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Calculating and Understanding: Formal Models and Causal Explanations in Science, Common Reasoning and Physics Teaching Ugo Besson Published online: 8 August 2009 Ó Springer Science+Business Media B.V. 2009 Abstract This paper presents an analysis of the different types of reasoning and physical explanation used in science, common thought, and physics teaching. It then reflects on the learning difficulties connected with these various approaches, and suggests some possible didactic strategies. Although causal reasoning occurs very frequently in common thought and daily life, it has long been the subject of debate and criticism among philosophers and scientists. In this paper, I begin by providing a description of some general tendencies of common reasoning that have been identified by didactic research. Thereafter, I briefly discuss the role of causality in science, as well as some different types of explanation employed in the field of physics. I then present some results of a study examining the causal reasoning used by students in solid and fluid mechanics. The differences found between the types of reasoning typical of common thought and those usually proposed during instruction can create learning difficulties and impede student motivation. Many students do not seem satisfied by the mere application of formal laws and functional relations. Instead, they express the need for a causal explanation, a mechanism that allows them to understand how a state of affairs has come about. I discuss few didactic strategies aimed at overcoming these problems, and describe, in general terms, two examples of mechanics teaching sequences which were developed and tested in different contexts. The paper ends with a reflection on the possible role to be played in physics learning by intuitive and imaginative thought, and the use of simple explanatory models based on physical analogies and causal mechanisms. 1 Physical Reasoning and Causal Explanations in Common Thought Students do not arrive at physics classes with their minds empty of ideas about the physical phenomena they are about to study. To the contrary, they already possess interpretative schemata, and personal ideas and conceptions concerning many physical situations. Beginning in the seventies, a vast and growing field of research has opened regarding U. Besson (&) Department of Physics ‘‘A. Volta’’, University of Pavia, Via A. Bassi 6, 27100 Pavia, Italy e-mail: [email protected] 123 Sci & Educ (2010) 19:225–257 DOI 10.1007/s11191-009-9203-9

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Page 1: Calculating and Understanding Formal Models

Calculating and Understanding: Formal Modelsand Causal Explanations in Science, Common Reasoningand Physics Teaching

Ugo Besson

Published online: 8 August 2009� Springer Science+Business Media B.V. 2009

Abstract This paper presents an analysis of the different types of reasoning and physical

explanation used in science, common thought, and physics teaching. It then reflects on the

learning difficulties connected with these various approaches, and suggests some possible

didactic strategies. Although causal reasoning occurs very frequently in common thought

and daily life, it has long been the subject of debate and criticism among philosophers and

scientists. In this paper, I begin by providing a description of some general tendencies of

common reasoning that have been identified by didactic research. Thereafter, I briefly

discuss the role of causality in science, as well as some different types of explanation

employed in the field of physics. I then present some results of a study examining the

causal reasoning used by students in solid and fluid mechanics. The differences found

between the types of reasoning typical of common thought and those usually proposed

during instruction can create learning difficulties and impede student motivation. Many

students do not seem satisfied by the mere application of formal laws and functional

relations. Instead, they express the need for a causal explanation, a mechanism that allows

them to understand how a state of affairs has come about. I discuss few didactic strategies

aimed at overcoming these problems, and describe, in general terms, two examples of

mechanics teaching sequences which were developed and tested in different contexts. The

paper ends with a reflection on the possible role to be played in physics learning by

intuitive and imaginative thought, and the use of simple explanatory models based on

physical analogies and causal mechanisms.

1 Physical Reasoning and Causal Explanations in Common Thought

Students do not arrive at physics classes with their minds empty of ideas about the physical

phenomena they are about to study. To the contrary, they already possess interpretative

schemata, and personal ideas and conceptions concerning many physical situations.

Beginning in the seventies, a vast and growing field of research has opened regarding

U. Besson (&)Department of Physics ‘‘A. Volta’’, University of Pavia, Via A. Bassi 6, 27100 Pavia, Italye-mail: [email protected]

123

Sci & Educ (2010) 19:225–257DOI 10.1007/s11191-009-9203-9

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students’ conceptions (representations, ideas…) about specific subjects in physics, with a

special emphasis on the cognitive status of these conceptions and their role in learning and

teaching. The terminology used has varied according to the time, the cultural area and the

interpretative framework: pre-conceptions, misconceptions, prior ideas, alternativeframeworks, children’s science, spontaneous or common reasoning… A veritable topog-

raphy of the more common conceptions is now available.

Together with ideas about specific subjects and phenomena of physics and other sci-

ences, research very quickly identified some general tendencies and transversal forms of

reasoning and explanation which are typical of common thought, and can pose problems

for learning.

Driver et al. (1985, cap. 10) describe some general characteristics found by analyzing

various studies on students’ conceptions:

• focusing on limited aspects of a particular physical situation and on the perceptually

salient characteristics of the objects and the phenomena;

• interpretation of phenomena in terms of absolute properties or qualities ascribed to

objects rather than in terms of interaction between elements of a system;

• focus on changes in time rather than to equilibrium or steady-state situations, for which

there would be nothing to explain, because nothing happens;

• linear causal reasoning, based on a logical and chronological sequence of one-cause-

one-effect chains, neglecting the reciprocity of interactions (cf. Rozier and Viennot

1991);

• use of undifferentiated notions, which have a range of connotations more extensive,

inclusive and global than those used by scientists and include properties of different

physical concepts;

• interpretations and explanations strongly context-dependent, calling upon different

ideas for situations that a scientist would consider as examples of the same type of

phenomena.

Viennot (1996) stresses the role of time in the students’ reasoning and the tendency to

focus on change and transitory rather than equilibrium or steady state situations, because

only changes are considered to require an explanation, and the tendency to transform

concepts in personages or things which change in a linear succession of events as in a

history or in an engine in which a piece acts on another and so on.

DiSessa (1983) defines some elementary explanatory structures (phenomenological

primitives, p-prims), which are constructed by abstraction and generalization of daily life

observations: springiness, squishiness, Ohm’s p-prim…Andersson (1986) has shown the relevance of an ‘‘experiential gestalt of causation’’,

based on an agent-instrument-patient scheme, as the common core of pupils’ physical

reasoning.

Ogborn (1993) stresses ‘‘the fundamental nature of causal action as an element of

reasoning that is not detached from objects and events, but considered as an essential part

of their meaning’’. Gutierrez and Ogborn (1992) have elaborated a framework for

understanding common causal reasoning based on a mechanistic mental model.

Brown (1992) shows the effectiveness of instruction proposing ‘‘visualizable, qualita-

tive, mechanist models which can help students to make sense of the more abstract

principles often invoked to explain the phenomena’’.

According to Psillos (1995), many difficulties in the understanding of electric circuits

stem from the fact that:

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‘‘The classical treatment of simple DC circuits is based on the study of functional

relationships between their basic macroscopic parameters such as current, resistance,

potential difference, energy etc… the circuit equations do not reveal a physical

process through which the circuit has reached its steady state and they cannot support

any form of causal mechanism… whilst students’ reasoning is causal, in the sense

that they are looking for causes which would lead a circuit to its new state.’’

Research in science education has shown the strong presence in common reasoning of

causal explanations, often conceived as a ‘‘mechanism’’ capable of accounting for physical

transformations. Formal laws and functional relationships, stating how things ‘‘have to be’’,

are insufficient for learning at school age and unsatisfactory for the student’s need to

understand. Pupils require a causal explanation, a mechanism, which can account for

the dynamics of facts and effects that have led to a given situation, how things are the way

they are.

It can be a question of simple causality (one cause, one effect), a linear chain of cause-

effect connections or of circular causality. The systemic reasoning used by physicists,

especially in stationary situations, is very far from the student’s intuition and way of

thinking. For this reason, it can be very difficult for students to give a physical sense to

many of the formulas studied.

Even in mathematics, where causality is certainly marginal and, to the contrary, thereasons for the regularities are sought, causality has been found to appear in student

reasoning. For example, in ‘‘everyday’’ logic the implication p ) q is often interpreted as

a causal and chronological relationship (Dumont 1985), and the concept of limit appears

closely linked to a dynamical and chronological conception, and is associated with

physical movement and a physical approach (Sierpinska 1985). This is a conception that

researchers in the didactics of mathematics consider an obstacle to a correct understanding

of the mathematical concept, in which time has no role.

2 Causality in Science

It is worth stressing that science education and the philosophy of science have different

aims and thus different points of view on these issues. The goals of the philosophy of

science are primarily epistemological. For this reason, it aims understand the role and

status of causality in scientific knowledge, and to determine whether causation is to be

understood instrumentally or not. On the other hand, the goals of research in science

education are understanding learning processes in science, and improving the under-

standing of scientific concepts and theories. The differences between these sets of

objectives necessarily lead to different evaluations of causation in science and science

learning.

My aim here is not to provide a deeper analysis of the various views of causality and

scientific explanation that are available, but to sketch an overview of the principal positions

of philosophers and scientists. This is to provide a useful point of reference and prob-

lematic background that can help us better to understand physics learning processes as well

as to delineate effective didactic strategies that exploit the connections among common

thought, student conceptions, and scientific reasoning.

As showed in Sect. 1, causal reasoning has an important role in common thought, and in

daily and social life. On the other hand, the idea of causality has long given rise to debate

and criticism among philosophers and scientists.

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After the elaborations of classical philosophy and the related controversies (Aristotle,

Hume, Kant…), different conceptions of causality have been supported by philosophers,

such as regularism, singularism, probabilistic causation, counterfactual theories, agency

theory… More recently the concept of cause has been questioned: by the positivists (Mach,

Duhem…), who argue that it is not possible to know the connections which exist in reality

and forbid physics from making conjectures about a level of reality which would be

subjacent to the phenomena; by the probabilistic or subjectivist interpretations of quantum

physics; by the chaos theorists, who dissociate causality and prediction; and by others, who

wish to restore the idea of finalism in science, and also of an intelligent design.

