The ‘thin lens’ in the light of idealisations

This paper argues that the simplifying idealisations assumed in making complex science tractable should be identified and discussed before model-building. It is suggested that alternative conceptions can persist, and learning difficulties arise for students when they do not adequately understand these idealisations. We argue that explicitly addressing idealisations for a particular physical phenomenon can ease students’ conceptual understanding. A theoretical framework for identifying and reflecting idealisations is introduced based on epistemological considerations from science philosophy. The thin lens approximation of geometric optics, a common topic taught in the introductory classroom, is used as an archetypal illustration.


Introduction
In teaching geometrical optics, forming a real image of an object by converging lenses plays a significant role.In this journal, the topic has been taken up on various occasions.Suggestions have been made for improving students' understanding, e.g., using calculation spreadsheets [1] and dynamic applets [2] for the construction of the following, the 'thin lens' illustrates the many idealisations that come into play in investigating the physical phenomenon of image formation.The theoretical considerations are then applied to explain widespread students' alternative conceptions in geometrical optics.Finally, we discuss the potential of explicitly addressing idealisations in the classroom to improve students' understanding.

Epistemology and idealisations
Epistemology denotes concepts and theories about the genesis, ontology, meaning, justification, and validity of knowledge.In the discourse of science education research, epistemological issues belong to the field of nature of science (NOS) [7,8].Thereby, the research focuses on a broad understanding of NOS, e.g., the epistemological significance of models and experiments in scientific knowledge acquisition [9,10].
We have discussed the importance of idealisations in this context at length elsewhere [11].Since we are talking about a frequently used model in this paper, we will only highlight the following consideration.In their sophisticated model of modelling, Gilbert and Justi [12] express two views: the models as representations view and the models as epistemic artefacts view.These views are not in contrast; the artefactual view expands on the commonly used focus on models as representations in science textbooks.Models can be seen as tools, performing more epistemic practices than just that of (external) representation.Thereby, also focussing on other practices is helpful such as 'making simplifications and idealisations; conceptualising imagined (or nondirectly observable) objects or processes; supporting arguments, explanations, and predictions' (ibidem., p 23). Models as representations can be dynamic simulations, graphics, or symbolic representations.
Our understanding of idealisation is based on considerations from the philosophy of science [13,14].It refers to the approach of simplifying a complex reality.For this purpose, individual properties of a thing, e.g., a natural phenomenon, are reinterpreted or even entirely excluded.Only those properties remain that are perceived as essential for the goal of an investigation.
The aim is to advance on being able to answer a question about nature.Consequently, idealisations are based on the claim of optimisation: they are deliberate substitutions in the search for knowledge.In this way, falsifying assumptions are consciously accepted.
In contrast to idealisations, didactic reductions prepare appropriate learning content for a target group, e.g., photosynthesis for a seventhgrade class (see also the model of educational reconstruction; [15]).The corresponding teaching decisions are usually hidden from the learners.On the other hand, idealisations are already included in the learning objects and can even become learning objects themselves.
Hüttemann [16] conducts a more extensive analysis of idealisations and their (epistemic) goals.In his work, he distinguishes between eight categories of idealisations.The categories differ concerning the objective pursued (table 1).The categories are not ranked.Nevertheless, the first four categories focus on the experimental preparation of phenomena and the theoretical analysis of experimental data.Categories five to eight are closer to the idealised properties of models.The fact that idealisations cannot always be assigned to only one of the categories is not unfortunate.In science teaching, the categories-one could also say teaching actions-are intended to initiate a process of reflection, that is, an intensive intellectual discourse.Physics teachers can use table 1 as a guide in the classroom to point out idealised aspects that may not have been thought of before.The categories presented in table 1 are further described in section 3 of the article, with the aid of a specific example from physics class, namely the image formation process by a thin lens.

Students' alternative conceptions
For a long time, it has been known from physics education research that knowledge about students' alternative conceptions and the adequate handling of these preconceptions is necessary to design physics lessons more successfully [17].The aim of physics lessons should be that the students know and understand also the physical concepts.Students hold alternative conceptions which are formed under the influence of everyday culture and language.The presence of Table 1.A framework for structuring idealisations based on [16].Reproduced from [11], with permission from Springer Nature.

