In the field of Nuclear Medicine, positron emission tomography (PET) plays an important role as one of the most common diagnostic tools in the area of medical imaging. However, various physical degradation factors occur when the data is recorded, which leads to a low signal-to-noise ratio. This makes the quality of the PET images less than optimal. The proposed method for solving this problem is to use a machine learning approach to find a sparse representation of the sinograms. Where a suitable sparse representation should retain as much of the signal and as little of the noise as possible. To accomplish this a sparse autoencoder was trained on wavelet decompositions of sinograms simulated from medical images in order to learn underlying structures. Three different wavelet families were tested, Daubechies 4, biorthogonal 4.4, and Haar. The trained model was able to find sparse representations of the input sinograms in the wavelet domain. Although the sparse autoencoder managed to learn the basic structures of the sinograms, it struggled with the more complex details. Compared to a conventional denoising method using hard thresholding the sparse autoencoder did not manage to produce as good of a result in terms of the reconstructed PET image quality.
In mathematics, students’ abilities to make transformations between mathematical representations are fundamental. The recent implementation of digital technologies, such as Dynamic Geometry Environments (DGEs), have changed students’ access to mathematical representations by providing a variety of different representations, available just by pressing a button. Students use of a DGE may change their mathematical communication to become more dynamic and active. However, it is not clear how the use of DGEs and the change in communication style relate to more epistemological aspects of students’ mathematical work. This study, therefore, investigates the interplay between students’ dynamic mathematical communication and their transformations of mathematical representations when using a DGE from a competency perspective. Based on analyses of instances of lower secondary school students’ dynamic mathematical communication, findings indicate that students’ mathematical communication reflect the new ways of engaging with mathematical representations when using a DGE. Hence, the communication becomes dynamic. Theoretically, the complexity of having “continuous” transformations of mathematical representations may reduce the readiness of the mathematical communication competency. The “continuous” transformation may in fact cause students to unintentionally ascribe dynamic properties to mathematical representation.
This chapter presents nine case studies in which school students engage in challenging mathematics outside their immediate classroom environment. In each case, students are encouraged to collaborate in investigations that go beyond the standard curriculum and creatively use the ingredients of the particular context. In Italy, students visit a mathematical laboratory to understand and utilize mathematical machines. Morning assembly at an Indian school brings students from many classes together in the solution of mathematical problems. Four of the projects are from France: students analyze the configuration of a heap of sand, pursue astronomical investigations with software, obtain a flavor of research by having secondary school teams investigate interesting problems, and are presented at all levels with open-ended research problems. There are three programs from the United States, the first, an advanced geometry sequence for secondary students completing the regular syllabus early, the second, activities arising from exhibits in an art museum, and the third, using the school lawn to deepen student understanding of geometric constructions. All such activities need to be evaluated for their effectiveness, so that they move from just being initiatives of dynamic individuals to serve as the foundation for systemic improvements in the way in which students learn, understand and use mathematics. In the early part of this chapter, we briefly mention how research into such activities might be approached.
This study uses a statistical method to identify verbal items among mathematical items from PISA 2003. The verbal items are preliminary analysed and compared to the non-verbal items concerning number of text lines, response types, cognitive level, and competences measured. The results show that the verbal items, to a higher percentage than the non-verbal items, measures the reproduction competency, are straightforward, and of open constructed-response type. These results and proposed further analyses are discussed.
This paper focuses on relationships between vocabulary in mathematical tasks and aspects of reading and solving these tasks. The paper contains a framework that highlights a number of different aspects of word difficulty as well as many issues to consider when planning and implementing empirical studies concerning vocabulary in tasks, where the aspect of common/uncommon words is one important part. The paper also presents an empirical method where corpora are used to investigate issues of vocabulary in mathematical tasks. The results from the empirical study show that there are connections between different types of vocabulary and task difficulty, but that they seem to be mainly an effect of the total number of words in a task.
In this paper we suggest a theoretical model of the connection between the process of reading and the process of solving mathematical tasks. The model takes into consideration different types of previous research about the relationship between reading and solving mathematical tasks, including research about traits of mathematical tasks (a linguistic perspective), about the reading process (a psychological perspective), and about behavior and reasoning when solving tasks (a mathematics education perspective). In contrast to other models, our model is not linear but cyclic, and considers behavior such as re-reading the task.