Comte (1830) had already supported the renunciation of the search for causes and the

mechanisms that produce the phenomena:

‘‘All good thinkers today recognize that our actual studies are strictly limited to an

analysis of phenomena aiming to discover their effective laws, i.e., their constant

relationships of succession and similarity, and can by no means concern their inti-

mate nature, whether their cause, first or final, or their essential mode of produc-

tion… For any scientific assumption to be really subject to judgment, it must relate

exclusively to the laws of the phenomena, and never to their modes of production.’’1

Mach (1883) wrote:

‘‘In nature there is neither causes nor effect… cause-effect connection exists only in

the abstraction which we carry out with the aim of reproducing facts… I endeavored

to replace the concept of cause by that of mathematical function’’. (1883, §4.4.3)

And B. Russell, with his typical English humor:

‘‘The reason why physics has ceased to look for causes is that in fact there are not

such things.The law of causality, like much that passes muster among philosophers,

is a relic of a bygone age, surviving, like the monarchy, only because it is errone-

ously supposed to do no harm’’. (1912, p. 1)

According to Wittgenstein (1921):

‘‘In no way can an inference be made from the existence of one state of affairs to the

existence of another entirely different from it. There is no causal nexus which

justifies such an inference. The belief in the causal nexus is superstition’’. (1921,

sections 5.135–5.1361).

In the article in which he formulated the uncertainty principle, Heisenberg (1927)

expressed himself in a drastic manner:

‘‘Since the statistical character of the quantum theory is closely connected with the

imprecision of all observations, one could be induced to suppose that behind the

perceived statistical world there would be a ‘‘real’’ hidden world in which the cau-

sality principle is valid. But such speculations, we emphasize it explicitly, seem to us

fruitless and meaningless. Physics has to formally describe only the complex of the

1 A. Comte (1830) Cours de philosophie positive, Paris, Rouen Freres, Tome 2, 28e lecon, pp. 435–436 e454. ‘‘Tous les bons esprits reconnaissent aujourd’hui que nos etudes reelles sont strictement circonscrites al’analyse des phenomenes pour decouvrir leurs lois effectives, c’est-a-dire leurs relations constantes desuccession et de similitude, et ne peuvent nullement concerner leur nature intime, ni leur cause, ou premiereou finale, ni leur mode essentiel de production. … Toute hypothese scientifique, afin d’etre reellementjugeable, doit exclusivement porter sur les lois des phenomenes, et jamais sur leurs modes de production.’’

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observations. Indeed… by means of the quantum mechanics the non validity of the

causality principle is definitively established’’.2

Currently, one tendency seems to have become dominant, that of skirting or avoiding the

idea of causality and of emphasizing, instead, functional relationships and systemic or holistic

descriptions. This way of thinking is often linked to a criticism of realism,3 considered as

more or less ‘naıve’, and to the assumption of an instrumentalist point of view. Of course, the

situation in the philosophy of science is more complex. A position which opposes causation is

not necessarily instrumentalist, nor is a position which favors causation necessarily realist: it

is possible to endorse realism and at the same time to be against causation. Nevertheless,

where causality is considered to be a ‘power’ attributed to objects, and not merely an epi-

stemic expedient, then the idea of causality implies realism in at least some form.

It is interesting to contrast two authoritative opposite citations.

Duhem (1908) sustains that the aim of physics is not to explain phenomena, but only

to represent and describe phenomena, and that physical theories are merely natural

classifications:

‘‘When Kepler or Galileo declared that Astronomy has to take as hypotheses proposi-

tions the truth of which is established by Physics, this assertion … could mean that the

hypotheses of Astronomy were judgments on the nature of things and on their real

movements, … that by testing the correctness of these hypotheses, experimental method

was going to enrich our cosmological knowledge with new truths … But, taken in this

sense, their assertion was false and harmful. … In spite of Kepler and of Galileo, we

believe today, as did Osiander and Bellarmin, that the hypotheses of Physics are only

mathematical artifices intended to save the phenomena’’ (1908–1990, pp. 139–140).

On the contrary, K. Popper complained that:

‘‘Today, the view of physical science founded by Osiander, Cardinal Bellarmin, and

Bishop Berkeley has won the battle without another shot being fired. Without any

further debate over the philosophical issue, without producing any new argument, the

instrumentalist view… has become an accepted dogma… it has become part of the

current teaching of physics.’’ (1968, §1, pp. 99–100)

However, there are also some scientists and philosophers who have reasserted the value

and the central, although not exclusive, role of causality in science and particularly in

physics (Bachelard 1934; Bunge 1959; Halbwachs 1971; Piaget 1971; Harre 1972; Salmon

1984).

2 However, in the same paper, Heisenberg uses expressions that clearly indicate a causal connection, even ifin the presence of a certain indeterminacy of some dynamic quantities. Referring to the Compton orphotoelectric effect, he writes: ‘‘When a light quantum that hits an electron is reflected or diffracted from itand then, again refracted through the microscope, it provokes the photoelectric effect‘‘; and ’’Of such light asingle quantum is enough to hurl the electron outside of its orbit’’. It seems to me that some physicists tendto generalize in a hurried and exaggerated way. Starting with some results of quantum physics that are incontrast with some characteristics of causality for some situations, they arrive at general philosophicalconclusions.3 ‘‘Realism is… A doctrine according to which being is independent of the knowledge that conscioussubjects may have of it at a given time: esse is not equivalent to percepi. Idealists hold that the intellectknows only its own states: see the commentaries on contemporary physics, which deny the existence of anygiven external to our representations (to measurements made by observers). Realism and idealism areopposed term for term, each asserting what the other denies. The first posits that thought is inside the being;the second posits that the being is contained in thought’’ (Largeault J. ‘‘Realisme’’, Encyclopaedia Uni-versalis 8, 2002).

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Max Planck wrote:

‘‘Scientific thought aspires to causality, indeed it is the same thing as causal thought,

and the final aim of any science must be to take the causal view-point to its last

consequences’’ (1949, chap. 7).

According to the philosopher R. Harre,

‘‘the discovery of the mechanism by which causes produce or generate their effects is

a central part of a scientific investigation… a scientific explanation is characterized

by the fact that it describes the causal mechanism which produces the phenomena’’

(1972, p. 118 e p. 181).

And according to W. Salmon ‘‘to give a scientific explanation is to show how events fit

into the casual structure of the world’’ (1984, p. 19).

Bachelard (1934):

‘‘We say, more generally, that the principle of causality subjects itself to the

requirements of objective thought and can be taken as the fundamental category of

objective thought… When heat dilates bodies or changes their color, the phenomena

give us a conclusive demonstration of the cause, even though they do not prove

determinism’’ (p. 115).

Piaget (1965):

‘‘In physics, one proscribes causality as an explanation in vain, ordering to limit itself

to laws: the research of the causal explanation remains more than ever an essential

need for the human thought’’ (1965, p. 53).

Similar ideas are currently supported from many physicists. I would cite as an example

Giuliani (2007, p. 274):

‘‘The ‘causality principle’, understood as a methodological commitment to the

searching for causes, has been one of the propulsive forces of scientific knowledge: a

discipline that, on the basis of hardly conclusive evidence, is really abandoning this

commitment, is doomed to drain its vital sources’’.

According to some authors, the instrumentalist point of view in its radical form is apaper position, that is, a position that can be only supported in a paper or book, but not in

real life or scientific practice (cf. for example Giuliani 1998). Something similar can be

said concerning causality, which atomic physics puts into question, but which we cannot

do without in daily and social life. As B. Brecht writes (1970):

‘‘The boldness of the physicists is often boasted about … What makes me laugh is

how they generalize their results, or refuse to generalize them. It is amusing how they

invite philosophers to draw consequences from the questionability of causality in

atomic physics, while at the same time assuring us that this is the case only in their

field, atomic physics, and does not have any influence over the broiling of steaks in

the houses of normal people’’ (pp. 227–228).

Indeed, who can deny that the HIV virus is the cause of AIDS, that the Colorado River

has caused the formation of the Grand Canyon, or that it is the flame of the stove that

makes water boil?

Bunge (1959) proposed an intermediate position:

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‘‘To declare that the sole verifiable relations are those obtaining among sense data,

concepts, and judgments, and to hold that it is vain to try to disclose autonomous

interconnections and real modes of production, is an anthropomorphic attitude

blocking scientific advance; it is a regressive attitude, even if most of its upholders

sincerely believe that they are in the van of modern thought… The right and pro-

gressive attitude is to face the fact that science has advanced to such a point that,

without dispensing entirely with the causal principle, it has assigned it a place in the

broader context of general determinism—a role that is neither the principal nor the

meanest, nor that of a ‘superstition’. The causal principle is one of the various

valuable guides of scientific research…’’ (pp. 334–335).

At the same time Bunge sustains realism:

‘‘Scientists explore the world and attempt to keep at arm’s length from the things

they handle or model, because they are intent on discovering what they are like in

themselves rather than for ourselves. … An analysis of any physical experiment

shows that the experimenter assumes the independent existence of the thing he

intends to observe, measure or alter… Were it not so… he would have to say that he

invented or constructed everything—which would sound either schizophrenic or

postmodern’’ (Bunge 2003, p. 464).

More recently, to conciliate the different behavior of microscopic and macroscopic

worlds, an interesting idea has been proposed of causality as a secondary quality (Menzies

and Price 1993) and/or an emergent property (Laudisa 2005), and causality is considered as

a category of thinking which is indispensable to us to interpret and act in the physical

macroscopic world.

3 Typologies of Physical Explanations and Cognitive Development

3.1 Explanations in Physical Sciences

Halbwachs (1971) distinguishes three types of explanation used in physics: causal or

heterogeneous, formal or homogeneous, and ‘bathygeneous’ explanations (bathygene in

the French text, from the Greek bathys, deep).

In causal or heterogeneous explanations, the change in the system is due to agents outside

the system. These produce an effect by means of delayed actions expressing real connec-

tions between things. Causality can be simple A?B, linear A?B?C?D…, reciprocal

A?B e B?A, or circular A?B?C?D?A. Moreover, often in the phenomena of real and

social life, there are many causes that concur and are entangled in a nearly inextricable way.

The Italian writer C.E. Gadda wrote, with efficacious artistic exaggeration:

‘‘Causes and effects… are never conceivable in the singular… The hypotyposis

[vivid rhetorical figure] of the chain of causes should be corrected and softened with

one of a mesh or net, but not of a mesh with two dimensions (surface) or three

dimensions (space-mesh, space-chain, three dimensional chain), but a mesh or net

with infinite dimensions. Every ring, or lump or tangle, of relations is bound by

infinite filaments to infinite lumps or tangles’’ (Gadda 1974, p. 79).

