Absolute value has often been taught procedurally. Many students struggle with solving absolute value equations and inequalities because they do not have an understanding of the underlying concepts. This study was designed to determine to what extent solving absolute value equations and inequalities by using the concept of distance on a number line is an effective method. The claim is that if students use the distance concept on a number line, they will develop the necessary conceptual understanding in addition to just a procedural knowledge that will lead to the success with and flexibility in the use of strategies for more challenging problems. The following questions were addressed in this study: How and to what extent can using a number line develop a conceptual understanding of absolute value equations and inequalities? What solution strategies do students tend to use to solve absolute value equations and inequalities? Does the strategy depend on the complexity of the problem? To what extent do they exhibit flexibility in their use of strategies? What extensions are students able to make? What misconceptions do they have? In this study, lessons and assessments were implemented based on the “best practice” of using multiple representations with a focus on conceptual understanding of absolute value. The lessons were consistent with current content standards. Students completed a pre and post assessment, and some students were selected to participate in a 15 minute interview based on their responses from their assessments. The results were analyzed qualitatively and show that students struggled with remembering the procedure for solving absolute value equations and inequalities. The results also show that students were more successful when using the distance concept on a number line.

The thesis focuses on how students work with the absolute value and on the strategies of students' search for equations from graphs of functions. The aim of the thesis is to describe solution processes and the most common mistakes of students in searching for the equations as well as finding out how students understand the absolute value and how they work with the absolute value in solving the problems. The first, theoretical part, is the analysis of textbooks, RVP and graduations tests on given topics and a summary of selected researches on this topic. The second, research part, is devoted to the description and analysis of own research that was carried out on high school students. The final chapter summarizes the results of the research and compares these results with the theoretical part of this work.

京都大学 / 新制・課程博士 / 博士(情報学) / 甲第23546号 / 情博第776号 / 新制||情||132(附属図書館) / 京都大学大学院情報学研究科数理工学専攻 / (主査)教授 山下 信雄, 教授 太田 快人, 教授 永持 仁 / 学位規則第4条第1項該当 / Doctor of Informatics / Kyoto University / DFAM

本研究的目的主要為探討學生在解決絕對值相關題目時所遇到的困難，進而瞭解學生在解此類問題時出現錯誤之原因，希望研究的結果能夠提供教師作為補救教學或改進教學策略的依據，增進教學成效，並作為未來教學及研究的參考。 　　本研究採調查法，並輔之以訪談蒐集資料。第一階段為問卷調查，經由對絕對值相關概念作文獻探討，以及與多位數學教師討論之後，研究者以自編之絕對值相關概念試題本進行施測，藉此瞭解學生在各向度的答題情況，並且作為選擇訪談對象的依據。第二階段為無結構開放式訪談，主要訪談學生作答時之想法與解題策略，所有訪談皆全程錄音，並轉錄成文字檔後進行內容分析，進一步瞭解學生在概念上錯誤的內涵，以及探討解題困難產生的原因。 　　研究結果發現，學生在絕對值相關概念之錯誤可歸納出五大原因：過度簡化絕對值定義之口訣、無法進行絕對值概念中「幾何概念」與「算術概念」之間的轉化、不瞭解絕對值概念中各同義詞之間的關係、以偏概全絕對值之定義以及文字符號概念之理解困難。文後尚有提供絕對值相關教學改善的建議。 / This study aims to explore the kinds of difficulties encountered by junior high school students in solving problems related to absolute value as well as analyzing and identifying the probable causes of such difficulties. It is hoped that the results from this attempt can provide teachers with useful information regarding how to improve their instructional practices and plan remedial instruction, thereby enhancing their teaching effectiveness. The main methodology for this study is survey design supplemented with clinical interviews that allowed for in-depth information collection regarding problem solving strategies and difficulties from selected respondents. During the first stage, a literature review was conducted on research studies that focused on absolute values. This was followed by discussions with several junior high school mathematics teachers relating to learning difficulties they observed. Subsequently, a paper and pencil test instrument on absolute values with three main dimensions was compiled by the author to test the learning status of the participating students. Their performances would form the basis for selecting them to participate in the second stage of the study, namely, the interview phase. All clinical interviews were unstructured and they were recorded and transcribed into verbal records. Analyses were then performed to identify the presence of conceptual misunderstandings and explored the causes of such difficulties. It was found that students’ conceptual errors on absolute values can be classified into five different types, namely, oversimplifying the definition of absolute value into mnemonic phrases, inability to perform inscriptional transformation between geometric properties and arithmetical concepts of absolute values, incomprehension of the relationships among the synonyms related to the concept of absolute value, over-generalizing the definition of absolute values and difficulties in understanding the connotation behind letter symbols. Several suggestions regarding instructional practices as well as future direction of research based on the present findings were provided at the end of this study.