In this study we analyse the communication competence included in two different frameworks of mathematical knowledge. The main purpose is to find out if mathematical communication is primarily described as communication of or about mathematics or if it is (also) described as a special type of communication. The results show that aspects of mathematics are mostly included as the content of communication in the frameworks but the use of different forms of representation is highlighted both in the frameworks and also in prior research as a potential cause for characterising mathematical communication differently than "ordinary" communication.
Det språk vi använder oss av i matematikklassrummet kan fokuseras på många olika sätt. Språket är också nödvändigt att förhålla sig till vid utvecklingen av sitt matematiska tänkande. Författarna diskuterar här relationer mellan språk och lärande.
There is much research on the role of theory in mathematics education research, at least from more overarching or theoretical perspectives. Micro analyses of the role of theory in particular research studies are rarer. We contribute by analysing one empirical study to allow for in-depth analyses and discussions around the role of theory in a specific case, concerning relationships between mathematics and reading. Our results show that studies that do not use an explicit theoretical model can still be strongly influenced by implicit theoretical assumptions. We conclude that it is important to identify existing theoretical assumptions in an empirical research study and try to convey them as clearly as possible, and we discuss specific issues concerning research on relationships between mathematics and reading.
The context of this paper is a project aimed at meeting the challenges of teaching an introduction-to-proof course at our own institutions. These courses are typically taken by second year students intending to major in mathematics or computer science and are taught in sections of 15 to 24 students. These students benefit from being given the freedom to explore different methods of proof without the presumption that they will use a certain method. This is the story of how what could have been a routine, closed problem became an open one.
American-style options are contracts traded on financial markets. These are derivatives of some underlying security or securities that in contrast to European-style options allow their holders to exercise at any point before the contracts expire. However, this advantage aggravates the mathematical formulation of an option's value considerably, explaining why essentially no exact closed-formed pricing formulas exist. Numerous price approximation methods are although available, but their possible areas of application as well as performance, measured by speed and accuracy, differ. A clearing house offering real-time solutions are especially dependent on fast pricing methods to calculate portfolio risk, where accuracy is assumed to be an important factor to guarantee low-discrepancy estimations. Conversely, overly biased risk estimates may worsen a clearing house's ability to manage great losses, endangering the stability of a financial market it operates.
The purpose of this project was to find methods with optimal performance and to investigate if price approximation errors induce biases in option portfolios' risk estimates. Regarding performance, a Quasi-Monte Carlo least squares method was found suitable for at least one type of exotic option. Yet none of the analyzed closed-form approximation methods could be assessed as optimal because of their varying strengths, where although the Binomial Tree model performed most consistently. Moreover, the answer to which method entails the best risk estimates remains inconclusive since only one set of parameters was used due to heavy calculations. A larger study involving a broader range of parameter values must therefore be performed in order to answer this reliably. However, it was revealed that large errors in risk estimates are avoided only if American standard options are priced with any of the analyzed methods and not when a faster European formula is employed. Furthermore, those that were analyzed can yield rather different risk estimates, implying that relatively large errors may arise if an inadequate method is applied.
I denna artikel beskriver en lärare och en forskare sitt arbete med att utveckla den muntliga kommunikationsförmågan inom matematik i en fjärdeklass. Aktionsforskning utgör bakgrunden för arbetet och likamedtecknet står i fokus när eleverna utmanas i sitt matematiska tänkande med hjälp av lappar.
We present a fluid simulation method adapted for stream parallelism on general purpose graphics processingunits (GPGPU). In this method the equations of Navier and Stokes are discretized using particles and kernelfunctions as in Smoothed Particle Hydrodynamics (SPH), but rather than using penalty methods or solving for a divergence free velocity field, incompressibility is enforced using holonomic kinematic constraints [1]. We useone constraint for each smoothed particle stating that the local density should be kept constant. Other constraintsare used for boundary conditions and multiphysics coupling. We also present a viscosity model in which theshear rate at each pseudo particle is constrained to satisfy a given constitutive law. The computation of theconstraint forces, namely, the pressure and the stresses, requires the solution system of linear equations whichhave a sparse, saddle point structure. These are solved using the Uzawa method of preconditioned conjugate gradients (CG) applied directly to the symmetric indefinite matrix. The overall simulation method has its rootsin a discrete variational principle and the SPOOK time stepping scheme for constrained mechanical systems [2].The SPOOK method is second order accurate on the positions and constraints violations, and is stable at largetime-steps, thus often allowing several orders of magnitude larger timesteps in our method compared to intraditional SPH methods. The numerical implementation on GPGPU that is the main result of this paper consistsof the following components: particle neighbour searches based on spatial decomposition; summation of kernel densities; construction of Jacobians representing the constraints on the density, boundary conditions, viscosityand multiphysics couplings; a Uzawa CG solver for the system of linear equations; and finally, discrete timestepping of velocities and positions. The CG solver is particularly suitable for stream computing since it is basedon matrix-vector multiplications. The sparse system data is stored in a compressed matrix format and the algorithms operating on this data on GPGPU are implemented in CUDA and OpenCL. Our simulation resultsinclude performance measurements, and validation of the method for benchmark problems. We achieve up totwo orders of magnitude speed-up from the GPGPU over traditional processors and together with the increased timestep efficiency of our method we arrive at interactive performance for systems with up to two million fluidparticles representing an incompressible fluid.
