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In French, Italian, and other Romance languages indefinite nominal phrases can be introduced by what appears to be the conflation of a genitive preposition and a definite article, the so-called “indefinite partitive articles” (e.g., Fr. Je cuisine de la soupe depuis deux jours. ‘I’ve been cooking soup for two days’). This is rather unexpected, since these nominal phrases are neither definite nor in a syntactic position in which we expect to find a genitive preposition. This led part of the literature to consider them as built by lexical items synchronically distinct from the genitive preposition/definite article but homophonous with them. This contribution shows how a constituent-based approach to the lexicon-syntax interface as nanosyntax, paired with a specific take on the sequence of syntactic functions, can capture their apparently conflicting distribution without stipulating multiple homophonous lexical items. The key factor in this proposal is a revised analysis of the Romance lexical item (LI) for (i) definite articles – linked to a constituent containing not only features of definiteness but also lower indefinite features and higher nominative/accusative case features – and (ii) the genitive preposition DE – linked to a constituent containing not only genitive features but also lower nominative/accusative features. Holding these LIs crosslinguistically stable, the variation attested in this domain is modeled as depending on the amount of functional structure lexicalized by the nominal root in the different languages.

The concept of fractions given in learning is the concept of part of whole and part of unit. The development of student's concepts of fractions can be built through fraction schemes. A partitive fraction scheme is a scheme that estimates the size of the fraction in the form of non-units to the whole that is not partitioned. The concept of fractions using a partitive fraction scheme uses a strong understanding of the concept of the part unit and part-whole fractions. This study aims to photograph the process of developing students' mental actions in constructing the concept of fractions using a partitive fraction scheme. This research method uses a qualitative approach with a case study type on students who are able to express partitive fraction schemes. The participants of this study were fifth-grade elementary school students who already understood the concept of part-whole and part of unit. The results showed that students used two types of mental action processes in the partitive fraction scheme: direct and indirect.

Adpositions localize an entity (Figure) with respect to another entity (Ground), designated by the complement of the adposition. Most Finnic adpositions are postpositions with a genitive Ground, while prepositions typically have a partitive Ground. This work is a cognitive-linguistic study of the synchronic and diachronic semantics of partitive-Ground adpositions. It is argued that adpositions with a partitive Ground select a proximal perspective to the locational relationship, while those with a genitive Ground select a distal perspective. Three alternative hypotheses are introduced and compared concerning the origin of two Finnish partitive-Ground adpositions, kohti ‘towards’ and päin ‘towards’: 1) These adpositions originated as instructive-case bodypart expressions of position (cf. selin ‘with one’s back at’), and their partitive Ground indicated a viewpoint person; 2) They were instructive forms with a meronymic-locative meaning, expressing a more precise target area within the Ground, together with a directionality toward that area; 3) The Ground of kohti and päin was originally a partitive object of ‘aiming’ or ‘shooting’ verbs, and the soon-to-be adpositions themselves were lexicalized adverbs of direction. Kokkuvõte. Tuomas Huumo: Milleks partitiiv? Kaassõnade laiendit markeeriva partitiivkäände motiividest läänemeresoome keeltes. Kaassõnad suhestavad trajektori orientiiriga, mida väljendab kaassõna laiend. Enamik läänemeresoome keelte kaassõnadest on tagasõnad, mille laiend on genitiivis, eessõnade laiend aga on tavaliselt partitiivis. Käesoleva töö eesmärgiks on selgitada partitiivis laiendiga kaassõnade sünkroonilist ja diakroonilist semantikat kognitiivse lingvistika vaatenurgast. Autor väidab, et partitiivlaiendiga kaassõnad valivad asukohasuhtele proksimaalse perspektiivi, erinevalt genitiivlaiendiga kaassõnadest, mis valivad distaalse ja holistilise perspektiivi. Autor pakub välja kolm alternatiivset hüpoteesi, mis puudutavad kahe soome keele partitiivlaiendiga kaassõna, kohti ja päin (mõlemad tähendusega ’suunas, poole’), päritolu: 1) kaassõnad grammatiseerusid instruktiivis kehaosaväljenditest (vrd. selin ’seljaga millegi poole’) ja nende partitiivlaiend väljendas esialgselt vaatleja asukohta; 2) vormid olid lokatiivsed instruktiivid, mis väljendasid meronüümiliselt täpsemat sihtala orientiiri sees; 3) partitiivlaiend oli esialgselt objekt ning väljendas suunatud tegevuse (nt. ’sihtima’, ’laskma’) sihtkohta, tulevased kaassõnad aga olid suunda täpsustavad adverbid.

