Math and Computer Science

Listed below are the math and computer science courses offered in 2012, and at which site and session each course is offered. If you are unfamiliar with our site codes, please see the site key below. The course title links will take you to the appropriate catalog course description and links to sample syllabi for the course.

If you would like to read about science, writing, or humanities courses, select the appropriate discipline in the following drop down menu.

Math and Computer Science Course Offerings

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Math Course Descriptions and Sample Syllabi

Math Problem Solving

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Can five 20' by 18' carpets lying flat with no overlap fit in a 40' by 50' room? How can you precisely measure two liters of water using only a four-liter pitcher and a three-liter pitcher? How many different ways can you add up four even, positive numbers to get a sum of 16?

Problem solving in mathematics is far more complex than translating a word problem into numbers and symbols and applying an established method. It involves finding a path to a solution when there is no clear place to begin. In this course, students learn general strategies for solving problems that involve a wide range of mathematical concepts. Challenging problems lead students to use varied approaches such as drawing diagrams and graphing, making lists, estimating, eliminating unreasonable possibilities, identifying patterns, guessing and checking, and manipulating variables.

Working individually or in small groups, students learn to ask precise and thought-provoking questions, to match appropriate strategies to particular problems, and to effectively communicate their thought processes along the way. Demonstrations, activities, games, and explorations are incorporated to nurture students as critical thinkers and creative problem solvers, strengthening their mathematical-reasoning abilities and preparing them for future study in discrete math, probability, cryptology, and other growing fields of mathematics.

By the end of the course, students are able to demonstrate several methods for determining that the carpets can’t fit, to measure out exactly two liters of water, and to show that there are five different ways to get 16 by adding four even, positive numbers.

Sample text: Materials compiled by the instructor.

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Geometry and Spatial Sense

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Spatial understanding is necessary for interpreting, understanding, and appreciating our inherently geometric world. Many tasks a person performs—whether rearranging furniture in a room, solving a jigsaw puzzle, or laying out a garden—require good spatial reasoning abilities. 

Through interactive investigation and discussion, students discover mathematical relationships such as congruence, symmetry, and reflection. They learn geometric formulas to calculate area, surface area, perimeter, circumference, and volume and are asked to explain their findings using geometric terminology. Students explore the geometric properties of regular polygons and create their own polyhedra using paper-folding techniques. Tessellations, similarity, measurement, polygons, polyhedra, and curved shapes are investigated through two- and three-dimensional drawings and physical models. Students also construct classic geometric figures using a straight edge and a compass as they are introduced to the essential elements of geometry. Through explorations and demonstrations, students leave this course with greater spatial sense as well as a solid understanding of foundational geometric concepts.

Sample text: Materials compiled by the instructor.

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Numbers: Zero to Infinity

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What does a subatomic particle measured in femtometers have in common with a galaxy measured in light years? Both are a part of the uniquely human effort to quantify the world around us. In this course, students explore numbers, from the very small to the unimaginably large, and learn how numeric representations help to explain natural phenomena such as time, distance, and temperature.

Moving beyond traditional arithmetic, this course centers on hands-on activities that develop understanding of the scope and scale of numbers. Students consider such questions as: if your dog were the size of a dinosaur, how much dog food would you need? They develop approximation and computational strategies for learning scientific notation, and determine whether answers to problems are reasonable. In examining the diversity of measurement systems, students learn the origins of some familiar and unfamiliar methods of measurement, and invent their own units of measurement. Additionally, students use dimensional analysis to investigate conversions between different scales or systems of measurement. They apply concepts of ratio and proportion by constructing and analyzing scale models of our solar system, the human body, and other objects in our natural world.

Sample texts: Materials compiled by the instructor; a supplemental text such as Gulliver’s Travels, Swift.

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Inductive and Deductive Reasoning

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Reasoning, logic, and critical thinking skills are the building blocks of intellectual inquiry. This course focuses on developing these skills through problem solving, puzzles, and exposure to a wide range of topics in mathematics. Students learn to distinguish between inductive and deductive reasoning and examine the roles played by each in mathematics. 

