Comparing degrees of ‘publicness’ and ‘privateness’ in school systems: the development and application of a public-private index

ABSTRACT The process of privatising services historically provided by the state has blurred the boundaries between what is considered to be ‘private’ and ‘public’. However, few efforts have been made in the educational arena to develop tools to measure this process. Most of the previous research has relied on narrow definitions about what is private and what is public. This work proposes a tool to measure the degree of publicness-privateness of school systems, avoiding binary separation of the concepts. We develop an index and test this tool in two different landscapes: London and Santiago. In these cases, it serves to illustrate major changes in the levels of public-private participation in both school systems, reflecting differences between the systems and over time. We conclude that the index has potential for development and use in the analysis of public and private dimensions in education in broader international contexts.


Introduction
Over recent decades, numerous countries have seen increasing involvement of nongovernmental organisations and actors in educational activities, formerly the remit of the state, perhaps especially in the provision of schooling, which is our focus in this paper (Waldow, Falkenberg, and Rothe 2017). This process has implied the participation of new private actors and blurred the boundaries between the notions of what is considered to be private or public (Mockler et al. 2020). This paper develops a conceptual framework to understand these concepts and tests a measure of the extent of 'publicness' and 'privateness' of school systems for use in comparative studies.
There is considerable global interest in learning from different countries' experiences implementing hybrid models, not least in order to understand in what ways different kinds of systems can be considered 'better' or more likely to produce particular outcomes, and there is a growing number of cross-national studies (Dronkers and Robert 2008;Brewer and Hentschke 2009).
However, at the current time there exists no standard framework to describe and measure privatisation, either to support comparative studies of the extent of the phenomenon, nor to assess its possible determinants or outcomes. Most comparative studies of privatisation in school systems are qualitative in nature (Wiborg 2015;West and Nikolai 2017). Quantitative studies tend to rely on limited measures such as the volume of private education expenditure or levels of enrolment in fee-paying institutions. For example, the OECD considers a school to be 'private' when it is managed by a nongovernmental organisation, regardless of the source of the funding resources (OECD 2013). 1 Some research relies on conventional definitions of private and public, emphasising issues of provision and financing (Patrinos and Sosale 2007), other reports capture the opinions of stakeholders regarding the process and degree of privatisation in their contexts (e.g. The European Trade Union Committee for Education [ETUCE]). While the first approach limits the understanding of the multiple manifestations of privatisation, the second is not appropriate for making comparisons between dissimilar cultural contexts and over time (Winchip, Stevenson, and Milner 2019).
The objective of this paper is to develop such a framework: a conceptually broad-based measurement tool that will enable the description of the public/privateness features of a landscape. We call this a public-private index, and it allows comparison between units (neighbourhoods, districts, cities, etc.) and over time. A key point is that the index is developed from a review of the conceptual literature and not from the available data. We therefore start by identifying the key ways in which public and private are conceptualised in relation to education systems, and implications for measurement. We then construct a measurement framework (index) with domains and dimensions of publicness and privateness reflecting, to the extent that is possible, these key concepts. Not all dimensions of public and private can be captured in quantitative measures that use publicly available data (a key requirement for a comparative tool of this kind) and we discuss these limitations. To test whether the index can be operationalised and whether it has the potential to deliver useful insights, we trial it through a limited study of the secondary school systems in Santiago and London, comparing them with each other and at different times. This reveals interesting findings of the effects of changing policy regimes affecting these cities. We conclude by reflecting on these points. Concepts of public and private in school systems and implications for measurement This study was originally motivated by our attempts to compare the systems of schooling in our two countries, Chile and England, and the ways in which they are evolving over time, becoming more or less private or public. We quickly found that it was not always obvious that whether institutions, policies or practices were 'more private' than others, by how much or in what way. Our assumptions about what is public and what private were conditioned by our country-specific experiences but also by our reading of different bodies of literature emanating from different traditions. This led us to a review of underpinning concepts and issues across different disciplinary literatures (and within education, in different fields of research), and what they imply for measurement of the public and private.
A key issue is that distinctions between what is public and what is private can be made in many different aspects of education. Within school systems, so-called privatisation has taken multiple forms. Among others, these include the charging of fees; the provision of schools by for-profit providers; their provision by non-state organisations on a non-profit basis; consumer choice of schools; removing schools from democratic oversight; and liberalising decisions about admissions, curriculum, and pedagogy. Forms of 'endogenous' privatisation (Ball and Youdell 2008), introducing private sector norms and logics into educational practice, have also proliferated, including 'corporatisation' (Courtney 2015) and 'commercialisation' (Hogan and Thompson 2017).
There are multiple 'grey areas' in which public-private distinctions are unclear. Recent, mainly 'privatising' developments have been overlaid on existing state education systems. These themselves varied in their reach (e.g. whether they had taken over all previously existing forms of private educational provision or whether some of these had been retained alongside new state systems) and the 'publicness' of their activities (e.g. the actors involved, governance, and accountabilities). Thus the boundaries between 'public' and 'private' have gradually become blurred, leaving a fuzzy area in which these separate dimensions are hard to distinguish (Robertson et al. 2012;Power & Taylor, 2013). Some authors have underlined that this process is unavoidable, with the main challenge for states being the definition of rules and systems to regulate increasing private participation (Rizvi 2016). As Burchardt (2013) acknowledges, the school system in England challenges the idea of using a state/non-state binary for measuring public and private distinctions. First, the great diversity of providers might be regarded as more or less public (for example, faith organisations, universities, housing associations, and educational charities established as philanthropic arms of companies). Second, due to the extent to their capacity for private action depends on the regulatory regime, which changes over time.
