Developing mathematics understanding and abstraction : the case of equivalence in the elementary years

Generalising arithmetic structures is seen as a key to developing algebraic understanding. Many adolescent students begin secondary school with a poor understanding of the structure of arithmetic. This paper presents a theory for a teaching/learning trajectory designed to build mathematical understanding and abstraction in the elementary school context. The particular focus is on the use of models and representations to construct an understanding of equivalence. The results of a longitudinal intervention study with five elementary schools, following 220 students as they progressed from Year 2 to Year 6, informed the development of this theory. Data were gathered from multiple sources including interviews, videos of classroom teaching, and pre-and post-tests. Data reduction resulted in the development of nine conjectures representing a growth in integration of models and representations. These conjectures formed the basis of the theory.

Drawn to abstraction

Three paintings exploring and testing the area of abstraction through gesture, colour and canvas structures. The paintings interrogate how abstraction can, through affect and sensation as well as historical reference, construct meaning.

Isotactics as a foundation for alignment and abstraction of behavioral models

There are many use cases in business process management that require the comparison of behavioral models. For instance, verifying equivalence is the basis for assessing whether a technical workflow correctly implements a business process, or whether a process realization conforms to a reference process. This paper proposes an equivalence relation for models that describe behaviors based on the concurrency semantics of net theory and for which an alignment relation has been defined. This equivalence, called isotactics, preserves the level of concurrency of aligned operations. Furthermore, we elaborate on the conditions under which an alignment relation can be classified as an abstraction. Finally, we show that alignment relations induced by structural refinements of behavioral models are indeed behavioral abstractions.

Imaging affect : abstraction and the echo of the unknowable

As an artist my primary interest is in the abstract, that is in images of the imageless. I am curious about the emergence of pictorial significance and content from this unknowable space. To speak of the significance of an imageless image is also to speak of its affect. I aim to explore this both theoretically and practically. Theoretically I will explore affect through the late work of Lyotard and his notion of the affect-phrase. This is an under-examined aspect of Lyotard and demarcates a valuable way to look at the origins, impact and ramifications of affect for art. Practically I will apply these understandings to the development of my own creative work which includes both painting and digital work. My studio practice moves towards exploring the unfamiliar through the powerful and restless silence of affect.In this intense space each work or body of work 'leaks' into the next occasioning a sense of borderlessness, or of uncertainty. This interpenetration and co-mingling of conceptual and material terrains combines to present temporal and spatial slippages evident within the works themselves and their making, but it is also evident in bodies of work across the chronology of their making. Through a mapping of my own painting and digital arts practice and the utilisation of Lyotard’s notion of the affect -phrase I aim to describe the action of this ‘charged emptiness’ on creativity and explore and explain its significance on that we call image and its animation of what we call critical discourse.

Abstraction as noise : enhancing the poetic

Abstraction in its resistance to evident meaning has the capacity to interrupt or at least provide tools with which to question an overly compliant reception of the information to which we are subject. It does so by highlighting a latency or potentiality inherent in materiality that points to the possibility of a critical resistance to this ceaseless flow of sound/image/data. This resistance has been remarked on in differing ways by a number of commentators such as Lyotard, in his exploration of the avant-garde and the sublime for example. This joint paper will initially map the collaborative project by Daniel Mafe and Andrew Brown, Affecting Interference which conjoins painting with digital sound and animations into a single, large scale, immersive exhibition/installation. The work acts as an interstitial point between contrasting approaches to abstraction: the visual and aural, the digital and analogue. The paper will then explore the ramifications of this through the examination of abstraction as ‘noise’, that is as that raw inassimilable materiality, within which lays the creative possibility to forge and embrace the as-yet-unthought and almost-forgotten. It does so by establishing a space for a more poetic and slower paced critical engagement for the viewing and receiving information or data. This slowing of perception through the suspension of easy recognition runs counter to our current ‘high performance’ culture, and it’s requisite demand for speedy assimilation of content, representing instead the poetic encounter with a potentiality or latency inherent in the nameless particularity of that which is.

