The Mysterious
Arithmetic of Lexicographic Codes
John H. Conway, Princeton University
Designing the pre-service teacher curriculum to better meet the needs of our future teachers
David Ausubel claimed that “the single most important factor influencing learning is what the learner already knows. Ascertain this and teach him accordingly.” How do we determine what a student already knows and then use the evidence of learning that we collect to improve our teaching and student learning? What classroom experiences foster the development of mathematical thinking—pattern recognition, generalization, abstraction, problem solving, careful analysis, rigorous argument and flexible thinking? Designing a curriculum that builds on students’ prior knowledge to develop these skills effectively and at appropriate levels for all students is one of the biggest and most important challenges we face. To address these questions, we will analyze some mathematical tasks that illustrate how mathematical knowledge for teaching differs from knowledge of mathematics
Role assignments (otherwise known as role
colorings) for graphs were introduced in 1991 by Everett and Borgatti to model
social role in social network theory, and were defined earlier in terms of homomorphisms
and vertex partitions by Sailer in 1978 and White andReitz in 1983. They arise from the idea that if social role
is defined properly, then individuals with the same role will relate in the
same way to other individuals playing counterpart roles. Let r(x) denote the role assigned to vertex
x, where r(x) is an integer between 1 and k.
Let N(x) be the open neighborhood of x.
We say that r defines a role assignment if whenever r(x) = r(y), then
r(N(x)) = r(N(y)). In other words, in a
role assignment, if two individuals have the same role, they are related in the
social network to individuals with the same sets of roles. This concept gives
rise to some challenging mathematical problems with applications ranging from sociology
to ecology to counter-terrorism. We discuss the mathematics of role assignments,
presenting algorithms, approximations, and connections to areas of mathematics
besides graph theory. We also discuss the applications and present some open
problems.
General Background
Twenty-three years teaching mathematics at junior high school, high school, undergraduate and graduate courses.
Primary responsibility: Since 1991, my major focus has been on the mathematical content preparation of teachers.
Primary research interest: the nature of mathematical thinking and how students think about/learn mathematics.
Academic Background
Ph.D., University of Warwick, Coventry, U.K. (Mathematics Education: Dr. David O. Tall, Advisor). 1998.
M.A. DePaul University, Chicago, Il. (Mathematics Education). 1988.
B.A. College of Saint Teresa, Winona, Minn. (Major: Physics, Minor: Mathematics). 1961.
Teaching Experience
William Rainey Harper College, Palatine, Illinois.1990-2002.
School District U-46, Elgin, Illinois (Mathematics). Elgin H. S. 1982-1990; Canton Jr.H. S. 1979-1982.
Awards and Recognition
NSF Two-Year College Award: Exemplary Program in Teacher Preparation
Illinois Community College Board Award for Excellence in Teaching and Learning. .
Presidential Award for Excellence in Mathematics Teaching, Illinois State Award.
Tandy Technology Outstanding Teacher Award.
National/State Committees and Task Forces, incluing:
National Research Council (NRC) Mathematics Learning Study II Planning Committee. Washington, D.C.
CBMS National Summit on Teacher Preparation, Washington, D.C. Steering Committee.
MSEB Task Force: Mathematical Preparation of Elementary School Teachers.
NSF/AMATYC Conference on The Role of Two-Year Colleges in Teacher Preparation.
U.S. Department of Education: The Mathematical Preparation of Elementary and Middle School Teachers.
MAA President’s Task Force on NCTM Standards.
IL Governor’s Conference on the Mathematics Preparation of Teachers.
MAA Committees and Task Forces
Committee on the Undergraduate Program in Mathematics (CUPM). 1999-2005.
Task Force on preparation for Calculus.
Task Force on the first college mathematics course.
First College-level Mathematics Course Working Conference. West Point, NY.; Washington, D.C.
Task Force on Precalculus. Washington, D.C.
Mathematics Preparation of Undergraduates Working Conference. Washington, D.C.
CRAFTY/Michigan State University: Mathematics Preparation of Teachers K-12.
Committee on the Teaching of Undergraduate Mathematics (CTUM). 1996-2002.
Mini-Conference on Professional Development
Subcommittee on Developmental Mathematics.
Committee on Professional Development.
Recent Invited Colloquiums & Presentations include:
MAA
Invited Paper Session on the Use of Hand-Held Technology in College and
University Developmental
Algebra Classrooms. Phoenix, AZ.
MAA (CUPM) Conference on the First College-level Mathematics Course. West Point, NY.
CRAFTY/Michigan State University Conference on Mathematics Preparation of Teachers K-12. E. Lansing, MI.
