Showing papers in "Mathematics and Computer Education in 2012"

Applied Partial Differential Equations with Fourier Series and Boundary Value Problems

Abstract: This title is part of the Pearson Modern Classics series. Pearson Modern Classics are acclaimed titles at a value price. Please visit www.pearsonhighered.com/math-classics-series for a complete list of titles. Applied Partial Differential Equations with Fourier Series and Boundary Value Problems emphasizes the physical interpretation of mathematical solutions and introduces applied mathematics while presenting differential equations. Coverage includes Fourier series, orthogonal functions, boundary value problems, Green's functions, and transform methods. This text is ideal for readers interested in science, engineering, and applied mathematics.

Ted: Ideas Worth Spreading

Abstract: TED: IDEAS WORTH SPREADING Website: www.ted.com/ Contact: TED Conferences LLC, 250 Hudson Street, Suite 1002 New York, NY 10013 Phone: 212-346-9333, Price: Free website As a college instructor, I try to keep an eye out for any good ideas that might help me become a better mathematics teacher or simply a better person. Like many of us, that is why I attend periodic conferences, meetings, and professional development activities. But as useful as these are, they require a lot of time and, often, quite a bit of money. I have recently begun watching a number of wonderful presentations on the Internet that are as good as some of the best presentations that I've attended at national conferences. If you haven't heard of TED yet, you certainly will in the future. This website has been around a little less than five years, and is sweeping the world as a truly international collection of quality ideas by thoughtful professionals. TED, which stands for Technology, Entertainment, and Design, started out as a series of in-person conferences on creative thinking has now grown into an impressive free, non-profit Internet site that collects high-quality idea presentations covering a wide range of topics. More than 1000 of these presentations or "TEDTalks" are now available, with more added each week. The talks are presented or subtitled in English, and many are also available in a variety of translations. The talks, under a Creative Commons license, can be freely shared, reposted, or downloaded for use. What I like most about the TEDTalks is that they represent many truly interesting ideas. They are also of relatively short duration (usually not longer than about 1 8 minutes) that can be watched easily on an iPhone or iPad. Many of the presentations relate directly to the interests of the MACE community. For example, I recently watched excellent videos on: * reforming mathematics education * fractal mathematics * the mathematics of coral reefs * the mathematics of cities * mathematical models of war * teaching mathematics with computing * origami mathematics; and * mathemagic and there are many more videos directly related to mathematics. There are also some "out there" ideas, such as how to defend the Earth against asteroids; understanding extra dimensions; time travel possibilities; the future of mobile computing; and many other interesting topics. As far as I could tell, these videos are very well peer-reviewed, and only the best quality presentations are stored at the site. To me that is a real strength of TED, that there is no need to wade through lots of junk videos to find the really useful ones, as you have to do on so many other video sharing sites. …

Game Theory and the Humanities

Abstract: Game theory is a mathematical theory of strategy that has been applied to the analysis of conflict and cooperation in such fields as economics, political science, and biology. In this seminar, we discuss more unusual applications—to the humanities, including history, literature, philosophy, the Bible, theology, and law—as well as some of the usual ones. No mathematical background beyond high school mathematics is assumed, but a willingness to learn and apply sophisticated reasoning to analyze the interactions of players in games is essential.

A Comparison of Performance and Attitudes between Students Enrolled in College Algebra vs. Quantitative Literacy

