T HIS CALENDAR lists meetings of the Society which have been approved by the Council at which papers
may be presented. Programs of Annual Meetings appear in the Notices and on the AMS website; programs
for sectional meetings appear on the AMS Web pages in the Meetings & Conferences section, and are
electronically archived in the Notices section on the AMS website.
M EET ING #
976
977
978
979
980
981
982
983
DAT E
May 3{ 5, 2002
June 12{ 16, 2002
June 20{ 22, 2002
October 5{ 6, 2002
October 12{ 13, 2002
October 26{ 27, 2002
November 9{ 10, 2002
January 15{ 18, 2003
(109th Annual Meeting
PLACE
Montr¶
eal, Qu¶
ebec, Canada
Pisa, Italy
Portland, Oregon
Boston, Massachusetts
Madison, Wisconsin
Salt Lake City, Utah
Orlando, Florida
Baltimore, Maryland
Abstracts of the 5002nd Meeting.
0
ABST RACT
DEADLINE
March 12
Expired
Expired
August 13
August 20
September 4
September 17
October 1
ABST RACT
ISSUE
Vol. 23, No. 3
None
Vol. 23, No. 3
Vol. 23, No. 4
Vol. 23, No. 4
Vol. 23, No. 4
Vol. 23, No. 4
Vol. 24, No. 1
Invit ed Addresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Creat ive Use of Technology in Teaching Mat hemat ics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
E-Learning Mat hemat ics Courses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
Independent Learning Experiences for Undergraduat es in Mat hemat ics . . . . . . . . . . . . . . . . .
7
T he Use of Recent Hist ory of Mat hemat ics in Teaching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
Recreat ional Mat hemat ics in t he Classroom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
Using Popular Cult ure in t he Mat hemat ics and Mat hemat ics Educat ion Classroom . . . . .
13
RUME SIGMAA Session . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
Innovat ive Met hods in Courses for Non-Majors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
Enlivening Mult ivariat e Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
T he Role of Proof in Teaching Mat hemat ics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
Mat h & Society . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
General Cont ribut ed Paper Session . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
Discret e Met hods in Geomet ry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
Hist ory of Mat hemat ics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
T he present er of each t alk is indicat ed by an ast erisk (* ) in t he abst ract .
1
5002-A 0-3
L aszl o L ovasz* (l ovasz@mi cr osof t . com), M icrosoft Research, One M icrosoft Way,
Redmond, WA 98052. Graphs, Ei genvalues, and Geometr ic Representati ons Par t I : Graphs
and Ei genvalues.
A graph can be described by it s adjacency mat rix. T his simple connect ion bet ween graph t heory and linear
algebra becomes much deeper when we discover t hat purely graph-t heoret ic propert ies of graphs and purely
linear algebraic propert ies of t he mat rix (like it s eigenvalues) are closely connect ed. (Received A pril 30, 2002)
5002-A 0-4
L aszl o L ovasz* (l ovasz@mi cr osof t . com), M icrosoft Research, One M icrosoft Way,
Redmond, WA 98052. Graphs, Ei genvalues, and Geometr i c Representati ons Par t I I :
Geometr ic representati ons of graphs.
T here is a t hird unexpect ed connect ion bet ween eigenvalues and geomet ric represent at ions: adjacency mat rices
lead t o embeddings of graphs in Euclidean spaces, re° ect ing graph t heoret ic propert ies. For example, for planar
graphs, t his met hod gives represent at ions as polyt opes, and all polyt opal represent at ions can be obt ained t his
way. (Received A pril 30, 2002)
5002-A 0-5
L aszl o L ovasz* (l ovasz@mi cr osof t . com), M icrosoft Research, One M icrosoft Way,
Redmond, WA 98052. Graphs, Ei genvalues, and Geometr i c Representati ons Par t I I I :
Ei genvalues and geometr i c representati ons.
Various geomet ric represent at ions of graphs have been st udied and used in a variety of ways: for example,
planar graphs can be represent ed by convex polyhedra. Ot her represent at ions (like ort hogonal represent at ions)
have applicat ions t o graph algorit hms t hat t est propert ies of graphs which have not hing t o do wit h geomet ry.
(Received A pril 30, 2002)
5002-A 0-132
C ol i n A d am s* , W illiams College. \ Blown away: W hat knot to do when sai li ng" by Sir
Randolph \ Ski pper " Bacon I I I .
Being a t ale of advent ure on t he high seas involving great risk t o t he t ale t eller, and how an underst anding of
t he mat hemat ical t heory of knot s saved his bacon. (Received June 05, 2002)
5002-A 0-133
A l an E d el m an * , M assachuset t s I nst it ut e of Technology. W hy are random matr i ces cool?
Come and ¯nd out . (Received June 05, 2002)
5002-A 0-134
C at h er i n e G ol d st ei n * , CNRS-Universit y of Paris Sud. Fer mat' s ar thmetic i n
seventeenth-centur y context.
Several enigmas surround t he life and work of Pierre Fermat . Present ed as a t ypical problem-solver by some,
he is considered by ot hers t o be t he founder of modern number t heory. He is supposed t o have had a disdain
for proofs, but is best known for having claimed one. He was an amat eur, but one who seemed at ease wit h
all t he mat hemat ical issues of his t ime. Focusing on arit hmet ic, t he t alk will explain how a cont ext ualizat ion
of Fermat ' s work wit hin t he mat hemat ical craft and scient i¯c milieu of his t ime can resolve t hese apparent
paradoxes. (Received June 05, 2002)
5002-A 0-135
J i m L ew i s* , Universit y of Nebraska at L incoln. T he mathemati cs education of teacher s.
One year ago, t he Conference Board of t he M at hemat ical Sciences (CBM S) released T he M at hemat ical Educat ion
of Teachers. T his report is t he mat hemat ics communit y' s response t o a series of report s t hat call for improving
mat hemat ics educat ion in our K -12 schools and which argue t hat invest ing in good t eachers is t he key st ep in
improving K -12 educat ion in A merica. A re we making progress in t he st ruggle t o improve t he mat hemat ical
educat ion of t eachers? We will consider t his quest ion from bot h a nat ional and local point of view. (Received
June 05, 2002)
5002-A 0-136
R ob i n L ock * , St . L awrence Universit y. Fun and games for teachi ng statisti cs.
How can we capt ure t he at t ent ion of st udent s who t hrive on video games and compet it ive sport s? Perhaps
by int roducing a bit of gaming and compet it ion int o our regular classroom rout ines. W hile we can' t expect t o
mat ch t he fascinat ion of a sophist icat ed elect ronic game or t he excit ement of t he conference playo®s, we might
liven up a daily class hour and sneak in some import ant st at ist ical ideas along t he way. We describe several
games and compet it ive act ivit ies t hat can be used t o st imulat e int erest and help st udent s underst and concept s
2
CREAT IV E USE OF T ECHNOL OGY IN T EACHING M AT HEM AT I CS
3
such as correlat ion, con¯dence int ervals, least squares lines, propert ies of est imat ors and experiment al design.
(Received June 05, 2002)
5002-A 0-137
I som H er r on * , Rensselaer Polyt echnic I nst it ut e. Random walk, di ®usi on, and energy
decay.
Random walk, a discret e process, has a st riking connect ion t o Brownian mot ion which is governed in t he cont inuous limit by t he di®usion equat ion. T he energy is one propert y of a solut ion t o t his equat ion. T he energy
rat e of decay will be examined as a means of charact erizing solut ions t o t he equat ion. T he met hods employed
have import ant applicat ions in cont inuum mechanics. (Received June 05, 2002)
5002-A 0-138
Fr an k M or gan * , Depart ment of M at hemat ics and St at ist ics, W illiams College,
W illiamst own, M A 01267. Soap bubbles: Open problems.
Despit e much recent progress by many mat hemat icians, including undergraduat es, many simple open problems
remain. T his present at ion will also include a lit t le cont ext wit h demonst rat ions, explanat ions, and prizes. No
prerequesit es. (Received June 05, 2002)
5002-A 0-139
A n n i e Sel d en * , Tennessee Technical Universit y. T wo research tradi ti ons separated by a
common subj ect: M athematics and mathemati cs educati on.
\ T here are no proofs in mat hemat ics educat ion." W hile t his is t rue, claims are made in mat hemat ics educat ion
research and evidence is provided for t hem. I n t his t alk, I will explore t he nat ure of such research, t he kinds
of claims and evidence, and what such research might have t o o®er t eachers of mat hemat ics, especially at t he
undergraduat e level. A long t he way, I will point out di®erences bet ween t he ways research is done in t he t wo
¯elds. (Received June 05, 2002)
5002-A 1-20
M i ch ael R . H u b er * (mi chael - huber @usma. edu), Depart ment of M at hemat ical Sciences,
Unit ed St at es M ilit ary A cademy, T hayer Hall Room 223, West Point , New York 10996.
Getti ng SM A RT About Technology i n the Calculus Classroom. Preliminary report .
At t he Unit ed St at es M ilit ary A cademy, one of our goals is t o develop st udent s int o crit ical t hinkers and compet ent problem solvers. We have been using t he SM A RT Board, an int eract ive whit eboard, for t he past year in
our mat hemat ics classrooms, in bot h t he core and t he elect ives courses. T his t alk will o®er several examples of
real world problems (such as modeling pollut ion in a reservoir wit h di®erent ial equat ions, solving relat ed rat es
and opt imizat ion problems in di®erent ial calculus, physics applicat ions of int egral calculus) and show how t he
SM A RT Board is used t o link I nt ernet sit es, comput er algebra syst em out put (such as M at hemat ica and M aple),
and course web pages t o enhance st udent learning. St udent s can concent rat e on problem solving in t he classroom
and re-read course not es any t ime out side of t he classroom, in any order (looking at all not es, reviewing a page
of not es from t he end of class, et c.). At USM A , we have st udent s work in groups almost every day, solving
problems on t he SM A RT Board. T hey t hen don' t have t o wast e t ime in class copying down solut ions, as t he
solut ions will be available aft er class. T his allows us t o work more problems. St udent feedback from SM A RT
Board use will be present ed, as well as inst ruct or feedback. (Received M ay 17, 2002)
5002-A 1-25
P et er G av i n L aR ose* (gl ar ose@umi ch. edu), Depart ment of M at hemat ics, t he University
of M ichigan, East Hall, 525 E. Universit y, A nn A rbor, M I 48109-1109, and R ob er t
M eggi n son (meggi n@mat h. l sa. umi ch. edu), Depart ment of M at hemat ics, t he Universit y of
M ichigan, East Hall, 525 E. Universit y, A nn A rbor, M I 48109-1109. I mplementation and
A ssessment of On-li ne Gateway Testi ng.
Gat eway t est s provide a means of assuring t hat st udent s in reformed precalculus and calculus courses acquire
t he algebraic and comput at ional skills needed in courses following t hese, while allowing t he focus of t he course t o
be on t he concept ual underst anding of mat erial int rinsic t o t he reformed courses. However, logist ical di± cult ies
plague t he pencil-and-paper administ rat ion of t hese t est s, signi¯cant ly decreasing t heir usefulness. To address
t hese di± cult ies we have implement ed an on-line version of t hese t est s which allows st udent s t o pract ice t he
skills on t heir own and t ake t he t est for a grade in a proct ored environment . We report on t he det ails of
our implement at ion and assessment of t he syst em, including our administ rat ion procedures and lab use. Our
assessment included evaluat ion of st udent s' percept ions of t he e®ect iveness of t he program and t heir act ual skill
acquisit ion, and show t hat t he on-line gat eway t est ing is successful in developing st udent s' skills. (Received
M ay 20, 2002)
4
CREAT IV E USE OF T ECHNOL OGY IN T EACHING M AT HEM AT I CS
5002-A 1-28
B ar t D . St ew ar t * (ab8146@usma. edu), 3078 B Webb Place, West Point , NY 10996. Don' t
K now Java? You Can Sti l l Excel: On Bui ldi ng Complete I nteractive Envi ronments.
Preliminary report .
W it h modern advances, t echnology cont inues t o weave it self wit hin our classrooms. T hese t echnological advances,
while cert ainly able t o enhance a st udent ' s abilit y t o learn object ives and concept s, come wit h an associat ed cost
- speci¯cally, t he responsibility of learning some non-user friendly comput er soft ware. I n an e®ort t o reduce t he
soft ware learning curve, it is possible t o creat e a t ot ally int eract ive environment t hat rivals some popular Java
A pplet s in mere minut es using not hing more t han M icrosoft O± ce.
W hy an int eract ive environment ? Dynamic environment s promot e opport unit y for self-explorat ion and discovery. T he explorat ion fost ers a deeper underst anding of mat erial rat her t han simply rest ing on t he periphery.
I n pract ice, t hese simple-t o-make environment s have proven more t han wort hwhile. W it h t he " point and
click" t echnology, my st udent s were able t o invest igat e a myriad of Discret e Dynamical Syst em behaviors,
bot h linear and nonlinear, t hrough observing t he e®ect of varying paramet ers in a numerical and graphical
fashion simult aneously. Creat ing t his t ype of environment not only added new dimension t o my st udent s' focus,
creat ivit y, and willingness t o explore, but it also present ed an easy, adapt able t ool for all of us. (Received M ay
21, 2002)
5002-A 1-70
C ar ol G . C r aw for d * (cgc@usna. edu), M at hemat ics Depart ment , U.S. Naval A cademy,
572 Holloway Road, A nnapolis, M D 21402. I nteracti ve W eb Based Labs for the U.S. Naval
Academy - Real W or ld A ppli cations Desi gned wi th Java for the 3-semester Calculus
Sequence.
T he aut hor present s a package of int eract ive, web-based calculus labs designed for t he 3-semest er calculus
sequence at T he Unit ed St at es Naval A cademy. Joint work wit h M ark M eyerson, t he aut hor developed t hese
modules over a two-year period as a curriculum proj ect sponsored by t he A cademy. T he aut hors ¯rst present ed
t hese labs at t he 1999 A M S-M A A Nat ional M eet ings in Washingt on, DC.
T hese labs were designed using Java applet s and t he web. Nine" labs" were produced, each one cont aining a
Java language applet t hat simulat ed some real world (milit ary or indust rial) applicat ion of calculus. T he applet s
are int eract ive, allowing t he user t o experiment wit h adjust ing paramet ers t o see how t hat a®ect s t he result s.
Each lab consist s of t he int eract ive applet , a self-quiz, a mat h review and addit ional applicat ions.
T he aut hor will present how t hese labs are current ly being int egrat ed int o t he 3-semest er calculus curriculum
at t he U.S. A cademy. T he discussion will also include comment s on how t hese labs can be linked t o as part of
an " elect ronic syllabus" in calculus wit h links t o mult iple sources for t he st udent . A nyone on t he web can experience t hese labs at :ht t p:/ / mat hweb.mat hsci.usna.edu/ facult y/ meyersonmd/ labs/ index.ht ml (Received M ay
30, 2002)
5002-A 1-71
M i ch el l e M . M cC assey * (am5410@usma. edu), T hayer Hall, Rm 227B, 646 Swift Road,
Unit ed St at es M ilit ary A cademy, West Point , NY 10996, and A l ex J . H ei d enb er g
(aa5178@usma. edu), T hayer Hall, Rm 225, 646 Swift Road, Unit ed St at es M ilit ary
A cademy, West Point , NY 10996. I ntegration Of Laptops I nto T he Classroom A t USM A .
Preliminary report .
Port able Not ebook Comput ers provide an even great er t echnological resource t hat facilit at es t he need t o once
again reexamine our goals for educat ion. St orage and organizat ion coupled wit h powerful graphical, analyt ical,
and numerical capabilit ies allow st udent s t o enhance t heir learning. Beginning wit h t he Class of 2006, cadet s at
USM A will be issued lapt op comput ers rat her t han deskt ops. T his t ransit ion t o lapt ops has generat ed changes
in t he way USM A t hinks about t he t eaching and assessment of learning in mat hemat ics. A pilot course in
t he Spring 2002 t erm experiment ed wit h t he use lapt ops in t he classroom. T he pilot class int roduced discret e
calculus concept s in preparat ion for t he Fall 2002 t erm when 1000 new cadet s will t ake t he same course. T his
t alk will int roduce some of t he changes made t hus far t o incorporat e not ebook comput ers. We will conduct a
demonst rat ion of several creat ive exercises t hat allow t he st udent exposure t o mat hemat ical modeling concept s
wit h t he abilit y t o explore how t hey may det ermine pat t erns and connect ions which facilit at e t he process of
const ruct ive underst anding. A ddit ionally, since it is well known t hat what we t est is what st udent s learn, we will
provide insight int o how our met hods of assessment will adapt t o t he changes in t echnology. (Received M ay
30, 2002)
CREAT IV E USE OF T ECHNOL OGY IN T EACHING M AT HEM AT I CS
5
5002-A 1-84
C l ar k P. W el l s* (wel l sc@gvsu. edu), 2267 M ackinac Hall, Grand Valley St at e Universit y,
1 Campus Drive, A llendale, M I , 49401. T he M athemati cs of M ove: Usi ng M aple and
Netscape to Solve L i near Algebrai c Problems in Chemistr y W eb Page Desi gn. Preliminary
report .
T he shape of a molecule plays a large role in it s chemical act ivit y, so underst anding t he geomet ry of a molecule
is very import ant . I n chemist ry, just as in mat hemat ics, discovery helps st udent s \ own" knowledge. W hen a
colleague from chemist ry approached me wit h a problem involving manipulat ing models of molecules via t he
Web, I saw an opport unit y t o combine discovery in bot h chemist ry and mat hemat ics.
M DL Soft ware' s Chi me plug-in for Net scape allows t he user t o manipulat e a molecular model, an almost
\ hands-on" experience. T he problem was t o ¯nd a way t o provide a st udent wit h a \ hint " | moving t he molecule
from it s current posit ion t o a prede¯ned posit ion. T he di± cult ies were t hat Chi me:
1. only accept s inst ruct ions t o rot at e in t he form of rot at ions about coordinat e axes in ¯xed order,
2. report s posit ions in t he opposi te order of t hat needed, and
3. animat es rot at ions by assuming t hat composit ion of rot at ions is commut at ive.