University of New Mexico. Los Cruces, NM. .
National Summit on Mathematics Education of Preservice Teachers. Washington, D.C.
CBMS National Summit on Mathematics Education of Preservice Teachers. Washington, D.C.
NSF/CUPM Conference on Rethinking Precalculus. Washington, D.C.
NSF-sponsored New Jersey Algebra Initiative. Middlesex County College, Middlesex, NJ.
NSF/AMATYC Conference on the Role of Two-Year Colleges in Teacher Preparation. Washington, D.C.
Phi Theta Kappa/NSF Conference on Teacher Preparation. Washington, D.C.
Arizona State University. Tempe, AZ.
California State University Conference on Developmental Algebra: Planning for Changes. Los Angeles, CA.
University of New Mexico. Los Cruces, NM.
Publications
Textbooks
DeMarois, McGowen, & Whitkanack. Applying Algebraic Thinking to Data. 3rd Edition. (in press). Dubuque, IA: Kendall Hunt Publishers.
2nd Edition, 2001; 1st Edition, 1998; Preliminary Edition , 1996.
DeMarois, McGowen,
& Whitkanack. Mathematical
Investigations: Concepts and Processes for the Introductory Algebra Student.
3rd Edition. (in preparation). Dubuque, IA: Kendall Hunt Publishers.
2nd Edition, 2001; 1st Edition, 1998; Preliminary Edition,
1996.
Papers recently published/in press in refereed publications.
Davis, Gary E. and McGowen, Mercedes A.(2002). Function Machines & Flexible Algebraic Thought. Proceedings of the 26th International Group for the Psychology of Mathematics Education. University of East Anglia, Norwick, U.K. Vol. 2, 273-280.
Davis, Gary E. and McGowen, Mercedes A. (2001). Jennifer’s Journey: Seeing and Remembering Mathematical Connections in a Pre-service Elementary Teachers Course. In M. van den Heuval-Panhuizenm (Ed.) Proceedings of the 25th International Group for the Psychology of Mathematics Education. Utrecht, The Netherlands: Vol. 2, 305-312.
McGowen, Mercedes. (in press). Who are the students who take pre-calculus? MAA Notes: A Fresh Start for Collegiate Mathematics.
McGowen, Mercedes. (in press). Developmental Algebra: The First Mathematics Course for Many College Students. MAA Notes: A Fresh Start for Collegiate Mathematics.
McGowen, Mercedes A. and Davis, Gary E. (2002). Growth and Development of Pre-Service Elementary Teachers’ Mathematical Knowledge. In Denise S. Mewborn (Ed.) Proceedings of the XIX Annual Meeting, North American Chapter of International Group for the Psychology of Mathematics Education. Athens, GA.: Vol. 3, 1135- 1144.
McGowen, Mercedes A. and Davis, Gary E. (2001). Changing Pre-Service Elementary Teachers’ Attitudes to Algebra. In Chick, H., Stacey, K., Vincent, J. & Vincent, J (Eds.) Proceedings of the 12th ICMI Study on The Future of the Teaching and Learning of Algebra. University of Melbourne, Australia: Vol. 2, 438-335.
McGowen, M., DeMarois, P. and Tall, D. (2000). Using the Function Machines as a Cognitive Root. Proceedings of the 22nd Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Tucson, AZ: Vol. 1, 247-254.
Tall, D., Gray, E., Ali, M. Crowley, L., DeMarois, P., McGowen, M., Pitta, D., Pinto, M., Thomas, M., Yusof, Y. (2000). Symbols and the Bifurcation between Procedural and Conceptual Thinking. Canadian Journal of Mathematics, Science and Technology Education, Vol 1, No. 1, 81-104.
Other Professional Activities
Facilitator:
Professional Development Workshops
AMATYC NSF Summer Institutes Enumclaw, WA.; Grand Rapids, MI.
Elmhurst School District Middle Schools Workshop. Elmhurst, IL.
New Jersey NSF Algebra Initiative Workshop. Middlesex County College, Middlesex, NJ.
University of Nebraska at Omaha: Omamha, NB.
University of Illinois-Community College Collaborative for Excellence in Teacher Preparation Summer Institutes. Chicago, IL.
Presentations & Panels
AMS/MAA Joint Meetings, Psychology of Mathematics Education International Group for the Mathematics Education (PME); AMATYC, ICTCM, NCTM, ICME (Spain), regional, state and and local conferences.
