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Analogical prediction
 Proceedings of the 9th International Workshop on Inductive Logic Programming, volume 1634 of Lecture Notes in Artificial Intelligence
, 1999
"... Abstract. Inductive Logic Programming (ILP) involves constructing an hypothesis H on the basis of background knowledge B and training examples E. An independent test set is used to evaluate the accuracy of H. This paper concerns an alternative approach called Analogical Prediction (AP). AP takes B, ..."
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Abstract. Inductive Logic Programming (ILP) involves constructing an hypothesis H on the basis of background knowledge B and training examples E. An independent test set is used to evaluate the accuracy of H. This paper concerns an alternative approach called Analogical Prediction (AP). AP takes B, E and then for each test example 〈x, y 〉 forms an hypothesis Hx from B, E, x. Evaluation of AP is based on estimating the probability that Hx(x) = y for a randomly chosen 〈x, y〉. AP has been implemented within CProgol4.4. Experiments in the paper show that on English past tense data AP has significantly higher predictive accuracy on this data than both previously reported results and CProgol in inductive mode. However, on KRK illegal AP does not outperform CProgol in inductive mode. We conjecture that AP has advantages for domains in which a large proportion of the examples must be treated as exceptions with respect to the hypothesis vocabulary. The relationship of AP to analogy and instancebased learning is discussed. Limitations of the given implementation of AP are discussed and improvements suggested. 1
Learning and Problem Solving Strategies of ESL Students
"... The mathematics problem solving approaches of a group of elementary and secondary ESL students were investigated through a performance assessment accompanied by thinkaloud procedures. Students were enrolled in ESL mathematics classes in a Title VII project implementing the Cognitive Academic Learni ..."
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The mathematics problem solving approaches of a group of elementary and secondary ESL students were investigated through a performance assessment accompanied by thinkaloud procedures. Students were enrolled in ESL mathematics classes in a Title VII project implementing the Cognitive Academic Learning Approach (CALLA). In this approach, curriculum content is used to develop academic language and learning strategies are taught explicitly to increase students metacognitive awareness and to facilitate their learning of both content and language. Participating teachers were identified either as high implementation teachers (extensive involvement in staff development and other project activities) or low implementation teachers (limited involvement in project activities). The study was designed to identify learning and problem solving strategies of students at high, average and low mathematics achievement levels, and to compare strategic approaches of students in high implementation and low implementation classrooms. The results indicated that significantly more students in high implementation classrooms were able to solve the problem correctly than were students in low implementation classrooms. As expected, students rated high in math performance also performed significantly better on finding the correct problem solution. Of greater interest was the finding that there were no differences in the actual number 1 2 Bilingual Research Journal, 16:3&4, Summer/Fall 1992 of problem solving steps used by students in the two implementation levels, but that significant differences for high implementation classrooms were found for correct sequence of problem solving steps, which has been featured in instruction in the high implementation classrooms. This seemed to indicate that ex...
Partially Isomorphic Generalization and Analogical Reasoning
 Proceedings of European Conference on Machine Learning (1994), Lecture Notes in Artificial Intelligence 784
, 1994
"... Analogical reasoning is carried out based on an analogy which gives a similarity between a base domain and a target domain. Thus, the analogy plays an important role in analogical reasoning. However, computing such an analogy leads to a combinatorial explosion. This paper introduces partially isomor ..."
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Analogical reasoning is carried out based on an analogy which gives a similarity between a base domain and a target domain. Thus, the analogy plays an important role in analogical reasoning. However, computing such an analogy leads to a combinatorial explosion. This paper introduces partially isomorphic generalizations of atoms and rules which make it possible to carry out analogical reasoning without computing the analogy, and also gives a relationship between our generalization and the analogy. Then, we give a procedure which produces such a generalization in polynomial time with respect to the length of a given atom or rule, and realize it as a Prolog program. 1 Introduction Analogical reasoning is an important paradigm of machine learning [1, 5, 6]. it acquires unknown knowledge by computing an analogy, which gives a similarity between a base domain and a target domain. In analogical reasoning, we first detect an analogy, and then project the wellknown knowledge in the base domai...
Construction of Conceptual Knowledge: The Case of ComputerAided Exploration of Period
 Doubling, Proceedings of the British Society for Research into Learning Mathematics
, 2000
"... This research focuses on students using an experimental approach with computer software to give visual meaning to symbolic ideas and to provide a basis for further generalisation. They use computer software that draws orbits of x = f(x) iteration and are encouraged to investigate the iterations of f ..."
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This research focuses on students using an experimental approach with computer software to give visual meaning to symbolic ideas and to provide a basis for further generalisation. They use computer software that draws orbits of x = f(x) iteration and are encouraged to investigate the iterations of fλ(x)=λx(1Ðx) as λ increases. The iterations pass through successive acts of perioddoubling as λ=λ0, λ1, λ2,...; they are invited to estimate the values of λ and to compare their experimental results with the theory of geometric convergence. The supervisor acts as a mentor, using various styles of questioning to provoke links between different ideas. Data is collected in various ways to give evidence for the ways in which students develop conceptual links between symbolic theory and the visual and numeric aspects of computer experiment.
