drive to create new meanings and new ways to represent these meanings. Learners struggling to create good concept maps are themselves engaged in a creative process, and this can be challenging, especially to learners who have spent most of their life learning by rote. Rote learning contributes very little at best to our knowledge structures, and therefore cannot underlie creative thinking or novel problem solving.

As defined above, concepts and propositions are the building blocks for knowledge in any domain. We can use the analogy that concepts are like the atoms of matter and propositions are like the molecules of matter. There are only around 100 different kinds of atoms, and these make up an infinite number of different kinds of molecules. There are now about 460,000 words in the English language (most of which are concept labels), and these can be combined to form an infinite number of propositions. Although most combinations of words might be nonsense, there is still the possibility of creating an infinite number of valid and meaningful propositions. Poets and novelists will never run out of new ideas to express in new ways. We shall never run out of opportunities to create new knowledge! As people create and observe new or existing objects or events, the creative people will continue to create new concents and new knowledge. Creating new methods of observing or recording events usually opens up new opportunities for new knowledge creation. For example, the creation of the concept mapping method for recording subject’s understandings has led new opportunities to study the process of learning and new knowledge creation.

While there is value in studying more extensively the process of human learning and human knowledge creation, this is beyond the scope of this document. The reader is invited to peruse some of the references cited. Some important considerations for construction of better concept maps and facilitation of learning will be discussed further below.

Constructing Good Concept Maps

In learning to construct a concept map, it is important to begin with a domain of knowledge that is very familiar to the person constructing the map. Since concept map structures are dependent on the context in which they will be used, it is best to identify a segment of a text, a laboratory or field activity, or a particular problem or question that one is trying to understand. This creates a context that will help to determine the hierarchical structure of the concept map. It is also helpful to select a limited domain of knowledge for the first concept maps.

A good way to define the context for a concept map is to construct a Focus Question, that is, a question that clearly specifies the problem or issue the concept map should help to resolve. Every concept map responds to a focus question, and a good focus question can lead to a much richer concept map. When learning to construct concept maps, learners tend to deviate from the focus question and build a concept map that may be related to the domain, but which does not answer the question. It is often stated that the first step to learning about something is to ask the right questions.

Given a selected domain and a defined question or problem in this domain, the next step is to identify the key concepts that apply to this domain. Usually 15 to 25 concepts will suffice. These concepts could be listed, and then from this list a rank ordered list should be established from the most general, most inclusive concept, for this particular problem or situation at the top of the list, to the most specific, least general concept at the bottom of the list. Although this rank order may be only approximate, it helps to begin the process of map construction. We refer to the list of concepts as a parking lot, since we will move these concepts into the concept map as we determine where they fit in. Some concepts may remain in the parking lot as the map is completed if the mapmaker sees no good connection for these with other concepts in the map.

The next step is to construct a preliminary concept map. This can be done by writing all of the concepts on Post-its(TM), or preferably by using the IHMC CmapTools (Cañas et al., 2004b, http://cmap.ihmc.us) computer software program described below. Post-its allow a group to work on a whiteboard or butcher paper and to move concepts around easily. This is necessary as one begins to struggle with the process of building a good hierarchical organization. Computer software programs are even better in that they allow moving of concepts together with linking statements and the moving of groups of concepts and links to restructure the map. When CmapTools is used in conjunction with a computer projector, two or more individuals can easily collaborate in building a concept map and see changes as they progress in their work. CmapTools also allows for collaboration between individuals in the same room or anywhere in the world, and the maps can be built synchronously or asynchronously, depending on the mapmakers’ schedules.

It is important to recognize that a concept map is never finished. After a preliminary map is constructed, it is always necessary to revise this map. Other concepts can be added. Good maps usually result from three to many revisions. This is one reason why using computer software is helpful.

Once the preliminary map is built , cross-links should be sought. These are links between concepts in different segments or domains of knowledge on the map that help to illustrate how these domains are related to one another. Cross-links are important in order to show that the learner understands the relationships between the sub-domains in the map.

