# Mathematics

Mathematics is perhaps the most precise of human languages.

With it, scientists and engineers have been able to describe and understand complex physical and social phenomena. As a language, together with its axiomatic underpinnings, it is a much-explored structure in itself. These two views of Mathematics, its application to things external and to things internal, represent a distinction between Applied and Pure Mathematics.

Students who desire to major or minor in mathematics do so for a number of reasons. Some of these include

- those who wish specifically to become teachers of mathematics in high school;
- those whose interests lie primarily in the study of mathematics as a science, the purpose of such students being usually to continue their studies at the graduate level;
- those whose interests lie in the field of engineering and/or physics, with emphasis on applied mathematics;
- those whose interests lie in the fields of biology, chemistry, or economics;
- those who wish to study mathematical statistics;
- those whose interests lie primarily in computers and computational mathematics.

## Prerequisites to the Major

Students desiring to major in mathematics are required to have successfully completed:

Code | Title | Credit Hours |
---|---|---|

MATH 115 | Calculus I | 4 |

MATH 116 | Calculus II | 4 |

MATH 215 | Calculus III | 4 |

MATH 216 | Intro to Diff Equations | 3-4 |

or MATH 228 | Diff Eqns with Linear Algebra | |

MATH 227 | Introduction to Linear Algebra | 3 |

Select one of the folllowing: | 3-4 | |

Computing Environ for Math | ||

Computer Science I | ||

Total Credit Hours | 21-23 |

## Dearborn Discovery Core Requirement

The minimum passing grade for a Dearborn Discovery Core (DDC) course is 2.0. The minimum GPA for the program is 2.0. In addition, the DDC permits any approved course to satisfy up to three credit hours within three different categories. Please see the General Education Program: The Dearborn Discovery Core section for additional information.

### Foundational Studies

Written and Oral Communication (GEWO) – 6 Credits

Upper Level Writing Intensive (GEWI) – 3 Credits

Quantitative Thinking and Problem Solving (GEQT) – 3 Credits

Critical and Creative Thinking (GECC) – 3 Credits

### Areas of Inquiry

Natural Science (GENS) – 7 Credits

- Lecture/Lab Science Course
- Additional Science Course

Social and Behavioral Analysis (GESB) – 9 Credits

Humanities and the Arts (GEHA) – 6 Credits

Intersections (GEIN) – 6 Credits

### Capstone

## Foreign Language Requirement

Complete a two-semester beginning language sequence.

Ancient Greek I and II | MCL 105 and MCL 106 |

Arabic I and II | ARBC 101 and ARBC 102 |

Armenian I and II | MCL 111 and MCL 112 |

French I and II | FREN 101 and FREN 102 |

German I and II | GER 101 and GER 102 |

Latin I and II | LAT 101 and LAT 102 |

Spanish I and II | SPAN 101 and SPAN 102 |

## Major Requirements

A total of at least 33 credit hours of coursework must be elected in mathematics (MATH) and cognate areas at the upper level (300-400 level courses). Students are required to elect 27 hours of coursework in mathematics (MATH) including:

Code | Title | Credit Hours |
---|---|---|

Required Courses | ||

MATH 300 | Math Lang Proof & Struct | 3 |

Courses chosen in accordance with one of the following options: Algebra Option or Analysis Option: | 12 | |

Algebra Option: | ||

First Course in Modern Algebra | ||

Linear Algebra | ||

Advanced Calculus I | ||

Select at least one of the following: | ||

Advanced Calculus II | ||

Introduction to Topology | ||

Func of a Complex Var with App | ||

Analysis Option: | ||

First Course in Modern Algebra | ||

Advanced Calculus I | ||

Advanced Calculus II | ||

or MATH 492 | Introduction to Topology | |

Select at least one of the following: | ||

Survey of Geometry | ||

Elementary Number Theory | ||

Linear Algebra | ||

Func of a Complex Var with App | ||

Applied Courses | ||

Select at least two applied mathematics courses (CADM): | 6 | |

Applied Combinatorics | ||

Probability | ||

Computing with Mathematica | ||

Dynamical Systems | ||

Stochastic Processes | ||

Mathematical Statistics | ||

Fourier and Boundary | ||

Func of a Complex Var with App | ||

Introduction to Wavelets | ||

Mathematical Modeling | ||

Intro to Numerical Analysis | ||

Matrix Computation | ||

Linear Algebra w/Applications (Program Adviser approval required) | ||

Math Electives | ||

Any two other MATH courses numbered 300 through 499 approved for Mathematics majors. | 6 | |

Cognates | ||

Select 6 credits upper level (300/400 and 3000/4000) from the following: | 6 | |

CCM | ||

Economic Statistics | ||

Introduction to Econometrics | ||

IMSE (except IMSE 334) | ||

ME | ||

Symbolic Logic | ||

Philosophy of Science | ||

PHYS | ||

Total Credit Hours | 33 |

^{1} | Courses joined with |

**Notes:**

- Students who wish to use graduate-level courses, numbered 500 or higher, as part of the 27 credit hours of upper-level MATH coursework required for the major, must submit a Petition to obtain the approval of the faculty Program Advisor in Mathematics.

