IMAGERY AND FORMAL THINKING: APPROACHES TO INSIGHT AND UNDERSTANDING IN PHYSICS EDUCATION

Manfred Euler, Leibniz-Institute for Science Education (IPN), University of Kiel, Germany

This lecture is dedicated to the memory of the late GIREP president Karl Luchner. Physik ist überall (Physics is everywhere)[1]

K. Luchner

1. Deficits in science education: the quest for understanding and insight

This is the first GIREP seminar held between the biannual meetings. In focusing on the develop-ment of formal thinking the present conference addresses a problem, which is of vital importance for the future of physics education. Since long, we can observe a declining interest in physics. This trend is ubiquitous, and many reasons can be found to account for this phenomenon. In close corre-spondence with the motivational deficits the cognitive outcomes of physics teaching are far from what is expected. An adequate unfolding of formal thinking is undoubtedly one of the crucial prob-lems of learning and understanding physics. The development of formal thinking can be seen as the acquisition of networks which assign meaning to symbolic elements and which allow students to navigate in the world of physics [2]. Formal thinking, creating symbolic descriptions that model structures and processes in the real world, is at the heart of the methodology of physics. As formal thinking obviously is not the normal mode of mental activity, it is also at the heart of the difficulties of learning physics. Thus, finding better ways of promoting formal thinking and of moving from the concrete to the abstract is essen-tial for restructuring the ways we teach and learn physics. It is useful to consider the subject of the present seminar in the context of international efforts to monitor scientific literacy. The status and the effectiveness of science education in many countries are not considered satisfactory and sufficient to master the challenges of the future. We are amidst a transformation process from the post industrial to the knowledge society. There is the general fear that scientific literacy and the public awareness of science do not comply satisfactorily with the needs of a global knowledge societ [3]. The OECD Programme for International Student Assessment (PISA) has addressed some questions which are considered vital in view of the rapid global changes [4]. Are students well prepared to meet the challenges of the future? Are they able to analyse, reason and communicate their ideas effectively? Do they have the capacity to continue learning throughout life? The term “literacy” is used in a metaphoric way to describe a broad conception of knowledge and skills for life, which are broken down to various processes, including, among many other aspects, the ability to apply knowledge from science in more or less authentic real world situations. Due to the comprehensive approach of assessing student performance and of collecting ample context information, PISA pro-vides the empirical framework for a better understanding of the causes and possible consequences of observed skill shortages.

Among other findings, PISA confirms the results of earlier studies like TIMSS (Third International Mathematics and Science Study [5,6]). With respect to the consequences, I can speak only for my country: The TIMSS results have shattered the long-held beliefs about the high standards of physics education. Science education is more or less efficient only with respect to imparting the knowledge of facts. However, broad deficits exist on the level of more demanding science processes, e.g. applying knowledge to new situations. Broadly speaking, physics teaching focuses on conveying factual knowl-edge ("know what”). Approaching the “know how” and the “know why” poses big problems. Physics education falls short of attaining more challenging goals like flexible application of knowledge in new contexts, and of fostering insight and understanding. Formal thinking, as such, is not addressed in these studies but their results clearly point out deficits in applying what has been learned. This deficit is closely related to formal thinking, as learners are unable to abstract sufficiently and transfer knowledge from one context of experience to a related domain. However, international comparisons do show that there are specific differences between countries in attaining these more demanding goals.

Fig. 1: Insights in insight: How to circumvent self referentialcircles?

Under what conditions can one expect physics education to promote insight and understanding? This article approaches the elusive phenomenon of insight from various perspectives and discusses its specific meaning for cognitive processes by using examples from physics and mathematics. Intuitive pictures, mental transformations of imagery and reflections about these processes are con-sidered important for learning and for doing physics as well. In this context, images represent the (more or less) concrete symbolic substrate from which formal thinking emerges. All learning theo-ries agree on learning being an active process. Active transformation processes of mental imagery are necessary to build up more and more abstract representations. How can we link the mental images and their transformation processes with concrete experiences in a meaningful way? On a more general level the question of the relation between instruction and autonomous construc-tion is touched upon. Insight and understanding result from genuine individual mental construction processes. From an external perspective these processes are difficult to investigate as they are accessible only indirectly and incompletely. Is it possible to stimulate processes of insight and to support meaningful learning by suitable instructional means? We are far from being able to offer a patent solution but the present article intends to focus on these issues and to initiate broader discussions. 2. The elusive nature of insight: is it something special?

