abstract: Researchers have described two fundamental conceptualizations for division, known as partitive and quotitive division. Partitive division is the conceptualization of a÷b as the amount of something per copy such that b copies of this amount yield the amount a. Quotitive division is the conceptualization of a÷b as the number of copies of the amount b that yield the amount a. Researchers have identified many cognitive obstacles that have inhibited the development of robust meanings for division involving non-whole values, while other researchers have commented on the challenges related to such development. Regarding division with fractions, much research has been devoted to quotitive conceptualizations of division, or on symbolic manipulation of variables. Research and curricular activities have largely avoided the study and development of partitive conceptualizations involving fractions, as well as their connection to the invert-and-multiply algorithm. In this dissertation study, I investigated six middle school mathematics teachers’ meanings related to partitive conceptualizations of division over the positive rational numbers. I also investigated the impact of an intervention that I designed with the intent of advancing one of these teachers’ meanings. My findings suggested that the primary cognitive obstacles were difficulties with maintaining multiple levels of units, weak quantitative meanings for fractional multipliers, and an unawareness of (and confusion due to) the two quantitative conceptualizations of division. As a product of this study, I developed a framework for characterizing robust meanings for division, indicated directions for future research, and shared implications for curriculum and instruction. / Dissertation/Thesis / Doctoral Dissertation Mathematics Education 2019

Important to understanding bilingualism and second language (L2) learning are L2 morphological processing and acquisition of tense and aspect. This study used narrative elicitation to examine the expression of boundedness and definiteness in Finnish and English by first language (L1) Finnish speakers who speak English as an L2 and L1 English speakers who speak Finnish as an L2. In Finnish, boundedness and definiteness were largely portrayed by using partitive and accusative cases, though tense and aspect conjugation also played a role. In English, boundedness was largely conveyed through tense and aspect conjugation and definiteness through article usage. Both L1 speaker groups appeared to demonstrate first language transfer as well as form following meaning in acquisition, meaning that a given form will be acquired first in contexts where the meaning of the form is inherent. There was also evidence pointing to avoidance by L2 speakers. Notably, varying interpretations of what the images used portrayed also seemed to play a role in some of the differences in responses across groups. The narrative elicitation methodology was useful in producing meaningful and easily comparable results.

This dissertation is a first look at English-speaking children’s acquisition of the syntax of the partitive. It presents four experiments that contrast three types of structures and examines how they interact with adjectival modification: the partitive, the pseudopartitive and complex nouns with prepositional adjuncts. The experimentation investigates whether children recognize that the Determiner Phrase (DP) in the partitive is a barrier to adjectival modification. The partitive is contrasted with the pseudopartitive –a minimal pair structure that lacks an internal DP. The data shows that children under the age of six do not distinguish between the partitive and the pseudopartitive. They allow adjectives preceding the partitive to modify the second noun; this is standardly considered licit for the pseudopartitive structure, but not the partitive. This result is evidence that children are under-representing the syntax of the partitive and of DP. Syntactic representations of minimal DP and minimal partitive structures are suggested and it is argued that these structures may persist as an option in the adult grammar. Chapter 2 discusses multiple layers in DP, DP’s status as a barrier/phase and how children acquire its syntax (Abney 1987, Cinque 1994, de Villiers & Roeper 1995, Kupisch 2006, Bošković 2008). This chapter also includes evidence for an underrepresented DP in the grammar of some adult English speakers (Schafer & de Villiers 2000, Carlson et al 2006). Chapter 3 presents background literature on the syntax of the partitive (Jackendoff 1977, Hoeksema 1996), introduces the pseudopartitive structure (Selkirk 1977, Stickney 2004 and Alexiadou, Haegeman & Stavrou 2007) and presents acquisition hypotheses. Chapters 4 & 5 present a pilot experiment and three picture choice tasks. The experimental data shows that children and a subset of adults do not distinguish between partitive and pseudopartitive and yet they maintain a clear distinction between pseudopartitive and other similar complex nouns. Chapter 6 presents two syntactic analyses of the data. One uses a split-DP structure (Zamparelli 2000, Laenzlinger 2000) to explain the lack of barrier in children’s partitives. The other suggests a reduced partitive structure (Rutkowski 2007). Both analyses require a reanalysis of the features of DP in children’s partitives.

