Rearranging equations to develop physics reasoning

Researchers generally agree that physics experts use mathematics in a way that blends mathematical knowledge with physics intuition. However, the use of mathematics in physics education has traditionally tended to focus more on the computational aspect (manipulating mathematical operations to get numerical solutions) to the detriment of building conceptual understanding and physics intuition. Several solutions to this problem have been suggested; some authors have suggested building conceptual understanding before mathematics is introduced, while others have argued for the inseparability of the two, claiming instead that mathematics and conceptual physics need to be taught simultaneously. Although there is a body of work looking into how students employ mathematical reasoning when working with equations, the specifics of how physics experts use mathematics blended with physics intuition remain relatively underexplored. In this paper, we describe some components of this blending, by analyzing how physicists perform the rearrangement of a specific equation in cosmology. Our data consist of five consecutive forms of rearrangement of the equation, as observed in three separate higher education cosmology courses. This rearrangement was analyzed from a conceptual reasoning perspective using Sherin’s framework of symbolic forms. Our analysis clearly demonstrates how the number of potential symbolic forms associated with each subsequent rearrangement of the equation decreases as we move from line to line. Drawing on this result, we suggest an underlying mechanism for how physicists reason with equations. This mechanism seems to consist of three components: narrowing down meaning potential, moving aspects between the background and the foreground and purposefully transforming the equation according to the discipline’s questions of interest. In the discussion section we highlight the potential that our work has for generalizability and how being aware of the components of this underlying mechanism can potentially affect physics teachers’ practice when using mathematics in the physics classroom.


Introduction
Physics, as a discipline, uses a wide range of representations in order to construct and communicate its knowledge.In an undergraduate physics textbook, for example, a student will come across a wide range of representational formats such as diagrams, sketches, images, written language and mathematical expressions.In this respect, the use of mathematics has been extensively researched and discussed within the Physics Education Research (PER) literature [1].
How exactly is mathematics used in physics and what does this mean for physics education?
In the PER community the consensus seems to be the following: in order for someone to become a physics expert, they must learn to blend mathematical knowledge with physics intuition.However, the teaching of the use of mathematics in physics has traditionally tended to focus almost exclusively on problem solving-teaching students how to work mathematically with an equation in order to obtain a numerical result.It is generally agreed that focusing on mathematical operations in this way can lead to a lack of development of the conceptual understanding and physics intuition, which is necessary to become a physics expert [2,3].
A number of authors have explored the various ways physics students reason with equations, in an attempt to better understand their thought processes-see section 2 for details.However, to our knowledge there is not an equivalent body of empirical work regarding how physics experts themselves use mathematics for physical reasoning.In the work that has been done, researchers seem to agree on the general idea that physicists know how to organize knowledge more effectively than novices [4] and that they blend physical intuition with mathematical knowledge [5][6][7].These arguments, while logical, are either axiomatic in nature, or, as pointed out by Tuminaro and Redish [4] focus on algorithmic analysis aimed at problem solving that does not provide insight into how a physicist thinks and functions.In particular, there is a lack of work that deals with the reasoning of physicists when they manipulate and rearrange equations in order to better understand the physics of a situation.
In this paper, we outline an underlying mechanism that we believe characterizes physicists' manipulation of equations in order to facilitate physics reasoning.We suggest that for physics faculty, an understanding of this mechanism has the potential to have important implications for the teaching and learning of physics.