Le travail présenté dans cette thèse porte sur le rôle du cortex prémoteur dorsal (PMd) au sujet de la prise de décision (sélection d’une action parmis nombreux choix) et l'orientation visuelle des mouvements du bras. L’ouvrage décrit des expériences électrophysiologiques chez le singe éveillé (Macaca mulatta) permettant d’adresser une fraction importante des prédictions proposées par l'hypothèse des affordances concurrentes (Cisek, 2006; Cisek, 2007a). Cette hypothèse suggère que le choix de toute action est l’issue d'une concurrence entre les représentations internes des exigences et des atouts de chacune des options présentées (affordances; Gibson, 1979). Un intérêt particulier est donné au traitement de l'information spatiale et la valeur des options (expected value, EV) dans la prise de décisions. La première étude (article 1) explore la façon dont PMd reflète ces deux paramètres dans la période délai ainsi que de leur intéraction. La deuxième étude (article 2) explore le mécanisme de décision de façon plus détaillée et étend les résultats au cortex prémoteur ventral (PMv). Cette étude porte également sur la représentation spatiale et l’EV dans une perspective d'apprentissage. Dans un environnement nouveau les paramètres spatiaux des actions semblent être présents en tout temps dans PMd, malgré que la représentation de l’EV apparaît uniquement lorsque les animaux commencent à prendre des décisions éclairées au sujet de la valeur des options disponibles. La troisième étude (article 3) explore la façon dont PMd est impliqué aux “changements d'esprit“ dans un procès de décision. Cette étude décrit comment la sélection d’une action est mise à jour à la suite d'une instruction de mouvement (GO signal). I II Les résultats principaux des études sont reproduits par un modèle computationnel (Cisek, 2006) suggérant que la prise de décision entre plusieurs actions alternatives peux se faire par voie d’un mécanisme de concurrence (biased competition) qui aurait lieu dans la même région qui spécifie les actions. / This thesis examines the role of the dorsal premotor cortex (PMd) in the process of decision making (action selection) and visual guidance of arm movements. The work describes electrophysiological experiments conducted in awake monkeys (Macaca mulatta) and tests a number of important predictions suggested by the affordance competition hypothesis (Cisek, 2006; Cisek, 2007a). This hypothesis suggests that decisions can be viewed as the result of a competition between internal representations of conflicting demands and opportunities for actions or affordances (Gibson, 1979). Specific interest is given to the interaction between spatial information and expected value (EV) in a proposed affordance competition mechanism for action selection. The first study presented (article 1) explores how EV is represented during the delay period in PMd. This study also describes how this area reflects the spatial metrics of the options and examines the interaction between value and spatial information. The second study (article 2) explores the mechanism of action selection in more detail and extends the results to ventral premotor cortex (PMv). This study also addresses the nature of value and spatial representations from a learning perspective. In a novel environment the spatial metrics of the actions seem to be invariably present in PMd, meanwhile EV representations appear only once the animals make behaviorally informed decisions about the value of the available options. The third study (article 3) explores how PMd is involved in “changes of mind” in which action selection is updated following a movement instruction (GO signal). III IV The major findings in all these studies are reproduced by a computational model (Cisek, 2006) suggesting that decisions between actions can be made through a biased competition process that takes place in the same region that specifies the actions.