By insight one generally means the ability of immediately knowing and of seeing clearly by mental activities which are not fully transparent. This mode of thought is felt in sharp contrast to rational and analytical reasoning. In the context of physics and mathematics, insight means the ability to "see" the solution even without running logically through all the steps of problem solving. In

insights, the essence of phenomena or processes is intuitively grasped or linked with other phenomena without fully decoding the path of argumentation. There is a close connection to crea-tive thinking, when new solutions to problems are generated. Insight describes the transition process of our minds from the state of ignorance to the state of clarity about the solution to the problem [7]. There are many metaphors for insight; the “sudden beginning to see the light” is one, the “flash of genius” is another. In a letter to a friend the famous mathematician Gauss describes the process in the following way: Just like lightning strikes, the puzzle is solved. I myself would not have been able to prove the guiding line between what I already knew ... and how it was possible to prove it finally [8]. A common characteristic in almost all of the reports about sudden insight in mathematics and the sciences is – in addition to the affective moment – its visual character. Insight depends on pictures that entangle concrete experience and visual imagination with abstract, symbolic elements. An overview of the current state of psychological research may be found in the volume entitled The Nature of Insight [9], which also gives some useful references for insights relevant to physics. As everyone will know from more or less intensive personal introspective experience, processes of insight are associated with the following characteristics [10]: • suddenness: A discontinuity or an abrupt transition from ambiguity to understanding the prob-

lem and its solution is perceived instead of a gradual steady transition process. • spontaneity: Insight cannot be forced, but rather happens intrinsically and spontaneously with-

out an apparent agent. • unexpectedness: Insights can emerge even when one is not consciously involved in the problem. • correctness: Often, insights are accompanied by an immediate feeling of correctness, although a

full logical justification is missing. • satisfaction: A deep feeling of satisfaction may accompany or even precede the moment of

insight. Dissolving the tension of hitherto unsolved cognitive conflicts may result in exclaiming a loud Eureka!, as we know from folklore reports about Archimedes and his bathtub insight on buoyancy. A more quiet everyday form of satisfaction is the Aha! experience. In any case we must not underestimate the strong affective element of the moment of insight, which drives our mental activity and finally rewards tedious thoughts and hard labour. Generally, one finds two basic assumptions about the character of insight which express two con-tradictory points of view [11]: • Insight is not a special process. It is rather considered as a smooth extension of common

perception, learning and thinking processes. This point of view emphasises the continuity, the cumulativity and the associativity of the process: The way, how a new problem is solved only depends on prior knowledge. One produces new solutions by combining what one already knows. Thus, insight is nothing more than a chain of associations of elements available to the somehow prepared mind. It is doubtful whether it is possible to solve new problems according to a kind of "flash of genius" independent of previous experience.

• Insight is a special process. It can be understood as an acceleration or - in the extreme - even as a circumvention of the normal, conscious chain of arguments. Insight is considered an uncon-scious process, restructuring knowledge in such a way that new connections emerge. Insights are seen in close analogy to visual perception processes. Similar to the emerging of visual perceptions from optical input signals by means of an active structuring process, knowledge is restructured in creative thought processes. This can take place in various ways, e.g. by complet-ing missing information, by reformulating the problem, by eliminating thought blockades, by separating an object from its functional context, or by transforming it to an analogous problem for which the solution is already known.

While the first point of view developed out of traditional psychological research, the second approach came from the field of Gestalt psychology. It was scorned for a long time for its presumed lack of methodological strictness and theoretical clarity. This point of view, it was argued, is based on anecdotic evidence and not on facts that can be proved by experiments. Emotionally, however, it is very attractive as it describes characteristics of insight that everyone can perceive introspectively. Can one's own intuition (and the intuition of many great spirits of the scene) be so wrong? From the perspective of modeling cognitive processes within the framework of neural networks, the differences in the two points of view become irrelevant. Interestingly enough, many features of the phenomenology of Gestalt perception can be modelled by neural networks [12]. Instead of an exclusive "either - or" it seems to be better to view the two perspectives as the extremes of an approach to insight which allow a continuum of possible intermediate forms depending on the com-plexity of the problem and the level of expertise the problem solver has. 3. Metaphors and models for insight: metamorphoses of internal images

