Some techniques for building mathematical intelligent tutoring systems

Abstract

Some techniques that are useful in building intelligent tutoring systems for mathematics are described. These techniques have been implemented in Integration-Kid, an intelligent tutoring system for integral calculus. Its production system is a language which helps the problem-solver to interact with students. For the mathematical problem-solver, we have adopted term-rewriting rules which represent basic mathematical knowledge. The situation map is a representation which lays out all possible situations in the process of solving a complex problem. We shall also describe an algorithm which transforms a linear mathematical expression to a readable, two-dimensional expression.     

Why computer science students find formal reasoning frightening

Abstract  This paper focuses on difficulties which first-year undergraduate computer science students experience in learning formal reasoning techniques. The main findings from an empirical investigation are reported. These indicated that, in the student population studied, many of those difficulties stemmed from a combination of the mathematical content of the courses and the students' unfamiliarity with formal notations and language. The paper concludes by discussing a selection of computer-based tools which have been selected to address these difficulties.     

Teaching mathematics and the Inuit community

Abstract

In the spring of 2000, the Inuit community and the Kativik School Board were pondering the difficulties encountered by students in mathematics and the measures that could be taken to help these students. One significant fact that it seemed might help explain these difficulties was that Inuit students learn Inuit mathematics (for example, a base‐20 numeral system) in their own language in the first three years of their schooling and then go on to study in either French or English. A collaboration between researchers, teachers from the Inuit community, and curriculum developers was established. In this article, we present facets of the environment, language, and culture of the Inuit that influenced the development of the Inuit numeration system, their ways of measuring (length, distance, time, etc.), and their great aptitude in spatial representations. We then discuss the current collaborative project that aims at the development of teaching situations adapted to Inuit classrooms.     

Une étude ethnomathématique au Burkina Faso: L'arithmétique au quotidien

Abstract

This study is based on the observation of a divide between mathematics taught in school and the social and cultural context in which they apply. These mathematics, which present significant difficulties for students, appear to be far-removed from the needs and concerns of society. However, many social practices require daily recourse to mathematical concepts. To better understand how these concepts—the nature of which is not fully understood in schools—are used in context, an ethnographic study was carried out to examine various practices of daily life. Here we address practices involving the sale and purchase of agricultural products.     

The role and uses of technologies in mathematics classrooms: Between challenge and modus Vivendi

Abstract

Drawing support from theoretical frameworks, this article analyses the role and uses of technology in the teaching of mathematics, with specific reference to the complexity of integrating technology into teaching. Four aspects are discussed here: the changes that technology brings about in classroom mathematical activity, the difficulties students encounter in solving problems in a technology‐rich environment, the evaluation of potential assignments and the creation of conditions appropriate for learning, and the difficulties faced by teachers who use technology in deciding when and how to intervene. This analysis is illustrated by examples taken primarily from dynamic geometry, but similar analyses could be conducted for computerized environments of other kinds.     

关于C语言教学过程优化的探讨

就优化C语言教学过程提出了一些具体作法。根据教学对象和课时资源情况,对教材内容和结构进行再安排,以实现精讲多练目的;根据C语言学习中遇到的困难,课下通过网络进行师生互动,以弥补传统互动方式的不足;根据C语言特点,精心设计有难度梯度的上机实验,使学生克服畏难情绪,不断提高编程能力。     

Social Choice And Formal Language Theory

Abstract

This paper defines concepts of grammar, language and automata and their application to both structural and axiomatic representations of the social choice process. A general factorization of the social choice process is given and it is also shown that the formal language model has the power to include mathematical axiomatic characterizations as a special case. In addition, the general model can be extended to explicitly describe uncertainty in the social decision process.     

Identifying Difficulties in Learning Uml

Abstract

Despite its recognition as a standard object-oriented modeling language, Unified Modeling Language (UML) has been criticized for such deficiencies as semantic inconsistencies, vagueness, and conflicting notations. the relationship between these deficiencies and the difficulties in the learning process is the focus of this study. A concept mapping technique is used to unveil the learning difficulties and suggestions for alleviating them are provided.     

Classification and Concept Consistency

Abstract

This article investigates the extent to which undergraduates consistently use a single mechanism as a basis for classifying mathematical objects. We argue that the concept image/concept definition distinction focuses on whether students use an accepted definition but does not necessarily capture the more basic notion that there should be a fixed basis for classification. We examine students’ classifications of real sequences before and after exposure to definitions of increasing and decreasing; we develop an abductive plausible explanations method to estimate the consistency within the participants’ responses and suggest that this provides evidence that many students may lack what we call concept consistency.     

Language Repair Strategies in Bilingual Tutoring of Mathematics Word Problems

Abstract: This study explores the language repair strategies (aimed at repairing communication problems) of two bilingual speakers during mathematics word problem tutoring sessions. Bilingual repair was shown to gradually shift from a linguistic to an epistemic focus during problem solving (i.e., communication became more conceptually focused over time). Linguistic repair involved varied translation techniques (literal and free), whereas epistemic repair entailed gradual refinement of specialized mathematical meanings through intralinguistic communicative strategies such as simplified rewording. It is suggested that understanding language repair may help tutors develop more effective dialogic instructional approaches to the teaching of mathematical word problems to English learners.     