Food webs represent trophic (feeding) interactions in ecosystems. Since the late 1970s, it has been recognized that food-webs have a surprisingly close relationship to interval graphs. One interpretation of food-web intervality is that trophic niche space is low-dimensional, meaning that the trophic character of a species can be expressed by a single or at most a few quantitative traits. In a companion paper we demonstrated, by simulating a minimal food-web model, that food webs are also expected to be interval when niche-space is high-dimensional. Here we characterize the fundamental mechanisms underlying this phenomenon by proving a set of rigorous conditions for food-web intervality in high-dimensional niche spaces. Our results apply to a large class of food-web models, including the special case previously studied numerically.
Studies of cooperation have traditionally focused on discrete games such as the well-known prisoner’s dilemma, in which players choose between two pure strategies: cooperation and defection. Increasingly, however, cooperation is being studied in continuous games that feature a continuum of strategies determining the level of cooperative investment. For the continuous snowdrift game, it has been shown that a gradually evolving monomorphic population may undergo evolutionary branching, resulting in the emergence of a defector strategy that coexists with a cooperator strategy. This phenomenon has been dubbed the ‘tragedy of the commune’. Here we study the effects of fluctuating group size on the tragedy of the commune and derive analytical conditions for evolutionary branching. Our results show that the effects of fluctuating group size on evolutionary dynamics critically depend on the structure of payoff functions. For games with additively separable benefits and costs, fluctuations in group size make evolutionary branching less likely, and sufficiently large fluctuations in group size can always turn an evolutionary branching point into a locally evolutionarily stable strategy. For games with multiplicatively separable benefits and costs, fluctuations in group size can either prevent or induce the tragedy of the commune. For games with general interactions between benefits and costs, we derive a general classification scheme based on second derivatives of the payoff function, to elucidate when fluctuations in group size help or hinder cooperation.
Adaptive dynamics is a mathematical framework for studying evolution. It extends evolutionary game theory to account for more realistic ecological dynamics and it can incorporate both frequency- and density-dependent selection. This is a practical guide to adaptive dynamics that aims to illustrate how the methodology can be applied to the study of specific systems. The theory is presented in detail for a single, monomorphic, asexually reproducing population. We explain the necessary terminology to understand the basic arguments in models based on adaptive dynamics, including invasion fitness, the selection gradient, pairwise invasibility plots (PIP), evolutionarily singular strategies, and the canonical equation. The presentation is supported with a worked-out example of evolution of arrival times in migratory birds. We show how the adaptive dynamics methodology can be extended to study evolution in polymorphic populations using trait evolution plots (TEPs). We give an overview of literature that generalises adaptive dynamics techniques to other scenarios, such as sexual, diploid populations, and spatially-structured populations. We conclude by discussing how adaptive dynamics relates to evolutionary game theory and how adaptive-dynamics techniques can be used in speciation research.
Equivariance of linear neural network layers is well studied. In this work, we relax the equivariance condition to only be true in a projective sense. Hereby, we introduce the topic of projective equivariance to the machine learning audience. We theoretically study the relation of projectively and linearly equivariant linear layers. We find that in some important cases, surprisingly, the two types of layers coincide. We also propose a way to construct a projectively equivariant neural network, which boils down to building a standard equivariant network where the linear group representations acting on each intermediate feature space are lifts of projective group representations. Projective equivariance is showcased in two simple experiments. Code for the experiments is provided in the supplementary material.