Abstract: INTRODUCTION In the 2006-2007 academic year, the University of South Dakota formed a mathematics task force to evaluate student success in introductory mathematics courses and to investigate the question of quantitative literacy and its meaning for graduates of a liberal arts institution. Questions about this paper or study may be directed to Dan Van Peursem at dpuersem@usd.edu. The need for Quantitative Literacy courses was established in reports from the Mathematical Association of America (MAA) [21], American Mathematical Association of Two- Year Colleges (AMATYC) [2, 4], and the College Board [25]. National focus groups have been formed around this topic including the Special Interest Group of the MAA on Quantitative Literacy (SIGMAA-QL) and the National Numeracy Network (NNN), both formed in 2004. In the ensuing years, numerous reports and courses have been developed centering on the topic of Quantitative Literacy [8, 14, 15, 16, 17, 22, 23, 24, 25, 26, 27]. As recently as 2010, a fair amount of discussion continues about the assessment of a Quantitative Literacy course and the specific learning goals that such a course may include [6]. This paper does not resolve the difficulty of assessing Quantitative Literacy course outcomes, but evaluates the effectiveness of a Quantitative Literacy course in preparing students for outcomes typically expected from a traditional College Algebra course. One of the main goals stemming from our mathematics task force for the newly created Quantitative Literacy course was to have students from this course use and understand the mathematics that they would encounter in everyday situations such as evaluating data presented in newspaper articles, understanding different voting procedures, and using basic mathematics to understand implications of interest rates in personal finance decisions. In developing the course, we received advice from Dr. Bernie Madison (University of Arkansas), who had developed a similar course on his campus. It was a challenge to find an appropriate Quantitative Literacy textbook to include a variety of course material emphasizing realistic and meaningful problem solving investigations, included topics of mathematics in art and in music, and which did not duplicate Finite mathematics course material. As a result, some supplementary course material was developed by the department. Another goal in developing this course was to provide students with an alternative to college algebra as a terminal mathematics course. To satisfy that purpose, the course must fulfill the Board of Regents requirements for mathematical competency in the system general education core, with an expectation that students completing courses in the core have sufficient understanding to be successful on the Collegiate Assessment of Academic Proficiency (CAAP) exam which is published by American College Testing (ACT) and is designed to assess students' proficiency in solving mathematical problems encountered in postsecondary curricula. In the South Dakota BOR system, all students seeking a four year degree are required to take and pass the exam after completing forty-eight hours and students seeking a two year degree are required to take and pass the exam after completing thirty-two hours of course work. The traditional CAAP exam is weighted much more toward the algebraic manipulation of a College Algebra course rather than the many application problems that one would find in a Quantitative Literacy course. This could be problematic for both institution and students, as data for the institutional performance on the proficiency exams is reported to the public and passing the proficiency exam is a required component for continued enrollment for students. Thus, we needed assurance that our institutional CAAP scores would not suffer as a result of this newly created course, and that students completing the course would not be disadvantaged in progression toward a degree. Several other institutions within the system also requested permission to offer the course, but only Black Hills State University followed the study guidelines which would allow us to make meaningful comparisons between the College Algebra and Quantitative Literacy courses. …

Interesting Subseries of the Harmonic Series

Abstract: (ProQuest: denotes formulae omitted) INTRODUCTION Usually, the first nontrivial divergent series shown to students is the harmonic series (1) It is the purpose of this paper to show that examining subseries of this harmonic series will give us a rich source of both convergent and divergent series This should be especially interesting to good students since our methods show the novel role played by prime numbers It is easy to show that subseries like , using only even integers, and , using only odd integers, also diverge We will make extensive use of the geometric series which converges for |x| (2) Here, series (2) is the subseries of (1) consisting of numbers n having only one prime p in their prime power factorization Now let q be another prime and write (3) Multiplying (2) and (3) we get (4) Here the series on the right of (4) is the subseries of (1) consisting of all numbers n having only the two primes p and q in their prime factorization For series of positive numbers, as we are examining here, the product of two convergent series converges Thus, we have determined that this subseries converges and sums to In the remainder of this paper we will examine subseries of (1) like the above example This material would be suitable for good students in calculus after they have seen infinite series And is especially useful for the study of real analysis CONVENIENT NOTATION AND GENERALIZATIONS Let {p^sub 1^,p^sub 2^,p^sub 3^, } denote a set of distinct prime numbers and Z{p^sub 1^, p^sub 2^, p^sub 3^, } denote the set of all natural numbers with only these primes as factors For example, Z2,3} is the set of all numbers of the form 2ry , where the powers r and s are non negative integers Thus Z{2,3} = {1,2,3,4,6,8,9,12,16,24,27,32,48, } Then the subseries of the harmonic series (1) with only denominators from Z{2,3} is denoted by From the analysis of (4) in the previous section, we see that this series converges and sums to In the previous section we examined the subseries of (1) over the integers Z{p,q} and obtained (using our new notation) from (4) We can easily generalize this result to any number of prime factors Z{p^sub 1^,p^sub 2^,p^sub 3^, } and get (5) This last result is the key to our investigations of subseries of the harmonic series Notice that if the number of primes in {p^sub 1^,p^sub 2^,p^sub 3^, } is finite, then the left side of (5) is a finite product and the series on the right converges to the value calculated on the left However, if the number of primes is infinite, then the product on the left of (5) is infinite, and thus might diverge We will investigate these cases in the next section EXAMPLES OF CONVERGENT AND DIVERGENT SUBSERIES Example 1: Let p^sub n^ denote the n^sup th^ prime number Then the set {p^sub 1^, p^sub 2^, p^sub 3^, } *s me set °f aM prime numbers and by the fundamental theorem of arithmetic, Z{p^sub 1^, p^sub 2^, p^sub 3^, …