- Students seeking secondary teacher certification must take MATH 331, MATH 486, EDD 450 and EDD 451. Also, MATH 395 and a course in statistics (STAT) are recommended for such students.
None of the following MATH courses may be used to fulfill any requirements of either a Mathematics major or a Mathematics minor:
Course List Code Title Credit Hours MATH 385 Math for Elemen Teachers I 3 MATH 386 Math for Elem Teachers II 3 MATH 387 Math for Elem Teachers III 3 MATH 442 Geometry for Teachers 3 MATH 443 Algebra for Teachers 3 MATH 444 Data Anlsys,Prob&Stat forTchrs 3 MATH 445 Number & Prop'l Rsng for Tchrs 3 MATH 446 Discrete Math/Modeling for Tch 3 MATH 447 Micro in Math for Teachers 2 MATH 449 Concepts of Calc for Teachers 3 MATH 486 Sec School Math for Teachers 3 - Applied Statistics courses (STAT) cannot be used to fulfill the Math major or minor/concentration requirements.
- At least 12 of the 27 upper level credit hours in mathematics (MATH) must be elected at UM-Dearborn in order to graduate.

- In order to enroll in a mathematics class, a student must have earned a grade of at least C- in all prerequisite mathematics courses; a grade below C- signals that the student should
*immediately repeat*the class in order to build a stronger foundation for subsequent study. The same principle applies when a mathematics course is a prerequisite for courses of other disciplines.

## Credit by Examination

The department grants credit for Calculus I to those students who have received a score of three, four, or five on the AB Exam or a score of three on the BC Exam of the Advanced Placement Program Tests of the College Entrance Examination Board. Credit is granted for both Calculus I and Calculus II to those students who have received a score of four or five on the BC Exam of the Advanced Placement Program Tests. In each case, the student is then eligible to elect the next calculus course in the calculus sequence.

## Minor or LIBS Concentration

A minor or concentration consists of 12 credit hours in mathematics (MATH) courses approved for upper-level credit in the mathematics major .

**MATH 080 Introductory Algebra 3 Credit Hours**

The Developmental Mathematics sequence (MATH 080, MATH 090) is offered as a service to students who need extra preparation in mathematics. MATH 080 is for students who are likely to need two semesters of additional preparation in mathematical computation and symbol manipulation, communication, and conceptual understanding. Topics in the two-course sequence include: arithmetic readiness, real numbers and expressions, linear equations and inequalities, lines and functions, systems of linear equations, rational expressions and equations, radicals and complex numbers, quadratic equations and functions, function operations and inverses. Students are required to have Internet-ready devices available for each class meeting. Skill development takes place online and outside scheduled class meetings. The course is graded on an A, B, C, NC (not completed) basis. This course is offered for additive credit.

**Prerequisite(s): ** MPLS with a score of 080

**MATH 090 Intermediate Algebra 3 or 6 Credit Hours**

The Developmental Mathematics sequence (MATH 080, MATH 090) is offered as a service to students who need extra preparation in mathematics. MATH 090 is for students who (1) have successfully completed MATH 080 or (2) are likely to require only one semester of additional preparation in mathematical computation and symbol manipulation, communication, and conceptual understanding. Topics in the two-course sequence include: arithmetic readiness, real numbers and expressions, linear equations and inequalities, lines and functions, systems of linear equations, rational expressions and equations, radicals and complex numbers, quadratic equations and functions, function operations and inverses. Students are required to have Internet-ready devices available for each class meeting. Skill development takes place online and outside scheduled class meetings. The course is graded on an A, B, C, NC (not completed) basis. This course is offered for additive credit.

**Prerequisite(s): ** MATH 080 or MPLS with a score of 090

**MATH 104 College Algebra 4 Credit Hours**

Primary purpose of this course is to prepare students for sucess in MATH 113. Topics include equations and inequalities, linear, quadratic, polynomial, rational, logarithmic and exponential functions along wiht their graphs; application of these functions, systems of linear inequalities. This course does not cover trigonometric functions and cannot be used as a prerquisite for MATH 115. Students electing this course should have at least taken two years of High School Algebra and one year of HIgh School Geometry or MATH 090. Students cannot receive credit for both MATH 104 and MATH 105. (F, W, S)

