¿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

Rebuilding Defenses: Explorations in Norman Fairclough’s Theory of Critical Discourse Analysis

Tutkielmassa eritellään Norman Faircloughin kriittisen diskurssianalyysin teoriaa ja siihen kohdistettua kritiikkiä. Pyrkimyksenä on sovittaa näitä erilaisia näkemyksiä keskenään ja tarjota ratkaisuja yhteen kiriittisen diskurssianalyysin keskeiseen ongelmaan eli emansipaation (sosiaalisten epäkohtien tunnistamisen ja ratkaisemisen) puutteellisuuteen. Teoriaosuudesta esiin nousevia mahdollisuuksia sovelletaan tekstianalyysiin. Tutkimuksen kohteena on teksti Rebuilding America’s Defenses: Strategy, Forces and Resources For a New Century ja jossain määrin sen tuottanut järjestö Project for the New American Century. Näitä tarkastellaan ennen kaikkea sosiaalisina ilmiöinä ja suhteessa toisiinsa. Faircloughin mallin suurimmiksi ongelmiksi muodostuvat perinteinen käsitys kielestä, jonka mukaan kielen järjestelmän abstraktit ja sisäiset suhteet ovat tärkeimpiä, sekä ideologinen vastakkainasettelu kritiikin lähtökohtana. Ensimmäinen johtaa kielellisten tutkimustulosten epätyydyttävään kykyyn selittää sosiaalisia havaintoja ja jälkimmäinen poliittiseen tai maailmankatsomukselliseen väittelyyn, joka ei mahdollista uusia näkemyksiä. Tutkielman lopputulema on, että keskittymällä asiasisältöön kielen rakenteen sijasta ja ymmärtämällä tekstin tuottaja yksittäisenä, rajattuna sosiaalisena toimijana voidaan analyysiin saada avoimuutta ja täsmällisyyttä. Kriittiinen diskurssianalyysi kaipaa tällaista näkemystä kielellisten analyysien tueksi ja uudenlaisen relevanssin löytääkseen.

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.

Perspectives interculturelles et interlinguistiques sur le discours académique – Crosscultural and Crosslinguistic Perspectives on Academic Discourse. Volume 1

The Department of French Studies of the University of Turku (Finland) organized an International Bilingual Conference on Crosscultural and Crosslinguistic Perspectives on Academic Discourse from 2022 May 2005. The event hosted specialists on Academic Discourse from Belgium, Finland, France, Germany, Italy, Norway, Spain, and the USA. This book is the first volume in our series of publications on Academic Discourse (AD hereafter). The following pages are composed of selected papers from the conference and focus on different aspects and analytical frameworks of Academic Discourse. One of the motivations behind organizing the conference was to examine and expand research on AD in different languages. Another one was to question to what extent academic genres are culturebound and language specific or primarily field or domain specific. The research carried out on AD has been mainly concerned with the use of English in different academic settings for a long time now – mainly written contexts – and at the expense of other languages. Alternatively the academic genre conventions of English and English speaking world have served as a basis for comparison with other languages and cultures. We consider this first volume to be a strong contribution to the spreading out of researches based on other languages than English in AD, namely Finnish, French, Italian, Norwegian and Romanian in this book. All the following articles have a strong link with the French language: either French is constitutive of the AD corpora under examination or the article was written in French. The structure of the book suggests and provides evidence that the concept of AD is understood and tackled to varying degrees by different scholars. Our first volume opens up the discussion on what AD is and backs dissemination, overlapping and expansion of current research questions and methodologies. The book is divided into three parts and contains four articles in English and six articles in French. The papers in part one and part two cover what we call the prototypical genre of written AD, i.e. the research article. Part one follows up on issues linked to the 13 Research Article (RA hereafter). Kjersti Fløttum asks wether a typical RA exists and concentrates on authors’ voices in RA (self and other dimensions), whereas Didriksen and Gjesdal’s article focuses on individual variation of the author’s voice in RA. The last article in this section is by Nadine Rentel and deals with evaluation in the writing of RA. Part two concentrates on the teaching and learning of AD within foreign language learning, another more or less canonical genre of AD. Two aspects of writing are covered in the first two articles: foreign students’ representations on rhetorical traditions (Hidden) and a contrastive assessment of written exercices in French and Finnish in Higher Education (Suzanne). The last contribution in this section on AD moves away from traditional written forms and looks at how argumentation is constructed in students’ oral presentations (Dervin and Fauveau). The last part of the book continues the extension by featuring four articles written in French exploring institutional and scientific discourses. Institutional discourses under scrutiny include the European Bologna Process (Galatanu) and Romanian reform texts (Moilanen). As for scientific discourses, the next paper in this section deconstructs an ideological discourse on the didactics of French as a foreign language (Pescheux). Finally, the last paper in part three reflects on varied forms of AD at university (Defays). We hope that this book will add some fuel to continue discussing diverse forms of and approches to AD – in different languages and voices! No need to say that with the current upsurge in academic mobility, reflecting on crosscultural and crosslinguistic AD has just but started.

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.

