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Bilingualism was implemented in the Community of Madrid (Spain) more than ten years ago, through the incorporation of the English language in the teaching methods of certain schools. Since that time, various research projects have been carried out, with the objective of comparing the academic performance of students in bilingual schools with those in nonbilingual schools. The present paper makes use of primary education data from the Department of Education and Research for the Community of Madrid in an effort to analyze whether or not bilingualism results in the relative improvement of educational outcomes achieved in primary schools in the Region of Madrid, in Spain. More specifically, the data used is from sixth grade classrooms, given that, generally speaking, in this grade all schools give a standardized test which measures academic performance in Math, Science and Technology, Spanish Language Arts and English Language Arts. Our assessment makes use of a multinomial logit model, and includes the most common variables found in the research on the determination of educational outcomes (variables related to whether or not schools are bilingual, which is the main focus of this paper), as well as other less common variables considered to be relevant. These include absenteeism, satisfaction levels among families and students, and the percentage of students in second chance programs. The results show that bilingualism does not lower children performance in the subjects taught in English or in the subjects taught in Spanish. Academic performance in Mathematics, Science and Technology, and Spanish Language Arts is similar with respect to those schools which are not bilingual. However, results in English are significantly higher in bilingual schools when compared to non-bilingual schools.

We study the harmonic polytope, which arose in Ardila, Denham, and Huh's work on the Lagrangian geometry of matroids. We show that it is a (2n − 2)-dimensional polytope withvertices and 3 n − 3 facets. We give a formula for its volume: it is a weighted sum of the degrees of the projective varieties of all the toric ideals of connected bipartite graphs with n edges; or equivalently, a weighted sum of the lattice point counts of all the corresponding trimmed generalized permutahedra.

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