Welcome
to another issue of Stright Lines. For any of you receiving this as a first issue and thinking
those IUP Mathematics Faculty can't spell “straight”, I remind you that the
Mathematics Department is located in Stright Hall.
I
am sad to report that we have received no letters from graduates to include in
this newsletter. We still hope to hear from you.
In
the last issue Joe Kirchner recalled several humorous stories from his days at
IUP. We hoped to get some more stories from alumni or retired faculty. Joe
thought “it might be a little tough getting funny stories from a bunch of math
majors” and he seems to have been right since we have received no
contributions. We still would welcome humorous stories about your days at IUP. Jim
Reber, Editor.
__________________
IUP Graduates are Involved!
In
one of the first issues of Stright Lines I
extolled the professional activities of IUP graduates in mathematics education.
IUP graduates continue to make a difference in education. For example, check
out the web page of the Mathematics Council of Western Pennsylvania at
www.mcwp.org. It is maintained by David Taylor, an IUP graduate. David taught
for 3.5 years in Maryland and then returned to Western Pennsylvania to teach mathematics
at South Fayette Township Jr.-Sr. High School. One of his responsibilities at
the school has been the Cooperative Satellite Learning Project, a cooperative
effort among the school, NASA, Goddard Space Flight Center, and Allied Signal
Technical Services Corporation. This fall David became Director of Information
Technology at South Fayette.
This
year from March 15 - 17, 2001, the 50th Annual Meeting of the Pennsylvania
Council of Teachers of Mathematics (PCTM) will be held in Pittsburgh at
Greentree’s Radisson and Holiday Inn hotels. IUP graduates are certainly
prominent among committee chairs, presenters and presiders. Dave Depner is
Co-chair of Local Arrangements and Susan Stonebraker is Chair of Meals and
Functions. Many alumni who have earned their degrees at IUP are sharing their
knowledge and expertise by presenting programs. These alumni include Linda
Brecht, Elaine Carbone, Patty Flach, Rhonda Fedyk-Foust, Nina Girard, Bill
Hadley, Jennifer Landsman, Peggy Lunardini, Majory Maher, Rita McMinn, Mary Lou
Metz, Mary Lynn Raith, Shannon Relihan-Rieger, Cathy Schloemer, Eli Shaheen,
Anita Smith, Kirstie Trump, John Uccellini, and Mark Zelinskas. Many of the
presenters are also presiding over sessions, as are Adrienne Kapisak, Dorothy
Mullin, and nine student teachers. (I wonder who twisted the arms of the latter!)
Anyway, these student teachers deserve mention; they are Tracy Birchall, Leah
Drane, Jessica Feerst, Melissa Luckey, Shawn Moorhead, Doug Murdoch, John
Nelson, Denise Shade, and Jane Shumaker. (If I have forgotten anyone, please
advise me, and I will give you proper coverage in the next Stright Lines.)
If
you are attending the PCTM meeting, please look on the message board by the
registration table for IUP announcements. We will try to plan some
get-togethers.
You
may be interested in knowing where some of our recent graduates have taken positions.
We are now fortunate to have many of our graduates getting their first
positions in Pennsylvania. Recent graduates who are now teaching in
Pennsylvania are as follows: Shelly Huston, Shady Side Academy, Pittsburgh;
Ralph Santilli, Butler; Matt Rodkey, Homer Center in Homer City; Jeff Ziegler,
Pittsburgh Public Schools; Brad Baker, Beaver; LeeAndrea McCullough, Quaker
Valley, Sewickly; Karin Rabenold, Marion Center; and Lisa Sargent, Manheim
Twp., Lancaster. Among those who have gone to other states are Kim White,
Concord, North Carolina; Janel Hartzok, Westminster, MD; and Joyce George,
Ocean City, MD.
Periodically
we get e-mail messages from former students who are recruiting mathematics
teachers for their schools. Two of these have come from Chris Clark at Manassas
Park, Virginia and Chris O’Rourke at McLean, Virginia. Although our mathematics
department does not operate a placement bureau, we are always happy to share
job postings. If you are searching for a job or trying to fill a position,
please forward information to us.
