Jeff Johannes                    Head picture

Picture with Hopf Link     Picture at Bryce Canyon  


Teaching (Day / Time Grid Schedule)

Monday 5-6p in South 309
Tuesday 1-2p in South 309
Wednesday 12:30-1:20p in South 309
Thursday 8-9p in South 336
Friday 2:30-3:30p in South 309

In this course we develop an adult-level perspective and insight into the nature of mathematics taught in elementary school.  In 140 we focus on arithmetic.  Our two themes for this course are the nature of mathematics and the practice of problem solving (the act of doing mathematics).  Students often work together to develop and deepen their understanding of the mathematical concepts.  Frequently this involves the use of manipulatives to incorporate a tangible explanation for the methods studied.  This course is the next step toward a career of explaining concepts (here mathematics) that you have taken for granted before.  

I will assume that university students do not need to be taught elementary school mathematics.  We will not be reviewing, but rather justifying that material.  For a reference on the content of elementary school mathematics, here are the New York State Standards for Mathematics.


More Mathematics for Elementary Education
M 3:30 - 4:20p:  Fraser 116
Prerequisite:  MATH 140 and MATH 141
For anyone who wants to continue exploring the mathematics for Elementary Education after completing 140 and 141, all activites, including bringing and directing your own. 


Introduction to Mathematical Proof
MWF 1:30 - 2:20p:  Fraser 119
Prerequisite:  MATH 222
It is often said that mathematics is a language.  In this class you will begin to learn to speak this language.  Just like in an introductory language course, we will start with the most fundamental concepts and grammar rules.  After we have some familiarity with the language of formal mathematics, we will practice this language in the setting of counting problems of different types.  More like an advanced language class, it will not suffice just memorizing the vocabulary (in fact, hopefully we can keep vocabulary to a minimum), but rather you will be required to understand and speak clearly in this language.  The material learned here will help you understand the mathematics you read and clarify the mathematics you write.  Because we are learning how to write mathematics, exposition will also be a component in your evaluation.

Real Analysis
MWF 11:30a - 12:20p:  South 328
Prerequisite:  MATH 223 and MATH 239

What if you look back at Calculus I - II and focus all your attention on the proofs and reasons?  This is what we do in Real Analysis.  If you paid attention to all the reasons when you took the courses originally, this will all be familiar.   If not, you will wish you had.  We will start with "what is a real number" and end with the Fundamental Theorem(s) of Calculus.

  • Past Classes Taught at Geneseo
  • INTERDEPARTMENTAL 101:  First Semester Seminar - Does Anybody Really Know What Time, Day, Month or Year It Is?
    INTERDEPARTMENTAL 101:  First Semester Seminar - Infinity, the Fourth Dimension and Geneseo
    INTERDEPARTMENTAL 101:  First Semester Seminar - Reality Bites
    INTERDEPARTMENTAL 301:  Topics in Secondary Mathematics
    MATHEMATICS 101: Welcome to the Math Major
    MATHEMATICS 104: Mathematical Ideas
    MATHEMATICS 113:  Finite Mathematics
    MATHEMATICS 140:  Mathematical Concepts for Elementary Education I
    MATHEMATICS 141:  Mathematical Concepts for Elementary Education II
    MATHEMATICS 160:  Media Statistics
    MATHEMATICS 221:  Calculus I
    MATHEMATICS 222:  Calculus II
    MATHEMATICS 223:  Calculus III
    MATHEMATICS 228:  Calculus II for Biologists
    MATHEMATICS 233:  Elementary Linear Algebra
    MATHEMATICS 239:  Introduction to Mathematical Proof
    MATHEMATICS 301:  Mathematical Logic
    MATHEMATICS 315:  Combinatorics
    MATHEMATICS 324:  Real Analysis
    MATHEMATICS 330:  Abstract Algebra
    MATHEMATICS 333:  Linear Algebra II
    MATHEMATICS 335:  Geometry
    MATHEMATICS 338:  Topology
    MATHEMATICS 348:  Oral Presentation & Research Seminar
    MATHEMATICS 350:  Vector Analysis
    MATHEMATICS 371:  Complex Analysis
    MATHEMATICS 381 (380):  Galois Theory
    MATHEMATICS 380:  Differential Geometry
    MATHEMATICS 380:  Symmetry
    MATHEMATICS 380:  Visual Mathematics
    MATHEMATICS 390:  History of Mathematics
    MATHEMATICS 470:  History and Fundamental Concepts of Mathematics


    I am no longer chair of the department's curriculum committee.  I am also the chair of the colloquium committee for 2002-2007 and 2008-.  Please visit our schedule of colloquium talks for this year.  I am furthermore serving on the graduate advisement committee and overseeing the mathematics GRE subject exam preparations #2 #2and#4answers #3 #4 (practice).  