Adding further complexity, the terms' privatisation', 'private' and 'public', are used differently within different disciplinary traditions, although they are often taken for granted within the different disciplines and sub-disciplines in which they are used (Starr 1988;Belfield and Levin 2002), To take a simple example, Starr (1988) considers one concept of 'public' as visible (happening in public), contrasted with 'private' as meaning out of view. With this notion of what is public, policy technologies such as inspection and reporting of policies and results would ostensibly render schools more public. However, in many countries, these technologies have been introduced in order to facilitate the operation of school markets and would be deemed to be a feature of privatisation. In relation to school systems, the literature falls into two main categories. One, broadly speaking, has its origins in political economy or public policy traditions. Here public-private distinctions tend to concern how policies are delivered and services are organised. Debates are concerned with state/market relationships and how these affect notions of public-private goods and the distribution of those goods. Thus, publicness or privateness might be determined by who pays for schooling, who provides it, how it is allocated, and the extent of choice. Obolenskaya and Burchardt (2016) operationalise such an approach in a framework for measuring publicness and privateness in different public services in the UK. This has three dimensions: finance (who pays); provision (who provides), and decision (who decides about what service is taken up). By contrast, a second category of literature originates specifically within the field of education and is concerned with issues such as 'in whose interests is education?', 'which aspects of education should be considered private matters or public matters?', and 'what goals do societies have for their school systems and how are these secured by private or public provision?' (e.g. Marginson 2018). Much of the literature on the privatisation of education addresses these kinds of questions, considering how systems differ in educational matters such as curriculum content, school organisational features such as ability grouping, the professional orientations of teachers and school leaders, and student selection and exclusion policies. For example, Courtney's (2015) analysis of privatisation in England produces a typology of schools (within the state-funded sector) based on variables such as curricular specialisation and pupil selection. Questions about 'provision' and 'decision' appear in his framework under a broader consideration of 'legitimation', addressing issues such as 'who gets a say?' and 'whose interests are being served?'. These are reflected in variables including the authority appointing the governing body; ownership of land and buildings; ownership of the 'brand'; accountabilities to stakeholders other than pupils, parents, and communities; and the values of these stakeholders. Recently, Mockler et al. (2020) developed a typology for understanding the provision of schooling. Acknowledging the blurred boundaries between 'public' and 'private', the authors sought to organise factors mentioned in the privatisation literature to shed light on the complexities of school provision in different educational contexts. They propose four dimensions: Control, Access, Funding, and Teaching. Interestingly, the authors present their work as a 'hybrid heuristic/empirical typology' and stress the importance of conceiving each component as a gradual scale.
This wide conceptual range in relation to the concepts' public' and 'private' in education suggest that a tool for measurement must be multi-dimensional and that a priori there is no sound basis to favour one concept of the public or private over another, either because it dominates one disciplinary tradition or because it has particular resonance in a certain country.
Two other considerations emerge from the public-private literature. One is that the term 'privatisation' is typically used normatively -'not only to describe but to celebrate and condemn' (Starr 1988, 1). In much of the literature on education, it is seen as a bad thing (Brighouse 2004), although, as we previously highlighted, that view is contested (both philosophically and empirically). Even amongst advocates of public education, the idea that 'public = good' and 'private = bad' has been challenged as overly simplistic and obscuring some of the nuances of meaning. As both Gerrard (2015) and Newman (2007) point out, states tend to represent particular interests (or publics), and those of more marginalised, minority communities (or publics) can go unserved. Thus, public education (run by the state/public) is not necessarily the same thing as education in the public interest. As Starr puts it: 'Any serious inquiry into the meaning of privatisation must begin, therefore, by unloading the complex freight that the public-private distinction carries'. Moreover, most contemporary studies are about privatisation, that is, the process of things becoming more private. The focus tends to be on what is changing and why (for instance, different actors, interests, behaviours, discourses) and the outcomes of these changes, rather than on documenting the extent of 'publicness' of former systems or the 'privateness' of emerging systems at any given time. Analysis of the extent to which different systems were really 'public' in the first place tends to be overlooked and, in fact, many studies focus entirely on privatisation trends in state systems, disregarding the elements of a country's education system which are private in any case. 'Private' and 'public' can be used loosely when the overwhelming direction of travel is obviously more private. The Chilean case highlights the need for different kinds of enquiries. After decades of 'privatisation', Chile's system has recently been undergoing a process of 'publicisation', with reforms reinstating more public features (such as the abolition of selection, fees, and profit-making), albeit within a system which retains the use of a voucher, the notion of school choice, and the widespread provision of schooling by non-state actors. With a different direction of travel, different questions need to be asked -not only how and why this is happening and how stakeholders are responding, but to what extent a public system has been restored, and whether (and in what ways) it is now more public than systems in other countries.
A further issue is that there are ontological and epistemological differences in approaches to policy reality -and therefore to what should be measured. One prominent tradition of critical policy studies is concerned with policy processes, power relations, and shifts in the political and discursive climates in which policies are made and enacted (Reay 2006;Braun, Maguire, and Ball 2010). By contrast, empirical social policy and political economy traditions tend to capture 'realised policy' in the form of spending or institutional behaviours (e.g. how many schools vary the curriculum), since they would argue that only these direct actions can influence outcomes and their distribution between groups of people or areas. Variations in practice within overarching discursive, political or regulatory frames are also of interest to policy sociologists and geographers of education, particularly in relation to the operation of local markets (Bowe, Gold, and Ball 1992;Ball 2007;Taylor 2009). The former approach would tend to suggest that the extent of privateness in the state education system of a country should be assessed by the possibility for private action and not by the extent of private action at any one time. For example, it might be relevant to measure the 'freedom' allowed to schools to determine their own curriculum and not how many schools actually determine their own curriculum, since the fact of the curriculum being removed from public/state power and placed in private hands is what denotes privateness. Indeed, a discursive shift towards regarding curriculum autonomy as 'freedom' (i.e. good) rather than, for example, depoliticisation (i.e. bad) might be identified as a dimension of privateness (Gunter 2018). The latter approach would tend to suggest that the unit of analysis should be the individual educational institution and that local variations in institutional structure or practice are of empirical interest.