**Prerequisite(s): ** MATH 090 or MPLS with a score of 105

**MATH 105 Pre-Calculus 4 Credit Hours**

Primary purpose of this course is to prepare students for success in Calculus. Topics include equations and inequalities; linear, quadratic, polynomial, rational, logarithmic, exponential and trigonometric functions along with their graphs; application of these functions. Students electing this course should have taken at least two years of High School Algebra and one year of High School Geometry or MATH 090. Students cannot receive credit for both MATH 104 and MATH 105. (F.W.S)

**Prerequisite(s): ** MATH 090 or MPLS with a score of 105

**MATH 113 Calc I for Biology & Life Sci 4 Credit Hours**

This course develops basic concepts of Calculus from the perspectives of Biology and Life Sciences. Topics include differential and integral calculus of algebraic/logarithmic/exponential functions of one variable, limits, continuity, differentiation, integration, graphing, optimization, related rates and area. Applications include modeling biological problems of medicine, genetics, Biomechanics, ecology, population growth and decay. (This course does not fulfill the calculus requirements for concentration in chemistry, physics, biochemistry, engineering, or mathematics) Student cannot receive credit for both Math 113 and Math 115.

**Prerequisite(s): ** MATH 105 or MPLS with a score of 115 or MATH 104

**MATH 114 Calc II for Biology & Life Sci 4 Credit Hours**

The topics of this course include advanced methods of integration (integration by parts, partial fraction), modeling with differential equations, some elementary differential equations, matrix algebra, systems of linear equations using matrix method, introduction to probability, conditional probability, discrete and continuous random variables (exponential and normal random variables). Problems in biology, medicine and physiology are used to illustrate how computation and mathematics can improve and enhance the understanding of these problems. Students cannot receive credit for both Math 114 and Math 116.

**Prerequisite(s): ** MATH 113 or MATH 115 or MPLS with a score of 116

**MATH 115 Calculus I 4 Credit Hours**

Functions and their graphs; limits and continuity of functions, differentiation, algebraic and trigonometric functions, applications of derivatives, definite and indefinite integrals, and applications of definite integral. This course includes computer labs. Students cannot receive credit for both MATH 113 and MATH 115. (F,W,S).

**Prerequisite(s): ** MATH 105 or MPLS with a score of 115

**MATH 116 Calculus II 4 Credit Hours**

Transcendental functions, techniques of integration, improper integral, infinite sequences and series, Taylor's theorem, topics in analytic geometry, polar coordinates, and parametric equations. This course includes computer labs. Students cannot receive credit for both MATH 114 and MATH 116. (F,W,S).

**Prerequisite(s): ** MATH 115 or MPLS with a score of 116

**MATH 131 Conceptual Mathematics 4 Credit Hours**

The purpose of Math 131 is to develop an awareness of the use of mathematics in the world around us. Students are encouraged to understand organizational tools of mathematics, including set theory and the use of deductive logic. Areas of application may include: consumer Mathematics, Probability, Statistics, social decision making, apportionment, graph theory, and mathematical modeling. Students intending to elect this course should have taken the equivalent of one year of high school algebra and one year of high school geometry. This course is not open to mathematics concentrators. (F,W,S).

**MATH 205 Calc III for Engin Students 3 Credit Hours**

Vectors in the plane and space, topics from multivariable calculus including partial differentiation and multiple integration, with an emphasis on applications, and line integrals and Green's theorem. This course includes computer labs. Students cannot receive credit for both MATH 205 and MATH 215. (F,W,S).

**Prerequisite(s): ** MATH 116 or MPLS with a score of 215

**MATH 215 Calculus III 4 Credit Hours**

Vectors in the plane and space, vector-valued functions and curves, functions of several variables including limits, continuity, partial differentiation and the chain rule, multiple integrals and coordinate transformations, integration in vector fields, and Green's and Stokes' theorems. This course includes computer labs. Students cannot receive credit for both MATH 205 and MATH 215. (F,W).

**Prerequisite(s): ** MATH 116 or MPLS with a score of 215

**MATH 216 Intro to Diff Equations 3 Credit Hours**

Solutions and applications of differential equations of the first and second order, linear equations with constant coefficients, solutions by means of power series, Laplace transforms, and numerical methods for solution of differential equations. (F,W,S).

**Prerequisite(s): ** MATH 116

**MATH 217 Intro to Matrix Algebra 2 Credit Hours**

Systems of equations, matrices, determinants, the n-dimensional real vector spaces, orthonormal basis, linear transformations, and eigenvalues and eigenvectors. Students cannot receive credit for both MATH 217 and MATH 227. (F,W,S).

**Prerequisite(s): ** MATH 114 or MATH 116 or MPLS with a score of 215

**MATH 227 Introduction to Linear Algebra 3 Credit Hours**

An introduction to the theory and methods of linear algebra with matrices. Topics include: systems of linear equations, algebra of matrices, matrix factorizations, vector spaces, linear transformations, eigenvalues and eigenvectors, science and engineering applications, and computational methods. Students cannot receive credit for both MATH 227 and MATH 217. (F,W,S).

