Using meta-level inference to constrain search and to learn strategies in equation solving

This thesis addresses two questions:- How can search be controlled in domains with a large search space?- How can this control information be learned?It is argued that both problems can be tackled with the aid of a technique called meta-level inference.In this technique, the control information is separated from the factual information. The control information is expressed declaratively, i.e. the control information is represented as explicit rules. These rules are axioms in the meta-theory of the domain. This gives rise to a two level program, the factual information forms the object-level and the control information forms the meta-level. Inference is performed at the meta-level. and this induces inference at the object-level. Search at the object-level is replaced by search at the meta-level. This has several advantages, one of the most important being that the meta-level search space is usually much smaller than the object-level space, so the search problem is greatly reduced.Two programs are presented in this thesis to support this claim. Both programs operate in the domain of symbolic equation solving. However, the techniques used can be applied to a wide variety of domains.The first program. PRESS, solves symbolic, transcendental, non-differential equations. PRESS makes extensive use of meta-level inference to control search. This overcomes problems experienced by other approaches. For example, systems that apply rewrite rules exhaustively usually only use the rules one way round, to avoid looping. However, this often makes the system incomplete, and the techniques for completing this set are not easily mechanized. PRESS is able to use rules in both directions, using inference to decide which direction is appropriate.The second program, LP is also an equation solving program, but, unlike PRESS, it is capable of learning new equation-solving techniques. It embodies a new learning method, called Precondition Analysis. Precondition Analysis combines meta-level inference with concepts from the field of planning, and allows the program to learn even from a single example. This learning technique seems particularly suitable in domains where the operators don't have precisely defined effects and preconditions. Equation solving is such a domain

A conceptual approach to the early learning of Algebra using a computer

This thesis describes an investigation into the conceptual understanding of algebra by early learners (age 11-13 years) and how a computer-based approach may be used to improve such, without any consequent loss of manipulative skills. The psychological framework for the investigation centred on the importance of the individual child's construction of a cognitive framework of knowledge and the relevance of the current state of this to the facilitating of concept acquisition. As such it incorporates elements of the developmental psychology of Piaget, Ausubel and Skemp. Furthermore, in order to assist in the synthesis of a sufficiently broad psychological theory of education it was necessary to postulate the formulation of a new integrated bi-modal model of learning. This is described, along with details of its application and significance to a theory of cognitive integration which is designed to promote versatile learning (after Brumby, 1982) in mathematics through a relational linking of global/holistic and serialist/analytic schemas. The research comprised two initial investigations followed by the main experiment. The results of the initial investigations with early learners of algebra showed that the dynamic algebra module written for the research produced a significant improvement in the children's conceptual understanding of algebra. The main experiment sought to further clarify this improvement and to compare and contrast it with that produced by a traditional skill-based algebra module. In order to facilitate this comparison, the performance of 57 matched pairs of pupils from two groups of three parallel forms of the first year of a 12+ entry co-educational secondary school was analysed. The results of the investigation confirmed the value of the dynamic algebra module as a generic organiser (in the sense of Tall, 1986) for the understanding of algebraic concepts, producing a significant difference in conceptual understanding, without any detrimental effect on manipulative skills. Furthermore, the beneficial effects of the programme were such that its results showed that it had provided a better base than the skill-based approach for the extension of algebraic understanding past the initial stages and into more involved areas such as linear equations and inequalities. The findings of this research show that the use of a module based on a computer environment, with its many advantages for conceptual learning, prior to the more formal introduction of algebraic techniques, is of great cognitive value. They also provide evidence for the theoretical model of learning proposed in the thesis, and suggest that for the production of a versatile learner in mathematics, more attention should be paid to the integration of the global/holistic abilities of the individual with his/her serialist/analytic abilities. The implications for the future are that such abilities, and hence mathematical competence may well be improved in other areas of the curriculum by the use of the computer within a similar theoretical framework

The Development of a Semantic Model for the Interpretation of Mathematics including the use of Technology

The semantic model developed in this research was in response to the difficulty a group of mathematics learners had with conventional mathematical language and their interpretation of mathematical constructs. In order to develop the model ideas from linguistics, psycholinguistics, cognitive psychology, formal languages and natural language processing were investigated. This investigation led to the identification of four main processes: the parsing process, syntactic processing, semantic processing and conceptual processing. The model showed the complex interdependency between these four processes and provided a theoretical framework in which the behaviour of the mathematics learner could be analysed. The model was then extended to include the use of technological artefacts into the learning process. To facilitate this aspect of the research, the theory of instrumentation was incorporated into the semantic model. The conclusion of this research was that although the cognitive processes were interdependent, they could develop at different rates until mastery of a topic was achieved. It also found that the introduction of a technological artefact into the learning environment introduced another layer of complexity, both in terms of the learning process and the underlying relationship between the four cognitive processes

Computational aerodynamics and artificial intelligence

The general principles of artificial intelligence are reviewed and speculations are made concerning how knowledge based systems can accelerate the process of acquiring new knowledge in aerodynamics, how computational fluid dynamics may use expert systems, and how expert systems may speed the design and development process. In addition, the anatomy of an idealized expert system called AERODYNAMICIST is discussed. Resource requirements for using artificial intelligence in computational fluid dynamics and aerodynamics are examined. Three main conclusions are presented. First, there are two related aspects of computational aerodynamics: reasoning and calculating. Second, a substantial portion of reasoning can be achieved with artificial intelligence. It offers the opportunity of using computers as reasoning machines to set the stage for efficient calculating. Third, expert systems are likely to be new assets of institutions involved in aeronautics for various tasks of computational aerodynamics

Beliefs, autonomy, and mathematical knowledge

The purpose of this study was to investigate the apparent effects of students\u27 beliefs about mathematics and autonomy on their learning of mathematics. The study utilized a multiple-case study design with analysis by and across cases. The cases represented six high school students enrolled in either Algebra II or Algebra II/Trigonometry. Data was collected in three phases: (a) classroom observations and assessment of the teacher\u27s perception of her role in the learning process, (b) an assessment of students\u27 beliefs about mathematics and autonomy, and (c) an assessment of students\u27 newly formed mathematical constructs on functions. The beliefs\u27 assessment included observing and questioning students while they solved mathematics problems. Follow-up probes explored the students\u27 rationale for their strategies and their dependency on rules and algorithms when solving problems, particularly problem-solving situations. To further corroborate the beliefs expressed by the students or those inferred from their solutions, the students also marked and discussed a mathematics topics ranking grid and vocabulary lists, graded a sample algebra test, and responded to scenarios on student\u27s/teacher\u27s roles. From the data, a detailed portrait of each individual\u27s beliefs about mathematics was developed. These portraits were compared with the students\u27 understanding of functions and the classroom expectations. The results from this research investigation suggest three hypotheses concerning students\u27 beliefs about mathematics, autonomy, and mathematical knowledge. First, students\u27 beliefs about mathematics rather than being dichotomous form a continuum from strongly conceptual in outlook to strongly procedural. Second, students\u27 autonomy augments their beliefs about mathematics and often meditates them. Third, students\u27 beliefs and autonomy appear to concur with their problem-solving strategies and with their knowledge of mathematics. Collectively these hypotheses suggest that students\u27 beliefs and autonomy are an integral component of students\u27 conception of mathematics and influence both how problems are approached and how mathematics is learned. Further study needs to be done on how and when these beliefs are formed and under what conditions these beliefs are modified and changed. Finally, the interplay among beliefs, autonomy, and learning needs to be investigated in the actual classroom context