hubble ultra deep field



The Background arguments (Part One):

The purpose of this paper is to identify those areas of cosmology which lead to misconceptions in both the specialized and general public and examine the culpability of visual analogies or models in these misconceptions and their role as teaching aids.  It is also the aim of this paper to lay down parameters for visual models addressing expansion, and to undertake a preliminary evaluation of the standard visual models in the light of such criteria.  Finally one alternative visual model will be presented and its conformity to the previous parameters examined.

Keywords: misconceptions in cosmology, analogy, visual model, local expansion, global expansion, cosmological redshift.



“You do not understand something until you can explain it to your grandmother.”  Albert Einstein.


1.1   Many students have considerable difficulty in absorbing concepts of astronomy.   They are resistant to some ideas, as their preconceived notions somehow block the incorporation of new concepts, making it impossible to progress as desired.  This is more than apparent in “A Private Universe”, a video documentary produced by Schnepps and Sadler (1988) for the Harvard-Smithsonian Center for Astrophysics, where interviews held with high school students and Ivy League graduates showed that even the brightest students in the class have false ideas based on enduring misconceptions that traditional instructional methods cannot overcome.  They find that “students will graduate from college with the same scientific misconceptions that they had on entering grade school.”  In their survey, 21 of 23 students, alumni and faculty members failed to give satisfactory answers as to the reason for the seasons.  This misconception is attributed in the video to the standard visual model where a highly exaggerated ellipse is shown in many text books due to perspective drawings.

 Figure 1: The Cause of the Trouble?

figure 1

Such perspective drawings prevent students from learning that the earth’s orbit is nearly round and that the seasons result from the inclination of the earth and the difference between direct and indirect light. 
Other misconceptions are common.  For example, one very bright ninth-grader, Heather, when asked to explain what indirect light was, was unable to let go of her own theory that sunlight somehow performs an impromptu ninety-degree turn as it bounces off “things in the way” between the sun and the earth, despite one-on-one teaching to the contrary. 

Figure 2.   Heather’s personal theory of “indirect light”

figure 2

“Her own personal theory is so deeply ingrained that despite our attempts she never abandons it”.  The class teacher sees that “They have experiences and ideas that they associate with other things and which closes [sic] off their minds to what it is you are trying to get across to them.”  The interviewers conclude that new concepts compete with the preconceived ideas of our listeners.  All students hold these preconceived ideas but they are unaware of their private theories.  “We must make them aware – only then can we enable them to learn and free them from this private universe.”
From this and other studies (see Cartelli (2003) and Zirbel (2004) for example) modern educators are becoming conscious of the importance of using correct and unambiguous visual models as teaching aids.  Misconceptions created in the minds of young students may well stay with them for the rest of their lives, impeding them from comprehending further and more complex concepts.
Visual models and analogies used to explain expansion in cosmology are vital to a correct understanding of the subject.  Unfortunately many of the current visual models used are out-of-date and no longer appropriate for use in twenty-first century classrooms and lecture theatres. 
One of the most difficult concepts to assimilate if one is visualizing the universe with the “expanding balloon” model, where galaxies are spots drawn on the surface of the balloon with a felt-tip pen, is that local groups and clusters of galaxies do not expand with the Hubble flow.  In this model everything is expanding, and it would clearly be impossible for small parts of the balloon to “overcome” expansion.  Since most of us begin our cosmological studies with this visual model, we are inclined to reject the concept that local expansion does not take place.   
Other important concepts are the cosmic microwave background, (CMB), the existence of dark matter, the existence of voids, and the recent experimental evidence for dark energy.  None of these concepts can be integrated easily in the current visual models and analogies which are usually used to portray expansion.
Many recent scientific papers [see for example Barnes, Francis, Berian and Lewis (2006); Clavering (2005); Davis, Lineweaver and Webb (2002); Grøn and Elgarøy (2006) and Peacock (2008)] refer to what has now become the tethered galaxy problem.  The object of such studies is to predict what would happen to a galaxy which is released into the Hubble flow on one side of a cluster.  Generally it is thought that, contrary to what one might expect, the galaxy will not join the Hubble flow directly, but will fall towards the cluster, pass through it and rejoin the Hubble flow on the other side of the cluster.
Unfortunately if we attempt to visualize such an event in the expanding balloon analogy, or the rubber sheet model we have all seen in many text books, it is impossible to explain or comprehend.  The model simply does not work for this type of movement.  It has become necessary to update the existing analogies or visual models used to incorporate new concepts and eliminate confusion.
The aim therefore of this paper is to list the parameters that a good visual model should meet, in order to allow the student to achieve a reasonable intuitive understanding of cosmological concepts without a detailed knowledge of special or general relativity.  Such a model should enable the student to grasp the essentials of expansion without causing him difficulties in the future when he wishes to deepen his knowledge of the subject. 