Conceptual development of the measurement framework
It is clear that not all of this complexity can be reflected in a quantitative measurement tool. 2 However, we suggest that some of it can, and that this would be an advance on the limited approaches that currently exist. We, therefore, proceeded to develop a measurement framework responding, as far as practically possible, to these conceptual issues and challenges. This required some decisions about definition and scope.
An initial decision was to dispense with the term 'privatisation', implying a direction of travel and a value judgment, and to make the object of comparative study the degree and nature of 'publicness' and 'privateness'.
Second, we decided to build the framework on school-level measurement. This can be seen as a bottom-up approach to understanding the extent of publicness or privateness in a system. Each school's public and private features are measured and then aggregated to different relevant geographies (municipality, city, region, country). An alternative (topdown) approach would be to measure system-level features. We opt for the school-level approach as a stronger empirical measure of the actual extent of private and public activity at any given time, and because of its value in permitting analysis of different local markets and the influence of political and administrative factors at different sub-national scales, as well as providing (through aggregation) a system-wide measure. The consequences are that features of the system that can only be identified at the system-level (perhaps, for example, the discursive climate) are not captured, and the focus of the measure is on the empirical outcomes of public-private shifts at a snapshot in time, not on the broader political and policy moves. This, we suggest, is appropriate for a measurement tool while acknowledging that it does not reflect all of the ways in which public and private can be conceptualised and are important. The extent to which school-level scores can vary will depend on the structure of school systems. It is possible to imagine a country (or any other relevant unit of analysis) in which all schools are of the same type and all subject to the same central regulations, in which case schools would all have the same score. Changes to the system-wide score could only come from system-wide decisions affecting the privateness or publicness of all schools. Alternatively, a country might have schools of different types (some more 'private' and some more 'public'), with different scores on a public-private measure. Changes in a system-wide score could come about overtime if more of one type of school opened or closed or new types were created. Or (most likely) a country might both have different types of schools and make changes over time, making certain types (or all types) more public in nature or more private. Thus, change in a system-wide score could come about because of changes in the composition of the system (for example that more 'private' schools are set up) or via changes in regulation and practice (for example, that existing public schools are allowed to adopt more private features). Ideally, a school-level measure should enable these different drivers of change to be identified. In connection with this, we decided not to limit measurement to variations within state school (government) systems but to include all mainstream schools, inclusive of those provided privately before any privatising changes to state systems. This enables the traditional private school to be included on a continuum with other schools in degrees of public-privateness.
Third, we decided to develop a multidimensional framework, reflecting as many of the different perspectives identified in the literature as practicable, and constructed in such a way as to highlight which elements identified as public or private are 'doing the work' in influencing the measure. In other words, while there should be an overall score, it should also be possible to analyse changes in the different dimensions. Since there is no sound basis for weighting some elements as more important than others, we decided that all should have equal value.
Reflecting the different perspectives identified in the literature, we decided on four broad domains of publicness-privateness at the school level. These were: (a) Delivery, or 'who provides and regulates the school?'. (b) Access/inclusion, or 'who can go to the school?'. (c) Finance, or 'who pays for the education provided in the school?' (d) Decision, or 'who determines what happens in the school'?
The delivery and finance domains capture the state/market dimensions that dominate public policy/political economy traditions, and which are also prominent in education literature. The access/inclusion and decision domains capture more of the issues regarding educational content that are prominent in educational literature. 'Decision' here is used not in the way that Burchardt et al. (2013) employ it, but instead as an identifier of who makes decisions regarding the issues affecting teaching and learning.
We identified distinct dimensions to reflect the key aspects of each domain discussed in the literature 3 and a continuum from public to on each dimension (Table 1). For example, in the delivery domain, we consider who the provider is, what regulations govern their entry into the market, and to what extent are they controlled by the state in terms of inspection or the ability to close the school. On the first of these dimensions, a school provided by the state would be deemed the most public form, and one provided by a non-state profit-making provider would be the most private form. Other kinds of provision, such as charities, would lie in the middle. For 'decision', we observe the governance of decisions on admissions, the curriculum, teaching qualifications and terms of employment. On the last of these dimensions, a school in which qualifications to teach and terms and conditions of employment are mandated by the state would be considered the most public form and one with the freedom to employ who it chooses on the terms it chooses would be considered the most private form. A school with a hybrid of these conditions would lie in the middle, partially public and partially private. We decided that we could not include criteria where there was no consensus in the literature about whether something was more public or less public. For example, whether educational charities associated with A public authority directly appoints the provider (or is the provider) Providers can apply and a public authority decides based on limited criteria Inspection/ Intervention over quality "How is the quality of the school overseen?" The state can intervene The state cannot intervene and the school is not mandated to report publicly on quality Access/ Inclusion Pupils' selection "Who can attend the school" Anyone in the age band Limited by gender, religion or academic ability Socioeconomic inclusion "Does income and wealth directly influence access?" No -education is free Yes -fees are charged and there is no obligation to ensure socioeconomic diversity Finance Family co-payment "Who pays for the education provided" The state (in full) The family (in full) businesses are more private than those related to religious organisations. In each domain, there will be issues of debate and contention, resolution of which could inform further development of the measures. It can be observed that some of the dimensions tend to be closely correlated in practice. In England the provider dimension might be seen to drive many of the others, since different types of schools operate under different regulations. However, that would not necessarily the case in a different system Fourth, we decided that the index needed to be widely applicable in order to be useful. This meant that we needed to construct the dimensions in ways that were not based solely on specific characteristics only relevant in particular countries, but which could reflect the range of institutions and practices observed in educational systems around the world. It also meant that the index would need to be capable of being applied in many countries using public data, without the need for bespoke data collection. Thus, although concepts of public and private drove the construction of the measurement tool, consideration of data availability influenced what could practically be included. This led us to adjust some of the decision criteria more in the direction of 'possibilities and constraints' than 'realised policy', even though our intention was the latter. For example, Chilean public data do not reveal whether schools actually make a profit, only that they are authorised to do so. Although we would argue that schools that do operate for profit are more private than schools which choose not to, we cannot make this distinction empirically without a survey of individual school financial records. Thus, the 'provider' dimension only covers whether the provider is authorised to make a profit or not.
Our approach differs from previous efforts to describe and classify school provision in three main ways. First, we offer a tool to measure the level of public-privateness of different landscapes. Even those previous works that provide a more sophisticated conceptualisation to understand what is public or private have not developed a framework for its measurement. Moreover, the existing tools for understanding the private and public features of schooling focus on the system level. On the other hand, our framework can be applied to different administrative or geographical units and 'aggregated' to the system level. Second, we operationalise the dimensions of what is publicness and privateness and propose a way of capturing the graduality of this continuum. This means that we provide a framework to understand and measure the extent to which publicness or privateness is expressed in a unit of analysis (school, district, city, etc.). Third, we offer a method based on school-level characteristics. Other conceptualisations (e.g. Mockler et al. 2020) also include system-level features (such as school-choice or unionisation of teachers) that cannot be observed at the school level. Although those are relevant factors, they do limit the possibilities for measurement in some units (district, city).