To resolve the conflict between the "business as usual" standpoint and the view of insight being something special, we discuss optical pattern recognition processes. They can serve as a metaphor or even an analogy of higher cognitive processes. This analogy suggests that there is nothing special about insight processes. They are compatible with conventional sensory, learning and thinking processes. At the level of perception analogous processes are continually taking place almost unnoticed. Nevertheless, we as our own internal observers perceive them as having the char-acteristics described in the second point of view. This is one of the oddities of the internal and external view of complex systems, and our head is undoubtfully such a complex system. The internal observer's view and the descriptions of the "participant" can be incompatible with those of the "detached observer", a fact that is just as impor-tant for physics as for cognitive science [13]. The view from within the system compared to the view from the outside, the endoperspective and the exoperspective, are conflicting. This incompati-bility expresses more than the normal observer dependency of descriptions of reality, in which the results of measurements may depend on the individual frame of reference. In the latter case, it is possible to construct smooth transformations that relate consistently the outcomes of an experiment in one system to the descriptions of reality given in another system. The problem of internal observers is more deeply rooted and cannot be reduced to smooth transfor-mations from one perspective to the other. It is closely connected to the emergence phenomenon, where new entities are generated by a spontaneous re-ordering process, as soon as the (open) sys-tems are complex enough. In physics, these processes occur in dynamical systems far from equilib-rium (cf. par. 7). Two perception experiments are presented that elucidate by analogy how this puzzle of conflicting discreteness and continuity might be solved. On the one hand, one can maintain the belief gained from the view of the internal observer that insight is a discontinuous process. On the other hand, in spite of the emergent discontinuity, the underlying processes can be regarded as fully continuous, building upon what one already knows. In order to accept this unifying view, one has to change the perspective and compare the view from within the system with the description from the outside. In general, such a switch from the internal view to the reflective meta-perspectives is also a decisive step moving from learning facts to conceptual understanding.

Experiment 1: Gestalt-Perception - spontaneous and induced switching processes

Fig. 2: The Necker cube. Bistability in visual perception can be perceived as aspontaneous process, but it can also be stimulated externally or induced voluntarily byguiding the eye movements.

Have a look at Fig. 2 on the left (Necker Cube). You clearly perceive a cube in two possible spatial configurations. Although the drawing is one dimensional and unique, the percept is spatial and bistable. The actions of the observer’s conscious mind appear to animate the image. In case that you have no prior experience with this figure, the switching between the two arrangements of cubes is completely spontaneous. This subjective impression of spontaneity is fully correct as psycho-physical experiments confirm! When the switching sequence is registered (by pushing a button, whenever the percept snaps into another configuration), a time sequence of events results that will pass all the tests for randomness. The switching sequence corresponds to a Poisson-process which, for instance, also shows up in spontaneous emission of light quanta from excited states or in radioactive decay, both phenomena being physical archetypes for spontaneous processes [14]. The observed spontaneous switching processes from one percept to the other exhibits many characteristics of insight that were described earlier. Snap! The insight shows up spontaneously and the subject appears to have no possibility of controlling the process. From the external view, however, the processes are transparent and they can be controlled (theo-retically at least). In fact, an "expert" with enough background knowledge can voluntarily induce the transition from the one state to the other. To such an expert, the spontaneity vanishes. A simple trick suffices to turn a novice into an expert for switching Necker cubes. A change in the direction of gaze indicated by the two dots in Fig. 2 (right) helps to break the symmetry of the figure. Focussing on one of the two dots will induce the switch. One can even move the tip of a pencil periodically back and forth along the lines, and ask the subject, to follow the tip. In this case, the subject will perceive the switching in synchrony with the periodic motion. This little experiment turns the spontaneous switching into an induced transition process, that can even be guided externally. Transferred to the insight-experience, we can conclude that, whenever a sufficient level of knowl-edge is available, the insight-feeling is turned into nothing special. There is "business as usual", and the feeling of "not being able to do anything about it" disappears. The mystery of insight becomes demystified in so far as a fully regular and continual association with other mental or bodily activi-ties can be produced. In the present experiment these activities (eye movements) are rather trivial. The next demonstration focuses on the crucial role of prior knowledge.

Experiment 2: When do Pictures say More Than a 1000 Words?

Do you recognise the two rather coarse grained faces in Fig. 3? A hint: Squint your eyes and have a blurry look or look from a distance - that helps! Optical pattern recogni-tion - like insight - requires at least in certain phases oversimplifying and generalising. After a short "incuba-tion period" the insight should come abruptly. All the faces shown are those of physicists. Insight should come quickly in the case of the right-hand picture, since the person is well known and almost everyone has seen tperson once stuck his tongue out at obously acquired knowledge and the accproblem can be solved. Recognising the face on the left is moreexist in the classical sense. It is a superMann. As the prior knowledge is not avknowledge is apparently not possible acA reflection of the processes that genertions under which pictures work effecwords is incorrect in the naive, literal vtures (just think of the many kilobytes puter memory). The effective informatiand activating internal rearrangement phand, quite limited. In the example of shades of grey), i.e. about 600 bits - fwords - suffice to describe the person anBoth experiments tell us something aboon the function of mental imagery. Thelife are not, they happen [15]. Along thto images and insight: Images are not,spond to dynamic processes and are linkmentally. When do images occur? The nevertheless they can be triggered or stdemonstrates the "happening", that is tcan be used as metaphors for insights. Although deep insights and visual patteresponse and the complexity of the peracceptable, one cannot circumvent the ctered and stimulated to some extent - and learning. In science teaching, imagethings and situations statically. We hahappen. Fostering insightful learning prsuitable learning arrangements) and coapproaches). Deficiencies in meaningful mental imachievements of students in the "know