Figure 6. A “string” map created by a fourth grade student following a class field trip to a paper mill. The class identified concepts in the parking lot on the left, but this student was not successful in using many of these and her map makes little sense. This student was a good oral reader, but she had very poor reading comprehension and was a committed rote learner (see Novak & Gowin, 1984, page 108).

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After a preliminary map is constructed, cross-links should be sought. These are links between concepts in different segments or domains of knowledge on the map that help to illustrate how these domains are related to one another. Cross-links are key to show that the learner understands the relationships between the sub-domains in the map.

It is important to help students recognize that all concepts are in some way related to one another. Therefore, it is necessary to be selective in identifying cross-links, and to be as precise as possible in identifying linking words that connect concepts. In addition, one should avoid “sentences in the boxes”, that is, full sentences used as concepts, since this usually indicates that a whole subsection of the map could be constructed from the statement in the box. “String maps” illustrate either poor understanding of the material or an inadequate restructuring of the map. Figure 6 shows an example of a string map.

Students often comment that it is hard to add linking words onto the “lines” of their concept map. This is because they poorly understand the relationship between the concepts, or the meanings of the concepts, and it is the linking words that specify this relationship. Once students begin to focus-in on good linking words, and on the identification of good cross-links, they can see that every concept could be related to every other concept. This also produces some frustration, and they must choose to identify the most prominent and most useful cross-links. This process involves what Bloom (1956) identified as high levels of cognitive performance, namely evaluation and synthesis of knowledge. Concept mapping is an easy way to encourage very high levels of cognitive performance, when the process is done well. This is one reason concept mapping can also be a very powerful evaluation tool (Edmondson, 2000).

Finally, the map should be revised, concepts re-positioned in ways that lend to clarity and better over-all structure, and a “final” map prepared. When computer software is used, one can go back, change the size and font style, and add colors to “dress up” the concept map.

Thus, we see that concept maps are not only a powerful tool for capturing, representing, and archiving knowledge of individuals, but also a powerful tool to create new knowledge.

The CmapTools Software Toolkit

The CmapTools (Cañas et al., 2004b) software (available for download at: http://cmap.ihmc.us) developed at the Institute for Human and Machine Cognition brings together the strengths of concept mapping with the power of technology, particularly the Internet and the World Wide Web (WWW). The software not only makes it easy for users of all ages to construct and modify concept maps in a similar way that a word processor makes it easy to write text, it allows users to collaborate at a distance in the construction in their maps, publish their concept maps so anybody on the Internet can access them, link resources to their maps to further explain their contents, and search the WWW for information related to the map.

The software allows the user to link resources (photos, images, graphs, videos, charts, tables, texts, WWW pages or other concept maps) located anywhere on the Internet or in personal files to concepts or linking words in a concept map through a simple drag-and-drop operation. Links to these resources are displayed as icons underneath the concepts, as shown in Figure 7. Clicking on one of these icons will display a list of links from which the user can select to open the linked resource. Using CmapTools, it is possible to use concept maps to access any material that can be presented digitally, including materials prepared by the mapmaker. In this way, concept maps can serve as the indexing and navigational tools for complex domains of knowledge, as will be illustrated later with NASA materials on Mars (Briggs et al., 2004). By facilitating the linking between concept maps, learners can construct Knowledge Models (Cañas et al., 2003b; Cañas et al., 2005), which are collections of concept maps with linked resources about a particular topic, demonstrating that their understanding about a domain is not limited to a single concept map.

Figure 7. A concept map about birds constructed by a high-school student. Icons under the concepts provide links to resources (e.g., images, pictures, web pages, videos, other concept maps), some of which are shown in the Figure.

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Facilitating Collaborative and Distance Learning

There is a growing body of research that shows that when students work in small groups and cooperate in striving to learn subject matter, positive cognitive and affective outcomes result (Johnson et al., 1981; Berk & Winsler, 1995). Vygotsky (1978) introduced the idea that language and social dialogue can support learning, especially when members of the social group are at about the same Zone of Proximal Development (ZPD). He describes the ZPD as that level of understanding for a given subject where the learner can progress on her/his own, with minimal aid from a tutor. When students work cooperatively in groups and use concept maps to guide their learning, significantly greater learning occurs (Preszler, 2004). In our work with both teachers and students, small groups working cooperatively to construct concept maps have proven to be useful in many contexts. In the early 1990s, Latin America, students using the IBM Net (before the Internet) were very successful in creating concept maps both with students in their classroom and with students in other countries (Cañas et al., 2001). In our own classes and workshops, and in classes taught by our students and colleagues, small groups of students working collectively to construct concept maps can produce some remarkably good maps.