Construction of the measure
Having identified the conceptual bases of the framework, we combined them in a numerical index to enable a score to be assigned to each dimension and domain (allowing them to be identified separately), a total school score and an aggregate score for any larger units (e.g. municipalities, cities, or countries).
The dimensions are not, in the main, variables that have any objective numerical value at the school level. That is to say that it is not possible to construct them in such a way that a value of 10 implies twice the level of privatisation of a value of five. While some could have been constructed in that way (e.g. selection or family co-payment) others could not, so we took the decision to construct the decision criteria ordinally, with each dimension having a score of between 0 (fully public) and 5 (fully private). The decision to allocate these particular scores (rather than 0-1 or 1-10, for example) was arbitrary.
The reasons for opting for a numerical summative index are twofold. First, we have highlighted that privatisation has multiple manifestations and it cannot be considered a single phenomenon. As the conceptual limits between public and private are blurred and their implementation is context-specific, we do not expect that all of the variables included will be correlated in all cases. Second, and more importantly, we emphasise the relevance of the graduality of the schools' publicness and privateness. Therefore, starting from a qualitative description, we use categorical variables to express the degree to which a school may be considered public or private. This conceptualisation separates us from methods commonly used for construction of indices (such as factor analysis). Alternatively, we could have produced standardised dimension scores based either on numerical variables where possible or the distribution of occurrence of an attribute within a country. For example, we could have estimated the proportion of 'non-state agency providers' but, instead, we stress the gradualness by assigning different scores to schools authorised to make a profit and those that are not. From our perspective, the latter appear to be 'more public' than the former. Therefore, we are interested in how often a specific feature is present and how extensively each of the schools represents the notion of publicness or privateness.
To calculate the school-level scores in the index, we followed the next steps. First, we allocated a score (0 to 5) to each dimension based on the decision criteria (table 2). The details on how scores were allocated to the different types of schools are presented in the appendix. 4 Second, we calculated each of the domains' score by averaging the scores of the dimensions within it. This means that each of the dimension within a domain has an equal contribution to the domain score. Third, we sum the domain scores to produce the final school-level index. A totally private school (scoring 5 in each of the four domains) would have a value of 20 on the index, and a totally public school would have a value of zero. Similarly, mean scores for larger geographical units may be calculated by averaging school scores, producing municipality, city, or country scores of 0-20 (weighting the contribution of each school to the value of the general index, based on the proportion of the total enrolment of the geographical unit that the school represents).
We return to some of these conceptual and methodological issues in the discussion. However, two key tests were whether the index could be operationalised in two different countries in practice and whether it would produce any meaningful results. It is to these issues that we now turn.

