Photograph of George Ashline           George Ashline         

   George Ashline

Chair, Department of Mathematics, Saint Michael's College

Colchester, Vermont USA 05439

You can find below some information about my current classesinformation about other classes that I have taught, eCollege information, advising information, my papers, and ways to reach me.  I have also compiled links in a number of areas in which I am interested, including mathematics education, fractal geometry and chaos theorythe history of mathematics, finite mathematics, photo links, and Saint Michael's College.

Material for my Fall 2013 classes.


  • Calculus I, MA 150.



Material for other classes that I teach.

  • Elementary Statistics, MA 120.
  • Probability and Statistics, MA 251.
  • Mathematics Education Seminar, MA 381.
  • Junior/Senior Seminar, MA 410.

Introductory eCollege information for students

  1. The URL location of eCollege is and you should place this in their Favorites folder.  This URL is also found at many places in the Saint Michael’s College web,  You can find the URL under the Current Students section of our SMC web.

  2. Your Login ID is your mikenet name, the same as you use to login to the mikenet system (for example, Gregory Santos may be gsantos2)

  3. Your initial password is your SMC ID#.  You may change this password using the My Profile button located at the top right of the screen after they log into eCollege (before they access a class).  If you forget your password in the future, there is a “Forgot your Password” option at the login screen or you may contact the eCollege Help Desk.

  4. There is a Student Orientation Tutorial that you may access, available on the screen after you log in.  This tutorial may be helpful if you have never used eCollege before, but it is somewhat oriented to a fully online distance education course, which is not what is used at Saint Michael’s College.  (Note that there are 1-2 distance education classes per semester at SMC.)

  5. If you have questions on how to use eCollege, then you should contact the eCollege Help Desk available 24/7 at 303.873.0005 or .  The 303 number is a toll number, but the eCollege email is free to them.  Response time from eCollege is quick.  You should contact the eCollege Help Desk and not contact the Saint Michael’s College Help Desk since the SMC Help Desk will only send them to eCollege.

  6. If you are adding a course after the semester has started, then you should submit your paperwork to the Registrar quickly so you will get access to your eCollege courses.  With the hecticness at the beginning of a new semester, the Registrar to Datatel to eCollege process may take 1 to 3 days. Once you have been added into your new course by the Registrar, your names will appear in eCollege within a day.

  7. You will be able to see the announcements and find the syllabus and other course materials through the Doc Sharing portion of our eCollege site.

  8. Since eCollege is relatively new to SMC and new to your professor, please be patient with the system and with my use of it.  Hopefully, you will find that this site beneficial over the course of the semester.

  9. The eCollege system has remarkable uptime statistics.  This is a very reliable system hosted on a server in Denver, Colorado.    Thus, the excuse: “I couldn’t get into eCollege” is really not a very viable explanation as to why they could not access our course site.. 

  10. Finally, be wary of the need to use the GO button to choose various categories for Doc Sharing.  You should use the down arrow to find your category choice and click GO to go to that choice.  For example, you will need to do this  when accessing documents from various categories for our course.  The selection of the category does not work without the GO.




Advising and Departmental information.

·         General Information

               Registrar's Website


               LSR Checklist

               Study Abroad

·         Math Majors

                Mathematics Dept Website

                Mathematics Dept Newsletter

                Mathematics Handbook

                Mathematics Traditional and Interdisciplinary Tracks

                Why Study Mathematics?

                Mathematics Major Checklist

                Mathematics Prerequisites Tree

·         Education Majors

                Education Dept. Website

                Elementary Education Handout

                Secondary Education Handout

                Education Dept. Advising Resources

·         Math Career Information

                AMS Mathematics Careers Information

                Employment Information for Mathematical Sciences (EIMS)

                UNC Wilmington Mathematics Careers Information

                BYU We Use Math site

                Research Experiences for Undergraduates (REUs) Summer Programs

                Actuarial Information

                American Academy of Actuaries and Society of Actuaries (SOA)

                Plus Magazine, with a variety of Mathematical career interviews


My papers.

Copies of any of these manuscripts are available upon request.