**Prerequisite(s): ** MATH 116 or MPLS with a score of 215

**MATH 228 Diff Eqns with Linear Algebra 4 Credit Hours**

Full Title: Differential Equations with Linear Algebra This course provides an introduction to ordinary differential equations. Emphasis is placed on the development of abstract concepts and applications for first-order and linear higher-order differential equations, systems of differential equations, introductory numerical methods, matrix algebra, and Laplace transform techniques. Students cannot receive credits for both MATH 228 and MATH 216 and MATH 217 (F,W,S).

**MATH 276 Discrete Math Meth Comptr Engr 4 Credit Hours**

An introduction to fundamental concepts of discrete mathematics for computer engineering. Topics will be chosen from: set theory, partially ordered sets, lattices, Boolean algebra, semi-groups, rings, graphical representation of algebraic systems, graphs and directed graphs. Applications in various areas of computer engineering will be discussed. (F,W,S).

**Prerequisite(s): ** MATH 116 or MPLS with a score of 215

**MATH 297 The Nature of Mathematics 3 Credit Hours**

Mathematics will be presented in a way so that Honors Program students (including nonscience majors) can learn what makes mathematics a fascinating field of study rather than a collection of dry formulas. A few "Great Theorems" will be studied in their historical context, inter-connections between mathematics and science will be studied, and some famous personalities will be presented. Open only to students in the CASL Honors Program.

**MATH 300 Math Lang Proof & Struct 3 Credit Hours**

A required course for students completing a Mathematics concentration, this course is also a prerequisite for many upper-level Mathematics courses. The course focuses on developing the following: an understanding of, and facility with, the logic and syntax of mathematical statements; and ability to recognize and propose appropriate strategies and outlines for proving given statements; facility in writing mathematical proofs; a knowledge base/toolbox of foundational material including basic concepts and terminology related to naive set theory.

**MATH 315 Applied Combinatorics 3 Credit Hours**

An introduction to methods and applications of enumerative and configural combinatorics. Students study several elegant and useful techniques for counting and/or generating the elements in large and unwieldy finite sets. Students will also study topics in graph theory that are applicable to real world problems. Topics include basic counting principles, the principle of inclusion-exclusion, generating functions and recurrence relations. Topics from graph theory include graph models, paths, circuits, cycles, connectedness; additional topics include the theory and applications of planarity, coloring, directed graphs, and networks and network flows.

**Prerequisite(s): ** (MATH 200 and (MATH 227 or MATH 300) or MATH 217)

**MATH 325 Probability 3 Credit Hours**

Brief overview of summary and display of data, probability concepts, discrete and continuous random variables and associated probability models, expectation, independent random variables, probability generating functions and moment generating functions, sampling distributions, the central limit theorem, the t-distribution, properties of estimators, and interval estimation. Previously taught as Mathematical Statistics I. (F).

**MATH 331 Survey of Geometry 3 Credit Hours**

A development of Euclidean geometry as a formal axiom system and an introduction to non-Euclidean geometries and to Transformational Geometry. Geometric models and the history of geometry are stressed. Development of students' geometric intuition as well as their ability to work in a formal axiom system are emphasized. (F).

**MATH 372 Computing with Mathematica 3 Credit Hours**

The course explores a variety of topics from different areas of undergraduate mathematics including calculus, matrix algebra, number theory, geometry, and discrete mathematics. Students learn to design customized Mathematica functions to solve specific problems in these areas using the symbolic, computational, graphics and programming tools provided within Mathematica. (AY,W).

**MATH 385 Math for Elemen Teachers I 3 Credit Hours**

The purpose of this course and the Math 386 and Math 387 courses is to provide future teachers with foundational knowledge of mathematics they will teach. An inquiry approach is emphasized involving problem solving, problem posing, pattern seeking, reasoning, justification, representations, and communications. Topics in Math 385 include numeration, meaning of operations, the reasoning behind procedures, and the rational number system, including fractions and decimals. (F,W)

**Restriction(s): **

Can enroll if College is Education, Health, and Human Services

**MATH 386 Math for Elem Teachers II 3 Credit Hours**

The purpose of this course and the Math 385 and Math 387 courses is to provide future teachers with foundational knowledge of mathematics they will teach. An inquiry approach is emphasized involving problem solving, problem posing, pattern seeking, reasoning, justification, representations, and communications. Topics in Math 386 include number theory, proportional reasoning, the geometry of two-dimensional shape and measurement, integers, and the real number system. (F,W)