Evolution of cooperation has traditionally been studied by assuming that individuals adopt either of two pure strategies, to cooperate or defect. Recent work has considered continuous cooperative investments, turning full cooperation and full defection into two opposing ends of a spectrum and sometimes allowing for the emergence of the traditionally-studied pure strategies through evolutionary diversification. These studies have typically assumed a well-mixed population in which individuals are encountered with equal probability. Here, we allow for the possibility of assortative interactions by assuming that, with specified probabilities, an individual interacts with one or more other individuals of the same strategy. A closely related assumption has previously been made in evolutionary game theory and has been interpreted in terms of relatedness. We systematically study the effect of relatedness and find, among other conclusions, that the scope for evolutionary branching is reduced by either higher average degree of, or higher uncertainty in, relatedness with interaction partners. We also determine how different types of non-linear dependencies of benefits and costs constrain the types of evolutionary outcomes that can occur. While our results overall corroborate the conclusions of earlier studies, i.e. higher relatedness promotes the evolution of cooperation, our investigation gives a comprehensive picture of how relatedness affects the evolution of cooperation with continuous investments.
Abstract Microbes produce many molecules that are important for their growth and development, and the exploitation of these secretions by nonproducers has recently become an important paradigm in microbial social evolution. Although the production of these public-goods molecules has been studied intensely, little is known of how the benefits accrued and the costs incurred depend on the quantity of public-goods molecules produced. We focus here on the relationship between the shape of the benefit curve and cellular density, using a model assuming three types of benefit functions: diminishing, accelerating, and sigmoidal (accelerating and then diminishing). We classify the latter two as being synergistic and argue that sigmoidal curves are common in microbial systems. Synergistic benefit curves interact with group sizes to give very different expected evolutionary dynamics. In particular, we show that whether and to what extent microbes evolve to produce public goods depends strongly on group size. We show that synergy can create an "evolutionary trap" that can stymie the establishment and maintenance of cooperation. By allowing density-dependent regulation of production (quorum sensing), we show how this trap may be avoided. We discuss the implications of our results on experimental design.
ObjectiveStructural MRI (sMRI) increasingly offers insight into abnormalities inherent to schizophrenia. Previous machine learning applications suggest that individual classification is feasible and reliable and, however, is focused on the predictive performance of the clinical status in cross‐sectional designs, which has limited biological perspectives. Moreover, most studies depend on relatively small cohorts or single recruiting site. Finally, no study controlled for disease stage or medication's effect. These elements cast doubt on previous findings’ reproducibility.
MethodWe propose a machine learning algorithm that provides an interpretable brain signature. Using large datasets collected from 4 sites (276 schizophrenia patients, 330 controls), we assessed cross‐site prediction reproducibility and associated predictive signature. For the first time, we evaluated the predictive signature regarding medication and illness duration using an independent dataset of first‐episode patients.
ResultsMachine learning classifiers based on neuroanatomical features yield significant intersite prediction accuracies (72%) together with an excellent predictive signature stability. This signature provides a neural score significantly correlated with symptom severity and the extent of cognitive impairments. Moreover, this signature demonstrates its efficiency on first‐episode psychosis patients (73% accuracy).
ConclusionThese results highlight the existence of a common neuroanatomical signature for schizophrenia, shared by a majority of patients even from an early stage of the disorder.
In the competitive automobile manufacturing industry only the most efficient and profitable companies will survive. Companies search for ways to cut costs and increase production without lower their standards of quality. In the field of optimization research such problems can be formulated as combinatorial optimization problems. The Car Sequencing problem (CSP) tries to find the optimal production sequence which increases the number of produced units. Parrello et al. (1986) where the first one to formulate the problem with respect to an assembly line in a automobile factory. Volvo GTO in Umeå desires to minimize their paint consumption in the paint shop. By formulating paint process in the top coat, which is the last part of the paint shop, as a Integer Linear Program the number of color changes can be minimized. Hence the paint consumption is reduced which leads to lower costs. Using a greedy approach while simulating the process the number of color changes can be reduced based on historical data. This result is based on a somewhat simplified model of a real life problem. The number of parameters in the model can be increased to reflect the complexity of the paint shop. The purpose of this thesis is to present a way to decrease the paint consumption
Despite the digital revolution much of the mathematics practiced in schools is still tightly bound to two-dimensional texts. This emphasis on text is neither surprising, nor inadequate, since mathematics has developed through a long history with the use of written text, consisting of natural language, mathematical notation and images. Natural language is our native language consisting of letters and words (see e.g., www.oed.com). Different features of the mathematics text are also important in written tests, since reading the text is part of the assessment. If the text is hard to read, that difficulty can be relevant as part of assessing the communicative competence in mathematics. Crucial is, however, whether potentially difficult textual features are part of what the assessment aims at. This issue is investigated in the current study, using a synthesis of statistical results and qualitative analyses of task text.