Formal or homogeneous explanations, on the contrary, consist of simultaneous func-

tional relations between quantities which describe the system and are all internal to the

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system. Some examples are: the law of hydrostatics p = po ? qgh, the ideal gas equation

pV = nRT, the mechanical energy conservation Ec ? Ep = E, Bernoulli’s theorem, etc. In

fact, these formal laws should be considered more descriptions than explanations. How-

ever, many scientists consider that a phenomenon is sufficiently and satisfactorily

explained when it is well described by this kind of law, in the sense that the law makes the

calculation of physical quantities possible.

The last category, ‘bathygeneous’ explanations, includes explanations built on a deeper

level or an underlying structure such as a smaller scale (atoms, electrons, cells, grains,

fibers…) or a more general, deeper theory (for example, electromagnetism explaining

classical optics, kinetic theory explaining the behavior of gases).4

Halbwachs (1971, pp. 73–76) considers that reasoning in terms of reciprocal and cir-

cular causality is a useful bridge towards systemic reasoning and that in any case causal

reasoning remains an implicit support underlying formal explanations. The progressive

movement from simple to reciprocal and circular causality may constitute a necessary

cognitive path in order to prepare students to understand homogenous laws:

‘‘Thus, we have proceeded to a homogeneous explanation, which gives us a func-

tional law, but about which one can only say ‘that is how it is’… However, such laws

will continue to have an explanatory value, as the residue of simple, then mutual

causality, which lead to the homogeneous formulations, but continue to underlie

them.’’

After all, formal laws, as pV = nRT, Ohm’s law and others, describe steady state

situations which have been determined by dynamic interactions between the parts of the

system and they are the result of causal laws governing these interactions. Also the

Maxwell’s equations of electromagnetism can be derived from a causal law (Jefimenko

2004).

This distinction between formal and causal laws corresponds to the one between laws ofco-existence and laws of succession, a more common distinction among philosophers of

science. Dorato (2000, p. 80) discusses the problem of reducing the former type of laws to

the latter (§2.4), concluding that complete reducibility does not seem possible.5

Duhem (1906) makes the drastic proposal that physics should not claim to supply

explanations, that a physical theory is not an explanation, but a description and a classi-

fication which can provide a useful economy of thought: ‘‘a system of mathematical

4 It is possible to compare these three types of physical explanation to three of the four kinds of causesindicated by Aristotle (350 B.C.). The efficient cause, ‘‘the causes whence the principle of change occurs, allthat acts’’. The material and formal causes, ‘‘the causes out of which, of these some as substrate [material],others as concept [formal] e.g., ‘‘the letters of syllables, the material of man-made objects, fire and otherelements of bodies, the parts of the whole, and the premises of the conclusion.’’ Aristotle speaks also aboutreciprocal causes, which act ’’as fatigue causes vigor and vigor, fatigue‘‘. In Physics, book II (B), 3.5 Here it seems me that the author commits a rather common error in the interpretation of the laws of co-existence (or formal laws). An equation of the type pV = nRT represents a relationship at a given timebetween the indicated quantities for an ideal gas in a state of equilibrium. However, the author thinks thatthe equation further asserts that if at a given time the pressure varies in a region of the gas, at the same timethe temperature and/or density will vary so as to maintain the relationship continuously valid. To thecontrary, this is not in fact the case. The law establishes a relationship between quantities for a state ofequilibrium. If something changes in a part of the gas, the gas is no longer in equilibrium; therefore theequation is no longer applicable. It only says to us that if and when a state of equilibrium is re-established,the indicated quantities will assume the values necessary to satisfy the equation once again. The passagefrom the initial to the second equilibrium state happens in a finite time and as a result of local interactionprocesses in the situation of non-equilibrium. The same reasoning holds good for Coulomb’s theorem ofelectrostatics E = r/e, to which the author also makes reference.

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statements, deduced from a small number of principles, whose aim is to represent

experimental laws as simply and completely as possible’’. According to the Hempel’s

(1965) deductive-nomological model, a scientific explanation is characterized by its ability

to unify different empirical phenomena into a single category of facts described by the

same law. Further, an event is explained when its description can be obtained as a

deductive conclusion of a set of general laws and particular conditions. In this model, a

causal relation simply reflects the relations of explanatory dependency connected to the

unifying deductive structure of scientific knowledge. A causal relation is subordinated to

this structure and has a merely epistemic character.

According to the ‘‘statement view’’ of scientific theories, developed by logical empir-

icism and named also ‘‘received view’’ because of its broad acceptance in philosophy until

sixties, a scientific theory is a system of theoretical statements (axioms), expressing the

relations between theoretical terms in a formalized symbolic language, and a set of cor-

respondence rules that interpret these terms empirically (Grandy 2003; Develaki 2007).

The statement view provides the frame of reference for a number of epistemological

positions, and is linked to the ideas of Duhem and Hempel.

Since seventies, a new view emerged which interprets scientific theories as sets of

models. This ‘‘model-based view’’ unifies different philosophical approaches.6 ‘‘sharing

the common characteristic of ascribing particular importance to the concept of model,

understood as a basic structural element of the theories and as a mediator between theory

and reality’’ (Develaki 2007, p. 728). The problem of how physical explanation is con-

ceived according to the model-based view is a complex topic which cannot be explored

here. Nonetheless, it is clear that model-based reasoning and explanations are not limited to

logical conclusions and deductions, as asserted by the statement view.

3.2 Models in Science Education

The model-based view has had a great impact on science education research and psycholo-

gists recognized the ubiquity of models in human reasoning. Several science education

researches were devoted to the role of models in learning and teaching.7 According to

Koponen (2007, pp. 766–768), ‘‘in science education, models are currently considered a

means for a more authentic education, facilitating a scientific way to describe, explain and

predict the behavior of the world and acquire knowledge’’: the model-based view of science

and science education should focus on the ‘‘empirically reliable models’’ that ‘‘are our bridges

to reality’’ and on the notion of ‘‘the empirical reliability of models’’, which is established ‘‘in

the process of matchmaking’’ with experimental data, theory and phenomena. Portides (2007,

p. 721) stresses that ‘‘scientific models and the processes operative in their construction are

essential to science learning and to understand how theory relates to experiment’’.

‘Model’ is a term with manifold meanings. According to Grandy and Dushl (2007), it is

a ‘functional kind’ concept, i.e., ‘‘defined in terms of the function [it] perform[s] and not of

structural similarities.’’ The authors provide the example of the concept of a clock: there is

little physical similarity among water, mechanical, and digital clocks, but they all serve the

same function. ‘‘The common element to all models is that they are external aids to

6 This fact is reflected by the various other names attached to this view, e.g. ‘‘semantic view’’ (in contrast tothe linguistic-syntactic approach of the statement view), ‘‘non-statement view’’, ‘‘model-theoretic view’’,‘‘structuralist’’ (see Develaki 2007; Grandy 2003).7 For example, a special issue of the journal Science & Education was recently devoted to this subject:Science & Education (2007), 16 (7–8): 647–881, Special issue: Models in Science and in Science Education.

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reasoning. They are primarily cognitive prostheses, but they also serve social and epistemic

ends’’ (p. 149).

Nevertheless, a common core shared by the different possible meanings is that a model

in some way represents a ‘target’ system by means a ‘source’ system, mapping some

attributes and structures from one system to the other. (In education it is worth empha-

sizing that only some elements, and not all, can be represented by a model).

Norman (1983) distinguishes between personal mental models (‘‘what people have in

their heads and what guides their use of things’’) and shared conceptual models (‘‘tools for

the understanding or teaching of physical systems’’). Analogously Tiberghien (1994),

distinguishes between models of students and models proposed in instruction. She indicates

two features of student models that sharply differentiate them from instructor models: a

close and direct relation with the material world and its objects, and the frequent use of

causal links. By contrast, causal connections are often absent from the reasoning proposed

in teaching, which tends to privilege functional relationships. It is a matter of facilitating

the student’s passage from one type of model to the other by means of a conceptual change.

Gilbert and Boulter (1998) distinguish four types of models in science education: mentalmodels (personal, private representations of a target), expressed models (which are

expressed by an individual through actions, speech or writings), consensus models(expressed models which have been subjected to testing by a social group, in our case by

the scientific community, and are considered valid or important), and teaching models(specially-constructed models used to aid student understanding).

In constructing, evaluating and implementing models for teaching, they have to be

placed in this last category. That is, teaching models should be considered to be a bridge or

mediation between personal and consensual models, without attempting to situate them toutcourt and totally in the model that is accepted at the level of scientific research. It is

important to be aware that a didactical transposition (Chevallard 1991) of scientific content

is necessary which involves reconstructions, adaptations and alterations of sense and scope.

Grandy and Dushl (2007) distinguish five types of models: mathematical, physical,computer, visual and analogical models. Mathematical models provide the means to

manipulate data and information to produce predictions and evaluations. However, all the

different kinds of models do this by different means.

According to many researchers, including Hestenes, Grandy, and Tiberghien, physics

education requires mathematical models in which the properties of physical systems are

represented by quantitative and qualitative relationships between appropriate variables.

According to many physicists, to have a model means to have a set of equations describing

a problem.

By contrast Duit and Glynn (1996), emphasize the importance of analogical models in

science education (e.g., the planetary model of the atom, the corpuscular model of light, the

particle model of gas, the hydraulic model of electric circuits, etc.). On their view, some

analogical models are constructed and then explicitly used as teaching and learning aids to

provide a bridge to conceptual scientific models. In certain cases, once the conceptual

models have been learned, the analogies are no longer needed: ‘‘the bridge may be pulled

down’’. These authors (see also Duit 1991) summarize as follows some findings of didactic

research on the use of analogies:

• Students use analogies spontaneously.

• For analogies to be fruitful, students need to be guided in their use and interpretation.

• Analogies facilitate learning only in specific areas, therefore multiple analogies are

often necessary in order to facilitate learning in broader domains.

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• Analogies based on superficial similarities have little inferential power; structural

analogies are needed for this purpose.

• Analogies are accepted as teaching aids by students only when they find the target

subject matter sufficiently challenging.

According to these authors, analogies are valuable tools for conceptual change because

they can open up new perspectives, stimulate interest and increase motivation, permit the

visualization of abstract ideas, and facilitate the understanding of abstractions by means of

similarities between different real systems. At the same time, there are certain associated

risks. For example, students frequently over-extend analogies or may take them for

identities. Further, difficulties and misunderstandings associated with the source can be

projected onto the target.