Ann
Massey asmassey@grove.iup.edu
News about Graduates
Dr. Buriok received a note from John A. Miller (J.Miller@connect.xerox.com) who graduated from IUP in 1977 and is now Managing Principal, Document Management and Imaging, with Xerox Connect in Pittsburgh. John noted that he gave one of the commencement speeches in the department. He also mentioned that Xerox Connect is growing quickly and hires many college seniors.
Mark
Rayha (Class of
1993) resigned his position as a Business Systems Analyst with Citistreet
(formally known as the Copeland Companies) and accepted a position as a Lead
Systems Analyst at Schering-Plough, a pharmaceutical company. His home email
address is m.rayha@gte.net.
Tracie
A. Moreland
(Class of 1996) finished the Applied Statistics graduate program at Villanova
University (while working full time). She is currently with Merck & Co. as
a marketing analyst.
Dr.
Rebecca Stoudt received a note from Aurele Houngbedji (amhst44+@pitt.edu) who graduated
from IUP with an M.S. degree in August, 1996. Aurele graduated with his Ph.D. on
April 30, 2000. He will be working at Ohio Savings Bank in the Capital Markets
Department as a Quantitative Analyst. The position is related directly to Aurele’s
research, which is stochastic modeling in Finance. He will be doing
quantitative research, financial data analysis, derivatives trading and risk
management.
Cindy
Venturino Biedrycki
wrote Dr. Massey from Prince William County in Virginia where she and Stephanie
Clifton are
teaching. Both finished their masters degrees in Curiculum and Instruction at
Virginia Tech last August.
__________________
Alumni Bulletin Board
Available at our Web site
If
you go to the IUP Mathematics Department web site, http://www.ma.iup.edu/, you
can leave a message on the Alumni Bulletin Board. One recent posting is from
Kirstie Trump (MDteach4u2@aol.com)
on 09/27/00 :
Hello
everyone! I am currently teaching 8th grade math and algebra in Carroll County,
Maryland. IUP prepared me well for teaching and I am grateful to all of my
professors and classmates for always supporting me. Carroll County is always
looking for good math teachers and loves to recruit IUP graduates. Please email
me if you would like more info!
Mullin Receives Award
Last
year Dorothy Mullin received the Award for Outstanding Contributions to MCWP (Mathematics
Council of Western Pennsylvania). Dorothy has served as a member of the MCWP
board and chairperson of many committees for PCTM and NCTM regional meetings as
well as for MCWP meetings. Always she has been willing to give of herself to
make professional events successful.
Dorothy
received her bachelor’s degree in mathematics education from IUP and both her
masters in mathematics and her doctorate in mathematics education at the
University of Pittsburgh. She taught at Penn State, McKeesport for more than 20
years. We were fortunate to have Dorothy in our department when she returned to
her Alma Mater as a temporary instructor for a year.
__________________
Obituaries
Dale
Shafer died in Florida on March 21, 1999. His master’s degree was from Columbia
University and his doctorate of education degree was from the University of
Oklahoma. He taught for two years in the Oley Valley School District, for three
years at Slippery Rock College, and for 30 years in the IUP Mathematics
Department from 1964 - 1994. He was the executive secretary of the School,
Science and Math Association for 10 years. At IUP he often taught statistics
courses.
Richard
“Dick” Wolfe died in South Carolina on January 24, 2000 from injuries suffered
in a traffic accident. His master’s and doctorate degrees were from the University
of Illinois in Champaign-Urbana. He taught at Waynesboro High School and then
here at IUP from 1967 until his retirement in 1991. He taught mathematics
education courses and supervised numerous student teachers over the years.
I.
“Ike” Leonard Stright died on February 9, 2000. He received his Ph.D. degree
from Case Western Reserve University. He taught mathematics in high school and
at Baldwin Wallace College and Northern Michigan University. He became Professor
of Mathematics at IUP in 1947 and was Dean of the Graduate School from 1957
until 1971. The building which houses the Mathematics Department, and hence
this newsletter, is named for Dr. Stright.
Word from Daniel Griffith,
Class of 1970
Dr.
Daniel A. Griffith, now Professor of Geography at Syracuse University, sent us
two recent publications. One article appeared in the Journal of Statistical
Planning and Inference. He noted that his IUP mathematics education prepared him very well
for earning an M.S. in statistics (1985). The other article appeared in Linear
Algebra and Its Applications. This article draws upon his undergraduate and graduate
work in mathematics at IUP (B.S., 1970; graduate work 1970-72). Daniel observes
that training by three of his IUP instructors - Mr. D. McBride (retired), Dr.