Abstract The partitive (one of the fifteen Finnish cases and a typical case characterizing Finnic languages) developed from the Uralic separative locative into a grammatical case. In modern Finnish, it is one of the object, existential subject and predicative cases, representing that side of the case alternations that expresses unboundedness and negative polarity. Probably because the three case alternations differ in certain respects and clear-cut grammar rules cannot always be formulated, the use of the partitive remains a constant struggle for learners of Finnish. This study investigates the use of partitive objects, subjects and predicatives in Estonian, German and Dutch learners of Finnish as a foreign language. By comparing groups of learners from L1 backgrounds closely related and non-related to the target language (TL), it is aimed to explore the role of presence versus lack of relevant prior linguistic knowledge. The use of the partitive is namely largely similar in the closely related Estonian language. However, the purpose of the study is not only to gain valuable insights into the phenomena of L1 influence and intralingual influence but also to identify (common and L1 background-specific) stumbling blocks in the use of the partitive case, and to draw pedagogical implications based upon the findings. Research materials were selected from the Estonian, German and Dutch subcorpora of the International Corpus of Learner Finnish (ICLFI), aligned to the CEFR proficiency levels, and analyzed based on combined error-frequency analyses, involving partitive over- and underuse errors and partitive-requiring contexts (PRCs). As will be shown, the study reveals conspicuous differences between the learner corpora. In general, the Estonian learner corpus not only shows significantly fewer partitive errors than the other corpora, but also some specific error patterns attributable to subtle L1-L2 differences and, unlike the remaining corpora, a lack of overgeneralization of L2 grammar rules. The findings do not only indicate that -and how- prior linguistic knowledge matters, but also suggest that stumbling blocks could potentially be turned into stepping stones by emphasizing L1-L2 differences in the case of Estonian learners of Finnish, and by highlighting similarities and differences from within the TL in cases of learners from non-related L1 backgrounds. / Tiivistelmä Itämerensuomalaisille kielille tyypillinen partitiivi on aikaa myöten kehittynyt separatiivi-nimisestä uralilaisen kantakielen paikallissijasta syntaktisia funktioita ilmaisevaksi sijamuodoksi. Nykysuomessa partitiivi on yksi objektin, eksistentiaali-subjektin ja predikaativin sijoista, jolla ilmaistaan rajaamattomuutta ja kielteisyyttä. Partitiivin käyttö aiheuttaa suomen kielen oppijoille usein ongelmia, luultavasti koska objektin, subjektin ja predikatiivin sijanvalinnassa on vaihtelua ja ratkaisevien kieliopillisten sääntöjen muodostus on toisinaan erittäin vaikeaa. Tässä tutkimuksessa tarkastelen virolaisten, saksalaisten ja hollantilaisten suomea vieraana kielenä opiskelevien partitiiviobjektin, -subjektin ja -predikatiivin käyttöä lähde- ja kohdekielen samanlaisuuden ja erilaisuuden näkökulmasta. Lähde- ja kohdekielen roolin selvittämisen lisäksi tutkimuksen tavoitteina on identifioida partitiivin (yleiset ja lähdekielikohtaiset) ongelmakohdat ja yhdistää tutkimustulokset vieraan kielen oppimiseen ja -opetukseen. Tutkimusaineistoina on virolaisten, saksalaisten ja hollantilaisten opiskelijoiden kirjoittamia tekstejä, jotka on poimittu Kansainvälisestä oppijansuomen korpuksesta (ICLFI) ja arvioitu Eurooppalaisen viitekehyksen (CEFR) kielitaitotasojen mukaan. Virhe- ja frekvenssianalyyseissä aineistoista analysoidaan muun muassa partitiivin yli- ja alikäyttövirheet ja partitiivin vaatimat kontekstit. Tutkimuksesta käy ilmi, että virolaisten oppijoiden ja ei-sukukieliä puhuvien oppijoiden tuotoksissa on silmiinpistäviä eroja. Virolaisten aineistosta löytyy yleisesti tilastollisesti vähemmän partitiivivirheitä kuin kahdesta muusta osakorpuksesta ja lisäksi myös virhekategorioita ja -rakenteita, jotka johtunevat lähdekielen vaikutuksesta. Tämän lisäksi saksalaisten ja hollantilaisten tuotoksissa on selvästi enemmän kohdekielen sääntöjen yliyleistämistä kuin virolaisten osakorpuksessa. Tutkimustulokset siis osoittavat, että lähdekieli vaikuttaa kohdekielen oppimiseen, ja sen, miten se vaikuttaa. Lisäksi tutkimustulosten avulla on mahdollista kehittää sellaisia opetuksen apuvälineitä, joilla voidaan selventää virolaisille suomenoppijoille L1:n ja L2:n partitiivin käytön eroja ja yhtäläisyyksiä ja tehostaa oppimista; ei-sukukielisten oppijoiden opetuksessa ovat puolestaan kohdekielen ja sijanvaihteluiden sisäiset tunnusmerkit tärkeitä.