Not only are t he problems less t rivial t han t hey appear at ¯rst , but t he connect ion t o chemist ry and comput ers
provide an applied set t ing for some import ant point s arising in t he st udy of linear t ransformat ions, eigenvect ors,
and eigenvalues. (Received June 03, 2002)
5002-A 1-100
T i m ot hy D . C om ar * (t comar @ben. edu), Depart ment of M at hemat ics, Benedict ine
Universit y, 5700 College Road, L isle, I L 60532. V isuali zation of M bi us Transfor mati ons i n
T wo and T hree Di mensions Using a CA S.
We present several comput er algebra act ivit ies t hat are designed t o help st udent s develop a visual underst anding
of t he beaut iful geomet ry of M bius t ransformat ions in t he complex plane and in t he upper-halfspace model of
t hree-dimensional hyperbolic space. Our t echniques depend heavily on applying basic analyt ic geomet ry and
t he graphing capabilit ies of a comput er algebra syst em. T hese explorat ions address several import ant issues for
st udent s including t he appropriat e and e®ect ively usage of a comput er algebra syst em t o visualize geomet ric
phenomena and t he recognit ion of t he int erplay between algebraic represent at ions and geomet ric renderings.
T he culminat ing explorat ions can be used as part of capst one experiences, which bring t oget her not ions from
complex variables, geomet ry, algebra and t opology. (Received June 03, 2002)
5002-A 1-101
K ar en L . Shu m an * (kshuman@mat h. ui uc. edu), Depart ment of M at hemat ics, Universit y of
I llinois at Urbana-Champaign, 1409 W . Green St reet , Urbana, I L 61801. An anonymous
web-based questi on for m for begi nni ng actuar i al stati stics students. Preliminary report .
St udent s in t he spring 2002 act uarial st at ist ics course at t he Universit y of I llinois Urbana-Champaign submit t ed
quest ions anonymously over t he web wit h a cgi script . T he inst ruct or post ed t he quest ions wit h answers on t he
class web page. St udent s were surveyed mid-t erm about t heir use of t he form and t heir react ions were posit ive.
We will discuss t he t ypes of quest ions asked, t he frequency wit h which t he form was used, and t he way t he form
changed t he way t he inst ruct or int eract ed wit h t he st udent s. (Received June 03, 2002)
5002-A 1-103
Sean L . For m an * (sf or man@sj u. edu), Dept . M at h and Comput er Science, Saint Joseph' s
Universit y, 5600 Cit y A ve, Philadelphia PA 19119. On-li ne Educati onal Generati on of
T SP-i nstances. Preliminary report .
Hamilt onian Circuit s and by ext ension t he Traveling Salesman Problem (T SP) are oft en covered as part of
Graph T heory sect ions in Cont emporary M at hemat ics courses. T SP requires t he det erminat ion of t he short est
circuit (a t our t hat begins and ends in t he same cit y) connect ing a given set of cit ies. T ypically, t he st udent s are
int roduced t o t he T SP and shown several common heurist ics t hat can be used t o ¯nd an approximat e (somet imes
opt imal) solut ion.
T his paper describes a web applicat ion, T SP Generat or, t hat t akes as an input a list of user-provided cit ies (up
t o 30), and produces a variet y of out put s useful t o an inst ruct or or st udent covering t his subject . A descript ion
of how t his applicat ion has been used in Saint Joseph' s Topics in Cont emporary M at hemat ics course will be
given as well.
T SP Generat or will produce t he cit y-t o-cit y dist ance mat rix, will ¯nd approximat e solut ions using t wo common heurist ics, t he Cheapest L ink A lgorit hm and t he Repet it ive Nearest Neighbor A lgorit hm, and in some
cases, will ¯nd t he opt imal circuit among all possible circuit s. Finally, a map of t he input t ed cit ies is displayed.
ht t p:/ / www.sju.edu/ ~sforman/ research/
usa t sp.ht ml (Received June 03, 2002)
6
CREAT IV E USE OF T ECHNOL OGY IN T EACHING M AT HEM AT I CS
5002-A 1-125
M au r a B M ast * (maur a. mast @umb. edu), Depart ment of M at hemat ics, Universit y of
M assachuset t s Bost on, Bost on, M A 02125-3393. Usi ng technology i n a Quanti tati ve
Reasoning Cour se. Preliminary report .
I n t his t alk, I will describe how inst ruct ors at t he Universit y of M assachuset t s Bost on use t echnology in t eaching a
new course called Quant it at ive Reasoning. I n t his course, t he st ress is on reasoning, rat her t han on mat hemat ical
manipulat ion and comput at ion. St udent s learn how t o use t he algebraic and t echnological t ools employed in
t he social, physical and life sciences t o analyze quant it at ive informat ion. T hey st udy some basic st at ist ics, and
explore linear, exponent ial and quadrat ic models; t hey use t hese t ools t o analyze real dat a. T his course is t aught
in a comput er lab (use of t echnology is an explicit part of t he course syllabus), and st udent s use M icrosoft Excel,
I nt ernet browsers, and special course-speci¯c soft ware. T he goal, and t he challenge, is t o use t he t echnology in
a meaningful way on a daily basis. I will discuss how we are meet ing t his goal, and how we are dealing wit h t he
challenges. (Received June 04, 2002)
5002-B1-87
B . L y n n B od n er * (bodner @monmout h. edu), Depart ment of M at hemat ics, M onmout h
Universit y, West L ong Branch, NJ 07764, and R i ch ar d A K u nt z (kunt z@monmout h. edu),
Depart ment of M at hemat ics, M onmout h Universit y, West L ong Branch, NJ 07764. A
Preli mi nar y A ssessment of a W eb-based Per sonali zed System of I nstr uction ( PSI )
Envi ronment Created for a Pre-A lgebra Cour se.
Since t he Fall 2000 semest er, t he M at hemat ics depart ment of M U has o®ered 12 sect ions of a Pre-A lgebra course;
9 of t he sect ions were run in a t radit ional lect ure format , but t he remaining 3 used a web-based Personalized
Syst em of Inst ruct ion (PSI) environment . T he PSI syst em, called M UTest er, was developed by Richard K unt z
and is a comput erized version of t he plan put fort h in Good-bye Teacher by Fred K eller in 1968. T he M UTest er
syst em allows each st udent t o access pract ice exercises and t ake t est s t hrough I nt ernet Explorer. I t also facilit at es
individual st udent and class progress-t racking, as well as dynamic problem dat abase maint enance. A more
complet e descript ion of M UTest er was present ed at t he 2001 M A A meet ing in San A nt onio by Richard K unt z and
is available on his webpage. T his paper will present our preliminary out come-assessment ¯ndings regarding t he
Pre-A lgebra st udent s as t hey enrolled in successive mat h courses. We will also demonst rat e recent improvement s
made t o M UTest er since it s incept ion and discuss our fut ure plans. (Received June 03, 2002)
5002-B1-88
L i n d a B ecer r a* (becer r al @dt . uh. edu), Depart ment of M at hemat ics, Universit y of
Houst on-Downt own, Houst on, T X 77058, and O n gar d Si r i saen gt ak si n
(ongar ds@dt . uh. edu), Depart ment of M at hemat ics, Universit y of Houst on-Downt own,
Houst on, T X 77058. Desi gn and I mplementati on of E-Lear ni ng Col lege Algebra.
We demonst rat e an e-learning mat hemat ics applicat ion t hat includes writ ing, comput at ional and graphical act ivit ies. St udent s may access t his applicat ion and may also submit t heir responses t o t he inst ruct or online. In t he
applicat ion, links t o comput at ional and graphical t ools are provided as st udent s need t hem. T he applicat ion may
also include inst ruct or not es and online quizzes. T his applicat ion is designed and implement ed using M icrosoft
A ct ive Server Pages (A SP) wit h a dat abase and Java applet s. A SP is ut ilized t o creat e an online laborat ory
front end t hat cont ains inst ruct ions, HT M L element s, and a script ing code (Received June 03, 2002)
5002-B1-90
P. M an oh ar an * , Universit y College, Universit y of M aryland, College Park, M D 20742. A
Global Classroom on W ebtycho platfor m.
A bst ract : Universit y of M aryland Universit y College (UM UC), is a " virt ual Universit y" wit hin t he Universit y
Syst em of M aryland, recognized as a leader in dist ance educat ion, part icularly in courses delivered online via t he
I nt ernet . A dhering t o it s mission of bringing convenient and relevant learning opport unit ies t o working adult
learners, UM UC has developed more t han 300 online courses, making UM UC t he largest online universit y in
t he count ry wit h more t han 60,000 online enrollment . I n our present at ion, we brie° y discuss a sample of a web
based mat hemat ics course t hat is current ly o®ered at UM UC. (Received June 03, 2002)
5002-B1-91
A n an d a G u n aw ar d en a* , Depart ment of M at hemat ics, Carnegie-M ellon Universit y. A
TextCentr i c communi cati on model for teaching M athemati cs.
Communicat ion is a crit ical element of any learning environment . A bilit y t o communicat e e®ect ively and accurat ely in t he cont ext of core cont ent makes learning more enjoyable. St udies have shown t hat st udent s learn
more when cont ent is present ed and discussed in an annot at ed mode. I n t his t alk we will demonst rat e a new
I NDEPENDENT L EA RNI NG EX PERI ENCES FOR UNDERGRA DUAT ES I N M AT HEM AT I CS
7
eL earning syst em for building learning object s around t he core t ext cont ent and a communicat ion model for
exchanging t hese learning object s wit h ot her learners. (Received June 03, 2002)
5002-B1-92
P r am od K anw ar * (pkanwar @mat h. ohi ou. edu), Depart ment of M at hemat ics, Ohio St at e
Universit y. Exper i ences in Teachi ng an onli ne L i near Algebra Cour se.
T he present er will share his experiences in t eaching a linear algebra course using dist ance-learning t ools. A mong
ot her t hings, t he present er will focus on delivery of course mat erial, using comput er algebra syst ems and ot her
elect ronic t ools in t he course, cont act wit h t he st udent s, t est ing opt ions, and st rengt hs and weaknesses of such
an approach of t eaching. (Received June 03, 2002)
5002-B1-93
N k ech i M ad on n a* (nagwu@bmcc. cuny. edu). I ntroductor y Statisti cs: E-di str i buted Cour se
Development, Teachi ng and Lear ni ng.
T his present at ion will share wit h t he audience t he present er' s experience in developing and t eaching, and in
ment oring facult y t o develop and t each, t wo e-dist ribut ed st at ist ics courses at t he Borough of M anhat t an Communit y College (BM CC), Cit y Universit y of New York (CUNY ), wit hin t he auspices of t he BM CC T it le I I I
and Dist ance L earning Facult y Development Programs. BM CC is a minorit y-serving t wo-year urban commut er
college wit h a diverse st udent populat ion of approximat ely 16,000 st udent s. T wo part -t ime facult y members are
current ly being ment ored by t he present er t o enhance and t each t he current courses, and t o develop t heir own
e-dist ribut ed courses, given t hat part -t ime facult y have fewer opport unit ies t o avail t hemselves of t he professional development avenues. T he present er will also share wit h t he audience t he impact of t his t ype of ment oring
collaborat ion on t he professional growt h and development of t he present er and her part -t ime facult y ment ees.
(Received June 03, 2002)
5002-B1-95
J am es W h i t e* (mat hwr i g@gt e. net ), M at hwright . I ntroducing M athwr ight M i crowor lds.
T he aim of t his t alk is t o describe a new t ype of web document t hat recent ly made it s appearance at t he New
M at hwright L ibrary and Caf (ht t p:/ / www.mat hwright .com) and t o discuss it from t wo perspect ives: from t he
point of view of it s readers (st udent s of mat hemat ics), and from t he point of view of it s prospect ive aut hors
(t eachers of mat hemat ics). (Received June 03, 2002)
5002-B1-96
E r i n M H od gess* (hodgesse@uhd. edu), Depart ment of M at hemat ics, Universit y of
Houst on-Downt own. Stats A li ve! T he Development of an Online Stati sti cs Cour se.
We discuss t he development and pilot phase of a ¯rst semest er Business St at ist ics course. We consider t he
posit ive and negat ive aspect s of t his mode of present at ion. We elaborat e on t he grading rubric and t he int eract ive
element s of t his discipline. Finally, we discuss fut ure plans. (Received June 03, 2002)
5002-C1-15
M . E l i zab et h M ay ¯ el d * (mayf i el d@hood. edu), Dept . of M at hemat ics and Comput er
Science, Hood College, 401 Rosemont A venue, Frederick, M D 21701. An i nter nshi p at the
MAA.
One of my advisees, a senior mat hemat ics major, recent ly complet ed an int ernship at t he nat ional o± ce of t he
M A A in Washingt on. I will describe t he work she did t here, t he skills needed for t his int ernship, t he role of t he
on-sit e supervisor and my role as int ernship advisor, t he port folio t he st udent submit t ed for her ¯nal grade, and
what t he st udent learned on t he job. (Received M ay 15, 2002)
5002-C1-31
Z su zsan n a M . K ad as* (zkadas@smcvt . edu), Depart ment of M at hemat ics, Saint M ichael' s
College, W inooski Park, Colchest er, V T 05439. I ndependent Study on a Shoestr ing.
For over t en years, all mat hemat ics majors at St . M ichael' s College have been required t o t ake t he Senior
Seminar. T he Seminar meet s once a week and provides st udent s wit h t he opport unit y t o pursue a t opic t hat
may not arise in t he regular course o®erings, t o writ e an exposit ory research paper, and t o present t he t opic t o
st udent s and facult y. I n t his t alk we will describe t he hist ory of t he Senior Seminar, t he mechanics of how it
runs, t he Seminar as a collegial undert aking, and t he role of writ ing and peer assessment . We will cont rast t he
Senior Seminar wit h ot her independent learning opt ions at St . M ichael' s: t he Honors T hesis and t he Readings
and Research course. Examples will be provided as well as an overall assessment of t he success of t his course for
majors. (Received M ay 21, 2002)
8
I NDEPENDENT L EA RNI NG EX PERI ENCES FOR UNDERGRA DUAT ES I N M AT HEM AT I CS
5002-C1-33
K ay B . Som er s* (mekbs01@mor avi an. edu), Depart ment of M at hemat ics and CS,
M oravian College, 1200 M ain St reet , Bet hlehem, PA 18018. An Over view of Recent Field
Study Exper i ences.
T his t alk will include a descript ion of a variet y of ¯eld st udy experiences undert aken by M oravian College
st udent s over t he past t en years. T he ¯eld st udy experience normally t akes t he place of one course unit and
requires t hat st udent s spend at least t en hours per week t hroughout t he semest er working on a project or
project s at a local company. St udent s keep a j ournal, meet weekly wit h t he College supervisor and give writ t en
and oral report s at t he end. Various aspect s of t he ¯eld st udy experience will be discussed, including College
requirement s, companies' and st udent s' expect at ions, how t o obt ain such placement s, and guidance given t o
st udent s. (Received M ay 22, 2002)
5002-C1-45
R ay m on d F T en n ant * (Raymond. Tennant @zu. ac. ae), Zayed Universit y, P.O. Box 4783,
A bu Dhabi, Unit ed A rab Emirat es. I ndependent Research Proj ects for M athematics
Students at Zayed Univer si ty. Preliminary report .
Zayed Universit y is a modern universit y for nat ional women in t he Unit ed A rab Emirat es. Each st udent is
required t o have a year-long research experience, which includes one semest er of independent research direct ed
by a facult y advisor and culminat ing in a capst one project t he following semest er. T he capst one project is
designed t o int egrat e t he st udent ' s academic experience in t he major by focusing on a part icular t heme, problem,
or project and t o allow her t he opport unit y t o design and implement a creat ive, original research st udy.
I n t his paper, t he aut hor describes experiences direct ing undergraduat es at a universit y honors program in
t he U.S. and how t his knowledge has been useful in developing undergraduat e research project s for mat hemat ics
st udent s at Zayed University. Current proj ect s are being designed which connect t o t he cult ure of t he A rab
world. One project views mat hemat ics from t he hist orical perspect ive of A rab mat hemat icians. A not her project
const ruct s I slamic t ilings and ornament at ion using st raight edge and compass met hods along wit h comput er
imaging programs. T his project also classi¯es t he di®erent t ypes of ornament at ion from mosques, palaces, and
ot her buildings in t he region. Examples of st udent project s will be given. (Received M ay 28, 2002)
5002-C1-49
K ev i n W . D en n i s* , Saint M ary' s Universit y of M innesot a, 700 Terrace Height s, 59,
W inona, M N 55987. Seni or Seminar . Preliminary report .
At Saint M ary' s Universit y of M innesot a, all senior mat hemat ics majors are required t o t ake a 1-credit course
ent it led " Senior Seminar" . T he maj or component of t his seminar is an independent learning experience, which
st udent s must present orally and in writ t en form. We will address how t he course was st ruct ured, t he role of
t he inst ruct or and t he nat ure of t he st udent s' work. We will also present t he rubric used in grading t he oral
present at ions and papers. (Received M ay 28, 2002)
5002-C1-59
D or een R . N . D e L eon * (ddel eon@mat h. ucl a. edu), Depart ment of M at hemat ics,
Universit y of California, Box 951555, L os A ngeles, CA 90095-1555, and J oh n J . W est m an
(j west man@mat h. ucl a. edu), Depart ment of M at hemat ics, Universit y of California, Box
951555, L os A ngeles, CA 90095-1555. I ntroduction to Sci enti ¯c Research: A Group
A pproach to I ndi vi dual Undergraduate Research.
I nt roduct ion t o Scient i¯c Research is a non-t radit ional course at UCL A t hat provides undergraduat es wit h
a research experience and is feat ured in a successful NSF V I GRE grant . T he t arget audience is freshman
and sophomore st udent s, and t he course requires only a quart er or t wo of calculus and basic programming
skills. T herefore, M aple is used t o help explore mat hemat ical models analyt ically, graphically, and numerically.
T he format for t he class feat ures formal lect ures, group discussions, st ruct ured comput er demonst rat ions, and
brainst orming sessions. T his course provides a format in which a number of st udent s can experience research
in a more e± cient manner t han working one-on-one wit h a facult y member. T he goal is st udent involvement
in performing individual research. St udent s select t heir own t opics for research, which aids in ret ent ion and
complet ion of t he course, and t hey are encouraged t o ¯nd facult y ment ors. I nnovat ion is encouraged in st udent
project s. T he course is cent ered around communicat ion skills, development al knowledge set s, and t ask set s,
using various resources such as LATEX , M icrosoft PowerPoint , and M icrosoft Word. T he grade for t his course is
based on complet ion of t he t ask set s, ¯nal present at ion, and class part icipat ion. (Received M ay 29, 2002)
5002-C1-62
M at t h ew J H ai n es* (hai nes@augsbur g. edu), A ugsburg College, 2211 Riverside A ve, Box
28, M pls, M N 55454. Pr i mi ng Students to be T hrown i nto M ore I ndependent Lear ni ng
Exper i ences.