Fig. 3: Pictures of famous physicists. Who is who?

his specific picture. Additionally, when you know that this trusive photographers, then the task is an easy one. Previ-ess to available information determine how quickly a new

difficult; it is practically impossible! This person does not position of the faces of two physicists, Feynman and Gell-

ailable, no insight can arise. Insight without some form prior cording to this visual analogy. ate meaning in these cases give some hints about the condi-tively. The proverb a picture says more than a thousand ersion. There is a plethora of potential information in pic-or even megabytes that uncompressed images use in com-on that is actually involved in making the image “happen" rocesses that finally switch to the percept is, on the other the Einstein picture about 200 pixels with 3 bits each (23 ar less than the information contained in 1000 meaningful d even what is happening in the scene. ut dynamical processes in visual perceptions that shed light theoretical biologist von Bertalanffy remarked The forms of e same line on could reformulate the statement with respect images happen. Images are no static objects. They corre-ed with prior experience. Images are models that can be run first experiment shows, that they occur spontaneously - but imulated by suitable arrangements. The second experiment he linking process with prior experience. Both experiments

rn recognition differ considerably by the time scale of the cepts it is not unreasonable to assume close links. If this is onclusion that insights (like visual perceptions) can be fos-

a conclusion that has far reaching implications for teaching s are often considered from a passive perspective, showing ve to do more in teaching and learning to make images ocesses requires a delicate balance of instruction (providing nstruction (giving ample space to explore and reflect own

agery can be considered one of the causes for the poor how" tasks reported earlier. The following selection of ex-

amples is guided by the assumption that intuitive pictures, their metamorphoses, or, to be more spe-cific, their active transformation processes, and the conscious reflection of these transformations are crucial for learning and for doing physics. 4. No insight without internal images: imagery and scientific imagination

What role do insights and images play in learning? Certainly their role will not be very different from research processes. Thus a short glance at the latter will be helpful. Although research in sci-ence and mathematics is generally considered as being especially analytic, logical and rational, the creative processes in these fields are inspired by imagery and intuition. Much evidence for this can be found in the biographies and correspondence of well known researchers, although such reports have to be read with a certain amount of critical distance. An especially notable example is W. Pauli, who is generally considered one of the most rational and critical minds in physics [16]. In spite of his rational attitude he has always tried to account for what he called “underground physics” giving way for creative mental processes. The Pauli-Jung correspondence documents his effort to understand the role of unconscious processes in creative scientific thought and demonstrates the meaning the attached to symbols and symbolism, light as well as dark symbolism [17,18]. Pauli's reports show the difficulty of reconstructing the mostly unconscious creative processes rationally, due to the interweaving of dreams, concrete pictures, abstract ideas, symbols (scientific as well as religious) and highly emotional personal experiences. There is no insight without internal images! This is how the theoretical chemist Primas saw creative processes in the field of mathematics and the sciences [19]. As he emphasises, the pictorial intuition is, however, only one part of the creative process. What is intuitively seen must be corroborated and critically questioned by rational reconstruction, otherwise intuition is nothing more than shallow fantasies. An adequate interplay between intuition and rational reconstruction is crucial not only for doing physics but also for learning physics. The evidence that images promote insight and understanding in science (and mathematics as well) is overwhelming. So we have to ask, why only little effort is made in physics education to account for the role of imagery for a better understanding of physics. Apparently, visual approaches have been banned by a formalistic tradition. The relations between the concrete and the abstract are not sufficiently balanced. While the formalistic approach tries to prevent misconcepts based on false intuitions it often falls short of giving the abstract symbols and formal operations a concrete mean-ing which we need to operate successfully with these concepts. We must reconcile ourselves and our students with the fact, that we need concrete images and sym-bols of physical entities, even though physical reality is not fully imaginable in terms of everyday reality. We are forced to use abstractions, but there remains some important residual concreteness in the abstract symbols. There are physical as well as psychological grounds for these "elements of reality" in the abstract formalism. Although many textbooks use images on a superficial level, only few encourage visualisation and even fewer reflect the role of imagery and imagination. The Feynman lectures are a notable excep-tion. In a chapter on the solution of Maxwell’s equations in free space Feynman lines out the de-mand on scientific imagination in the context of classical field theory. He conveys a vivid picture of his own struggle for a visualisation of the invisible and the untouchable that mixes abstract symbols with concrete actions and perceptions. He confesses that I have... no picture of this electromagnetic field that is in any sense accurate. .... When I start describing the magnetic field moving through space, I speak of the E- and B-fields and wave my arms and you may imagine that I can see them. I’ll tell you what I see. I see some kind of vague shadowy, wiggling lines – here and there is E and B written on them somehow, and perhaps some of the lines have arrows on them – an arrow here or there which disappears, when I look too closely at it. .... I have a terrible confusion between the symbols I use to describe the objects and the objects themselves [20]. Retreating to the purely mathematical view is not a solution either. On the one hand, any attempt to make the electromagnetic field fully “touchable” is bound to fail. On the other hand, the electro-magnetic field is more than merely an abstract and arbitrary mental construction. Although our

