Querying Geometric Figures Using a Controlled Language, Ontological Graphs and Dependency Lattices

Dynamic geometry systems (DGS) have become basic tools in many areas of geometry as, for example, in education. Geometry Automated Theorem Provers (GATP) are an active area of research and are considered as being basic tools in future enhanced educational software as well as in a next generation of mechanized mathematics assistants. Recently emerged Web repositories of geometric knowledge, like TGTP and Intergeo, are an attempt to make the already vast data set of geometric knowledge widely available. Considering the large amount of geometric information already available, we face the need of a query mechanism for descriptions of geometric constructions. In this paper we discuss two approaches for describing geometric figures (declarative and procedural), and present algorithms for querying geometric figures in declaratively and procedurally described corpora, by using a DGS or a dedicated controlled natural language for queries.Comment: 14 pages, 5 figures, accepted at CICM 201

Symbols and the bifurcation between procedural and conceptual thinking

Symbols occupy a pivotal position between processes to be carried out and concepts to be thought about. They allow us both to d o mathematical problems and to think about mathematical relationships. In this presentation we consider the discontinuities that occur in the learning path taken by different students, leading to a divergence between conceptual and procedural thinking. Evidence will be given from several different contexts in the development of symbols through arithmetic, algebra and calculus, then on to the formalism of axiomatic mathematics. This is taken from a number of research studies recently performed for doctoral dissertations at the University of Warwick by students from the USA, Malaysia, Cyprus and Brazil, with data collected in the USA, Malaysia and the United Kingdom. All the studies form part of a broad investigation into why some students succeed yet others fail

Towards an Intelligent Tutor for Mathematical Proofs

Computer-supported learning is an increasingly important form of study since it allows for independent learning and individualized instruction. In this paper, we discuss a novel approach to developing an intelligent tutoring system for teaching textbook-style mathematical proofs. We characterize the particularities of the domain and discuss common ITS design models. Our approach is motivated by phenomena found in a corpus of tutorial dialogs that were collected in a Wizard-of-Oz experiment. We show how an intelligent tutor for textbook-style mathematical proofs can be built on top of an adapted assertion-level proof assistant by reusing representations and proof search strategies originally developed for automated and interactive theorem proving. The resulting prototype was successfully evaluated on a corpus of tutorial dialogs and yields good results.Comment: In Proceedings THedu'11, arXiv:1202.453

Course Portfolio for Math 407 Mathematics for High School Teaching: Refining Conceptual Understanding in a Mathematics Course for Pre-service Teachers

My intention in this portfolio is to present my approach to teaching an upper-level mathematics course for pre-service secondary level mathematics teachers. Several teaching strategies are discussed in the context of designing a coherent approach to this course, which emphasizes the need for conceptual reasoning above all other goals. These strategies are evaluated and assessed in connection to the learning outcomes using samples of student work from the course. Also presented are samples of course materials that were used to lead students through an organized discussion of the relevant concepts. These materials convey some basic mathematical knowledge and therefore may suited to other courses as well. Additionally, this portfolio includes a survey of students perceptions and attitudes towards conceptual mathematics at the beginning of the course, which can be viewed as base- line information, as well as a sample of student work production and self-reflections at the end of the curse, which establish a certain growth in confidence and abilities

Proof, proving, and teacher-student interaction: theories and contexts

This chapter focuses on the role of the teacher in teaching proof and proving in the mathematics classroom. Within an over-arching theme of diversity (of countries, curricula, student age-levels, teachers' knowledge, and so forth), we review three carefully-selected relevant theories: socio-mathematical norms, teaching with variation, and instructional exchanges. Each of these theories starts from the abstraction of observations in existing school mathematics classrooms and uses those observations to probe into the rationality of teachers in order to understand what sustains those classroom contexts and also how such contexts might be changed. We argue that each theory offers insight into the role of the teacher in the teaching and learning of proof and proving. In informing future research, this chapter provides support for meeting the challenge of theorising about the role of the teacher in the teaching and learning of proof and proving in mathematics classrooms across the diverse contexts worldwide

Kajian terhadap kualiti air bag i sistem pengumpulan air hujan menggunakan penapis biopasir