Category of idealisation
Meaning of the category 1st Preparation Determination of the objects to be investigated to measure or to enable a theoretical treatment 2nd Isolation or coverage Also the determination of the objects to be investigated to measure or to enable a theoretical treatment 3rd Extrapolation and adjustment of data Representation of the object and its behaviour, to obtain a theoretical understanding 4th Mathematical simplification To solve a formula in an easy way, e.g., summation instead of integration 5th Abstraction of a physical system Abstraction of a physical system into several parts to be considered or calculated 6th Abstraction of properties Deprivation of particular properties 7th Idealisation, more specifically (attribute properties to a physical system that it obviously does not have) Aiming at mathematical and graphical manageability and increase predictive power 8th Simplification or neglection Development of the functional dependency between several properties or physical quantities these preconceptions can extend up to the university level.Since they often do not correspond to physically correct concepts (e.g., image formation in geometrical optics), learning difficulties can arise during physics classes.Furthermore, students can also develop (scientifically) false conceptions in physics class through misunderstanding given explanations.The terminology for these (so-called) misconceptions, alternative or naïve conceptions differs in the literature [18].This paper talks about alternative conceptions; no distinction is made between the different nuances of the named terms.One goal of physics classes should be fostering conceptual change [19][20][21], hence, giving the students opportunities for a change or, to be exact, growth in their conceptual understanding of a given subject or phenomenon.
Students' conceptions describe dispositions to interpret physical concepts in a certain way or to describe phenomena in a certain way that differs from technical physics presentations [22].Since alternative conceptions can work well at first, achieving a conceptual change in the classroom is sometimes difficult.The indicated methods based on which adequate conceptual understanding can be promoted (e.g., confrontation or integration) varies with the different theories of conceptual change [e.g.19,20,23].Regardless of the specific method, it is necessary to support the rearrangement of their existing knowledge structures and the acquisition of new fragments of knowledge by appropriate teaching actions.
Individual factors play a role in shaping these cognitive processes, as conceptual change is influenced by the society and environment of the individual learner [24].Students have to perceive the scientific concepts as credible and plausible to be receptive to (in their point of view) new ideas.Basic scientific ideas, such as idealisations, can support students in these cognitive processes.Explicitly addressing the underlying idealisations of, for example, a model provides insight into the nature of the presented knowledge, thereby enhancing its credibility.A further understanding of the assumptions made during the construction of scientific knowledge can also prevent misunderstandings of its key aspects and reduce the likelihood of forming alternative conceptions at an early stage.Idealisations should therefore be introduced primarily guided by direct teacher instructions.

The 'thin lens' as an idealisation
One standard idealisation in geometrical optics is the thin lens-sometimes even called the ideal lens-used to investigate the image formation process.If we want to reconstruct how a converging lens forms an image, we use models as representations (e.g., figures 1(a) and (b)).Figure 1(a) shows the construction of the position of one image point behind the lens using special light rays.We draw two special light rays through the lens starting from the same point on the object, e.g., one light ray parallel to the optical axis and another passing through the centre of the lens (shown in blue).This procedure can be repeated for different object points to understand where and how the image is constructed.The thin lens itself is an idealisation.However, looking at this idealisation, one recognises even more underlying idealisations.The most significant is undoubtedly the idealisation of using individual light rays or one specific bundle of light (abstraction of properties), using special light rays to predict the position and size of the image (simplification or neglection), and reducing the lens to its central plane (idealisation more specifically).They are crucial for predictions about image formation, like the position and size of the image.Another idealisation in this model is focusing on refraction and scattering (abstraction of a physical system).In the context of an experimental investigation of light propagation on a thin lens, light scattering on the table surface or the board is used to make light bundles visible (preparation).Of course, when light hits a lens, there is also reflection.Typically, however, this observation is excluded from explaining the formation of the image.To observe the light bundles better, the room is darkened a little (isolation or coverage).To avoid chromatic and spherical aberrations, light bundles close to the optical axis are used; paraxial light bundles (extrapolation and adjustment of data).
The following listing illustrates the multitude of idealisations involved in this example and summarises the meaning of the respective idealisation in the right column of table 1.The list numbers correspond to those in figures 1(a) and (b).
x Preparation Light scattering on a bright background.y Isolation or coverage Darkening of the room.z Extrapolation and adjustment of data Paraxial rays of light-Error consideration for rays far away from the optical axis.

