# Department ofMathematics

### In-Depth Course Descriptions:

#### MATH 105 - Mathematics of Personal Finance

This course emphasizes mathematics behind the financial transactions that will likely be important to students and their families, such as percentages and their applications: car loans, home mortgages, savings, and investments such as stocks and bonds. The concept of the time value of money is discussed as it relates to many of these applications. The course is designed to increase awareness in some of the most useful applications of math, and add to enjoyment and self-confidence in math ability.

#### MATH 107 - Mathematical Explorations

This course emphasizes numerical and logical reasoning and the impact that mathematics has on modern society. The topics are chosen by the instructor, and may include voting methods, basic probability, basic statistics, modeling networks, or basic financial mathematics. The course is designed to build enjoyment and self-confidence in math ability, and to sharpen critical thinking.

#### MATH 115 - College Algebra and Trigonometry

This course reviews algebra, emphasizing problem-solving skills including factoring, working with polynomials, using the laws of exponents; and graphing common functions – lines and quadratics. The idea of a function is introduced using polynomial, rational, trigonometric, logarithmic, and exponential functions. We also look at inverse trigonometric functions, identities, graphing and the trigonometry of right triangles. The course is appropriate as preparation for MATH 123, MATH 128, or MATH 140.

#### MATH 123 - Modern Elementary Math I – Number Sense and Algebraic Sense

This course is intended for preservice K-8 teachers. Elementary teachers need a strong conceptual understanding of mathematics in order to be successful at teaching children mathematics. This course gives future elementary teachers an opportunity to explore elementary school mathematics in depth, through hands-on learning while considering children’s strategies. The mathematical focus of this course is on two of the five formal strands of school mathematics: number sense and algebraic sense. The topics studied include:

• Extending and generalizing geometric and number patterns;
• Understanding place value;
• Algorithms for operations on whole numbers, integers and rational numbers;
• Understanding and teaching about fractions, percents, and ratios.
Special emphasis is placed on the process standards of the National Council of Teachers of Mathematics: problem solving, proof and reasoning, communication, connections, and representation. In particular, students learn to explain Why rather than just explaining How, continually explaining their reasoning to the instructor and their peers, thus developing the ability to explain mathematical concepts to their future students.

The prerequisite for this course is successful completion of MATH 115, or alternatively, an appropriate score on the Math Placement Exam.

#### MATH 124 - Modern Elementary Math II – Measurement, Geometric Sense, Statistics and Probability

This course, as a continuation of MATH 123, is intended for preservice K-8 teachers. Its focus is on enhancing the conceptual understanding of the remaining three formal strands of school mathematics: geometry, measurement, and probability and statistics. As in MATH 123, students engage in hands-on learning while considering children’s strategies. Some of the topics that may be discussed in the class are:

• Properties of polyhedra;
• Classification of triangles and quadrilaterals;
• Properties of transformations;
• Discovering area formulas for special quadrilaterals;
• Discovering surface area and volume formulas for special three-dimensional solids;
• Appropriate graphical representations of data;
• Basic probability formulas.

Special emphasis is placed on the process standards of the National Council of Teachers of Mathematics: problem solving, proof and reasoning, communication, connections, and representation. In particular, students learn to explain Why rather than just explaining How, continually explaining their reasoning to the instructor and their peers, thus developing the ability to explain mathematical concepts to their future students.

The prerequisite for this course is successful completion of MATH 123 is the prerequisite for this course.

#### MATH 128 - Linear Models and Calculus, An Introduction

Modeling quantitative relationships is an important skill for working in the business world. Imagine owning a business and wondering how many units you should produce in order to minimize costs, or to maximize profit. When taking out a loan, can you determine the required monthly payment in order to amortize the loan in 15 years? What functions are used to best represent the time it takes for a population to double? Triple? A basic understanding of functions and problem-solving are necessary to answer all of these questions.

Students will learn to solve practical linear programming problems and progress to working with quadratic, exponential and logarithmic functions. Students will be introduced to the time value of money – interest and annuities – a topic that is useful not just to those in the business world, but to anyone who plans to buy a house or a car, or to set up a retirement fund. The course concludes with an introduction to differential calculus, whose most obvious applications are in understanding marginal profit, cost and revenue.

