The main goal of this study was to improve students’ outcomes and perception in Mathematics. For this, 12 out of 34 voluntary students were involved in an international contest: European Space Agency (ESA) Mission Space Lab. The experience was organized as STEM, under a guided PjBL. Students identified an environmental problem, executed a way to monitor it from the International Space Station (ISS) and interpreted the data received. Students’ final report was awarded by ESA. Additionally, participants increased their performance in their math final exams compared to the control group. Furthermore, the perception of students and their families about the usefulness of mathematics was very positive. The only drawback detected was the increase of workload. Thus, Green STEM, using direct instruction and guide in PjBL, may be a good tool to improve students’ grades and opinion about the importance of mathematics.

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Recent research in machine teaching has explored the instruction of any concept expressed in a universal language. In this compositional context, new experimental results have shown that there exist data teaching sets surprisingly shorter than the concept description itself. However, there exists a bound for those remarkable experimental findings through teaching size and concept complexity that we further explore here. As concepts are rarely taught in isolation we investigate the best configuration of concepts to teach a given set of concepts, where those that have been acquired first can be reused for the description of new ones. This new notion of conditional teaching size uncovers new insights, such as the interposition phenomenon: certain prior knowledge generates simpler compatible concepts that increase the teaching size of the concept that we want to teach. This does not happen for conditional Kolmogorov complexity. Furthermore, we provide an algorithm that constructs optimal curricula based on interposition avoidance. This paper presents a series of theoretical results, including their proofs, and some directions for future work. New research possibilities in curriculum teaching in compositional scenarios are now wide open to exploration.

In curriculum learning the order of concepts is determined by the teacher but not the examples for each concept, while in machine teaching it is the examples that are chosen by the teacher to minimise the learning effort, though the concepts are taught in isolation. Curriculum teaching is the natural combination of both, where both concept order and the set of examples can be chosen to minimise the size of the whole teaching session. Yet, this simultaneous minimisation of teaching sets and concept order is computationally challenging, facing issues such as the “interposition” phenomenon: previous knowledge may be counter-productive. We build on a machine-teaching framework based on simplicity priors that can achieve short teaching sizes for large classes of languages. Given a set of concepts, we identify an inequality relating the sizes of example sets and concept descriptions. This leverages the definition of admissible heuristics for A* search to spot the optimal curricula by avoiding interposition, being able to find the shortest teaching sessions in a more efficient way than an exhaustive search and with the guarantees we do not have with a greedy algorithm. We illustrate these theoretical findings through case studies in a drawing domain, polygonal strokes on a grid described by a simple language implementing compositionality and recursion.