¿Para qué sirven los problemas en la clase de matemáticas? = What is the use of problems in Mathematics lessons?

The results obtained in several yield tests, at an international level (mainly the famous PISA 2003 report, by the OCDE), have raised a multiplicity of performances in order to improve the students' yield regarding problem solving. In this article we set a clear guideline on how problems should be used in Mathematics lessons, not for obtaining better scores in the yield tests but for improving the development of Mathematical thinking in students. From this perspective, the author analyses, through eight reflections, how the concept of problem, transmitted both in the school and from society, influences the students

Una visión actualizada de la didáctica de la matemática en educación infantil = An up-to-date vision of pre-school mathematics teaching

This article aims to investigate pre-school mathematics teaching from an uptodate perspective. To pursue this contemporary vision we focus on four key questions: what kind of maths is being worked on, who is doing it, how it is being done, and why it is being done

Proceedings of JAC 2010. Journées Automates Cellulaires

The second Symposium on Cellular Automata “Journ´ees Automates Cellulaires” (JAC 2010) took place in Turku, Finland, on December 15-17, 2010. The first two conference days were held in the Educarium building of the University of Turku, while the talks of the third day were given onboard passenger ferry boats in the beautiful Turku archipelago, along the route Turku–Mariehamn–Turku. The conference was organized by FUNDIM, the Fundamentals of Computing and Discrete Mathematics research center at the mathematics department of the University of Turku. The program of the conference included 17 submitted papers that were selected by the international program committee, based on three peer reviews of each paper. These papers form the core of these proceedings. I want to thank the members of the program committee and the external referees for the excellent work that have done in choosing the papers to be presented in the conference. In addition to the submitted papers, the program of JAC 2010 included four distinguished invited speakers: Michel Coornaert (Universit´e de Strasbourg, France), Bruno Durand (Universit´e de Provence, Marseille, France), Dora Giammarresi (Universit` a di Roma Tor Vergata, Italy) and Martin Kutrib (Universit¨at Gie_en, Germany). I sincerely thank the invited speakers for accepting our invitation to come and give a plenary talk in the conference. The invited talk by Bruno Durand was eventually given by his co-author Alexander Shen, and I thank him for accepting to make the presentation with a short notice. Abstracts or extended abstracts of the invited presentations appear in the first part of this volume. The program also included several informal presentations describing very recent developments and ongoing research projects. I wish to thank all the speakers for their contribution to the success of the symposium. I also would like to thank the sponsors and our collaborators: the Finnish Academy of Science and Letters, the French National Research Agency project EMC (ANR-09-BLAN-0164), Turku Centre for Computer Science, the University of Turku, and Centro Hotel. Finally, I sincerely thank the members of the local organizing committee for making the conference possible. These proceedings are published both in an electronic format and in print. The electronic proceedings are available on the electronic repository HAL, managed by several French research agencies. The printed version is published in the general publications series of TUCS, Turku Centre for Computer Science. We thank both HAL and TUCS for accepting to publish the proceedings.

Teaching mathematics and programming : new approaches with empirical evaluation

Programming and mathematics are core areas of computer science (CS) and consequently also important parts of CS education. Introductory instruction in these two topics is, however, not without problems. Studies show that CS students find programming difficult to learn and that teaching mathematical topics to CS novices is challenging. One reason for the latter is the disconnection between mathematics and programming found in many CS curricula, which results in students not seeing the relevance of the subject for their studies. In addition, reports indicate that students' mathematical capability and maturity levels are dropping. The challenges faced when teaching mathematics and programming at CS departments can also be traced back to gaps in students' prior education. In Finland the high school curriculum does not include CS as a subject; instead, focus is on learning to use the computer and its applications as tools. Similarly, many of the mathematics courses emphasize application of formulas, while logic, formalisms and proofs, which are important in CS, are avoided. Consequently, high school graduates are not well prepared for studies in CS. Motivated by these challenges, the goal of the present work is to describe new approaches to teaching mathematics and programming aimed at addressing these issues: Structured derivations is a logic-based approach to teaching mathematics, where formalisms and justifications are made explicit. The aim is to help students become better at communicating their reasoning using mathematical language and logical notation at the same time as they become more confident with formalisms. The Python programming language was originally designed with education in mind, and has a simple syntax compared to many other popular languages. The aim of using it in instruction is to address algorithms and their implementation in a way that allows focus to be put on learning algorithmic thinking and programming instead of on learning a complex syntax. Invariant based programming is a diagrammatic approach to developing programs that are correct by construction. The approach is based on elementary propositional and predicate logic, and makes explicit the underlying mathematical foundations of programming. The aim is also to show how mathematics in general, and logic in particular, can be used to create better programs.

