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The Essential: systems that obey simple laws
particles of a solid mutually inform one another very quickly when anything
occurs (if the reader will forgive this anthropomorphic image). It therefore
seems unnecessary to specify that “one can neglect internal propagation of
effects”. Nevertheless, the transfer of information sometimes takes an
amount of time that is not negligible relative to the time scale of
characteristic events: there can be waves in solids, too.
Box 1 illustrates some simple examples of quasistatic analyses taken
from the fields of mechanics and electricity.
The notes accompanying each example show that many quantities are
involved: situations of this sort are generally called multi-variable, or
multifunctional, problems. Here, complexity is reduced, but not eliminated.
Then there are the laws. Some describe, in a phenomenological fashion,
each of the parts (the “subsystems”) involved. Others express relationships
between subsystems – e.g., those expressing conservation, or a fundamental
law, such as the law of reciprocal actions.
The general practice is to make no mention of time. But the remarks in
chapter 4 apply even more strongly here. Many of the quantities involved are
Quasistatic or causal changes
modified in the course of the transformation; everything can “change” at the
same time, since everything is “informed” at the same time. Here, again,
events are simultaneous.
In this type of “quasilegal” description, the laws remain unchanged
throughout the process of “multiple-transformations.” They can therefore be
written in more explicit form, with a reference to time alongside each
physical quantity (see box 1).
NATURAL REASONING: MORE STORIES
Thinking in terms of simultaneity does not come naturally to us, as has
been shown. Moreover, the idea of an object is ever-present in natural
thought. It is logical that these elements of common reasoning should crop
up again here. And they do: an analysis of the evolution of systems is very
often structured like a story, in the sense that there is a temporal link
between events. Although in a quasistatic analysis one should think in terms
of “at the same time,” common reasoning prefers the storyteller’s “later on,”
and, sometimes, “further on.”
The idea of something happening “further on” presupposes that there is a
clear spatial structure in the system, which is not always the case. In the first
examples discussed below, the situations are ones where there is a “natural
path” in space to guide reasoning; the subsequent examples show that the
story can also take place “on the spot.”
SYSTEMS WITH A CLEAR SPATIAL
Springs connected end to end
Let us imagine a “system” made up of two springs, and
and spring constants and
These springs are connected end to
end, as shown in box 2. The variables characterising each spring are given
there, as well as their interactions, both with each other and with the
environment. Some describe the state of each spring (extensions
the sum of which gives the total elongation ) and others the forces
magnitude of the force acting on the lower end of spring
the same for
magnitude of the force exerted by the experimenter
on the lower end of the set of two springs). These quantities change
simultaneously during a transformation of the system, if it is stretched, for
example. But the relationships given in box 1 remain valid at every moment
of the transformation. This implies something that is not always obvious at
first: the transformation is quasistatic; all of the elements of the system are
informed about what is taking place at the same time. In short, “one neglects
Each state of the system can be described as if it were fixed, without
consideration of what brought it to that point or of what will follow: in that
sense time does not enter into this problem.
This situation was presented in a survey questionnaire (Fauconnet, 1981)
to evaluate the impact of the temporal aspect of events on the reasoning of
students. One version (box 2) proposes a comparison between two systems,
both made up of the same elements – two springs that are identical to
respectively but which are in different states of elongation: the first
system is free, the second is stretched. The second version (box 3) proposes
another system made up of elements that are identical to
time they have undergone a transformation: the system is being pulled on.
In both versions, the same data is given, i.e., the total elongation in the
“stretched” state, and the student is asked to find the displacement of the
junction point and the external force exerted in this state.
Quasistatic or causal changes
The quantities involved and the relationships between them are exactly
the same in both versions of the question (“states” or “transformation”). Do
the students’ responses clearly reflect this similarity?
Each participant was only given one version of the question. So only the
frequencies of characteristic responses in groups that dealt with one of the
two proposed situations can be compared.
The students’ diagrams (box3) show that the force
exerted by the
external agent is “opposed,” in a sort of “balance,” by the two other forces,
each acting upon one of the springs. Newtonian physics tells us that such a
balance cannot provide information about the acceleration of a given body
(“a system”), since the forces acting are not all external to the system being
considered. But the diagrams indicate that students envisage this situation
(the author speaks of how they “read” the situation) in the by-now familiar
terms of an anthropomorphic conflict: each element of the two-spring system
resists in its own way the aggressive action of the external agent – using “its
own force” (even though this means ignoring the law of reciprocal actions).
