Proceso de transformación de las concepciones del profesorado sobre la resolución de problemas matemáticos

The aim of this study is to analyze the transformation of Primary School teachers’ conceptions about mathematical problem solving. We performed a study with 18 teachers from three public schools: in each class (from 1º to 6º) there were two interventions, and we were interviewed teachers before and after them. The results have show identified changes in: 1) teacher’s expectations about students’ abilities; classroom management; perception of diversity; mathematical strategies used by students; communication in the classroom; causes of the problems encountered; and relevance process of problem solving in mathematics teaching. The transformation of teachers’ conceptions is due to the following factors: a) awareness of the practice; b) systematic reflection; c) the contrast between different ways to work solving problems in math class

Hållpunkter för lärande : småbarns möten

Developed from human activities, mathematical knowledge is bound to the world and cultures that men and women experience. One can say that mathematics is rooted in humans’ everyday life, an environment where people reach agreement regarding certain “laws” and principles in mathematics. Through interaction with worldly phenomena and people, children will always gain experience that they can then in turn use to understand future situations. Consequently, the environment in which a child grows up plays an important role in what that child experiences and what possibilities for learning that child has. Variation theory, a branch of phenomenographical research, defines human learning as changes in understanding and acting towards a specific phenomenon. Variation theory implies a focus on that which it is possible to learn in a specific learning situation, since only a limited number of critical aspects of a phenomenon can be simultaneously discerned and focused on. The aim of this study is to discern how toddlers experience and learn mathematics in a daycare environment. The study focuses on what toddlers experience, how their learning experience is formed, and how toddlers use their understanding to master their environment. Twenty-three children were observed videographically during everyday activities. The videographic methodology aims to describe and interpret human actions in natural settings. The children are aged from 1 year, 1 month to 3 years, 9 months. Descriptions of the toddlers’ actions and communication with other children and adults are analyzed phenomenographically in order to discover how the children come to understand the different aspects of mathematics they encounter. The study’s analysis reveals that toddlers encounter various mathematical concepts, similarities and differences, and the relationship between parts and whole. Children form their understanding of such aspects in interaction with other children and adults in their everyday life. The results also show that for a certain type of learning to occur, some critical conditions must exist. Variation, simultaneity, reasonableness and fixed points are critical conditions of learning that appear to be important for toddlers’ learning. These four critical conditions are integral parts of the learning process. How children understand mathematics influences how they use mathematics as a tool to master their surrounding world. The results of the study’s analysis of how children use their understanding of mathematics shows that children use mathematics to uphold societal rules, to describe their surrounding world, and as a tool for problem solving. Accordingly, mathematics can be considered a very important phenomenon that children should come into contact with in different ways and which needs to be recognized as a necessary part of children’s everyday life. Adults working with young children play an important role in setting perimeters for children’s experiences and possibilities to explore mathematical concepts and phenomena. Therefore, this study is significant as regards understanding how children learn mathematics through everyday activities.

Quasiconformal mappings and inequalities involving special functions

This PhD thesis in Mathematics belongs to the field of Geometric Function Theory. The thesis consists of four original papers. The topic studied deals with quasiconformal mappings and their distortion theory in Euclidean n-dimensional spaces. This theory has its roots in the pioneering papers of F. W. Gehring and J. Väisälä published in the early 1960’s and it has been studied by many mathematicians thereafter. In the first paper we refine the known bounds for the so-called Mori constant and also estimate the distortion in the hyperbolic metric. The second paper deals with radial functions which are simple examples of quasiconformal mappings. These radial functions lead us to the study of the so-called p-angular distance which has been studied recently e.g. by L. Maligranda and S. Dragomir. In the third paper we study a class of functions of a real variable studied by P. Lindqvist in an influential paper. This leads one to study parametrized analogues of classical trigonometric and hyperbolic functions which for the parameter value p = 2 coincide with the classical functions. Gaussian hypergeometric functions have an important role in the study of these special functions. Several new inequalities and identities involving p-analogues of these functions are also given. In the fourth paper we study the generalized complete elliptic integrals, modular functions and some related functions. We find the upper and lower bounds of these functions, and those bounds are given in a simple form. This theory has a long history which goes back two centuries and includes names such as A. M. Legendre, C. Jacobi, C. F. Gauss. Modular functions also occur in the study of quasiconformal mappings. Conformal invariants, such as the modulus of a curve family, are often applied in quasiconformal mapping theory. The invariants can be sometimes expressed in terms of special conformal mappings. This fact explains why special functions often occur in this theory.