The mathematics of McTaggart's paradox

Mc Taggart's celebrated proof of the unreality of time is a chain of implications whose final step asserts that the A-series (i.e. the classification of events as past, present or future) is intrinsically contradictory. This is widely believed to be the heart of the argument, and it is where most attempted refutations have been addressed; yet, it is also the only part of the proof which may be generalised to other contexts, since none of the notions involved in it is specifically temporal. In fact, as I show in the first part of the paper, McTaggart's refutation of the A-series can be easily interpreted in mathematical terms; subsequently, in order to strengthen my claim, I apply the same framework by analogy to the cases of space, modality, and personal identity. Therefore, either McTaggart's proof as a whole may be extended to each of these notions, or it must embed some distinctly temporal element in one of the steps leading up to the contradiction of the A-series. I conclude by suggesting where this element might lay, and by hinting at what I believe to be the true logical fallacy of the proof.

HILBERT BETWEEN THE FORMAL AND THE INFORMAL SIDE OF MATHEMATICS

Abstract: In this article we analyze the key concept of Hilbert's axiomatic method, namely that of axiom. We will find two different concepts: the first one from the period of Hilbert's foundation of geometry and the second one at the time of the development of his proof theory. Both conceptions are linked to two different notions of intuition and show how Hilbert's ideas are far from a purely formalist conception of mathematics. The principal thesis of this article is that one of the main problems that Hilbert encountered in his foundational studies consisted in securing a link between formalization and intuition. We will also analyze a related problem, that we will call "Frege's Problem", form the time of the foundation of geometry and investigate the role of the Axiom of Completeness in its solution.

Memristive Computing

Memristive computing refers to the utilization of the memristor, the fourth fundamental passive circuit element, in computational tasks. The existence of the memristor was theoretically predicted in 1971 by Leon O. Chua, but experimentally validated only in 2008 by HP Labs. A memristor is essentially a nonvolatile nanoscale programmable resistor — indeed, memory resistor — whose resistance, or memristance to be precise, is changed by applying a voltage across, or current through, the device. Memristive computing is a new area of research, and many of its fundamental questions still remain open. For example, it is yet unclear which applications would benefit the most from the inherent nonlinear dynamics of memristors. In any case, these dynamics should be exploited to allow memristors to perform computation in a natural way instead of attempting to emulate existing technologies such as CMOS logic. Examples of such methods of computation presented in this thesis are memristive stateful logic operations, memristive multiplication based on the translinear principle, and the exploitation of nonlinear dynamics to construct chaotic memristive circuits. This thesis considers memristive computing at various levels of abstraction. The first part of the thesis analyses the physical properties and the current-voltage behaviour of a single device. The middle part presents memristor programming methods, and describes microcircuits for logic and analog operations. The final chapters discuss memristive computing in largescale applications. In particular, cellular neural networks, and associative memory architectures are proposed as applications that significantly benefit from memristive implementation. The work presents several new results on memristor modeling and programming, memristive logic, analog arithmetic operations on memristors, and applications of memristors. The main conclusion of this thesis is that memristive computing will be advantageous in large-scale, highly parallel mixed-mode processing architectures. This can be justified by the following two arguments. First, since processing can be performed directly within memristive memory architectures, the required circuitry, processing time, and possibly also power consumption can be reduced compared to a conventional CMOS implementation. Second, intrachip communication can be naturally implemented by a memristive crossbar structure.

