The Official Newsletter of the IUP Mathematics Department
January, 2001___________________Volume 4, Issue 1
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 firstname.lastname@example.org
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 email@example.com.
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 (firstname.lastname@example.org) 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 Curriculum 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.
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.
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:
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 -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 indeterminate 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] ,
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:
Department of Mathematics
Indiana University of PA
Indiana, PA 15705
FAX (724) 357-7908
We will get to your era soon enough!
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