J. Hoyt (retired) and Mr. C. Maderer - helped make this second article
possible. In closing he notes that he continues to appreciate the mathematics
training he receive at IUP that has enabled him to both publish in statistics
journals and contribute to the linear algebra literature.
__________________
The SPIRAL Project
By Rebecca A. Stoudt and Roberta
M. Eddy
SPIRAL
(Science/Mathematics/ Technology Preparation Involving Real-world Active
Learning) is a teacher professional development project funded by the
Eisenhower Professional Development Program and IUP matching funds. SPIRAL is a
multi-disciplinary program that SPIRALs concepts from K through 12 and out
across the disciplines. The disciplines involved are Mathematics, Biology,
Chemistry, Geoscience, and Physics. The use of a wide variety of technology is
woven throughout the program.
The
project is co-directed by Rebecca Stoudt (Mathematics) and Roberta Eddy
(Chemistry). Other SPIRAL faculty are Janet Walker and Gary Stoudt
(Mathematics), Terry Peard (Biology), John Wood (Chemistry), Connie Sutton
(Geoscience), and Norman Gaggini and Ken Hershman (Physics). Kent Jackson
(Special Education), Mary Ann Rafoth (Educational and School Psychology), and
Len Lehman (Curriculum Consultant) complete the SPIRAL staff.
The
central focus of this project is an 8-day, intensive, residential, summer
institute (SI) where preservice teachers, inservice teachers, and
administrators come together to learn instructional strategies and to conduct
field-tested activities consistent with state and national standards. The SI
emphasizes two SPIRAL models, LIGHT and ECOSYSTEM. An awareness of special
needs students and diverse learning styles in science and mathematics is
stressed throughout the SI. Furthermore the incorporation of SPIRAL activities
into the school district’s curricula is facilitated by two SI synthesis and
curriculum incorporation sessions. SPIRAL also includes ongoing professional
development activities such as follow-up workshops (fall and spring),
development of portfolios, and a joint ARIN/SPIRAL Academic Alliance for
educators of mathematics and science.
A
5-member SPIRAL school district team ideally consists of an administrator (can
be a principal, assistant principal, curriculum director, or head of
department), a special needs or learning support instructor, and three K-12
teachers of mathematics and science (specifically an elementary teacher, a
middle school teacher, and a high school teacher). When each team arrives at
the SI, it is linked with two IUP preservice teachers, one elementary and one
secondary. The preservice teachers
are majoring or concentrating in mathematics and/or science.
SPIRAL
participants use standard-based models of teaching that emphasize the inquiry
approach and cooperative learning. As a result, the participants’ content
knowledge in all SPIRAL disciplines has increased significantly in every SI
since the beginning of SPIRAL (1998). This significant increase was measured by
the pre/post-test scores of 93 inservice and 51 preservice teachers. In fact,
for each SI, the post-test score mean was at least double the pre-test score
mean.
The
Eisenhower Professional Development Program has awarded SPIRAL approximately
$597,000 since the projects beginning. These awards have been matched with
approximately $349,000 from IUP (College of Natural Sciences and Mathematics,
College of Education, Graduate School and Research), Texas Instruments, and
ARIN IU-28. Hence, SPIRAL is almost a $1 million project to date.
A
large portion of the grant money is spent on supplies and materials for the
teams to take back to their home schools so that they can easily implement
SPIRAL activities in their curricula. Each team receives over $4000 of
equipment which includes but is not limited to: (1) TI-83 Plus
calculator/viewscreen; (2) CBL2 kit with set of probes--biology gas pressure
sensor, dissolved oxygen, colorimeter, pH system; (3) CBR system; (4) digital
camera; (5) various CD-ROMs; (6) aquatic kick net; (7) Silica Gel GF thin layer
chromatography plates; (8) UV lamp; (9) HACH Color Cube kits (iron,
nitrogen-nitrate, phosphorous orthophosphate); (10) HACH Color Disc Kits (iron,
nitrogen-nitrate, phosphorous orthophosphate); (11) light, image, shadow kits;
(12) topographic and geologic maps; (13) Guide Book to Rocks and Soil; (14) rock/mineral
set; (15) fossil set;
(continued on page 4)
(continued from page 3)
(16) pocket gem field magnifier; (17) pH tester; (18) fluorescent experiment
kit, (19) lightsticks; (20) cool
blue light and goofy glowing gel kits; (21) color filters; (22) mirror set;
(23) soil percolation kit; (24) bar magnet set; (25) student clinometer; (26)
refracting telescope kit; (27) Ecneics kit; (28) solar system floor puzzle;
(29) star chart; (30) solar system/planet poster; (31) spectrum analysis chart;
(32) spectroscope and (33) numerous activity books.