Le travail présenté dans cette thèse porte sur le rôle du cortex prémoteur dorsal (PMd) au sujet de la prise de décision (sélection d’une action parmis nombreux choix) et l'orientation visuelle des mouvements du bras. L’ouvrage décrit des expériences électrophysiologiques chez le singe éveillé (Macaca mulatta) permettant d’adresser une fraction importante des prédictions proposées par l'hypothèse des affordances concurrentes (Cisek, 2006; Cisek, 2007a). Cette hypothèse suggère que le choix de toute action est l’issue d'une concurrence entre les représentations internes des exigences et des atouts de chacune des options présentées (affordances; Gibson, 1979). Un intérêt particulier est donné au traitement de l'information spatiale et la valeur des options (expected value, EV) dans la prise de décisions. La première étude (article 1) explore la façon dont PMd reflète ces deux paramètres dans la période délai ainsi que de leur intéraction. La deuxième étude (article 2) explore le mécanisme de décision de façon plus détaillée et étend les résultats au cortex prémoteur ventral (PMv). Cette étude porte également sur la représentation spatiale et l’EV dans une perspective d'apprentissage. Dans un environnement nouveau les paramètres spatiaux des actions semblent être présents en tout temps dans PMd, malgré que la représentation de l’EV apparaît uniquement lorsque les animaux commencent à prendre des décisions éclairées au sujet de la valeur des options disponibles. La troisième étude (article 3) explore la façon dont PMd est impliqué aux “changements d'esprit“ dans un procès de décision. Cette étude décrit comment la sélection d’une action est mise à jour à la suite d'une instruction de mouvement (GO signal). I II Les résultats principaux des études sont reproduits par un modèle computationnel (Cisek, 2006) suggérant que la prise de décision entre plusieurs actions alternatives peux se faire par voie d’un mécanisme de concurrence (biased competition) qui aurait lieu dans la même région qui spécifie les actions. / This thesis examines the role of the dorsal premotor cortex (PMd) in the process of decision making (action selection) and visual guidance of arm movements. The work describes electrophysiological experiments conducted in awake monkeys (Macaca mulatta) and tests a number of important predictions suggested by the affordance competition hypothesis (Cisek, 2006; Cisek, 2007a). This hypothesis suggests that decisions can be viewed as the result of a competition between internal representations of conflicting demands and opportunities for actions or affordances (Gibson, 1979). Specific interest is given to the interaction between spatial information and expected value (EV) in a proposed affordance competition mechanism for action selection. The first study presented (article 1) explores how EV is represented during the delay period in PMd. This study also describes how this area reflects the spatial metrics of the options and examines the interaction between value and spatial information. The second study (article 2) explores the mechanism of action selection in more detail and extends the results to ventral premotor cortex (PMv). This study also addresses the nature of value and spatial representations from a learning perspective. In a novel environment the spatial metrics of the actions seem to be invariably present in PMd, meanwhile EV representations appear only once the animals make behaviorally informed decisions about the value of the available options. The third study (article 3) explores how PMd is involved in “changes of mind” in which action selection is updated following a movement instruction (GO signal). III IV The major findings in all these studies are reproduced by a computational model (Cisek, 2006) suggesting that decisions between actions can be made through a biased competition process that takes place in the same region that specifies the actions.