What is the next term of the sequence 1, 5, 12, 22, 35? How do these numbers relate to triangular and square numbers? The students’ introduction to inductive reasoning begins with a search for patterns in data and creating recursive and explicit formulas to describe those patterns. Students master material by considering puzzles, algebraic and geometric concepts, patterns, and real-world questions that can be answered using induction.

As they move on to topics in deductive reasoning, students learn to use a system of logic to draw conclusions from statements that are accepted as true. Students encounter a variety of classic problem types as they explore symbolic logic, truth tables, axiomatic systems, matrix logic, syllogisms, Venn diagrams, knights and knaves problems, and Euler circuits. Emphasis is placed on the importance of proving conclusions using valid arguments and developing the ability to recognize fallacious reasoning.

Sample texts: Materials compiled by the instructor; a supplemental text such as The Number Devil, Enzensberger.

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Great Discoveries in Mathematics
(This course was formerly known as Problems, Strategies and Solutions: History of Mathematics)

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From ancient to modern times, mathematics has been instrumental in the development of science, engineering, and philosophy. In this math course, students consider the questions and problems that have fascinated humans across cultures since the beginning of recorded history.

Throughout the course, students explore number systems and mathematical concepts first considered by early cultures, including the Egyptians, Greeks, Mayans, Babylonians, and Chinese. Additionally, they consider this newfound conceptual knowledge in its historical context. Students examine number systems with different bases, and in doing so, gain a much deeper understanding of the base ten number system. Through hands-on explorations, they learn about great mathematical discoveries throughout time, such as Pascal’s Triangle, the Pythagorean Theorem, the Golden Ratio, pi, and Zeno’s paradoxes.

By examining the historical development of major mathematical ideas, students leave the course with a greater awareness of a wide range of topics within mathematics, including number theory, algebra, and geometry. They acquire a solid background in mathematical concepts they will encounter in more advanced course work.

Sample text: Materials compiled by the instructor.

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Data and Chance

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Behind only one of three doors is a fabulous prize. After you choose door #1, the host reveals door #2 has nothing behind it. She then offers you the opportunity to change your selection. Should you switch to door #3? This classic example of conditional probability is not as simple as it seems.

In this course, students develop a greater understanding of data and chance, two areas of mathematics that easily transfer from the classroom to the real world. Students conduct experiments and generate data which they display in graphs, charts, and tables in order to compare the effects of particular variables. For example, students might analyze data to examine how various design characteristics of a paper airplane, such as weight or length, affect the distance it will travel. In addition, students consider other data sources, including newspapers and journals, and identify examples of incorrectly gathered or misrepresented data that have been used to mislead consumers or influence voters.

Students also explore probability, the study of chance, to learn how to use numerical data to predict future events. Students examine permutations and combinations; develop strategies for calculating the number of possible outcomes for various events; calculate probabilities of independent, dependent, and compound events; and learn to distinguish between theoretical and experimental probability.

By the way, you should switch.

Sample text: Materials compiled by the instructor.

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Individually Paced Math Sequence (7th Grade and Above)

www.cty.jhu.edu/summer/mathsequence.html

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 our 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.

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.

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Discrete Math

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Can any given map be colored in with just four different colors such that no two regions sharing a common edge are the same color? Mathematicians took more than 100 years to answer this question in the affirmative, establishing the result known as the Four-Color Theorem. Students investigate problems such as this as they study an area of math beyond algebra and geometry. Discrete math introduces students to set theory, as well as combinatorics and graph theory—each with a range of important real-world applications such as determining the number of ways to create a password of a given length or finding the shortest path between multiple locations using GPS navigation.