Physics reasoning or calculating?
The role of mathematics in physics has been widely discussed in the literature.Historically, the centrality of mathematics in physics has been stressed, by a number of well-known figures within the physics community.Feynman [8] argued that it is impossible to construct meaningful physical laws without a deep mathematical understanding.Einstein talked about nature being the realization of the most basic mathematical thoughts and Hertz referred to equations as a tool where physicists get more out than they put in (see [9]).Mathematics has also been described as the language of physics-an intrinsic part of every physicist's discourse [3].
This centrality of mathematics in physics is something that subsequently affects how mathematics is used in physics education.A recurring theme in the literature is the dichotomy between mathematics as a technical, computational tool and as a conceptual, physical reasoning tool.Several authors claim either that students struggle when it comes to blending physics conceptual understanding with mathematical skills [10] or that it is particularly challenging for physics educators themselves to come up with problem solving scenarios that emphasize physics meaning making rather than computational problem solving [4,5].Various studies have also highlighted this problem, for example, Johansson et al describe how students were specifically instructed to calculate quantum physics problems without attempting to understand what was going on conceptually [11].In a study by Airey and Linder [12] which used interviews to probe students' conceptual understanding of the mathematical formalism in their coursework, students were able to calculate the curl of an electric field, but did not appreciate its meaning.These types of problems are what led to the introduction of Conceptual Physics [2].The idea behind Conceptual Physics was that because the use of mathematics in physics education can be seen as a barrier to conceptual understanding, there is a need to teach physics in a way where all problem solving starts with conceptual understanding-mathematics should only be introduced once a problem has been sufficiently modelled and understood [2,7].
On the other hand, others have argued that separating conceptual understanding and the use of mathematics in this way is not always appropriate.Various authors have tried to bridge the mathematical/conceptual gap by examining how mathematics itself can be used for conceptual reasoning.Some of the proponents of this work have suggested a set of scientific thinking skills (epistemic games) that students engage with when they use mathematics in physics problem solving [4,5,10].Taking a different tack, other authors have explored what students mean when they say they understand a physics equation, from an epistemological perspective [13,14].Expanding on these ideas, Uhden et al [9] claim that it is not appropriate to separate physical (conceptual) and mathematical (quantitative) models in physics education.They propose their own model of physical-mathematical reasoning where the two coexist.Redish has also proposed his own model for the use of mathematics in science [15], focusing on four main steps: mapping physical quantities to mathematical expressions, processing the mathematics, interpreting the results into the physical world and then evaluating whether the result is adequate.Bing and Redish's model describes how students blend physics and mathematics by combining elements of a 'mathematical machinery' and a 'physical world' input space [16].Schermerhorn has also suggested a model of how students construct and interpret equations by blending symbolic, mathematical processing and physics concepts elements [17] and Van den Eynde et al constructed their own 'Dynamic Blending Diagram' to provide insight on how students move between the physical, mathematical and blended spaces when they interpret an equation to describe a physical system [18].Drawing from the mathematical modeling cycle (MMC), Ye et al [19] developed an extended MMC model to describe the range of cognitive and metacognitive strategies that students engage in, when dealing with problems in chemical kinetics.Van den Eynde et al [20] explored how reasoning through graphs can function as a catalyst to facilitate student's mathematical and physical blending in the context of partial derivatives in thermodynamics.
To summarize, a study of the current literature about the role of mathematics in physics yields two main ideas: either that conceptual understanding needs to be initially separated from mathematics (with the goal that conceptual understanding should come before the mathematical), or that the two are inseparable and they therefore need to be taught simultaneously.

Rearranging equations
One notable aspect of studies of mathematics in physics, is that they rarely examine the specific roles of different mathematical processes, instead preferring to refer to the use of mathematics in physics in more general terms.Most researchers agree that an ability to rearrange equations is highly valuable for the computational aspect of mathematics in physics.However, according to Tuminaro and Redish students of physics often resort to 'plug and chug' [4].Here, 'plug' refers to inserting numerical values into an equation, and the 'chug' refers to executing mathematical operations in order to arrive at a numerical solution.In mathematics education research similar problems have been noted by Duval, who identified two aspects of using mathematics: conversion and treatment [21].Duval's conversion refers to transferring meaning from the real world to mathematics.He claims that this is the much more difficult and cognitively challenging function for students.In Duval's view, treatmentthe line-to-line transformation of an equation-while important from a mathematical point of view (solving equations) plays little or no role in students' conceptual understanding.
In a few rare cases in the literature, however, we can find examples where students suggest that rearranging equations is not only a tool for calculating, but also a marker of conceptual understanding.Airey et al found that many students believe they understand an equation when they can rearrange it and use it in different formats [13].An even more in-depth treatment can be found in Eichenlaub and Redish, in a student's answer about using the equation of nuclear decay of a sample of particles over time [5].The authors argue that this student performed a rearrangement of the 'formal' equation, not in procedural terms, but by blending physical intuition and mathematical reasoning to arrive at a logical solution.
For our part, we are interested in the underlying mechanisms at work when it comes to using mathematics for physical reasoning.We chose to 'dig deeper' into the process of rearranging physics equations, analyzing our data from the perspective of physics reasoning.