Using Dynamic Geometry Software to Develop Problem Solving Skills

Abstract: (ProQuest: ... denotes formula omitted.) 1. INTRODUCTION Currently, much of the research in the area of Dynamic Geometry Software (DGS) is focused on how students experiment and generate conjectures about relationships in geometry. But how students use DGS as a problem solving tool and how DGS can strengthen the user's problem solving capacity and heuristics are not generally emphasized. The purpose of this study is to examine how students interact with particular features of a software tool while solving geometric construction problems. We hope to provide insight about students' use of various software features and whether or not these uses support different methods of problem solving. Geometry education has been strongly linked with dynamically interactive software such as Cabri and Geometers Sketchpad [1, 2]. A user can drag elements of a construction while maintaining necessary constraints. This provides an environment in which students can experiment quickly and freely. They can check their intuitions and conjectures in the process of looking for patterns and checking the invariant properties of figures [3]. Luthuli [4] calls this type of instruction "research-based geometry". The ease of creating a variety of results has enabled the transition from particular to general cases in a dynamic geometry environment [5]. The primary goal of this type of instruction is to discover or generalize geometric relationships via induction. Some authors use DGS differently. For example, HoIz [6] states that the general uses of DGS are often limited to verification, in the sense that students are expected only to vary or confirm empirically at the computer. He concludes that we must go beyond confirmation and suggests increasing the use of DGS to support heuristic approaches to problem solving. Guven [7] with his own experimentation shows that DGS can be used for insight to a proof. As a result of his own explorations in an interactive geometrylearning environment, Baki [8] concludes that CabrVs dragging, tabulating, and immediate-feedback capabilities greatly assisted him in determining properties, special cases, and counter-examples that could be linked to form a conjecture and a justification. 1.1 Problem Solving and DGS The design of technology tools has the potential to dramatically influence students' mathematical understanding and problem solving methods [9]. For example, Harskamp and Suhre [10] show that computer programs based on a direct instructional approach to learning or constructivist views of learning help students to improve the quality of their problem solving analysis and verification. Schoenfeld [11] contributes a framework of different factors that affect students' abilities to solve problems. In his framework, four components comprise the major directions of students' problem solving: Resources: Body of knowledge that an individual is capable of bringing to bear in a particular mathematical situation. They are the facts, definitions, procedures, rules, and intuitive understandings of mathematics. Heuristics: Rules of thumb for effective problem solving. They are strategies and techniques for approaching a problem. Control: The ways in which individuals monitor their own problem solving process, use their observations of partial results to guide future problem solving actions, and decide how and when to use available resources and heuristics. Belief systems: One's mathematical world view, the perspective with which one approaches mathematics. [Note: For the purpose of this paper, we focused on the first three aspects.] Students need to understand how DGS can affect the problem solving process. Thus, the definition of resources should be expanded to include both mathematical resources and mathematical ability with available technological tools [9]. Healy and Hoyles [12] found, in the context of a dynamic geometry environment, that the presence of particular technological features can afford or constrain the available heuristics students may use for problem solving. …

Roads to Infinity: The Mathematics of Truth and Proof

Abstract: ROADS TO INFINITY: THE MATHEMATICS OF TRUTH AND PROOF by John Stillwell A. K. Peters, 2010, 203 pp. ISBN: 978-1-56881-466-7 Roads to Infinity: The Mathematics of Truth and Proof 'is an account of the discovery of the uncountably infinite; the interaction between set theory and logic in the realm of the irifinite; and the mathematical consequences of accepting the infinite levels of infinity. The book follows essentially two roads to infinity: Cantor's diagonal argument and Cantor's construction of the ordinals. Stillwell shows how these two themes intertwine and influence a wide range of mathematical questions including consistency, provability, computability, and existence. Roads to Infinity comprises seven chapters, each based upon a mathematical question. The historical responses to the question are explored and the concepts and theorems resulting from these responses are explained in essentially non-technical language. However, the abiiity to read mathematical symbolism and understand mathematical argumentation is required. Still well begins with Cantor's diagonal argument. His focus is the uncountability of the real continuum and he includes in the discussion the ever-amazing uncountability of transcendental numbers, an application of the diagonal argument to the rate of growth of functions, the paradoxes of set theory, and the axioms of Zermelo-Fraenkel set theory. Next, the book examines the transfinite ordinals, the continuum hypothesis, the axiom of choice and well-ordering, measurability of sets, and Cohen's technique of forcing. Also included here is a discussion of how Cantor's set theory arose from his investigation Fourier series. In the third chapter, Stillwell turns his attention to questions of computability and provability. Here we encounter Godei' s first and second incompleteness theorems, Turing machines, the Halting Problem (which Stillwell finds lurking in Cervantes' Don Quixote^!), and Hubert's Entscheidungsproblem for predicate logic. The chapter leads nicely into Chapter 4 on the consistency and completeness of propositional and predicate logic. A major theme in this chapter on logic is "cut-elimination", a way of inference in logic that replaces modus ponens by falsification trees. Chapter 5 focuses on arithmetic. Here we find a detailed discussion of Peano Aritiimetic, and an infinite extension of Peano Arithmetic and how the extension may be used to prove the consistency of Peano Arithmetic (which cannot prove its own consistency). The diagonal argument theme is reinforced in this chapter as Stillwell shows how the unprovability of consistency of Peano Arithmetic within Peano Arithmetic is related to the argument that 2N" is uncountable. …