”Kielestä kiinni”. Kuudesluokkalaisten maahanmuuttajaoppilaiden suomen kielen tason vaihtelut

Tämän tutkimuksen tarkoituksena on selvittää, minkälaista vaihtelua esiintyy maahanmuuttajaoppilaiden suomen kielen taidoissa peruskoulun kuudennella luokalla. Tutkimuksen tavoitteena on myös selvittää, minkälainen yhteys taustamuuttujilla (sukupuoli, äidinkieli, maahantuloikä, maahantulon syy, maassaoloaika ja vanhempien koulutausta) ja opetusjärjestelyillä, kuten perusopetukseen valmistavalla opetuksella, suomi toisena kielenä -opetuksella ja oman äidinkielen opetuksella, on suomen kielen taidon tasoon. Lisäksi tutkimuksen tavoitteena on selvittää oppilaan käyttämän kielen (suomen kieli ja äidinkieli) yhteyttä suomen kielen taidon tasoon. Tutkimusmetodina toimi mixed methods -tutkimus, ja tutkimuksen lähestymistapoja olivat kvantitatiivinen survey-tutkimus ja kvalitatiivinen sisällön analyysi. Tutkimukseen osallistui 219 maahanmuuttajaoppilasta 20:stä Turun koulusta. Tutkimusaineisto kerättiin Turun erityisopettajien ja suomi toisena kielenä -opettajien laatiman kielitestipaketin avulla. Oppilaan suullista ja kirjallista tuottamista arvioivat lasta opettavat opettajat eurooppalaisen viitekehyksen kielitaitotasojen kriteereitä käyttäen. Oppilaat arvioivat omaa äidinkielen ja suomen kielen taitoaan. Lisäksi oppilaat ja vanhemmat täyttivät tutkijan laatimat taustatietolomakkeet. Kielitestien tulosten mukaan oppilaista yli puolella oli tyydyttävä suomen kielen taito. Kaikista neljästä kielellisestä osiosta maahanmuuttajataustaiset oppilaat menestyivät parhaiten rakennekokeessa ja sanelussa, kun taas kuullun ja luetun ymmärtämisen tulokset olivat heikompia. Opettajien arviointien perusteella oppilaiden suulliset taidot vastasivat keskimäärin itsenäisen kielenkäyttäjän osaajan tasoa (B2) ja kirjoittamistaidot kynnystasoa (B1). Oppilaiden suomi toisena kielenä -arvosanan keskiarvo oli 7,26. Suomessa asumisen kestolla, maahantulon syyllä, äidinkielellä, maahantuloiällä, ja vanhempien koulutaustaustalla oli tilastollisesti merkitsevä yhteys suomen kielen taidon tasoon. Mitä kauemmin oppilaat olivat asuneet Suomessa ja mitä nuorempina he olivat tulleet Suomeen, sitä paremmin he menestyivät kielitesteissä. Paluumuuttajat menestyivät kielitaitotehtävissä kaikkein parhaiten ja pakolaiset heikoiten. Somalinkieliset erottuivat muista kieliryhmän edustajista heikoimpina suomen kielen taidon tasoltaan. Venäjänkieliset ja vietnaminkieliset saavuttivat parhaat tulokset kaikissa mittareissa. Erityisesti äidin korkeampi koulutustaso oli yhteydessä oppilaiden korkeampaan suomen kielen taidon tasoon. Oppilaat arvioivat suomen kielen taitonsa omaa äidinkieltään paremmaksi puhumisessa, lukemisessa ja kirjoittamisessa. Parhaiten eri mittareissa menestyivät oppilaat, jotka eivät olleet osallistuneet perusopetukseen valmistavaan opetukseen eivätkä erilliseen suomi toisena kielenä -opetukseen. Omaa äidinkieltään enemmän opiskelleet menestyivät kielitaitotehtävissä paremmin kuin vähän aikaa omaa äidinkieltään opiskelleet, mutta yhtä hyvin kuin ne, jotka eivät olleet opiskelleet omaa äidinkieltään lainkaan. Oppilaat, jotka puhuivat kaveriensa kanssa sekä omaa äidinkieltään että suomen kieltä, osoittautuivat kielitaidon tasoltaan paremmiksi kielitesteissä ja opettajien arvioinneissa.

Yläkoululaisten ymmärrys yhtäsuuruusmerkistä ja sen vaikutukset oppilaiden algebralliseen osaamiseen

Tämä tutkielma on kirjallisuuskatsaus, joka keskittyy yläkoululaisten ymmärrykseen yhtäsuuruusmerkistä. Tarkoitus on selvittää, millainen näkemys yhtäsuuruusmerkistä yläkoululaisilla on, ja toisaalta myös, miten erilaiset näkemykset vaikuttavat oppilaiden algebralliseen osaamiseen. Tärkeimpiä lähdejulkaisuja ovat olleet Journal for Research in Mathematics Education ja Research in Mathematics Education. Tavoite on ollut löytää kaikki aiheeseen liittyvä aineisto ja sen pohjalta esitellä kattavasti yläkoululaisten näkemyksiä yhtäsuuruusmerkistä sekä niiden vaikutuksia muuhun matemaattiseen osaamiseen. Tutkielman keskeinen tulos on, että suuri osa yläkoululaisista omaa operationaalisen näkemyksen eli näkee yhtäsuuruusmerkin vastausta ilmaisevana symbolina. Yhtäsuuruusmerkin ilmaisema relaatio eli sen eri puolien välinen suhde jää usein huomioimatta oppilailta, mikä vaikuttaa negatiivisesti esimerkiksi yhtälönratkaisutaitoon. Operationaalisen näkemyksen yleisyyteen löydettiin syitä käytetystä oppimateriaalista ja sen painottumisesta yhtälöihin, joissa vasemmalla puolella yhtäsuuruusmerkkiä ovat laskutoimitukset ja oikealla puolella vastaus. Relaation ymmärtämistä eli relationaalista näkemystä edistäisivät yhtälöt, joissa molemmilla puolilla yhtäsuuruusmerkkiä on laskutoimituksia. Saatujen tulosten pohjalta yhtäsuuruusmerkin opettamiseen ja sen ilmaiseman relaation ymmärtämiseen tulisi kiinnittää enemmän huomiota sekä suosia erilaisia relationaalista näkemystä vahvistavia yhtälötyyppejä.