Symbolic forms
In order to carry out our analysis, we chose to use a specific theoretical framework-Sherin's symbolic forms [22].This framework describes how physics students-and we believe experts-reason with equations in physics.Introduced by Bruce Sherin as a 'vocabulary of elements' in terms of which 'students learn to understand physics equations', a symbolic form consists of a conceptual schema, and a symbol template-an arrangement of mathematical symbols that serves as a knowledge element to which the conceptual meaning is assigned [22].Following these ideas, Sherin analyzed several groups of undergraduate physics students engaging with physics equations in problem solving sessions and created a semi-exhaustive list of symbolic forms that students draw upon when they make meaning with physics equations.A copy of this list can be found in table 1.Since its publication, this framework has been expanded to other areas of physics in order to examine student meaning making with specific mathematical expressions [23][24][25][26][27].Moreover, Schermerhorn and Thompson [28] combined the symbolic forms and conceptual blending frameworks to suggest a model that describes how students in upper-secondary physics blend mathematical knowledge with contextual physics knowledge when constructing a physics equation.

Aim and research questions
Drawing on the ideas described in the introduction, our aim for this paper is to examine one aspect of how physicists use mathematics for physical reasoning.We aim to do this by examining the line-by-line rearrangement of an equation in Cosmology, which, we believe displays conceptual understanding and physical reasoning functions.These rearrangements have been performed by physicists who are experts in Cosmology.We believe these rearrangements to be of high value to the discipline of Cosmology and an integral part of a novice astronomer's education.
Specifically, we will look at rearranging the Friedmann equation in Cosmology.We aim to demonstrate how this line-by-line rearrangement displays an underlying meaning-making mechanism that physicists appear to 'follow' in such situations.Glimpses of similar ideas can be found in Eichenlaub and Redish [5].In order to analyze the equation and its rearranged forms in terms of its physical meaning, we use Sherin's framework of symbolic forms.Our research questions are formulated as follows: Table 1.Sherin's semi-exhaustive list of symbolic forms [22].

Cluster
Symbolic form Symbol pattern

Competing terms
Competing terms RQ 1: What aspects of physics reasoning can be identified when we use Sherin's framework of symbolic forms to analyze the line-by-line rearrangement of the Friedmann equation?
RQ 2: What does this analysis suggest about the way that physicists rearrange equations in order to reason about physics?
We also aim to showcase some important implications that we believe our work can have in physics teaching and learning.Initially, these lectures were observed as part of a larger project involving a semiotic audit of representations used in Higher Education Astronomy [29].During the data collection, in all three courses we observed the lecturers performing different variations of the rearrangement of the Friedmann equation for the dynamics of the Universe.Subsequently, this rearrangement was given as a task for students to perform as well.Despite variations, which could be attributed to the individual differences of the three lecturers and the different levels and demands of each course, we suggest that the directionality and the underlying logic behind the rearrangement was common: to change the initial equation into a simpler form and then use it to reason about the geometry and contents of the Universe.As our data we chose the following forms of the equation, combining data from all three courses, that we believe showcase the main aspects of different stages of rearrangement adopted by the three lecturers.

Data and methodology
• Line 1: initial form This is the initial form of the Friedmann equation, as it was presented to students.The ( ) a t symbol is the scale factor; a dimensionless parameter that indicates the magnitude of the expansion of the Universe.The ( ) e t symbol is the energy density and indicates the density of the different contents of the Universe (matter, radiation, and the cosmological constant).The G, R 0 and c symbols are constants (gravitational constant, curvature radius and the speed of light) and k is a number that takes the values of − + 1, 0, 1 and is indicative of the three potential curvatures of the Universe: closed, flat, open (k = -+ 1, 0, 1).
From the first to the second form, the only change that occurs is the introduction of the Hubble parameter: The Hubble Parameter indicates the rate of expansion of the Universe and was first introduced by the astronomer Edwin Hubble following his famous discovery of the relationship between radial velocity and distance in 1929.Estimating the current value of the Hubble Parameter (H 0 ) is an open problem in Cosmology.
• Line 3: third form The introduction of new terms and a few extra steps are required in order to obtain this third form.First of all, lecturers introduced the concept of critical density-the energy density required for the Universe to be flat: Using this notion of critical density, cosmologists introduce the density parameter Ω as the ratio of the energy density of the Universe at any given time, over the critical density: Ω is a dimensionless parameter that indicates how 'far' the Universe is from being flat (k = 0).Now, dividing the second line of the equation by the critical density, followed by multiplying the resultant terms by p c G 3 8 2 and moving the energy density term to the lefthand side, we arrive at the third form.This form was used by lecturers to reason about the curvature of the Universe, in relation to Ω. From this form of the equation, it is now easy to compare the left and right side of the equation and make the following statements: For the fourth form of the equation, we have the following changes: firstly, the density parameter term is broken down into the sum of all the different energy densities that every component of the Universe (matter, radiation, cosmological constant) contributes to the overall energy density.Therefore, 'behind' the summation symbol in this form, we practically have Moreover, the curvature of the Universe is now associated with a density parameter: Line 5: To arrive at this fifth form of the Friedmann equation a simple rearrangement is needed: the curvature density term is moved to the right-hand side.Despite its obvious nature, compared to the previous form, this fifth form was also shown to students in all three courses.A summary of all the steps of the rearrangement can be found in table 2 below.