Operationalising the index: Santiago and London
Santiago and London were chosen as test sites for the index as we were already collecting and analysing data for these cities and because they provided contrasting dynamics of privatisation. As is well documented elsewhere (Bellei 2015), Chile's school system was one of the most 'privatised' in the world from the early 1980s to 2014. Under a voucher scheme, school provision was undertaken by a range of state and non-state providers, including for-profit schools. Private subsidised schools could also charge fees and select students and had autonomy over curriculum and pedagogical matters and staff terms and conditions. In 2014, legislation was passed introducing a gradual overhaul of this system, including the abolition of co-payment and selection and the introduction of a central admissions system (Valenzuela and Montecinos 2017). By contrast, England had a more 'public' system until the early 2000s, with most schools run and funded by state providers including faith schools incorporated into the state system in different ways. Government The provider is the state 0 The provider is a non-state agency not authorised to make a profit 3 The provider is a non-state agency authorised to make a profit 5 Opening decision Public authority appoints the provider (or is the provider) 0 Application by providers assessed by public authority according to both demand and qualifying criteria for providers 3 Application by providers assessed by public authority according to one factor (demand/ qualifying criterion) or other.

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Inspection/ Intervention over quality School can be intervened in by the state based on their quality 0 School is mandated to provide information to the state regarding their quality, but cannot be intervened in by the state 3 School is not mandated to provide information regarding their quality to the state and cannot be intervened in by the state 5 Access/ Inclusion Pupils' selection School does not restrict the admission of students from any specific group (gender, religion, or academic ability) 0 School only admits students from one or more specific groups (gender, religion, or academic ability) 5