Husserl's approach to transcendental intersubjectivity

It is our intention in the course of the development of this thesis to give an account of how intersubjectivity is "eidetically" constituted by means of the application of the phenomenological reduction to our experience in the context of the thought of Edmund Husserl; contrasted with various representative thinkers in what H. Spiegelberg refers to as "the wider scene" of phenomenology. That is to say, we intend to show those structures of both consciousness and the relation which man has to the world which present themselves as the generic conditions for the possibility of overcoming our "radical sol itude" in order that we may gain access to the mental 1 ife of an Other as other human subject. It is clear that in order for us to give expression to these accounts in a coherent manner, along with their relative merits, it will be necessary to develop the common features of any phenomenological theory of consdousness whatever. Therefore, our preliminary inquiry, subordinate to the larger theme, shall be into some of the epistemological results of the application of the phenomenological method used to develop a transcendental theory of consciousness. Inherent in this will be the deliniation of the exigency for making this an lIintentional ll theory. We will then be able to see how itis possible to overcome transcendentally the Other as an object merely given among other merely given objects, and further, how this other is constituted specifically as other ego. The problem of transcendental intersubjectivity and its constitution in experience can be viewed as one of the most compelling, if not the most polemical of issues in phenomenology. To be sure, right from the beginning we are forced to ask a number of questions regarding Husserl's responses to the problem within the context of the methodological genesis of the Cartesian Meditations, and The Crisis of European Sciences and Transcendental Phenomenology. This we do in order to set the stage for amplification. First, we ask, has Husserl lived up to his goal, in this connexion, of an apodictic result? We recall that in his Logos article of 1911 he adminished that previous philosophy does not have at its disposal a merely incomplete and, in particular instances, imperfect doctrinal system; it simply has none whatever. Each and every question is herein controverted, each position is a matter of individual conviction, of the interpretation given byaschool, of a "point of view". 1. Moreover in the same article he writes that his goal is a philosophical system of doctrine that, after the gigantic preparatory work. of generations, really be- . gins from the ground up with a foundation free from doubt and rises up like any skilful construction, wherein stone is set upon store, each as solid as the other, in accord with directive insights. 2. Reflecting upon the fact that he foresaw "preparatory work of generations", we perhaps should not expect that he would claim that his was the last word on the matter of intersubjectivity. Indeed, with 2. 'Edmund Husserl, lIPhilosophy as a Rigorous Science" in Phenomenology and theCrisis6fPhilosophy, trans". with an introduction by Quentin Lauer (New York.: Harper & Row, 1965) pp. 74 .. 5. 2Ibid . pp. 75 .. 6. 3. the relatively small amount of published material by Husserl on the subject we can assume that he himself was not entirely satisfied with his solution. The second question we have is that if the transcendental reduction is to yield the generic and apodictic structures of the relationship of consciousness to its various possible objects, how far can we extend this particular constitutive synthetic function to intersubjectivity where the objects must of necessity always remain delitescent? To be sure, the type of 'object' here to be considered is unlike any other which might appear in the perceptual field. What kind of indubitable evidence will convince us that the characteristic which we label "alter-ego" and which we attribute to an object which appears to resemble another body which we have never, and can never see the whole of (namely, our own bodies), is nothing more than a cleverly contrived automaton? What;s the nature of this peculiar intentional function which enables us to say "you think just as I do"? If phenomenology is to take such great pains to reduce the takenfor- granted, lived, everyday world to an immanent world of pure presentation, we must ask the mode of presentation for transcendent sub .. jectivities. And in the end, we must ask if Husserl's argument is not reducible to a case (however special) of reasoning by analogy, and if so, tf this type of reasoning is not so removed from that from whtch the analogy is made that it would render all transcendental intersubjective understandtng impos'sible? 2. HistoticalandEidetic Priority: The Necessity of Abstraction 4. The problem is not a simple one. What is being sought are the conditions for the poss ibili:ty of experi encing other subjects. More precisely, the question of the possibility of intersubjectivity is the question of the essence of intersubjectivity. What we are seeking is the absolute route from one solitude to another. Inherent in this programme is the ultimate discovery of the meaning of community. That this route needs be lIabstract" requires some explanation. It requires little explanation that we agree with Husserl in the aim of fixing the goal of philosophy on apodictic, unquestionable results. This means that we seek a philosophical approach which is, though, not necessarily free from assumptions, one which examines and makes explicit all assumptions in a thorough manner. It would be helpful at this point to distinguish between lIeidetic ll priority, and JlhistoricallJpriority in order to shed some light on the value, in this context, of an abstraction.3 It is true that intersubjectivity is mundanely an accomplished fact, there havi.ng been so many mi.llions of years for humans to beIt eve in the exi s tence of one another I s abili ty to think as they do. But what we seek is not to study how this proceeded historically, but 3Cf• Maurice Natanson;·TheJburne in 'Self, a Stud in Philoso h and Social Role (Santa Cruz, U. of California Press, 1970 . rather the logical, nay, "psychological" conditions under which this is possible at all. It is therefore irrelevant to the exigesis of this monograph whether or not anyone should shrug his shoulders and mumble IIwhy worry about it, it is always already engaged". By way of an explanation of the value of logical priority, we can find an analogy in the case of language. Certainly the language 5. in a spoken or written form predates the formulation of the appropriate grammar. However, this grammar has a logical priority insofar as it lays out the conditions from which that language exhibits coherence. The act of formulating the grammar is a case of abstraction. The abstraction towards the discovery of the conditions for the poss; bi 1 ity of any experiencing whatever, for which intersubjective experience is a definite case, manifests itself as a sort of "grammar". This "grammar" is like the basic grammar of a language in the sense that these "rulesil are the ~ priori conditions for the possibility of that experience. There is, we shall say, an "eidetic priority", or a generic condition which is the logical antecedent to the taken-forgranted object of experience. In the case of intersubjectivity we readily grant that one may mundanely be aware of fellow-men as fellowmen, but in order to discover how that awareness is possible it is necessary to abstract from the mundane, believed-in experience. This process of abstraction is the paramount issue; the first step, in the search for an apodictic basis for social relations. How then is this abstraction to be accomplished? What is the nature of an abstraction which would permit us an Archimedean point, absolutely grounded, from which we may proceed? The answer can be discovered in an examination of Descartes in the light of Husserl's criticism. 