**Prerequisite(s): ** MATH 385

**Restriction(s): **

Can enroll if College is Education, Health, and Human Services

**MATH 387 Math for Elem Teachers III 3 Credit Hours**

The purpose of this course and the Math 385 and Math 386 courses is to provide future teachers with foundational knowledge of mathematics they will teach. An inquiry approach is emphasized involving problem solving, problem posing, pattern seeking, reasoning, justification, representations, and communications. Topics in Math 387 include data analysis; probability; the geometry of three-dimensions including shape, spatial visualization, and measurement; geometric concepts of similarity and congruence; coordinate geometry; and transformational geometry. Algebraic reasoning is integrated throughout. (F,W)

**Prerequisite(s): ** MATH 386

**Restriction(s): **

Can enroll if College is Education, Health, and Human Services

**MATH 390 Topics in Mathematics 1 to 3 Credit Hours**

A course designed to offer selected topics in different areas of mathematics. The specific topic or topics will be announced together with the prerequisites each term. Course may be repeated for credit when specific topics differ.

**MATH 390A Topics in Mathematics 3 Credit Hours**

TOPIC TITLE: Mathematics for Middle School and High School Teachers. Students involved in this topics course will be engaged in activities which deepen their understanding of middle school and high school mathematics and the teaching of these topics to students from underrepresented groups in urban schools. These activities will include presentations by outside speakers, reading of professional journals, student and teacher presentations, and weekly field trips to a high school in Detroit or a middle school in Dearborn. The course is intended for students involved in an Eisenhower grant which focuses on the recruitment of teachers of mathematics in urban schools.

**Restriction(s): **

Can enroll if Class is Junior or Senior

**MATH 390B Topics in Mathematics 1 to 3 Credit Hours**

TOPIC TITLE: Introduction to Wavelets This course will introduce the students to the theory and the applications of wavelet. Topics will include discrete Fourier analysis, the Fast Fourier Transform, linear transformations, orthogonal decomposition, discrete wavelets analysis, the filter bank, Haar Wavelet family, and applications.

**MATH 390E Topics in Mathematics 3 Credit Hours**

TOPIC TITLE: Preparation for Industrial Careers PIC Math prepares mathematical science students for industrial careers by engaging them in research problems that come directly from industry. A strong component of PIC Math involves students working as a group on a semester-long undergraduate research problem from business, industry, or government. Undergraduate research is a high impact teaching and learning practice and has been shown to improve students abilities in Problem solving. Critical thinking, Independent thinking, and Communicating.

**Prerequisite(s): ** MATH 200 or MATH 205 or MATH 215 or MATH 216 or MATH 217 or MATH 227 or MATH 276

**MATH 391 Topics in Mathmatics Education 1 to 3 Credit Hours**

A course designed to offer selected topics in mathematics related to K-12 education. The specific topic or topics will be announced together with the prerequisites each term. Course may be repeated for credit when specific topics differ. (OC).

**MATH 391A Topics in Mathmatics Education 1 to 3 Credit Hours**

TOPIC TITLE: Teaching Mathematics in Urban Middle Schools and High Schools Students involved in this Topics course will be engaged in activities which deepen their understandings of middle school and high school mathematics and the teaching of these topics to students from underrepresented groups in urban schools. These activities will include presentations by outside speakers, reading of professional journals, student and teacher presentations, selecting units from reform curricula and deepening their understanding of the mathematics involved inpreparation to teach leassons from the unit, and extensive field experiences in high schools and middle schools in Detroit and Dearborn. Activities relating to thr project start the second half of the Fall 1999 term with about one-half of the Winter 2000 term MATH 391 course spent in schools teaching mathematics. This course is intended for students involved in an Eisenhower grant which focuses on the recruitment of teachers of mathematics for urban schools.

**MATH 391B Topics in Mathematics and Stat 1 to 3 Credit Hours**

Topic: Number and Proportional Reasonning in Middle School Mathematics Teachers. This course is designed to deepen the teachers of middle school mathematics understanding of the rational number system and its extension to the real number system in a way that models appropriate pedagogy and raises curriculum issues relevant to teaching number concepts for conceptual understanding and computation fluency. A particular emphasis will be on understanding and applying concepts of proportional reasoning. Topics related to this emphasis include analyzing connections between fraction concepts and ratios and proportions; describing the relationship between proportions and direct and indirect variation; analyzing and applying the connections between proportions and similar figures, probability and sampling; and modeling and solving problems involving rations and proportions. Other major topics incude analyzing number theoretic concepts such as prime numbers and divisibility; and comparing and contrasting models of operations across number systems. Calculator and computer technology will be used as problem solving tools and for support in conceptual understanding. Curriculum resources and materials that support conceptual understanding are considered.

**MATH 395 Elementary Number Theory 3 Credit Hours**

Properties of the integers, the division algorithm, Euclid's algorithm, Fermat's theorems, unique factorization of integers into primes, congruences, arithmetic functions, Diophantine equations, continued fractions, quadratic reciprocity. (W).