FRED S. ROBERTS received his A.B. in mathematics from
Dartmouth
College in 1964 and
his M.S. and Ph.D. in mathematics from Stanford
University in 1967
and 1968. After a postdoctoral
fellowship at the
University of
Pennsylvania, he joined the professional staff of the
RAND Corporation in
1968. He was a postdoctoral fellow at
the
Institute for
Advanced Study in Princeton in 1971-72, and then joined
the faculty at
Rutgers University, where he is currently a Professor
of Mathematics and a
Fellow of RUTCOR, the Rutgers Center for
Operations
Research. At Rutgers, he is a member of
five graduate
faculties, in
Mathematics, Operations Research, Computational
Molecular Biology,
BioMaPS (Interdisciplinary Ph.D. Program at the
Interface between
the Biological, Mathematical, and Physical
Sciences) and
Education. He has held visiting
positions at Cornell
University, AT\T
Bell Laboratories and Northeastern University.
At
Rutgers, he has
chaired the Applied Mathematics Committee and has
been Director of the
Rutgers Center for Operations Research.
In
January 1996, he was
named the Director of DIMACS, the Center for
Discrete Mathematics
and Theoretical Computer Science.
DIMACS, with
administrative offices
at Rutgers, was founded as a National Science
Foundation Science
and Technology Center and is a joint project of
AT&T Labs -
Research, Bell Labs/Lucent Technologies, Telcordia
Technologies
(formerly Bellcore), NEC Research, Princeton University,
and Rutgers. Dr. Roberts had previously been the
Associate Director
of the center and
was Acting Director on two occasions. He has been
asked by Rutgers to
chair the Rutgers University Homeland Security
Research Initiative,
coordinating homeland security research efforts
on all Rutgers'
campuses.
Professor Roberts' major research interests
are in
mathematical models
in the social, behavioral, biological, and
environmental
sciences and of problems of communications and
transportation;
graph theory and combinatorics and their
applications;
measurement theory; utility, decisionmaking, and social
choice; and
operations research. His first book,
Discrete
Mathematical Models,
with Applications to Social, Biological, and
Environmental
Problems, has been called a classic in the field, and
was translated into
Russian in 1986. He has also authored
three
other books: Graph
Theory and its Applications to Problems of
Society; Measurement Theory, with Applications to Decisionmaking,
Utility, and the
Social Sciences; and Applied Combinatorics.
Professor Roberts is
also the editor of fifteen other books covering
such varied topics
as energy modeling, reliability of computer and
communication
networks, mathematical psychology, computational
biology, and
precollege discrete mathematics, and the author of some
150 scientific
articles.
Professor Roberts has been a leader in
focusing the
mathematical
sciences community on outreach to areas outside of
mathematics. He is currently on the editorial board of
six
scientific journals
in discrete mathematics, mathematical and
computer modeling,
mathematical social sciences, computational
biology, and
mathematical psychology. He has been an
organizer of 44
scientific
conferences, including the 6-year DIMACS ``Special Year"
on Mathematical
Support for Molecular Biology, during which he was
instrumental in
fostering lasting collaborations between mathematical
and biological
scientists. He recently launched the
DIMACS Special
Focus on
Computational and Mathematical Epidemiology. Roberts has
been an active
member of a variety of professional organizations, and
has held such
positions as Secretary, Vice President, and
President-Nominee of
SIAM (The Society for Industrial and Applied
Mathematics),
Secretary and Member of the Board of the Societal
Institute for the
Mathematical Sciences, member of the COMAP
Consortium Council,
member of the Committee on Applications of
Mathematics of the
National Research Council, and member of the Board
of Visitors of the
Office of Naval Research.
Professor Roberts is a frequent lecturer all
over the world.
Among his more
noteworthy talks have been a 14-lecture series to Le
Troisieme Cycle
Romand in Operations Research in Grimentz,
Switzerland, an
address to the Beijing Mathematical Society, a talk
at the International
Congress on Mathematics Education, and an
address to the Royal
Nepal Academy of Science.
Among his honors and awards, Professor
Roberts has been the
CBMS-NSF Research
Conference Lecturer at Colby College, the
Outstanding
Mathematician Lecturer at the University of New Haven,
and holder of the
Robert G. Stone Chair at Northeastern
University. He has
been the recipient of a Humboldt Fellowship (which
he declined), a
University Research Initiative Award from the Air
Force Office of
Scientific Research, the Commemorative Medal of the
Union of Czech
Mathematicians and Physicists, and the Distinguished
Service Award of
ACM-SIGACT (Association of Computing Machinery
Special Interest
Group on Algorithms and Computation Theory). He
recently received
the National Science Foundation Science and
Technology Centers
Pioneer Award in a ceremony at NSF conducted by
NSF Director Rita
Colwell.