CmapTools provides extensive support for collaborative work during concept map construction. The concept maps built using CmapTools can be stored on servers (CmapServers, see: Cañas et al., 2003a) where anybody on the Internet can access them. Many of the CmapServers are “public”, allowing anybody (no authorization needed) to publish their collections of concept maps and resources (Cañas et al., 2004a). Through CmapServers, users of all ages and working in many disciplines have published thousands of maps on all topics and domains. While concept maps on these public servers are only a sample of concept maps submitted by persons using CmapTools, and some do not meet our criteria of good concept maps, they nevertheless serve to illustrate diverse applications. When a concept map is saved to a CmapServer, a “web page” version of the map is also stored, so a WWW browser is sufficient to browse through all the published concept maps.

Through the storing of concept maps in CmapServers, CmapTools encourages collaboration among users constructing the maps. When maps are stored in a server on the Internet, users with appropriate permissions (Cañas et al., 2003c) can edit shared concept maps at the same time (synchronously) or at their convenience (asynchronously). “Discussion threads” and “Annotations” in the form of electronic “Post-It” notes can be used to make anecdotal comments on concept maps or during map construction. The high degree of explicitness of concept maps makes them an ideal vehicle for exchange of ideas or for the collaborative construction of new knowledge. We have also found that the obstacles deriving from personal insecurities and fear of embarrassment are largely circumvented, since critical comments are directed at the concept map, not at the person(s) building the map. Having learners comment on each other’s concept maps, whether they are in the same classroom or in different schools, is an effective form of peer-review and collaboration.

The extensive support that CmapTools provides for the collaborative construction of concept maps by groups, whether they are at the same location or in distant locations, has encouraged the increasing use of collaboration during map building. In a variety of educational settings, concept mapping in small groups has served us well in tasks as diverse as understanding ideas in assimilation learning theory to clarifying job conflicts for conflict resolution in profit and non-profit corporations (e.g., Beirute & Mayorga, 2004). Concept maps are now beginning to be used in corporations to help teams clarify and articulate the knowledge needed to solve problems ranging from the design of new products to marketing to administrative problem resolution.

A New Model for Education

A Concept Map-Centered Learning Environment

CmapTools provides a variety of features that make it possible for teachers to use concept maps for a variety of the tasks that students perform (Cañas & Novak, 2005). In addition to a network environment that fosters collaboration and the possibility of constructing knowledge models, the software allows users, among other features, to (a) search for information based on a concept map (Carvalho et al., 2001), by which a student can use the Cmap to research information to learn more about the topic, leading to an improved map with linked resources, and iteratively proceed on another search; (b) record the process of constructing a Cmap for later playback, providing support to the teacher in what is considered to be a key aspect of concept mapping: the process of constructing a map; (c) piece-wise display a concept map and associated resources full-screen for oral presentations; (d) graphically compare two Cmaps, allowing the teacher to compare the student’s map to his/hers for an initial evaluation. The concept map can thus become an artifact around which the various activities of the learning process can be centered, as shown in Figure 8.

Based on the features provided by CmapTools, the student can use the concept map prepared as a pre-test as an initial step towards learning the pieces of knowledge that he/she needs to better understand, as the basis on which to perform the research that leads to this understanding, as a way to organize the various sources from which the student will construct this understanding, as the artifact with which to collaborate with peers, and as the means to present his/her findings at the end of the unit. Furthermore, the concept maps constructed by the student can become the foundation for a portfolio evaluation (see Vitale & Romance, 2000) of his/her performance.