The prerequisite for this course is successful completion of MATH 115, or alternatively, an appropriate score on the Math Placement Exam. Although the course is primarily taken by business majors, the material is applicable to students outside of the School of Business. Note that business students who anticipate emphasizing finance or who are considering a double major in economics should take MATH 151 rather than MATH 128.

#### MATH 140 - Precalculus I

Put simply, functions are machines that take a numerical input and give a numerical output. This idea is commonly used in many applications of mathematics: for example, we recognize that the number of people who will buy a product depends on the price of the product. In this class, we investigate functions, including their algebraic properties and graphs, and explore how to use them to solve problems. We will study trigonometric, exponential, logarithmic, and rational functions, all of which are useful for further investigations in mathematics, especially, in calculus. Additional topics, such as complex numbers, may be covered.

The prerequisite for this course is successful completion of MATH 115, or alternatively, an appropriate score on the Math Placement Exam. This course is the natural precursor to MATH 151.

#### MATH 145 - Statistics for Biologists

This course introduces biology students to the basic ideas of statistics and statistical inference in the context of their applications to biology. This course is intended to be an alternative to Math 151 (Calculus I), and it satisfies the minimum mathematics requirement for the biology major. The course emphasizes intuition and interpretation in context, employing technology to handle the computations, and downplaying the algebraic manipulation of formulae. The principal topics include:

• Methods for describing data numerically and graphically;
• Randomness and experimental design;
• Basic introduction to probability;
• Introduction to the binomial and normal probability distributions;
• Correlation and simple linear regression, including checking regression assumptions;
• Confidence intervals and hypothesis testing (emphasis on types of tests and interpretation of results, rather than on theory behind the test process)

The main focus of the course should be testing and interpreting real data sets from the biological sciences. Students will learn to use some of the basic tools in a statistical software package such as Minitab.

The prerequisite for this course is successful completion of MATH 140, or alternatively, a score on the Math Placement Exam that places the student into MATH 151 or a higher level course.

#### MATH 151 - Calculus I

Slope tells something about the rate of change in a line. This is an extremely useful concept, but has the shortcoming of only being applicable to lines. In Calculus I this problem is overcome by the derivative, essentially a concept of slope that can be applied to functions other than lines. Armed with the derivative, we can answer questions about the rate of change of many functions, allowing us to find maxima or minima of functions, study velocity and acceleration of physical bodies, chemical reactions and population growth. We can graph complex curves and describe the relative efficiency of rival computer algorithms. Indeed, the calculus provides a universal language to precisely describe and compute rates of growth and corresponding changes in amount.

#### MATH 152 - Calculus II

Nearly everyone knows that the area of a circle is πr², and so on. But few think about where these formulas come from. In Calculus II we use the concept of the integral to study the area under curves. This naturally generalizes to the study of volumes of solids in space. But this same concept, combined with the derivative (from Calculus I) can be used in many unexpected and powerful ways. Quantities as diverse as the GNP (gross national product) and total run time of a computer program can be described as an area under a curve on a graph. Calculus II provides tools to compute these quantities and relate them to the functions that describe their rates of change.

It is possible for infinitely many numbers to sum to a finite value. For example, it can be shown that 1+½+¼+...=2. The integral and derivative are used as tools to help us understand such infinite series. In turn, these series help us to understand several functions better. For example,  and can be written as infinitely long polynomials and can be approximated reasonably well by, say, polynomials of degree four or five.

#### MATH 203 - History of Mathematics

Often, when we learn mathematics, we learn it without the story of who developed it, and when and why.  In the History of Mathematics, we look at the stories behind the mathematics.

These stories take us to many places on the earth and through a long period of time.  We begin about 4000 years ago with the ancient civilizations of Egypt and Mesopotamia, where there was already a good deal of mathematics known, particularly algebra and the art of computation.  We also explore the early mathematical discoveries of China and India.  Next we go to the amazing flowering of mathematics that occurred in ancient Greece: geometry, astronomy, trigonometry and much more.