Kajian ini telah mengkaji rekabentuk Sistem Pengumpulan Air Hujan (SP AH) sebagai satu sistem yang boleh menghasilkan satu altematif sumber air dari segi kualiti aimya bagi kegunaan domestik. SP AH telah dibangunkan di kampus KUiTTHO dan setiap pemilihan serta susunan komponen SPAH dikenalpasti sama ada boleh menyumbang kepada peneemaran terhadap kualiti air yang dikumpul. Komponen SP AH terdiri daripada permukaan tadahan, saluran, tangki eurahan pertama, tangki simpanan dan rawatan. Kajian ini juga telah membuat penambahan pada SP AH iaitu bagi komponen rawatan dengan menggunakan penapis biopasir. Sebanyak 3 jenis kedalaman telah digunakan iaitu 20 em, 40 em dan 70 em bagi medium pasir halus (saiz diameter, d = 0.2-0.35 mm, pemalar keseragaman, UC = 2.68, keporosan, 8 = 0.40, ketumpatan, Pp = 2.65g/ml). Sampel kualiti air diambil dalam tempoh Februari 2005 hingga April 2005. Ini termasuk tempoh pematangan penapis bagi membentuk lapisan schmutzdecke selama 3 minggu. Didapati, pengaruh persekitaran dan peneemaran terhadap kualiti air tadahan dapat diatasi. Kualiti air tadahan sebelum rawatan berada didalam Kelas IIA - V dapat dikurangkan kepada Kelas I - IIA selepas melalui rawatan penapis biopasir. Peratus penyingkiran bagi kesemua parameter berada dalam nilai yang amat baik iaitu sehingga 34 % bagi pH, 97 % bagi COD dan BOD, 98 % bagi kekeruhan, 100 % bagi TSS, 99 % bagi jumlah koliform dan 95 % - 100 % bagi kandungan logarn berat. Kedalaman penapis biopasir mempengaruhi kualiti air dengan hirarki kualiti air selepas rawatan 70 em > 40 em > 20 em. Manakala, pengaruh tempoh kering di antara kejadian hujan memberikan kesan pada keseluruhan kualiti air terkumpul. Data kualiti air hujan ini merupakan langkah awal dalam pengaplikasian SPAH di karnpus KUiTTHO seterusnya implikasi kejadian banjir kilat dan seumpamanya dapat dielakkan pada masa akan datang

Identifying Structure in Introductory Topology: Diagrams, Examples, and Gestures

Despite the prevalence of research in calculus, linear algebra, abstract algebra, and analysis in undergraduate mathematics, the teaching and learning of general topology is a largely unexplored area of research. Although enrollment in courses like linear algebra is often higher than that of topology, the study of students’ learning and understanding of topology is of great significance to the Research in Undergraduate Mathematics Education (RUME) community. Courses in topology present many students with their first experience in axiomatic reasoning and explicit interactions with mathematical structure, itself. I present a thorough case study of Stacey, an undergraduate taking a first course in undergraduate topology. Through the lenses of mathematical structuralism, constructivism, embodied cognition, and commognition, I investigated Stacey’s proving behaviors. Papers 1, 2, and 3 present a top-down description of Stacey’s behaviors as she sought to identify the key ideas of proofs in general topology. In Paper 1, I described Stacey’s proving behaviors using vocabulary borrowed from the literature on problem solving and showed that she used diagrams to arrive at the key idea. In Paper 2, I observed that Stacey seldom produced specific examples, but she reasoned about her diagrams as examples and manipulation of these examples led her to the key ideas of several proofs and to identify appropriate counterexamples when necessary. In Paper 3, I used the theories of embodied cognition and commognition to argue that Stacey’s use of diagrams to ground abstract structures in the external world gave her the ability to manipulate those structures spatially, ultimately leading her to the key idea. These three papers are three perspectives on the same theme, each digging more deeply than the one before. Stacey’s behaviors in Papers 1, 2, and 3 describe her search for and investigation of abstract mathematical structures. Stacey’s recognition of structure and her ability to work with it helped her to succeed in writing proofs. I conclude this dissertation with suggestions for teaching, including incorporation of the theories of embodied cognition and commognition, as well as directions for future research