“Analogies prove nothing, that is true, but they can make one feel
 more at home” Sigmund Freud.


2.1   Visual models are key elements in explaining cosmological concepts both to the general and to the specialized public.  They help to form our intuitive grasp of the principles involved, and our understanding of further mathematical concepts will be based on this foundation, which means that such models should also allow the visualization of new ideas such as dark energy or dark matter, wherever possible.  Where new concepts cannot be integrated into previously assimilated models, they may be dismissed as absurd, or unintelligible.
“Analogy” (2000) is defined as: a) similarity in some respects between things that are otherwise dissimilar and b) a comparison based on such similarity. The term visual model has been used in this study to include such diagrams as figures 1 and 2 above, where the only intention is to portray a natural phenomenon in two dimensions, with the inherent problems that might cause.  “Visual model” is exactly defined in part two of this section, but may be thought of as encompassing both the analogical models defined above and similarity models, such as figure 1,  in which the things compared are not inherently unalike.
Visual models are frequently used to present expansion of the universe.  Barnes et al (2006) find that “There is more to applying physical laws than simply solving equations.  In order to make physical laws more transparent and accessible, we use physical concepts that develop an intuition or a mental picture of the scenario.  A successful physical concept allows us to short-cut the mathematics, qualitatively understanding a scenario without having to solve the equations.”  However, the academic argument surrounding the expansion of space is not as clear as standard explanations suggest.
Buffer, Lubben, Ibrahim and Pillay (2008) state “Conceptual models have particular importance in physics education since they are representative of the physical system and the translation between different conceptual models has been shown to play an important role in the learning of physics”.
Paris and Glynn (2004) find that “carefully crafted analogies can help learners to make correct conceptual inferences without causing them to make incorrect ones”.  They claim that their study suggests that the learning of science concepts was enhanced by means of analogies … which can facilitate the cognitive process of building new relations between what is already known and what is new.
Spezzini (2010) found in her survey that 87% of former master’s students on her course “Phonology for ESL Teachers” found that visual analogies had exerted a positive or very positive effect on knowledge and 84% perceived a positive or very positive effect on motivation.  She found that course evaluations improved and exam scores increased significantly.  Her data suggested that the use of visual analogies positively influenced satisfaction and learning.
Francis, Barnes, Berian and Lewis (2007) consider the question of the value of analogies.  They conclude that they are useful so long as we are aware of what they successfully illustrate and what constitutes pushing the analogy too far.  They might correctly demonstrate the effects of the expansion of the universe, but not the mechanism.  However, they also ask why we should be seeking Newtonian analogues when we know general relativity describes the situation well, and can be described in simple terms, when any Newtonian view will break down at a non-trivial limit.
Francis et al also point out that when presenting new ideas, lecturers and textbooks often resort to analogies of stretching rubber sheets or cooking raisin bread to allow students to visualize how galaxies are moved apart, and how waves of light are stretched by the expansion of space, saying that these kinds of analogies are apparently thought to be useful in giving students a mental picture of cosmology, before they have the ability to directly comprehend the implications of the formal general relativistic description.  They also agree that misuse of the term “expansion of space” is dangerous, but argue that throwing the baby of an intuitive framework out with the bathwater of misconceptions leaves us only with bare mathematics, which in the case of general relativity is particularly daunting for the uninitiated and useless as a conceptual device.”  However, they say that: “The alternative is either to give up on a physical concept entirely, so that the only rationale for the cosmological facts is that “That’s what the maths tells us” or to formulate a new framework into which these facts and more can be accommodated.”  They think that the first option is unsavoury and the second unlikely, unless only Newtonian ideas are used.
Sarantopoulos and Tsaparlis (2004) find that “From the affective perspective, we have demonstrated that analogies have a positive effect for most students.” However, they warn that “analogies should be used only when they contribute significantly to acquiring new concepts and processes.”  They conclude that analogies should be used as an aid to understanding when the target is difficult to understand.
The famous scientist Robert Oppenheimer, when addressing the American Psychological Association had this to say about the subject:
Analogy is indeed an indispensable and inevitable tool for scientific progress.  … Whether or not we talk of discovery or of invention, analogy is inevitable in human thought, because we come to the new things in science with what equipment we have, which is how we have learned to think, and above all how we have learned to think about the relatedness of things. (1956)
Orgill and Bodner (2003) find that students argue that good analogies use analog concepts with which the students are familiar.  Students also suggested that visuals be used, because they thought that visuals add to the explanation of the analogy and appeal to more visually-based learning styles and memory.  This last was quantified in a stud of university students of biochemistry, where their findings show that when visuals were used students were more likely to remember the analogy and less likely to misinterpret it.  Their conclusions in this study were that analogies are entertaining and helped the students understand course information.  They state, however, that “It is clear, though, that not all analogies are good and that not all ‘good’ analogies are useful to all students.” 
Davies and Goel (2001) find that visual knowledge on its own, with no explicit representation of causal knowledge, is sufficient for enabling analogical transfer, and they conclude that they can now confidently conjecture that visual knowledge alone can enable retrieval, mapping and transfer in analogy.   
Glynn (2007) adds that analogical thinking is efficient; it helps us to understand new phenomena and solve new problems by drawing upon our past experiences.  However, he cautions that the analogies used are double-edged swords. He says that “An analog can be used to explain and even predict some aspects of the target concept; however, at some point, every analogy breaks down and, at that point, misconceptions may begin.”
Hewson (1992) considers that learning involves an interaction between new and existing conceptions with the outcome being dependent on the nature of the interaction. 