3. The Impulse for Scientific Philosophy. The Method to which it Gives Rise. 6. Foremost in our inquiry is the discovery of a method appropriate to the discovery of our grounding point. For the purposes of our investigations, i.e., that of attempting to give a phenomenological view of the problem of intersubjectivity, it would appear to be of cardinal importance to trace the attempt of philosophy predating Husserl, particularly in the philosophy of Descartes, at founding a truly IIscientific ll philosophy. Paramount in this connexion would be the impulse in the Modern period, as the result of more or less recent discoveries in the natural sciences, to found philosophy upon scientific and mathematical principles. This impulse was intended to culminate in an all-encompassing knowledge which might extend to every realm of possible thought, viz., the universal science ot IIMathexis Universalis ll •4 This was a central issue for Descartes, whose conception of a universal science would include all the possible sciences of man. This inclination towards a science upon which all other sciences might be based waS not to be belittled by Husserl, who would appropriate 4This term, according to Jacab Klein, was first used by Barocius, the translator of Proclus into Latin, to designate the highest mathematical discipline. . 7. it himself in hopes of establishing, for the very first time, philosophy as a "rigorous science". It bears emphasizing that this in fact was the drive for the hardening of the foundations of philosophy, the link between the philosophical projects of Husserl and those of the philosophers of the modern period. Indeed, Husserl owes Descartes quite a debt for indicating the starting place from which to attempt a radical, presupositionless, and therefore scientific philosophy, in order not to begin philosophy anew, but rather for the first time.5 The aim of philosophy for Husserl is the search for apodictic, radical certitude. However while he attempted to locate in experience the type of necessity which is found in mathematics, he wished this necessity to be a function of our life in the world, as opposed to the definition and postulation of an axiomatic method as might be found in the unexpurgated attempts to found philosophy in Descartes. Beyond the necessity which is involved in experiencing the world, Husserl was searching for the certainty of roots, of the conditi'ons which underl ie experience and render it pOssible. Descartes believed that hi~ MeditatiOns had uncovered an absolute ground for knowledge, one founded upon the ineluctable givenness of thinking which is present even when one doubts thinking. Husserl, in acknowledging this procedure is certainly Cartesian, but moves, despite this debt to Descartes, far beyond Cartesian philosophy i.n his phenomenology (and in many respects, closer to home). 5Cf. Husserl, Philosophy as a Rigorous Science, pp. 74ff. 8 But wherein lies this Cartesian jumping off point by which we may vivify our theme? Descartes, through inner reflection, saw that all of his convictions and beliefs about the world were coloured in one way or another by prejudice: ... at the end I feel constrained to reply that there is nothing in a all that I formerly believed to be true, of which I cannot in some measure doubt, and that not merely through want of thought or through levity, but for reasons which are very powerful and maturely considered; so that henceforth I ought not the less carefully to refrain from giving credence to these opinions than to that which is manifestly false, if I desire to arrive at any certainty (in the sciences). 6 Doubts arise regardless of the nature of belief - one can never completely believe what one believes. Therefore, in order to establish absolutely grounded knowledge, which may serve as the basis fora "universal Science", one must use a method by which one may purge oneself of all doubts and thereby gain some radically indubitable insight into knowledge. Such a method, gescartes found, was that, as indicated above by hi,s own words, of II radical doubt" which "forbids in advance any judgemental use of (previous convictions and) which forbids taking any position with regard to their val idi'ty. ,,7 This is the method of the "sceptical epoche ll , the method of doubting all which had heretofor 6Descartes,Meditations on First Philosophy, first Med., (Libera 1 Arts Press, New York, 1954) trans. by L. LaFl eur. pp. 10. 7Husserl ,CrisiS of Eliroeari SCiences and Trariscendental Phenomenology, (Northwestern U. Press, Evanston, 1 7 ,p. 76. 9. been considered as belonging to the world, including the world itself. What then is left over? Via the process of a thorough and all-inclusive doubting, Descartes discovers that the ego which performs the epoche, or "reduction", is excluded from these things which can be doubted, and, in principle provides something which is beyond doubt. Consequently this ego provides an absolute and apodictic starting point for founding scientific philosophy. By way of this abstention. of bel ief, Desca'rtes managed to reduce the worl d of everyday 1 ife as bel ieved in, to mere 'phenomena', components of the rescogitans:. Thus:, having discovered his Archimedean point, the existence of the ego without question, he proceeds to deduce the 'rest' of the world with the aid of innate ideas and the veracity of God. In both Husserl and Descartes the compelling problem is that of establ ishing a scientific, apodictic phi'losophy based upon presuppos itionless groundwork .. Husserl, in thi.s regard, levels the charge at Descartes that the engagement of his method was not complete, such that hi.S: starting place was not indeed presupositionless, and that the validity of both causality and deductive methods were not called into question i.'n the performance of theepoche. In this way it is easy for an absolute evidence to make sure of the ego as: a first, "absolute, indubitablyexisting tag~end of the worldll , and it is then only a matter of inferring the absolute subs.tance and the other substances which belon.g to the world, along with my own mental substance, using a logically val i d deductive procedure. 8 8Husserl, E.;' Cartesian 'Meditation;, trans. Dorion Cairns (Martinus Nijhoff, The Hague, 1970), p. 24 ff.

