Investigating functional thinking in the elementary classroom : foundations of early algebraic reasoning

This paper examines the development of student functional thinking during a teaching experiment that was conducted in two classrooms with a total of 45 children whose average age was nine years and six months. The teaching comprised four lessons taught by a researcher, with a second researcher and classroom teacher acting as participant observers. These lessons were designed to enable students to build mental representations in order to explore the use of function tables by focusing on the relationship between input and output numbers with the intention of extracting the algebraic nature of the arithmetic involved. All lessons were videotaped. The results indicate that elementary students are not only capable of developing functional thinking but also of communicating their thinking both verbally and symbolically.

Exploring young student's functional thinking (Exploración del pensamiento funcional de estudiantes jóvenes)

The Early Years Generalizing Project (EYGP) involves Australian years 1 to 4 (age 5 to 9) students and investigates how they grasp and express generalizations. This paper focuses on data collected from 6 Year 1 students in an exploratory study within a clinical interview setting that required students to identify function rules. Preliminary findings suggest that the use of gestures (both by students and interviewers), self-talk (by students), and concrete acting out, assisted students to reach generalizations and to begin to express these generalities. It also appears that as students became aware of the structure, their use of gestures and selftalk tended to decrease.

Exploring young students' functional thinking

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Two of everything: developing functional thinking in the primary grades through children's literature

Traditionally algebra has been regarded as the domain of the secondary school years in Australia and many other countries. Non-mathematics teachers, parents and students often narrowly regard algebra as the manipulation of symbols adhering to tightly prescribed rules (Serow, Callingham & Muir, 2013). It is now recognised, however, that foundational ideas associated with algebraic thinking can, and should be, included in mathematics curricula in the pre-school and primary years (Bobis, Mulligan & Lowrie, 2009). This stance is reflected in the Australian Curriculum: Mathematics (Australian Curriculum Assessment & Reporting Authority, 2012) which extends key algebraic ideas to patterns and generalisations, and acknowledges that number and algebra are developed together as each enriches the study of the other. This article explores the concept of functional thinking and demonstrates how the story, ‘Two of Everything’ (Hong, 1993) is employed as a springboard for developing functional thinking with students from the early years through to upper primary schooling.

An exploration of young students' ability to generalise function tasks

The Early Years Generalising Project involves Australian students, Years 1-4 (age 5-9), and explores how the students grasp and express generalisations. This paper focuses on the data collected from clinical interviews with Year 3 and 4 cohorts in an investigative study focusing on the identifications, prediction and justification of function rules. It reports on students' attempts to generalise from function machine contexts, describing the various ways students express generalisation and highlighting the different levels of justification given by students. Finally, we conjecture that there are a set of stages in the expression and justification of generalisations that assist students to reach generality within tasks.

O desenvolvimento do pensamento algébrico de alunos do 4º ano de escolaridade : uma experiência de ensino

Tese de doutoramento, Educação (Didática da Matemática), Universidade de Lisboa, Instituto de Educação, 2014

The functional significance of EEG microstates-Associations with modalities of thinking