{ Mathematical simplification
Is not used in this example.

| Abstraction of a physical system
Concentrate on light refraction.Ignore light scattering Ignore light reflection.

} Abstraction of properties
Individual rays of light instead of light bundles.Geometrical optics instead of wave or quantum models.~Idealisation, more specifically (attribute properties to a physical system that it obviously does not have) Refraction at the central plane: Ease of drawing, Potentially infinitely large lens diameter.

Simplification or neglection
Selected light paths (e.g., through the centre or focal point of the lens).
Position of the bottom of the object on the optical axis.

Implications for teaching physics
Since idealisations are crucial for epistemology in natural science, we feel that dealing with idealisations is also essential in physics education.Especially with regard to the emergence of alternative concepts and their further development towards adequate concepts, we consider an explicit discussion of the underlying idealisations helpful.At this point, two examples should illustrate how misunderstood idealisations can make learning more difficult.During the explanation process of image formation sensibly, the focus is on refraction, not reflection-see also number 5 of our list.However, ignoring reflection can lead to alternative conceptions about pictures' brightness or light propagation.Students sometimes have the alternative concept that light cannot travel very far [25] or there is suddenly more light behind the lens at the bright focal point than in front of it, 'light can become more or less' [26].In a model approximation, it should become clear that the amount of light remains constant, but the area under consideration is smaller.For example, if a sheet of paper is held in the focal length of a converging lens, an area of the paper smaller than the diameter of the lens is brightly illuminated.Even less light is involved in creating this light spot than initially hits the lens because a part of it was reflectedbut ignored for explaining the image.
The misunderstanding of idealisations, for example, the meaning of the central plane (number 7 on our list), can even lead to the fact that the formation of an image through a lens can not be explained at all.Students usually work through image formations similar to the way in which one works through a recipe; first, they draw a parallel beam from the ends of the object.The learners think that a lens can only fully image objects if the objects are not more extensive than the lens itself.If an object is larger than the lens diameter, the construction fails, and the learners conclude that no image can be formed.Similar difficulties arise when the imaging lens is half covered [27,28].
The students' alternative concept can be described as 'the diameter of the lens determines the size of the image' [26].It has often been discussed that the concept of special light rays is misunderstood [28,29].In the example above, in particular, the role of the central plane of the lens should be discussed.The central plane is not only used for better drawing handling, for which two boundary layers between air and the lens and between the lens and air are reduced to a lightrefracting line.This central plane is a line that is first imagined and then drawn, the length of which is independent of the diameter of the lens.It allows us to use special light rays in the model, such as the parallel light ray, even if, in reality, this light ray would not go through the lens at all as the object is taller than the lens.Even half covering the lens is no longer a problem for forming the real image in the model.For example, the light ray through the middle of the lens can still be drawn, even through the obstacle, where real light rays are stopped (or better scattered or absorbed).Figure 2 illustrates how helpful a meaningful understanding of the central plane and the special light rays as idealisations is to manage the hurdle of a tall object and a partly covered lens.
In (re-)constructing a model (as a representation), the importance of underlying idealisations should be made known [11].Identifying idealisations can offer an introduction to reflection (see table 1).From our point of view, dealing with idealisations can increase learners' model understanding and thus overcome students' alternative conceptions.This is not only true in optics but also in every physics or science domain.
An empirical test of this proposal is still pending.In a current study, we investigate how physical explanations become effective when they explicitly or implicitly address idealisations.The explanations of concrete phenomena conceived for this purpose address the fields of optics, magnetism, mechanics, and electricity.

Figure 1 .
Figure 1.(a) Model as a representation of a real image formed by a 'thin lens'.Refraction occurs at the idealisation 'central plane'-model as an epistemic artefact.(b) Representation of the image formation process using the light bundle contributing to it.

Figure 1 (
b) is the commonly used representation of the selected light bundle contributing to the image of one specific image point.Due to refraction, the light intersects behind the lens, and an image point is created (the tip of the arrow).

Figure 2 .
Figure 2.Even a partly covered lens images the object completely.In the model as a representation the refraction occurs at the idealisation 'central plane' and the two special light rays can be drawn.