minds are unable to conceive an adequate picture of the electromagnetic field, like, for instance a mechanical model that we can grasp, the instruments that we build according to the laws of physics can “grasp” the field in a predictable way. An analysis of Feynman’s struggle for appropriate visualisation points out some reasons, why it is so difficult to convey images of physical processes. On the psychological side there is the absolute privacy of mental imagery. Images are one’s own individual constructions that depend on prior ex-perience. Even though two persons see the same picture, the meaning they attach to it may be very different. Images cannot simply be “implanted”. Images are often highly resistant to teaching and changes of mental imagery is a long process, as overwhelming evidence from research on conceptual change and reasoning in physics shows [21]. On the physical side, meaningful images are everything but arbitrary inventions. They must be con-sistent with the part of reality that we try to model. This requires sufficient complexity of our images to map the essentials of the target domain on the structural and functional level. Such a requirement still leaves ample space for concrete, engineering type of models, for instance models used in graphical model building that are based upon state and rate variables. These models can be viewed as having some concrete underlying quasi mechanical or fluid-dynamical machinery. On a more abstract level, however, our images must be in accord with general principles of physics like symmetry and invariance. Ultimately, such general principles will "kill" all concrete images of some underlying machinery and will leave us alone with abstract mathematical structures. Somehow, the role of mental imagery in the modelling process of physics can be compared to a ladder. We climb up the ladder from concrete pictures to increasingly formal and abstract representations. Sometimes we even forget about the ladder, that we used to climb up. From time to time it is good to step down the ladder, reflect the situation, and link the formal and abstract ideas of physics with the concrete experiences, from which they emerged. Therefore, a glimpse on concrete imagery for problem solving at basic levels of mathematics and physics is helpful to reflect the role of imagery for formal thinking. 5. Linking the concrete with the abstract: examples for image driven problem solving

The potential of concrete images for problem solving and the richness of individual images and strategies can be demonstrated by the problem in Fig 4. The item is taken from the exposition of the PISA framework [22]. Two arrangements of T-shirts and drinks are shown and the problem is to find out the respective prices. I exposed students and teachers with the item at various occasions. Usually, they start writing down two equations and begin to solve them formally. They have to be instructed to solve the problem mentally without writing down equations. After this hint most people are fascinated by the variety of successful methods of solving the problem and of “seeing” the solution. One way of solving the problem is the following. Because of the symmetry of the above arrange-ment, one knows the price of one T-shirt and one drink. This amount can be taken off mentally from the lower arrangement. From the remaining parts on both sides one knows the price for two drinks and the problem is solved. This procedure combines visual elements (symmetry) with mental arithmetic and algebra (substi-tution, carrying out the same operation on both sides of the equation).

Fig.4: How much is a T-shirt, how much is a drink? Explain how you found the solution.

There exist other solutions which are even more "visual". Some stu-dents have argued in the way shown in Fig. 5, which comes very close to a "proof without words". Both images are extrapolated as a logical sequence (both on the side of the objects and the respective prices). This approach is a very convincing example for the creative role of visual schemata for non-routine problem solving. This type of problem and the encouragement of the non-formal yet highly insightful way of problem solving is very far away from the standard repertoire of textbook problems, however. Another example for insight refers to a problem that elementary school children are able to solve spontaneously. The parallelogram puzzle, investigated by Wertheimer is a classical example from psychological research of insight [23] and the underlying processes are basic to modelling in physics on various levels of abstraction. Children explore how the area of a rectangle can be measured by counting the number of small squares, that cover the rectangle. After the children have understood the method, they are confronted with the problem of measuring the area of a parallelogram with the same width as the rectangle. Of course, the original method fails as the oblique sides do not fit the squares (Fig. 6).