2.2   According to Glynn (1994), (2007) and (2008) analogies consist of two parts, the target concept and the analog concept.

Figure 3

figure 3

The target concept in this case is the scientific concept which it is hoped the student will acquire, and the analog concept is the simple day-to-day example they are provided with in order for them to draw parallels.  The connection the student makes between the two is known as the connection, or mapping (Glynn 2008).  Analogies should promote elaboration, which is the cognitive process of constructing relations between what is already known and what is new.  Apart from the target domain and the analog domain, analogies can be classified into visual or verbal.
The Teaching-With-Analogies Model originally proposed by Glynn, Duit, and Thiele and explained in Methods and Strategies (Glynn 2007) suggests that the following steps should be used: (i) Introduce the target concept, (ii) Remind the students what they know of the analog concept, (iii) Identify the relevant features, (iv) Connect the similar features of the target concept and the analog concept, (v) Indicate where the analogy breaks down, and (vi) Draw conclusions.
These six steps have worked extremely well in the classroom.  Unfortunately a completely visual model, which the student might have come across in a text-book or on the Internet, may not have the back-up of a real teacher to develop these steps in a logical fashion, so the model should itself encourage the student to intuitively cover these steps for himself.  It should therefore be clear what the connections between the known and unknown concept are.
Analogies were defined above as consisting of an analog concept, a connection and a target concept.  One term which includes simpler types of conceptual models is “similarity model”.  This type of model is described by Gentner and Markman (1997) when they say: “... in analogy, only relational predicates are shared, whereas in literal similarity, both relational predicates and object attributes are shared.”  They go on to explain that “Analogy occurs when comparisons exhibit a high degree of relational similarity with very little attribute similarity.  As the amount of attribute similarity increases, the comparison shifts toward literal similarity.”
Similarity models can be described, therefore, as pictures or diagrams in which the analog concept is only a more intuitive and/or elementary version of the target concept.  In such cases the analog concept is referred to as the source concept.  Since the source concept and target concept are not inherently dissimilar, they do not fit the definition of an analogy given earlier.  Thus a similarity model can be considered to map onto itself since the source concept is intended to represent reality, but in a much simpler way than the target concept.  In such models the student must be able to link the concepts well, and understand the limitations of the model.  Figure 1 of the earth’s orbit was a classic example of this.  The diagram is merely an attempt to draw a real, three-dimensional phenomenon in two dimensions, using perspective drawing.  It cannot be classified strictly as an analogy, but fits into the definition of a similarity model.  However, it apparently failed dismally in its objectives, since students were unable to link the two concepts correctly, leading them to assume a high ellipticity of the earth’s orbit.   
“Model” (2005) is defined as “A systematic description of an object or phenomenon that shares important characteristics with the object or phenomenon.  Scientific models can be material, visual, mathematical or computational.”  In cosmology some of the visual scientific models used are analogies, and others are similarity models, so for the purposes of this study visual model will be defined as a scientific model which is visual, consisting of a source concept which maps or connects to a target concept, thus giving a general term which can refer to either analogical or similarity modelling. 
In “A Framework for Conceptual Change” by Zirbel (2004), a misconception is described in its simplest form as a concept that is not in agreement with our current understanding of natural science.  She points out that individuals whose ideas conflict with new information might disregard or discount the new information in favor of their existing beliefs. The student attending class is not a “blank slate”.  She also finds that concepts can be viewed as a consequence of the process of observing and processing.  Zirbel thinks that a common problem is that a half-understood concept might appear to look like a full-blown conceptual understanding of a situation, but in fact only a fraction of it makes sense and it has not been exposed to thorough critical review.  Zirbel says “When we consciously think about a concept and try to verify it in our minds, these concepts can get grounded regardless of whether they are correct.”  
Chi and Roscoe (2002) distinguish between two types of naïve knowledge: on the one hand preconceptions that can be easily and readily revised through instruction and on the other misconceptions which are robust and highly resistant to change, even when not supported by observations. 
Surprisingly, such misconceptions are not confined to novices.  Even faculty members and science graduates can retain misconceptions fostered when they were much younger.  As in the Harvard-Smithsonian study “A Private Universe” cited above, Cartelli (2003) also concludes that the “expert versus novice paradigm” was not supported, as in his study expert people showed the same wrong ideas the students evidenced when they were asked to answer some specific questions.  In his study he found that 100% of pupils in primary school gave wrong answers, but – even more surprisingly – between 65% and 85% of High School and University students also gave the same wrong answers.  His further investigation showed in fact that those university students who had attended specific physics course didn’t use the physics paradigms they had been taught in the analysis of the above situations.  This implies that despite formal training to the contrary, their initial misconceptions had not been corrected by more advanced study.  They continued to be unaware of the misconceptions which had been formed in high school or earlier.
