The use of data-mining for the automatic formation of tactics

This paper discusses the usse of data-mining for the automatic formation of tactics. It was presented at the Workshop on Computer-Supported Mathematical Theory Development held at IJCAR in 2004. The aim of this project is to evaluate the applicability of data-mining techniques to the automatic formation of tactics from large corpuses of proofs. We data-mine information from large proof corpuses to find commonly occurring patterns. These patterns are then evolved into tactics using genetic programming techniques

Generalisation strategies and representation among last-year primary school students

Recent research has highlighted the role of functional relationships in introducing elementary school students to algebraic thinking. This functional approach is here considered to study essential components of algebraic thinking such as generalization and its representation, and also the strategies used by students and their connection with generalization. This paper jointly describes the strategies and representations of generalisation used by a group of 33 sixth-year elementary school students, with no former algebraic training, in two generalisation tasks involving a functional relationship. The strategies applied by the students differed depending on whether they were working on specific or general cases. To answer questions on near specific cases they resorted to counting or additive operational strategies. As higher values or indeterminate quantities were considered, the strategies diversified. The correspondence strategy was the most used and the common approach when students generalised. Students were able to generalise verbally as well as symbolically and varied their strategies flexibly when changing from specific to general cases, showing a clear preference for a functional approach in the latter

Transforming the mathematical practices of learners and teachers through digital technology

This paper argues that mathematical knowledge, and its related pedagogy, is inextricably linked to the tools in which the knowledge is expressed. The focus is on digital tools and the different roles they play in shaping mathematical meanings and in transforming the mathematical practices of learners and teachers. Six categories of digital tool-use that distinguish their differing potential are presented: i. dynamic and graphical tools, ii. tools that outsource processing power, iii. tools that offer new representational infrastructures for mathematics, iv. tools that help to bridge the gap between school mathematics and the students’ world; v. tools that exploit high-bandwidth connectivity to support mathematics learning; and vi. tools that offer intelligent support for the teacher when their students engage in exploratory learning with digital technologies Following exemplification of each category, the paper ends with some reflections on the progress of research in this area and identifies some remaining challenges

Interpretable Deep Learning: Beyond Feature-Importance with Concept-based Explanations

Deep Neural Network (DNN) models are challenging to interpret because of their highly complex and non-linear nature. This lack of interpretability (1) inhibits adoption within safety critical applications, (2) makes it challenging to debug existing models, and (3) prevents us from extracting valuable knowledge. Explainable AI (XAI) research aims to increase the transparency of DNN model behaviour to improve interpretability. Feature importance explanations are the most popular interpretability approaches. They show the importance of each input feature (e.g., pixel, patch, word vector) to the model’s prediction. However, we hypothesise that feature importance explanations have two main shortcomings concerning their inability to describe the complexity of a DNN behaviour with sufficient (1) fidelity and (2) richness. Fidelity and richness are essential because different tasks, users, and data types require specific levels of trust and understanding. The goal of this thesis is to showcase the shortcomings of feature importance explanations and to develop explanation techniques that describe the DNN behaviour with greater richness. We design an adversarial explanation attack to highlight the infidelity and inadequacy of feature importance explanations. Our attack modifies the parameters of a pre-trained model. It uses fairness as a proxy measure for the fidelity of an explanation method to demonstrate that the apparent importance of a feature does not reveal anything reliable about the fairness of a model. Hence, regulators or auditors should not rely on feature importance explanations to measure or enforce standards of fairness. As one solution, we formulate five different levels of the semantic richness of explanations to evaluate explanations and propose two function decomposition frameworks (DGINN and CME) to extract explanations from DNNs at a semantically higher level than feature importance explanations. Concept-based approaches provide explanations in terms of atomic human-understandable units (e.g., wheel or door) rather than individual raw features (e.g., pixels or characters). Our function decomposition frameworks can extract specific class representations from 5% of the network parameters and concept representations with an average-per-concept F1 score of 86%. Finally, the CME framework makes it possible to compare concept-based explanations, contributing to the scientific rigour of evaluating interpretability methods.The author would like to appreciate the generous sponsorship of the Engineering and Physical Sciences Research Council (EPSRC), The Department of Computer Science and Technology at the University of Cambridge, and Tenyks, Inc

Models of modelling: genres, purposes or perspectives

The number of papers and research reports addressing the theory and/or practice of mathematical modelling with some form of connection to education is growing astronomically. Small wonder then that educational publications featuring articles emerging from this field, present such a plethora of views that even those experienced in the field can become disoriented, let alone those feeling their way in a new area. This paper joins a conversation that concerns itself with meanings, approaches, priorities, and intentions associated with the use of the term ‘mathematical modelling’ as it occurs in education. For example it will be argued that there are essentially two generic approaches to modelling within education: modelling that acts primarily as a ‘vehicle’ for the attainment of other curricular priorities, and modelling as ‘content’ that seeks first to nurture and enhance the ability of students to solve authentic real world or life-like problems. Within these approaches there are particular purposes and perspectives, but the latter are just that – they are not (as sometimes suggested) additional modelling genres. The paper visits areas of relevance to its theme: such as stated priorities of educational authorities in curriculum statements; types of activity that make up the two modelling genres; a selection of writings that canvass a rich array of issues, challenges, and research foci that are currently engaging interest and activity within the field; and the implications of criticisms of modelling, both appropriate and misplaced

Knowledge-based design support and inductive learning

Designing and learning are closely related activities in that design as an ill-structure problem involves identifying the problem of the design as well as finding its solutions. A knowledge-based design support system should support learning by capturing and reusing design knowledge. This thesis addresses two fundamental problems in computational support to design activities: the development of an intelligent design support system architecture and the integration of inductive learning techniques in this architecture.This research is motivated by the belief that (1) the early stage of the design process can be modelled as an incremental learning process in which the structure of a design problem or the product data model of an artefact is developed using inductive learning techniques, and (2) the capability of a knowledge-based design support system can be enhanced by accumulating and storing reusable design product and process information.In order to incorporate inductive learning techniques into a knowledge-based design model and an integrated knowledge-based design support system architecture, the computational techniques for developing a knowledge-based design support system architecture and the role of inductive learning in Al-based design are investigated. This investigation gives a background to the development of an incremental learning model for design suitable for a class of design tasks whose structures are not well known initially.This incremental learning model for design is used as a basis to develop a knowledge-based design support system architecture that can be used as a kernel for knowledge-based design applications. This architecture integrates a number of computational techniques to support the representation and reasoning of design knowledge. In particular, it integrates a blackboard control system with an assumption-based truth maintenance system in an object-oriented environment to support the exploration of multiple design solutions by supporting the exploration and management of design contexts.As an integral part of this knowledge-based design support architecture, a design concept learning system utilising a number of unsupervised inductive learning techniques is developed. This design concept learning system combines concept formation techniques with design heuristics as background knowledge to build a design concept tree from raw data or past design examples. The design concept tree is used as a conceptual structure for the exploration of new designs.The effectiveness of this knowledge-based design support architecture and the design concept learning system is demonstrated through a realistic design domain, the design of small-molecule drugs one of the key tasks of which is to identify a pharmacophore description (the structure of a design problem) from known molecule examples.In this thesis, knowledge-based design and inductive learning techniques are first reviewed. Based on this review, an incremental learning model and an integrated architecture for intelligent design support are presented. The implementation of this architecture and a design concept learning system is then described. The application of the architecture and the design concept learning system in the domain of small-molecule drug design is then discussed. The evaluation of the architecture and the design concept learning system within and beyond this particular domain, and future research directions are finally discussed