Halloun and Hestenes (Hestenes 1987, 1992; Halloun 2007) develop a modeling theory

of physics instruction, aimed to promote student learning of model-laden theory and

inquiry in science education. In this connection, their primary examples are the mathe-

matical models of classical mechanics. Hestenes (1987, 1992) affirms that a theory can be

put into relation with experience only via the construction of models. He employs New-

tonian mechanics as a starting-point to develop a didactic approach to physics instruction

based on modeling activity, and then proposes to generalize this approach to other areas of

physics (for example, electrical circuits):

‘‘The great game of science is modeling the real world, and each scientific theory

lays down a system of rules for playing the game. The object of the game is to

construct valid models of real objects and processes. Such models comprise the

content core of scientific knowledge. To understand science is to know how scientific

models are constructed and validated. The main objective of science instruction

should therefore be to teach the modeling game’’ Hestenes (1992, p. 732).

According to Hestenes (1987, p. 441), models in physics are surrogate objects, i.e.,

conceptual representations of real systems. They are mathematical models with four

components:

• A set of names for the objects and agents which constitute the system and which

interact with it;

• A set of descriptive variables representing properties of the system;

• The equations of the model which describe its structure and evolution over time;

• An interpretation which relates the descriptive variables to the properties of the system.

Halloun (2007) describes a pedagogical framework for science instruction based on a

modeling learning cycle. This cycle includes the phases of exploration, model adduction,

model formulation, model deployment, and paradigmatic synthesis. The teacher guides

students as they progress through the learning cycle, which presents an evolutionary

interaction among their own ideas, the physical world, and scientific models. Progress

through the cycle is facilitated by the use of appropriate tools, most notably modelingschemata, which are organizational templates used to ensure that all concepts and models

are constructed in a way that is comprehensive, and also consistent with a given theory.

3.3 Piaget and the Genesis of Idea of Causality

Piaget (1971) studied the characteristics and genesis of the concept of causality in the

context of cognitive development, relating it to the attribution to objects of a set of

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operations. He distinguishes between functional laws and causality. The former concern

observation of perceivable, general regularities and imply operations performed on objects.

By contrast, causality concerns connections which go beyond the observable, which are

necessary, and implies operations attributed to objects.

Operations correspond to what the subject can make by manipulating objects, while

causality expresses what the objects can make by acting each other and on the subject.

Piaget thinks that the genesis of the concept of causality proceeds in interaction with the

genesis of operations, each favoring the other by means of convergences and conflicts. The

operations of the subject, elaborated by means of reflective abstractions, are first applied to

objects and then subsequently attributed to them. This occurs on the basis of an isomor-

phism between operational structures and objective properties (which requires that the

properties be considered as pre-existing their discovery by the subject). By permitting such

an attribution, this isomorphism ‘‘supplies, precisely for this reason, the principle of an

explanation that satisfies the conditions of cognitive assimilation, that is, of the under-

standing of objects by the subject’’ (p. 68). Reciprocally, the application and attribution of

operational structures to different physical situations produces different morphisms and

causal explanations, and thereby leads to restructuring and differentiating the previous

operational and conceptual structures of the subject (p. 72). Operations and causality have

a common origin in the subject’s actions. Initially they are undifferentiated, and then

become reinforced and progressively differentiated during development. It is precisely the

persistence of non-differentiation and confusion between subject operations, on the one

side, and the causality inherent to objects, on the other, that would seem to be at the origin

of the typical undifferentiated notions of common thought (see above Sect. 1). Piaget

makes reference, for example, to the mixed notion of force-thrust-movement. At the stage

where ‘‘thrust’’ (elan) is considered both as a source and a result of the movement, the

force or thrust remain internal to the object in motion. Only at a more mature stage is a

differentiation made in which the force becomes the external cause of movement. Other

examples include those of the mixed notion of weight as action and as property-amount,

and the relative non-differentiation of time and speed.

4 Some Particular Features of Causal Physical Explanations in Common Thought

In a study (cf. Besson 2004a, b), conducted using interviews and questionnaires involving

high school pupils, university students and teachers, I highlighted three aspects, related to

one another, which can create problems in physics learning:

• a tendency to displace or delocalize forces and causes, skipping intermediate objects;

• confusion between efficient and contingent causes;

• a difficulty in connecting local causes and global effects.

4.1 Delocalization of Forces

In the situation of Fig. 1, block B is placed on block A, which is pulled by exerting a force

F on it. There is friction between the two blocks. Students are asked to indicate all the

forces acting on block B.

Many students (37% of 253 Italian high school students) considered that the force Facted also on block B, thus ‘displacing’ the cause of movement beyond block A, because

they imagined some kind of link between the two blocks due to friction or adherence:

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« Since F acts upon A, it also acts upon B, since there is friction between A and B…»

«Because between the two boxes there is friction, F acts, with smaller intensity andindirectly, also on box B. »

« The force F that acts on block A is transmitted to B through friction ».

These students consider there is a ‘‘transmission’’ of forces from one object, which

forces actually are acting on, to another object in contact with it (moreover many students

think that no horizontal forces act on the upper block, whose movement is explained

without forces, by the idea of ‘‘drag’’ or ‘‘adherence’’ between blocks, see also Caldas and

Saltiel 1995).

I observed a similar type of reasoning in other situations, for example with the question

‘‘Pressure of liquid and atmospheric pressure’’ (Fig. 2), submitted to 148 first university

year students in Belgium.

Sixty six percent answered, correctly, that the pressure changes and decreases, 27% that

it does not change. What it is interesting here are the justifications given for the correct

answers by many of these students; they express the idea that air acts directly on the bottom

of the container.

« The atmospheric pressure exerts pressure on the top of the liquid and therefore on thebottom of the container »

« The pressure exerted by the water itself on the bottom is identical, but the atmosphericpressure is added to that. »

According to these students, there are two distinct forces which act on the bottom of the

container, one exerted by the water, and the other by the air over the liquid, instead of a

single force, exerted by the water which is in contact with the bottom of the container. In

the students’ reasoning, the formula of hydrostatics law p = po ? qgDh, converted in

terms of forces, is transformed into an addition involving two forces exerted on the bottom

of the container by two different bodies, giving: p = po ? qgDh ) ptot = pair ?

pwat ) Ftot = Fair ? Fwat. In this way the air above the water, is considered to act directly

on the internal part of the bottom of the container.

Fig. 1 The two blocks question

Fig. 2 ‘Pressure force of liquids and atmospheric pressure’ question. A cylindrical container is filled withwater. If the container is moved to a place where the atmospheric pressure is weaker, but gravity is the same,does the force exerted by the water on the bottom of the container change or remain unchanged? Why?

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4.2 Efficient Cause and Contingent Cause

The examples given above can also be interpreted as a confusion between what ‘‘triggers’’

an event (the pressure of the air, or the external force F) and what effectively acts to

produce it (the pressure of the water, or the friction force). I call this confusion, which is

relatively frequent in common reasoning, the confusion between efficient cause and con-tingent cause (or triggering cause).

Similarly Bachelard (1949, p. 209), speaks of a ‘‘trigger (declic, in French) causality,

which sets objective causalities into action.’’ Halbwachs makes a distinction between the

‘‘conditions of the production of a phenomenon’’ (1971, p. 26) and the ‘‘true’’ cause, which

produces the phenomenon. From a philosophical point of view, it is difficult to distinguish

clearly between conditions and causes, and some determining factors of a given situation

can in fact be considered both as conditions or constraining relations and as concomitant

causes of the phenomenon. However, from the physical point of view it is important to

distinguish the different nature and significance of each.

Indeed, usually when we ask ‘‘why’’ in daily life, we are not looking for ‘‘causes,’’ but

for ‘‘reasons’’, meanings, aims, or responsibility. Often conditions or factors considered

normal are not considered to be among the causes of an event, because they constitute the

so-called causal field, which remains in the background as a set of conditions that, for

pragmatic reasons, are not treated as causes (Benzi 2003, p. 39). What interest us are the

factors that are at the origin of a divergence from the customary course of events, or that

determine the peculiar characteristics of a specific event.

You are holding an object in your hand; you open your hand and the object falls. What

is the cause of the object’s fall: that you opened your hand, or the gravitational force of

Earth? One might answer: both, there is no single cause, but various causes, yet one cannot

do without taking into account the different nature of the two ‘‘causes’’ considered.

Opening your hand enables the effect of the force of gravity to manifest itself as downward

acceleration, but it is the force of gravity that makes the object fall downwards.

If a person accused of a crime argues that he was not the real cause of his victim’s death,

since all he did was push him out of the window, and that gravity was what had caused him

to fall, the judges will not hesitate to sentence him.

This shows that in common life usually we take into account what triggers or activates

efficient cause, removes barriers or sparks off events.

It can also be said that in these practical cases, gravity is considered to be one of the

background conditions of the causal field of the situation. The problem is that physicists

often study precisely the processes that regulate the background and general way the

situation works. For example, when studying the motion of a ball, the student concentrates

on the launch phase and its modalities, establishing direct cause-effect relationships

between launch modalities and the point of arrival of the ball. To the contrary, the physics

teacher proposes to study the phenomena and forces at work during the flight of the ball,

and considers launch characteristics only as initial conditions for the development of

examples. Common thought tends to concentrate on the contingent cause and to identify it

with the efficient cause.

This ambiguity shows up clearly in the case of fluids in the presence of gravity. How is

the change in pressure with depth or altitude to be explained? What is the cause of this

variation? The force of gravity, weight, triggers an increase in the pressure forces, but it is

the interactions among parts of the fluid which relate directly to pressure, and it is the

interactions between the parts of the fluid and the solid sides of the container that ‘‘pro-

duce’’ the pressure forces, which are only the resultants, on a macroscopic level, of these

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interactions. The solid sides of the container prevent the fluid from falling, and they act as

boundary conditions, as a contingent cause. But here gravity is also a contingent cause, a

condition for the change in pressure to be occur, whereas it is the local, superficial

interactions, whose nature is electromagnetic, that produce this change and, therefore, are

the efficient cause.