This study reports on the relation between commonness of the vocabulary used in mathematics tasks and aspects of students’ reading and solving of the tasks. The vocabulary in PISA tasks is analyzed according to how common the words are in a mathematical and an everyday context. The study examines correlations between different aspects of task difficulty and the presence of different types of uncommon vocabulary. The results show that the amount of words that are uncommon in both contexts are most important in relation to the reading and solving of the tasks. These words are not connected to the solution frequency of the task but to the demand of reading ability when solving the task.
Effective mathematics teaching for Indigenous language speaking students needs to be based on fair expectations of both students and teachers. Concepts of ‘age-appropriate learning’ and ‘school readiness’ structure assessment expectations that entire cohorts of Indigenous language speaking students are unable to meet. This institutionalises both student and teacher failure, as both are exhorted to meet unachievable expectations. The voices of teachers teaching in a very remote school provide insight into teachers’ responses to the mismatch between the system expectations and the teaching context. Teacher interviews in a small Northern Territory school, conducted within an ethnographic study, showed that teachers’ decisions regarding the level of mathematics curriculum taught were informed by students’ prior learning and by the language dynamic in their classrooms. The need and pressure to teach Standard Australian English also affected how mathematics was taught. This leads to a reformulation of the concept of school readiness to ask how schools can be more ready for their Indigenous language speaking students in terms of preparing and supporting teachers.
Starting from basic financial mathematics, we cover the mathematics of pricing swaptions, options on interest rate swaps. We then continue to the topic of obtaining an approximately pure volatility exposure. This exposure to volatility, which in practice enables us to trade volatility according to our perceptions of the market, is obtained by buying or selling swaptions and appropriate amounts of the underlying interest rate swap contract. Taking offsetting positions in the underlying contract is called hedging and is covered in depth. We note that hedging can primarily be done in two ways, and discuss the advantages and disadvantages of each of them. After deriving the value formulas for such a swaption strategy aimed at isolating volatility exposure we end with a discussion on the transition from theory to practice.We find that this way of trading volatility is conceptually simple, but that pre-trade profitability analysis is difficult due to the sometimes poor availability of the sophisticated data needed to simulate such a swaption strategy. Despite the possible limitations in the data necessary to translate this theory into an experimental setup, this thesis serves as a good basis for further research on the profitability of a volatility trading strategy using interest rate swaptions.
This report discusses analysis of data collected as participant observer during doctoral work (Gade, 2006), examined more recently with Engeström’s version of activity theory (Gade, 2014). The empirical data under consideration refer to two particular group tasks conducted by two experienced teachers at the commencement of the academic term, with all students assembled into small groups. I consider the conduct of these tasks as an Activity setting (Tharp & Gallimore, 1988), in realising which teachers exemplified the pedagogical category of Joint productive activity (Dalton & Tharp, 2002). These two constructs draw on cultural historical activity theory (CHAT) perspectives which explain how human consciousness and personality develop in goal-directed activities.
The appropriation of cultural resources in any social environment is of theoretical interest in order to understand Vygotsky's contention of how skills that are interpsychological become intrapsychological, allowing the individual to direct attention to elements that are necessary to carry out a particular task at hand. The analysis of appropriation is of methodological interest to be able to obtain explanatory power and understand as Rogoff argues how students comprehend their roles, transform practice through appropriation and relate their participation from one activity to another. Combining a theoretical and methodological interest, this paper discusses two instances of appropriation by students who utilise a textbook and a mnemonic in meeting instructional goals in the classroom teaching-learning of mathematics. Leont'ev distinguished the process of appropriation as unique to human ontogenetic development, one that takes place in an objective world of cultural resources. In analysing embedded goals, mastery of cultural resources and transformation of practice, an understanding is sought of the complex, collaborative and situated conditions that facilitate the process of appropriation and the cultural nature of human development.