Methodology
The methodology we followed for our analysis falls within the scope of abductive reasoning [30].In abductive reasoning, the data are collected in an exploratory way and subsequently fitted into an already existing framework.For us, the data consisted of the rearranged forms of the Friedmann equation which we attempted to fit to Sherin's framework of symbolic forms.
It is important to note here, that in a diversion from standard abductive reasoning methodology, we did not aim to expand the already existing framework, but rather applied it to our data in order to search for new, emerging theoretical claims.For this reason, we did not follow the subsequent steps of abductive reasoning methodology (collection of new data, discrimination of plausible explanations, checking for consistency).The analysis was initially carried out by the first author and reformulated after discussions with the second author.
Regarding our data analysis, we employed the following approach: we took each line of the rearrangement and tried to identify every possible symbolic form that could be assigned to it.
This was done on the basis of the different symbol templates that could be associated with each form of the equation.Since each symbolic form can potentially be assigned to a different conceptual schema, this means that the potential conceptual meanings of each line of the transformation can be 'mapped out'.We argue that when attempting to engage with this equation, students could potentially try to reason with any one of these schema.
For illustration purposes, we will now demonstrate how we applied our methodology to one specific line of the equation.For this purpose, we have chosen the fourth rearrangement of the equation: According to the symbolic forms framework, as presented in table 1, this equation could be associated with two different symbolic forms: competing terms and parts of a whole.This means that students have two possibilities to choose between when attempting to make meaning with this equation.If students apply the 'competing terms' conceptual schema, then they make sense of the different density parameters as competing influences-if one increases, the other decreases.If, however, they apply the 'parts of a whole' symbolic form, Table 2. Line to line rearrangement of the Friedmann equation, as occurred in our data collection.

Lines
Equation forms New terms 1 ( ) then they understand the left side of the equation as a whole that consists of different 'ingredients'-the different density parameters.In this way, the symbolic forms framework helped us analyze the rearranged forms in terms of the potential for physics reasoning.A more in-depth description of this process follows in the next section, where we present our data analysis.In this section we present the first part of our analysis, which corresponds to our first research question: What aspects of physics reasoning can be identified when we use Sherin's framework of symbolic forms to analyze the line-by-line rearrangement of the Friedmann equation?As described in the Methodology section, we highlight how different symbolic forms can be assigned to different rearrangements of the equation.In order to do this, we present all the potential symbolic forms that we identified in our analysis, along with their conceptual schemas and symbol templates as introduced in Sherin's work (see the appendix).