For schools t hat do not require independent learning experiences, such experiences can st ill play a meaningful
role in t he educat ion of some st udent s. Brief exposures can lead st udent s t o more independent learning. Such
I NDEPENDENT L EA RNI NG EX PERI ENCES FOR UNDERGRA DUAT ES I N M AT HEM AT I CS
9
experiences can occur in various forms: challenge problems in class or a newslet t er, compet it ion problems, or
independent st udies t hat are st udent init iat ed.
T his present at ion discusses a mini-proj ect which led a st udent t o read about t he hist ory of project ile mot ion.
Her mot ivat ion was t wo-fold: 1) she was int erest ed in t he hist ory of mat hemat ics and 2) she agreed t o be part of
a present at ion about t rebuchet s. T he t rebuchet proj ect was init iat ed by a colleague who is a medieval hist orian.
He invit ed a physicist and me t o assist in creat ing an int erdisciplinary learning act ivit y. I n my present at ion I
will describe t he facult y collaborat ion, t he work carried out by t he st udent , and analyze t he pot ent ial for such
experiences t o mot ivat e st udent s t o pursue furt her independent st udy. (Received M ay 29, 2002)
5002-C1-72
J oh n C oh n , IBM , Essex Junct ion, V T , J oan n a E l l i s-M on agh an * , Depart ment of
M at hemat ics, St . M ichael' s College, W inooski Park, Colchest er, V T 05439, D an N ar d i ,
Depart ment of Comput er Science, Universit y of Vermont , Burlingt on, V T 05405, and
R ob er t Sn ap p , Depart ment of Comput er Science, Universit y of Vermont , Burlingt on, V T
05405. I ndependent Studi es wi th I ndustr y Par tner ship. Preliminary report .
I ndust ry applicat ions can provide a rich source of rewarding independent st udy project s for bot h undergraduat e
and graduat e st udent s. I n t his t alk, we will share our experience in building a relat ionship wit h t he local I BM
plant and solicit ing problems of mut ual int erest . We will present an example of a current project , developing
pat t ern-recognit ion soft ware using graph t heory t echniques. (Received M ay 30, 2002)
5002-C1-79
B r u ce F T or r en ce* (bt or r enc@r mc. edu), Dept . of M at hemat ics, Randolph-M acon
College, A shland, VA 23005. Undergraduate Research - A M odel for Success.
At Randolph-M acon, I have direct ed several successful st udent research project s. One way t o measure success is
by out comes: publicat ions, present at ions, and awards.
Because I feel t hat it is import ant t o give st udent s opport unit ies t o present t heir result s, I require t hat
every project st art wit h t he goal of a present at ion bot h at our college and at a regional M A A meet ing. T his
immediat ely set s in t he st udent s' minds a not ion of t he seriousness wit h which t hey are expect ed t o proceed. I n
pract ice, it also set s a ¯rm t imet able for complet ing various component s of t he project , and oft en leads nat urally
t o t he st udent s ent ering nat ional compet it ions, and present ing in larger forums. L ast year a pair of my st udent s
ent ered and received t he \ out st anding" designat ion in COM A P' s M at hServe compet it ion (one of t wo in t he
nat ion). Our regional M A A sect ion helped send t hem t o M adison t o present at last summer' s M at h Fest . Ot her
project s have led t o publicat ions in journals such as M at h Horizons and t he Pi M u Epsilon Journal.
I n t his t alk I will discuss how t o st ruct ure st udent direct ed research wit h t hese out comes in mind. (Received
June 02, 2002)
5002-C1-89
D ai sy C ox M cC oy * (Dai sy. mccoy@l sc. vsc. edu), PO Box 190, L yndon Cent er, V T ,
05850, and P at r i ci a L aR ose (pat t y11512@aol . com), 165 Richardson Rd, Orange, V T ,
05641. T he I mpact of an I ndependent Lear ni ng Exper ience on a Student and Her Professor .
Facult y usually believe t hat independent learning experiences are valuable for st udent s. I n t his t alk we will
explore some of t he ways t hat a part icular independent st udy impact ed t he st udent and t he inst ruct or. We will
also discuss ways t he experience could have been improved and t he long t erm e®ect s as t he st udent re° ect s on
her experiences during t he st udy and her subsequent explorat ions in mat hemat ics. (Received June 03, 2002)
5002-C1-120
L i on el R osen * (l r osen@l ynn. edu), Professor & Chair, Depart ment of M at hemat ics,
College of A rt s and Sciences, L ynn Universit y, 3601 Nort h M ilit ary Trail, Boca Rat on,
Florida 33431. I nnovati on, Cooperati on and Exper imentati on; T he Cor ner stones of an
Honor s Col lege Algebra Cour se for Non-M athemati cs M ajor s.
At L ynn Universit y a new policy has been int roduced t hat requires all undergraduat e st udent s t o t ake college
algebra. I n response t o t his requirement , t he Honors Program has developed a new college algebra course. I t
covers advanced t opics beyond ordinary college algebra and is int ended t o prepare honors st udent s for calculus
and ot her advanced mat hemat ics courses. I t is also int ended t o encourage st udent s t o pursue successful careers
in mat hemat ically orient ed ¯elds. I n t his t alk I will describe t he cont ent of t he new honors college algebra course
and t he roile of ¯eld t rips and limit ed int ernships in t he course. I will also discuss t he research component of t he
course and t he role of out side evaluat ors in helping t o det ermine t he grade for t he research paper. (Received
June 04, 2002)
10
T HE USE OF RECENT HI ST ORY OF M AT HEM AT ICS I N T EACHI NG
5002-D1-11
M at t D L u n sfor d * (ml unsf or @uu. edu), Union Universit y, 1050 UU Drive, Jackson, T N
38305. T he I deas of Evar iste Galoi s: Recover i ng M oti vati on in A bstract Algebra.
How can t he ideas of Evarist e Galois as published in his famous memoir, along wit h a t horough discussion of t he
hist orical perspect ive of his work, enhance t he t eaching of undergraduat e abst ract algebra? T his t alk will answer
t hat quest ion in t wo ways. First , t he ideas of Galois in his famous paper M emoir on t he Condit ions for Solvabilit y
of Equat ions by Radicals (published post humously in 1846) will be discussed. T his memoir includes t he ¯rst
use of t he t erm " group" and t he origin of t he concept of normal subgroup. A lso present in primit ive form are
t he not ions of ext ension ¯elds, split t ing ¯elds, polynomial rings, and quot ient rings. Secondly, t he applicat ion
of t hese ideas t o t he current pedagogy of abst ract algebra will be explored. In part icular, t he t alk will focus on
t he placing maj or t opics in t heir proper hist orical cont ext , mot ivat ing major t opics t hrough t he examinat ion of
concret e mat hemat ical problems from which t hey arose, and present ing major t opics in an order t hat respect s
t he origins of t he subject mat t er while meet ing t he pedagogical needs of t he st udent s. To accomplish t hese
goals, t he t alk will support a " rings-¯elds-groups" approach t o a ¯rst course in abst ract algebra as opposed t o
t he st andard " groups-rings-¯elds" approach. (Received M ay 14, 2002)
5002-D1-12
L aw r en ce D ' A nt on i o* (l dant @r amapo. edu), 24 M eadoway, Dobbs Ferry, NY 10522. T he
Group Law for El li pti c Cur ves. Preliminary report .
Ellipt ic curves play a cent ral role in t he recent hist ory of mat hemat ics, for example, t he proof of Fermat ' s L ast
T heorem. T his t alk will focus on t he t opic of t he addit ion of rat ional point s on ellipt ic curves. T here is a lovely
int eract ion bet ween algebraic and geomet ric ideas in t his t opic. Speci¯cally it gives st udent s a visual example
of a group operat ion, but one t hat also uses ideas from number t heory.
Furt hermore, t he t heory of ellipt ic curves has a very rich hist ory, for which t his t alk will t ry t o give an
overview. Part icular highlight s are it s origins in t he work of Diophant us, t he development of t he chord and
t angent met hod by Newt on, t he connect ion wit h ellipt ic int egrals in t he work of Euler and Jacobi, culminat ing
in t he monument al paper of 1901 by Poincare. (Received M ay 14, 2002)
5002-D1-13
R on B ar n es* (Bar nesR@uhd. edu), CM S Depart ment , Universit y of Houst on-Downt own,
One M ain St reet , Houst on, T X 77002, and L i n d a B ecer r a (Becer r aL@uhd. edu), CM S
Depart ment , Universit y of Houst on-Downt own, One M ain St reet , Houst on, T X 77002. New
Paradi gms for the Hi stor y of M athematics Cour se wi th an Emphasis on I mpor tant I deas
and I ndi vi duals of the 20th Centur y M athemati cal and Computer Sci ences. Preliminary
report .
T his t alk consist s of a brief overview of an earlier paper " Ret hinking t he Hist ory of M at hemat ics Course"
and ext ends it by giving some new ways of int egrat ing more recent ideas and individuals int o t he course.
Topics considered include: (a) T imelines - of more recent development s wit h parallel snapshot s of 20t h cent ury
development s in M at hemat ics and Comput er Science; (b) Individuals - brief biographies of import ant men,
women and minorit ies in Comput er Science and St at ist ics (various resources including books and videos are
not ed); (c) I deas and Concept s - earliest development s in CS t o quant um comput ing, various t imelines can be
developed (hist ory of comput er science and st at ist ics t ext s and ot her resources are used t o highlight import ant
ideas and concept s); (d) M inorit ies and Women Cont ribut ions and Progress in t he Profession - various resources
including t hose of AW M are considered. A lso, a review of t he recent NSF Report on Equal Opport unit ies in
Science and Engineering is discussed. T his report covers a number of separat e issues ranging from diversit y t o
fut ure project ions and concerns. Several examples will be present ed from each of t hese cat egories, including a
number of t imelines t o illust rat e t he described paradigm. (Received M ay 14, 2002)
5002-D1-16
A m y Sh el l -G el l asch * (aa7423@usma. edu), Depart ment of M at hemat ical Sciences, West
Point , NY 10996. Usi ng the Hi stor y of Your Col lege and Depar tment to Bui ld a Hi stor y of
M athematics Cour se. Preliminary report .
M ost st udent s graduat e from college wit h lit t le or know knowledge of t he hist ory of t he inst it ut ion of which t hey
so proudly where t he sweat shirt s. T hey also learn very lit t le about t he depart ment t hey are st udying in and t he
people t hey are learning from and what t hey do. Since no one semest er hist ory of mat hemat ics course can come
close t o covering all t opics, some way of focusing t he course must be found. To remedy t hese t wo issues, t his
year I will be o®ering a course ent it le T he Hist ory of M at hemat ics from t he West Point Perspect ive. T his course
will use t he hist ory of: t he A cademy, t he Depart ment of M at hemat ical Sciences, and t he curriculum t o mot ivat e
t he t opics covered. I will discuss how t he t opics are chosen as well as st udent assignment s t hat merge t opics in
t he hist ory of mat hemat ics wit h t he hist ory of t he inst it ut ion and it s facult y. (Received M ay 15, 2002)
T HE USE OF RECENT HI ST ORY OF M AT HEM AT ICS I N T EACHI NG
11
5002-D1-19
J e® J oh an n es* (j ohannes@member . ams. or g), M at hemat ics Depart ment , 1 College Circle,
Geneseo, NY 14454-1401. M oder n Geometr y. Preliminary report .
Once we get beyond Euclid and A rchimedes most of t he geomet ry and all of t he t opology we discuss is 19t h
and 20t h cent ury mat hemat ics. T he 19t h cent ury opens wit h t he birt h of algebraic geomet ry and t he rise of
nonEuclidean geomet ry. Di®erent ial geomet ry and axiomat ic geomet ry are also bot h receiving serious at t ent ion. T he 20t h cent ury feat ures t he invent ion and innovat ions of t opology. I n t his t alk we will survey t hese
development s and weave t hem t hrough t he appearances of visual mat hemat ics in t he undergraduat e curriculum.
(Received M ay 16, 2002)
5002-D1-38
A l ex an d er F K l ei n er * (al exander . kl ei ner @dr ake. edu), M at hemat ics and Comput er
Science, Drake Universit y, 2507 Universit y, Des M oines, I A 50311. W hat is a Poi nt of
I n° ecti on? A Preli mi nar y Repor t. Preliminary report .
A bst ract : A survey of 20t h cent ury, English language t ext s (most ly at t he level of freshman calculus) provides
a cat alog of t hings we t ell our st udent s t o avoid. Some aut hors give a formal de¯nit ion, ot hers use an example
or an informal discussion. M oreover, at least 10 di®erent de¯nit ions can be found in t hese books, somet imes
di®erent de¯nit ions are implied in t he same t ext . On t he ot her hand, several art icles, writ t en over t he last
cent ury, have shown t hat t hese de¯nit ions are, in general, not equivalent .
T his t opic, a basic part of almost every Calculus I class, can be used t o show st udent s t he import ance of
precise mat hemat ics and t he value of well chosen examples and graphs.
T he paper will discuss t he spect rum of de¯nit ions of point of in° ect ion and provide examples and t heorems t o
show t heir di®erences. It will also point t o st art ing places for st udent s who wish t o explore t he issue t hemselves.
(Received M ay 23, 2002)
5002-D1-82
W i l l i am C . C al h ou n * (wcal houn@bl oomu. edu), Depart ment of M at h., Comp. Sci. and
St at s., Bloomsburg Universit y, 400 East Second St reet , Bloomsburg, PA 17815-1301. From
Hi lber t' s Program to Computer Programming. Preliminary report .
T he impact of comput ers on our lives is obvious t o everyone, while mat hemat ical logic is an esot eric subject
for most people. Yet t he hist ories of mat hemat ical logic and comput ers are t ight ly int erwoven. I bring t he
connect ions bet ween logic and comput ing int o my Discret e M at hemat ics and T heory of Comput at ion courses.
M y st udent s are surprised and int erest ed t o learn t hat work on abst ract quest ions in logic led t o ideas t hat were
fruit ful in t he design of comput ers.
T he pat h from mat hemat ical logic t o comput ers includes some of t he most famous names in 20t h cent ury
mat hemat ics. We can t race t he genesis of " comput er code" from Hilbert ' s formal proof t heory, t o Go̧del' s coding
of proofs by numbers, t o t he codes for Turing machines and t he inst ruct ions for von Neumann' s st ored program
comput er.
I have my st udent s give brief present at ions in class. I n addit ion t o t he four mat hemat icians list ed above,
st udent s have chosen t o speak on Cant or, K leene, Chomsky and Julia Robinson. T he st udent s have found t he
present at ion t o be a valuable and enjoyable assignment . T hey appreciat e t he " general int ellect ual int erest " of
t he mat hemat ics and relat e t o t he human challenges faced by t hese brilliant mat hemat icians. (Received June
03, 2002)
5002-D1-116
L i n d a M cG u i r e* (l mcgui r e@hal . muhl ber g. edu). Usi ng Recent Hi stor y in a
Combi nator ics and Graph T heor y Cour se. Preliminary report .
T he t alk I propose giving would focus on a pedagogical approach t hat I have implement ed in a junior-senior
level course in combinat orics and graph t heory. Each st udent in t he class was assigned t o ¯nd out about at
least one researcher, mat hemat ician or ot herwise, whose work is classi¯ed as being in t he combinat orics or
graph t heory realm. T he st udent was t hen responsible for " int roducing" t he researcher t o t he class via an oral
present at ion/ discussion. T his present at ion would precede t he discussion of t he course mat erial t hat relat ed t o
t he work t his person had done. A ft er t he st andard class mat erial was covered, st udent s would need t o " wrap
up" t he session by discussing t he current or fut ure work of t he researcher in quest ion and any open problems
t hat might be accessible t o t he class. At t he end of t he t erm we compiled a port folio of t he work t hat t he ent ire
class had produced. St udent s were t aken wit h t heir roles as hist orians, set t ing t he st age for t he mat hemat ics t o
come. M any import ant writ ing issues regarding reliable hardcopy sources, proper cit at ion t echniques, and t he
int egrit y of int ernet sources arose and were addressed. (Received June 04, 2002)
12
T HE USE OF RECENT HI ST ORY OF M AT HEM AT ICS I N T EACHI NG
5002-D1-141
D i ck J ar d i n e* (r j ar di ne@keene. edu), K eene St at e College, A ssist ant Professor of
M at hemat ics, K eene, New Hampshire 03435. Numer i cal M ethods Hi stor y Proj ects.
St udent project s wit h a signi¯cant hist orical component can humanize t he numerical analysis experience, and
such project s are t he focus of t his present at ion. Various comput at ional project s wit h hist orical feat ures are
described which have been successfully employed t o enliven a numerical met hods course. A ddit ionally, t he
hist ory of mat hemat ics is t he foundat ion of a ¯nal course project which serves as a course review, and includes
t he work of such mat hemat icians from t he last 200 years as Runge, A dams, Birkho®, Richardson, and ot hers.
T hese project s are appropriat e for an undergraduat e mat hemat ics or comput er science course in numerical
analysis. (Received June 06, 2002)
5002-E1-23
K at e G . M cG i v n ey * (kgmcgi @shi p. edu), M at hemat ics Depart ment , Shippensburg
Universit y, Shippensburg, PA 17257. Usi ng Simulati ons to Solve Expected Value Problems.
Preliminary report .
We will look at a variet y of int erest ing probabilit y problems such as t he " Hat Check Problem" and t he " Chinese
Birt h Problem" and will explore expect ed value concept s. I n each case we will see how simulat ions, using
t echnology, can be used t o experiment ally ¯nd a solut ion. T hese rich problems can be used in int roduct ory
probabilit y and discret e mat h courses as well as serve as an applicat ion t o series in calculus. (Received M ay
20, 2002)
5002-E1-35
St ep h an i e F i t ch et t * (sf i t chet @f au. edu), Florida At lant ic Universit y, Honors College,
5353 Parkside Drive, Jupit er, FL 33458. Group T heor y wi th a Rubik' s Cube. Preliminary
report .