The Welland Canals Foundation, Honourable William Hamilton Merritt and the St. Catharines Historical Society and Museum, 1971-1990, n.d.

‘The Father of Canadian Transportation’ is a term commonly associated with William Hamilton Merritt. Although he is most known for being one of the driving forces behind the building of the first Welland Canal, he was many things throughout his life; a soldier, merchant, promoter, entrepreneur and politician to name a few. Born on July 3, 1793 at Bedford, Westchester County, N.Y. to Thomas Merritt and Mary Hamilton, Merritt’s family relocated to Canada shortly after in 1796. The move came after Merritt’s father petitioned John Graves Simcoe for land in Upper Canada after serving under him in the Queen’s Rangers during the American Revolution. The family quickly settled into their life at Twelve Mile Creek in St. Catharines. Merritt’s father became sheriff of Lincoln County in 1803 while Merritt began his education in mathematics and surveying. After some brief travel and further education Merritt returned to Lincoln County, in 1809 to help farm his father’s land and open a general store. While a farmer and merchant, Merritt turned his attention to military endeavours. A short time after being commissioned as a Lieutenant in the Lincoln militia, the War of 1812 broke out. Fulfilling his duty, Merritt fought in the Battle of Queenston Heights in October of 1812, and numerous small battles until the Battle of Lundy’s Lane in July 1814. It was here that Merritt was captured and held in Cheshire, Massachusetts until the war ended. Arriving back in the St. Catharines area upon his release, Merritt returned to being a merchant, as well as becoming a surveyor and mill owner. Some historians hypothesize that the need to draw water to his mill was how the idea of the Welland Canals was born. Beginning with a plan to connect the Welland River with the Twelve mile creek quickly developed into a connection between the Lakes Erie and Ontario. Its main purpose was to improve the St. Lawrence transportation system and provide a convenient way to transport goods without having to go through the Niagara Falls portage. The plan was set in motion in 1818, but most living in Queenston and Niagara were not happy with it as it would drive business away from them. Along with the opposition came financial and political restraints. Despite these factors Merritt pushed on and the Welland Canal Company was chartered by the Upper Canadian Assembly on January 19, 1824. The first sod was turned on November 30, 1824 almost a year after the initial chartering. Many difficulties arose during the building of the canal including financial, physical, and geographic restrictions. Despite the difficulties two schooners passed through the canal on November 30, 1829. Throughout the next four years continual work was done on the canal as it expended and was modified to better accommodate large ships. After his canal was underway Merritt took a more active role in the political arena, where he served in various positions throughout Upper Canada. In 1851, Merritt withdrew from the Executive Council for numerous reasons, one of which being that pubic interest had diverted from the canals to railways. Merritt tried his hand at other public works outside transportation and trade. He looked into building a lunatic asylum, worked on behalf of War of 1812 veterans, aided in building Brock’s monument, established schools, aided refugee slaves from the U.S. and tried to establish a National Archives among many other feats. He was described by some as having “policy too liberal – conceptions too vast – views too comprehensive to be comprehensible by all”, but he still made a great difference in the society in which he lived. After his great contributions, Merritt died aboard a ship in the Cornwall canal on July 5, 1862. Dictionary of Canadian Biography Online http://www.biographi.ca/EN/ShowBio.asp?BioId=38719 retrieved October 2006 Today numerous groups carry on the legacy of Merritt and the canals both in the past and present. One such group is the Welland Canals Foundation. They describe themselves as: “. . . a volunteer organization which strives to promote the importance of the present and past Welland Canals, and to preserve their history and heritage. The Foundation began in 1980 and carries on events like William Hamilton Merritt Day. The group has strongly supported the Welland Canals Parkway initiative and numerous other activities”. The Welland Canals Foundation does not work alone. They have help from other local groups such as the St. Catharines Historical Society. The Society’s main objective is to increase knowledge and appreciation of the historical aspects of St. Catharines and vicinity, such as the Welland Canals. http://www.niagara.com/~dmdorey/hssc/dec2000.html - retrieved Oct. 2006 http://www.niagara.com/~dmdorey/hssc/feb2000.html - retrieved Oct. 2006

