Contributed Paper and Poster Sessions

Contributed Paper and Poster Sessions

SarahLawrenceCollege

Sunday, May 6, 2007

Research Session:

3:30 – 3:50

1) TITLE: The Importance of Completing the Square of Some Rational function to Simplify Some Integral Computation

ABSTRACT: In this presentation we propose a clever and very simple way to help Calculus students to integrate some relatively difficult integral involving rational functions. Completing the square of a rational function will be used. The extension of the computation to a larger family of functions will be presented and compared with the classical method of computation which is usually based on partial fraction decomposition.

PRESENTER: Dr. Chokri Cherif

Borough of ManhattanCommunity College, CUNY

3:50 – 4:10

2) TITLE: The Taylor Series for e and the Primes 2, 5, 13, 37, 463: a Surprising Connection

ABSTRACT: We explain a surprising connection between the partial sums of the Taylor series e = Sum 1/n! and a certain sequence of prime numbers 2, 5, 13, 37, 463, ... . Using Mertens's Theorem on the series of prime reciprocals, we give a heuristic argument that our sequence of primes should be infinite but very sparse. See for a preprint.

PRESENTER: Dr. Jonathan Sondow

4:10 – 4:30

3) TITLE: Title: Log(-1)=0??!

ABSTRACT: The simplistic "proof" of this result is: 2log(-1)=log((-1)2)= log 1 = 0. Therefore log(-1)=0. This talk shows how this result and method, properly interpreted, is nevertheless true. We work with the 2-adic metric over the rationals, in which ||r/s|| = 2-k where k is the highest power of 2 in the factorization of r/s. Starting with the familiar series

we can substitute x = 2, and the series converges because ||2||=1/2. Using the above “proof”, we prove the result. In this case, the log function cannot be considered the inverse of the exponential function, but we show that log(-1)=0 using the above “proof.” The result can be interpreted number theoretically.

PRESENTER: Emeritus Professor Melvin Hausner, NYU

4:30 – 4:50

4) TITLE: Converging on the Eye of God

(This represents joint work with Francisco Rangel)

ABSTRACT: If one takes a golden rectangle and excises the largest square contained within it (and sharing a corner with it), one gets a smaller golden rectangle. If one iterates this process, one gets ever-smaller golden rectangles in an infinite chain. Connecting alternate corners of the excised squares yields a golden spiral. The point into which this spiral appears to vanish is called the “Eye of God.” As there are four different orientations in which to carry out this iterated excision process, there are, in fact, four Eyes of God for any golden rectangle. In this presentation, I will show that the x and y coordinates of the four Eyes of God (when the original golden rectangle is placed in “standard position” in the plane) are all of the form:

Here Fn represents the nth Fibonacci number. More surprising, I will show that, for any integer k, this value corresponds to the x or y coordinate of some Eye of God in some golden rectangle: a golden offspring or a golden ancestor (in the sense of the excision process described above) of the original, given one.

PRESENTER: D.N. Seppala-Holtzman

St. Joseph's College

4:50 – 5:10

5) TITLE: Two Lucasian Gems

ABSTRACT: The mathematical fortunes of Cambridge were put on a solid foundation by Henry Lucas, secretary to chancellor of the University of Cambridge, when in June 1663, a month before his death, he drew up a will directing his executors to provide for an endowment of a mathematical professorship at the University. Many notable mathematicians have held the Lucasian Chair including Isaac Newton, Charles Babbage, Paul Dirac, and Stephen Hawking. This talk focuses on two Lucasian results: Nicholas Saunderson’s method to determine the greatest common divisor of two positive integers and John Colson’s promiscuous scheme to express positive integers without using the digits 6,7,8, and 9.

PRESENTER: Jim Tattersall

ProvidenceCollege

Education Session:

3:30 – 3:50

1) TITLE: It has to be right, that's what my calculator ‘says’!!!

ABSTRACT: Through the use of some well-chosen examples, this presentation shows how students can easily be misled by the use of technology (both calculators and math software), and in some cases get wrong answers! Thus, emphasizing the need for a deeper understanding of the mathematical principles involved.

PRESENTER: Abraham S. Mantell

NassauCommunity College

3:50 – 4:10

2) Title: Teaching Mathematics Teachers: A Comparison of Mathematics Teacher Education Programs in Taiwan and the U.S.

ABSTRACT: Past and current research in comparative mathematics education has continually emphasized student achievement and cognitive differences. Most studies seem to conclude that Asian students usually outperform their American counterparts. There are many factors that account for this result. One important factor, however, that seems to have been overlooked in the research is the extent to which teacher education programs in mathematics influence student achievement. In this presentation, we examine the origin of the Mathematics Program at the National Taiwan Normal University (NTNU) and its influences on the mathematics education trends in Taiwan. In tracing the evolution of the Mathematics Program at NTNU from its establishment in 1946 to the present time, we have identified the internal and external influences and the major curricular trends that may have affected Taiwanese mathematics education. We then examine the mathematics teacher education trends in the U.S., using New York state as an example. Within the last few years, community colleges throughout New York have offered content-specific courses leading to an education major for students who wish to transfer to four-year colleges and universities. We summarize our findings by highlighting the similarities and differences of both countries’ trends in mathematics teacher education.

