Department of Physics
University of Auckland
Students, and indeed all of us, suffer misconceptions about the physical world. These show up readily, in the students' case, when they are assessed on the conceptual level. For example, a situation in geometric optics:
This picture shows a situation characteristic of Goldberg and McDermott's work. A lens forms an image of a lit candle on a screen, and Anton stands a considerable distance behind it. The screen could be made of something like ground glass or tracing paper, allowing Anton to see the image as well. The test question is as follows: if the screen is taken away, where does Anton see the image - on the lens, inside the lens, or behind the lens? Few students think that the image will be in the same place as the screen was. Moreover, most students are confident that if the screen is kept but half of the lens is cut away, then a corresponding half of the image on the screen will disappear. What would Anton now see, if the screen were removed? One very common answer is that Anton would see only the image-half that did not previously disappear.
My interest lies in understanding why and how students arrive at these conclusions. The conclusions themselves are less of a problem, because they're just symptoms of an underlying disorder, or what we regard as a disorder: some would have us treat these alternative theories as equal contenders to our more orthodox ideas, and I do not mean to endorse that level of relativism. From the students' viewpoint, they are correct. Truth might not be relative, but belief certainly is. In light of that, the teachers' job can be construed as to get students to see things from the orthodox viewpoint.
Researchers in this field have found widespread acceptance - amidst students - of a theory that, interestingly enough, echoes the ancient Epicurean theory of vision. According to the Epicureans, objects radiate thin concentric shells or 'skins' (Greek eidola). The skins carry the image of the object, and the object is seen when its skins strike the eye. Some students reportedly conceive of images as being thrown off objects, but unlike the Epicurean skins, these images don't expand concentrically. They travel intact, as if well-defined ghosts of the objects themselves. When such an image-ghost enters a lens, it is flipped upside-down, and exits the lens on the other side to continue its journey. So a lens, in this belief-system, is conceptualised as a black-box image-inverter.
To correctly understand the function of lenses requires another fundamental idea, but one that most students don't use: the principle that every point on an object radiates light independently of other points on the object. The medievals had an allied notion called "cross-multiplication of species"; Huygens had a similar idea in making every point on a wavefront into a distinct source of wavelets. The principle is crucial in understanding the relationship between points on an object and its image, and in appreciating the ability of lenses to produce images that resemble their objects. Unfortunately, the lack of this principle in student thinking impedes any understanding of the type that we want to teach.
Another kind of problem involves the vocabulary of optics: the words "image" and "focus" are not necessarily interpreted according to their physics meanings - these and many other technical words can be vague at best, having been blurred through their metaphorical uses in other contexts. When a lecturer speaks about "foci" and "images", the message is not necessarily clear - the technical meanings of these words may be unknown to the audience. The problem is how to make the words meaningful.
One final example of a problem is the apparent obviousness of mechanics and optics, since it is so easy for an audience to simply switch off if they think that they've seen it all before. If an audience is convinced that the material is known to them, and even easy, the task of convincing them otherwise can easily become even harder. The problem is a classic one: how to catch the attention of a passive audience.
What I would like to do is understand how these unorthodox ideas of physics develop in students' minds, and how we can teach students to think in more orthodox terms.
It is commonplace now to model student learning on ideas from the history of science, drawing in particular on Kuhn's notion of 'paradigm shifts'. The relevant part of Kuhn's work looks not at the ultimate origins of science, but at the transition from one theory to its successor. An advantage of such a model is that it focusses on the shift of the paradigm, thus matching the intended outcome of teaching: given that students already have beliefs, it is too naïve to approach them as blank and pure minds - tabulae rasae - awaiting the implantation of noble truths. Instead, the process must be one of changing what is already there, to make the student able to understand the orthodox outlook, and perhaps to show its advantages over various unorthodox alternatives.