Simplifying process model abstraction : techniques for generating model names

The increased adoption of business process management approaches, tools and practices, has led organizations to accumulate large collections of business process models. These collections can easily include hundred to thousand models, especially in the context of multinational corporations or as a result of organizational mergers and acquisitions. A concrete problem is thus how to maintain these large repositories in such a way that their complexity does not hamper their practical usefulness as a means to describe and communicate business operations. This paper proposes a technique to automatically infer suitable names for business process models and fragments thereof. This technique is useful for model abstraction scenarios, as for instance when user-specific views of a repository are required, or as part of a refactoring initiative aimed to simplify the repository’s complexity. The technique is grounded in an adaptation of the theory of meaning to the realm of business process models. We implemented the technique in a prototype tool and conducted an extensive evaluation using three process model collections from practice and a case study involving process modelers with different experience.

Abstraction : on ambiguity and semiosis in abstract painting

Drawing on the fields of philosophy, phenomenology, art history and theory as well as the candidate's own painting practice, this PhD explores the nature of ambiguity and semiosis in contemporary abstract painting. The thesis demonstrates how the aesthetic qualities of pause and rupture, transition and slippage work emergently to break established clichés, habits and intentions in the experiencing of abstract painting and artistic practice.

Structural abstraction of process specifications

Software engineers constantly deal with problems of designing, analyzing, and improving process specifications, e.g., source code, service compositions, or process models. Process specifications are abstractions of behavior observed or intended to be implemented in reality which result from creative engineering practice. Usually, process specifications are formalized as directed graphs in which edges capture temporal relations between decisions, synchronization points, and work activities. Every process specification is a compromise between two points: On the one hand engineers strive to operate with less modeling constructs which conceal irrelevant details, while on the other hand the details are required to achieve the desired level of customization for envisioned process scenarios. In our research, we approach the problem of varying abstraction levels of process specifications. Formally, developed abstraction mechanisms exploit the structure of a process specification and allow the generalization of low-level details into concepts of a higher abstraction level. The reverse procedure can be addressed as process specialization.

Business process model abstraction

In order to execute, study, or improve operating procedures, companies document them as business process models. Often, business process analysts capture every single exception handling or alternative task handling scenario within a model. Such a tendency results in large process specifications. The core process logic becomes hidden in numerous modeling constructs. To fulfill different tasks, companies develop several model variants of the same business process at different abstraction levels. Afterwards, maintenance of such model groups involves a lot of synchronization effort and is erroneous. We propose an abstraction technique that allows generalization of process models. Business process model abstraction assumes a detailed model of a process to be available and derives coarse-grained models from it. The task of abstraction is to tell significant model elements from insignificant ones and to reduce the latter. We propose to learn insignificant process elements from supplementary model information, e.g., task execution time or frequency of task occurrence. Finally, we discuss a mechanism for user control of the model abstraction level – an abstraction slider.

On application of structural decomposition for process model abstraction

Real world business process models may consist of hundreds of elements and have sophisticated structure. Although there are tasks where such models are valuable and appreciated, in general complexity has a negative influence on model comprehension and analysis. Thus, means for managing the complexity of process models are needed. One approach is abstraction of business process models-creation of a process model which preserves the main features of the initial elaborate process model, but leaves out insignificant details. In this paper we study the structural aspects of process model abstraction and introduce an abstraction approach based on process structure trees (PST). The developed approach assures that the abstracted process model preserves the ordering constraints of the initial model. It surpasses pattern-based process model abstraction approaches, allowing to handle graph-structured process models of arbitrary structure. We also provide an evaluation of the proposed approach.

The triconnected abstraction of process models

Companies use business process models to represent their working procedures in order to deploy services to markets, to analyze them, and to improve upon them. Competitive markets necessitate complex procedures, which lead to large process specifications with sophisticated structures. Real world process models can often incorporate hundreds of modeling constructs. While a large degree of detail complicates the comprehension of the processes, it is essential to many analysis tasks. This paper presents a technique to abstract, i.e., to simplify process models. Given a detailed model, we introduce abstraction rules which generalize process fragments in order to bring the model to a higher abstraction level. The approach is suited for the abstraction of large process specifications in order to aid model comprehension as well as decomposing problems of process model analysis. The work is based on process structure trees that have recently been introduced to the field of business process management.