A failure to understand can be devastating to a student who has carefully thought about a concept in the framework he has been taught, and yet is not aware that the original framework was faulty, or indeed that part of his intuitive grasp of the subject is based on erroneous building blocks.  Zirbel (2004) tells us that many studies [she cites, for example, Hewson & Hewson (1983) or Trumper (1997)] have shown a negative approach to facilitating conceptual change, that conceptual change is largely unsuccessful.  Students are likely, if faced with data which conflicts with their preconceived notions, to ignore it and to discount the data.  They may protect their existing perceptions when faced with conflicting data, as Heather did in the Harvard-Smithsonian survey.  It is therefore of prime importance for the future of cosmology that the initial visual models used to teach such ideas as expansion should be as accurate and as flexible as possible.  They should enable a student to visualize the current concept without limiting his understanding of future, more complicated, concepts.  A flawed model will cause damage rather than enlighten.
The difficulty of conceptual change is highlighted in the paper “Hints of a Fundamental Misconception in Cosmology” by Prather, Slater and Offerdahl (2002).  The authors establish that only 1% of first year university astronomy majors who identified the Big Bang as a theory describing creation of the universe thought that The Big Bang was an explosion from nothing, whereas 80% of these students thought that it was an explosion of pre-existing matter.  It appears that phenomenological primitives may explain this result.  Phenomenological primitives, also known as p-prims, are simple ideas that have been accepted by a person uncritically because it is probably all he has ever known on the subject.  The p-prim “ex-nihilo” would seem to prevent students from accepting that something can be made from nothing.  Prather et al find that students misapply the established p-prim of ex-nihilo to reject other possibilities when talking about the Big Bang.  They conclude that:
It appears that students do harbor pre-existing and often scientifically inaccurate ideas, or alternatively they may inappropriately activate phenomenological primitives on the spot to make sense of contemporary astronomy topics.  In either case, these recurrent patterns in student thinking need to be explicitly addressed for meaningful learning to take place.
According to Sherin (2006) p-prims, beyond serving as heuristic cues to formal knowledge as diSessa speculated, can drive problem solving in a fairly direct manner.  Southerland, Abrams, Cummins and Anselmo (2000), in their study which tests both the conceptual framework perspective and the p-prim perspective, identify the p-prim as a deeply rooted aspect of knowledge, an organizing core intuition.  Mislevy and Braun (2001) claim that the distinguishing feature of intuitive physics or indeed intuitive reasoning in any field is that the p-prims are the bottom line.  Suping (2003) concludes that though there are competing views of how conceptual change occurs, there seems to be no argument about whether conceptual change occurs; it is central to learning in science. 
Sadler (1996) has several useful comments to make.  Firstly he points out that recent advances in cognitive psychology show that learning is a process of constructing knowledge, not simply of transmission of knowledge from expert to novice.  He concludes that “Students hold preconceptions when they enter their courses and learning many astronomical concepts is dependent on mastery of easier concepts.”  Additionally, he finds that “Many of these ideas are well-developed misconceptions that have been constructed by students from their own experiences and thoughts.  These ideas are very resistant to change, but once changed are almost impossible to remember.”  And finally he says that:
Powerful ideas in science are often hierarchical in nature; they build upon one another.  Students who do not have the foundation stones firmly in place will find it very difficult to construct an understanding of astronomy.  It is important that teachers realize the difference between exposure to an idea and the mastery of it.
The main properties of p-prims were defined by diSessa (2002) as being: (i)  small and monolithic; (ii) many; (iii) work by recognition; (iv) feelings of naturalness;  (v) explanatorily primitive; (vi) fluid; (vii) a problematic connection to language; (viii) origins in minimal abstractions and (ix) development by reorganization.  diSessa further points out that many studies give no status at all to elements like p-prims, and claims that intrinsically difficult-to-change entities like concepts theories and ontologies provide no explanation whatsoever for commonplace cognitive phenomena.  He also introduces coordination classes, another knowledge type.  Coordination classes are large and complex systems which are probably made up of p-prims.  Coordination classes are intended to constitute a model of a certain type of scientific concept.  He says that p-prims play a role in both expert and naïve thought, whereas coordination classes may well not exist in naïve thinking.
Chi and Roscoe (2002) define conceptual change as the processes of removing misconceptions, which they claim are concepts categorized into an inappropriate category.  They also believe that this conceptual shift process itself is not inherently difficult, but is instead challenging mainly when students lack awareness of their misconceptions, and or lack the alternative categories to which they should reassign their misconceptions.
Posner, Strike, Hewson and Gertzog (1982) suggested that there are four requirements for successful conceptual change when the students’ current concepts are inadequate to allow him to grasp some new phenomenon successfully.  These are:


These four criteria should be taken into account when drawing up the list of parameters to be met by visual models in cosmology, whether the student has previous misconceptions or not.
Hewson (1992) highlights three different stages of conceptual change teaching: (i) diagnosis, (ii) status change and (iii) evidence of outcome.  Diagnosis is used to elicit students’ existing conceptions and reasons why they are held, status change consists in the use of strategies designed to help students lower the status of existing, problematic knowledge, and raise the status of other, competing ideas, and evidence of outcome requires that evidence be found if the students’ leaning outcomes are based, in part, on an explicit consideration of their prior knowledge.  He says that two characteristics of alternative (pre) conceptions are that they are often significantly different from and thus alternative to, generally accepted views of the subject, i.e. they conflict with ideas teachers want students to learn; and they are surprisingly resistant to change as a result of traditional instruction.  Learning involves changing a person’s conceptions in addition to adding new knowledge to what is already there, and involves an interaction between new and existing conceptions with the outcome being dependent on the nature of the interaction.  Hewson also confirms the Posner et al (1982) postulates above, saying that learners use their existing knowledge to determine whether different conditions are met; that is whether a new conception is intelligible, plausible and fruitful.  If the new conception is all three, learning proceeds without difficulty.
Cosmology, and especially expansion, with its need for a background in general relativity for formal comprehension of the mathematical concepts, must be considered one of the most difficult target domains there is in science today, and so is therefore a prime candidate for the use of visual models to aid the student.
Glynn (2007) points out that as students develop cognitively and learn more science, they will evolve beyond these simple analogies, adopting more sophisticated and powerful mental models.