Students show deficient understanding on fraction division and supporting that understanding remains a challenge for mathematics educators. This article aims to describe primary students' understanding of partitive fraction division (PFD) and explore ways to support their understanding through the use of sequenced fractions and context-related graphical representations. In a design-research study, forty-four primary students were involved in three cycles of teaching experiments. Students' works, transcript of recorded classroom discussion, and field notes were retrospectively analyzed to examine the hypothetical learning trajectories. There are three main findings drawn from the teaching experiments. Firstly, context of the tasks, the context-related graphical representations, and the sequence of fractions used do support students' understanding of PFD. Secondly, the understanding of non-unit rate problems did not support the students' understanding of unit rate problems. Lastly, the students were incapable of determining symbolic representations from unit rate problems and linking the problems to fraction division problems. The last two results imply to rethink unit rate as part of a partitive division with fractions. Drawing upon the findings, four alternative ways are offered to support students' understanding of PFD, i.e., the lesson could be starting from partitive whole number division to develop the notion of fair-sharing, strengthening the concept of unit in fraction and partitioning, choosing specific contexts with more relation to the graphical representations, and sequencing the fractions used, from a simple to advanced form.

In this paper, we focus on partitive articles (PAs), i.e., determiners which, generally, have an indefinite interpretation, and on one of their potential correlates, i.e., invariable de, in Francoprovençal, a non-standardized, highly endangered Gallo-Romance language (cf. Zulato, Kasstan & Nagy 2018), and show the fine-grained spatial distribution of these elements in the Swiss and Aosta Valley (Italy) varieties. Presenting several maps based on fieldwork data from Valais (Switzerland) and the Aosta Valley (Italy), we demonstrate that the spatial distribution of PAs and de is more complex than reported in the literature: we complement the basic subdivision of Francoprovençal into two types, Francoprovençal A and B (cf. Kristol 2014, 2016), with a more nuanced picture, in which the morphosyntactic features of PAs play a crucial role: in Francoprovençal A, the presence of PAs depends on the syntactic context whereas, in Francoprovençal B, their presence is limited mainly to two areas, in which singular and plural PAs do not occur together (one area only has singular PAs whereas the other one only has plural PAs). We also show that there is no correlation between phonologically overt plural marking on nouns and absence of PAs; however, we found a correlation between overt sigmatic number marking on nouns and absence of PAs.

This paper presents findings from a study that examined the strategies that children, who had only been taught the part-whole fraction sub-construct at school, used for finding the fraction associated with solving varied partitive quotient problems. A qualitative, microgenetic research design was used involving nine year 5 (aged 9-10) children engaged in eight individual task-based interviews over a 6-week period. The data analyzed showed that across the eight tasks, six of the nine children used more than one strategy for quantifying each person's share but, by the third task, in general, each child had settled into a regular pattern of strategy use. The analyzed variations in children's approaches to solving the partitive quotient problems revealed instances of when and how the part-whole fraction sub-construct interfered with children's engagement with the partitive quotient problems. Considering that, internationally, the part-whole sub-construct is still the first fraction sub-construct that many children learn in schools, the findings are significant, since they provide new, in-depth insights into emerging approaches to solving partitive quotient tasks that are influenced by children's existing part-whole knowledge. The findings provide new evidence for intra- and inter-individual variation in strategy use and strategy selection in tasks related to the partitive quotient meaning of fractions. The paper highlights for education, the need for teaching that introduces different fraction sub-constructs to learners early in their schooling so that one meaning of fractions does not become representative of all fraction knowledge.