Put crudely, Kuhn's paradigm shifts run roughly as follows: the scientific community subscribes to a paradigm that explains both the old observations, and also new observations that are being made. Alternative theories might be proposed from time to time, but their advantages are not sufficient to allow them to take root. But eventually, observations begin to exceed the explanatory ability of the orthodox paradigm: experimenters find small anomalies, and theorists tweak the models to account for them. Initial adjustments might involve no more than the slight refinement of fitted parameters, or a more precise numerical approximation. As anomalies become more problematic, 'correction factors' become necessary, special cases are created as exceptions to the general rule, and the orthodox theory grows increasingly chimeric. Thus it becomes unwieldy and unattractive, and prone to abandonment in favour of a more powerful and elegant contender. Along with a new theory comes a whole new way of generating and approaching data - the old measurements are all re-interpreted in the new context, and experiments are done in a new way, according to new standards, in pursuit of new ideals. This is the new paradigm.
The Copernican Revolution is widely cited as an example of a paradigm shift, where an earth-centred understanding of the cosmos was rejected in favour of a sun-centred understanding. Along with this came Newton's Law of Universal Gravitation, and his insistence that he was only producing descriptions, not discovering the ultimate causes of nature. But Newton's gravitational law cannot, as is widely known, fully explain the orbital motions of the solar system. General relativity offers a better alternative, but in reality, most of us still take the Newtonian path: it's easily good enough for working out how far a rock will fly from its trebuchet. Likewise, the earth-centred universe was not wholly abandoned: it is still useful in astronomy, often far more so than either sun-centred or galaxy-centred coördinate systems.
What popular accounts miss is that the old theories remain active, though their influence is restrained. Physics by nature entertains many different world views, reconciling them through such intermediaries as Ehrenfest's theorem and the correspondence principle, the clear-cut perpendicularity between Huygenian wavefronts and geometric rays, the Taylor polynomials that bridge linear approximations to their non-linear superiors.
The Kuhnian approach to history of science accords much importance to the role of observation. At first sight, it could be thought that observations motivate the overthrow of a theory. But observation alone is not enough; we need one small refinement: conscious awareness of anomalies is the motivator.
Unfortunately for us, there are several ways for dealing with observations: they could be believed and somehow accommodated into the prevailing theory; they could be ignored in the hope that they will go away - as they sometimes do. And even before that problem arises, there are already difficulties arising from the subjectivity of observation. Kuhn's model, as do many others, positions the paradigm as an intermediary through which nature is observed, and the paradigm thus stands as a possible bias, much like Francis Bacon's 'idols': our personal biases, cultural biases, prior learning, and so on. Some of us are colour-blind, tone-deaf, or unable to handle the apparatus through a lack of fine motor skills. We may often be well aware of such biases or the risk thereof, thus warranting extra care in deciding whether anomalous observations are worth worrying about. So far, I have put forth ideas from three general areas: problems in student understanding, an approximately-Kuhnian model of scientific progress, and the importance and difficulty of empirical observation. Now I will propose a way of combining these ideas into a useful mode of teaching and learning.
These ideas can all be incorporated into an extension of the standard "predict-observe-explain" (POE) sequence. In its earliest incarnation, the POE was nothing more than a set of carefully structured exercises that led the student infallibly to the desired conclusion. Having been subjected to that method in secondary school, I can say that it is not guaranteed to work. It would be more useful if students could be engaged to interpret observations in terms of their prevailing ideas, and then to re-assess their ideas in terms of those new observations. Hence the following pattern:
A sequence such as this must be guided, for two reasons: firstly, the students have to see the anomaly. Without guidance, the Baconian idols may move in and give observations that reinforce their prior beliefs - if their unorthodox theories are confirmed in the orthodox laboratories, the task of teaching them to think otherwise could become much harder. And secondly, the introduction to the orthodox theory - which may well be 'new' to the student - needs guidance to ensure that it isn't learnt in a distorted form, or substituted by yet another unorthodox theory.
A variation on this extended POE begins with a prediction, to be countered by simple logic - a proof by reductio ad absurdum, for instance, or a counterexample.
Let us return to the case of Anton standing behind a real image produced by a converging lens.
The prediction question is, "What happens to the image on the screen if you cover up half of the lens with cardboard?"
The standard answer: "Half of the image disappears."
