Två vanliga räkneregler som elever lär sig om i matematikundervisningen är prioriteringsregeln och vänster-till-höger-principen. Tidigare forskning har dock visat att elever också använder påhittade regler som vanligtvis inte brukar användas inom matematiken. Syftet med den här studien är att undersöka dessa ”egenskapade” regler. Syftet uppnås genom att studera vad det är för mindre kända räkneregler som eleverna tillämpar samt om hur konsekventa eleverna är i sin användning av en typ av räkneregel.  I studien gjorde 55 elever i årskurs 5 ett arbetsblad bestående av fem numeriska uttryck. Av de 55 eleverna använde 16 av dem någon form av regel som gick ut på att tal i de numeriska uttrycken parades ihop. 13 av de här 16 eleverna blev intervjuade om hur de hade tänkt när de löste uppgifterna. Data för studien utgörs därför av elevernas arbetsblad såväl som transkriberingarna från intervjuerna. Studien visar tre olika slags ”regler” som eleverna använder, förutom de vanliga räknereglerna vänster-till-höger-principen och prioriteringsregeln. De tre räknereglerna bygger alla på att tal paras ihop på ett eller annat sätt. Trots att nästan ingen av de 13 eleverna hade fått undervisning om de vanliga räknereglerna, så använder eleverna egna regler som följer logiska strukturer. Dessutom visar studien att de flesta eleverna inte är speciellt konsekventa när det kommer till valet av regel. Många av eleverna väljer att använda olika slags räkneregler för att beräkna uttryck som är uppbyggda på nästan samma sätt. / Two common rules of arithmetic that students learn about in education are the order of operations and the counting from left to right. However, previous research has shown that students also use made-up rules which are not usually used in mathematics. The aim of this study is to investigate the rules of arithmetic created by the students themselves. The aim is achieved by examine what kind of less-known rules of arithmetic that students apply and also how consistent students are in their use of a type of rule.   In the study, 55 students did a worksheet consisting of five tasks. In total, 16 of the 55 students used some kind of rule where numbers in the numerical expressions were paired in some way. Furthermore, 13 of the 16 students were interviewed about their way of thinking when solving the tasks. The data therefore consists of the students’ worksheets and transcriptions from the interviews.  The study shows that, in addition to the usual conventions left-to-right and order of operations, students use three different kinds of rules of arithmetic. The three rules of arithmetic are based on the principle that numbers are paired in one way or another. Despite that almost none of the 13 students had been taught the conventional rules of arithmetic, most students use own rules that follow logical structures. In addition, the study shows that most students are not particularly consistent when it comes to choosing strategy. Many students choose to use different kind of rules of arithmetic when they are calculating expressions that are structured in almost the same way.

For the development of this work, in the introduction we present a few topics which motivated it, as well as the research problem and its reason. The main objective of this dissertation is to research on the possible reasons for the hierarchy of the four mathematical operations. With this purpose, we attempted to verify if there were some explanations for such hierarchy. Thus, we researched on national and international sites. On this search, we found two articles of interest, namely: “the order of operation in elementary arithmetic” and the thesis “the school mathematics knowledge: operations with the natural numbers in the grade and middle school”, which were formulated comments about them. This research is classified as qualitative and bibliographical descriptive. In the chapter about the theory we presented how the subject numerical expression is explained in some official documents, as well as in textbooks of the Middle School. Since we have not found any reasonable and plausible explanation for such hierarchy, we included a chapter about a historical retrospective on the order of operations and the use of the parenthesis which, in turn, prepared the way for the chapter on a proposal to justify the why of the hierarchy for the four mathematical operations. / Para o desenvolvimento deste trabalho apresentamos na introdução alguns tópicos motivadores da pesquisa, bem como a sua problemática e justificativa. Esta dissertação tem como objetivo principal pesquisar as possíveis justificativas para a hierarquia das quatro operações aritméticas nas expressões numéricas. Para isso buscamos verificar se existia alguma proposta para a justificativa da hierarquia das operações na resolução de expressões numéricas. Assim, realizamos buscas tanto em sites nacionais, como também internacionais. Nessas buscas os trabalhos de interesse que encontramos foram: o artigo Order of operations in elementar arithmetic e a tese “O conhecimento matemático escolar: operações com números naturais (e adjacências) no Ensino Fundamental” os quais foram realizadas descrição e comentários cabíveis a respeito. O tipo de pesquisa é qualitativa, bibliográfica descritiva e, de certa forma, também explicativa. No referencial teórico apresentamos como o conteúdo expressão numérica é colocado em alguns documentos oficiais e livros didáticos do Ensino Fundamental. Como não foi encontrada nenhuma justificativa plausível e completa para a hierarquia das quatro operações nas expressões numéricas, realizamos um capítulo denominado retrospectiva histórica do uso das quatro operações e dos parênteses, neste capítulo descrevemos sobre as quatro operações, e sobre os parênteses que servirá para embasar o próximo capítulo denominado: hierarquia das quatro operações, buscando uma justificativa.

Nos travaux portent sur les Expressions Numériques Approximatives (ENA), définies comme des expressions linguistiques impliquant des valeurs numériques et un adverbe d'approximation, telles que "environ 100". Nous nous intéressons d’abord à l’interprétation d’ENA non contextualisées, dans ses aspects humain et computationnel. Après avoir formalisé des dimensions originales, arithmétiques et cognitive, permettant de caractériser les ENA, nous avons conduit une étude empirique pour collecter les intervalles de plages de valeurs dénotées par des ENA, qui nous a permis de valider les dimensions proposées. Nous avons ensuite proposé deux modèles d'interprétation, basés sur un même principe de compromis entre la saillance cognitive des bornes des intervalles et leur distance à la valeur de référence de l’ENA, formalisé par un front de Pareto. Le premier modèle estime l’intervalle dénoté, le second un intervalle flou représentant l’imprécision associée. Leur validation expérimentale à partir de données réelles montre qu’ils offrent de meilleures performances que les modèles existants. Nous avons également montrél’intérêt du modèle flou en l’implémentant dans le cadre des requêtes flexibles de bases de données. Nous avons ensuite montré, par une étude empirique, que le contexte et les interprétations, implicite vs explicite, ont peu d’effet sur les intervalles. Nous nous intéressons enfin à l’addition et à la multiplication d’ENA, par exemple pour évaluer la surface d’une pièce d’"environ 10" par "environ 20 mètres". Nous avons mené une étude dont les résultats indiquent que les imprécisions liées aux opérandes ne sont pas prises en compte lors des calculs. / Approximate Numerical Expressions (ANE) are imprecise linguistic expressions implying numerical values, illustrated by "about 100". We first focus on ANE interpretation, both in its human and computational aspects. After defining original arithmetical and cognitive dimensions allowing to characterize ANEs, we conducted an empirical study to collect the intervals of values denoted by ANEs. We show that the proposed dimensions are involved in ANE interpretation. In a second step, we proposed two interpretation models, based on the same principle of a compromise between the cognitive salience of the endpoints and their distance to the ANE reference value, formalized by Pareto frontiers. The first model estimates the denoted interval, the second one generates a fuzzy interval representing the associated imprecision. The experimental validation of the models, based on real data, show that they offer better performances than existing models. We also show the relevance of the fuzzy model by implementing it in the framework of flexible database queries. We then show, by the mean of an empirical study, that the semantic context has little effect on the collected intervals. Finally, we focus on the additions and products of ANE, for instance to assess the area of a room whose walls are "about 10" and "about 20 meters" long. We conducted an empirical study whose results indicate that the imprecisions associated with the operands are not taken into account during the calculations.