PRESENTERS: Daniel Ness, DowlingCollege

Chia-ling Lin, NassauCommunity College

4:10 – 4:30

3) Title: LEARNING BASIC MATHEMATICS TROUGH KNITTING

ABSTRACT: A knitting project is a wonderful combination of Art and Mathematics. Fractions, rate, proportion, percent, linear relationship, and basic geometric shapes are among the basic math concepts that we will encounter in the knitting projects. An example of such a project is an Afghan blanket – a traditional eastern blanket made from knitted rectangular blocks of different colors. In our presentation we will discuss an approach of teaching basic math topics through knitting and share our experience of using “Designing Afghan” project in “Introduction to Algebra” course.

PRESENTERS: Marina Dedlovskaya, Ph.D.

LaGuardiaCommunity College, CUNY

Cecilia Macheski, Ph.D.

LaGuardiaCommunity College, CUNY

4:30 – 4:50

4) Title: Methods of Differentiation of Functions of two Variables

ABSTRACT: Originally when Newton found derivatives of implicit functions he used the process of partial differentiation and the difference quotient (Cajori, 1923). One recommendation for mathematics at all levels is for increased attention to integration of topics throughout the curriculum (AMATYC, 1995; NCTM, 1989; CBMS, 2000). Within the calculus curriculum, there is a strong separation between differentiation of functions of a single variable and functions of more than one variable. Usually, when students are taught implicit differentiation, each term of a function is differentiated separately and no attempt is made to generalize the difference quotient to the entire function. In this talk, we will examine student reactions to four different methods of differentiation: 1.) term-by-term implicit differentiation with respect to, 2.) term-by-term implicit differentiation with respect to, 3.) Newton’s method of ratios of partial derivatives, and 4.) Newton’s method of application of the difference quotient.

PRESENTER: George McCormack, Ed.D.

LaGuardiaCommunity College, CUNY

4:50 – 5:10

5) Title: Experimenting ePortfolio with Mathematics

ABSTRACT: An ePortfolio is a tool for students to collect their academic work, reflect on their learning, and share their portfolios on the World Wide Web. It includes projects from classes, images, and a synthesis of a student's academic career. It offers a rich and textured view of students' learning and development. Personal essays encourage students to explore their changing sense of themselves. It helps students to connect classroom, career, and personal goals and experiences.

The concept of ePortfolio is new to us. The novelty is not just technical. For students, it is a break from their general expectation of solving routine problems for completing assignments and tests and encourages creative experiences. For teachers, it is a new way to draw attention to the fundamental concepts and applying the learned knowledge to solve real life problems. Particularly, the publication aspect of ePortfolio introduces a kind of awareness, for teachers as well as for students, which was seldom possible before ePortfolio.

My experience in working with students for ePortfolio projects has been very rewarding. It provided an insight into teaching math in a more creative way, which helped in bringing a higher level of enthusiasm into the class. As a math instructor, ePortfolio means creating additional assignments and projects, which encourage students to go beyond the surface of the subject, produce presentable and publishable material that demonstrates their learning. ePortfolio requires additional grading time so that the instructor can give constructive feedback. Moreover, introducing students to ePortfolio reduces lecture time. In a presumably difficult subject like math, initially this approach did not seem to flow naturally. I had to be very creative in my teaching approach focusing on writing and expressing oneself in mathematics as well as problem solving.

PRESENTER: Prabha Betne

LaGuardiaCommunity College

Student Session:

3:30 – 3:50

1)TITLE: The St. Petersburg Paradox

ABSTRACT: The St. Petersburg Paradox, published by Bernoulli, describes the following lottery game. A player keeps flipping a coin for as many times as it shows heads, but once it shows tails the game is over. The longer the player can stay in the game the greater his reward will be. If the game ends after the first flip, the reward will me $1, after the second flip the reward will be $2, after the third it will be $4, doubling with every next throw.

The question is what price should the lottery set? To gain a profit the lottery must make the price greater than the average amount of money a player wins after playing the game, which is the expectation of the reward. By finding it I was able to determine the price.

In my paper I proved that if the reward in each round increases at a rate of 2 or more times, the expectation of the reward in the game turns out to be infinity. However, if the reward in each round increases at a slightly slower rate, such as 1.8 or 1.9, the average reward will not be infinity and the price can be calculated.

The original St. Petersburg Paradox is a problem where the price for the game can not be calculated, from the “theoretical” point of view, because the expectation for the reward is infinity. However, in my paper I proved that from the “practical” point of view, the problem could in most cases be solved.

PRESENTER: Vadim Chanyshev

The BronxHigh School of Science

3:50 – 4:10

2)Title: Sums of Powers of Natural Numbers

ABSTRACT: The formula for the sum of the first n natural numbers is well known, as is the formula for . What happens when we move up to powers greater than 2? Is there a formula for ? Or ? In general, how can we generate formulas for in terms of n and k, and what methods can be used to generate them?