Generating finite integral relation algebras

Relation algebras and categories of relations in particular have proven to be extremely useful as a fundamental tool in mathematics and computer science. Since relation algebras are Boolean algebras with some well-behaved operations, every such algebra provides an atom structure, i.e., a relational structure on its set of atoms. In the case of complete and atomic structure (e.g. finite algebras), the original algebra can be recovered from its atom structure by using the complex algebra construction. This gives a representation of relation algebras as the complex algebra of a certain relational structure. This property is of particular interest because storing the atom structure requires less space than the entire algebra. In this thesis I want to introduce and implement three structures representing atom structures of integral heterogeneous relation algebras, i.e., categorical versions of relation algebras. The first structure will simply embed a homogeneous atom structure of a relation algebra into the heterogeneous context. The second structure is obtained by splitting all symmetric idempotent relations. This new algebra is in almost all cases an heterogeneous structure having more objects than the original one. Finally, I will define two different union operations to combine two algebras into a single one.

Myths of Modern Education

My focus is on assessment criteria of language proficiency in community college education. To demand clear writing is an application of scientism; it seeks to keep separate the fact/value distinction of positivism. This dangerously undermines the democratizing possibilities of education, since clear writing, taken to its extreme, is ultimately anonymous and dehumanizing. The active student-as-citizen is, therefore, subsumed under the neoliberal dictate of the passive student-as-consumer. The process of language acquisition is reduced to a fictitious act of knowledge transmission and regurgitation, and, therefore, those subversive aspects of language learning, such as creativity and critical inquiry, are undermined. An initial overview of the tenets of modernity will provide a conceptual framework for this examination.

The Monastery

The Monastery by Sir Walter Scott, was prsented to margaret Woodruff for proficiency in tennis singles, June 17th, 1913. The book is bound in a leather cover which is embossed with the St. Margaret's College, Toronto insignia. A full text version is available at the following link: https://ia800300.us.archive.org/34/items/themonastery06406gut/mnsry10.txt

The History of the Consulate & The Empire of France Under Napoleon

The book is stamped in the front and back with the "Upper Canada College" crest. There is also a plate within the book indicating that it was presented to H.K. Woodruff by the senate as an acknowledgement of merit and as a reward for general proficiency in Upper Canada College in June, 1878. On the top of this plate it says: 1st Form Prize, Upper Modern Form. The full text is available in the Brock University Special Collections and Archives or view the full text at the following link: http://www.archive.org/details/consulateempire1and2thie

Didactique des grandeurs en mesure et élèves en difficulté d'apprentissage du 2e cycle du primaire

