Modeling and analysis of cyber-physical systems are still challenging. One reason is that cyber-physical systems involve many different parts (cyber or physical), of different nature (discrete or... Show moreModeling and analysis of cyber-physical systems are still challenging. One reason is that cyber-physical systems involve many different parts (cyber or physical), of different nature (discrete or continuous), and in constant interaction via sensing and actuating.For instance, consider a group of robots, each running a program that takes decision based on the sequence of sensor readings. The sensors that equip a robot return the current position of the robot and the position of any adjacent obstacle. The interactionoccurring between each robot in the group cannot be derived solely from the specification of individual robots. If the field on which the robots roam changes its property, the same group of robots might sense different values, and therefore take different actions. Also, the time at which a robot acts and senses will affect the decision of each controller and will change the resulting collective behavior.This thesis proposes a compositional approach to the design and programming of interacting cyber-physical components. We present an algebra that provides a novel perspective on modeling interaction of cyber-physical components. Using our algebraic framework, one can design a complex cyber-physical system by first designing each part, and then specifying how the parts interact. We formalized the relation between our abstract semantic model and an implementation written in Maude, a programming language based on rewriting logic. We present some applications, including safety and liveness properties of a system consisting of a set of robots, each equipped with a battery, running on a shared field. Show less
For many students, algebraic formulas are abracadabra: they lack symbol sense. Symbol sense includes identifying the structure of algebraic formulas, giving meaning to them, and reasoning with and... Show moreFor many students, algebraic formulas are abracadabra: they lack symbol sense. Symbol sense includes identifying the structure of algebraic formulas, giving meaning to them, and reasoning with and about formulas. Besides basic skills, symbol sense is needed to solve algebraic problems and it is not known how to teach symbol sense systematically.In this research we have studied how teaching graphing formulas by hand (i.e. sketching a graph of a formula) could promote grade 11- and 12-students’ symbol sense. Our GQR-design (Graphing formulas through recognition and qualitative reasoning) is based on expert research and pays explicit attention to recognition of basic function families and features, and to qualitative reasoning, that focuses on the global shape of a graph, with global descriptions and ignoring what is not relevant. These aspects get little attention in regular education, that often focuses on algebraic manipulation.We found that students improved their insight into formulas and found a positive correlation between students’ abilities to graph formulas and their abilities to solve non-routine algebra problems with symbol sense. The students were able to use essential aspects of symbol sense, learned in the context of graphing formulas, such as taking a global view for recognition, qualitative reasoning, and questioning a formula, when solving algebra problems.In the future, when technology will take over the manipulation of algebraic formulas, symbol sense will become even more important. Graphing formulas could promote students’ symbol sense in upper secondary school, and therefore deserves a prominent place in mathematics curriculum. Show less
Kop, P.M.G.M.; Janssen, F.J.J.M.; Drijvers, P.H.M.; Driel, J.H. van 2020
Student insight into algebraic formulas, including the ability to identify the structure of a formula and its components and to reason with and about formulas, is an issue in mathematics education.... Show moreStudent insight into algebraic formulas, including the ability to identify the structure of a formula and its components and to reason with and about formulas, is an issue in mathematics education. In this study, we investigated how 16- and 17-year-old pre-university students’ insight into algebraic formulas can be promoted through graphing formulas by hand. In an intervention of five 90-min lessons, 21 grade 11 students were taught to graph formulas by hand. The intervention’s design was based on experts’ strategies in graphing formulas, that is, using a combination of recognition and qualitative reasoning, and on principles of teaching complex skills. To assess the effect of this intervention, pre-, post-, and retention tests were administered, as well as a post-intervention questionnaire. Six students were asked to think aloud during the pre- and posttests. The results show that all students improved their abilities to graph formulas by hand. The think-aloud data suggest that the students improved both on recognition and reasoning, and give a detailed picture of how students used recognition and qualitative reasoning in combination. We conclude that graphing formulas by hand, based on the interplay of recognition and qualitative reasoning, might be a means to promote students’ insight into algebraic formulas. Show less