T he group of permut at ions of a Rubik' s cube, while ext remely large, is an int erest ing and concret e example
of a ¯nit e group. To solve a Rubik' s Cube, most people develop or learn sequences of moves (operat ors) t hat
do speci¯c t hings like swapping t wo edges or t wist ing t wo corner pieces in opposit e direct ions. We will discuss
how st udying subgroups of t he cube group can lead t o ident i¯cat ion of useful operat ors, and how conjugat ion of
operat ors can be used as an aid in solving t he cube. (Received M ay 22, 2002)
5002-E1-42
R i ch ar d G L aat sch * (l aat scr g@muohi o. edu), 407 Sandra Drive, Oxford, OH 45056. A
Cour se i n M athemati cal Recreati ons.
Since 1979 M iami Universit y has o®ered a t opics course in M at hemat ical Recreat ions t o advanced undergraduat e
st udent s, most of whom have been secondary educat ion majors. T he t alk will discuss t he genesis, cont ent , and
requirement s of t he course as t aught by one of it s inst ruct ors, and provide samples of problems and some
st udent cont ribut ions t o t he ¯eld of recreat ional mat hemat ics. A print ed out line of t he course will be provided.
(Received M ay 24, 2002)
5002-E1-114
I l h an M . I zm i r l i * (i xi @st r ayer . edu), 1025 15t h St reet NW , Washingt on, D.C. 20005.
Some Problems in T he T heor y of Equati ons and Number T heor y. Preliminary report .
Problem 1:T here are 81 t rees such t hat each year t he ¯rst t ree yields 1 unit of produce, t he second 2 unit s,
and 81st 81 unit s. How should t hese be divided among 9 people so t hat each person get s 9 t rees and an equal
amount of yearly produce? T he solut ion can be obt ained by const ruct ing a t able where each column sum is
369. A ssume now we have m = n ¤ n t rees, such t hat each year one get s 1 unit of produce from t he ¯rst t ree, 2
unit s from t he second t ree,and m unit s from t he m t h t ree. How should t hese t rees be divided among n people
so t hat each person get s n t rees and an equal amount of yearly produce? T his leads us t o an n by m syst em
of equat ions a solut ion of which can be obt ained by const ruct ing an \ almost magic" square t hat has some very
int erest ing propert ies. Problem 2: A rrange successive numbers in a t riangle so t hat t he di®erence of t he t wo
adjacent numbers in a row equals t he number bet ween t hem in t he lower row. I s t here a procedure t o const rcuct
such t riangles wit h n rows? Problem 3: A pair of nat ural numbers will be called osculat ory numbers if t hey are
perfect squares and t he digit s of one is a permut at ion of t he digit s of t he ot her. For example, 169 and 196 are
osculat ory numbers. Can we ¯nd a rule t o const ruct all such numbers? (Received June 04, 2002)
5002-E1-131
H om er S. W h i t e* (hwhi t e0@geor get owncol l ege. edu), Depart ment of M at hemat ics,
Campus Box 311, Georget own College, Georget own K Y , 40324. Checker s, I ncompleteness,
and Di ®erential Equations.
Conway, Berlekamp and Guy have a neat t heorem about a checker-like gamet hat received at t ent ion in a recent
M at hemat ics M agazine art icle. T heir proof is simple and delight ful , but more import ant ly, it looks just like
USI NG POPUL A R CULT URE I N T HE M AT HEM AT I CS A ND M AT HEM AT I CS EDUCAT ION CL A SSROOM
13
proofs in several di®erent " serious" branches of mat hemat ics. In part icular, it resembles a classical proof of
L efschet z t hat a cert ain class of second -order di®erent ial equat ions have bounded solut ion curves. Covering t he
checker game in a DE class provides a nice way t o acquaint t he class wit h t he mot ivat ion for t he boundedness
proof. T his t alk will do somet hing similar wit h a connect ion bet ween t he checker game and an incomplet eness
t heorem in logic. (Received June 05, 2002)
5002-F1-27
B ar t D . St ew ar t * (ab8146@usma. edu), 3078 B Webb Place, West Point , NY 10996.
M oti vati ng M ath Concepts in 3, 2, 1. Preliminary report .
W it h t he advent of t echnological advances, not only are t he rigors of " chalk and blackboard" drills cont inuing t o
fade, but classroom learning environment s are also changing. M uch more t han number crunching, mat hemat ical
science programs are shift ing, or should be, t o produce creat ive t hinkers who can relat e mat hemat ical t heory
and pract ice t o real-world applicat ions. I n t he bat t le t o educat e t oday' s st udent accordingly, mot ion pict ures
can serve as force mult ipliers!
M ot ion pict ures t ap our imaginat ion, and our imaginat ion helps t o make abst ract concept s clear. Concept s,
when grasped, promot e creat ive and crit ical t hinkers. Hence, incorporat ing mot ion pict ures in t o t he classroom
appears t o be a good idea. It is a good idea for many reasons, t o include, a®ect ing classroom at mosphere,
sparking our imaginat ion, and present ing mat h' s applicabilit y t o real-world phenomena.
Our imaginat ion, t oo, helps us t o make sense of t he ambiguous. In fact , " imaginat ion is more import ant t han
knowledge, for knowledge is limit ed whereas imaginat ion embraces t he ent ire world...." Consequent ly, ignit ing
st udent s' imaginat ion should come ¯rst and foremost , and t hen t he expect at ion t hat t he mat hemat ical knowledge
will follow may not be t oo far fet ched. (Received M ay 21, 2002)
5002-F1-29
T om m y R at l i ®* (t r at l i f f @wheat oncol l ege. edu), Depart ment of M at hemat ics, W heat on
College, Nort on, M A 02766. Car toon Character s and Calculus.
One of t he goals in all of my classes is t hat t he st udent s do not view mat hemat ics as isolat ed from t he rest of t heir
academic and non-academic lives. In t his t alk, I will discuss several writ ing assignment s from my single variable
and mult ivariable calculus courses t hat are writ t en as let t ers from cart oon charact ers asking for t he st udent s'
advice. One of t he common t hemes is t hat t here is almost no mat hemat ical not at ion in t he assignment s, but
t hey require signi¯cant mat hemat ics t o solve. (Received M ay 21, 2002)
5002-F1-41
Sar ah J . G r eenw al d * (sj g@mat h. appst at e. edu), 121 Bodenheimer, Depart ment of
M at hemat ics, A ppalachian St at e Universit y, Boone, NC 28608, and A n d r ew N est l er
(nest l er _andr ew@smc. edu), Depart ment of M at hemat ics, Sant a M onica College, 1900 Pico
Blvd, Sant a M onica, CA 90405-1628. r dr r : Engagi ng Students wi th Signi ¯cant
M athematical Content from \ T he Si mpsons" . Preliminary report .
Now in product ion of it s 14t h season, \ T he Simpsons" is an award-winning global pop cult ure phenomenon.
But did you know t hat " T he Simpsons" also cont ains over one hundred mat hemat ical moment s, wit h mat erial
ranging from arit hmet ic t o calculus t o Riemannian geomet ry? Since \ T he Simpsons" has been on t he air for
most of our college st udent s' lives, t hey likely are familiar wit h t he program and it s large cast of charact ers,
including a resident mat hemat ician named Professor Frink. For t hese reasons, and because it cont ains so many
inst ances of mat hemat ics, t his program is an ideal source of fun ways t o int roduce import ant concept s and
mot ivat e st udent s t owards deep underst anding.
During t his t alk, we will discuss how use of \ T he Simpsons" led t o signi¯cant explorat ion of t he Pyt hagorean
T heorem, calculus concept s, t he irrat ionalit y of ¼, Fermat ' s L ast T heorem, and t he shape of space. We will
discuss t he bene¯t s of st udying \ T he Simpsons" in t he classroom, e®ect s on t he t eaching and learning dynamics,
di± cult ies encount ered, and speci¯c examples of how t his mat erial can be successfully implement ed. Web links
t o a guide t o mat h on \ T he Simpsons" and numerous relat ed classroom worksheet s will be available. (Received
M ay 24, 2002)
5002-F1-44
B r i an H op k i n s* (hopki ns_b@spc. edu), Depart ment of M at hemat ics, Saint Pet er' s
College, 2641 K ennedy Blvd., Jersey City NJ 07306. K evin Bacon and Graph T heor y.
Preliminary report .
Graph t heory, t he mat hemat ics of connect ion, can be int roduced t o st udent s at many levels of mat hemat ical
mat urit y. Sociologist St anley M ilgram' s idea of " six degrees of separat ion" can be expressed in t erms of graph
USI NG POPUL A R CULT URE I N T HE M AT HEM AT I CS A ND M AT HEM AT I CS EDUCAT ION CL A SSROOM
14
t heory, wit h people as vert ices and edges connect ing people who know each ot her. But t he result ing graph is
huge and poorly de¯ned. T here is a large collect ion of solid dat a t hat is readily available for st udy: t he I nt ernet
M ovie Dat abase. T he Universit y of V irginia' s Oracle of Bacon allows you t o ¯nd any act or' s connect ion t o
K evin Bacon, in fact , t o any ot her act or. W ho' s " far away" from K evin Bacon? W hy K evin Bacon? I s anot her
act or a " bet t er" choice for t he " cent er" of t he Hollywood universe? T hese quest ions help mot ivat e graph t heory
concept s, are easily addressed by t he UVa sit e, and use st udent knowledge t o develop mat hemat ics beyond t he
familiar realm of formulas. I have used t his mat erial in junior level discret e mat h and freshman ¯nit e mat h,
wit h very posit ive result s ? st udent s were highly engaged, especially some who had not dist inguished t hemselves
before. (Received M ay 26, 2002)
5002-F1-106
Sar ah L M ab r ou k * (smabr ouk@f r c. mass. edu), Framingham St at e College, M at hemat ics
Depart ment , 100 St at e St reet , PO Box 9101, Framingham, M A 01701-9101. M issi on
I mpossi ble, Baywatch, Per r y M ason, James Bond, and Tool T i me I n Calculus?
Taking advant age of charact ers and sit uat ions from movies and t elevision can make a project seem more enjoyable
and less st ressful. Use of a narrat ive in present ing a project provides an avenue t hrough which t o assess t he
st udent ' s underst anding of t he mat hemat ics applied. Requiring t he cont inuat ion of t he narrat ive as t he st udent
present s t he analysis of and t he solut ion for t he underlying problem provides a way in which t o evaluat e t he
st udent ' s comprehension of and abilit y t o explain mat hemat ics.
For t his project , each narrat ive hides an opt imizat ion problem t hat t he st udent s must ¯nd, solve, and explain
in a project paper t hat must be a writ t en cont inuat ion of t he original narrat ive. St udent s are required t o explain
all aspect s of t he underlying problem and it s solut ion t oget her wit h t he mat hemat ics t hat t hey use t o solve t he
problem using t he charact ers present ed in t he narrat ive.
I n t his paper, I will discuss my use of t hese narrat ives in Calculus I as well as present some of t he web pages
t hat st udent s creat ed as part of t he project . I will present some of t he narrat ives and discuss t he underlying
opt imizat ion problems. I n addit ion, I will discuss st udent react ion t o t he use of t hese project narrat ives in
Calculus I . (Received June 03, 2002)
5002-F1-124
W i l l i am T B u t t er w or t h * (wbut t er w@bar at . edu), M at hemat ics Depart ment , 700 E
West leigh Rd., L ake Forest , I L 60045, and P au l R C oe (CoePaul @emai l . dom. edu),
M at hemat ics Depart ment , 7900 W Division St reet , River Forest , I L 60305. Come On
Down* T he Pr i ze is Ri ght i n Your Classroom.
T he Price is Right (T PI R) is a rich source of examples of applied probabilit y, combinat orics, game t heory and
comput er simulat ion. W hile some of t he games played on st age by individual cont est ant s st ress a knowledge of
pricing, many are also heavily based on probabilit y. T PI R st age games are a t reasury of int erest ing modules
t hat can be e®ect ive learning t ools in a wide range of classrooms, from L iberal A rt s and M at hemat ics Educat ion
t o Discret e M at h and upper-level Probabilit y. We' ll show how st udent s explore import ant mat hemat ics and
improve t heir problem-solving skills by analyzing some of t hese games; and because t hese examples are drawn
from a familiar source, t hey provide special mot ivat ion for st udy, and oft en reduce mat h anxiet y.
A ft er a general discussion of how t hese games can be adapt ed for classroom use, we' ll explore up t o t hree
di®erent games, " M oney Game," " Punch-A -Bunch," and " Spelling Bee," using short video clips. Each of t hese
games has it s own mat hemat ical and pedagogical value. W hile t he probabilit y comput at ions of " M oney Game"
are relat ively st raight forward, t hose of " Punch-A -Bunch" and " Spelling Bee" int roduce more complex element s
of st rat egy and decision-making. (Received June 04, 2002)
5002-G1-51
I r i n a A . C h er n i k ova* (i r i na@uakr on. edu), 2068 Jennifer St reet , A kron, OH 44313, Set h
I . H i r sh or n (shi r sh@umi ch. edu), 108 SOE, Dearborn, M I 48128, and D eb or ah S.
W eb er (weber @uakr on. edu), 426 Barret t Circle, A kron, OH 44691. M ulti-Level I nteracti ve
Strategy for Teaching and Lear ning i n I ntroductor y Col lege-Level M athemati cs Cour ses.
St udent s in college int roduct ory mat hemat ics courses bring di®erent levels of preparat ion t o t he classroom, negat ive at t it ude t oward mat hemat ics and disbelieve hey will be able t o succeed. A mult i-level int eract ive st rat egy
t o t eaching and learning in such courses was developed t o address t hese st udent -cent ered problems. W it hin t he
project st udent s have t he opport unit y t o build t heir mat hemat ical skills individually and increment ally, wit hout
falling behind on new mat erial, and t o develop more posit ive at t it udes t oward mat hemat ics. Element s of t he new
st rat egy include reorganizat ion of course out line t o re° ect a st rong logical st ruct ure of course t opics; development
of a four-level syst em of individual class and homework assignment s; spiral t est ing wit h immediat e t est result s
RUM E SI GM A A SESSI ON
15
and feedback; development of supplement al, int eract ive course mat erial. St udent performance levels are est ablished using t he placement t est s. St udent s work on individual project s at home and under t he st ruct ured and
syst emat ic supervision of t he inst ruct or. T hese project s are gradually increased in di± cult y. Quizzes are used
t o det ermine if a st udent is ready t o move t o t he next level. Universit y of A kron, Summer Teaching Fellowship,
funded project . (Received M ay 28, 2002)
5002-G1-73
F l or en ce S. G or d on * (f gor don@nyi t . edu), 61 Cedar Road, E. Nort hport , NY 11731.
Tradi ti onal Col lege Algebra/ Tr i g ver sus a M odeli ng A pproach: Compar i ng Student
Per for mance.
NY I T conduct ed a comprehensive st udy comparing a modeling approach in college algebra/ t rig t o t he t radit ional course t hat emphasized developing algebraic skills. T he modeling course feat ures a more act ive learning
environment , st ressing t he behavior of families of funct ions and t he not ion of ¯t t ing funct ions t o dat a as t wo
unifying t hemes. St udent s in t he modeling sect ions were challenged wit h non-rout ine concept ual problems and
realist ic applicat ions t hroughout . T hey also conduct ed several individual project s and wrot e project report s
applying t he mat hemat ical ideas and met hods t o set s of real-world dat a on t opics of int erest t o t hem. T hey did
not get an explicit emphasis on developing algebraic skills. T he major issues st udied in t he college algebra/ t rig
courses were how t he st udent s in bot h groups react ed t o t he t wo approaches and how t hey performed on a
series of common problems on t he respect ive ¯nal exams in t erms of algebra skills. T he st udy cont inued on int o
calculus. Based on t heir college algebra/ t rig background, t he st udy examined how t he st udent s performed in
calculus on quizzes, class t est s, t he ¯nal exam and t heir overall persist ence and success rat es. T he st udent s from
a modeling approach out performed in every cat egory. (Received M ay 31, 2002)
5002-G1-97
C ar ol E . Seam an * (seaman@uwosh. edu), M at hemat ics Depart ment , Universit y of
W isconsin Oshkosh, 800 A lgoma Blvd., Oshkosh, W I 54901, J en n i fer E ar l es Szy d l i k
(szydl i k@uwosh. edu), M at hemat ics Depart ment , University of W isconsin Oshkosh, 800
A lgoma Blvd., Oshkosh, W I 54901, J am es M ar t y (mar t y@uwosh. edu), M at hemat ics
Depart ment , Universit y of W isconsin Oshkosh, 800 A lgoma Blvd., Oshkosh, W I 54901, and
St ep h en D . Szy d l i k (szydl i ks@uwosh. edu), M at hemat ics Depart ment , Universit y of
W isconsin Oshkosh, 800 A lgoma Blvd., Oshkosh, W I 54901. Compar i son of Preser vi ce
Elementar y Teacher s' Beli efs: 1968 and 1998. Preliminary report .
We describe t he beliefs of a large sample of element ary educat ion st udent s as t hey progress t hrough t heir t eacher
educat ion program. T he st udy replicat es Collier' s (1972) work of t hree decades ago. T he st udy focuses on t he
beliefs of t eachers about bot h t he nat ure of mat hemat ics and t he t eaching of mat hemat ics using scales t hat
measure what Collier t ermed an " formal-informal" dimension of belief and what t oday might be charact erized
as an inst rument alist -problem solving (Ernest , 1998) dimension of belief. We report dat a from bot h t he 1968
and t he 1998 replicat ion st udy and draw comparisons. Dat a for bot h st udies were collect ed from st udent s
at four st ages of mat hemat ical preparat ion: 1) prior t o t aking t he ¯rst mat hemat ics cont ent course; 2) aft er
complet ion of one cont ent course; 3) aft er complet ion of at least t wo cont ent courses; and 4) aft er complet ion
of all mat hemat ics cont ent courses and t heir mat hemat ics met hods course. I n our present at ion we will describe
di®erences in t he quest ionnaire responses of st udent s at each st age of t heir program and compare t he beliefs of
t oday' s preservice t eachers t o t hose of t he past . (Received June 03, 2002)
5002-G1-111
P am C r aw for d * (pcr awf o@j u. edu), Depart ment of M at hemat ics, Jacksonville Universit y,
2800 Universit y Boulevard Nort h, Jacksonville, Florida 32211, and N el l R ay b u r n
(r aybur nn@apsu. edu), Depart ment of M at h & Comput er Science, A ust in Peay St at e
Universit y, P. O. Box 4626, Clarksville, Tennessee 37044. Pedagogi cal i mpli cations of
college students' under standi ngs of scale.
