Formal Analysis of Linear Control Systems using Theorem Proving

Control systems are an integral part of almost every engineering and physical system and thus their accurate analysis is of utmost importance. Traditionally, control systems are analyzed using paper-and-pencil proof and computer simulation methods, however, both of these methods cannot provide accurate analysis due to their inherent limitations. Model checking has been widely used to analyze control systems but the continuous nature of their environment and physical components cannot be truly captured by a state-transition system in this technique. To overcome these limitations, we propose to use higher-order-logic theorem proving for analyzing linear control systems based on a formalized theory of the Laplace transform method. For this purpose, we have formalized the foundations of linear control system analysis in higher-order logic so that a linear control system can be readily modeled and analyzed. The paper presents a new formalization of the Laplace transform and the formal verification of its properties that are frequently used in the transfer function based analysis to judge the frequency response, gain margin and phase margin, and stability of a linear control system. We also formalize the active realizations of various controllers, like Proportional-Integral-Derivative (PID), Proportional-Integral (PI), Proportional-Derivative (PD), and various active and passive compensators, like lead, lag and lag-lead. For illustration, we present a formal analysis of an unmanned free-swimming submersible vehicle using the HOL Light theorem prover.Comment: International Conference on Formal Engineering Method

Attitudes, Beliefs, Motivation and Identity in Mathematics Education: An Overview of the Field and Future Directions

Mathematics Education; Educational Psycholog

From memories of the past to anticipations of the future:pre-service elementary teachers’ mathematical identity work

Abstract This study explored mathematical identity work by drawing on the cases of Finnish and Slovenian pre-service elementary teachers. All cases reported having had negative experiences with mathematics during their school years. These experiences were shown to have a central meaning for pre-service teachers’ mathematical identities. However, identity also extends to the future. For this reason, pre-service teachers’ anticipations of the future were also explored. The concepts of narrative identity (Ricoeur, 1992) and possible selves (Markus & Nurius, 1989) were applied in the context of mathematics education. The overall narrative perspective of the study enabled a psycho-social understanding of identity. The special interest of the study was confined to an understanding of the role that educational contexts play in pre-service teachers’ mathematical identity work. Narrative inquiry was applied as a research methodology. In-depth interviews invited pre-service teachers to construct narratives of their mathematics-related experiences. These narratives were analysed holistically and categorically, as well as in terms of content and form. The findings showed striking similarities in pre-service teachers’ school-time memories. The cases in question felt like victims of their own mathematical experiences. The anticipations of mathematics teaching were also underlined by the challenges rooted in their school-time experiences. However, a surprising finding was that the identity work in which the Finnish and Slovenian cases engaged during their teacher education differed substantially. The main reasons for the differences in identity work seemed to stem from different emphases and pedagogical practices in mathematics education courses within the Finnish and Slovenian teacher education settings. The study argued that identity work can be facilitated during teacher education. To begin such a process, it would be central to focus on pre-service teachers’ biographical context through narrative pedagogical tools. The findings also showed that neglecting issues from school-time experiences might engender further challenges for pre-service teachers’ future mathematics teaching. Finally, the study argued for the need to openly address identity during teacher education. The significant theoretical contribution of the study is the conceptualisation of ‘mathematical identity work’.Tiivistelmä Tutkimuksessa tarkasteltiin matemaattista identiteettityötä suomalaisten ja slovenialaisten luokanopettajaopiskelijoiden kokemusten kautta. Opiskelijoiden mukaan heillä oli ollut omana kouluaikanaan kielteisiä matematiikan opintoihin liittyviä kokemuksia, joilla osoitettiin olevan negatiivisia vaikutuksia opiskelijoiden matemaattisiin identiteetteihin. Koska tutkimuksessa korostuu identiteetin tulevaisuusaspekti, tarkastelun kohteina olivat opiskelijoiden tulevaisuuteen liittyvät toiveet ja odotukset. Tutkimuksessa sovellettiin narratiivisen identiteetin (Ricoeur, 1992) ja mahdollisten minuuksien (Markus & Nurius, 1989) käsitteitä matematiikan opetuksen kontekstissa. Identiteetin ymmärtämisen psyko-sosiaalisena ilmiönä mahdollisti narratiivinen näkökulma. Erityinen huomio kohdistettiin siihen, millainen merkitys kasvatuksellisilla konteksteilla on luokanopettajaopiskelijoiden matemaattisessa identiteettityössä. Tutkimusmetodologiana käytettiin narratiivista tutkimusta. Opiskelijat kertoivat syvähaastatteluissa matematiikkaan liittyvistä kokemuksistaan. Nämä narratiivit analysoitiin holistisesti ja kategorisesti ottaen huomioon myös niiden sisältö ja muoto. Tuloksista ilmenee merkittävää samankaltaisuutta luokanopettajaopiskelijoiden omaan kouluaikaan liittyvissä muistoissa. Monet esimerkiksi kuvailivat itsensä uhreiksi. Myös tulevaan matematiikan opetukseen liittyvät ennakko-odotukset olivat värittyneet opettajaopiskelijoiden omaan kouluaikaan liittyvien haasteellisten kokemusten kautta. Yllättävä tulos oli se, että suomalaisten ja slovenialaisten opiskelijoiden luokanopettajakoulutuksen aikainen identiteettityö erosi huomattavasti toisistaan. Erojen pääsyynä ovat nähtävästi erilaiset painotukset ja käytänteet opettajankoulutuksen matematiikan pedagogisissa opinnoissa. Tutkimus osoittaa, että identiteettityötä voidaan pyrkiä edistämään opettajankoulutuksen aikana. Prosessin aloittamiseksi olisi tärkeää kohdentaa huomio opettajaopiskelijoiden elämäkerrallisiin konteksteihin soveltamalla narratiivisia pedagogisia työkaluja. Sillä että omaan kouluaikaan liittyviä kokemuksia ei oteta huomioon, voi olla kielteisiä heijastuksia opettajaopiskelijoiden tulevaan matematiikan opetukseen. Tutkimuksen mukaan identiteetti on syytä ottaa avoimesti tarkasteluun opettajankoulutuksen aikana. Tutkimuksen teorian kannalta merkittävä anti on termin matemaattinen identiteettityö käsitteellistäminen