For
more information, pictures, sample activities, syllabi for inservice/preservice
academic credit, and links to electronic portfolios of SPIRAL teams, we invite
you to visit the SPIRAL website at http://www.iup.edu/smetc/spiral/
__________________
IUP's Curriculum Through the
Years, Part 3
by Gary Stoudt
In
the last two issues we looked at the opening of the Indiana Seminary and Normal
School and the State Normal School of the Ninth District, in Indiana,
Pennsylvania. One of the texts used in the curriculum of the Indiana Seminary
and Normal School was Ray’s Algebra. Thanks to Dr. Ed Donley who loaned me a
copy of the book, I can tell you something about the book in order to help you
get a feel for what the mathematical studies at the Normal School where like.
Unless otherwise stated, quotes in this article are from this book.
Dr.
Donley’s copy is of the 1875 edition, so it is most likely that this was the
text used at the time of the school’s founding in 1875. The full title of the
book is Elements of Algebra for Colleges, Schools, and Private Students, Second
Book. The author is Joseph Ray, M.D., professor of mathematics at Woodward
College. Woodward College was located in Cincinnati but no longer exists. The
publisher was Wilson, Hinkle and Co. in Cincinnati. There are very few diagrams
in the text, although it is typeset using modern notation. According to Miami Valley Vignettes, by
George C. Crout (http://www.middle-america.org/crout/ mvvig/pioneers.html):
Ray wrote a series of texts which
made arithmetic understandable to elementary pupils. Joseph Ray was a professor
at Woodward College, later becoming its president. In addition to his work at
the Cincinnati college, he was a state leader in education. Ray compiled a set
of three texts in mathematics, taking the student from simple processes to
advanced ones. His third book was used in both high school and colleges. The
series was published in Cincinnati. Even after his death in 1865, the Ray
textbook series dominated the textbook field in mathematics until the early
1900's.
The text has a wonderful Preface, part of which is
reproduced here.
Algebra is justly regarded one of
the most interesting and useful branches of education, and an acquaintance with
it is now sought by all who advance beyond the more common elements. To those
who would know Mathematics, a knowledge not merely of its elementary principles,
but also of its higher parts, is essential; while no one can lay claim to that
discipline of mind which education confers, who is not familiar with the logic
of algebra.
It is both a demonstrative and a
practical science - a system of truths and reasoning, from which is derived a
collection of Rules that may be used in the solution of an endless variety of
problems, not only interesting to the student, but many of which are of the
highest possible utility in the arts of life.
Those were the days! This sentiment is still alive today
in the current debate concerning “algebra for all.” Of course, we also still make the claim that algebra is
“useful.”
The
text starts with definitions, notation, and the fundamental rules of
arithmetic, including operations with polynomials, all in Chapter 1. The
description of operations with monomials is much like a modern text with the
exception of the use of the vinculum (a horizontal bar) along with
parentheses. For example, 
There is no mention of FOIL, but there is an interesting
method of multiplying and dividing polynomials called the “method of detached
coefficients.” Ray states “this method is applicable where the powers of the
same letter increase or decrease regularly.” For example, to multiply 
by 
:
1
- 3 + 0 + 1
1
+ 0 - 1
1
- 3 + 0 + 1
- 1 + 3 - 0
- 1
1
-3 -1 + 4 -0 -1
the answer is 
.
In
the next two chapters we move into factoring (factoring of quadratic trinomials
is done “by inspection”) and working with algebraic fractions, which we would
call rational expressions. This is all done in the fairly standard “modern”
way. The lone exception is the work done on converting fractions into infinite
series. For example, ( 1 - x ) / (1 + x ) is written as an infinite series
using long division.