To test whether in this situation there is confusion between efficient and contingent

causes, I submitted to 46 students (first university year) and 36 teachers (in initial or

in-service training) the question entitled ‘‘Interactions’’, in which they were asked about

which of the four fundamental types of force or interaction (gravitational, electromagnetic,

strong or weak) a swimmer feels on his eardrums, when deep under water. The results were

very clear and homogeneous: 94% answered that it was a gravitational force:

‘‘The force is due to the weight of the water situated above’’

‘‘Because it is a phenomenon linked to weight force, and so to the attraction exerted byEarth on the water.’’‘‘Because it depends on the acceleration of gravity (Stevin’s law).’’‘‘The force exerted is weight.’’

The justifications show the shift in meaning from descriptions involving the influence of

gravity on the value of pressure, towards an identification of the pressing force with weight

or gravity: pressure is due to weight, weight causes, creates pressure, the pressing force isweight, that is a gravitational force.8

A similar problem appears in the analysis of contact forces between solids. In the

question entitled ‘‘A book on a table’’, I proposed the simple situation of a book placed

upon a table, and students are asked to identify all the forces acting on the table. A great

majority of pupils mention the weight of the book in their answers (72% of 153 high school

pupils).

‘‘The table is subject to the force of gravity and to the weight force that the book exertson it.’’‘‘The weight force of the book acts on the table…’’‘‘The table, interposing itself between the book and the ground, receives the action of theweight force.’’

The attention of these students is centered on weight, which is supposed to trigger a

force between the table and the book, and contact interactions between the book and the

table are forgotten.

In an interview, a student (final year of secondary school), asked to be more specific as

to the origin and nature of this force, provided further details, for example:

‘‘It is like gravity, it is gravity… it is exerted by the book, no, by Earth, by both… Ifthere were no gravity, the book wouldn’t exert any pressure on the table; therefore itis Earth which exerts that force on the table, indeed on the Moon that force would besmaller… No, no, it isn’t Earth, but the force depends on Earth, therefore,

8 It could be objected that the question is a trap question, because of an abrupt change of explanatory level,passing from a macroscopic and phenomenological description to one microscopic which refers to funda-mental theories. However, short talks, focusing on the contact interaction between water and ear, had beensufficient for the teachers who answered the question to understand their error. This confirms that it was acustomary and automatic reasoning scheme and logical short circuit, therefore really representing a spon-taneous tendency of thought.

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indirectly… Earth exerts a force on the book, therefore, indirectly, on the table aswell…’’

One can see the confusion this student is experiencing, he oscillates between different

typologies of explanations, he seems conscious of the problem but he does not arrive to

solve it by himself.

A similar problem arises in the explanations in which force causes the acceleration of a

car: is it the force of the engine or the friction force of the wheel-ground interaction?

Obviously, the engine triggers the appearance of a forward friction force in the wheel-

ground interaction, but it is the friction force that directly produces the acceleration, as it is

the quantity F in Newton’s law F = ma for the car.

In an article in a French scientific review, this dilemma is expressed by the authors as

follows:

‘‘The overly dogmatic physicist … asserts that it is the road that accelerates the

vehicle. Surely this answer is not satisfactory… What is the role of the will of the

pilot? … If we think of the deformation of the surface layer of the tire… It is this

modification that is responsible for the interactions between the road and the vehi-

cle’’. (Roux and Seigne 2001, p. 506)

In this case, a clear distinction between what triggers and what produces could solve the

contradiction and uneasiness expressed by the authors of the article. Moreover, precisely

the evocation of the local deformations of the tire layers in contact with the road could

have led the authors to consider the forces of interaction involved in this contact as the

forces that directly provoke the acceleration of the car.

4.3 Local Causes and Global Effects

The above observations lead us to think that these types of reasoning may also be linked to

the fact that the student (and more generally the common thought) has a global and

undifferentiated view of the physical system, and does not go into a more detailed analysis

considering local properties and interactions. Focused on global forces acting on the whole

bodies, students do not consider what happens locally, in the little spots of the bodies in

contact, to understand the behavior of the physical system. This brings to another important

characteristic of common reasoning: neglecting to connect (or having difficulty in con-

necting) global descriptions, in terms of formal laws and quantities concerning the whole

system, and local analysis, on a smaller scale, in terms of local interactions and causal

laws.

For example, Stevin’s law, mentioned above, tells us that the pressure in a liquid

increases with depth. To explain this, one usually takes into account the weight of the

liquid above, and in this way, to determine the pressure at a given point one has to consider

the total height of the liquid above, as far as the surface that is in contact with the

atmosphere. That is a form of global reasoning. The pressure at one point is linked to what

is happening in very remote parts of the liquid. But one might ask oneself in what way the

liquid below is different from the liquid above, to account for this increase in pressure. If

nothing changes (temperature, density…), this difference becomes a mystery. One might

say that the liquid at one point does not know what is happening at a distance from it. To

take another example, if one takes a sample of New York air in a container with an

absolute manometer, and takes this container to the mountains, the manometer still indi-

cates the same pressure as in New York. It is easy enough to accept that the air in New

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York is not like the air in the mountains, and not just because it is more polluted; likewise,

one must accept that the water at the bottom of the container has something different than

the water at the top, and that, locally, is where one must look for the origin of what

produces the difference in pressure. This problem stems clearly in many answers to the

‘‘Two fish’’ question (Fig. 3), where students are asked to compare the pressure for two

fish—one in the open sea, the other in an underwater cave—at the same depth.

Many students show that they know the Stevin’s law, the formal rule, according to

which at the same depth the pressures are equal, but they don’t understand how this can

happen, because they are not able to coordinate the global effects with the local property,

in order to obtain a systemic reasoning:

‘‘It is true that the fish are at the same depth, so they ought to feel the same pressure.But the water above the fish in the sea is a greater mass than the mass of water that isabove the fish in the cave. The fish in the sea therefore feels greater pressure than theother fish.’’

The difficulty to connect global effects and local actions appears clearly in the case of

the Archimedes’ upthrust. In the ‘‘Ball in water’’ situation (Fig. 4) students were asked,

among other, if the pressure forces exerted by water on the ball have something to do with

the Archimedes’ upthrust.

Only 11% (of 111 high school and 214 university students) clearly and explicitly

connected the buoyant force with the pressure forces, 31% answered that the buoyant force

has nothing to do with these forces, and 30% said it concerned only, or coincided with the

pressure force at the bottom of the ball. Many students stressed the ‘global’ nature of

Archimedes’ upthrust, in contrast with pressure forces, which are the local, contact forces,

acting upon each part of the ball surface.

« No, because the buoyant force is proportional to the volume of the object and thedensity of the liquid, whereas the pressure acts on the surface and is not affected by thevolume of the object. »« No, the buoyant force only pushes from the bottom up; it has nothing to do with thepressure a liquid exerts around an object in all directions. »« At the bottom there are in fact two forces that are added together: the pressure forcesand buoyancy. »

Fig. 3 ‘‘Two fish’’ question. Students are asked to compare the pressure in the position of two fish—one inthe open sea, the other in an underwater cave—at the same depth

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It is interesting to note that similar answers were given by some French teachers in

initial training (6/17 answers). For example: ‘At equilibrium, it is: FC ? FD ? FA ?

FB ? p ? P = 0, where P = weight of ball, p = buoyant force’.

5 Implications and Proposals for Teaching

As described above, the characteristics of causal reasoning and physical explanation used

by students and common thought differ in several ways from those employed by scientists

and teachers. As a consequence, certain didactic issues arise:

(a) how to introduce the problem of explanation in science, and the role of laws, theories

and models;

(b) how to manage the problem of the consistency of what is proposed and practiced in

teaching with current philosophical elaborations;

(c) how effectively to exploit the types of reasoning and physical explanation typical of

students and common thought to overcome cognitive and affective obstacles, as well

as conceptual and methodological errors.

In the following, I will focus primarily on problem (c), and limit myself to some brief

comments on problems (a) and (b).

Extensive research has been devoted to the problem of how to introduce into the

classroom themes concerning the logic and methodology of science, and the nature of

science.9 As already stressed in Sect. 2, we need to remember that the goals of the

philosophy of science are different from those of science education. Science education

research does not aim to solve theoretical epistemological problems, nor to define the most

logically coherent epistemological system. Science education research should aim to

design learning paths which take into account the varying characteristics of students. These

paths should include a range of philosophical approaches from which students can pick and

choose according to their own specific cultural tendencies. Instruction should put students

in a position to choose their own philosophical positions in a rational manner, based on

knowledge of the available philosophical options, and on the mastery of appropriate rea-

soning tools. The didactic objective is to supply resources useful for a better understanding

of scientific facts and theories, and for building a rational methodology and modern image

Fig. 4 The ‘ball in water’question

9 See for example the books Matthews (1994) and McComas (1998). The journal Science & Educationpublished many articles on these themes, as an example a recent issue is dedicated to the teaching of ’’natureof science‘‘ (2008, Vol. 17, n. 2–3, Special Issue: Teaching and Assessing the Nature of Science). In USA,the National Science Education Standards (http://www.nap.edu/readingroom/books/nses/overview.html)and the Standard for Science Teachers Preparation (http://www.nsta.org/pdfs/NSTAstandards2003.pdf andhttp://www.msu.edu/*dugganha/NOS.htm), include a section dedicated to ‘‘Nature of Science’’.

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of science. Sections 2 and 3 have shown that a universally accepted philosophical con-

ception does not exist. On the contrary, a heated debate prevails among scientists and

philosophers. As a consequence, correct deontology entails that the teaching objective

should be to provide knowledge that problematizes the different approaches, rather than

privileging the one particular conception considered ‘‘sound epistemology’’ by the teacher

or researcher.10

Concerning problem c), the differences between reasoning and explanation preferred by

students and common thought, on the one side, and those employed by scientists and

teachers, on the other, can be summarized, with some simplifications, as follows.

Students mainly:

• reason in terms of changes;

• are interested in the transitory phases, and in the processes leading to a given situation;

• consider causes and effects in temporal succession;

• need intuitive models and physical analogies.

While often instead teachers:

• reason in terms of equilibrium or steady-state situations;

• are interested to the maintenance conditions for a situation;

• consider relationships between quantities at a given time;

• propose formal laws and abstract mathematical models.