The momentary, global functional state of the brain is reflected by its electric field configuration. Cluster analytical approaches consistently extracted four head-surface brain electric field configurations that optimally explain the variance of their changes across time in spontaneous EEG recordings. These four configurations are referred to as EEG microstate classes A, B, C, and D and have been associated with verbal/phonological, visual, attention reorientation, and subjective interoceptive-autonomic processing, respectively. The present study tested these associations via an intra-individual and inter-individual analysis approach. The intra-individual approach tested the effect of task-induced increased modality-specific processing on EEG microstate parameters. The inter-individual approach tested the effect of personal modality-specific parameters on EEG microstate parameters. We obtained multichannel EEG from 61 healthy, right-handed, male students during four eyes-closed conditions: object-visualization, spatial-visualization, verbalization (6 runs each), and resting (7 runs). After each run, we assessed participants' degrees of object-visual, spatial-visual, and verbal thinking using subjective reports. Before and after the recording, we assessed modality-specific cognitive abilities and styles using nine cognitive tests and two questionnaires. The EEG of all participants, conditions, and runs was clustered into four classes of EEG microstates (A, B, C, and D). RMANOVAs, ANOVAs and post-hoc paired t-tests compared microstate parameters between conditions. TANOVAs compared microstate class topographies between conditions. Differences were localized using eLORETA. Pearson correlations assessed interrelationships between personal modality-specific parameters and EEG microstate parameters during no-task resting. As hypothesized, verbal as opposed to visual conditions consistently affected the duration, occurrence, and coverage of microstate classes A and B. Contrary to associations suggested by previous reports, parameters were increased for class A during visualization, and class B during verbalization. In line with previous reports, microstate D parameters were increased during no-task resting compared to the three internal, goal-directed tasks. Topographic differences between conditions concerned particular sub-regions of components of the metabolic default mode network. Modality-specific personal parameters did not consistently correlate with microstate parameters except verbal cognitive style which correlated negatively with microstate class A duration and positively with class C occurrence. This is the first study that aimed to induce EEG microstate class parameter changes based on their hypothesized functional significance. Beyond, the associations of microstate classes A and B with visual and verbal processing, respectively and microstate class D with interoceptive-autonomic processing, our results suggest that a finely-tuned interplay between all four EEG microstate classes is necessary for the continuous formation of visual and verbal thoughts, as well as interoceptive-autonomic processing. Our results point to the possibility that the EEG microstate classes may represent the head-surface measured activity of intra-cortical sources primarily exhibiting inhibitory functions. However, additional studies are needed to verify and elaborate on this hypothesis.

The role of relevant information in functional counterfactual thinking

Although counterfactual thinking is typically activated by a negative outcome, it can have positive effects by helping to regulate and improve future behavior. Known as the content-specific pathway, these counterfactual ruminations use relevant information (i.e., information that is directly related to the problem at hand) to elicit insights about the problem, create a connection between the counterfactual and the desired behavior, and strengthen relevant behavioral intentions. The current research examines how changing the type of relevant information provided (i.e., so that it is either concrete and detailed or general and abstract) influences the relationship between counterfactual thinking and behavioral intentions. Experiments 1 and 2 found that counterfactual thinking facilitated relevant intentions when these statements involved detailed information (Experiment 1) or specific behaviors (Experiment 2) compared to general information (Experiment 1), categories of behavior, or traits (Experiment 2). Experiment 3 found that counterfactuals containing a category of behavior facilitated specific behavioral intentions, relative to counterfactuals focusing on a trait. However, counterfactuals only facilitated intentions that included specific behaviors, but not when intentions focused on categories of behaviors or traits (Experiment 4). Finally, this effect generalized to other relevant specific behaviors; a counterfactual based on one relevant specific behavior facilitated an intention based on another relevant specific behavior (Experiment 5). Together, these studies further clarify our understanding of the content-specific pathway and provide a more comprehensive understanding of functional counterfactual thinking.

Functional oligosaccharides: application and manufacture.

Oligosaccharides are attracting increasing interest as prebiotic functional food ingredients. They can be extracted or obtained by enzymatic hydrolysis from a variety of biomass sources or synthesized from simple oligosaccharides by enzymatic transfer reactions. The major prebiotic oligosaccharides on the market are inulin, fructo-oligosaccharides, and galacto-oligosaccharides. They have been evaluated using a range of in vitro and in vivo methods, although there is a need for more large-scale human trials using modern microbiological methods. Prebiotics are being studied for their effects on gut health and well being and specific clinical conditions, including colon cancer, inflammatory bowel disease (IBD), acute infections, and mineral absorption. Developing understanding of the functional ecology of the human gut is influencing current thinking on what a prebiotic might achieve and is providing new targets for prebiotic intervention.

Contribuição do design thinking para o processo de inovação de produtos

Este trabalho visa identificar práticas de design thinking (DT), isto é, as abordagens características da disciplina do design para criação de inovações, que possam ser incorporadas ao processo de desenvolvimento de novos produtos (DNP). Cinco dessas práticas – abordagem centrada em pessoas, estilo de trabalho colaborativo, pensar fazendo, visualização e combinação de abordagens divergente e convergente – foram analisadas por meio do estudo de caso em profundidade de três empresas polares: um escritório profissional de design, uma empresa de produtos para cuidados pessoais e um fabricante de ferramentas profissionais. A discussão teórica apoiou-se em autores de DNP e de DT que estudaram a evolução desses campos em gerações de maturidade, trazendo também visões bem recentes que apontam para a relevância das mudanças em curso. Entre os achados da pesquisa, observou-se que o modelo de gestão e a estrutura organizacional – orientada por projetos com times multifuncionais ou por processos com estrutura funcional – tem influência na adoção das práticas de DT. Outro achado, esse menos evidente, mostrou que a forma como a empresa se relaciona com clientes e usuários, guarda semelhança com a forma como trata seus profissionais de DNP. Uma descoberta interessante é a função moderadora do alinhamento de propósitos entre empresa e colaborador, para a adoção dessas práticas. Como contribuição ao campo de estudo, apresenta-se um diagrama de cinco gerações de evolução da oferta de novos produtos (ONP), termo utilizado para incorporar não só os produtos com inovações desenvolvidas e de propriedade da empresa focal, mas também aquelas desenvolvidas pelos próprios usuários e as inovações construídas nas redes sociais e oferecidas pela empresa focal.