A complet e underst anding of Cart esian graphical represent at ion is mult ifacet ed and complex. One aspect of t he
int erpret at ion and const ruct ion of meaningful graphs is t he realizat ion of t he role t hat scale plays. Yet many
college algebra, precalculus and calculus t ext books t ake for grant ed st udent s' facilit y wit h issues of scale in t hat
oft en scale is given for paper-and-pencil work, is aut omat ically det ermined by graphing t echnology, or is t he same
on bot h axes in t ext book examples. T his paper present s t he aut hors' st udy of t he relat ionship bet ween st udent s'
underst anding of scale and t heir A POS (A ct ion-Process-Object -Schema) funct ion level. Fift een college st udent s
at various mat hemat ical levels were quest ioned t o assess t heir underst anding of graphical represent at ions of
funct ions. T he aut hors t hen furt her explored t he at t it udes, underst anding, and knowledge of t wo of t he st udent s
regarding t heir concept s of scale as re° ect ed in Cart esian graphs. T hese st udent s were chosen because of t he
richness of informat ion in t heir int erview responses and because t he aut hors found various aspect s of t heir
16
RUM E SI GM A A SESSI ON
responses surprising. T he pedagogical implicat ions of t he aut hors' st udy of st udent s' underst anding of scale will
be discussed. (Received June 04, 2002)
5002-G1-128
C i n d y St en ger * (cst enger @mnu. edu). I mproving M athemati cal T hinki ng in
Undergraduates T hrough I nvolvement i n a Cooperati ve Research Project.
T his present at ion highlight s a research project int o an applied st at ist ics course. T he NCT M St andards est ablished t hinking, reasoning, and problem solving as cent ral goals of mat hemat ics educat ion and described ways
t o implement t hese goals t hrough cooperat ive learning, project s, group work, and t echnology implement at ion.
A cent ral goal of mat hemat ics educat ion research cont inues t o be t hat of underst anding t he nat ure of mat hemat ical learning and mat ching inst ruct ion t o t his developing underst anding. M at hemat ical t hinking has been
de¯ned bot h in t erms of t he views and skills which st udent s possess. St enger found t hat mat hemat ical t hinking
de¯ned according t o Schoenfeld was rare among universit y mat hemat ics st udent s at all levels. I n order t o obt ain
a snapshot of undergraduat e mat hemat ical t hinking, St enger ident i¯ed t he best and worst case st udent s wit h
regard t o mat hemat ical skills and views. T hese st udent s were ident i¯ed as mat ure and immat ure and t he skills
and views t hat provided t he great est di®erent iat ion bet ween t he t wo groups were ident i¯ed. For example, it
was found t hat immat ure st udent s did not underst and t he import ance of mat hemat ics t o t hem personally or
t he commit ment , personal responsibilit y, and perseverance involved in doing mat hemat ics. (Received June 05,
2002)
5002-H1-17
C at h er i n e A . R ob er t s* (cr ober t s@hol ycr oss. edu), Depart ment of M at hemat ics &
Comput er Science, Box 100A , College of t he Holy Cross, 1 College St reet , Worcest er, M A
01610. A cour se cal led T he Science of A r t.
T he t alk will discuss t he course \ T he Science of A rt " , designed as a liberal st udies course for sophomores and
juniors. I t was t eam t aught by t hree professors A F a mat hemat ician, a chemist and an art hist orian. T he
seminar-st yle course examined myriad int erfaces bet ween art and science. Sample t opics included art forgery,
t he chemist ry of st ained glass, pat t erns in Islamic t ile art and t he mat hemat ical t echnique of drawing in onepoint perspect ive. T his t alk will present t he nut s-and-bolt s of designing and delivering t his web-enhanced course,
including a discussion of inst it ut ional support and t he challenges and rewards of t eam t eaching. (Received M ay
15, 2002)
5002-H1-22
H ol l y S Z u l l o* (hzul l o@car r ol l . edu), M at hemat ics Depart ment , Carroll College, Helena,
M T 59625. Teachi ng Di ®erence Equati ons to Busi ness M aj or s.
Business majors are t ypically t aught t o calculat e compound int erest and mort gage payment s t hrough t he use
of formulas t hat have lit t le or no meaning t o t he st udent s. W it h di®erence equat ions, st udent s can learn t o
writ e relat ively simple models t hat perfect ly describe t he changing balances of a bank account or a loan. T he
models can t hen be st udied numerically and analyt ically, giving st udent s t he capability of calculat ing a mont hly
balance, t ot al int erest earned/ paid, and t ime unt il a loan is paid o®. T he numerical solut ions can be generat ed
using Excel, an import ant t ool for business majors. T he mat hemat ics and t he spreadsheet perfect ly complement
one anot her, wit h di®erence equat ions being an ideal sit uat ion in which t o use relat ive addressing for moving
from one t ime period t o t he next , and absolut e addressing t o allow easy changes t o a cert ain paramet er such as
int erest rat e. T he graphical capabilit ies of Excel are used t o generat e graphs of account balances over t ime. T his
present at ion will illust rat e how t hese t opics are t aught in a 100-level course t hat has only high school algebra as
a prerequisit e. (Received M ay 19, 2002)
5002-H1-24
M i ch ael A . J on es* (j onesma@pegasus. mont cl ai r . edu), Depart ment of M at hemat ical
Sciences, M ont clair St at e Universit y, Upper M ont clair, NJ 070431. Optimi zati on of
K naster ' s I nher i tance Procedure and How to Use I t as a Transi ti on to Game T heor y i n a
M ath for L i beral Ar ts Cour se.
K nast er' s I nherit ance Procedure was proposed by Bronislaw K nast er in 1945 t o divide any number of discret e
goods bet ween any number of players; it requires t he players t o place monet ary values or bids on all of t he
goods. T his procedure is oft en discussed in M at h for L iberal A rt s courses t hat concent rat e on cont emporary
applicat ions of mat hemat ics for non-major st udent s. T his fair division procedure provides an opport unit y t o
int roduce opt imizat ion t o st udent s who will never learn t he calculus. A simple analysis of t he procedure can lead
st udent s t o det ermine t he opt imal monet ary values t o place on t he it ems, given t he bids of t he ot her players. T he
opt imizat ion problem nat urally leads t o pure st rat egy Nash equilibria of t he K nast er' s I nherit ance Procedure
I NNOVAT I V E M ET HODS I N COURSES FOR NON-M A JORS
17
when viewed as a game, t hereby providing a t ransit ion bet ween fair division procedures and game t heory in
M at h for L iberal A rt s courses. (Received M ay 20, 2002)
5002-H1-37
R ach el W . H al l * (r hal l @sj u. edu), Depart ment of M at h and C.S., Saint Joseph' s
Universit y, 5600 Cit y A ve., Philadelphia, PA 19031. M ulticultural M athemati cs.
Preliminary report .
T he st udy of mult icult ural mat hemat ics aims t o st rengt hen and expand st udent s' underst anding of fundament al
mat hemat ics, including arit hmet ic, algebra, and geomet ry, t hrough comparat ive st udy of t he mat hemat ics of
world cult ures; t o encourage st udent s t o appreciat e t he cont ribut ions of all cult ures t o t he development of
mat hemat ics; t o explore t he connect ions bet ween mat hemat ics, art , and music; and t o increase diversity in
mat hemat ics enrollment and course o®erings. T his t alk det ails a course t aught in mult icult ural mat hemat ics
for liberal art s maj ors at Saint Joseph' s Universit y. I will present a sample syllabus, reading list , and ideas for
st udent project s, and discuss st udent response t o t he course. (Received M ay 23, 2002)
5002-H1-54
A gn es M . R ash * (ar ash@mai l host . sj u. edu), Dept . of M at hemat ics and Comput er
Science, Saint Joseph' s Universit y, 5600 Cit y A venue, Philadelphia, PA 19131. M athemati cs
M odels i n Chemi str y { An I nnovation for Nonmathemati cs and Nonsci ence M aj or s.
Preliminary report .
At many colleges and universit ies, st udent s in t he humanit ies, social sciences and business majors are not required
t o t ake a laborat ory science course nor a calculus course. However, t here are many bright st udent s in t hese
programs who are capable of underst anding t he int erconnect ion bet ween mat hemat ics and science and bene¯t ing
from an int erdisciplinary course wit h dept h as well of breadt h of t opics. T his present at ion discusses a course
designed t o ful¯ll t he core requirement s in mat hemat ics and science for non-mat hemat ics and non-science majors.
One course opt ion for t hese bright st udent s is a t wo-semest er course, wit h a lab component , in mat hemat ics and
chemist ry t aught in t he Honors Program. T he course, M at hemat ical M odels in Chemist ry, involves select ed t opics
in mat hemat ics from discret e mat hemat ics and calculus t hat have applicat ions in chemist ry. T he list of t opics will
be given wit h a det ailed explanat ion of t hree of t he unit s and t he associat ed laborat ory experiment s. Examples
of t he mat hemat ics component include mat rix operat ions, mult ivariat e funct ions, level curves, derivat ives and
t heir applicat ions in chemist ry. (Received M ay 28, 2002)
5002-H1-76
R al p h A C zer w i n sk i * (r czer wi nski @mai l . mi l l i ki n. edu), M at h and Comput er Science
Dept , M illikin Universit y, 1184 W M ain St , Decat ur, I llinois 62522. A Honor s Cour se:
M athematics and the V i sual Ar ts.
T he st udent s t aking t his course were sophomore honors st udent s sat isfying a requirement in mat hemat ics and
t he nat ural sciences. Present ers of t hese courses were encouraged t o be innovat ive and int erdisciplinary while
maint aining a high level of expect at ion. T he st udent s in my course were majors in t heat re art s, music, biology,
polit ical science, and physics. I n my course descript ion I indicat ed t hat we would explore several of t he many
bridges bet ween t he visual art s and mat hemat ics. Topics t hat I planned for us t o st udy included mat hemat ics
as a subject mat t er for art , t he import ance of pat t ern and symmet ry in bot h mat hemat ics and art , t he role of
aest het ics in t he development of mat hemat ics, and t he mat hemat ical approach in cont emporary art . I proposed
t o ¯nd int erplay and commonalit y bet ween t he visual art s and mat hemat ics. T he st udent s wrot e short papers,
solved a wide variet y of mat hemat ics exercises, lead discussions on t he readings, wrot e one long paper, creat ed
a mat h/ art object , and t ook exams. I will discuss t he organizat ion of t he course; t he challenges, successes, and
failures of t he init ial o®ering of t he course; and planned revisions for next year. (Received M ay 31, 2002)
5002-H1-109
E v el y n C B ai l ey * (ebai l ey@emor y. edu), M at hemat ics Depart ment , Oxford College of
Emory Universit y, 100 Hamill St reet , Oxford, GA 30054, and Fan g C h en
(f chen2@emor y. edu), M at hemat ics Depart ment , Oxford College of Emory Universit y, 100
Hamill St reet , Oxford, GA 30054. Statisti cs Exper i ments T hat W or k, T hat Eli ci t
Di scussion, and T hat Requi re Problem-Solvi ng Ski l ls. Preliminary report .
Experiment at ion should be an int egral part of int roduct ory st at ist ics courses. T hese experiment s are eit her
open-ended for st udent s t o design or algorit hmic where a procedure is provided. T his session will explore t he use
of bot h format s of experiment s using t he Chi-Square goodness-of-¯t model. Experiment s using t his model can
lead t o class discussions about random sampling, about appropriat e procedures, and about possible ext ensions of
experiment s. A second set of experiment s provides opport unit ies for st udent s t o use t he linear regression model
which can lead t o class discussions of misconcept ions relat ed t o linear regression. St udent s become excit ed and
act ive learners t hrough t he use of open-ended experiment at ion. T hree speci¯c, ¯eld-t est ed st udent experiment s
will be discussed in t he present at ion. (Received June 03, 2002)
18
I NNOVAT I V E M ET HODS I N COURSES FOR NON-M A JORS
5002-H1-123
J an et L A n d er sen * (j ander sen@hope. edu), Dept of M at hemat ics, Hope College, Holland,
M I 49422-9000. Connecti ng Sci ence and M athemati cs General Educati on Cour ses: One
A pproach to Quanti tative L i teracy.
St udent s oft en see general educat ion requirement s as disconnect ed courses t o be checked o®. T his percept ion
is reinforced when general educat ion science courses are primarily qualit at ive and mat hemat ics courses ignore
applicat ions (scient i¯c or ot herwise). At Hope College, wit h t he part ial support of NSF, we have developed 3
general educat ion courses. T he mat hemat ics course (M at hemat ics in Public Discourse) is a co- or pre-requisit e for
t he science courses (T he At mosphere and Environment al Change and Populat ions in a Changing Environment ).
T he courses are connect ed by cont ent t hemes (use of funct ions, graphs, and st at ist ical analysis of dat a); pedagogy
(explorat ory labs, invest igat ive worksheet s, cooperat ive learning); and t echnology (T I -83 and CBL ). I n t he
mat hemat ics course, st udent s complet e reading assignment s prior t o class. A bout 75% of class t ime is spent
doing group act ivit ies using mat erials developed from newspaper art icles, magazines, and ot her mat erials (such
as advert isement s and web pages). T he T I -83 and CBL allow us t o bring scient i¯c experiment at ion and dat a
analysis int o t he mat h classroom. T he mat hemat ics course was ¯rst t aught in spring 1998 and is current ly t aken
by about 35% of t he st udent body. (Received June 04, 2002)
5002-H1-126
M ar y E . Sear cy * (sear cyme@appst at e. edu), Depart ment of M at hemat ical Sciences,
A ppalachian St at e Universit y, Walker Hall, Boone, NC 28608. Helping Non-M aj or s V i ew
M athematics as an I l l-Str uctured Disci pli ne.
A powerful t ool of mat hemat icians is being able t o view mat hemat ics as an " ill-st ruct ured discipline" , where
int erpret at ion is int egral t o t he process of doing and underst anding mat hemat ics and debat es about int erpret at ion
and implicat ions occur nat urally. However, st udent s in undergraduat e mat hemat ics courses, part icularly nonmajors, do not oft en share t his view. I n t hese cases, it may be helpful t o provide st udent s wit h an int roduct ory
project t hat allows t hem t o broaden t heir perspect ives of mat hemat ics. One such proj ect involves having st udent s
answer t he quest ion, " How many barrels of wat er did Columbus t ake wit h him on his voyage t o t he New World?"
T here is no " answer" for st udent s t o ¯nd because t he voyage manifest s have been lost . T he mat hemat ics involved
is simple arit hmet ic, which allows accessibilit y and helps keep t he focus on t he process. T hrough t his project
st udent s experience t he need for researching a given sit uat ion, st at ing assumpt ions, t ranslat ing knowledge and
assumpt ions int o mat hemat ical st ruct ures, encount ering several " possible" solut ions, and evaluat ing and re¯ning
t heir solut ion process. T he experience provides a foundat ion in t he cognit ive sca®olding for a given course and
challenges st udent s' not ions regarding t he nat ure of mat hemat ics. (Received June 04, 2002)
5002-I 1-57
J e®r ey A G r ah am * (gr aham@susqu. edu), Depart ment of M at hemat ical Sciences,
Susquehanna Universit y, 514 Universit y A venue, Selinsgrove PA 17870. T he M yster y of
f x x f y y ¡ (f x y ) 2 Solved.
M ost calculus books give t he formula in t he t it le and some condit ions for det ermining whet her a crit ical point of a
real-valued funct ion of t wo variables is a maxima, minima, saddle, or unknown. T here is usually no explanat ion
of why t he formula works nor is t here given any met hod for det ermining maxima or minima for real-valued
funct ions of more t han t wo variables. I n t his t alk, we will show how a lit t le bit of linear algebra and our
experience from one dimensional calculus can ¯ll bot h of t hese holes. (Received M ay 29, 2002)
5002-I 1-85
J am es R . B ozem an * (j ames. bozeman@l sc. vsc. edu), Depart ment of M at hemat ics,
L yndon St at e College, L yndonville, V T 05851. Polar M yster i es and Eccentr i ci ti es.
I n t his t alk I will present examples of project s assigned each t ime I t each mult ivariable calculus. T hese project s
are designed t o encourage st udent s t o furt her explore t opics which have been covered in t he class. I especially
t ry t o give project s in which t echnology, in part icular graphing calculat ors and comput ers, can be used. T he
project s t hat t he audience will work on in t his present at ion are: Polar M yst eries I n t his project st udent s are
asked t o graph t wo polar curves and visually check for point s of int ersect ion. T hey are t hen asked t o solve
t he t wo equat ions simult aneously. T he answers t o t hese t wo quest ions are di®erent and t he st udent s are asked
t o complet ely explain why. T his project helps st udent s t o see some of t he subt let ies of polar coordinat es as
compared t o rect angular coordinat es. Polar Eccent ricit ies I n t his project st udent s cont inue t heir explorat ion
int o polar graphs, t his t ime polar equat ions of conics t hat represent orbit al mot ion. Using K epler' s laws, t he
role of t he eccent ricit y, e, is invest igat ed bot h analyt ically and graphically. For example, what must be t rue
of orbit al speeds for di®erent values of e? I n part icular, what happens in t he case of a circular orbit vis-µ
a-vis
K epler' s L aws. (Received June 03, 2002)
ENL I V ENI NG M ULT I VA RI AT E CA L CUL US
19
5002-I 1-86
J am es R . B ozem an * (j ames. bozeman@l sc. vsc. edu), Depart ment of M at hemat ics,
L yndon St at e College, L yndonville, V T 05851. Tr i ple I ntegrals and the Four th Di mension.