Moreover, I have emphasized some particular characteristics of common reasoning

(Sect. 4): a tendency to delocalize forces, confusion between efficient and contingent

causes and the difficulty in connecting local causes and global effects. These aspects are

related to one another. Failing to analyze the local details of interactions and properties

may result in a global view, one which does not properly identify the specific, localized

spots where interactions are at work, and therefore misplaces or displaces forces and

causes. Moreover, this sort of approximation and global analysis may lead to, or be related

to, a confusion between what triggers or initiates a phenomenon, and what produces it

directly.

The students’ need to understand is often connected to the ability to form a mental

model of the physical situation which would allow them to imagine a causal connection, a

mechanism that could explain the concatenation of events. A purely formal and mathe-

matical model expressed in terms of equations, rules, and constraints, does not seem

sufficient for effective learning at the school level. Thus it is matter of finding the

coherence between, on the one side, the laws that establish ‘‘how things must be’’ and, on

10 This is not always what happens. For example Guilbert and Meloche (1993), hold that many teachershave what they term an ‘‘empiricist-realist’’ conception of science. This supposedly ingenuous, antiquatedview involves, among other things, the myth of the progressive construction of knowledge, and the idea thatscience concerns the description of the world as it really is: ’’it seems, according to some students, that it iseven possible to distinguish the true from the false‘‘. Against this conception, these authors counter thatstudents should be led to a subtle, modern constructivist model, according to which theories are merelyspeculative constructions allowing a more systematic collection of observations, and reality does not existindependently of us. It is odd that precisely the authors who support instrumentalist and relativist con-ceptions, and who maintain that true or false, better or worse theories do not exist (each one with its ownfield of validity and effectiveness, depending on context and aims), at the same time introduce their ownconception as the proper and correct epistemological model, and contrast it to others denigrated as ingen-uous and antiquated. If no scientific theory is better than the others and there is no truth to discover, then tobe consistent, epistemological theories as well would have to be considered all at the same level, with no onebetter than others.

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the other, an explanation of ‘‘how it comes about that things effectively are thus,’’ how a

new state of equilibrium is stabilized, and by means of which mechanisms and processes.

According to Silva (2007, p. 845):

‘‘People understand and reason … by constructing mental models with mechanical

features… To appeal to material entity … is a necessary intermediate step towards

the construction of abstract concepts [author refers particularly to the electromag-

netic field]… It is not sufficient that students are able to deal with the mathematical

aspects of a theory. … [Moreover] the historical analogies can help students in

constructing mental models of difficult scientific concepts and developing qualitative

understandings of mathematical expressions … [and] in constructing mental models

of physical phenomena which will be as close as possible to currently accepted

scientific models.’’

The different approaches discussed in Sect. 3 show that authoritative scientists use

various types of explanations, models and reasoning patterns in their work, and refer to

different philosophical positions. This suggests that in order to be complete, science

education needs to present all these different possibilities. Moreover, the research of Piaget

and Halbwachs quoted in Sect. 3 indicates that mature scientific reasoning is constructed

progressively. This cognitive development needs some intermediate steps using simpler

types of reasoning in order to evolve towards more complex and abstract reasoning pat-

terns. For this reason, even quite elementary, naıve models and forms of reasoning can be

useful, and perhaps necessary, in order to achieve the stable acquisition of knowledge.

According to many researchers, some student ideas and conceptions—even those that

are incorrect or do not coincide with scientific ones—need not be considered merely as

obstacles to oppose and surmount. Rather, they can also constitute a useful basis, raw

material with which to construct a cognitive structure nearer to the objectives of instruc-

tion. In this connection, we find discussions of anchoring conceptions (Clement et al.

1989), conceptual clay (Ogborn 1993) and cognitive resources (Hammer 2000).

Brown (1992) showed the effectiveness of instruction based on ‘‘visualizable, qualita-

tive, mechanist models to help students to make sense of the more abstract principles often

invoked to explain the phenomena’’.

According to Chabay et Sherwood (1999):

‘‘Reasoning from a mechanistic mental model involves ‘running’ the model in one’s

head, and observing the consequences. An auxiliary benefit is a concrete sense of the

process by which the system moves from one state to another. Psychologists have

found that reasoning from runnable mental models is often more natural than is

constraint-based reasoning’’.

Matthews (2007, p. 648) stresses that:

‘‘Good teachers start instruction with what is familiar and known, and build to what

is unknown… As models, analogies and metaphors are central to this bridging from

the known to the unknown … education researchers have focused on various aspects

of model utilization in children’s learning.’’

Some researchers proposed using common causal reasoning as a basis for teaching-

learning sequences in electricity (Steinberg 1983; Psillos and Koumaras 1993; Sherwood

and Chabay 1993; Gutwill et al. 1996). They propose teaching situations involving changes

(the transient states in electrical circuits for example), more accessible to pupils’ intuition

and in which pupils are encouraged to reason in terms of causality, before broaching steady

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state descriptions, as a first step towards acquiring a physicist’s systemic reasoning. Psillos

stress that:

‘‘the classical treatment of simple DC circuits is based on the study of functional

relationships between their basic macroscopic parameters. The reasoning which is

expected presupposes a systemic view and is a-causal… [whilst] students’ reasoning

is causal… they are looking for causes which would lead a circuit to its new state’’.

(Psillos 1995, pp. 67–68)

Physical, analogical models and formal, mathematical models are both useful in science

education, much as are reasoning in terms of changes and mechanisms, and reasoning in

terms of stationary situations and simultaneous relationships. Formal, mathematical models

are widely used in common teaching, textbooks and research (see, for example, Hestenes

1987, 1992; Tiberghien 1996; Grandy 2003). Here I focus primarily on the role of physical

analogical models, and reasoning in terms of changes and causal mechanisms. With this

aim, in Sects. 6 and 7 I briefly describe two research projects elaborating teaching

sequences which rely on spontaneous causal reasoning and the common need for expla-

nations based on visual models and appeals to intuition.

The examples concern two topics already mentioned in the first part of this paper

(Sect. 4): pressure in liquids and solid friction.

6 Fluid Statics

A study involving a few hundred high school and college students (Besson 2004a) dem-

onstrates many of the difficulties involved in understanding pressure and forces in a static

fluid.

We take as an example the case of three containers with equal bottoms but different

shapes, all filled with water to the same level (Fig. 5). Here a majority of students think

that the force exerted by the water on the bottom is greater in the case of the container (c),

because it contains more water.

Only slightly fewer than half the students correctly described the magnitudes and

directions of the forces exerted by the water on the four discs, A, B, C, D drawn on an

immersed football (see Fig. 4; the question did not ask for calculations, but only to list the

forces in order of magnitude and to indicate their direction by an arrow) and only a small

minority clearly connected such forces to Archimedes’ upthrust. Moreover, the majority of

high school students thought that water pressure is different at two points at the same

elevation, one in the open sea and the other within an underwater cave (Fig. 3). The most

frequent reasoning was: pressure depends upon depth; pressure depends upon the amount

or weight of the water directly above the considered point; in a smaller space, the pressure

is larger. Above all, we observed a strong persistence of reasoning based on the idea that

pressure depends on the weight of the water directly above the considered point (‘the water

column above’, or the ‘overhead water’, according to the actual words of one pupil).

Moreover, many students believe that the roof of an underwater cave does not exert any

force on the water:

‘‘The rock does not push on the water, also without water the rock does not fall.’’‘‘The rock does not rest on the water, its weight does not exert any force; it is static.’’

These results as well as others show that knowledge of formal rules (e.g., Stevin’s Law of

Hydrostatics or the famous and nearly mythic rule of Archimedes’ principle) is not

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sufficient to produce a satisfactory understanding of the proposed situations and to connect

such rules with a local analysis of what happens in the various small parts of the fluid, and of

the interactions of these small parts between themselves and with the solid walls. It

sometimes happens that students give a correct answer on the basis of known formulas, but

then in their discussions they wonder ‘‘how’’ this result can be possible, how ‘‘to explain it’’;

and even change their answer, unsatisfied with the clear indications given by the formula

(see Sect. 4.3). There emerges the need for a ‘‘mechanism’’ which could explain how an

equilibrium situation has ‘come about’. A global view of the fluid is not sufficient to satisfy

this demand. What is required is an analysis in terms of the transmission of changes which

can reconcile global properties and rules with the local behavior of the fluid. This neces-

sitates moving to a smaller scale of description to construct an explanation based on the

behavior of the ‘‘elements’’ or particles into which we imagine the fluid to be decomposed.

This analysis has led to propose a mesoscopic model for describing the behavior of a fluid

in the presence of gravity (Besson and Viennot 2003, 2004). In the model, a liquid is

imagined to be composed of many small elements, analogous to little rubber balls, which fill

the container. The balls are in contact with each other, pushing against their neighbors, and

are only slightly compressible. In this analogy, the little balls do not correspond to the

molecules of the liquid, but rather to the ‘‘fluid elements’’. In fact, the mesoscopic approach

is the standard method in fluid mechanics and more generally in continuum mechanics. In it,

the kinetic aspects remaining hidden inside the mesoscopic elements and their effects are

represented by average macroscopic quantities (pressure, temperature, etc.). Usually the

decomposition into small parts is carried out in an abstract and mathematical way. By

contrast, the idea of our model is to make this mesoscopic breakdown concrete and acces-

sible to the students’ intuition by suggesting objects that can be taken to behave analogously

in the narrowly circumscribed field of fluid statics in the presence of gravity. The small

‘elementary quantities’ typical of the language of calculus, the vanishing quantities evoked

by Newton,11 become objects that are capable of interacting, pushing and being compressed.

Unlike molecules, these mesoscopic elements keep the essential properties of macroscopic

objects, and can therefore be treated as small pieces, small objects possessing temperature,

density, elasticity, and able to act on one another through contact interactions. One could say

that they do not resemble so much the atoms of Democritus as the ‘‘homeomeries’’ of

Anaxagoras, who held that bodies are composed of similar small parts.12

Fig. 5 ‘The three containers’question. The bottoms of thecontainers are equal

11 I. Newton, Philosophiae Naturalis Principia Mathematica, book I, section I, especially the final Scho-lium. The adjective ’’vanishing‘‘ and the verb ‘‘to vanish’’ (evanescens—evanescentes and evanescere, inLatin) are also attributed by Newton to parallelograms, angles, segments, arches, triangles (cf. Lemmas III,VI, VII, VIII, XI etc.).12 Anaxagoras, Greek philosopher (500–428 B.C.): ‘‘In his treatise On Nature, he explains the origin ofbodies without positing elements like water, fire, etc., but by means of homeomeries (a term introduced byAristotle): material particles that join together in order to form the bodies but which, in contrast to atoms,have the same qualities as the bodies that they constitute’’ (Durozoi G. & Roussel A. Dictionnaire dePhilosophie, Paris, Nathan, 1997, pp. 19–20).