This aspect of the answers is generally observed for the “states” version (box
In other cases,
(the “external” force) is linked to the displacement of
the lower end of the spring as if the extension concerned only the lower
This typical response is often associated with a diagram of
forces that is compatible with the correct analysis: all the force magnitudes
are equal, which was not the case previously, when the external force alone
had to compensate for two resistances. Here the image is that of a
transmission (dare one say “propagation”?) of forces, rather than of a static
opposition. This characteristic is found mostly in answers for the
“transformation” version (see box 2).
Quasistatic or causal changes
Taking this analysis further, the author points out that, in the erroneous
the fact that the junction point of the two springs is
also displaced by a downward pull is not taken into account. Writing
probably indicates a very local analysis of
events, focusing only on the spot where the external action is applied. Only
later is the transmission of the action to the rest of the system envisaged.
This gives curious results, as overestimating the lower spring’s extension
entails overestimating the intensity of the forces as well, and the extension of
the upper spring turns out to be greater than the displacement of the lower
end (see the student’s answer included in box 3).
But what is most striking is the role of time in this type of solution. The
commentary in box 3 clearly shows that the external force is seen as having
first a local action, and then, “after a while, it is transmitted to the top
spring”. This is a clear denial of the premises of quasistatic analysis. It is
hardly surprising that this sort of denial occurs most often in a
The recurrent nature of these elements in students’ answers makes it clear
that the spatio-temporal content of the situations and questions is important
in natural reasoning. This is confirmed through other interesting examples in
the same study.
A physical situation involving communicating vessels is proposed
(Fauconnet, 1981) in questions comparable to those we have just described;
again, there are two versions of the questions. For this situation, the results
are surprisingly similar as regards both the typical elements found in the
answers and the frequency with which they arise in the “states” and
As we shall now see, investigations in another, much broader field have
led us to conclusions which complement remarkably this pioneering study.
Sequential reasoning in electric circuits
Electric circuits are structures that have a clear spatial content. Box 4
shows a series of situations that all have a common structure. In each
situation, two identical components (elements of a circuit with terminals)
frame another, different circuit element, and a generator. The equations that
The proposed situation is as follows: a liquid is poured into two vessels of different crosssections; the bottoms of the vessels communicate. A quantity of non-miscible liquid of
is added in one vessel. The volume
of the “external” transformation agent
is often associated with the cylindrical volume between the initial and final positions of the
separating surface. In other words, it is seen as indicating a local transformation; the
displacement in the level of the liquid on the other side of the system is not taken into
apply to series circuits like these in quasistationary analysis are contained in
box 1. They clearly show that the order of the elements is irrelevant. If one
dipole is exchanged for another, the values of the affected quantities do not
change; neither are the other components affected in any way.
Quasistatic or causal changes
The comments of the pupils and students very often deviate from this
analysis. In the first situation, for instance, they often say that the second
bulb shines less brightly than the first, which is, nevertheless, identical. In
the next situation, they state that the second capacitor is charged less rapidly
than the first after the circuit has been closed (from the first billionth of a
second, and even earlier, quasistationary analysis is amply justified),
although the two capacitors are charged at the same rate.2 In the question
concerning the black box, they are asked if bulb 1 shines in the same way as
bulb 2 no matter what is in the box (which is indeed the case), or if it
depends on the contents of the box. Several conditions are stated: “No
battery in the box,” “No resistor...” When there is a resistor at either end of
a coil, the difference in phase of the potential difference across the resistors
relative to the potential difference across the generator is said to be different:
“There is a change of phase for the second resistor, the potential difference
before the coil is not affected.” And in the situations on alternating current,
the circuit in which two resistances frame a diode often gives rise to the idea,
shown in the diagrams, that the potential difference is rectified after the
diode and not before it, whereas the diode affects the current in the circuit as
a whole; and therefore also affects the potential difference across all the
resistances in series.
All these answers are described as “sequential” in the box, because they
can be interpreted as stemming from the following reasoning:3
This is true even when the capacitors are not identical.