Incorporating ExplanationBased Generalization with Analogical Reasoning
 Bulletin of Informatics and Cybernetics
, 1994
"... The EBG system builds an explanation and learns a concept definition as its generalization provided a domain theory is complete. It does not work when a domain theory is incomplete. Then we introduce a notion of generalizations by an analogy which makes it possible to construct rules necessary for d ..."
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The EBG system builds an explanation and learns a concept definition as its generalization provided a domain theory is complete. It does not work when a domain theory is incomplete. Then we introduce a notion of generalizations by an analogy which makes it possible to construct rules necessary for domain theories. Furthermore, we develop EBG by analogical reasoning which copes with the incompleteness by using generalizations by an analogy. We first formulate EBG in a mathematical way, construct EBG by analogical reasoning in terms of our formulation, and realize EBG by analogical reasoning system as a Prolog program. 1 Introduction EBG(ExplanationBased Generalization) takes as inputs a domain theory, a training example, a goal concept and an operationality criterion. It constructs an explanation in terms of the domain theory that proves how the training example satisfies the goal concept definition. Then it determines a set of operationally sufficient conditions for the goal concept ...
Fullerenes and Coordination Polyhedra versus HalfCubes Embeddings
, 1997
"... A fullerene F n is a 3regular (or cubic) polyhedral carbon molecule for which the n vertices  the carbons atoms  are arranged in 12 pentagons and ( n 2 \Gamma 10) hexagons. Only a finite number of fullerenes are expected to be, up to scale, isometrically embeddable into a hypercube. Looking fo ..."
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A fullerene F n is a 3regular (or cubic) polyhedral carbon molecule for which the n vertices  the carbons atoms  are arranged in 12 pentagons and ( n 2 \Gamma 10) hexagons. Only a finite number of fullerenes are expected to be, up to scale, isometrically embeddable into a hypercube. Looking for the list of such fullerenes, we first check the embeddability of all fullerenes F n for n ! 60 and of all preferable fullerenes C n for n ! 86 and their duals. Then, we consider some infinite families, including fullerenes with icosahedral symmetry, which describe virus capsids, onionlike metallic clusters and geodesic domes. Quasiembeddings and fullerene analogues are considered. We also present some results on chemically relevant polyhedra such as coordination polyhedra and cluster polyhedra. Finally we conjecture that the list of known embeddable fullerenes is complete and present its relevance to the Katsura model for vesicles cells. Contents 1 Introduction and Basic Properties 2 1...
Philosophies And Pedagogies Of Mathematics
"... The paper discusses major philosophical stances on the nature of mathematics as held by foundationalists and quasiempiricalism supporters. It is argued that the contrasting philosophical views between the two groups parallels in many respects the pedagogical debate between behaviourism and socioco ..."
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The paper discusses major philosophical stances on the nature of mathematics as held by foundationalists and quasiempiricalism supporters. It is argued that the contrasting philosophical views between the two groups parallels in many respects the pedagogical debate between behaviourism and socioconstructivism. It is also argued that behaviourism has been influenced by foundationalist conceptions of mathematics while socioconstructivism has been influenced by quasiempirical philosophies.
THINKERS
"... May 2008Me parece que la inteligencia no consiste en resolver problemas sino en encontrarlos. [It seems to me that intelligence is not about solving problems but about finding them.] (Poniatowska, 2001, p. 452) Preface This dissertation consists of three articles. In Section 1, the leading article i ..."
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May 2008Me parece que la inteligencia no consiste en resolver problemas sino en encontrarlos. [It seems to me that intelligence is not about solving problems but about finding them.] (Poniatowska, 2001, p. 452) Preface This dissertation consists of three articles. In Section 1, the leading article is The Discourse of Bilingual Mathematization. In this research study I first conceptualize and then explore bilingualism and experience as cognitive resources for how bilingual students mathematize various kinds of problems in Spanish and English. In this paper I also raise the concern that, in general, research dealing with bilingual students in mathematics has failed to adequately investigate issues that pertain to how bilinguals reason in mathematics. This concern is the focus of the article in Section 2, Bilingualism and Learning Mathematics: A Literature Review. In this article I review and organize literature according to two prevailing views: (a) the view of language as a tool that constructs mathematical knowledge and (b) the view of language and mathematical knowledge as simultaneously constructing each other. The third article, presented in Section 3, is Using What Matters to Bilingual Students in Mathematics Instruction. In
PROOFS
"... Can we do something to improve the teaching of firstyear calculus? Michel Helfgott, SUNY Oswego After teaching mathematics for many years I have come to the conclusion that the above question has an affirmative answer. As is often mentioned, teaching is an art rather than a science, and I do not pr ..."
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Can we do something to improve the teaching of firstyear calculus? Michel Helfgott, SUNY Oswego After teaching mathematics for many years I have come to the conclusion that the above question has an affirmative answer. As is often mentioned, teaching is an art rather than a science, and I do not presume to have the cure to all the ills that plague mathematics education. However, it is my belief that something can be gained by paying special attention to five basic guidelines when teaching calculus. They are as follows: 1. Try to strike a balance with regard to what to prove and what to accept without proof. 2. Convey the idea that sometimes there is more than one way to solve a problem. 3. Discuss significant applications in the classroom, not relegating them to the end as optional materials. 4. Place the subject in a historical perspective whenever possible.