Series Active Filters (SAF) are designed for harmonic isolation and load voltage regulation of single-phase and three-phase voltage harmonic source type nonlinear loads. The novel Absolute Value Method (AVM) for load voltage harmonic extraction is proposed and applied in the control algorithm of SAF. The SAF compensated systems are represented by simplified linear models such that SAF controller gains can be easily determined. Harmonic isolation and load voltage regulation performances of 2.5 kW single-phase and 10 kW three-phase SAF compensated systems are evaluated by detailed simulations. Laboratory prototype single-phase and three-phase SAFs and loads are designed and manufactured. Digital signal processor based control platform is employed. Exclusive laboratory tests are conducted. Via laboratory experiments and simulations it is shown that AVM yields superior harmonic isolation and load voltage regulation performance compared to the conventional low/high pass filtering method. Theory, simulations, and experiments are well correlated and illustrate the feasibility of the proposed method.

Power system state estimation (PSSE) is an energy management system function responsible for the computation of the most likely values of state variables viz., bus voltage magnitudes and angles. The state estimation is obtained within a network at a given instant by solving a system of mostly non-linear equations whose parameters are the redundant measurements, both static such as transformer/line parameters and dynamic such as, status of circuit breakers/isolators, transformer tap positions, active/reactive power flows, generator active/reactive power outputs etc. PSSE involves solving an over determined set of nonlinear equations by minimizing a weighted norm of the measurement residuals. Typically, the L1 and L2 norms are employed. The use of L2 norm leads to state estimation based on the weighted least squares (WLS) criterion. This method is known to exhibit efficient filtering capability when the errors are Gaussian but fails in the case of presence of bad data. The method of hypothesis testing identification can be incorporated into the WLS estimator to detect and identify bad data. Nevertheless, it is prone to failure when the measurement is a leverage point. On the other hand state estimation based on the weighted least absolute value (WLAV) criterion using L1 norm, has superior bad data suppression capability. But it also fails in rejecting bad data measurements associated with leverage points. Leverage points are highly influential measurements that attract the state estimator solution towards them. Consequently, much research effort has focused recently, on producing a LAV estimator that remains robust in the presence of bad leverage measurements. This problem has been addressed in the thesis work. Two methods, which aims development of robust estimator that are insensitive to bad leverage points, have been proposed viz., (i) The objective function used here is obtained by linearizing L2 norm of the error function. In addition to the constraints corresponding to measurement set, constraints corresponding to bounds of state variables are also involved. Linear programming (LP) optimization is carried out using upper bound optimization technique. (ii) A hybrid optimization algorithm which is combination of”upper bound optimization technique” and ”an improved algorithm for discrete l1 linear approximation”, to restrict the state variables not to leave the basis during optimization process. Linear programming optimization, with bounds of state variables as additional constraints is carried out using the proposed hybrid optimization algorithm. The proposed state estimator algorithms are tested on 24-bus EHV equivalent of southern power network, 36-bus EHV equivalent of western grid, 205-bus interconnected grid system of southern region and IEEE-39 bus New England system. Performances of the proposed two methods are compared with the WLAV estimator in the presence of bad data associated with leverage points. Also, the effect of bad leverage measurements on the interacting bad data, which are non-leverage, has been compared. Results show that proposed state estimator algorithms rejects bad data associated with leverage points efficiently.