DISAGREEING OVER EVALUATIVES: PREFERENCE, NORMATIVE AND MORAL DISCOURSE

Abstract Why would we argue about taste, norms or morality when we know that these topics are relative to taste preferences, systems of norms or values to which we are committed? Yet, disagreements over these topics are common in our evaluative discourses. I will claim that the motives to discuss rely on our attitudes towards the standard held by the speakers in each domain of discourse, relating different attitudes to different motives -mainly, conviction and correction. These notions of attitudes and motives will allow me to claim that different domains of evaluative discourse have a different distribution of disagreements driven by them.

A Bridge Builder or a Raconteur? How Japan Manoeuvres the Global Development Discourse within the TICAD Process

Japan has been a major actor in the field of development cooperation for five decades, even holding the title of largest donor of Official Development Assistance (ODA) during the 1990s. Financial flows, however, are subject to pre-existing paradigms that dictate both donor and recipient behaviour. In this respect Japan has been left wanting for more recognition. The dominance of the so called ‘Washington Consensus’ embodied in the International Monetary Fund (IMF) and the World Bank has long circumvented any indigenous approaches to development problems. The Tokyo International Conference on African Development (TICAD) is a development cooperation conference that Japan has hosted since 1993 every five years. As the main organizer of the conference Japan has opted for the leading position of African development. This has come in the wake of success in the Asian region where Japan has called attention to its role in the so called ‘Asian Miracle’ of fast growing economies. These aspirations have enabled Japan to try asserting itself as a major player in directing the course of global development discourse using historical narratives from both Asia and Africa. Over the years TICAD has evolved into a continuous process with ministerial and follow-up meetings in between conferences. Each conference has produced a declaration that stipulates the way the participants approach the question of African development. Although a multilateral framework, Japan has over the years made its presence more and more felt within the process. This research examines the way Japan approaches the paradigms of international development cooperation and tries to direct them in the context of the TICAD process. Supplementing these questions are inquiries concerning Japan’s foreign policy aspirations. The research shows that Japan has utilized the conference platform to contest other development actors and especially the dominant forces of the IMF and the World Bank in development discourse debate. Japan’s dominance of the process is evident in the narratives found in the conference documents. Relative success has come about by remaining consistent as shown by the acceptance of items from the TICAD agenda in other forums, such as the G8. But the emergence of new players such as China has changed the playing field, as they are engaging other developing countries from a more equal level.

"It's Barack Obama Who's Soft" Political Bias on BBC, CNN, and Fox News: A Critical Discourse Analysis of News Websites

Tutkielmassa analysoidaan kolmen internetsivuston uutisartikkeleita kielitieteen näkökulmasta. Tavoitteena on selvittää esiintyykö internetsivustojen BBC, CNN ja Fox News uutisoinnissa politiikkaan liittyviä ennakkoasenteita tai puolueellisuuksia ja miten ne käytännössä näkyvät uutisartikkeleiden kielessä. Kriittiseen diskurssianalyysiin pohjautuen tutkielma esittelee jokaisen uutissivuston taustaan (esimerkiksi rakenteeseen ja rahoitukseen) liittyviä seikkoja sekä mediadiskurssiin ja politiikkaan liittyvät taustatiedot, jolla taataan Norman Fairclough'n kolmivaiheisen menetelmän mahdollisimman perusteellinen toteuttaminen. Uutissivustoja analysoidaan kriittiselle diskurssianalyysille sopivan funktionaalisen kieliopin ja muiden lingvististen välineiden avulla. Koko aineiston (404 artikkelia) otsikot analysoidaan ensin, minkä jälkeen analysoidaan yhdeksän kokonaista artikkelia kolmeen eri aihealueeseen liittyen niin, että jokaiselta internetsivustolta analysoidaan yksi artikkeli jokaista aihetta kohden. Analyysikeinoina käytetään ensisijaisesti systeemis-funktionaalisen kieliopin tekstuaalisen metafunktion välineitä (thematic structure). Myös ideationaalisen metafunktion välineitä (transitivity), referenssiketjuja (referential identity chains) ja leksikaalista analyysia käytetään hyväksi. Lähtökohtaisesti tavoitteena on analysoida uutissivustoja vertailevasti, jolloin analyysin tulokset ovat paremmin havainnoitavissa ja perusteltavissa. Hypoteesi aikaisempien tutkimusten ja yleisen mielikuvan perusteella on, että CNN uutisoi demokraattipuolueelle ja Fox News taas republikaanipuolueelle edulliseen sävyyn. Tutkimustulokset vaihtelivat hypoteesia tukevista ja sen vastaisista tuloksista niihin, jotka eivät olleet tarpeeksi tuettuja kumpaankaan suuntaan. Vahvimmat tulokset ovat kuitenkin hypoteesia tukevia, joten tässä tutkielmassa todetaan, ettei uutisointi ole puolueetonta ainakaan näiden kolmen internetsivuston kohdalla. Lisäksi muutaman aihealueen kohdalla uutisointi on niin toistuvaa tietystä näkökulmasta, että luonnollistumisteorian mukaista aatteiden luonnollistumista saattaa tapahtua. Tutkielmassa käytettyjen menetelmien menestyksen perusteella suositellaan, että tekstuaalisen metafunktion analyysivälineitä käytetään enemmän. Lisäksi suositellaan meta-analyysin harkitsemista, jotta voitaisiin selvittää, mitkä analyysimetodit parhaiten sopivat minkäkinlaisen aineiston analysointiin.

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.

Irruptive discourse and conflicted curiosity

Kirjallisuusarvostelu