Socioeconomic inclusion
The school does not charge family fees 0 The school charges family fees but is under public regulation to ensure socioeconomic diversity 3 The school charges family fees and it is not under public regulation to ensure socioeconomic diversity 5 Finance Family co-payment Individuals do not pay 0 Individuals pay in part, the state pays in part 3 Individuals pay in full 5 Decision Admission The pupils are allocated to schools by an Educational Authority (or based on public rules) and neither parents nor schools express preferences 0 The pupils are allocated to the school by an Educational Authority but parents may express preference 1 The pupils are not allocated to the schools by an Educational Authority, but schools must comply with public rules to decide the admission 3 The pupils are not allocated to the schools by an Educational Authority, and schools are not mandated to comply with public rules to decide the admission 5 Curriculum The school follows a national/federal/state curriculum 0 The school determines its own curriculum but has to follow some nationally-determined criteria 3 The school determines its own curriculum 5 Teachers Teachers' qualifications are mandated and their terms and conditions comply with a state-determined code 0 Either teachers' qualifications are mandated OR their terms and conditions comply with a state-determined code 3 Teachers' qualifications are not mandated and schools can follow their own terms and conditions of employment 5 funding went to schools rather than families (albeit on a per-pupil basis and under local management after 1988), and state schools could neither charge fees nor profit. There was a national curriculum and frameworks for teacher qualification and pay. This system began to change in 2001 with the introduction of academy schools (independent statefunded schools). Much more rapid changes came after 2010 with widespread academisation and greater freedom for schools (Exley 2012;Wiborg 2015). Therefore, we would expect the index to show that i) Chile had a more private system than England, and ii) that Chile's system was becoming more public at the end of the period and England's more private.
For reasons of manageability in this test exercise, we restricted our enquiry to the largest city in each country, Santiago and London. Although London is larger than Santiago, the cities have very similar numbers of municipalities/boroughs. Santiago has a higher proportion of children and also a much larger number of schools, with a lower average school size (Table 3). It has also had much faster growth in both population and schools over the period observed.
Because we were interested in the ability of the index to record changes over time due to changing policy regimes, we made a decision to capture data at points in time relevant to policy development in each country, rather than at the same points in time for each country. For Santiago, data was captured for 2007 (reflecting the system generated by the major privatising reform of 1981) and 2015 (reflecting changes arising from several policy reforms started in 2008 and which we discuss later). 5 We also model data for 2025, based on projections of the implications of the reform process that started in 2016, assuming that no additional changes affecting the index will be implemented and that enrolments by type of school remain stable. 6 Alternatively, this exercise may be understood as an exploration of how the index would look like for Santiago at present if the ongoing reform were already fully implemented. For London, data was captured for 2001 (prior to the opening of the first academy schools), 2010 (before the introduction of wider academisation), and 2015 (the latest data available at the time of the study and providing a point of comparison with Santiago).
Values for each of the dimensions of the index were generated in two ways: I) from administrative data sources provided by the Educational Authorities in each country; ii) from attributing characteristics applicable, by legislation or regulation, to schools of particular types (e.g. that academy schools may vary their curriculum) at the relevant point in time. Unless stated, the municipality-and city-level estimates have been adjusted based on the proportion of students enrolled in each school. The information regarding the school size was obtained from the aforementioned administrative records.
It is evident from this that changes to the index value for a municipality, city, or country over time can be driven in three ways: • New schools of particular types may open or existing schools may close. • Existing schools may gain or lose pupils. • Existing schools may become more or less private-public as laws and regulations affecting them are changed.
In our analysis, however, we do not focus on identifying which of these ways drive the changes in publicness or privateness of the cities. Our goal, in this example, is to test whether the tool captures variation over time and which dimensions register changes.

Results
The adjusted results for the city level show an upward trend in privatisation in London from 2007 to 2015 (Figure 1). Although in 2007 the four domains of the index -Delivery, Access, Finance, and Decision -showed very similar values, for the following years certain differences emerge.

Total index value and scores by domain
The adjusted scores show that Santiago displays a downward trend, mainly driven by the reductions in the scores in the 'Delivery' and 'Decision' domains from 2007 to 2015, and in all the domains from 2015 to 2025. The decrease in the first period is associated with the introduction of the 'Quality of Education Assurance system' (2010) which established a new institutional arrangement to inspect and assess schools' performance in several areas. One of the principal changes associated with this law was the categorisation of schools based on a set of educational outcomes and the potential closure of schools that repeatedly failed the assessment. This change directly impacts the 'Delivery Domain', given that national authorities are empowered to intervene in schools based on their performance. Although certain other laws addressing issues of school admissions and socioeconomic inclusion were enacted in this period (e.g. the Preferential School Subsidy Law, General Education Act), they do not significantly affect the index. 7 The most important change in the second period (2015-2025) is associated with the Inclusion Law (2014), a reform banning student selection in all subsidised schools, eliminating the use of family co-payment, and forbidding profit-making entities from participating as education providers. 8 As a result of these changes, and considering a scenario where all the other factors remain constant, in 2025, Santiago will present lower privatisation levels than London in 2001. This is a relevant finding, as the enrolment in private subsided schools steadily increased from 1981 to 2018. Therefore, the downward trend reflected in the index seems to be associated with the new regulations changing how the subsidised sector operates and not with an enrolment trend. The index thus serves as a tool for comparing cities (or countries). It also helps understand the heterogeneity of privateness within a city (in other words, the 'geography of privatisation) and to what extent the degree of privateness has varied over time.
As shown in Figure 2, in 2001 London had a broadly homogeneous landscape characterised by low levels of privatisation. The main exception was the City of London, which had a very small number of schools (all of them 'independent'). To describe the picture shown in the maps, five categories of the same size were used to characterise the degree of privateness in the boroughs/municipalities, ranging from 'Mainly Public' to 'Mainly Private'. In 2001 the northeast boroughs (municipalities) were mainly public, while the northwest and south boroughs had somewhat higher levels of privateness. The number of boroughs classified as 'Mainly Public' decreased from 13 in 2001 to only two in 2010. In 2015, more than half of the boroughs were in the intermediate group of privateness and four were in the two higher levels. Although the privatisation process has affected all of the boroughs in the city, the northeast area remains less private. In 2007, the most highly private municipalities in Santiago were located in the northeast of the city, which concentrates the wealthiest sector of the population and where a significant proportion of the non-subsidised private schools are located. Although somewhat less private, a considerable portion of the municipalities (16 out of 33) showed high levels of privateness (relatively evenly distributed throughout the city). This implies that the offer of highly private education was not confined to wealthy areas but was also present in some of the most deprived zones. In the later years, the reduction in the degree of privatisation was seen in all municipalities. In 2025, none of the municipalities will belong to the highest level of privatisation, and only five out of 34 will remain in the two higher levels. The significant levels of privatisation in those areas may be explained by the presence of non-subsidised schools and gentrification processes in some municipalities which display particular characteristics in their education offer (e.g. concentration of highly selective schools). While in 2007, none of the 34 municipalities was classified in the 'Mainly Public' group, in 2025, almost two thirds will belong to that category.