Password Generation by the Intersection of Two Functions

Abstract: (ProQuest: ... denotes formulae omitted.) INTRODUCTION This paper presents a novel method of generating a strong, high entropy (pseudo-random), recoverable password based on the intersection of two functions that is approachable by students who have studied Dihedral Groups, and have had, at least, an introductory computer programming course. Although it is possible to use a small output hash function such as Adler32 [1] to generate recoverable passwords (i.e., recoverable: in the event of a lost password it can be easily reconstructed), most hash functions available will convert a seed value to a fixed character length. For a further discussion of hash functions see [2] and [3]. The intersection of two functions method is expandable to create a password of any character length. A major strength of the method outlined here ensures that passwords are recoverable in addition to high entropy, and unlike many well-known hash functions, this password method can be extended to produce a character string of any desired length. Teachers can use this method as an example for considering why the intersection of functions might be useful and relevant to daily life. Students can use this method as a starting point for examining other functions that might generate a computer password. This method can be used as a problem for multiple computer programming assignments: to generate discussion on what constitutes strong computer passwords, on issues of creating and forgetting computer passwords with the necessity for recoverable passwords, for questions on how to code place value, dates, and matrices, and can lead to an introduction of other types of data structures. Consider the intersection C of an exponential function and a line determined by points A and B, where A > 1 on the v-axis and point B > 0 on the x-axis as in Figure 1, below. Then the ith character of the password is generated from the x-coordinate of C, with each value of B generated as described below. The overall method is illustrated in Figure 2. The password seed is generated by selecting the «th time a particular day of the week occurs in a selected month that willl always be used and not easily forgotten. The particular date of the nth occurrence of that day of the week in the selected month along with the year is used to construct a date string mmddyyyy. The seed value v can be calculated by letting ... where cf. is the repeated sum of the date string, mmddyyyy. ds =m + m + d + d + y + y + y + y (and the sum is repeated until a single digit results), and ? is the truncated result of the calculation keeping only the leading 9 digits. Next, ? is concatenated with itself to form an 18 digit string w. This completes the first step, password seed value generation. Consider the string w as a set of consecutive order pairs. In each ordered pair(x^sub i^, y^sub i^) of the string w, x^sub i^ represents the row and y^sub i^ represents the column of D^sub 5^ for the ordered pair i (see Table 1, in Appendix). Let s^sub i^ - x^sub i^y^sub i^ represent the group product in D^sub 5^ to create the set S of filtered values s^sub 1^,s^sub 2^,s^sub 3^,...s^sub 9^ that will be used to form the password generating equations. This completes the second step, generating the filtered seed. Nine linear equations of the form ... are generated from the filtered values s^sub 1^, s^sub 2^, ... ,s^sub 9^. The exponential function is one of ... with ... Therefore each point of intersection C^sub i^, and thus each password character, is generated by the same exponential function and one of 9 linear functions, (see Figure 2, above). This completes the third and fourth steps, generate a set of functions. Lastly, let C^sub x^ be the x-coordinate of the intersection point C, and consider the four digit number c^sub 1^c^sub 2^c^sub 3^c^sub 4^ consisting of the first four decimal digits of C^sub x^. Let α^sub 1^^sub 1^^sub 2^ = c^sub 1^c^sub 2^c^sub 3^c^sub 4^ (mod 78) and use the symbol stored in cell α^sub 1^α^sub 2^, where α^sub 1^α^sub 2^ ∈ {00, 01, 02, . …