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Current Research Activities

Euclid and His Twentieth Century RivalsReasoning with Geometric Diagrams - It has long been assumed in mathematical communities that diagrams can be properly used only as aids to human intuition which have no place in formal proofs, reducing diagrams to second-class citizens of the mathematical world.  My research aims to show that diagrams can be used rigorously and fruitfully in formal mathematics, and, indeed, that the formal proofs that one gets using diagrams are more understandable and correspond better to  normal informal proofs (of the kind that one finds in Euclid's Elements, for example) than traditional linguistic formal proofs.

  • The most complete version of my work with diagrams is found in my book Euclid and His Twentieth Century Rivals: Diagrams in the Logic of Euclidean Geometry, available here from the University of Chicago Press.  It is also available from most online booksellers. A preprint pdf version of the book can be downloaded here: euclid20thcenturyrivalsmiller.pdf.
  • A beta version of my computer program CDEG: Computerized Diagrammatic Euclidean Geometry, which is described in this book, is available here.
  • "The Philosophical and Pedagogical Implications of a Computerized Diagrammatic System for Euclidean Geometry" briefly describes CDEG and some of its philosophical and pedagogical implications.  It appears in the MAA Notes Volume Using the Philosophy of Mathematics in Teaching Undergraduate Mathematics, © 2017 The Mathematical Association of America.  This whole volume is currently available for free as an ebook for MAA members.
  • "CDEG: Computerized Diagrammatic Geometry 2.0" is a brief paper about this computer system that I presented as a poster at the Diagrams 2012 conference.  A preprint of this paper can be downloaded here: cdeg_diag2012_poster.pdf.  The poster can be downloaded as a powerpoint file here: D2012Poster.ppt.  An extended (unpublished) version of the paper can be downloaded here: cdeg_diag2012.pdf.
  • An extended abstract of my book was presented at the Diagrams 2006 conference at Stanford University and appeared the Springer-Verlag Lecture Notes in AI Series volume 4045  Diagrammatic Representation and Inference, Barker-Plummer, Cox, and Swoboda, eds., © 2006 Springer-Verlag.  You can download it here: extabstract.pdf.
  • "On the Inconsistency of Mumma's Eu" is a paper that was published in the Notre Dame Journal of Formal Logic (2012) 53: 27-52, and is available here. A preprint of this paper can be downloaded here: onmummaseundjfl.pdf.  John Mumma's Eu is another diagramatic formal system for giving proofs in Euclidean geometry which is similar to my formal system FG, but has several key differences that, as it turns out, make his system unsound and inconsistent. This article explains these problems with his formal system.
  • The most technical results about the computational complexity of using diagrams in Euclidean geometry can be found in my article "Computational Complexity of Diagram Satisfaction in Euclidean Geometry," Journal of Complexity 22 (2006) 250-274, which is available here.  A preprint version of this article can be downloaded here: complexity_case_analysis_diagrams.pdf.
  • "A Brief Proof of the Full Completeness of Shin's Venn Diagram Proof System" was published in the Journal of Philosophical Logic (2006) 35: 289-291, and is available here. A preprint version of this paper can be downloaded here: compactvenn.pdf.
  • My 2001 Cornell Ph.D. dissertation is A Diagrammatic Formal System for Euclidean Geometry.  You can download it in pdf form here:  thesis.pdf.  In addition to describing my formal system FG for giving diagrammatic proofs in Euclidean Geometry, it contains: a short history of diagrams, logic, and geometry, from 1700 B.C. up to the present;  examples of how to use weaker subsystems of FG to determine the logical structure of geometry; discussion of the unique role of lemmas in diagrammatic proofs in geometry;  sample transcripts from CDEG, the computer system implementation of FG; proofs that the question of whether or not a given diagram is satisfiable (that is, whether or not it represents a physically realizable situation) is decidable, but is NP-hard, and therefore not solvable in a reasonable amount of time;  a discussion of  earlier work by Isabel Luengo developing a different diagrammatic formal system for geometry, and an explaination of why her system in unsound; and a general discussion of why a formal system like FG is useful.
  • This is a three page non-technical overview of some of my work, in pdf format: overview.pdf.  It was published in the Springer-Verlag Lecture Notes in AI Series volume Theory and Application of Diagrams, Anderson, Cheng, and Haarslev, eds., © 2000 Springer-Verlag. Here is the paper "Case Analysis in Euclidean Geometry," which is refered to in this overview: caseanal.pdf.