Data analysis
It is important to note here, that for our analysis we combined symbolic forms from different clusters, as proposed by Sherin [22].Regarding symbolic forms from the competing terms cluster, Sherin mentions that they are often activated with concepts such as forces, momentum or acceleration, while the signs between terms are more likely to be associated with directions on a diagram.For our part, we believe that the different concepts appearing in the Friedmann equation (e.g.matter, radiation, curvature density parameters) can also be conceptualized as different influences working against each other, despite the absence of any diagrammatic representations.Hence, we also incorporated symbolic forms from the competing terms cluster into our analysis.
• Line 1: Due to the presence of several symbols and different terms in this equation, there are a large number of ways one can try to make physics sense with this equation.In particular, it is interesting to point out that a lot of the symbols act as coefficients, multiplying other terms and that there are also a lot of terms that could potentially be recognized as dependent on each other.Hence, we have marked out that the dependence and coefficient symbolic forms can be used in multiple ways on this first line.The Ratio symbolic form also works in the same way as we can associate many symbols in the ratio symbol template (see table 1 for reference).
In the second rearrangement of the equation, we can identify the emergence of a new symbolic form, identity.This occurs because the left side of the equation has been transformed to just one symbol.Moreover, the ratio on the left side has 'disappeared', and we thus have fewer ratio symbol templates in the equation.Since nothing else changed from line 1, the other symbolic forms remain the same.
The appearance of a new term under the 'Ω' symbol has removed several symbols from the equation, therefore lowering the number of possible dependence and coefficient symbolic forms that could be activated.The appearance of unity as a term in the equation has taken away the possibility of the left side of the equation being conceptualized as 'opposition' or 'competing terms', since these require two competing influences that oppose each other (see [22], p. 532).
• Line 4: Potential Symbolic forms: Parts of a Whole, Competing Terms.After the introduction of multiple 'Ω' terms, we now only have two possible ways of interpreting this form of the equation.As explained in the example given in our methodology section, this can be either the parts of a whole or the competing terms symbolic form.
With the simple movement of the W k symbol to the right-hand side of the equation, two new symbolic forms emerge; a single symbol is left on the left side of the equation, thus reactivating the identity symbolic form.The equation can also be interpreted using the dependence symbolic form (the left-hand side depends on what is on the right).
As a conclusion regarding our first research question, our analysis clearly shows that as we move from line to line in this rearrangement, the number of symbolic forms associated with each line decreases.Initially, moving from Line 1 to Line 2, does not affect the number of symbolic forms significantly, but as the rearrangement progresses, the number decreases to just two symbolic forms.Therefore, as an answer to our first research question, about which aspects of physics reasoning that can be identified when we use Sherin's framework of symbolic forms to analyze the line-by-line rearrangement of the Friedmann equation, we can see that there is a clear reduction in the number of symbolic forms.This result is summarized in table 3.

Research question 2: how do physicists rearrange equations to reason about physics?
With our second research question, we wanted to dig deeper into the underlying mechanism that drives the procedure we described above.In our answer to research question 1, we stated that when analyzing this rearrangement in terms of symbolic forms, we identify a clear reduction in the number of forms as we move from line to line.With research question 2, we are interested in the consequences of this reduction in symbolic forms in terms of meaning making.Put simply, what can we say about the way physicists rearrange equations in order to use mathematics for physics reasoning?
We identify three components that describe this underlying mechanism: (i) the narrowing down of meaning potential, (ii) the movement between foreground and background, (iii) the purposefulness of the procedure.In this section, we discuss each of these components.

5.2.1.
The narrowing down of meaning potential.In order to illustrate our first finding about the underlying mechanism at work when physicists' rearrange equations, we draw on the work of Airey and Linder.In their social semiotic description of the teaching and learning of undergraduate physics, they point out that disciplinary-specific semiotic resources (or representations, as they are more commonly called in PER), do not have a single, predetermined meaning, but rather possess a range of meaning potentials [12].This means that the representations presented in physics lectures have the potential to mean a range of different things to physics students depending on a number of factors, such as the task at hand, the students' prior knowledge of the discipline, the teacher's skill in dealing with any expected misconceptions, etc.Moreover, as pointed out by Podolefsky and Finkelstein, most representations have a number of non-disciplinary 'surface features' that students may incorrectly focus on.Thus, the meaning potential that is actually activated in an individual student might be anything from a range of disciplinary and non-disciplinary meanings [31].From a disciplinary point of view, the equation that we examined illustrates Airey and Linder's [12] range of possible meanings that are available through the number of possible symbolic forms that could potentially be assigned to each line of the rearranged equation.As an example, we will again refer to Line 4 of the rearrangement, and the two symbolic forms that can potentially be assigned to it: competing terms and parts of a whole.Here we see that this form of the equation can be physically interpreted in two different ways; one where the different parameters are thought of as competing influences that 'try to balance or cancel each other out' (competing terms) or the Universe as a whole, consists of different 'ingredients' (parts of a whole).
What does this suggest, then, about the reduction in the number of symbolic forms from line to line, that we demonstrated in RQ1?We believe this reduction to represent a narrowing down of the meaning potential.If each symbolic form represents a different aspect of the disciplinary meaning potential of the equation, then, by starting with a larger number of symbolic forms in the first line and reducing them to two in the final rearrangement, what the physicists have effectively done is narrow down the meaning potential (figure 1).This makes the equation clearer and therefore easier to reason with in the 'correct' manner.Therefore, we believe one of the features of the underlying mechanism that drives physicists' rearrangement of equations is the narrowing down of meaning potential.