·        “Vermont Middle Level Mathematics Initiative: Courses and Materials for Mathematics Educators,”                               (with Marny Frantz), In C. Beaver, L. Burton, M. Fung, and K. Kruczek (Eds.), Resources for Preparing Middle School Mathematics Teachers, MAA Notes, Volume 80, MAA: Washington, DC (2013) 169-179.

·        “Closing the Achievement Gap: Systemic Collaboration for Equity in Mathematics,”(with Marny Frantz, Kendra Gorton, Sandy Hepp, and Stephanie Ratmeyer), In J. Bay-Williams and W. Speer (Eds.), Professional Collaborations in Mathematics Teaching and Learning: Seeking Success for All, NCTM 74th Yearbook, NCTM: Reston, VA (2012) 31-45.

·       Multiple Ways to Solve Proportional Reasoning Problems” (with Leslie Ercole and Marny Frantz), Mathematics Teaching in the Middle School, 16:8, (April 2011), 482-490.

·        Math and Bio 2010 (with Joanna Ellis-Monaghan, Zsuzsanna Kadas, and Declan McCabe). MAA FOCUS centerfold poster and accompanying article, 30: 3, (Jun.-Jul. 2010), 15-17.

·        Modeling Seashell Morphology” (with Joanna Ellis-Monaghan, Zsuzsanna Kadas, and Declan McCabe). In UMAP/ILAP Modules 2009: Tools for Teaching (UMAP Modules in Undergraduate Mathematics and Its Applications, Bedford, MA: COMAP), Module 801, (2009) 101-139.

·        “Proportional Reasoning and Probability:  A Professional Development Context for Mathematics Educators” (with Marny Frantz), Connect: Innovations in K-8 Science, Math, and Technology, 23: 2, (Nov.-Dec. 2009), 8-10.

·        Using Checklists in Project Assessment,” MAA FOCUS, “Teaching Time Savers” article, 29: 5, (Oct.-Nov. 2009), 13.

·        Using Mathematically-Rich Tasks to Deepen the Pedagogical Content Knowledge of Primary Teachers,” (with Regina Quinn), In B. Clarke, B. Grevholm and R. Millman (Eds.), Tasks in Primary Mathematics Teacher Education: Purpose, Use and Exemplars, Mathematics Teacher Education, Springer Science+Business Media LLC, 4 (2009) 197-214.

·         “What We Learned…By Organizing and Hosting an MAA Sectional Meeting,” (with Karl-Dieter Crisman, Jo Ellis-Monaghan, Zsu Kadas, Greta Pangborn, and Lloyd Simons) MAA FOCUS, 29: 3, (Apr./May 2009) 16-18

·         “Alumni Profiles of Saint Michael’s College,” Math Horizons, XV: 4, (Apr. 2008) 28-29.

·         “How High? How Long? How Fast?  Modeling with Water Rockets and Calculus,” With J. Ellis-Monaghan, PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies, XVI: 2, (Jun. 2006) 121 - 137.

·         Integrating Exit Questions into Instruction,” NCTM News Bulletin, 42, 3 (Oct. 2005) 6.

·         A House of Your Own: The Mathematics of the American Dream (Part 2), With J. Ellis-Monaghan, The New Jersey Mathematics Teacher, 63: 2, (May 2005) 20-26.

·         Credit Cards and Cars: The Mathematics of the American Dream (Part 1), With J. Ellis-Monaghan, The New Jersey Mathematics Teacher, 63: 1, (January 2005) 16-23.

·        The Lottery: A Dream Come True or a Tax on People Who are Bad at Math, With J. Ellis-Monaghan, PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies, XIV: 4, (December 2004) 303 - 314 .

·         Water Rockets in Flight: Calculus in Action, (with J. Ellis-Monaghan and A. Brizard), UMAP/ILAP Modules 2002-2003: Tools for Teaching, pp. 151-188.  Interdisciplinary Lively Applications Projects are designed to motivate mathematical concepts and skills in the context of a practical application in another field. 

·         Home Sweet Home: A Financial Incentive for the Lower Level Mathematics Course, with Jo Ellis-Monaghan. PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies, XI: 1, (March 2001) 16-26.