I n t his t alk I will present an example of a project assigned each t ime I t each mult ivariable calculus. T his project is
designed t o encourage st udent s t o furt her explore t opics which have been covered in t he class. T he problem also
helps st udent s t o bet t er underst and t hese ideas, and can lead t o lively discussions. T he project t hat t he audience
will work on in t his present at ion is: T he Hypervolume of t he 4-dimensional Ball In t his project st udent s are
asked t o use t riple int egrals t o ¯nd t he hypervolume of t he 4-dimensional ball. T his problem requires ext ending
t he ideas used in ¯nding t he volume of t he 3-dimensional ball wit h double int egrals. I t is necessary t o use
int egral t ables t o solve t he t riple int egrals in bot h spherical and cylindrical coordinat es, and t he st udent s are
also required t o writ e t he formula in rect angular coordinat es. T his project helps t he class t o bet t er underst and
t hese coordinat e syst ems and t heir uses in evaluat ing int egrals. T he fact t hat we are ¯nding a result about 4dimensions leads t o many int erest ing discussions. T he book Flat land is used as a resource for t his invest igat ion
and ot her examples of hypervolumes over 3-spheres are exhibit ed. (Received June 03, 2002)
5002-I 1-107
Sar ah L M ab r ou k * (smabr ouk@f r c. mass. edu), Framingham St at e College, M at hemat ics
Depart ment , 100 St at e St reet , PO Box 9101, Framingham, M A 01701-9101. W elcome To
3-Space.
For most st udent s, ot her t han considering t he solids of revolut ion examined during Calculus I , working in and
visualizing mat hemat ics in 3-space is foreign t o t hem. W hile previously t hey considered only t he x-y plane and
lines and curves t hat t hey could draw in t his plane, in M ult ivariat e Calculus, st udent s are asked t o examine and
t o graph lines and planes t hat no longer must lie ° at on t heir deskt op and t o analyze equat ions t hat generat e
surfaces t hat could correspond t o shapes t hat t hey creat ed using Play-Doh when t hey were children.
I n an e®ort t o help st udent s t o become comfort able wit h t heir st udy of mat hemat ics in 3-space, I creat ed an
ongoing assignment t hat provides st udent s wit h opport unit ies t o explore mat hemat ics in 3-space as t hey learn
M aple and explore some physical sit uat ions. Part of t he assignment requires st udent s t o recognize and t o model
t he mat hemat ics t hat t hey are st udying in t he world around t hem. I n t his paper, I will discuss some of t he
aspect s of t his assignment , present some of t he M aple Worksheet s t hat t he st udent s will use, and discuss some
simple t ools t hat can be used t o help st udent s t o visualize mat hemat ics in 3-space. (Received June 03, 2002)
5002-I 1-122
J oh n B . G ed d es* (j bgeddes@ci suni x. unh. edu), Depart ment of M at hemat ics and
St at ist ics, 33 College Road, Universit y of New Hampshire, Durham NH 03824, and K el l y
B l ack (kel l y. bl ack@unh. edu), Depart ment of M at hemat ics and St at ist ics, 33 College
Road, Universit y of New Hampshire, Durham NH 03824. Usi ng a Force Table to M otivate
Systems.
We have adapt ed a st andard force t able used in t he int roduct ory physics courses int o a dynamic experiment
which allows st udent s t o explore some key concept s in mult ivariat e calculus. We use t he apparat us t hroughout
t he course t o: const ruct syst ems of equat ions for bot h t he st at ic and dynamic problem; ¯nd root s for t he st at ic
problem when t he balance point is unknown; develop syst ems of di®erent ial equat ions and corresponding solut ion
curves t o describe t he dynamic problem; mot ivat e t he not ion of pot ent ial funct ion. We revisit t his experiment
t hroughout t he course, and st udent s are required t o writ e a report and make explicit connect ions bet ween many
of t he cent ral ideas examined. During t he present at ion we will provide an overview of t he di®erent act ivit ies for
which we have used t he apparat us, as well as a discussion on how it is used t o t ie t oget her a variet y of t opics in
mult ivariat e calculus. (Received June 04, 2002)
5002-I 1-127
R ob er t J H esse* (r hesse@csbsj u. edu), Saint John' s Universit y, Collegeville M N,
56321-3000. Arch M athemati cs.
I n t he spring of 1999 and 2001 I incorporat ed a series of M at hemat ica project s or \ labs" int o a vect or analysis
course. T hese labs used t he t echnology of a Comput er A lgebra Syst em (CA S) t o furt her develop a part icular
concept , solve an applicat ion, or bring t oget her several disj oint ideas. By far t he most popular lab explored
t he mat hemat ics behind t he Je®erson Nat ional Expansion M emorial, commonly called t he A rch. Besides being
a must -see for any t ourist visit ing Saint L ouis, t he A rch has some nice mat hemat ical propert ies. T he cent er
of t he A rch follows t he pat h of an upside-down cat enary curve: A cosh(t =A ) and cross-sect ions of t he A rch are
equilat eral t riangles. I n t he lab, st udent s found a paramet erizat ion of t he A rch and used t his paramet erizat ion
t o comput e it s surface area. I n t his t alk I will go over a lit t le of t he mat hemat ics involved t o solve t he problem
as well as t he impact t he project had wit h t he st udent s. (Received June 04, 2002)
20
T HE ROL E OF PROOF I N T EACHI NG M AT HEM AT ICS
5002-J1-32
T ed Su n d st r om * (sundst r t @gvsu. edu), Depart ment of M at hemat ics, Grand Valley St at e
Universit y, A llendale, M I 49401. Explorati on, W r iti ng, and Proof in M athematics.
A t ransit ion course from t he problem solving orient at ion of calculus courses more abst ract upper level courses
is an import ant course in t he mat hemat ics major at many universit ies. At Grand Valley St at e Universit y, a
dist inguishing charact erist ic of t his course is t hat it is a part of t he Universit y' s Supplement al Writ ing Skills
Program. A s such, t he inst ruct or is required t o work wit h t he st udent s on revising draft s of papers before ¯nal
submission. T his requirement is sat is¯ed wit h t he Port folio of Proofs, which consist s of t en proposit ions t o be
proven or disproven. St udent s may hand in each proof t o t he professor t wo t imes t o be crit iqued. T he goal
is t hat each st udent will have a complet ed Port folio of Proofs t hat will illust rat e t he various proof t echniques
t hat st udent s may encount er in lat er courses. We at t empt t o achieve t his and ot her course object ives wit h an
act ive learning environment by incorporat ing t wo or t hree preview act ivit ies t hat must be complet ed before each
class. We also use classroom act ivit ies t hat can be done individually or in a collaborat ive learning set t ing where
st udent s work in groups t o brainst orm, make conject ures, t est each ot hers' ideas, reach consensus, and hopefully,
develop sound mat hemat ical argument s t o support t heir work. (Received M ay 21, 2002)
5002-J1-34
D or a C ar d en as A h m ad i * (d. ahmadi @mor eheadst at e. edu), Depart ment of M at hemat ical
Sciences, L appin Hall, M orehead St at e Universit y, M orehead, K Y 40351, and K at h r y n M .
L ew i s* (k. l ewi s@mor eheadst at e. edu), Depart ment of M at hemat ical Sciences, L appin Hall,
M orehead St at e Universit y, M orehead, K Y 40351. Cooperati ve Lear ni ng and Explorati ons
to Engage Students i n Lear ni ng Proof. Preliminary report .
M orehead St at e Universit y o®ers a course t it led \ I nt roduct ion t o M at hemat ical Proof " , which is required for
mat hemat ics majors and for st udent s majoring in middle grades educat ion wit h a mat h component . T his course
is a prerequisit e for some of t he upper level mat hemat ics courses. I t is import ant because it provides a bridge
bet ween courses wit h most ly comput at ional problems (such as calculus) and courses wit h most ly t heoret ical
proofs (such as abst ract algebra). T he present ers will t alk about t he use of cooperat ive learning and explorat ions
in helping st udent s learn t o writ e and read proofs and t o get t hem involved during class. (Received M ay 22,
2002)
5002-J1-43
Su san n a S. E p p * (sepp@condor . depaul . edu), Depart ment of M at hemat ical Sciences,
DePaul Universit y, 2320 N. K enmore, Chicago, I L 60614. I ssues in teaching the pr i nci ples
of logic.
Proof is somet imes regarded as a somewhat pedant ic ¯nal check of st at ement s already known t o be t rue. T his
view is a reason given for present ing mat hemat ics informally in t he ¯rst t wo college years. However t he t hought
processes needed for polished proofs are also needed for t heir discovery. I n a formal proof, logical principles are
used syst emat ically t o develop a coherent , complet e line of reasoning from hypot hesis t o conclusion. I n discovery,
one allows oneself t o make assumpt ions, leave gaps, jump bet ween point s, reason backward from a conclusion t o
what would make it t rue, et c. But t hroughout t he discovery process, one needs an inst inct for logical reasoning.
Unfort unat ely st udent s do not t ypically ent er college wit h t his inst inct . T hrough close work wit h st udent s in
t ransit ion-t o-higher-mat h and discret e mat h courses, I have come t o believe t hat an e®ect ive way t o improve
most st udent s' logical reasoning skills is t o spend some t ime t eaching logical principles explicit ly. However, when
t he focus is on t eaching logic as a subj ect , st udent s may learn t o obt ain correct answers wit hout developing t he
inst inct for t hinking logically. I will discuss met hods t o address t his problem and help st udent s deepen t heir
underst anding of mat hemat ical reasoning. (Received M ay 24, 2002)
5002-J1-47
J e® J oh an n es* (j ohannes@member . ams. or g), M at hemat ics Depart ment , 1 College Circle,
Geneseo, NY 14454-1401. W elcome to mathemati cs: a cor ner stone exper ience. Preliminary
report .
A n int roduct ion t o proofs course is t he cornerst one of t he undergraduat e mat hemat ics major. I n t his course,
we have t he opport unit y t o int roduce our st udent s t o t he language of mat hemat ics, t o preview upper division
mat hemat ics courses and t o prepare st udent s for st udying more advanced mat hemat ics. T his t alk will view t he
course from t he perspect ives of t he st udent , of t he inst ruct or, and of subsequent inst ruct ors. (Received M ay
28, 2002)
T HE ROL E OF PROOF I N T EACHI NG M AT HEM AT ICS
21
5002-J1-66
N an cy L i n ek en H agel gan s* (NHagel gans@Ur si nus. edu), M at hemat ics and Comput er
Science Depart ment , Ursinus College, Collegville, PA 19426. Proof i n Abstract Algebra.
T his t alk addresses an A bst ract A lgebra course t hat follows, and builds on, a Discret e M at hemat ics course t hat
serves as t he t ransit ion t o upper level mat hemat ics courses at Ursinus College. T he main t opics in t he t ransit ion
course are logic and met hods of proof, but most proofs involve uncomplicat ed assert ions, and t hese proofs have
a simple st ruct ure employing just one proof met hod. Despit e t hese limit at ions, t he course helps st udent s acquire
t he vocabulary t o discuss proofs as well as a sense of when t o apply various met hods of proof. I n t he A bst ract
A lgebra course I use act ivit ies t hat t ake advant age of t his knowledge t o ext end st udent s' abilit y t o analyze more
complicat ed assert ions and proofs. St udent s read a few pages of new mat erial before class, and aft er a brief class
lect ure, t hey work in t eams on proving relat ed st at ement s. We frequent ly ¯rst discuss a st rat egy for proving a
complicat ed assert ion and develop an out line for a proof wit h ellipsis t o indicat e where det ails are needed for
complet ion. St udent s can t hen concent rat e on t he new algebraic ideas as t hey complet e t he proof. (Received
M ay 29, 2002)
5002-J1-74
O l y m p i a N i cod em i * (ni codemi @geneseo. edu), Depart ment of M at hemat ics, 325B Sout h
Hall, SUNY Geneseo, Geneseo, NY 14454, and M or r i s O r zech
(or zechm@mast . queensu. ca), Depart ment of M at hemat ics and St at ist ics, Queen' s
Universit y, K ingst on, Ont ario K 7L 3N6, Canada. Teaching Proof { Some Naggi ng
Questi ons. Preliminary report .
T he last hour of our session will be devot ed t o an audience-based discussion of issues raised or suggest ed by
t he t alks. T he present at ions will have dealt wit h t he rat ionale, cont ent , pedagogy and success of a " proofs"
course. St ill, import ant quest ions remain. W hat are t he barriers t o st udent s accept ing proof as necessary for
mat hemat ical underst anding? Does our professional pract ice as mat hemat icians somet imes lead us as t eachers
t o set t oo high a st andard for explanat ions t hat would make us say, " you underst and" t o t he st udent ? Do
we somet imes persist t oo early, or beyond a point of diminishing ret urns, in t rying t o convince st udent s t hat
proof, in t he argot of t he discipline, should be t heir st andard for mat hemat ical t rut h? M ight we t hus neglect
or even block ot her avenues of discovery and invest igat ion, such as visualizat ion, or qualit at ive descript ion, or
comput er-based explorat ion t hat would lead t o a more st udent -generat ed and st udent -perceived need for proof ?
We hope t hat t hese quest ions, and t hose of audience members, will lead t o a lively exchange of views. (Received
M ay 31, 2002)
5002-J1-77
C on n i e M . C am p b el l * (campbcm@mi l l saps. edu), Depart ment of M at hemat ics, M illsaps
College, Jackson, M S 39210. I s that what you real ly meant? Helpi ng students develop
preci se mathemati cal language. Preliminary report .
I have part icipat ed in a t hree-year st udy which was designed t o clarify how st udent s learn t o underst and and
const ruct mat hemat ical proofs. A s part of t his st udy, groups of st udent s in a t ransit ion course were videot aped
as t hey const ruct ed several element ary mat hemat ical proofs. T hose proof sessions were analyzed t o evaluat e t he
met hods ut ilized by t he st udent s as t hey developed and wrot e proofs. T he videos revealed several misconcept ions t he st udent s held concerning t he relat ionship bet ween logic and proof. I n addit ion, t here were frequent
inconsist encies bet ween what t he st udent s said and what t hey wrot e in t heir proofs. I will discuss some insight s
gained from hearing and wat ching st udent s work, and will describe some classroom act ivit ies designed t o help
st udent s develop t heir abilit y t o const ruct mat hemat ical proofs. (Received M ay 31, 2002)
5002-J1-94
M ar gar et L M or r ow * (mar gar et . mor r ow@pl at t sbur gh. edu), M at hemat ics Depart ment ,
SUNY Plat t sburgh, Broad St , Plat t sburgh, NY 12901. Students' concepti ons of proof.
T his present at ion will focus on mat erial from t he research lit erat ure on st udent s' beliefs about proof. Findings
t hat will be discussed include st udent s' misconcept ions about t he nat ure and role of proof, and t heir percept ion
of t he formal aspect s of proof. Some of t he research ¯ndings raise quest ions about whet her we should be doing
more t o address st udent s' beliefs about proof as we t ry t o assist t hem t o make t he t ransit ion t o proof based
courses. (Received June 03, 2002)
5002-J1-118
J am es D . T ook e* (j t ooke@eou. edu), School of Educat ion and Business, East ern Oregon
Universit y, One Universit y Blvd., L a Grande, OR 97850. Di scussi ng or De¯ni ng { How
Recogni zi ng A nother van Hi ele Level Can Prepare Students for Proof. Preliminary report .
T he van Hiele levels are a t heoret ical foundat ion many high school geomet ry books use t o analyze st udent
readiness for proofs. T his t alk will report on research t hat support s a modi¯cat ion of t he t radit ional levels.
A wareness of t his modi¯cat ion, and of t he reason for it , could help overcome t he ident i¯cat ion of su± cient
and necessary condit ions t hat bedevils many st udent s' approach t o proof. A n examinat ion of t he de¯nit ion of
22
T HE ROL E OF PROOF I N T EACHI NG M AT HEM AT ICS
rect angle will make t he point . A t ext book might say t hat a rect angle is a parallelogram wit h four right angles.
However, a parallelogram wit h just one right angle is a rect angle. I t is a nice observat ion or discussion t hat t he
ot her t hree angles be right angles, but it not part of t he de¯nit ion. T he t radit ional van Hiele levels analysis is
t hat at L evel 3 st udent s recognize necessary and su± cient condit ions, hence are ready for proof. However, aft er
a K -9 experience of seeing discussions t hat are called de¯nit ions, many st udent s can const ruct a st at ement about
a concept using necessary at t ribut es, but fail t o recognize when t hose at t ribut es are su± cient . I f anot her level
were est ablished, say 3B, t hen t eachers would t each t hrough it as t hrough t he ot hers. L evel 3B would call for
recognizing su± ciency, 3A for recognizing necessit y. (Received June 04, 2002)
5002-J1-119
P ad r ai g M .M . M cL ou gh l i n * (pmcl ough@mor ehouse. edu), Depart ment of M at hemat ics,
M orehouse College, At lant a, GA 30314. Usi ng a Fusion of M odi ¯ed M oore, Tradi ti onal and
Refor m M ethods to Teach Students to Do Proofs.
I n t his t alk I will argue t hat t he experience of doing a mat hemat ical argument is as good a reason for t he exercise
as t he ¯nished proof. I will discuss what t he inst ruct or can do t o guide t he st udent t hrough an experience
consist ent wit h t his point of view, while keeping psychological and philosophical underpinnings in mind. T he
aim of t he inst ruct ional set t ing I will survey (for t he t eaching of proof ) is t o int egrat e experient ial process and
¯nal product so as t o realize John Dewey' s vision of t he ends and means becoming t he same. (Received June
04, 2002)
5002-K 1-39
St ev en J B r am s* (st even. br ams@nyu. edu), Dept . of Polit ics, New York Universit y, 715
Broadway, 4t h Fl., New York, NY 10003. Fai r Di visi on of I ndi vi si ble Goods.
Dividing up t he propert y in an est at e, assigning draft players t o sport s t eams, allocat ing cabinet posit ions
t o polit ical part ies in a parliament ary government { t hese are all examples wherein indivisible goods (including
people) must be apport ioned. Problems t hat arise in making such a division fair, such as t he envy caused when
one person t hinks t hat anot her person received more t han he or she did, will be discussed. Procedures for
ameliorat ing such problems will also be considered. (Received M ay 23, 2002)
5002-K 1-48
K en n et h A . R oss* (r oss@mat h. uor egon. edu), Depart ment of M at hemat ics, Universit y of
Oregon, Eugene, OR 97403-1222. W hat are the odds?