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We focus attention on local interactions and the transmission of changes so as to help

students see the consistency of formal rules and laws with the local behavior of the fluid.

To this end, it is important to avoid asserting that liquids are incompressible. This is in part

because it is not true, the compressibility of liquids constituting an important field of

scientific research (see Aitken and Tobazeor 1998; Bridgman 1958). Moreover, incom-

pressibility would exclude any local modification in a fluid, and therefore any difference in

its action on the wall of the container or on surrounding fluids. How could an element of a

fluid transmit the effect of a disturbance to adjacent elements if the element itself is not

changed in any way? As a matter of fact, the variations in the volume of fluids are often

negligible in calculations, but they are essential to understanding.13 Calculating well does

not always mean understanding well and in physics there can be a very important dif-

ference between a small quantity and a null quantity, that is, between zero and very small.

Obviously, the proposed model has certain limitations. For example, it does not take into

account phenomena due to molecular attractive forces (surface tension, adhesion) and

molecular kinetics (diffusion, thermal dilatation). Furthermore, it could create confusion

between liquids and granular materials. Nonetheless, the model can constitute a productive

didactic instrument within a limited field of application. It is important here and in general for

students to become aware not only of the possibilities but also of the limitations of the models

they use.

A short teaching sequence based on this model was tested with first year university

students in Belgium, after a traditional course of fluid statics involving lectures and lab-

oratory activities. The short sequence was inserted into this context with only minor

changes in the usual curriculum. The aim was to promote a progressive construction of

reasoning based on discussion and analysis of diagrams and figures. The sequence began

with a mechanical consideration of the rubber balls. Three situations were studied: first a

series of balls aligned horizontally and pushed against a vertical barrier; second, the balls

set in a vertical position and pushed downwards; and finally, the balls set vertically and

given an upward push. The forces of interaction between the balls and the wall were

analyzed, as well as how these forces vary with harder pushing, taking small deformations

into account. Next, the analogy between liquids and the rubber balls in a container with a

piston was proposed (Fig. 6). The students were asked to answer questions of the type ‘‘In

which points is the pressure equal to the pressure at a point close to the piston, or at a point

in the low or high part of the container?’’ In the final session, the students were divided into

groups of two or three and asked to compare the forces exerted by the water on the bottoms

(of equal areas) of three differently-shaped containers (|__|, /__\ and \__/) filled to the same

level (Fig. 5).

The written answers to the final questions and the recorded discussions showed that

many students had modified their view of the physical situations and enriched their rea-

soning with more elaborated and productive arguments. For example, in the situation of the

two fish (Fig. 3):

« The pressure is equal … if you consider the rubber ball model, the ball at the farleft is submitted to forces, which are reproduced from ball to ball till they reach

13 Arons (1990, p. 64) thinks that ‘‘Students need explicit help and guidance in learning to visualize effectsthat elude direct sense perception. The deformation of apparently rigid objects in the context now underconsideration is usually the first opportunity in a physics course, and its importance should not be under-estimated. Later, such visualization is essential to understanding what happens…’’ in many physicalsituations.

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those in the cave ». «The pressure is equal, because the difference in height iscompensated for by the force which the wall exerts ».

In the situation of the three containers (Fig. 5):

« Because the balls have weight they tend to be squashed and therefore to exert alateral … horizontal force on the balls next to them, and that force is transmitted tothe walls and as the wall is sloping it induces a force towards the bottom. »

These quotations show that a significant change was produced in the reasoning of many

students by use of the model, which motivated a systematically more articulate analysis

directed towards finding coherence among the different aspects of the proposed situation. It

can be said that the model favors coordination among the three types of physical expla-

nations described in Sect. 3: the formal one, i.e., of the law of hydrostatics, with the causal

one, and partially also with that one ‘‘bathygeneous’’, introduced by the transmission of the

interactions between the mesoscopic elements.

7 Friction Between Solids

In secondary education and in introductory physics courses friction is generally presented

as a marginal topic, briefly dealt with in the chapter on mechanics, in an abstract and

schematic way: apart from a brief mention of the effect of the ‘roughness’ of surfaces, the

Fig. 6 The analogy between a liquid in a container and the rubber balls

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solid bodies between which friction takes place are nearly always considered as rigid

bodies. The excessive schematization hinders any attempt at creating an image of the

underlying microscopic phenomena that may be the basis for a causal explanation of

friction, an image and an explanation that students need in order to be able to understand

physical situations, as previous research has proved. This schematization may be sufficient

to calculate, on the basis of simplified laws, the physical quantities necessary to solve a

problem, but it cannot make one understand the physical situation.

Research on students’ conceptions has singled out a few specific difficulties with

friction between solids (Caldas and Saltiel 1995; Caldas 1999). Students rarely acknowl-

edge that friction can play a motive role: friction is almost exclusively considered as

resistive. It is often represented by one force only, opposed to the ‘‘actual’’ motion and not

to the relative motion between the two solids in contact. It is also commonly thought that a

solid can be dragged by another solid by ‘‘adhesion’’, without necessarily requiring the

presence of a force to act on it explicitly (dragging effect). There is also a tendency to

identify normal force with weight, a trend favored by the examples in handbooks, which

focus too much on situations of horizontal motion or motion on an inclined plane in which

normal force is equal to weight or to a component of weight.

A teaching learning sequence on friction phenomena has been elaborated and tested at

the University of Pavia, Italy (Besson et al. 2007, 2009) with the aim of overcoming

some common difficulties and erroneous or overly simplistic conceptions concerning the

topic. This sequence is also designed to help students acquire the elements of an

explanatory model which will allow them to construct an image of the mechanisms

producing friction.

The objective is for students to acquire some elements of explanatory models based

on properties and processes on a mesoscopic scale, not directly visible to the naked eye.

This will permit them to construct a simplified but functional image of the mechanisms

producing friction, and to elaborate reasoning, explanations and qualitative predictions.

The proposed models attempt to satisfy the dual requirement of didactic effectiveness, on

the one hand, and adequacy to current physical knowledge of the material structures and

processes involved, on the other. In fact, besides recognizing the importance of models

in human reasoning and for an effective learning, ‘‘what educators and scientists are

interested in is what mental models are true, which ones more accurately reflect the

world and its processes, which models are conducive to genuine knowledge’’ (Matthews

2007, p. 649).

To show the variety of phenomena and situations involving friction and to motivate the

student interest, the sequence begins with simple qualitative experiments illustrating the

different typologies of friction, in different practical situations, in which friction is con-

sidered either an obstacle and a disturbance or a useful and desired phenomenon. Then

more systematic and quantitative experiments are proposed, concerning first vertical then

horizontal friction forces, leading students to obtain the classical quantitative relations,

discussed as phenomenological laws deriving from a multitude of underlying processes on

surfaces in contact. The problem of the surfaces topography is discussed with the dis-

tinction between apparent and real area of contact, which is propaedeutic to any approach

to the mechanisms of friction. Figures and drawings are shown which illustrate the

irregular nature, on a micrometric scale, of the surface of apparently smooth objects, with

protuberances and hollows (the asperities of the surface) (Fig. 7). The analysis of the

topography and behavior of micrometric scale surfaces is interesting, also from a point of

view of the method and nature of physics investigation. It shows, in fact, two typical

characteristics of science: going beyond what is visible to the naked eye, revealing new

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details, new invisible worlds and new entities; looking for the explanation of phenomena in

mechanisms involving entities on a smaller scale, such as microorganisms, cells,

molecules.

The typical behavior of the surface asperities under load is discussed (increase of the

area of each contact and of the number of contacts, small approach of the two bodies in

contact), together with the dependency of real contact area on the load and the role of the

elastic and plastic deformations.

As for the mechanisms producing friction, it is to be explained that there is a variety

of phenomena, the relevance of which varies according to the situations and materials

considered. Some mechanisms are presented in a descriptive, simplified and intuitive

form: adhesion between the asperities of surfaces; the deformation, tracking or scratching

of surfaces; the impact and interlocking among asperities; the wear due to the relative

motion of the two contact surfaces; the effect due to particles trapped between the

surfaces (third body). Some models presented by scientists in different periods to explain

sliding friction phenomena are illustrated. Three examples are showed in Fig. 8.

The testing of the sequence among teachers at the initial training stage has provided

encouraging results, both from the point of view of overcoming some of the typical

difficulties with the topic and from that of activating new and richer lines of reasoning.

Many reasoning produced by the students, although incomplete, raise the topic to a much

more refined level than the simple repetition of fixed and abstract rules based on ide-

alized objects. Often, the specific characteristics of proposed models have been the basis

for quite articulate explanations. For example, in the answers to questions concerning the

situation of Fig. 9, many students drew sketches of surface roughness in which the

asperities were represented as deformed in different ways according to the different

accelerations of the system. Thus in this case the model were not merely a passive

figurative representation but assumed an operational explanatory role, even if only in a

very simplified and schematic way.

Fig. 7 Topography of surfaces. Apparent area A and real area Ar of contact

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8 Conclusion: Calculating and Understanding

Causality has long given rise to controversy and criticism, and is not always highly

regarded among philosophers and scientists. In any case, there is an active debate on these

issues, as different ideas are set out authoritatively.

On the other side, didactic research has shown the extensive presence and role of causal

explanations in common sense reasoning. Although from the philosophical and ontological

Fig. 8 Models of friction. From above: Coulomb’s model (1785), schematic representation of the Bowdenand Tabor’s adhesive junction model (1950) and a more recent spring-like model (Persson 1998, p. 291)

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point of view the causal connection and the idea of cause may be evaluated very differ-

ently, even to the point of being considered myths without sense, from the psycho-cog-

nitive and didactic point of view, the importance of causal reasoning in cognitive and

learning processes makes carrying out a deeper study impossible to avoid.

Didactic research has shown that the reasoning and physical explanations preferred by

students and common thought differ in many cases from those accepted in science and in

physics teaching.