See also Shipstone (1985).
There is an entity referred to somewhat interchangeably as “electricity”
or “electrons,” to which students associate various physical quantities, such
as current, potential difference (or voltage), or even phase. It leaves the
generator through one terminal and heads out into the circuit. Its progress is
more or less affected as it passes each component; some students explain, for
example, that “the current gets used up in the resistor,” or that “the current is
rectified by the diode”, but there is no action “backwards” from the end
point to the source. Generally, the entity returns to another end of the
generator, though the adventures described do not always end that way.
There are variations on this theme. Sometimes the local character of this
type of reasoning is so strong that each component and the quantities
associated with it are considered independently from the other components.
Another particularly tenacious idea is that the current leaving the generator is
the same no matter what the circuit, like a spring that always yields the same
amount of water regardless of whatever dams there may be further down the
That such an idea should outlive higher education casts doubt on the
simplistic interpretations of Bachelard’s “epistemological break”5 – an idea
echoed (perhaps rather dubiously) by the recent interest in “conceptual
change” in the Anglophone literature (see Posner et al., 1982). The transition
from common knowledge to scientific knowledge is held to involve crises,
cognitive conflicts that bring about decisive or even definitive restructuring.
The survey results presented above scarcely bear out the theory that such
cognitive overhauls occur.
Of course, advanced students no longer believe that two identical bulbs in
series do not shine equally brightly; they have mastered that highly familiar
situation. But, faced with an unfamiliar question, such as the one concerning
the “black box” (box 4, situation 3), a great many of them rejoin the ranks of
sequential reasoners. As J.L. Closset (1983) put it, “Sequential reasoning
does not disappear, it is suppressed.” For these advanced students, then,
there has been no dramatic restructuring, but, rather, local learning. Even
teachers sometimes show that sequential reasoning is their most readily
available cognitive tool. Such reasoning certainly does work quite often. For
proof that a systemic vision of a circuit is not a requisite for everyone, one
has only to listen to electronic engineers talking about frequencies that
“start”, “pass” or “do not pass,” without ever mentioning that anything
All these difficulties are now explicitly addressed in France. On this subject, see the French
National Curriculum for grade 8 (Bulletin Officiel, 1992a, applied in 1993) – see appendix
to chapter 2 and Couchouron et al. (1996). For a compilation of consciousness-raising
questions, see Courdille (1991).
A notion which resembles that of the “change in paradigm” that Kuhn introduced in
Quasistatic or causal changes
“comes back” through the electrical Earth or that the circuit is closed. The
comment of Bernard Schiele seems to fit the data: “The difference between
scientific knowledge and common knowledge is a matter of degree, not of
kind” (Schiele, 1984, p 91; see also Viennot 1989b).
These two examples of systems with a clear spatial structure indicate that
reasoning is influenced by the way one looks at things. With electric circuits,
in particular, time is introduced naturally in the typical analysis of the
proposed situations. In such an analysis, students follow the progress of the
main character, “electricity,” around the circuit suggested by the diagram. It
must be pointed out that in this case, natural reasoning is “based”, so to
speak, on an artificial support, i.e., the way circuits are represented in
diagrams at school. Children who have never seen such diagrams do not
make the same mistakes. When they have never seen a closed circuit, they
imagine other routes, or reason along totally different lines, asserting, for
instance, that two identical bulbs will shine equally brightly, because “they
take what they need”.6
From Rozier (1988)
Other situations also show how spatial structure can determine reasoning.
The conduction of heat along bars, for example, often gives rise to sequential
reasoning. In the survey question presented in box 5 (Rozier, 1988),
regarding the transfer of heat along a conductor made up of two sections
with different conductivities, students are asked about the consequences of
replacing the second (or “downstream”) part.
As might have been expected, the most common answer is that only the
quantities concerning the “downstream” part change. Yet the system of
equations applying to this problem (a quasistationary situation) indicates
clearly enough that the quantities in the two parts cannot be calculated
separately: the “downstream” part affects the “upstream” part. But, once
again, “conduction” brings to mind “forward movement.” This time, the
leading actor in the story is “heat,” seen as an almost local entity whose
evolution in space takes time and differentiates the downstream from the
See Closset, 1983, pp 202-203.