CTY’s mathematics, science, and computer science courses are dedicated to Dr. Richard P. Longaker, Provost of Johns Hopkins University from 1979 to 1987, in recognition of his advocacy and guidance through CTY’s initial years. Mathematics can be described as a language, tool, science, and art. Computer Science is an area of study that continues to gain importance for its wide applications. CTY math and computer science courses aim to give students the skills, knowledge, and perspective to understand the multifaceted nature of these fields. Students move beyond basic skills to gain greater understanding of both the underlying meanings of mathematical concepts and the intriguing ways math can be applied and extended in a range of contexts. CTY mathematics and computer science courses offer students the opportunity to enrich or accelerate their studies, strengthen their problem-solving skills, and explore challenging material. Students investigate advanced concepts through a process of discovery and engagement that promotes a lifelong interest in these disciplines. The courses ask students not only to master concepts and theoretical material, but also to apply their new knowledge to real-world questions and problems. Through hands-on approaches and thought-provoking exercises, students learn to make connections between abstract ideas and their use in a range of fields, including science, engineering, economics, and advanced mathematics. Students who wish to receive credit or placement for CTY courses need to work closely with their home schools to determine the appropriate steps to be taken. Please refer to our Eligibility web page for minimum test score requirements for math courses. Sample syllabi for all courses are also available.
Cryptology
Information is power. Even before the first written word, the need to safeguard information created an ongoing evolutionary battle between codemakers and codebreakers. Cryptology is the study of secret writing such as codes and ciphers. In this math course, students begin their journey with an exploration of many early techniques for creating secret writing, such as cipher wheels, the Caesar shift, monoalphabetic substitution, and the Vigenère cipher. They move on to learn about modern techniques including RSA public key cryptography. Delving deeper into modern techniques, students explore how data transmitted by computer can be secured with digital encryption. Discussions about the vulnerabilities of each encryption system enable students to attack and decrypt messages using techniques such as frequency analysis and cribbing. Students apply what they learn to encrypt and decrypt their own secret writing. Though the course’s central focus is on the mathematics of cryptology, the historical context of cryptography and cryptographic devices is provided to further develop understanding of this branch of mathematics. For example, students examine the design and fallibility of the Enigma Machine, one of the most important cryptographic devices in history. Sample text: The Code Book, Singh. Field Trip Fee (Carlisle and Lancaster only): $65. Students visit the National Security Agency’s National Cryptologic Museum. Session 1: Carlisle, Lancaster, Los Angeles, Saratoga Springs
Session 2: Carlisle, Lancaster, Los Angeles, Saratoga Springs Top
Advanced Cryptology Prerequisite: CTY’s Cryptology. In today’s environment, data transmission and data security play an increasingly critical role in the global marketplace and in national security. Picking up where CTY’s Cryptology leaves off, this course delves further into the rich mathematics behind the science of secret keeping. Since it forms the basis of much of modern RSA style cryptography, students begin this course with a review of key concepts in number theory. Students then progress into the study of topics such as non-repudiation and the connection between security and authentication. In addition, they learn cipher techniques not covered in the first level course. Further study into historical cryptography systems leads students to a new understanding of the techniques used in breaking the Enigma devices of World War II. Students also investigate the inner workings of the M-209, a device primarily used by the United States military in World War II. Finally, students expand beyond number theory and combinatorics to examine other mathematical techniques that can be used in the processes of code making or code breaking. Students employ computational techniques to learn the fundamentals of statistical attacks on cryptosystems. In addition, they may learn the fundamentals of abstract algebra in an effort to grasp the elements of elliptic curve cryptography. Students leave this course with an advanced understanding of the mathematical basis of cryptology. Sample text: New course. Session 1: Lancaster
Session 2: Not offered
Probability and Game Theory
Prerequisite: Algebra I. The study of probability and game theory is an excellent way for students to apply math to real-world situations. Game theory is a very specific branch of mathematics focusing on the application of probability to competitive behavior. Students investigate a variety of topics including voting patterns, coalition building, and bargaining. In this class, students use mathematical modeling as an important foundation from which they can pursue further investigations. The models of game theory are abstract representations of real-life situations. For example, Nash equilibria have been used to study political competition, and the Prisoner’s Dilemma has been used to analyze the social networks of different populations. The mathematics covered in this course includes concepts of probability and linear algebra. Class exercises involve individual and group work as well as possible class tournaments. Sample texts: Game Theory and Strategy, Straffin; Thinking Strategically, Dixit. Session 1: Carlisle, Lancaster, Los Angeles, Saratoga Springs
Session 2: Carlisle, Lancaster, Los Angeles, Saratoga Springs Top
Mathematical Logic
Prerequisite: Algebra I. In this course, students learn about and practice what most mathematicians spend their time doing: proving or disproving conjectures. Students are introduced to the techniques of formal proof and counterexample. While the focus is on mathematical proofs, the logical reasoning skills developed are the building blocks of serious intellectual inquiry in almost every academic discipline. Students learn about the key concepts of logic, including validity, soundness, consistency, and satisfiability. They develop systems of logic in formal symbolic languages, including the propositional calculus and first-order quantified logic. These systems allow students to test the validity of arguments and write formal proofs with precision. Students explore the syntax (rules of grammar) and semantics (meanings of the symbols) of these languages. They then engage in metalogic: reasoning logically about a system of logic. They examine soundness, asking whether all proofs in the system really prove their conclusion; and completeness, considering whether every valid conclusion can actually be proven using the rules of that system. By the end of the course, students become proficient at a skill essential to mathematicians—the ability to write proofs accurately and rigorously. Most importantly, students develop strong problem-solving skills and learn to think analytically, traits vital for rigorous inquiry in any field. Sample texts: Logic: Techniques of Formal Reasoning, Kalish et al.; Formal Logic: Its Scope and Limits, Jeffrey. Session 1: Baltimore, Lancaster