Le programme -Une école adaptée à tous ses élèves-, qui s'inscrit dans la réforme actuelle de l'éducation au Québec, nous a amenée à nous intéresser aux représentations dans les grandeurs en mesure en mathématiques des élèves en difficulté d'apprentissage. Nous nous sommes proposés de reconduire plusieurs paramètres de la recherche de Brousseau (1987, 1992) auprès de cette clientèle. La théorie des champs conceptuels (TCC) de Vergnaud (1991), appliquée aux structures additives, a été particulièrement utile pour l'analyse et l'interprétation de leurs représentations. Comme méthode de recherche, nous avons utilisé la théorie des situations didactiques en mathématiques (TSDM), réseau de concepts et de méthode de recherche appuyé sur l'ingénierie didactique qui permet une meilleure compréhension de l'articulation des contenus à enseigner. Grâce à la TSDM, nous avons observé les approches didactiques des enseignants avec leurs élèves. Notre recherche est de type exploratoire et qualitatif et les données recueillies auprès de 26 élèves de deux classes spéciales du deuxième cycle du primaire ont été traitées selon une méthode d'analyse de contenu. Deux conduites ont été adoptées par les élèves. La première, de type procédural a été utilisée par presque tous les élèves. Elle consiste à utiliser des systèmes de comptage plus ou moins sophistiqués, de la planification aux suites d'actions. La deuxième consiste à récupérer directement en mémoire à long terme le résultat associé à un couple donné et au contrôle de son exécution. L'observation des conduites révèle que les erreurs sont dues à une rupture du sens. Ainsi, les difficultés d'ordre conceptuel et de symbolisation nous sont apparues plus importantes lorsque l'activité d'échange demandait la compétence "utilisation" et renvoyait à la compréhension de la tâche, soit les tâches dans lesquelles ils doivent eux-mêmes découvrir les rapports entre les variables à travailler et à simuler les actions décrites dans les énoncés. En conséquence, les problèmes d'échanges se sont révélés difficiles à modéliser en actes et significativement plus ardus que les autres. L'étude des interactions enseignants et élèves a démontré que la parole a été presque uniquement le fait des enseignants qui ont utilisé l'approche du contrôle des actes ou du sens ou les deux stratégies pour aider des élèves en difficulté. Selon le type de situation à résoudre dans ces activités de mesurage de longueur et de masse, des mobilisations plurielles ont été mises en oeuvre par les élèves, telles que la manipulation d'un ou des étalon(s) par superposition, par reports successifs, par pliage ou par coupure lorsque l'étalon dépassait; par retrait ou ajout d'un peu de sable afin de stabiliser les plateaux. Nous avons également observé que bien que certains élèves aient utilisé leurs doigts pour se donner une perception globale extériorisée des quantités, plusieurs ont employé des procédures très diverses au cours de ces mêmes séances. Les résultats présentés étayent l'hypothèse selon laquelle les concepts de grandeur et de mesure prennent du sens à travers des situations problèmes liées à des situations vécues par les élèves, comme les comparaisons directes. Eles renforcent et relient les grandeurs, leurs propriétés et les connaissances numériques.

Grothendieck et les topos : rupture et continuité dans les modes d'analyse du concept d'espace topologique

La thèse présente une analyse conceptuelle de l'évolution du concept d'espace topologique. En particulier, elle se concentre sur la transition des espaces topologiques hérités de Hausdorff aux topos de Grothendieck. Il en ressort que, par rapport aux espaces topologiques traditionnels, les topos transforment radicalement la conceptualisation topologique de l'espace. Alors qu'un espace topologique est un ensemble de points muni d'une structure induite par certains sous-ensembles appelés ouverts, un topos est plutôt une catégorie satisfaisant certaines propriétés d'exactitude. L'aspect le plus important de cette transformation tient à un renversement de la relation dialectique unissant un espace à ses points. Un espace topologique est entièrement déterminé par ses points, ceux-ci étant compris comme des unités indivisibles et sans structure. L'identité de l'espace est donc celle que lui insufflent ses points. À l'opposé, les points et les ouverts d'un topos sont déterminés par la structure de celui-ci. Qui plus est, la nature des points change: ils ne sont plus premiers et indivisibles. En effet, les points d'un topos disposent eux-mêmes d'une structure. L'analyse met également en évidence que le concept d'espace topologique évolua selon une dynamique de rupture et de continuité. Entre 1945 et 1957, la topologie algébrique et, dans une certaine mesure, la géométrie algébrique furent l'objet de changements fondamentaux. Les livres Foundations of Algebraic Topology de Eilenberg et Steenrod et Homological Algebra de Cartan et Eilenberg de même que la théorie des faisceaux modifièrent profondément l'étude des espaces topologiques. En contrepartie, ces ruptures ne furent pas assez profondes pour altérer la conceptualisation topologique de l'espace elle-même. Ces ruptures doivent donc être considérées comme des microfractures dans la perspective de l'évolution du concept d'espace topologique. La rupture définitive ne survint qu'au début des années 1960 avec l'avènement des topos dans le cadre de la vaste refonte de la géométrie algébrique entreprise par Grothendieck. La clé fut l'utilisation novatrice que fit Grothendieck de la théorie des catégories. Alors que ses prédécesseurs n'y voyaient qu'un langage utile pour exprimer certaines idées mathématiques, Grothendieck l'emploie comme un outil de clarification conceptuelle. Ce faisant, il se trouve à mettre de l'avant une approche axiomatico-catégorielle des mathématiques. Or, cette rupture était tributaire des innovations associées à Foundations of Algebraic Topology, Homological Algebra et la théorie des faisceaux. La théorie des catégories permit à Grothendieck d'exploiter le plein potentiel des idées introduites par ces ruptures partielles. D'un point de vue épistémologique, la transition des espaces topologiques aux topos doit alors être vue comme s'inscrivant dans un changement de position normative en mathématiques, soit celui des mathématiques modernes vers les mathématiques contemporaines.