(continued on page 5)
(continued from page 4)
In Chapters 4 and 5 Ray moves into solving equations,
starting with the “simple equation” (linear equation). This is done in the
usual way, but Ray includes some interesting word problems, as in this example:
“A smuggler had a quantity of brandy, which he expected would sell for 198
shillings; after he had sold 10 gallons, a revenue officer seized one third of
the remainder, inconsequence of which, what he sold brought him only 162
shillings. Required the number of gallons he had, and the price per gallon.”
There are also included many problems that we would recognize (Plus ça
change...). Classic problems such as division of items (“a sum of money is to
be divided among five persons so that ...”); work problems (“If A does a piece
of work in 10 days...”); traveling problems (“There are two places, 154 miles
distant from each other, from which two persons, A and B, set out at the same
instant...”); number problems (“There are three numbers whose sum is 187...”);
and purchasing problems (“If 10 apples cost a cent, and 25 pears cost 2 cents,
...”). Ray then discusses systems of two linear equations (no solution by
graphing, though) and literal equations.
We
now move on to powers and roots in Chapter 6. Interestingly, the binomial
theorem is stated (as Newton’s Theorem) but Pascal’s triangle is nowhere to be
found. Ray shows how to find square roots and cube roots of numbers and
polynomials. (For the younger folks out there, send me an email if you want to
know the method!) The sections that follow deal with radicals, including
fractional exponents and “imaginary, or impossible quantities.” The chapter
ends with a section on simple inequalities.
The
solution of quadratic equations begins in Chapter 7. First we solve the pure
quadratic, which “contains only the second power of the unknown quantity, and
known terms” and then the “affected quadratic,” which “contains the first and
second power of the unknown quantity, and known terms.” The affected quadratic
is first solved by completing the square. Next the affected quadratic is solved
by the “Hindoo [sic] Method.” This method was known to Brahmagupta (b. 598) and
Ray describes it much as Brahmagupta did, except Ray uses modern notation.
1st. Reduce
the equation to the form 
2nd. Multiply
both sides by four times the coefficient of 
.
3rd. Add the
square of the coefficient of x to each side, extract the square root, and
finish the solution.
As an example, consider 
.
Multiply both sides by 8: 
.
Add 25 to both sides: 
.
Extract the root: 4x - 5 = 
, etc.
Next
in the text is a discussion of the theory of quadratic equations, a look at
equations that are quadratic in form, theorems concerning the roots of
quadratic equations, theorems concerning imaginary roots, and so on. The
chapter ends with a discussion of the solution of two simultaneous quadratic
equations in two variables.
Chapter
8 is concerned with ratios, proportion and progressions. Included here is a
discussion of the mean proportion of two numbers, alternation, inversion, and
composition of proportions, harmonic proportions, arithmetical, geometrical,
and harmonic progressions, including the sums of arithmetic and geometric
series. In Chapter 9 Ray discusses permutations, combinations, and the binomial
theorem. The notation 
for combinations is not used. Instead Ck is used, where it is assumed n
is known.
Infinite
series is the topic covered in Chapter 10, along with the general Binomial
theorem and decomposition of fractions into partial fractions, which Ray calls
“decomposition of rational fractions.” This topic is in this chapter because of
its relationship to the technique of indeterminant coefficients for finding the
terms of a series expansion. Work with series is done in the spirit of Newton:
treating infinite sums as finite sums with respect to performing algebraic
operations on them. Here is an example.
Thus it is required to develop 1/
(3x - x2) and we
assume the series to be 
, etc., we have after clearing of
fractions [multiply both sides by 3x - x2] ,
, etc.
from which, by equation the
coefficients of the same powers of x, 1 = 0, 3A = 0, etc.
The first equation, 1 = 0, being
absurd, we infer that the expression cannot be developed under the assumed
form. But, 
Putting 
,etc., clearing of fractions, and
equating the coefficients of the like powers of x, we find
, 
,
, 
, etc. Hence 
Or, since the division of 1 by the first term of the
denominator gives 
, or 3 x -1 we ought to have assumed 
, etc.
(continued on page 6)
(continued from page 5)
Work done with series is also done in the spirit of
Leibniz, using the so-called “differential method of series.” This method is
based on sequences of differences.