Multidisciplinary approach for the aesthetic treatment of maxillary lateral incisors agenesis: thinking about implants?

Missing maxillary lateral incisors create an esthetic problem with specific orthodontic and prosthetic considerations. Implants are commonly used to replace congenitally missing lateral incisors in adolescent orthodontic patients. However, an interdisciplinary approach should be observed during the diagnosis, prognosis, and treatment plan to provide a result with good predictability and meet the esthetic and functional expectations of the patient. The present study describes a case of a young patient with tooth agenesis of maxillary lateral incisors, which was conducted with an integrated planning. After 5-year follow-up of 2 fixed implant-supported prostheses, clinical and radiographic examination showed the treatment to be successful. (Oral Surg Oral Med Oral Pathol Oral Radiol 2012;114:e22-e28)

Functional taxonomy of bacterial hyperstructures.

The levels of organization that exist in bacteria extend from macromolecules to populations. Evidence that there is also a level of organization intermediate between the macromolecule and the bacterial cell is accumulating. This is the level of hyperstructures. Here, we review a variety of spatially extended structures, complexes, and assemblies that might be termed hyperstructures. These include ribosomal or "nucleolar" hyperstructures; transertion hyperstructures; putative phosphotransferase system and glycolytic hyperstructures; chemosignaling and flagellar hyperstructures; DNA repair hyperstructures; cytoskeletal hyperstructures based on EF-Tu, FtsZ, and MreB; and cell cycle hyperstructures responsible for DNA replication, sequestration of newly replicated origins, segregation, compaction, and division. We propose principles for classifying these hyperstructures and finally illustrate how thinking in terms of hyperstructures may lead to a different vision of the bacterial cell.

Neural correlates of wishful thinking.

Wishful thinking (WT) implies the overestimation of the likelihood of desirable events. It occurs for outcomes of personal interest, but also for events of interest to others we like. We investigated whether WT is grounded on low-level selective attention or on higher level cognitive processes including differential weighting of evidence or response formation. Participants in our MRI study predicted the likelihood that their favorite or least favorite team would win a football game. Consistent with expectations, favorite team trials were characterized by higher winning odds. Our data demonstrated activity in a cluster comprising parts of the left inferior occipital and fusiform gyri to distinguish between favorite and least favorite team trials. More importantly, functional connectivities of this cluster with the human reward system were specifically involved in the type of WT investigated in our study, thus supporting the idea of an attention bias generating WT. Prefrontal cortex activity also distinguished between the two teams. However, activity in this region and its functional connectivities with the human reward system were altogether unrelated to the degree of WT reflected in the participants' behavior and may rather be related to social identification, ensuring the affective context necessary for WT to arise.

Functional and effective connectivity in MEG. Application to the study of epilepsy