Students in this course begin by building a foundation in set theory and proof. They then explore combinatorics, examining the number of possible configurations of different sets of objects. Students move on to investigate graph theory, an area that introduces them to both historic problems such as the Seven Bridges of Königsberg and the Travelling Salesman, as well as more modern applications such as the analysis of social networks or traffic patterns.

Students leave the course not only with a familiarity with a flourishing branch of mathematics, but also with an enriched mathematical vocabulary and an improved ability to understand and create mathematical arguments.

Sample text: Materials compiled by the instructor.

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Mathematical Modeling

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Mathematics is more than just numbers and symbols on a page. Applications of mathematics are indispensable in the modern world. Math can be used to determine whether a meteor will impact Earth, predict the spread of an infectious disease, or analyze a remarkably close presidential election. In this course, students learn how to create mathematical models to represent and solve problems across a broad range of disciplines, including political science, economics, biology, and physics.

Students in this class investigate voting systems by constructing mathematical models of how groups make decisions and how elections are conducted. They consider how goods, property, and even political power can be fairly divided and apportioned. Students learn how to use Euler and Hamilton circuits to find the optimal solutions in a variety of real-world situations, such as determining the most efficient way to schedule airline travel. In investigations of growth and symmetry, students develop linear and exponential growth models and explore fractals and the Fibonacci numbers. Students leave this course with the ability to use the seemingly abstract language of mathematics to gain a greater understanding of the world around them.

Sample text: Excursions in Modern Mathematics, Arnold and Tannenbaum.

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The Mathematics of Money

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From managing one’s personal investments to examining the profitability of a multibillion-dollar global corporation, the mathematics of money is at the heart of successful financial endeavors. Why are round-trip fares from Orlando to Kansas City higher than those from Kansas City to Orlando? How do interest rate adjustments made by the Federal Reserve affect the real estate market? How does one calculate the price-earnings ratio of a stock and use that to help predict that stock’s future performance? Mathematics is an indispensable part of the answer to each of these questions.

This course provides students with a mathematical grounding in central concepts of business and finance. Students investigate the mathematics of buying and selling, and apply these principles to real- world situations. They gain fluency with the concepts of simple and compound interest and learn how these affect the present and future value of loans, mortgages, and interest-bearing accounts. Students investigate various forms of taxes, considering their impact on personal and governmental budgets. In their examination of these topics, students manipulate and solve algebraic expressions, and also learn to apply a range of mathematical concepts including direct and indirect variation and arithmetic and exponential growth. Through simulations, entrepreneurial projects, and classroom investigations, this course provides students with the foundation required to be more secure in their own personal financial management, and enhances their understanding of the broader economic conditions that shape investments in the public and private sector.

Sample texts: Materials compiled by the instructor.

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Probability and Game Theory

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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, and may also include class tournaments.

Sample texts: Game Theory and Strategy, Straffin; Thinking Strategically, Dixit.

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Game Theory and Economics

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Thomas J. Watson, the founder of IBM, once said, “Business is a game—the greatest game in the world if you know how to play it.” In today’s global marketplace, understanding game theory, the branch of mathematics which focuses on the application of probability to competitive behavior, is crucial to understanding business and economics. 

In this course, students use game theory as a framework from which to analyze a variety of real-world economic situations. Students begin the course by analyzing simple games, such as two-person, zero-sum games, and learn how these games can be used to model actual situations encountered by entrepreneurs and economists. For instance, students may apply the concept of Nash equilibria to find the optimum strategy for the pricing of pizza in the competition between Domino’s and Pizza Hut.

As they acquire an understanding of more complex games, students apply these methods to analyze a variety of economic situations, which may include auctions and bidding behavior, fair division and profit sharing, monopolies and oligopolies, and bankruptcy. Through class discussions, activities, research, and mathematical analysis, students learn to predict and understand human behavior in a variety of real-world contexts in business and economics.

Sample texts: Game Theory and Strategy, Straffin; Thinking Strategically, Dixit.

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Paradoxes and Infinities

New course for 2012!

