This work is a summary of an R&D&I research project, funded by the Ministry of Economy, Industry and Competitiveness of the Government of Spain, which we are developing in the Autonomous Community of Andalusia. The aim of the project is to design programs and actions for school improvement based on the findings of a case study that will be carried out in Primary Education schools of very high and very low effectiveness. In order to achieve this, we will use the results of the schools with respect to two basic instrumental competencies (mathematical reasoning and linguistic communication) in the Diagnostic Assessments (DA) carried out by the Andalusian Agency for Educational Evaluation (AGAEVE, for its acronym in Spanish). 

The field of research of this study brings together the study of school effectiveness and school improvement (SESI). Although the work of Reynolds, Hopkins and Stoll (1993) and Creemers and Reezigt (2005) can be considered groundbreaking in this field, perhaps the most relevant work is the International Handbook of School Effectiveness and Improvement (Townsend & Avalos, 2007), which is an excellent synthesis of the state of the art in this line of research.

Currently, the most important developments are taking place thanks to increasingly relevant statistical tools, such as growth or profit measures and hierarchical linear models, also known as multilevel models or mixed effects models, which allow for added value measures (Ferrao & Couto , 2014; Stoll & Sammons, 2007). This set of statistical techniques respects the nested structure of data in two or more levels, which is the usual one in the educational context, where students are grouped into classes and these into schools. This allows for the combined study of the effects of the variables of each level (Goldstein, 2011, Raudenbush & Bryk, 2002, Snijders, 2011).

From a longitudinal point of view, it is especially noteworthy the work of Bryk, Sebring, Allensworth, Easton and Luppescu (2010), which describes the study carried out in urban schools in Chicago. In the field of equalization of scores, the work and contributions of Kolen and Brennnan (2004) are a must.

More pertinent to our research are Joaristi, Lizasoain and Azpillaga (2014), carried out in schools in the Basque Autonomous Community of Spain, Curry, Pacha and Baker (2007), focused in the study of successful practices in schools, and Creemers, Stoll and Reezigt (2007), which presents an international comparative analysis of good practices using case studies.

In addition, mixed methodological approaches that combine quantitative analysis of school effectiveness with case studies of effective schools are becoming more frequent (MacBeath, 2007, MacBeath & Mortimore, 2001).

The starting hypothesis of our research is that it is possible to design more focused and precise school improvement plans and programs, based on knowledge of the praxis, characteristics and context of schools with very high and very low efficiency. The study of low efficacy schools helps to increase the internal validity of the conclusions and to design more adequate school improvement plans.

In order to contrast this hypothesis, three specific objectives are established:

1. To identify and select schools with very high and very low effectiveness from the perspective of cross-contextualized and longitudinal studies, using statistical multilevel modeling techniques.
2. To gain in-depth knowledge of the reality of these schools through a multiple case study.
3. To formulate school improvement programs and actions based upon the differential analysis of these schools.

The project assumes a mixed methodology, with the use of quantitative and qualitative techniques. Population The data will be analyzed at the census level. As a guideline, the last of the DAs carried out (2016-2017) was applied to 93,961 students in the 2nd year of Primary Education, grouped in the 2531 schools that teach this educational stage in Andalusia. Variables The scores obtained in mathematical reasoning and linguistic communication by the Andalusian students of the second year of primary school, during the academic years 2011-2012, 2012-2013, 2013-2014, 2014-2015 and 2016-2017, will constitute the dependent variables in our study. On the other hand, we would have the time variable (the five years of application of the DA) and the strictly contextual covariates at the student level (N1): gender, previous performance and ISEI (International Socio-Economic Index of Occupational Status). At the center level (N2), the type of school (public, private or charter-subsidized private schools) and the aggregate variables of N1 are included. Design We will use multilevel modeling applying hierarchical linear models (HLM). This set of statistical techniques is the most appropriate to fulfill the first of the objectives of our study, because it respects the nested structure of the data. In our case, students (N1) and centers (N2) for contextualized cross-sectional analyses (according to denomination proposed by OECD, 2011) and times (N1) and centers (N2) for longitudinal models. In addition, these techniques allow us to study the joint effects of the variables at each level (Goldstein, 2011; Raudenbush and Bryk, 2002; Snijders, 2011). A center will be considered to be of very high or very low effectiveness when: - the average of their residues is very high or very low (contextualized cross-sectional model). - its scores show a sharp upward or downward trend (longitudinal model). - its scores reach maximum or minimum (ceiling or floor effect model) - its residues show a sharp upward or downward trend (contextualized longitudinal model) Taking these criteria as a reference, 32 schools will be selected and a multiple case study will be conducted, which will provide evidence from very different sources, media and types: official information provided by the inspection and educational administration, internal documentation of the school itself, semi-structured interviews with various agents, discussion and observation groups. Once the information is collected, and on the basis of the scheme of categories and subcategories previously prepared, the qualitative analysis will be carried out using the usual techniques.

The results of the models applied to solve objective 1 will provide a very robust and reliable orderly relationship between schools with very high and very low effectiveness, in which both equity and excellence will have been taken into account. From the differential study of the characterization of the 32 centers that finally participate (objectives 2 and 3), a comprehensive relationship of the characteristics, context and good practices of schools with very high effectiveness will emerge. Analogously, with regards to the schools with very low effectiveness, the characteristics, specific contexts, difficulties and any inappropriate practices will be identified. All this has a high explanatory potential of the results of these centers and, therefore, it seems reasonable that the design of actions and improvement programs should be based on this differential and comprehensive catalog. The expected results of the case study will be reflected in a document which will describe the main lines of the proposed school improvement intervention plans. It will include a self-assessment instrument for schools that are eventually incorporated, an internal and external evaluation plan of the impacts of these plans, and a draft with the characteristics of an online platform that will serve as a support and as a meeting place for all agents involved.

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