Cette thèse examine deux modèles alternatifs de prises de décision motrice à travers des données comportementales humaines et des données électrophysiologiques de singes obtenues lors d'une tâche de décision multi-attributs. Les théories psychologiques classiques suggèrent que la prise de décision soit une fonction de l'exécutif central (EC). En accord avec cela, de nombreuses études ont montré des modulations neuronales concernant les décisions dans le cortex préfrontal (PFC), renforçant la notion que les décisions sont prises à un niveau abstrait dans l'exécutif central du cerveau : le PFC. Cependant, de telles corrélations neuronales se trouvent également dans les régions sensorimotrices, qui étaient traditionnellement considérées externes à l’EC. Cela a conduit à un modèle alternatif de prise de décision dans un EC, impliquant plusieurs zones cérébrales, y compris les zones exécutives et sensorimotrices. Ce second modèle suggère qu'une décision est prise lorsque les compétitions au sein et entre les aires cérébrales arrivent à une résolution, ce qui permet d'atteindre un consensus distribué (CD). L'objectif principal de cette thèse est de tester les prédictions faites par ces deux modèles. Pour ce faire, nous avons conçu une tâche d'atteinte basée sur la valeur d'attributs multiples et créé une situation dans laquelle les deux modèles font des prédictions neuronales distinctes. Dans cette tâche, deux attributs visuels indépendants indiquaient le montant de la récompense associé à chaque cible. L'un était un degré de luminosité, information ascendante (BU pour "bottom-up"), ciblant le réseau de saillance par le biais de la voie visuelle dorsale. L'autre était un indice d'orientation de ligne, information descendante (TD pour "top-down"), ciblant le réseau de catégorisation basé sur la connaissance par le biais de la voie visuelle ventrale. Nous avons effectué des enregistrements dans la région d’atteinte pariétale (PRR) et le cortex pré-moteur dorsal (PMd) du singe, dont les activités neuronales ont été précédemment impliquées comme étant modulées par des attributs BU et TD similaires. Dans la plupart des essais, les deux attributs étaient congruents – tous les deux favorisant la même cible. Cependant, un sous-ensemble d'essais avait des cibles avec la même valeur de récompense totale, mais où les deux attributs étaient en conflit (les caractéristiques BU et TD favorisant des cibles opposées). Le modèle de l'EC prédit que dans ce cas, l’activité neuronale la plus précoce doit apparaître dans une région exécutive et que les régions sensorimotrices doivent recevoir la diffusion de cette décision. Ainsi, ce modèle prédit que la différence du temps de réaction entre le PRR et le PMd sera constante, quelle que soit la manière dont la décision est prise. En revanche, le modèle CD prédit que l’intervalle de décision doit refléter le rôle d'une région dans la décision en cours. Plus précisément, si PRR et PMd font tous deux parties du réseau de décision distribué et jouent un rôle dans l'évaluation des attributs BU et TD, un choix en faveur de l'attribut BU devrait apparaître d'abord dans le PRR et par la suite dans le PMd, tandis qu'un choix en faveur de l'attribut TD devrait apparaître dans l'ordre inverse. Notre étude démontre que le temps de réaction des participants humains était plus rapide dans les essais congruents et lors de l'utilisation de l'information BU par rapport à l'utilisation de l'information TD. La distribution ne reflétait pas linéairement la complexité de l'attribut et semblait plutôt suggérer une intégration incomplète des informations disponibles. Ainsi, le résultat n'était pas entièrement explicable par un modèle d'EC pur. Le temps de réaction des participants était également plus rapide lorsqu'ils choisissaient entre deux options de grande valeur par rapport aux options de faible valeur, ce qui suggère que la loi de Weber ne s'applique pas aux attributs visuels indiquant des informations de valeur. La distribution du temps de réaction de notre premier singe était similaire à celle des participants humains. Sur le plan neuronal, l’intervalle de décision du PMd était presque toujours plus rapide que celle du PRR et le PRR ne précédait jamais le PMd; aussi, la différence de l’intervalle de décision entre ces régions n'était pas constante. Le PMd a montré un biais de base pré-stimulus dans les essais de choix libre, alors que ce n’était pas le cas pour le PRR. La distribution de l’intervalle de décision dans le PMd variait également en fonction des conditions d'essai, tandis que celle du PRR ne distinguait que les cibles uniques des cibles multiples. Une tendance similaire a été observée dans les analyses préliminaires des potentiels de champ locaux (LFP). Enfin, les résultats préliminaires suggèrent des effets plus cohérents de la micro-stimulation dans le PMd que dans le PRR. Nos résultats soutiennent le rôle causal du PMd, mais pas celui du PRR. Nos résultats sont cohérents avec les rapports précédents sur l'activité neuronale liée au choix dans les régions pariétales, car