published_or_final_version / Mathematics / Master / Master of Philosophy

Let 1, 2, 3, 5, 8, … denote the Fibonacci sequence beginning with 1 and 2, and then setting each subsequent number to the sum of the two previous ones. Every positive integer n can be expressed as a sum of distinct Fibonacci numbers in one or more ways. Setting R(n) to be the number of ways n can be written as a sum of distinct Fibonacci numbers, we exhibit certain regularity properties of R(n), one of which is connected to the Euler φ-function. In addition, using a theorem of Fine and Wilf, we give a formula for R(n) in terms of binomial coefficients modulo two.

Alternate numeration systems are common in preservice teacher (PST) mathematics curricula, but there is limited research on how to leverage alternate systems to promote the development of mathematical knowledge for teaching. I analyzed the role of alternate numeration systems in three ways. I conducted a thematic analysis of current PST textbooks to consider the role of alternate numeration systems in written curricula. I conducted a teaching experiment to analyze PSTs' mathematical activity as they engaged with a base five task sequence to reinvent an algorithm for multiplication. And I introduced problematizing mathematical contexts as a design heuristic, situating this within the design theory of Realistic Mathematics Education. I found that alternate numeration systems can be leveraged to create opportunities for PSTs to (a) engage in guided reinvention of an algorithm, (b) improve understanding of base ten by comparing it to other numeration systems, and (c) reflect on their learning experience and the learning experiences of children.

The Fan Assessment Numeration System (FANS) is an improved air velocity traverse method for measuring in situ fan performance. The FANS has been widely used, but variations of its test procedure are commonly employed to accommodate physical or operational barriers encountered in the field. This laboratory study evaluated the use of transitions to connect a 1.37m FANS unit to two smaller fans (1.22m and 0.91m diameter) and one 1.37m diameter fan. Tests were conducted with the FANS unit positioned on both intake and discharge sides of the fans. Three different transition angles (30o, 45o and 60o) and the use of no transition were evaluated. Discharge tests were also performed with no enclosed connection between FANS and fan housings. A different experiment was conducted for each fan size. Data was analyzed by comparing test results to the control with Dunnett’s procedure. Results showed significant differences as much as 5.3% ± 1.20% for intake treatments, 17.2% ± 3.04% for sealed discharge treatments and 37.1% ± 12.24% for discharge treatments with no enclosed connection. All transition angles produced similar fan test results. Differences between test results from the discharge and control treatments increased as differences between FANS and fan dimensions increased.

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Les travaux de cette thèse s'inscrivent dans la théorie des automates et des langages formels. ils peuvent se diviser en deux parties qui donnent également deux visions différentes de manipuler les langages dans la théorie des automates. La première partie s'intéresse à la notion de dérivation des expressions qui permet de faire passer le formalisme des quotients de langages au niveau des expressions rationnelles. en particulier cette thèse étudie les termes dérivés cassés d'une expression rationnelle. ces termes dérivés cassés permettent, sous certaines circonstances, et à l'aide d'autres opérations, une réversibilité de la transformation d'un automate en une expression rationnelle. Dans la seconde partie, la théorie des automates est utilisée pour traiter des problèmes sur les systèmes de numération. les systèmes de numération représentent des nombres par des mots. il est possible d'utiliser des automates et des transducteurs afin d'être capable de 'compter' sur un langage rationnel représentant les entiers. plus précisément ces automates sont étudiés pour le cas des systèmes de numération abstraits qui associent à chaque entier un mot d'un langage rationnel, ordonné par l'ordre radiciel. dans un tel système, la fonction qui permet de calculer le mot suivant est une fonction co-séquentielle par morceaux, c'est-à-dire qu'il suffit de lire deux fois le mot d'entrée de la droite vers la gauche pour qu'une machine calcule son image. / The works in this thesis lies in the automata and formal languages theory. in the first part, the notion of derivation of rational expressions is studied. more precisely the broken derived terms of a rational expressions. Theses broken derived terms allow, under certain circumstances, with some other operations on automata, to have the reversibility of the transformation of an automaton into a rational expression. In the second part, automata and tranducers allow to 'count' on a numeration system, where integers are represented by words on a rational language. more precisely, this part adress the problem of counting in an abstract numeration systems, which maps to any word of a rational language, ordored by radix order, the integer corresponding to the order of the word. in such a numeration system, the function which computes the successor of a word is a piecewise co-sequential function: it can be realised by a machine which reads the input two times to give the output.