Foreground and background.
The second component driving this rearrangement is closely connected with the new terms that are introduced in almost every line of the equation.From Line 1 to Line 2, Hubble's parameter ( ) H t is introduced by grouping up two other terms of the equation ( ) From Line 2 to Line 4, we have the introduction of the different energy density (Ω) parameters that occur after grouping up several constant and time dependent terms (see table 2 for reference).It is essentially this grouping up of terms and the introduction of new ones that results into the final two forms of the equation, with reduced meaning potential.
What is interesting here is that during this whole rearranging procedure, some aspects of physical concepts that are present in the first form of the equation (speed of light 'c', gravitational constant 'G', the scale factor ' ( ) a t ', the Hubble Parameter ' ( ) H t ', etc), have completely disappeared by the time we reach the last two forms of the equation.Here, we essentially only see the Ω parameters, which were not even present in the first equation.At first glance, one would say that the first and final forms are completely different equations.One could argue that the physicists who performed this rearrangement chose which aspects of the equation they wanted to remove and which aspects they wanted to highlight in the final form of the equation.
Essentially, what we identify here is that this creation of new terms by grouping up other terms is carried out in order to move certain aspects to the background whilst bringing other aspects into the foreground of the equation.This can be thought about like taking several terms of the equation and putting them in a sealed box with a new symbol written on the outside: Ω.Now, when someone (a student or even another physicist) looks at the equation, all they can see is the box labeled Ω, not what is on the inside.Of course, all the grouped up terms are still there, inside the box (in the background) but what is noticeable by someone engaging with the equation is now only the box itself labeled Ω (in the foreground).
What do physicists gain by hiding some terms in the background whilst introducing new terms into the foreground?It is evident that the density parameter, as a concept, indicates a very specific physical meaning (in this case, each density parameter gives us an estimation of the values of different physical aspects of the Universe: amount of matter, radiation, the cosmological constant, etc).Therefore, by hiding all the other terms and creating an equation with just these useful parameters, it now becomes easier to reason with the equation in order to make specific physics arguments regarding the relationship between the distribution of the contents of the Universe in terms of matter, radiation, cosmological constant and curvature.This results in the simple mathematical reasoning: 5.2.3.Purposefulness.So far, we have discussed two components that we claim demonstrate how rearrangement of the equation helps change it into a form that is easier to reason with.The third component we identified is different than the first two, in the sense that it gives us information about the directionality of this rearrangement.The series of operations carried out on the equation are far from random, rather they have a certain direction, that is they are purposeful.
This purposefulness is also drawn from the process of grouping up terms that are present in the equation and the introduction of new terms.In the second component, where we described the interplay between foreground and background, we discussed how this process of grouping terms brings certain aspects of the physical relationships into the foreground making them much more noticeable.The question is why certain aspects are chosen to be brought to the foreground instead of others.Put simply, if physicists can potentially make noticeable any aspect of the equation they want, then why do they choose only very specific aspects in this case, represented by the symbol Ω?
Mathematically speaking, a great many other rearrangements could potentially be performed, as long as algebraic rules are not broken.However, despite the possibilities, we see that this specific rearrangement, with these newly introduced terms, is one that is preferred by the physics community.The Ω parameters are widely used by physicists to this day, in order to compare different cosmological models (see for example, the graph presented by Suzuki et al [32]).We believe that an analysis of our data shows that, in general, this rearrangement follows a direction that is dictated by the discipline's needs, questions of interest and overall agenda.
In this case, the newly introduced Ω terms have a number of characteristics that make them valuable for cosmologists.First of all, they are non-dimensional, something that physicists value in many cases, when working with cross-scale problems, since they are scale invariant [33].Furthermore, the energy density parameters can be calculated using a specific procedure from other observables.Calculating the Ω parameters from different sets of data from various astrophysical sources is an important area of astrophysics and cosmology.
Perhaps the most important characteristic that we can identify with this grouping of terms and the rearrangement procedure, is that the final form of the equation seems to clearly answer a number of specific physical questions.For example, one of the main questions in cosmology pertains to the curvature of the Universe.The distribution between the different 'ingredients' of the Universe (matter, radiation, cosmological constant) is also of great importance since it gives us a lot of information about the history and the evolution of the Universe.The last two forms of the equation manage not only to provide insights into how to successfully answer those questions, but also put them in relation to each other, exactly in line with what the discipline of cosmology aims to achieve.More importantly, for our purposes this is a task that the initial form of the equation would be quite unsuccessful at doing.
To summarize, we have argued that this rearrangement is directed towards the needs of the discipline of cosmology, whether that be to create dimensionless parameters in order to deal with cross-scale problems, or to create a parameter that can be calculated from observables; both of these aspects are important to successfully answer questions that are of high interest in the discipline's agenda.Hence, this is why we claim that this rearrangement is far from random, rather it is purposeful, and directed by the discipline's specific needs and questions of interest.
This point is easy to illustrate with a counterargument.What would happen if we performed a different rearrangement of the equation, without having in mind the characteristics described above?Here is an example of such a purposeless rearrangement that mathematically follows similar procedures to the actual purposeful transformation discussed above: Here we will introduce some new terms: So, the final form of our equation in this rearrangement is as follows: So, just like the earlier rearrangement, we have hidden some terms in the background, introduced new terms and reduced the number of symbolic forms.But how successful is this equation?Firstly, the newly introduced quantity under the symbol X(t) has an unclear physical meaning; how could we describe, in physics terms, the product of the energy density (ε(t)) and the scale factor squared ( ( )  a t 2 )?Compared to the energy density parameter, Ω, whose physical meaning is much clearer (indicates how far the Universe is from being flat), our term X(t) is lacking in physical meaning and therefore it currently has no use in expressing shared ideas or concepts within cosmology.
Consequently, we did not end up with a relationship that could provide us with a way to answer some of cosmology's questions of interest about the contents and the curvature of the Universe.Looking at the final form, we have a 'fuzzy' quantity ( ( ) X t ) on the left-hand side, that depends on the energy density and the scale factor and two terms on the right that have to do with the rate of expansion of the Universe ( ( )  a t ) and a constant B that is related to the curvature.Also notice that under the 'B' term we have hidden the κ parameter that corresponds to the curvature of the Universe, so someone who engages with the equation at first glance would not be able to notice the existence of a curvature term in the equation.
Overall, despite following all mathematical rules, we have not created dimensionless parameters and we also have not provided ways of answering cosmology's questions of interest.This is because our rearrangement, though correct, was without physical purpose compared to the purposeful nature of the rearrangement that we observed in the courses and analyzed.