Fig. 5: A visual solution of the problem

Wertheimer gives a very detailed and vivid account of the case of a little girl (age 5.5 years!). Con-fronted with the parallelogram problem she says, “I certainly don't know how to do that." Then after a moment of silence: “This is no good here," pointing to the region at the left end; "and no good here," pointing to the region at the right. “It's troublesome, here and here." Hesitatingly she said: "I could make it right here . . . but . . . “Suddenly she cried out, "May I have a pair of scissors? What is bad there is just what is needed here. It fits”. She took the scissors, cut the figure vertically, and placed the left end at the right ([23], p. 49). Other children bend the parallelogram to a ring so that the oblique parts fit together and find the solution that way. This problem solving task demonstrates that visual imagination is connected with concrete experience (cutting out areas, putting them together). Yet there is much more involved that goes beyond mechanically carrying out actions. Some children are able to anticipate their actions, finding ways to solve the problem without ever having seen the solution before. They go beyond their past experience and create something new. The underlying visually driven problem solving processes reach out deeply into a type of mental modelling which is relevant both in physics and in mathematics. In an abstract sense the little girl's insight is an early example for seeing invariants and constructing conserved quantities by suitable mental transformation processes (which have to be grounded by concrete experience). The require-ments on mental modelling are closely related to modelling processes in physics, creating a deeper insight into the concept of torque and of angular momentum. For instance, in order to understand Kepler's second law one must find out why in any central force field planets sweep equal areas in equal times. As Newton's geometrical derivation of Kepler's second law shows, this requires the transformation of the areas of parallelograms (respectively of triangles) [24, 25].

The little girl's insight opens up a broad road towards mental modell-ing and a deeper understanding of heavenly and earth-bound motion. This line of thought shows that great ideas and achievements in the his-tory of science are not very different from everyday insights that even children can develop at an early age. As a consequence, we must make our teachers sensible for the poten-tials of visual literacy and foster cognitive development along these lines at a much earlier age. The potential of images is not suffi-ciently exploited in physics and mathematics teaching. Images do have an important function not only as and external medium of representation but also as an internal medium of problem solving. In general, teachers do only very little to challenge the mental imagery of pupils. They do not support systematically multiple ways of problem solving, in which images and their reflections would quite naturally play a stronger role. This can be seen from video-analyses of physics lessons [26]. The education of future physics teachers has to address these problems adequately. At present, however, the formal way of physics teaching at universities is, in my view, one of the biggest obstacles for making future teachers susceptible to more appropriate approaches to formal thinking in schools. It is quite natural to copy bad examples!

Fig. 6: The parallelogram problem. Children who know how to measure the area of a rectangle are asked to measure the area of a parallelogram.

6. From concrete actions to general principles: insights require reflections

Fig. 7 shows a mechanical balance at equilibrium. Similar to the T-shirt item shown in Fig. 4, this problem can be solved more effectively by using concrete pictorial operations rather than by solving an equation. Removing mentally one brick from each side of the scales is a good strategy and a good indicator for insight, although the argumentation - in comparison to the formal algebraic solu-tion - "only" takes place at a concrete level.

The example demonstrates the close interplay between the pictorial, concrete, physical access and the symbolic, formal mathematical one. The physical object "balance" is a possible realisation of the mathe-matical object "equation". Algebraic operations at the abstract level correspond to physical processes at a concrete level. The equilibrium is not disturbed when the same processes take place on both sides of the physical or mathematical object. Our mental images and the abstract mathematical symbols that we use are "embodied" in such a way that they are based on concrete experi-ence about physical systems.

Fig. 7. The scale is in equilibrium. On the left scale is a 1kg piece and half brick. On the right scale is a full brick. What is the mass of the full brick?

It is interesting to note that the physical system "balance" corresponds to a basic mathematical metaphor giving birth to algebra, an abstract subject within mathematics. Algebra comes from the Arabic word "al jabr", the meaning of which is connected with balance and compensation [27]. As we all know that mathematical formalism develops a life of its own, many students (and teachers too) often forget about this correspondence. In his criticism of the pure formalist approaches to mathematics, the famous Russian mathematician Arnold opens an article on teaching mathematics with the statement Mathematics is a part of physics and he goes on saying Mathematics is the part of physics where experiments are cheap [28]. Penrose, a master in visualizing abstract ideas, promotes an evolutionary view of our mathematical abilities, which he considers an incidental fea-ture, a by-product of evolution [29]: For our remote ancestors, a specific ability to do sophisticated mathematics can hardly have been a selective advantage, but a general ability to understand could well have. An important feature of understanding is mental model making, including reflections about models and the modeling process. Models are executable mental images, mapping essential features of reality and allowing to anticipate the future. They can be run at practically no cost and their biological relevance for survival is evident. To some, Arnold’s criticism may sound too strong but in my personal view it is important to keep in mind that the mathematical enterprise is based primarily upon physical experience and gains its potential in a social context (communicate an externalize ideas). Our conceptual system (including the formal mathematical system) is embodied. Most abstract ideas in the domain of science and mathematics arise via conceptual metaphors. This is a mechanism that projects concrete, embodied (that is, sensory-motor) reasoning to abstract reasoning. Thus, many abstract inferences can be considered as extensions or projections of sensory-motor inferences. Such a line of thought about formalism has been developed from linguistic approaches [30]. It may sound heretic to a formalist, but for learning and doing physics it is important to keep in mind that formal thinking has a human face and builds upon concrete experience. Nevertheless, the abstraction processes (as the development of physics shows) transcend the con-crete experience on which they are based. How is that possible? There is a creative element in making models. Going from the level of concrete experience to the level of generalisations (or, more formally, to the level of axioms) always involves a creative jump. This is very clearly expressed in Einstein’s view on the modelling cycle. His EJASE-scheme connects the level of experience (E) with the level of axioms (A) and the conclusions (S), formally derived from the axioms, by a creative process depicted by J for jump [31] for a more detailed exposition). In many pedagogical discussions on the modelling cycle this creative element is completely ignored!