We see some more development of algebra with the Arabic mathematicians of Medieval times; in fact our word, algebra, is from the Arabic.  We next move to Europe to see algebra in Italy, analytic geometry in France, logarithms in Scotland and the beginnings of calculus almost everywhere.  We follow the development of calculus and see how it changed from around 1600 to around 1800.  We then look at the surprising story of non-Euclidean geometry in the 1800's.  We can only survey more recent discoveries briefly because they are more difficult and there are so many of them.

We study the biographies of a number of mathematicians, and look at the special problems encountered by women mathematicians.  Many of the students in this class intend to become mathematics teachers so we examine the histories of specific areas of mathematics taught in the schools, such as number systems, algebra, geometry and trigonometry.

#### MATH/STAT 242 - Introduction to Mathematical Statistics

As the title suggests, we will apply mathematical techniques to develop some of the fundamental ideas of statistics. So just what is statistics? Statistics is the art of extracting patterns from data. This might consist of summarizing complicated data, whether numerically, graphically or by constructing a simple mathematical model that connects pieces of data to one another. Whereas mathematics uses a language of certainty, theorems and proofs, statistics has developed precisely to deal with uncertainty, estimates, bounds and probabilities.

In this course we will examine answers to several important questions in statistics. How do you describe a data set so as to capture its ‘center' and its variation? This will lead to topics such as the mean and the variance of a sample. What is probability and how do we model it mathematically? This will lead to the classical distributions: binomial, Poisson, exponential and normal. How do you decide whether your preconceptions about a large population are in agreement with the data obtained from a sample? This will lead us to confidence intervals and hypothesis testing.

Throughout the course, we will see that statistics is much more than just the application of mathematical techniques. We will see that, before we can apply the mathematics, we must have good data and reasonable models. After we have done our mathematical analysis, we must still decide whether we have enough certainty to make conclusions. In short, we will be the lawyers, judge and jury in the court of data analysis.

We will apply the techniques of algebra and calculus to investigate probability, to develop models and to explore their properties and understand why some estimation techniques have better properties than others do. We will apply Minitab statistical software to real world data sets and to simulated data sets.

Successful completion of MATH 151 is a prerequisite for this course. This course is cross-listed under both mathematics and statistics. Students can take this course for the mathematics major and minor, the statistics minor and the actuarial science minor.

#### MATH 245 - Discrete Structures

The possession of logical reasoning skills is essential for anyone interested in computer science.  In this class, students enhance these skills by studying a variety of mathematical topics related to the study of computer science, which may include propositional logic, set theory, relations, functions, combinatorics, graph theory, and applications of these topics.   Students also learn proof-techniques such as induction (a "domino" technique that allows one to prove that a statement relating to a variable n is true for all positive integers n) and proof by contradiction (in which one proves a desired result by showing that if it isn't true, nonsensical things happen), thereby increasing their mathematical maturity and their ability to make reasoned arguments, prerequisites for programming. Topics vary from term to term, and may depend on student interest.  Here is a sample of things students may learn in this class:

(1)  The logical difference between the statements, Not all people have red hair and All people do not have red hair;

(2)  How to show that 1+2+3+...+n= n(n+1)/2, for any positive integer n;

(3)  How to show that the set of integers and the set of rational numbers have the same "size", but the set of real numbers is "bigger";

(4)  How to compute the probability of getting a royal flush in poker.

The course is intended primarily for computer science majors and math majors.

Note: though this class has Math 152 as a prerequisite, to ensure the mathematical preparedness of its students, its material is not directly related to that learned in the calculus sequence.

#### MATH 253 - Calculus III

Most things are related to more than just one factor. For example, your minimum monthly credit card payment depends on the total you owe and your interest rate. The amount you actually pay depends on the minimum payment due and the amount you have available to pay. The growth rate of a deer population depends on the size of the population, its age distribution, the food supply and predation. The pressure exerted by gas in a cylinder depends on the amount of gas, its temperature and the volume of the cylinder.