**MATH 396 Introduction to Cryptography 3 Credit Hours**

This course discusses ways of encrypting information, a function which is vital to economics, defense and the empowerment of society. It is more crucial now than ever before to be able to securely transfer information in this age of electronic communication. After discussing primitive ways of encrypting information and explaining the need for more sophisticated encoding methods, this course explores the mathematics (number theory, finite fields and probability) behind both historic and more recent cryptosystems that have been developed for the secure transmission of data along non secure channels. This course continues with symmetric and public key cryptosystems, elliptic curves, digital signatures, zero knowledge protocols and other more advanced methods. This course does not assume any prior knowledge of number theory or probability. (YR)

**Prerequisite(s): ** MATH 205 or MATH 215 or MATH 216 or MATH 217 or MATH 227 or MATH 276

**MATH 399 Independent Studies in Math 1 to 3 Credit Hours**

Independent study in mathematics for topics at the junior level. Topics and objectives chosen by agreement between student and instructor.

**MATH 4000 Capstone in Mathematics 3 Credit Hours**

Math 4000 is the Capstone course in Mathematics, covering an advanced topic in Mathematics determined by the instructor. Topics may include, but are not limited to, algebraic geometry, functional analysis, functions of several complex variables, and aspects of the study of numerical analysis, partial differential equations ,cominatorics, probability, number theory, or topology. Students are expected to complete a research project in the area of the particular topic. (F, W)

**Prerequisite(s): ** MATH 217 or MATH 227

**Restriction(s): **

Can enroll if Class is Junior or Senior

**MATH 404 Dynamical Systems 3 Credit Hours**

The aim of this course is to survey the standard types of differential equations. This includes systems of differential equations, and partial differential equations, including for each type, a discussion of the basic theory, examples of applications, and classical techniques of solutions with remarks about their numerical aspects. Also included are autonomous and periodic solutions, phase space, stability, perturbation techniques and Method of Liapunov. Students cannot receive credit for both MATH 404 and MATH 504. (AY).

**Prerequisite(s): ** (MATH 216 or MATH 217 or MATH 228) and MATH 227

**MATH 405 Integral Equations 3 Credit Hours**

Origin and classification of integral equations, connections with differential equations, integral equations of convolution type, method of successive approximations, single kernels, elements of Hilbert space, linear operators, resolvents, Fredholm theory and Hilbert-Schmidt theory. Students cannot receive credit for both MATH 405 and MATH 505. (OC).

**MATH 412 First Course in Modern Algebra 3 Credit Hours**

Introduction to groups, subgroups, group homomorphisms, factor groups, simple groups, cyclic groups. Sylow theorems, rings, ideals, integral domains, fields, polynomial rings, Kronecker's theorem, also properties of the integer, rational, real, and complex numbers. Students cannot receive credit for both MATH 412 and MATH 512. (W).

**Prerequisite(s): ** (MATH 227 or MATH 217 or MATH 228) and (MATH 200 or MATH 300)

**MATH 413 Linear Algebra 3 Credit Hours**

Vector spaces, linear transformations and matrices, determinants, inner product spaces, bilinear and quadratic forms, Hamilton-Cayley theorem, eigenvalues and eigenvectors, and spectral theorem. Students cannot receive credit for both MATH 413 and MATH 513. (F)

**Prerequisite(s): ** (MATH 200 and MATH 216 or MATH 300) and (MATH 217 or MATH 227)

**MATH 420 Stochastic Processes 3 Credit Hours**

Review of distribution theory. Introduction to stochastic processes, Markov chains and Markov processes, counting, and Poisson and Gaussian processes. Applications to queuing theory. Students cannot receive credit for both MATH 420 and MATH 520. (AY,W).

**MATH 425 Mathematical Statistics 3 Credit Hours**

Interval estimation and pivotal quantities, maximum likelihood estimation, hypothesis tests, linear models and analysis of variance, bivariate normal distribution, regression and correlation analysis, and nonparametric methods. Students cannot receive credit for both MATH 425 and MATH 525. Previously taught as Mathematical Statistics II. (AY,S).

**Prerequisite(s): ** MATH 325

**MATH 442 Geometry for Teachers 3 Credit Hours**

Properties of two and three-dimensional figures are covered, including congruence, symmetry, transformation, and measurement. Trigonometry from a geometric perspective and the use of trigonometry in problem solving are included. Topics also include coordinate geometry and visualization as well as the nature of axiomatic reasoning and the role it has played in the development of mathematics. An investigative approach involving problem solving, reasoning and proof, connections, and communication will be emphasized. Calculator and computer technology will support the investigation of these topics. Classroom resources and materials are considered. Different levels of geometric thinking will be explored. No credit for CASL concentration, minor, or area of focus. Open only to certified teachers or elementary education students. Student cannot receive credit for both MATH 442 and MATH 542.