2.3   CONCLUSION   Visual models should be accepted as a valid and necessary part of scientific teaching in order to enable a student to gradually increase his understanding of new concepts.  As such they should be carefully crafted and examined minutely to determine their efficacy in transmitting correctly the target domain.  There is much room for further study in this area.
Intelligibility, plausibility and fruitfulness will determine the success of conceptual change.  In order to be effective source concepts should be simple.  Models should consist of a source domain which is immediately clear to the student, and with which he is familiar.  The target domains of models should be formed according to the current paradigms in science.  The connections between the two should be intuitive for the student. 
The studies cited above indicate that students recur to already existing knowledge in order to understand new concepts.  Such already existing knowledge may initially begin as simple p-prims, core concepts which the student has derived himself from observation and experience.  In order for a model to be “intuitively” understandable, it must be a simple bridging experience from such p-prims.  A student will then find little difficulty in extending his already defined framework to encompass the new concepts.  If, however the visual model is constructed on overly complex structures, the student will detect a difficulty, and will reject this new concept as anomalous to his already existing data base.  Ausubel said, over thirty years ago, “The unlearning of preconceptions might very well prove to be the most determinative single factor in the acquisition and retention of subject-matter knowledge.” (as cited by Sadler,1996)
Analogies consist of three parts:  Analog concept, connection and target concept.  However, the target concept should be embedded in hierarchical concepts so that the student may extend the model to successfully incorporate more advanced ideas.  Ideally, hierarchical concepts should be taken into account when developing parameters for analogical and similarity models.  If the target concept is not predisposed to incorporate hierarchical learning the student may later classify more advanced concepts as implausible and reject them.  For visual models in general, the structure should be that of figure 4.

Figure 4

figure 4

We can now see that Figure 1, of the earth’s orbit, is not only misleading because of perspective; it is badly designed.  The model only addresses the hierarchical concept of Keplerian orbits rather than the target concepts of indirect light when eccentricity approaches zero, as in the solar system.
Badly-constructed models can prevent learning and introduce misconceptions which can survive more formal teaching, leading to confusion and poor results for the student.  A model can be evaluated by examining its source, target and hierarchical concepts and connectivity.  To date, such analysis has not been formally applied to models of expansion in cosmology.  Visual models of expansion have been used for almost a century with little or no evaluation of their effectiveness as a teaching tool. 
Well-constructed visual models are essential when the target domain requires mathematical knowledge beyond the scope of the student, as he has no other means of acquiring the basis on which to build hierarchical concepts.  Source and target domains and the connectivity between them should be clear, and the student should be aware of where the visual model breaks down




“The greatest problem of communication is the illusion that it has been achieved”.  George Bernard Shaw.