Focus Question, Parking Lots and Expert Skeleton Maps

A concept map-centered learning environment implies that concept maps are used throughout the development of a learning unit or module. Concept maps within this environment are likely to be used as the mechanism to determine the level of understanding students have about the topic being studied before the topic is introduced. The maps are then developed, extended and refined as the students develop other activities on the topic and increase their understanding, possibly concluding with complex knowledge models that link resources, results, experiments, etc., and that can be used if desired as a final presentation by the students.

Just as there are many possible uses of concept maps within the classroom activities, there are a variety of “starting points” for the construction of the initial concept maps by students.

Figure 8. The whole spectrum of learning activities can be integrated using CmapTools, incorporating various learning activities recorded via the software creating a digital portfolio as a product of the learning.

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Each student can construct the initial concept map individually, giving the teacher feedback on the level of understanding of every student. Within the option of individual construction of the map, the students can be allowed to collaborate through a Knowledge Soup (Cañas et al., 1995; Cañas et al., 2001), where students are able to share propositions but not see each other’s maps (see Figure 9). The concept map can be constructed by students working in couples or small groups, where the teacher must pay attention to the level of participation of every student. CmapTools has a recorder feature tht allows recording and playback of steps in map construction, including identification of each contributor.

The concept map can also be a class effort, using a projector, where all students give their opinion and participate in the construction of the map. Teachers must be alert to evaluate the individual participation of every student.

Likewise, the starting point from which the map is constructed can vary depending on the expected previous understanding by the students, the difficulty and novelty of the topic, and the teacher’s confidence in mastering the topic.

Focus Question

Figure 9. Concept map that is part of a collaborative Knowledge Soup. The list of propositions on the top right window are automatically derived from the Cmap, and those with a “pin” have been “published”. The lower right window shows propositions from other participants in Soup, some of which have discussion threads attached questioning or commenting on the proposition.

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The starting point for constructing a concept map can consist of only the focus question. For example, “How do we measure time?” can be given to the students as the question to answer through the construction of the concept map. The type of focus question makes a difference in the type of concept maps that the student builds. A question like “What are plants?” will lead to a declarative, more classificatory concept map than the question “Why do we need plants?” Experiments show that not only the focus question, but also the root concept of a concept map have a strong influence on the quality of the resulting concept map (Derbentseva et al., 2004, 2006). It is important that a question be given and not just a topic (e.g. “make a concept map about plants”), since answering the question helps the students focus on their maps. Whenever a concept map is made with CmapTools and then saved, the maker is asked to provide a focus question, as well as key concepts for this concept map.

Parking Lot

We refer to a list of concept waiting to be added to a concept map as the parking lot of concepts. The staring point for the construction of the concept map can be a list of concepts that the teacher wants to make sure all students include in their map. An example of this was given in Figure 6 above. Figure 10 presents the focus question and parking lot for the focus question “What is the structure of the Universe?” The student, group of students, or class is expected to build a concept map that answers the question and includes at least the concepts in the list. Experienced concept mappers agree with researchers that the most challenging and difficult aspect of constructing a concept map is constructing the propositions; that is, determining what linking phrases will clearly depict the relationship between concepts. So giving the student some of the concepts does not take away from the difficulty in the map construction, although it may somewhat limit the creativity of the student in selecting the concepts to include. It does provide the teacher with insight into which concepts the student(s) had trouble integrating into the concept map, indicating little or no understanding of these concepts.

Figure 10. The beginning of a concept map with a focus question and a parking lot with concepts to be included in the map.

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Expert Skeleton Maps

For difficult topics – whether difficult for the students as determined by the teacher’s previous experience, or difficult for the teacher because of his/her background – using an “expert skeleton” concept map is an alternative. An “expert skeleton” concept map has been previously prepared by an expert on the topic, and permits both students and teachers to build their knowledge on a solid foundation. “Expert skeleton” concept maps serve as a guide or scaffold or aid to learning in a way analogous to the use of scaffolding in constructing or refurbishing a building.

Figure 11 is an “expert skeleton” concept map that corresponds to the same topic as the “parking lot” in Figure 10. Observe that in this example, some of the concepts were left in the “parking lot” for the student to add to the concept map.