The Teaching and Learning of Mathematics using Inquiry-Based Methods - Most of my teaching is done using inquiry-based methods, and I have had several publications and presentations in this area.

  • "Teaching Inquiry to High School Teachers Through the Use of Mathematics Action Research Projects" describes the use of mathematics Action Research Projects (ARPs) as a capstone experience in a Master’s program for in-service secondary teachers. It argues that these projects have had the effect of encouraging teachers to decide to try more inquiry-based teaching methods in their own classrooms, and at the same time, have given them the knowledge and tools to be successful in doing so. It was published in the PRIMUS special issues on Teaching Inquiry, currently available for free online. A preprint version of this article can be downloaded here.
  • "Teaching Writing and Proof-Writing Together" describes a First-Year Writing Seminar that I developed and taught in the math department at Cornell University as part of Cornell's Writing Across the Curriculum program.  It appears in the MAA Notes Volume Beyond Lecture: Resources and Pedagogical Techniques for Enhancing the Teaching of Proof-Writing Across the Curriculum, © 2016 The Mathematical Association of America.  This whole volume is currently available for free as an ebook for MAA members.
  • "A Mentoring Program for Inquiry-Based Teaching in a College Geometry Class," written with Nathan Wakefield, describes our experiences in implementing an intensive year-long mentoring program for training a novice instructor (Nathan) in using inquiry-based teaching methods.  This paper was published in 2014 in the International Journal of Education in Mathematics, Science and Technology 2(4), 266-272.
  • "Mathematical Modeling" is a set of notes for a project-based mathematical modeling class for in-service secondary teachers that I developed and taught from as part of UNC's MathTLC grant.  They have been published in the Journal of Inquiry-Based Learning in Mathematics.
  • "Notes for R H Bing’s Plane Topology Course" is a set of notes for an inquiry-based, Moore method plane topology class originally written by R H Bing.  I edited these notes, and added some discussion of how to teach from them. They have been published in the Journal of Inquiry-Based Learning in Mathematics
  • Multiply-Modified Moore/Miller Methods: The Many Faces of Inquiry-Based Learning in my Classes is a talk about how my teaching has evolved over time.  This talk was given at the 14th annual Legacy of R. L. Moore conference in Washington, DC, in June, 2011.
  • "Modern Geometry II" is a set of annotated course notes for my inquiry-based Modern Geometry II class that was published in the Journal of Inquiry-Based Learning in Mathematics. These are notes for a second semester class that explores Euclidean and non-Euclidean geometries from multiple perspectives, with an emphasis on developing problem solving, communication, and logical reasoning skills, directed primarily at pre-service secondary math teachers. The non-Euclidian geometries include taxicab, hyperbolic, and spherical geometry. This is an inquiry-based class in which students spend most of their time in class working on problems in groups and then write them up. The materials are meant to be accompanied by the book Experiencing Geometry [Henderson & Taimina, 2005].
  • "Modern Geometry I" is a set of annotated course notes for my inquiry-based Modern Geometry I class that was published in the Journal of Inquiry-Based Learning in Mathematics.  These are notes for a class that explores Euclidean and non-Euclidean geometries from multiple perspectives, with an emphasis on developing problem solving, communication, and logical reasoning skills, directed primarily at pre-service secondary math teachers and pre-service elementary teachers specializing in math. It is an inquiry-based class in which students spend most of their time in class working on problems in groups and then write them up. The materials are loosely based on those in the first several chapters of the book Experiencing Geometry [Henderson & Taimina, 2005].
  • "Visualization on Cones and Pool Tables Using Geometer's Sketchpad" describes several activities that I have developed for use in an undergraduate modern geometry course for pre-service K-12 teachers.  It was published in PRIMUS, Vol. XVI, No. 3 (September 2006), pp. 257 - 274, and can be downloaded here: primus.pdf.  (Unfortunately, it is a very large file.) This paper is also included in the instructor version of my Modern Geometry I notes above.

Other Research:

  • Generalized Baseball Curves: Three Symmetries and You're In!, written with Dean Allison and Ricardo Diaz, was published in the online MAA journal Loci.  The curve traced out on a baseball by its seam has many symmetries.  This article explores the class of closed spherical curves with the same symmetries, which we call generalized baseball curves.