Over decades of t eaching element ary probabilit y, I have emphasized t hat t he not ion of odds is an hist orical
accident , because of gambling, and t hat t he \ right " way t o look at t hings is via probabilit y. I n t his cont ext , I
have st ressed t hat t he road t o underst anding is a one-way st reet from societ y' s point of view (odds) t o t he point
of view of mat hemat ics (probabilit y). However, I recent ly realized t hat t here are examples in probabilit y where
t he ideas become simpler, especially for gamblers and freshmen, when expressed in t erms of odds. (Received
M ay 28, 2002)
5002-K 1-60
J oh n C . M acel i * (macel i @i t haca. edu), Depart ment of M at hemat ics & Comput er Science,
I t haca College, It haca, New York 14850. A Contemporar y General Educati on Cour se.
We will discuss t he curriculum for a general educat ion course t hat has been t aught for t he past t en years at
I t haca College. T his course is t aken by st udent s from all disciplines. T he course t eaches many of t he ideas
t hat people call \ quant it at ive reasoning" in t he cont ext of problems arising in t he everyday experiences of t he
st udent s. Topics have included: lot t eries, polling, mat h in t he media, and elect ions. We will brie° y describe
some of t he unit s from t he course along wit h some act ivit ies and project s. (Received M ay 29, 2002)
5002-K 1-61
St an Sel t zer * (sel t zer @i t haca. edu), Depart ment of M at hemat ics & Comput er Science,
I t haca College, It haca, New York 14850. A M ath Honor s Cour se for L i beral Ar ts Students.
How do you exemplify \ t he spirit of inquiry" (t he mot t o of our Honors Program) in a mat hemat ics course when
many of t he st udent s are at t he precalculus level? I will describe M at h & Polit ics, a course t hat deals wit h
apport ionment and vot ing wit h a st rong emphasis on bot h current event s and original sources. (Received M ay
29, 2002)
5002-K 1-68
C at h er i n e A . G or i n i * (cgor i ni @mum. edu), M aharishi Universit y of M anagement ,
Fair¯eld, I A 52557{ 1066. Number s, Codes, and Soci ety.
Codes and t he mat hemat ics behind t hem have become an int egral, if invisible, part of societ y. T his paper
describes a general educat ion course at M aharishi Universit y of M anagement t hat int roduces st udent s t o modular
GENERA L CONT RI BUT ED PA PER SESSION
23
arit hmet ic, t he met hodology of modern mat hemat ics, and t he applicat ion of mod arit hmet ic t o check digit s, errorcorrect ing codes, and crypt ology. St udent s are encourage t o develop and just ify t heir opinions about t he uses of
t hese t echnologies in societ y. (Received M ay 29, 2002)
5002-K 1-113
D ay l en e Z i el i n sk i * (dzi el i nski @bel l ar mi ne. edu), M at hemat ics Depart ment , Bellarmine
Universit y, 2001 Newburg Road, L ouisville, K Y 40205. Desi gni ng And Using Rubr ics To
A ssess And I mprove Student W r i ti ng. Preliminary report .
W hile t eaching a writ ing course at Bellarmine Universit y, I have searched for a way t o improve st udent writ e
and make assessing st udent writ ing more t ime e± cient for me and useful for t he st udent s.
Because grading writ ing is essent ially di®erent from grading mat h, I have had t o develop t echniques t hat
allowed me t o learn and execut e t his new t ype of grading. One of t he most helpful t echniques for bot h my
st udent s and me has been rubrics. I will present t echniques for developing and using rubrics for assessing st udent
writ ing which not only work, but also improve st udent writ ing and decrease grading t ime. (Received June 04,
2002)
5002-L 1-14
D av i d J am es H al i t sk y * (davi d. hal i t sky@cst - hsv. com), 2928 St ewart Campbell Point e,
T hompsons St at ion, T N 37179. T he Human Acqui si ti on of Car tesian Coordinates and the
Loss of I n° exion T wi xt Lati n and French: A Case of Unconsci ous Peircean Abduction?
Preliminary report .
T he human acquisit ion of analyt ic geomet ry occurred while French was losing/ had lost t he case in° ect ions of it s
ancest or L at in. A lt hough t he loss of case (L OC) and t he acquisit ion of analyt ic geomet ry (A AG) occurred in t wo
di®erent cognit ive domains, an import ant analogy bet ween L OC and A AG can nonet heless be seen by comparing
t he ways in which: a) L at in and French indicat ed t he grammat ical funct ion of nouns pre- and post -L OC; b)
mat h- emat icians de¯ned t he t rignomet ric funct ions sine and cosine pre- and post -A AG.
Pre-A AG, sine and cosine could only be de¯ned by AT TACHI NG L A BEL S (O,A ,H) t o t he sides of a right
t riangle. Post -A AG, sine and cosine could be de¯ned by orient ing t he t riangle relat ive t o t he EX T RI NSI C
FRA M EW ORK o®ered by a Cart esian coordinat e syst em, rendering t he pre-A AG labels unnecessary.
But since t he post -A AG " Cart esian coordinat e syst em" is an EX T RI NSI C FRA M EW ORK paralleled by
t he post -L OC EX T RINSI C FRA M EW ORK of " ¯xed word-order" , and since t he pre-A AG L A BEL S O,A ,H
are paralleled by t he PRE-L OC case-ending L A BEL S su± xed t o nouns t o indicat e grammat ical funct ions (e.g.
Nominat ive, A ccusat ive, Dat ive), A AG and L OC would seem t o be relat ed by a simple analogy of t he t ype which
was t o Peirce t he wellspring of successful human abduct ion. (Received M ay 14, 2002)
5002-L 1-21
G eor ge L . A sh l i n e* (gashl i ne@smcvt . edu), Box 187, Depart ment of M at hemat ics, St .
M ichael' s College, 1 W inooski Park, Colchest er, V T 05439. Secondar y Educati on Semi nar
i n M athematics: A Capstone Exper i ence. Preliminary report .
Preparing st udent s t o t each mat hemat ics at t he secondary level is of crit ical import ance, especially given t he
current nat ional and local short ages in quali¯ed mat hemat ics t eachers. I n t his present at ion, I will describe
t he seminar I creat ed t o provide a capst one experience for our st udent s int ending t o t each mat hemat ics at t he
secondary or middle school levels. T his course o®ers st udent s a cont ext in which t o prepare lessons and t ry
di®erent met hods of t eaching mat hemat ical concept s at various grade levels using various curricula. (Received
M ay 17, 2002)
5002-L 1-30
A l an K och * (akoch@agnesscot t . edu), Depart ment of M at hemat ics, A gnes Scot t College,
141 E. College A ve., Decat ur, GA 30030. A Case For T he T hree Dollar Bi l l.
W hy do we use 1, 5, 10, 20, 50, and 100-dollar bills? Is t here a bet t er choice for denominat ions? By using some
element ary mat hemat ics we will const ruct what we call an opt imal currency. A n opt imal currency is a choice of
denominat ions t hat minimizes, on average, t he number of bills needed t o purchase an it em. T he const ruct ion of
t he denominat ions will be subject t o t wo axioms in an at t empt t o simplify t he calculat ions for t he cust omer. One
int erest ing feat ure of t his opt imal currency is t hat it does not use t he ¯ve, t went y, or ¯ft y-dollar bill. (Received
M ay 21, 2002)
24
GENERA L CONT RI BUT ED PA PER SESSION
5002-L 1-36
A l l an K r i l l * (al l an. kr i l l @geo. nt nu. no), Dept of Geology and M ineral Engineering,
Nowegian Universit y of Science and Technology, 7491 Trondheim, Norway. Pseudonumerals
and pseudonumer i cs: better symbols and words for number s.
Roman numerals are un¯t for arit hmet ic, and A rabic numerals for memory. Our words for numbers (eg \ one
hundred and t wo" for 102) are cumbersome, and zero is missing. We use symbols when we work wit h numbers
on paper, but we use bot h symbols and words when we work wit h t hem in our minds. Our words do not in¯x t he
symbols. T he modern plague of unmemorable I D-numbers (t elephones, PI Ns, et c) gives mat h-numbers a bad
name. M et iculous st udent s, pot ent ial mat hemat icians, may shun illogical numbers t hat cannot be remembered.
I n 1634, Herigone made a code for memorizing numbers as words. Today' s code gives users great number-memory
abilit ies, and t en new symbols (pseudonumerals) can universalize t he code. T he symbols are st rike-t hrough let t ers
0-S 1-T 2-N 3- M 4-R 5-L 6-J 7-K 8-F 9-P and represent t he t en groups of consonant sounds. T hey are numbers,
but t heir sounds can be made int o words (pseudonums) by adding vowel sounds. T he word " pseudonumeral"
is a pseudonum for 012345 = ST NM RL (pSeuToNuM eRaL ). A pseudonum for M A A ' s phone number, 800 741
9415 = FSS K RT PRT L , is \ he faces a cardboard hill" (he FaSeS a K aRT PoaRT hiL ) Pseudonumerics are
t he simplest numbers ever invent ed. A s a pseudonumeric, 102 is T SN (symbols) and \ yiT iSiNy" (word.) See
ht t p:/ / pseudonumerology.com (Received M ay 22, 2002)
5002-L 1-40
K y l e L . R i l ey * (kyl e. r i l ey@sdsmt . edu), K yle Riley, Dept . of M at h and Comput er
Science, Sout h Dakot a School of M ines and Technology, Rapid Cit y, SD 57701-3995.
Conducti ng an Exter nal Revi ew of a M athemati cs Program.
T he M A A Guidelines for Programs and Depart ment s in Undergraduat e M at hemat ical Sciences recommends t he
use of a periodic ext ernal review as a t ool t o assess t he depart ment and t he mat hemat ics program. T he ext ernal
review should also play a role in t he st rat egic planning of t he depart ment . T his t alk will present some of t he
bene¯t s of using an ext ernal review and provide an out line on how t o conduct an ext ernal review. (Received
M ay 24, 2002)
5002-L 1-50
J am es N B oy d * (j nboyd19@i 2020. net ), St . Christ opher' s School, Richmond, VA 23226.
M AT HEM AT I CA as a Lever .
T his paper will present applicat ions of M AT HEM AT I CA which will enable t he user t o develop ideas in problem
solving. Solut ions as pursued which in t he past would not have been followed up because of t heir comput at ional
di± cult y. Problems in geomet ry, probabilit y, algebra, and mechanics will be considered. T he t hemes of t his
paper are t he learning, t eaching, and doing of mat hemat ics t hrough problem solving when ideas are not held
host age t o t he drudgery of comput at ion. (Received M ay 28, 2002)
5002-L 1-56
A l ex J . H ei d enb er g, Depart ment of M at hemat ical Sciences, Unit ed St at es M ilit ary
A cademy, West Point , New York 10996, E l i zab et h W . Sch ot t , Depart ment of
M at hemat ical Sciences, Unit ed St at es M ilit ary A cademy, West Point , New York 10996, and
J oh n A . W ask o* , Depart ment of M at hemat ical Sciences, Unit ed St at es M ilit ary A cademy,
West Point , New York 10996. Graphi ng Calculator ; Fr i end or Foe? Preliminary report .
A dvances in t echnology, from graphing calculat ors t o comput er algebra syst ems and handheld devices, have
signi¯cant ly changed t he college classroom. T hese advances have forced changes bot h in t he way mat erial is
present ed and in how and what st udent s learn. For inst ruct ors, it has raised t he quest ion of how t o inspire
a concept ual and t heoret ical underst anding of mat h concept s when t hese complex problems can oft en now
be solved wit h a simple command. To furt her explore t hese quest ions, a 4-week classroom experiment was
conduct ed in t he int roduct ory Calculus course at t he Unit ed St at es M ilit ary A cademy (USM A ). I n conduct ing
t his experiment , our goal was t o det ermine if placing more emphasis on t he t heory and less on t he import ance
of t he correct answer would lead t o long-t erm knowledge. We measured st udent performance via 5 di®erent
assessment opport unit ies over t he course of t wo semest ers. I n t he experiment al group t he graphing calculat or
was used ext ensively t o graphically reinforce t he t heoret ical and concept s in solving t hese problems. Cont rol
group cadet s, by comparison, were free t o ut ilize all aspect s of t he graphing calculat or. T his present at ion will
discuss t he result s and ¯ndings of our experiment . (Received M ay 29, 2002)
5002-L 1-58
J e®r ey A G r ah am * (gr aham@susqu. edu), Depart ment of M at hemat ical Sciences,
Susquehanna Universit y, 514 Universit y A venue, Selinsgrove PA 17870. I ntegrati on i n
L 2 [0; 1] i s Almost Easy.
I n t his paper, we will show t hat t he set of polynomials such t hat
Z 1
p(x)dx = p(1) ¡ p(0)
is dense in L 2 [0; 1].
0
(Received M ay 29, 2002)
GENERA L CONT RI BUT ED PA PER SESSION
25
5002-L 1-63
R u t h I B er ger * (ber ger r @l ut her . edu), Depart ment of M at hemat ics, L ut her College,
Decorah, I A 52101. Hidden Group Str ucture. Preliminary report .
You may have seen a problem like:" Show t hat 5, 15, 25, 35 is a group under mult iplicat ion modulo 40" in
Gallian' s A bst ract A lgebra book. T his is quit e puzzling t o st udent s since t here is no obvious ident it y. Even
aft er e= 25 has been ident i¯ed, t he quest ion of det ermining inverses requires some t hought . T he above set is in
fact 5* U(8), but t hat does not immediat ely explain it s group st ruct ure. T his t alk will provide det ails on why
k* U(n) is a group under mult iplicat ion modulo k* n, for cert ain k. T his will allow you t o produce many more
such challenging examples for your A bst ract A lgebra st udent s. (Received M ay 29, 2002)
5002-L 1-78
M i zan R K h an * (khanm@East er nCT. edu), Depart ment of M at hemat ics, East ern
Connect icut St at e Universit y, W illimant ic, CT 06226. On the distr i buti on of i nver ses
modulo n.
We use uniform dist ribut ion t o clarify 2 result s on t he di®erence of an element and it s mult iplicat ive inverse mod
n. T his is j oint work wit h J. Beck. (Received M ay 31, 2002)
5002-L 1-81
Sh ak i r M an sh ad * (smanshad@nmsu. edu), General St udies Depart ment , Dona A na Branch
Communit y College, New M exico St at e Universit y, L as Cruces, NM 88003. Geometr i c
Cor respondence between Poi nts i n the Scene and Poi nts i n the I mage. Preliminary report .
A major problem in image analysis and underst anding is how t o represent or model object s and images. To
solve t his problem, suit able dat a st ruct ures for represent ing object s and images must be select ed. Graphical
represent at ion scan keep su± cient informat ion about t he image.
Comput er graphics, which requires t wo-dimensional represent at ions of t hree-dimensional scenes, has revived
t he int erest in t hese subject s. T he maj or mat hemat ical t ools necessary for creat ing displays t hat appear t hreedimensional include met hods for t he solut ion of geomet rical problems, posit ional t ransformat ions, and proj ect ions. Unfort unat ely, in comput er graphics, one must follow some precise rule because comput er programs can
only implement mat hemat ical formulas. I f we are not pleased wit h t he result s of t he applicat ion of t he st rict
perspect ive rules, we must amend t hem in some mat hemat ically explicit way [Pavlidis, 1982].
I n t his present at ion I will discuss a met hod of comput ing t he pict ure plane window. T he image of t he scene
is obt ained by proj ect ion ont o t he pict ure plane. T he met hod of project ion is perspect ive. (Received June 03,
2002)
5002-L 1-83
M ar gar et M . B al ach ow sk i * (mmbal ach@mt u. edu), M ichigan Technological Universit y,
Depart ment of M at hemat ical Sciences, 1400 Townsend Drive, Hought on, M I 49931-1295,
and Sh ar i L . St ock er o (st ocker o@mt u. edu), M ichigan Technological Universit y,
Depart ment of M at hemat ical Sciences, 1400 Townsend Drive, Hought on, M I 49931-1295.
T he I mpact of Fi r st Year Teachi ng A ssistants i n a Fir st Year Classroom. Preliminary
report .
Graduat e t eaching assist ant s have t radit ionally played a vit al role in t he ¯rst year mat h programs at M ichigan
Technological Universit y. I n t he past , ¯rst year GTA s were involved in non-inst ruct ional lab and recit at ion
act ivit ies. Second year st udent s advanced t o inst ruct ional dut ies in a freshman level class, wit h as much aut onomy
as part -t ime inst ruct ors, but supervised by a facult y member. M any GTA s ¯nd t hat balancing t he roles of
st udent and t eacher, and performing bot h jobs well, is a daunt ing t ask. A ddit ionally, many undergraduat es
place unrealist ic demands on t heir t eachers, seeing t hemselves as cust omers who demand t he best product for
t heir dollar. Recent research has shown t hat e®ect ive t eaching is a learned skill, developed t hrough inst it ut ional
support , personal int rospect ion, and analysis of t eaching st rengt hs and weaknesses. At M T U we have developed
new GTA orient at ion and t raining programs t hat we believe bene¯t bot h t he GTA s and t heir st udent s. Based on
informal st udent evaluat ions, a drop in t he number of st udent complaint s, and t he self-evaluat ion of t he GTA s,
we believe t he ¯rst year of our program has been quit e successful. We will present an overview of t he program
and discuss changes and improvement s we are programming for t he fut ure. (Received June 03, 2002)
5002-L 1-98
A nt h ony Y A i d oo* (ai dooa@east er nct . edu), Depart ment of M at hemat ics, East ern
Connect icut St at e Universit y, W illimant ic, CT 06226, USA . On M athematical M odels i n
Biology. Preliminary report .
T he complexit y of Biological phenomena demands t he applicat ion of int erdisciplinary met hods t o t he solut ion of
biological problems. A s biology becomes more quant it at ive, t here has been a signi¯cant upsurge in t he reliance
on mat hemat ical models and concept s t o analyze and predict biological phenomena. I n t his paper, we consider
examples of mat hemat ical models in biology. Our examples will be t aken from populat ion dynamics, molecular
event s and nerve conduct ion. We will also examine t he mat hemat ical t ools applied in t he development and
analysis of such quant it at ive models in biology. (Received June 03, 2002)
26
GENERA L CONT RI BUT ED PA PER SESSION
5002-L 1-104
E d w ar d J . G r i ggs* (edl ynn@execpc. com), 21355 Clarion L ane, Waukesha, W I
53186-5409. Simple Acti viti es for Stati sti cs.