We will answer this question in a number of ways: using analysis, calculus, and combinatorics. A ‘telescoping method’ was also found in the literature, in which telescoping sums are used to create a recursive formula for each sum individually. I was able to generalize this method, but in practice the other methods we will examine are more efficient. It is fascinating to see how this one simple topic relates to so many different branches of mathematics.

PRESENTER: Kevin Sackel

BethpageHigh School

4:10 – 4:30

3)Title: Pi of the Needle

ABSTRACT: Mathematicians and students alike have always regarded pi as a fascinating number. Pi is defined as the ratio between the circumference of a circle to its diameter, however it has appeared in scenarios that at first glance appear to have no relation to circles. One example of such a scenario is tossing a needle on lined paper and determining the probability that it would land on a line. This experiment has been first associated with a French naturalist named Buffon in 1777. A few other researchers have continued upon his connecting pi to probability of a needle toss. The theory is that if one were to toss a needle of l length onto paper with parallel lines spaced at d length apart, if l=d, then the probability of the needle landing on a line is π/2. (This is based on Buffon’s derived formula P (probability of touching a line) = 2l/πd.) I found this interesting and curious, as I would not have guessed Pi’s affiliation with this sort of scenario. After I did the experiment for myself with different lengths of needles to create comparative data, I plugged my numbers into Buffon’s equation with the purpose of proving that his theory is an accurate method of calculating pi. My results proved affirmative and I recorded all of my data and calculations to prove this. Pi is directly related to Buffon’s experiment because the probability of the needle landing on a line is directly related to the angle it makes with the nearest line. Angles can be measured in both degrees and radians and for the convenience of this experiment radians are used. Thus, when graphed, a sine curve is produced and the probability of the needle landing on a line can be linked to the number π.

PRESENTER: Kiera C. Galloway

BronxHigh School of Science

4:30 – 4:50

4)Title: A Canon of Canonical Forms

ABSTRACT: In this paper I partition the polynomials into equivalence classes such that the elements in each class are aesthetically ‘similar’. This ‘similarity’ is precisely defined by an algebraic relation. First, I define the equivalence relation and show several properties for it. Then I derive the equivalence classes for power functions. Next, I investigate the relationships between the previously defined relation and isometries and similarities. I then derive the equivalence classes for quadratic, cubic, and quartic polynomials, as well as for a general polynomial. Finally, I conclude with a few applications.

The results are as follows. The defined relation is shown to be an equivalence relation and linear. It is shown that a polynomial lies in the equivalence class of a power function iff all other coefficients are a certain function of the first two. Only under certain restrictions can the relation be an isometry or similarity. There is one equivalence class for the quadratics, one for the cubics, and an infinite number of equivalence classes for polynomials of degree at least four. Applications include extensions to other functions and computer data storage and retrieval.

PRESENTER: Christopher Lopez

The BronxHigh School of Science

4:50 – 5:10

5)Title: Converging on the Eye of God

(This represents joint work with D.N. Seppala-Holtzman)

and

These values happen to be the x and y coordinates of the lower right-hand corner Eye of God, where Fn is the nth Fibonacci number. I will then show how this conjecture led to finding a more general formula and understanding.

PRESENTER: Francisco R. Rangel

St. Joseph's College

Poster Session:

3:30 – 5:00

1)TITLE: Setduko and Logicduko – Teaching Symbols with Games

ABSTRACT: Set theory and Logic is a mathematics course often assigned to liberal arts students. Liberal arts students benefit from seeing a nonverbal method of argumentation but are challenged by the abstractness of the material. One of the particular challenges for such students is the new symbols they must learn in these courses. Many of these students may have not learned any new symbols since preschool. To make learning the symbols of set theory and logic simple, I have developed a game based on Sudoku (see figures 1 and 2).

/
/
/
/
Each of these symbols: ,,, can only appear once in each of the rows, columns, and 2 by 2 boxes.
Figure 1 A Setduko / / /
/ /
/ / /
/ / /
/ /
/ / /
Each of these symbols: ,,,,, can only appear once in each of the rows, columns, and 2 by 3 boxes.
Figure 2 A Logicduko

These puzzles are not supposed to be challenging. They lead the students to use the symbols in an enjoyable activity. When going over the answer students, I require the students to give the correct names for the symbols, such as intersection for . Hence, the students learn the correct names for the connectives of set theory and logic. Once the students complete a puzzle, they are ready to learn how set theory and logic respectively use their connectives.

Once you have a single setduko at the appropriate difficulty, it is easy to generate many more one’s through symmetries. For example, switching the first two rows generates another valid setduko, and so does switching the last two rows. The same can be done with columns. Switching the first two rows with the last two rows also leads to a valid setduko. Permuting the symbols, for example replacing every with an and every with a also leads to a valid setduko. Thus, there are 8 row permutations, 8 column permutations, and 24 permutations of the symbols leading to 1536 setduko permutations. Even if not all the permutations are all unique, each setduko, like that in figure 1, generates over a 1000 different setukos that you can distribute to your students. Each logicduko generates far more permutations (over 1,000,000).