Two teachers, Olaf and Knut, conducted two group-tasks in succession, early in the academic year at a gymnasium or upper secondary school in Norway. In doing so they steered classroom practice away from traditional instruction, with Olaf alone as teacher, to cooperative learning in small groups with guidance from both. While the first group-task titled When Together initiated cooperative learning by students in small groups, the second titled How Heavy initiated student groups to build upon group cooperation and work with other groups in a collaborative classroom practice. It was Olaf and Knut’s intention to have their students cooperate in small groups at all times and collaborate with students from other groups on occasion. A few months into the year, Olaf and Knut’s students’ groups had opportunity to discuss rules of cooperation whereupon their collaborative classroom practice became the norm. Using examples of students’ attempts at both group-tasks, I portray Olaf and Knut’s initiation of such a practice. Using Engeström’s activity model I shed light on how students' participation was transformed to meet with their intentions.
This paper reports on a classroom study of meaning being made by students working in small groups in a mathematics classroom at an upper secondary school. A naturalistic study, it observes student activities within the classroom from a socio-cultural-historical perspective. It addresses the artefact mediated classroom practice, wherein student participation and meaning making is guided and apprenticed by the teachers. Mediated by artefacts within intentional practices, the mathematics classroom constitutes a culture which is also the medium for meaning making.
This paper discusses the potential of Vygotskian units in researching classroom practices. Adopting a close-to-practice approach and action research where possible, an attempt is made to shed light on ongoing practices across grades 4-6, 7-9 and the gymnaisum. The theory/practice approach adopted keeps alive the relationship between theory-which-informs and theory-being-built, as well as existing-practice and steered-practice in these studies. The potential to inform researcher reflexivity and guide educational action along with the ability of units to grasp complexity, from as many perspectives as possible, and ascend to the concrete is brought forward. Such a strategy is deliberate, geared towards informing practitioner inquiry in ongoing classrooms.
Beyond understanding the Vygotskian construct of zone of proximal development or ZPD with reference to an individual student, this paper explores the formation of ZPD within the pedagogical constructs of cooperation, wherein students cooperate with each other within their groups; as well as collaboration, wherein students from different groups that constitute the classroom collaborate with each other. Identified on the basis of functions that are in the process of maturing, the formation of either ZPD is exemplified from a socio-cultural-historical study at an upper secondary mathematics classroom in Norway. An emphasis on what distinguishes events in instruction that are educational from those that are not is also explored, before illustrating what nature of ZPD is established. The role of guidance received, imitation and cultural resources in the development of higher mental functions is understood as these ZPD are formed, enabling students to act independently within the classroom teaching-learning of mathematics.
Practitioner action and reflection are understood as central to practitioner research in this paper. While Vygotskian based premise highlight instructional practices and intervention as central to practitioner understanding, Bakhtin based dialogical premise draw on relational constructs and attend to practitioner being, answerability and authorship. Vygotskian and Bakhtinain units of analysis used are viewed as means with which to conceive practitioner action/reflection and build a community of practitioners. Taken together practitioner knowing is found inextricable with practitioner ontology, making practitioner research of ongoing instructional practices more human in line with Dewey.
Drawing on narrative inquiry as well as activity theory perspectives, in this chapter I examine three aspects in relation to research conducted with a mathematics teacher Lotta, as Charlotta is known, at a Grade 4-6 school in Sweden. First, the nature of relational knowing which, born of practitioner narratives, led us to collaborate and take action (Hollingsworth, Dybdahl & Minarik, 1993).
The notions of praxis, or practical action, and phronesis, or practical wisdom, are engaged with from two perspectives in this paper – with examples from author's classroom studies across various Grade schools in Sweden on one hand and the theoretical exposition of these notions across a selection of existing literature on the other. A persuasive case is made for having praxis and phronesis be regarded as units of analysis suitable for grasping as well as furthering practitioner inquiry. The results of such research, conducted as a practice, are aimed at informing the vicarious and the experiential, upon which any reader can in turn base actions. In their ability to examine the practical, particular and local, the pursuit of such a social science is argued as being one that matters to the practitioners themselves.
Three perspectives are brought together in this paper, towards recognising the centrality of participant agency in school change and educational reform. First, observations made by Sarason in relation to school change and educational reform. Second, Engeström's activity theory based interventionist methodology, which seeks recognition of agency in the conduct of local interventions. Finally an empirical case of teacher-researcher collaboration and project based intervention at a Grade four mathematics classroom in Sweden. Herein lies the possibility of viewing school change and educational reform in terms of students' interests and teacher agency within classroom interventions.