Other functions may only depend on one variable, but give an output that is more than just one number. For example, a person traveling around the world has, at any given time, a latitude and a longitude (and perhaps an altitude too if s/he is in an airplane). Thus position can be considered a function of time but it cannot be represented by a single value; it must be given as a doublet (or triplet) of numbers representing latitude and longitude (and altitude). Such a doublet or triplet can be represented as a vector.

Calculus III extends the ideas of Calculus I and II by considering derivatives and integrals of functions with more than one variable, or of vector-valued functions. Along the way, other possible coordinate systems (such as polar coordinates) are discussed.

#### MATH 317 - Introduction to Proof in Mathematics

In mathematics we accept a statement as true only if we have a proof that it is true. Since the method of proof is so basic to mathematics, anyone who seriously wants to learn mathematics beyond a fairly elementary level must be able to understand proofs and be reasonably proficient at constructing them. The purpose of this class is to teach you how to understand proofs and to develop your skills at constructing proofs. Skill at proving develops over a long period of time; this class is only a beginning. The best way to learn to do proofs is to do them, so you will be given plenty of opportunity to practice proving things.

We will begin with an introduction to logic. Logic is a tool that we will use to analyze proofs to see if they are correct and to help us to construct proofs. We will practice writing proofs in a number of areas of mathematics: set theory, including infinite sets, inequalities and functions. We will study the whole numbers using mathematical induction. In addition to the usual lecture format, a good deal of class time will be spent with students presenting their proofs to the class or constructing proofs together.

This class is good preparation for Linear Algebra (MATH 331), Abstract Algebra (MATH 433), Analysis (MATH 455) and other advanced mathematics classes.

#### MATH 321 - Geometry

The geometry most of us learned in high school is based on Euclid's famous 5 Postulates and works well for describing things in or on a flat surface. However, the surface of our world is not flat and any pilot or ship's navigator must understand the rules of spherical geometry.

The discovery of two-dimensional non-Euclidean geometries early in the nineteenth century by Gauss, Bolyai and Lobachevski allowed us to ask for the first time, "Could the geometry of the three-dimensional universe in which we live also be non-Euclidean?" The work of Riemann and, later, Minkowski provide a geometric structure for Einstein's theory of relativity and modern theories of cosmology where the ultimate collapse or expansion of the universe is related to the curvature of space itself.

The discovery of two-dimensional non-Euclidean geometries also initiated a momentous shift in our view of the entire mathematical enterprise. The question of axiomatic foundations raised by the non-Euclidean geometries now pervades all branches of the subject and forms the acid test of mathematical validity.

This class examines the foundations of geometry that lead to Euclid's geometry in the plane and to other possible geometries, most notably spherical and hyperbolic, and concentrates on exploring the rules of geometric logic that are universal.

#### MATH 331 - Linear Algebra

Why algebra? Algebra was invented because of the limitations of our geometric intuition. In applications ranging from business to engineering to the social sciences, it is often useful to work with data that naturally correspond to points in the plane, or in three-dimensional space, or even in fifty-dimensional space. Certainly we could draw pictures or build models to avoid algebra for points in the plane or in three-dimensional space, but what pictures or models could help us to "see" in fifty dimensions? This obstacle motivates the development of vectors and the development of algebraic rules and techniques for manipulating them. In this course we pursue two intimately related subjects: matrix theory and linear algebra.

Matrix theory is concerned with vectors and matrices. Vectors are the n-dimensional generalizations of the ordered pairs representing points in the plane. We will investigate how our geometric concepts naturally imbed in algebraic concepts. We will learn how the geometry of lines and planes, lengths and angles is replaced by systems of equations and operations on vectors. Further, we will see how systems of equations can be analyzed in terms of the properties of a single algebraic object: the matrix.

Linear algebra is the study of sets of vectors and how operations on individual vectors can be applied to entire sets. Linear algebra is the abstraction of the fundamental properties displayed by vectors and matrices. This abstraction allows us to use the knowledge and skills developed working with vectors and matrices to answer questions about the behavior of wave functions in Fourier analysis or about the nature of solutions to important families of differential equations.