Process model abstraction : a slider approach

Process models provide companies efficient means for managing their business processes. Tasks where process models are employed are different by nature and require models of various abstraction levels. However, maintaining several models of one business process involves a lot of synchronization effort and is erroneous. Business process model abstraction assumes a detailed model of a process to be available and derives coarse grained models from it. The task of abstraction is to tell significant model elements from insignificant ones and to reduce the latter. In this paper we argue that process model abstraction can be driven by different abstraction criteria. Criterion choice depends on a task which abstraction facilitates. We propose an abstraction slider - a mechanism that allows user control of the model abstraction level. We discuss examples of combining the slider with different abstraction criteria and sets of process model transformation rules.

Theorizing in design science research : an abstraction layers framework

For Design Science Research (DSR) to gain wide credence as a research paradigm in Information Systems (IS), it must contribute to theory. “Theory cannot be improved until we improve the theorizing process, and we cannot improve the theorizing process until we describe it more explicitly, operate it more self-consciously, and decouple it from validation more deliberately” (Weick 1989, p. 516). With the aim of improved design science theorizing, we propose a DSR abstraction-layers framework that integrates, interlates, and harmonizes key methodological notions, primary of which are: 1) the Design Science (DS), Design Research (DR), and Routine Design (RD) distinction (Winter 2008); 2) Multi Grounding in IS Design Theory (ISDT) (Goldkuhl & Lind 2010); 3) the Idealized Model for Theory Development (IM4TD) (Fischer & Gregor 2011); and 4) the DSR Theorizing Framework (Lee et al. 2011). Though theorizing, or the abstraction process, has been the subject of healthy discussion in DSR, important questions remain. With most attention to date having focused on theorizing for Design Research (DR), a key stimulus of the layered view was the realization that Design Science (DS) produces abstract knowledge at a higher level of generality. The resultant framework includes four abstraction layers: (i) Design Research (DR) 1st Abstract Layer, (ii) Design Science (DS) 2nd Abstract Layer, (iii) DSR Incubation 3rd Layer, and (iv) Routine Design 4th Layer. Differentiating and inter-relating these layers will aid DSR researchers to discover, position, and amplify their DSR contributions. Additionally, consideration of the four layers can trigger creative perspectives that suggest unplanned outputs. The first abstraction layer, including its alternative patterns of activity, is well recognized in the literature. The other layers, however, are less well recognized; and the integrated representation of layers is novel.

The Coming of Post-reflexive Society: Commodification and Language in Digital Capitalism

Language is a unique aspect of human communication because it can be used to discuss itself in its own terms. For this reason, human societies potentially have superior capacities of co-ordination, reflexive self-correction, and innovation than other animal, physical or cybernetic systems. However, this analysis also reveals that language is interconnected with the economically and technologically mediated social sphere and hence is vulnerable to abstraction, objectification, reification, and therefore ideology – all of which are antithetical to its reflexive function, whilst paradoxically being a fundamental part of it. In particular, in capitalism, language is increasingly commodified within the social domains created and affected by ubiquitous communication technologies. The advent of the so-called ‘knowledge economy’ implicates exchangeable forms of thought (language) as the fundamental commodities of this emerging system. The historical point at which a ‘knowledge economy’ emerges, then, is the critical point at which thought itself becomes a commodified ‘thing’, and language becomes its “objective” means of exchange. However, the processes by which such commodification and objectification occurs obscures the unique social relations within which these language commodities are produced. The latest economic phase of capitalism – the knowledge economy – and the obfuscating trajectory which accompanies it, we argue, is destroying the reflexive capacity of language particularly through the process of commodification. This can be seen in that the language practices that have emerged in conjunction with digital technologies are increasingly non-reflexive and therefore less capable of self-critical, conscious change.