This paper reports the nature of classroom practice that afforded students' meaning making at an upper secondary mathematics classroom in Norway. The participation of both teachers and students in the collaborative classroom practice they jointly establish is outlined. A longitudinal and person-in-practice view sheds light not only on the meaning producing foreground that was initiated, but also the nature of its growth. An artefact of instructional practice of two teachers Olaf and Knut is thus evidenced. In this there is opportunity to appreciate mathematical content, pedagogy and students' thinking in an integrated manner – making such knowledge useful and usable by practising as well as prospective teachers of mathematics.
Three students Dan, Levi and Thor, attempt a group-task containing worksheets A and B. While worksheet A asks students to calculate and compare surface area to volume ratio of a sphere for six successive units, worksheet B asks them to consider the metabolism of living cells and the bearing the ratio has on their functioning and size. While Levi and Thor own the group-task, follow its instructions and deliberate on its questions, Dan declares his intention of observing Levi and Thor and takes a free ride. Based on students inscriptions and transcript of audio-recordings, I show how Levi and Thor work through calculations required in worksheet A with ease, even coming up with conjectures. In attempting worksheet B they are able to correlate better metabolism in cells with a smaller radius, yet question if that model is indeed borne out in reality. The concept of activity in Vygostkian theory is used to study students attempts at the group-task. Three constructs from cultural historical activity theory and/or CHAT namely leading activity, germ cell of activity and learning activity are utilised to shed light on attempts by Dan, Levi and Thor at their group-task.
Extended teacher-researcher collaboration is reported in this paper, by drawing upon cultural historical activity theory (CHAT) perspectives. Teaching is herein conceived as dialectical practice in which teachers are both shaped by and shape instructional practices. Three instructional interventions conducted at a Grade four mathematics classroom in Sweden constitute and exemplify the construct of Formative Intervention. Teacher-researcher collaboration which paralleled such conduct next exemplifies the construct of Expansive learning activity. Such transformation and change sheds light on how mid-level taken for granted phenomena in schools can be worked with and around, besides contributing to the motivational sphere of students and teachers.
This paper details researcher collaboration with Lotta, a mathematics teacher at a Grade 4-6 school. This began with a six month study during which I observed Lotta's classroom teaching at Grade six. My pursuit of narrative perspectives in this study, created room for intersubjectivity and endeared Lotta and me to collaborate thereafter. Lotta's application for project funding from national school authorities, then paved way for researcher intervention in Lotta's teaching, at her now Grade four. Our conceiving teaching-learning in Vygotskian terms allowed for two kinds of interventions. Talk based interventions geared towards correlating mathematics and communication as part of Lotta's project and specific interventions relating to say mathematical problem posing and students' use of the mathematical = sign. In Lotta next being employed by national school authorities themselves, our collaboration shifted to reflecting and reporting on our interventions by means of co-authorship. The trajectory from participant observation to co-authorship in our collaboration thus corresponds to my reference to Lotta as Lea in my first study, to her theorising and facing up to public scrutiny as herself. Lotta is since Rektor of a school at which we intend to widen our collaboration.
While recognising agentive understanding as teacher and researcher to be a result of intervening in Lotta's instructional practice, it is purpose, goals and outcomes of teacher-researcher collaboration that is of interest. With its transformative manner as backdrop and in line with writings led by Anna Stetsenko I examine three specific aspects. First, in what way did our collaboration qualify as leading activity, enabling research to capture our selves as evolving in our interventions. Second, how did our actions exemplify a transformative activist stance that promoted human knowing and becoming, rooted in the ideals of social justice. Finally, how did our attempts at collaborative purposeful transformation stem out of social practice, through social practice and for social practice we set out to change. In line with a practical-theoretical endeavor, such analysis sheds light on a transformatory vision of human development in terms of a dialectical, non-reductionist, non-dualist, non-additive relational ontology as played out in our case of teacher-researcher collaboration.
This article portrays action research conducted in relation to students’ faulty use of the equality sign at a Grade four mathematics classroom in Sweden. Substantial background on teacher-researcher collaboration that prepares ground for the action research is offered. Drawing on CHAT perspectives the conduct of activities mediated by diagrammatic objects (lappar) as cultural artefacts and students' inscriptions in these is also provided. As semiotic activity the action research cycle allows students to reflect on signs, meanings and their interrelationships while working with knowledgeable others. A three stage conclusion discusses relevant research in mathematics education, the nature of practical theory realised, and actionable knowledge thereby possible in terms of trustworthiness.