Teaching element ary st at ist ics present s a paradox: while st at ist ics is, in pract ice, a " hands-on" discipline, t he
st udy of st at ist ics is rat her abst ract , wit h most of t he t heory beyond t he level of a t ypical sophomore business or
engineering st at ist ics class. T hrough a series of simple and inexpensive " hands-on" classroom act ivit ies, st at ist ics
st udent s can be int roduced in a concret e manner t o t he concept s of 1) t reat ment of dat a, 2) relat ive frequency
int erpret at ion of probabilit y, 3) t he Cent ral L imit T heorem, 4) con¯dence int erval and 5) hypot hesis t est ing.
T his paper present s t hese act ivit ies along wit h some of t he result s of t heir use in a sophomore engineering
st at ist ics course. (Received June 03, 2002)
5002-L 1-105
Sar ah L M ab r ou k * (smabr ouk@f r c. mass. edu), Framingham St at e College, M at hemat ics
Depart ment , 100 St at e St reet , PO Box 9101, Framingham, M A 01701-9101. W i reless
Phone Plan Analysi s - An Oppor tuni ty for A ppli cati on of Pi ecewi se Functi ons.
A nalysis of long-dist ance and wireless phone plans provides an opport unit y for st udent s t o generat e dat a, t o
creat e and t o compare graphs, t o det ermine t he symbolic represent at ion for a funct ion which in many cases
is a piecewise funct ion, and t o apply t echniques for solving syst ems of linear equat ions as t hey analyze a real
sit uat ion t o which t hey can relat e. Taking advant age of t hose regularly ringing, highly int errupt ive, at t imes
irrit at ing but commonly used " boxes" helps st udent s t o underst and t he usefulness of t he mat hemat ics t hat t hey
are learning. For some st udent s, t his is t he ¯rst t ime t hat t hey have modeled a sit uat ion t hat is meaningful t o
t hem.
I n t his paper, I will discuss my use of bot h long-dist ance and wireless phone plans in College A lgebra. I will
discuss t he special considerat ions for modeling t hese funct ions as well as some of t he di± cult ies t hat st udent s
have in int erpret ing wireless phone plans present ed on company web sit es. I n addit ion, I will discuss t he result s
of my wireless phone plan project , st udent react ion t o t he project , and present some of t he st udent -creat ed web
pages. (Received June 03, 2002)
5002-L 1-108
E v el y n C B ai l ey (ebai l ey@emor y. edu), M at hemat ics Depart ment , Oxford College of
Emory Universit y, 100 Hamill St reet , Oxford, GA 30054, and Fan g C h en *
(f chen2@emor y. edu), M at hemat ics Depart ment , Oxford College of Emory Universit y, 100
Hamill St reet , Oxford, GA 30054. Graphi ng Por tfoli os i n Calculus: Rei nforci ng Concepts
and I nvi ti ng Creati vi ty. Preliminary report .
Since t he fall of 1993, st udent s in calculus I and I I have creat ed graphing port folios. T he primary goal of t his
assignment is t o reinforce graphing concept s creat ively. St udent s use t he soft ware GRA PHM AT I CA t o prepare
at least 15 dist inct ly di®erent pages of graphs. A minimum of t hree are designed by t he st udent s t o demonst rat e
t he import ant aspect s of graphing using algebraic, t rigonomet ric, logarit hmic, exponent ial funct ions, and polar
coordinat es (calculus I I ). T he remaining pages use combinat ions of graphs t o creat e pict ures around a t heme
chosen by t he st udent . Evaluat ion of t he port folio is based on t he select ion of graphs, t he document at ion
associat ed wit h t he graphs, t he complet eness of graphing aspect s in t he t echnical graphs, t he creat ivit y of t he
t hemat ic graphs, and t he oral present at ion of t he t hemat ic port ion of t he port folio. Examples of st udent work
will be shown at t he present at ion. (Received June 03, 2002)
5002-L 1-110
A l l i son M . P acel l i * (pacel l i @mat h. br own. edu), 112 Wat erman St . A pt . 4, Providence,
RI 02906. Class Groups of Global Functi on Fi elds.
A funct ion ¯eld in one variable over a ¯eld F is a ¯eld K cont aining F and at least one element T , t ranscendent al
over F , such t hat K is a ¯nit e, algebraic ext ension of F (T ). If t he const ant ¯eld F is ¯nit e, we say K is a global
funct ion ¯eld. Global funct ion ¯elds are very similar in nat ure t o t he ot her class of global ¯elds, namely algebraic
number ¯elds. I n t his t alk, we will discuss t he st ruct ure of t he class group, and show, among ot her t hings, t hat
every ¯nit e abelian group occurs as a subgroup of t he class group of some global funct ion ¯eld. (Received June
04, 2002)
5002-L 1-117
T ony J u l i an el l e* (t j ul i ane@emba. uvm. edu), Dept of M at hemat ics and St at ist ics,
Universit y of Vermont , Burlingt on, V t . 05405. One M il li on Calculus Problems: Gradi ng
the A P Calculus Exam.
A n overview of t he organizat ion and st ruct ure of t he grading of t he A P calculus exam. (Received June 04,
2002)
DI SCRET E M ET HODS I N GEOM ET RY
27
5002-L 1-140
D er ek O ' N eal B r u ®* (der ek. br uf f @vander bi l t . edu), Vanderbilt Universit y,
Depart ment of M at hemat ics, 1326 St evenson Cent er, Nashville, T N 37240. T he Vander bi lt
Uni ver sity Undergraduate Semi nar i n M athemati cs: Exper i ences of a Graduate Student
Organi zer and Presenter .
T he Vanderbilt University Undergraduat e Seminar in M at hemat ics is designed t o show t he undergraduat e communit y at Vanderbilt some fascinat ing and excit ing sides of mat hemat ics. During t he course of a semest er,
t he seminar feat ures t alks by graduat e st udent s on a variet y of t opics in mat hemat ics, from set t heory and
abst ract algebra t o analysis and applied mat hemat ics. Each t alk is designed t o be int erest ing and accessible t o
undergraduat es, whet her t hey' re mat h majors or not .
One goal of t he seminar is t o show st udent s t he beaut y and power of mat hemat ical ideas t hey might not see
in t heir regular coursework. T he seminar organizers' hope is t hat t hose same st udent s will develop a bet t er idea
of, and a keener int erest in, what mat hemat ics is and what mat hemat ics can do.
I n t his present at ion, one of t he graduat e st udent founders of t he seminar will describe t he impet us for t he
seminar, t he organizat ion and cont ent of t he seminar, feedback from st udent at t endees, as well as fut ure plans
for t he seminar. (Received June 06, 2002)
5002-M 1-2
J oel S. Foi sy * (f oi syj s@pot sdam. edu), M at hemat ics Depart ment , SUNY Pot sdam,
Pot sdam, NY 13676. A su± cient conditi on for a graph to be i ntr i nsi cal ly knotted.
We call a graph G i ntr insi cally knotted if every (t ame) spat ial embedding of t he graph cont ains a knot t ed cycle.
I n t he early 80' s, Conway and Gordon showed t hat every spat ial embedding of K 7 , t he complet e graph on
7 vert ices, cont ains a knot t ed Hamilt onian cycle. We have adapt ed t he met hods of Conway and Gordon t o
est ablish t hat if a graph sat is¯es a cert ain linking condit ion for every t ame spat ial embedding, t hen t he graph
must be int rinsically knot t ed. Using t his su± cient condit ion for int rinsic knot t ing, we have been able t o show
t hat K 3;3; 1; 1 is int rinsically knot t ed (K ohara and Suzuki had demonst rat ed t hat Conway and Gordon' s proof
cannot be applied direct ly t o K 3; 3; 1;1 ), and t hat t here is an int rinsically knot t ed graph wit h a vert ex t hat can
be removed, and t he result ing graph is not int rinsically linked (cont ains a pair of non-split linked cycles in every
spat ial embedding). (Received A pril 26, 2002)
5002-M 1-9
M at t h i as B eck * (mat t hi as@mat h. bi nghamt on. edu), Depart ment of M at hemat ical
Sciences, St at e Universit y of New York, Binghamt on, NY 13902-6000. Computi ng the
conti nuous di scretely: the quest for the volume of the Bi r kho® polytope. Preliminary report .
T he n t h Bi r kho® polytope B n is t he set of all doubly stochasti c n £ n mat rices, t hat is, t hose mat rices wit h
nonnegat ive real ent ries in which every row and column sums t o one. A famous open problem concerns t he
volume of B n , which is only known up t o n = 9.
T he Ehr har t polynomi al of a polyt ope wit h int egral vert ices count s t he int eger point s in t he polyt ope as it get s
dilat ed by an int eger fact or. One reason for being int erest ed in t his count ing funct ion is t hat it s leading t erm gives
t he volume of t he polyt ope. We will present a new way of comput ing Ehrhart polynomials for polyt opes using
complex-analyt ic met hods. T heir applicat ion t o t he Birkho® polyt ope provides an example showing t hat pure
mat hemat ics and comput at ionally e± cient algorit hms are not mut ually exclusive. (Received M ay 08, 2002)
5002-M 1-52
M i ch ael O A l b er t son * (al ber t son@mat h. smi t h. edu), Depart ment of M at hemat ics,
Smit h College, Nort hampt on, M A 01063. Realizing Fi ni te Groups as Euclidean I sometr i es.
A ¯nit e set W ½ Rd is said t o realize t he group G if t he isomet ry group of W is isomorphic t o G. T he isomet ry
dimension ±(G) of a group is t he minimum dimension of a realizat ion. A lbert son and Bout in showed t hat
realizat ions exist for all ¯nit e groups and in part icular ±(G) < jGj. I n t his t alk we will visit a few highlight s and
open quest ions. (Received M ay 28, 2002)
5002-M 1-55
B r i gi t t e Ser vat i u s* , M at h. Sci., W PI , Worcest er, M A 01609-2280. Dual-Euler i an graphs.
A dual-Eulerian graph is a plane graph which has an ordering de¯ned on it s edge set which forms simult aneously
an Euler circuit in t he graph and an Euler circuit in t he dual graph. Dual-Eulerian graphs were de¯ned and st udied in t he cont ext of silicon opt imizat ion of cmos layout s. We examine t he connect ions bet ween t he dual-Eulerian
propert y, Pet rie walks, and t he connect ivit y of t he graph. We will also consider t he dual-Eulerian propert y for
graphs embedding in surfaces of higher genus. M oreover, a polynomial t ime algorit hm for det ermining zero
Euler-Pet rie genus of an Eulerian graph is provided. (Received M ay 28, 2002)
28
DI SCRET E M ET HODS I N GEOM ET RY
5002-M 1-69
H eat h er M J oh n st on * (hej ohnst on@vassar . edu), Depart ment of M at hemat ics, Vassar
College, Poughkeepsie, NY 12604. Polygonal knot theor y and stuck unknots.
I magine a polygon in t hree dimensions formed from a series of rigid st icks joined end t o end by universally
° exible joint s. T wo polygons are t he same if one can be moved t o t he ot her wit hout passing t he st icks t hrough
each ot her and wit hout changing t he lengt h or number of t he st icks.
We st udy bot h open and closed chains of st icks. We prove t hat if t he edge lengt hs of a ¯ve st ick open chain
sat isfy an inequalit y t hen t here are t hree kinds of polygons. Ot herwise t here is only one.
T he main quest ion in knot t heory is whet her or not a loop of st ring can be unknot t ed. We give an example
of a polygon which could be unknot t ed if it were made of st ring. Our polygon is st uck in it s t angled posit ion,
but only because we are forbidden t o bend or break it s st icks. (Received M ay 30, 2002)
5002-M 1-75
C ol i n C A d am s* (cadams@wi l l i ams. edu), Depart ment of M at hemat ics, Bronfman Science
Cent er, W illiams College, W illiamst own, M A 01267. Combi nator i cs of Alter nati ng K nots
and thei r Generalizati ons.
A variet y of powerful result s are known t o hold for alt ernat ing knot s. We will discuss t he ongoing e®ort s t o
generalize t hese result s t o an assort ment of ot her sit uat ions. (Received M ay 31, 2002)
5002-N1-7
J u n e E B ar r ow -G r een * (J. E. Bar r ow- Gr een@open. ac. uk), Cent re for t he Hist ory of
M at hemat ics, T he Open Universit y, Walt on Hall, M ilt on K eynes M K 7 6A A , England. T he
Role of Br i ti sh M athemati ci ans in the Fi r st W or ld W ar .
A s a result of t he ant i-war st ance adopt ed by Bert rand Russell, which led t o his sacking from Trinit y College
and subsequent imprisonment , t here is probably a great er familiarit y wit h what Brit ish mat hemat icians did not
do during t he First World War rat her t han wit h what t hey did. However, several Brit ish mat hemat icians were
act ively involved in t he war e®ort , t he most coherent group being t hose gat hered t oget her by t he physiologist
AV Hill t o work on t he problems of ant i-aircraft gunnery. Hill' s group-known as ' Hill' s brigands' -was predominant ly made up of Cambridge men, it s leading ¯gures being RH Fowler, EA M ilne and HW Richmond. Brit ish
mat hemat icians also worked on ot her t opics relat ing t o t he con° ict such as ballist ics, ant i-submarine warfare
and wireless t elegraphy. Codebreaking provided t he except ion. I t engaged t he e®ort s of several highly int elligent
individuals, but -in st ark cont rast t o t he Second World War-t here was hardly a mat hemat ician amongst t hem.
I n t he t alk I shall describe how Brit ish mat hemat icians came t o be employed in war work and explain some of
t heir achievement s. (Received M ay 01, 2002)
5002-N1-8
C r ai g G . Fr aser * (cf r aser @chass. ut or ont o. ca), I nst . Hist . Phil. Sci. Tech., V ict oria
College, Universit y of Toront o, Toront o, Ont . M 5S 1K 7, Canada. T he Concept of Var i ati on
i n Ni neteenth-Centur y Analysi s. Preliminary report .
W it h t he invent ion of t he calculus in t he sevent eent h cent ury, t he concept s of change and variat ion became
est ablished in mat hemat ics. T he not ion of t he variat ion of a funct ion or a curve was basic t o t he calculus of t he
variat ions, alt hough it was oft en dealt wit h only implicit ly. T he paper examines how t he variat ion was de¯ned
and used during t he ninet eent h cent ury in t he researches of Carl Jacobi, A lfred Clebsch, A dolph M ayer and
ot hers. A comparison of t heir researches wit h subsequent t went iet h-cent ury work will help t o illuminat e bot h
t echnical and hist orical facet s of concept s of variat ion as t hey have been deployed in t he t heory of opt imizat ion.
(Received M ay 06, 2002)
5002-N1-10
V . Fr ed er i ck R i ck ey * (f r ed- r i ckey@usma. edu), Depart ment of M at hemat ical Sciences,
U. S. M ilit ary A cademy, West Point , NY 10996-1786. George Baron, One of Amer ica' s
Fi r st M athemati ci ans. Preliminary report .
George Baron t aught at West Point in 1801, but was ¯red before t he Unit ed St at es M ilit ary A cademy was
founded on M arch 16, 1802. He used a Black Board in his t eaching and may have been t he ¯rst in t he count ry
t o do so. A ft er leaving West Point he founded t he ¯rst mat hemat ical periodical in t he Unit ed St at es, t he short
lived M at hemat ical Correspondent . T he cont ent s of t his journal and what lit t le is known of Baron' s life will be
described. (Received M ay 13, 2002)
HI ST ORY OF M AT HEM AT ICS
29
5002-N1-65
A m y E Sh el l -G el l asch * (aa7423@usma. edu), Unit ed St at es M ilit ary A cademy,
Depart ment of M at hemat ical Sciences, West Point , NY 10996. Old Books and Hi dden
Gems. Preliminary report .
Old books, in part icular st udent and inst ruct or t ext s can give an int erest ing, t elling, and at t imes amusing look
int o t he t eaching and social environment of an inst it ut ion in t he past . T he West Pont library has an ext ensive
collect ion of such t ext s for t he 19t h cent ury. I will present a sampling of t he gems found in t hese books as well
as t alk about what can be gleaned from t hem. (Received M ay 29, 2002)
5002-N1-102
H ar d y G r ant * (har dygr ant @yahoo. com), 539 Highland A venue, Ot t awa, ON K 2A 2J8,
Canada. M athemati cs in the T hought of Ni cholas of Cusa. Preliminary report .
T he famous 15t h-cent ury cardinal appears on at least one modern list of " great " mat hemat icians; on t he ot her
hand his cont emporary Regiomont anus dismissed his e®ort s in mat hemat ics as " ridiculous" . But what ever his
t echnical compet ence, it is quit e cert ain t hat Nicholas' s percept ion of mat hemat ics colored deeply his in° uent ial
views on such issues as t he limit s of human knowledge and t he relat ion of man t o God. I shall t ry t o sket ch
from bot h perspect ives { t he t echnical and t he philosophical { t he place of mat hemat ics in t he world-view of
t his fascinat ing ¯gure. (Received June 03, 2002)
5002-N1-130
K en n et h I . G r oss* (gr oss@emba. uvm. edu), Depart ment of M at hemat ics, Universit y of
Vermont , Burlingt on, V T 05405. Elementar y teacher s as mathemati cians { tur ning
challenges i nto oppor tunities.
I n t he lat e 1980s a revolut ion in K -6 mat hemat ics inst ruct ion was quiet ly t aking place across t he Unit ed St at es under t he heading of \ st andards based inst ruct ion." T he curriculum was reorganized according t o a new paradigm,
new t eaching and classroom st rat egies were int roduced, more advanced and diverse t opics were incorporat ed int o
t he curriculum, and new so-called \ st andards-based" mat erials were published in support of t he new philosophy
{ all of which placed enormous pressure on t he K -6 classroom t eacher. T he t heme of t his t alk is mat hemat ics
inst ruct ion at t he element ary level, as seen t hrough a mat hemat ical lens focused on Vermont . Topics will include
t he st andards-based revolut ion; mat hemat ics cont ent needs of element ary t eachers; and t he Vermont M at hemat ics I nit iat ive (V M I ), a heavily cont ent int ensive, mast ers degree grant ing, comprehensive st at ewide mat hemat ics
professional development program for element ary t eachers in Vermont . (Received June 05, 2002)