This course is very different from calculus. In calculus there are relatively fewer theoretical ideas, and most of the course is devoted to applying those ideas and the associated techniques to specific computations. In MATH 331 students learn a large variety of new ideas and, while calculations are important, they are primarily tools for understanding the examples that motivate the theory. Consequently much of the work in this course is focused on explaining why certain relationships between ideas are true or why certain sets have specified properties rather than on simply producing a slope or an integral or a number. Calculus is a prerequisite for this course primarily because students rarely have adequate facility with mathematical thinking, working with equations, working extensively with symbols, thinking about exceptions or using technical language-prior to completing the calculus sequence.

#### MATH/STAT 342 - Probability and Statistical Theory

This is a continuation of MATH/STAT 242 (Introduction to Mathematical Statistics, previously MATH/STAT 341). In this class, students will expand their basic knowledge from MATH/STAT 242 into broader and deeper probability and statistics theory. For instance, students will learn about conditional distributions of multiple random variables, limiting distributions, moment generating functions and higher moments than mean and variance. Students will learn more methods for testing statistical hypotheses, such as the two-sample T test, the F-test and non-parametric methods. There will also be an introduction to analysis of variance (ANOVA).

To insure that students learn more than just theoretical ideas a term project applying class knowledge to solving real world problems is usually assigned. Minitab will be used for the data analysis.

Students are required to complete MATH/STAT 242 prior to enrolling in this class. MATH/STAT 342 is cross listed under mathematics and statistics. Students can take it for the mathematics major (or minor), the statistics minor and the actuarial science minor.

#### MATH/STAT 348 - Applied Regression Analysis and ANOVA

Regression analysis of data is a powerful statistical tool that is widely used in biology, psychology, management, engineering, medical research, government and many other fields. It provides a technique for building a reasonable mathematical model that relates the mean value of a response (e.g., profit) to various independent variables or predictors (e.g., advertising budgets, size of inventory, etc.).

Any prediction or estimation based on a random sample of data will contain a certain unknown error. In this course, students will learn various methods to build a best regression model for a given set of data under certain constraints so that the error is minimized.

When the relation between the dependent and independent variables is linear, we call it linear regression. Students will also learn about nonlinear regression, where there can be a nonlinear relationship (such as quadratic or exponential). Real world problem solving skills are emphasized. Minitab is used extensively for the data exploration and data analysis. A term project (with open topics) is normally assigned for students to explore knowledge beyond the classroom.

Students are required to complete MATH/STAT 341 prior to enrolling in this class. MATH/STAT 348 is cross-listed under mathematics and statistics. Students can take it for the mathematics major (or minor), the statistics minor and the actuarial science minor.

#### MATH 351 - Differential Equations

Differential equations are a powerful tool in constructing mathematical models for the physical world. Their use in industry and engineering is so widespread and they perform so well that they are among the most successful of modeling tools.

For example, a cup of hot coffee is initially at and is left in a room with an ambient temperature of . Suppose that initially it is cooling at a rate of  per minute. Then the model for the cup's temperature is . This is an example of a differential equation. We are interested in predicting the temperature, T, of the coffee at any time t. We can also ask, "How long does it take the coffee to cool to a temperature of, say,

#### MATH 356 - Numerical Analysis

When one pushes the square root button on a calculator to compute the square root of 2, one should ask, "How does the calculator do it?" Numerical analysis deals with implementing numerical methods to answer questions like this one.

While numerical methods have always been useful, since the invention of computers, the role of numerical methods in scientific research has become essential. No modern applied mathematician, physical scientist or engineer can be properly trained without some understanding of numerical methods. There is more involved here than just knowing how to use the methods. One needs to know how to analyze their accuracy and efficiency. Numerical analysis is a broad and challenging mathematical activity, whose central theme is the effective constructability of various kinds of approximations.

