Preservice elementary school teachers' knowledge of fractions: a mirror of students' knowledge?

This research analyses preservice teachers' knowledge of fractions. Fractions are notoriously difficult for students to learn and for teachers to teach. Previous studies suggest that student learning of fractions may be limited by teacher understanding of fractions. If so, teacher education has a key role in solving the problem. We first reviewed literature regarding students' knowledge of fractions. We did so because assessments of required content knowledge for teaching require review of the students' understanding to determine the mathematics difficulties encountered by students. The preservice teachers were tested on their conceptual and procedural knowledge of fractions, and on their ability in explaining the rationale for a procedure or the conceptual meaning. The results revealed that preservice teachers' knowledge of fractions indeed is limited and that last-year preservice teachers did not perform better than first-year preservice teachers. This research is situated within the broader domain of mathematical knowledge for teaching and suggests ways to improve instruction and student learning

Knowledge Representation with Ontologies: The Present and Future

Recently, we have seen an explosion of interest in ontologies as artifacts to represent human knowledge and as critical components in knowledge management, the semantic Web, business-to-business applications, and several other application areas. Various research communities commonly assume that ontologies are the appropriate modeling structure for representing knowledge. However, little discussion has occurred regarding the actual range of knowledge an ontology can successfully represent

Topological correction of hypertextured implicit surfaces for ray casting

Hypertextures are a useful modelling tool in that they can add three-dimensional detail to the surface of otherwise smooth objects. Hypertextures can be rendered as implicit surfaces, resulting in objects with a complex but well defined boundary. However, representing a hypertexture as an implicit surface often results in many small parts being detached from the main surface, turning an object into a disconnected set. Depending on the context, this can detract from the realism in a scene where one usually does not expect a solid object to have clouds of smaller objects floating around it. We present a topology correction technique, integrated in a ray casting algorithm for hypertextured implicit surfaces, that detects and removes all the surface components that have become disconnected from the main surface. Our method works with implicit surfaces that are C2 continuous and uses Morse theory to find the critical points of the surface. The method follows the separatrix lines joining the critical points to isolate disconnected components

The role of language in mathematical development: Evidence from children with specific language impairments

A sample (n=48) of eight year olds with Specific Language Impairments is compared with age-matched (n=55) and language matched controls (n=55) on a range of tasks designed to test the interdependence of language and mathematical development. Performance across tasks varies substantially in the SLI group, showing profound deficits in production of the count word sequence and basic calculation and significant deficits in understanding of the place-value principle in Hindu-Arabic notation. Only in understanding of arithmetic principles does SLI performance approximate that of age-matched-controls, indicating that principled understanding can develop even where number sequence production and other aspects of number processing are severely compromised

The development and use of variables in mathematics and computer science

There are a wide variety of uses of variables in mathematics which we cope with in practice through conventions and tacit assumptions. Experience with computers has made us articulate, criticise and develop these assumptions much more carefully. Historically the term 'variable quantity' was introduced in the context of describing and calculating changing quantities which corresponded to phenomena in the observable world (e.g. the velocity of fluxion of a body moving under the inverse square law). The evolution of the concept has divorced it from these routes of reference and required us to establish the formal apparatus of interpretation and valuation. While the changes considered are highly structured this may be satisfactory, but computing power invites us to cope with change in vastly more complex, unstructured situations such as in simulation of 'real world' processes. We relate this challenge to the distinctive differences in the use of variables in mathematics and practical computing, and we develop a general framework in which all uses of variables can be described in a unified way

Object oriented design of a thermo-mechanical FEM code

An object oriented design is presented for a computer program that can perform\ud thermo-mechanically coupled analyzes. The target of the design is a \ud exible and robust\ud computer program. It should be easy to adapt and extend, re-using existing code, without\ud interfering with already established algorithms.\ud The program uses publicly available toolkits that are currently emerging as C++ pack-\ud ages. First of all the Standard C++ Library (formerly Standard Template Library) is\ud used for packing items in container classes. Secondly the matrix and vector operations\ud are derived from the Template Numerical Toolkit (TNT) and �nally (not essentially for\ud the numerical part) a graphical user interface is made, based on the wxWindows package,\ud that can generate a GUI for Motif and MS-Windows with the same code.\ud Attention is given to the design of classes such as speci�c elements and material classes\ud based on more general classes. A hierarchy of classes is constructed where general behavior\ud is put high in the hierarchy and speci�c behavior low. The choice between inheritance and\ud aggregation is made at several levels

Procedural Noise Adversarial Examples for Black-Box Attacks on Deep Convolutional Networks

Deep Convolutional Networks (DCNs) have been shown to be vulnerable to adversarial examples---perturbed inputs specifically designed to produce intentional errors in the learning algorithms at test time. Existing input-agnostic adversarial perturbations exhibit interesting visual patterns that are currently unexplained. In this paper, we introduce a structured approach for generating Universal Adversarial Perturbations (UAPs) with procedural noise functions. Our approach unveils the systemic vulnerability of popular DCN models like Inception v3 and YOLO v3, with single noise patterns able to fool a model on up to 90% of the dataset. Procedural noise allows us to generate a distribution of UAPs with high universal evasion rates using only a few parameters. Additionally, we propose Bayesian optimization to efficiently learn procedural noise parameters to construct inexpensive untargeted black-box attacks. We demonstrate that it can achieve an average of less than 10 queries per successful attack, a 100-fold improvement on existing methods. We further motivate the use of input-agnostic defences to increase the stability of models to adversarial perturbations. The universality of our attacks suggests that DCN models may be sensitive to aggregations of low-level class-agnostic features. These findings give insight on the nature of some universal adversarial perturbations and how they could be generated in other applications.Comment: 16 pages, 10 figures. In Proceedings of the 2019 ACM SIGSAC Conference on Computer and Communications Security (CCS '19