The second sentence is true. The first sentence is false. Are these sentences true or false? How is it that observing an orange pumpkin is seemingly evidence for the claim that all ravens are black? Students in this course explore conundrums like these as they analyze a range of mathematical and philosophical paradoxes.

Students begin by considering Zeno’s paradoxes of space and time, such as The Racecourse in which Achilles continually travels half of the remaining distance and so seemingly can never reach the finish line. To address this class of paradoxes, students are introduced to the concepts of infinite series and limits. Students also explore paradoxes of set theory, self-reference, and truth, such as Russell’s Paradox, which asks who shaves a barber who shaves all and only those who do not shave themselves. Students analyze the Paradox of the Ravens as they study paradoxes of probability and inductive reasoning. Finally, they examine the concept of infinity and its paradoxes and demonstrate that some infinities are bigger than others.

Through their investigations students acquire skills and concepts that are foundational for higher-level mathematics. Students learn and apply the basics of set theory, logic, and mathematical proof. They leave the course with more nuanced problem-solving skills, an enriched mathematical vocabulary, and an appreciation for and insight into some of the most perplexing questions ever posed.

Sample text: New course.

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Geometry Through Art

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“Geometry is the right foundation of all painting.” In this way, the German artist Albrecht Dürer described a connection between mathematics and art that can be found in every culture. In this introductory geometry course, students learn about geometric figures, properties, and constructions, and use this knowledge to analyze works of art ranging from ancient Greek statues to the modern art of Salvador Dalí. 

Beginning with the foundations of Euclidean geometry, including lines, angles, triangles, and other polygons, students examine tessellations and two-dimensional symmetry. Using what they learn about points, lines, and planes, students investigate the development of perspective in Renaissance art. Next they venture into three dimensions, analyzing the geometry of polyhedra and considering their place in ancient art. Finally, students explore non-Euclidean geometry and its links to twentieth-century art, including the drawings of M. C. Escher.

Through lectures, discussions, hands-on modeling, and small group work, students gain a strong foundation for the further study of geometry, as well as an appreciation of the mathematical aspects of art.

Sample text: Squaring the Circle: Geometry in Art and Architecture, Calter.

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Mathematical Logic

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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.

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Cryptology

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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 the codes and ciphers used to create secret writing. 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.

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Advanced Cryptology

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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 course leaves off, this class delves further into the rich mathematics behind the science of secret keeping.

Students begin with a review of key concepts covered in the first-level course, then progress into the study of more complicated cipher techniques and additional topics from number theory. Students also examine how other areas of mathematics and computer programming are applied to cryptology. For example, they employ computational techniques to learn about statistical attacks on cryptosystems.

In addition, students deepen their knowledge of historical cryptography systems. They develop a more nuanced understanding of the techniques used in breaking the Enigma devices used during World War II and investigate the inner workings of the M-209, a device of the same era that was used primarily by the United States military. Students leave this course with an advanced understanding of the mathematical basis and history of cryptology. 

Sample text: Materials compiled by the instructor.

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Fundamentals of Microeconomics

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Alfred Marshall described economics as “the study of mankind in the ordinary business of life.” How much are you willing to pay for ice cream on a hot summer day? Who is responsible for the cost of pollution? Is there such a thing as a perfectly competitive market? Why has Google been accused of monopolistic practices? Microeconomics provides insights into these questions as it examines how individual buyers and sellers make decisions about allocating limited resources.

In this course, students analyze microeconomic theory and consider it in the context of today’s economic climate. They begin by studying the fundamental concepts of supply and demand curves, price elasticities, market structure, public goods, and externalities. Students build on this foundation to explore topics from the broad range of microeconomics: competition, consumer choice, monopoly, oligopoly, and the role of government in promoting greater efficiency and equity.

By applying mathematical concepts and critical analysis to economic theory, students uncover how economists analyze and predict changes in the behavior of both consumers and producers. Students leave the course with a firm foundation in central topics in microeconomics and a better understanding of practical economic issues that affect us all.