**Prerequisite(s): ** MATH 387

**Restriction(s): **

Can enroll if College is Education, Health, and Human Services

**MATH 443 Algebra for Teachers 3 Credit Hours**

Algebraic structure is emphasized, especially as it relates to arithmetic. Emphasis is on the development of algebraic reasoning and generalizations with the appropriate pedagogy. Curriculum issues relevant to teaching algebra for conceptual understanding are included. Major topics include algebraic representations of linear, exponential, power and quadratic patterns, systems of equations, and applications. An investigative approach involving problem solving, reasoning and proof, connections and communications will be emphasized. Classroom resources and materials are considered as well as calculators and computer technology as problem-solving tools to aid in algebraic thinking. No credit for CASL concentration, minor or area of focus. Students cannot receive credit for both MATH 443 and MATH 543. (F, W, S).

**Prerequisite(s): ** MATH 386

**Restriction(s): **

Can enroll if College is Education, Health, and Human Services

**MATH 444 Data Anlsys,Prob&Stat forTchrs 3 Credit Hours**

Concepts of probability using both experimental and theoretical models are considered with an emphasis on the use of probability models to describe physical phenomena and to make and interpret predictions. Topics in data analysis and statistics include drawing inferences from visual displays of data, applying techniques of inferential statistics, sampling and simulations to generate solutions to problems, and making appropriate inferences using best fit techniques. Evaluating data and arguments to establish validity, interpreting, calculating and solving problems related to correlation, distributions, percentiles and standard scores are also included. An investigative approach involving problem solving, reasoning and proof, connections, and communication will be emphasized. Calculator and computer technology will support the investigation of these topics. No credit for CASL concentration, minor, or area of focus. Open only to certified teachers or elementary education students. Student cannot receive credit for both MATH 444 and MATH 544.

**Prerequisite(s): ** MATH 387

**Restriction(s): **

Can enroll if College is Education, Health, and Human Services

**MATH 445 Number & Prop'l Rsng for Tchrs 3 Credit Hours**

This course deepens previous work on rational number ideas and applications, and explores the concepts of ratio and proportion. Content includes a variety of situations involving proportions, for example, real-world problems involving ratios, rates, and percents, geometry involving similarity, algerbra involving linearity, probability involving assigning a probability to an event, and trigonometry involving slope. Distinguishing proportional situations from those that are not and reasoning proportionally in appropriate situations are emphasized. The course includes problem solving, reasoning and proof, connections, communication, and multiple representations. No credit for CASL concentration, minor, or area of focus. Open only to certified teachers or elementary education students or by permission of instructor. Students cannot receive credit for both MATH 445 and MATH 545. (AY).

**MATH 446 Discrete Math/Modeling for Tch 3 Credit Hours**

This course interweaves the ideas of discrete mathematics with the approaches and strategies of mathematical modeling. It gives pre- and inservice teachers opportunities to deepen their understanding and use of mathematical models based on the concepts of discrete mathematics. Topics include recurrence, induction, permutations, combinations, binomial distributions, circuits, critical paths, minimal spanning trees, adjacency matrices, algorithm design and optimization. Systems thinking and multiple representations are emphasized. No credit for CASL concentration, minor, or area of focus. Open only to certified teachers or elementary education students. Students cannot receive credit for both MATH 446 and 546. (AY).

**MATH 447 Micro in Math for Teachers 2 Credit Hours**

Use of the microcomputer in the mathematics classroom with an emphasis on the LOGO programming language. Problem solving, hands-on activities, and a cooperative learning environment are emphasized. Students cannot receive credit for both MATH 447 and MATH 547.

**Prerequisite(s): ** MATH 385

**MATH 448 Problem Solving for Teachers 2 Credit Hours**

Problem-solving skills important to the mathematics classroom are introduced. Both the development of these skills for those in the course and the implementation of them in the classroom are pursued. Resource materials for the classroom are considered. (F).

**Prerequisite(s): ** MATH 386

**Restriction(s): **

Can enroll if Level is Rackham or Graduate

**MATH 449 Concepts of Calc for Teachers 3 Credit Hours**

Concepts of Calculus for Teachers focuses on calculus concepts appropriate for middle school mathematics teachers and teacher-candidates. The course provides a deep understanding of the major concepts of calculus: rates of change, accumulation (net change), area, and limits. Students will experience concrete approaches to the various topics using problem solving, manipulatives and technology as appropriate, with the intent being to help the learners discover how the ideas of calculus are useful in a variety of settings. Visual, numeric and commonsense approaches are used. No credit for CASL concentration, minor, or area of focus. Open only to certified teachers or elementary education students. Students cannot receive credit for both MATH 449 and 549. (AY)

**MATH 451 Advanced Calculus I 3 Credit Hours**

Properties of the real number system; point set theory for the real line including the Bolzano-Weierstrass theorem; sequences, functions of one variable: limits and continuity, differentiability, Reimann integrability. Students cannot receive credit for both MATH 451 and MATH 551. (F).