Summary of findings
We will now summarize our findings.Our data analysis highlighted the following themes: • For RQ1, we asked what we would observe if we analyzed a line-by-line rearrangement of a physics equation using Sherin's framework of symbolic forms.By noting the potential symbolic forms for each line, we noticed that moving from Line 1 to Line 5, involves a reduction in the number of symbolic forms.
• For RQ2, we asked what we can say about the ways physicists rearrange an equation for physical reasoning purposes.By incorporating the analysis for RQ1, we identified three distinct components in this procedure: (i) the narrowing down of meaning potential, (ii) hiding certain aspects in the background and introducing others into the foreground, and (iii) a purposeful transformation towards the discipline's needs, questions of interest and agenda.

Discussion
In this section, we examine the potential implications of our analysis for PER and the actual practice of teaching physics in a university classroom.

Generalizability and overall value
We would like to begin the discussion by stating that, despite our analysis being a case study of the rearrangement of just one equation, we believe from our experience that there is the potential for generalizability.Looking into equations from other areas of physics, we find examples of rearrangements that display the components that we have described.For example, in high-school kinematics, the derivation of the work-energy theorem started from the kinematic formulas (for distance, velocity and acceleration) and Newton's second law.These were rearranged with the sole purpose of introducing the concepts of work and energy in the equation.Here, the final form is where the kinematic concepts of distance, time, velocity, acceleration are no longer visible.There are strong parallels with the rearrangement of the Friedmann equation: the meaning potential of the initial kinematic equations was narrowed down; aspects were pushed into the background in order for others (work and kinetic energy) to come forth; every move was geared to answering one specific physical question: how does the kinetic energy of a physical system relate to the work done by the forces acting on it?
We believe that the value of our work lies first of all in revealing the underlying mechanism that drives the way physicists rearrange equations for physical reasoning.Starting from the longstanding debate about the role of mathematics in physics and physics education, our work showcases an example of how physicists use mathematics and manipulate equations, not just for computational reasons, but also for physics reasoning.We have shown how introducing new terms into an equation and conceptualizing them with a clear physical meaning is an integral part of the process that cannot be isolated or seen separately from the use of mathematics.