It is important to reflect the transition from concrete experience to abstraction in detail. Again, we can use a simple example, closely related to the scale problem and transformations of mechanical equilibrium. Fig. 8a shows a visual approach to the lever principle which is attributed to Archimedes [32]. We start with a symmetric configuration. We accept that it is in balance due to the symmetry principle. With a few transformations, that add additional loads and shift them, leaving the system in equilibrium, we arrive at an asymmetric configuration: One unit of load at triple dis-tance from the center is in balance with three units of load at one unit of distance. The sequence of pictures speaks by itself. It acts quasi like "proof without words". But nevertheless it is not a proof. Why?

Fig. 8: Insight into the lever principle (a) and the center of mass principle (b).

There is a circularity in the argumentation. Some of the transformations (shifting the loads by one unit in one direction and compensating the effect by an equal shift in the opposite direction) take for granted the principle that is intended to be proved by the whole procedure! The same holds for the demonstration of the center of mass (Fig. 8b). In rigid body mechanics, there are three principles that are closely interrelated: The lever principle, the superposition principle of torques and the center of mass - principle. They are part of experience that we take over to the system of axioms. They cannot be proved. But taking one of these principles for granted, the others can be derived. There is a certain freedom of constructing the system of axioms that requires a creative jump from experience to theory. Making this transition explicit is one of the most difficult stumbling blocks in model making and formal thinking. 7. Metamorphoses of complex open systems: metaphors for insight?

In our attempts to foster understanding and insightful learning of physics we have to put more emphasis on the nature of human understanding and the concrete “embodied” base of abstraction and formalisation. Although understanding has the appearance of being a simple and common-sense quality, it is impossible to define it. In his popular writings on atomic physics and human knowl-edge Bohr noted that an analysis of the very concept of explanation would, naturally, begin and end

with a renunciation as to explaining our own conscious activity [33]. While our experience starts with concrete concepts, the development of physics has led to the insight that physical reality is not comprehensible in terms of concrete touchable systems. Questions, that have arisen in the early days of quantum physics are still on the agenda today and will persist as we cannot circumvent the abstractness of physics. The way to abstract and formal thinking is unavoidable, but what can we do not to lose too many students on that way? Making physics more meaningful to the inquiring mind is important. This in-cludes, among many other aspects, the question, what kind of physics we have to learn, in order to make the "physics in our heads" more tangible. The embodied mind, our mental imagery and intuition are based considerably on mechanical experience. In accordance with the con-crete nature of experience we have re-stricted the present discussion to exam-ples that are more or less mechanical in"visual" and "comprehensible". Promotingpromising. It is surprising how far we casimple mechanical systems of our everydayin surprisingly complex ways [3] and the eserve as models for complex dynamical properties emerge, like adaptive behavior amental realm. The more we understand thmore we also learn about the workings of oAlong this line of thought we can even trphenomenon of bistability that is found in mswitching bistable percepts like the Neckerto a thermally driven process, giving risespontaneous switching of the cube. This spontaneous to the induced switching modeWhen the eyes follow a periodically movinprocess locks to the periodical signal in ssynchronized with the external drive signastochastic resonance that is ubiquitous in noin that phenomenon on many levels of biol[34]. These models of complex phenomenon emlead to concepts that are no longer mechanchanical substrate" they originated from. Tfor the metamorphoses of pictures or for systems can be modelled from first principlextent, universal behavior. In my personal

Fig. 9: Transformations of inner landscapes. A model for mentally switching bistable figures.

nature. Mechanical systems can be grasped, they are mechanical imagery and its critical reflection is highly n get by using mechanically inspired imagery. Rather world like driven oscillators or coupled clocks behave xperiments described there). These mechanical systems processes of open systems far from equilibrium. New nd perception like qualities, that we usually refer to the e complex behavior of matter far from equilibrium, the ur mind.

y to model the metamorphoses of internal images. The any complex systems can serve as a model for mentally

cube in Fig. 9. The spontaneous switching corresponds to a Poisson sequence of events that compares to the model even allows predictions for the transition from . g dot as the experiment in Fig. 2 shows, the switching pite of the spontaneous noise. The switching becomes l. There is a close connection with the phenomenon of nlinear dynamics. Noise plays a highly constructive role

ogical information processing and in neural computation

anate from "mechanical" intuition. Nevertheless, they istic in the strict sense. One hardly recognizes the "me-hey even supply models for "physics in the head", e.g. other complex dynamic processes in our brains. These es by field-theoretical approaches and show, to a certain view the universality of dynamical processes far from

equilibrium is one important factor why our brain can represent relevant aspects of reality. Considering mental images through the looking glass of physics creates new insights on the working of our brains, whose collective effects over the centuries finally set up the program of physics. Is the theoretical frame of today's physics sufficient to model internal observers?