Figure 11. An expert skeleton concept map dealing with a key concept that needs to be understood as a foundation for learning science, based on the parking lot from Figure 10. Some concepts were left in the parking lot for the student to add to the Cmap.

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The use of “expert skeleton” concept maps is a research topic we are pursuing, and for which we don’t have as much experience as with the focus question and parking lot starting points. O’Donnell, Dansereau, & Hall (2002) have shown that “knowledge maps” can act as scaffolds to facilitate learning.

It is important to note that the “expert skeleton” concept maps should be built by an expert on the topic. The intention is that the expert will be better at selecting the small number of concepts that are key to understanding the topic, and express accurately the relationships between these concepts. In general, it is much more difficult to build a good, accurate concept map about a topic with a small number of concepts (e.g., four or five) than with fifteen to twenty concepts.

There is no predetermined size that an “expert skeleton” concept map should have. But the expected final number of concepts in the map is a function of the number of concepts in the “skeleton”. For example, a “skeleton map” that consists of five concepts should be expanded by the student to a map with 15 to 20 concepts. If the “skeleton” map contains 20 concepts, which makes it more of a complete map, the final map could be expected to contain about 50 to 60 concepts. In this case, we are probably referring to using a relatively complete (not skeleton) map as a scaffold, expecting students to go deeper into the topic by creating several submaps that are linked to the starting point map.

We foresee a program of using “expert skeleton” maps to scaffold learning beginning with the development of a series of concept maps in a discipline, starting with the most general, most inclusive ideas and then gradually moving to more specific concept maps that will guide the learners. For example, Figure 11 shows a “expert skeleton” concept map for the sciences that encompasses key major concepts needed to understand science. Learners can begin with such a map, add concepts from the parking lot, link digital resources and also construct more specific submaps. More specific expert concept maps can also be provided, such as that shown in Figure 12. Here we also see a submap that might be created by a group of learners, and a sample of two resources that could be accessed via icons on the submap.

Figure 12. An Energy transformation Cmap that could be accessed by linking it to the “Energy” concept in the concept map in Figure 11, and a Photosynthesis Cmap that may be linked to it.

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One of the advantages in using CmapTools for scaffolding learning is the search function mentioned above, which permits access to WWW resources that are screened to fit the context of meanings defined by the concept map (Carvalho et al., 2001; Leake et al., 2004). Thus if one clicks on a concept such as “electrical energy” in Figure 12 and selects one of the “search” menu options, CmapTools will retrieve WWW resources that not only deal with electricity, but also relate to other concepts in the map. The program tries to figure out what the Cmap is about and prepare a query for Web search engines that will generate results that are relevant to the ideas being developed in the concept map. Of course, the learner still needs to select new concepts from the material and construct new propositions on the concept map that add meanings and clarity to the map. Thus, the learner or team of learners is very actively engaged in the meaning building process, an essential requirement for meaningful learning to occur.

Learners can also engage in laboratory or field studies that will add important concrete experiences needed for developing fuller meanings to concepts, and sometimes the excitement that comes with discovering new ideas or relationships.

The extent of materials and ideas that can be built into knowledge structures using “expert skeleton” concept maps, CmapTools, and WWW resources far surpass what any textbook or any teacher could provide. In fact, teachers supervising this kind of study are likely to learn as many new things as their students. Moreover, beginning with the “expert skeleton” maps as starting points reduces the chance that misconceptions or faulty ideas held by learners or teachers will be reinforced and maximize the chance that they will build knowledge structures that in time remove or diminish misconceptions (Novak, 2002).

The World of Science Project

In 1966, Bobbs-Merrill published an elementary science textbook series, The World of Science, written largely by Novak with the objective of introducing basic science concepts to elementary school students and teachers. Unlike most elementary science textbooks, this series presents in-depth instruction in basic concepts at all grade levels, including instruction in concepts dealing with the nature of science, nature of matter, energy and energy transformations. The books have been scanned and a DVD of all six books is available. Our plan is to use The World of Science books as a starting point for a demonstration project for A New Model for Education. To begin, “expert skeleton” concept maps have been prepared for some sections of the grade two book and the whole of the grade four book of the World of Science entitled The Expanding World of Science. All of these concept maps are publicly available on the CmapTools Network[2]. The “expert skeleton” concept maps would serve as a starting point for students and teachers for each section illustrated in the book, and then students would use these Cmaps together with CmapTools to search the WWW for pertinent resources and ideas. Figure 13 illustrates one of the “expert skeleton” concept maps that could be used as the starting point for building a knowledge model, preferably students working in teams and sharing ideas.