Implications for physics teaching and learning
What we described above is an important finding with clear implications for the use of mathematics in the physics classroom.As highlighted in the research overview, blending mathematics with physical intuition is part of a physicist's toolbox and we feel our work highlights this blending.We also believe that there are a number of consequences of our findings for physics teachers.We suggest that being aware of the underlying mechanisms we describe in this paper will almost certainly affect how teachers explain what they do when they are working with mathematics.
For example, in our observations of physics lectures, the Friedmann equation rearrangement was presented without especially motivating each of the steps-it was the final form of the equation that was of interest.However, armed with our hypothesis of an underlying mechanism, we suggest teacher practice has the potential to subtly change.Therefore, at this point we would like to highlight how each of the components that we have described can be leveraged by physics teachers in their practice.
7.2.1.Leveraging the purposefulness of the procedure.Having in mind the purposeful direction towards the disciplinary question of interest, gives a teacher the opportunity to begin by introducing the task at hand.Once the students are presented with the initial form of the equation, then a good starting point would be to begin by introducing a question.For example: How can we reason about the curvature of the Universe with the Friedmann equation?Being aware of the reduced number of symbolic forms in the final form of the equation gives the teacher the opportunity to introduce the emerging conceptual schema of the final form (in our example, competing terms and parts of a whole) at an earlier stage of the lecture.This could help the students build a mental model (for more details about student's mental modelling, see [34][35][36]) of the subject in question (for example, the Universe as a whole consists of different influences in competition) that would then have to be expressed mathematically by rearranging the initial equation.7.2.3.Leveraging the movement between foreground and background.At this point, a teacher could discuss what conceptual aspects of the initial equation would need to be present in the final form and which ones could be hidden in the background, on the basis of answering the overall question.In this way, the introduction of new terms does not come out of nowhere, but on the basis of rearranging the equation in order to bring it to a final form that would be useful for answering the task at hand.Additionally, a discussion about the characteristics of the newly introduced terms (e.g.dimensionlessness) could also be useful in order to motivate the need to introduce them.From this point on, the algebraic part of the rearrangement can follow.
A summary of the basic stages of how to leverage our suggested mechanism in an educational setting can be found in figure 2. In summary, we argue that the mechanisms we have highlighted in this paper are unlikely to be unique to this specific rearrangement of the Friedmann equation.We believe that the three aspects of mathematical rearrangements that we identify: purposeful transformation, narrowing down of meaning potential and moving terms from foreground to background, represent an important contribution to our understanding of the ways in which mathematics can be used to reason about physical phenomena.We suggest that understanding these underlying mechanisms gives teachers a new way of thinking about their own line-by-line transformation of equations when using mathematics to reason about physics in the classroom.

4. 1 .
Data collectionOur data collection for this case study began by the first author attending three different courses in Cosmology at various levels of Higher Education:(i) Nuclear &Particle Physics, Astrophysics & Cosmology (Undergraduate level-Stockholm University, Physics Department).(ii) Cosmology (Undergraduate & Postgraduate students-Stockholm University, Astronomy Department).(iii) Cosmology & Multi-messenger Astrophysics (Master & PhD students-Uppsala University, Physics & Astronomy Department).

Figure 1
Figure 1 Reduction in the number of symbolic forms means narrowing down the meaning potential.Each of the circular areas represent the meaning potential of the shown equation form, which decreases as we move towards the final lines of the rearrangement.Notice that in the final two lines, the number of symbolic forms does not reduce, but the meaning potential changes.

7. 2 . 2 .
Leveraging the reduction of symbolic forms/narrowing down of meaning potential.

Figure 2 .
Figure 2. A suggested teaching sequence inspired by our proposed mechanism of physics reasoning with mathematics.