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 mathematics. 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 accelerate need to work closely in advance with their local schools to negotiate appropriate placement and/or credit. 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
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. Unlike mathematical modeling, which is a broad field, 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, the theory of 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, from philosophical argumentation to scientific methodology. 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; students reason logically about a system of logic. Specifically, they examine both its soundness and completeness: that all proofs in the system really prove their conclusions and that 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 with elegance and assurance. 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
Set Theory
Prerequisite: Geometry. In the town of Seville, there is a male barber who shaves all men, and only those men, who do not shave themselves. Does the barber shave himself? This is an applied form of Russell's paradox, which served as a crucial turning point in the development of set theory. Just as atoms are fundamental to the study of matter, sets may be viewed as the building blocks of mathematics. A set is a collection of objects, and set theory studies the properties of sets and their relationships. Despite this seemingly simple subject matter, set theory is a vibrant branch of mathematics, as well as its most common foundation. In this course, students move beyond the use of formulas and equations and into the realm of proof as they examine the essential components of set theory, such as functions, relations, and orderings. They explore cardinality, learning how one infinite set can be larger than another. By the end of the class, students consider complex topics in set theory, such as the axiom of choice, transfinite arithmetic, and the continuum hypothesis. Students leave the course with not only a razor-sharp understanding of the central concepts of set theory and an enriched mathematical vocabulary, but also a firm foundation for advanced exploration in all branches of mathematics. Sample text: New course. Session 1: Lancaster
Session 2: Lancaster, Los Angeles Top
Number Theory
Prerequisites: Geometry and Algebra II. Called “the queen of mathematics” by Karl Friedrich Gauss, number theory is the study of the counting numbers, the most basic of all number systems. Despite the simplicity of the counting numbers, many accessible problems in number theory remain unsolved. For example, the Goldbach Conjecture formulated in 1742 posits that every even integer larger than 2 is the sum of two prime numbers. 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 including 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 Top
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 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, Lancaster
Session 2: Baltimore Top
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