Figure 13. Schema showing the New Model for Education with an “expert skeleton” concept map that can serve as the “backbone” for an emerging portfolio in science.

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The science books provide relevant readings and suggested activities. It would be important for the teacher to help students perform these activities, and similar related activities, some of which might by suggested in WWW resources. Learners would also add their own concepts to the “expert skeleton” concept map, as well as resources identified in readings and from the Internet. Figure 14 illustrates a stage in this process[3].

Figure 14. Schema showing the New Model for Education with concepts and resources added to the “expert skeleton” concept map, plus a page from a World of Science book providing relevant reading and activities.

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Obviously, it would be a very deficient science program that did nothing more than have students copy and do some building on the “expert skeleton” concept maps provided for grade two, or for any other grade. Students need concrete, hands on experiences with real things and to observe real phenomena to put meaning into the concept labels provided in the concept maps and other resources.

A pilot program effort is already in progress in Italy, where Giuseppe Valittuti (2004) and his colleagues are now working to translate The World of Science books into Italian. Valittuti and his colleagues have obtained funding from the Italian Ministry of Education for teacher training and a number of elementary school teams began working with the World of Science concept maps and other resources during the 2005-2006 year. The plan is to have four sets of schools focus on different aspects of The World of Science series and produce photos and videos of students doing projects that illustrate and utilize the various science concepts. There will be much feedback from classrooms helping the teams to refine their work, sharing “electronic portfolios” using CmapTools. This feedback should help us to rapidly refine concept maps, techniques and approaches for improving practice of the New Model for Education. The CmapTools Network may serve as a clearinghouse for some of these efforts through its Public servers in Italy and other countries. We anticipate that an abundance of both anecdotal and empirical data will flow from these efforts in a few years. Based on the solid theoretical and related research findings now available, there is every reason to be optimistic that these innovative efforts will be successful. Progress of this project can be followed at: www.leparoledellascienza.it.

Problems of Implementation

The greatest challenge we may expect is to change the school situational factors in the direction of teacher as coach and learner from the prevailing model of teachers as disseminator of information. We know that we need to engage teachers and administrators in training programs that can model the new educational approaches, and we need to seek their counsel on ways to improve on the New Model for Education. There is also the challenge of changing assessment practices that now rely primarily on multiple-choice tests that measure mainly rote recall of information, to performance-based tests that require students to demonstrate that they understand basic concepts and can use these concepts in novel problems solving, and that they can use Internet resources to grow and modify their concepts and learn new concepts. There remains in the New Model plenty of room for acquisition of specific facts and procedures, but now these should be learned within the context of powerful conceptual frameworks. Research (Bransford et al., 1999) has shown that factual information acquired in a context of meaningful learning is not only retained longer, but this information can be used much more successfully to solve new problems.

We might expect some oppositioin to implementation of the New Model of Education from individuals who believe that "inquiry" learningis the only way to improve education. In fact, research overwhelmingly supports the value of "guided learning", such as that involved in A New Model of Education (Mayer, 2004; Kirschner et al., 2006; Sweller et al., 2007).

There is an enormous job of teacher education that needs to be done before the New Model can be implemented in schools. Teachers need to become familiar with the use of CmapTools software and the various tools it contains. They also need to learn about the theory underlying concept mapping, including the ideas in this paper. Teacher education programs should model the kind of learning we are recommending, and we might use as “expert skeleton” concept maps some of the concept maps available from Novak’s (1998) book accessible at the “IHMC Public Cmaps (2)” CmapServer, reachable through the CmapTools Places, under the folder “World of Science”. Teachers should work collaboratively to build on some of the simpler concept maps dealing with education ideas and perhaps add resources to some of the more complex concept maps. Even with the current state of technology and pedagogical understandings, it is possible for schools, states or countries to mount a New Model for Education.