Pre-service mathematics teachers’ narrated failure:stories of resilience

Abstract In educational research, failure has often been touched upon as a by-product, but rarely has failure been investigated without attempting to find ways around it instead of attempting to simply understand it better. This paper provides insight into a pilot study on understanding math failure through narrative. I analyse two pre-service mathematics teachers’ narratives about failure and identity in order to illustrate their personal understandings of their own failures. The analysed stories provide insight into the subjects and their failure resilience, and initiate a discussion on subjective understanding of resilience. These stories also illustrate the need to further research regarding the subjective understandings of failure in general and in math specifically

How failure shapes teacher identities:pre-service elementary school and mathematics teachers’ narrated possible selves

Abstract This study explored pre-service teachers’ possible teacher selves with respect to how they have been shaped by their experiences of math failure. The study contributes to identity research by applying the theory of possible selves and by comparing and contrasting narrated possible teacher selves of pre-service elementary school teachers and pre-service mathematics teachers. Three categories of possible selves were identified: teacher traits and actions, student strategies, and teacher self-development. How possible teacher selves may inform teacher identity development and teacher preparation in the context of teaching mathematics is discussed, as are methodological considerations for examining narrated possible selves

Towards conceptualising failure in mathematics as an autobiographical experience

Abstract Studies have shown that failure experiences play a role in pre-service teachers’ development. Given that autobiographical experiences are a foundation of learning and that failure is a wide-spread experience, particularly in mathematics, we need further insight into what kind of experience failure actually is. This paper draws on 59 pre-service teachers’ written experiences of failure in order to map them out and provide insight into what counts as failure from the perspective of the future teachers of mathematics, that is, pre-service mathematics and pre-service elementary school teachers. The findings, alongside the earlier research on negative experiences, form a basis for conceptualising failure in mathematics as an autobiographical experience and distinguishing it from a negative experience. A theoretical insight into the nature of failure is gained; the failure experiences seem to be less relational than anticipated. The paper also discusses the relevance of failure experiences for teacher education

How to select reading for application of pedagogical bibliotherapy?:insights from prospective teachers’ identification processes

Abstract There is compelling evidence on the effectiveness of bibliotherapy for facilitating one’s development. Arguably, in teacher education, bibliotherapy has been shown to facilitate prospective teachers’ professional development. However, teacher educators may experience difficulties in finding relevant reading to stimulate prospective teachers’ identification process, which is essential if the application of pedagogical bibliotherapy is to be successful. This study is situated in a mathematics education course for prospective elementary school teachers and explores the features of relevant pedagogical reading via analysis of four prospective elementary school teachers’ identification processes with two case descriptions. The identification here relates to the reading of the so-called true stories; prospective teachers read stories of other prospective teachers from the same teacher education context. We outline the recommendations for the selection of reading material when applying pedagogical bibliotherapy in order to assist prospective teachers’ professional development

An elementary teacher’s narrative identity work at two points in time two decades apart

Abstract The number of studies on how the process of identity work takes place in pre- and in-service teacher training contexts has recently increased. This narrative study contributes to this body of work by examining one elementary teacher’s identity work in the context of teaching mathematics at two points in time—the present, as an experienced teacher, and two decades prior, as a pre-service teacher. As part of our narrative approach, this study introduces a biography-stimulated recall method and exemplifies its use. The findings account for continuity and change in a teacher’s identity over time and broaden understanding of influences on identity in pre- and in-service teaching stages. The central role of crisis in teachers’ identity development, particularly when initiated within teacher education contexts, is highlighted. Methodological considerations in narrative follow-up studies are discussed

Future directions in research on mathematics-related teacher identity

Abstract Mathematics education research has placed great emphasis on teacher identity, examining both pre- and in-service teachers, and within these cohorts, specialised mathematics teachers and non-specialists such as elementary teachers. Extensive research has already been done; hence, this paper discusses possible future directions for research on teacher identity in mathematics education. Among other issues, we highlight that general education research on identity has infrequently informed research on mathematics-related teacher identity. This not only limits the transfer of knowledge but also isolates mathematics education from general education research. We suggest that connecting these lines of research and their findings may not only strengthen mathematics education research and mathematics teaching and learning but also contribute to less isolation within the discipline as a whole