    Professional Activities

    I am an active member of the Mathematical Association of America.  In particular, I am the Representative to the Congrees, liaison coordinator and the co-chair of the Program Committee for the Seaway Section.  I am also a member of the American Mathematical Society. 

    Areas of research

    Current projects

    I am currently pursuing several research projects.  The newest project is translating an 1820 article by Simeon Poisson about complex line integrals.   I am also researching the first 103 years of history of mathematics at Geneseo.  Another project is an exploration of the role of Euclid's Fourth Postulate:  "All right angles are equal."  The older of these projects consists of  investigating how the Casson-Walker-Lescop 3-manifold invariant changes when modifying the presenting link for a 3-manifold. This project has evolved into studying questions of the Ohtsuki invariants of rational homology spheres, and questions of the space of finite type invariants for links of three or more components.  Another long-term project is to study symmetries of links.  In particular I am examining a refinement of unlinking number accounting for which components are involved in each of the crossing changes, a so-called coloured unlinking number. Finally, I am examining comparisons and connections between mathematician Evariste Galois and composer Hector Berlioz.

    Areas of interest

    Prepared Talks


    A Type 2 Polynomial Invariant of Links Derived from the Casson-Walker Invariant, Journal of Knot Theory and Its Ramifications, Vol. 8, No. 4 (1999) 491-504.

    The Casson-Walker-Lescop invariant and link invariants, Journal of Knot Theory and Its Ramifications, Vol. 14, No. 4 (2005) 425-433.

    Bandpass moves and the Casson-Walker-Lescop invariant, New York Journal of Mathematics, Vol. 10 (2004), 231-247.

    Modern Geometry and the End of Mathematics, in MAA notes #68 From Calculus to Computers: Using the Last 200 Years of Mathematics History in the Classroom, 2005

    Academic history

    The Past

    Project NExT

    Seaway NExT Steering Committee Chair 2008-
    Project NExT 2000-2001 fellow (aka "Gold Dot" ). 
    Co-organized "Creating and Teaching Interdisciplinary Courses" session at Joint Meetings in New Orleans.  January 11, 2001.


    AMS national meeting in New Orleans, January 10 - 13, 2001.

    "Colored Unlinking", special session on geometry and topology of low dimensional manifolds.
    "Temperament of Mind: Hector Berlioz and Evariste Galois", contributed paper.

    AMS sectional meeting in Las Vegas, April 21 - 22, 2001.
    Co-organized special session on the topology of links.

    MAA national MathFest in Madison, August 2 - 4, 2001.
    "Infiltrating Preservice Elementary School Mathematics with History", contributed paper session on the use of history in the teaching of mathematics.

    MAA national MathFest in Burlington, VT, July 31 - August 4, 2002

    "Modern Geometry", contributed paper session on the use of recent history of mathematics in teaching.
    "Welcome to Mathematics:  A Cornerstone Experience", contributed paper session on the role of proof in teaching mathematics.

    MAA national MathFest in Boulder, CO, July 30 - August 2, 2003

    "Days are Numbers:  The Mathematics of the Calendar", general contributed paper session.
    "Honesty is the Best Philosophy", contributed paper session on innovations in quantitative literacy. 

    MAA national MathFest in Providence, RI, August 11 - 15, 2004

    Co-organised session on "Extracurricular Mathematics"

    MAA national MathFest in Albuquerque, NM, August 3 - 6, 2005

    "Why Are We Math Majors?", contributed paper session on current issues in mathematics education courses.
    "Greatest Hits of Mathematics", general contributed paper session.

    MAA national MathFest in Knoxville, TN, August 9 - 12, 2006

    "A Little Moore Probability", contributed paper session on current issues in mathematics education courses.
    "Kaleidoscope", general contributed paper session.