·         Microcosm to Macrocosm: Population Models in Biology and Demography,  With Jo Ellis-Monaghan, Tools For Teaching 1999, 39- 80, UMAP unit 777.

·         The Defect Relation for Meromorphic Maps on Parabolic Manifolds, Memoirs of the American Mathematical Society, 139, 165, May 1999. Abstract (postscript file).

·         How Many People are in Your Future? Elementary Models of Population Growth, With Jo Ellis-Monaghan, Updated version republished in Kuntz, S., et. al., "Case Studies in Ecology." New York: WCB McGraw-Hill, (1999). Also published in Making Meaning: Integrating Science through the Case Study Approach to Teaching and Learning, a compilation of case studies published by McGraw-Hill Primis. 42- 80 (1999).

·         Interdisciplinary Population Projects in a First Semester Calculus Course, With Jo Ellis-Monaghan, PRIMUS, IX, 39-55, March 1999.

Here are some interesting sites on mathematics education:

Through the NCTM, a number of resources are available, such as the electronic version of  Principles and Standards of School Mathematics. The Journal of Research in Mathematics Education (JRME) offers other mathematics education links, including those with curricular materials, databases, electronic communities, forums, mailing lists, electronic publications, and professional organizations. Also, Mathematics Education Dialogues is the NCTM-sponsored "open forum for the exchange of points of view about issues in mathematics education."

The Math Forum is an online "Math Education Community Center" with links to Web resources, teaching materials, online discussion groups with other teachers, problems of the week (for grades 3-12), Ask Dr. Math (for math questions), and Teacher2Teacher (for teaching questions). You can browse and/or search K-12 lesson plans according to topic or grade level.

Math Archives offers online teaching materials and links, with such focal areas as Calculus Resources On-line, Graphing Calculators, JAVA and Other Interactive WWW Pages, Lessons, Tutorials, and Lecture Notes, Sites with K-12 Teaching Materials in Mathematics, and Visual Calculus. At its K-12 Teaching Materials, information can be obtained about Lesson Plans, Software, Topics in Mathematics, Contests and Competitions, and other topics of interest. At its Topics in Mathematics, useful information is accessible in such areas as Arithmetic, Algebra, Geometry, Trigonometry, Pre-calculus, Calculus, Statistics, and other areas.

Math Central is "a service for people involved in mathematics education from kindergarten to grade twelve". It is "a meeting place for teachers to share resources, a service to teachers, students and parents who may need an answer to a mathematical question and a facility to allow teachers to carry on a dialogue amongst themselves" (through Teacher Talk).

The Eisenhower National Clearinghouse was created to "identify effective curriculum resources, create high-quality professional development materials, and disseminate useful information and products to improve K-12 mathematics and science teaching and learning." Lesson plans and activities in mathematics are also provided. 

Coordinated by Scott Mandel, Teachers Helping Teachers is intended to is to provide some basic mathematics teaching tips, new ideas in methodologies, and a forum for sharing expertise with colleagues.  Lesson plans and activities for mathematics teachers are also offered.

The Geometry Center at the University of Minnesota offers a repository for materials and projects relationg to the computation and visualization of geometric structures. Links to educational materials can be found at

Created with funding through the NASA Dryden Flight Research Center in Edwards, California, math activities based on NASA projects and aeronautical themes are provided for K-12 teachers.

Though the NSA’s Mathematics Education Partnership Program (MEPP) outreach program promoting mathematics and science education, learning units are available that were created through NSA-sponsored teacher workshops (Summer Institutes for Mathematics and Elementary Teachers). 

Launched in December of 1999, Figure This – Math Challenges for Families campaign is intended to involve parents in their children’s education and help parents encourage their 10 to 14-year old children to do math problems together after school. Sample challenges and an index of the mathematics covered in the challenges can be obtained at the site . Also, there is a related Math is Power site.

Part of a research and development effort to deliver a full range of information resources to educators and students by the University of Kansas, the Explorer can be found at This site offers browsable and searchable curricular materials in mathematics.

At, the Columbia Education Center provides lesson plans created by teachers for use in their own classrooms. Materials are offered for the elementary (K-5), intermediate (6-8), and high school (9-12) levels.