Conceptions of mathematics teacher education : thoughts among teacher educators in Tanzania

This study addresses the question of teacher educators’ conceptions of mathematics teacher education (MTE) in teacher colleges in Tanzania, and their thoughts on how to further develop it. The tension between exponents of content as opposed to pedagogy has continued to cause challenging conceptual differences, which also influences what teacher educators conceive as desirable in the development of this domain. This tension is connected to the dissatisfaction of parents and teachers with the failure of school mathematics. From this point of view, the overall aim was to identify and describe teacher educators’ various conceptions of MTE. Inspired by the debate among teacher educators about what the balance should be between subject matter and pedagogical knowledge, it was important to look at the theoretical faces of MTE. The theoretical background involved the review of what is visible in MTE, what is yet to be known and the challenges within the practice. This task revealed meanings, perspectives in MTE, professional development and assessment. To do this, two questions were asked, to which no clear solutions satisfactorily existed. The questions to guide the investigation were, firstly, what are teacher educators’ conceptions of MTE, and secondly, what are teacher educators’ thoughts on the development of MTE? The two questions led to the choice of phenomenography as the methodological approach. Against the guiding questions, 27 mathematics teacher educators were interviewed in relation to the first question, while 32 responded to an open-ended questionnaire regarding question two. The interview statements as well as the questionnaire responses were coded and analysed (classified). The process of classification generated patterns of qualitatively different ways of seeing MTE. The results indicate that MTE is conceived as a process of learning through investigation, fostering inspiration, an approach to learning with an emphasis on problem solving, and a focus on pedagogical knowledge and skills in the process of teaching and learning. In addition, the teaching and learning of mathematics is seen as subject didactics with a focus on subject matter and as an organized integration of subject matter, pedagogical knowledge and some school practice; and also as academic content knowledge in which assessment is inherent. The respondents also saw the need to build learner-educator relationships. Finally, they emphasized taking advantage of teacher educators’ neighbourhood learning groups, networking and collaboration as sustainable knowledge and skills sharing strategies in professional development. Regarding desirable development, teacher educators’ thoughts emphasised enhancing pedagogical knowledge and subject matter, and to be determined by them as opposed to conventional top-down seminars and workshops. This study has revealed various conceptions and thoughts about MTE based on teacher educators´ diverse history of professional development in mathematics. It has been reasonably substantiated that some teacher educators teach school mathematics in the name of MTE, hardly distinguishing between the role and purpose of the two in developing a mathematics teacher. What teacher educators conceive as MTE and what they do regarding the education of teachers of mathematics revealed variations in terms of seeing the phenomenon of interest. Within limits, desirable thoughts shed light on solutions to phobias, and in the same way low self-esteem and stigmatization call for the building of teacher educator-student teacher relationships.