As different from building theory from research the ability of theorising for action research at two Grade eight classrooms is discussed, by deploying cultural studies and problematising how students and teachers are (dis)empowered by cultural, political, global and economic forces. Moving beyond adoption of a neutral outlook to schooling and mathematics, the practice of mathematics education research necessitates careful and critical thinking through of content, context and agency. An attempt is made to narrate a better story of what could transpire within four walls of a mathematics classroom - situated as classroom and school themselves are in wider societal contexts.
This paper deliberates upon two small stories offered by Sara and Eva at a grade 4-6 Swedish school. The two narratives allow research to examine the notion of students' subjectivity within teaching-learning of mathematics and schooling more broadly. Offered as talk-in-interaction these are analysed with respect to what Sara and Eva did with their talk, as well as the nature of self that either was able to accomplish. There is opportunity to examine the role played by these stories in mediating the lived world and told world that either experienced. While Sara examined writing in story books, Eva examined babysitting her siblings. The positioning of either as social beings in their contextual tellings, informs agency and actions of either in their respective lives. Such analysis informs research about students' identity or their who in addition to the how of their participation, vital to initiating any educational action.
Long association with a mathematics teacher at a Grade 4-6 school in Sweden, is basis for reporting a case of teacher-researcher collaboration. Three theoretical frameworks used to study its development over time are relational knowing, relational agency and cogenerative dialogue. While relational knowing uses narrative perspectives to explore the experiential and relational nature of collaboration; relational agency, draws on activity theory perspectives and identifies the change in the purpose of collaboration, from initially conducting classroom interventions to co-authoring research. Finally, cogenerative dialogue, deploys hermeneutic-phenomenological perspectives and investigates the dialogue that transpired between Lotta and the author, as they co-authored their research report. Such analysis sheds invaluable light on a case of expansive learning activity.
Drawing upon cultural historical activity theory (CHAT) perspectives,in this chapter we portray a classroom practice of problem posing that evolved with a cohort of students across Grades 4 and 5 in Sweden. In line with a language and literacy pedagogy, the classroom practice in which students utilised textbook vocabulary handed out on slips of paper (lappar in Swedish) advanced through three distinct stages namely: formulating written questions, problem posing as dyads and actively posing problems to one another. Mediated explicitly by lappar, such a practice provided social and public opportunities for students to attribute personal meaning and make conscious use of words in semiotic activity, as well as appropriate cultural meaning and valid norms of use. The increasing gain and display of agency by students in this practice, informed by student(s)-acting-with-lappar-as-mediational-means as unit of analysis, was indicative of their self-regulation, volition and independence. Developmental in approach, such classroom practice was born through teacher–researcher collaboration.
We report an exploratory talk based, whole class plenary intervention, in relation to students' understanding of everyday measures and measurement, in a grade four classroom at a grade 4-6 school in Sweden. Extended, project related, teacher-researcher collaboration forms basis for such cultural historical activity theory or CHAT based efforts. As formative intervention, the conduct of the plenary is not pre-determined but embedded in ongoing curricular realities, with the agency of students and teacher promoted, pedagogical ideas reutilised and the role of researcher viewed as supporting design and growth of the intervention. Under Charlotta's guidance as teacher, the plenary is opportunity for her students to examine improbable scenarios such as, Can Eva and Anton measure the length of Sweden on foot, Can Lars and Iris measure their age in decimeters. A zone of proximal development is created, in which students make the transition from spontaneous to scientific concepts and learn how various units of measurement are objects-that-can-be-used-for-certain-purposes. With opportunity for critical and reflective inquiry, in a plenary designed to lead development, Charlotta's students look beyond the making of rote measurements and articulate a theory of measure in nascent terms. Such a landscape of teaching-learning is finally understood in terms of the nature of talk that was facilitated, the manner of pedagogy utilised, the style of teaching exercised and the kind of learning that was demanded of her students.
Cultural historical activity theory (CHAT) perspectives are used to shed light on an extended teacher-researcher collaboration, at a Grade 4-6 school in Sweden. Beginning with participant observation and emerging forms of engagement like co-authorship of research reports, the collaboration is understood as expansive learning activity. Treating the practices of teaching and research as distinct yet collaborating activity systems within this, provides an opportunity to analyse the manner in which joint conduct of project related instructional interventions became shared object. This also enabled teacher and researcher to become active stakeholders in each other's practice. Dialectical realisation of stakeholdership and shared object led to reconceptualisation and transformation of the very horizons of our work.