The home page for the University of British Columbia’s Electronic Games for Education in Math and Science or E-Gems, with links to several of their "current research projects", in which they "focus on the human-computer interaction issues associated with learning in an electronic game environment." One of their goals is to "engage learners (9-14) in conscious reflective exploration of math concepts" which can be integrated with other forms of education.

UVM’s MathMoments is intended to develop connections between Vermont school children and the UVM mathematics department. It offers Elementary School, Middle School, and High School "Problems of the Moment". These are intended to encourage work on some fun math problems at various grade levels.

George Pólya’s problem solving tips in “How to Solve It”.

Here are some interesting sites on fractal geometry and chaos theory:

 As described in the last chapter of their text Fractals, Graphics, and Mathematics Education, Michael Frame and Benoit Mandelbrot have created a Panorama of Fractals and Their Uses.  This is intended to be a detailed wordbook or lexicon that orders alphabetically a wide mixture of fractal uses for the classroom.  Frame and Mandelbrot also provide information on their introductory course in fractal geometry at Yale University, including some details on possible fractal software.

Kenneth Monks at the University of Scranton provides a couple of versions of his chapter Fractal Themes at Every Level, as well as his fractal course software and some of his pertinent student papers.

Robert Devaney at Boston University offers an interactive version of his chapter on The Fractal Geometry of the Mandelbrot Set: Periods of the Bulbs.   Also, see his NSF-sponsored Dynamical Systems and Technology Project for ideas on effectively using technology to integrate fractals and chaos into the classroom.  

Cynthia Lanius at Rice University details a fractals unit for elementary and middle school students that “adults are free to enjoy”.  The unit describes various fractal properties and involves creating and interpreting such figures as the Sierpinski triangle, Jurassic Park fractal, and Koch snowflake.

Florida Atlantic University and the Broward County School District offer Pattern Exploration: Integrating Math and Science for the Middle School, containing a rich collection of fractal geometry teacher enhancement materials created through a joint NSF grant. 

Eric Weisstein’s World of Mathematics (a web resource of Wolfram, the maker of the CAS Mathematica) contains information about fractals and links to pertinent topics and resources.  Also, Alexander Walz provides information on fractals and their creation using the CAS Maple V.

Mathematics Archives and the Math Forum both provide a variety of fractal information and links to other online resources.

The Maryland Chaos Group provides a Chaos Gallery with  interesting fractal graphics, chaos animations, and links to pertinent mathematics.

Robert Doerner's Interacting with Chaos deals with non-linear dynamics and chaos and discusses four sample systems, starting with the logistic map and ending with the driven damped pendulum.

Tom Stoppard's Arcadia may also be of interest.

The journal FRACTALS publishes peer-reviewed work in the form of full-length research papers, short communications, reviews of both technical and pedagogical nature, and popular articles.  The table of contents and paper abstracts for previous issues are accessible.

As suggested by a number of the contributors to the text, one particularly effective means to generate fractal graphics is through the freeware fractal generator Fractint, created for IBM PCs and compatible computers.   

There are many pages that offer interesting facts about mathematics and mathematicians. For example, the following sites may be of interest:

Harvey Mudd College's Fun Facts, an archive of interesting mathematical facts.

Information on various mathematical constants.

Jeff Miller's information on various mathematical symbols, mathematical words, and mathematicians on stamps.

Mark Woodard's examples of mathematician quotes.

Keith Lynn's examples of mathematician pictures.

Here are some interesting sites on the history of mathematics:

Some useful links for the History of Mathematics include:

Here are some interesting sites on number theory:


Here are some interesting sites on lotteries and using lotteries in liberal arts mathematics course:

Here are some Saint Michael's-related links.

For more information, you may want to check out these.

Photo Links.

Here are links to several photos of the Saint Michael's campus and buildings:

Campus Aerial     Campus Beauty    St. Edmund's Hall        Library     Founders Hall  Hoehl Welcome Center

Reaching Me.

My office is Jeanmarie 261. My phone number is (USA area code 802) 654-2434. My postal mail address is Box 285, Saint Michael's College, Colchester, VT USA 05439. My email address is [insert first initial and last name]