3.1   In this study Zirbel’s definition of misconception: “concepts not in agreement with our current understanding of natural science” (Zirbel 2004), will be used, which indicates that any study of misconceptions in cosmology will require previous review of the current paradigms, and also that misconceptions, like paradigms, evolve over time.
The fact that misconceptions evolve may be seen anecdotically by the cosmological constant, which Einstein found himself forced to introduce.  He was laboring under the misconception that the universe was static.  This was not a misconception at the time, since the understanding of natural science before 1920 was that of a static universe.  He felt obliged to adapt his field equations, which had indicated expansion as a natural outcome.   When understanding of natural science developed and observational evidence made expansion of the universe the paradigm, Einstein rapidly withdrew his cosmological constant, with the famous “biggest mistake” comment.  Current understanding, however, is that the universe is accelerating in its expansion due to dark energy, which brings back the need for a term similar to the cosmological constant.  The cosmological constant has progressed from paradigm to misconception and back to paradigm.  This type of flux is typical of the science of cosmology, as it is a science with limited experimental and even observational possibilities, and one of the few disciplines in which the word “untestable” is not necessarily a criticism.  As seen, even the greatest cosmologists may find that today’s paradigm will be tomorrow’s misconception, and possibly even vice versa.  It may merely be a question of time.
In cosmology there are various levels of misconceptions: some start when the student is very young, and hears the expression “Big Bang” for the first time.  Others are introduced in high school when the balloon analogy is used, and still more become ingrained in higher education when the student is asked to master more advanced ideas.  Misconceptions would seem to come in layers, rather like those of an onion. 
There is confirmation of this opinion in the paper by Sadler (1996).  Sadler speaks about the hierarchy of concepts, explaining that ideas in science are often hierarchical in nature.  Students who do not have the foundation stones in place find it very hard to understand new concepts.  These ideas are very resistant to change, and although studies have shown that misconceptions can be changed, conventional courses do little in this aspect.  Sadler explains that:
There is another approach that should be considered, i.e.  choosing concepts on the basis of their structural relationship to learning other concepts.  In this way, concepts that are required for understanding more difficult ideas should be taught first.  When mastery of these ideas is achieved, one can move to harder topics that depend on these prerequisite ideas.
Sadler also finds that courses which cover too much content appear to leave many students with reinforced misconceptions and a decreased ability to answer many astronomical questions.
Some of the principal problems in the field of cosmology are detailed in an article in Scientific American: “Misconceptions about the Big Bang” (Lineweaver and Davis 2005), which was based on a previous paper (Davis and Lineweaver 2003).  In the article the authors talk about various misconceptions on the subject of expansion.  Their article highlights the following six areas of difficulty, where the answers they give have been paraphrased and abbreviated:
1.  What is expansion, anyway?   They explain that Galaxies are not fragments of a Big Bang bomb; rather the space between us and the galaxies is expanding. 
2.  Ubiquitous cosmic traffic jam.  They use this title to describe the Big Bang, and explain that there was no one particular location for the Big Bang, rather it happened everywhere.
3.  Receding faster than light.  Davis and Lineweaver explain that Hubble’s Law predicts that galaxies beyond a certain distance, known as the Hubble distance, recede faster than the speed of light.  People may think that this contradicts Einstein’s special theory of relativity which says that nothing can have a velocity exceeding that of light.  However, Einstein’s theory applies to “normal” velocities – motion through space.  The Hubble flow is not a motion through space; it is a motion of space itself.
4.  Stretching and cooling.  As space expands, light becomes stretched.  As photons travel through expanding space, they lose energy and their temperature decreases.  In this way, the universe cools as it expands.
5.  Running to stay still.  A distant light beam is moving toward us at the speed of light with respect to its local space, but its local space is receding from us faster than the speed of light.  Although the light beam is traveling toward us at the maximum speed possible, it cannot keep up with the stretching of space.  However, since the Hubble constant changes with time, the photons can then find themselves in a region of space that is receding slower than the speed of light.  Thereafter they can approach us.
6.  Is Brooklyn expanding?  No, it isn’t.  Expansion by itself produces no force.  Photon wavelengths expand with the universe because, unlike atoms and cities, photons are not coherent objects whose size has been set by a compromise among forces.  Recently it has been determined that the universe is accelerating its rate of expansion.  If acceleration (of expansion) is constant the planet Earth simply settles into a static equilibrium size slightly larger than the size it would have attained with no acceleration.  However, if acceleration is not constant, it could eventually grow strong enough to tear apart all structures, leading to a “big rip”.  But this rip would occur not because of expansion or acceleration per se but rather because of an accelerating acceleration. 
Zackrisson (2009), in his “Seminar 1: Common Misconceptions about modern Cosmology” identifies five main areas of confusion.  He gives these as the Big Bang, cosmic curvature, expansion of the universe, size of the universe, and distances in cosmology. 
Berman in “Misconceptions on Relativity, Gravitation and Cosmology” (2008) looked through elementary physics textbooks, and found many inclusions of misconceptions.  Amongst these were: “the universe has a centre”, “dark matter and dark energy are the same thing, because of Einstein’s mass-energy relation” and “the universe expands, so it accelerates”.
Davis and Lineweaver (2003) cite 25 examples of confusing comments about expansion from scientific works written by even renowned scientists such as Feynman, Lovell and even Hubble himself.
Peacock also comments on common misconceptions in cosmology in his lecture notes at Caltech (1999).  Here he typifies such misconceptions into four groups: the initial singularity, the origin of the redshift, the nature of the expansion, and the empty universe.
There are also many diametrically opposed statements about the subject in textbooks and popular science literature.  For example, in 1993 Weinberg states: “… space does not expand.  Cosmologists sometimes talk about expanding space, but they should know better.”  And Rees agrees: “Expanding space is a very unhelpful concept.  Think of the universe in a Newtonian way – that is simply in terms of galaxies exploding away from each other” (Chown, New Scientist, 1993).  Yet Carrol and Ostlie in their textbook “An introduction to modern Astrophysics” (2007) say:
… a galaxy’s recessional velocity is not due to its motion through space; instead, the galaxy is being carried along with the surrounding space as the universe expands.   …  In the same manner, a galaxy’s cosmological redshift is produced by the expansion as the wavelength of the light emitted by the galaxy is stretched along with the space through which the light travels.
Indeed, true experts on the subject are not immune to their own mathematical misconceptions.  Harrison (1993) criticizes two of his colleagues for falling into errors of this type.  He says that Sandage remarks that Hubble’s redshift-distance law is valid for all distances, when he should say that the velocity-distance law is valid for all distances.  Harrison goes on to say that Peebles et al tell us that the velocity-distance law requires relativistic correction when the velocity approaches the velocity of light, when this remark is true for the redshift-distance law, but not the velocity-distance law.
Lastly, in their comments about the big bang, NASA in its “Universe 101” page (2010) tells us that in order to avoid misconceptions we should keep in mind the following points:  a) The Big Bang did not occur at a single point in space as an “Explosion” b) It is not logically necessary or sensible to consider the question of what the universe is expanding into.  c) It is beyond the realm of the Big Bang Model to say what gave rise to the Big Bang. 
If we examine the above studies, the one constant factor in all of the papers is expansion.  It forms the basis of the current cosmological paradigm, and is one of the most difficult concepts to explain. 