On the calculus of positively constructed formulas for automated theorem proving

The paper deals with an expressive logic language LF and its calculus. Formulas of this language consist of some large-scale structural elements, such as type quantifiers. The language LF contains only two logic symbols—∀ and ∃, which form the set of logic connectives of the language. The logic calculus JF and complete strategy for automated proof search based on a single unary rule of inference are considered. This calculus has a number of other features which lead to the reduction of the combinatorial complexity of finding the deductions in comparison to the known systems for automated theorem proving as the Resolution method and Genzen calculus. Problems of effective implementation of JF as a program system for automated theorem proving are considered.     

Language Is Physical

Some aspects of the physical nature of language are discussed. In particular, physical models of language must exist that are efficiently implementable. The existence requirement is essential because without physical models no communication or thinking would be possible. Efficient implementability for creating and reading language expressions is discussed and illustrated with a quantum mechanical model. The reason for interest in language is that language expressions can have meaning, either as an informal language or as a formal language associated with mathematical or physical theories. It is noted that any universally applicable physical theory, or coherent theory of physics and mathematics together, includes in its domain physical models of expressions for both the informal language used to discuss the theory and the expressions of the theory itself. It follows that there must be some formulas in the formal theory that express some of their own physical properties. The inclusion of intelligent systems in the domain of the theory means that the theory, e.g., quantum mechanics, must describe, in some sense, its own validation. Maps of language expressions into physical states are discussed. A spin projection example is discussed as are conditions under which such a map is a Gödel map. The possibility that language is also mathematical is very briefly discussed. PACS: 03.67–a; 03.65.Ta; 03.67.Lx     

Software verification and graph similarity for automated evaluation of students’ assignments

ContextThe number of students enrolled in universities at standard and on-line programming courses is rapidly increasing. This calls for automated evaluation of students assignments.ObjectiveWe aim to develop methods and tools for objective and reliable automated grading that can also provide substantial and comprehensible feedback. Our approach targets introductory programming courses, which have a number of specific features and goals. The benefits are twofold: reducing the workload for teachers, and providing helpful feedback to students in the process of learning.MethodFor sophisticated automated evaluation of students’ programs, our grading framework combines results of three approaches (i) testing, (ii) software verification, and (iii) control flow graph similarity measurement. We present our tools for software verification and control flow graph similarity measurement, which are publicly available and open source. The tools are based on an intermediate code representation, so they could be applied to a number of programming languages.ResultsEmpirical evaluation of the proposed grading framework is performed on a corpus of programs written by university students in programming language C within an introductory programming course. Results of the evaluation show that the synergy of proposed approaches improves the quality and precision of automated grading and that automatically generated grades are highly correlated with instructor-assigned grades. Also, the results show that our approach can be trained to adapt to teacher’s grading style.ConclusionsIn this paper we integrate several techniques for evaluation of student’s assignments. The obtained results suggest that the presented tools can find real-world applications in automated grading.     

The teacher's responsibility in whole‐class debriefing of mathematical activity

Abstract

Prompted by recent moves in the United Kingdom to guide teachers’ practices in whole‐class, direct interactive teaching, in this article, we offer an opportunity for North American mathematics educators to reflect on possibilities for whole‐class teaching of mathematics. We focus particularly on the plenary aspect of lessons—what might be considered the debriefing of mathematical activity—and specifically on the teacher's responsibility during those sessions, both to his or her students and to the authenticity of the discipline of mathematics. Drawing on data from a Grade‐3 classroom and invoking complexity science as a theoretical lens to explore the classroom as a complex learning system, we present implications for teaching in whole‐class debriefings of mathematical activity.     

Images and the Growth of Understanding of Mathematics-for-Working

Abstract

Some of mathematics teaching is routine, like an exercise from a textbook for which you have received instruction and already know what to do. On other occasions, however, teaching mathematics is challenging, involving problems of teaching for which the solutions may not be readily apparent. These situations require the application of mathematical knowledge in concert with other types of knowledge for teaching. In this article, we explore the idea that teachers of mathematics act as applied mathematicians in applying mathematical knowledge to the resolution of teaching problems. This task involves the complex interplay of mathematical and teaching knowledge and processes of problem solving with success judged according to how well students learn. The article discusses these ideas through an examination of seven scenarios.     

Learning new programming languages: an analysis of the process and problems encountered

Abstract

Experienced programmers transferring to a new language have a far easier time than the novice learning a first language. However, they still experience considerable difficulties. The objective in this exploratory study was to characterize the kinds of learning and transfer that take place in the early stages of using a new programming language and where difficulties develop. ‘Think-aloud’ protocols were videotaped as subjects went about trying to write a program in a new programming language. Subjects used One of two unfamiliar languages, one similar to their known language (Pascal) and the other dissimilar. Three types of analyses were done on the recorded protocols: a procedural analysis showing the activities the subjects engaged in as they learned the new language, a programming knowledge analysis showing in which areas of program development difficulties were encountered, and a solutions analysis showing how successful the programmers were at using unique features of the new language. We found that the procedures programmers used to learn a new language were independent of the language being learned. The slight differences that existed in procedures were between levels of expertise. Programmers spent the majority of their time reading a language textbook. The programming knowledge analysis showed that programmers main area of concentration was planning how to implement their approach given the constructs available in the language. We observed many iterations of programmers trying to implement plans, failing and having to revise their plans. Examination of the subjects solutions and implementation approaches in Pascal led us to believe that programmers learning a new language are often biased by their implementation of algorithms in previous languages.