The purpose of this study was to find some remedy to frustrations engendered when children fall to grasp the essential principle of place value after several attempts at reteaching. It was hypothesized that these children must have failed to acquire understanding of some numerical principle(s) prerequisite to understanding the place value aspect of the numeration system. Four plausible prerequisite principles were identified (1) synthesis of ordinal and cardinal properties of the numeration system, (2) both the addition and subtraction operations, (3) understanding of counting by groups, and (4) understanding of exchange equivalences such as one ten for ten ones, etc. It was hypothesized that understanding of analog clock reading was also dependent upon understanding of the same four prerequisite principles. By conducting four pilot studies, six interview protocol instruments were developed to measure levels of understanding for the four prerequisite principles and the place value and clock reading criterion principles. Three levels of understanding: no understanding, transitional understanding, and competence were designated to correspond with Plagetian stages in the development of a new operation. Forty-eight children, twenty with second grade completed and twenty-eight with third grade completed, were tested on all six instruments. Hypotheses tested were: (1) if the four identified prerequisite principles are necessary to understanding of place value, then subjects will demonstrate a level of understanding on the place value measure no higher than their lowest level of understanding achieved on the four prerequisite measures; and (2) if the four identified prerequisite . principles are necessary to understanding of clock reading, then subjects will demonstrate a level of understanding on the clock reading measure no higher than their lowest level of understanding achieved on . the four prerequisite measures. The findings were that both hypotheses were supported at the .01 probability level. Analysis of the research design and examiner observations suggested possible explanations for anomalous aspects of the obtained data. Limitations, directions for further research, and implications for teachers were also discussed. / Ph. D.

Ce travail de thèse porte sur les micro-décisions (Comiti, Grenier & Margolinas, 1995) c’est-à-dire les décisions didactiques de l’enseignant en classe, l’objectif est de contribuer au développement d’un système informatique capable d’accompagner l’enseignant dans ses prises de décisions.Nous avons choisi d’étudier ces décisions en entrant par un savoir : la numération en cycle 2 (Tempier, 2013). Le cadre théorique retenu pour la description de ce savoir est la Théorie Anthropologique du Didactique (Chevallard, 1998) plus spécifiquement une version implémentable : T4TEL (Chaachoua, Ferraton, & Desmoulins, 2013), (Chaachoua & Bessot, 2016).Afin de modéliser l’activité du professeur au sein d’une situation didactique et de prendre en compte ses activités en dehors de cette situation nous utilisons la structuration du milieu (Margolinas, 2004). Les micro-décisions de l’enseignant sont ainsi étudiées en rapport avec son projet d’enseignement, ses observations de l’activité des élèves, ses connaissances de type épistémiques et de type histoire didactique.Notre méthode de recherche est une ingénierie didactique dont la spécificité est d’impliquer des enseignants dans les phases d’analyse et de conception. Dans le cadre de cette ingénierie nous avons conçu : (1) une simulation du matériel de numération « bûchettes » : « SimBûchettes » ; (2) une base d’exercice pour « SimBûchettes » et (3) un dispositif expérimental. Ce dispositif expérimental est composé d’un outil de simulation côté élève dont les fondements sont didactiques - « Simbûchettes » - et d’un outil d’orchestration, côté enseignant, qui lui permet de consulter et d’organiser l’activité des élèves en temps réel - instanciation du Framework Chao (Wang, 2016) pour « Simbûchettes » -. Via notre dispositif nous avons accès aux actions de l’enseignant sachant les informations consultées concernant la production de l’élève et pouvons inférer ses micro-décisions.Ce dispositif nous a permis d’observer les décisions didactiques d’un enseignant d’une classe de CE1 pendant une année scolaire et d’affiner ainsi notre modèle des micro-décisions de l’enseignant. / This thesis work deals with micro-decisions (Comiti, Grenier & Margolinas, 1995), namely teachers’ decisions in class in relation to the subject they have to teach. Our aim is to contribute to the development of a TEL (Technology Enhanced Learning) that can guide teachers in their decisions.These decisions are analyzed through a specific field: decimal number system in cycle 2 (Tempier, 2013). For the description of this field we have chosen the Anthropological Theory of Didactics (Chevallard, 1998), more specifically an implementable version: T4TEL (Chaachoua, Ferraton, & Desmoulins, 2013), (Chaachoua & Bessot, 2016).Margolinas’s model about structuring the environment (2004) is used to take into account different learning activities during a teaching session. So, teachers’ micro-decisions are studied in relation to their teaching project, their observations of pupils’ activities, their knowledge of epistemic and didactic history type.Our research method is a didactical engineering whose specificity is to involve teachers in the analysis and design stages. In this engineering we have designed (1) a simulation of counting material “counting rods”: “SimBûchettes” ; (2) a bank of exercises for “SimBûchettes” and (3) an experimental device. This device is composed of, a simulation tool whose fundations are didactic – “Simbûchettes” -, on the pupils’ side, and on the teachers’side a classroom orchestration tool which allow teachers to consult and organize pupils’ activities in real time – Chao Framework’s instantiation (Wang, 2016) for “Simbûchettes”.Via our device we have access to teachers’ actions, and we can know what information has been consulted in pupils’ work. Then we can infer the micro-decisions teachers have made.This device has allowed us to observe teachers’ decisions in a primary class (CE1, 7 years-olds) during one school year and refine our teachers’ micro-decisions model.