Concept Maps for Evaluation

We are now beginning to see in many science textbooks the inclusion of concept mapping as one way to summarize understandings acquired by students after they study a unit or chapter. Change in school practices is always slow, but it is likely that the use of concept maps in school instruction will increase substantially in the next decade or two. Other innovative practices for assessing student understanding of subject matter are also available (Mintzes et al., 2000). When concept maps are used in instruction, they can also be used for evaluation. There is nothing written in stone that says multiple choice tests must be used from grade school through university, and perhaps in time even national achievement exams will utilize concept mapping as a powerful evaluation tool. This is a chicken-and-egg problem because concept maps cannot be required on national achievement tests if most students have not been given opportunities to learn to use this knowledge representation tool. On the other hand, if state, regional, and national exams would begin to include concept maps as a segment of the exam, there would be a great incentive for teachers to teach students how to use this tool. Hopefully, in the next two decades, this will come to pass. Currently there are a number of projects in the USA and elsewhere that are doing research to see if better evaluation tools can be developed, including the use of concept maps. We should begin to see significant advances in this area in the next several years. Some features of the latest versions of CmapTools also facilitate the use of concept maps for assessment. For example, the “Compare concept maps” tool allows the comparison of an “expert” concept map for a topic with maps constructed by students, and all similar or different concepts and propositions are shown in color.

Concept Maps and Curriculum Planning

In curriculum planning, concept maps can be enormously useful. They present in a highly concise manner the key concepts and principles to be taught. The hierarchical organization of concept maps suggests more optimal sequencing of instructional material. Since the fundamental characteristic of meaningful learning is integration of new knowledge with the learners’ previous concept and propositional frameworks, proceeding from the more general, more inclusive concepts to the more specific information usually serves to encourage and enhance meaningful learning. Thus, in curriculum planning, we need to construct a global “macro map” showing the major ideas we plan to present in the whole course, or in a whole curriculum, and also more specific “micro maps” to show the knowledge structure for a very specific segment of the instructional program. Faculty working independently or collaboratively can redesign course syllabi or an entire curriculum. For example, faculty working together to plan instruction in veterinary medicine at Cornell University constructed the concept map shown in Figure 15.

Using concept maps in planning a curriculum or instruction on a specific topic helps to make the instruction “conceptually transparent” to students. Many students have difficulty identifying the important concepts in a text, lecture or other form of presentation. Part of the problem stems from a pattern of learning that simply requires memorization of information, and no evaluation of the information is required. Such students fail to construct powerful concept and propositional frameworks, leading them to see learning as a blur of myriad facts, dates, names, equations, or procedural rules to be memorized. For these students, the subject matter of most disciplines, and especially science, mathematics, and history, is a cacophony of information to memorize, and they usually find this boring. Many feel they cannot master knowledge in the field. If concept maps are used in planning instruction and students are required to construct concept maps as they are learning, previously unsuccessful students can become successful in making sense out of science and any other discipline, acquiring a feeling of control over the subject matter (Bascones & Novak, 1985; Novak, 1991, 1998).

Figure 15. A concept map prepared cooperatively by the faculty of the College of Veterinary Medicine at Cornell University to show the over-all structure for a revised curriculum.

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Capturing and Archiving Expert Knowledge

One of the uses of concept maps that is growing at a fast rate is the use of concept maps to capture the “tacit” knowledge of experts. Experts know many things that they often cannot articulate well to others. This tacit knowledge is acquired over years of experience and derives in part from activities of the expert that involve thinking, feeling and acting. Often experts speak of a need to “get a feeling for what you’re working on”. In fact, the biography of one Nobel Lauriat in biology (Barbara McClintock) was entitled, A Feeling for the Organism (Keller, 1983). Nonaka and Takeuchi (1995) stress the importance of capturing and using the knowledge of corporate expert’s tacit knowledge if a company wants to become “the knowledge creating company”.