Students show deficient understanding on fraction division and supporting that understanding remains a challenge for mathematics educators. This article aims to describe primary students’ understanding of partitive fraction division (PFD) and explore ways to support their understanding through the use of sequenced fractions and context-related graphical representations. In a design-research study, forty-four primary students were involved in three cycles of teaching experiments. Students’ works, transcript of recorded classroom discussion, and field notes were retrospectively analyzed to examine the hypothetical learning trajectories. There are three main findings drawn from the teaching experiments. Firstly, context of the tasks, the context-related graphical representations, and the sequence of fractions used do support students’ understanding of PFD. Secondly, the understanding of non-unit rate problems did not support the students’ understanding of unit rate problems. Lastly, the students were incapable of determining symbolic representations from unit rate problems and linking the problems to fraction division problems. The last two results imply to rethink unit rate as part of a partitive division with fractions. Drawing upon the findings, four alternative ways are offered to support students’ understanding of PFD, i.e., the lesson could be starting from partitive whole number division to develop the notion of fair-sharing, strengthening the concept of unit in fraction and partitioning, choosing specific contexts with more relation to the graphical representations, and sequencing the fractions used, from a simple to advanced form.

Adpositions localize an entity (Figure) with respect to another entity (Ground), designated by the complement of the adposition. Most Finnic adpositions are postpositions with a genitive Ground, while prepositions typically have a partitive Ground. This work is a cognitive-linguistic study of the synchronic and diachronic semantics of partitive-Ground adpositions. It is argued that adpositions with a partitive Ground select a proximal perspective to the locational relationship, while those with a genitive Ground select a distal perspective. Three alternative hypotheses are introduced and compared concerning the origin of two Finnish partitive-Ground adpositions, kohti 'towards' and päin 'towards': 1) These adpositions originated as instructive-case bodypart expressions of position (cf. selin 'with one's back at'), and their partitive Ground indicated a viewpoint person; 2) They were instructive forms with a meronymic-locative meaning, expressing a more precise target area within the Ground, together with a directionality toward that area; 3) The Ground of kohti and päin was originally a partitive object of 'aiming' or 'shooting' verbs, and the soon-to-be adpositions themselves were lexicalized adverbs of direction. [ABSTRACT FROM AUTHOR]

The primary objective of this article is to expand the discussion on partitive‐related phenomena diffusion in the prehistoric Circum‐Baltic area by considering the role of early Germanic languages. The central message is that early Germanic languages have been historically overlooked in the study of partitive phenomena in the Circum‐Baltic area, and this paper aims to address this oversight. After briefly discussing the main hypotheses related to the development and spread of partitive functions, the focus will shift to two understudied phenomena in early Germanic languages, namely the 'genitive of quantification' and the 'genitive of negation'. It will be shown that these two phenomena were present in early Germanic languages, suggesting that they should be included in further studies concerned with contact in the prehistoric Circum‐Baltic area. [ABSTRACT FROM AUTHOR]

The concept of fractions given in learning is the concept of part of whole and part of unit. The development of student's concepts of fractions can be built through fraction schemes. A partitive fraction scheme is a scheme that estimates the size of the fraction in the form of non-units to the whole that is not partitioned. The concept of fractions using a partitive fraction scheme uses a strong understanding of the concept of the part unit and part-whole fractions. This study aims to photograph the process of developing students' mental actions in constructing the concept of fractions using a partitive fraction scheme. This research method uses a qualitative approach with a case study type on students who are able to express partitive fraction schemes. The stages of data analysis consist of three stages, namely; (1) the results of task 1 data are corrected and determined by students who can solve fraction problems with part-whole and partitive unit fraction schemes, (2) students who can complete task 1 are selected as participants (3) students complete task 2 to see their problem-solving abilities fractions using a partitive fraction scheme, (3) students who can use a partitive fraction scheme were interviewed to see the mental action processes used in solving problems. The participants of this study were 35 students of fifth-grade elementary school who already understood the concept of part-whole and part of unit. The results showed that students used two types of mental action processes in the partitive fraction scheme: direct and indirect. [ABSTRACT FROM AUTHOR]