Nuestro cerebro contiene cerca de 1014 sinapsis neuronales. Esta enorme cantidad de conexiones proporciona un entorno ideal donde distintos grupos de neuronas se sincronizan transitoriamente para provocar la aparición de funciones cognitivas, como la percepción, el aprendizaje o el pensamiento. Comprender la organización de esta compleja red cerebral en base a datos neurofisiológicos, representa uno de los desafíos más importantes y emocionantes en el campo de la neurociencia. Se han propuesto recientemente varias medidas para evaluar cómo se comunican las diferentes partes del cerebro a diversas escalas (células individuales, columnas corticales, o áreas cerebrales). Podemos clasificarlos, según su simetría, en dos grupos: por una parte, la medidas simétricas, como la correlación, la coherencia o la sincronización de fase, que evalúan la conectividad funcional (FC); mientras que las medidas asimétricas, como la causalidad de Granger o transferencia de entropía, son capaces de detectar la dirección de la interacción, lo que denominamos conectividad efectiva (EC). En la neurociencia moderna ha aumentado el interés por el estudio de las redes funcionales cerebrales, en gran medida debido a la aparición de estos nuevos algoritmos que permiten analizar la interdependencia entre señales temporales, además de la emergente teoría de redes complejas y la introducción de técnicas novedosas, como la magnetoencefalografía (MEG), para registrar datos neurofisiológicos con gran resolución. Sin embargo, nos hallamos ante un campo novedoso que presenta aun varias cuestiones metodológicas sin resolver, algunas de las cuales trataran de abordarse en esta tesis. En primer lugar, el creciente número de aproximaciones para determinar la existencia de FC/EC entre dos o más señales temporales, junto con la complejidad matemática de las herramientas de análisis, hacen deseable organizarlas todas en un paquete software intuitivo y fácil de usar. Aquí presento HERMES (http://hermes.ctb.upm.es), una toolbox en MatlabR, diseñada precisamente con este fin. Creo que esta herramienta será de gran ayuda para todos aquellos investigadores que trabajen en el campo emergente del análisis de conectividad cerebral y supondrá un gran valor para la comunidad científica. La segunda cuestión practica que se aborda es el estudio de la sensibilidad a las fuentes cerebrales profundas a través de dos tipos de sensores MEG: gradiómetros planares y magnetómetros, esta aproximación además se combina con un enfoque metodológico, utilizando dos índices de sincronización de fase: phase locking value (PLV) y phase lag index (PLI), este ultimo menos sensible a efecto la conducción volumen. Por lo tanto, se compara su comportamiento al estudiar las redes cerebrales, obteniendo que magnetómetros y PLV presentan, respectivamente, redes más densamente conectadas que gradiómetros planares y PLI, por los valores artificiales que crea el problema de la conducción de volumen. Sin embargo, cuando se trata de caracterizar redes epilépticas, el PLV ofrece mejores resultados, debido a la gran dispersión de las redes obtenidas con PLI. El análisis de redes complejas ha proporcionado nuevos conceptos que mejoran caracterización de la interacción de sistemas dinámicos. Se considera que una red está compuesta por nodos, que simbolizan sistemas, cuyas interacciones se representan por enlaces, y su comportamiento y topología puede caracterizarse por un elevado número de medidas. Existe evidencia teórica y empírica de que muchas de ellas están fuertemente correlacionadas entre sí. Por lo tanto, se ha conseguido seleccionar un pequeño grupo que caracteriza eficazmente estas redes, y condensa la información redundante. Para el análisis de redes funcionales, la selección de un umbral adecuado para decidir si un determinado valor de conectividad de la matriz de FC es significativo y debe ser incluido para un análisis posterior, se convierte en un paso crucial. En esta tesis, se han obtenido resultados más precisos al utilizar un test de subrogadas, basado en los datos, para evaluar individualmente cada uno de los enlaces, que al establecer a priori un umbral fijo para la densidad de conexiones. Finalmente, todas estas cuestiones se han aplicado al estudio de la epilepsia, caso práctico en el que se analizan las redes funcionales MEG, en estado de reposo, de dos grupos de pacientes epilépticos (generalizada idiopática y focal frontal) en comparación con sujetos control sanos. La epilepsia es uno de los trastornos neurológicos más comunes, con más de 55 millones de afectados en el mundo. Esta enfermedad se caracteriza por la predisposición a generar ataques epilépticos de actividad neuronal anormal y excesiva o bien síncrona, y por tanto, es el escenario perfecto para este tipo de análisis al tiempo que presenta un gran interés tanto desde el punto de vista clínico como de investigación. Los resultados manifiestan alteraciones especificas en la conectividad y un cambio en la topología de las redes en cerebros epilépticos, desplazando la importancia del ‘foco’ a la ‘red’, enfoque que va adquiriendo relevancia en las investigaciones recientes sobre epilepsia. ABSTRACT There are about 1014 neuronal synapses in the human brain. This huge number of connections provides the substrate for neuronal ensembles to become transiently synchronized, producing the emergence of cognitive functions such as perception, learning or thinking. Understanding the complex brain network organization on the basis of neuroimaging data represents one of the most important and exciting challenges for systems neuroscience. Several measures have been recently proposed to evaluate at various scales (single