    MAA national MathFest in San Jose, CA, August 2 - 5, 2007

    "Where are we from? - An entire class project", contributed paper session on getting students to discuss and to write about mathematics.
    "Four dimensional tic-tac-toe on a torus - the game of SET", general contributed paper session

    5th annual Midwest Undergraduate Mathematics Symposium at Simpson College, Indianola, IA.  April 2008

    "Game:  SET - Math", invited plenary address

    MAA national MathFest in Madison, WI, July 30 - August 2, 2008

    "Vedic Decimal Patterns", contributed paper session on interesting topics in history of mathematics that enchance the teaching and learning of mathematics.
    "Place Value Airthmetic via Polynomials", contributed paper session on innovations in mathematics education.

    MAA national MathFest in Portland, OR, August 5 - 8, 2009

    "Counting Student Groupings", contributed paper session on fascinating examples from combinatorics, number theory, and discrete mathematics.
    "The Mathematical Fiction and Exposition of Rudy Rucker", general contributed paper session

    MAA national MathFest in Pittsburgh, PA, August 4 - 7, 2010

    "Euclid's Neglected Postulate", contributed paper session on history of mathematics uses in the classroom.
    "Four different experiences", contributed paper session on first year seminar / experience mathematics courses.

    MAA national MathFest in Lexington, KY, August 3 - 6, 2011

    "A Mile Wide and an Inch Deep - 4000 BCE to 1950", general contributed paper session.
    "Probability and Strategy in Farkle", contributed paper session on recreational mathematics.  

    MAA national MathFest in Madison, WI, August 1 - 4, 2012

    "The Flat Earth: The Mathematics of Mapmaking", general contributed paper session.
    "Editing Episodes -  Magazine, Chapter, and More:  Diverse Writing and Editing Projects", contributed paper session on incorporating writing and editing into mathematics classes. 

    MAA national MathFest in Hartford, CT, July 31 - August 3, 2013

    "Tinkering with a Mathematical Goldmine" about this website, contributed paper session on recreational mathematics.
    "Spherical Triangle Area and Angle Sum", contributed paper session on geometry proofs.
    MOVES conference at Museum of Mathematics, NYC, August 4 - 6, 2013

    "Game:  SET - Math" talk
    "Kaleidoscopic Mathematics" family activity

    MAA national MathFest in Portland, OR, August 6 - 9, 2014

    "On the Other Side of the World, or Right Next Door?",  general contributed paper session.
    "Geometry Knows Topology:  The Gauß-Bonnet Theorem", contributed paper session on geometry proofs.

    MAA national MathFest in Washington, DC, August 5 - 8, 2015

    "Kaleidoscopic Journey", contributed paper session on Mathematics and Art.

    MAA national MathFest in Columbus, OH, August 4 - 6, 2016

    "Take a Ride on the Parallel Transport", general contributed paper session.

    MAA national MathFest in Chicago, IL, July 27 - 29, 2017

    "Galois Theory for High School Students", contributed paper session on Exploring Zeroes of Polynomials.

    MAA national MathFest in Denver, CO, August 2 - 4, 2018

    "Writing Across the Mathematics Curriculum", contributed paper session Developing Mathematical Thinking and Communication Through Writing.

    Finger Lakes Master Teacher Program, Geneseo, NY, February 9, 2019

    "Episodes in Multicultural Mathematics" workshop.

    MAA national MathFest in Cincinnati, OH, July 31 - August 3, 2019

    "Enhancing Classes with In-Class Group Work and Presentations" panel presentation for Project NExT.
    "19th Century Normal Mathematics", contributed poster.

    Finger Lakes Master Teacher Program, Geneseo, NY, January 11, 2020

    "Mathematics and Music" workshop.

    MAA Seaway Section Panel, October 8, 2021

    "Coordinating Groupwork Remotely" panelist.

    MAA Seaway Section Panel, February 24, 2021

    "A Seaway Conversation on the Course - History of Mathematics" panelist.

    MAA national virtual MathFest, August 4 - 7, 2021

    "Needlepoint, Toplogy, Geometry, and Beyond", contributed paper session MathArt - ArtMath

    MAA national MathFest in Philadelphia, PA, August 3 - 6, 2022

    "Poisson's exploration of complex line integrals", contributed paper session Complex Analysis: History, Pedagogy, Innovation, and Research

    MAA national MathFest in Tampa, FL, August 2 - 5, 2023

    "Analsis vs. Algebra:  What is Abstract?", contributed paper session Analysis and Algebra:  Convincing Students They Can Do It

    The Future

    Among many other things, perhaps a personal web site.

    Return to:  Mathematics Department,   SUNY Geneseo.