Comportement linguistique des immigrants allophones et caractéristiques du quartier de résidence, région métropolitaine de recensement de Montréal, 2006

Les immigrants allophones qui s’établissent dans la région métropolitaine de recensement (RMR) de Montréal sont vraisemblablement confrontés à la concurrence qui y existe entre le français et l’anglais. À l’aide de données agrégées du recensement canadien de 2006, nous explorons le rôle que pourrait jouer l’environnement linguistique résidentiel dans l’adoption de deux comportements linguistiques; le transfert linguistique vers le français ou l’anglais et la connaissance des langues officielles chez ceux n’ayant pas effectué de transfert, tout en tenant compte de leurs caractéristiques individuelles. Des liens initiaux existent entre la composition linguistique des 56 quartiers de la RMR et les comportements linguistiques des immigrants allophones. De plus, des caractéristiques individuelles similaires mènent à des orientations linguistiques similaires. Sans séparer ces deux effets, des régressions linéaires nous permettent de croire que la connaissance de l’anglais et/ou du français n’est pas déterminée par la composition linguistique du quartier, alors que cette dernière ne peut être écartée lorsque nous analysons la langue d’usage à la maison (transferts).

Construction négociée par la triade de formation en stage d'un savoir-enseigner les mathématiques au primaire

Dans le contexte actuel de l’éducation au Québec où la réforme des programmes de formation des jeunes appelle un renouvellement des pratiques d’enseignement, notre recherche s’intéresse au développement de la dimension didactique de la pratique liée à l’enseignement des mathématiques qui est considéré comme l’un des éléments clés des nouvelles orientations. Nous abordons la question par le biais de la collaboration de formation initiale pour l’enseignement des mathématiques au primaire qui se vit en stage entre des praticiennes en exercice et en formation et une didacticienne des mathématiques. Cette rencontre sur le terrain des stages au primaire entre praticiennes et didacticienne, longtemps réclamée et rendue possible à l’UQAT , nous a amené à formuler une première question de recherche touchant ce qui se construit à travers les échanges de ces partenaires de la formation au cours des supervisions pédagogiques conjointes qui les réunissent en stage. Nous avons cadré ce questionnement à partir des balises théoriques de la didactique professionnelle qui proposent modèle et concepts pour expliciter l’activité professionnelle et traiter des phénomènes de développement des compétences professionnelles en contexte de travail et de formation. La didactique professionnelle attribue un rôle essentiel à la communauté de pratique et au processus d’analyse de l’expérience dans le développement professionnel des novices et dans l’explicitation d’un savoir d’action jugé pertinent et reconnu. Nous y faisons donc appel pour poser le potentiel que représentent les échanges issus de la collaboration quant à leur contribution à l’établissement d’un savoir de référence pour l’enseignement des mathématiques. La didactique professionnelle propose également le recours au concept de schème pour décrire l’activité professionnelle et à l’idée de concepts organisateurs comme élément central de l’activité et comme variable de la situation professionnelle concernée. Nous recourons à ces mêmes concepts pour expliciter le savoir de référence pour l’enseignement des mathématiques qui émerge à travers les échanges des partenaires de la formation. Dans le cadre d’une étude de cas, nous nous sommes intéressée aux échanges qui se déroulent entre une stagiaire qui effectue son troisième et avant dernier stage , l’enseignante-associée qui la reçoit et la chercheure-didacticienne qui emprunte le rôle de superviseure universitaire. Les échanges recueillis sont issus de trois cycles de supervision conjointe qui prennent la forme de rencontres de préparation des situations d’enseignement de mathématique; d’observation en classe des séances d’enseignement pilotées par la stagiaire auprès de ses élèves; et des rencontres consacrées à l’analyse des situations d’enseignement observées et de l’activité mise en œuvre par la stagiaire. Ainsi les objets de discussion relevés par les différents partenaires de la formation et la négociation de sens des situations professionnelles vécues et observées sont analysés de manière à rendre visibles les constituants de l’activité professionnelle qui sont jugés pertinents par la triade de formation. Dans un deuxième temps, en partant de cette première analyse, nous dégageons les concepts organisateurs des situations professionnelles liées à l’enseignement des mathématiques qui sont pris en compte par la triade de formation et qui constituent des variables de la situation professionnelle. Les constituants de l’activité et des situations professionnelles qui résultent de cette analyse sont envisagés en tant que représentations collectives qui se révèlent à travers les échanges de la triade de formation. Parce que ces représentations se sont trouvées partagées, négociées dans le cadre des supervisions pédagogiques, elles sont envisagées également en tant que savoir de référence pour cette triade de formation. Les échanges rendus possibles entre les praticiennes et la didacticienne placent ce savoir de référence dans une dynamique de double rationalité pratique et didactique. Enfin, partant de l’apport déterminant de la communauté de pratique et de formation de même que du savoir de référence que cette dernière reconnait comme pertinent dans le développement professionnel des novices, les résultats de cette recherches peuvent contribuer à réfléchir la formation des futures enseignantes en stage en ce qui a trait à l’enseignement des mathématiques au primaire.