Near-Threshold Computing

Valmistustekniikoiden kehittyessä IC-piireille saadaan mahtumaan yhä enemmän transistoreja. Monimutkaisemmat piirit mahdollistavat suurempien laskutoimitusmäärien suorittamisen aikayksikössä. Piirien aktiivisuuden lisääntyessä myös niiden energiankulutus lisääntyy, ja tämä puolestaan lisää piirin lämmöntuotantoa. Liiallinen lämpö rajoittaa piirien toimintaa. Tämän takia tarvitaan tekniikoita, joilla piirien energiankulutusta saadaan pienennettyä. Uudeksi tutkimuskohteeksi ovat tulleet pienet laitteet, jotka seuraavat esimerkiksi ihmiskehon toimintaa, rakennuksia tai siltoja. Tällaisten laitteiden on oltava energiankulutukseltaan pieniä, jotta ne voivat toimia pitkiä aikoja ilman akkujen lataamista. Near-Threshold Computing on tekniikka, jolla pyritään pienentämään integroitujen piirien energiankulutusta. Periaatteena on käyttää piireillä pienempää käyttöjännitettä kuin piirivalmistaja on niille alunperin suunnitellut. Tämä hidastaa ja haittaa piirin toimintaa. Jos kuitenkin laitteen toiminnassa pystyään hyväksymään huonompi laskentateho ja pienentynyt toimintavarmuus, voidaan saavuttaa säästöä energiankulutuksessa. Tässä diplomityössä tarkastellaan Near-Threshold Computing -tekniikkaa eri näkökulmista: aluksi perustuen kirjallisuudesta löytyviin aikaisempiin tutkimuksiin, ja myöhemmin tutkimalla Near-Threshold Computing -tekniikan soveltamista kahden tapaustutkimuksen kautta. Tapaustutkimuksissa tarkastellaan FO4-invertteriä sekä 6T SRAM -solua piirisimulaatioiden avulla. Näiden komponenttien käyttäytymisen Near-Threshold Computing –jännitteillä voidaan tulkita antavan kattavan kuvan suuresta osasta tavanomaisen IC-piirin pinta-alaa ja energiankulusta. Tapaustutkimuksissa käytetään 130 nm teknologiaa, ja niissä mallinnetaan todellisia piirivalmistusprosessin tuotteita ajamalla useita Monte Carlo -simulaatioita. Tämä valmistuskustannuksiltaan huokea teknologia yhdistettynä Near-Threshold Computing -tekniikkaan mahdollistaa matalan energiankulutuksen piirien valmistaminen järkevään hintaan. Tämän diplomityön tulokset näyttävät, että Near-Threshold Computing pienentää piirien energiankulutusta merkittävästi. Toisaalta, piirien nopeus heikkenee, ja yleisesti käytetty 6T SRAM -muistisolu muuttuu epäluotettavaksi. Pidemmät polut logiikkapiireissä sekä transistorien kasvattaminen muistisoluissa osoitetaan tehokkaiksi vastatoimiksi Near- Threshold Computing -tekniikan huonoja puolia vastaan. Tulokset antavat perusteita matalan energiankulutuksen IC-piirien suunnittelussa sille, kannattaako käyttää normaalia käyttöjännitettä, vai laskea sitä, jolloin piirin hidastuminen ja epävarmempi käyttäytyminen pitää ottaa huomioon.

On Energy Efficient Computing Platforms

In accordance with the Moore's law, the increasing number of on-chip integrated transistors has enabled modern computing platforms with not only higher processing power but also more affordable prices. As a result, these platforms, including portable devices, work stations and data centres, are becoming an inevitable part of the human society. However, with the demand for portability and raising cost of power, energy efficiency has emerged to be a major concern for modern computing platforms. As the complexity of on-chip systems increases, Network-on-Chip (NoC) has been proved as an efficient communication architecture which can further improve system performances and scalability while reducing the design cost. Therefore, in this thesis, we study and propose energy optimization approaches based on NoC architecture, with special focuses on the following aspects. As the architectural trend of future computing platforms, 3D systems have many bene ts including higher integration density, smaller footprint, heterogeneous integration, etc. Moreover, 3D technology can signi cantly improve the network communication and effectively avoid long wirings, and therefore, provide higher system performance and energy efficiency. With the dynamic nature of on-chip communication in large scale NoC based systems, run-time system optimization is of crucial importance in order to achieve higher system reliability and essentially energy efficiency. In this thesis, we propose an agent based system design approach where agents are on-chip components which monitor and control system parameters such as supply voltage, operating frequency, etc. With this approach, we have analysed the implementation alternatives for dynamic voltage and frequency scaling and power gating techniques at different granularity, which reduce both dynamic and leakage energy consumption. Topologies, being one of the key factors for NoCs, are also explored for energy saving purpose. A Honeycomb NoC architecture is proposed in this thesis with turn-model based deadlock-free routing algorithms. Our analysis and simulation based evaluation show that Honeycomb NoCs outperform their Mesh based counterparts in terms of network cost, system performance as well as energy efficiency.