l'activité du PRR reflétait le choix du singe dans notre tâche. Nos résultats sont également cohérents avec d'autres études montrant l'absence de preuves du rôle causal des régions pariétales dans la prise de décision, car l'ordre relatif de l'activité prédictive du choix dans le PRR et le PMd ne variait pas entre les différentes conditions. À la lumière de ces deux modèles, nos résultats suggèrent une troisième alternative, qui inclut potentiellement le PMd en tant que partie du réseau de décision, mais pas le PRR. / This thesis examines two alternative models of action decisions through human behavioural and monkey electrophysiological data obtained during a multi-attribute decision task. Classic psychological theories suggest that decision-making is a function of the Central Executive (CE). In line with this, many studies showed neural correlates of decision variables in the prefrontal cortex (PFC), strengthening the notion that decisions are made at an abstract level in the brain’s central executive: PFC. However, such neural correlates are also found in sensorimotor areas, which were traditionally considered outside the CE. This has led to an alternative model to the decision making in a CE, involving multiple brain areas including both executive and sensorimotor areas. This second model suggests that a decision is made when competitions within and across brain areas come to a resolution, thus a Distributed Consensus (DC) is achieved. The main objective of this thesis is to test the predictions made by these two models. To do so, we designed a multi-attribute value-based reaching task, and created a situation in which the two models made distinct neural predictions. In this task, two independent visual attributes indicated the amount of reward associated with each reach target. One was a “bottom-up” (BU) brightness, targeting the saliency network through the dorsal visual pathway. The other was a “top-down” (TD) line orientation cue, targeting the knowledge-based categorization network through the ventral visual pathway. We recorded from monkey parietal reach region (PRR) and dorsal premotor cortex (PMd), whose activities have previously been implied to be modulated by similar BU and TD attributes. In most trials, the two attributes were congruent – both favoring the same target. However, a subset of trials consisted of a conflict between the two attributes (BU and TD features favoring opposite targets), but the targets had the same total reward values. Here, the CE model predicted that the earliest choice-predictive activity should appear in an executive region, and sensorimotor regions were expected to be receiving this decision broadcast. Thus, the model predicted the latency difference between PRR and PMd to be constant, regardless of how the decision is made. In contrast, the DC model predicted choice latency should reflect a region’s role in the ongoing decision. Specifically, if both PRR and PMd are part of the distributed decision network and play a role in evaluating the BU and TD attributes, a choice in favor of the BU attribute should appear first in PRR and then in PMd, whereas a choice in favor of the TD attribute should appear in the opposite order. We report that human participants’ reaction time (RT) was faster in congruent trials and when using the BU information compared to when using the TD information. The RT distribution did not linearly reflect the attribute complexity, and instead suggested an incomplete integration of available information. Thus, the result was not fully explainable with a pure CE model. Their RT was also faster when choosing between two high-valued options compared to low-valued options, suggesting that Weber-Fechner law does not apply to visual attributes that indicate value. Our first monkey’s RT distribution was similar to that of human participants. Neurally, choice latency of PMd was almost always faster than that of PRR and PRR never preceded PMd, and the latency difference between these regions was not consistent. PMd showed a pre-stimulus baseline bias in free-choice trials, whereas PRR did not. The distribution of choice latency in PMd also varied with trial conditions, whereas that of PRR only discriminated single versus multiple targets. A similar trend was seen in preliminary analyses of local field potentials. Finally, preliminary results suggest more consistent effects of microstimulation in PMd than in PRR. Our results support the causal role of PMd, but do not support that of PRR. This is consistent with previous reports of choice-related neural activity in the parietal regions, as PRR activity did reflect the monkey’s choice in our task. Our results are also consistent with other studies showing the absence of evidence for parietal regions’ causal role in decision-making, as the relative order of choice-predictive activity in PRR and PMd did not vary between different conditions. In light of the two models, our results suggest a third alternative, which potentially includes PMd, but not PRR, as part of the decision network.