This study is according to Scientific Program of Generative Grammar, more specifically on minimalist assumptions of the Principle and Parameters Theory (CHOMSKY, 1993, 1995, 1998, 2001) and try to describe and analyse the subject-verb agreement into partitive constructions in Brazilian Portuguese (BP) and European Portuguese (EP) grammars. The main aim is provid an explanation for the type of overt morphologically agreement verified in sentences like: (i). The most of students do-past. 3pl. the homework. The most of students did the homework . For that, this research proposes as central hypothesis the idea that operation Agree according to Béjar (2003, 2008) can satisfactorily explain the partial agreement in question, taking the pressupost idea on specification/underspecification of phi-features ÉJAR, 2003, 2008; CARVALHO, 2008), as well the approach in which agreement is understood as feature sharing, according to idea elaborated by Frampton & Gutmann (2000a). In this thesis, it were applied some tasks to BP and EP speakers, aiming to verify the acceptability of these speakers with relation to the partial agreement in partitive constructions. The tasks results point out that in some contexts, Brazilian speakers accept more easily partial agreement than Portuguese ones. / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / O presente estudo insere-se no Programa de Investigação da Gramática Gerativa, mais especificamente nas assunções minimalistas da Teoria de Princípios e Parâmetros (CHOMSKY, 1993, 1995, 1998, 2001) e busca descrever e analisar a concordância sujeito-verbo nas construções partitivas no Português Brasileiro (PB) e no Português Europeu (PE). O objetivo central consiste em prover uma explicação para o tipo de concordância morfologicamente visível verificada em frases do tipo: (i) A maioria dos estudantes fizeram o trabalho. Para isso, toma como hipótese central de trabalho a ideia de que a Operação Agree conforme reformulada por Béjar (2003, 2008) dá conta de explicar satisfatoriamente a concordância parcial em questão, tomando como pressuposta a ideia de especificação/subespecificação de traços-Ф (BÉJAR, 2003, 2008; CARVALHO, 2008), bem como a visão de concordância como compartilhamento de traços, conforme proposta elaborada por Frampton & Gutmann (2000a). Para tanto, procedeu à aplicação de alguns testes de preferência a falantes do PB e do PE, a fim de averiguar a aceitabilidade desses falantes em relação à concordância mencionada. Os resultados dos testes apontam uma maior aceitação, em alguns contextos, por parte dos falantes do PB do que por parte dos falantes do PE.

Denna systematiska litteraturstudie syftar till att med hjälp av forskning identifiera avgörande faktorer för framgångsrika undervisningsstrategier av stambråk i grundskolans matematikundervisning. Studien baseras på elva vetenskapliga artiklar som bearbetats systematiskt med hjälp av innehållsanalys för att besvara forsknings-frågorna om vilka avgörande faktorer som forskningen visar för undervisningen av stambråk samt vilka framgångsrika undervisningsstrategier som finns. Forskningen visar att areamodellen som representationsform dominerar undervisningen av bråk vilket innebär att stambråk får lite plats i undervisningen. Stambråket är en viktig del för att kunna tillägna sig avgörande faktorer av bråk. Resultatet visar att en undervisning med linear measurement (linjära representationsformer) betonar stambråkets roll som tolkningsverktyg för att kunna jämföra andra bråk samt det omvända förhållandet där en större nämnare utgör en mindre andel. Resultatet visar också att undervisningen av stambråk etablerar grundläggande principer för rationella tal och mer avancerade matematiska områden som proportionalitet och algebra. Därmed är lärares val av undervisningsstrategier och representationsformer samt deras kunskaper inom dessa områden vitala för vad eleverna kan tillägna sig i samband med bråkundervisningen.