What are the opportunities and challenges of cloud computing technology in the healthcare information systems

This thesis discusses the opportunities and challenges of the cloud computing technology in healthcare information systems by reviewing the existing literature on cloud computing and healthcare information system and the impact of cloud computing technology to healthcare industry. The review shows that if problems related to security of data are solved then cloud computing will positively transform the healthcare institutions by giving advantage to the healthcare IT infrastructure as well as improving and giving benefit to healthcare services. Therefore, this thesis will explore the opportunities and challenges that are associated with cloud computing in the context of Finland in order to help the healthcare organizations and stakeholders to determine its direction when it decides to adopt cloud technology on their information systems.

Video transcoding in a distributed cloud computing environment

Video transcoding refers to the process of converting a digital video from one format into another format. It is a compute-intensive operation. Therefore, transcoding of a large number of simultaneous video streams requires a large amount of computing resources. Moreover, to handle di erent load conditions in a cost-e cient manner, the video transcoding service should be dynamically scalable. Infrastructure as a Service Clouds currently offer computing resources, such as virtual machines, under the pay-per-use business model. Thus the IaaS Clouds can be leveraged to provide a coste cient, dynamically scalable video transcoding service. To use computing resources e ciently in a cloud computing environment, cost-e cient virtual machine provisioning is required to avoid overutilization and under-utilization of virtual machines. This thesis presents proactive virtual machine resource allocation and de-allocation algorithms for video transcoding in cloud computing. Since users' requests for videos may change at di erent times, a check is required to see if the current computing resources are adequate for the video requests. Therefore, the work on admission control is also provided. In addition to admission control, temporal resolution reduction is used to avoid jitters in a video. Furthermore, in a cloud computing environment such as Amazon EC2, the computing resources are more expensive as compared with the storage resources. Therefore, to avoid repetition of transcoding operations, a transcoded video needs to be stored for a certain time. To store all videos for the same amount of time is also not cost-e cient because popular transcoded videos have high access rate while unpopular transcoded videos are rarely accessed. This thesis provides a cost-e cient computation and storage trade-o strategy, which stores videos in the video repository as long as it is cost-e cient to store them. This thesis also proposes video segmentation strategies for bit rate reduction and spatial resolution reduction video transcoding. The evaluation of proposed strategies is performed using a message passing interface based video transcoder, which uses a coarse-grain parallel processing approach where video is segmented at group of pictures level.

Mathematics inspired by Darwin. Adaptive dynamics of dispersal and cooperation

In 1859, Charles Darwin published his theory of evolution by natural selection, the process occurring based on fitness benefits and fitness costs at the individual level. Traditionally, evolution has been investigated by biologists, but it has induced mathematical approaches, too. For example, adaptive dynamics has proven to be a very applicable framework to the purpose. Its core concept is the invasion fitness, the sign of which tells whether a mutant phenotype can invade the prevalent phenotype. In this thesis, four real-world applications on evolutionary questions are provided. Inspiration for the first two studies arose from a cold-adapted species, American pika. First, it is studied how the global climate change may affect the evolution of dispersal and viability of pika metapopulations. Based on the results gained here, it is shown that the evolution of dispersal can result in extinction and indeed, evolution of dispersalshould be incorporated into the viability analysis of species living in fragmented habitats. The second study is focused on the evolution of densitydependent dispersal in metapopulations with small habitat patches. It resulted a very surprising unintuitive evolutionary phenomenon, how a non-monotone density-dependent dispersal may evolve. Cooperation is surprisingly common in many levels of life, despite of its obvious vulnerability to selfish cheating. This motivated two applications. First, it is shown that density-dependent cooperative investment can evolve to have a qualitatively different, monotone or non-monotone, form depending on modelling details. The last study investigates the evolution of investing into two public-goods resources. The results suggest one general path by which labour division can arise via evolutionary branching. In addition to applications, two novel methodological derivations of fitness measures in structured metapopulations are given.