Note: Students who have taken CTY’s Principles of Microeconomics should not enroll in this course.

Sample text: Principles of Microeconomics, Mankiw.

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Macroeconomics and the Global Economy

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What are the key indicators of an economy’s performance? How do governments craft monetary and fiscal policy to promote economic growth? What does it mean for a country to have a trade deficit? Analyzing economies at an aggregate level, macroeconomics—the study of economic 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: Macroeconomics, Krugman and Wells.

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Number Theory

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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 proof attempts.

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: Number Theory: A Lively Introduction with Proofs, Applications, and Stories, Pommersheim, Marks, and Flapan.

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The Mathematics of Competitive Behavior

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Thomas J. Watson, the founder of IBM, once said, “Business is a game—the greatest game in the world if you know how to play it.” In today’s global economy and the international political arena, the study of games and strategy continues to be a vital part of the education of historians, economists, and politicians. In this course, students learn how to use principles of probability, statistics, and combinatorics to make strategic decisions based on another party’s actions and reactions. With these tools, students investigate the applications of game theory, learning not only how different strategies helped to define historical events, but also how they are applied today in the fields of economics and politics.

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Computer Science Course Descriptions and Sample Syllabi

Introduction to Robotics

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In the field of robotics, computer science and engineering come together to create machines that can perform a variety of tasks from manufacturing microchips to exploring Mars.

In this course, students develop familiarity with computer science concepts. For example, they explore topics such as algorithms, sequential control flow, and Boolean operators. Students also survey basic principles of mechanical engineering, such as simple machines and locomotion; and basic principles of electrical engineering, such as circuits and sensor feedback. Using LEGO® robotics equipment, they work together to construct, program, and test their robots in an object-oriented programming environment.

For their culminating project, students design, build, and program robots to complete a complex task. The project demonstrates the basic computer science and engineering principles that underlie everything from the space shuttle to the average home toaster. Students gain a foundation in computer programming and engineering that will become increasingly important in the highly technical twenty-first century.

Sample text: Materials compiled by the instructor.

Lab Budget:
$1200 — $1400 per 3-week session (depending on enrollment)
(This budget is slightly higher than other courses due to material and equipment costs).

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Foundations of Programming

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Students in this course gain insight into the methods of computer programming and explore the algorithmic aspects of computer science. They learn the theoretical constructs common to all high-level programming languages by studying the syntax and basic commands of a particular programming language such as Java, C, C++, or Python*. Building on this knowledge, students move on to study additional concepts of programming, including arrays, testing, and debugging. By solving a variety of challenging problems, students learn to start with a concept and work through the steps of writing a program: defining the problem and its desired solution, outlining an approach, encoding the algorithm, and debugging the code.

Through a combination of individual and group work, students complete supplemental problems, lab exercises, and various programming projects in order to reinforce concepts learned in class. By the end of the course, students can develop more complex programs and are familiar with some of the standards of software development practiced in the professional world. Students leave with an understanding of how to apply the techniques learned to other high-level programming languages.

*Note: The programming language learned may change based on the instructor’s preference.

Sample text: An introductory computer programming text.

Lab Budget:
$780 — $975 per 3-week session (depending on enrollment)

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Fundamentals of Computer Science

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Not a programming course, Fundamentals of Computer Science introduces students to three major areas of the discipline: theory and algorithms, hardware systems, and software systems. In the theoretical component of the course, students learn about algorithms, Boolean algebra, binary mathematics, and the theory of computation. While studying hardware systems, they gain familiarity with the physical components of computers, digital logic, computer architecture, and computer networks. As students investigate software systems, they are introduced to elements of programming languages, compilers, computer graphics, and operating systems.

Sample text: Materials compiled by the instructor.

Lab Budget:
$780 — $975 per 3-week session (depending on enrollment)

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Data Structures and Algorithms

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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.

Sample text: Data Structures and Algorithm Analysis, Weiss.

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