3.2   The basic misconceptions of the general public would seem to arise from the following areas:

The third problem is a thorny issue, but one which, at this moment, has no particular relevance as a misconception.  The definition of a misconception given above requires that the subject be in disagreement with current understanding of natural science.  Most cosmologists will simply derive the universe backwards to the Planck time and state that nothing can be said about anything previous to this.  Others, such as Penrose (2010) have put forward theories of cyclic universes, where a previous universe existed before the Big Bang.  Current multiverse theory would also indicate that the ex-nihilo premise is unnecessary in modern cosmology. 
The three main misconceptions of the general public are therefore all concerned with global expansion.  The first misconception is that the universe expands from one privileged point, the second that the Big Bang was an explosion, and the third that it is expanding into something else.  All of these points may be incorporated easily into the principal misconceptions for advanced students, enunciated below.


3.3   The principal areas which are leading to misconceptions on the part of more advanced students would seem to be:

The question of superluminal galaxies is a misconception which does not need any visual model to solve.  It is simply a question of definition.  Einstein’s theory of relativity does not allow travel at greater than the speed of light.  However, the intrinsic expansion of space cannot be defined as travel.  In essence this is the solution to the superluminal question.  It is still true that nothing could overtake a light beam.  A verbal analogy could be made by alluding to the fact that each of us is currently travelling at over two million kilometers per hour relative to the CMB, but a traffic policeman can still only fine us if we break the 40 kilometer per hour limit in towns. 


3.4   CONCLUSION   Misconceptions in cosmology arise mainly from the concept of expansion.  On one level students are confused about where the universe expands from, where it expands to, and why it expands at all.  On another level students are uncertain about how it expands, how much of it expands and why we observe a redshift.
The main areas of misconceptions to be treated in this paper are therefore global expansion and the related concepts of redshift and whether or not things expand locally.  According to Zirbel’s definition of misconception cited above, current understanding of these concepts must be examined in order to determine parameters necessary in any visual model which claims to be of aid to the student in his understanding of these topics. 
A paradigm is based on majority opinion in any one scientific area, so we should not expect to find absolute agreement in any bibliographic study.  Cosmology has significantly less than absolute agreement on the subject of expansion, but many of the difficulties are due to different mathematical models used in the calculations, and the different status of the considered observers.