Session 2: Baltimore, Lancaster, Los Angeles Top
Number Theory
Prerequisites: Geometry and Algebra II. Called “the queen of mathematics” by the great mathematician Karl Friedrich Gauss, number theory is the study of the natural numbers, the number system from which all others are derived. Despite the simplicity of the counting numbers, many accessible problems in number theory remain unsolved. For example, the Goldbach Conjecture, formulated in 1742, which posits that every even integer larger than 2 is the sum of two prime numbers, has defied all attempts to be proven. In this course, students are introduced to the major ideas of elementary number theory and the historical framework in which these concepts were developed. While strengthening their ability to analyze and construct formal proofs, students explore topics such as the Euclidean Algorithm and continued fractions, Diophantine equations, Fibonacci numbers and the golden ratio, modular arithmetic, Fermat’s Little Theorem, RSA public key cryptography, and Fermat’s Two Square Theorem. Students leave the course with an appreciation of the elegance of theoretical mathematics. Sample text: Materials compiled by the instructor. Session 1: Lancaster
Session 2: Lancaster, Los Angeles Top
Macroeconomics and the Global Economy Prerequisite: Algebra I. Students who are eligible for humanities courses may take this course, so long as they have satisfied the Algebra I prerequisite. What are the key indicators of an economy’s performance? How do governments craft monetary and fiscal policy to promote economic growth? How do economic systems affect one another? Analyzing economies at an aggregate level, macroeconomics—the study of economics systems—explores questions such as these, providing a bird’s-eye view of economic activity. Students in this course explore fundamental concepts in macroeconomics including national income, economic growth, inflation, employment, money, banking, financial markets, and the role of public policy. Building upon this foundation, students consider the global economy and issues in international trade and finance. Students examine comparative advantage and balance of payments, along with exchange rates and foreign currencies. By applying mathematical concepts to economic theory, students explore how economists analyze and predict changes in the economy. Through lectures, readings, discussions, simulations, and research, students gain a firm grounding in macroeconomics and an introduction to central concepts in international trade and finance. Throughout the course, they draw from this knowledge to better understand the state of the US and world economies today. Sample text: New course. Session 1: Baltimore
Session 2: Not offered
Individually Paced Mathematics Sequence
Individually Paced Math Sequence includes courses in Algebra I, Geometry, Algebra II, and pre-calculus topics including Functions, Trigonometry, Discrete Math, and Analytic Geometry. It is designed for students with the interest and ability to accelerate their study of mathematics, perhaps to lay the groundwork to take Calculus at an earlier grade level or to begin studying college mathematics while still in high school. Inspired by research conducted by Johns Hopkins University’s Study of Mathematically Precocious Youth (SMPY), this course is designed for students with strong independent learning skills, high self-motivation, and high mathematical ability. Math Sequence allows students to work independently at a pace commensurate with their individual abilities. At the beginning of the session, instructors assess the knowledge base of each student with diagnostic testing. The results of this testing, together with information provided by students and their schools, determine the appropriate starting point for each student in CTY’s curriculum and allow students to focus their attention on unfamiliar material. For most of the day, students work on their own, solving problems from their textbooks. The instructor and teaching assistant monitor each student’s progress carefully and offer support and individualized instruction as needed. Students must demonstrate a mastery of the concepts and skills within each subject area before moving on to the next topic. Students who have not consulted with their home school regarding credit and placement should not register for this course. For more information, visit the Math Sequence Web Page
Sample texts: Algebra, Smith; Geometry, Jurgensen; Algebra and Trigonometry, Smith; Advanced Mathematics, Brown; Precalculus with Trigonometry, Foerster. Session 1: Baltimore
Session 2: Baltimore Top
Fundamentals of Computer Science Prerequisite: Algebra I. This course, unlike a traditional programming course, introduces students to three major areas of computer science: theory and algorithms, hardware systems, and software systems. The theoretical component of the course covers the study of algorithms, Boolean algebra, binary mathematics, and the theory of computation. While studying hardware systems, students learn about the physical components of computers, digital logic, and computer architecture. In investigating software systems, students are introduced to elements of programming languages, compilers, and computer graphics. Students also explore operating systems, a key link between hardware and software, and computer networks. Note: Learning a particular programming language is not a goal of the course. Rather, students apply and illustrate some concepts they learn through work on programming projects. Sample text: An Invitation to Computer Science, Schneider and Gersting. Lab Fee: $65 Session 1: Lancaster, Los Angeles, Saratoga Springs
Session 2: Lancaster, Los Angeles, Saratoga Springs Top
Data Structures and Algorithms Prerequisite: CTY’s Fundamentals of Computer Science or an equivalent computer science programming experience. In order for a computer to find a solution to a particular problem, it is necessary to formalize the problem in terms of a mathematical model, find a suitable algorithm to solve the problem in that model, and then implement the algorithm in a particular programming language. In this class, students learn how to design, analyze, and implement these algorithms. Students begin by studying data structures such as arrays, lists, stacks, queues, trees, and sets in order to learn different ways of organizing data. Students then analyze many sorting, searching, and graphing algorithms to determine their run-time efficiency. By examining these fundamental algorithms, students learn how design decisions can affect the efficiency of an algorithm. A series of programming assignments helps students learn how to put these abstract ideas into practice. By the end of this course, students acquire the conceptual tools necessary to model and analyze computational problems. Note: Programming experience is required for this course. Sample text: Data Structures and Algorithm Analysis, Weiss. Session 1: Lancaster
Session 2: Not offered Top
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