**Prerequisite(s): ** (MATH 300 or MATH 200) and (MATH 216 or MATH 217 or MATH 228) and MATH 227

**MATH 452 Advanced Calculus II 3 Credit Hours**

Includes the rigorous study of functions of two and more variables, partial differentiation and multiple integration. Special topics include: Taylor Series, Implicit Function Theorem, Weierstrass Approximation Theorem, Arzela-Ascoli Theorem. Students cannot receive credit for both MATH 452 and MATH 552. (AY,W).

**Prerequisite(s): ** MATH 451

**MATH 454 Fourier and Boundary 3 Credit Hours**

Fourier series and integrals. Their use in solving boundary value problems of mathematical physics by the method of separation of variables. Sturm-Liouville theory and generalized Fourier series, including those involving Bessel functions and Legendre polynomials, with applications. Students cannot receive credit for both MATH 454 and MATH 554. (F).

**Prerequisite(s): ** (MATH 216 or MATH 217 or MATH 228) and MATH 227

**MATH 455 Func of a Complex Var with App 3 Credit Hours**

Complex number system. Functions of a complex variable, their derivatives and integrals. Taylor and Laurent series expansions. Residue theory and applications, elementary functions, conformal mapping, and applications to physical problems. Students cannot receive credit for both MATH 455 and MATH 555. (W).

**Prerequisite(s): ** (MATH 216 or MATH 217 or MATH 228) and MATH 227

**Restriction(s): **

Can enroll if Level is Undergraduate

**MATH 458 Introduction to Wavelets 3 Credit Hours**

This course will introduce the students to theory and application of wavelets using linear algebra. Topics will include the discrete Fourier transform, the fast Fourier transform, linear transformations, orthogonal decomposition, discrete wavelet analysis, the filter bank, Haar Wavelet family, Daubechies's Wavelet family, and applications. Students cannot receive credit for both MATH 458 and MATH 558. (OC)

**Prerequisite(s): ** (MATH 216 or MATH 217 or MATH 228) and MATH 227

**Restriction(s): **

Can enroll if Class is Sophomore or Junior or Senior

**MATH 462 Mathematical Modeling 3 Credit Hours**

The processes of constructing, implementing, and evaluating mathematical models of "real world" phenomena are investigated. Models involving continuous and discrete mathematical constructs are considered. Deterministic and stochastic models are compared. Examples are taken from genetics, epidemiology, queuing theory, and other fields. Students cannot receive credit for both MATH 462 and MATH 562. (F).

**Prerequisite(s): ** (MATH 216 or MATH 217 or MATH 228) and MATH 227

**MATH 472 Intro to Numerical Analysis 3 Credit Hours**

Solution of linear systems by Gaussian elimination, solution of non-linear equations by iterative methods, numerical solution of ordinary differential equations, data fitting with spline functions, numerical integration, optimization. Students cannot receive credit for both MATH 472 and MATH 572. (F).

**MATH 473 Matrix Computation 3 Credit Hours**

A study of the most effective methods for finding the numerical solution of problems which can be expressed in terms of matrices, including simultaneous linear equations, orthogonal projections and least squares, eigenvalues and eigenvectors, positive definite matrices, and difference and differential equations. Students cannot receive credit for both MATH 473 and MATH 573. (AY, W).

**MATH 480 History of Mathematics 3 Credit Hours**

A unified view of the rise of mathematics from ancient times to the present, as seen in its conceptual developments and developers, its major themes and its applications (including computers). Students cannot receive credit for both MATH 480 and MATH 580. (OC).

**MATH 486 Sec School Math for Teachers 3 Credit Hours**

Basic concepts, relationships, generalizations, and applications from the secondary school mathematics curriculum are discussed both from an advanced viewpoint and from the standpoint of the learner. Included are the roles of technology, problem solving, and current thinking on the teaching of secondary mathematics topics. Students cannot receive credit for both MATH 486 and MATH 586. (F).

**MATH 492 Introduction to Topology 3 Credit Hours**

Metric spaces, topological spaces, continuous maps, connectedness, compactness, separation axioms. Students cannot receive credit for both MATH 492 and MATH 592. (AY,W).

**Prerequisite(s): ** MATH 451

**MATH 499 Independent Studies in Math 1 to 3 Credit Hours**

Independent study in mathematics for topics at the senior level. Topics and objectives chosen by agreement between student and instructor. (OC).

* | An asterisk denotes that a course may be taken concurrently. |

Frequency of Offering

The following abbreviations are used to denote the frequency of offering: (F) fall term; (W) winter term; (S) summer term; (F, W) fall and winter terms; (YR) once a year; (AY) alternating years; (OC) offered occasionally