The use of velocity traverses to measure in-situ air flow rate of ventilation fans can be subject to significant errors. The Fan Assessment Numeration System (FANS) was developed by the USD-ARS Southern Poultry Research Laboratory and refined at the University of Kentucky to measure air flow of fans in-situ. The procedures for using the FANS unit to test fans in-situ are not completely standardized. This study evaluated the effect of operating fan positions relative to the FANS unit for ten 1.22 m diameter fans in two types of poultry barns, with fans placed immediately next to each other and 1.6 m apart. Fans were tested with the FANS unit placed near both the intake and discharge sides of the tested fans. Data were analyzed as two Generalized Randomized Complete Block designs (GRCB), with a 2 (FANS inside or outside) x 6 (operating fan combinations) factorial arrangement of treatments. Results showed significant differences as much as 12.6 ± 4.4% between air flow values obtained under conditions of different operating fan combinations. Placing the FANS unit outside provided valid fan test results. A standardized procedure for using the FANS unit to test fans in-situ was elaborated and presented in this work.

Using the Material Engagement Theory of Cognitive Archaeologist Lambros Malafouris as its framework, the thesis offers a unique synthesis of data from neuroscience, ethnography, linguistics, and archaeology to outline how number concepts are realized, manipulated, and elaborated. The process is described as an interactivity of psychological processes like numerosity, behaviors that manipulate objects into concept-generating stimuli, and material objects with semiotic qualities distinct from those of language and agency distinct from that of brains and bodies. The counting technologies of the Ancient Near East (ANE) are then analyzed through archaeological and textual evidence spanning the late Upper Paleolithic to the Bronze Age, from the first realization of number concepts in a pristine original condition to their elaboration into one of the ancient world's greatest mathematical traditions, a foundation for mathematical thinking today. Insights from the way numbers are realized through psychological-behavioral-material interactivity are used to challenge three dominant conceptualizations of ANE numbers: first, the idea that the ANE numerical lexicon would have counted only to very low numbers; second, that Neolithic tokens were the first counting technology; and third, that numbers were 'concrete' before they became 'abstract'. Considering archaeological evidence from the Epipaleolithic Levant and drawing on linguistic and ethnographic evidence to characterize the regional prehistory, the thesis suggests that the numerical lexicon would have included relatively high numbers prior to the Neolithic; that finger-counting (linguistically attested) and tallies (archaeologically attested) would have preceded tokens; and that numbers are 'abstract' concepts whose content changes in conjunction with the incorporation and use of different material forms. The evidence provided to support these alternatives implies that numbers may have originated in the late Upper Paleolithic and arithmetic early in the Neolithic, pushing the onset of these capabilities further back than is commonly held. In addition to tallies and tokens, the thesis explores fingers and numerical notations as material artifacts, enabling an analysis of how materiality might structure numerical concepts, influence a number system's capabilities, limitations, and elaboration potential, and affect brains and behavior over cultural spans of time. Insights generated by the case study are then applied to the role of materiality in cognition more generally, including how concepts become distributed across multiple material forms; the reasons why materiality might be transparent (or invisible) in cognition; and the differences between thinking through and thinking about materiality.