Most of the methods used prior to concept maps consisted of various forms of interviews and analyses with experts, including case studies of how experts accomplished some remarkable achievement (Hoffman et al., 1995; Klein & Hoffman, 1992). In fact, these methods continue to be highly popular with many cognitive scientists, most of whom are unfamiliar with Ausubel’s work and the kind of epistemological ideas on which concept mapping is based. We also used “clinical interviews” in our early work, as noted above, but we found it necessary to invent a better way to represent what our learners knew and how their knowledge was changing over time. At IHMC, we began using interviews to identify expert knowledge needed to interpret computer readings from computer outputs of a machine designed to assess problems with heart functions, following the injection of a bolus of radioactive solution, and to diagnose coronary dysfunction (Ford et al., 1991; Ford et al., 1996). However, when we began to concept map the expert knowledge of a cardiologist who literally “wrote the book” on this technology, it was evident that there were concepts missing in the map and that the “tacit knowledge” of our expert was not fully expressed in his book or in our interviews. Thus, the concept map not only allowed us to represent the expert’s knowledge, but also to find gaps in the knowledge structure we were procuring through interviews.

While we expect that interviews, case study analyses, “critical incident” analyses and similar techniques will have value in extracting and representing expert knowledge, it is likely that the end product of these studies might still be best represented in the form of concept maps, perhaps with some of the interview data and other information presented through icons on maps.

At IHMC we continue to be very active in the area of capturing and representing expert knowledge (Coffey et al., 2002). As the CmapTools software has evolved, it has become an increasingly useful tool for this work, as illustrated by the remarkable resources on Mars prepared at NASA Ames’ Center for Mars Exploration (Briggs et al., 2004). Figure 16 shows a “Home” concept map for the knowledge portfolio that Briggs created and Figure 17 shows one of the many submaps he created. The entire set of concept maps can be viewed at: http://cmex.ihmc.us. In addition to submaps, a wide variety of digital resources can be accessed via the concept maps. Many other projects are represented in the IHMC Public CmapServer accessible through CmapTools, including projects dealing with weather forecasting (Hoffman et al., 2000, see: http://www.ihmc.us/research/projects/StormLK/), electronic technicians (Coffey et al., 2003), and Thai fabric crafts.

Figure 16. A "Home" concept map for the knowledge portfolio created by NASA for Mars Exploration.

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Figure 17. An example of a concept map that can be accessed via clicking on one of the resources attached to a concept on Figure 16.

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Conclusions

In this paper we have tried to present the theoretical foundations and the origins of what we call concept maps. While at first glance concept maps may appear to be just another graphic representation of information, understanding the foundations for this tool and its proper use will lead the user to see that this is truly a profound and powerful tool. It may at first look like a simple arrangement of words into a hierarchy, but when care is used in organizing the concepts represented by the words, and the propositions or ideas are formed with well-chosen linking words, one begins to see that a good concept map is at once simple, but also elegantly complex with profound meanings. Concept mapping has been shown to help learners learn, researchers create new knowledge, administrators to better structure and manage organizations, writers to write, and evaluators assess learning. As with any tool, it can also be misused, and we have illustrated some examples of this.

We also wish to use this document as a foundation for further experimentation, critique, and dialogue regarding the use of this tool. The CmapTools web site provides opportunities for lively exchanges among users and researchers. This document itself should be a “living” document, with revisions occurring periodically as we gain new knowledge and experiences with the use of this tool. We invite all users of concept mapping and CmapTools to participate in this dialogue.

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[1] Revised January 22, 2008. Cite as: “Novak, J. D. & A. J. Cañas, The Theory Underlying Concept Maps and How to Construct Them, Technical Report IHMC CmapTools 2006-01 Rev 01-2008, Florida Institute for Human and Machine Cognition, 2008", available at: http://cmap.ihmc.us/Publications/ResearchPapers/TheoryUnderlyingConceptMaps.pdf.

[2] Go to http://cmapdp.ihmc.us, Click on "IHMC Public Cmaps (2)", and then select the "The World of Science" folder.

[3] The World of Science books were published in 1966 and some of the figures are dated.