#### MATH 411 - The Mathematics of Risk

In this course, which is central to the financial mathematics major, we examine how the formulae that populate finance books are developed. We investigate the relationship between income and expense streams through time, and the present value of an investment with those cash flows. We will investigate what calculus can tell us about the sensitivity of that valuation to changes in market interest rates, and how securities can be designed to make the securities insensitive to small changes. We will develop the basic theory of geometric brownian motion for the pricing of securities such as stocks, and we will develop the binomial tree model for pricing derivative securities such as call options. We will learn about Lagrange multipliers, and use them to understand Markowitz optimal portfolio theory. While it would be very helpful to have a basic understanding of stocks and bonds before starting this course; it is much more important to have a solid command of the big ideas of calculus - rates of change, accumulation of changes, optimization, partial derivatives - and to have a solid command of basic probability and expected values.

#### MATH 433 - Abstract Algebra

If you can tell time, you already know some abstract algebra: you just don't know you know it! Suppose you have lunch every day at 1:00pm. Then you'll have lunch at 1:00pm today and at 1:00pm tomorrow. We just called both of those times ‘1:00pm', but they're not really the same moment in time, since they're occurring on different days! It turns out they both can be thought of as representatives of a coset of  in ; this coset, in turn, is an element of the factor group .

Huh, you ask? What's a coset? What's ? What's ? What's a factor group?! Take this class and find out! Abstract algebra is the study of algebraic structures such as groups, rings and fields. (You don't know what these objects are yet, but if you take this class you will!) You encounter such objects everywhere in math:  the coordinate plane is an example of a group; the set of all  matrices over the real numbers is an example of a ring; the set of all real numbers is a field. By studying these structures abstractly, we can give one proof for many results that hold for wildly different objects, instead of proving each result for each object separately.

Abstract algebra is a beautiful and powerful area of mathematics and it is an essential part of any mathematics curriculum. It has applications in many sciences, from physics to chemistry, in addition to having extremely important uses in areas such as cryptography.

While the concepts in this class require minimal prerequisite knowledge of topics such as calculus, this class is heavily proof-based and requires a large amount of mathematical maturity. The ability to write grammatically and make logical arguments is extremely important, while the ability to differentiate will be of little, if any, use. Conceptual understanding, not a calculator, is at the heart of this course!

#### MATH/EDUC 446 - Mathematics in Secondary Education

This course has been designed for prospective teachers of middle school and high school mathematics and reflects the recommendations of the National Council of Teachers of Mathematics (NCTM). The following excerpt is from the NCTM Principles and Standards book:

"The Teaching Principle"

Effective mathematics teaching requires understanding what students know and need to learn and then challenging and supporting them to learn it well.

Teachers need to know and use ‘mathematics for teaching' that combines mathematical knowledge and pedagogical knowledge. They must be information providers, planners, consultants and explorers of uncharted mathematical territory. They must adjust their practices and extend their knowledge to reflect changing curricula and technologies and to incorporate new knowledge about how students learn mathematics. They also must be able to describe and explain why they are aiming for particular goals."

The course takes the art of teaching through a series of motivational ideas suitable for many grade levels and abilities and includes a discussion of activities, materials and manipulatives suitable for classroom use. Problem solving and heuristics is a major theme in the course. Other topics covered include cooperative learning, questioning techniques, technology, lesson planning, homework options, mini-discovery lessons and technology lessons.

#### MATH 455 - Mathematical Analysis

Why does calculus work?  In this course we examine the foundations of calculus.  What properties of the real numbers distinguish them from the rational numbers?  What role do these differences play in the development of such fundamental concepts as limits and convergence?  What does continuity really mean, and why do we need it?  Along the way, we will study sequences, series and limits, first of numbers, and then of functions.  One consequence of our study will be a better appreciation of the central role of power series in many of the results of calculus.

This course is strongly recommended for anyone considering a graduate degree in pure or applied mathematics, statistics, theoretical physics or operations research.  Surprisingly, a deep understanding of the theoretical underpinnings of calculus is necessary to make progress in such applied areas as optimization, numerical analysis, financial modeling, probability and differential equations.

This course is almost entirely focused on formal definitions and rigorous proofs.  Students are encouraged to have as much exposure to proofs as possible prior to enrolling in this course.

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