An understanding of partitive division is foundational for numerous other mathematics topics, including unit rate, slope, and probability. However, research has shown that learners tend to have a limited understanding of partitive division when the divisor is a proper fraction. To extend research on conceptions of partitive division in this study, we used Moschkovich's (1999) transitional conceptions perspective to examine how conceptions of partitive division evolve. This article reports on preservice teachers' (PSTs) conceptions of partitive division with proper-fraction divisors before and after they explored partitive division in a mathematics content course for elementary teachers (N = 17). Our analysis of pre- and post-interviews revealed an initial transitional conception and two levels of refinement of their conceptions. Furthermore, we identified two perturbations that PSTs experienced during refinement of their conceptions. By identifying ordered levels of refinement and associated perturbations, this exploratory study extends the transitional conceptions perspective. Furthermore, this study adds new insights into the conceptual complexities that the partitive model for division of fractions presents to PSTs (and to learners in general) and suggests new hypotheses about ways that conceptions of partitive division undergo refinement.

The purpose of this study was to examine and elaborate upon elementary prospective teachers' (PSTs) conceptions of partitive division with fractions. We examined the degree to which PSTs' conceptions were connected (i.e., capable of translating between representations correctly; aware that partitive division generates a unit rate for its quotient) and flexible (i.e., capable of differentiating between opportunities to partition or iterate (or both) when solving a partitive division task; aware that partitioning or iterating (or both) could be associated with the operation of division, as appropriate). Seventeen PSTs participated in task-based interviews prior to instruction in a mathematics content course for teachers. These PSTs demonstrated disconnected conceptions of partitive division with fractions when they incorrectly translated between representations and either inconsistently or did not express awareness that the purpose of the task was to generate a unit rate. These PSTs demonstrated rigid conceptions of partitive division with fractions such that they did not express awareness that the process of iterating could be associated with the operation of division, even when they obtained a correct answer by iterating. Results extend prior research by looking beyond PSTs' performance on tasks to elaborate upon PSTs' conceptions of the operation of partitive division. This study contributes new insights into PSTs' conceptions that can be used by mathematics teacher educators to inform the design of future instructional interventions.

As Koptjevskaja-Tamm points out, the division of labor between the elative and the partitive case in partitive constructions largely corresponds to the distinction between partitive nominal constructions (elative) and pseudo-partitive constructions (partitive), as shown in Figure 2 (from [50]: 533). 2.2 Partitive elements Partitive constructions may also develop into "faded" partitives, a term introduced by [51] to indicate seemingly partitive constructions in which, however, there is only a vague hint to a possible part-whole relation, as discussed in De Hoop ([34]: 193-206; see also [75]). As observed in Section 1, the term "partitive", "partitives" has been used with different meanings, comprising partitive and pseudo-partitive constructions, faded partitives and indefinite or "generalized" partitives. Section 3.2.2 is devoted to so-called "generalized" partitives, i.e. partitive cases, partitive genitives, partitive articles that express notions other than partivity, such as indefiniteness or unboundedness. Partitive constructions and their developments In this section we present some basic notions related to partitivity, focusing on types of constructions connected with the notion of part/whole relation, i.e. the partitive nominal construction, or "proper partitive", and the pseudo-partitive construction (Section 2.1). [Extracted from the article]

The article takes a closer look at the partitive genitive constructions in Latvian, their structure, semantics, and syntactic functions. With the help of a corpus analysis, an attempt has been made to find out what determines gender and number agreement variations between the partitive genitive constructions in subject position and the predicate, if it comprises a declinable participle. Attention was paid to such features as subject animacy, voice of the predicate, word order (SV / VS), quantifier lexeme and grammatical number of the genitive. 320 examples with partitive genitive constructions were selected from The Balanced Corpus of Modern Latvian LVK2018, including such quantifier lexemes as daļa 'part', skaits 'number', vairums 'quantity', vairākums 'majority', puse 'half', daudzums 'amount / quantity', lērums 'bagful', simts 'hundred', tūkstotis 'thousand', daudz 'much/many', maz 'little / few'. The best results are given by the combination of values 'animate', 'active' and 'SV', in which 93.24% of the examples are partitive genitive agreement. Data analyzed so far suggest that quantifiers involving the semantic element of partitivity (e.g., daļa 'part') favor partitive genitive agreement, whereas quantifiers lacking this semantic element favor quantifier agreement. This is an exploratory quantitative study on agreement tendencies in Latvian. [ABSTRACT FROM AUTHOR]