Trends of privatisation in London (boroughs) and Santiago (municipalities)
To test the impact of our conceptual and operational decisions, we constructed two further versions of the index. First we calculated a new version of the index in which each dimension contributes equally to the final index score (in this case, each indicator has a weight of .11 of the total score). The decision to apply equal domain weights was based on the assumption that each domain represents an equally substantial concept defining what is public and private. However, not all domains contain the same number of dimensions. If these individual dimensions are themselves conceptually important, it could be argued that their contribution has been excessively levelled out in the original model. Figure 3 contrasts the results. It is notable that only small differences can be seen between the two versions. Some minor changes are observed for London (2015). Data suggests that those variations are mainly driven by the 'Decision' domain (particularly regarding 'Admissions'). These figures suggest that with the currently conceptual construction of the index, weighting is not a major concern, but this could change should substantial revisions be made to the domain or dimension content. The stability of the result between the two versions of the index is something that must be observed carefully. While the way in which the weights are allocated does not produce greater changes in London nor Santiago, this may not be the case in other educational systems where particular patterns of changes in enrolment may be taking place (especially when observing small units such as districts or municipalities).

Effect of using equal dimension weights on the index values
Second, we tested the difference in results using weighted and non-weighted enrolment scores. Our concern is to test how critical the enrolment size of the school is to define the final score of the index. For example, large enrolment-size schools may be associated with certain domains of the index (e.g. provider). Not taking into account this factor may lead to over or underestimating the values of the index. The data suggest that using weights corrects the gross scores for most of the boroughs in London. When enrolment-based weighting is not used, the index tends to overestimate the level of privateness of most of the boroughs in London (Figure 4), since there is a sizeable number of small 'more private' schools. Considering school enrolment seems to be crucial in 2001 and 2010 (with the average reduction of the scores reaching almost 15%) and somewhat less relevant in 2015 due to the greater homogeneity of the schools in terms of privateness in the latter period. Greater uniformity in the level of privateness across schools translates into fewer variations in the scores due to enrolment. The adjustment for the Chilean municipalities is smaller than for the London boroughs in all years. Unlike London, the effects of not taking the size of the enrolment of the schools into account are mixed. Indeed, while in some municipalities the scores are underestimated in others there is an overestimation of the degree of privateness (albeit the differences are low-magnitude). Values for 2007 show somewhat greater differences between the two measurement (relative to 2015 and 2025). This is likely to be an effect of the new regulations that have gradually equated the rules under which the state-controlled and private subsidised schools operate (making both of them 'more public'). In that scenario, the size of the enrolment becomes less important. Using weights to consider the enrolment may be more relevant when analysing small units or scenarios where certain types of schools are closely associated with 'more private or public' features.

Effect of controlling for school enrolment on the index values
In each of the graphs, the x-axis represents the index values when weighting is applied, and the y-axis shows the value without using weighting. The 45° line denotes the line of equality for the two measurements.