Let the series [Ray’s term]
be a, b, c, d, e,... ; then the
respective orders of differences are,
first order b - a, c - b, d - c, e - d, ...
second order c - 2b + a, d - 2c + b, e - 2d + c, ...
third order d - 3c + 3b - a, e - 3d +3c - b, ...
fourth order e - 4d + 6c - 4b +
a, ....
If we denote the first terms of
the 1st, 2nd, 3rd, 4th, etc., orders of differences by D1, D2, D3, D4, etc., and invert the order of
the letters we have D1
= - a + b; D2
= a - 2b + c;
D3 = - a + 3b - 3c + d; D4 = a - 4b + 6c - 4d + e,
etc. Here, the coefficients of a,
b, c, d, etc., in the nth order of differences, are evidently those of the
terms of a binomial raised to the nth power; and their signs are alternately
positive and negative.
From this the author shows how to find the nth term of a
series a, b, c, d, e,... using differences:
D1 = - a + b; whence b = a + D1
D2 = a - 2b + c; whence c = a + 2D1+D2
D3 = - a + 3b - 3c + d; whence
d = a + 3D1+3D2+D3
D4 = a - 4b + 6c - 4d; whence
e = a + 4D1+6D2+4D3+D4.
This technique is then applied to counting the number of
balls in triangular and rectangular piles of cannon balls. This is the only
place in the book where illustrations appear; there are illustrations of piles
of cannon balls! The chapter concludes with a look at “recurring series,” what
we would call recursive sequences.
In
Chapter 11 Ray discusses continued fractions, logarithms, exponential
equations, interest, and annuities. In the logarithm sections, time is spent on
computing common logarithms using a table of logarithms. Next there is a brief
section on the rules of single and double position. These are techniques for
solving linear equations of the form ax + b = m. These techniques were known to
the ancient Egyptians and were used in medieval Europe under the name “regla
falsi,” or “false position.” The
technique involves making two guesses x1 and x2 and finding the differences e1 and e2 between
ax1
+ b and m and ax2
+ b and m. This section is placed here in the text because this technique is
used to solve exponential equations of the form xx = a. An example is given, to solve xx = 100.
Begin by rewriting as x log x = 2.
First supposition Second supposition
x = 3.5; log x = .544068 x = 3.6;
logx = .556303
x log x = 1.904238 x
log x = 2.002690
a = 2 a = 2
error = -.095762 error = .002690
Diff results : diff assumed nos.
: :
Error 2nd result : Its cor.
.098452 : 0.1 : : .002690 :
0.00273
Hence x = 3.6 - .00273 nearly.
The sections on interest and annuities are very similar
to what is in books now, with the exception that all the formulas are derived
using properties of series, instead of just being given.
In
Chapter 12 the general theory of equations is discussed, including the
relationship between the coefficients and roots of equations, the factor
theorem, the Fundamental Theorem of Algebra, Descartes’ rule of signs, the
transformation of equations, and Sturm’s theorem. Chapter 13 ends the book with
a discussion of numerical solutions of polynomial equations, including Horner’s
and Newton’s methods. Also included is Cardano’s rule for solving cubics!
This
is quite a text. We do not know how much of the book was covered in the course
that used it. It is important to keep in mind that this course was required
of all students at Indiana Normal.
I
hope you enjoyed this look into the past. It would be interesting to learn how
many of these topics were covered in later years. You can help by going through
your old textbooks (or just going through your memories) and dropping us a
note. As always, let me know what you think and please feel free to get
involved. Send (via email, FAX or U.S. Mail) what mathematics/education courses
you took, the professors’ names, what textbooks you used, and when to:
Gary Stoudt
Department of Mathematics
Stright Hall
Indiana University of PA
Indiana, PA
15705
gsstoudt@grove.iup.edu
FAX (724) 357-7908
We will get to your era soon enough!
__________________
Write to Us
Send
us your comments and suggestions on the newsletter or let us know what you are
doing. You can write us at:
Department of Mathematics
Indiana University of Pennsylvania
233 Stright Hall
Indiana, PA 15705-1072
You can visit our web page at
www.ma.iup.edu
and send email to us at:
jburiok@grove.iup.edu
or
jreber@grove.iup.edu