In performing the experiment, many students actually claim to see part of the image disappear from the screen. Admittedly, a little of it does vanish, because the light path is vignetted, though I have never seen the image lose anything more than a tiny strip along its edge. The fact that the image gets dimmer is may be missed completely - note that the standard prediction makes no reference to brightness, only to shape. Many students do notice the dimming, but disregard it as unimportant. For these reasons, it is important for a teacher to be there, guiding them through the observations, making sure that they don't miss anything important. Once the important features are noticed, it becomes possible to break down some of the old ideas that fulfilled the university entrance requirements.
This situation could be extended to dispel the common misconception that there are only three rays (the three principal rays) that travel from the object through the lens to form the image: if there were only three rays, and all three passed through the lens, there would be none left by which to see the object directly. If this were the case, then the whole world would be plunged into darkness every time someone held a lens up to catch all of the rays.
One possible counterargument is that only two rays are needed to form an image - the third ray is extraneous. This can be rebutted by asking how many rays are involved when two lenses are used to cast two images, or when pieces of paper are used to cover the parts of the lens where the principal rays travel. This demonstrates that three rays aren't enough. But trouble could arise if the student answers that the lenses themselves cause the objects to radiate, or else that the lenses are the sources of the rays. Getting out of such situations can be tough - not only are we seldom prepared for them, but the arguments can quickly exceed the students' ability, interest and patience. In such circumstances, the teacher may well lose by default.
Both fortunately and unfortunately, I have never had a physics student argue to that depth.
I mentioned also the ambiguity or vagueness of some of the technical terms used in physics. The word "focus" sometimes seems meaningless in student writing, and I have seen it used to mean things other than what we expect. The word itself is a formidable one: in physics alone, it can be an adjective, a noun, and a verb. The noun, obviously, is the place where rays intersect or converge. This meaning is, historically speaking, its primary meaning, since it was coined by Kepler to denote the place where a conic section concentrated parallel rays from the sun. Kepler's original context was mirrors used for fire-lighting, and he chose "focus" because, in Latin, it means "hearth" or "fireplace". The corresponding English term was "point of burning", and its use persisted well into the eighteenth century. An advantage of defining the word this way is that a great many students are already familiar with the concept: a magnifying glass collects the sun's rays into a small point that has brought them countless hours of entertainment, painstakingly trained onto scraps of paper and small insects.
From the idea of a "point of burning", one can generalise to the foci of diverging lenses and convex mirrors. The Latin word "focus" thus becomes a tool of abstraction, facilitating an extension from the familiar English into new territory.
Another problematic belief is that converging lenses have to be biconvex, and diverging lenses must be biconcave. One approach to countering this, now proven successful in our first-year teaching laboratories, is to mount meniscus lenses in pairs, hiding the edges inside the mount, so that the biconvex pair is diverging, and the biconcave pair, converging. These were placed amidst the 'normal' lenses, thus providing students with a classification exercise more complicated than usual. Naturally, most were confused, and many insisted that there was something wrong with these lenses, regarding them as deceitful tricks. Some students were drawn into argument, becoming agitated and frustrated enough to pull out textbooks in support of their case. Often it is said that physics students don't spend enough time discussing and pondering their ideas; our laboratory trials showed that this need not be the case.
We know that students come to us with preconceived physics ideas that disagree with ours, and the teachers' task can be interpreted as one of showing them a different viewpoint: the viewpoint of orthodox physics. The aim, so viewed, is not simply to override their old beliefs by imposing a cognitive change: it is to guide the students so that they take themselves into intellectual conflict, and launch into conceptual change of their own accord - with a teacher guiding the formation of a new theory, ensuring that they arrive within the scope of the orthodoxy. By taking this approach, they engage significantly in the entire problem, and we introduce them to physics not as dogma, but as a justified doctrine that is capable of explaining phenomena where their previously held theories fail.
Goldberg, F.M. and McDermott, L. C. "An investigation of student understanding of the real image formed by a converging lens or concave mirror." Am. J. Phys. 55, 108-119 (1987).
Kuhn, T.S. The Copernican Revolution. Cambridge, MA: Harvard University Press, 1957.
NOTE: Another important theme in this project, not discussed in this paper, developed the language-based approach in terms of Vygotsky's cognitive psychology.
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