Common thought has a preference for causal reasoning based on a logical and chro-

nological linear sequence of one cause–one effect chains. This is similar to what takes

place in a story or novel with its unfolding of events, or in a ‘‘mechanism’’ in which one

‘‘piece’’ acts on another and so on. By contrast, physics teachers more often focus on

equilibrium or steady state situations, using formal explanations and functional relation-

ships between physical quantities at a given instant. Pupils require a causal explanation, a

mechanism, which can account for the physical phenomena.

Common thought is mostly interested in transient phases during which ‘‘something

happens’’, as well as the dynamics of the facts and effects that have led to a given situation,

that is, to how things are the way they are. Science as well concerns itself with transitory

phenomena, studying processes of change and evolution with time. However, the need to

introduce adequate simplifications results in instruction that neglects these aspects, limiting

itself instead to the study of how a system is maintained and behaves in ‘‘the regular’’

phase. These differences can create difficulty in learning physics, especially if the sim-

plifications employed are not made explicit or discussed, but rather remain, as often

happens, among the implicit elements of the content taught.

The didactic experiments described in Sects. 6 and 7, as well as others present in the

literature, propose to use certain tendencies and preferences of common thought so as to

favor a connection between physical contents and students’ conceptions. In fact, for

learning to be meaningful, the new contents proposed by instruction have to be connected

to and interact with the students’ world. This can take place even via contrast and cognitive

conflict, but with methods and reasoning that have meaning and value for students.

Otherwise, there is the risk that what is taught will slide away without leaving stable

effects, and will give a feeling of abstractness and extraneousness.

In particular, the experiences described briefly above propose the didactic use of

explanatory models based on images and simple mechanisms, and appeal to intuition and

analogies. Models and physical analogies of this type can change the students’ view of the

physical situation and encourage new forms of reasoning and more articulated analysis

aimed at establishing coherence between global and local aspects. It has been observed

that, independently of the particularities of the model, the representation of mechanisms

and structural details at a smaller scale can promote higher quality student reasoning.

The appeal to intuition and analogies is often criticized and considered dangerous,

because it can favor wrong conceptions and unsuitable identifications between different

Fig. 9 A question proposed to students. A cart with a dish placed on it is put in motion with a smallacceleration, then moved at uniform motion and finally slowed down. Students are asked to indicate theforces acting on the dish during each phase of motion and explain their answers

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phenomena. However, in their speeches and reasoning on specialized subjects, even

physicists and engineers use a particular expert intuition that allows them to obtain pre-

dictions and interpretations before proceeding to calculations and rigorous deductions.

Such intuition has been formed progressively with the habit of dealing with certain subject

matters. When knowledge and procedures have begun to sediment into a network of mental

schemes at the unconscious level, it constitutes the physical intuition that plays such a large

role in the difference between novices and experts in the resolution of physics problems.

An objective of instruction should be to develop new schemes of physical intuition in the

student, approaching a semi-expert intuition on specific subjects and situations. Without

these physical references, the manipulation of laws and formulas runs the risk of becoming

a syntactic game.

Maxwell (1856) thought the use of ‘‘physical hypotheses’’ was essential, but at the same

time cautioned against the risks of extrapolations:

‘‘The first process in the effective study of the science, must be one of simplification

and reduction of the previous investigation to a form in which the mind can grasp

them. This simplification may take the form of a purely mathematical formula or of a

physical hypothesis. In the first case we entirely lose sight of the phenomena to be

explained; and though we may trace out the consequences of given laws, we can

never obtain more extended views of the connections of the subject. If, on the other

hand, we adopt a physical hypothesis, we see the phenomena only trough a medium,

and are liable to that blindness to facts in assumptions which a partial explanation

encourages… In order to obtain physical ideas without adopting a physical theory we

must make ourselves familiar with the existence of physical analogies (pp. 155–

156).’’

By now it is clear that some spontaneous tendencies of reasoning can create difficulties

for students. However, it also seems to me to be opportune to take critical stock of the

excessively artificial character of some reasoning proposed in teaching. Purified of its

defects and naivety, common reasoning, based on processes of change and causality, seems

in some cases closer to physical reality than the still image described in certain situations

by the physics teacher.

For example Rozier and Viennot (1991), observe that when a teacher wants to explain

the expansion of a heated gas at constant pressure, he or she usually thinks of a stationary

situation described by means of mathematical relationships such as pV = nRT, where the

quantities V and T vary simultaneously. By contrast, many students imagine a causal chain

with effects that follow successively in the time: heat is given, therefore T increases, then pincreases and then (after a short time) also volume increases. In this way they also con-

tradict the given condition that pressure remains constant.

Actually, it is difficult to understand how a piston can be put in motion, and increase the

volume, without there also being an increase of pressure in the zone of the gas close to the

wall where the heat is given. This pressure increase would then be transferred (in a finite,

albeit short, time) to the other parts of the gas and to the zone close to the piston, thus

provoking its movement. To be sure, we often suppose that it is matter of a quasi-statictransformation at constant pressure, but such transformations do not exist in reality, and in

any case it is difficult to imagine a static transformation, an expression which is itself

oxymoronic, even if it is corrected with a wiser quasi. To what extent can reasoning

neglect this quasi, and what is hidden behind it? To be sure, this works in calculations, but

for a student who is approaching physics, to be able to calculate does not always mean to

have understood. Finally, these two typologies of explanations, the one focused on the

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description of the equilibrium situations and the other on the dynamic processes leading

from one equilibrium situation to another, are not to be considered conflicting but com-plementary. They both have their usefulness and meaning, while at the same time being

both incomplete (look at note 5). Freed from imprecision, errors, and naiveties, they should

both constitute objectives of instruction. This is necessary to obtain a coherent and suf-

ficiently complete understanding, within the limits of the desired level of investigation.

Moreover, this expulsion of time and becoming can appear a perverse negation of

reality and give the feeling that much of science does not concern the real world and life,

but only a separated and stranger universe. If what we want is instruction connected to

student thought and the student’s world, perhaps it is necessary to try to explain change

without denying it, looking for a way to reconcile global and formal reasoning, based on

stationary situations, with local, causal explanations based on the transitory and change.

Obviously, it is not a matter of renouncing the idealizations and schematizations by which

physics lives and prospers, but of introducing them into the classroom with caution,

pointing out the approximations and simplifications, and showing and discussing how they

are connected with real phenomena and processes.

Duhem (1906, chap. IV) contrasts two types of mentalities: that of strong and limited

minds, who prefer abstract and deductive reasoning, and descriptions in terms of mathe-

matical equations; and that of the weak and wide (and imaginative) minds, who tend to

build physical hypotheses on material structures and on underlying mechanisms and

processes, in order to form a mental image of the phenomena.14 We should take into

consideration that many students belong to the second category, even though actually it is

matter of a preference more than of an absolute characteristic of individuals. On the other

hand, in a sense this difference resemble to that one between the intuitive and heuristic

logic of discovery and the objective and rigorous logic of demonstration, two modalities of

thought that are both present in the activity of physicists and mathematicians.

Bruner (1963) stresses the importance for students of intuitive, rather than formal

understanding,15 even if he emphasizes the pedagogical difficulties tied to the solicitation

of the intuitive thought during instruction, because

‘‘it requires a sensitive teacher to distinguish an intuitive mistake—an interestingly

wrong leap—from a stupid or ignorant mistake, and it requires a teacher who can

14 Duhem develops many examples to demonstrate that the first typology supposedly prevails among theFrench (and Germans), and the second among the English (apart from Napoleon, who is classified among thesecond category). In particular, he strongly criticises the exaggerated details of mechanical models elabo-rated in that period to explain electric phenomena: ‘‘the theory of electrostatics constitutes a set of abstractnotions and general statements, expressed in the clear and precise language of geometry and algebra,connected by the rules of a strict logic … this set totally satisfies the thought of a French physicist… It is notthe same for an Englishman; these abstract notions do not satisfy his need to imagine concrete, material,visible things… It is in order to satisfy this need that he creates a model… The use of similar models… is aconstant in English physics; some make only a moderate use of these models, others appeal to them at eachstep. Here is a book that aims to expound modern theories of electricity; we find only ropes that move onpulleys… tubes that pump water… cog-wheels that mesh into one on another; we thought we were enteringinto the calm and ordered realm of the deductive reason, and we find ourselves in a workshop.’’ The book towhich Duhem refers, published in French translation in 1891, was obviously of an Englishman, O. Lodge.15 The distinction must be considered more operational and didactic (intuitive?) than psychological andtheoretical, because a rigorous distinction between intuitive and analytical knowledge presents many dif-ficulties and ‘‘it is not even clear what we mean by intuitive knowledge’’ (Bruner 1963). In the same book,Bruner gives a short definition of intuition: ’the intellectual technique of arriving to plausible but tentativeformulations without going through the analytical steps by which such formulations would be found to bevalid or invalid conclusions’.

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give approval and correction simultaneously to the intuitive student. To know a

subject so thoroughly that he can go easily beyond the textbook is a great deal to ask

of a high school teacher.’’ (1963, chap. 3)

In any case, the existence of these two cognitive channels, intuitive-imaginative and

analytical-formal, has to be assumed, and both should be considered essential to learning.

Physics education must therefore cover and cultivate both, with their mutual connections,

as well as the ability to pass from one to another. Similarly, descriptions of both stationary

and transitory situations should be used and coordinated with one another. Precisely this

variety in the modality of description, reasoning and explanation constitutes the strength of

scientific thought and can contribute to making scientific instruction more welcoming to

students with various cognitive styles, and therefore to contrast the widespread idea that the

physics studied at school is an abstract and distant thing, that is not truly concerned with

the real world and life.

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Aristotle. (350 B.C). Physics. An English translation can be found in R.P. Hardie and R.K. Gaye, TheInternet Classics Archive, http://classics.mit.edu//Aristotle/physics.html by D.C. Stevenson, WebAtomics, 1994–2000.

Arons, A. B. (1990). A guide to introductory physics teaching. New York, NY: Wiley.Bachelard, G. (1934). Le nouvel esprit scientifique. Paris: PUF.Bachelard, G. (1949). Le rationalisme applique. Paris: PUF.Benzi, M. (2003). Scoprire le cause. Milano: Franco Angeli.Besson, U. (2004a). Students’ conceptions of fluids. International Journal of Science Education, 26(14),

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