Fig. 10: Images of physics: from everyday objects to strange realities.

How far can such imagery guide our intuition? Although concrete pictures are necessary for our understanding, physical reality transcends concrete imagery. Behind all our attempts to create con-crete pictures "lurks" the paradox. Our visual imagination is powerful enough even to handle paradoxical situations. Fig. 10 shows a strange picture of reality, which is also a metaphor for gaining insight into the modelling method of physics. We make categories and models for certain aspects of reality that appear consistent as long as they are viewed separately. The two shelves with the objects on them might correspond to the particle or the field approach in physics. However, when we try to put both systems together (as it happened in the development of quantum theory), we arrive at a paradox (the impossible shelf). Such impossible pictures link physics with arts. Combining physics and arts is one important subject. Physics as the art of model making is another one. The real challenge, however, is to convey how much our naive conceptions of reality are challenged by the program of physics! The perception of physics in the future will depend critically on how we can convey to a broader public that Physics is everywhere, how the late GIREP president Karl Luchner put it in his book. There is much to be done to make physics education more attractive and effective! References [1] K. Luchner, Physik ist überall, München, (1991). [2] M. Michelini, Introduction to the First GIREP Seminar, Udine, (2001). [3] M. Euler, Physics and Physics Education Beyond 2000: Views, Issues and Visions. In, R. Pinto, S. Surrinach (Eds.),

Physics Teacher Education Beyond 2000, Proc. Int. GIREP Conference Barcelona 2000, Paris, (2001). [4] Knowledge and Skills for Life. First Results from the OECD Programme for International Student Assessment

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[6] A.E. Beaton et al., Science Achievement in the Middle School Years: IEA’s Third Int. Mathematics and Science Study (TIMSS), Chestnut Hill, (1996).

[7] R.E. Mayer, Thinking, Problem Solving, Cognition, New York, (1992). [8] C.F. Gauss, Brief an Olbers, Werke, Vol. X.1, Leipzig (1917), 24. [9] R.J. Sternberg, J.E. Davidson (Hrsg), The Nature of Insight, Cambridge, Mass., (1995). [10] C. M. Seifert, D. E. Meyer, N. Davidson, A. L. Patalano, I. Yaniv, Demystification of Cognitive Insight:

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Chaos, 4, (1994), 1383 [15] L. von Bertalanffy, Theoretische Biologie, Francke, Bern, (1951). [16] W. Pauli, Physik und Erkenntnistheorie, Braunschweig, (1984). [17] C. A. Meier (Hrsg), Wolfgang Pauli und C.G. Jung: Ein Briefwechsel 1932-1958, Berlin, (1992). [18] H. Atmanspacher, H. Primas, E. Wertenschlag-Birkhäuser (Hrsg.), Der Pauli-Jung-Dialog und seine Bedeutung für

die moderne Wissenschaft, Berlin, (1995). [19] H. Primas, Es gibt keine Einsicht ohne innere Bilder, GAIA 1 (1992), 311 [20] R.P. Feynman, R.B. Leighton, M. Sands, The Feynman Lectures on Physics, Addison-Wesley, Reading, II, Ch. 20-

3, (1963). [21] See articles on students’ knowledge and learning in: A. Tiberghien, E.L. Jossem, J. Barojas (Eds.), Connecting

Research in Physics Education with Teacher Education, ICPE-Book, 1997; available for download at http://physics.ohio-state.edu/~jossem/ICPE/BOOKS.html

[22] Measuring Student Knowledge and Skills: A new framework for Assessment, OECD, (1999). [23] M. Wertheimer, Productive Thinking, Harper & Row, New York, (1959). [24] J. Fauvel, R. Flood, M. Shortland, R. Wilson (Eds.), Let Newton be! A New Perspective on his Life and his Works,

Oxford Univ. Press, Oxford, (1992). [25] D.L. Goodstein, J.R. Goodstein, Feynman's Lost Lecture, Norton, New York, (1996). [26] T. Seidel, M. Prenzel, R. Duit, M. Euler, H. Geiser, L. Hoffmann, M. Lehrke, C. Müller, R. Rimmele, Lehr-

Lernskripts im Physikunterricht und damit verbundene Bedingungen für individuelle Lernprozesse, Unterrichtswissenschaft, 30, (2002), 52

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