Discussion
It appears to us that despite the complexities of the underlying concepts and the contested nature of any attempt to combine and capture them in a single variable, it is possible to construct a broad-based measurement framework that captures multiple dimensions. The index works in practice, at least in the cases tested here. It identifies Santiago as more private than London at the start of the period and at the current time, but not after enacting the recent reforms -a significant finding. It also identifies potential drivers of privatisation. In the case of London, privatisation has occurred through a combination of the transfer of existing schools into different forms of more private provision and the gradual deregulation of these providers. In the case of Santiago, with a rapidly growing population, the entry of new, more private providers into the market is the main driver.
Significantly, while these differences in policies and effects may already be known qualitatively, the index provides the ability to quantify them. The fact that it works in these two cities suggests the potential for substantial new enquiries in the field and major new empirical contributions that have not been possible to date. These could include: • ranking countries in terms of public or privateness; • creating typologies of geographies that share similar forms and dynamics of privatisation or publicisation in a more extensive manner than can be done from the country case study approach that dominates the existing literature; • applying measures of public or privateness as variables in studies seeking to assess the outcomes of changes towards greater or lesser privatisation; • modelling the effects of planned or emerging policy developments.
As the first attempt to create such an index, the current version doubtless has conceptual, methodological, and operational limitations which remain to be tested in subsequent work. For example, when we presented the draft index to social policy audiences, they tended not to be convinced by the access/inclusion domain, since it appeared to them to be a feature of educational systems related to historical/cultural choices and not state/ market relationships. Educational audiences tended to find this domain noncontroversial, having stronger notions of the meaning of public being 'for all the public', as well as 'provided by the public' and recognising the tendency of private provision to cater for particular interests. Subsequent authors may wish to develop alternative versions which refine the concepts or which prioritise some over others. However, we note that given the complexity of the phenomenon known as privatisation in school systems, the development and use of a cross-national measurement will inevitably be an exercise in compromise. Similarly, the scores assigned to each variable within a dimension are arbitrary and can only be interpreted as a method of expressing the graduation in a public-Private continuum. Although this is a limitation, our focus is more general, attempting to identify dimensions where the public and private spheres can be measured and compared, going beyond the traditional administrative-oriented classifications.

Conclusion
In this paper we have attempted to develop a measurement of the expression of 'publicness' and 'privateness' of educational systems, going beyond traditional approaches based on administrative or funding characteristics.
We recognise that the components and construction of the index will inevitably be challenged, and helpfully so. Understandings and definitions of privatisation are contested within and across countries and disciplines. Addressing these contestations with an attempt to develop a comparative measurement is in itself, we argue, a valuable addition to the literature on privatisation. The application of the index in more cities and countries with different policy contexts and different data infrastructure may also provide challenges regarding the details of its construction and operationalisation.
We conclude, however, that the index as constructed offers strong potential for international, historic, dynamic, and forward-looking study of educational privatisation. In our comparison, the index reflected a steady trend of privatisation in London, whereas, in Santiago, it reveals a significant process of re-publicisation. Importantly, it enables the comparison of changes in terms of the extent of private or publicness and identification of the major factors driving changes and differences. For example, while in certain countries privatisation may be explained mainly by an increase in private providers, in other educational systems, the same index value could be mostly related to regulations on the curriculum or teaching qualifications Notes 1. Similarly, UNESCO considers schools to be private when 'not operated by a public authority but are controlled or managed, whether for profit or not, by private bodies such as nongovernment organizations, religious bodies, special interest groups, foundations or business enterprises' (UNESCO 2017, 429). 2. For example, Carrasco and Gunter (2019) suggest that some manifestations of privatization are also 'private', meaning that they correspond to decisions and actions made by individuals or families in the pursuit of self-interest (and out of the sight of the public). 3. Further details of how each of these dimensions relates to the publicness or privateness of the educational systems are presented in Lubienski (2006), Ball and Youdell (2008), and Bellei (2015). In particular, Mockler et al. (2020) offer a detailed overview of 21 components to produce a typology of schooling. 4. The final databases used for the analysis are available upon request to the corresponding author. 5. Unfortunately, records from previous years are incomplete and impedes estimates of the index values in periods closer to the 1981 reform. Similarly, the analysis for London starts in 2001 due to data availability.
6. Alternatively, the outcome of this projection may be interpreted as a description of how Santiago would look (at present) if the reform had been fully implemented. This caveat is important as the time implementation framework of the reform may change due to political decisions. 7. The lack of impact of those regulations on the index values may be explained by their weakness (for the case of admission policies in the General Education Law see Carrasco, Gutiérrez, and Flores 2017) or due to their gradual implementation. 8. The 'Inclusion Law' is part of the Educational Reform passed during the presidency of Michelle Bachelet. Other legal acts included in the reform also affect the index in the domains of 'Delivery' and 'Decision'.

Disclosure statement
No potential conflict of interest was reported by the author(s). Alejandra Rasse is an associate professor at the Social Work school at the Pontificia Universidad Católica de Chile (PUC). She is also an associate researcher at the Centre for Sustainable Urban Development (CEDEUS) and the Millennium Nucleus Authority and Power Asymmetries (NUMAAP). Her research focuses on urban transformation, territorial inequities, socio-spatial segregation and poverty.