cells, cortical columns, or brain areas) how the different parts of the brain communicate. We can classify them, according to their symmetry, into two groups: symmetric measures, such as correlation, coherence or phase synchronization indexes, evaluate functional connectivity (FC); and on the other hand, the asymmetric ones, such as Granger causality or transfer entropy, are able to detect effective connectivity (EC) revealing the direction of the interaction. In modern neurosciences, the interest in functional brain networks has increased strongly with the onset of new algorithms to study interdependence between time series, the advent of modern complex network theory and the introduction of powerful techniques to record neurophysiological data, such as magnetoencephalography (MEG). However, when analyzing neurophysiological data with this approach several questions arise. In this thesis, I intend to tackle some of the practical open problems in the field. First of all, the increase in the number of time series analysis algorithms to study brain FC/EC, along with their mathematical complexity, creates the necessity of arranging them into a single, unified toolbox that allow neuroscientists, neurophysiologists and researchers from related fields to easily access and make use of them. I developed such a toolbox for this aim, it is named HERMES (http://hermes.ctb.upm.es), and encompasses several of the most common indexes for the assessment of FC and EC running for MatlabR environment. I believe that this toolbox will be very helpful to all the researchers working in the emerging field of brain connectivity analysis and will entail a great value for the scientific community. The second important practical issue tackled in this thesis is the evaluation of the sensitivity to deep brain sources of two different MEG sensors: planar gradiometers and magnetometers, in combination with the related methodological approach, using two phase synchronization indexes: phase locking value (PLV) y phase lag index (PLI), the latter one being less sensitive to volume conduction effect. Thus, I compared their performance when studying brain networks, obtaining that magnetometer sensors and PLV presented higher artificial values as compared with planar gradiometers and PLI respectively. However, when it came to characterize epileptic networks it was the PLV which gives better results, as PLI FC networks where very sparse. Complex network analysis has provided new concepts which improved characterization of interacting dynamical systems. With this background, networks could be considered composed of nodes, symbolizing systems, whose interactions with each other are represented by edges. A growing number of network measures is been applied in network analysis. However, there is theoretical and empirical evidence that many of these indexes are strongly correlated with each other. Therefore, in this thesis I reduced them to a small set, which could more efficiently characterize networks. Within this framework, selecting an appropriate threshold to decide whether a certain connectivity value of the FC matrix is significant and should be included in the network analysis becomes a crucial step, in this thesis, I used the surrogate data tests to make an individual data-driven evaluation of each of the edges significance and confirmed more accurate results than when just setting to a fixed value the density of connections. All these methodologies were applied to the study of epilepsy, analysing resting state MEG functional networks, in two groups of epileptic patients (generalized and focal epilepsy) that were compared to matching control subjects. Epilepsy is one of the most common neurological disorders, with more than 55 million people affected worldwide, characterized by its predisposition to generate epileptic seizures of abnormal excessive or synchronous neuronal activity, and thus, this scenario and analysis, present a great interest from both the clinical and the research perspective. Results revealed specific disruptions in connectivity and network topology and evidenced that networks’ topology is changed in epileptic brains, supporting the shift from ‘focus’ to ‘networks’ which is gaining importance in modern epilepsy research.

Using an experiential business game to stimulate sustainable thinking in marketing education

Recent years have seen a significant increase in the importance of environmental protection and sustainability to consumers, policy makers, and society in general. Reflecting this, most organizations are at least aware of this new agenda and wish to be seen as taking steps to improve behaviors in this regard. However, there appears to be a gap between this evolving agenda and the comparatively low level of knowledge that marketing managers actually have of the environmental impact of their own functional decisions. We suggest that this low knowledge level may be due, in part, to the marketplace focus of foundational marketing educational programs, and we attempt to show how broadening the horizons of marketing courses can help students (i.e., future managers) more deeply understand the environmental consequences of their actions. We demonstrate the use of a novel business game, based on the Life Cycle Assessment method, as the foundational cornerstone for the development of a broad understanding of the environmental impact of marketing decisions and actions for the entire life cycle of a product—from raw material extraction to ultimate disposal. The results of an empirical study show that this approach increases students’ appreciation for, and understanding of, these fundamental environmental sustainability concepts.